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\section{Introduction}
Electron spin confined in a semiconductor quantum dot is a promising qubit candidate because of both the long dephasing time and the relative convenience for scalability~\cite{Loss1998,Hanson2007}. The spin dephasing time can be as long as a millisecond in isotopically purified Si quantum dot~\cite{veldhorst2014,veldhorst2015}. While in III-V semiconductor quantum dots, such as GaAs, the spin dephasing time is in the microsecond region~\cite{petta2005}, limited mainly by the hyperfine interaction between the electron and lattice nuclear spins~\cite{Yao2006,Cywinski2009}. Single qubit manipulation in the quantum dot can be achieved via either the electron spin resonance~\cite{koppens2006} or the electric-dipole spin resonance (EDSR)~\cite{Rashba2003,Golovach2006,LiRui2013,nowack2007,Nadj2012}. Two qubit manipulation can be naturally achieved by using the exchange interaction in a double quantum dot~\cite{Burkard1999,Hu2000}.
The manipulation time $T_{\rm Rabi}$ and the dephasing time $T^{*}_{2}$ are two important time scales for the qubit~\cite{Buluta2011}. The values of these two quantities determine whether a qubit candidate is suitable for quantum computing. An ideal quantum computer requires that enough number (about 1000) of single qubit manipulations should be completed in the qubit dephasing time~\cite{ladd2010}. Dephasing is a leading obstacle limiting all potential applications of the qubit. In order to alleviate the qubit suffering from dephasing caused by environmental noises, we should first understand various possible dephasing mechanisms~\cite{chan2018}.
There are both no internal spin-orbit coupling and negligible lattice nuclear spins in isotopically purified 28Si, such that Si quantum dot is expected to be one of the most feasible platforms for quantum computing~\cite{veldhorst2014,veldhorst2015}. The spin qubit in Si quantum dot is so separate from the external environment that single qubit manipulation becomes relatively inconvenient. Electron spin resonance in a quantum dot is proved to be technically challenging~\cite{koppens2006}. A feasible way is to integrate the quantum dot with a slanting magnetic field~\cite{pioro2008,Brunner2011,kawakami2014,Chesi2014,Forster2015,Scarlino2015}, such that single spin manipulation can be achieved via EDSR. However, as observed in experiments, the slanting field also brought the 1/f charge noise to the spin qubit~\cite{Kawakami2016,yoneda2018}. 1/f charge noise commonly exists in many nano-structures~\cite{Dutta1981,Weissman1988,Paladino2014}, and it has also been regarded as the main noise source that causes the dephasing of the qubit, such as Josephson qubit~\cite{Astafiev2004,You2007,bylander2011}, quantum dot charge qubit~\cite{Petersson2010,Shi2013}, spin qubit~\cite{chan2018,Kha2015}, singlet-triplet qubit~\cite{Culcer2009,Hu2006,Gamble2012}, etc.
In this paper, we study the slanting field mediated spin manipulation and spin dephasing in a Si quantum dot. In the spin manipulation via EDSR, the transverse slanting field mediates a transverse driving term which contributes to the periodic oscillation of the spin population inversion, while the longitudinal slanting field mediates a longitudinal driving term which gives a modulation to the spin population inversion. Fortunately, the effect of the modulation can be reduced by applying a large Zeeman field to the quantum dot. The pure dephasing is caused by the longitudinal spin-1/f-charge noise interaction, which is also mediated by the longitudinal slanting field. We propose prolonging the spin dephasing time by reducing the quantum dot size, lowering the experimental temperature, reducing the longitudinal slanting field, or using a dynamical decoupling scheme~\cite{Uhrig2007}. Under eight pulse sequences, the spin dephasing time $T_{2}$ can be prolonged to the sub-millisecond region. Finally, because the upper bound of the 1/f charge noise spectrum is usually less than the qubit level spacing in the quantum dot, the 1/f charge noise cannot contribute to the spin relaxation.
\section{The model}
\begin{table}
\centering
\caption{\label{tab}The parameters of the Si quantum dot used in our calculations. The values are taken from Ref.~\onlinecite{yoneda2018}}
\begin{ruledtabular}
\begin{tabular}{ccccccc}
$m/m_{0}$\footnote{$m_{0}$ is the free electron mass}&$g$&$B_{0}$~(T)&$\omega_{0}$~(THz)\footnote{$r_{0}=\sqrt{\hbar/(m\omega_{0})}=20$ nm}&$b_{t}$\footnote{in unit of (mT/nm), $z_{0}=g\mu_{B}b_{t}/(2m\omega^{2}_{0})=2.431\times10^{-2}$nm}&$b_{l}$\footnote{in unit of (mT/nm), $y_{0}=g\mu_{B}b_{l}/(2m\omega^{2}_{0})=0.4862\times10^{-2}$nm}&{\it T} (mK)\\
$0.2$&$2$&$0.5$&1.447&$1.0$&$0.2$&100
\end{tabular}
\end{ruledtabular}
\end{table}
We consider a realistic quantum dot model which is intimately related to the experimental situations demonstrated recently~\cite{yoneda2018,Yoneda2014}. The quantum dot has a two-dimensional harmonic confining potential on the $yz$ plane and is exposed to both static and slanting magnetic fields. The slanting field, which is used to assist the spin manipulation via an external electric-field, is created by covering a Co micromagnet on the quantum dot~\cite{yoneda2018,Yoneda2014,Neumann2015,Wu2014}. The model under consideration reads
\begin{equation}
H=\frac{p^{2}_{y}+p^{2}_{z}}{2m}+\frac{1}{2}m\omega^{2}_{0}(y^{2}+z^{2})+\frac{g\mu_{B}({\bf B}_{0}+{\bf B}_{\rm m})\cdot\boldsymbol{\sigma}}{2},
\end{equation}
where $m$ is the effective electron mass, $\omega_{0}$ is the frequency of the harmonic confining potential [the quantum dot characteristic length $r_{0}=\sqrt{\hbar/(m\omega_{0})}$], ${\bf B}_{0}=(0,0,B_{0})$ is an in-plane static field applied long the $z$ direction, and ${\bf B}_{\rm m}=(B^{x}_{\rm m},B^{y}_{\rm m},B^{z}_{\rm m})$ is the stray field induced by the Co micromagnet. One can expand the stray field up to the linear terms using Taylor's formula
\begin{eqnarray}
B^{x}_{\rm m}(z)&=&B^{x}_{\rm m}(0)+b_{t}z,\nonumber\\
B^{y}_{\rm m}(z)&=&B^{y}_{\rm m}(0)+b_{l}z,\nonumber\\
B^{z}_{\rm m}(x,y)&=&B^{z}_{\rm m}(0)+b_{l}y+b_{t}x,\label{eq_slantingfield}
\end{eqnarray}
where $b_{t}$ and $b_{l}$ are the slopes of the transverse and longitudinal fields~\cite{yoneda2018,Yoneda2014}, respectively. One can check that the above stray field does not violate Maxwell's equations $\nabla\cdot{\bf B}_{\rm m}=0$ and $\nabla\times{\bf B}_{\rm m}=0$ for a static system. The small $x,y$-components $B^{x,y}_{m}(0)$ of the stray field are neglected from consideration and the $z$-component $B^{z}_{m}(0)$ can be absorbed to the static Zeeman field $B_{0}$. After the above linear approximation, the quantum dot Hamiltonian can be written as
\begin{eqnarray}
H&=&\frac{p^{2}_{y}+p^{2}_{z}}{2m}+\frac{m\omega^{2}_{0}}{2}(y^{2}+z^{2}+2y_{0}y\sigma^{z})+\Delta\sigma^{z}\nonumber\\
&&+m\omega^{2}_{0}z\sqrt{z^{2}_{0}+y^{2}_{0}}(\sigma^{x}\cos\theta_{0}+\sigma^{y}\sin\theta_{0}),\label{eq_model2}
\end{eqnarray}
where $y_{0}=g\mu_{B}b_{l}/(2m\omega^{2}_{0})$ and $z_{0}=g\mu_{B}b_{t}/(2m\omega^{2}_{0})$ characterize the length scale of the longitudinal and transverse gradient fields, respectively, $\theta_{0}=\arctan(y_{0}/z_{0})$ and $\Delta=g\mu_{B}B_{0}/2$ is half of the Zeeman splitting. It should be noted that the vector potential $\textbf{A}=\textbf{B}_{0}\times\textbf{r}/2$ is perpendicular to the $yz$ plane, such that there are no vector potential components in the Hamiltonian (\ref{eq_model2}), i.e., $p_{y}=-i\hbar\partial_{y}$ and $p_{z}=-i\hbar\partial_{z}$. Also, we have assumed the quantum dot lies on the $x=0$ plane.
In line with the experimental investigation~\cite{yoneda2018}, here we choose Si as our quantum dot material. In our following calculations, unless otherwise stated, the parameters chosen are listed in table~\ref{tab}.
\section{Slanting field mediated electric-dipole spin resonance}
\begin{figure}
\centering
\includegraphics[width=8.5cm]{EDSR.eps}
\caption{\label{fig_Rabi_Si}EDSR described by the driving Hamiltonian (\ref{eq_edsr}) under the resonant condition $\hbar\omega=2\Delta$. The qubit population inversion is defined as $|c_{\Uparrow}(t)|^{2}-|c_{\Downarrow}(t)|^{2}$ for state: $|\varphi(t)\rangle=c_{\Uparrow}(t)|\!\!\Uparrow\rangle+c_{\Downarrow}(t)|\!\!\Downarrow\rangle$. The qubit is initially in state $|\varphi(0)\rangle=|\!\!\Uparrow\rangle$ and the driving strength is chosen as $E_{y}=E_{z}=4000$ V/m. The results at the external fields of $B_{0}=0.005$ T (a), $B_{0}=0.02$ T (b), and $B_{0}=0.5$ T (c).}
\end{figure}
The manipulation of the quantum-dot spin qubit is usually achieved via EDSR. Quantum-dot EDSR can be mediated by internal spin-orbit coupling~\cite{Rashba2003,Golovach2006,LiRui2013,nowack2007,Nadj2012,Khomitsky2012,Nowak2013,Romhanyi2015}, electron-nuclear hyperfine interaction~\cite{Laird2007,Rashba2008,LiRui2016}, and external slanting magnetic field~\cite{Tokura2006,Rancic2016}. In the earlier seminal work of Tokura and co-workers~\cite{Tokura2006}, only a transverse slanting field is proposed to mediate the EDSR. However, under realistic experimental circumstance, the micromagnet brings no only the transverse but also the longitudinal slanting fields to the quantum dot~\cite{pioro2008,Brunner2011,yoneda2018,Yoneda2014} [see Eq.~(\ref{eq_model2})]. Here we examine the impacts of the longitudinal slanting field on the spin manipulation.
Under the external electric-field driving, an additional electric-dipole interaction term $e\textbf{E}\cdot\textbf{r}\cos\omega\,t$ should be added to Hamiltonian (\ref{eq_model2}). When we focus only on the qubit Hilbert space spanned by $|\!\!\Uparrow\rangle\equiv|\Psi_{0,0,\uparrow}\rangle$ and $|\!\!\Downarrow\rangle\equiv|\Psi_{0,0,\downarrow}\rangle$, the electric-driving Hamiltonian can be reduced to the form of a two-level atom interacting with a classical field~\cite{scully1999quantum} (for details see Appendix \ref{appendix_A})
\begin{eqnarray}
H_{\rm dr}&=&\Delta\tau^{z}-eE_{y}y_{0}\tau^{z}\cos\omega\,t\nonumber\\
&&-eE_{z}\sqrt{z^{2}_{0}+y^{2}_{0}}(\tau^{x}\cos\theta_{0}+\tau^{y}\sin\theta_{0})\cos\omega\,t,\label{eq_edsr}
\end{eqnarray}
where $\tau^{z}=|\!\!\Uparrow\rangle\langle\Uparrow\!\!|-|\!\!\Downarrow\rangle\langle\Downarrow\!\!|$, $\tau^{x}=|\!\!\Uparrow\rangle\langle\Downarrow\!\!|+|\!\!\Downarrow\rangle\langle\Uparrow\!\!|$, $\tau^{y}=-i|\!\!\Uparrow\rangle\langle\Downarrow\!\!|+i|\!\!\Downarrow\rangle\langle\Uparrow\!\!|$, $E_{y}$ and $E_{z}$ are the $y$ and $z$ components of the driving-field, respectively, and $\omega$ is the frequency of the driving-field. This Hamiltonian is slightly different from the standard Rabi oscillation Hamiltonian in quantum optics~\cite{scully1999quantum} because of the presence of the second term, which is induced by the longitudinal slanting field given in Eq.~(\ref{eq_model2}).
Let us examine the influence of the longitudinal driving term [the second term in Eq.~(\ref{eq_edsr})] on the spin manipulation. Similar to the standard Rabi oscillation, the qubit is initially prepared in state $|\varphi(0)\rangle=|\!\!\Uparrow\rangle$. When the frequency of the driving field matches the qubit level spacing $\hbar\omega=2\Delta$, the spin population inversion is obtained by numerically solving the time-dependent Schr\"odinger equation governed by Hamiltonian (\ref{eq_edsr}). We find that, at small external magnetic field such as $B_{0}=0.005$ T, there is an apparent modulation on the spin population inversion [see Fig.~\ref{fig_Rabi_Si}(a)]. When the magnetic field is increased to $B_{0}=0.02$ T, the modulation becomes relative small[see Fig.~\ref{fig_Rabi_Si}(b)]. When the external magnetic field is large enough, such as $B_{0}=0.5$ T, the modulation becomes negligible (almost invisible) [see Fig.~\ref{fig_Rabi_Si}(c)]. Anyway, one can reduce the modulation via increasing the external magnetic field $B_{0}$. This is very reasonable, the longitudinal driving term can be regarded as a time-dependent Zeeman field applied to the spin qubit $(\Delta-eE_{y}y_{0}\cos\omega\,t)\tau^{z}$. The larger the static magnetic field, the smaller the relative ratio $eE_{y}y_{0}/\Delta$, hence the smaller the effect of the longitudinal driving term.
Next, let us analyze the strength of the Rabi frequency, which characterizes the qubit manipulation time. Note that the qubit is encoded to the lowest two energy levels of the quantum dot. Although the qubit Hilbert space is well separated from the other higher orbital levels in the quantum dot, i.e., the Zeeman splitting $2\Delta_{B_{0}=0.5~{\rm T}}$ (0.058 meV) is much smaller than the orbital splitting $\hbar\omega_{0}$ (0.95 meV), there still exist leakages from the qubit Hilbert space to the higher orbital states under the strong field driving. The spin dynamics in this case are totally nontrivial, and one has to consider the multi-level effects in the EDSR~\cite{Khomitsky2012}. In order to avoid the electron being excited to higher orbital states, here the electric field strength is constrained to $|\textbf{E}|\ll(\hbar\omega_{0})/(er_{0})=4.769\times10^{4}$ V/m. This result gives an upper bound on the Rabi frequency in our model $\Omega_{R}\ll\,eE_{\rm max}\sqrt{z^{2}_{0}+y^{2}_{0}}/h=286$ MHz, and agrees qualitatively well with the experimental observations~\cite{yoneda2018,Takedae2016}.
\section{\label{sec_IV}Charge noise induced pure-dephasing}
1/f charge noise has been observed in many quantum nano-structures~\cite{Dutta1981,Weissman1988,Paladino2014}, and it has also been regarded as the main noise limiting the dephasing time of many qubit candidates~\cite{Astafiev2004,You2007,bylander2011,Petersson2010,Shi2013,Kha2015,Culcer2009,Hu2006,Gamble2012,lirui2018a}. The physical origin of the charge fluctuation spectrum with 1/f distribution is still unclear, and many theoretical models have been proposed~\cite{Paladino2014}. Here we just assume that the charge field has a spectrum function $A^{2}/\omega$, and the value of $A$ is chosen to fit well with the experimental observation.
We assume the fluctuating charge field has a similar form as that of the vacuum electromagnetic field~\cite{scully1999quantum}
\begin{equation}
\textbf{E}(\textbf{r})=\sum_{k}\Xi_{k}\vec{e}_{k}(a_{k}e^{i\vec{k}\cdot\vec{r}}+a^{\dagger}_{k}e^{-i\vec{k}\cdot\vec{r}}),\label{eq_chargefield}
\end{equation}
where $\Xi_{k}$ is the charge field in the wavevector space, $\vec{e}_{k}$ is a unit vector, and $\vec{k}$ is the wavevector. The transverse character of the electromagnetic field gives rise to $\vec{e}_{k}\cdot\vec{k}=0$~\cite{scully1999quantum}. In order to simplify the complexity of the problem, we further assume the wave is propagating along the $x$ direction: $\vec{k}=k\vec{e}_{x}\perp\,yz$ plane, such that $\vec{e}_{k}$ is an in-plane unit vector, hence $\textbf{E}(\textbf{r})=\sum_{k}\Xi_{k}\vec{e}_{k}(a_{k}+a^{\dagger}_{k})$ (the quantum dot is confined on the $x=0$ plane). Replacing the classical field in Eq.~(\ref{eq_edsr}) with the above quantized electric-field, we obtain the total Hamiltonian describing the interaction between the spin qubit and the charge noise
\begin{eqnarray}
H_{\rm tot}&=&\Delta\tau^{z}-\sum_{k}e\Xi_{k}y_{0}\tau^{z}(a_{k}+a^{\dagger}_{k})\cos\Theta+\sum_{k}\hbar\omega_{k}a^{\dagger}_{k}a_{k}\nonumber\\
&&-\sum_{k}e\Xi_{k}(z_{0}\tau^{x}+y_{0}\tau^{y})(a_{k}+a^{\dagger}_{k})\sin\Theta,\label{eq_decoherence}
\end{eqnarray}
where $\Theta$ is the azimuth of the charge field on the $yz$ plane. The exact value of $\Theta$ is unknown, such that it is reasonable to average over all possible angle $\Theta$ for the obtained physical quantities, e.g., $\Gamma(t)\equiv\langle\Gamma(t)\rangle_{\Theta}=\int^{2\pi}_{0}\Gamma(t)d\Theta/2\pi$.
\begin{figure}
\centering
\includegraphics{dephasing.eps}
\caption{\label{fig_dephasing}The pure dephasing of the spin qubit due to the $1/f$ charge noise. We have chosen the noise spectrum strength $A_{r_{0}=20~{\rm nm},T=100~{\rm mK}}=35$ MHz in order to fit well with the experimental observation~\cite{yoneda2018}.}
\end{figure}
The pure-depasing of the qubit is caused by the longitudinal coupling between the qubit and the charge noise as illustrated by the second term in Eq.~(\ref{eq_decoherence}). This term can been traced back to the longitudinal slanting term in Eq.~(\ref{eq_model2}). If we model the qubit dephasing as ${\rm exp}\left[-\Gamma_{\rm ph}(t)\right]$, the decaying factor be written as~\cite{Palma1996}
\begin{equation}
\Gamma_{\rm ph}(t)=2\frac{y^{2}_{0}}{r^{2}_{0}}\int^{\omega_{\rm max}}_{\omega_{\rm min}}d\omega\,S(\omega)\frac{\sin^{2}(\omega\,t/2)}{(\omega/2)^{2}},\label{eq_dephasingrate}
\end{equation}
where $\omega_{\rm min (max)}$ is the lower (upper) bound of the noise frequency, and the spectrum function is defined as
\begin{eqnarray}
S(\omega)&=&\sum_{k}\frac{e^{2}r^{2}_{0}\Xi^{2}(\omega)[2n(\omega)+1]}{2\hbar^{2}}\delta(\omega-\omega_{k})\nonumber\\
&\approx&\sum_{k}\frac{e^{2}r^{2}_{0}\Xi^{2}(\omega)k_{B}T}{\hbar^{3}\omega}\delta(\omega-\omega_{k})\equiv\frac{A^{2}_{r_{0},T}}{\omega},\label{eq_noisespectrum}
\end{eqnarray}
with $A_{r_{0},T}$ being a parameter characterizing the strength of the charge noise. The lower bound of the noise spectrum is about $\omega_{\rm min}\approx\,10^{-2}$ Hz~\cite{yoneda2018}, and the upper bound of the noise spectrum is about $\omega_{\rm max}\approx\,5\times10^{5}$ Hz~\cite{yoneda2018}. We have also included the temperature effect in deriving Eq.~(\ref{eq_noisespectrum}) by writing the Bose occupation number as $n(\omega)=1/\left[{\rm exp}(\hbar\omega/k_{B}T)-1\right]\approx\,k_{B}T/(\hbar\omega)\gg1$, under the realistic temperature~\cite{yoneda2018} ($T=100$ mK) for all the low frequency noise modes ($\omega_{\rm max}\sim0.004$ mK). Note that $A_{r_{0},T}$ has the dimension of the frequency, in order to fit well with the experimental observed dephasing time $T^{*}_{2}\approx20\,\mu$s~\cite{yoneda2018}, we have chosen $A_{r_{0}=20~{\rm nm},T=100~{\rm mK}}=35$ MHz (see Fig.~\ref{fig_dephasing}). It is instructive to see for the time scale $t<1/\omega_{\rm max}=2 \mu$s, we can write the dephasing factor as the following simple form (a similar version of Ref.~\onlinecite{Schriefl2006})
\begin{equation}
\Gamma_{\rm ph}(t)=2A^{2}_{r_{0},T}t^{2}\frac{y^{2}_{0}}{r^{2}_{0}}\ln\frac{\omega_{\rm max}}{\omega_{\rm min}}.\label{eq_gaussdecay}
\end{equation}
Thus, the qubit dephasing at short time must be a Gauss decay. Actually, for time scale larger than $t>1/\omega_{\rm max}$ in our model, we find that the difference between the Gauss decay (\ref{eq_gaussdecay}) and the exact decay (\ref{eq_dephasingrate}) is very small (see Fig.~\ref{fig_dephasing}).
Let us discuss on the spectrum function defined in Eq.~(\ref{eq_noisespectrum}). Although our derivation of the spectrum function with 1/$\omega$ distribution has been made plausible, the difficulty lies in choosing reasonable $\Xi_{k}$ such that the second expression can be written as the third expression in the last line of Eq.~(\ref{eq_noisespectrum}). Actually, the physical mechanism of the charge spectrum with 1/$\omega$ distribution is still unclear~\cite{Paladino2014}. Here, we give a simple argument to realize the 1/f spectrum function. Note that the wavevector $\vec{k}$ is perpendicular to the $yz$ plane, and for the electromagnetic wave we have the dispersion relation $\omega_{k}=ck$, where $c$ is the speed of light. We make the following replacement in Eq.~(\ref{eq_noisespectrum}) $\sum_{k}\rightarrow\int\,d\omega_{k}L/(\pi\,c)$, where $L$ is the length of the space in the $x$ dimension ($V=L^{3}$). It is suggested that the charge field of wavevector $\Xi_{k}$ should be a constant $\Xi_{k}\equiv\Xi$, which is in stark contrast with that of the vacuum electromagnetic field~\cite{scully1999quantum}. Hence the spectrum function can be written as
\begin{equation}
S(\omega)=\frac{e^{2}r^{2}_{0}\Xi^{2}Lk_{B}T}{\pi\,c\hbar^{3}\omega},
\end{equation}
which is indeed of the 1/$\omega$ form. Note that the linear temperature dependence of the spectrum function is consistent with both theoretical~\cite{Dutta1981,Culcer2009} and experimental~\cite{Jung2004} investigations. Although we only study the low-frequency 1/f charge noise, it is still of interest to discuss the spectrum function in the high-frequency region under this argument. Note that the first line of Eq.~(\ref{eq_noisespectrum}) is valid in all frequency range. For the high-frequency noise modes $\hbar\omega\gg\,k_{B}T$, $n(\omega)=1/\left[{\rm exp}(\hbar\omega/k_{B}T)-1\right]\approx0$. Hence, the spectrum function in the high-frequency region should be
\begin{equation}
S(\omega)=\frac{e^{2}r^{2}_{0}\Xi^{2}L}{2\pi\,c\hbar^{2}}.
\end{equation}
This spectrum function is irrelevant to the frequency $\omega$. In the noise theory, noise with this kind of spectrum is called white noise~\cite{Paladino2014}.
\section{Prolong the dephasing time}
The dephasing time $T^{*}_{2}$ is an important time scale for the qubit~\cite{Buluta2011}. A long dephasing time is always appreciated for almost all qubit candidates. Based on the spin dephasing theory built in the above section, here we study how to prolong the spin dephasing time in a Si quantum dot.
The first intuitional approach is to reduce the quantum dot characteristic length $r_{0}$~\cite{Bermeister2014}. The characteristic length is related to the electric dipole moment of the quantum dot, such that reducing $r_{0}$ obviously reduces the effective coupling between the spin and the charge noise in Eq.~(\ref{eq_decoherence}). However, the coupling between the spin and the classical field, i.e., the Rabi frequency in Eq.~(\ref{eq_edsr}), is reduced simultaneously. Therefore, reducing $r_{0}$ not only increases the dephasing time $T^{*}_{2}$ [see Fig.~\ref{fig_dephasingvsr0}(b)] but also increases the Rabi manipulation time $T_{\rm Rabi}$ [see Fig.~\ref{fig_dephasingvsr0}(a)]. The $r_{0}$ dependence of the dephasing can be roughly written as ${\rm T}^{*}_{2}\propto\,r^{-4}_{0}$. From this viewpoint, reducing $r_{0}$ may not be an effective way to prolong the dephasing time. Note that the spin dephasing time $T^{*}_{2}$ is obtained by solving $\Gamma_{\rm ph}( T^{*}_{2})=1$ in Eq.~(\ref{eq_dephasingrate}).
\begin{figure}
\includegraphics{T2vsr0.eps}
\caption{\label{fig_dephasingvsr0}The spin manipulation time (a) and the spin dephasing time (b) as a function of the quantum dot characteristic length $r_{0}$. The manipulation time is defined as $T_{\rm Rabi}=h/(2eE_{z}\sqrt{z^{2}_{0}+y^{2}_{0}})$, where $E_{z}=4000$ V/m, and the dephasing time $T^{*}_{2}$ is solved from $\Gamma_{\rm ph}(T^{*}_{2})=1$.}
\end{figure}
The second approach is to lower the environmental temperature $T$~\cite{Culcer2009}. Lower the temperature can remarkably reduce the average occupation number $n(\omega)\approx\,k_{B}T/(\hbar\omega)$ in the low frequency noise mode. The typical temperature in experiment is about $100$ mK~\cite{yoneda2018}. The effects of lowering the temperature are shown in Fig.~\ref{fig_dephasingvstemperature}(a). The temperature dependence of the dephasing can be roughly written as $T^{*}_{2}\propto1/\sqrt{T}$. A substantial improvement in the dephasing time is achievable if the experimental temperature can be lowered to the micro-Kelvin region.
\begin{figure}
\includegraphics{T2vsT.eps}
\caption{\label{fig_dephasingvstemperature}(a) The spin dephasing time as a function of the environment temperature $T$. (b) The spin dephasing time as a function of the longitudinal field gradient $b_{l}$.}
\end{figure}
The third approach is to engineer the slanting fields~\cite{Goldman2000,Neumann2015,Yoneda2015}. As can be seen from Eqs.~(\ref{eq_edsr}) and (\ref{eq_decoherence}), the longitudinal field gradient $b_{l}$ is detrimental to both the spin manipulation and the spin dephasing. While the transverse field gradient $b_{t}$ contributes to the Rabi frequency in EDSR. Thus, it is desirable to design a proper micromagnet structure, that can give rise to both an increased transverse slanting field (shorter $T_{\rm Rabi}$) and decreased longitudinal slanting field (longer $T^{*}_{2}$). The dependence of the dephasing $T^{*}_{2}$ on the longitudinal field slope $b_{l}$ is shown in Fig.~\ref{fig_dephasingvstemperature}(b). This dependence can be roughly written as $T^{*}_{2}\propto\,1/b_{l}$.
Of great interest is designing a proper micromagnet-quantum-dot structure such that the longitudinal field gradient is reduced. Let us consider a cuboid micromagnet, the dimensions of which along $x$, $y$, and $z$ are $W$, $D$, and $L$, respectively (see Fig.~\ref{fig_micromagnet}). The external magnetic field is applied along the $z$ direction, and we assume the micromagnet is fully polarized. The origin of the coordinate system is located at the geometric center of the micromagnet. We give two possible structures with one micromagnet involved [see Fig.~\ref{fig_micromagnet}(a)] and two micromagnets involved [see Fig.~\ref{fig_micromagnet}(b)]. The $y$-dimension of the micromagnet $D$ should be large enough such that there is no $y$-component of the field ($B^{y}_{m}=0$) near the quantum dot, only $x$ and $z$-components of the field are retained ($B^{x}_{m}\neq0$ and $B^{z}_{m}\neq0$). From Eq.~(\ref{eq_slantingfield}), these ideal structures give $b_{l}=0$.
\begin{figure}
\includegraphics[width=8.5cm]{micomagnet.eps}
\caption{\label{fig_micromagnet}Possible micromagnet-quantum-dot structures giving rise to reduced longitudinal field gradient ($d>W/2$). (a) The quantum dot is placed below the micromagnetic~\cite{Goldman2000}. (b) The quantum dot is placed below two identical micromagnets~\cite{yoneda2018}. }
\end{figure}
\begin{figure}
\includegraphics{micromagnet_field.eps}
\caption{\label{fig_micromagnet_field}The stray field and its gradient near the quantum dot. The structure parameters are $L=3$ $\mu$m, $W=0.3$ $\mu$m, $D=3.76$ $\mu$m and $s=0.2$ $\mu$m. (a) The results for single micromagnet design given in Fig.~\ref{fig_micromagnet}(a). (b) The results for two micromagnets design given in Fig.~\ref{fig_micromagnet}(b).}
\end{figure}
The Co micromagnet has a Curie temperature $T_{C}\approx1400$ K and a saturation magnetization $M_{s}=1.467\times10^{6}$ A/m~\cite{Neumann2015}. Assuming full polarization and neglecting the edge fluctuations of the micromagnet, one can obtain the field distribution using the analytical method given in Ref.~\onlinecite{Goldman2000}. Because the quantum dot is placed on the symmetrical line of the proposed micromagnet structure, from symmetry analysis, the stray field at
$(-d,0,0)$ in Fig.~\ref{fig_micromagnet_field}(a) or $(-d,0,(L+s)/2)$ in Fig.~\ref{fig_micromagnet_field}(b) must parallel with $\hat{z}$, i.e., $\bf{B}_{m}//\hat{z}$, and its strength depends on $d$. Hence, there is a transverse field gradient $b_{t}=\partial\,B^{z}_{m}/\partial\,x=\partial\,B^{x}_{m}/\partial\,z$. While the longitudinal field gradient $b_{l}=\partial\,B^{y}_{m}/\partial\,z=\partial\,B^{z}_{m}/\partial\,y=0$ is guaranteed by the large dimension $D$ of the micromagnet. In the single micromagnet design, the maximal transverse field gradient is about $0.6$ mT/nm (see Fig.~\ref{fig_micromagnet_field}(a)). Of course, a larger field gradient is achievable by reducing $L$. In the two micromagnets design, the transverse field gradient can be as large as $10\sim20$ mT/nm (see Fig.~\ref{fig_micromagnet_field}(b)). The structure with two micromagnets more easily produces a larger transverse slanting field.
The forth promising way is to use the dynamical decoupling scheme~\cite{Uhrig2007,Lee2008,Yang2008,Cywinski2008} as has also been used in experiments. The spirit of dynamical decoupling is to frequently flip the spin using pulse sequences, such that the effective spin-noise interaction is eliminated as being of high-order small. Certainly, the performance of dynamical decoupling depends on how many pulses are applied~\cite{Medford2012}. Consider $n$ pulses applied to the qubit at a serious instant time $0<\delta_{1}t<\delta_{2}t<\ldots<\delta_{n}t<t$, i.e., at each instant time the spin qubit is flipped by the pulse, we want to determine the qubit phase coherence at the time $t$. Note that here we only consider ideal pulses, i.e., each pulse has a delta-function shape, so that the spin flip is accomplished at the instant time of the pulse applied~\cite{Uhrig2007}.
Under $n$-pulse sequences, the dephasing of the spin qubit due to 1/f charge noise reads as~\cite{Uhrig2007}
\begin{equation}
\Gamma^{d}_{\rm ph}(t)=\frac{y^{2}_{0}}{2r^{2}_{0}}\int^{\omega_{\rm max}}_{\omega_{\rm min}}d\omega\,S(\omega)\frac{|y_{n}(\omega\,t)|^{2}}{(\omega/2)^{2}},
\end{equation}
where
\begin{equation}
y_{n}(\omega\,t)=1+(-1)^{n+1}e^{i\omega\,t}+2\sum^{n}_{l=1}(-1)^{l}e^{i\delta_{l}\omega\,t}.
\end{equation}
For Carr-Purcell-Meiboom-Gill (CPMG) pulse sequence, the $n$ pulses are applied at the following serious instant time $\delta_{l}=(l-1/2)/n$~\cite{Carr1954,Meiboom1958}, while for Uhrig pulse sequence $\delta_{l}=\sin^{2}(\frac{\pi\,l}{2n+2})$~\cite{Uhrig2007} ($l=1,\cdots\,n$). In principle, dynamical decoupling can prolong the qubit dephasing time to any desired time scale as long as enough number of pulses are applied~\cite{Medford2012}. The practical performance of dynamical decoupling is often limited by the fact that realistic pulses are impossible in delta-function shape, i.e., the flip of the spin must cost a finite time. The phase coherence of the spin qubit under dynamical decoupling is shown in Fig.~\ref{fig_dyndcoup}. As can be seen from the figure, the phase coherence time under spin echo is about $T^{\rm echo}_{2}\approx100$ $\mu$s. Under eight-pulse sequences, the spin dephasing time can be prolonged to $T_{2}\approx260$ $\mu$s. We also find that the CPMG-pulse sequences [see Fig.~\ref{fig_dyndcoup}(a)] perform a little better than the Uhrig-pulse sequences [see Fig.~\ref{fig_dyndcoup}(b)] in our model.
\begin{figure}
\includegraphics{dd.eps}
\caption{\label{fig_dyndcoup}The phase coherence of the spin qubit under dynamical decoupling. The noise spectrum strength is chosen as $A_{r_{0}=20~{\rm nm},T=100~{\rm mK}}=35$ MHz. (a) CPMG-pulse sequences. (b) Uhrig-pulse sequences.}
\end{figure}
\section{Relaxation of the spin qubit}
The relaxation time $T_{1}$, i.e., the lifetime, is also an important characteristic time of the qubit~\cite{Khaetskii2001}. Even if there is no pure-dephasing for the qubit, the phase coherence time $T_{2}$ can still be limited by the qubit relaxation $T_{2}=2T_{1}$~\cite{Golovach2004}. Here we examine whether the 1/f charge noise will give rise to spin relaxation~\cite{Huang2014} in our model. The possible relaxation mechanism comes from the third term in Hamiltonian (\ref{eq_decoherence}). There is no exact method in calculating the relaxation rate, instead, the Fermi golden rule is usually used to calculate this quantity~\cite{Khaetskii2001}
\begin{equation}
\Gamma_{\rm relax}=\frac{\pi}{\hbar^{2}}\int^{\omega_{\rm max}}_{\rm \omega_{min}}\,d\omega\rho(\omega)e^{2}\Xi^{2}(z^{2}_{0}+y^{2}_{0})\delta\left(\omega-\frac{2\Delta}{\hbar}\right),
\end{equation}
where $\rho(\omega)$ is density of state of the charge noise mode. It should be noted that the qubit level spacing is about 80 GHz and the maximal charge noise frequency is about $0.5$ MHz~\cite{yoneda2018}, such that there is no charge noise frequency can match the level spacing of the spin qubit. Here arises the problem of whether the upper bound of the charge noise spectrum is indeed in the MHz range~\cite{chan2018,Kawakami2016,bylander2011,kuhlmann2013}? Our simple argument in Sec.~\ref{sec_IV} suggests $\omega_{\rm max}\ll\,k_{B}T/\hbar$ ($\sim13$ GHz for $T=100$ mK). An upper bound of 20 KHz in a SiMOS quantum dot is reported in Ref.~\onlinecite{chan2018}. Even if the qubit level spacing lies in the range of the charge noise spectrum, i.e., $\omega_{\rm min}<2\Delta/\hbar<\omega_{\rm max}$, our following calculation shows that the spin relaxation time is actually very long. By making the replacement $\sum_{k}\rightarrow\int\,d\omega_{k}\rho(\omega_{k})$ in Eq.~(\ref{eq_noisespectrum}), we have $\rho(\omega)e^{2}r^{2}_{0}\Xi^{2}\equiv\hbar^{3}A^{2}_{r_{0},T}/(k_{B}T)$. Hence, the relaxation rate can be written as
\begin{equation}
\Gamma_{\rm relax}=\frac{\pi\hbar\,A^{2}_{r_{0},T}}{k_{B}T}\times\frac{z^{2}_{0}+y^{2}_{0}}{r^{2}_{0}}.
\end{equation}
For a Si quantum dot with the parameters given in Table~\ref{tab}, we have $\Gamma_{\rm relax}=0.4519$ Hz, hence ${\rm T}_{1}=2.2$ s, indeed is a very long relaxation time. Thus, based on the above analysis, we suggest that 1/f charge noise does not limit the spin relaxation time in a Si quantum dot integrated with a slanting field.
\section{Summary}
In summary, we have studied in detail the spin manipulation and the spin dephasing in a Si quantum dot integrated with a slanting magnetic field. The longitudinal slanting field not only gives rise to a modulated Rabi oscillation in the spin manipulation, but also mediates a longitudinal spin-charge interaction which leads to spin dephasing. Several practical strategies are also proposed to alleviate the spin dephasing. Also, 1/f charge noise does not limit the spin relaxation time due to the mismatching between the qubit level spacing and the charge noise frequency. Our study can help clarify the spin dephasing mechanism in Si quantum dot.
\section*{Acknowledgements}
This work is supported by the National Natural Science Foundation of China Grant No.~11404020, the Postdoctoral Science Foundation of China Grant No.~2014M560039, and Doctoral Fund of Yanshan University Grant No. BL18043.
|
{
"timestamp": "2019-04-30T02:17:30",
"yymm": "1804",
"arxiv_id": "1804.05476",
"language": "en",
"url": "https://arxiv.org/abs/1804.05476"
}
|
\section{introduction}
\label{sec:intro}
We consider the problem of approximating a binary matrix $X \in \{0,1\}^{m{\times}n}$ as the product of two other binary matrices $U \in \{0,1\}^{m{\times}p}$ and $V \in \{0,1\}^{n{\times}p}$ plus a third \emph{residual} matrix $E$,
\begin{equation}
X = UV^\intercal + E.
\label{eq:mf}
\end{equation}
\color{blue}
\mnote{1.1}The problem of Binary Matrix Factorization (BMF) arises naturally in many problems, in particular in the so called ``association matrices'' where a $X_{ij}=1$ indicates that some object $i$ belongs to group $j$~\cite{bmf-app-biclustering}, or some entity (e.g., a gene) $i$ is present in some species $j$~\cite{bmf-app-diversity,bmf-app-microbial}, or are related to some type of disease~\cite{bmf-app-tumor}. The latter examples show the increasing relevance of this type of data in genomics. Other similar, growing applications include recommender systems (client-product preferences, etc.).
\color{black}
The BMF problem dates back to at least the 1960s~\cite{bmf-oldest} and has been treated extensively
in the last three decades by various research communities, under quite different names. It was first
studied as a combinatorial problem as a particular case of the classic \emph{set covering} problem
(see~\cite{monson95} and references therein). It then received great attention from the data mining
community.
\color{blue}\mnote{2.2-3} Long known to be an NP-hard problem~\cite{asso}\color{black}, the earlier works in this field developed heuristics such as the \emph{tiling} or \emph{tile matching/searching} methods, where binary matrices are decomposed as Boolean or modulo-2 superpositions of rectangular tiles~\cite{proximus,tiling}; the BMF problem was later formulated as a matrix factorization problem in~\cite{bmf07}, with several works following that line since then.
\color{blue}
Being an NP-hard problem, the quality of the decompositions $(U,V)$ is commonly \mnote{1.2}measured in terms of their \emph{interpretability}, that is whether the columns of $U$ and $V$ exhibit patterns that are intuitive in some sense, or reflect \emph{a priori} knowledge about the problem. For example, in data mining problems, where the pairs $(U_i,V_i)$ are interpreted as \emph{association rules} (for example, a group of clients -- indicated by non-zeros in $U_i$ -- prefers a certain group of products -- indicated by non-zeros in $V_i$);
the examples in this paper are designed to show this interpretability in a visual way, by studying the results on visual patterns obtained on sets of images.
\color{black}
A thorough survey of BMF methods is beyond the scope of this paper; we refer the reader
to~\cite{bmf-comp} for a more in-depth review. We will however mention some works which are
representative of the diversity of formulations and tools that surround the treatment of this
problem, as well as the shortcomings that are common to the current state of the art and that
motivate the development of the tools that we present in this work.
\subsection{Brief overview of Binary Matrix Factorization}
We begin with the ASSO algorithm proposed in~\cite{asso}. The method begins by constructing the so
called \emph{association matrix} $C$, which is a thresholded version of a particular normalization
of the correlation matrix $X{^\intercal}X$. It then produces a series of increasing rank approximations
by adding to $U$ a column taken from $C$ and searching for a corresponding binary row $V_k$ whose
outer product with $U_k$ minimizes the number of non-zeros in the current approximation error
$E$. This method can be efficiently implemented with bitwise and integer operations. It is also a
popular and simple method with good performance. However, the complexity of each ASSO step is
$O(kn^2m)$, so that it does not scale well in applications where $n$ (the number of samples) is very
large, something very common in current data science problems, which is the target of our work.
\color{blue}
\mnote{0.1}
\mnote{2.2-3}
Many works approximate the BMF problem by a relaxed (non-convex) Non-negative Matrix Factorization (NMF) problem where $U$ and $V$ are allowed to take on real values, and then map the resulting (approximate) solution to the binary domain using some predefined rule. Examples of this are~\cite{bmf07,bmf13,bmf-nmf2}. In particular, the work~\cite{bmf13} develops a set of BMF \emph{identifiability} conditions, that is, conditions under which the binary factorization of $X$ is unique (up to permutations). However, as in~\cite{bmf07,bmf13}, their solution is an approximation based on a local minima of the NMF problem, so there are no guarantees that the binarized pair $(U,V)$ obtained coincides with the unique solution even if the identifiability conditions are satisfied.
Being based on non-linear optimization, the NMF methods are significantly more computationally demanding than binary methods such as ASSO.
\color{black}
The work~\cite{bmf-mp} stands out as an interesting alternative to BMF which formulates the
decomposition of $X$ as a Bayesian denoising problem with a particular prior on the
unobserved \emph{clean} matrix $\hat{X}$ ($X=\hat{X}+E$) and uses a \emph{Message Passing} algorithm
to find the maximum a posteriori estimation of $\hat{X}$. Message Passing is a mature technique
which in the form presented in this work can be quite computationally demanding. Approximate Message
Passing \cite{amp,gamp} techniques have since been developed which may provide significant
efficiency gains to the technique proposed in~\cite{bmf-mp}, but we are currently unaware of any
development in this direction.
An example of a tile-searching heuristic is the Proximus method~\cite{proximus}, which approximates
the first principal left and right binary components of the binary matrix $X$. The method can be
extended to produce a hierarchical representation of the matrix with further rank-$1$ components,
although these do not coincide with additional factors in a rank-$k$ factorization. Contrary to the
above methods, the focus of Proximus is on speed an scalability, and indeed is much faster and
scales better than any of the above methods. Also, as we will see in the next subsection, finding
the first principal component of a binary matrix is closely related to a crucial step in one of the
main dictionary learning methods. e will describe this method in detail later in this document.
\color{blue}
\subsection{Dictionary learning}
\emph{Dictionary learning} methods were first introduced in \cite{olshausen97}
and later adopted as a powerful extension of the \emph{transform analysis} concept, ubiquitous in
signal processing since the introduction of Fourier analysis and their related discrete variants
(DFT, DCT). In this setting, the matrix $X \in \ensuremath{\mathbb{R}}^{m{\times}n}$ consists of $n$ columns of
dimension $m$, and the decomposition obtained $X=DA+E$ is comprised of a \emph{dictionary} $D$, a
matrix of \emph{linear coefficients} $A$, plus a residual matrix $E$. Dictionary learning methods
are adaptive: given sufficient data samples, they can be trained to efficiently represent such
samples as a linear combination of very few basis elements or ``atoms'' (see~\cite{dl-review} for a
review). Despite coming from a very different community, dictionary learning methods can be seen as
matrix factorization methods which are specially tailored to the where $X$ is either extremely
``fat'' ($n \gg m$) or ``tall'' ($n \ll m$). Furthermore, many of these methods can be implemented on-line, that is,
they can process new data samples as they arrive, and adapt the dictionary along the
way~\cite{online-dl}. Another important aspect of dictionary learning methods is that they are not
restricted to low-rank decompositions; if fact, the solutions can be \emph{overcomplete}, meaning that the
number of columns $p$ in $D$ may be (much) larger than $m$, the dimension of the samples to be
represented.
\mnote{2.2-3}As with all BMF methods, dictionary learning problems are non-convex and their solution cannot be obtained exactly. Nevertheless, similarly to what happens with BMF, their enormous practical success in a wide range of signal processing and machine learning problems, and their ability to produce human-interpretable patterns, has led to their widespread adoption.
\color{black}
\subsection{Model selection}
As with any statistical model, the problem of model selection, (in this case, choosing $p$) is of
paramount importance to BMF. Various works~\cite{bmf-mdl,bmf-tiling-mdl,bmf-sel,panda} have
addressed this particular problem. In particular,~\cite{bmf-mdl} and~\cite{bmf-tiling-mdl} are based
on the Minimum Description Length (MDL) principle~\cite{mdl1,mdl2,mdl3}, which forms the basis of
our model selection strategy as well. As a side note, the work~\cite{bmf-tiling-mdl} represents a
recent example of the tiling approach. This problem has also been addressed in dictionary learning
in~\cite{dl-mdl}.
\subsection{Main contribution}
\color{blue}
To the best of our knowledge, the works in the existing literature on BMF make no particular
assumptions on the \emph{shape} of the matrices to be decomposed. In particular, many methods which
are efficient for the $n \approx m$ case do not scale well for $n \gg m$ and vice versa. Also, most
methods deal with the \emph{offline} analysis of readily available matrices, which makes them
unsuitable to many recent data processing tasks involving \emph{online} adaptation of the models.
The main motivation behind this work is the ability of dictionary learning methods to cope with the challenges mentioned above. We propose two dictionary learning-based BMF methods which are particularly suited to the treatment of extremely fat (or tall) matrices, and which are also suitable to online processing of samples. The first one, named Method of Binary Directions (MOB), is an adaptation of the method proposed in~\cite{online-dl}, itself an adaptation of the Method of Optimum Directions (MOD)~\cite{mod} (hence the name). The second method is based on the idea of sequential rank-one updates of the K-SVD method~\cite{aharon06}; we adopt the Proximus~\cite{proximus} algorithm as a fast approximation to the mentioned operation; we thus bring together tools from two inherently related but otherwise disconnected fields to obtain a novel formulation to the binary matrix factorization. Finally, both methods rely on a third novel algorithm which we call Binary Matching Pursuit (BMP). This is a binary adaptation of the Matching Pursuit method~\cite{mp} for approximating the sparsest solution to a least squares regression problem. \mnote{2.1}This adaptation, although conceptually analogous to MP, is non-trivial, as its efficiency relies on a careful combination of different algebraic operations.
\color{black}
Our methods construct binary dictionaries and binary coefficients matrices using efficient bitwise and a few integer operations. This is particularly relevant to the efficiency of our methods as recent processor architectures incorporate the ability to handle large number of bits through SIMD (Single Instruction Multiple Data) instructions. For example, a current off-the-shelf processor can perform a \emph{popcount} instruction (which counts the number of $1$s in a binary array) on a 256-bit register. This allows, for example, to compute the dot product between two binary vectors of dimension $256$ with just two processor instructions. \color{blue} \mnote{2.4}As we show in Section~\ref{sec:bdl}, our methods are also linear both in $n$ and $m$ and thus scale well for large matrices.\color{black}
\color{blue} The main objectives of this paper are to present our proposed methods, assess their interpretability on different datasets, to analyze their computational properties such as computational complexity and convergence rate, and to see how these properties are affected by the initial conditions. We thus focus our experiments on a small set of easily-interpretable datasets for which the patterns obtained can be easily recognized as salient features in the data. \color{black}
The rest of this document is organized as follows: Section~\ref{sec:background:dictionary-learning} provides the notation and background on the methods on which our methods are based. The proposed methods themselves are described in Section~\ref{sec:bdl}.
Section~\ref{sec:model-selection} describes the proposed MDL-based model selection algorithm for searching the best model order $p$. %
We present and discuss our results in Section~\ref{sec:results}, and provide concluding remarks in Section~\ref{sec:conclusion}.
\section{Background}
\label{sec:background}
\subsection{Notation}
\label{sec:background:notation}
\def\mathbf{1}{\mathbf{1}}
\def\mathrm{bool}{\mathrm{bool}}
\def\circ{\circ}
\def\lor{\lor}
\def\bigwedge{\bigwedge}
\def\bigvee{\bigvee}
\def\oplus{\oplus}
\def\otimes{\otimes}
\def\mathrm{mod}{\mathrm{mod}}
\newcommand{\iter}[1]{^{(#1)}}
\newcommand{\ensuremath{\quad\mathrm{s.t.}\quad}}{\ensuremath{\quad\mathrm{s.t.}\quad}}
\newcommand{\norm}[1]{\ensuremath{\left\|#1\right\|}}
\newcommand{\support}[1]{\mathrm{supp}(#1)}
\newcommand{\rankf}[1]{\mathrm{rank}(#1)}
\def\mathrm{rank}{\mathrm{rank}}
\newcommand{\fun}[1]{\mathrm{#1}}
\newcommand{\abs}[1]{\ensuremath{\left|#1\right|}}
\newcommand{\setdef}[1]{\ensuremath{\left\{#1\right\}}}
\newcommand{\ensuremath{\mathrm{span}}}{\ensuremath{\mathrm{span}}}
\newcommand{\svec}[1]{_{[#1]}}
\newcommand{\row}[1]{_{#1:}}
\newcommand{\col}[1]{_{:#1}}
\newcommand{\havg}[1]{\langle\!\langle{#1}\rangle\!\rangle}
We begin this section by establishing the notation to be used throughout the paper.
Standard operations such as addition or subtraction are denoted as usual, $1+1=2$, $1-1=0$,
etc. Given two binary values $a$ and $b$, we use $a \land b$ to denote their logical AND (Boolean product),
$a \lor b$ is their OR (Boolean sum), $a \oplus b$ is modulo-2 addition (Boolean eXclusive OR), and
$\neg a$ is the 1's complement (Boolean negation) of $a$. The same notation is used for element-wise operations between vectors and matrices of the same dimension.
Let $x$ and $y$ be two binary vectors and $A$ and $B$ two binary matrices.
We denote the standard inner and outer vector-vector, matrix-vector and matrix-matrix products as $x{^\intercal}y$, $xy^\intercal$, $Ax$ and $AB$.
The Boolean inner product between two vectors, $z = x^\intercal \circ y$ is defined as $\bigvee x_i \land y_i$. Similarly, the $C_{ij}$ element of the matrix product $C=A \circ B$ is defined as the Boolean product between the $i$-th row of $A$, $A\row{i}$, and the $j$-th column of $B$, $B\col{j}$. We define the modulo-2 inner product of $x$ and $y$ as $x \otimes y = (x+y)\;\mathrm{mod} \;2$. Finally, similarly to the Boolean case the $C_{ij}$ element of the modulo-2 product between two matrices, $C=A \otimes B$ is given by $A\row{i} \otimes B\col{j}$.
The cardinality of a set $J$ is denoted by $|J|$. Given a set of indexes $J$, $B\col{J}$ denotes the
sub-matrix of columns of $B$ indexed by $J$, and $B\row{J}$ denotes a subset of its rows. These can
be combined with single indexes or other sets, e.g., $B_{iJ}$ contains the elements of row $i$ and
column indexes in $J$. For a
vector $x$, its Hamming weight $h(x)$ is defined as the number of non-zero elements in $x$, that is
$h(x)=|\{i: x_i \neq 0\}|$. The same notation $h(A)$ is applied to count the non-zero elements of
matrices. The Hamming weight is usually referred to as the $\ell_0$ pseudo-norm,
$\|x\|_0=h(x)$. Note that, for binary vectors, $h(x)=\|x\|_0=\|x\|_1=\norm{x}_2^2$. Likewise, for
binary matrices, $\|A\|_0=\|A\|_{1,1}=\|A\|_F^2$. The function $\mathbf{1}(\cdot)$ is defined so
that $\mathbf{1}(cond)=1$ if $cond$ is true, and $0$ otherwise.
\color{blue}
\subsection{A note on computational complexity}
\mnote{2.4}For each algorithm we provide a brief computational complexity analysis using a
simplified form of the familiar ``big O'' notation. Here a procedure which has order $O(m)$ is said
to have \emph{linear complexity} in $m$, that is, it requires at most $am + b$ operations where $a$
and $b$ do not depend on $m$. Similarly $O(m^2)$ indicates \emph{quadratic complexity}, and
$O(m \log m)$ requires about $a(m \log m) + b$ operations in the worst case. This use is slightly
different than the formal Bachmann-Landau definition of $O(\cdot)$, which is defined in asymptotic
terms. For example, we might write $O(m \log m + p)$ as a shorthand to $O(m \log m) + O(p)$ to
indicate that a function requires about $am \log m + bp + c$ operations.
On the other hand, in all cases we will make a distinction as to the nature -- floating point,
integer, bitwise -- of the operations, as the constants involved in each case can vary greatly. For
example, as pointed out in the introduction, a bitwise inner product of two binary vectors of length
$256$ can computed in just \emph{two} CPU instructions, whereas it might require over $512$ instructions to compute the same operation on floating point or integer vectors.
\color{black}
\subsection{Dictionary Learning and sparsity}
\label{sec:background:dictionary-learning}
\color{blue}
As described before, dictionary learning methods seek a to decompose $X$ as $X=DA+E$, where $D$ is such that we can achieve $\|E\col{j}\| \ll \|X\col{j}\|$ by using just a few atoms of $D$, that is, with a number of non-zero elements in $A\col{j}$ much smaller than $p$. The latter requirement is called \emph{sparsity}. Thus, dictionary learning is often also referred to as \emph{sparse modeling}, and finding $A\col{j}$ given $D$ as \emph{sparse coding}.
A typical approach to the dictionary Learning problem is to obtain a local solution by \emph{alternate minimization} in $D$ and $A$ of a cost function $f(D,A) + g(A)$,
\begin{eqnarray}
A\iter{t+1} =& \arg\min_{A} \{ f(D\iter{t},A) + g(A) \} \\
D\iter{t+1} =& \arg\min_{D} \{ f(D,A\iter{t+1}) + g(A\iter{t+1}) \},
\label{eq:dl}
\end{eqnarray}
where $f(\cdot)$ is a \emph{data fitting} term, usually $\|DA-X\|_F^2$ (here $\|X\|_F$ denotes the Frobenius norm of matrix $X$), and $g(\cdot)$ is a \emph{regularization} term which promotes sparsity in the columns of $A$. We now describe the two methods on which our methods are inspired.
\color{black}
\subsubsection{Method of Optimal Directions (MOD)}
\label{sec:mod}
For the case $f(D,A)=\|DA-X\|_2^2$ and $g(A)=\sum_{j}\|A\col{j}\|_1$ the \emph{Method of Directions} (MOD)~\cite{mod}, is given by
\begin{eqnarray}
A_j\iter{t+1}\!\! &=&\!\! \arg\min_{a \in \ensuremath{\mathbb{R}}^p} \{ \|x_j - D\iter{t}a \|_2^2 + \|a\|_1 \}, \\
D_r\iter{t+1}\!\! &=&\!\! U_r/\min\{1,\|U_r\|_2\},\nonumber\\
\!\!&&U=\!\!X(A\iter{t+1})^\intercal\left(A\iter{t+1}({A}\iter{t+1})^\intercal\right)^{-1},\,
\label{eq:mod}
\end{eqnarray}
The first step corresponds to an $\ell_1$-regularized least squares regression problem on each column of $A$, also known as LASSO~\cite{lasso}, a non-differentiable convex problem whose solution has been extensively studied in recent years, with several efficient algorithms designed specifically for the task (see e.g.~\cite{fista}).
In the second step, each atom $D\col{r}$ of the dictionary corresponds to a normalized down version of the least squares solution $u_r$.
The MOD algorithm is well suited for online dictionary adaptation, as both $A{A}^\intercal$ and
$XA^\intercal$ can be efficiently updated when new columns are added to $X$. Furthermore, if new
samples arrive one at a time, the inverse of the Hessian matrix $({A}A^\intercal)^{-1}$ can be
efficiently updated via the Matrix Inversion Lemma~\cite{matrix-inv-lemma}. Moreover, as shown
in~\cite{online-dl}, excellent results can be still obtained if the Hessian is approximated by its
diagonal (in which case computing its inverse requires just $O(p)$ operations).
\color{blue}
\paragraph*{Computational complexity} The offline version of the MOD dictionary \mnote{2.4}update step presented in Algorithm~\ref{eq:mod} requires $O(mnp) + O(p^2n) + O(p^3)$ floating point operations, which will generally be dominated by the $(p^2n)$ term.
\color{black}
\subsubsection{Matching Pursuit and the K-SVD algorithm}
\label{sec:ksvd}
In this algorithm, proposed in~\cite{aharon06}, $f(E)=\norm{E}_F^2$ and $g(A)=h(A)$. The columns of $A$ are computed using a greedy method known as OMP (Orthogonal Matching Pursuit)~\cite{omp}, which under certain conditions can be shown to provide the actual solution to the corresponding $\ell_0$-penalized least squares problem (see~\cite{tropp07}). A simpler variant of this step uses the (non-orthogonal) Matching Pursuit (MP)~\cite{mp}, which is described next in Algorithm~\ref{alg:mp},
\begin{algorithm}[ht]
\caption{\label{alg:mp}Matching Pursuit}
\KwData{vector to encode $x$, dictionary $D$, maximum residual norm $\epsilon$, maximum coefficients weight $h_{\max}$}
\KwResult{Coefficients vector $a$}
Set iteration $t \leftarrow 0$, residual $r\iter{0} \leftarrow x$, initial coefficients $a\iter{0} \leftarrow 0$\;
Set $g\iter{0} \leftarrow D^\intercal{r\iter{0}}$, $G \leftarrow D^\intercal{D}$ \;
\While{$\norm{r\iter{t}} \geq \epsilon$ and $h(a) < h_{\max}$}{
$i = \arg\max \left\{g\iter{t} \right\} $ \;
${\Delta} \leftarrow D\col{i}^\intercal r\iter{t}$ \;
$a_i\iter{t+1} \leftarrow a_i\iter{t} + \Delta$ \;
$r\iter{t+1} \leftarrow r\iter{t} - {\Delta}D\col{i} $\;
$g\iter{t+1} \leftarrow g\iter{t} - {\Delta}G\col{i} $\tcc*{(a)}
$t \leftarrow t+1 $ \;
}
\Return $a \leftarrow a\iter{t}$ \;
\end{algorithm}
What MP does at each iteration is to project the residual onto the atom that is most correlated to it, and then remove the projection from the residual. For this to work well, the atoms must be normalized to have $\ell_2$ norm 1.
The vector $g$ keeps track of the correlation between the residual $r$ and the dictionary $D$. Its update $(a)$ is derived as follows:
\begin{eqnarray}
g\iter{t+1}
&=& D^\intercal{r\iter{t+1}}=D^\intercal(r\iter{t}-{\Delta}D_i)\nonumber\\
&=& g\iter{t}-{\Delta}D^\intercal{D}\col{i} =g\iter{t}-{\Delta}G\col{i}.
\label{eq:corr-up}
\end{eqnarray}
Note that, by means of \refeq{eq:corr-up}, updating $g$ requires only $O(p)$ floating point operations, whereas the na\"{\i}ve update requires $O(mp)$ operations. This same trick, with a few modifications, will also be useful in the binary case.
\color{blue} \mnote{2.4}As the Gramm matrix is computed only once, the overall complexity of MP for processing all $n$ samples is $O(p^2m + p + kmn)$ floating point operations.\color{black}
The dictionary update of K-SVD performs a simultaneous update of each atom $D\col{r}$ and the row of $A$ associated to it. This update is described in Algorithm~\ref{alg:ksvd}.
\begin{algorithm}
\KwData{Current iterate $(D\iter{t}$, $A\iter{t})$}
\KwResult{Next iterate $(D\iter{t+1}$,$A\iter{t+1}$}
\For{$r=1,\ldots,p$}{
$J \leftarrow \{j: A_{rj}\iter{t} \neq 0\}$ \;
$R \leftarrow X\col{J} - D\iter{t}A\col{J}\iter{t} + D\col{r}\iter{t}(A_{rJ}\iter{t})$ \;
$U{\Sigma}V^\intercal \leftarrow \mathrm{SVD}(R)$ \;
$D\col{r}\iter{t+1}\leftarrow U\col{1}$ \;
$A_{rJ}\iter{t+1} \leftarrow V\col{1}$ \;
}
\caption{\label{alg:ksvd}K-SVD Dictionary update.}
\end{algorithm}
\color{blue}
\paragraph*{Computational complexity} Each of the $p$ updates of $D$ requires $O(mn)$
\mnote{2.4} operations for updating the residual plus another $O(n^2)$ operations for computing the first
pair of singular eigenvectors. This gives a total $O(pn^2) + O(pmn)$ operations, which in general
will in our case ($n \gg m \approx p$) will be significantly more expensive than the $O(p^2n) + O(pmn) + O(p^3)$ complexity of MOD.
\color{black}
\section{Binary Dictionary Learning}
\label{sec:bdl}
Below we describe our dictionary learning methods, which assume a given fixed dictionary size $p$
and employ the traditional alternate descent approach to obtain the best model $(D,A,E)$ for that
$p$. We leave the description of the top-level model selection algorithm for choosing the best model
order $p$ to Section~\ref{sec:model-selection}.
Both BMF algorithms share a common coefficients update step, the Binary Matching Pursuit (BMP)
algorithm, and two choices for the dictionary update step: MOB (a binarized version of MOD) and
K-PROX (combining ideas of K-SVD and Proximus); these are detailed next.
\subsection{Coefficients update via Binary Matching Pursuit (BMP)}
\label{sec:bdl:bmp}
\begin{algorithm}
\KwData{sample to encode $x$, dictionary $D$, initial coefficients $a\iter{0}$, maximum coeffs. weight $h_{\max}$, maximum residual weight $w_{\max}$.}
\KwResult{Optimum coefficients for $x$, $a$}
Set iteration $t=0$, coefficients $a\iter{0}=a_0$ ,residual $r\iter{0}=x \oplus D{\otimes}a\iter{0}$\;
Set modulo-2 Gramm matrix $G \leftarrow D^\intercal \otimes D$\tcc*{(a)}
Set residual correlation $g\iter{0} \leftarrow D^\intercal{r\iter{0}}$\tcc*{(b)}
\While{$h(r\iter{t}) \geq w_{\max}$ \textbf{and} $t < h_{\max}$ }{
$k \leftarrow \arg \max_l \{\;|g\iter{t}_l|\;/\;\|D\col{l}\|_0\;\} $ \tcc*{(c)}
\If{$g\iter{t}_{k} = 0$} {
\Return $a \leftarrow a\iter{t}$ \;
}
$r\iter{t+1} \leftarrow r\iter{t} \oplus D\col{k} $\;
\If { $h(r\iter{t+1}) \geq h(r\iter{t})$ }
{
\Return $a \leftarrow a\iter{t}$ \;
}
\eIf {$a\iter{t+1}_{k} = 1$} {
$a\iter{t+1}_{k} \leftarrow 0$; $g\iter{t+1} \leftarrow g\iter{t} - G\col{k}$\tcc*{(d)}
} {
$a\iter{t+1}_{k} \leftarrow 1$; $g\iter{t+1} \leftarrow g\iter{t} + G\col{k}$ \tcc*{(d')}
}
}
\Return $a \leftarrow a\iter{t}$ \;
\caption{\label{alg:bmp} Binary Matching Pursuit.}
\end{algorithm}
In essence, BMP is a binarized version of the Matching Pursuit Algorithm~\ref{alg:mp}. For a given
sample $x$, we begin ($t=0$) with an initial coefficients vector $a\iter{0}=a_0$, a residual
$r\iter{0}_j=x \oplus D \otimes a\iter{0}$ and an initial vector $g\iter{0} = D^\intercal r\iter{0}$
which keeps track of the correlation between the columns of $D$ and $r\iter{t}$. Then, at each
iteration $t$ we determine the atom $D\col{k}$ which is most correlated to $r\iter{t}$. We
then \emph{toggle} the coefficient corresponding to that atom, $a_k\iter{t}$, and update
$r\iter{t}$ and $g\iter{t}$ accordingly. The pseudocode is given in Algorithm~\ref{alg:bmp}.
Some steps of this algorithm, marked as $(a)$, $(b)$, $(c)$ and $(d)$ (appearing twice) in
Algorithm~\ref{alg:bmp}, are not obvious from the overall description of the algorithm and need to
be clarified. In $(a)$, the \emph{modulo-2 Gramm matrix} of $D$ is computed; this matrix is used in
an analogous way in Algorithm~\ref{alg:mp} for the fast update of the correlations vector $g$ as
described in \refeq{eq:corr-up}. In $(b)$, we compute the standard correlation between the columns of
$D$ and $r$. In $(c)$, since the atoms are not normalized, the best candidate is chosen using a
form of normalized correlation called \emph{association accuracy}~\cite{association-accuracy},
$g\iter{t}_l / \|D\col{l}\|_2^2 = g\iter{t}_l / \|D\col{l}\|_0$.
Finally, in $(d)$ and $(d')$, despite the correlation vector $g$ being initially computed using
the \emph{standard} matrix-vector product, its updated has exactly the same form
as \refeq{eq:corr-up}, but the Gramm matrix $G$ is actually computed using the modulo-2;
$(d)$ corresponds to the case when $\Delta=1$, that is, when $a_k$ is switched from $0$ to $1 $, and
$(d')$ corresponds to $\Delta=-1$, when $a_k$ is switched off. (This curious result is easily verified by writing down the corresponding arithmetic.)
\color{blue}
\paragraph*{Computational complexity} The initialization of BMP requires $O(p^2m)$ binary
operations for computing the Gramm matrix and integer ones $O(p^2m)$ for the \mnote{2.4}correlation vector
$g$. Then, for each sample $X\col{j}$ a maximum of $h_{\max}$ iterations can be required, each one
requiring $O(p)$ floating point operations for finding the most correlated atom, another $O(p)$
integer operations for updating the residual $r$, and another $O(p)$ integer ones for updating $g$, for a total of $O(h_{\max}p)$ operations per sample. Overall, a whole pass over the $n$ data samples requires $O(p^2m)+ O(p) + O(h_{\max}pn)$ operations.
\color{black}
\subsection{MOB: Method Of Binary Directions}
\label{sec:bdl:mod}
Here we want to update $D$ so as to minimize the Hamming weight $h(E)$ of the residual matrix $E =
X \oplus D \otimes A$. Suppose we want to update the $r$-th atom at iteration $k$. The
affected columns will only be those for which the coefficients in $A$ corresponding to that atom
are non-zero. We define $J=\{j : A_{rj} \neq 0 \}$ to be the set of indexes of those columns. What
we want is to update $D\col{r}$ so that the weight of the columns of the residual
affected by it is minimized,
\begin{eqnarray}
D\col{r}\iter{t+1} &=& \arg\min_{d \in \{0,1\}^m} \sum_{j \in J} h(E\iter{t}\col{j} \oplus d) \nonumber\\
&=& \arg\min_d \sum_{j \in J} (\sum_i E\iter{t}_{ij} + |J| d_i).
\label{eq:mob1}
\end{eqnarray}
According to~\refeq{eq:mob1}, the optimization of $D\col{r}\iter{t+1}$ is separable in each of its elements,
\begin{eqnarray}
D_{ir}\iter{t+1} &=& \arg\min_{u \in \{0,1\}} \sum_j E_{ij}\iter{t} \oplus u \nonumber\\
&=& \arg\min_{u \in \{0,1\}} \left|\;\sum_{j\in J} \;E_{ij}\iter{t} - |J|u\;\right| \nonumber\\
&=& \arg\min_{u \in \{0,1\}} |\,h( E_{iJ}\iter{t} ) - |J|u\,|\nonumber\\
&=& \mathbf{1}\left( \frac{ h( E_{iJ}\iter{t} ) }{|J|} > \frac{1}{2} \right)
\label{eq:mob2}
\end{eqnarray}
Note that when $h(E_{iJ})/|J| = 1/2$ both $0$ and $1$ are optimum, in which case we use $0$. Also, note that \refeq{eq:mob2} can be rewritten as
$$D_{ir}\iter{t+1} = \mathbf{1}\left( \frac{h(EA\row{r}^\intercal)} {h(A\row{r})} > \frac{1}{2} \right).$$
As these statistics can be easily updated as new samples arrive, it follows that MOB is suited for fast online dictionary adaptation. (Our implementation of this feature is still work in progress.)
\color{blue}
\paragraph*{Computational complexity of MOB} Our current (offline) \mnote{2.4} implementation requires $O(mnp)$ bitwise operations for the modulo-2 product $E A^\intercal$, $O(np)$ integer operations for computing the weights of the rows of $A$, and $O(mp)$ integer comparisons for updating $D$, for a total of $O(mnp)$ binary plus $O(mp)$ integer operations.
\color{black}
\subsection{K-PROX: Dictionary Update via Proximus}
\label{sec:bdl:k-prox}
In this case, following the K-SVD concept, we want to obtain the best rank-one approximation to the
residual $E$ obtained after removing the contribution of $D\col{r}$. Let $J = \{j: A_{rj} \neq 0 \}$
and $R_{J}=D\col{r}{\otimes}A_{rJ} \oplus E_{J}$. We then have,
\begin{equation}
(D\col{r}\iter{t+1},A_{rJ}\iter{t+1}) = \arg\min_{u,v} h\left( R\col{J}\iter{t} \oplus uv^\intercal \right).
\label{eq:bsvd1}
\end{equation}
As the name K-PROX implies, our approximation to the NP-hard problem \refeq{eq:bsvd1} is based on the Proximus algorithm~\cite{proximus}, summarized in Algorithm~\ref{alg:proximus},
\begin{algorithm}
\caption{\label{alg:proximus} Proximus}
\KwData{matrix $X \in \{0,1\}^{m{\times}n}$, $u\iter{0} \in \{0,1\}^m$, $v\iter{0} \in \{0,1\}^n$ }
\KwResult{Vectors $u$, $v$ so that $X \approx uv^\intercal$}
Set iteration $k=0$\;
\Repeat {$u\iter{t+1}(v\iter{t+1})^\intercal = u\iter{t}(v\iter{t})^\intercal$} {
$u\iter{t+1}_i\!\! \leftarrow \mathbf{1} \left( X\row{i} v\iter{t} > h(v\iter{t})/2 \right),\;i=1,\ldots,m$ \;
$v\iter{t+1}_j\!\! \leftarrow \mathbf{1}\left( X\col{j}^\intercal{u\iter{t+1}}\!>\!h(u\iter{t+1})/2 \right),j=1,\ldots,n$ \;
$k \leftarrow k+1 $ \;
}
\end{algorithm}
Interestingly, Algorithm~\ref{alg:proximus} provides a local optimum to the rank-one approximation that we seek. This is stated in Proposition~1 below.
\begin{proposition}
The output $(u,v)$ of the Proximus Algorithm~\ref{alg:proximus} is a local optimum of the problem $\min \|X - uv^\intercal\|_0$.
\end{proposition}
\begin{proof}
Given $v\iter{t}$, it is easy to check that the update $u\iter{t+1}$ in Algorithm~\ref{alg:proximus} is the value of $u$ that \emph{globally} minimizes $\|X \oplus uv\iter{t+1} \|$ (if $ s\iter{t}_i = w\iter{t}/2$, both $0$ and $1$ are equally optima; in such case, we default to $0$). The same happens with the update $v\iter{t+1}$ given $u\iter{t+1}$. Therefore, $h(E\iter{t})=h(X \oplus u\iter{t}(v\iter{t})^\intercal)$ cannot increase with $k$. As $h(E\iter{t}) \geq 0$ is bounded, non-increasing, and the iterates can take on a finite number of values, the sequence $h(E\iter{t})$ must converge after a finite number of steps. Let $(u,v)$ be the arguments at which the stopping condition is satisfied. By definition of the algorithm, no change in $u$ or $v$ decreases the objective. This guarantees that $(u,v)$ is a local minimum in a Hamming ball of radius at least $1$.\footnote{We cannot guarantee that a simultaneous change in a single coordinate of $u$ and a single coordinate of $v$ will not decrease the cost function!.}
\end{proof}
\color{blue}
\paragraph*{Computational complexity of K-PROX} As with the K-SVD algorithm, we update $D$ one atom at a time. This requires $O(mn)$ operations for computing \mnote{2.4}the residual in \refeq{eq:bsvd1} and running Algorithm~\ref{alg:proximus}, which takes a finite number of iterations requiring $O(mn) + O(np)$ bitwise operations and the same number of integer comparisons.
\color{black}
\subsection{Initialization}
\label{sec:bdl:init}
Initialization is of paramount importance to the success of any non-convex matrix factorization method. At the same time, there is no provably optimum way of doing so, otherwise we would be contravening the NP-hard nature of the factorization problem itself. We are left with heuristics based on intuition and prior information, if any. Ultimately, \emph{initialization is an art}, and also an engineering decision which may depend on several aspects. As an example, we could use the resulting factorization obtained with \emph{any} of the existing methods mentioned in Section~\ref{sec:intro} as an initial point. In this case, to maintain scalability and simplicity, we experiment with two simple but generally effective methods drawn also from dictionary learning literature: given dictionary size $p$, the first draws $p$ atoms using a pseudo-random Bernoulli$(\theta)$ distribution; we use $\theta=1/2$, but other values could be used to reflect prior information on the problem.
The second method is to draw $p$ columns from $X$ at random and use them as the initial atoms. WE report on both strategies on Section~\ref{sec:results}.
\section{Model selection}
\label{sec:model-selection}
Beyond theoretical results and simulations, when confronted with real data,
the true underlying model governing the generation of data is rarely
available. In such situations, models can only be assumed to be tools for
understanding the data at hand, and the best choice is dictated by how much
regularity or structure each candidate model is capable to grasp from that
data. In particular, the complexity of a model is limited by the amount of
data available to estimate the different parameters of the model. An overly
complex model will tend to \emph{overfit} the data, whereas an overly simple
one will miss important details. The problem of model selection is that of
finding the best model for a given data set. Typically, this is done by
seeking a balance between the \emph{goodness of fit} of a model, usually
expressed in terms of the likelihood of the data given the model, and a
measure of the \emph{model complexity}; some popular examples in this
category are the Bayesian Information Criterion (BIC)~\cite{bic}, the
Akaike's Information Criterion~\cite{aic}, and the Minimum Description
Length (MDL) principle~\cite{mdl1,mdl2,mdl3}.
MDL translates the model
selection problem as one of data compression, where both the data and the
model have to be (hypothetically) transmitted and perfectly recovered using
some encoding mechanism. The tension between complexity is then resolved by
the number of bits (\emph{codelength}) required to describe the data in
terms of the model (\emph{stochastic complexity}) and the number of bits
required to describe the model itself (\emph{model complexity}). The
original version of MDL~\cite{mdl1} made this division explicit, and is
asymptotically equivalent to BIC. However, the modern version of
MDL~\cite{mdl2,mdl3} uses the more recent
information-theoretic \emph{universal coding theory} to make the
aforementioned balance implicit by producing an optimum, joint description
of both the model and the data which is furthermore independent of arbitrary
choices of, for example, the way the model is parameterized.
These advantages make MDL an appealing method for model selection.
In our case, the data $X$ is described by the triplet $(D,A,E)$. We describe
each of these components separately using universal codes, so that
$L(X)=L(D)+L(A)+L(E)$ is the total codelength of describing $X$. Here the
tension between a good fit and a simple model is represented respectively by
$L(E)$ and $L(D)+L(A)$.
\subsection{\revTwo{Forward selection algorithm}}
\color{blue}
Model selection is usually formulated as two nested problems. The inner
problem is how to search for the best model $(D,A,E)$ among all models of
\mnote{2.5} order $p$. The second problem is to choose the best model for $X$ among all
possible models of order $p \geq 0$. Given a codelength function $L(X)$,
our method uses a \emph{forward selection} strategy to sweep over all
models. Starting with a initial order $p=p_0$, we approximate the best model
given $p$ using one of our dictionary learning algorithms, and then
gradually increase $p$ until the codelength $L(X)$ is no longer diminished.
\mnote{2.4}
In going from $p$ to $p+1$ we employ a \emph{warm restart} strategy, that is, atoms and coefficients learned for model order $p$ are used as the starting point for learning the $p+1$-th order model. The overall procedure is summarized in Algorithm~\ref{alg:fwd},
\begin{algorithm}
\caption{\label{alg:fwd}MDL-based Forward Selection Algorithm}
\KwData{matrix $X \in \{0,1\}^{m{\times}n}$, initial order $p_0$ and model $(D\iter{0},A\iter{0},E\iter{0})$ }
\KwResult{Selected model $(D^*,A^*,E^*)$}
$p \leftarrow p_0$ \; $(D\iter{p},A\iter{p},E\iter{p}) \leftarrow
(D\iter{0},A\iter{0},E\iter{0})$ \; $L\iter{p} \leftarrow
L(D\iter{p})+L(A\iter{p})+L(E\iter{p})$ \;
\Repeat {$L\iter{p} \geq L\iter{p-1}$} {
Initialize the rank-$1$ model $(d,a)$ using $E\iter{p}$ as input \;
$(\tilde{D},\tilde{A},\tilde{E}) \leftarrow
([D\iter{p}|d],\,[(A\iter{p})^\intercal|a^\intercal]^\intercal\,,\,E\iter{p}-da^\intercal) $ \;
Adapt $(D\iter{p+1},A\iter{p+1},E\iter{p+1})$ using $(\tilde{D},\tilde{A},\tilde{E})$ as the starting point\;
$L\iter{p+1} \leftarrow L(D\iter{p+1})+L(A\iter{p+1})+L(E\iter{p+1})$ \;
$p \leftarrow p+1$ \;
}
$(D^*,A^*,E^*) \leftarrow (D\iter{p-1},A\iter{p-1},E\iter{p-1})$ \;
\end{algorithm}
\color{black}
\subsection{Codelength computation}
It remains to describe how we compute $L(X)$. The universal compression of
binary sources has been extensively studied in the literature. Moreover, we
do not need to perform a real encoding; we need only to compute the
codelength. One particularly simple method, for which the codelength is easy
to compute, is \emph{enumerative coding}~\cite{enum}. Given a binary string
$x$ of length $n$ and $r=h(x)$, its enumerative code is
composed of two parts. The first one describes $r$ with
$\lceil\log_2(n)\rceil$ bits, and the second one describes the index of $x$
in the lexicographically ordered list of all binary strings of length $n$ and weight $r$, which requires $\lceil\log_2{r \choose n}\rceil$ bits. The
total codelength is then
\begin{equation}
L(x) = \lceil\,\log_2(n)\,\rceil + \left\lceil\log_2{r \choose n}\right\rceil.
\label{eq:codelength}
\end{equation}
(Note that $\log{r \choose n}$ can be accurately approximated by using
Stirling's formula, i.e., requiring $O(1)$ operations.) As prior
information, we expect the different columns of $D$ to represent particular
patterns in the data, the corresponding rows of $A$ their appearance in $X$,
and the different rows (corresponding to different variables of the data
samples) of $E$ to exhibit different patterns also. Accordingly, we encode
each column of $D$ and each row of $E$ and $A$ with its own code,
\begin{equation}
L(X) = \sum_{i=1}^m L(E\row{i}) + \sum_{k=1}^p L(D\col{k}) + \sum_{k=1}^p L(A\row k)
\label{eq:model-codelength}
\end{equation}
\color{blue}
\paragraph*{Computational complexity} The cost of each forward selection step is \mnote{2.4-5}clearly dominated by the dictionary learning algorithms. The codelength evaluation is negligible, requiring $O(mn) + O(mp) + O(np)$ operations to count the number of zeros in each component $(D,E,A)$.
\color{black}
\section{Results and discussion}
\label{sec:results}
\color{blue}
The main objectives of the following experiments are three. The first is to
demonstrate the interpretability of the resulting models; we expect to
\mnote{1.2}\mnote{2.6} obtain atoms which exhibit recognizable patterns of the input data. The
second is to see the effect of different initialization strategies on the
final result. The third is to study the numerical properties of the methods, in particular their convergence rate and computational cost.\footnote{The version used in this
paper can be downloaded from \url{http://iie.fing.edu.uy/~nacho/bmf/bmf.zip}; this
includes scripts and data to reproduce all the results shown in this section
and many more not discussed in this paper for lack of space. The latest
version is available as a GIT
repository \url{https://gitlab.fing.edu.uy/nacho/bmf}.}
\color{black}
The first dataset is a binarized version of MNIST~\cite{mnist}, a set of $n=10000$ $17{\times}17$ images of handwritten digits;
here each columns of $X$ contains the vectorized version of a digit
($m=289$). The second consists of all the $m=4088$ non-overlapping
blocks of size $16{\times}16$ of the \emph{halftone Einstein}, a
binary $1160{\times}896$ image; the columns of $X$ are the $4088$
vectorized blocks ($m=256$) of the image. As can be seen in
Figure~\ref{fig:datasets}, both datasets have easily recognizable patterns.
\begin{figure*}[t]
\centering%
\includegraphics[height=1.8in]{fig/mnist_bin.png} %
\includegraphics[height=1.8in]{fig/einstein-and-stripe.png} %
\includegraphics[height=1.8in]{fig/einstein-patches-and-stripe-crop.png}
\caption{\label{fig:datasets} \color{blue} Left to right: a few samples of the MNIST dataset; halftone image of Einstein with the stripe used in Figure~\ref{fig:einstein-kprox} highlighted in magenta; detail of Einstein's left eye and the $16{\times}16$ blocks partition; including part of the stripe. the first case, we expect the atoms in the final dictionary to resemble numbers of different shapes (see e.g.~\cite{cvpr10}). In the case of Einstein, we expect the dictionary atoms to resemble the halftoning patterns observable in the $16{\times}16$ blocks shown on the right picture.\color{black}}
\end{figure*}
\begin{figure*}[t]
\centering%
\includegraphics[width=\textwidth]{fig/mnist-mob.pdf}%
\caption{\label{fig:mnist-mob}\color{blue} MNIST model obtained using Forward selection with MOB at each step. The model was initialized with $p_0=16$ random samples. The final dictionary, with $p=79$ atoms, is shown to the left as a mosaic from top to bottom and left to right. The three rows to the right of the dictionary show $40$ samples from $X$ (top), their final residual from $E$ (middle), and the corresponding coefficients from $A$ (bottom). Each column of $A$ is represented as a vertical pattern of $79$ bands: the top band corresponds to the first atom, and the bottom one to the 79th; a black band on the $i$-th row indicates that atom $i$ is being used to represent the sample on top of the same column, giving rise to the residual shown in the middle. %
In general, coefficients are not easy to interpret. However, in some cases the correspondence is clear. In this case the four digits ``0'' use the third atom (painted in red). Something similar happens with the ``6''s: three of them use the 5th atom (green) and one uses the 9th (blue). The corresponding coefficients are marked as red, green and blue bands in the bottom-right picture.
\color{black}}
\end{figure*}
\begin{figure*}[t]
\centering%
\includegraphics[width=1.0\textwidth]{fig/einstein-kprox.pdf}
\caption{\label{fig:einstein-kprox}\color{blue} Einstein model obtained using forward selection and K-PROX. Here too we used random samples for the initialization. As in Figure~\ref{fig:mnist-mob}, we show the resulting dictionary on the left, and the samples (top), residuals (middle) and coefficients (bottom) for the patches contained in the stripe highlighted in Figure~\ref{fig:datasets}. \color{black} }
\end{figure*}
\begin{figure}
\centering%
\includegraphics[height=1.0in]{fig/einstein-init-rand-init.png}
\includegraphics[height=1.0in]{fig/einstein-init-rand-mob.png}
\includegraphics[height=1.0in]{fig/einstein-init-rand-kprox.png}\\[1ex
\includegraphics[height=1.0in]{fig/einstein-init-samp-init.png}
\includegraphics[height=1.0in]{fig/einstein-init-samp-mob.png}
\includegraphics[height=1.0in]{fig/einstein-init-samp-kprox.png}\\[1ex
\caption{\label{fig:init-einstein} \color{blue} Effect of initialization on Einstein. \color{blue} Top row: initial dictionary drawn from a Bernoulli($1/2$) process (left), resulting MOB (center) and K-PROX (right) models for $p=36$. Bottom row: initial dictionary using random columns from $X$ (left), resulting MOB (center) and K-PROX (right) dictionaries. Both initialization methods worked well in this case using both MOB and K-PROX. In all cases, the dictionaries evolved so that some atoms resemble halftoning patterns. Other atoms were not changed, indicating that they were never used.
\color{black}}
\end{figure}
\begin{figure}[t]
\centering%
\includegraphics[height=1.0in]{fig/mnist-init-rand-init.png}
\includegraphics[height=1.0in]{fig/mnist-init-rand-mob.png}
\includegraphics[height=1.0in]{fig/mnist-init-rand-kprox.png}\\[1ex
\includegraphics[height=1.0in]{fig/mnist-init-samp-init.png}
\includegraphics[height=1.0in]{fig/mnist-init-samp-mob.png}
\includegraphics[height=1.0in]{fig/mnist-init-samp-kprox.png}\\[1ex
\caption{\label{fig:init-mnist} \color{blue} Effect of initialization on MNIST. Top row: initial pseudo-random dictionary using a Bernoulli($1/2$) model (left), result of MOB (center) and of K-PROX (right) for $p=36$. Bottom row: initial dictionary using random samples from the dataset (left), MOB (center) and K-PROX (right) dictionaries. Here the pseudo-random initialization does not work; no atom is ever used, and so both algorithms stop at iteration $1$. In the case of the samples-based initialization, both final models show some adaptation. \color{black}}
\end{figure}
\color{blue}
\subsection{Interpretability of the resulting models}
Figure~\ref{fig:mnist-mob} shows the MNIST model obtained using MOB and forward selection, The dictionary and a few columns from the coefficients and residual \mnote{1.3}\mnote{2.6}matrices $A$ and $E$; atoms from $D$ and samples from $E$ are represented as mosaics with each sample/atom displayed as a tile. As can be seen, many atoms look like numbers; other atoms exhibit number-like silhouettes. The results obtained with K-PROX (shown in the supplementary material) are very similar in all aspects. Figure~\ref{fig:einstein-kprox} shows the model obtained for the Einstein image using K-PROX; we show the dictionary, a few samples; again, the results are clearly interpretable in terms of the patterns that can be observed in the data. In this case too the results obtained with MOB (in the supplementary material) are very similar.
\subsection{Sensitivity to initialization}
Figures~\ref{fig:init-einstein} and~\ref{fig:init-mnist} show initial and final dictionaries of fixed size $p=36$ for MNIST and Einstein, using both dictionary learning methods, but using each of the two initialization methods described in Section~\ref{sec:bdl:init}. In this case, the random samples initialization does a good job in both cases, whereas the pseudo-random initialization fails miserably on the MNIST case. These results should be taken with a grain of salt, only to show how different initialization methods can work (or fail) under different circumstances.
\subsection{Numerical results}
\begin{figure}
\centering%
\includegraphics[width=0.235\textwidth]{fig/einstein_bps_vs_atoms_comp.pdf}\hspace{2ex}%
\includegraphics[width=0.23\textwidth]{fig/einstein_time_vs_atoms.pdf}\\[1.5ex]
\includegraphics[width=0.23\textwidth]{fig/einstein_bps_vs_atoms_detail.pdf}\hspace{2ex}%
\includegraphics[width=0.23\textwidth]{fig/einstein_bps_vs_atoms_kprox_detail.pdf}%
\caption{\label{fig:numerical-results-atoms}
\color{blue}%
Convergence of Forward Selection on Einstein. Top to bottom, left to right: MDL cost function (in avg. bits per sample) for MOB and K-PROX; computational cost (in seconds) for both variants; break-down of the cost function in its three parts ($E$, $A$ and $D$) for MOB; same for K-PROX. %
Both MOD and K-PROX produced similar models in the end. K-PROX, however, required more time to run ($14$s) than MOB ($4$s). Thanks to the warm-restart strategy, the cost per forward selection step is small ($0.11$s for MOB and $0.03$s for K-PROX) and approximately constant despite the growing model size $p$. Finally, both break-downs show typical MDL curves: as $p$ increases, the stochastic complexity ($L(E)$) decreases while the model cost $L(A)+L(D)$ increases.\color{black}}
\end{figure}
\begin{figure}
\centering%
\includegraphics[width=0.23\textwidth]{fig/einstein_bps_vs_iter.pdf}\hspace{2ex}%
\includegraphics[width=0.23\textwidth]{fig/einstein_time_vs_iter.pdf}\\[1.5ex]
\includegraphics[width=0.23\textwidth]{fig/einstein_change_vs_iter_mob.pdf}\hspace{2ex}%
\includegraphics[width=0.23\textwidth]{fig/einstein_change_vs_iter_kprox.pdf}%
\caption{\label{fig:numerical-results-iters}
\color{blue}%
Convergence of MOB and K-PROX on Einstein. Top to bottom, left to right: MDL cost function; execution time; change in the arguments $A$,$D$ and $E$ for MOB; same for K-PROX. Both methods converge quickly, with K-PROX reaching its near-optimum in just one iteration. In terms of execution time, however, just one iteration of K-PROX required $0.6$s (this is much larger than the $0.11$s reported in Figure~\ref{fig:numerical-results-atoms} since the model has to be learned from scratch), whereas all $10$ MOB iterations take an accumulated time smaller than $0.1$s. Note that the convergence is exact, not asymptotic: both algorithms stop when the change of all arguments is $0$. \color{black}}
\end{figure}
\color{blue}
In this set of experiments we are interested mainly in three aspects of the proposed methods. First, the convergence of the forward selection method in \mnote{1.3}\mnote{2.4-5}terms of overall cost function, the convergence rate of the dictionary learning algorithms, and the empirical computational complexity of both the selection and learning methods as measured in running time (seconds), both per iteration and accumulative.\footnote{The timing results were obtained on a Lenovo V310-14IKB notebook with an Intel i5-7200U (4 cores) processor, 8GB of RAM, running Lubuntu 18.04 64bits, with executables compiled from C++ code using GCC 7.3.0 with maximum optimization (``-O3''), multicore support (``-fopenmp'') and all SIMD functions enabled.} %
Figure~\ref{fig:numerical-results-atoms} shows that forward selection converges exponentially to a local minimum which is very close in cost function for both MOB and K-PROX. The forward selection mechanics shown in the break-down of the arguments are typical of methods such as MDL, which shows the correctness of the procedure.
Figure~\ref{fig:numerical-results-iters} shows that both MOB and K-PROX converge quickly both in cost function and the respective optimization variables (we recall that convergence is exact here -- there is no further change in the arguments). This same behavior was observed for all model sizes in Figure~\ref{fig:numerical-results-atoms}.
In terms of execution time, both Figure~\ref{fig:numerical-results-atoms} and ~\ref{fig:numerical-results-iters} show MOB as the fastest of the two methods in terms of computing speed. It should be noted however that our implementation for MOB is reasonably optimized, whereas that of K-PROX is not. By comparing the average time for learning a fixed order model in figures~\ref{fig:numerical-results-atoms} and \ref{fig:numerical-results-iters} it should be clear that the warm-restarts strategy used is crucial for efficiently searching through the different model sizes.
\color{black}
\section{Concluding remarks}
\label{sec:conclusion}
\color{blue}
In this paper we have presented two novel and efficient Binary Factorization Methods based in dictionary learning techniques through the combination of three novel algorithms, BMP for learning coefficients, and MOB and K-PROX for updating the dictionaries. We have provided theoretical guarantees on their convergence to local minima, and a simplified but rigorous analysis of their computational complexity.
\mnote{1.3}\mnote{2.4-5}Through experimentation on two very different datasets, we have demonstrated that our methods produce interpretable results in both cases, requiring very few iterations to converge, and with an overall computational complexity which is very competitive (even though we did not employ any speed-up strategies such as online or mini-batch learning), requiring as little as $0.1$s to learn a complete dictionary on a dataset of $10000$ $289$-dimensional samples from scratch. We have also shown the scalability of our model in terms of growing model size, allowing us to search over a large family (hundreds) of candidate models (dictionaries) in as little as $4$ seconds on a modest notebook.
In a follow up of this work, which is currently underway, we will present on-line implementations of the MOB and K-PROX algorithms developed here, with the objective of employing them on huge genomic datasets. Other possible lines of work include finding conditions under which our methods (or some of them) can be guaranteed to recover a given underlying model.\mnote{2.2-3}
\color{black}
\bibliographystyle{IEEEtran}
|
{
"timestamp": "2018-07-27T02:03:42",
"yymm": "1804",
"arxiv_id": "1804.05482",
"language": "en",
"url": "https://arxiv.org/abs/1804.05482"
}
|
\section{Introduction}
\label{sec:intro}
Peaks in the underlying mass density field are the most likely sites for the formation of halos where gas is expected to accrete and form galaxies
\citep{White1978}.
In the classical picture of \cite{PS}, matter in regions with linear density contrast above a threshold $\delta_c$ is assigned to
halos of mass larger than $M$, where $M$ defines the smoothing of the density field. This implies that halos of mass $M$ form at
peaks with $\delta=\delta_c$ in the smoothed density contrast $\delta$.
Naturally, most studies have focused on peaks associated with halos. Indeed,
statistical properties of local extrema \citep[e.g.][]{Adler1981} have gained a great deal of attention in cosmology \citep{BBKS} (hereafter BBKS).
Correlations of halos and their distribution in relation to the mass density field of the gravitationally dominant dark matter (DM), i.e. biasing
\citep[][]{Kaiser1984a}, have been studied
extensively with analytic methods and numerical simulations.
For Gaussian initial conditions and on sufficiently large scales, halos follow a linear biasing relation, $\delta_\mathrm{h}=b\delta$ between the
halo number density contrast, $\delta_\mathrm{h}$ and the mass density contrast $\delta$.
The bias factor $b$ depends on the height of the peaks associated with halos and on their mass.
An important result obtained in simulations \citep{Dalal:2007cu}, and confirmed by analytic techniques
\citep{grinstein/wise:1986,Dalal:2007cu,Matarrese:2008nc,Slosar:2008hx}, is that the presence
of initial local-type non-Gaussianity introduces a peculiar scale dependence in the bias factor dubbed ``non-Gaussian bias''.
The specific form of $b(k)$ ($k$ is the wavenumber of a given scale) opens the window for probing initial non-Gaussianity based on the clustering
properties of galaxies in planned large redshift surveys, e.g. Euclid \citep{EuclidRB} and DESI \citep{DESICollaboration2016a}.
It is well known that non-Gaussianity strongly affects the tails of density probability distributions \citep{Adler1981,catelan/etal:1988}.
Several authors have further specialized these results to local density maxima of non-Gaussian density fields, where the non-Gaussianity is either of a
generic form
\cite[e.g.][]{Catelan1988,Gay2012,codis/etal:2013,uhlemann/etal:2018} or developed via non-linear gravitational evolution of initial gaussian conditions
\citep[e.g.][]{Suginohara1991,matsubara:1994}.
In particular, \cite{Gay2012,codis/etal:2013} considered the effect of a generic non-Gaussianity on extrema counts and Minkowski functionals of the dark
matter density field.
In this work, we consider peaks and dips in cosmological density field smoothed on scales much larger than those of galactic and galaxy cluster halos
($\lower.5ex\hbox{$\; \buildrel < \over \sim \;$} 10$ Mpcs).
Using N-body simulations in large cosmological boxes, we focus on the total number of local extrema for density fields constructed from the {\it halo}
distribution, as a proxy for a galaxy catalogue. Earlier analyses (\cite{Croft:1997rv,desoma,De:2009uz}) have used this type of statistics for constraining parameters related to the linear matter power spectrum on smaller scales ($\lower.5ex\hbox{$\; \buildrel < \over \sim \;$} 10$ Mpcs).
Our goal is to assess the extent to which the abundance of extrema in three-dimensional (3D) fields inferred from current and forthcoming large galaxy redshift surveys can be used as
a cosmological tool and, more specifically, a probe of local primordial non-Gaussianity.
As we shall see, our main findings have a straightforward interpretation in terms of the non-Gaussian bias.
We adopt standard notation. The mean total and baryonic mass densities (in units of the critical density) are denoted by
$\Omega_\mathrm{m}$ and $\Omega_\mathrm{b}$, respectively. The Hubble constant is $H_0$ and $h=H_0/[100 \mathrm{\, {\rm km }\, {\rm s}^{-1}\; Mpc^{-1}} ]$.
The linear growth factor (normalized to unity at the present time) at redshift, $z$, is $D(z)$.
The outline of the paper is as follows. In \S\ref{sec:basics} we lay out known relations between the number of extrema and the underlying power spectrum for Gaussian fields.
A description of the N-body simulations is provided in \S\ref{sec:simulations} and the corresponding results for the abundance of local extrema identified in
smoothed density fields derived from the DM and halo distributions are in \S\ref{sec:results}.
In \S\ref{sec:prospects} we discuss the prospects for the application of the number of extrema as a test of cosmological parameters and conclude with a summary in
\S\ref{sec:discussion}.
\section{Definitions and Theoretical Expectations}
\label{sec:basics}
We define local maxima (peaks) in a smoothed random field, $f$, as points in space where the spatial gradient is $\partial_\alpha f=0$ and the Hessian
$\partial_\alpha\partial_\beta f$ is negative definite. Local minima (dips) are defined similarly but with a positive definite Hessian.
For a random Gaussian field, peaks and dips have an equal total number \textit{per unit volume}, which was computed by BBKS to be
\begin{equation}
n_{0}\approx 0.016 R_*^{-3}\; ,
\label{eq:npk}
\end{equation}
where
\begin{equation}
R_*=\sqrt{3} \frac{\sigma_1}{\sigma_2}\; ,
\label{eq:rst}
\end{equation}
and the spectral moments
\begin{equation}
\sigma^2_j=\int \frac{k^2\textrm{d} k}{2\pi^2}P(k) W_R^2(k) k^{2j}\; .
\label{eq:sigi}
\end{equation}
The expression for $n_0$ is independent of the clustering amplitude and it depends only on the shape of the power spectrum, $ P(k)$ of the field, and the smoothing
Kernel $W_R(k)$.
For $P(k)\sim k^n$, and a Gaussian smoothing window $W_R^2(k)=\exp(-k^2R^2)$, it is easy to see that,
\begin{equation}
\frac{R}{R_*}=\left(\frac{n+5}{6}\right)^{1/2}\; .
\label{eq:Rn}
\end{equation}
The total number of peaks is preserved under a local monotonous one-to-one mapping, $F(\delta)$, of the density field.
Thus we expect this quantity to be independent of time in the quasi-linear scales. On smaller scales, local extrema tend to merge and diffuse,
leading to deviations from expression Eq.~(\ref{eq:npk}) above.
In addition to the DM density field, we also examine peaks and dips in the smoothed distribution of halos.
The corresponding spectral moments $\sigma_j\equiv\sigma_{j,h}$ are given by
\begin{equation}
\label{eq:sigmahi}
\sigma_{j,h}^2 = \int\frac{k^2 dk}{2\pi^2}\, k^{2j} W_R^2(k) \Big[b^2(k) P(k) + \frac{1}{\bar n_h}\Big] \;.
\end{equation}
The expression in square brackets is a model for the power spectrum of the halo distribution
where $P(k)$ here refers to the underlying density field and $b(k)$ describes the scale-dependent halo bias.
The term $1/ {\bar n_h}$ is due to the finite number of halos and approximated as a Poisson discreteness noise.
Using the simulations described below we have found that the added discreteness variance is strongly suppressed for large smoothing and
is actually sub-Poissonian, in agreement with the findings of \cite{casas-miranda/etal:2002,Hamaus:2010im}.
On linear scales, the halo bias $b(k)$ is constant for Gaussian initial conditions but depends on the halo mass i.e. $b(k)= b^\mathrm{G}(M)$.
We also consider local-type non-Gaussianity \citep{Salopek1990,Gangui,KomatsuSpergel} for which
the Bardeen potential $\Phi$ deep in matter domination is expanded around a random Gaussian field $\phi$ as
\begin{equation}
\Phi(\mathbf{x}) = \phi(\mathbf{x})+f_{\rm NL} \left( \left[\phi(\mathbf{x})\right]^2-\langle \phi^2\rangle\right) \;.
\end{equation}
The bispectrum of $\Phi$ induces the following scale dependence in the bias factor,
\begin{equation}
\label{eq:bNG}
b(k) = b^\mathrm{G}(M) + \frac{\alpha(f_{_{\rm NL}})}{k^2T(k)}\; ,
\end{equation}
where
\begin{equation}
\alpha(f_{_{\rm NL}}) \equiv 3 f_{_{\rm NL}} \frac{\partial{\rm ln}\,\bar n_h}{\partial{\rm ln}\,\sigma_8} \frac{\Omega_m H_0^2}{D(z)c^2} \;,
\end{equation}
and $\bar n_h(M)$ is the abundance of halos (per unit $M$) computed for the Gaussian field without the $f_{_{\rm NL}}$ terms.
When implementing Eq. \eqref{eq:sigmahi} to compare it to data (see Section \S\ref{sec:results}), we use the following approximation
\begin{equation}\label{eq:universal}
\frac{\partial{\rm ln}\,\bar n_h}{\partial{\rm ln}\,\sigma_8} \approx \delta_c (b^{\rm G}(M)-1),
\end{equation}
with $\delta_c=1.687$, which is valid for universal mass functions and the spherical collapse model~\footnote{See \cite{Biagetti:2016ywx} for a
quantitative analysis on this approximation on the same set of simulations, sim 1, used here.}.
We do not include expressions \citep[e.g.][]{Gay2012} for the theoretical corrections to Eq.~(\ref{eq:npk}) due to $f_{_{\rm NL}}$ non-Gaussianity. Indeed, we will
see below that the expression remains accurate provided that the appropriate $\sigma_i$ is used.
\section{Simulations}
\label{sec:simulations}
Two sets of simulations, respectively in a $2h^{-1}\,{\rm Gpc}$ and a $3h^{-1}\,{\rm Gpc}$ box, are available for initial conditions generated from $\Lambda$CDM initial power spectra
with slightly different cosmological parameters, as described in the Table.
The simulations were run with the Gadget2 \citep{Gadget2} N-body code on the Baobab cluster at the University of Geneva.
The initial particle displacements were implemented at $z_i=99$ using the public code 2LPTic \citep{Crocce2006} for realizations with Gaussian initial conditions
and its modified version \citep{Scoccimarro12} for non-Gaussian initial conditions of the local type.
The transfer function for the smaller box (simulations 1, see Table) was obtained using the CLASS code \citep{Blas2011}.
This set contains runs for Gaussian initial conditions and two for local-type non-Gaussianity respectively, with $f_{_{\rm NL}}=250$ and $f_{_{\rm NL}}=-250$.
For each of these initial conditions, we obtain 8 random realizations corresponding to different random seeds.
The transfer function of the second set, simulations 2, was obtained using the CAMB code \citep{Lewis:1999bs}.
This set includes 3 types of models: Gaussian initial conditions ($f_{_{\rm NL}}=0$) and non-Gaussian initial conditions, respectively,
with $f_{_{\rm NL}}=100$ and $f_{_{\rm NL}}=-100$. For each type of models, we have 3 simulations corresponding to different random realizations of the initial conditions.
The Rockstar \citep{Behroozi2013} algorithm is employed to identify halos, with linking length $\lambda=0.28$.
\begin{table}
\centering
\begin{tabular}{c c c c c c c}
\hline \hline
& $L$ & $N_\mathrm{p}$& $M_\mathrm{halo}$& $\sigma_8 $ & $\Omega_m$& $\Omega_b $ \\
\hline
sim 1 & 2& $1536^3$& $3.67$ & 0.85 & 0.3 & 0.0455 \\
sim 2& 3 & $1024^3$& $37.9$ & 0.81 &0.272 & 0.0455 \\
\hline
\end{tabular}
\caption{ Simulation parameters, where $L$ is the box size (in unit of $h^{-1}\,{\rm Gpc}$), $N_\mathrm{p}$ number of simulation particles,
and $M_\mathrm{halo}$ is the minimum halo mass identified in the simulation (in unit of $10^{12}h^{-1}{\rm M}_{\odot}$).
Both, simulations 1\&2, include Gaussian and two choices for non-Gaussian initial conditions.
Outputs of simulations 1
are available at $z=0$ and $z=1$, while only the output at $z=0$ is available for simulations 2.
In all simulations the Hubble parameter is $h=0.7$ and the spectral index of the initial power spectrum at large scales is $n_s=0.967$.}
\end{table}
Density fields are interpolated from the DM and halo distributions in the simulation box
on a $512^3$ cubic grid using the Clouds-in-Cells (CIC) scheme.
The grid spacing is thus $3.9h^{-1}\,{\rm Mpc} $
and $5.85h^{-1}\,{\rm Mpc}$, for simulations 1 \& 2, respectively. The density fields were additionally smoothed with a Gaussian window
of 8 different widths in the range $20h^{-1}\,{\rm Mpc}$ to $500h^{-1}\,{\rm Mpc}$. For each smoothed field, local maxima (minima) were identified as grid points surrounded by
grid points with lower (higher) density values.
Fig.~\ref{fig:box} shows the total number of maxima
in the smoothed DM density field in the full boxes of simulations 1 \& 2 at $z=0$.
The theoretical predictions obtained from the BBKS expression (Eqs.~\ref{eq:npk}-\ref{eq:rst}) using the linear power spectrum $P(k)=P_\mathrm{L}(k)$
for the two models are also shown, as indicated in the figure\footnote{In performing the integration in Eq.~(\ref{eq:sigi}), it is important to impose a low $k$ cutoff
corresponding to the finite box size of the simulations. }.
The shaded area encompasses the expected range of (1$\sigma$) shot-noise for simulations 2.
The number drops like $R^{-3}$, consistently with Eq.~(\ref{eq:npk}) since $R_*\propto R$ upto a factor of $\mathcal{O}(1)$ which depends on the shape of the power
spectrum at scale $R$ (cf. Eq.~\ref{eq:Rn}).
The figure refers to the Gaussian simulations only. A similar figure can be found in \citep{2011MNRAS.413.1961L},
but for comparison of the theoretical expression with
peaks identified in the initial conditions of their simulations.
\begin{figure}
\includegraphics[width=0.45\textwidth]{Fig1.pdf}
\vskip 0.0in
\caption{Total number of maxima versus the smoothing length, from the DM distribution in simulations 1 \& 2 for Gaussian initial conditions at redshift $z=0$
The lines represent the corresponding theoretical prediction using eq. \ref{eq:npk} and the shaded area
represents the $1\sigma$ shot-noise for
the larger simulation. }
\label{fig:box}
\end{figure}
\section{Results}
\label{sec:results}
\subsection{Total number of minima and maxima}
Gaussian initial conditions imply equal probability of producing peaks and dips, up-to fluctuations due to the finite box size.
However, on scales $\lower.5ex\hbox{$\; \buildrel < \over \sim \;$} 10$s of Mpcs, non-linear gravitational evolution breaks the initial symmetry through
the merging and smearing of dips and peaks.
For non-Gaussian initial conditions, the statistical symmetry between maxima and minima is already broken
initially.
We choose to first analyze the (total) number $n_\mathrm{1}$, per unit volume, of minima, $n_\mathrm{min}$, and maxima,
$n_\mathrm{max}$ in the simulations. The differences between the abundance of minima and maxima will be discussed at a later stage.
More precisely, we consider
\begin{equation}
\label{eq:n1}
n_\mathrm{1} =\frac{1}{2}\big(n_\mathrm{min}+n_\mathrm{max}\big) \; ,
\end{equation}
which is computed from the smoothed density fields for the various simulations. An advantage of $n_1$ is that it boosts the statistical significance of the measured abundance. For a Gaussian field, $n_1=n_0$ given in Eq.~(\ref{eq:npk}).
Inclusion of non-Gaussian terms modify the abundance of either the minima or maxima by a leading-order
correction proportional to the skewness of the density field and its derivatives \citep{Gay2012}.
The combined leading order correction for both minima and maxima cancel out in the expression of $n_\mathrm{1}$.
Consequently, the BBKS prediction Eq.~(\ref{eq:npk}) remains valid up to a small correction of order $f_{_{\rm NL}}^2$.
According to Fig.~\ref{fig:box}, differences in $n_1$ between the simulations are visually hard to examine directly.
Thus, we consider the statistic,
\begin{equation}
\label{eq:Rfromn}
\Upsilon\equiv \frac{n_1R^3}{0.016}
\end{equation}
where $R$ is the width of the smoothing window. According to Eq.~(\ref{eq:npk}), for a Gaussian field $\Upsilon=({R}/{R_*})^3$.
The three-panel Fig.~\ref{fig:panels} summarizes the main results.
The top panel plots $\Upsilon$, averaged over the 8 random realizations in simulations 1, against the smoothing length, $R$, for the DM density field.
The shaded area represents the 1$\sigma$ shot-noise in $n_1$ corresponding to the finite number of
peaks and dips in the simulation box.
It is estimated as $\sqrt{n_0 L^3/2}$ where $n_0$ is the theoretical
value according to Eq.~(\ref{eq:npk}) and the factor of $1/2$ arises from the definition of $n_1$ which involves both minima and maxima.
We have checked that the scatter from the 8 individual runs (not shown for clarity) is consistent with this estimate of the shot-noise.
For our Gaussian as well as non-Gaussian simulations, the results in the top panel
for $z=1$ and $z=0$ are almost identical.
The dotted line shows $(R/R_*)^3$ computed according to the theoretical expression Eq.~(\ref{eq:rst}) derived for Gaussian fields, where $\sigma_i$ are computed
using Eq.~(\ref{eq:sigi}) with the initial power spectrum $P_\mathrm{L}(k)$.
There is a reasonable match between the dotted curve and $(R/R_*)^3$ derived from $n_1$ for the Gaussian simulations (black and red solid curves).
Overall, the impact of $f_{_{\rm NL}}$ is very small, in agreement with the fact that, for dark matter, $n_1$ depends on $f_{_{\rm NL}}$ only at order $f_{_{\rm NL}}^2$.
The middle panel refers to results obtained from the halo distribution in simulations 1.
The solid curves corresponding to the Gaussian simulations at $z=1$ and $z=0$ are similar.
In great contrast to the upper panel, both $f_{_{\rm NL}}=250$ and $f_{_{\rm NL}}=-250$ models (dashed and dash-dotted lines) at the two redshifts are substantially
different.
It is interesting to check how well the BBKS expression in Eq.~(\ref{eq:rst}) fits the $\Upsilon$ computed from the halos in the non-Gaussian simulations.
To do that we compute $(R/R_*)^3$ using Eq.~(\ref{eq:rst}) for $\sigma_1$ and $\sigma_2$ computed directly from the halo
density fields.
The results are plotted as the plus signs and circles, respectively, for the $f_{_{\rm NL}}=250$ and $f_{_{\rm NL}}=-250$ simulations.
We present the $z=1$ case only but the excellent agreement of $\Upsilon$ with $(R/R_*)^3$ computed from $n_\mathrm{1}$ also holds at $z=0$ .
The bottom panel summarizes results for simulations 2 ($z=0$) of the larger box.
The halos in these simulations have a larger mass and therefore follow a different biasing relation than halos in simulations 1, yielding different
quantitative results.
For these simulations also, the BBKS expression (computed with $\sigma_i$ measured in the simulations), shown as the plus signs and circles, furnishes an
excellent match.
Therefore, despite the fact that relations Eqs.~(\ref{eq:npk}) and (\ref{eq:rst}) are formally obtained for Gaussian fields,
they remain accurate for the non-Gaussian fields considered here, provided the actual $\sigma_i$ are used.
In Fig.~\ref{fig:fnltheory} we compare the theoretical expectation of Eq. \eqref{eq:sigmahi} against $\Upsilon $ measured from the non-Gaussian simulations.
The theoretical curve fits good the data on scales $R \lesssim 100$Mpc/h and provides a qualitatively good description at all scales.
Deviations may be due in part to our approximation Eq. \eqref{eq:universal} and, especially at large scales, to the finite box size of the simulations.
\begin{figure}
\includegraphics[height=1.2\textwidth]{Fig3Panel.pdf}
\caption{The quantity $\Upsilon$ as estimated from Eq.~(\ref{eq:Rfromn}).
\textit{Top:} from the number of peaks and throughs in the dark matter distribution of simulations 1.
\textit{Middle: } The same the {\it Top}, but for the halo distribution.
\textit{Bottom:} For DM and halos for simulations 2, at $z=0$ only.
In all panels, the grey area represents the shot-noise estimated from the expression with using the theoretical linear power spectrum.}
\label{fig:panels}
\end{figure}
\begin{figure}
\includegraphics[width=0.45\textwidth]{Fig3.pdf}
\caption{A test of the analytic prediction for the non-Gaussian model. The dashed and dash-dotted curves are taken from the middle panel in the previous figure.
The curves with the circles plot $(R/R_*)^3$ computed with the approximate $\sigma_i$ given in Eqs.~(\ref{eq:sigmahi}-\ref{eq:bNG}) }
\label{fig:fnltheory}
\end{figure}
To conclude this Section, we note that the effect of $f_{_{\rm NL}}$ on $\Upsilon$ is only weakly degenerate with that of $\sigma_8$ because $n_1$ primarily depends on the
ratio of spectral moments $\sigma_{1,h}/\sigma_{2,h}$.
\subsection{Asymmetry and height distribution}
So far we have considered $n_1$, without distinguishing between minima and maxima.
In Fig.~\ref{fig:asymm}, we examine the asymmetry between the abundances of minima and maxima as a function of the smoothing width for simulations 2 at redshift $z=0$.
There is a clear excess of $N_\mathrm{max}$, which is significantly above the level of the shot-noise (grey area).
The trend is reversed at larger scales for both DM non-Gaussian models, but it becomes immersed in the shot-noise.
Results of the three individual runs for the Gaussian DM simulation are also shown.
It is clear that the shot-noise estimated theoretically as described above (grey area) is consistent with the scatter in the individual runs.
We explore the probability density distribution (PDF) of the value of the densities at the minima and maxima.
We define,
\begin{equation}
\nu=\frac{\delta}{\sigma_0}\quad {\rm and} \quad \nu_\mathrm{ln}=\frac{{\rm ln}\,(1+\delta)-\mu}{\sigma_\mathrm{ln}}
\end{equation}
where $\sigma_0$ is the rms of density field all over space while
$\mu $ and $\sigma_\mathrm{ln}$ are the mean and rms of the values of ${\rm ln}\,(1+\delta)$ at either the minima or maxima.
The quantity $\nu_\mathrm{ln}$ is motivated by the result that the PDF of the density field is well approximated by a log-normal distribution \citep[e.g.][]{Coles91,Kofman}.
In Fig.~\ref{fig:PDFnu} and \ref{fig:PDFnu80} we plot the PDF of $\nu $ (top) and
$\nu_\mathrm{ln}$ (bottom) for a smoothing of $R=20h^{-1}\,{\rm Mpc}$ and $80h^{-1}\,{\rm Mpc}$ for simulations 2.
The 3 curves of each line-style correspond to the Gaussian and 2 non-Gaussian simulations.
It is evident that the
PDF of densities at either maxima or minima is weakly sensitive to whether the initial conditions were Gaussian or not.
This is expected given that corrections arise at order $f_{\rm NL}^2$ as noted above.
Thus, for clarity, the plot does not indicate which of the simulations is shown.
For $R=20h^{-1}\,{\rm Mpc}$, the BBKS theoretical prediction for $P(\nu)$ (expression 4.3 in their paper) shown as the black in the top panel, is a poor fit to any of the PDFs measured in the simulations.
However, $P(\nu_\mathrm{ln}) $ for the DM density field (dotted), exhibit only minor differences at the tails, where the PDF for maxima is slightly skewed to positive values
relative to the Gaussian (black in the bottom panel), the distribution at minima is negatively skewed. The rather small differences between
the PDF from the halos and the corresponding DM are due to deviations from linear biasing.
Fig.~\ref{fig:PDFnu80} shows the same results, but for $R=80h^{-1}\,{\rm Mpc}$. This large smoothing greatly reduces the effect of non-linear evolution, bringing
the BBKS theoretical PDF (black curve, top panels) closer
to the measured PDF than it is for $R=20h^{-1}\,{\rm Mpc}$.
The log-normal curve (black, bottom) remains a good fit to $P(\nu_\mathrm{ln} $ for the DM although not as good as in the smaller smoothing. It is interesting that the log-normal describes the halo PDF fairly well for this smoothing.
At $R=20h^{-1}\,{\rm Mpc}$ and $80h^{-1}\,{\rm Mpc}$ the halo bias in the Gaussian and non-Gaussian simulations are small (cf. Eq.~(\ref{eq:bNG}). This explains the similarity between the halo PDFs in the simulations irrespective of the initial statistic.
\begin{figure}
\includegraphics[width=0.45\textwidth]{Fig4.pdf}
\caption{ The relative difference between the total number of maxima and minima in simulations 2, versus the smoothing width, at redshift $z=0$.}
\label{fig:asymm}
\end{figure}
\begin{figure}
\includegraphics[width=0.45\textwidth]{PDFnu.pdf}
\includegraphics[width=0.45\textwidth]{PDFnulog.pdf}
\caption{ \textit{Top:} The PDF of $\nu$ at minima and maxima in simulations 2, as indicated in the figure.
The black curve is the theoretical prediction for $P(\nu)$ given in BBKS.
\textit{Bottom:} The PDF of $\nu_\mathrm{ln}$ at minima and maxima for simulations 2. Here
the black line is a Gaussian with zero mean and unit variance. }
\label{fig:PDFnu}
\end{figure}
\begin{figure}
\includegraphics[width=0.45\textwidth]{PDFnu80.pdf}
\includegraphics[width=0.45\textwidth]{PDFnulog80.pdf}
\caption{ The same as the previous figure, but for $R=80h^{-1}\,{\rm Mpc}$. }
\label{fig:PDFnu80}
\end{figure}
\section{Abundance of extrema as a cosmological test}
\label{sec:prospects}
We offer a preliminary assessment of using total number of peaks and dips as a test of cosmological models.
A proper analysis should take into account the covariance between the abundances corresponding to different smoothing scales.
However, this task is beyond the scope of the current paper.
Instead, we will focus on the expected discriminatory power of extrema abundance at distinct scales.
As an example, we consider the {\small Euclid} mission \citep{EuclidRB}, which will target emision line galaxies in
the redfshift range $0.9<z<1.8$ across $\sim 35\%$ of the sky.
For Planck's cosmological parameters, the corresponding survey volume is $48 (h^{-1}\,{\rm Gpc})^3$. Furthermore,
the typical host halo mass is $\sim 10^{11-12}h^{-1}{\rm M}_{\odot}$, in broad agreement with the minimum halo mass resolved
in simulations 1.
We wish to assess the ability that a measured total number $N$ of
extrema in a survey can reject a certain model given the hypothesis of an assumed fiducial underlying model.
For this purpose, we assume that $N$ follows a Poisson distribution
\begin{equation}
\label{eq:Poisson}
P_{\bar N}(N)= \frac{{\bar N}^N}{N!} \mathrm{e}^{-\bar N}\; ,
\end{equation}
where $\bar N$ is the mean number expected in a particular given model.
Given an observed $N$,
the preferred of two competing models with expected mean numbers ${\bar N}_1$ and ${\bar N}_2$, respectively, is determined by
\begin{eqnarray}
\label{eq:D}
D_{_{\bar N_1 \bar N_2}}&=&-2{\rm ln}\, \frac{P_{{\bar N}_1}}{P_{{\bar N}_2}}\nonumber \\
&=& 2 N{\rm ln}\,\frac{\bar N_2}{\bar N_1}+2({\bar N_1-\bar N_2})\; .
\end{eqnarray}
The mean value of $D$ over all measurements, which we loosely denote by $\Delta \chi^2$ is
\begin{eqnarray}
\label{eq:chis}
\Delta \chi^2&=&\sum_N P_{\bar N} D_{_{\bar N_1 \bar N_2}} \nonumber \\
&=&2\bar N {\rm ln}\,\frac{\bar N_2}{\bar N_1}+2({\bar N_1-\bar N_2})\; ,
\end{eqnarray}
where we have used $\sum_N P_{\bar N}(N)=1$ and $\sum_N N P_{\bar N}(N)=\bar N$.
For $\bar N_2=\bar N$, the quantity $\Delta \chi^2$ yields the confidence level with which
a model with $\bar N_1$ can be rejected if the underlying model is $\bar N$.
We use this statistic to assess whether the abundance of dips and peaks
can be used to reject certain models given a Gaussian cosmological model with fiducial cosmological parameter.
We focus on $\Omega_m$ and $f_{_{\rm NL}}$, separately.
Fig.~\ref{fig:chiomega} examines $\Delta \chi^2$ as a function of the matter density $\Omega_\mathrm{m}$.
Here, $\bar N $ is computed using Eq.~(\ref{eq:npk}-\ref{eq:sigi}) for fiducial DM linear
power spectrum with the cosmological parameters corresponding to simulations 1.
The same parameters with the exception of $\Omega_\mathrm{m}$ are used in the same expression to derive $\bar N_1 $.
This figure, therefore, refers to a Gaussian model ($f_{_{\rm NL}}=0$) and, in addition to DM density fields,
it is also relevant for halos with linear constant bias with respect to the DM.
Only two filtering scales are considered, as indicated in the figure.
It is remarkable that for $R=50h^{-1}\,{\rm Mpc}$ the $1\sigma $ level ($\Delta \chi^2=1$) is at $\Delta \Omega\approx\pm 0.01$ from the fiducial $\Omega_m=0.3$.
It should be pointed out that for the $\Lambda$CDM linear power spectrum the abundance on a filtering scale given in $h^{-1}\,{\rm Mpc}$ is degenerate with respect
to $\Omega_m h$ and $\Omega_b h$.
This sensitivity to $\Omega_m$ declines rapidly at $R=100h^{-1}\,{\rm Mpc}$ due to the $1/R^3$ dependence of the number of dips and peaks.
The sensitivity to $f_{_{\rm NL}}$ is demonstrated in Fig.~\ref{fig:chifnl} plotting $\Delta \chi^2$ with $\bar N$ from the fiducial model and
$\bar N_1$ for $f_{_{\rm NL}}\ne 0$ but with all other parameters fixed at the fiducial values. These curves refer to filtered halo distribution
where the theoretical expressions in
Eqs.~(\ref{eq:sigmahi}-\ref{eq:bNG}) are used in Eq.~(\ref{eq:npk}) to derive the mean number of dips and peaks $\bar N$ in a Euclid
volume survey at $z=1$.
In these calculations, we consider a halo mass distribution consistent with simulations 1, with a minimum mass of $3.67\times 10^{12}h^{-1} M_\odot$.
For this mass threshold, we have seen in the previous section that the theoretical
predictions are in reasonable agreement with the simulations.
The sensitivity to $f_{_{\rm NL}}$ is improved for the larger filtering widths, $R$ thanks to the stronger $f_{_{\rm NL}}$-dependence of halo bias
on larger scales. For $R=300h^{-1}\,{\rm Mpc}$, we find $\Delta \chi^2=1$ for deviations $\Delta f_{_{\rm NL}}\approx \pm 25 $.
This is encouraging especially if combined with measurements as a function of filtering scales and different halo masses.
\begin{figure}
\includegraphics[width=0.45\textwidth]{Fig6.pdf}
\caption{ Abundance of dips and peaks as a cosmological test for estimating $\Omega_\mathrm{m}$ from a survey like {\small Euclid}.
Values of $\Delta \chi^2=1$ correspond to $1\sigma$ limits from the fiducial value of $\Omega_\mathrm{m}$. }
\label{fig:chiomega}
\end{figure}
\begin{figure}
\includegraphics[width=0.45\textwidth]{Fig7.pdf}
\caption{ The same as the previous figure but for $f_{_{\rm NL}}$ instead of $\Omega_\mathrm{m}$. }
\label{fig:chifnl}
\end{figure}
\section{Discussion and Conclusions}
\label{sec:discussion}
Locating points of maxima and minima is straightforward even for 3D density fields estimated from realistic galaxy redshift surveys.
Since the total abundance is computed irrespective of height,
it should be robust against the details of how the density field is estimated from the data.
The total abundance is also insensitive to redshift space distortions,
which in any case can be modeled with standard perturbation theory for smoothing widths $R\gtrsim 50h^{-1}\,{\rm Mpc}$ \citep{codis/etal:2013}
\citep[see also][]{lam/etal:2010}.
Further, it depends explicitly only on the shape of the power spectrum.
Any dependence on the amplitude (e.g. $\sigma_8$) is indirectly encoded
in non-linear corrections to the shape of the gravitationally evolved power spectrum.
The lack of sensitivity of this abundance statistics on the amplitude thereby implies
that it breaks most of the degeneracy
between $f_{_{\rm NL}}$ and the primordial amplitude of scalar perturbations, which arises in measurements of
galaxy clusters counts and shear peaks in weak lensing maps for instance
\citep[this degeneracy can also be broken by combining clusters and voids, see][]{kamionkowski:voids}.
We have demonstrated that a primordial non-Gaussianity of the local-$f_{_{\rm NL}}$ type imprints a strong signal in the abundance of peaks
and dips of the halo density field owing to the non-Gaussian bias.
An important result of the current paper is that the BBKS prediction derived for Gaussian density field can account for this effect reasonably well,
provided that the matter power spectrum is replaced by the halo power spectrum.
Therefore, the abundance of peaks and dips (a 1-point statistics) is sensitive to the scale-dependent bias in the halo power spectrum (a
2-point statistics), like the covariance of cluster counts \citep{cunha/etal:2010}.
This effect disappears when the density field perfectly traces the matter distribution as is the case for shear peaks for instance.
We have made a preliminary assessment of the applicability of the total abundance statistics as a test of $f_{_{\rm NL}}$ for a survey with specifications similar to those of the {\small Euclid}
mission \citep{EuclidRB}.
From a measurement at a single smoothing scale $R$, we obtain an uncertainty of $\Deltaf_{_{\rm NL}}=25$ (for $R=300h^{-1}\,{\rm Mpc}$) and 40
(for $R=50h^{-1}\,{\rm Mpc}$).
This suggests that a measurement combining different smoothing scales and halo masses should be able to achieve a sensitivity of
$\Deltaf_{_{\rm NL}}\lesssim 10$. While the sensitivity of this approach will likely be worse than the limits set by the latest CMB measurements
from Planck, $f_{_{\rm NL}}=0.8\pm 5$ \citep{PlanckPNG}, this approach should be competitive with galaxy clusters and shear peak counts in
weak-lensing maps,
for which the forecasted uncertainty is $\Deltaf_{_{\rm NL}}\sim 9$ \citep[e.g.][for a galaxy survey like {\small eROSITA}]{pillepich/etal:2012}
and $\Deltaf_{_{\rm NL}}\sim 13$ \citep[e.g.,][for a weak-lensing survey with Euclid specifications]{marian/etal:2011}, respectively.
However, our approach may also be affected by the Eddington bias that plagues galaxy cluster counts or shear peaks. Namely, additive
noise in the data will presumably increase the number of peaks while reducing the number of dips, which would mimic a small positive
$f_{_{\rm NL}}$. We will defer a more detailed study of this effect to future work.
The abundance of extrema depends on the cosmological parameters of the background cosmology. Here we explored the dependence on $\Omega_m$ alone with very encouraging results of an accuracy at the level of $\Delta \Omega_\mathrm{m}\sim 0.01$.
For a given filtering scale given in $h^{-1}\,{\rm Mpc}$, the abundance depends is nearly degenerate with the combination $\Omega_\mathrm{m}h$. Thus, this result regarding $\Omega_\mathrm{m}$ could alternatively by expressed as an accuracy of
$0.7\, {\rm km }\, {\rm s}^{-1}$ on $H_0$ if all other parameters are fixed.
\section*{Acknowledgements}
This research was supported by the I-CORE Program of the Planning and Budgeting Committee,
THE ISRAEL SCIENCE FOUNDATION (grants No. 1829/12 and No. 203/09 for AN; No. 1395/16 for VD)
and the Asher Space Research Institute.
M.B. acknowledges support from Delta ITP consortium, a program of the Netherlands Organisation for Scientific Research (NWO)
that is funded by the Dutch Ministry of Education, Culture and Science (OCW).
\bibliographystyle{mnras}
|
{
"timestamp": "2018-04-23T02:02:47",
"yymm": "1804",
"arxiv_id": "1804.05328",
"language": "en",
"url": "https://arxiv.org/abs/1804.05328"
}
|
\section{Introduction}\label{sec1}
In recent years, precision medicine has become an important topic in both industry and academia. It aims to find a mechanism of treatment assignment such that the patient can benefit the most. A lot of researchers have conducted vast investigations through different approaches. For example, the framework of outcome weighted learning is proposed to identify the individualized treatment rule (ITR) in \cite{Zhao2012}. Alternatively, Zhang et al. \cite{Zhang2012a} \cite{Zhang2012b} propose another general framework of estimating optimal treatment regimes with robustness from the perspective of classification. Fu, Zhou and Faries \cite{Fu2016} develop a comprehensive binary search approach, which is easy to interpret and apply in clinical study. Chen \cite{Chen2017} further extends the outcome weighted learning to other models and loss functions. However, these methods only focus on binary treatment recommendation. To handle multiple treatments, Zhang et al. \cite{Zhang2017} develop a multi-category outcome weighted learning approach using angle based classifiers.
Their model essentially assumes that each patient takes one out of multiple treatments. However, in reality, patients can benefit from taking multiple treatments simultaneously. Therefore, it is of great interest to develop methods for precision medicine in the context of combination therapies.
The adoption of combination therapies in medication is inevitable, especially in chronic diseases. For example, patients with type 2 diabetes often receive multiple medications, because single treatment may be insufficient to effectively control the blood glucose level. Hence, medications exerting their effects through different mechanisms are needed. For instance, DPP4 increases incretin level, which inhibits glucagon release. Sulfonylurea increases insulin release from $\beta$-cell in pancreas. DDP4 and Sulfonylurea function through different biological pathways, while serving the same purpose of reducing blood glucose. Thus, we can expect better control in blood glucose when taking both medications. In addition, diabetes may also cause other complications which require additional medications. Due to these two reasons, patients with type 2 diabetes are very likely to receive a combination of multiple treatments, and so is the case in other chronic diseases such as cancer, and chronic heart failure (CHF).
However, existing algorithms may not be adequate to precision medicine with combination therapies. Algorithms for multi-class classification are not scalable with the increasing number of treatments, that is, if we have $K$ treatments, there are $2^K$ possible combinations, which implies at least $2^K-1$ classifiers. This is the consequence of treating each combination individually \adapt{without considering the underlying structures among those treatments}. For example, DDP4 and Sulfonylurea can reduce A1c by $0.5\%$ and $1.0 \%$, respectively. \adapt{If the treatment effects of DDP4 and Sulfonylurea are additive, combining these two medications together can reduce A1c by $1.5\%$ . If small interaction exists, the reduction is expected to lie between $1.0\%$ and $2.0\%$.} {\adapt Our proposed method is able to leverage this information to improve the learning efficiency, which will be elaborated in Section \ref{sec5} and \ref{sec6}.
Different from the traditional supervised learning, the estimand is the optimal treatment assignment which is not directly observed from the patients. For example, in a classification problem, the correct label $Y$ is observed for each observed covariates $X$. Similarly, in a regression problem, an outcome $Y$ is also observed for each observed covariates $X$. In our treatment assignment problem, information about optimal treatment assignment rule is only available indirectly through the outcome due to the fact that only one potential outcome can be observed. In this paper, we extend deep learning algorithms, and propose an outcome weighted loss function to estimate optimal treatment assignment rule for combination therapies.
Our paper is organized in the following manner. Section \ref{sec2} provides a brief review to outcome weighted learning and multi-label classification problem. It explains the necessity of using multi-label classification rather than multi-class classification, and also {\adapt addresses some requirements that a good method needs to achieve}. Section \ref{sec3} incorporates multi-label classification with outcome weighted learning, and illustrates basic properties of our proposed method. Section \ref{sec4} further provides theoretical justifications on the fisher consistency under general cases and a special case with additive treatment effect and small interactions. Section \ref{sec5} and \ref{sec6} show the advantages of our method through simulations and real data analysis. In section \ref{sec7}, our method is further extended to a family of loss function which can be adaptive to the treatment interactions. We also discuss other possible future extensions in section \ref{sec7}.
\section{Review}\label{sec2}
\label{review:multi-label}
In this section, we will review some literature related to precision medicine and multi-label classification. In the meantime, the keys to solve this problem are identified.
Precision medicine has been a hot topic for years. On the one hand, outcome weighted learning proposed in \cite{Zhao2012} is one of the most popular method. Outcome weighted learning finds the optimal decision rule that maximizes the conditional expectation of the outcome given the decision rule. Let $X$ be the covariates, function $D(\cdot)$ be the decision rule, which is a mapping from space of covariates $\mathcal{X}$ to the space of treatment $\mathcal{A}$. The conditional expectation of the outcome $R\in \mathbb{R}$ given treatment assignment $D(X)$ can be written as:
\begin{equation}\label{eq:owl}
E^{D}(R)=E\left[\frac{R}{\pi_A}I\{A=D(X)\}\right],
\end{equation}
where $\pi_a=\Pr(A=a|X)$, where $A$ is the random variable of treatment assignment and $a\in \mathcal{A}$ is a treatment. In this framework, searching for the best treatment assignment $D$ to maximize $E^{D}(R)$ is converted into a classification problem. {\adapt Similar to the multi-class classification problem, the outcome weighted learning framework \eqref{eq:owl} can also be applied to multiple treatments problems using angel based learning \cite{Zhang2017}.} On the other hand, researchers also propose another general framework in \cite{Zhang2012a,Zhang2012b}, which estimates the contrast function of the treatment effect directly. The proposed methods in \cite{Zhang2012a} and \cite{Zhang2012b} also enjoy the advantage of double robustness which allows for misspecification on either potential outcome model or propensity score model. Chen \cite{Chen2017} further extends the ideas in \cite{Zhao2012} and \cite{Zhang2012b} to more general loss functions.
However, different from traditional multi-class classification problem where each patient is assigned to one of the treatments, multi-label classification problem allows patients to be assigned to a combination of multiple treatments. A naive implementation called Label Powerset (LP) transforms a multi-label classification problem with $K$ treatments (or classes) into a multi-class classification problem with $2^K$ classes, therefore, the dimension increases exponentially with the number of treatment $K$. To avoid the curse of dimensionality and to increase efficiency, two strategies are often adopted. The first strategy is Binary Relevance (BR) \cite{Luaces2012}. It transforms the original problem to a $K$ independent binary classification problem. For example, suppose $\mathcal{A}=\{a_1,a_2,\cdots,a_K\}$, BR type of methods build $K$ classifiers to individually and independently decide whether $a_k$, $k=1,\cdots,K$, should be adopted.
In this case, assumed linear classification rule with $p$-dimensional covariates, at least $Kp$ coefficients need to be estimated, which is much smaller than that of LP. However, it is broadly criticized by its independent assumption on treatments (or classes).
Therefore, ranking by pairwise comparison \cite{Hüllermeier2008} is proposed. {\adapt However, this method only provides a ranking of the treatments. It does not provide any `zero' point to separate treatments which are beneficial from those are harmful.}
Besides those attempts to convert the multi-label classification problem into other problems,
an alternative is adopting an appropriate loss. One of the commonly adopted loss function is Hamming loss.
Let $A=(A_1,\cdots,A_K)\in \mathcal{A}$ represents a vector of length $K$, where $\mathcal{A}=\{-1,1\}^K$. If the $k$th treatment is adopted, then $A_k=1$; If not, then $A_k=-1$. Accordingly, decision rule $D(X)=(D_1(X),\cdots,D_K(X))\in \{-1,1\}^K$. Hamming loss of a decision rule $D(\cdot)$ given the sample $(X,A)$ is defined as
\begin{equation}
\frac{1}{K}\sum_{k=1}^{K}I\{A_k\not=D_k(X)\}.
\end{equation}
And it can be interpreted as the proportion of the misclassified labels. Say there are only 3 treatments (labels), the relationship between Hamming loss and 0-1 loss is shown in Figure~\ref{fig:compare_loss}.
\begin{figure}[ht]
\centering
\includegraphics[scale=0.62]{compare_loss.eps}
\caption{The left is the 0-1 loss, the right is the Hamming loss. Hamming loss is a step function and bounded by 0-1 loss.\label{fig:compare_loss}}
\end{figure}
Another interesting topic is the shared subspace.
The idea proposed by \cite{Yan2007} assumes that all the decision rules are directly depend on the same subspace of covariates. Similar to central mean space estimation in sufficient dimension reduction \cite{Cook2005},
shared subspace essentially assumes that for certain $B$, there exist $\bar{D}_k$'s such that $D_k(X)=\bar{D}_k(B^\top X)$, for all $k\in 1,\cdots,K$.
Through shared subspace, we can always borrow some efficiency from other treatments, and thus facilitate the estimation of unknown non-linear relationship.
In addition, scalability of the algorithm is also important because heavy computation complexity can undermine its practical application.
Multi-label classification enables the scalable computation with respect to the number of treatments. The algorithm still needs to be scalable with the increasing sample size.
From this review of the literature, it's clear that
the method we are looking for should fully address the following issues:
\begin{enumerate}
\item \adapt{Applicability to precision medicine}.
\item Appropriate loss with the framework of multi-label classification.
\item Shared subspace.
\item Scalable computation with both number of treatments and sample size.
\end{enumerate}
\section{Method}\label{sec3}
In this section, a loss function is proposed within outcome weighted learning framework, which takes advantage of multi-label classification. Our classifier based on neural networks (NN) is introduced and combined with the proposed loss function. NN as a classifier naturally satisfies the requirement of shared subspace. And its algorithm is also scalable with the sample size.
\subsection{Outcome weighted learning with multi-label classification}
Let $X_i\in \mathcal{X}$ be the column vector of $p$-dimensional covariates for $i$th patient among total $n$ patients, $A_i=(A_{i1},\cdots,A_{iK})\in \{-1,1\}^K$ is the treatment assignment of the $i$th patient, and $R_i\in \mathbb{R}$ is the observed outcome of the $i$th patient. Again, decision rule is denoted as $$D(X)=(D_1(X),\cdots,D_K(X))\in \{-1,1\}^K.$$
Considering the following loss function for a given decision rule $D(X)$,
\begin{equation}\label{eq:proposed_loss}
L(D)=\frac{1}{n}\sum_{i=1}^{n}\frac{R_i}{\pi_{A_i}}\frac{1}{K}\sum_{k=1}^KI\{A_{ik}\not=D_k(X_i)\},
\end{equation}
where $\pi_{A_i}$ may be known in clinical trial or estimated in observational study.
When the problem is a multi-class classification problem, the loss above is exactly equal to the outcome weighted 0-1 loss proposed in \cite{Zhao2012}. When the problem is a multi-label classification problem, the proposed loss \eqref{eq:proposed_loss} is upper bounded by the outcome weighted 0-1 loss in outcome weighted learning. In addition, if $\frac{R_i}{\pi_{A_i}}$ is a constant, it reduces to Hamming loss. The loss function proposed in \eqref{eq:proposed_loss} combines the loss in outcome weighted learning and Hamming loss. Thus, we call it outcome weighted Hamming loss.
One of the difficulty in optimizing the proposed loss \eqref{eq:proposed_loss} is the non-smoothness and non-convexity of indicator functions. Hinge loss \cite{Cortes1995} or logistic loss can be adopted as surrogate losses for indicator functions. The fisher consistency of proposed loss in \eqref{eq:proposed_loss} and its surrogate loss are provided in the following sections. As proved in Section \ref{sec4}, our proposed loss is fisher consistent under small amount of interactions.
\adapt{
Another difficulty of the proposed method comes from the estimation of $\pi_{A_i}$. We provide two solutions to this issue when $K$ is large. First solution assumes that the treatment assignment $A$ is independent with the covariates $X$. In this case, $P(A=a|X)=P(A=a)$ can be estimated by the proportion of the patients with $A=a$ in the sample. Second solution assumes that each treatment assignment $A_k$ are independent with each other given covariates $X$. In this case, $P(A=a|X)=\prod_{k=1}^K P(A_k=a_k|X)$ and each $P(A_k=a_k|X)$'s can be estimated by logistic regression because $A_k$ are binary. When $K$ is small, multinomial regression can also be used to estimate $P(A=a|X)$ directly. For simplicity, the first solution is adopted in Section \ref{sec5} and Section \ref{sec6}.}
\subsection{Decision rule with deep learning}
In the previous section, outcome weighted Hamming loss is defined. Admittedly any suitable classifier can fit into our framework, the classifier adopted in this paper is the Neural Network (NN) for its advantages in shared subspace and scalable computation.
To simply illustrate our idea, we start from a $3$-layer NN. The first layer on the bottom is the input layer where $X_i=(X_{i1},\cdots,X_{ip})^\top$ is the input vector of covariates. The layer in the middle is the hidden layer with $d$ hidden variables. The top layer consists of $K$ output variables. An example of the graph structure of this NN is presented in Figure~\ref{fig:NN}. In this toy example, only 3 treatments are considered, say treatment $E$, treatment $F$, and treatment $G$. Thus, $\mathcal{A}=\{-1,1\}^3$. For example, $(1,1,-1)$ represents $EF$ which is the combination of treatment $E$ represented by $(1,-1,-1)$, and $F$ represented by $(-1,1,-1)$. The $k$th output in the top layer is $\tilde{D}_k(X_i)$ given $X_i$ in the input layer. The sign of $\tilde{D}_k(X_i)$ indicates the treatment assignment of $k$th treatment, which is $D_k(X_i)$. As shown in Figure~\ref{fig:NN}, the adjacent layers are fully connected and no variables in the same layer are connected.
Given the graphic structure described above, many NNs can fit into this framework. For example, Deep Belief Nets (DBN) proposed in \cite{Hinton2006}, which consists of stacks of Restricted Boltzmann Machines (RBMs) and a classifier based on the very top hidden layer. Another common choice is \adapt{Deep Neural Network (DNN)}, which is a large NN without probabilistic modeling. DNN can be obtained by firstly training a DBN and then fine-tuning by back-propagation. This approach of building DNN often times can help avoid local minimizers and obtain better generalization error. However, the topologies of DBN and DNN are slightly different. Although the skeleton for both are the same as shown in Fig~\ref{fig:DNN_DBN}, the directions of the connected lines can be different. In DBN, besides the top two layers, all connections between hidden layer and hidden layer or hidden layer and visible layer are bi-directional (or undirected), and only connections between top two layers are directed from the hidden layer to the output. In DNN, all connections are directed from lower layers to higher layers. Directed connections are defined by relationship similar to \eqref{eq:nn_build1} and \eqref{eq:nn_build2}, which have no probabilistic framework. Bi-directional (or undirected) connections are defined through undirected graphical models, such as RBM \cite{Goodfellow-et-al-2016}. Thus, the nodes connected by bi-directional (or undirected) connections are random variables. The value of the node is either a random sample from the defined conditional distribution or the conditional expectation given other nodes. For example, if the lower layer on the right in Fig~\ref{fig:DNN_DBN} is defined by RBM and denote the input layer as $v$, the first hidden layer as $h$, the joint density function of $(v,h)$ is defined as
\begin{equation*}
f_{(v,h)}(v,h)=\frac{1}{Z(\theta)}{\exp\{-E(v,h;\theta)\}},
\end{equation*}
where $\theta$ is unknown parameter, $Z(\theta)$ is a normalization constant, $E(v,h;\theta)$ is energy function which has certain forms depending on the type of the model \cite{Goodfellow-et-al-2016}. If $\theta$ is known, given an input $v$, the conditional distribution of $h$ given $v$ can be calculated. Either a random sample from this conditional distribution or conditional expectation of $h$ given $v$ can be used as the input for next layers. The density function of $v$ is
\begin{equation*}
f_{v}(v)=\int\frac{1}{Z(\theta)}{\exp\{-E(v,h;\theta)\}}dh.
\end{equation*}
Then $\theta$ can be estimated by the maximize likelihood estimation (MLE) given observed data $v$. However, In this paper, we focus on DNN in both simulations and real data analysis.
DNN is defined in the following fashion. For clarification, we explicitly define those variables and weights of connections given toy example in Fig \ref{fig:NN}. Let the input layer in the very bottom be $v=(v_1,\cdots,v_p)\in \mathbb{R}^p$, the hidden layer (consists of $d$ hidden variables) in the middle be $h=(h_1,\cdots,h_d)\in\mathbb{R}^d$, and the output layer on the very top be $\tilde{D}=(\tilde{D}_1,\cdots,\tilde{D}_K)\in\mathbb{R}^K$. Let $W_1$ be the $p\times d$ matrix representing the weights on connections between input layer and hidden layer. For example, the entry of $W_1$ on Row $3$ and Column $4$ is the weight on connection between $v_3$ and $h_4$. Similarly, let $W_2$ be the $d\times K$ matrix representing the weights on connections between hidden layer and output layer. For example, the entry of $W_2$ on Row $3$ and Column $2$ is the weight on connection between $h_3$ and $\tilde{D}_2$. Now, with these notations, the relationship between these variables can be defined as the following:
\begin{eqnarray}\label{eq:nn_build1}
h&=&ReLU(W_1^\top v+h_0),\\ \label{eq:nn_build2}
\tilde{D}&=&W_2^\top h+\tilde{D}_0,
\end{eqnarray}
where $ReLU(\cdot)$ is ReLU function defined as $ReLU(t)=t1\{t>0\}$, for any $t\in \mathbb{R}$. $ReLU(O)$ with a vector $O$ represents a vector with ReLU function applied to each entry of $O$. $h_0$ is a constant $d$-dimensional vector and $\tilde{D}_0$ is a constant $K$-dimensional vector. Apparently, from the above relationship, $\tilde{D}$ can be written as $\tilde{D}(v)=(\tilde{D}_1(v),\cdots,\tilde{D}_K(v))$. Having the indicator function replaced by Hinge loss \cite{Cortes1995}, we can formulate the optimization problem as follows,
\begin{equation}\label{eq:optimization}
\textrm{minimize}_{\theta}\frac{1}{n}\sum_{i=1}^{n}\frac{R_i}{\pi_{A_i}}\frac{1}{K}\sum_{k=1}^K\left[1-A_{ik}\tilde{D}_k(X_i)\right]_+,
\end{equation}
where $\theta=(W_1,h_0, W_2,\tilde{D}_0)$ and $[\cdot]_+$ represents taking non-negative part. This optimization problem can be solved directly by back-propagation and SGD. Note that the structure of DNN and the activation function can be modified based on real data. The final estimated decision rule, $\hat{D}$, is the sign of $\tilde{D}$ given the minimizer of \eqref{eq:optimization}, $\hat{\theta}$.
\begin{figure}
\centering
\includegraphics[scale=0.5]{DNNDBNFig.eps}
\caption{The left is the structure of DNN and the right is that of DBN. The solid connections represent a directed edge from the lower layer to the top layer. The dashed connections represent bi-directional (or undirected) connections.\label{fig:DNN_DBN}}
\end{figure}
NN is a natural choice in this setting for the following reasons. First, the optimization of NN given the proposed loss in \eqref{eq:proposed_loss} can be implemented by back-propagation \cite{Rumelhart1986} and stochastic gradient descent (SGD) \cite{Bottou2010}, which is scalable in terms of sample size due to the nature of SGD. Second, the decisions are made based on hidden layers which depend on some shared linear directions of $X_i$ and a pre-specified non-linear activation functions \cite{LeCun2015}. Thus, the subspace of these $K$ decision rules are shared. Third, the hidden variables can capture complicated correlations among treatments, which is quite clear in the point of graphical model that given $X_i$, $\tilde{D}_k$'s are not independent with each other.
Based on universal approximation theorem proved in \cite{Barron1993}, hidden variables also introduce more flexibility into the model. Without the hidden layers, NN is equivalent to a linear model.
In most applications, the structure of hidden layers such as the number of hidden layers and the number of hidden variables in each hidden layer is adjustable. More complicated the hidden structure is, more flexible the decision rule can be. Thus, our NN decision rule with proposed loss function in \eqref{eq:proposed_loss} can fully satisfy our requirements. Another ad hoc view of the hidden layers in Fig~\ref{fig:NN} is that those hidden variables may represent certain biological pathways through which certain treatment can affect the outcome. Thus, the decision rule is reasonable to depend on certain hidden variables.
\begin{figure}
\centering
\includegraphics[scale=0.5]{NNFig.eps}
\caption{The first layer on the bottom is the input layer where $X_i=(X_{i1},\cdots,X_{ip})^\top$ is the input. The layer on the top is hidden layer with $d$ hidden variables. The top layer consists of $K$ output random variables.\label{fig:NN}}
\end{figure}
\subsection{Overfitting}
Due to strong representative power of DNN, how to avoid over-fitting has become a big issue. To avoid over-fitting, a commonly used strategy is regularization through penalization. Mathematically, we can consider to solve the following problem:
\begin{equation}
\textrm{minimize}_{\theta}\frac{1}{n}\sum_{i=1}^{n}\frac{R_i}{\pi_{A_i}}\frac{1}{K}\sum_{k=1}^K[1-A_{ik}\tilde{D}_k(X_i)]_++\lambda p(W_1,W_2),
\end{equation}
where $\theta=(W_1,h_0, W_2,\tilde{D}_0)$. $p(W_1,W_2)$ is the penalty on $W_1$ and $W_2$. For example, $p(W_1,W_2)=\|W_1\|_F^2+\|W_2\|_F^2$ is the ridge penalty, where $\|\cdot\|_F$ is the Frobenius norm, and $p(W_1,W_2)=\|W_1\|_1+\|W_2\|_1$ is lasso penalty, where $\|\cdot\|_1$ represents the sum of absolute value of all entries. Both ridge penalty and lasso penalty can shrink the weights towards $0$, but the advantage of lasso penalty is that it can produce sparse solutions. Another popular method to prevent over-fitting is the so-called dropout proposed in \cite{Srivastava2014}. Wager et al. \cite{Wager2013} argue that dropout is closely related to adaptive penalization and they also clarify its relationship with ridge penalty.
In general, both regularization and dropout are possible choices to prevent over-fitting.
\adapt{In both regularization and dropout, how to choose the tuning parameter has been investigated for years\cite{Srivastava2014, tibshirani1996regression,zou2005regularization}. Cross-validation or training-validation-testing data split can be used to evaluate the prediction error and select tuning parameters with the best performance. }
\adapt{Other parameters such as number of layers and number of nodes each layer also play an important role in balancing under-fitting and over-fitting. Of course, cross-validation or training-validation-testing data split can be applied to tune the NN's structure, but it is extremely computational intensive and impractical some times. In general, how to design NN's structure is still an open problem, but extensive researches have been done in this field. On the one hand, some researchers are working on using genetic algorithm and reinforcement learning to decide these parameters \cite{leung2003tuning,zoph2016neural}. On the other hand, sparsity induced by lasso penalty can partially solve this issue. Suppose that all weights connected to a particular variable are $0$, it is equivalent to excluding this variable in the structure of NN. Additionally, other penalties, for example, in \cite{Scardapane:2017:GSR:3067301.3067328}, weights connected to a particular variable can be forced to be $0$ and thus the node is excluded in the structure. Thus, in practice, we may suggest a structure as more complex as possible considering the sample size and employ lasso penalty or other penalties to help select nodes included in the structure automatically.}
\section{Algorithm}
In this section, we introduce the algorithm to solve the proposed optimization problem. A straightforward solution is back-propagation with stochastic gradient descent (SGD). Back-propagation proposed in \cite{Goodfellow-et-al-2016} is an efficient algorithm to numerically compute the derivatives with respect to certain weight given a NN. SGD provides an computational effect alternative to gradient descent. Beyond directly implementing SGD, pre-training can also be used to prevent local minimizers and facilitate convergence of the algorithm. One of the pre-training procedure utilizes Restricted Boltzmann Machine (RBM) \cite{Goodfellow-et-al-2016}. In this procedure, we firstly train stacks of RBM in such fashion that the first RBM is trained on observed covariates and next RBM is trained on the top of the first one, taking the hidden layer in the first RBM as the visible layer in the next RBM, and so on. Then, for the two layers on the very top, a multi-label classifier based on outcome weighted Hamming loss is trained. After this pre-training procedure, the whole network is fine-tuned by back-propagation and SGD.
\subsection{Stochastic gradient descent}
\adapt{
In this section, we briefly introduce the stochastic gradient descent (SGD) under general empirical risk minimization. Say we want to minimize the following loss function
\begin{equation}
\min_{\theta} \frac{1}{n}\sum_{i=1}^{n} l(Y_i, X_i;\theta),
\end{equation}
where $\theta$ is the general notation for all parameters of interest, $(X_i, Y_i) ,~i=1,\cdots,n$ is the observed data. In gradient descent, we have
\begin{equation}
\theta^{(m+1)}\leftarrow \theta^{(m)}-t\frac{1}{n}\sum_{i=1}^{n}\nabla_{\theta}l(Y_i, X_i;\theta),
\end{equation}
where $\theta^{(m)}$ is the value of $\theta$ at the $m$ step, and $t>0$ is a scalar called learning rate (step size). In stochastic gradient descent, we have
\begin{equation}
\theta^{(m+1)}\leftarrow \theta^{(m)}-t\frac{1}{|\mathcal{S}^{(m)}|}\sum_{i\in \mathcal{S}^{(m)}}\nabla_{\theta}l(Y_i, X_i;\theta),
\end{equation}
where $\mathcal{S}^{(m)}$ is a random sample from $\{1,\cdots, n\}$ with or without replacement, and $|\mathcal{S}^{(m)}|$ is the cardinality of the set $\mathcal{S}^{(m)}$, which is a pre-specified batch size. It is easy to see that
\begin{equation}
E_{\mathcal{S}}\left [\frac{1}{|\mathcal{S}^{(m)}|}\sum_{i\in \mathcal{S}^{(m)}}\nabla_{\theta}l(Y_i, X_i;\theta)\right ]=\frac{1}{n}\sum_{i=1}^{n}\nabla_{\theta}l(Y_i, X_i;\theta),
\end{equation}
where $E_{\mathcal{S}}[\cdot]$ is the expectation of random sampling.} Because each update only depends on a small size of random sample, it is scalable with the increase of sample size.
\adapt{In each update of SGD, gradient is computed on a small batch based on a sub-sample of training dataset. On the one hand, the stochastic induced by sub-sampling prevents the algorithm from falling into local minimizers. On the other hand, it reduces the computation complexity dramatically compared with usual gradient descent. In general, SGD converges faster than usual gradient descent, especially for statistical problems \cite{bousquet2008tradeoffs}. However, due to the stochastic nature of SGD, the calculated gradient based on a batch is mostly not zero even at the global minimizer. Thus, the variance of sampling small batches has an impact on the updates of each iteration. To reduce this variance, SVRG and other techniques \cite{Johnson2013} are proposed. Another simple strategy is to gradually increase the size of the batch in gradient calculation and decrease learning rate (step size) exponentially in gradient update. The convergence of SGD has been proved in strictly convex problem \cite{bousquet2008tradeoffs}, while generally the convergence of SGD is still an open problem.
\subsection{Implementation}
\adapt{In this section, we introduce some packages to implement SGD. A well known package in Python is called Tensorflow developed by Google. Tensorflow provides an extremely powerful tool to customize NN (number of layers and number of nodes each layer) and SGD (batch size and step size). It also includes some well-developed advanced algorithm for SGD and keeps updating. Compared with Tensorflow, Keras is a high-level NN API which is more user-friendly. It allows researchers to build a very flexible NN with a few lines of code. The down side is that it may not be easy to customize everything using Keras. Recently, R initiated an access to Tensorflow, which is more friendly to statistical programmers. An experimental R package to implement the Keras is also available on GitHub. Readers can get more information on R versions of Tensorflow and Keras on https://tensorflow.rstudio.com.}
\section{Theoretical result}\label{sec4}
In this section, the theoretical justification of our proposed loss is provided. In the first part, we prove the fisher consistency of outcome weighted Hamming loss, which implies that minimizing the proposed loss is equivalent to minimizing the original outcome weighted 0-1 loss. In the second part, surrogate loss for outcome weighted Hamming loss is proposed and its consistency with outcome weighted Hamming loss is also provided, which indicates that minimizing the surrogate loss is also equivalent to minimizing the outcome weighted Hamming loss. Combining these two parts of theoretical results provides an overall theoretical support for our proposed method.
Without loss of generality, we assume that outcome $R$ is non-negative.
\subsection{Fisher consistency of Hamming loss}
We establish the fisher consistency of outcome weighted Hamming loss in this section, indicating the equivalence of minimizing outcome weighted Hamming loss and outcome weighted 0-1 loss. Let $D^*(X)=(D_1^*(X), \cdots,D_K^*(X))\in \{-1,1\}^K$ be the decision rule that minimizes outcome weighted 0-1 loss:
\begin{equation}
\mathcal{R}(D)=E\left[\frac{R}{\pi_A}I\{A\not = D(X)\}\right].
\end{equation}
It is easy to see that $D^*(X)=\{a:\max_a E[R|A=a, X]\}$. Again, outcome weighted Hamming loss is defined as
\begin{equation}
\mathcal{R}_H(D)=E\left[\frac{R}{\pi_A}\frac{1}{K}\sum_{k=1}^KI\{A_k\not = D_k(X)\}\right].
\end{equation}
\begin{theorem}[Fisher consistency of outcome weighted Hamming loss]\label{thm1}
Suppose $$D_k^*(X)=\textrm{sign}\left\{\sum_{\{a:a_k=1\}}E[R|A=a,X]-\sum_{\{a:a_k=-1\}}E[R|A=a,X]\right\},$$
then any function $f$ such that $$\mathcal{R}_H(f)=\inf_D\{\mathcal{R}_H(D)\}$$ satisfies
$$\mathcal{R}(f)=\inf_D\{\mathcal{R}(D)\}.$$
\end{theorem}
Note that in many cases, condition $D_k^*(X)=\textrm{sign}\left\{\sum_{\{a:a_k=1\}}E[R|A=a,X]-\sum_{\{a:a_k=-1\}}E[R|A=a,X]\right\}$ holds. For example, a sufficient condition is when treatment effects are additive and there is no interaction, i.e. $E[R|A=a, X]=\sum_{\{k:a_k=1\}}T_{e_k}(X) + m(X)$, where $T_{e_k}, k=1,\cdots,K$, is the treatment effect given only treatment $k$, $e_k$ is a $K$-dimensional vector having the $k$th entry equals $1$ and all other entries equal $-1$.
Moreover, small interactions are tolerable, as stated in Theorem \ref{thm2}.
\begin{theorem}[Fisher consistency in special case]\label{thm2}
Suppose that
\begin{equation*}
E[R|A=a, X]=\sum_{\{k:a_k=1\}}T_{e_k}(X) + r_a(X) + m(X),
\end{equation*}
where $T_{e_k}(X)$ is the treatment effect of only $k$th treatment being adopted, $r_a(X)$ is the additional interaction, and $m(X)$ is the main effect. If $2\sup_a|r_a(X)|< \inf_k|T_{e_k}(X)|$ for all $X$, then any function $f$ such that $$\mathcal{R}_H(f)=\inf_D\{\mathcal{R}_H(D)\}$$ satisfies
$$\mathcal{R}(f)=\inf_D\{\mathcal{R}(D)\}.$$
\end{theorem}
Theorem \ref{thm2} provides a sufficient condition for fisher consistency of outcome weighted Hamming loss. It essentially provides a theoretical guarantee of the robustness of our method against small interactions. When the condition is violated, our method is not necessarily consistent and its performance depends on the magnitude of interactions and amounts of patient affected. In simulation, we compare the performance our method with other methods under small violation of this condition
\subsection{Multi-label consistency of surrogate loss}
Minimizing the proposed loss \eqref{eq:proposed_loss} is still very hard due to the non-smoothness and non-convexity of indicator functions. Therefore, it is natural to replace the indicator functions in outcome weighted Hamming loss with some surrogate loss. For example, a common choice for outcome weighted Hamming loss is
\begin{equation}\label{eq:surrogate_hamming}
\Phi_H(\tilde{D})=E\left[\frac{R}{\pi_A}\frac{1}{K}\sum_{k=1}^K\phi(A_k\tilde{D}_k(X))\right],
\end{equation}
where $\phi$ is a pre-defined convex function. Still, it is critical that minimizing the surrogate loss is equivalent to minimizing the original outcome weighted Hamming loss. Unfortunately, not every $\phi$ satisfies this condition. As shown in the following theorem, \eqref{eq:surrogate_hamming} is consistent with outcome weighted Hamming loss if $\phi$ is one of the following:
\begin{enumerate}
\item Exponential: $\phi(x)=e^{-x}$;
\item Hinge: $\phi(x)=(1-x)_+$;
\item Least squares: $\phi(x)=(1-x)^2$;
\item Logistic Regression: $\phi(x)=\ln(1+e^{-x})$.
\end{enumerate}
Formally, we have the following theorem.
\begin{theorem}[Multi-label consistency of surrogate loss]\label{thm3}
Suppose $$\phi'(0)<0,$$
for any function $\tilde{f}$ such that $$\Phi_H(\tilde{f})=\inf_{\tilde{D}}\{\Phi_H(\tilde{D})\},$$ let $f=\textrm{sign}(\tilde{f})$, then $f$ satisfies
$$\mathcal{R}_H(f)=\inf_D\{\mathcal{R}_H(D)\}.$$
\end{theorem}
\section{Simulation}\label{sec5}
\subsection{Simulation with correctly specified model}\label{subsec5.1}
\label{Sec: correctmodel}
In this section, we will illustrate the performance of two DNN approaches under outcome weighted Hamming loss in \eqref{eq:optimization} by comparing with a naive method through simulations. In the following simulation settings, $K=5$, so in total there are $2^5=32$ combinations of treatments. The dimension of covariates is set to be $p=30$, which is common in clinical trials. As explained in Section \ref{review:multi-label}, the $K$ treatment multi-label problem can be decomposed to a multi-class problem with $2^K$ classes. The naive method further converts the multi-class problem to a series of two-class classification problem, where for each, it directly learns a binary classifier with linear decision rule through outcome weighted learning. To form a multi-class classifier with $32$ different classifiers with intercepts, $2^K(2^K-1)(p+1)/2=15376$ coefficients in $2^K(2^K-1)/2=496$ different linear decision rules have to be estimated. The first DNN approach, DNN-simple, is a DNN with only input and output layer. For any monotone activation function, DNN-simple is equivalent to $K=5$ binary linear classifiers with no shared subspace. In general, the only difference between DNN-simple and naive method is how to form the classifier for multi-label classification. In DNN-simple, only $K\times p+K=155$ parameters needs to be estimated. The comparison of DNN-simple and naive method essentially shows the efficiency boosting by adopting our proposed outcome weighted Hamming loss. The other DNN approach, DNN-1hdd, adds one hidden layer between input layer and output layer as shown on the left in Figure \ref{fig:DNN_DBN}. The hidden layer is fully connected with the input layer and the output layer.The NN structure allows for more flexibility and hence more parameters. In DNN-1hdd method, suppose the number of hidden variable is $n_h$, the total number of parameters to be estimated is $p(n_h+1)+n_h(K+1)=p+n_h(p+K+1)=30+36n_h$. By comparing DNN-simple and DNN-1hdd, it is easy to tell the loss and gain to accommodate a more flexible model. In addition, the Bayes rule is also evaluated, simply to quantifies the signal-to-noise ratio in our simulation settings. The Bayes rule is the treatment assignment which gives the largest conditional expectation of the potential outcomes, i.e.
\begin{equation}
D_{\rm B}(X)=\arg \max_aE\left[R|A=a,X\right].
\label{Eq:bayesrule}
\end{equation}
The Bayes rule is impossible to implement in reality, but in simulations, since we know the data generating procedure, we can directly evaluate $E\left[R|A=a,X\right]$.
In the simulation, every patient has the same probability to receive one of the treatment combination, and for each treatment, the treatment effect is generated from a NN with one hidden layer as shown in the left of Figure~\ref{fig:DNN_DBN}. The number of the hidden variables in the hidden layer is $n_h=45$. Firstly, we define the treatment effect of the $k$th treatment for $i$th patient $T_{k,i}$. Let
\begin{eqnarray}
h&=&ReLU(W_1^\top X_i),\\
T_{k,i}&=&\textrm{sign}\left (W_{2,\cdot,k}^\top h\right )\left\{\frac{0.05\exp\left\{W_{2,\cdot,k}^\top h\right\}}{1+\exp\left\{W_{2,\cdot,k}^\top h\right\}}+2.0\right\},
\end{eqnarray}
where $h$ is the hidden layer which is a $n_h$-dimensional vector, $W_{2,\cdot,k}$ is the $k$th column of $W_2$, and $\textrm{sign}(\cdot)$ is the function of taking sign coordinate-wise. All entries in $W_1$ and $W_2$ are generated from a standard normal distribution independently. Secondly, the main effect is defined by the following:
\begin{equation}
M_i=0.05\frac{\exp\left\{\gamma^\top X_i\right\}}{1+\exp\left\{\gamma^\top X_i\right\}}-2.05,
\end{equation}
where $\gamma$'s are the coefficients for main effect, whose entries are also generated from a standard normal distribution independently. Let $A_i\in\{-1,1\}^K$ represents the combination of assigned treatment to $i$th patient. Note that $R_{i,A_i}$, the potential outcome when given $A_i$ is defined by the following:
\begin{equation}
R_{i,A_i}=\sum_{\{k: A_{ik}=1\}}T_{k,i}+M_i+\sigma\epsilon_{i,A_i},
\end{equation}
where $\epsilon_{i,A_i}$ follows standard normal distribution and is independent with any other random variables. It can be simply verified that $\forall \beta_{p,k}, \beta_{n,k}, $ and $\gamma$, given any $A_i$, $\sum_{\{k: A_{ik}=1\}}T_{k,i}+M_i\in[-12,8]$. Let $A_i^{\rm opt}$ represents the selected combination of treatment, given which the potential outcome is maximized. In our setting, $\sigma$ is chosen to be $1.1$. In fact, it can be observed that the Bayes rule as defined in Equation \ref{Eq:bayesrule} is the sign of the $T_{k,i}$ for each $k$. In addition, based on Theorem \ref{thm2} and \ref{thm3}, our proposed loss is consistent with outcome weighted 0-1 loss in this setting.
In both DNN approaches, the activation function between hidden layer and output layer, or input layer and output layer is fixed to be a simple centered monotone transformation. The activation function between input layer and hidden layer in DNN-1hdd is chosen to be ReLU. $L_1$ penalty is applied to all the weights in both of DNN approaches.
To compare different methods, three scores are defined to quantify their performance. Two of the scores are based on misclassification rate. Let $\hat{A}_i$ be the predicted combination of treatments. Usually, the misclassification rate is defined as
\begin{equation}
MCR=\frac{1}{n}\sum_{i=1}^n 1\{\hat{A}_i\not = A_i^{\rm opt}\}.
\end{equation}
To account for the fact that $P(A_i^{\rm opt}=a)$ is not the same, for all $a\subset\{1,\cdots,K\}$, the average of proportion of misclassification rate is also proposed as following
\begin{equation}
AMCR=\frac{1}{2^K}\sum_{a\in\{-1,1\}^K}\frac{\sum_{i=1}^n 1\{\hat{A}_i\not=a, A_i^{\rm opt}=a\}}{\sum_{i=1}^n 1\{A_i^{\rm opt}=a\}}.
\end{equation}
To adjust MCR and AMCR based on the total number of combinations of treatments and make their performance comparable with binary classifier for binary classification problem, adjusted MCR and AMCR is calculated by $1-(1-MCR)^{1/K}$ and $1-(1-AMCR)^{1/K}$, respectively. Because in the framework of personalized medicine, the ultimate goal is not classification, but better clinical outcome. Thus, it is primary to consider the average benefit which defines as following
\begin{equation}
\label{Eq:AB}
AB=\frac{1}{n}\sum_{i=1}^n R_{i,\hat{A}_i}.
\end{equation}
For MCR (or adjusted MCR) and AMCR (or adjusted AMCR), lower is better. However, higher AB is preferred.
The simulation procedure works as follows. First, a training set and a validation set with the same sample size $n_{train}K$ are generated separately, and a sequence of candidate tuning parameters are pre-specified, in this case, 0.1, 0.01, 0.001, 0.0001. Then for each candidate tuning parameter, the Naive method, DNN-simple, and DNN-1hdd are trained on the training dataset and tested on the validation set, where we compute the misclassification rate on the validation dataset for each method. The tuning parameter gives the lowest misclassification rate on the validation set is selected, respectively for each method. At last, we evaluate the three scores (MCR, AMCR, AB) under the selected tuning parameter on testing dataset which is independently generated with sample size $n_{test}=10n_{train}K$. The above procedure is repeated 100 times. The mean and standard error (SE) of the scores are reported in Table \ref{tab2}. Figure \ref{Fig:comparison} shows a summary for adjusted scores and AB over 100 repeats.
Overall, DNN-1hdd performs the best in terms of MCR (or adjusted MCR) and AB. Note that Bayes method is infeasible in practice, its superiority is because it takes advantage of the true model. The Bayes method serves as a reference to quantify the best possible performance we could get in this problem. Although DNN-simple can only produce linear decision rules, which don't include the true decision rule, its performance is still competitive with DNN-1hdd because linear decision rule can explain the true non-linear model to some extent, especially in small samples when the information is limited. The fact that DNN-1hdd allows more flexibility also introduces additional variation. In general, DNN-1hdd performs slightly better than DNN-simple with this bias and variance trade-off. In terms of AMCR (or adjusted AMCR), DNN-simple performs the best among all three methods. From Table \ref{tab2} and Figure \ref{Fig:comparison}, as the sample size increases, the misclassification decreases and average benefit increases for all methods, which agrees with our consistency result proved in Theorem \ref{thm2}. However, the decreasing in MCR (or adjusted MCR) becomes slower as sample size increases, because (1) the scores are lower bounded by Bayes rule; (2) the algorithm is terminated after a certain times of iterations and takes the final update as the minimizer. In practice, one possible strategy is to use a validation dataset to monitor the algorithm and pick the update with the best performance in validation dataset.
\begin{center}
\begin{table}[t]%
\centering
\caption{Comparison of naive method, DNN-simple, and DNN-1hdd by adjusted scores.\label{tab2}}%
\begin{tabular*}{500pt}{@{\extracolsep\fill}ccccc@{\extracolsep\fill}}
\toprule
$n_{train}$ & Method & adjusted MCR & adjusted AMCR & AB\\
\midrule
200 & Naive &0.4708(0.0026) &0.4652(0.0012) &-1.5171(0.1004)\\
200 & DNN-simple & 0.2824(0.0022) &0.3555(0.0022)& 0.5765(0.1172)\\
200 & DNN-1hdd & 0.2782(0.0028) &0.3709(0.0025)& 0.6188(0.1200)\\
200 & Bayes & 0.0657(0.0001)&0.1108(0.0027) &3.1173(0.1143)\\ \hline
1000 & Naive &0.3517(0.0024) &0.3729(0.0012) &-0.0897(0.0948)\\
1000 & DNN-simple & 0.2125(0.0016) &0.3008(0.0028)& 1.3609(0.1153)\\
1000 & DNN-1hdd & 0.2142(0.0015) &0.3240(0.0041)& 1.3376(0.1145)\\
1000 & Bayes & 0.0657(0.0001)&0.1111(0.0027) &3.0871(0.1133)\\ \hline
4000 & Naive &0.3083(0.0029) &0.3396(0.0018) &0.3502(0.0883)\\
4000 & DNN-simple & 0.1907(0.0014) & 0.2902(0.0034) & 1.7257(0.1166)\\
4000 & DNN-1hdd & 0.1806(0.0012) & 0.2991(0.0050) & 1.8435(0.1134)\\
4000 & Bayes & 0.0656(0.0000)&0.1140(0.0029) &3.1965(0.1127)\\ \hline
10000 & Naive &0.2996(0.0035) &0.3305(0.0015) &0.4934(0.0766)\\
10000 & DNN-simple & 0.1846(0.0014) & 0.2865(0.0033) &1.7272(0.1008)\\
10000 & DNN-1hdd & 0.1617(0.0012) &0.2889(0.0063) &1.9858(0.0986)\\
10000 & Bayes & 0.0657(0.0001)&0.1111(0.0027) & 3.1272(0.0977)\\
\bottomrule
\end{tabular*}
\end{table}
\end{center}
\begin{figure}
\centering
\includegraphics[height=9cm, width=12cm]{Comparison_new}
\caption{Comparison of naive method, DNN-simple, and DNN-1hdd by adjusted MCR, AMCR, and AB.\label{Fig:comparison}}
\end{figure}
\begin{figure}[ht]
\centering
\includegraphics[height=9cm, width=12cm]{Comparison_mis}
\caption{Comparison of naive method, DNN-simple, and DNN-1hdd by adjusted MCR, AMCR, and AB with model mis-specification.\label{Fig:comparison_mis}}
\end{figure}
\subsection{Simulation with model mis-specification}
In this section, we compare our two DNN approaches with a naive method when the condition in Theorem \ref{thm2} is not satisfied. The simulation procedure is the same as in Section \ref{Sec: correctmodel} except for the way to generate $R_{i, A_i}$. In this simulation, given $T_{k,i}$ and $M_i$, the potential outcome is generated by
\begin{equation}
R_{i,A_i}=\sum_{\{k: A_{ik}=1\}}T_{k,i}-\gamma (\sum_{\{k: A_{ik}=1\}}T_{k,i})^2+M_i+\sigma\epsilon_{i,A_i},
\end{equation}
where $\gamma=0.1$ and $\sigma=0.2$. In this setting, it is easy to see that $\sup_A|\gamma (\sum_{\{k: A_{ik}=1\}}T_{k,i})^2|\approx10$ and $\inf_k|T_{k,i}|\approx2$. Thus, the condition in Theorem \ref{thm2} is violated. The simulation procedure is the same as that described in Section \ref{subsec5.1}.
The averages of Adjust MCR, AMCR, and AB over 100 repeats with their standard deviations are reported in Table \ref{tab2_mis} and Figure \ref{Fig:comparison_mis}, which show that our proposed methods DNN-1hdd and DNN-simple outperform Naive method with respect to all three scores. Overall, DNN-1hdd has the best performance with respective to AB and adjusted MCR, while DNN-simple performs the best when considering the adjusted AMCR. Although the Naive method is consistent, it performs very bad under finite sample because of the high dimensionality of this problem. Our two DNN approaches still perform well when the small interaction assumption is violated.
\begin{center}
\begin{table}[t]%
\centering
\caption{Comparison of naive method, DNN-simple, and DNN-1hdd by adjusted scores with model mis-specification.\label{tab2_mis}}%
\begin{tabular*}{500pt}{@{\extracolsep\fill}ccccc@{\extracolsep\fill}}
\toprule
$n_{train}$ & Method & adjusted MCR & adjusted AMCR & AB\\
\midrule
200 & Naive &0.4863(0.0024) &0.4720(0.0017) &-2.6427(0.0921)\\
200 & DNN-simple & 0.3048(0.0053) &0.3703(0.0029)& -0.9197(0.0516)\\
200 & DNN-1hdd & 0.2994(0.0060) &0.3829(0.0028)& -0.8746(0.0491)\\
200 & Bayes & 0.1157(0.0064)&0.1726(0.0027) &0.2124(0.0199)\\ \hline
1000 & Naive &0.4074(0.0063) &0.4096(0.0022) &-1.2951(0.0470)\\
1000 & DNN-simple & 0.2432(0.0063) &0.3244(0.0036)& -0.4939(0.0308)\\
1000 & DNN-1hdd & 0.2433(0.0067) &0.3462(0.0043)& -0.4885(0.0307)\\
1000 & Bayes & 0.1127(0.0065)&0.1732(0.0029) &0.2136(0.0164)\\ \hline
4000 & Naive &0.3761(0.0070) &0.3898(0.0044) &-1.1061(0.0424)\\
4000 & DNN-simple & 0.2323(0.0064) & 0.3166(0.0040) & -0.3592(0.0267)\\
4000 & DNN-1hdd & 0.2254(0.0070) & 0.3275(0.0051) & -0.3151(0.0247)\\
4000 & Bayes & 0.1199(0.0066)&0.1761(0.0028) &0.2293(0.0161)\\ \hline
10000 & Naive &0.3890(0.0084) & 0.4059(0.0020)& -1.1827(0.0370)\\
10000 & DNN-simple &0.2210(0.0053) &0.3102(0.0037)&-0.3050(0.0279)\\
10000 & DNN-1hdd &0.2016(0.0060)&0.3128(0.0060)&-0.1966(0.0247)\\
10000 & Bayes &0.1146(0.0054) &0.1761(0.0027)& 0.2354(0.0172)\\
\bottomrule
\end{tabular*}
\end{table}
\end{center}
\section{Real data analysis}\label{sec6}
In this section, we apply our method to an electronic health record (EHR) data for type 2 diabetes patients from Clinical Practice Research Datalink (CPRD). 1139 patients are included in the dataset. For each patient, 21 covariates are collected before treatment assignment, which include demographical variables such as gender, BMI, HDL, and LDL, and also indicator of complications such as stroke and hypertension. The primary endpoint is change in A1c. Because A1c typically drops after applying the treatments, the primary endpoint is always negative and smaller is preferable. Thus, the proposed method can be easily implemented by considering negative change in A1c as patient outcome.
In this dataset, 4 treatments are considered for each patient, DDP4, sulfonylurea (SU), metformin (MET), and TZD. These 4 treatments function via four different biological processes. DDP4 increases incretin levels, which inhibits glucagon release. SU increases insulin release from $\beta$-cell in pancreas. MET decreases glucose production by the liver and increase the insulin sensitivity of body tissue. TZD makes cells more dependent on oxidation of carbohydrates. These 4 treatments target on different cells or functional organs, so it is reasonable to assume little treatment interaction among them and additive treatment effects.
In this real dataset, $K=4$, resulting in $16$ possible combinations of treatments. Because the original average benefit score can not be directly calculated in real settings, the naive method, DNN-1hdd, and DNN-simple are evaluated under a weighted version of average benefit defined as follows,
\begin{eqnarray}
T&=&\frac{\sum_{i=1}^{n} w_iR_i}{\sum_{i=1}^{n} w_i},\nonumber
\end{eqnarray}
where $w_i=\frac{I\{\hat{A}_i=a_i\}}{P(A_i=a_i|X_i)}$, $a_i$ is the observed treatment assignment, $R_i$ is the negative change in A1c, $\hat{A}_i$ is the predicted treatment assignment for the $i$th patient. Intuitively, $T$ is the weighted average of change in A1c over all the patients with the same treatment assignment as our predicted treatment assignment, which estimates the change in A1c if the fitted treatment assignment is adopted. If our proposed methods can recover the underlying optimal decision rule to some extent, their $T$ are expected to be lower than that of the naive method, indicating higher efficacy of our treatment recommendations. In addition, the number of patients $N$ satisfying $a_i=\hat{A}_i$ is also reported.
Multiple imputation is adopted to deal with missing data. Because of the randomness of the multiple imputation, $5$ different imputed datasets are analyzed following the same procedure, and then the scores from these $5$ imputed data are summarized by average.
For each imputed dataset, we do the following. Firstly, the whole dataset is randomly split into two datasets, training set and testing set. Training set contains $912$ patients, and testing dataset contains $227$ patients. All three methods are fitted on training set respectively, and the score for each of the method is calculated based on the testing set. This procedure is repeated $100$ times, each time the score is recorded. After summarizing these scores across $5$ imputed datasets, the mean of $T$, $N$ and their standard errors (SEs) are reported in Table~\ref{tab3}, and all results over $100$ repeats are shown by boxplots in Figure~\ref{Fig:realdata}.
It can be observed that both DNN-simple and DNN-1hdd have lower $T$ compared with naive method. Moreover, both DNN methods show significant effect on A1c while naive method does not. Comparing between two DNN methods, DNN-1hdd performs slightly better than DNN-simple in terms of $T$. In addition, $N$, which is the size of the subgroup with the same treatment assignment as predicted for our proposed methods, is much larger than that for naive method. Overall, our proposed methods have better performance than naive method, and DNN-1hdd is slightly better than DNN-simple.
\begin{center}
\begin{table}[t]
\caption{Comparison of Naive method, DNN-simple, and DNN-1hdd by scores in real data example. $T$ is the weighted average of change in A1c over all the patients with the same treatment assignment as the method suggests. $N$ is the number of the patients whose treatment assignment coincide with the predicted.\label{tab3}}%
\begin{tabular*}{500pt}{@{\extracolsep\fill}ccc@{\extracolsep\fill}}
\toprule
Method & $T$ & $N$ \\ \midrule
Naive & -1.534(0.078) & 4.410(0.107)\\
DNN-simple & -2.605(0.058) & 23.058(0.371) \\
DNN-1hdd & -2.695(0.057) & 25.790(0.394)\\
\bottomrule
\end{tabular*}
\end{table}
\end{center}
\begin{figure}
\centering
\includegraphics[height=9cm, width=12cm]{realdata}
\caption{Boxplots of scores for Naive method, DNN-simple, and DNN-1hdd over 100 repeats .\label{Fig:realdata}}
\end{figure}
\section{Conclusions and discussions}\label{sec7}
In this paper, an outcome weighted deep learning framework is proposed to estimate optimal combination therapies. Both simulation and real data analysis provide solid evidence on the power of our proposed method. Essentially, the proposed loss, outcome weighted Hamming loss, can be applied to any occasion, even when deep learning is not a desired classifier. For example, linear classifiers can be used in the case when efficiency and convexity is very important. Although other nonlinear classifiers can also be adopted, when adopting these methods, advantages of deep learning such as sharing subspace may disappear unless inducing certain techniques such as dimension reduction. Deep learning framework can also be applied under other losses. For example, partial ranking loss proposed in \cite{Gao2013} can also be combined with deep learning approach. However, for ranking based loss function, one of the significant drawback is the lack of `zero' point to distinguish between good treatments and bad treatments. In other word, ranking based loss function can only provide a rank of treatments instead of recommending treatment. In this paper, the advantages of adopting deep learning approach and the loss function has been articulated . Our method enjoys all of these advantages.
Furthermore, the proposed method is critical for future research. It can be extended in multiple directions. The first possible extension focuses on loss functions. In previous sections, it is easy to observe that Hamming loss provides an approximation to the original 0-1 loss. As we have shown, this approximation is done by overlooking strong interactions among treatments. While original 0-1 loss is very flexible so that it is generally consistent, the huge number of parameters to be estimated may undermine the efficiency and lead to huge computation costs. Thus, a natural extension of our method is to design a family of loss functions such that our method can be adaptive to certain amount of interactions among treatments. For example, Hamming loss is the proportion of mis-classified labels, and can be rewritten into the following
\begin{equation*}
1-\frac{1}{K}\sum_{k=1}^K 1\left\{A_k=D_k\right\}.
\end{equation*}
0-1 loss can also be rewritten into the following
\begin{equation*}
1-\prod_{k=1}^K 1\left\{A_k=D_k\right\}.
\end{equation*}
Naturally, a family of loss functions can be defined as
\begin{equation*}
1-\frac{1}{\binom{\tau}{K}}\sum_{\{k_1\cdots,k_{\tau}\}\subset\{1,\cdots,K\}} \prod_{q=1}^{\tau}1\left\{A_{k_q}=D_{k_q}\right\},
\end{equation*}
where $\tau$ is a parameter. We call it $\tau$th order Hamming loss. When $\tau=1$, it is the same as Hamming loss, which has the most strict conditions on treatment interactions in order to guarantee its fisher consistency. When $\tau=K$, it is actually 0-1 loss, which has the fewest conditions on treatment interactions in order to guarantee its fisher consistency. With the increasing of $\tau$, the loss function can accommodate more and more interactions, but may lead to more and more parameters to be estimated. Thus, the existence of $\tau$ allows us to choose loss function adaptively to the data.
The second possible extension focuses on the dynamic assignment of multiple treatments. Combination therapies considered in this paper do not involve treatment transition problem. For the real data analysis and simulation, all the treatment are assumed to be applied to the patient at the same time. However, this is not always true in real life. Patients typically change from one treatment to another. Thus, instead of deciding a static treatment assignment, a dynamic treatment assignment with transition scheduling is a more realistic and reasonable solution to precision medicine. During this process, multiple outcomes may also be involved in the analysis. The process of single outcome over time may also play an important role in this extension.
The third extension focuses on how to combine our proposed technique with methods in \cite{Zhang2012a} and \cite{Zhang2012b} such that we can directly estimate the contrasts of treatment effects. In this case, it is critical to model the potential outcomes and propensity scores in an efficient way, especially for combination therapies. In general, our work in this paper is the keystone to all these potentials.
One of the limitations in our proposed method is that our method may fail under large interactions between treatments. When the interactions between treatments share the same direction of treatment recommendations, our method still holds, even the condition in \ref{thm2} fails. For example, in the simulation with model mis-specification, if $\gamma\leq 0$, it can be shown that our method is still consistent but the condition in \ref{thm2} fails. When the interactions between treatments are large and have the opposite direction of treatment recommendations, for example, it is extremely harmful to take two good medications simultaneously. Each combination therapy should be considered as a totally new treatment whose effect is irrelevant to the treatment effects when taking medications separately. In this case, treatment recommendation should be considered as a multi-class classification problem rather than a multi-label classification problem, and no information can be borrowed to improve efficiency.
\section*{Acknowledgments}
This work was one of research topics in Eli Lilly and Company summer intern program. All supports were provided by Eli Lilly and Company. Thank Yebin Tao for suggestions on this work.
|
{
"timestamp": "2018-04-17T02:11:15",
"yymm": "1804",
"arxiv_id": "1804.05378",
"language": "en",
"url": "https://arxiv.org/abs/1804.05378"
}
|
\section{Introduction}
\label{sec:introduction}
Temporal networks can be seen as an extension of the static network paradigm to include information about when interactions happen, not only between whom~\cite{TemporalNetworkReview1,TemporalNetworkReview2,masudalambiotte}. Just as for static networks, it is interesting to study dynamic phenomena happening on temporal networks, and how the structure of the interaction affects these. In the literature, there has been a great focus on disease spreading~\cite{masuda_holme_rev}. Somewhat less commonly, researchers have studied random walks~\cite{Delvenne2010,Perra2012,Starnini2012,Hoffmann2012,Ribeiro2013,Lambiotte2013,Rocha2014,Mata2014,Speidel2015,Delvenne2015} and threshold models of social spreading phenomena~\cite{taka,PhysRevE.89.062815,KARIMI20133476}. Another fundamental dynamic problem on networks is navigation~\cite{Kleinberg2000,SHLee2012}. This concerns agents traveling on the network with given starting points and destinations, but with incomplete knowledge of the network, like the sense of direction in spatially embedded networks~\cite{SHLee2012}. The problem of navigation on temporal networks has so far not been explored in the literature. The goal of this paper is to establish this research question and investigate solutions in form of an extension of spatially navigating agents.
The basic setting is a stream of contacts---triples $(i,j,t)$ of two nodes $i$ and $j$ and a time $t$---representing an interaction event between the two nodes. Then we assume an agent, as a walker in a random walk, can move to another node at the time of a contact. When a contact happens, we assume that the agent can make the decision whether or not to take a step, based on the history of the temporal network. In line with the assumption of incomplete information, we assume that future contacts are not known to a node. For simplicity, however, we assume the last observed time from destination to target is obtainable for all nodes. In this setting, we test three strategies. One is called \textit{greedy navigation} (GN) where agents jump from $i$ to $j$ at a contact $(i,j,t)$ if the previously observed time to reach the target is shorter from $j$ than $i$. The other two strategies are for reference and not using any available information. The second strategy is the \textit{greedy walk} (GW) of Ref.~\cite{Saramaki2015} where agents move at every contact. The third strategy is to simply wait at the origin until there is a contact with the target, we call it the \textit{wait for target} (WFT) strategy.
We explore the strategies on six empirical temporal networks. The reason why we use empirical contact data rather than temporal-network models as the basis of our work is twofold. First, there is a very large number of possible structures and correlations in temporal networks compared with static networks, so that one cannot simply scan through them in models~\cite{TemporalNetworkReview2}. It is also very challenging to identify the most important structures for the dynamic process in question~\cite{TemporalNetworkReview2}. Second, studying empirical networks contributes to the understanding of the original system itself. Such an analysis enables us how different data sets differ with respects to navigating agents.
In the remainder of this paper, we will present and motivate the model of navigation, present the empirical data sets, our analysis procedure, and our simulation results.
\section{Preliminaries}
\label{sec:model}
\subsection{Temporal network representations}
There are different ways to incorporate information about the timing of contacts into network modeling. In this work, we will use so-called the \textit{contact list} (sometimes the \textit{link stream}) framework~\cite{TemporalNetworkReview1}. In that setting, the basic unit of interaction is a \textit{contact} (or \textit{event})---a triple $(i,j,t)$ showing that nodes $i$ and $j$ are in contact at time $t$. The time from the first to last time in a data set is the \textit{duration} $T$. Other descriptive quantities are the time resolution (minimal time between two contacts) $\delta t$, the number of contacts $C$, and the number of nodes $N$.
\begin{figure}
\includegraphics[width=\columnwidth]{schematic.pdf}
\caption{Schematic diagram of greedy navigation in temporal networks.
Each horizontal line component corresponds to a node, horizontal axis represents time,
and vertically connected nodes represent contacts at that time~\cite{TemporalNetworkReview1,TemporalNetworkReview2,masudalambiotte}.
At time $t = t_0$, node $3$ (the source)
wants to send a unit of package to node $4$ (the target).}
\label{fig:schematic}
\end{figure}
\subsection{Navigation}
Our model for temporal network navigation is inspired by our work on network navigation with spatial information~\cite{SHLee2012}. The idea for our \textit{greedy navigators} is that the agents know the direction of their target, and then takes steps as close as possible (angularly) to its direction at every step.
In that case of spatial (and static) networks, therefore, the available information is embedded in the location of the nodes.
In the case of temporal networks, illustrated in Fig.~\ref{fig:schematic}, we consider the entire past or history as available information, and, for simplicity, assume that it is accessible to every node.
During the run, we keep updating the information of the shortest path from a certain node to another.
For instance, in Fig.~\ref{fig:schematic}, node $3$ at $t = t_0$ estimates, based on its experience, that it takes one step from node $1$ to
node $4$ and three steps (via node $3$ and $1$) from node $2$ to node $4$ in terms of the shortest hopping distance (the number of vertical jumps in the diagram),
while $\tau_{1 \to 4}$ from node $1$ to node $4$
and $\tau_{2 \to 4}$ from node $1$ to node $4$ in terms of the shortest time duration (the horizontal time duration in such a diagram, assuming that
all of the interactions are instantaneous),
based on the history up to $t = t_0$.
The way our agents exploit the information---i.e.\ the way they are greedy---is realized by the following rule: if a walker at node $3$ tries to move to node $4$ at time $t = t_0$, as the node ``closest'' (the closest node is chosen uniformly at random in case of ties) to
the target (node $4$) is node $1$, node $3$ indefinitely waits for an interaction with node $1$ ($t = t_{13}$), even if the interaction between node $3$ and
node $2$ happens (at $t = t_{23}$) prior to the interaction between node $3$ and node $1$. Once the walker reached from node $3$ to node $1$, node $1$ will wait for
the direct interaction with node $4$ (one step or $\tau = 0$ to the target) and finalize the active navigation.
Our strategy assumes that historical interaction patterns
will happen repeatedly or periodically and does no better job than random hopping in the absence of such temporal patterns (e.g., the Markovian Poisson process). On the other hand, in many empirical temporal networks
such periodical patterns exist due to the circadian rhythm and plays a crucial role in human interactions~\cite{HHJo2012}.
Note that we can define such a greedy navigation strategy based either on the hopping distance (distance-based temporal greedy navigation) or on the time (time-based temporal greedy navigation), as the node with the shortest distance to the target and the node with the shortest time to the target can be different. We will use both strategies as specified below and, when specificity is needed, denote the hop-based one by GNH and the time based one by GNT.
\subsection{Data}
\label{sec:data}
\begin{table}
\caption{Description of the basic properties of the data sets: the number of nodes $N$, the number of contacts $C$, the duration $T$, the number of samples $n$, and the time resolution $\delta t$. \label{tab:props}}
\begin{tabular}{lllllll}
Data set & $N$ & $C$ & $T$ & $n$ & $\delta t$ & Ref. \\ \hline
Gallery & $159.01$ & $6027.71$ & $1$ day & $69$ & $20$ seconds & \cite{gallery} \\
Primary School & $242$ & $125773$ & $1$ day & $1$ & $20$ seconds & \cite{school} \\
High School 1 & $126$ & $28561$ & $4$ days & $1$ & $20$ seconds & \cite{hschool} \\
High School 2 & $180$ & $45047$ & $7$ days & $1$ & $20$ seconds & \cite{hschool} \\
Conference & $113$ & $20818$ & $2.5$ days & $1$ & $20$ seconds & \cite{conference} \\
Hospital & $75$ & $32424$ & $4$ days & $1$ & $20$ seconds & \cite{Vanhems2013} \\
\end{tabular}
\end{table}
In this section, we will discuss the empirical data sets we use. All our data sets come from the SocioPatterns project \url{sociopatterns.org} where human proximity interactions are tracked by radio-frequency identification devices
in various types of locations such as a hospital, a primary school, a high school an art gallery, and conference. Some of the data sets come from repeated experiments over $n$ days. In this case, we run our analyses on the separate days and average the results. A summary of the basic properties of these data sets and references to the original studies can be found in Table~\ref{tab:props}.
\section{Results}
\label{sec:results}
We will start with an example---navigation on the \textit{Hospital} data set---then continue to aggregate properties of all data sets.
\subsection{Different strategies applied to the Hospital data}
\label{sec:hospital_data}
\begin{figure}
\includegraphics[width=0.9\columnwidth]{reachability_hospital.pdf}
\caption{A plot of the reachability versus average time to all of the other nodes from a certain node (denoted as ``node $0$'') of the hospital ward dynamic contact network at each time. For all of the cases, the time unit is day.}
\label{fig:reachability_hospital}
\end{figure}
\begin{figure}
\includegraphics[width=0.9\columnwidth]{distance_hospital.pdf}
\caption{A plot corresponding to Ref.~\ref{fig:reachability_hospital} but for the average distance in the navigation.
}
\label{fig:distance_hospital}
\end{figure}
\begin{figure}
\includegraphics[width=0.9\columnwidth]{time_hospital.pdf}
\caption{A plot corresponding to Ref.~\ref{fig:reachability_hospital} but for the average time to the target.}
\label{fig:time_hospital}
\end{figure}
Figure~\ref{fig:reachability_hospital} shows the reachability---the fraction of reachable nodes in the future that can be connected by time-respecting paths~\cite{TemporalNetworkReview1}---for a given starting nodes as a function of the time of the beginning of the navigation for the Hospital network~\cite{Vanhems2013}. We see that the greedy navigators (the hop-based version) has the largest reachability for intermediate values of the time. The time based greedy navigation is very similar and omitted for clarity. The waiting-for-target strategy is monotonically decreasing, which is trivial as those agents do nothing to improve their chance of reaching the target. Greedy navigators are always more efficient than the greedy walkers, whereas WFT is the best strategy for early times. At the early times greedy navigators have not assembled enough information to outperform greedy walkers. Thus, a yet more efficient strategy might thus be to wait for the target in the beginning of the time period, then switch to greedy navigation after a while. At the end of the data, the reachability goes down for all strategies simply because the number of time-respecting paths goes to zero.
The results for the average hopping distances (the number of hops to reach the target, given that the target is reached) are plotted in Fig.~\ref{fig:distance_hospital}. In this case the WFT strategy is trivially one so we omit it from the figure. The greedy navigators are always just a little over one, indicating that the mostly just wait for the target. The greedy walkers have a complex pattern that comes from that their character of always moving in combination that the movement can be almost deterministic for walks with few ties.
Next, we turn to the time-based greedy navigators. We plot the time to reach the target (given that the target is reached) for the Hospital data set in Fig.~\ref{fig:time_hospital}. This is the corresponding quantity to hop-distance of Fig.~\ref{fig:distance_hospital} for this time-based picture. This figure show the typical saw-tooth pattern of latency (the minimal possible time to reach from one node to another). The time goes down linearly until a key contact is passed, then it jumps discontinuously upward by the time to the next contact on a path to the target.
\begin{figure*}
\includegraphics[width = 0.5\textwidth]{averages.pdf}
\caption{Average values of descriptive quantities of the navigation for our six data sets. Panel (a) plots the average reachability of hop-based greedy walkers as a function of the average reachability of greedy navigators. The diagonal line shows where the two quantities equal each other. Panel (b) is the corresponding figure for the hopping-count distance of the navigations. Panel (c) shows the average reachability for time-based navigations. In this panel, we also include values for the WFT strategy (in addition to greedy walkers). Panel (d) shows the time to reach the target for the time-based navigators. This is the corresponding measure to the distance of panel (b).
}
\label{fig:averages}
\end{figure*}
\begin{figure*}
\includegraphics[width = 0.6\textwidth]{corr_hop_reachability.pdf}
\caption{The Spearman correlation between quantities describing the positions of nodes in the network---degree, strength, duration, age and burstiness---and the reachability. The outlined bars corresponds to GNT. Panel (a) shows results for greedy navigators; (b) the corresponding for greedy walkers and (c) for wait-for-target walkers.
}
\label{fig:corr_hop_reachability}
\end{figure*}
\begin{figure}
\includegraphics[width = \columnwidth]{corr_hop_distance.pdf}
\caption{Plots corresponding to Fig.~\protect{\ref{fig:corr_hop_reachability}} but for the average hopping distance.} This quantity is trivially one for WFT and thus not shown.
\label{fig:corr_hop_distance}
\end{figure}
\begin{figure*}
\includegraphics[width = 0.6\textwidth]{corr_time_distance.pdf}
\caption{Plots corresponding to Fig.~\protect{\ref{fig:corr_hop_reachability}} but for the average time to the target.
}
\label{fig:corr_time_distance}
\end{figure*}
\subsection{Average navigation performance}
In Figs.~\ref{fig:reachability_hospital}, \ref{fig:distance_hospital} and \ref{fig:time_hospital}, we see the navigation performance from one node in one networks. Next, we plot the average values of these quantities over all nodes and all data sets. See Fig.~\ref{fig:averages}. We plot the three quantities of the greedy walks and wait-for-target strategy as a function of the corresponding values for greedy navigation. We see that for all the data sets, except one (Gallery), the greedy navigation outperforms greedy walks. The exception has a peculiar interaction structure. Since it records visitors coming and going to an art gallery, the people present early in the data are not present at the end. This means that relying on the past reachability information is not only useless, it is misleading---if the best path to the target went via node $i$ before, then $i$, by a large chance left the gallery already. We can see that the hop-based and time-based strategies (of panel (c)) are very similar.
In Fig.~\ref{fig:distance_hospital}(b), we can see the distance (number of contact followed during the navigation) for the two non-trivial strategies GN and GW (WFT is constantly one). Although it is hard to prove that there is no temporal network data set where the distance is longer for GN than GW, it sounds very unlikely that an empirical (relatively well-behaved) data set would have that feature. Interestingly, the GN values are closer to one than two. This means that greedy navigators usually wait for the target, and only occasionally exploits their information to improve the navigation. This makes the increase in performance as seen in Fig.~\ref{fig:distance_hospital}(a) quite remarkable.
In Fig.~\ref{fig:distance_hospital}(c), we plot the time to the target, given that the target is reached. This quantity is quite similar for all three strategies (which we could also see in Fig.~\ref{fig:time_hospital}). One explanation is that if the agents find a path, they typically find the same one.
\subsection{Structural explanations of navigability}
\label{sec:correlation}
Navigability, as reflected in our three measured quantities, can change abruptly. The variability among nodes is also very large. In this section, we explore if this variability can be explained by temporal-network measures quantifying the position of nodes. We try a mix of static and temporal quantities: The number of other nodes a node is in contact with (\textit{degree}), the number of contacts a node participates in (\textit{strength}), the time between the first and last contact a node participates in (\textit{duration}), the time of the first contact of a node (\textit{first time}). Finally, we measure the \textit{burstiness} as defined in Ref.~\cite{KIGoh2008}:
\begin{equation}
B \equiv \frac{\sigma_\tau - m_\tau}{\sigma_\tau + m_\tau} \,,
\label{eq:GB_burtiness}
\end{equation}
where $m_\tau$ and $\sigma_\tau$ are the mean and the standard deviation of $P(\tau)$ (the distribution of times $\tau$ between contacts of a node). We use the absolute value $|S|$ of the Spearman rank correlation since several of the quantities have heavy-tailed distributions.
In Fig.~\ref{fig:corr_hop_reachability}, we plot the $S$ for the reachability values and the five positional descriptors. We see that the values are rather large, so these simple structures can explain the behavior of the navigators to a fairly large extent. Then we notice that degree and burstiness give slightly larger values of the correlation compared to other structural measures; however, the variation is not extremely large---other measures are also correlated. This can to some extent be explained by them being correlated to each other---if e.g.\ the degree is large, then so is probably also the strength. It is interesting that degree and burstiness are better predictors since they measure two very different aspects. The first time shows a bit lower values of $|S|$. The navigation problem is thus dependent on both temporal and topological structures. This is similar other dynamics on the network~\cite{masuda_holme_rev}. We also note that GNH and GNT give very similar values. In practice there seems to be no point in separating them.
Greedy walks (Fig.~\ref{fig:corr_hop_reachability}(b)) are less correlated with the structural measures, whereas WFT agents show a similar behavior to GN. Of the data sets, Hospital has the highest correlation values for all structural measures. The data set with the weakest correlations vary with the structural measures. For the first time statistics, Gallery shows high correlations. As mentioned, this could be understood from the time stretched nature of this datasets---the first time correlates with the presence of a node in the end and beginning of the data.
In Fig.~\ref{fig:corr_hop_distance}, we show the correlations between the hopping distance and the structural quantities. These are somewhat weaker than the reachability (ranging between 0 and 0.4 as opposed to 0 to 0.8 for reachability. The correlations for WFT are very similar to those of GN; see Fig.~\ref{fig:corr_hop_distance}(c). Also in this case, the Gallery and Hospital data sets are the ones with highest correlations.
In our final analysis plot, Fig.~\ref{fig:corr_time_distance}, we investigate correlations with the time to reach a reachable target. Over all, this case gives low correlations. This could be understood since many of the actual paths leading to the target are rather few and the waiting time between the contacts depend on many outer, effectively random factors. In this case, the Gallery data shows spectacularly large values of the correlations for GW and WFT with the first times. Once again, this can be understood from the time-stretched nature of the network. There are relatively long paths from sources in the early times of the data to targets in the end. Since greedy navigation bases its paths on (in this case, misinformed) data, it shows weak correlations.
\section{Discussion and Conclusions}
\label{sec:discussion}
In this paper, we have introduced the problem of navigation in temporal networks as the problem to decide whether or not to follow a contact given what is known of past interactions. We have contrasted a temporal-network navigation strategy---greedy navigation---that assumes the past predicts the future, with two more simplistic strategies.
Greedy navigation is a very simple strategy, just taking steps that would have worked well in the past, disregarding any trends in the activity of the nodes, etc. Still one would expect real agents in a temporal network context to have a similarly simple intuitive approach (rather than some strategy with heavy computational overhead). On the other hand, it is hard to think of real processes that are very well described by our scenario. A subway passenger without a map, could be a mental scenario, though thoroughly unlikely. Some distributed computing problems could probably also be candidate applications---perhaps opportunistic networking where data is transferred through devices in close proximity~\cite{huang2008survey}.
Even though the information accessible to the greedy navigators is quite elementary (the list of temporal distance between the target and all of the other nodes), we have demonstrated that they can actually augment the path-finding process by exploiting temporal correlations. There is thus information encoded in the past contacts that can be exploited in future navigation. There is not one structural quantity that can explain the behavior of our navigators---some structures (degree and burstiness) do correlate more strongly with the quantities describing the navigation; some data sets have consistently stronger correlations than other. Finally, it is easier to explain the reachability in terms of the position of nodes, than it is to describe the number of hops, or the time to reach the target (if it is reached).
We believe this is the beginning of an interesting research direction of temporal network research. One can extend our research by finding more efficient, simple navigation strategies. It could be interesting to vary the accessible information, or to tune the structures in a systematic way in a model-based study.
\bibliographystyle{abbrv}
|
{
"timestamp": "2018-04-17T02:06:25",
"yymm": "1804",
"arxiv_id": "1804.05219",
"language": "en",
"url": "https://arxiv.org/abs/1804.05219"
}
|
\section{Introduction} \label{intro}
As the demand for localization services increases, indoor localization technology based on fingerprint recognition has become the prevailing positioning technology due to its high precision and minimal hardware requirements. In addition to high accuracy, an indoor positioning system should have low complexity and require little processing time to accommodate mobile devices. Fingerprint-based indoor localization is an effective method that can satisfy these requirements; however, the received signal strength (RSS) or channel state information (CSI) from surrounding access points must be measured at each reference point to build a fingerprint database \cite{3}.
Most of the fingerprint-based localization systems are based on wireless local area networks (WLANs), which are available in public places \cite{8353839}, \cite{8326317}. Fingerprint-based localization consists of two basic phases: 1) an offline phase (collect data at each reference point to construct the fingerprint data and train the classification model for the online phase) and 2) an online phase (receive data online, compare the received data with the fingerprint database to achieve localization) \cite{4}. The training phase is used to construct the fingerprint database by collecting and preprocessing survey data related to each reference point's position. During the online phase, a mobile device records real-time data and compares the received data with the database. The reference point that most closely matches the received data is assumed to be device's location.
Many existing indoor localization systems use RSS for fingerprints due to simplicity and low hardware requirements. For example, the \textit{Horus} system uses a probabilistic method to estimate location with RSS data \cite{5}. However, RSS data has high variability over a fixed location due to multipath effects in indoor environments \cite{6}. This high variability can introduce significant localization errors. Additionally, RSS values consist of relatively coarse information that does not fully exploit the many subcarriers in an orthogonal frequency-division multiplexing (OFDM) system. Instead, in the widely used OFDM systems, CSI provides more precise multipath information than RSS does by exploiting the different signal strengths and phases in different subcarriers \cite{8304587}. Some commercial off-the-shelf network interface cards with IEEE 802.11n standard provide detailed subcarrier amplitude and phase information in the form of CSI. Thus, this paper considers the CSI-based fingerprint for indoor localization.
One major issue of fingerprint-based localization is that it is difficult to determine how much data is needed to obtain the desired accuracy during the offline phase. In the next section, we show that sampling more fingerprint data provides better results. Thus, building a sufficiently large fingerprint database is vital to high localization performance. However, this task is time consuming and labor-intensive \cite{3}. As an example, it takes half an hour to collect the fingerprint data at one reference point in our experiments. This excessive time to perform fingerprint data collection affects the popularity and applications of fingerprint-based localization.
To reduce the data collection cost, several methods have been proposed \cite{3}, \cite{7}, \cite{8}. In particular, in \cite{3}, the authors propose a method based on compressive sensing to recover absent fingerprints. Their approach shows the hidden structure and redundancy characteristics of fingerprints in a merging matrix. In \cite{7}, the authors present a semi-supervised manifold learning technique for building a fingerprint database from partially labeled data, where only a small portion of the signal strength measurements must be marked with the corresponding coordinates. Note that these methods only reduce the number of reference points or try to recover fingerprints; they fail to reduce the number of samples to be collected and still require a significant amount of time for data collection.
To expand the fingerprint database while reducing the human effort, we present a method to increase the amount of training data collected at each reference point based on generative adversarial nets (GAN) \cite{goodfellow2014generative}, \cite{cao2019recent}. Specifically, we first transform the CSI data collected at each reference point into amplitude feature maps. Then, through pixel transformation, each amplitude feature map is transformed into an image with the same resolution to construct the initial fingerprint database. Because the initial database is constructed by mapping the amplitude feature maps of the reference points, it contains the location information of all reference points. Then, we propose an Amplitude-Feature Deep Convolutional Generative Adversarial Network (AF-DCGAN) model as well as a corresponding training algorithm that generates images similar to the original amplitude feature maps. Finally, the generated amplitude feature maps are merged into the initial fingerprint database, resulting in an expanded database. As described in the following section, the localization accuracy improves as the number of samples in the fingerprint database increases. We also evaluate the proposed fingerprint construction method through extensive experiments in a typical indoor classroom environment. The accuracy of the initial database reaches 1.34~m, while the accuracy of the expanded database reaches 0.92~m, indicating the effectiveness of the proposed method.
The main contributions of this paper are as follows:
\begin{enumerate}[1.]
\item We build a fingerprint database by converting processed CSI data into amplitude feature maps. This approach visualizes the position of the sampling points and allows us to visually determine the locations of the testing points.
\item Based on the nature of the amplitude feature maps, an AF-DCGAN model is proposed that converges quickly and generates samples with improved diversity. We use the AF-DCGAN model to generate additional amplitude feature maps of the sampling points' positions, which reduces the collection time associated with each single sample point and saves human effort.
\item Extensive experiments are conducted and the results show that the proposed scheme can provide better performance compared to state-of-the-art schemes.
\end{enumerate}
The remainder of this paper is organized as follows. Section~\ref{related_work} reviews the related work. Section~\ref{basic_idea} briefly describes the basic idea underlying this paper. Section~\ref{csi_collection_feature_map} introduces the CSI and the amplitude feature map conversion technique. Section~\ref{af_dcgan_model_method} presents the method for expanding the amplitude feature maps based on AF-DCGAN, including the training algorithm and data-generating steps. Simulation and experimental results are shown in Section~\ref{experiment}. Finally, Section~\ref{conclusion} concludes the paper.
\section{Related work} \label{related_work}
WiFi localization technology can be divided into two categories: positioning based on propagation models and positioning based on fingerprint. Although the propagation model-based methods do not require signal sampling, fingerprint-based positioning achieves higher accuracy \cite{9}, \cite{8307353}, \cite{11}, \cite{12}. Therefore, WiFi indoor positioning based on fingerprints is gaining popularity.
Although building a fingerprint database for localization purposes is highly efficient, collecting the data to construct the fingerprint database usually requires significant human effort. Many researchers have proposed solutions for the construction of the fingerprint database to reduce human effort. In \cite{3} and \cite{7857072}, a novel approach based on compressive sensing is presented to recover absent fingerprints. Jun \emph{et al.} \cite{16} present the design, implementation, and evaluation of AP-Sequence. This fingerprint-based localization system achieves extremely low overhead in fingerprint map construction and maintenance. The method in \cite{17} leverages a more stable RSSI gradient to build a gradient-based fingerprint map by comparing the absolute RSSI values at nearby positions. A novel fingerprint collection technique is proposed in \cite{18} that detects WiFi APs to form WiFi fingerprints from the signals collected by ZigBee interfaces. In \cite{19}, an FM-based indoor localization system that does not require proactive site profiling is presented to construct the fingerprint database based solely on an estimate of indoor RSS distribution.
The authors of \cite{7980032} propose a fingerprint-based device-free localization system named \textit{iUpdater} to significantly reduce the labor cost and increase the accuracy. It is able to accurately update the whole database with RSS measurements at a small number of reference locations, thus reducing the human labor cost. Milioris \emph{et al.} \cite{21} use the Matrix completion framework to build complete training maps from a part of the reference fingerprints by learning the relevant fingerprint structure. In \cite{8003484}, the authors propose \textit{AcMu}, an automatic and continuous radio map self-updating service for wireless indoor localization that exploits the static behaviors of mobile devices. By accurately pinpointing mobile devices with a novel trajectory matching algorithm, they use stationary mobile devices as reference points to collect real-time RSS samples. The authors of \cite{23} propose the Enriched Training Database (ETD), which is a web-service that enables the management and storage of training fingerprints and includes additional “enriching” functionality. The user can automatically generate virtual fingerprints based on propagation modeling of the virtual training points through the enriching functionality. The same authors also propose a new method to acquire training fingerprint locations that eliminates the burden of manually defining training points and covers areas with insufficient density to train fingerprints. In \cite{24}, the authors propose a novel method to construct a comprehensive fingerprint database by using the radio propagation model. To address the limitations of indoor dynamic measurements, the authors in \cite{25} propose a Gaussian process regression for fingerprint-based localization that uses realistic and virtual indoor dynamic measurement data.
However, most of these methods use modeling to reduce the number of reference points or recover fingerprints based on existing partial fingerprints. The accuracy of localization systems decreases dramatically when many reference points are omitted to reduce labor cost. In this paper, we attempt to sample data at all the reference points and then generate additional data using an AF-DCGAN model, which effectively expands the number of fingerprints in the database. Thus, we obtain better positioning performance with the expanded fingerprint database, while also reducing the human effort.
\section{Basic idea} \label{basic_idea}
In a fingerprint-based indoor localization system, the more data the fingerprint database contains, the higher its positioning accuracy is. We first conduct experiments to show this relationship. We use different methods to construct fingerprint databases and then perform localization experiments in a classroom. The environment and floor plan are illustrated in Fig.~\ref{fig_experiment_environment}. We deploy a TL-WR742N wireless router as the transmitter, equipped with one transmit antenna, and a ThinkPad laptop equipped with Intel Wireless Link 5300 NICs (IWL5300) with three receive antennas. We divide the classroom into 49 $(7\times 7)$ sections and set the center point in each section as the corresponding reference point.
\begin{figure}[htb]
\centering
\subfloat[The receiving antennas (MP) are placed behind the measurement area to cover all the experiment area.]{
\includegraphics[width=2.5in]{figure/fig_deployment_mp}\label{fig_deployment_mp}}\hfill
\subfloat[The router is placed in front of the measurement area.]{
\includegraphics[width=2.5in]{figure/fig_deployment_ap}\label{fig_deployment_ap}}\hfill
\subfloat[Floorplan of the experiment area.]{
\includegraphics[width=2.5in]{figure/fig_deployment_floorplan}\label{fig_deployment_floorplan}}\hfill
\caption{Illustration of the experimental enviroment (a classrom). The router is placed at the front of the measurement area. The receiving antennas are placed behind the measurement area to cover the measurement area.} \label{fig_experiment_environment}
\end{figure}
At each reference point, we collect 800 samples and test several different fingerprint database construction methods, including the Horus method \cite{5}, Gaussian process regression (GPR) \cite{26}, Low-Rank matrix fill (LR-M) method \cite{27} and the thin spline interpolation method (SPL-M) \cite{28}. The Horus system requires reference point data uses location-clustering techniques to reduce the computation requirements of the algorithm. The GPR method uses half the reference points and infers the posterior received RSS mean and variance at other points to build a fingerprint database. The LR-M represents the distribution of wireless fingerprints as a low-rank matrix and constructs a dense radio map from relatively sparse measurements using a revised low-rank matrix completion method. The SPL-M method explores the use of different interpolation functions to complete the fingerprint mapping necessary to achieve the required accuracy. During the experiments, we use 200, 400 and 800 samples at each reference point to build three fingerprint databases of different sizes. Then, we use a k-means clustering algorithm to perform positioning. The results are shown in Fig.~\ref{fig_localization_error_database_size}. The Y-axis indicates the mean error of localization. These experiments show that, as the size of the fingerprint database for each sampled data location increases, the localization accuracy gradually increases. Larger training datasets achieve more satisfactory results than smaller training datasets. As the number of samples used in the database increases, the mean error decreases.
\begin{figure}[htb]
\centering
\includegraphics[width=3.5in]{figure/fig_localization_error_database_size}
\caption{Localization accuracy of different fingerprint database sizes. As the number of samples increases, the mean localization error gradually decreases.}
\label{fig_localization_error_database_size}
\end{figure}
However, it requires tremendous labor to sample a large amount of wireless fingerprints. Inspired by unsupervised learning \cite{Shrivastava_2017_CVPR}, we use GAN to generate more sample data for each reference point, to expand the fingerprint database and to improve the accuracy of the localization system. In the following sections, we convert the CSI data into amplitude feature maps and then extend the fingerprint database by the proposed AF-DCGAN model. With this model, the convergence process in the training phase is accelerated and the diversity of the generated CSI amplitude feature map is increased dramatically. Based on the extended fingerprint database, the accuracy of the indoor localization system can be improved with reduced human effort.
\section{CSI collection and feature map conversion} \label{csi_collection_feature_map}
In the field of wireless communication, Channel State Information (CSI) is a channel attribute of the communication link. It describes the signal’s attenuation factor on each transmission path, that is, the value of each element in the channel gain matrix $H$, such as signal scattering, multipath fading, shadow fading or power decay of distance.
In the widely used OFDM system, CSI provides more multipath information than does RSS, by including the signal strength and phase of different subcarriers. Recently, some IEEE 802.11n standard commercial off-the-shelf network interface cards provide access to detailed subcarrier amplitude and phase information via CSI. Specifically, with the Intel 5300 NIC, a sample of Channel Frequency Response (CFR) over WiFi bandwidth can be obtained as CSI information, including the number of transmit antennas $N_t$, the number of receive antennas $N_r$, the number of subcarriers $N_S$, the packet transmission frequency $f$ and the CSI matrix $H$ \cite{29}, which is a $N_t\times N_r\times N_S$ imaginary number matrix as follows:
\begin{equation}
H=(H_{uv})_{N_t\times N_r}
\end{equation}
Each pair of transmit-receive antennas (a TX-RX pair) is a link, and $H_{uv}$ is the CSI data of the link formed by TX $u$ and RX $v$, containing the information of $N_S$ subcarriers.
\begin{equation}
\begin{split}
& H_{uv}=(h_1^{uv},\dots,h_k^{uv},\dots, h_{N_s}^{uv})^T, \\
& 1 \leq u \leq N_t, 1 \leq v \leq N_r, 1 < k < N_S
\end{split}
\end{equation}
Each $h_k$ characterizes the amplitude and phase of the corresponding subcarrier which can be expressed as
\begin{equation}
h_k^{uv}=\left | h_k^{uv} \right |e^{j\angle h_k^{uv}}, 1 \leq k \leq N_S
\end{equation}
where $\left | h_k^{uv} \right |$ denotes the amplitude response and $e^{j\angle h_k^{uv}}$ denotes the phase response of subcarrier $k$. It has been shown that the raw phase values reported by the wireless network card are inaccurate, and the phase values vary greatly in some frequency bands even in static environments \cite{Xie2015}. Thus, currently in most CSI-based localization algorithms the phase values are not used to determine the positions of the target. In contrast, the amplitude value for a given sub-carrier maintains a good stability at a certain location, which is thus adapted in CSI-based localization approaches \cite{XIAO201773}. To show the superiority of our algorithmic approach, and to directly compare with related work, we also limit ourselves to use amplitude information only. Generalization to phase information is left for future work.
In a defined space as shown in Fig.~\ref{fig_experiment_environment}, the CSI data clearly differs when people stand in different places. Fig.~\ref{fig_csi_packets} shows the amplitude ($\left | h_k^{uv} \right |$) of all subcarriers in one link over time/packets when a person stands at three different positions in a classroom. Fig.~\subref*{fig_csi_packets_a} is the three-dimensional amplitude map of the reference position, and the distance of the position of Fig.~\subref*{fig_csi_packets_b} and Fig.~\subref*{fig_csi_packets_c} from Fig.~\subref*{fig_csi_packets_a} is 1m and 5m, respectively. As the figure shows, the difference in the amplitude of the CSI increases as the distance between the measurement locations increases. These characteristics of CSI amplitude changes are exploited in building a fingerprint database to perform indoor localization.
\begin{figure}[htb]
\centering
\subfloat[]{
\includegraphics[width=3in]{figure/fig_csi_packets_a}\label{fig_csi_packets_a}}\hfill
\subfloat[]{
\includegraphics[width=3in]{figure/fig_csi_packets_b}\label{fig_csi_packets_b}}\hfill
\subfloat[]{
\includegraphics[width=3in]{figure/fig_csi_packets_c}\label{fig_csi_packets_c}}\hfill
\caption{ Large changes occur in CSI amplitudes at different locations: (a), (b) and (c) show three-dimensional patterns of amplitude, subcarriers and packets at three positions separated by 1 m and 5m.} \label{fig_csi_packets}
\end{figure}
\subsection{CSI collection} \label{csi_collection}
First, we evenly divide the designated indoor space into $M$ sampling spaces and use the centers of the sampling spaces as reference points to form a reference point (RP) set as follows.
\begin{equation}
RP=[RP_1, \cdots, RP_i, \cdots, RP_M]
\end{equation}
where $RP_i$ denotes the reference point in the $i$-th ($1 < i < M)$ square grid.
Assume $N_{ap}$ wireless access points (wireless routers in IEEE 802.11) are deployed in the indoor space. Then, each CSI sample has $N_{ap}\times N_t\times N_r \times N_S$ dimensions. During the collection stage, we obtain $N_X$ samples of CSI data at a fixed rate when people stand at different reference points, forming a time series set of the $i$-th reference point as follows.
\begin{equation}
CSI_{N_X}^{i}=\{csi_{1,N_X} ^i,\cdots, csi_{m,N_X}^i,\cdots, csi_{N_t\times N_r,N_X}^i\}
\end{equation}
where $csi_{m,N_X}^i$ denotes the $N_X$ WiFi signals of the $m$-th ($1 < m <N_t\times N_r$) link received by the $i$-th reference point, and $csi_{m,N_X}^i$ is a two-dimensional imaginary number matrix with size $N_X\times N_S$.
\subsection{Convert CSI to amplitude feature maps}
The CSI amplitude varies significantly in different positions, which can be exploited to perform indoor localization. To obtain a fingerprint corresponding to a specific location, we represent the characteristics of that location by plotting the amplitude of the CSI as a feature map.
We randomly select 100 out of $N_X$ rows of the two-dimensional imaginary number matrix $csi_{m,N_x}^i$ for $m=1\cdots T$ to form $T$ two-dimensional matrices of $100\times N_S$, to reconstruct the position information set at the $i$-th reference point $CSI_i^{'}$ as follows.
\begin{equation}
CSI_i^{'}=\{csi_{1,T}^{'i},\cdots,csi_{m,T}^{'i},\cdots, csi_{N_t\times N_r,T}^{'i}\}
\end{equation}
where $csi_{m,T}^{'i}$ denotes $T$ imaginary number matrices of $100\times N_S$ of the $m$-th link at the $i$-th reference point $RP_i$.
The real and imaginary parts of the imaginary number matrices in the reconstructed position information set $CSI_i^{'}$ are selected to create amplitude feature maps to obtain the amplitude feature maps set $\Phi_i$ of $N_t\times N_r$ links at $RP_i$. Further, we obtain a set of $\Phi$ of amplitude feature maps at $M$ reference points.
\begin{equation}
\Phi_i=\{\phi_{1,n}^{i} \cup \cdots\cup \phi_{k,n}^{i} \cup \cdots \cup \phi_{N_t\times N_r,n}^{i}\}
\end{equation}
\begin{equation}
\Phi=\{ \Phi_1,\cdots, \Phi_i, \cdots, \Phi_M\}
\end{equation}
where $\phi_{k,n}^{i}$ denotes $n$ amplitude feature maps of the $k$-th link at the $i$-th reference point, and $\Phi_i$ denotes $n$ amplitude feature maps of $N_t\times N_r$ links at the $i$-th reference point.
As shown in Fig.~\ref{fig_afm_subcarrier}, the amplitude feature maps of two positions in $M$ reference points obtained through data processing are created so that the position information indicated by the CSI data can be visualized.
\begin{figure}[htb]
\centering
\subfloat[]{
\includegraphics[width=3.5in]{figure/fig_afm_subcarrier_a}\label{fig_afm_subcarrier_a}}\hfill
\subfloat[]{
\includegraphics[width=3.5in]{figure/fig_afm_subcarrier_b}\label{fig_afm_subcarrier_b}}\hfill
\caption{Different colors represent different links: (a) and (b) are feature maps of two sampling points from the $M$ sampling points. Given three antennas at the receiver and one antenna at the transmitter, each link represents a path between the transmitting end and the receiving end. Each subcarrier of these three links has different amplitudes, thus resulting three lines in the figures.} \label{fig_afm_subcarrier}
\end{figure}
In order to adapt to the image classification model and to improve the accuracy of image classification, we transform the data in the $\Phi_i$ of $RP_i$ into a picture with a resolution of $256\times 256$. A training set of the $i$-th reference point is obtained as follows. Then, we identify the initial fingerprint database $\Phi^{'}$.
\begin{equation}
\Phi_i^{'}=\{\phi_{1,n}^{'i}\cup \cdots\cup \phi_{k,n}^{'i} \cup \cdots \cup \phi_{N_t\times N_r,n}^{'i}\}
\end{equation}
\begin{equation}
\Phi^{'}=\{ \Phi_1^{'},\cdots, \Phi_i^{'}, \cdots, \Phi_M^{'}\}
\end{equation}
where $\phi_{k,n}^{'i}$ denotes the $n$ amplitude feature maps of the $k$-th link at $RP_i$ after the resolution transform, and $\Phi_i^{'}$ denotes the $n$ amplitude feature maps of the $N_t\times N_r$ links at $RP_i$.
\section{AF-DCGAN model and training method} \label{af_dcgan_model_method}
To reduce the sampling time and human effort, we use GAN to generate additional amplitude feature maps for each reference point, thus expanding the initial fingerprint database. In this way, we obtain additional samples similar to the initial fingerprint database to expand the fingerprint database and to improve the localization accuracy without additional human effort.
\subsection{AF-DCGAN model} \label{af_dcgan_model}
GAN is inspired by two-player game theory. The two players in the GAN model are the generative model ($G$) and discriminative model ($D$) \cite{30}. $G$ captures the distribution of sample data to generate a sample similar to the training data with added noise that obeys a certain distribution (uniform, Gaussian, etc.). Using this approach, the generated samples approximate real samples taken at the same location. $D$ is a binary classifier that estimates the probability that a sample comes from the training data. If the sample comes from the real training data, $D$ outputs a high probability; otherwise it outputs a small probability.
During training, one side is fixed, and the network weights at the other side are updated and alternately iterated. During the training process, both sides attempt to optimize their networks, forming a rivalry that continues until the two parties reach a dynamic balance (Nash Equilibrium). $G$ restores the distribution of the training data and creates samples increasingly similar to the real data until $D$ can no longer discriminate the results at an accuracy above 50\%, at which point the discriminator and generator have reached Nash equilibrium. $D$ and $G$ play the following two-player minimax game with the value function $V(D,G)$ as follows:
\begin{equation}
\begin{split}
\min_{G} \max_{D}V(D,G)= & E_{x\sim P_{data}(x)}\{logD(x)\}+ \\
& E_{z\sim P_z(z)}\{log(1-D(G(z)))\}
\end{split}
\end{equation}
where $P_{data}(x)$ denotes the real sample set, $z$ denotes the signal with a uniform distribution. $P_z(z)$ denotes the fake sample set, $G(z)$ is the output of the generator, and $D(x)$ is the output of the discriminator.
Convolutional neural networks perform well at supervised learning tasks but have been rarely used for unsupervised learning. DCGAN \cite{31} combines a CNN using supervised learning with a GAN using unsupervised learning. In this paper, we extend the fingerprint database by using DCGAN to generate additional amplitude feature maps. DCGAN is a well-established GAN model; however, in our application, it converges slowly when generating amplitude feature maps and the model collapses during the training process. To resolve these problems, the AF-DCGAN model is proposed in the following section. Using AF-DCGAN accelerates convergence and increases the diversity of the generated CSI amplitude feature maps dramatically.
The AF-DCGAN model is depicted in Fig.~\subref*{fig_generative_model} and Fig.~\subref*{fig_discriminative_model}. And it is worth noting that in AF-DCGAN, the final deconvolution layer size of the generator is $128\times 128$, which allows us to directly generate images with $256\times 256$ pixels.
\begin{figure*}[htb]
\centering
\subfloat[]{
\includegraphics[width=7in]{figure/fig_generative_model}\label{fig_generative_model}}\hfill
\subfloat[]{
\includegraphics[width=7in]{figure/fig_discriminative_model}\label{fig_discriminative_model}}\hfill
\caption{Network structure of the AF-DCGAN model: (a) $G$ is the generative model, implemented by a deconvolutional network; $Z$ is the signal, which has a uniform distribution; $deconv$ represents the deconvolution layer in the CNN model; $\Psi$ denotes the feature maps generated by the deconvolutional layers in the generator. (b) Discriminative model, implemented by a convolutional network. $\Phi'$ are feature maps from the training set; $\Psi$ are the generated samples; $conv$ represents the convolution layer in the CNN model; $D(x)$ indicates the probability that the input sample stemmed from the training set corresponding to $x$ or $G(z)$ as the input; $FC$ denotes the fully connected layers in the CNN models.}
\end{figure*}
\subsection{Training process}
From WGAN \cite{32}, we know that the GAN has problems such as difficult training, loss of generators and discriminators, lack of diversity of training samples and in generating training samples. Therefore, generating a large number of CSI amplitude feature maps by GAN is not easy. We test the performances of WGAN and DCGAN, and the results are shown in Fig.~\ref{fig_afm_dcgan_wgan}.
\begin{figure}[htb]
\centering
\subfloat[]{
\includegraphics[width=3in]{figure/fig_afm_dcgan}\label{fig_afm_dcgan}}\hfill
\subfloat[]{
\includegraphics[width=3in]{figure/fig_afm_wgan}\label{fig_afm_wgan}}\hfill
\caption{CSI amplitude feature maps generated by GAN model: (a) Maps from DCGAN in the 100th epoch. It can be seen from the figure that all the subgraphs are exactly the same and the sample diversity is very poor. (b) Maps from WGAN in the 100th epoch. The model is difficult to converge during the training process.} \label{fig_afm_dcgan_wgan}
\end{figure}
When the traditional DCGAN is applied to expand the CSI fingerprint database, the diversity of the generated amplitude feature maps is poor and yields no performance gains for the indoor localization. Although WGAN reduces the difficulty of GAN training, convergence is not reached in some settings, and the generated pictures are worse than those of DCGAN. When WGAN is used for our amplitude feature maps, the model collapses quickly during the training process, and the generated data seems to be random, as shown in Fig.~\subref*{fig_afm_wgan}.
WGAN proposes to use the Wassertein distance as an optimization method to train GAN, but this still leaves difference between the mathematical and real implementation. The Wassertein distance needs to satisfy the strong continuity condition: Lipschitz continuity. To satisfy this condition, the authors forced the Lipschitz continuity to be met by limiting the weight to a range, but it also creates hidden dangers. The Lipschitz constraint is in the sample space, and the discriminator function $D(x)$ gradient value is required to be no more than a finite constant $K$. The weighted value constraint ensures the boundedness of the weight parameter and indirectly limits the gradient information. To solve these problems, we imitate part of WGAN to improve DCGAN in AF-DCGAN to improve the diversity of the generated samples. We remove the Sigmoid function from the last layer of the discriminator and return the normalized fully connected layer to make sure the value of $D(x)$ in loss function locates with [0,1]. Specifically, after sending the first batch of training data into discriminator, we normalize the output of full connection layer $D(x)$ with function $D_{norm}(x)=\frac{D(x)-D_{1b\_min}(x)} {D_{1b\_max}(x)-D_{1b\_min}(x)}$, where $D(x)$ is the output value of full connection layer, $D_{1b\_min}(x)$ and $D_{1b\_max}(x)$ are the minimum and maximum output value in the first batch of training data respectively. The other batches are normalized in the similar way with the same $D_{1b\_min}(x)$ and $D_{1b\_max}(x)$, clipping to 0 and 1 when out of range. Next, in our experiments, if Adam is used, the discriminator's loss sometimes collapses. When it collapses, the cosine of the angle between the update direction and the gradient direction given by Adam becomes negative. It means that the discriminator's loss gradient is unstable, so it is not suitable to use a momentum-based optimization algorithm such as Adam. Since the RMSProp optimization algorithm is suitable for gradient instability, we change the adaptive optimization algorithm from Adam to RMSProp to make our model stable for amplitude feature maps from WGAN. In addition, we do not use the Wassertein distance as an optimization method to train the GAN model. The training process is shown in Algo. \ref{alg_afdcgan_training}.
\begin{algorithm}
\caption{Training of AF-DCGAN Model.}\label{alg_afdcgan_training}
\begin{algorithmic}[1]\label{algo}
\Require $\Phi_i^{'}$ from initial fingerprint database $\Phi^{'}$, the learning rate $LR$, the clipping parameter $c$, the batch size $bs$, and the number of iterations of the discriminator per generator iteration $f_{d}$.
\State Randomly initialize discriminator parameters $w_0$, generator's parameters $\theta _0$
\ForAll {training iterations}
\ForAll {$t$ in $f_{d}$}
\State Sample mini-batch of $b_s$ examples $\{x^{(1)}, \cdots, x^{(b_s)}\}$ from the real data $\Phi _{i} ^{'}$
\State Sample mini-batch of $b_s$ examples $\{z^{(1)}, \cdots, z^{(b_s)}\}$ from the noise prior $p_g(z)$
\State $d_w\leftarrow \bigtriangledown _w[\frac{1}{bs}\sum_{i=1}^{bs}f_w(x^{(i)})-\frac{1}{bs}\sum_{i=1}^{bs}f_w(g_\theta(z^{(i)}))]$
\State $w \leftarrow w +LR \times RMSProp(w,d_w)$
\State $w \leftarrow clip(w,-c,c)$
\EndFor
\State Sample mini-batch of $bs$ examples $\{z^{(1)}, \cdots, z^{(b_s)}\}$ from the noise prior $p_g(z)$
\State $g_\theta \leftarrow -\bigtriangledown _\theta\frac{1}{bs}\sum_{i=1}^{bs}f_w(g_\theta(z^{(i)}))$
\State $\theta \leftarrow \theta - LR \times RMSProp(\theta,g_{\theta})$
\EndFor
\State In the generator $G$, output $l$ feature maps with $256\times 256$ resolution constitute a set of $\Psi_i$.
\end{algorithmic}
\end{algorithm}
We enter $\Phi_i^{'}$ sequentially into the AF-DCGAN model to generate amplitude feature maps for all the locations to obtain the set of generated amplitude feature maps as follows:
\begin{equation}
\Psi=\{\Psi_1,\cdots,\Psi_i,\cdots,\Psi_M\}
\end{equation}
where $\Psi_i$ denotes the amplitude feature maps of the $N_t\times N_r$ links generated by the generator corresponding to the $i$-th reference point. Example amplitude feature maps generated by the generator of a well-trained model are shown in Fig.\ref{fig_afm_afdcgan_a}. They look roughly similar, however, if we look at the details of generated feature maps, we can notice a lot of differences.
\begin{figure}[htb]
\centering
\includegraphics[width=3in]{figure/fig_afm_afdcgan_a}
\caption{CSI amplitude feature maps generated by the AF-DCGAN in the 60th epoch. It can be seen from the figure that there are obvious differences among all the subgraphs, which greatly improves the diversity of the samples.} \label{fig_afm_afdcgan_a}
\end{figure}
Thus far, we have used the AF-DCGAN to generate more amplitude feature maps, extending the initial fingerprint database and greatly reducing the manual labor. We eventually obtain the expanded fingerprint database $\Gamma=\{\Phi^{'},\Psi\}$.
\section{Validation Experiment} \label{experiment}
Experiments are conducted to evaluate the performance of the proposed fingerprint database extension method.
\subsection{Experiment Methodology}
The experiment is conducted in the classroom, as shown in Fig.~\ref{fig_experiment_environment}. The circumscribed rectangle of the classroom is taken as the indoor positioning area ($7m \times 7m$), which is evenly divided into $49$ square grids. The center point of each cell is used as a reference point to form a set of reference points. The distance between two adjacent reference points is one meter. This distance is selected for the following reasons. As shown in Fig.~\subref*{fig_csi_packets_a} and Fig.~\subref*{fig_csi_packets_b}, the two positions are one meter apart, and we can see the difference in their amplitude maps, although they are similar. When the distances are too close, the receiving paths of the adjacent locations are similar; consequently, the features of the amplitude feature maps in the fingerprint database will be highly similar, which will reduce the localization accuracy. As shown in Fig.~\subref*{fig_csi_packets_a} and Fig.~\subref*{fig_csi_packets_c}, the two positions are five meters apart, and we can see that their amplitude maps are significantly different. When the distances are too far apart, the feature differences of the amplitude feature maps of adjacent locations will become larger, making certain features difficult to match and resulting in reduced localization accuracy.
In the classroom, we deployed a TL-WR742N wireless router as the transmitter, operating in IEEE 802.11n AP mode and equipped with one transmitting antenna, and a ThinkPad x201 laptop equipped with Intel Wireless Link 5300 NICs (IWL5300) as receivers, with three receive antennas. The Laptop’s operating system is Ubuntu 16.04, installed on a custom system kernel with a modified network driver. The firmware and driver of IWL5300 are modified to export the CSI of each packet's predicted IEEE 802.11 packet delivery from wireless channel measurements made by the ‘Linux 802.11n CSI Tool’ \cite{33} and containing information about all subcarriers. The router was placed at the front of the measurement area. The receiving antennas were placed behind the measurement area to cover the whole area. To collect the CSI data, \emph{Ping} commands were executed on the laptop every 0.4 s to generate network traffic \footnote{The captured CSI data can be downloaded at https://github.com/quheng54/dataset-for-AF-DCGAN}. The AF-DCGAN model was implemented on TensorFlow and accelerated by a GPU (GeForce GTX 1060).
During the experiments, we collected CSI data first. Then, after processing it into amplitude feature maps, we use AF-DCGAN to generate the additional amplitude feature maps. We set the learning rate of the model $LR=0.0002$, cutting parameter $c=0.01$, batch size $bs=49$, and the number of iterations of discriminator per generator iteration $f_{d} = 2$. When transmitting CSI data, the volunteers stood at the reference points (shown in Fig.~\ref{fig_experiment_environment_layout} as the green circles). At each reference point, 5000 data samples are collected. There are three links and each link contains 30 groups of subcarriers. Thus, each CSI sample has $1\times 3\times 30$ dimensions. For each reference point, we randomly select 100 samples from the $5000$ samples for $10000$ times to draw the amplitude feature maps. After pixel transformation, the obtained amplitude feature maps of the three links form the initial fingerprint database. Then, we use the AF-DCGAN model corresponding to each position to generate $10000$ additional amplitude maps for each reference point that reflect the same type of reference point to extend the initial database. In this way, the diversity of samples generated at a single location can be increased, and the localization accuracy can be improved. We call the extended fingerprint database consisting of the newly generated feature maps GOAFM.
During the localization step, a deep learning method is used to classify the amplitude feature maps, similar to device-free indoor localization in \cite{7765094} and Deepfi in \cite {35}, \cite{7438932}, which are commonly used for localization. For clarity, the architecture is illustrated in Fig.~\ref{fig_AF_cnn_model}, and is named Amplitude Feature CNN (AF-CNN).
We treat the AF-CNN as a classification model to determine the reference point to which a test point belongs. During the experiments, we select $20$ points in the experimental area as test points that differ from the reference point (shown as the red stars in Fig.~\ref{fig_experiment_environment_layout}). Each point has $5000$ samples. The test-sample point data are processed in the same manner as explained above to obtain the amplitude feature maps required for the test. Then, we put the amplitude feature maps into the trained classification model for testing. The test results output the first four best-matching reference points. The first four best-matching reference points are selected because the test points are often surrounded by four reference points; thus, using the center of the resulting first four highest matching reference points improves the localization accuracy. Finally, we calculate the geometric center of the first four highest matching points as the localization result.
\begin{figure}[htb]
\centering
\includegraphics[width=2.5in]{figure/fig_experiment_environment_layout}
\caption{Layout of the classroom for training/test positions. }
\label{fig_experiment_environment_layout}
\end{figure}
\begin{figure}[htb]
\centering
\includegraphics[width=3.5in]{figure/fig_AF_cnn_model}
\caption{Architecture of AF-CNN model.}
\label{fig_AF_cnn_model}
\end{figure}
Finally, we verify the superiority of the extension fingerprint database by comparing the localization results using the initial fingerprint database and the fingerprint database expanded by AF-DCGAN. We also compare the localization results using the initial fingerprint database and the fingerprint database expanded by adding noise (ADNOI). During the experiments, we simply add some Gaussian random noise to the sampled data in original amplitude feature maps, with , to generate an augmented fingerprint database. We name it 100\% ADNOI when the same amount of feature maps are supplemented, and 200\% ADNOI when adding twice the amount of maps.
Additionally, we compare our method to existing methods for building a fingerprint database with GPR and with the existing localization methods FIFS \cite{34} and DeepFi \cite{35} to demonstrate our method's competitiveness.
\subsection{Localization performance}
(1) Performance on different localization methods: We use AF-CNN to conduct the localization experiment. Fig.~\subref*{fig_localization_cdf_initial} shows the cumulative distribution function (CDF) of the localization error for the initial fingerprint database and the extended fingerprint database by adding noise. The red line represents the CDF curve using the initial fingerprint database, which has an average error of $1.34$m. For this fingerprint database, the minimum error distance is $0.12$m, and the maximum error distance is $2.68$m. The probability of an error distance within $1$m is 47\%, within $2$m, 62\%, and within $3$m, 100\%. The green line represents the CDF curve using the initial database with an additional 100\% of ADNOI, and its average error is $1.25$m. In this condition, the minimum error distance is $0.05$m, and the maximum error distance is $2.47$m. The probability of an error distance within $1$m is 49\%, within $2$m, 73\%, and within $3$m 100\%. The yellow line represents the CDF curve when the test points are located using a fingerprint database with an added 200\% of fingerprints of amplitude feature maps generated by adding noise for each point, and its average error is $1.23$m. For this fingerprint database, the minimum error distance is $0.06$m, and the maximum error distance is $2.45$m. The probability of an error distance within $1$m is 50\%, within $2$m, 73\%, and within $3$m 100\%.
These results demonstrate the validity of the database generated using ADNOI. We can use ADNOI to augment the database without expending human effort while also improving the localization accuracy. With 200\% ADNOI is added, the sample size of the fingerprint database is saturated and the error is no longer reduced.
As shown in Fig.~\subref*{fig_localization_cdf_goafm}, the red line represents the CDF curve using the initial fingerprint database. The blue line represents the CDF curve using the initial database with an additional 50\% of GOAFM, and its average error is $1.21$m. In this condition, the minimum error distance is $0.06$m, and the maximum error distance is $2.63$m. The probability of an error distance within $1$m is 50\%, within $2$m, 80\%, and within $3$m 100\%. The black line represents the CDF curve when the test points are located using a fingerprint database with an added 100\% of fingerprints of amplitude feature maps generated by AF-DCGAN for each point, and its average error is $1.18$m. For this fingerprint database, the minimum error distance is $0.04$m, and the maximum error distance is $2.23$m. The probability of an error distance within $1$m is 53\%, within $2$m, 90\%, and within $3$m 100\%. The green line represents the CDF curve with 150\% of generated fingerprints of amplitude feature maps added to the initial database, and its average error is $0.92$m. In this condition, the minimum error distance is $0.03$m, and the maximum error distance is $1.98$m. The probability of an error distance within $1$m is 59\%, within $2$m, 100\%, and within $3$m, 100\%. The yellow line represents the CDF curve with 200\% of generated fingerprints of amplitude feature maps added to the initial database, it has an average error of $0.92$m. For this fingerprint database, the minimum error distance is $0.03$m, and the maximum error distance is $1.96$m. The probability of an error distance within $1$m is 59\%, within $2$m, 100\%, and within $3$m, 100\%. The location results are shown in Tables \ref{table_localization_error} and \ref{table_localization_cdf}. Error is represented by the mean error distance. As shown in Fig.~\subref*{fig_localization_cdf_goafm}, for the initial fingerprint database, having more fingerprint database samples available improves the positioning accuracy. After adding the generated amplitude feature maps to the initial database, the localization accuracy improve. After adding 50\% GOAFM to the initial database, the accuracy is $1.21$m, an improvement of 9.70\%. After adding 100\% GOAFM to the initial database, the accuracy is $1.18$m, an improvement of 16.00\%, and after adding 150\% GOAFM to the initial database, the accuracy is $0.92$m, an improvement of 31.30\%. After 150\% GOAFM is added, the sample size of the fingerprint database is basically saturated, and the error is no longer reduced. So, after adding 200\% GOAFM to the initial database, the accuracy is the same as the 150\% GOAFM added to the initial database.
These results demonstrate the validity of the database generated using AF-DCGAN. In other words, we can use AF-DCGAN to augment the database without expending human effort while also improving the localization accuracy. As shown in Fig.~\subref*{fig_localization_cdf_initial_goafm}, adding GOAFM generated by the AF-DCGAN is better than adding ADNOI to the initial database.
\begin{figure}[htb]
\centering
\subfloat[Initial database.]{
\includegraphics[width=3in]{figure/fig_localization_cdf_initial}\label{fig_localization_cdf_initial}}\hfill
\subfloat[Initial database and GOAFM.]{
\includegraphics[width=3in]{figure/fig_localization_cdf_goafm}\label{fig_localization_cdf_goafm}}\hfill
\subfloat[ADNOI and GOAFM.]{
\includegraphics[width=3in]{figure/fig_localization_cdf_initial_goafm}\label{fig_localization_cdf_initial_goafm}}\hfill
\caption{As the number of samples increases, the localization accuracy of the fingerprints database increase.} \label{fig_localization_cdf}
\end{figure}
We also evaluate the performance of our localization method based on our fingerprint database by comparing it with the existing localization methods FIFS and DeepFi. We conduct these three experiments in the classroom environment as illustrated in Fig.~\ref{fig_experiment_environment}. As shown in Fig.~\subref*{fig_localization_performance_inital}, Fig.~\subref*{fig_localization_performance_inital_GOAFM}, Tables \ref{table_localization_error} and \ref{table_localization_cdf}, our method is superior to FIFS and DeepFi. When we used the initial database for positioning, compared to FIFS, the localization accuracy improved by 3.60\%. When we added all the GOAFM to the initial database, the localization accuracy improved by 15.11\%. When we added 150\% of the GOAFM to the initial database, the localization accuracy improved by 33.81\%. In other words, we are able to significantly improve the localization accuracy without expending additional labor to collect more CSI data. As shown in Fig.~\subref*{fig_localization_performance_inital} and Fig.~\subref*{fig_localization_performance_inital_GOAFM}, the localization accuracy of the initial database is not better compared with DeepFi, but after adding 150\% GOAFM, its positioning accuracy increases by 24.59\%, even without consuming additional labor to collect more CSI data.
\begin{figure}[tb]
\centering
\subfloat[Initial database with FIFS and DeepFi.]{
\includegraphics[width=3in]{figure/fig_localization_performance_inital}\label{fig_localization_performance_inital}}\hfill
\subfloat[Initial database and GOAFM with FIFS and DeepFi.]{
\includegraphics[width=3in]{figure/fig_localization_performance_inital_GOAFM_2}\label{fig_localization_performance_inital_GOAFM}}\hfill
\caption{Comparison to other localization methods.} \label{fig_database_construction_performance}
\end{figure}
\begin{table}[htbp]
\centering
\caption{Localization Error of Different Methods}\label{table_localization_error}
\centering
\begin{tabular}{c|c|c|c}
\hline
Method & Min. & Max. & Mean \\\hline
Initial library & 0.12m & 2.68m & 1.34m \\
Initial library and 100\% ADNOI & 0.05m & 2.47m & 1.25m \\
Initial library and 200\% ADNOI & 0.06m & 2.45m & 1.23m \\\hline
Initial library and 50\% GOAFM & 0.06m & 2.63m & 1.21m \\
Initial library and 100\% GOAFM & 0.04m & 2.23m & 1.18m \\
Initial library and 150\% GOAFM & 0.03m & 1.98m & 0.92m \\
Initial library and 200\% GOAFM & 0.03m & 1.96m & 0.92m \\\hline
FIFS & 0.27m & 2.71m & 1.39m \\\hline
DeepFI & 0.29m & 2.47m & 1.22m \\\hline
\end{tabular}
\end{table}
\begin{table}[htbp]
\centering
\caption{Localization Range Probability of Different Methods}\label{table_localization_cdf}
\centering
\begin{tabular}{c|c|c|c}
\hline
Method & 1m & 2m & 3m \\\hline
Initial library & 47\% & 62\% & 100\% \\
Initial library and 100\% ADNOI & 49\% & 73\% & 100\% \\
Initial library and 200\% ADNOI & 50\% & 73\% & 100\% \\\hline
Initial library and 50\% GOAFM & 50\% & 80\% & 100\% \\
Initial library and 100\% GOAFM & 53\% & 90\% & 100\% \\
Initial library and 150\% GOAFM & 59\% & 100\% & 100\% \\
Initial library and 200\% GOAFM & 59\% & 100\% & 100\% \\\hline
FIFS & 23\% & 73\% & 100\% \\\hline
DeepFI & 33\% & 88\% & 100\% \\\hline
\end{tabular}
\end{table}
(2) Performance on different database construction methods: In this subsection, we evaluate the performance of our AF-DCGAN-based fingerprint database by comparing it with existing methods of fingerprint database construction based on GPR. We conduct these two experiments in the research classroom using the same CSI dataset. For GPR modeling, we select 4 of the 30 carrier models that are more accurate. Then, we determine the geometric center of these 4 model positioning results as the final positioning result. The results shown in Fig.~\subref*{fig_database_construction_performance_intial_GPR}, Fig.~\subref*{fig_database_construction_performance_intial_GOAFM_GPR}, Tables \ref{table_localization_contruction_error} and \ref{table_localization_construction_cdf} reveal that our fingerprint database construction method is superior to GPR. When we use the initial database for positioning, compared to GPR , the localization accuracy improved by 23.42\%. When we add the 150\% GOAFM to the initial database, the localization accuracy improved by 47.43\%. Moreover, the localization accuracy is significantly improved without expending more labor to collect the CSI data because of the AF-DCGAN.
\begin{figure}[htb]
\centering
\subfloat[Initial database.]{
\includegraphics[width=3in]{figure/fig_database_construction_performance_intial_GPR}\label{fig_database_construction_performance_intial_GPR}}\hfill
\subfloat[GOAFM.]{
\includegraphics[width=3in]{figure/fig_database_construction_performance_intial_GOAFM_GPR}\label{fig_database_construction_performance_intial_GOAFM_GPR}}\hfill
\caption{Comparison with the other database construction methods.} \label{fig_database_construction_performance}
\end{figure}
\begin{table}[htbp]
\centering
\caption{Localization Error of Different Database Construction Methods}\label{table_localization_contruction_error}
\centering
\begin{tabular}{c|c|c|c}
\hline
Method & Min. & Max. & Mean \\\hline
Initial library and 50\% GOAFM & 0.06m & 2.63m & 1.21m \\
Initial library and 100\% GOAFM & 0.04m & 2.23m & 1.18m \\
Initial library and 150\% GOAFM & 0.03m & 1.98m & 0.92m
\\\hline
GPR & 0.34m & 3.34m & 1.75m \\\hline
\end{tabular}
\end{table}
\begin{table}[!t]
\renewcommand{\arraystretch}{1.3}
\caption{Localization Range Probability of Different Database Construction Methods}\label{table_localization_construction_cdf}
\centering
\begin{tabular}{c|c|c|c}
\hline
Method & 1m & 2m & 3m \\\hline
Initial library and 50\% GOAFM & 50\% & 80\% & 100\% \\
Initial library and 100\% GOAFM & 53\% & 90\% & 100\% \\
Initial library and 150\% GOAFM & 59\% & 100\% & 100\% \\\hline
GPR & 24\% & 63\% & 83\% \\\hline
\end{tabular}
\end{table}
\section{CONCLUSION} \label{conclusion}
In this paper, a novel approach was proposed to reduce the collection of WiFi fingerprints. The new AF-DCGAN model based on GAN was used to generate additional amplitude feature maps similar to those in the reference database, which effectively increased the number of samples in the training set. We conducted exhaustive tests to demonstrate the performance of the proposed method, and the results showed the superiority of AF-DCGAN over the existing methods of building fingerprint database and localization methods, i.e. the accuracy of indoor WiFi positioning was improved without increasing the labor involved in building the fingerprint databases. The proposed scheme has limitations to be addressed in the future. For example, CSI data is highly susceptible to environmental changes, and localization accuracy will decrease when the indoor environment changes significantly.
We envision localization to play a vital role in the future society by its ability to offer enriched applications. To make the localization more accurate we will consider more types of information such as phase information in CSI data besides the amplitude of CSI data in the future. Pretrained networks and more complex CNN will be investigated for further performance improvement. Besides, the GAN-based scheme will be compared with other popular data generation method such as SMOTE \cite{articleSMOTE} in terms of accuracy and computational complexity.
\section*{Acknowledgment}
The corresponding author of this manuscript is Jie Li. This research is supported in part by National Natural Science Foundation of China, Grant No. 51877060, ANHUI Province Key Laboratory of Affective Computing \& Advanced Intelligent Machine, Grant No.ACAIM180102, and the Fundamental Research Funds for the Central Universities, Grant No. JZ2018HGTB0253, JZ2019HGTB0089 and PA2019GDQT0006, and State Grid Science and Technology Project (Research and application of key Technologies for integrated substation intelligent operation and maintenance based on the fusion of heterogeneous network and heterogeneous data).
\bibliographystyle{IEEEtran}
|
{
"timestamp": "2019-10-14T02:05:15",
"yymm": "1804",
"arxiv_id": "1804.05347",
"language": "en",
"url": "https://arxiv.org/abs/1804.05347"
}
|
\section{Detailed Results for individual pipelines of BFSS}
\label{sec:appendix}
As mentioned in section \ref{sec:expt}, $\textsc{bfss}$ is an ensemble of two pipelines, an AIG-NNF pipeline and a BDD-wDNNF pipeline.
These two pipelines accept the
same input specification but represent them in two different ways.
The first pipeline takes the input formula as an AIG and builds an NNF
(not necessarily a wDNNF) DAG, while the second pipeline first builds an ROBDD
from the input AIG using dynamic variable reordering, and then obtains a wDNNF representation from the ROBDD
using the linear-time algorithm described in~\cite{darwiche-jacm}.
Once the NNF/wDNNF representation is built, the same algorithm is used to generate skolem functions, namely,
Algorithm ~\ref{alg:easy} is used in Phase 1 and CEGAR-based synthesis using
$\textsc{UniGen}$\cite{unigen2} to sample counterexamples is used in Phase 2. In this section, we give the individual results of the two pipelines.
\subsection{Performance of the AIG-NNF pipeline}
\begin{table}[ht]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
\hline {Benchmark} & {Total } & {\# Benchmarks} & {Phase 1} & {Phase 2} & {Solved By} \\
{Domain} & {Benchmarks} & {Solved} & {Solved} &{Started} & {Phase 2} \\
\hline
{QBFEval} & 383 & 133 & 122 & 110 & 11 \\
\hline
{Arithmetic} & 48 & 31 & 31 & 12 & 0 \\
\hline
{Disjunctive} & & & & & \\
{Decomposition} & 68 & 68 & 66 & 2 & 2 \\
\hline
{Factorization} & 5 & 4 & 0 & 5 & 4 \\
\hline
\end{tabular}
\caption{$\textsc{bfss}$: Performance Summary for AIG-NNF pipeline }
\label{tab:bfss2}
\end{center}
\end{table}
In the AIG-NNF pipeline, $\textsc{bfss}$ solves a total of $236$ benchmarks, with $133$ benchmarks in QBFEval, $31$ in Arithmetic, all the $68$ benchmarks of Disjunctive Decomposition and $4$ benchmarks in Factorization.
Of the $254$ benchmarks in QBFEval (as mentioned in Section \ref{sec:expt}, we could not build succinct AIGs for the remaining benchmarks and did not run our tool on them), Phase 1 solved $122$ benchmarks and Phase 2 was started on $110$ benchmarks, of which $11$ benchmarks reached completion. Of the $48$ benchmarks in Arithmetic, Phase 1 solved $31$ and Phase $2$ was started on $12$. On the remaining $5$ Arithmetic benchmarks, Phase 1 did not reach completion. Of the $68$ Disjunctive Decomposition benchmarks, $66$ were successfully solved by Phase 1 and the remaining $2$ by Phase 2. Phase 2 had started on all the $5$ benchmarks in Factorization and reached completion on $4$ benchmarks.
\subsubsection{Plots for the AIG-NNF pipeline}
\begin{figure}[h]
\begin{subfigure}{2.3in}
\hspace{-1cm}
\includegraphics[angle=-90,scale=0.28] {op-time_BFSSvsCADETQBF.eps}
\end{subfigure}
\begin{subfigure}{2.3in}
\includegraphics[angle=-90,scale=0.28] {op-time_BFSSvsCADETTACMIS.eps}
\end{subfigure}
\caption{$\textsc{bfss}$ (AIG-NNF Pipeline) vs $\textsc{Cadet}$. Legend : \texttt{A}: Arithmetic, \texttt{F}: Factorization, \texttt{D}: Disjunctive decomposition \texttt{Q} QBFEval. \texttt{TO}: benchmarks for which the corresponding algorithm was unsuccessful.}
\label{fig:bfsscadetAIG}
\end{figure}
Figure \ref{fig:bfsscadetAIG} shows the performance of $\textsc{bfss}$ (AIG-NNF pipeline) versus $\textsc{Cadet}$ for all the four benchmark domains. Amongst the four domains, $\textsc{Cadet}$ solved $53$ benchmarks that $\textsc{bfss}$ could not solve. Of these, $52$ belonged to QBFEval and $1$ belonged to Arithmetic. On the other hand, $\textsc{bfss}$ solved $58$ benchmarks that $\textsc{Cadet}$ could not solve. Of these, $1$ belonged to QBFEval, $10$ to Arithmetic, $3$ to Factorization and $44$ to Disjunctive Decomposition. From Figure \ref{fig:bfsscadetAIG}, we can see that while $\textsc{Cadet}$ takes less time than $\textsc{bfss}$ on many Arithmetic and QBFEval benchmarks, on Disjunctive Decomposition and Factorization, the AIG-NNF pipeline of $\textsc{bfss}$ takes less time.
\begin{figure}[h]
\centering
\begin{subfigure}{2.3in}
\hspace{-1cm}
\includegraphics[angle=-90,scale=0.28] {op-time_BFSSvsParSynQBFEval.eps}
\end{subfigure}
\begin{subfigure}{2.3in}
\includegraphics[angle=-90,scale=0.28] {op-time_BFSSvsParSynTACMIS.eps}
\end{subfigure}
\caption{$\textsc{bfss}$ (AIG-NNF Pipeline) vs $\textsc{parSyn}$ (for legend see Figure \ref{fig:bfsscadetAIG})}
\label{fig:bfssparsynAIG}
\end{figure}
Figure \ref{fig:bfssparsynAIG} shows the performance of $\textsc{bfss}$ (AIG-NNF pipeline) versus $\textsc{parSyn}$. Amongst the $4$ domains, $\textsc{parSyn}$ solved $22$ benchmarks that $\textsc{bfss}$ could not solve, of these $1$ benchmark belonged to the Arithmetic domain and $21$ benchmarks belonged to QBFEval. On the other hand, $\textsc{bfss}$ solved $73$ benchmarks that $\textsc{parSyn}$ could not solve. Of these, $51$ belonged to QBFEval, $17$ to Arithmetic and $4$ to Disjunctive Decomposition. From \ref{fig:bfsscadetAIG}, we can see that while the behaviour of $\textsc{parSyn}$ and $\textsc{bfss}$ is comparable for many QBFEval benchmarks, on most of the Arithmetic, Disjunctive Decomposition and Factorization benchmarks, the AIG-NNF pipeline of $\textsc{bfss}$ takes less time.
\begin{figure}[h]
\centering
\begin{subfigure}{2.3in}
\hspace{-1cm}
\includegraphics[angle=-90,scale=0.28] {op-time_BFSSvsRSynthQBF.eps}
\end{subfigure}
\begin{subfigure}{2.3in}
\includegraphics[angle=-90,scale=0.28] {op-time_BFSSvsRSynthTACMIS.eps}
\end{subfigure}
\caption{$\textsc{bfss}$ (AIG-NNF Pipeline) vs $\textsc{RSynth}$ (for legend see Figure \ref{fig:bfsscadetAIG})}
\label{fig:bfssrsynthAIG}
\end{figure}
Figure \ref{fig:bfssrsynthAIG} gives the comparison of the AIG-NNF pipeline of $\textsc{bfss}$ and $\textsc{RSynth}$. While $\textsc{RSynth}$ solves $8$ benchmarks that $\textsc{bfss}$ does not solve,
$\textsc{bfss}$ solves $193$ benchmarks that $\textsc{RSynth}$ could not solve. Of these $106$ belonged to QBFEval, $20$ to Arithmetic, $66$ to Disjunctive Decomposition and $1$ to Factorization. Moreover, on most of the benchmarks that both the tools solved, $\textsc{bfss}$ takes less time.
\begin{figure}[h]
\centering
\begin{subfigure}{2.3in}
\hspace{-1cm}
\includegraphics[angle=-90,scale=0.28] {op-time_BFSSvsAbsSyntheQBF.eps}
\end{subfigure}
\begin{subfigure}{2.3in}
\includegraphics[angle=-90,scale=0.28] {op-time_BFSSvsAbsSyntheTACMIS.eps}
\end{subfigure}
\caption{$\textsc{bfss}$ (AIG-NNF Pipeline) vs $\textsc{AbsSynthe-Skolem}$ (for legend see Figure \ref{fig:bfsscadetAIG})}
\label{fig:bfssabsAIG}
\end{figure}
Figure \ref{fig:bfssabsAIG} gives the comparison of of the performance of the AIG-NNF pipeline of $\textsc{bfss}$ and $\textsc{AbsSynthe-Skolem}$. While $\textsc{AbsSynthe-Skolem}$ solves $72$ benchmarks that $\textsc{bfss}$ could not solve, $\textsc{bfss}$ solved $91$ benchmarks that $\textsc{AbsSynthe-Skolem}$ could not solve. Of these $44$ belonged to QBFEval, $8$ to Arithmetic and $39$ to Disjunctive Decomposition.
\subsection{Performance of the BDD-wDNNF pipeline}
In this section, we discuss the performance of the BDD-wDNNF pipeline of $\textsc{bfss}$.
Recall that in this pipeline the tool builds an ROBDD
from the input AIG using dynamic variable reordering and then converts the ROBDD in a wDNNF representation.
In this section, by $\textsc{bfss}$ we mean, the BDD-wDNNF pipeline of the tool.
Table \ref{tab:bfss3} gives the performance summary of the BDD-wDNNF pipeline. Using this pipeline, the tool solved a total of $230$ benchmarks, of which $143$ belonged to QBFEval, $23$ belonged to Arithmetic, $59$ belonged to Disjunctive Decomposition and $5$ belonged to Factorization. As expected, since the representation is already in wDNNF, the skolem functions generated at end of Phase 1 were indeed exact (see Theorem \ref{lemma:init_sk_good}(b)) and we did not require to start Phase 2 on any benchmark. We also found that the memory requirements of this pipeline were higher, and for some benchmarks the tool failed because the ROBDDs (and hence resulting wDNNF representation) were large in size, resulting in out of memory errors or assertion failures in the underlying AIG library.
\begin{table}[ht]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
\hline {Benchmark} & {Total } & {\# Benchmarks} & {Phase 1} & {Phase 2} & {Solved By} \\
{Domain} & {Benchmarks} & {Solved} & {Solved} &{Started} & {Phase 2} \\
\hline
{QBFEval} & 383 & 143 & 143 & 0 & 0 \\
\hline
{Arithmetic} & 48 & 23 & 23 & 0 & 0 \\
\hline
{Disjunctive} & & & & & \\
{Decomposition} & 68 & 59 & 59 & 0 & 0 \\
\hline
{Factorization} & 5 & 5 & 5 & 0 & 0 \\
\hline
\end{tabular}
\caption{$\textsc{bfss}$: Performance Summary }
\label{tab:bfss3}
\end{center}
\end{table}
\subsubsection{Plots for the BDD-wDNNF pipeline}
\begin{figure}[h]
\centering
\begin{subfigure}{2.3in}
\hspace{-1cm}
\includegraphics[angle=-90,scale=0.28] {ap-time_BFSSvsCADETQBF.eps}
\end{subfigure}
\begin{subfigure}{2.3in}
\includegraphics[angle=-90,scale=0.28] {ap-time_BFSSvsCADETTACMIS.eps}
\end{subfigure}
\vspace{0.3cm}
\caption{$\textsc{bfss}$ (BDD-wDNNF Pipeline) vs $\textsc{Cadet}$ (for legend see Figure \ref{fig:bfsscadetAIG})}
\label{fig:bfsscadetBDD}
\end{figure}
Figure \ref{fig:bfsscadetAIG} gives the performance of $\textsc{bfss}$ versus $\textsc{Cadet}$. The performance of $\textsc{Cadet}$ and $\textsc{bfss}$ is comparable, with $\textsc{Cadet}$ solving $74$ benchmarks across all domains that $\textsc{bfss}$ could not and $\textsc{bfss}$ solving $73$ benchmarks that $\textsc{Cadet}$ could not. While $\textsc{Cadet}$ takes less time on many QBFEval benchmarks, on many Arithmetic, Disjunctive Decomposition and Factorization Benchmarks, the BDD-wDNNF pipeline of $\textsc{bfss}$ takes less time.
\begin{figure}[h]
\centering
\begin{subfigure}{2.3in}
\hspace{-1cm}
\includegraphics[angle=-90,scale=0.28] {ap-time_BFSSvsParSynQBFEval.eps}
\end{subfigure}
\begin{subfigure}{2.3in}
\includegraphics[angle=-90,scale=0.28] {ap-time_BFSSvsParSynTACMIS.eps}
\end{subfigure}
\vspace{0.3cm}
\caption{$\textsc{bfss}$ (BDD-wDNNF Pipeline)vs $\textsc{parSyn}$ (for legend see Figure \ref{fig:bfsscadetAIG})}
\label{fig:bfssparsynBDD}
\end{figure}
Figure \ref{fig:bfssparsynBDD} gives the performance of $\textsc{bfss}$ versus $\textsc{parSyn}$. While $\textsc{parSyn}$ could solve $30$ benchmarks across all domains that $\textsc{bfss}$ could not, the BDD-wDNNF pipeline of $\textsc{bfss}$ solved $75$ benchmarks that $\textsc{parSyn}$ could not.
\begin{figure}[h]
\centering
\begin{subfigure}{2.3in}
\hspace{-1cm}
\includegraphics[angle=-90,scale=0.28] {ap-time_BFSSvsRSynthQBF.eps}
\end{subfigure}
\begin{subfigure}{2.3in}
\includegraphics[angle=-90,scale=0.28] {ap-time_BFSSvsRSynthTACMIS.eps}
\end{subfigure}
\vspace{0.3cm}
\caption{$\textsc{bfss}$ (BDD-wDNNF Pipeline) vs $\textsc{RSynth}$ (for legend see Figure \ref{fig:bfsscadetAIG})}
\label{fig:bfssrsynthBDD}
\end{figure}
Figure \ref{fig:bfssrsynthBDD} gives the performance of $\textsc{bfss}$ versus $\textsc{RSynth}$. While $\textsc{RSynth}$ could solve $9$ benchmarks across all domains that $\textsc{bfss}$ could not, the BDD-wDNNF pipeline of $\textsc{bfss}$ solved $188$ benchmarks that $\textsc{RSynth}$ could not. Furthermore from Figure \ref{fig:bfssrsynthBDD}, we can see that on most benchmarks, which both the tools could solve, $\textsc{bfss}$ takes less time.
\begin{figure}[h]
\centering
\begin{subfigure}{2.3in}
\hspace{-1cm}
\includegraphics[angle=-90,scale=0.28] {ap-time_BFSSvsAbsSyntheQBF.eps}
\end{subfigure}
\begin{subfigure}{2.3in}
\includegraphics[angle=-90,scale=0.28] {ap-time_BFSSvsAbsSyntheTACMIS.eps}
\end{subfigure}
\vspace{0.3cm}
\caption{$\textsc{bfss}$ (BDD-wDNNF Pipeline) vs $\textsc{AbsSynthe-Skolem}$ (for legend see Figure \ref{fig:bfsscadetAIG})}
\label{fig:bfssabsBDD}
\end{figure}
Figure \ref{fig:bfssabsBDD} gives the performance of $\textsc{bfss}$ versus $\textsc{AbsSynthe-Skolem}$. While $\textsc{AbsSynthe-Skolem}$ could solve $39$ benchmarks across all domains that $\textsc{bfss}$ could not, the BDD-wDNNF pipeline of $\textsc{bfss}$ solved $52$ benchmarks which could not be solved by $\textsc{AbsSynthe-Skolem}$.
\subsection{Comparison of the two pipelines}
\begin{figure}[h]
\centering
\begin{subfigure}{2.3in}
\hspace{-1cm}
\includegraphics[angle=-90,scale=0.28] {time_BFSSvsBFSSQBF.eps}
\end{subfigure}
\begin{subfigure}{2.3in}
\includegraphics[angle=-90,scale=0.28] {time_BFSSvsBFSSTACMIS.eps}
\end{subfigure}
\vspace{0.3cm}
\caption{$\textsc{bfss}$ (AIG-NNF) vs $\textsc{bfss}$ (BDD-wDNNF) (for legend see Figure \ref{fig:bfsscadetAIG})}
\label{fig:bfssvsbfss}
\end{figure}
Figure \ref{fig:bfssvsbfss} compares the performances of the two pipelines.
We can see that while there were some benchmarks which only one of the pipelines could solve, apart from Factorization benchmarks, for most of the QBFEval, Arithmetic and Disjunctive Decomposition Benchmarks, the time taken by the AIG-NNF pipeline was less than the time taken by the BDD-wDNNF pipeline.
\end{document}
\section{Introduction}
\label{sec:intro}
\input{intro}
\section{Notations and Problem Statement}
\label{sec:prelim}
\input{preliminaries}
\section{Complexity-theoretical limits}
\label{sec:eth}
\input{hardness}
\section{Phase 1: Efficient polynomial-sized synthesis}
\label{sec:nnf}
\input{initial_part1}
\input{initial_part2}
\section{Phase 2: Counterexample-guided refinement}
\label{sec:cegar}
\input{cegar}
\section{Experimental results}
\label{sec:expt}
\input{experiments}
\section{Conclusion}
\label{sec:concl}
\input{conclusion}
\bibliographystyle{splncs03}
|
{
"timestamp": "2018-05-21T02:08:01",
"yymm": "1804",
"arxiv_id": "1804.05507",
"language": "en",
"url": "https://arxiv.org/abs/1804.05507"
}
|
\section{Introduction}
\label{sec:intro}
The field of similarity computation is more than a century old, e.g.,
see the review at \cite{DaDa11}; it offers many measures to compute
similarity depending on how the inputs are modelled. In this paper, we
focus on computing the similarity of ranked lists.
Among the many list similarity measures in the literature, Spearman's
footrule and Kendall's tau are commonly used. In this paper, we
generalize these measures to include weights and also to work for
partial lists as well as permutations. The main contribution of this
paper is to prove the equivalence of these weighted versions, by
generalizing a proof by Diaconis and Graham for the unweighted
versions. Here equivalence means these measures are within small
constant multiples of one another.
\section{Related Work}
\label{sec:related}
A detailed related work on similarity in general is given in
\cite{DaDa11}. For more recent work, refer to
\cite{ChLiFe14,FaToMi12,FaMiPu17,KuVa10}. For other ways of
incorporating weights with the similarity measures in question, refer
to \cite{ChLiFe14,FaToMi12,FaMiPu17,KuVa10,Sh98,ShBaTs00}
The work in this paper and that in \cite{DaDa11} were actually part of
a comprehensive multi-year project on all forms of similarity for web
search metrics at Yahoo! Web Search; we started the project around
2007. An internal version of \cite{DaDa11} from the year 2009 with our
proof was cited in \cite{KuVa10}.
\section{Rank Assignment}
\label{sec:rank-def}
Consider the ordered or ranked lists $\sigma=(a,b,c,d,e,f)$ and
$\pi=(b,f,a,e,d,c)$. These lists are {\em full} in that they are {\em
permutations} of each other, which in turn means these lists contain
the same items or elements, possibly in a different order. The rank of
an element $i$ in $\sigma$ gives its order and is denoted by $\sigma(i)$. For
example, the rank of $e$ in $\sigma$ is $\sigma(e)=5$ whereas its rank in $\pi$
is $\pi(e)=4$.
Now consider the ordered lists $\sigma=(a,b,c)$ and $\pi=(b,e,c,f)$. These
lists are {\em partial} in that they are not permutations of each
other. This means one list may contain elements that the other list
does not. Moreover, their sizes may be different too. For example, $a$
is present in $\sigma$ but absent from $\pi$; also, $\sigma$ has a length of
$3$ whereas $\pi$ has a length of $4$. For each list, the rank of an
element in that list is well defined.
For comparing ranked partial lists, it is convenient to consider that
these partial lists are actually permutations of each other, with
missing elements somehow added at the end of each partial list. The
motivation for this view comes from web search engines in that a
search result that is missing from the first page of shown (usually
10) results is most probably still in the search index, which is huge,
but not shown.
To figure out how to add the missing elements, consider again our
example partial lists $\sigma=(a,b,c)$ and $\pi=(b,d,c,e)$. Completing
these lists means having each list contain all the elements from their
union, while preserving the rank of their existing elements. The union
of $\sigma$ and $\pi$ is equal to $\{a,b,c,d,e\}$, where we use the set
notation to imply unordering. Comparing these lists to their union
shows that $\sigma$ is missing $\{d,e\}$ whereas $\pi$ is missing
$\{a\}$. Let $\sigma'$ denote the completion of $\sigma$. Here we have two
options for $\sigma'$: 1) $\sigma'=(a,b,c,d,e)$ or 2) $\sigma'=(a,b,c,e,d)$. For
$\pi$, we have a single option: $\pi'=(b,d,c,e,a)$. In the
literature~\cite{FaKuSi03}, another option is to put $d$ and $e$ at
the same rank; we will not consider it here due to the same search
engine analogy above, i.e., if the search engine returns more results,
they will again be ranked.
Now consider the similarity between $\sigma'$ and $\pi'$. Since $d$ is
before $e$ in $\pi$ as well as $\pi'$, the first option for $\sigma'$
increases the similarity whereas the second option decreases the
similarity. Either option can be used for completing partial lists to
permutations; our choice in this paper is the first option yet again
due to the same search engine analogy: Assuming a competitor search
engine has the same elements at the same rank will provide motivation
to speed up innovation to beat the competition.
Now that we know how to transform partial lists to become full lists,
we will focus only on full lists in the sequel. We will do one more
simplification in that without loss of generality, we represent
the lists using the ranks of their elements rather than the elements
themselves. For example, $\sigma=(a,b,c,d,e,f)$ and $\pi=(b,f,a,e,d,c)$
become $\sigma=(1,2,3,4,5,6)$ and $\pi=(2,6,1,5,4,3)$. Without loss of
generality, we could also have used $\pi$ as the reference list to
determine the integer names of the elements in $\sigma$.
\section{Weighted Measures}
We now define the weighted versions of Spearman's footrule and
Kendall's tau. We also provide examples.
Note that the normalized forms of these measures map to the interval
$[0,1]$ where $0$ means the two input lists are ranked in the same
order and $1$ means the two input lists are ranked in the opposite
order. If a mapping to $[-1,1]$ is desired, where $1$ and $-1$
indicate the same and opposite orders respectively, the normalized
value $v$ needs to be transformed to $1-2v$.
\begin{figure}
\centering
\includegraphics[width=0.6\textwidth]{spearman.png}
\caption{Weighted Sperman's footrule in the Python programming
language. Use unity weights to derive the unweighted versions.}
\label{fig:spearman}
\end{figure}
\subsection{Weighted Spearman's Footrule}
\label{sec:footrule}
We define the weighted version of Spearman's footrule
\cite{DiGr77,Spearman1906} for lists of length $n$ as
\begin{equation}
\label{eq:footrule}
S_w(\sigma,\pi) = \sum_{i\in\sigma\cup\pi} w(i) |\sigma(i) - \pi(i)| .
\end{equation}
where $w(i)$ returns a positive number as the weight of the element
$i$.
The measure $S_w$ can be normalized to the interval $[0,1]$ as
\begin{equation}
\label{eq:norm-s-w}
s_w(\sigma,\pi) = \frac{S_w(\sigma,\pi)}{\sum_{i=1}^{n} w(i) |(i) - (n - i + 1)|}
\end{equation}
where the denominator reaches its maximum when both lists are sorted
but in opposite orders.
Fig.~\ref{fig:spearman} shows the algorithms, written in the Python
programming language, to compute both the numerator and the
denominator of $s_w(\sigma,\pi)$ in Eq.~\ref{eq:norm-s-w} as well as
$s_w(\sigma,\pi)$ itself as their ratio. Using unity weights lead to the
unweighted versions of these algorithms.
\begin{figure}
\centering
\includegraphics[width=0.6\textwidth]{kendall.png}
\caption{Weighted Kendall's tau in the Python programming
language. Use unity weights to derive the unweighted versions.}
\label{fig:kendall}
\end{figure}
\subsection{Weighted Kendall's Tau}
\label{sec:kendall}
The unweighted Kendall's tau is the number of swaps we would perform
during the bubble sort to reduce one permutation to another. As we
described the way we determine the ranks of the extended lists
(\S~\ref{sec:rank-def}), we can always assume that the first list $\sigma$
is the identity (increasing from 1 to $n$), and what we need to
compute is the number of swaps to sort the permutation $\pi$ back to
the identity permutation (increasing). Here, a weight will be
associate to each swap.
We define a weighted version of Kendall's tau
\cite{Kendall1938,Sievers1978} for lists of length $n$ as
\begin{equation}
\label{eq:kendall}
K_w(\sigma=\iota,\pi)=\sum_{1\leq i<j\leq n} \frac{w(i)+w(j)}{2}[\pi(i)>\pi(j)]
\end{equation}
where $[x]$ is equal to 1 if the condition $x$ is true and 0
otherwise.
The measure $K_w$ can be normalized to the interval $[0,1]$ as
\begin{equation}
\label{eq:norm-k-w}
k_{w} = \frac {K_w(\sigma,\pi)}{\sum_{\{i,j\in\sigma\cup\pi:i<j\}} \frac{w(i)+w(j)}{2}}
\end{equation}
where the value of the denominator is exactly the maximum value that
the numerator can reach when both lists are sorted but in opposite
orders.
Fig.~\ref{fig:kendall} shows the algorithms, written in the Python
programming language, to compute both the numerator and the
denominator of $k_w(\sigma,\pi)$ in Eq.~\ref{eq:norm-k-w} as well as
$k_w(\sigma,\pi)$ itself as their ratio. Using unity weights lead to the
unweighted versions of these algorithms.
\subsection{Examples}
\label{sec:examples}
For the sake of simplicity, let us assume that $w(i)=w$ for all $i$ in
this section. We will have three examples. Let us compute $S_w$,
$K_w$, $s_w$, and $k_w$ for each example.
\mypar{Example 1} Given $\sigma=(a,b,c,d,e)$ and $\pi = (a,b,c,d,e)$, we have
$\sigma' = (a,b,c,d,e) \sim (1,2,3,4,5)$ and $\pi' = (a,b,c,d,e) \sim
(1,2,3,4,5)$. Then,
\[
S_w = 0w=0\mbox{ and }K_w = 0w=0.
\]
Also,
\[
s_w = \frac{0w}{12w} = 0\mbox{ and }k_w = \frac{0w}{10w} = 0.
\]
\mypar{Example 2} Given $\sigma=(a,b,c,d,e)$ and $\pi = (e,d,c,b,a)$, we have
$\sigma' = (a,b,c,d,e) \sim (1,2,3,4,5)$ and $\pi' = (e,d,c,b,a) \sim
(5,4,3,2,1)$. Then,
\begin{align*}
S_w =& w(|1-5| + |2-4| + |3-3| + |4-2|+ |5-1|)\\
=& w(4+2+0+2+4)\\
=& 12w
\end{align*}
and
\begin{align*}
K_w =& w([5 > 4] + [5 > 3] + [5 > 2] + [5 > 1]\\
& + [4 > 3] + [4 > 2] + [4 > 1]\\
& + [3 > 2] + [3 > 1] + [2 > 1])\\
=& 10w.
\end{align*}
Also,
\[
s_w = \frac{12w}{12w} = 1\mbox{ and }k_w = \frac{10w}{10w} = 1.
\]
\mypar{Example 3} Given $\sigma=(a,b,c)$ and $\pi=(b,d,c,e)$, we first extend
them to full lists and replace elements by their ranks with $\sigma$ as
the reference list: $\sigma'=(a,b,c,d,e) \sim \sigma'=(1,2,3,4,5)$ and
$\pi'=(b,d,c,e,a) \sim \pi'=(2,4,3,5,1)$. Then,
\begin{align*}
S_w =& w(|1-2| + |2-4| + |3-3| + |4-5|+ |5-1|)\\
=& w(1 + 2 + 0 + 1 + 4)\\
=& 8w
\end{align*}
and
\begin{align*}
K_w =& w([2 > 1] + [4 > 3] + [4 > 1] + [3 > 1] + [5 > 1])\\
=& w(1 + 1 + 1 + 1 + 1)\\
=& 5w,
\end{align*}
where all the other comparisons return zero.
Then the normalized measures become
\[
s_w = \frac{8w}{12w} = \frac{2}{3}\approx 0.67\mbox{ and }k_w = \frac{5w}{10w} = \frac{1}{2}=0.5.
\]
Notice that $K_w\leq S_w \leq 2K_w$ because $5w\leq 8w\leq 10w$.
\section{Equivalence Between Measures}
\label{sec:equivalence}
We now prove that the Graham-Diaconis inequality between Spearman's
footrule and Kendall's tau is valid for weighted ranked lists
too. This inequality shows that these measures with or without weights
are within small constant multiples of each other, which is another
way of saying that these measures are equivalent~\cite{FaKuSi03}.
Denote by $S$ and $K$ the unweighted versions of $S_w$ and $K_w$,
respectively. The unweighted versions are obtained by setting each
weight in $S_w$ and $K_w$ to unity. This equivalence proof allows us
to use the simpler of these two measures, Spearman's footrule, as our
list similarity measure even for the weighted case.
For permutations, the equivalence between the unweighted versions $S$
and $K$ is well known from the following classical result~\cite{DiGr77}:
\begin{theorem}(Diaconis-Graham)
\label{th:DG77}
For every two permutations $\sigma$ and $\pi$
\begin{equation}
K(\sigma,\pi)\leq S(\sigma,\pi)\leq 2K(\sigma,\pi) .
\end{equation}
\end{theorem}
For a discussion on the equivalence of other unweighted list
similarity measures, see \cite{FaKuSi03}.
For weighted permutations, we generalize this result to the
equivalence between $S_w$ and $K_w$.
\begin{theorem}
\label{th:DDperm}
For every two permutations $\sigma$ and $\pi$,
\begin{equation}
K_w(\sigma,\pi)\leq S_w(\sigma,\pi)\leq 2K_w(\sigma,\pi)
\end{equation}
where every weight is a positive number.
\end{theorem}
Our proof below closely follows the notation and reasoning of the
original proof in \cite{DiGr77} and extends it to the weighted case.
\mypar{Proof} Before proving this theorem, we need the following
preliminary facts, which are the same as in \cite{DiGr77}. Note that
the element weights do not invalidate these facts.
Assume that all permutations below are defined on the same set,
where a permutation is a bijection (i.e., one-to-one and onto
function) from some set $X$ of size $n$ to $\{1, \dots, n\}$.
\mypar{Metric space} Both $S$ and $K$ are metrics, meaning that if
$d$ denotes one of these measures, then $d$ satisfies the metric
properties: $d(\sigma,\pi)\geq 0$ (i.e., non-negativity); $d(\sigma,\pi)=0$ if
and only if (iff) $\sigma=\pi$ (i.e., identity of indiscernible);
$d(\sigma,\pi)=d(\pi,\sigma)$ (i.e., symmetry); and, $d(\sigma,\pi)\leq
d(\sigma,\eta)+d(\eta,\pi)$ for some permutation $\eta$ (i.e., triangle
inequality).
\mypar{Right invariance} Both $S$ and $K$ are right invariant,
meaning if $d$ denotes one of these measures, then $d$ is right
invariant if a permutation $\eta$ exists such that
$d(\sigma,\pi)=d(\sigma\eta,\pi\eta)$. In particular,
$d({\bf \iota},\sigma)=d(\sigma^{-1},{\bf \iota})=d({\bf \iota},\sigma^{-1})$ where ${\bf \iota}$ stands for the
identify permutation on $X$ and $\sigma^{-1}$ is the permutation inverse
to $\sigma$ (i.e., $\sigma\A^{-1}={\bf \iota}$). To simplify the notation, we
abbreviate $d({\bf \iota},\sigma)$ by $d(\sigma)$, hence, $d(\sigma)=d(\sigma^{-1})$.
Now we come to the proof of the theorem. We divide the proof into
two parts. We prove first $S_w(\sigma)\leq 2K_w(\sigma)$ and then $K_w(\sigma)
\leq S_w(\sigma)$.
\mypar{The proof of $S_w(\sigma)\leq 2K_w(\sigma)$} Recall that
$K(\sigma)=K(\sigma^{-1})$ is the smallest number of pairwise adjacent
transpositions or swaps required to bring $\sigma$ to the identity
${\bf \iota}$. Note that bubble sort will make the same number of swaps to sort
its input list. Let $x_i$, $1\leq i\leq K(\sigma)$, be a sequence of
integers that indexes a sequence of transpositions transforming ${\bf \iota}$
to $\sigma_1$ to $\sigma_2$ to $\dots$ to $\sigma$. The $i$-th transposition
transforms $\sigma_i$ to $\sigma_{i+1}$ by interchanging $\sigma_i(x_i)$ to
$\sigma_i(x_i+1)$, i.e., the element at index $x_i$ and the next element
at index $x_i +1$. Without loss of generality, we may assume that
$\sigma_i(x_i)<\sigma_i(x_i+1)$. Consider the difference
\begin{align}
\label{eq:delta}
\Delta_{i+1}=&S_w(\sigma_{i+1})-S_w(\sigma_i)\\
=&(w(x_i+1)|x_i-\sigma_i(x_i+1)|+w(x_i)|x_i+1-\sigma_i(x_i)|)\nonumber\\
-&(w(x_i)|x_i-\sigma_i(x_i)|+w(x_i+1)|x_i+1-\sigma_i(x_i+1)|)\nonumber
\end{align}
where the contributions from the elements whose positions have not
changed, i.e., the elements at indices other than $x_i$ and $x_i+1$,
cancel out.
There are three possibilities, each of which removes the absolute
values in Eq.~\ref{eq:delta} to compute the difference.
\begin{itemize}
\item {\em Case 1.} If $\sigma_i(x_i)<\sigma_i(x_i+1)\leq x_i < x_i + 1$, then
$\Delta_{i+1}=-w(x_i+1)+w(x_i)$.
\item {\em Case 2.} If $x_i<x_i+1\leq \sigma_i(x_i)<\sigma_i(x_i+1)$, then
$\Delta_{i+1}=w(x_i+1)-w(x_i)$.
\item {\em Case 3.} If $\sigma_i(x_i)\leq x_i<x_i+1\leq\sigma_i(x_i+1)$, then
$\Delta_{i+1}=w(x_i+1)+w(x_i)$.
\end{itemize}
Thus $S_w(\sigma)=\sum_{i=1}^{K(\sigma)}\Delta_i\leq 2K_w(\sigma)$ because
$\Delta_{i+1}\leq 2((w(x_i+1)+w(x_i))/2)$ for each of the three
cases above. Note that the expression of $\Delta_{i+1}$ also
explains our choice of the additive weight aggregation in $K_w$.
\mypar{The proof of $K_w(\sigma) \leq S_w(\sigma)$}
To prove the left-hand side of the inequality, first denote the
inversion $\sigma(i)>\sigma(j)$ with $i<j$ by $[i;j]$. We then simplify
Eq.~\ref{eq:kendall} for $K_w(\sigma)$ as
\begin{equation}
\label{eq:Kw2}
K_w(\sigma)=\sum_{1\leq i<j\leq n} \frac{w(i)+w(j)}{2}[\sigma(i)>\sigma(j)]\nonumber
\end{equation}
or
\begin{align}
\label{eq:Kw}
2K_w(\sigma) & = & \sum_{1\leq i<j\leq n} w(i)[\sigma(i)>\sigma(j)] +\nonumber\\
& & \sum_{1\leq i<j\leq n} w(j)[\sigma(i)>\sigma(j)],
\end{align}
where $[x]$ is equal to 1 if the condition $x$ is true and 0
otherwise. Note that $2K_w$ is the sum of all inversions $[i;j]$ in
$\sigma$ and the weight for each inversion is added twice, once for
$\sigma(i)$ and another for $\sigma(j)$. For the proof, we will show that
$S_w$ upper-bounds the total number of inversions, hence, each term
in $2K_w$ separately.
Define two types of inversions: Type I and Type II. Call an inversion
$[i;j]$ a Type I inversion if $\sigma(i)\geq j$ and a Type II inversion if
$\sigma(i)\leq j$. Note that every inversion of $\sigma$ is either a Type I or
a Type II inversion or both. Similarly, the sum of Type I and Type II
inversions in $\sigma$ upper-bounds $K_w(\sigma)$.
For a fixed $i$, if $[i;k]$ is a Type I inversion, then we must have
$i<k\leq \sigma(i)$. Thus, the number of Type I inversions, or the
number of possible $k$, is at most $\sigma(i)-i$ and the total weight is
at most $w(i)(\sigma(i)-i)$. Similarly, if $[k;j]$ is a Type II
inversion, then we must have $\sigma(j)<\sigma(k)\leq j$. Thus, the number
of Type II inversions, or the number of possible $k$, is at most $j
- \sigma(j)$ and the total weight is at most $w(i)(j-\sigma(j))$. Then, it
follows that the first term in Eq.~\ref{eq:Kw} is at most the sum of
the number of Type I inversions and the number of Type II
inversions, which is further upper-bounded as
\begin{align}
\label{eq:temp}
\sum_{\sigma(i)\geq i} w(i)(\sigma(i)-i)+ \sum_{\sigma(j)\leq j} w(i)(j-\sigma(j)) & =
& \nonumber\\
\sum_{i} w(i)|\sigma(i)-i| & = & S_w(\sigma)\nonumber.
\end{align}
Similarly, for a fixed $j$, an argument similar to the case for $i$
and $w(i)$ can be carried out for the case for $j$ and $w(j)$ to prove
that the second term in Eq.~\ref{eq:Kw} is at most
$S_w(\sigma)$. Combined, these two arguments show that $2K_w(\sigma)\leq
2S_w(\sigma)$ or $K_w(\sigma)\leq S_w(\sigma)$.
QED.
We generalize this result to the equivalence between $S_w$ and $K_w$
for ranked partial lists.
\begin{theorem}
\label{th:DDpart}
For partial lists $\sigma$ and $\pi$ with the rank assignment for missing
elements as described in \S~\ref{sec:rank-def}.
\begin{equation}
K_w(\sigma,\pi)\leq S_w(\sigma,\pi)\leq 2K_w(\sigma,\pi)
\end{equation}
where every weight is a positive number.
\end{theorem}
\mypar{Proof} The proof directly follows from the way we assign ranks
of the missing elements in \S~\ref{sec:rank-def}. This is because the
resulting lists become permutations of each other. Thus,
Theorem~\ref{th:DDperm} applies. QED.
\section{Experimental Results}
\label{sec:results}
To provide more insight into these algorithms, we performed two sets
of experiments.
\begin{figure}
\centering
\includegraphics[width=0.7\textwidth]{dist.png}
\caption{Distribution of the ratio of Spearman's footrule to
Kendall's tau over all permutations of a 10-element unweighted
list.}
\label{fig:dist}
\end{figure}
The first set of experiments is about understanding the distribution
of the ratio of Spearman's footrule to Kendall's tau. We computed this
ratio over all permutations of a 10-element unweighted list. By
Theorem~\ref{th:DG77}, this ratio fits in the range from 1 to 2,
inclusive. Fig.~\ref{fig:dist} shows the outcome distribution with
orange lines marking the multiples of the standard deviation
(0.14). The distribution parameters are also shown in this
figure. Note that the median, mean, and mode are at or very close to
1.50, the middle value of the range; although this may almost imply a
balanced distribution, the distribution is actually slightly
right-skewed with a skewness of 0.42, which is also noticeable
visually.
\begin{figure}
\centering
\includegraphics[width=0.7\textwidth]{norm.png}
\caption{Distribution of the normalized Spearman's footrule and
Kendall's tau over all permutations of a 10-element unweighted
list.}
\label{fig:norm}
\end{figure}
The second set of experiments is about understanding the distributions
of the normalized measures. We again computed the normalized values
over all permutations of a 10-element unweighted list. The normalized
values fit in the range from 0 to 1, inclusive. Fig.~\ref{fig:norm}
shows the outcome distributions for both the normalized measures. As
is also obvious visually, the normalized Kendall's tau is distributed
in a balanced way with the median, mean, and mode are at the middle
value 0.50 of the range (with a standard deviation of 0.13) whereas
the normalized Spearman's footrule is distributed with a negative
skewness (i.e., left-skewed) of -0.18. The other parameters for the
normalized Spearman's footrule are 0.66, 0.66, and 0.68 for the
median, mean, and mode, respectively. The standard deviation is 0.14.
\section{Conclusions}
\label{sec:conclusions}
The field of similarity computation is more than a century old. In
this paper, we focus only on the similarity between two ranked
lists. Such lists may be full (permutations) or partial. For
permutations, the classical Graham-Diaconis inequality show their
equivalence using two well-known similarity measures: Spearman's
footrule and Kendall's tau.
In this paper, we consider ranked lists that may be partial and ranked
lists that may be weighted (with element weights) or both. We first
propose a rank assignment method to convert partial lists to
permutations. Next, we define weighted versions of Spearman's footrule
and Kendall's tau. Finally, we generalize the Graham-Diaconis
inequality to permutations with element weights. Due to the form of
our rank assignment, we also show that the same weighted
generalization applies to partial lists.
\section*{Acknowledgments}
I thank Paolo D'Alberto for a joint work that subsumed this work and
Ravi Kumar for discussions on the weighted form of Kendall's tau.
\bibliographystyle{plain}
|
{
"timestamp": "2018-04-17T02:11:52",
"yymm": "1804",
"arxiv_id": "1804.05420",
"language": "en",
"url": "https://arxiv.org/abs/1804.05420"
}
|
\section*{ACKNOWLEDGMENT}
{\footnotesize
This work has been partly funded by the Intelligence Advanced Research Projects Activity (IARPA). The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies, either expressed or implied, of IARPA, or the U.S. Government. The U.S. Government had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. The U.S. Government is authorized to reproduce and distribute reprints for governmental purposes notwithstanding any copyright annotation therein.
}
\addtolength{\textheight}{-5cm}
\bibliographystyle{ieeetr}
\section{Introduction}
Recent high-profile cyber attacks---the massive denial of service attack using Mirai botnet, infections of computers word-wide with WannaCry and Petya ransomware, the Equifax data breach---highlight the need for organizations to develop cyber crime defenses. Cyber threats are hard to identify and predict because the hackers that conduct these attacks often obfuscate their activity and intentions. However, they may still use publicly accessible forums discuss vulnerabilities and share tradecraft about how to exploit them. The behavior of the hacker community, as expressed in such venues, may provide insights into group's malicious intent. It has been shown that computational models based on various behavior learning theories can help in cyber security situational awareness \cite{dutt2013cyber}. While cyber situation awareness \cite{jajodia2009cyber, franke2014cyber} is critical for defending networks, it is focused on detecting cyber events.
In this paper, we describe a computational method that analyzes discussions on hacker forums to predict cyber attacks.
Opinion mining or sentiment analysis can be linked all the way back to Freud's 1901 paper on how slips of the tongue can reveal a person's hidden intentions \cite{freud19011960}. While sentiment analysis was originally developed in the field of linguistics and psychology, it has recently been applied to a number of other fields with the first seminal work in the computational sciences being Pang 2002 \cite{pang2002thumbs}. Historically, the context it has been applied to are social networks, comments (such as on news sites) and reviews (either for products or movies). In this work, we apply sentiment analysis to posts on Dark Web forums with the purpose of forecasting cyber attacks. The Dark Web consists of websites that are not indexed nor searchable by standard search engine and can only be accessed using a special browser service.
We further explore the link between community behavior and malicious activity. The connection between security and human behavior has been studied in designing new technology \cite{pfleeger2012leveraging}, here we look to reverse engineer by mapping the malicious events to hacker behavior. Social media has been shown to be a source of useful data on human behavior and used to predict real world events \cite{agarwal2015applying,asur2010predicting,kalampokis2013understanding}. Here, we inspect the ability of hacker forums to predict cyber events.
We consider each individual forum, applying sentiment analysis to each post in the forum. After computing a daily average per forum and a 7 day running average sentiment signal per forum, we test these signals against ground truth data. We determine some forums have significantly more predictive power and these isolated forums can beat the evaluation models in 36\% of the months under study using precision and recall of predictions within a 39-hour window of the event.
\section{Conclusions}
Malicious activity can be very devastating to national security, economies, businesses and personal lives. As such, cyber security professionals working with major organizations and nation states could use all the help they can get in preventing malicious activity. We present a methodology to predict malicious cyber events by exploiting malicious actor's behavior via sentiment analysis of posts on hacker forums. These forums on both Surface Web and Dark Web have some predictive power to be used as signals external to the network for forecasting attacks using time series models. Using ground truth data from two major organizations in the Defense Industrial Base across three different cyber event types, we show that sentiment signals can be more predictive than a baseline time series model. Additionally, they will often beat other state of the art external signals, in the 7 months under study across the 3 event types from the 2 organizations, sentiment signals performed the best 15 out of 42 times or 36\%. The signal parameters need to be tuned over significant historical data and the source forum could be shut off or taken down at any time; however, an automated implementation of this system would still be value added.
\section{Related Work}
Given the serious nature of cyber attacks, naturally there are a number of other research efforts to predict such attacks. As it relates to our efforts, the three main areas of research are sentiment analysis in cyber security, predictive methods for cyber attacks and leveraging dark web data in cyber security.
\subsection{Sentiment Analysis in Cyber Security}
The closest work which has applied sentiment analysis of hacker forums to cyber security is \cite{macdonald2015identifying}. While much research has investigated the specifics of cyber attacks, \cite{macdonald2015identifying} investigates the actual cyber actors via their communication activities. Their focus was the cyber-physical systems related to critical infrastructure and they developed an automated analysis tool to identify potential threats against such infrastructure. Despite recognizing that there are over 140 hacker forums on the public web, the authors chose only one forum to scrape the complete forum once. They leveraged the Open Discussion Forum Crawler to do the scrapping and then used OpenNLP to tag parts of speech, filtering on nouns. Those nouns were cross referenced with three list of malicious keywords to identify posts whose sentiment would be determined with SentiStrength. Contextual analysis of keyword pairings with sentiment scores allowed them to confirm current statistics about critical infrastructure cyber attacks. The main differences illustrated in our work is that we use looked at over 100 forums, not just one from both the Dark Web and Surface Web. In collecting posts for over a two year period, we found the sentiment of all posts applying Vader and LIWC for sentiment in addition to just SentiStrength. Furthermore, we were able to model our data against ground truth events from companies making our approach predictive in nature.
BiSAL \cite{al2015bisal} did sentiment analysis in English and Arabic on Dark Web forums with slight modification to cyber security terms. Other work such as \cite{chen2008sentiment} use sentiment in measuring radicalization. Remaining research in sentiment analysis, not specific to cyber security was presented earlier.
\subsection{Predicting Cyber Attack}
The issue of predicting cyber attacks is not new and their has been a considerable research effort in this field. The efforts split along two categories, using network traffic or non-network traffic. Forecasting methods such as \cite{park2012cyber,pontes2011applying,leslie2017statistical} analyze network traffic. Where \cite{park2012cyber} is specific to predicting attacks using IPV4 packet traffic, and \cite{pontes2011applying} looks at various network sensors at different layers to prevent unwanted Internet traffic, whereas \cite{leslie2017statistical} combines DNS traffic with security metadata such as number of policy violations and the number of clients in the network. Many researchers such as \cite{zhang2015predicting} based cyber prediction on open source information. They use the National Vulnerability Database. They highlight the difficulty in using public sources for building effective models. Other work has focused on detecting cyber bullying using graph detection models \cite{nahar2012sentiment} with success, but is limited in malicious activity and not a predictive model.
The closest to our research is Gandotra et al \cite{comptech} who outlined a number cyber prediction efforts using statistical modeling and algorithmic modeling. They highlight several significant challenges that we tried to address. The first challenge is that open source ground truth is incomplete and should be compiled from multiple sources and analysis doesn't scale to real world scenarios. We were able to get ground truth data from 2 companies that operate in the defense industrial base, this ground truth is across three different attack vectors and is over a two year time period. The additional challenges in \cite{comptech} focus on the volume, speed and heterogeneity of network data which we avoid since we are attempting to prevent cyber events specifically with non-network data. They also present two modeling approaches of statistical modeling and algorithmic modeling. We used statistical models not unlike what they present as classical time series models with auto-regressive, integrated moving average with historical data and external signals.
\subsection{Dark Web Research}
There has been a lot of research recently concerning the Dark Web or websites not indexed by major search engines. Typically the Dark Web refers to the TOR \cite{dingledine2004tor} network which is only accessible via specialized browsers. It has been shown by \cite{nunes2016darknet} that from an overall cyber security threat perspective, the Dark Web provides a valuable source of information for malicious activity. They developed a system that scrapes hacker forum and marketplace sites on the Dark Web to develop threat warnings for cyber defenders. We leverage the same data source but perform sentiment analysis to not only predict future threats, but to predict actual attacks. They also leverage the Deep Net which is the portion of the Surface Web not indexed by standard search engines.
While not using sentiment analysis, \cite{lacey2015s} offers insight to the trust establishment between participants in Dark Web forums. There may be behavioral patterns of malicious actors that provide insight to future activity. Dark Web conversations were shown to provide earlier insights than Surface Web conversations by \cite{sapienza2017early} indicating potential predictive power for cyber events. \cite{sapienza2017early} highlight two cases with a major DDoS attack and the Mirai attack. There may also be early insights on the Surface Web in many of the social media sites as illustrated in \cite{sabottke2015vulnerability}. Our work focused only on forums where it was likely that computer security items would be discussed, but does contain a mix of Dark Web and Surface Web. There has been work using natural language processing on Dark Web text for predictive method such as \cite{tavabi2018dark}. Other predictive approaches such as Cyber Attacker Model Profile (CAMP) \cite{watters2012characterising}, focus on the macro level of a country and financial cyber crimes, where we look at a wider range of malicious activity against specific target organizations.
\section{Results}
After generating ARIMAX models with each potential signal, they were scored as mentioned above for each month from July 2017 to January 2018. The following tables show the results for the months under study. By month, you can see the number of actual ground truth events (Evt), the number of warnings generated by each signal (Warn), and the precision (P), recall (R) and F1 score for each. The table is sorted by largest F1 score for each month with only the top five signals listed. Signals generated by sentiment analysis that were part of the top five for each month are highlighted in light blue.
\subsection{Organization A}
Table \ref{tab:output1} shows Organization A's endpoint-malware where sentiment signals dominated July, September and November and did reasonable well in the remaining months. Every month a sentiment signal beat at least on evaluation model. Malicious-Destination (Table \ref{tab:output2}) had periodic performance July, September, November and January but the case is not as strong as Endpoint-Malware. Lastly, Table \ref{tab:output3} shows Malicious-Email results which illustrate that sentiment signals did well in July to September with waning results for the later months. Upon further inspection this is believed to be due to some key forums going offline toward the end of the year.
\begin{table}[!t]
\centering
\caption{Results from Organization A's Endpoint-Malware
\label{tab:output1}}
\begin{tabular}{llccccc}%
\hline%
\textbf{Month}&\textbf{Evt}&\textbf{Warn}&\textbf{Signal}&\textbf{P}&\textbf{R}&\textbf{F1}\\%
\hline%
\rowcolor{LightCyan}July & 15 & 14 & forum211-Senti & 0.57 & 0.53 & 0.55 \\
\rowcolor{LightCyan}July & 15 & 29 & forum196-LIWC & 0.41 & 0.80 & 0.55 \\
\rowcolor{LightCyan}July & 15 & 27 & forum89-Senti & 0.41 & 0.73 & 0.52 \\
\rowcolor{LightCyan}July & 15 & 12 & forum111-LIWC & 0.58 & 0.47 & 0.52 \\
July & 15 & 9 & baseline & 0.67 & 0.40 & 0.50 \\
\hline
August & 19 & 14 & baseline & 0.71 & 0.53 & 0.61 \\
\rowcolor{LightCyan}August & 19 & 11 & forum111-LIWC & 0.82 & 0.47 & 0.60 \\
\rowcolor{LightCyan}August & 19 & 35 & forum8-Vader & 0.46 & 0.84 & 0.59 \\
August & 19 & 8 & daywise-baserate & 1.00 & 0.42 & 0.59 \\
\rowcolor{LightCyan}August & 19 & 23 & forum230-Senti & 0.52 & 0.63 & 0.57 \\
\hline
\rowcolor{LightCyan}September & 18 & 16 & forum111LIWC & 0.69 & 0.61 & 0.65 \\
\rowcolor{LightCyan}September & 18 & 32 & forum250LIWC & 0.50 & 0.89 & 0.64 \\
\rowcolor{LightCyan}September & 18 & 35 & forum211vader & 0.46 & 0.89 & 0.60 \\
\rowcolor{LightCyan}September & 18 & 41 & forum147LIWC & 0.41 & 0.94 & 0.58 \\
\rowcolor{LightCyan}September & 18 & 41 & forum194LIWC & 0.41 & 0.94 & 0.58 \\
\hline
October & 6 & 14 & daywise-baserate & 0.29 & 0.67 & 0.40 \\
October & 6 & 35 & baseline & 0.17 & 1.00 & 0.29 \\
\rowcolor{LightCyan}October & 6 & 29 & forum8vader & 0.17 & 0.83 & 0.29 \\
\rowcolor{LightCyan}October & 6 & 37 & forum111LIWC & 0.16 & 1.00 & 0.28 \\
\rowcolor{LightCyan}October & 6 & 43 & forum211vader & 0.14 & 1.00 & 0.24 \\
\hline
\rowcolor{LightCyan}November & 27 & 38 & forum6senti & 0.63 & 0.89 & 0.74 \\
\rowcolor{LightCyan}November & 27 & 42 & forum147LIWC & 0.60 & 0.93 & 0.72 \\
\rowcolor{LightCyan}November & 27 & 40 & forum111LIWC & 0.60 & 0.89 & 0.72 \\
\rowcolor{LightCyan}November & 27 & 41 & forum211senti & 0.59 & 0.89 & 0.71 \\
\rowcolor{LightCyan}November & 27 & 43 & forum121LIWC & 0.56 & 0.89 & 0.69 \\
\hline
December & 13 & 18 & arimax & 0.33 & 0.46 & 0.39 \\
December & 13 & 16 & dark-mentions & 0.31 & 0.38 & 0.34 \\
\rowcolor{LightCyan}December & 13 & 80 & forum121LIWC & 0.16 & 1.00 & 0.28 \\
\rowcolor{LightCyan}December & 13 & 73 & forum194LIWC & 0.16 & 0.92 & 0.28 \\
December & 13 & 10 & deep-exploit & 0.30 & 0.23 & 0.26 \\
\hline
January & 1 & 15 & dark-mentions & 0.07 & 1.00 & 0.13 \\
\rowcolor{LightCyan}January & 1 & 37 & forum6senti & 0.03 & 1.00 & 0.05 \\
\rowcolor{LightCyan}January & 1 & 61 & forum147LIWC & 0.02 & 1.00 & 0.03 \\
January & 1 & 64 & baseline & 0.02 & 1.00 & 0.03 \\
January & 1 & 19 & arimax & 0.00 & 0.00 & 0.00 \\
\hline%
\end{tabular}%
\end{table}
\begin{table}[!t]
\centering
\caption{Results from Organization A's Malicious-Destination
\label{tab:output2}}
\begin{tabular}{llllccc}%
\hline%
\textbf{Month}&\textbf{Evt}&\textbf{Warn}&\textbf{Signal}&\textbf{P}&\textbf{R}&\textbf{F1}\\%
\hline%
July & 4 & 5 & baseline & 0.40 & 0.50 & 0.44 \\
July & 4 & 3 & daywise-baserate & 0.33 & 0.25 & 0.29 \\
July & 4 & 17 & dark-mentions & 0.12 & 0.50 & 0.19 \\
\rowcolor{LightCyan}July & 4 & 42 & forum266-LIWC & 0.05 & 0.50 & 0.09 \\
July & 4 & 0 & arimax & 0.00 & 0.00 & 0.00 \\
\hline
August & 10 & 6 & baseline & 1.00 & 0.60 & 0.75 \\
August & 10 & 10 & daywise-baserate & 0.60 & 0.60 & 0.60 \\
August & 10 & 8 & dark-mentions & 0.50 & 0.40 & 0.44 \\
August & 10 & 0 & arimax & 0.00 & 0.00 & 0.00 \\
August & 10 & 0 & deep-exploit & 0.00 & 0.00 & 0.00 \\
\hline
\rowcolor{LightCyan}September & 4 & 15 & forum194LIWC & 0.20 & 0.75 & 0.32 \\
\rowcolor{LightCyan}September & 4 & 15 & forum210LIWC & 0.20 & 0.75 & 0.32 \\
\rowcolor{LightCyan}September & 4 & 15 & forum264LIWC & 0.20 & 0.75 & 0.32 \\
\rowcolor{LightCyan}September & 4 & 15 & forum6senti & 0.20 & 0.75 & 0.32 \\
\rowcolor{LightCyan}September & 4 & 15 & forum194LIWC & 0.20 & 0.75 & 0.32 \\
\hline
October & 2 & 0 & arimax & 0.00 & 0.00 & 0.00 \\
October & 2 & 0 & dark-mentions & 0.00 & 0.00 & 0.00 \\
October & 2 & 5 & daywise-baserate & 0.00 & 0.00 & 0.00 \\
October & 2 & 0 & deep-exploit & 0.00 & 0.00 & 0.00 \\
\hline
November & 1 & 5 & daywise-baserate & 0.20 & 1.00 & 0.33 \\
\rowcolor{LightCyan}November & 1 & 6 & forum111LIWC & 0.17 & 1.00 & 0.29 \\
\rowcolor{LightCyan}November & 1 & 6 & forum147LIWC & 0.17 & 1.00 & 0.29 \\
\rowcolor{LightCyan}November & 1 & 30 & forum210senti & 0.03 & 1.00 & 0.06 \\
November & 1 & 0 & arimax & 0.00 & 0.00 & 0.00 \\
\hline
December & 1 & 10 & daywise-baserate & 0.10 & 1.00 & 0.18 \\
December & 1 & 11 & dark-mentions & 0.09 & 1.00 & 0.17 \\
December & 1 & 0 & arimax & 0.00 & 0.00 & 0.00 \\
December & 1 & 0 & deep-exploit & 0.00 & 0.00 & 0.00 \\
\hline
\rowcolor{LightCyan}January & 2 & 24 & forum111LIWC & 0.08 & 1.00 & 0.15 \\
January & 2 & 0 & arimax & 0.00 & 0.00 & 0.00 \\
January & 2 & 10 & dark-mentions & 0.00 & 0.00 & 0.00 \\
January & 2 & 9 & daywise-baserate & 0.00 & 0.00 & 0.00 \\
January & 2 & 0 & deep-exploit & 0.00 & 0.00 & 0.00 \\
\hline%
\end{tabular}%
\end{table}
\begin{table}[!t]
\centering
\caption{Results from Organization A's Malicious-Email
\label{tab:output3}}
\begin{tabular}{llccccc}
\hline%
\textbf{Month}&\textbf{Evt}&\textbf{Warn}&\textbf{Signal}&\textbf{P}&\textbf{R}&\textbf{F1}\\%
\hline%
\rowcolor{LightCyan}July & 26 & 21 & forum210-LIWC & 0.76 & 0.62 & 0.68 \\
\rowcolor{LightCyan}July & 26 & 27 & forum250-LIWC & 0.67 & 0.69 & 0.68 \\
\rowcolor{LightCyan}July & 26 & 19 & forum147-LIWC & 0.74 & 0.54 & 0.62 \\
\rowcolor{LightCyan}July & 26 & 36 & forum159-Senti & 0.53 & 0.73 & 0.61 \\
\rowcolor{LightCyan}July & 26 & 17 & forum28-LIWC & 0.76 & 0.50 & 0.60 \\
\hline
\rowcolor{LightCyan}August & 11 & 17 & forum179-Vader & 0.59 & 0.91 & 0.71 \\
\rowcolor{LightCyan}August & 11 & 15 & forum250-LIWC & 0.60 & 0.82 & 0.69 \\
August & 11 & 7 & daywise-baserate & 0.86 & 0.55 & 0.67 \\
\rowcolor{LightCyan}August & 11 & 18 & forum210-Senti & 0.50 & 0.82 & 0.62 \\
\rowcolor{LightCyan}August & 11 & 25 & forum159-Senti & 0.44 & 1.00 & 0.61 \\
\hline
\rowcolor{LightCyan}September & 15 & 36 & forum264LIWC & 0.36 & 0.87 & 0.51 \\
September & 15 & 17 & daywise-baserate & 0.47 & 0.53 & 0.50 \\
\rowcolor{LightCyan}September & 15 & 18 & forum210senti & 0.44 & 0.53 & 0.48 \\
\rowcolor{LightCyan}September & 15 & 45 & forum147LIWC & 0.31 & 0.93 & 0.47 \\
\rowcolor{LightCyan}September & 15 & 46 & forum6senti & 0.28 & 0.87 & 0.43 \\
\hline
October & 11 & 14 & daywise-baserate & 0.50 & 0.64 & 0.56 \\
October & 11 & 8 & deep-exploit & 0.50 & 0.36 & 0.42 \\
\rowcolor{LightCyan}October & 11 & 42 & forum264LIWC & 0.17 & 0.64 & 0.26 \\
\rowcolor{LightCyan}October & 11 & 51 & forum194LIWC & 0.16 & 0.73 & 0.26 \\
\rowcolor{LightCyan}October & 11 & 102 & forum8vader & 0.11 & 1.00 & 0.19 \\
\hline
November & 50 & 16 & daywise-baserate & 0.69 & 0.22 & 0.33 \\
November & 50 & 4 & deep-exploit & 0.75 & 0.06 & 0.11 \\
November & 50 & 0 & arimax & 0.00 & 0.00 & 0.00 \\
November & 50 & 0 & dark-mentions & 0.00 & 0.00 & 0.00 \\
\hline
December & 17 & 22 & daywise-baserate & 0.55 & 0.71 & 0.62 \\
December & 17 & 10 & deep-exploit & 0.80 & 0.47 & 0.59 \\
December & 17 & 5 & dark-mentions & 0.80 & 0.24 & 0.36 \\
December & 17 & 0 & arimax & 0.00 & 0.00 & 0.00 \\
\hline
January & 40 & 18 & daywise-baserate & 0.94 & 0.43 & 0.59 \\
January & 40 & 8 & deep-exploit & 0.75 & 0.15 & 0.25 \\
January & 40 & 6 & dark-mentions & 0.83 & 0.13 & 0.22 \\
January & 40 & 0 & arimax & 0.00 & 0.00 & 0.00 \\
\hline%
\end{tabular}%
\end{table}
\subsection{Organization B}
Table \ref{tab:output4} shows that sentiment signals do best for July and October for Malicious-Destination. While baseline and daywise-baserate dominate the other months, sentiment signals perform better than the other evaluation models.
Similar to Organization A, the Malicious-Destination for Organization B (Table \ref{tab:output5}) does the best early in July in August and moderately well in September to November until degrading to below all evaluation models in December and January. This may be due the few number of events and perhaps sentiment signals do not perform the best under low frequency conditions. The performance for Malicious-Email (Table \ref{tab:output6}) is oddly cyclical; however, sentiment signals dominated December and beat at least one evaluation model for every month.
\begin{table}[!t]
\centering
\caption{Results from Organization B's Endpoint-Malware
\label{tab:output4}}
\begin{tabular}{llccccc}
\hline%
\textbf{Month}&\textbf{Evt}&\textbf{Warn}&\textbf{Signal}&\textbf{P}&\textbf{R}&\textbf{F1}\\%
\hline%
\rowcolor{LightCyan}July & 18 & 47 & forum264LIWC & 0.38 & 1.00 & 0.55 \\
\rowcolor{LightCyan}July & 18 & 50 & forum250LIWC & 0.36 & 1.00 & 0.53 \\
July & 18 & 43 & baseline & 0.37 & 0.89 & 0.52 \\
\rowcolor{LightCyan}July & 18 & 35 & forum8senti & 0.37 & 0.72 & 0.49 \\
\rowcolor{LightCyan}July & 18 & 50 & forum111LIWC & 0.32 & 0.89 & 0.47 \\
\hline
August & 28 & 39 & baseline & 0.67 & 0.93 & 0.78 \\
\rowcolor{LightCyan}August & 28 & 31 & forum264LIWC & 0.65 & 0.71 & 0.68 \\
\rowcolor{LightCyan}August & 28 & 32 & forum121LIWC & 0.63 & 0.71 & 0.67 \\
\rowcolor{LightCyan}August & 28 & 35 & forum211vader & 0.60 & 0.75 & 0.67 \\
\rowcolor{LightCyan}August & 28 & 33 & forum194LIWC & 0.61 & 0.71 & 0.66 \\
\hline
September & 31 & 40 & baseline & 0.60 & 0.77 & 0.68 \\
\rowcolor{LightCyan}September & 31 & 38 & forum210senti & 0.61 & 0.74 & 0.67 \\
\rowcolor{LightCyan}September & 31 & 37 & forum121LIWC & 0.57 & 0.68 & 0.62 \\
\rowcolor{LightCyan}September & 31 & 46 & forum219vader & 0.50 & 0.74 & 0.60 \\
\rowcolor{LightCyan}September & 31 & 30 & forum194LIWC & 0.60 & 0.58 & 0.59 \\
\hline
\rowcolor{LightCyan}October & 53 & 44 & forum210LIWC & 0.77 & 0.64 & 0.70 \\
October & 53 & 47 & baseline & 0.74 & 0.66 & 0.70 \\
\rowcolor{LightCyan}October & 53 & 41 & forum264LIWC & 0.78 & 0.60 & 0.68 \\
\rowcolor{LightCyan}October & 53 & 39 & forum250LIWC & 0.74 & 0.55 & 0.63 \\
\rowcolor{LightCyan}October & 53 & 40 & forum8vader & 0.73 & 0.55 & 0.62 \\
\hline
November & 37 & 52 & daywise-baserate & 0.62 & 0.86 & 0.72 \\
\rowcolor{LightCyan}November & 37 & 49 & forum121LIWC & 0.57 & 0.76 & 0.65 \\
\rowcolor{LightCyan}November & 37 & 53 & forum147LIWC & 0.55 & 0.78 & 0.64 \\
\rowcolor{LightCyan}November & 37 & 50 & forum111LIWC & 0.56 & 0.76 & 0.64 \\
\rowcolor{LightCyan}November & 37 & 50 & forum194LIWC & 0.56 & 0.76 & 0.64 \\
\hline
December & 35 & 30 & daywise-baserate & 0.67 & 0.57 & 0.62 \\
December & 35 & 27 & baseline & 0.63 & 0.49 & 0.55 \\
\rowcolor{LightCyan}December & 35 & 23 & forum250LIWC & 0.65 & 0.43 & 0.52 \\
\rowcolor{LightCyan}December & 35 & 28 & forum194LIWC & 0.57 & 0.46 & 0.51 \\
\rowcolor{LightCyan}December & 35 & 29 & forum147LIWC & 0.55 & 0.46 & 0.50 \\
\hline
January & 43 & 42 & baseline & 0.60 & 0.58 & 0.59 \\
January & 43 & 37 & daywise-baserate & 0.59 & 0.51 & 0.55 \\
\rowcolor{LightCyan}January & 43 & 35 & forum219vader & 0.60 & 0.49 & 0.54 \\
\rowcolor{LightCyan}January & 43 & 37 & forum111LIWC & 0.57 & 0.49 & 0.53 \\
\rowcolor{LightCyan}January & 43 & 37 & forum147LIWC & 0.57 & 0.49 & 0.53 \\
\hline%
\end{tabular}%
\end{table}
\begin{table}[!t]
\centering
\caption{Results from Organization B's Malicious-Destination
\label{tab:output5}}
\begin{tabular}{llccccc}
\hline%
\textbf{Month}&\textbf{Evt}&\textbf{Warn}&\textbf{Signal}&\textbf{P}&\textbf{R}&\textbf{F1}\\%
\hline%
\rowcolor{LightCyan}July & 6 & 8 & forum130vader & 0.63 & 0.83 & 0.71 \\
\rowcolor{LightCyan}July & 6 & 8 & forum8senti & 0.63 & 0.83 & 0.71 \\
\rowcolor{LightCyan}July & 6 & 8 & forum111LIWC & 0.50 & 0.67 & 0.57 \\
\rowcolor{LightCyan}July & 6 & 12 & forum194LIWC & 0.42 & 0.83 & 0.56 \\
\rowcolor{LightCyan}July & 6 & 9 & forum210senti & 0.44 & 0.67 & 0.53 \\
\hline
\rowcolor{LightCyan}August & 8 & 6 & forum210senti & 0.67 & 0.50 & 0.57 \\
August & 8 & 17 & daywise-baserate & 0.35 & 0.75 & 0.48 \\
\rowcolor{LightCyan}August & 8 & 13 & forum211senti & 0.38 & 0.63 & 0.48 \\
\rowcolor{LightCyan}August & 8 & 5 & forum210LIWC & 0.60 & 0.38 & 0.46 \\
\rowcolor{LightCyan}August & 8 & 21 & forum8vader & 0.29 & 0.75 & 0.41 \\
\hline
September & 6 & 11 & daywise-baserate & 0.55 & 1.00 & 0.71 \\
\rowcolor{LightCyan}September & 6 & 9 & forum210LIWC & 0.56 & 0.83 & 0.67 \\
\rowcolor{LightCyan}September & 6 & 10 & forum250LIWC & 0.30 & 0.50 & 0.37 \\
\rowcolor{LightCyan}September & 6 & 11 & forum121LIWC & 0.27 & 0.50 & 0.35 \\
\rowcolor{LightCyan}September & 6 & 1 & forum147LIWC & 1.00 & 0.17 & 0.29 \\
\hline
October & 9 & 8 & daywise-baserate & 0.25 & 0.22 & 0.24 \\
\rowcolor{LightCyan}October & 9 & 2 & forum121LIWC & 0.50 & 0.11 & 0.18 \\
\rowcolor{LightCyan}October & 9 & 114 & forum210senti & 0.03 & 0.33 & 0.05 \\
October & 9 & 0 & arimax & 0.00 & 0.00 & 0.00 \\
October & 9 & 0 & dark-mentions & 0.00 & 0.00 & 0.00 \\
\hline
November & 4 & 14 & daywise-baserate & 0.29 & 1.00 & 0.44 \\
\rowcolor{LightCyan}November & 4 & 5 & forum210LIWC & 0.20 & 0.25 & 0.22 \\
\rowcolor{LightCyan}November & 4 & 21 & forum219vader & 0.10 & 0.50 & 0.16 \\
\rowcolor{LightCyan}November & 4 & 9 & forum211vader & 0.11 & 0.25 & 0.15 \\
\rowcolor{LightCyan}November & 4 & 13 & forum210senti & 0.08 & 0.25 & 0.12 \\
\hline
December & 3 & 12 & daywise-baserate & 0.17 & 0.67 & 0.27 \\
December & 3 & 0 & arimax & 0.00 & 0.00 & 0.00 \\
December & 3 & 0 & dark-mentions & 0.00 & 0.00 & 0.00 \\
December & 3 & 0 & deep-exploit & 0.00 & 0.00 & 0.00 \\
\hline
January & 5 & 18 & daywise-baserate & 0.22 & 0.80 & 0.35 \\
January & 5 & 0 & arimax & 0.00 & 0.00 & 0.00 \\
January & 5 & 0 & dark-mentions & 0.00 & 0.00 & 0.00 \\
January & 5 & 0 & deep-exploit & 0.00 & 0.00 & 0.00 \\
\hline%
\end{tabular}%
\end{table}
\begin{table}[!t]
\centering
\caption{Results from Organization B's Malicious-Email
\label{tab:output6}}
\begin{tabular}{llccccc}
\hline%
\textbf{Month}&\textbf{Evt}&\textbf{Warn}&\textbf{Signal}&\textbf{P}&\textbf{R}&\textbf{F1}\\%
\hline%
\rowcolor{LightCyan}July & 24 & 49 & forum210LIWC & 0.33 & 0.67 & 0.44 \\
\rowcolor{LightCyan}July & 24 & 56 & forum210senti & 0.30 & 0.71 & 0.43 \\
July & 24 & 75 & baseline & 0.23 & 0.71 & 0.34 \\
July & 24 & 81 & daywise-baserate & 0.21 & 0.71 & 0.32 \\
\rowcolor{LightCyan}July & 24 & 81 & forum130vader & 0.21 & 0.71 & 0.32 \\
\hline
\rowcolor{LightCyan}August & 57 & 55 & forum111LIWC & 0.55 & 0.53 & 0.54 \\
August & 57 & 70 & baseline & 0.49 & 0.60 & 0.54 \\
August & 57 & 91 & daywise-baserate & 0.43 & 0.68 & 0.53 \\
\rowcolor{LightCyan}August & 57 & 107 & forum147LIWC & 0.39 & 0.74 & 0.51 \\
\rowcolor{LightCyan}August & 57 & 153 & forum6senti & 0.33 & 0.88 & 0.48 \\
\hline
September & 179 & 70 & daywise-baserate & 0.76 & 0.30 & 0.43 \\
\rowcolor{LightCyan}September & 179 & 102 & forum210senti & 0.58 & 0.33 & 0.42 \\
\rowcolor{LightCyan}September & 179 & 180 & forum210LIWC & 0.40 & 0.40 & 0.40 \\
\rowcolor{LightCyan}September & 179 & 100 & forum147LIWC & 0.54 & 0.30 & 0.39 \\
September & 179 & 76 & baseline & 0.57 & 0.24 & 0.34 \\
\hline
October & 71 & 125 & daywise-baserate & 0.50 & 0.87 & 0.63 \\
October & 71 & 118 & baseline & 0.49 & 0.82 & 0.61 \\
\rowcolor{LightCyan}October & 71 & 90 & forum211senti & 0.53 & 0.68 & 0.60 \\
\rowcolor{LightCyan}October & 71 & 142 & forum194LIWC & 0.44 & 0.89 & 0.59 \\
\rowcolor{LightCyan}October & 71 & 150 & forum210senti & 0.42 & 0.89 & 0.57 \\
\hline
November & 426 & 104 & daywise-baserate & 0.67 & 0.16 & 0.26 \\
\rowcolor{LightCyan}November & 426 & 205 & forum264LIWC & 0.39 & 0.19 & 0.25 \\
November & 426 & 118 & baseline & 0.55 & 0.15 & 0.24 \\
\rowcolor{LightCyan}November & 426 & 251 & forum210LIWC & 0.31 & 0.18 & 0.23 \\
\rowcolor{LightCyan}November & 426 & 579 & forum210senti & 0.20 & 0.27 & 0.23 \\
\hline
\rowcolor{LightCyan}December & 51 & 69 & forum210LIWC & 0.30 & 0.41 & 0.35 \\
\rowcolor{LightCyan}December & 51 & 329 & forum147LIWC & 0.09 & 0.55 & 0.15 \\
\rowcolor{LightCyan}December & 51 & 313 & forum111LIWC & 0.08 & 0.51 & 0.14 \\
\rowcolor{LightCyan}December & 51 & 249 & forum194LIWC & 0.08 & 0.41 & 0.14 \\
\rowcolor{LightCyan}December & 51 & 284 & forum211senti & 0.08 & 0.45 & 0.14 \\
\hline
January & 10 & 12 & deep-exploit & 0.25 & 0.30 & 0.27 \\
January & 10 & 103 & daywise-baserate & 0.10 & 1.00 & 0.18 \\
January & 10 & 186 & baseline & 0.05 & 1.00 & 0.10 \\
\rowcolor{LightCyan}January & 10 & 226 & forum111LIWC & 0.04 & 1.00 & 0.08 \\
\hline%
\end{tabular}%
\end{table}
\section{Data}
\subsection{Hacker Forum Texts}
We look at hacking forums from both the Surface Web and the Dark Web from 1 January 2016 to 31 January 2018. The Dark Web refers to sites accessible through The Onion Router private network platform \cite{dingledine2004tor}. The Surface Web refers to the World Wide Web accessible through standard browsers. In this paper, we focus only on English posts from 113 forums which were identified based on cyber security keywords consisting of 432,060 posts. The text from these forums were accessed using the methods proposed in \cite{robertson2017darkweb,nunes2016darknet}.
\subsection{Ground truth data}
We use ground truth data of cyber attacks from 2 major organizations in the Defense Industrial Base (DIB) industry. Henceforth, we will refer to them as Organization A and Organization B for anonymity. The ground truth comprises 3 event types:
\begin{itemize}
\item \textit{endpoint-malware:} a malicious software installation such as ransomware, spyware and adware is discovered on a company endpoint device.
\item \textit{malicious-destination} a visit by a user to a URL or IP address that is malicious in nature or a compromised website.
\item \textit{malicious-email} receipt of an email that contains a malicious email attachment and/or a link to a known malicious destination.
\end{itemize}
\section{Methodology}
In this section we document the methodology used and process workflow from the data processing to signal generation through warning generation and signal testing. Three cases studies are used to illustrate the process via example and Figure X provides a visual reference.
\subsection{Processing the Data}
Working with researchers at Arizona State University, we were able to develop a database of posts from forums on both the Dark Web and Surface Web which discuss computer security and network vulnerability topics. To protect the future utility of these sources, each forum has been coded with a number (forumid) from 1 to 350. The data consist of the forumid, date the post was made, and the text of the post. The data in this study was from 1 January 2016 to 31 January 2018. The data was collected by ASU and we used an API to pull and store the data in a local server and access it via Apache Lucene's Elastic Search engine.
\subsection{Evaluating Sentiment Analysis}
After a review of the sentiment analysis methods in SentiBench \cite{sentibench}, we decided to use Vader\cite{vader}, SentiStrenght\cite{senti} and LIWC15\cite{pennebaker2001linguistic}. For social networks, VADER and LIWC15 were found to be the best method for 3-class classification and SentiStrength was the winner for 2-class classification. \cite{sentibench} These three methods were used because they Vader has a Python module, SentiStrenght has a Java implementation and LIWC15 is a stand-alone program.
\subsection{Computing Daily Averages}
A sentiment score for each forum post was computed using the three sentiment methods outlined above. Since there can be multiple posts on a forum for a day, we characterization the overall sentiment of the day with a daily average. There can be a wide range of sentiment scores for any given day, especially if there are a lot of posts from on a popular forum. In order to understand the trend of sentiment over time, we compute running averages.
\subsection{Computing Running Averages}
A running daily average was computed in order to assess the trend of sentiment over time. The more days in the running average, the smoother the curve and the harder to detect a change. Whereas no using a running average or making it only 1 or 2 days would have many jump discontinuities and swings. We looked at adjusting the running average from 1 to 30 days and settled on 7 days primarily because that was our original prediction window. Figure \ref{fig:avgs} shows the average F1 score various signals computed with running averages of 3, 7, 10 and 14 days.
\begin{figure}[!t]
\includegraphics[width=\linewidth]{images/day_avg.png}
\caption{Average F1 Scores by Signal using Different Running Averages}
\label{fig:avgs}
\end{figure}
\subsection{Standardizing the Score}
To make the 3 sentiment scores more comparable, their scores were standardized. As previously mentioned, VADER generates sentiment scores on a scale of 0 to 1, SentiStrength goes from -4 to 4, and LIWC goes from 0 to 100 for Tone. While standardizing the scores do not affect the correlation any forum would have with the ground truth from our target organizations, it will be necessary when we potentially combine signals from various forums and sentiment methods to find more powerful predictors.
\subsection{Compute Correlations to Find Potential Signals}
As previously mention, we have ground truth events from 2 defense industrial base organizations of 3 different cyber event types. The event types are endpoint-malware, malicious-destination and malicious-email. Correlations were computed between all forum-sentiments against all event types from both organizations. Additionally, since we are looking for predictive signals, we computed correlations with a negative lag from 0 to 30 days with a lag of -30 meaning offset the sentiment signal 30 days before the organization's event occurrence. A number of signals stood out as being more correlated than others against certain event types as seen in Figure \ref{tab:corr}. This shows the LIWC sentiment on Forum 84 against Organization B's endpoint-malware events. The fact that multiple, consecutive lags have low p-values gives some indication that this might be a useful signal.
\subsection{Forecasting Models}
We also apply widely-used ARIMA model for forecasting events.
ARIMA stands for autoregressive integrated moving average. The key idea is that
the number of current events ($y_t$) depends on the past counts and forecast
errors. Formally, ARIMA($p$,$d$,$q$) defines an autoregressive model with $p$
autoregressive lags, $d$ difference operations, and $q$ moving average lags
(see~\cite{shumway2010time}). Given the observed series of events
$\mathcal{Y}=(y_1,y_2,\ldots,y_T)$, ARIMA($p$,$d$,$q$) applies $d$ ($\ge 0$)
difference operations to transform $\mathcal{Y}$ to a stationary series
$\mathcal{Y}^{\prime}$. Then the predicted value $y^\prime_t$ at time point
$t$ can be expressed in terms of past observed values and forecasting errors
which is as follows:
\begin{equation}
y^\prime_t = \mu_y + + \sum_{i=1}^p \alpha_i y^\prime_{t-i} + \sum_{j=1}^q \beta_j e_{t-j} + e_t\label{eq:arima}
\end{equation}
Here $\mu_y$ is a constant, $\alpha_i$ is the autoregressive (AR) coefficient at lag $i$, $\beta_j$ is the moving average (MA)
coefficient at lag $j$, $e_{t-j} = y^\prime_{t-j} - \hat{y}^\prime_{t-j}$ is the forecast error at lag $j$, and $e_t$ is assumed to be the white noise ($e_t\sim \mathcal{N}(0,\sigma^2)$). The
AR model is essentially an ARIMA model without moving average terms.
We use maximum likelihood estimation for learning the parameters; more
specifically, parameters are optimized with LBFGS
method~\cite{seabold2010statsmodels}. These models assume that $(p, d, q)$ are
known and the series is weakly stationary. To select the values for $(p, d, q)$
we employ grid search over the values of $(p, d, q)$ and select the one with
minimum AIC score.
\subsection{Testing Signals with ARIMAX}
Again, Table \ref{tab:corr} shows the signals that are better correlated with Organization B's ground truth events. The next step is to test these signals to see if they have any predictive power. To do this, the ARIMA model is used with the ground truth events to develop a baseline model from which to compare potential signals for the potential to have predictive power. Additionally, 4 other methods were used for comparison: Dark-Mentions, Deep-Exploit \cite{sapienza2017early}, ARIMAX with abuse.ch and a daywise-hourly-baserate model. Using ground truth events from both Organization A and Organization B, sentiment signals from the various forums, computed with the different methodologies were tested. Testing was done across the 3 event types for both Organizations with Precision, Recall and F1 computed to evaluate the signal. The timeseries of the sentiment for a given forum and sentiment method was used as the input to the timeseries forecasting model to predict future events. The model was trained on data from April 1, 2016 to May 31, 2017, in order to start generating warnings for the month of June 2017. After predictions were made for the month of June, they were scored against the actual ground truth and then the model was ran again to predict warnings for August 2017. This was done for all the way through January 2018.
\begin{table}[!t]
\centering
\caption{Best Signals for Organization B's Events
\label{tab:corr}}
\begin{tabular}{lrcccr}
\hline%
\textbf{Forum\#}&\textbf{Sent}&\textbf{Lag}&\textbf{Correlation}&\textbf{p Value}&\textbf{Events}\\%
\hline%
84 & LIWC & -11 & 0.2170 & 0.000055 & EP-Mal \\
84 & LIWC & -12 & 0.2221 & 0.000037 & EP-Mal \\
84 & LIWC & -14 & 0.2185 & 0.000052 & EP-Mal \\
219 & Vader & -18 & -0.2329 & 0.000079 & EP-Mal \\
264 & LIWC & -10 & 0.2472 & 0.000040 & EP-Mal \\
264 & LIWC & -12 & 0.2362 & 0.000095 & EP-Mal \\
264 & LIWC & -15 & 0.2380 & 0.000091 & EP-Mal \\
159 & Senti & -14 & 0.8498 & 0.000008 & Mal-Email \\
266 & Senti & -14 & -0.5517 & 0.000058 & Mal-Email \\
261 & LIWC & -3 & 0.2173 & 0.000043 & Mal-Dest \\
266 & Senti & -27 & -0.6243 & 0.000080 & Mal-Dest \\
\hline%
\end{tabular}%
\end{table}
\subsection{Scoring}
To determine how well the signals under study performed, a matching algorithm was used to compare the date occurrence of the predicted events with the actual events that occurred. Using the matching algorithm, we could consistently score which predicted events should be mapped to actual events and which predicted events did not occur as well as which actual events were not predicted. There is a window around the actual events which varies based on the event type. Endpoint-maleware has to be within 0.875 days, malicious-destination within 1.625 days and malicious-email within 1.375 days.
\subsection{External Signals}
Currently, there are other external signals that the data provider Organizations are currently evaluating for predictive potential. Again, external signals are timeseries information derived from open sources that are not based on information system network data. The other external signals under evaluation are:
\begin{itemize}
\item ARIMAX: is the same model outlined in \S 4.7; however, time series counts of malicious activity are acquired from \url{https://abuse.ch} and used in conjunction with historical data.
\item Baseline: is the exact same model in \S 4.7 with no external signal and using only historical ground truth data to predict the future rate of attack.
\item Daywise-Baserate: is the same as the ARIMAX model mentioned above; however, the model takes day of the week into consideration assuming that the event rate for each day of the week is not the same.
\item Deep-Exploit: is an ARIMA model that is based on the vulnerability analysis determined by \cite{tavabi2018dark}. This method referred to as DarkEmbed learns the embeddings of Dark Web posts and then uses a trained exploit classifier to predicted which vulnerabilities in Dark Web posts might be exploited.
\item Dark-Mentions: Is an extension of \cite{almukaynizi2017proactive} which predicts if a disclosed vulnerability will be exploited based on a variety of data sources in addition to the Dark Web using methods still being developed. These predictions are used to construct a rule based forecasting method based on keyword mentions in Dark Web forums and marketplaces.
\end{itemize}
\section{Sentiment Analysis}
The first effective use of sentiment analysis in a predictive sense was by Pang et. al. \cite{pang2002thumbs} in assessing movie reviews. Since then, sentiment analysis has expanded to other fields. Sentiment analysis can be done with or without supervision (label training data). Supervised methods can be adapted to create trained models for specific purposes and contexts. The drawback is that labeled data may be highly costly and often researchers end up using AMT - Amazon Mechanical Turk. The alternative is to use lexical-based methods that do not rely on labeled data; however, it is hard to create a unique lexical-based dictionary to be used for all different contexts. Deep learning methods allow for additional functions like taking into account order of words in a sentence like the Stanford Recursive Deep Model. Methods can either be 2 way (positive or negative) or 3 way (positive, neutral, negative). Furthermore, dictionary based sentiment algorithms are either polarity-based where sentiment is based only of the frequency of positive or negative words whereas valence-based methods factor the intensity of the words into polarity.
There are a number of issues with sentiment analysis which include: word pairs, word tuples, emoticons, slang, sarcasm, irony, questions, URLs, code, domain specific use of words (shoot an email, dead link), and inversions (small is good for portable electronics) which are difficult for computerized text analysis to handle.
Studies have found that a method’s prediction performance varies considerably from one dataset to another. VADER works well for some tweets, but not for others, depending on the context. SentiStrength has good Macro F1 values, but has low coverage because it tends to classify a high number of instances as neutral.
The choice of a sentiment analysis is highly dependent on the data and application, therefore you need to take into account prediction performance and coverage. There is no single method that always achieves a consistent rank position for different datasets. Therefore, in this paper we test multiple methods for sentiment analysis. Most languages themselves are biased positive and if a lexicon is built on data, the positive bias that data can lead to a bias in the lexicon. This is why most methods are better at classifying positive than neutral or negative methods meaning that they are biased, neutral are the hardest to detect \cite{sentibench}.
\subsection{Vader}
VADER: Valence Aware Dictionary for sEntiment Reasoning \cite{vader} is a rule-based sentiment model that has both a dictionary and associated intensity measures. It's dictionary has been tuned for microblog-like contexts and they incorporate 5 generalizable rules that goes beyond pure dictionary lookups:
\begin{enumerate}
\item Increase intensity due to exclamation point
\item Increase intensity due to all caps in the presence of other non-all cap words
\item Increase intensity with degree modifiers i.e. extremely
\item Negate sentiment with contrastive conjunction i.e. but
\item Examine the preceding tri-gram to identify cases where negation flips the polarity of the text.
\end{enumerate}
Therefore, VADER not only captures positive or negative, but also how positive and how negative beyond simple words counts. It is made further robust by the additional rules. It's "gold standard" lexicon was developed manually and with Amazon Mechanical Turk. Vader scores range from 0.0 to 1.0.
\subsection{LIWC}
Linguistic Inquiry and Word Count (LIWC) \cite{pennebaker2001linguistic} was a pioneer in the computerized text analysis field with the first major iteration in 2007, we used the updated version LIWC 2015. It has two components: the processing component and the dictionaries. The heart of LIWC are the dictionaries that contain the lookup words in psychometric categories which is able to resolve content words from style words. LIWC counts the inputted words in psychologically meaningful categories which produces close to 100 dimensions for any given text being analyzed. For the purposes of this research, we are only focused on Tone which bests maps to sentiment as we have defined it. The Tone scores range from 0 to 100. LIWC also ignores context, irony, sarcasm, and idioms.
\subsection{SentiStrength}
SentiStrength \cite{senti} is another lexicon-based sentiment classifier which leverages dictionaries and non-lexical linguistics information to detect sentiment. SentiStrength focuses on the strength of the sentiment and uses weights for the words in its dictionaries. Additionally, positive sentiment strength and negative sentiment strength is scored separately. Each is scored from 1 to 5, with 5 being the greatest strength. For our purposes, we seek overall sentiment so we subtract the negative sentiment from the positive sentiment so that strongly positive (5,1) becomes 4, neutral (1,1) becomes 0 and strongly negative (1,5) becomes -4. Therefore, SentiStrength scorese range from -4 to 4. SentiStrength is designed to do better with social media; however, it can't exploit indirect indicators of sentiment. It is also weaker for positive sentiment in news-related discussions.
|
{
"timestamp": "2018-04-17T02:08:39",
"yymm": "1804",
"arxiv_id": "1804.05276",
"language": "en",
"url": "https://arxiv.org/abs/1804.05276"
}
|
\section{Introduction}
Bicycle Sharing Schemes (BSS) have become increasingly vital elements of urban mobility due to their complementary effect to conventional modes and last-mile connectivity to transit systems \cite{Liu2012}. By now, there are more than 600 BSS globally with the largest systems in China, and successful deployments in Paris, London and Washington D.C. The health benefits of bicycle use in cities even outweighs accident risk \cite{Rojas-Rueda2012}. Furthermore, BSS offer sustainable solutions to urban transportation by contributing to resolving the thriving problems of congestion and pollution. In order to increase the expansion of BSS and attract new customers, it is vital to understand relevant spatial travel patterns, and adjust design and management strategies (e.g. pricing, marketing, expansions) to encourage adoption. For example, if bikesharing is utilized for last mile travel, then a transfer fare could increase its usage and simultaneously promote public transit adoption \cite{Guo2011}. Beyond that, a better understanding of trip patterns will allow for advanced bicycle relocation strategies and more reliable service provision which, in turn, will make the system more attractive to users. The key challenge for any shared mobility system lies in the respective network complexity, noise and the resulting operational implications. To overcome this, we propose a comprehensive and pervasive station-level characterization of the London network, based on spatio-temporal utilization features. Our framework extracts largely self-contained clusters which not only provide insight into mobility patterns, but also help with identifying bottlenecks and inefficiencies, and hence help decision makers to better understand supply and demand imbalances, plan operations and manage infrastructure. Comparing the explicitly non-spatial network model to the known geospatial structure of the system enables us to assess whether communities are a result of space (geography) or place (local features). We develop this approach deploying a dynamic community analysis from BSS rental data collected in London. We introduce a novel approach to community detection in BSS networks and assess interactions between communities as well as the convergence of communities during different hours of the day.
The remainder of this paper is organized as follows. Section 2 reviews relevant studies of BSS and community analysis. A detailed description of the London BSS dataset is provided in Section 3. Section 4 describes the main methodologies and procedures involved in the data-driven analyses. Section 5 contains the discussion and concluding remarks.
\section{Related Work}
With the proliferation of smart data related to BSS there has been a significant amount of research dedicated to either improving our understanding of the BSS to support evidence based policy making, or to perform logistic optimization methods for bicycle relocation. Based on a Barcelona BSS data, \cite{Froehlich2009} applied spatio-temporal analyses, clustering techniques and tested the performance of various machine learning algorithms. \cite{Kaltenbrunner2010} estimates station-level time-series using autoregressive predictive models on the same dataset. Other efforts include a characterization of the network based on the usage profiles of the stations. Vienna’s BSS was analyzed to obtain distinct clusters using partitioning algorithms on usage time-series data in addition to a predictive method to forecast ridership volume \cite{Vogel2011}. BSS stations in Paris were analyzed in respect to usage counts, using a novel Expectation Maximization (EM) model and relating the identified clusters according to their spatial relationships \cite{Etienne2014}. London’s BSS was examined to detect differences in the patterns of usage before and after the opening of the scheme to unregistered (casual) users \cite{Lathia2012} and to identify commuters \cite{Martin-Moral2017}. A big differentiating point between these studies is the type of data used: While some research only has access to availability data at station-level, others are able to use arrival and departure data, detailing every trip. This second set provides more detail as it captures information about periods of inactivity and activity, unlike availability data, which loses information when the net change in bikes at a station is small.
Another focus of analysis in BSS is the detection of communities, that is, the detection of groups of individual users or stations that exhibit a stronger interdependence between one another, as opposed to other members of the system. This allows for a spatial aggregation of the network and the extraction of patterns. \cite{ZaltzAustwick2013} analyze community structures in five urban BSS \textit{(London, UK; Boston, MA; Denver, CO; Minneapolis, MN; and Washington, DC)}. However, due to limited data availability, the researchers had to generate their own origin / destination (OD) matrices. Furthermore, their approach employs hierarchical community detection, which comes with some shortcomings, especially when resolving the boundaries of different communities or relating nodes that do not share any connections \cite{Newman2004}. \cite{BORGNAT2011a} perform community detection aggregation on the Lyon BSS dataset using the Louvain algorithm. Overall, we observe that existing literature on community structure detection in BSS lacks the use of non modularity based methods and more granular, empirical trip data. From a technical viewpoint, community detection using modularity maximization is known to be vulnerable to resolution restrictions, is limited to undirected information and assumes a process of endogenous network formation \cite{Fortunato2006}. Furthermore, the interaction between extracted communities and the evolution of communities over time remain largely unexplored fields. With our paper, we seek to address these gaps in research.
\section{Data}
\begin{figure}[htbp]
\includegraphics[,scale=0.6]{figure1.pdf}
\caption{Location of bikesharing stations in London}
\label{fig:fig1}
\end{figure}
The data for our analysis comes from the Transport for London \textit{(TfL)} Open Data API \footnote{Accessible via: https://api.tfl.gov.uk/}and contains information about the unique IDs for each bicycle, the names and IDs of the origin and destination stations, a unique transaction ID (per trip, not per user) and the start and end times of each rental. The dataset covers every recorded shared bicycle trip since 2012 and hence comprises millions of entries. Operating with such large amounts of data can be computationally expensive. Our further analysis will hence utilize an applicable subsample.
First, however, we address some minor problems regarding the collection of trip information, that translates into missing retrieval for destination station IDs or trips without information about bicycle IDs. These issues are not temporally consistent across the dataset, with some months exhibiting higher error rates than others. To adapt the analysis accordingly, we select a small, particularly accurate interval where a crude cleaning of the data, i.e. a removal of incomplete entries, does not result in significant non-response bias. We clean our subset by removing the following entries:
\begin{itemize}
\item Trips that start or end at a repair station.
\item Trips that do not report correct destinations and show a negative duration.
\item Trips that do not report the bicycle ID.
\end{itemize}
\begin{figure*}[htbp]
\centering
\includegraphics[,scale=0.6]{figure2.pdf}
\caption{Comparison of community detection algorithms: BSS stations are colored according to their respective community assignment across the four techniques}
\label{fig:fig2}
\end{figure*}
Our final dataset is comprised of 1,469,945 unique shared bicycle trips in June and July 2014, distributed over 750 stations in London (see Figure 1). Weekends are disregarded, as their varying trip patterns and added noise could harm our analysis. Aggregated on a station level, our data can be used to compile an Origin-Destination (OD) matrix. We can then formulate a graph $G$ with each station describing a network node $N_{\alpha}$, linked to every other station in the network by a set of directed edges $E_{\alpha}$ weighted by a flow $w_{\alpha}$ equal to the number of trips observed, given in the OD matrix.
\section{Methodology}
\subsection{Community detection}
Community detection techniques aim at reducing the complexity of a network to a degree that enables comprehensive insight into the underlying network structure. BSS are naturally suited for such applications. Simplifying BSS network descriptions and detecting clusters of stations that exchange many trips also have immediate operational implications by providing a valuable decision-support tool for network management and expansion. Having said that, reliable results depend on an appropriate choice of the community detection algorithm. The most popular methods rely on modularity maximization; however, they do not seem to be applicable in our case, as outlined in section \textit{II}. Most importantly, these methods assume an underlying process of network formation which, in the case of station-based BSS, is not present. This problem also motivated the development of the \textit{Infomap} algorithm, originally proposed by Rosvall \& Bergstrom \cite{Rosvall2008}. This method acknowledges that the system structure drives the flow in the system, leading to system-wide interdependencies. By partitioning the network, the length of the description of the movements can be longer or shorter (bigger and smaller cost of information). By choosing the partition that minimizes the description length, we find the division that provides the best representation of the community structures.
Deploying the \textit{Infomap} approach, we seek to partition the network nodes $N_{1,2,...,n}$ into $M_{1,2,...,m}$ modules by minimizing the information cost of describing the movements of a random walker or, if available, the empirical flow (here the trips) through the network. This is implemented by the \textit{map equation}:
\begin{equation}
L(M) = q_{\curvearrowright}H(\mathcal{Q}) + \sum^{m}_{i=1}p_{\circlearrowright}^{i}H(\mathcal{P}^{i}),
\end{equation}
where $q_{\curvearrowright}$ gives the probability that the random walker leaves the current module, $p_{\circlearrowright}^{i}$ gives the proportion the walker spends in the respective module $\mathcal{P}^{i}$, $H(\mathcal{Q})$ gives the index codebook entropy and $H(\mathcal{P}^{i})$ gives the module codebook entropy\footnote{Referring to Shannon's source coding theorem \cite{Shannon1948}}. Practically, this is carried out by assigning the modules $M_{i}$ to a neighboring module $M_{\beta}$, as long as this reduces $L(M)$. The \textit{Infomap} algorithm can then be applied as follows.
\begin{figure*}[htbp]
\centering
\includegraphics[,scale=0.4]{figure3.pdf}
\caption{Interactions and volume of London BSS communities: clusters are mapped at their geographic centroids. The size of point nodes and edges is scaled according to the observed flow within the community (nodes) and between the communities (edges)}
\label{fig:fig3}
\end{figure*}
\begin{algorithm}
\caption{\textit{Infomap} algorithm}\label{alg:infomap}
\begin{algorithmic}[1]
\Procedure{Infomap}{$G$}
\State $M(N_{\alpha}) \gets M_{\alpha}$
\While{$min(L(M))$}
\State Order $N_{1,2,...,n}$ randomly
\For{Each $N_{\alpha}$}
\If{$\exists M(N_{\alpha}) \gets M_{\beta}: L(M)\downarrow$}
\State $M(N_{\alpha}) \gets M_{\beta}$
\Else
\State $M(N_{\alpha}) \gets M(N_{\alpha})$
\EndIf
\EndFor
\EndWhile
\State \textbf{return} $M(N_{\alpha})$
\EndProcedure
\end{algorithmic}
\end{algorithm}
Dedicating a small fraction $\tau$ of the probability flow randomly links every node in the network to every other node and hence prevents the random walker from becoming stuck.
We run the \textit{Infomap} algorithm on our dataset and return an assigned module $M_{i}$ for each station (node) $N_{\alpha}$. As this paper is the first study to apply the \textit{Infomap} algorithm in the context of station-based BSS, we compare our results to three popular modularity based methods (\textit{Greedy modularity optimization, random walks and the Louvain algorithm}), as shown in Figure 2\footnote{For our computation we use the \textit{Infomap} software package (https://github.com/mapequation/infomap, v0.19.3) and \textit{R} (v3.4.2) with the \textit{igraph} package}. The technical differences between the four approaches---discussed extensively in previous research \cite{Yang2016a}---manifest in the respective output communities. Specifically, \textit{Infomap} is the only method to detect known physical structures in London, such as Hyde Park and Canary Wharf. While the three other methods converge at four clusters, the optimal solution from \textit{Infomap} returns six modules. The first and largest community is \textit{(1)} Central and East London, accounting for more than half of the trips in the network. It borders \textit{(2)} South-West London, \textit{(3)} Regent's Park and \textit{(4)} Hyde Park clusters in the West, which are the second, third and fourth largest clusters in terms of flow. These three clusters border the fifth largest community in \textit{(5)} Notting Hill. Finally, \textit{(6)} Canary Wharf contains the least flow of any cluster and is remotely located in the South-East, only bordering the Central cluster.
\subsection{Community interactions}
\begin{figure*}[htbp]
\centering
\includegraphics[,scale=0.4]{figure4.pdf}
\caption{Community evolution over time: cluster assignment for each station and hour-of-day is given using color codes (grey for no assignment). Communities are ordered and colored top-to-bottom by size.}
\label{fig:fig4}
\end{figure*}
Beyond the community detection, our results also enable insight into the interactions between the different communities. Around 75\% of the observed trips start and end within the same cluster. Nevertheless, investigating the exchange of trips between clusters provides a deeper insight into the underlying mechanics of the observed system, particularly for the smaller, more interactive communities. Our simplified community network and the flow between communities are displayed in Figure 3. The size of the communities and the links respectively highlight the observed flow. The exact numbers of trips (within, outbound and inbound) are given in Table 1.
\begin{table}[htbp]
\caption{Summary of the community cluster characteristics and interactions}
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline
\textbf{}&\textbf{}&\multicolumn{3}{|c|}{\textbf{Trips}} \\
\cline{3-5}
\textbf{Cluster} & \textbf{Stations}& \textbf{\textit{within}}& \textbf{\textit{out}}& \textbf{\textit{in}} \\
\hline
Central / East (1) & 408 & 760,404 & 112,263 & 118,146 \\
\hline
West (2) & 190 & 184,714 & 80,457 & 82,332 \\
\hline
Regents Park (3) & 71 & 48,259 & 75,618 & 71,990 \\
\hline
Hyde Park (4) & 26 & 80,354 & 63,617 & 60,289 \\
\hline
Notting Hill (5) & 35 & 17,481 & 30,443 & 29,084 \\
\hline
Canary Wharf (6) & 20 & 9,060 & 7,181 & 7,738 \\
\hline
\end{tabular}
\label{tab1}
\end{center}
\end{table}
The Central London community accounts for around 50\% of all trips in the network. Due to its large size, most trips occur within the community, while the connectivity with other clusters is particularly strong for the West London and Regents Park communities. This suggests that there might be additional hidden structures within the cluster that would allow for further simplification. The West London community behaves relatively similar, though having a higher share of trips interacting with other communities. The Regents Park and Notting Hill communities, both small in size, exhibit more interactive trips than within-cluster trips and hence suggest longer trip distance or special trip purposes. The last clusters---Hyde Park and Canary Wharf---are both small in size but nevertheless relatively isolated. For Hyde Park, a possible explanation is the specialized use of shared bicycles for leisures trips within the green-space. For Canary Wharf, this can be attributed to its remote geographic location on the Isle of Dogs, which reduces the attractiveness of bicycle trips to stations outside of the cluster.
The community detection and interactions analysis enables us to gain novel insights into the spatial usage patterns of London's BSS. The noisy Central London cluster does not seem to exhibit explicit community structures, which hints that the usage there is less community-driven and rather might be explained using temporal analyses (e.g. commuting peak times) or destination-based approaches. The smaller, more disconnected clusters, on the other hand, suggest a strong effect of community bounds on trip-making that might be explained looking into their respective location: while the Canary Wharf cluster is located in a business area where BSS trips might serve as last-mile connections to public transport, the emergence of the Hyde Park cluster within a large public green space suggests leisure activities. This seems especially likely as our observational period is during summer time, where mild weather conditions make parks particularly attractive. On the other hand, small but interactive clusters like Notting Hill and Regents park are based around mostly residential areas and suggest the use of the BSS for commuting. As such, community detection and interaction together reflect London's physical environment and land use.
\subsection{Community dynamics}
While our previous network analyses presents novel insights into general system structures, aggregating the usage in such a large timescale results in the loss of temporal information about the way the network behaves at different times of the day. To provide a deeper understanding of the emergence and collapse of communities over time, we split our dataset into one-hour intervals for further examination. Apart from this, the methodology remains unchanged. We present the \textit{Infomap} cluster assignments for each station and all 24 hours of the day in Figure 4. The results show the evolution of the communities over the course of the day. It is apparent that during some hours---especially at nighttime---the noise in our data is considerably larger which prevents the algorithm from detecting community structures, leaving several hundreds of hardly relevant communities and even unassigned stations, due to the very low flow. This suggests a limitation of our analysis, but also relates to the general lack of observation during those unusual travel hours. From 7am, one dominant community abruptly emerges that subsumes almost all stations. This is due to the morning commuting peak, characterized by long trips connecting stations in residential areas or close to public transport facilities with the business districts. As the peak cluster disintegrates around 10am, three to four stable clusters emerge that resemble the general community structure outlined in the previous sections. This period of stability contains the vast majority of observed trips and starts and ends with the respective morning and evening peak commuting times. From around 8pm, the daytime communities slowly disintegrate towards a full collapse at midnight.
Looking at the spatial dimension, we can observe that during night-time, location does not seem to be of importance. Only starting from 10am, Central London emerges as a community, alongside clusters in Canary Wharf, Hyde Park and West London. These communities vaguely correspond to those extracted from our previous analyses (see Figure 2 and 3) and are mostly stable during daytime. Again, Canary Wharf stands out as the most isolated community, with almost all stations assigned to the same cluster from 9am to 9 pm.
\section{Discussion and Conclusion}
Each of our three methodological sections comes with particular findings and implications relevant to BSS users, providers and public authorities. \textit{(A)} We present a new, information-theoretic method for BSS community detection that not only reflects known system features like directed links and exogenous network formation, but is also able to detect known urban built-environments like Hyde Park or Canary Wharf. Hence, we are able to infer valuable information on user behavior and the geographical boundaries of bikesharing trip-making. The findings may inform and motivate bikesharing service adoption according to the detected communities, or aim at connecting communities by incentivising users or expanding infrastructure. \textit{(B)} Our method enables us to explore bidirectional flow between communities. The imbalances in trip flows are a crucial challenge for any shared mobility scheme as they often result in vehicles getting stuck in areas characterized by high destination attractiveness and low origin attractiveness. While the communities themselves are mostly self-contained, the trips between communities and particularly their imbalances can shed new light on this issue and motivate novel relocation strategies. \textit{(C)} Lastly, we explore the emergence and collapse of communities during the course of a day, thus evaluating the noise and underlying mechanics in our system. We see that during nighttime---characterized by substantially less trips---the algorithm cannot detect clear community structures as the trips do not follow any distinct patterns. During the daytime, communities start to emerge with the clearly structured trips of the morning commuting peak and stabilize afterwards. There is some spatio-temporal fluctuations between the dominant clusters which again seem to be mostly driven by peak commuting hours. Contrarily, we find communities of remote geographic location (Canary Wharf) and specific leisure usage (Hyde Park) to be the most consistent. Beginning with the afternoon commuting peak, communities contract and eventually collapse around midnight. By exposing structure-over-time, we examine how community presence is driven by spatio-temporal dynamics. Time-sensitive events like commuting hours mostly effect certain areas---around public transport stations and business districts---while parks or leisure districts remain rather stable. These findings enable focused policies to address problems like supply shortages or congestion.
Altogether, our research reinforces the argument that BSS are inherently spatio-temporal systems of dynamic complexity. Our study also raises new questions regarding the driving factors of our observations. Further studies should extend the underlying questions from purely unsupervised learning problems to represent other layers of the urban system. For instance, research have recently shown that urban amenities \cite{Willing2017a} or weather data \cite{Reiss2015} can help with contextualizing patterns in shared mobility systems. Our findings also suggest a strong interconnection of said features with the bikesharing network, which implies that more holistic approaches are needed to draw meaningful operational and political conclusions. In future studies, this can be validated by comparing BSS to other large scale transportation networks and including further cities in those studies. Lastly, the methodological contribution of our work, while novel, could be expanded to address time-varying networks of bike stations and communities, where different motifs (loop, chain, star) and temporal evolution dynamics with extended time windows could potentially provide deeper insights into inherent relationships of spatially heterogeneous nodes (stations) or sub-networks (communities).
\section*{Acknowledgments}
The authors gratefully acknowledge funding from the UK Engineering and Physical Sciences Research Council, the EPSRC Centre for Doctoral Training in Urban Science (EPSRC grant no. EP/L016400/1); The Alan Turing Institute (EPSRC grant no. EP/N510129/1).
\bibliographystyle{IEEEtran}
|
{
"timestamp": "2018-08-20T02:01:24",
"yymm": "1804",
"arxiv_id": "1804.05584",
"language": "en",
"url": "https://arxiv.org/abs/1804.05584"
}
|
\section{Introduction}
\begin{figure}
\centering
\begin{subfigure}[b]{0.4\textwidth}
\includegraphics[width=\textwidth]{greatDodec}
\caption{Great dodecahedron}
\label{fig:great-dodec}
\end{subfigure}
\qquad
~
\begin{subfigure}[b]{0.4\textwidth}
\includegraphics[width=\textwidth]{dodecTorus}
\caption{Dodecahedral torus}
\label{fig:dodec-torus}
\end{subfigure}
~
\caption{Regular polygon surfaces with degree five faces}\label{fig:surfs}
\end{figure}
We study surfaces built by gluing regular and rigid Euclidean polygons together along their edges.
Recently surfaces built out of regular polygons with boundary have been used to build flexible metamaterials that can be deformed into various configurations \cite{o2017}. Very little is known about the space of shapes of these generalized polyhedra, called \emph{regular polygon surfaces} (RPSs) (see definition \ref{def:rps}), which are neither convex nor symmetric. We prove that under certain assumptions on the genus and face degrees, RPSs can be realized as a union of Platonic solids glued along common facets. Before giving a rigorous definition of a RPS, we introduce some terminology from graph theory.
\begin{defn}\label{def:surface-graph}
A \emph{surface graph} \((\Sigma, \Gamma)\) is a graph \(\Gamma\) embedded on a closed surface \(\Sigma\) in such a way that \(\Sigma\setminus \Gamma\) is a union of connected complementary components called \emph{faces} with each face homeomorphic to the \(2\)-cell. If in addition the closure of each face is homeomorphic to the closed \(2\)-cell, we call the surface graph a \emph{regular surface graph}. When the intersection of the closure of any two faces is either empty, a vertex in \(\Gamma\), or an edge in \(\Gamma\) we say that the surface graph is \emph{proper}.
\end{defn}
The \emph{degree} of a face in a surface graph is the number of edges incident to that face.
\begin{defn}\label{def:rps}
Let \((\Sigma,\Gamma)\) be a finite, regular, and proper surface graph in which \(\Sigma\) is a genus \(g\) surface. A genus \(g\) \emph{regular polygon surface} (RPS) is a triple \((\Sigma, \Gamma, \psi)\) whose \emph{geometric realization} \(\psi: \Sigma \to \mathbb{R}^3\) is continuous and maps a face of degree \(k\) to a regular Euclidean \(k\)-gon with unit edge lengths. To rule out degenerate RPSs, we assume that the intersection of the image under \(\psi\) of adjacent faces in the graph is either one vertex or one edge and its two incident vertices. If all of the face degrees are contained in the set \(\{k_1, \ldots, k_n\}\), then we call the surface a \((k_1, \ldots , k_n)\)-RPS.
\end{defn}
We allow geometric realizations which are not \emph{embeddings}, i.e., \(\psi\) may not be injective and the geometric realization of the surface may have self-intersections.
In figure \ref{fig:surfs} we show two examples of RPSs. The familiar Platonic solids as well as the Kepler-Poinsot polyhedra \cite{c1963} are all examples of RPSs. One way to build more complicated RPSs is to \emph{glue} two RPSs together along a common facet when both surfaces have facets with the same number of incident edges. To be precise, suppose \(P\) and \(Q\) are RPSs which both have a face of degree \(n\) and let \(f_p\) and \(f_q\) denote the respective faces. After cutting out the interior of \(f_p\) from \(P\) and the interior of \(f_q\) from \(Q\) we can glue the two surface graphs together along their boundaries (in an orientation-reversing way). The new RPS inherits a geometric realization in the obvious way from the geometric realizations of the original two RPSs. An example of a RPS constructed in this fashion is the \emph{dodecahedral torus} shown in figure \ref{fig:dodec-torus}.
When the genus is low enough the space of RPSs is constrained and we are able to prove the following three theorems.
\begin{thm}\label{pent}
Every oriented genus \(0\) or \(1\), \((5)\)-RPS can be realized as the boundary of a union of dodecahedra glued together along common facets.
\end{thm}
\begin{thm}\label{pent-n}
The only possible oriented genus \(0\), \((5,7,8,9,10)\)-RPSs are those which can be realized as the boundary of a union of dodecahedra glued together along common facets.
\end{thm}
\begin{thm}\label{square-oct}
Every oriented genus \(0\), \((4,8)\)-RPS can be realized as the boundary of a union of cubes and octagonal prisms glued together along common facets.
\end{thm}
\begin{wrapfigure}{r}{0.5\textwidth}
\begin{center}
\includegraphics[width=0.48\textwidth]{truncOctMol4d}
\caption{A \((4)\)-RPS of genus 49}
\label{truncOctMol4ds}
\end{center}
\end{wrapfigure}
Not all RPSs can be constructed by gluing together convex polyhedra. Figure \ref{truncOctMol4ds} shows a \((4)\)-RPS with genus 49 which cannot be realized as a union of cubes and prisms glued together. In section \ref{sec:counter} we explain how this surface is constructed and present two other examples of high genus RPSs which are not unions of cubes and prisms (figs. \ref{truncCubOctMol4d} and \ref{truncCubOctMolSqHex}). Note that the surfaces described in section \ref{sec:counter} cannot be embedded \(\mathbb{R}^3\).
The RPSs studied in this paper are examples of generalized polyhedra. While the five convex regular polyhedra known as \emph{Platonic solids} were described in Euclid's \emph{Elements}, there is no characterization of generalized polyhedra. New examples of polyhedra (in \(\mathbb{R}^3\)) are still being discovered (see \cite{gsw2014} for examples of regular polyhedra and \cite{gs2009} for examples of \emph{toroidal polyhedra}). Moreover, mathematicians have not reached a consensus on the definition of a generalized polyhedron. See \cite{g2003a} for a historical account of the study of polyhedra and a proposed definition of a generalized polyhedron. Although it is generally known that there are 13 convex \emph{Archimedean polyhedra}, whether regularity is a ``local'' or ``global'' condition has resulted in a mathematical error in many enumerations of these objects \cite{g2009}.
A few examples of generalized polyhedra were known classically. The Kepler-Poinsot great dodecahedron (figure \ref{fig:great-dodec}) is a genus four RPS with faces of degree five which is not a union of convex polyhedra since its vertex figures are the nonconvex star polygons known as pentagrams. It can be constructed in two steps by first \emph{stellating} (extending the faces symmetrically to form a new polyhedron) the dodecahedron to obtain the small stellated dodecahedron, then dualizing the polyhedron \cite{c1963}. The great dodecahedron was first depicted in a 1568 etching by Amman (see figure \ref{fig:jamnitzer}) of an engraving made by Jamnitzer \cite{j1568}.
\begin{wrapfigure}{l}{0.4\textwidth}
\begin{center}
\includegraphics[width=0.38\textwidth]{jamnitzer.pdf}
\end{center}
\caption{The Renaissance etching showing the earliest known depiction of the great dodecahedron \cite{j1568}}
\label{fig:jamnitzer}
\end{wrapfigure}
There has been some recent work on low genus polyhedra with rectangular faces. Donoso and O'Rourke \cite{do2001} proved that a polyhedron, of genus at most one with rectangular faces, has dihedral angles which are all integer multiples of \(\pi/2\). In addition they constructed a genus seven polyhedron with rectangular faces whose dihedral angles are not integer multiples of \(\pi/2\). Their result was later extended to genus two polyhedra with rectangular faces by Biedl et al. \cite{bcd2002}. Thurston developed a global theory that describes triangulations of the sphere with at most \(6\) triangles around a vertex \cite{t1998}. He found a natural bijection between these triangulations and a quotient space of a discrete lattice in \(\mathbb{C}^{1,9}\). See \cite{s2015} for a readable introduction to Thurston's paper which provides alternative proofs of the main theorems in \cite{t1998}.
RPSs also have applications to statistical mechanics. In the lattice formulation of quantum gravity, physicists are naturally led to an infinite dimensional integral over the space of Riemannian metrics. By approximating a manifold by piecewise linear manifolds, such as RPSs, with fixed edge lengths, one can replace the integral by a discrete sum, vastly simplifying the problem \cite{d1992}. Sampling large random piecewise linear manifolds is an important aspect of this theory \cite{ab2014}. It is conjectured that the associated metric space converges to the Brownian map. See \cite{lm2012} for a definition of the Brownian map and a survey of results about large random planar maps. Surface graphs which support a family of different geometric realizations can be used as a model for random surfaces. Our results imply that certain surface graphs do not have a non-trivial space of geometric realizations.
Although much is known about convex polyhedra in \(\mathbb{R}^n\) (see \cite{a2005} or \cite{g2003c}) and convex ideal polyhedra in \(\mathbb H^3\) (see \cite{r1996}), their techniques do not apply to the inherently nonconvex surfaces we study in this paper.
\nocite{gm1963,g1937,k1996n}
In order to prove the three main theorems we use inductive arguments that rely on a procedure for simplifying RPSs by removing certain subgraphs. Just as we can build more complicated RPSs by gluing two RPSs together, there is an inverse process where we can simplify a RPS by removing certain subgraphs and replacing them by others. We call the process \emph{polyhedral surgery} and it is defined as follows. Let \(P\) be a RPS with data \((\Sigma_1, \mathcal G_1, \psi_1)\) and \(Q\) a RPS with data \((\Sigma_2, \mathcal G_2, \psi_2)\).
Suppose both RPSs contain cycles of length \(n\), call them \(C_1 \) and \(C_2\) respectively. Label the vertices in \(P\) along \(C_1\) by \(v_1,\ldots,v_n\) and the vertices in \(C_2\) by \(w_1,\ldots ,w_n\). In addition, suppose there exists an isometry \(G\) of \(\mathbb{R}^3\) such that \(G\circ \psi_1(v_i)= \psi_2(w_i)\). Cutting \(\Sigma_1\) along the cycle disconnects it into two hemispheres, \(H_P^1\) and \(H_P^2\) and likewise for \(\Sigma_2\) with hemispheres \(H_Q^1\) and \(H_Q^2\). Now we can glue \(H_P^1\) to \(H_Q^1\) along their respective boundaries to form a new RPS with geometric realization defined by
\[ \psi(x) = \begin{cases} G \circ \psi_1(x)\qquad & \text{ if } x \in H_P^1 \\ \psi_2(x) \qquad & \text{ if } x \in H_Q^1 \end{cases}.\]
It is possible that there is a face \(f\) in \(H_P^1\) and a face \(g\) in \(H_Q^1\) that are adjacent in the surface graph of the new RPS and \(\psi(f)\) and \(\psi(g)\) intersect in more than just one edge. When two faces intersect in this manner we call them \emph{dangling faces}. However, a slight modification of \(C_1\) to include \(f\) prevents this from occurring.
Polyhedral surgery can always be performed when the geometric realization of the RPS formed by gluing the hemispheres \(H_P^2\) and \(H_Q^1\) forms a convex polyhedron. As an example we consider a case when the two hemispheres form a cube. Let \(P\) be a RPS containing the subgraph as shown on the left-hand side of figure \ref{sqsurg} and \(Q\) a RPS containing the subgraph shown on the right-hand side of the same figure. We can cut \(P\) along the cycle in green, remove the hemisphere containing \(v_1\) and glue in the hemisphere of \(Q\) which contains the subgraph shown in the right-hand side of figure \ref{sqsurg}. The resulting RPS has the same genus as \(P\) but has two fewer faces.
\begin{figure}
\begin{center}
\includegraphics[width=\textwidth]{sqsurg.pdf}
\caption{Surgery on a cube}
\label{sqsurg}
\end{center}
\end{figure}
\section{\((5)\)-RPSs}\label{sec:rps5}
When the degree of each face in a RPS is large and the genus is low, geometric constraints impose some rigidity on the structure of the geometric realization of the surface. In this section we use a discrete form of the Gauss-Bonnet theorem to prove theorem \ref{pent}.
\begin{thm}[Discrete Gauss-Bonnet Theorem]
For a RPS \(P\) with Euler characteristic \(\chi\) and curvature \(k_v\) at each vertex \(v\) of \(P\) we have
\begin{align*} \sum_{v\in P}^n k_v = 2 \pi \chi \end{align*}
where the \emph{vertex curvature} \(k_v\) is \(2\pi\) minus the sum over all faces containing \(v\) of the interior angle at \(v\) in the geometric realization of the face.
\end{thm}
A proof of this theorem can be found in many places including \cite{s2011}. While the Gauss-Bonnet theorem is not normally proved for the RPSs we study in this paper, the extension to our case is straightforward. One way to prove the theorem is by triangulating the RPS, counting the contribution to the curvature from each face \(\pi\), and applying Euler's formula for a graph on a surface with Euler characteristic \(\chi\).
We will find it more useful to assign curvature to faces instead of vertices. The \emph{facial curvature} \(k_f\) associated to face \(f\) is given by
\[ k_f=\sum_{v\in f}^n \frac{ k_{v}}{d_{v}} \]
where the sum is over all vertices incident to \(f\) and \(d_v\) is the degree of vertex \(v\).
For the remainder of this section we restrict our attention to pentagonal RPSs, those with all faces of degree five. Before proving theorem \ref{pent} we first prove a useful lemma.
\begin{lem}\label{pos-curv face}
Let \(f\) be a face of a RPS with all faces of degree five. If \(f\) has positive facial curvature then each vertex is of degree three. Moreover, if \(f\) has one negative curvature vertex then \(f\) has non-positive facial curvature.
\end{lem}
\begin{proof}
Since \(f\) has positive facial curvature there must be a vertex incident to \(f\) with positive vertex curvature. The curvature at a vertex of degree \(d\) is \(\pi/5 (10-3d)\) which is only positive if the vertex has degree three. This vertex contributes \(\pi/5 (10/d -3)\) to the facial curvature of each face containing it. If vertices with degrees \(d_1, \ldots d_5\) are incident to a face then its facial curvature is
\[\frac{\pi}{5} \sum_{i=1}^5 \left( \frac{10}{d_i}- 3 \right)=-3 \pi + \pi \sum_{i=1}^5 \frac{2}{d_i}.\]
This function is monotonically decreasing as a function of the degrees. If one vertex has degree \(6\) then it is non-positive. Checking the finitely many cases in which each vertex has degree less than or equal to \(6\) we find that the facial curvature is only positive when the face has at least \(4\) degree three vertices and a fifth vertex with degree at most \(5\).
Next assume that the face \(f\) has four vertices of degree three and a fifth vertex \(v\) with degree either four or five. Let \(g\) and \(h\) be the two faces incident to \(v\) which share an edge with \(f\). Since \(g\) shares an edge with \(f\) it must also share two vertices with \(f\) and since \(f\) only has one vertex with degree more than three, \(g\) and \(f\) must also share a degree three vertex. Thus the dihedral angle between \(g\) and \(f\) in the geometric realization is fixed to be that of the dodecahedron, and likewise for the dihedral angle between \(f\) and \(h\) in the geometric realization. This implies that the geometric realizations of \(g\) and \(h\) intersect along an edge. Therefore three is the maximum degree of \(v\). From this proof we find that when \(f\) has one negative curvature vertex it must have a second and any face with at least two negative curvature vertices has facial curvature of at most zero.
\end{proof}
\begin{proof}[Proof of theorem \ref{pent}]
First we prove theorem \ref{pent} for genus zero RPSs. Suppose that the set of counterexamples to the theorem is nonempty. Our RPSs are assumed to be finite thus there exists a lower bound on the number of faces in a surface which is an element of the set of counterexamples. Let \(n\) be this lower bound and \(P\) a member of the set of counterexamples with \(n\) faces. Let \((\Sigma, \Gamma, \psi)\) denote the data of \(P\). We use polyhedral removal surgery to construct a RPS with fewer than \(n\) faces whose realization is not a union of dodecahedra, therefore contradicting the assumption of minimality.
By assumption, \(P\) has genus zero and total curvature \(4\pi\) thus contains a face with positive facial curvature. Lemma \ref{pos-curv face} states that the degree of every vertex incident to a face with positive facial curvature is three. Let \(f\) be a face with positive facial curvature. To simplify notation, we label faces that share an edge with \(f\) as \emph{first-generation} faces and faces that share an edge with first-generation faces as \emph{second-generation} faces. If a first-generation face had positive curvature, then it would have to share a degree three vertex with some second-generation face. Let \(C_1\) be the cycle in \(\Gamma\) bounding the seven faces consisting of \(f\), the first-generation faces and the face in the second-generation which shares a degree three vertex with a first generation face. Cut the surface along \(C\) into two hemispheres \(H_P^1\) and \(H_P^2\), where \(H_P^1\) is the hemisphere with seven faces. Since all faces in \(H_P^1\) are connected by the same type of degree three vertices it can be realized as a hemisphere of a dodecahedron.
Let \(Q\) be a genus zero RPS with twelve faces. As can be seen from counting the curvature at every vertex, its geometric realization is a dodecahedron. Cut it along a curve \(C_2\) into two hemispheres such that one hemisphere has five faces and the other has seven. Label the hemisphere with five faces \(H_Q^1\) and the other \(H_Q^2\). Using polyhedral removal surgery we can glue \(H_Q^1\) and \(H_P^2\) along their boundaries to form a genus zero RPS \(P'\) with \(n-2\) faces. Since \(P\) cannot be realized as a union of dodecahedra and the faces we removed can be realized as a part of a dodecahedra, the new surface \(P'\) cannot be realized as a union of dodecahedra. However, \(P'\) has \(n-2\) faces which contradicts the assumption that \(n\) was the lower bound on the number of faces in a RPS which cannot be realized as a union of dodecahedra. It is possible that after gluing the hemispheres together, two adjacent faces have the same geometric realization. However, we can resolve this issue by changing \(C_1\) so that one of these faces is in \(H_P^1\). Likewise we modify \(C_2\) so that the boundaries of \(H_Q^1\) and \(H_P^2\) agree.
Now we argue by contradiction to establish that the surface must contain a face with positive facial curvature with an adjacent face that also has positive facial curvature. Assume that no face in the first-generation has positive facial curvature. No face in the second-generation can have positive facial curvature either. Otherwise it would have all degree three vertices thus share a degree three vertex with a first-generation face and we could apply the same argument as in the preceding paragraph to construct a counterexample to the theorem with \(n-2\) faces. Since each first generation face has two degree three vertices and the facial curvature is a monotonic function of the degrees, the facial curvature can be maximized by maximizing the number of degree three vertices. Each face has negative curvature so there are at most three degree three vertices. The only geometrically realizable configuration with three degree three vertices, subject to the constraint that the face has negative facial curvature, is the configuration with three vertices of degree three that are in different orientations. However in this case the remaining two vertices must have degree at least five giving this configuration curvature \(-\pi/5\), otherwise the surface would not have a valid geometric realization. If the face has two degree three vertices then its facial curvature is maximized with three degree four vertices and this configuration has facial curvature \(-\pi/6\). Since facial curvature is a monotonic function, any configuration with less than two degree three vertices will have less curvature than the configuration with two degree three vertices and three degree four vertices.
The central face \(f\) has facial curvature \(\pi/3\) which gives the region including \(f\) and the first generation faces, total facial curvature \(-2\pi/3\). Every positive curvature face must be contained in a region with total facial curvature at most \(-2\pi/3\), and these regions must be disjoint because second generation faces also have negative facial curvature. Thus, \(-2\pi s/3\) is an upper bound on the total curvature of the surface where \(s\) is the number of positive curvature faces. This contradicts our assumption that the surface has genus zero and positive total curvature. Therefore the RPS must have at least one face with positive facial curvature which is adjacent to a face with positive facial curvature. Using polyhedral removal surgery we can always build a counterexample to the theorem with fewer than \(n\) faces.
Finally, we extend the result to genus one RPSs. Arguing in the same manner as the genus zero case, suppose the set of genus one counterexamples to theorem \ref{pent} is non-empty and let \(n\) be a lower bound on the number of faces of a RPS in this set. Let \(P\) be an element of the set of counterexamples with \(n\) faces. Since the total curvature of \(P\) is zero, we divide the proof into two cases. In the first case, \(P\) has at least one face with positive facial curvature. The same argument in the preceding paragraph shows that \(P\) must have a region of seven contiguous faces on which we can use polyhedral surgery to build a counterexample with \(n-2\) faces, thus contradicting the assumption that \(n\) is a lower bound on the number of faces in a counterexample. In the second case, every face of \(P\) has zero facial curvature. Recall that the curvature of a face with vertices of degrees \(d_1,\ldots, d_5\) is
\[-3 \pi + \pi \sum_{i=1}^5 \frac{2}{d_i}.\]
This sum is negative when at least one of the vertices has degree greater than six. Checking the (finitely many) cases with each \(d_i\leq 6\) we find that the sum is zero either when the face has four degree three vertices and one degree six vertex or when the face has three degree three vertices and two degree four vertices. A RPS cannot have a face with four vertices of degree three and one vertex of degree six. Two adjacent faces incident to such a vertex would have 2-dimensional intersection in the surface's realization thus violating one of the conditions in the definition of a RPS. For the remainder of the proof we assume that every face has three degree three vertices and two degree four vertices.
In each face the two degree four vertices must be adjacent in order for the surface to have a geometric realization. This condition severely restricts the combinatorics of the underlying surface graph. In figure \ref{gzcase} we show a subgraph of a surface graph for which the three faces \(f_1, f_2\) and \(f_3\) satisfy this requirement on the degrees. Notice that there are three degree three vertices in the interior of the cycle shown in green, thus we can apply polyhedral removal surgery on the green cycle to reduce the number of faces in the surface. The hemisphere we glue in has five faces and comes from a hemisphere of a dodecahedron.
\begin{figure}
\begin{center}
\includegraphics[scale=0.4]{gzcase.pdf}
\caption{Subgraph of the surface graph of a RPS in which every face has zero facial curvature}
\label{gzcase}
\end{center}
\end{figure}
\end{proof}
\section{\((5,7,8,9,10)\)-RPSs}
In this section we restrict our attention to \((5,7,8,9,10)\)-RPSs and prove theorem \ref{pent-n}. Our assumption that a RPS has a geometric realization in \(\mathbb{R}^3\) places a restriction on the types of degree three vertices that may be present. A degree three vertex is only geometrically realizable when it has non-negative curvature. The curvature of a vertex at which two degree five faces and one degree \(n\) face meet is \((n-10)\pi/5n\) so the configuration is only realizable if \(n \leq 10\), justifying our restriction on the maximum degree of a face. Moreover, if two degree seven (or higher) faces meet at a vertex then the curvature is negative. Thus every degree three vertex is formed by the intersection of at least two degree five faces and one other face which may degree larger than five.
We exclude degree six faces because the existence of large combinatorial spheres with regular pentagonal and hexagonal faces, such as the truncated icosahedron, present an obstacle to our methods. Our methods are local arguments and these large combinatorial spheres imply that we must examine neighborhoods with many faces. Furthermore, vertices at which three degree six faces meet have zero vertex curvature which implies that the surface may have large regions of faces with zero facial curvature in between positive curvature faces. Nevertheless we conjecture that any genus zero RPS with faces of degree five or higher can be realized as a union of dodecahedra and truncated icosahedra glued together along common facets.
Before proving theorem \ref{pent-n} we introduce notation for the different types of vertices a face may have. A vertex at which \(k\) faces of degree \(m\) and \(l\) faces of degree \(n\) meet is denoted by \(m^k,n^l\). For a face with vertices \(v_1, \ldots v_n\) we use the product notation \(({m_1}^{k_1},{n_1}^{l_1}) \cdots ({m_t}^{k_t},{n_t}^{l_t}) \) to indicate that vertex \(v_i\) is of the form \(({m_i}^{k_i},{n_i}^{l_i})\). We always assume the vertices are ordered cyclically.
In order for a RPS to have a geometric realization only certain vertex combinations on a face are allowed. For example, a vertex of the form \((5^2,7)\) cannot be adjacent to a vertex of the form \((5^3)\). At a degree three vertex the degree of the faces incident to the vertex determine the dihedral angles between the images of the faces under the geometric realization. In table \ref{table:angles} we record the dihedral angles between faces incident to vertices with non-negative vertex curvature. Since the dihedral angles are different for different type vertices, we find that two degree three vertices can only be adjacent if both vertices have the same type.
\begin{center}
\begin{table}
\begin{tabular}{| l | l | l | }
\hline
Vertex type & \(5-5\) Dihedral angle & \(5-n\) Dihedral angle. \\ \hline
\((5^3)\) & \(116.57^{\circ}\) & \(116.57^{\circ}\) \\
\((5^2,7)\) & 142.65 & 132.43 \\
\((5^2,8)\) & 152.54 & 141.67 \\
\((5^2,9)\) & 162.27 & 153.22 \\
\((5^2,10)\) & 180 & 180 \\
\hline
\end{tabular}
\caption{Dihedral angles between faces incident to a \((5^2,n)\) vertex} \label{table:angles}
\end{table}
\end{center}
Suppose a face \(f\) is incident to two degree three vertices \(v\) and \(w\), both of which are adjacent to a third vertex \(u\). The dihedral angles between any two of the three faces incident to a degree three vertex are determined by the vertex type (see table \ref{table:angles}). Let \(g\) be the face incident to both \(u\) and \(v\) (which isn't \(f\)) and \(h\) the face incident to both \(u\) and \(w\) (which isn't \(f\)). If \(v\) and \(w\) do not have the same vertex type then the dihedral angle between \(g\) and \(h\) will not be one of those listed in table \ref{table:angles}, implying that the degree of \(u\) is at least \(4\). However, the vertex cannot have degree \(4\) because a regular polygon cannot be adjacent to both \(g\) and \(h\). Therefore the vertex must have degree at least \(5\).
Now we prove two lemmas which we use to prove theorem \ref{pent-n}.
\begin{lem}\label{verts n face}
Let \(f\) be a face of a RPS of degree \(n\) with \(n\geq 7\). If \(f\) has positive facial curvature then every vertex incident to \(f\) has the form \((5^2,n)\). Moreover, if \(f\) has one vertex with negative vertex curvature then \(f\) has negative facial curvature.
\end{lem}
\begin{proof}
The only positive curvature vertices are those of the form \((5^2,n)\) with vertex curvature \(\pi \left(2/n - 1/5 \right)\). The curvature of a degree four vertex incident to \(f\) is at most \(\pi \left(2/n - 1/5 \right)\) where the vertex has configuration \((5^3,n)\). If a face has \(k\) vertices of degree greater than \(3\) and \(n-k\) vertices of degree \(3\) then its facial curvature is \( -\pi (2n^2+4kn -20n + 5k)/30n \) which is negative when \(k >1 \) for \(n\geq 7 \).
Assume that \(f\) has \(n-1\) vertices of type \((5^2,n)\) and one vertex \(v\) which may have degree greater than 3. Let \(g\) and \(h\) be the two faces adjacent to \(f\) which are also incident to \(v\). Under the geometric realization of the surface, \(g\) and \(h\) intersect along an edge because the degree \(3\) vertices determine the dihedral angle between these two faces. If \(v\) had degree \(4\) then a face would be adjacent to both \(g\) and \(h\) but then the realization of this face could not be a regular polygon because two of its edges have the same geometric realization. This implies that \(v\) has degree at least five. However, if \(v\) had degree \(5\) then the geometric realization of two of the faces incident to \(v\) would overlap which is a contradiction. We find that \(v\) has degree at least \(6\).
A vertex of degree \(6\) has the largest vertex curvature in configuration \((5^5-n)\) with vertex curvature \(2\pi (-1+1/n)\). If a face has \(n-1\) vertices of type \((5^2-n)\) and one vertex of degree \(6\) then its facial curvature is at most \(-(n-1)(n-5)\pi/15 n\) which is negative for \( n>5\). Therefore when \(f\) has positive facial curvature every vertex must be of type \((5^2,n)\). Furthermore, this argument implies that a face with one negative curvature vertex must have a second vertex with negative vertex curvature and thus have negative facial curvature.
\end{proof}
\begin{lem}\label{pent-face-pos}
In a \((5,7,8,9,10)\)-RPS of genus zero, there exists a face of degree five with positive facial curvature.
\end{lem}
\begin{proof}
Suppose every face of degree five has non-positive curvature and let \(f\) be a degree \(n\) face with positive facial curvature. By lemma \ref{verts n face}, each vertex incident to \(f\) has configuration \((5^2,n)\) and so \(f\) is adjacent to \(n\) degree five faces, all of which have non-positive facial curvature. Each face in the first generation has at least two adjacent vertices of type \((5^2,n)\). As previously discussed, degree \(3\) vertices with different types cannot be adjacent and are separated by a vertex of degree at least \(5\). A configuration with two different types of degree three vertices has facial curvature at most \(\pi (160-47n)/75n\) when it is of type \((5^2,n)^2(5^4,n)(5^3)(5^4,n)\). Each first generation has at most \(4\) degree three vertices since the remaining vertex is incident to two degree \(n\) faces. The configuration with only one type of degree three vertex with the largest facial curvature is \((5^2,n)^4(5^2,n^2)\) with facial curvature \(\pi(110/n-17)/30 \). Any other facial configuration has less facial curvature and we find that the most curvature a face in the first generation can have is in the configuration \((5^2,n)^4(5^2,n^2)\).
The sum of the facial curvature from \(f\) and its \(n\) first generation faces is \(13 \pi/3- 19 \pi n /30\) which is negative for \(n \geq 7\). None of the faces in \(f\)'s second generation can have positive curvature because each either has degree five or is a degree \(n\) face incident to a vertex of degree \(4\) of the form \((5^2,n)^4(5^2,n^2)\) and as a result has negative facial curvature.
Thus, by summing the curvature over all faces with positive facial curvature and their first generation faces we find that the surface has negative total curvature. This contradicts the assumption that the surface has genus zero and we conclude that there exists a degree five face with positive facial curvature.
\end{proof}
\begin{proof}[Proof of theorem \ref{pent-n}]
The proof is similar to the proof of theorem \ref{pent}. Suppose that the set of counterexamples to the theorem is non-empty and let \(n\) be the lower bound on the number of faces of an element in the set. Let \(P\) be a counterexample with \(n\) faces. By lemma \ref{pent-face-pos} there exists a degree five face \(f\) with positive facial curvature. Since \(f\) has positive curvature, every vertex incident to \(f\) has positive vertex curvature. Moreover, \(f\) is incident to an odd number of vertices, implying that all vertices incident to \(f\) have the form \(5^3\). Thus all faces in the \(f\)'s first generation have degree five. If the second-generation faces were all degree five, then we could use the argument from the proof of theorem \ref{pent} to construct a counterexample with \(n-2\) faces. Thus, there exists a face with positive curvature such that a face in its second generation has degree larger than \(5\). Without loss of generality, assume that \(f\) is this face. If a face in \(f\)'s first generation had positive curvature then we could use polyhedral surgery, as in the proof of theorem \ref{pent}, to construct a counterexample to theorem \ref{pent-n} with \(n-2\) faces. The same argument implies that no degree five face in the second generation can have positive curvature.
Let \(g\) be a face with degree greater than five in the second-generation. This face shares two vertices with a face \(h\) in the first generation. Since \(h\) has degree five and two vertices of the form \(5^3\), the two vertices it shares with \(g\) must have degree at least four. Since \(g\) has two degree four vertices its facial curvature is \(-\frac{\pi (n-5) (n-1)}{15 n}\) which is negative for \(n \geq 6\). The vertex configuration of a first-generation face which maximizes the facial curvature of the face is the one with the fewest degree \(n\) vertices. The first generation faces contribute the most curvature with three of type \((5^3)^2(5^4)^3\), one of type \((5^3)^2(5^3,n)(5^2,n)(5^4)\), and one of type \((5^3)^2(5^4)(5^3,n)(5^4)\). The sum of the facial curvature of \(f\) and its first generation faces is \((5-2n )\pi/(3n)\) which is negative for \(n \geq 4 \). Since no second-generation face has positive curvature, the sum over all positive curvature faces, of the facial curvature of a face and its first-generation neighbors, gives an upper bound on the total curvature of the surface. However, this upper bound is negative which contradicts our assumption that the surface has genus zero.
\end{proof}
\section{\((4,8)\)-RPSs}
RPSs with faces of degree four or eight have both vertices with zero curvature and faces with zero facial curvature. Since positive curvature and negative curvature faces can be separated by large regions of zero curvature faces, we cannot use the curvature of local regions to rule out certain configurations as in the previous sections. However, these surfaces have additional structure which is not present in RPSs with faces of degree five. Let \(P\) be a RPS with data \((\Gamma, \Sigma, \psi)\). The geometric realization of each face in the graph is composed of pairs of parallel edges.
\begin{defn}
A \emph{band} \(B_{e,f}\) is a simple cycle in the dual graph \(\bar{ \mathcal G}\) of \(\mathcal G\) starting at edge \(e\) and face \(f\) of \(\mathcal G\) with the property that the geometric realizations of the primal edges associated to consecutive edges in the cycle are parallel translates (in \(R^3\)) of each other.
\end{defn}
The \emph{dual graph} \(\mathcal{ \hat G} \) of \(\mathcal G\) is the graph whose vertices correspond to faces of \(\mathcal G\) and where two vertices in \(\mathcal{\hat G}\) are adjacent exactly when the corresponding faces in the primal graph \(\mathcal G\) share an edge.
We can cut \(\Sigma\) along the edges in \(\mathcal G\) bounding a band \(B_{e,f}\). When the RPS has genus zero the cut disconnects \(\Sigma\) into two hemispheres \(H_1\) and \(H_2\), and an annulus. If every face in the band has degree four, then we can glue the two hemispheres together by identifying pairs of boundary edges, and their incident vertices, which were incident to the same face in the band \(B_{e,f}\) to form a new surface \(P'\). The geometric realization \(\psi'\) of \(P'\) is
\[\psi'(x)=\begin{cases} \psi(x)-e \qquad & \text{if } x\in H_1, \\ \psi(x) \qquad& \text{if } x \in H_2\end{cases}\]
where \(\psi\) is the realization of the original surface. We call the process of removing a band and gluing two hemispheres together \emph{band surgery}. The new surface \(P'\) satisfies all conditions of being a RPS except for one: two adjacent faces may have the same image under \(\psi\). Removing every such pair of dangling faces forms an actual RPS with the same genus as \(P\). If every face in a genus zero RPS has degree four then we can use band surgery to prove the following theorem.
\begin{thm}\label{thm: square}
Every genus zero RPS with faces of degree four can be realized as a union of cubes glued together along common facets.
\end{thm}
\begin{proof}
We use complete induction on the number of faces in the surface to prove the theorem. Since the total curvature is \(4\pi\) and the most curvature a vertex can have is \(\pi/2\), there are at least eight vertices in the surface. Likewise, the facial curvature can be as large as \(2\pi/3\) when all vertices are degree three. Thus there are at least six faces in the surface. The only possible geometric realization of a RPS with six faces is that of a cube. This proves the base case of the theorem. For our inductive hypothesis we assume the theorem is true for all surfaces with fewer than \(n\) faces with \(n \geq 6 \). Let \(P\) be a RPS with \(n\) faces. Let \(B_{e,f}\) be a band in the surface through face \(f\) with edges that are parallel transports of \(e\) (in their geometric realizations). Since all faces in the surface have degree four we can use band surgery to remove \(B_{e,f}\) and form a new surface with fewer than \(n\) faces. Remove all pairs of dangling faces until what remains is a RPS which we call \(P'\). By induction it is a union of cubes. Let \(\gamma_1\) and \(\gamma_2\) denote the boundary edges of \(B_{e,f}\) in \(P\) which are identified to form \(P'\). Since \(P'\) is a union of cubes, the arc \(\gamma_1\) can be realized by a \emph{cycle} in a surface built by gluing cubes together along common facets. A \emph{cycle} is a sequence of alternating edges and vertices starting and ending at the same vertex such that each edge is incident to the two vertices preceding and succeeding it in the sequence and without repeated edges. Splitting \(P'\) along \(\gamma_1\) and inserting a new layer of cubes forms a surface \(\tilde P\). For each pair of dangling faces that was removed we glue in a cube, possibly identifying faces of neighboring cubes. The resulting RPS \(Q\) can be realized as a union of cubes. Since \(Q\) has the same genus as \(P\) and the surface graphs of the two surfaces are isomorphic, we conclude that \(P\) can be realized as a union of cubes.
\end{proof}
When faces of degree eight are also present in the surface, the surface may not have a band in which every face has degree four. Band surgery doesn't work on bands with degree eight faces because after cutting the band out and gluing the two hemispheres together, the resulting surface may not have a valid geometric realization. However, we can still use bands to help us characterize the structure of these surfaces. A path in the dual graph has a \emph{turning point} at a primal face \(f\) if the two faces adjacent to \(f\) in the path intersect \(f\) along edges which are not parallel translates of each other in the geometric realization of the surface. A \emph{band bigon} is a simple cycle in the dual graph with exactly two turning points. Every band bigon is formed by two intersecting bands which intersect at the primal faces corresponding to the two turning points. The faces of the bigon are the primal faces corresponding to the dual vertices of the bigon. The geometric realization of the two turning points are faces in the RPS which lie in parallel planes in \(\mathbb{R}^3\) since both faces contain parallel transports of the edges determining the bands. At a turning point the dot product between the two unit vectors determining the bigon is either \(0, 1/\sqrt{2}\) or \(-1/\sqrt{2}\). Since the geometric realization of the RPS sends every edge of the graph to a unit vector in \(\mathbb{R}^3\), we will often abuse notation and identify an edge with its corresponding unit vector in \(\mathbb{R}^3\).
Cutting \(\Sigma\) along the boundary of a bigon disconnects the surface into two disks and an annulus. If one of the disks does not contain any bigons then the bigon is said to be \emph{minimal}. We call the subset of \(\Sigma\) corresponding to the union of the annulus and the disk which doesn't contain any bigons, the \emph{interior} of the bigon. The \emph{strict interior} of the bigon is just the disk which contains no bigons. It is an easy consequence of the Jordan curve theorem that on a genus zero RPS, the bands forming a minimal bigon pass through adjacent edges of the faces corresponding to the turning points of the bigon (see lemma \ref{lem: no sq oct}).
After a careful analysis of all minimal bigons, we show that the there are only a few possible geometric realizations of the interior of a minimal bigon. For any minimal bigon, the arc \(\gamma\) bounding the interior can be realized as a cycle on a surface built out of a union of cubes and octagonal prisms glued together along common facets.
This result is established in a sequence of lemmas and is crucial in the proof of theorem \ref{square-oct}. The first lemma in the sequence states that the turning points of a minimal bigon are either both squares or both octagons. To prove this we use the elementary fact that any band which passes through the interior of a minimal bigon must cross both bands forming the bigon. This is an easy corollary of the Jordan curve theorem combined with the assumption of minimality. A genus zero RPS cannot have a simple cycle in the dual graph with exactly one turning point (a \emph{monogon}) because no band can pass through two edges of a face which are not parallel to each other.
\begin{lem}\label{lem: no sq oct}
On a genus zero RPS, no minimal bigon can have a square at one turning point and an octagon at the other.
\end{lem}
\begin{proof}
Suppose two distinct bands \(B_v\) and \(B_h\) form a minimal bigon with one turning point a degree four face and the other a degree eight face. Label the faces \(S\) and \(O\) respectively. Since the two bands cross in a degree four face the dot product of their corresponding unit vectors must satisfy \(|v \cdot h|=1\). Thus the geometric realizations of \(S\) and \(O\) are parallel to the plane spanned by \(v\) and \(h\). The edges on \(O\) which are parallel transports of \(v\) and \(h\) cannot be adjacent which implies that there must be an edge between them and a band through this edge which passes through the interior of the minimal bigon. However, the RPS is topologically a sphere so by the Jordan curve theorem this band must exit the bigon and in the process cross either \(B_v\) or \(B_h\). In either case this contradicts the assumption of minimality.
\end{proof}
We are now able to classify bigons based on the types of their turning points. A minimal square bigon has squares at each of its two turning points and similarly a minimal octagon bigon has octagons at each of its two turning points.
\begin{lem}\label{lem: int oct}
Suppose \(B_v\) and \(B_h\) are two bands forming a minimal octagon bigon on a genus zero RPS. Two bands, neither of which are part of the bands forming the bigon, cannot cross in the interior of this minimal bigon.
\end{lem}
\begin{proof}
Let \(B_h\) and \(B_v\) be the two bands forming the minimal octagon bigon and let \(O_1\) and \(O_2\) be the two octagons at the turning points of the bigon. As previously noted, the bands of the bigon must pass through adjacent edges of the octagons at the turning points. Since the geometric realizations of \(O_1\) and \(O_2\) are regular octagons with unit edge lengths, the angle between \(v\) and \(h\) is fixed so that \(|h \cdot v | = 1/\sqrt{2}\). By changing coordinates we may assume \(v=(1,0,0)\) and \(h=1/\sqrt{2} (1,1,0)\). Since any band through the bigon must exit, the dot product of the unit vector associated to any band through the bigon with \(v\) or \(h\) is either \(0, 1/\sqrt{2}\) or \(-1/\sqrt{2}\).
First we show that the faces along \(B_h\) and \(B_v\) in the minimal bigon in between \(O_1\) and \(O_2\) all have degree four. Suppose there were an octagon on the boundary of the minimal bigon and without loss of generality that it's on \(B_v\). Let \(a,b\) and \(c\) denote the directions of the geometric realizations of the consecutive edges on this octagon with bands through them that enter the bigon. Since the realization of this face is a regular octagon we must have the following relations
\[ | a \cdot v|=|c \cdot v| = |a \cdot b| = | b \cdot c |= 1/\sqrt{2} \text{ and } |b \cdot v| =0.\]
However, one of the edges of the octagon is a parallel transport of \(v\) and since the realization of the octagon is a plane these equations cannot all be satisfied. For \(|b \cdot v|=0\) implies \(b=(0,b_2, \pm \sqrt{1-b_2^2})\) but \(B_b\) must cross \(B_h\) which implies \(b_1=0 \text{ or } \pm 1\). Then there are six possibilities for \(a\),
\[ a= 1/\sqrt{2}(1, \pm 1,0), 1/\sqrt{2}(1,0, \pm 1) \text{ or } 1/\sqrt{2}(0,1 \pm 1)\]
but none of these directions are allowed because \(B_a\) must cross \(B_h\).
Now suppose two bands cross in a face in the interior of the bigon. Since this face cannot lie on \(B_h\) or \(B_v\), there are two edges \(e \) and \(g\) on the face with \(e \cdot g=0\). By the assumption of minimality \(B_e\) crosses both \(B_h\) and \(B_v\). Likewise \(B_g\) crosses both \(B_h\) and \(B_v\) as well. Since there are no octagons along \(B_h\) and \(B_v\), these crossings must all occur at degree four faces. Thus,
\[ | e\cdot h| = |g \cdot h|= |e \cdot v| = | g\cdot v|=0\]
but there do not exist vectors in \(\mathbb{R}^3\) which satisfy both the above equations and \(e\cdot g =0\).
\end{proof}
The preceding lemma implies that the geometric realization of a minimal octagon bigon is in fact part of an octagonal prism. In other words, every face in the interior of the minimal bigon has degree four except for the two degree eight turning points. The dihedral angles between degree four faces are either \(\pi\) or \(3\pi/4\) and between degree eight and degree four faces the angles are \(\pi/2\). Moreover, every face in the interior of the bigon lies on one of the bounding bands and all bands that cross through the bigon are parallel transports of the same unit vector.
A minimal bigon is bounded by two cycles one of which is not adjacent to a face in the interior of the bigon. Let \(C\) be this cycle. Cutting \(\Sigma\) along \(C\) disconnects the surface into two hemispheres \(H^1\) and \(H^2\). Assume that \(H^1\) is the hemisphere containing the interior of the minimal bigon. As shown in the preceding paragraph, we can glue a hemisphere from a RPS which can be realized as an octagonal prism to \(H^1\) resulting in a RPS whose realization is an octagonal prism. Gluing the other hemisphere to \(H^2\) forms a new RPS with data \((\Sigma' , \mathcal G', \psi'\) in which \(\mathcal G'\) has two fewer degree eight faces than the original graph \(\mathcal G\). By construction, \(\Sigma'\) is homeomorphic to \(\Sigma\) so this operation does not change the genus of the surface. The geometric realization is defined from the original geometric realization \(\psi\) as \(\psi'(x)=\psi(x)\) for \(x\in H^2\) and extended by linearity to the complement \(\Sigma' \setminus H^1\). This operation is a special case of polyhedral surgery and we refer to it as \emph{octagon removal surgery}. It is possible that octagon removal surgery creates dangling faces, but after modifying \(C\) to include one of these faces there will be no dangling faces in the new surface. For later reference we record the content of this paragraph as the following lemma.
\begin{lem}\label{lem: oct red}
Octagon removal surgery removes two degree eight faces from any genus zero RPS with a minimal octagon bigon.
\end{lem}
Square bigons are potentially more complicated, however in some cases they turn out to be very simple to analyze.
\begin{lem}\label{lem: sq bigon}
If a genus zero RPS has a minimal square bigon without any degree eight faces on the boundary of the bigon, then the dihedral angle between any two faces in the interior of the bigon, in the realization of the surface, is \(\pi\). In particular, the realization of a minimal square bigon that does not contain an octagon consists of four faces of a rectangular prism.
\end{lem}
\begin{proof}
Let \(B_a\) and \(B_b\) be the two bands forming the minimal square bigon. Suppose that two adjacent faces have a dihedral angle which is not \(\pi\) and let \(B_c\) and \(B_d\) be the two bands that pass through these two faces. Since both \(c\) and \(d\) are orthogonal to \(a\) and \(b\), \(c\) must be parallel to \(d\) which implies that the dihedral angle between the faces is \(\pi\).
Now, suppose there is a face strictly in the interior of the bigon. Let \(B_c\) and \(B_d\) be the bands that cross at this face. Since both bands must pass through both sides of the bigon this creates four mutually perpendicular vectors in \(\mathbb{R}^3\) which is a contradiction. Thus the bigon consists of two bands that cross at two squares and every face in the interior of the bigon lies on one of the two belts, \(B_a\) or \(B_b\).
\end{proof}
When a minimal square bigon satisfies the conditions of \ref{lem: sq bigon}, its realization consists of four facets of a prism and we can use polyhedral removal surgery to remove these four facets and replace them by the other two facets of the prism in a manner analogous to octagon removal surgery. Polyhedral removal surgery applied to a prism reduces the number of faces in the surface's surface graph by \(2\). We call this special case of polyhedral surgery, \emph{prism removal surgery}.
\begin{lem}\label{lem: edge dirs}
Let \(f\) be a face in the interior of a minimal square bigon and let \(e\) be a unit vector determined by the geometric realization of an edge incident to \(f\). Choose coordinates so that the bigon is formed by two bands, \(B_v\) and \(B_h\), with \(h=(1,0,0)\) and \(v=(0,1,0)\). Then the vector \(e\) is a parallel translate of one of eight possible unit vectors:
\[\frac{1}{\sqrt{2}}\left (1,\pm 1,0 \right), \frac{1}{\sqrt{2}} \left(1,0, \pm 1\right)), \frac{1}{\sqrt{2}} \left(0,1,\pm 1\right)) \text{ or }\left (0,0,\pm 1\right)).\]
\end{lem}
\begin{proof}
Let \(B_e\) be the band through the face \(f\) starting at edge \(e\). Since the bigon is minimal, \(B_e\) must pass through both \(B_v\) and \(B_h\) and it must cross at either a square or an octagon. Thus the angle between \(e\) and \(v\) is either \(\pi/2\), \(3\pi/4\), \(5\pi/4\) or \(3\pi/2\). Likewise for the angle between \(e\) and \(h\). Therefore \(e\) is a parallel translate of the eight directions listed in the statement of the theorem.
\end{proof}
\begin{lem}\label{lem: adj bands}
Suppose there is a degree eight face on the boundary of a minimal square bigon on a genus zero RPS and let \(B_a,B_b\) and \(B_c\) denote the other three bands through this face in cyclic order. Each face on \(B_a\) in the interior of the bigon is adjacent to a face on \(B_b\). Similarly, each face on \(B_c\) in the interior of the bigon is adjacent to a face on \(B_b\).
\end{lem}
\begin{proof}
Choose coordinates so that the square bigon is formed by two bands, \(B_v\) and \(B_h\), with \(h=(1,0,0)\) and \(v=(0,1,0)\). Let \(O\) be the degree eight face in the statement of the theorem and without loss of generality assume that it lies on \(B_v\). Since the surface is topologically a sphere, by the Jordan curve theorem \(B_a, B_b\) and \(B_c\) must cross \(B_h\). Since \(a,b,c\) and \(v\) are the directions of four consecutive edges of a face whose realization is a regular octagon, we have
\[|a\cdot b| = | b \cdot c|= |c \cdot v| =|a \cdot v| = 1/\sqrt{2}\]
and
\[|a\cdot c| = | b \cdot v|= 0.\]
Moreover \(B_a,B_b\) and \(B_c\) all pass through \(B_h\), so the dot product of each with \(h\) is either \(0\), \(1/\sqrt{2}\) or \(-1/\sqrt{2}\). From lemma \ref{lem: edge dirs} we know that there are eight possible choices for the directions of the edges. The only possible directions for \(a,b,\) and \(c\) that satisfy all of these conditions are
\[
a = 1/\sqrt{2}(0,1,-1), \
b=(0,0,1), \text{ and }
c= 1/\sqrt{2} (0,1,1).\
\]
Now suppose that there is a face in the interior of the minimal square bigon between \(B_b\) and \(B_c\). This face could have degree four or eight, but in both cases there are two bands through this face determined by unit vectors which are orthogonal to each other. Let \(B_x\) and \(B_y\) denote these two bands. Since the bigon is minimal, \(B_x\) must cross \(B_c\) or \(B_a\), and likewise for \(B_y\). Of the eight possible directions for \(x\) and \(y\), the only directions consistent with the crossing condition is \(x=1/\sqrt{2}(0,1,-1)\) and \(y=1/\sqrt{2}(0,1,1)\). Computing \(x \cdot h\) we find that \(B_x\) crosses \(B_h\) at a degree eight face. Let \(O_1\) be this face and let the directions of the edges of the realization be \(a',b',\) and \(c'\). The fourth direction is a parallel translate of \(B_h\) because \(O_1\) is on \(B_h\). A similar calculation to the one determining the directions \(a,b\), and \(c\) shows that the unit vectors determining these directions are
\[
a' = 1/\sqrt{2}(1,0,-1), \
b'=(0,0,1), \text{ and } \
c'= 1/\sqrt{2} (1,0,1).
\]
Likewise \(B_y\) also crosses \(B_h\) at a degree eight face, which we label \(O_2\). An identical argument shows that the same unit vectors as above determine the directions of the realizations of the edges of \(O_2\). However, computing the dot products we find \(B_x\) crosses \(O_1\) along the edge in the direction of \(b'\) and \(B_y\) crosses \(O_2\) along the edge in the direction of \(b'\). This is a contradiction because \(x\) and \(y\) are orthogonal directions.
\end{proof}
There are two remaining operations to define before we can prove theorem \ref{square-oct}. In the proof we analyze all minimal bigons and use polyhedral surgery to decrease either the number of faces in the surface or the number of degree eight faces in the surface. When a minimal square bigon has degree eight faces along its boundary, the bigon can be very complex to analyze. The two operations we introduce are the \emph{cube flip} and the \emph{prism flip}. Both allow us to decrease the number of faces in the bigon, thereby reducing the complexity of the bigon.
First, we define the cube flip. Let \(h\) be a turning point of a minimal square bigon and let \(f\) and \(g\) be the two faces in the interior of the bigon which are adjacent to \(h\). If all three faces have degree four, then in their realization they form three faces of a cube. In an operation we call a \emph{cube flip}, we replace these three faces in the RPS by the three faces that form the other half of the cube. Figure \ref{cflip} shows a subgraph of a surface graph and the effect of a cube flip on this subgraph. The geometric realization of the new RPS is defined by extending the geometric realization of the original surface to the three faces \(f',g'\) and \(h'\) linearly so that the realization of each face is a Euclidean square with unit edge lengths.
\begin{figure}
\begin{center}
\includegraphics[scale=0.6]{sqflip.pdf}
\caption{Cube flip}
\label{cflip}
\end{center}
\end{figure}
Suppose that there exists a minimal bigon and that a cube flip can be performing on one of the turning points. Let \(B_h\) and \(B_v\) denote the bands determining the bigon and \(f_1\) and \(f_2\) the two faces corresponding to the bigon's turning points. After a cube flip at \(f_1\), the bigon formed by \(B_h\) and \(B_v\), which passes through \(f_2\), also passes through one of the new faces formed by the cube flip. The new bigon is still a minimal bigon but it contains fewer bands through its interior than the original minimal bigon.
The prism flip is defined similarly to the cube flip. In the prism flip we replace five faces in the RPS, whose realization forms part of an octagonal prism, with the five faces that form the other half of the prism. We skip a formal definition of the prism flip because it is so similar to the cube flip. A prism flip that involves the face at the turning point of the bigon reduces the number of bands that cross through the interior of the bigon. Finally, all of the tools are in place to prove theorem \ref{square-oct}.
\begin{proof}[Proof of theorem \ref{square-oct}.]
Let \(P\) be a RPS with data \((\Sigma, \Gamma, \psi)\). We use induction on the number of degree eight faces in the RPS. Let \(n\) be the number of degree eight faces in the surface. The base case, \(n=0\), is theorem \ref{thm: square}. Assume that the theorem is true for any RPS with fewer than \(n\) faces. Since the surface has a finite number of faces, there is always at least one minimal bigon. There are two cases depending on the type of the bigon. First, assume that the minimal bigon is an octagon bigon. Lemma \ref{lem: oct red} explains how octagonal removal surgery applied to this bigon produce a new RPS \(P'\) with two fewer degree eight faces than \(P\). By the induction hypothesis this surface can be realized as a union of cubes and prisms. Since the bigon that was removed can be realized as part of an octagonal prism, we can glue an octagonal prism to \(P'\) to form a surface which has the same surface graph and genus as \(P\). Thus \(P\) can be realized by a union of cubes and prisms.
In the second case, the minimal bigon is a square bigon. Suppose that this bigon is determined by two bands, \(B_h\) and \(B_v\), and has a turning point at a face \(f\). Let \(g\) and \(h\) be the two faces in the interior of the minimal bigon which are adjacent to \(f\). If both \(g\) and \(h\) have degree four then we can use a cube flip to decrease the number of bands through this bigon. If one of \(g\) and \(h\) has degree four and the other degree eight, then by lemma \ref{lem: adj bands} there are five faces, including \(f\), whose realization is a part of an octagonal prism. Thus we can always use a cube flip or a prism flip to decrease the number of bands through the bigon.
Since the bigon has finitely many faces, after finitely many flips the bigon will have one face which is adjacent to both turning points. If this face has degree four then we can remove a cube with polyhedral removal surgery and decrease the number of degree four faces in the surface by \(2\). If the face has degree eight then we could remove an octagonal prism with polyhedral removal surgery and decrease the number of degree eight faces in the surface by \(2\). This procedure reduces the number of degree four faces in the surface monotonically. However, we cannot remove all of the degree four faces from the surface. The surface has genus zero and thus has positive curvature vertices. At least two squares meet at every positive curvature vertex. Thus after removing finitely many faces from the surface, we must reach a surface \(P'\) in which the only minimal bigons are octagon bigons. Removing this octagon bigon and applying the induction hypothesis we find that the surface can be realized as a union of cubes and prisms. Since at every step we have removed either a part of a cube or a prism, we can glue these cubes and prisms back to \(P'\) to form a surface with the same surface graph and genus as \(P\). Therefore \(P\) can be realized as a union of cubes and prisms.
\end{proof}
\section{Examples of higher genus RPSs}\label{sec:counter}
In this section we construct three examples of high genus RPSs which are not unions of convex polyhedra. This can be seen by the absence of certain faces in the surfaces. All of the examples in this section can be constructed in two steps. First, place convex polyhedra at the vertices of a 3-cube or a 4-cube. Second, remove certain faces from each polyhedron and connect the boundary components using prisms with matching boundary components. Whether a 3-cube or a 4-cube is used depends on the structure of the convex polyhedra.
Figure \ref{truncOctMol4d} shows a genus 49 surface whose faces have degree four. It is constructed out of truncated octahedra placed at the vertices of a 4-cube and connected by hexagonal prisms. All hexagonal faces have been removed from the constituent truncated octahedra and hexagonal prisms. The Euler characteristic of the surface is
\begin{align*}
\chi & = 16\cdot 2-16\cdot 8-64\cdot 6+64\cdot 6 = -96.
\end{align*}
\begin{figure}[h]
\begin{center}
\includegraphics[width=\textwidth]{truncOctMol4d}
\caption{A \((4)\)-RPS of genus 49}
\label{truncOctMol4d}
\end{center}
\end{figure}
Figure \ref{truncCubOctMol4d} shows a genus 49 surface whose faces have degree four and eight. It is constructed out of truncated cuboctahedra placed at the vertices of a 4-cube and connected by hexagonal prisms. All hexagonal faces have been removed from the constituent truncated cuboctahedra and hexagonal prisms. The Euler characteristic of the surface is
\begin{align*}
\chi & = 16\cdot 2-16\cdot 8-64\cdot 6+64\cdot 6 = -96.
\end{align*}
\begin{figure}[h]
\begin{center}
\includegraphics[width=\textwidth]{truncCubOctMolSqHex4d}
\caption{A \((4,8)\)-RPS of genus 49}
\label{truncCubOctMol4d}
\end{center}
\end{figure}
Figure \ref{truncCubOctMolSqHex} shows a genus 17 surface whose faces have degree four and eight. It is constructed out of truncated cuboctahedra placed at the vertices of a 3-cube and connected by octagonal prisms. All octagonal faces have been removed from the constituent truncated cuboctahedra and hexagonal prisms. The Euler characteristic of the surface is
\begin{align*}
\chi & = 8\cdot 2 - 8\cdot 6 -24\cdot 8+24\cdot 8 = -32.
\end{align*}
\begin{figure}[h]
\begin{center}
\includegraphics[width=\textwidth]{truncCubOctMolSqHex}
\caption{A \((4,6)\)-RPS of genus 17}
\label{truncCubOctMolSqHex}
\end{center}
\end{figure}
\subsection*{Acknowledgments}
I thank Richard Kenyon for suggesting the study of RPSs and for his advice throughout this project. I thank Sanjay Ramassamy for suggesting a simplification of the proof in Section \ref{sec:rps5}. I also thank Ren Yi for many helpful conversations.
\bibliographystyle{../hep}
|
{
"timestamp": "2018-04-17T02:12:22",
"yymm": "1804",
"arxiv_id": "1804.05452",
"language": "en",
"url": "https://arxiv.org/abs/1804.05452"
}
|
\section{Introduction
}
\label{sec:intro}
Companies can market their respective products through several possible channels,
the most prominent being mass media advertising (television, radio, newspapers, etc.), fixed Internet banner ads, banner ads based on browsing history,
recommendations based on attributes
(location, age group, field of work, etc.), recommendations based on friends' purchases,
sponsored search ads,
social media,
sponsored Internet reviews, and so on. Potential customers or nodes also get indirectly influenced through their friends owing to word-of-mouth marketing.
In order to make optimal use of these channels, a company would want to make the decision of how to invest in each channel, based on the investment strategy of competitors who also market their products simultaneously.
\begin{comment}
Social networks have been an integral part of human lives for thousands of years, hence the term `social animal'. People who are linked in a social network discuss several issues and share various pieces of information, be it serious or casual, be it voluntarily or involuntarily, be it with an intention of diffusing it to a wider audience or keeping it private within a friend circle. In short, social networks play an important role in information diffusion and sharing. Given this property of a social network, it is natural for companies to exploit it to maximize their revenue. A primary method used by companies is based on viral marketing where the existing customers market the product among their friends. Campaigning is another example where a particular idea or a series of ideas is presented to some audience and hence, the idea is spread through the audience. It is clear that if one's objective is to diffuse some information and make it available to a wider audience for whatever reasons, one cannot ignore the possibility of using the social network. Moreover, in the present age of Internet and the ever-increasing popularity of various social networking sites, social networks have become an effective and efficient media for information diffusion.
Any company generally markets its product in order to reach and hence convince the potential customers to buy that product. In order to do this, the company would want to make optimal use of the various resources (such as time, effort, money, etc.). One of the most important resources is media, which can be used for various types of advertisements such as television and newspaper advertisements, search-based as well as social advertisements, word-of-mouth recommendations, etc. Moreover, in order to make optimal use of these resources, a company would ideally want to make the decision of how to invest on each resource, based on the strategy of competing companies (that is, how much they invest in each resource).
\end{comment}
This paper aims to present a framework for competitive influence maximization
in the presence of several marketing channels.
We focus on modeling three
channels, namely, viral marketing, mass media advertisement, and recommendations based on friends' purchases using social advertisement.
\subsubsection{Viral Marketing}
In our context, a social network can be represented as a weighted, directed graph, consisting of nodes which are potential customers.
The model we propose for influence diffusion in social network is a generalization of the well-studied linear threshold (LT) model
\cite{networkscrowdsmarkets}.
\begin{comment}
In LT model, every directed edge $(v,u)$ has weight $b_{u,v} \geq 0$, which is the degree of influence that node $v$ has on node $u$, and every node $u$ has an influence threshold $\chi_u$. The weights $b_{u,v}$ are such that $\sum_v b_{u,v} \leq 1$. Owing to lack of knowledge about the thresholds, which are held privately by the nodes, it is assumed that the thresholds are chosen uniformly at random from $[0,1]$. The diffusion process starts at time step 0 with the initially activated set of seed nodes, and proceeds in discrete time steps, one at a time. In each time step, a node is influenced or activated if and only if the sum of influence degrees of the edges incoming from activated neighbors (irrespective of the time of activation of the neighbors) crosses its own influence threshold, that is, when
$
\sum_v b_{u,v} \geq \chi_u
$.
Nodes, once activated, remain activated for the rest of the diffusion process.
The diffusion process stops when it is not possible to activate or influence any further nodes.
\end{comment}
Given such a model, a company would want to select a certain number of seed nodes to trigger viral marketing so that maximum number of nodes get influenced (buy the product)
\cite{kempe2003maximizing}.
\begin{comment}
In this paper, we assume that nodes are anonymous, given the network, that is, if we interchange any two nodes, the dynamics of the diffusion remains unchanged. In other words, we ignore any node-specific information such as online node profile and browsing behavior.
We focus on viral marketing in the presence of two other types of advertising channels, which are not explicitly based on the online node profile or browsing behavior:
\end{comment}
\subsubsection{Mass media advertisement}
This is one of the most traditional way of marketing where a company advertises its product to the masses using well-accessible media such as television, radio, and newspaper.
\begin{comment}
Even though the advertisement could be targeted towards a particular audience (like people watching a particular program on television or reading a particular page in newspaper), mass media generally allows potentially anyone to access the advertisement.
\end{comment}
The timing of when to show the ads is critical to ensure optimal visibility and throughput.
\subsubsection{Social advertisement based on friends' purchases}
While making purchasing decisions, nodes rely not only on their own preferences but also on their friends', owing to social correlation due to homophily (bias in friendships towards similar individuals) \cite{chua2012generative}. This, in effect, can be harnessed to suggest products to a node based on its friends' purchasing behaviors.
If a node has high influence on its friend (which is accounted for in diffusion models like LT), it is likely that the two nodes are similar. However, if the influence is low, it is not conclusive whether the nodes are dissimilar. So in addition to the influence parameter considered in LT-like models, marketing in practice requires
another parameter that quantifies similarity between nodes.
Note that since diffusion models do not consider this similarity, they alone cannot justify why two nodes having negligible influence on each other, display similar behaviors.
It has also been observed in Twitter that almost 30\% of information is attributed to factors other than network diffusion
\cite{myers2012information}.
The effect of such factors could hence be captured using the similarity parameter.
\vspace{-1.1mm}
\subsection{Related Work
\vspace{-1mm}
}
\label{sec:relevant}
The problem of influence maximization is well-studied in literature on social network analysis.
It is known that computing the exact value of the objective function for a given seed set (the expected number of influenced nodes at the end of diffusion that was triggered at that set), is \#P-hard under
the LT model \cite{chen2010scalablelt}.
However, the value can be well approximated using sufficiently large number of Monte-Carlo simulations.
Though the influence maximization problem under LT model is NP-hard,
the objective function is non-negative, monotone, and submodular; so greedy hill-climbing algorithm provides an
approximation guarantee for its maximization
\cite{kempe2003maximizing}.
There exist generalizations of LT model, e.g., general threshold model \cite{kempe2003maximizing}, extensions to account for time \cite{chen2012time}, and extensions to account for competition
\cite{borodin2010threshold,pathak2010generalized}.
State-of-the-art heuristics such as
LDAG \cite{chen2010scalablelt} and Simpath \cite{goyal2011simpath}
perform close to greedy algorithm while running several orders of magnitude faster.
There also exist algorithms
that provide good performance irrespective of the objective function being submodular
\cite{narayanam2010shapley}.
The problem of competitive influence maximization wherein multiple companies market competing products using viral marketing has also been studied
\cite{bharathi2007competitive,goyal2012competitive}.
Also, more realistic models have been developed, where influences not only diffuse simultaneously but also interact
with each other
\cite{myers2012clash,zarezade2017correlated}.
The impact of recommendations and word-of-mouth marketing on product sales revenue is well studied in marketing literature
\cite{godes2012strategic,van2010viral,aral2011creating}.
Biases in product valuation and usage decisions when agents consider a product that offers new features of uncertain value, have been investigated \cite{meyer2008biases}.
It has also been discussed
how marketers can apply latent similarities of customers for segmentation and targeting
\cite{braun2011scalable}.
\begin{comment}
\cite{myers2012information} develop a model in which information can reach a node either via links of the social network or through external sources and find that factors external to the network play a significant role in information diffusion.
\end{comment}
\begin{comment}
\subsection{Our Contributions}
\label{sec:contrib}
\begin{itemize}
\item A vectors-based model that considers both personal aggregation and clash of contagions of multi-feature products
\item Integration of multiple channels into the social network
\item Cross entropy method
\end{itemize}
\end{comment}
\vspace{-1mm}
\section{The Proposed Framework
\vspace{-1mm}
}
\label{sec:model}
\noindent
We propose a framework to facilitate study of different marketing aspects using a single model,
capturing several factors:
\begin{enumerate}[leftmargin=*]
\item
Companies market their products
using multiple channels;
\item
Diffusions of different products are mutually dependent;
\item
Each node aggregates the mass media advertisements, recommendations, and neighbors' purchasing decisions.
\end{enumerate}
We first describe LT model, followed by our competitive multi-feature generalization, and then integration of other channels into this generalized model.
Table \ref{tab:notation} presents notation.
In LT model, every directed edge $(u,v)$ has weight $b_{uv} \geq 0$, which is the degree of influence that node $u$ has on node $v$, and every node $v$ has an influence threshold $\chi_v$. The weights $b_{uv}$ are such that $\sum_u b_{uv} \leq 1$. Owing to thresholds being private information to nodes, they are assumed to be chosen uniformly at random from $[0,1]$. The diffusion process starts at time step 0 with the initially activated set of seed nodes, and proceeds in discrete time steps. In each time step, a node gets influenced if and only if the sum of influence degrees of the edges incoming from its already influenced neighbors
crosses its influence threshold, that is,
$
\sum_u b_{uv} \geq \chi_v
$.
The process stops when no further nodes can be influenced.
Formally, let $u\in \mathcal{N}(v)$ if and only if $b_{uv} \neq 0$.
Let $\mathcal{B}(t)$ be the set of nodes influenced by time $t$.
Then
\begin{table}[t!]
\vspace{-1.6mm}
\centering
\caption{Notation}
\label{tab:notation}
\vspace{-2mm}
\begin{tabular}{|p{.7cm}| p{7cm}|}
\hline
\T \B
$b_{uv}$ & influence weight of node $u$ on node $v$ \\ \hline
\T \B
$\mathcal{N}(v)$ & set of influencing neighbors of $v$ \\ \hline
\T \B
$h_{uv}$ & similarity between nodes $u$ and $v$ \\ \hline
\T \B
$\chi_v$ & threshold of node $v$ \\ \hline
\T \B
$\gamma^p$ & total budget for the marketing of product $p$ \\ \hline
\T \B
$k^p$ & budget for seed nodes for viral marketing of product $p$ \\ \hline
\T \B
$\beta_t^p$ & mass media advertising weight of product $p$ in time step $t$ \\ \hline
\T \B
$\alpha^p$ & social advertising weight of product $p$ \\ \hline
\T \B
$\mathcal{A}_v$ & final aggregate preference of node $v$ \\ \hline
\T \B
$\mathcal{P}_{v}$ & product bought by node $v$ \\ \hline
\T \B
$\mathcal{B}(t)$ & set of nodes influenced by time $t$ \\ \hline
\end{tabular}
\vspace{-4mm}
\end{table}
\begin{small}
\vspace{-3mm}
\begin{gather}
\hspace{-3mm}
v\in \mathcal{B}(t)\backslash \mathcal{B}(t-1) \text{\;\;iff\;}
\sum_{\substack{u\in \mathcal{N}(v) \\ u\in \mathcal{B}(t-2)}} \!\!\!\!\! b_{uv} < \chi_v \text{\;\;and} \sum_{\substack{u\in \mathcal{N}(v) \\ u\in \mathcal{B}(t-1)}} \!\!\!\!\! b_{uv} \geq \chi_v
\label{eqn:LT}
\end{gather}
\vspace{-3mm}
\end{small}
\vspace{-1mm}
\subsection{Competitive Multi-feature Generalization of LT Model
\vspace{-1mm}
}
Products these days, be they toothpastes or mobile phones, come with several features with different emphases on different features.
Let such an emphasis be quantified by a real number between 0 and 1.
That is, a product can be represented by a vector
of mutually
independent features $p=(p_1,\cdots,p_f)$, where $p_i \in [0,1]$.
Let the features of each product be suitably scaled such that $||p|| = 1$. Note that such scaling may not be feasible when there is a product $p$ which offers strictly better features than product $q$ ($\forall i:p_i>q_i$); so let the products be such that one of the features corresponds to `null'. So $p$ would have a lower null component as compared to $q$, thus making the scaling feasible (a higher null component would imply that the product has a poorer feature set).
\begin{comment}
One can argue that a node should have different thresholds for different features; but as thresholds are generally assumed to be private information, it would only add to the uncertainty in the dynamics of diffusion. So we consider a common threshold for all the features put together, and not individually.
\end{comment}
Our model for a node getting influenced is analogous to that used in classical mechanics to study the initial motion of a body placed on a rough horizontal surface, as a result of several forces acting on it. In our context, a node is analogous to the body, and its threshold is analogous to static frictional force stopping it from moving. Such a force is equal to $\mu_s mg$, where $m$ is mass of the body, $g$ is acceleration due to gravity, and $\mu_s$ is the coefficient of static friction between the body and surface.
For $\mu_s g = 1$ unit for all nodes, the frictional force and analogously, the threshold equals mass, which is chosen uniformly at random from $[0,1]$ (as assumed in the LT model).
A node $v$ gets influenced in time step $t$ when the net force on it crosses its threshold value $\chi_v$;
let the net force correspond to aggregate vector (say $\mathcal{A}_v$).
Let $d(\mathcal{A}_v,p)$ be the angular distance between $\mathcal{A}_v$ and the product vector $p$. Since $||p||=1$,
\begin{figure}[t]
\vspace{-1.6mm}
\centering
\includegraphics[scale=.32]{geometric_network_2.pdf}
\\
\includegraphics[scale=.55]{geometric_2.pdf}
\vspace{-2mm}
\caption{Geometric interpretation of the proposed model
}
\label{fig:geometric_rep}
\vspace{-5mm}
\end{figure}
\begin{small}
\vspace{-2mm}
\begin{displaymath}
d(\mathcal{A}_v,p)
= \arccos \left( \frac{\mathcal{A}_v \cdot p}{||\mathcal{A}_v|| } \right)
\end{displaymath}
\vspace{-3mm}
\end{small}
A node buys a product whose angular distance from its aggregate vector is the least
(it can be easily shown that such a product would have the least Euclidean distance as well).
If there exist multiple such products, one of them is chosen uniformly at random.
\begin{comment}
Note that one can as well model a node to buy a product whose Euclidean distance from its aggregate vector is the least. But studying angular distance suffices owing to the following result.
\begin{proposition}
\label{prop:equivalence}
A product's Euclidean distance from a vector is the least among all the products iff its angular distance from that vector is the least among all the products.
\end{proposition}
\begin{proof}
Consider a vector $r$. Let $d(r,p)$ and $d^E(r,p)$ be the angular distance and Euclidean distance between $r$ and a product $p$, respectively. So by cosine rule, using the fact that $||p||=1$ for any $p$,
$
d^E(r,p) = \sqrt{ ||r||^2 + 1 - 2||r|| \cos(d(r,p)) }
$.
Minimizing this is equivalent to maximizing $\cos(d(r,p))$, which is equivalent to minimizing $d(r,p)$
\begin{small}
\vspace{-2mm}
\begin{align*}
\argmin_p d^E(r,p)
&= \argmin_p \left\{ ||r||^2 + 1 - 2 ||r|| \cos(d(r,p)) \right\} \\
&= \argmax_p \cos(d(r,p))
= \argmin_p d(r,p)
\end{align*}
\vspace{-2mm}
\end{small}
(since the range of $d(r,p)$ is $[0,\pi]$, and $\cos(\cdot)$ is monotone decreasing in this range).
\end{proof}
\end{comment}
\begin{comment}
It is generally assumed in literature that nodes are equally enthusiastic about word-of-mouth marketing irrespective of the product's nature. However, a node's interest would naturally be low for products that are not publicly visible like toothpaste, while it would be high for products like mobile phone.
We model this nature with an {\em enthusiasm parameter} $\delta^p$ for product $p$.
Hence each edge weight $b_{uv}$ transforms to $\delta^p b_{uv}$.
\end{comment}
Hence the competitive multi-feature version of (\ref{eqn:LT}) is
\begin{small}
\vspace{-3mm}
\begin{gather*}
v\in \mathcal{B}(t)\backslash \mathcal{B}(t\!-\!1) \;,\; \mathcal{A}_v \!=\!\!\! \sum_{u\in \mathcal{N}(v)} \!\!\! {b_{uv}\mathcal{P}_u}
\;,\;
\mathcal{P}_v = \argmin_p d(\mathcal{A}_v,p)
\end{gather*}
\vspace{-5mm}
\begin{gather*}
\text{iff\;\;\;}
\norm{ \sum_{\substack{u\in \mathcal{N}(v) \\ u\in \mathcal{B}(t-2)}} \!\!\!\!\! b_{uv}\mathcal{P}_u} < \chi_v
\text{\;\;\;and\;\;\;}
\norm{ \sum_{\substack{u\in \mathcal{N}(v) \\ u\in \mathcal{B}(t-1)}} \!\!\!\!\! b_{uv}\mathcal{P}_u} \geq \chi_v
\end{gather*}
\end{small}
A geometric interpretation of the proposed model is presented
in Figure \ref{fig:geometric_rep}.
Consider 2 competing products having 2 features, say $p=(p_1,p_2),q=(q_1,q_2)$.
In time step 0, $S_p$ and $S_q$ are selected for seeding by products $p$ and $q$, respectively.
In time step 1, node $v$ aggregates the purchasing decisions of its neighbors $S^p$ and $S^q$, hence obtaining the aggregate vector $0.4p+0.2q$.
Say $\sqrt{(0.4p_1+0.2q_1)^2 + (0.4p_2+0.2q_2)^2} < \chi_v$, so $v$ is not influenced yet.
However, $u$ and $w$ purchase products $p$ and $q$ respectively (since the influence weights from $S^p$ to $u$ and $S^q$ to $w$ are 1). Hence in time step 2, node $v$ aggregates the purchasing decisions of $u$ and $w$, hence obtaining the aggregate vector $\mathcal{A}_v = (0.4p+0.2q)+(0.1p+0.2q)$.
Say $||\mathcal{A}_v||= \sqrt{(0.5p_1+0.4q_1)^2 + (0.5p_2+0.4q_2)^2} \geq \chi_v$, so $v$ is now influenced and it purchases product $p$ if $d(\mathcal{A}_v,p)<d(\mathcal{A}_v,q)$, $q$ if $d(\mathcal{A}_v,p)>d(\mathcal{A}_v,q)$, else it chooses randomly.
\vspace{-1mm}
\subsection{Properties of the Generalized LT Model
\vspace{-1mm}
}
The standard LT model is a special case of the proposed model, where there is a single product with one feature, i.e., $p=(1)$.
As the problem of influence maximization in the standard LT model is NP-hard, we
have that
the problem of influence maximization in the proposed model is also NP-hard.
We now explain the multi-feature (vector-based) model with an illustrative example, which will also throw light on the properties of the objective function under the proposed model.
Recollect that the threshold for any node is chosen uniformly at random from $[0,1]$.
Let $\mathbb{P}_v^p (S^p,S^q)$ be the probability that node $v$ gets influenced by product $p$ when $S^p$ and $S^q$ are selected for seeding by $p$ and $q$ respectively.
In Figure \ref{fig:countereg_monotone}, let the two products be $p=(1,0)$ and $q=(0,1)$.
\hspace{-10mm}
\begin{wrapfigure}{l}{27mm}
\vspace{-.3cm}
\includegraphics[scale=.31]{countereg_monotone_2.pdf}
\vspace{-.6cm}
\caption{
Example
}
\vspace{-.3cm}
\label{fig:countereg_monotone}
\end{wrapfigure}
Let $\sigma^p(S^p,S^q)$ be the expected number of nodes influenced by $p$ when $S^p$ and $S^q$ are selected for seeding by $p$ and $q$ respectively.
That is, $\sigma^p(S^p,S^q) = \sum_i \mathbb{P}_i^p (S^p,S^q)$.
With diffusion starting from $S^p$ and $S^q$ simultaneously,
we have
$\mathbb{P}_a^p (S^p,S^q) = 0$ and
$\mathbb{P}_a^q (S^p,S^q) = 0.60$.
Note that if $a$ is influenced by $q$, then it influences $v$ with probability 0.70 even before the influence of $p$ reaches it, starting from $S^p$; now even if influence of $p$ reaches it, it is impossible for its aggregate preference to be closer to $p$ than to $q$.
So node $v$ can get influenced by $p$ only if $a$ is not influenced by $q$.
So $\mathbb{P}_v^p (S^p,S^q) = 0.3(1-\mathbb{P}_a^q (S^p,S^q)) = 0.12$. So all 30 nodes which have $v$ as sure influencer get influenced by $p$ with probability 0.12.
Hence $\sigma^p (S^p,S^q) = 1+2+0.12(1+30) = 6.72$.
Now if the seed set for $p$ is $T^p = S^p \cup u$, $\mathbb{P}_a^p (T^p , S^q) = 0$ due to an incoming edge of 0.6 from $S^q$ (it is impossible for the aggregate preference of node $a$ to be closer to $p$ than to $q$). However,
$\mathbb{P}_a^q (T^p , S^q) = \sqrt{0.6^2+0.4^2} \approx 0.72$.
From the argument similar as above, $\mathbb{P}_v^p (T^p , S^q) = 0.3(1-\mathbb{P}_a^q (T^p , S^q)) < 0.084$.
So the 30 nodes
get influenced by $p$ with probability less than 0.084.
Hence $\sigma^p (T^p ,S^q) < 2+2+0.084(1+30) < 6.61$.
That is, $\sigma^p (T^p ,S^q) < \sigma^p (S^p,S^q)$.
Thus while the objective function in the standard LT model follows monotone increasing property,
adding a node to a set in the generalized model could decrease its value;
this proves {non-monotonicity}.
It can also be shown using counterexamples that $\sigma^p(\cdot)$ is neither submodular nor supermodular.
\begin{comment}
However, it was observed using simulations that the objective function $\sigma^p(\cdot)$ was close to being monotone increasing, that is, the monotonicity property was satisfied in most instances, while it was far from being either submodular or supermodular.
The above example also shows that even if products $p$ and $q$ are mutually orthogonally aligned in space, $p$'s influence may affect a node's decision regarding purchasing $q$ (since the threshold is an $f$-dimensional sphere and not cube).
\end{comment}
\begin{figure}[t]
\vspace{-1.2mm}
\centering
\includegraphics[scale=.42]{MM_SA_integration_2.pdf}
\vspace{-2mm}
\caption{Integration of mass media and social advertisements into the network
}
\label{fig:integration}
\vspace{-4mm}
\end{figure}
\vspace{-1mm}
\subsection{Integrating Mass Media \& Social Advertising into Network\vspace{-1mm}
}
Let $\beta_t^p$ be the investment for mass media advertising of product $p$ in time step $t$ and $\beta^p$ be the total investment over time $T$, that is, $\beta^p = \sum_{t=1}^T \beta_t^p$.
For social advertising, we consider that a company would recommend or advertise product $p$ to a node when any of its friends $u$ has bought the product. Let $\alpha^p$ be the effort invested in social advertising.
Let $h_{uv}$ (or $h_{vu}$) be the parameter that quantifies the similarity between nodes $u$ and $v$.
So $\alpha^p h_{vu}$ could be viewed as the influence of such a recommendation on $v$ owing to the purchase of product $p$ by $u$.
Since the total influence weight allotted by node $v$ for viral marketing is $\sum_{u \in \mathcal{N}(v)} b_{uv}$, the total weight that it can allot for other channels is $1-\sum_{u \in \mathcal{N}(v)} b_{uv}$.
Hence the weights allotted for other channels ($({\beta}_t^p)_{t=1}^T$ and ${\alpha}^p$) would be scaled accordingly to obtain the values of $(\hat{\beta}_t^p)_{t=1}^T$ and $\hat{\alpha}^p$ specific to node $v$. A simple scaling rule is:
\begin{small}
\vspace{-2mm}
\begin{gather*}
\frac{\hat{\beta}_t^p}{{\beta}_t^p} = \frac{\hat{\alpha}^p}{\alpha^p} = \frac{1-\sum_{u \in \mathcal{N}(v)} b_{uv}}{ \sum_p \left( \sum_{u \in \mathcal{N}(v)} \alpha^p h_{uv} + \sum_{t=1}^T \beta_t^p \right)}
\end{gather*}
\vspace{-2mm}
\end{small}
In order to integrate mass media and social advertisements into the network, we add pseudonodes and pseudoedges corresponding to them, as illustrated in Figure \ref{fig:integration}.
Pseudonode $p$ corresponds to the product company itself (the figure shows two separate copies of pseudonode $p$ for the two channels for better visualization; they are the same pseudonode).
Pseudonode $p$ and all seed nodes selected for viral marketing, are influenced in time step 0.
For integrating mass media advertisement,
we create a set of pseudonodes $\{p^{(t)}\}_{t=1}^T$ (where $p^{(1)}$ corresponds to pseudonode $p$), and pseudoedges $\{(p^{(t-1)},p^{(t)})\}_{t=2}^T$
of weight 1.
Hence $p^{(t)}$ gets influenced with probability 1 in time step $t-1$ (see Figure \ref{fig:integration}).
We further create pseudoedges $\{(p^{(t)},v)\}_{t=1}^T$ for node $v$ such that $b_{p^{(t)} v} = \hat{\beta}_t^p$.
Since $p^{(t)}$ gets influenced in time \mbox{$t-1$},
node $v$ receives influence of $\hat{\beta}_t^p$ from pseudonode $p^{(t)}$ in time step $t$; this is equivalent to mass media advertisement.
For integrating social advertisement, corresponding to edge $(u,v)$, we create an intermediary pseudonode $w$ with a fixed threshold $\chi_w>0$, and pseudoedges such that $b_{uw}=\epsilon \in (0,\chi_w), b_{pw}=\chi_w - \epsilon, b_{wv} = \hat{\alpha}^p h_{uv}$ (see Figure \ref{fig:integration}).
Now if the reference friend $u$ is influenced by some product $q$, where the angle between products $p$ and $q$ be $\theta$, the intermediary pseudonode $w$ gets influenced if and only if
$|| (\chi_w - \epsilon)p + \epsilon q || \geq \chi_w$. Since $||p|| = ||q|| = 1$, this is equivalent to
~
\begin{small}
\vspace{-6mm}
\begin{align*}
&(\chi_w - \epsilon)^2 + \epsilon ^2 + 2\epsilon (\chi_w - \epsilon) \cos \theta \geq \chi_w ^2 \\
\Longleftrightarrow \;&
2 \epsilon (\chi_w - \epsilon)(\cos \theta - 1) \geq 0 \\
\Longleftrightarrow \;&
\theta = 0 \;\;\; (\because \epsilon < \chi_w ) \\
\Longleftrightarrow \;&
q = p \;\;\; (\because ||p||=||q|| )
\end{align*}
\vspace{-4.2mm}
\end{small}
So $w$ gets influenced if and only if $u$ buys product $p$, after which $v$ is recommended to buy $p$ with influence weight $\hat{\alpha}^p h_{uv}$.
Also note the time lapse of one step between the reference friend $u$ buying the product and the target node $v$ receiving the recommendation. Hence the latency in recommendation using social advertising is implicitly accounted for.
\begin{comment}
It is to be noted that the original values of $b_{uv}$ assume that the influence on a node $v$ is entirely due to viral marketing, and do not account for influence through other channels.
With other influencing channels, the influence weight allotted by $v$ to decision of $u$ would reduce.
Since now the sum of incoming edges and pseudoedges is
$
\sum_{u \in \mathcal{N}(v)} b_{uv}+ \sum_p \left( \sum_{u \in \mathcal{N}(v)} \alpha^p h_{uv} + \sum_{t=1}^T \beta_t^p \right)
$,
the scaled down influence weights are
\begin{small}
\vspace{-2mm}
\begin{gather*}
\hat{b}_{uv} = b_{uv}\left( \frac{\sum_{u \in \mathcal{N}(v)} b_{uv}}{\sum_{u \in \mathcal{N}(v)} b_{uv}+ \sum_p \left( \sum_{u \in \mathcal{N}(v)} \alpha^p h_{uv} + \sum_{t=1}^T \beta_t^p \right)} \right)
\end{gather*}
\end{small}
\end{comment}
\vspace{-1mm}
\section{The Underlying Problem
\vspace{-1mm}
}
\label{sec:problem}
\begin{comment}
In the literature, it is generally assumed that nodes are equally enthusiastic about diffusing information irrespective of the nature of the product. However, the interest of nodes to diffuse information would naturally be low for products that are not publicly visible like toothpaste, while it would be high for products that are publicly visible like mobile phone.
We model this nature with an {\em enthusiasm parameter} $\delta^p$ for a product $p$. So it would be low for toothpaste, while high for mobile phone.
Owing to this factor, it is impossible to influence a node having a threshold greater than $\delta^p$. Moreover, even a node having threshold less than $\delta^p$ may not get influenced as the weighted sum of its incoming edges from influenced nodes, which are now scaled by $\delta^p$, may fall short of its threshold. So it may be necessary of a company to complement the viral marketing of the product with other kinds of marketing, in order to reach its potential customers.
In this paper, we study two other types of marketing, namely, mass media advertising and social advertising.
\end{comment}
The fundamental problem here is to distribute the total available budget among the three marketing channels under study.
Let $k^p$ be the number of free samples of product $p$ that the company would be willing to distribute.
Let $S^p$ be the corresponding set of nodes in the social network to whom free samples would be provided ($|S^p|=k^p$).
Let $c_p(\cdot)$ be the cost function for allotting effort of activating set $S^p$ to trigger viral marketing, $\alpha^p$ for social advertising, and $(\beta_t^{q})_{t=1}^T$ corresponding to each step of mass media advertising.
In general, $c_p(\cdot)$ would be a weighted sum of these parameters since the costs for adjusting parameters corresponding to different channels would be different.
Let $\gamma^p$ be the total budget for marketing of product $p$.
Let $\nu^p(\cdot)$ be the expected number of nodes (excluding pseudonodes) influenced by product $p$, accounting for the marketing strategies of $p$ and its competitors.
Hence the optimization problem for the marketing of product $p$ is
\begin{small}
\vspace{-3mm}
\begin{equation}
\label{eqn:optimization_problem}
\begin{split}
\text{Find } S^p,\alpha^p,(\beta_t^p)_{t=1}^T \text{ to maximize }\\
\nu^p(S^p,\alpha^p,(\beta_t^p)_{t=1}^T,(S^{q},\alpha^{q},(\beta_t^{q})_{t=1}^T)_{q \neq p}) \\
\text{ such that }
c^p(S^p,\alpha^p,(\beta_t^p)_{t=1}^T ) \leq \gamma^p
\end{split}
\end{equation}
\vspace{-2mm}
\end{small}
\begin{comment}
\textbf{\color{red}{
Given $S^p$, can get influence in terms of $\alpha^p$ and $(\beta_t^p)_{t=1}^T$.
The constraint is $c(k^p,\alpha^p,\beta^p ) \leq \gamma^p$.
As $k^p$ is fixed, obtain available $c(\alpha^p,\beta^p )$.
For single campaign,
If some relation among $(\beta_t^p)_{t=1}^T$'s assumed, then only two variables are $\alpha^p,\beta^p$ and with the constraint $c(k^p,\alpha^p,\beta^p ) \leq \gamma^p$.
As it will be monotone increasing, the constraint becomes $c(k^p,\alpha^p,\beta^p ) = \gamma^p$ and some $\beta^p$ is replaceable with $\alpha^p$.
Now maximize with respect to $\alpha^p$.
This is $\sigma^p(S^p)$.
Now use some algorithm.
}}
\end{comment}
\begin{comment}
With $\sigma^p(\cdot)$ not having any particular property, it is not guaranteed that the greedy algorithm, which performs well for optimizing monotone submodular functions, will perform well for optimizing $\sigma^p(\cdot)$.
In what follows, let $\mathcal{T}$ be the time taken for computing the objective function value for a given set.
\end{comment}
In the above optimization problem, we not only need to determine the optimal allocation among channels ($k^p,\beta^p,\alpha^p$), but the best $k^p$ nodes to trigger viral marketing $S^p$ such that $|S^p|=k^p$, and the optimal allocation over time for mass media advertising $(\beta_t^p)_{t=1}^T$ such that $\sum_{t=1}^T \beta_t^p = \beta^p$.
The problem hence demands a method for
multi-parametric optimization.
Methods such as Fully Adaptive Cross Entropy (FACE)
provide a simple, efficient, and general approach for simultaneous optimization over several parameters~\cite{de2005tutorial}.
In our context, the FACE method involves an iterative procedure where each iteration consists of two steps, namely,
(a) generating data samples according to a specified distribution and
(b) updating the distribution based on the sampled data to produce better samples in the next iteration.
Here, our sample is a vector consisting of
whether a node should be included in $S^p$,
budget allotted for each time step of mass media advertising $(\beta_t^p)_{t=1}^T$, and budget allotted for social advertising $\alpha^p$; each data sample satisfies the cost constraint $c^p(S^p,\alpha^p,(\beta_t^p)_{t=1}^T ) \leq \gamma^p$.
Initially, the data samples could be generated based on a random distribution.
The value of the objective function $\nu^p(\cdot)$ is computed for each data sample as per the proposed model using a sufficiently large number of Monte Carlo simulations.
The distribution is then updated by considering data samples which provide value of the objective function better than a certain percentile.
This iterative updating continues until convergence or for a fixed number of iterations.
The obtained terminal data sample acts as the best response allocation strategy for product $p$, in response to the strategies of competitors.
\begin{comment}
For the detailed FACE algorithm and the terminology used, the reader is referred to \cite{de2005tutorial}.
We initialize the method with distribution $(\frac{\gamma}{n},\ldots,\frac{\gamma}{n})$, that is, each node has a probability of $\frac{\gamma}{n}$ of getting selected in any sample set in the first iteration.
In any iteration, the number of samples (satisfying budget constraint) is bounded by
$N_{\text{min}}=n$ and
$N_{\text{max}}=20n$, and the number of elite samples is
$N_{\text{elite}}=\lceil \frac{n}{4} \rceil$.
We use a weighted update rule for the distribution, where in any given iteration, the weight of any elite sample is proportional to its function value.
The smoothing factor that we consider is $\alpha = 0.6$.
\end{comment}
\begin{comment}
\subsubsection{Random Sampling and Maximizing (RMax)}
Here,
we sample $O(n)$ number of sets that satisfy the budget constraint,
and then assign that set as the seed set which gives the maximum function value among the sampled sets.
Note that this method is different from the random set selection method~\cite{kempe2003maximizing},
where only one sample is drawn.
Its time complexity is $O(n\mathcal{T})$.
We consider this method as it is very generic and agnostic to the properties of an underlying objective function, and can be used for optimizing functions with arbitrary or no structure. This method is likely to perform well when the number of samples is sufficiently large.
\textbf{\color{red}{
In our simulations, we observed that in most cases, the FACE method converged in 5 iterations (extending till 7 at times) by giving a reliable solution ({\em reliable} refers to the case wherein the method deduces that it has successfully solved the problem). Also, the total number of samples drawn in any iteration was $n$ in almost all cases (it did not exceed $2n$ in any iteration).
So for direct comparison with FACE, we consider $5n$ sampled sets for RMax.
}}
\subsubsection{Greedy Algorithm}
The greedy algorithm for maximizing a function $\mathcal{F}$, selects nodes one at a time, each time choosing a node that provides the largest marginal increase in the value of $\mathcal{F}$, until the budget is exhausted.
Its time complexity is $O(kn\mathcal{T})$.
\end{comment}
\begin{comment}
\section{Experimental Results}
\label{sec:results}
We illustrate the results using collaboration network obtained from co-authorships in high-energy physics theory publications on the e-print arXiv, since it has been extensively argued to exhibit many of the structural features of large-scale social networks~\cite{newman2001structure}.
\subsection{Model for Influence Weights}
We normalize the weights derived in \cite{newman2001scientific}, thus arriving at the following expression for the influence weight of node $u$ on node $v$.
\begin{displaymath}
b_{vu} = \frac{ \sum_r \frac{\Delta_v^r \Delta_u^r}{\mu_r-1} }{ \sum_r \Delta_v^r }
\end{displaymath}
where $\mu_r$ is the number of coauthors of paper $r$, and $\Delta_v^r=1$ if node $v$ is a coauthor of paper $r$, else it is $0$.
Note that single author papers need to be excluded for the above expression to be well-defined.
The modified graph has $7610$ nodes and $31502$ directed edges.
\subsection{Model for Similarity}
For the purpose of our experiments, we consider the following natural model of homophily: the similarity between any two directly connected nodes is equal to the ratio of the influence weights among the two nodes and their common neighbors to the total influence weights that the two nodes are involved in. That is, the similarity between nodes $u$ and $v$, say $h_{uv}$ (or equivalently $h_{vu}$), is given by:
\begin{displaymath}
h_{uv} = \frac{ b_{uv} + b_{vu} + \sum_{i \in \mathcal{N}_u \cap \mathcal{N}_v} ( b_{iu} + b_{ui} + b_{iv} + b_{vi} ) }{ \sum_{i \in \mathcal{N}_u} (b_{iu} + b_{ui}) + \sum_{i \in \mathcal{N}_v} (b_{iv} + b_{vi}) }
\end{displaymath}
where $\mathcal{N}_v$ is the set of neighbors of node $v$.
\end{comment}
\vspace{-1.2mm}
\subsection*{A Note for Practical Implementation
\vspace{-1mm}
}
In order to implement the proposed framework in practice, a company would need to map its customers to the corresponding nodes in social network. To create such a mapping, it would be useful to get the online social networking identity (say Facebook ID) of a customer as soon as it buys the product.
This can be done using a product registration website (say for activating warranty) where a customer, when it buys the product, needs to login using a popular social networking website (such as Facebook), or needs to provide its email address which could be used to discover its online social networking identity.
Thus the time step when the node has bought the product, can also be obtained.
\begin{comment}
\section{Discussion}
\label{sec:discuss}
The problem of competitive influence maximization for a company is a complex one. There are several resources available and the company has to invest appropriately in the appropriate resources at appropriate times. The companies need to constantly keep on revising their strategies based on the past outcomes, which in turn, depend on the strategies of the competitors as well as the behavior of the potential customers (in the sense that how they aggregate the information they get from various channels). The emerging possibility of viral marketing through online social networks has added a new dimension to the problem. Our objective in this work was to present a framework for studying optimal strategies
in the presence of several marketing channels, so as to maximize the spread of influence in the social network.
\end{comment}
\begin{comment}
\subsection{What needs to be done?}
A primary objective of this work is to find influence maximizing strategy of investing in various marketing ways, given a certain total budget and the strategy of the competitors. The agenda will be as follows:
\begin{enumerate}
\item For a particular $k^p$, $S^p$ can be obtained
\item Find optimal strategy given the strategy of one other competitor
\item No constant factor approximation with polynomial algorithm
\end{enumerate}
\end{comment}
\begin{comment}
In order to solve the constrained optimization problem \eqref{eqn:optimization_problem}, one could utilize methods such as projection-based stochastic approximation (SA) algorithms \cite{kushner1978stochastic}, constrained simultaneous perturbation stochastic approximation (SPSA) algorithm \cite{sadegh1997constrained,wang2003stochastic}, cross entropy method for constrained optimization \cite{kroese2006cross}, etc. Projection-based SA algorithms are useful if we have simple constraints like interval or linear constraints, while constrained SPSA and cross-entropy based method are applicable when the constraints are non-linear.
Also, both of these methods are easy to implement.
\end{comment}
\begin{comment}
It is important to develop more scalable algorithms for maximizing the considered objective function.
It would be interesting to develop a model that considers clash of the contagions while still maintaining monotonicity and submodularity for using greedy for second mover and Nash equilibrium result, else prove that such a model is impossible.
\end{comment}
\vspace{-1mm}
\section*{References
\vspace{-1.25mm}
}
\bibliographystyle{IEEEtran}
\begingroup
\renewcommand{\section}[2]{}%
\begin{small}
|
{
"timestamp": "2018-04-17T02:14:02",
"yymm": "1804",
"arxiv_id": "1804.05525",
"language": "en",
"url": "https://arxiv.org/abs/1804.05525"
}
|
\section*{Acknowledgments}
Results presented in this work have been produced using the European Grid Infrastructure (EGI) through the National Grid Infrastructures $NGI \_ GRNET$ (HellasGrid) as part of the SEE Virtual Organisation. This research was supported by European Commission FP$7$-FET project Multiplex \#~$317532$. S.H.~acknowledges the ISF, ONR Global, the Israel-Italian collaborative project NECST, Japan Science Foundation, BSF-NSF, and DTRA (Grant \#~HDTRA-1-10-1-0014), together with the Israeli Ministry of Science, Technology and Space (MOST, grant \#3-12072) for financial support.
|
{
"timestamp": "2018-04-17T02:10:08",
"yymm": "1804",
"arxiv_id": "1804.05337",
"language": "en",
"url": "https://arxiv.org/abs/1804.05337"
}
|
\section{Introduction}
\label{sec:intro}
Soil surface is a starting point when investigating infiltration behavior of precipitation~\cite{brooks2015hydrological}.
In this context, knowledge about the soil-moisture dynamics is crucial to understand ecological, hydrological, and vegetation processes.
Traditionally, soil moisture is measured in-situ with high temporal resolution.
Time domain reflectometry (TDR) sensors represent such measuring devices.
Since it is extremely time-consuming and expensive to cover large areas with these devices, remote sensing techniques like hyperspectral sensors are deployed, even partly on drones~\cite{finn2011remote, stamenkovic2013estimation}.
To estimate soil-moisture with hyperspectral data, machine learning models are applied due to their potential to handle high-dimensional regression problems (cf.~\cite{stamenkovic2013estimation, riese2018introducing}).
These models are suitable to link hyperspectral image data as input data to soil-moisture data as target variable.
In addition to hyperspectral remote sensing, ground penetrating radar (GPR) measurements provide further data to model soil-moisture variations as described in~\cite{huisman2003measuring, allroggen20154d, xinbo2017measurement}.
In contrast to e.g. TDR probes, a GPR is feasible to consider the spatial variation of soil moisture~\cite{huisman2003measuring,jackisch2017form}.
Its measurements provide a more detailed spatial coverage and resolution.
The GPR data is manually acquired which is highly elaborative and therefore results in sparse datasets.
Clearly, hyperspectral remote sensing and GPR measurements, have their particular strengths in different scopes of estimating soil moisture.
Our objective is to investigate the fusion of both techniques to address soil-moisture estimation.
We take advantage of the area-wide coverage of hyperspectral snapshot and the spatially high-resolution information of subsurface variations obtained from GPR data.
In the proposed studies, we rely on a dataset which is introduced in~\cite{keller2018modeling}.
The GPR data acquisition is performed in a frequency that is \num{10} times lower than the TDR-based soil-moisture measurements.
As a consequence, data gaps occur and it is necessary to artificially extend the GPR data.
Solely with this extension, a comprehensive soil-moisture estimation based on machine learning models can be conducted (cf. ~\cite{keller2018modeling, keller2018is}).
In this contribution, we present two novel approaches which simulate GPR and TDR data to extend the given, sparse dataset.
Hyperspectral data combined with this simulated data is used to estimate soil moisture based on machine learning techniques.
The two approaches are entirely different.
In the first approach, we address the simulation of GPR data along the time axis by using TDR and hyperspectral data.
The results are temporal-simulated GPR data which are combined with hyperspectral data to estimate the target variable soil moisture.
In contrast, the second approach contains the simulation of TDR measurements along the GPR measurement area.
Based on the spatial resolution of the GPR measurement, the simulated TDR data is projected onto this area and is used as reference to estimate soil moisture solely with hyperspectral data.
We then investigate the potential of each approach with respect to the performance of the soil-moisture estimation.
In \cref{sec:data}, we briefly introduce the dataset on which the proposed approaches are based.
These approaches are described in \cref{sec:methods}.
We evaluate the performance of the regression task in \cref{sec:eval}.
In \cref{sec:conclusion}, we conclude our \mbox{studies}, respond to the underlying objective, and give an overview about future applications.
\section{Field campaign dataset}
\label{sec:data}
We use a dataset acquired during a multi-sensor field campaign in August 2017 in Linkenheim-Hochstetten, Germany~\cite{keller2018modeling}.
An undisturbed grassland site on loamy sand is divided into several plots.
Each plot has a size of $\SI{1}{\meter}\times\SI{1}{\meter}$.
They are monitored by a variety of sensors.
The plots are manually irrigated with a predefined irrigation schema~\cite{keller2018modeling}.
A hyperspectral snapshot camera performs the monitoring of the soil surface.
In addition, a GPR acquires profile measurements.
TDR probes measure hydrological parameters in various soil depths.
For this publication, the TDR data in a depth of \SI{5}{\centi\meter} represents the reference values.
The used hyperspectral snapshot camera is a Cubert\footnote{Cubert GmbH, Ulm, Germany} UHD 285.
It captures images consisting of $50\times 50$ pixels with \num{125} spectral bands from \SIrange{450}{950}{\nano\meter} and a spectral resolution of \SI{4}{\nano\meter}.
As one data preprocessing step, we dismiss the first five and the last five spectral bands of the hyperspectral bands to reduce the effect of sensor artifacts.
The GPR system pulseEKKO PRO GPR\footnote{Sensors \& Software Inc., Mississaugua, Canada} equipped with a pair of 1 GHz shielded antennas collects information about the subsurface heterogeneity.
It is moved along fixed axes on the measurement field at predefined times, described in detail in~\cite{keller2018modeling}.
For each GPR measurement, soil moisture variations $\Delta\theta$ are derived with a procedure described in~\cite{allroggen20154d} resulting in a spatial resolution of \SI{1}{\centi\meter} along the measurement line.
Finally, we receive GPR profiles with \num{100} $\Delta\theta$ values per measurement plot.
We merge the spatial resolution of the GPR data to the \SI{10}{\centi\meter} resolution of the hyperspectral data.
Thus, the resulting dataset includes a hyperspectral measurement for every GPR profile.
In our studies, four of eight measurement plots are included.
The dataset then consists of \num{94} datapoints with one measured GPR $\Delta\theta$ value and one TDR soil-moisture value.
\cref{fig:corr} shows the correlation between the soil moisture variations $\Delta\theta$ of the GPR and the soil-moisture values of the TDR.
The different markers symbolize the four plots where the data is acquired.
The Pearson correlation coefficient of all datapoints is $r_{\text{all}}=\SI{50}{\percent}$.
\cref{tab:corr} summarizes this coefficient for each of the four plots.
This behavior can be explained by the different irrigation schemes for the measurement plots (cf.~\cite{keller2018modeling}).
\begin{figure}[tb]
\centering
\includegraphics[width=0.40\textwidth]{corr_plot_gpr_sm.pdf}
\caption{Correlations between the soil-moisture variations $\Delta\theta$ measured by the GPR and the soil-moisture values measured by the TDR sensors of the regarded four different measurement plots.\label{fig:corr}}
\end{figure}
\begin{table}[tb]
\centering
\caption{Pearson correlation coefficient $r$ of the four plots.}
\begin{tabular}{lccccc}
\toprule
Plot & 1 & 2 & 3 & 4 & all\\
\midrule
$r$ in \% & 94 & 93 & 64 & -6 & 50\\
\bottomrule
\end{tabular}
\label{tab:corr}
\end{table}
\section{Simulation approaches}
\label{sec:methods}
We present two approaches to extend the given multi-sensor dataset and to perform a machine learning based estimation of soil moisture.
In \cref{fig:gpr_schema_all}, the GPR and TDR measurements are illustrated schematically with both simulation approaches.
The first approach originates from the \num{10} times finer temporal resolution of the TDR data in contrast to the GPR data (cf. \cref{fig:gpr_schema_meas}).
This approach generates simulated GPR data based on TDR data combined with measured GPR profiles.
The resulting data is spatially localized at the TDR positions of each plot (cf.~\cref{fig:gpr_schema_app1}).
In \cref{sec:methods:sub:app1}, we describe the first approach in detail.
The basis of the second approach are the GPR profiles along a line of each plot.
We simulate values of soil moisture (respectively TDR) along the GPR profiles by merging the measurements of TDR and GPR.
Schematically, the second approach is presented in \cref{fig:gpr_schema_app2} and described in detail in \cref{sec:methods:sub:app2}.
\begin{figure*}[tb]
\centering
\subfloat[]{\includegraphics[height=6.0cm, keepaspectratio]{GPR_Schema_5_1}\label{fig:gpr_schema_meas}}\qquad
\subfloat[]{\includegraphics[height=6.0cm, keepaspectratio]{GPR_Schema_5_2}\label{fig:gpr_schema_app1}}\qquad
\subfloat[]{\includegraphics[height=6.0cm, keepaspectratio]{GPR_Schema_5_3}\label{fig:gpr_schema_app2}}
\caption{Schematic overview of (a) the real measurements, (b) the GPR simulation as first approach, and (c) the TDR simulation as second approach. The vertical axes show the time with two exemplary points in time $t_1, t_2$ at which the GPR measurements (orange dashed lines) are performed. The horizontal axes show the \SI{1}{\meter} width of a measurement plot. The soil-moisture measurements performed by the TDR (blue dots) at a higher frequency. \label{fig:gpr_schema_all}}
\end{figure*}
\subsection{Simulation of GPR data}
\label{sec:methods:sub:app1}
For this first simulation approach, we propose two distinct simulations of the soil-moisture content as a GPR product.
First, we interpolate between different GPR measurements along the time axis (cf.~\cite{oliphant2007python}).
This interpolation ignores any TDR measurements, since they possibly bias the interpolation results.
Additionally, we calculate Gaussian noise to these results to reduce overfitting effects.
In sum, the simulated dataset consists of \num{481} datapoints with hyperspectral and GPR data.
An exemplary interpolation is shown in \cref{fig:app1_timeseries}.
\begin{figure}[tb]
\centering
\includegraphics[width=0.4\textwidth]{interpolated_gpr_timeseries_breakaxis}
\caption{Exemplary GPR time series of plot 2. The GPR measurements are orange and the GPR simulation with Gaussian noise are blue. The gap in the middle represents the night between to the two measurement days.\label{fig:app1_timeseries}}
\end{figure}
Next, we apply machine learning to link the GPR and TDR soil-moisture measurements.
We evaluate this approach with two regression models: linear regression and extremely randomized trees (ET)~\cite{geurts2006extremely}.
As before, the resulting dataset consists of \num{481} datapoints including simulated GPR data.
\subsection{Simulation of TDR data}
\label{sec:methods:sub:app2}
The second approach to extend the measured dataset is to simulate TDR soil-moisture values based on the GPR data and along the GPR profiles.
Our concept is to perform a regression with the GPR data at the TDR positions as input data and the soil-moisture values retrieved by the TDR measurements at simultaneous times as target variable.
We apply a linear interpolation (cf.~\cite{oliphant2007python}) along the (sorted) GPR measurements as well as a linear regression and an ET regression.
After the training phase, the regressor is able to estimate soil-moisture values with given GPR $\Delta\theta$ values.
As simulation, we use all GPR data except the ones located at the TDR probes.
In sum, we rely on \num{9} of \num{10} measured $\Delta\theta$ values of each plot.
Subsequently, we match the results of the simulated soil-moisture values with the respective pixels of the hyperspectral data.
The resulting dataset consists of $825$ datapoints including simulated TDR, GPR $\Delta\theta$ values, and the hyperspectral spectrum.
\section{Results \& evaluation}
\label{sec:eval}
In the following, we evaluate the performance of the soil-moisture estimation based on hyperspectral data combined with either simulated GPR as additional input feature or simulated TDR data as target variable.
The soil-moisture regression is performed with the ET regressor due to its best performance in~\cite{keller2018is}.
We split the three generated datasets (cf. \cref{fig:gpr_schema_all}) each into a training and a test subset at a ratio of $1:1$.
As an expression of the regression performance, we present the coefficient of determination $R^2$ and the root mean squared error RMSE.
\subsection{Evaluation of the first approach}
\label{sec:eval:sub:app1}
The regression results for the first approach with the simulated GPR values as one input data are shown in \cref{tab:app1_results}.
Adding any of the simulated GPR data as additional feature to the regression model increases the $R^2$ about at least \num{20}~p.p. compared to a regression performed without this data.
The GPR data estimated by the linear regression performs the best in estimating soil moisture and achieves an $R^2=\SI{83.3}{\percent}$.
In this specific approach, the simulated GPR data is a linear combination of several measured soil-moisture sample points.
We are aware of the TDR bias which is included in the linear and ET regression.
By contrast, the approach applying the interpolation and Gaussian noise is based exclusively on the time course of the GPR measurements.
The regression result based on the GPR interpolation is $R^2=\SI{74.4}{\percent}$.
This is a significant improvement and underlines the importance of GPR data combined with the hyperspectral data in this estimation.
\begin{table}[tb]
\centering
\caption{Results of the ET regression with the soil-moisture target variable and the simulated GPR data as input data. For the simulation, an interpolation, a linear regression, and an ET regression are evaluated. FI is the feature importance of the simulated GPR data in the ET regression.}
\begin{tabular}{lSSc}
\toprule
\multirow{2}{*}{GPR simulation} & {$R^2$} & {RMSE} & {FI of GPR}\\
& {in \%} & {in \si{\milli\meter}} & { in \%}\\
\midrule
No simulated GPR & 53.2 & 1.09 & {-}\\
Interpolation & 74.4 & 0.81 & 31\\
Linear regression & 83.3 & 0.65 & 39\\
ET regression & 77.9 & 0.75 & 36\\
\bottomrule
\end{tabular}
\label{tab:app1_results}
\end{table}
\subsection{Evaluation of the second approach}
\label{sec:eval:sub:app2}
As before, the evaluation is performed with an ET regression on simulated sensor-like soil-moisture data as target variable with solely hyperspectral data as input features.
The results are shown in \cref{tab:app2_results}.
Based on these regression results, we conclude that the regression models struggle to precisely link the simulated soil-moisture data and the hyperspectral data.
The small number of \num{94} measured GPR datapoints combined with a poor correlation of some plots as pointed out in \cref{sec:data}, cf. \cref{fig:corr}, explain these weak regression performances.
\begin{table}[tb]
\centering
\caption{Results of the ET regression with the simulated soil-moisture values as target variable and the hyperspectral data as input data. For the simulation, an interpolation, a linear regression, and an ET regression are evaluated.}
\begin{tabular}{lSS}
\toprule
{TDR simulation} & {$R^2$ in \%} & {RMSE in \si{\milli\meter}}\\
\midrule
Interpolation & 21.5 & 2.16\\
Linear regression & 37.5 & 0.76\\
ET regression & 35.1 & 0.81\\
\bottomrule
\end{tabular}
\label{tab:app2_results}
\end{table}
\cref{fig:sim_dist} illustrates an exemplary soil-moisture distribution based on the interpolation and the linear regression approaches.
The distribution as a result of the latter approach is much smoother than the distribution of the interpolation.
\begin{figure}[tb]
\centering
\includegraphics[width=0.48\textwidth]{model1_distributions_zone1_final}
\caption{Distributions of the measured $\Delta\theta$ of the GPR (top), the measured TDR values (center top), the linearly interpolated soil-moisture values (center bottom), and the simulated soil-moisture values of the linear regression (bottom) of measurement plot 1.\label{fig:sim_dist}}
\end{figure}
\section{Conclusion}
\label{sec:conclusion}
In this contribution, we evaluate the potential of hyperspectral data combined with either simulated GPR or simulated TDR data to estimate soil moisture.
This estimation is performed with a machine learning model.
Based on the two different approaches, we extend the underlying sparse dataset.
In the first approach, we simulate GPR data.
The second approach handles the simulation of soil-moisture values analog to TDR measurements.
Our results of the soil-moisture estimation reveal the potential of (simulated) GPR data as valuable input.
The simulated GPR, even as output of a common linear regression, enhances the overall performance of the soil-moisture estimation significantly.
The fusion of simulated soil-moisture values (TDR similar) and hyperspectral data as input data for the soil-moisture estimation performs the worst.
To conclude, the soil-moisture estimation is improved by the fusion of hyperspectral and ground penetrating radar data.
In future work, we plan to extend the field campaigns with many more GPR profiles.
Thus, we are able to enhance the GPR simulation based on measured data.
The area-wide estimation of soil moisture benefits from the proposed fusion of hyperspectral and GPR data.
\section{Acknowledgements}
\label{sec:ack}
We thank Niklas Allroggen for his tedious-acquired GPR profiles.
We also thank Conrad Jackisch for the monitoring of the hydrological parameters.
\bibliographystyle{IEEEbib}
|
{
"timestamp": "2018-08-08T02:10:09",
"yymm": "1804",
"arxiv_id": "1804.05273",
"language": "en",
"url": "https://arxiv.org/abs/1804.05273"
}
|
\section{Introduction}
The shape and the shape transitions of elastic nano-ribbons/polymers in a thermal environment is of interest in many disciplines, from the study of DNA and protein folding \cite{Bouchiat1998}, via the evolution of bio-molecular structures such as amyloids \cite{Adamcik2011} and cholesterol aggregates \cite{Smith2001}, to the way synthetic polymers and self assemblies may serve in drug deliveries, food additives or templates for the production of nano structures \cite{Zhan2005,Oda2008,Ziserman2011}. In a thermal environment, such slender structures fluctuate (gyrate) and their shape may vary widely depending on the temperature, size, chemical potential and other control parameters. Nano-ribbons and polymers are usually studied using simplified models such as the worm-like chain model, freely jointed chain and their variants {\cite{Sadowsky1930,Panyukov2000a,Giomi2010, Rubinstein2003}}, all assume no residual elastic stresses i.e., the ribbon/rod on its equilibrium configuration is stress free. However, many (in fact most) nano-ribbons and polymers are made of complex elements, which likely do not fit perfectly to each-other when forming extended aggregates such as sheets and ribbons. They are incompatible, and as such the resulting structure would be residually stressed. Recent theoretical and experimental works \cite{Armon2011,Armon2014,Guest2011,Levin2016} show that incompatible slender structure undergo various non trivial shape transformation and that their mechanics could differ a lot from seemingly similar, compatible, structures. These characteristics are likely to affect the resulting shape fluctuations of nano-metric self-assemblies \cite{Aggeli1997,Ziserman2011,Oda1999}.
In this work we apply the theory of incompatible elastic sheets to study the statistical mechanics of thermal ribbons that have spontaneous isotropic positive curvature. We use our recently developed effective 1D elastic model \cite{Grossman2016} to derive expressions for the persistence length $\ell_p$, the Kuhn length $\ell_k$ and the gyration radius $R_g$ and obtain the phase diagram, which characterizes different types of ribbon conformation statistics unique to such ribbons.
Ribbons with spontaneous positive curvature may be produced in the lab (see for e.g \cite{Guest2011,Hu2016,Zhang2017}), or arise naturally in self assembled systems with broken symmetry. Such systems may be uneven semi-solid bi-layers \cite{Yesylevskyy2014}, or asymmetric bola-amphiphile mono-layer with two distinct sides to it (say different size head groups with strong same-type affinity), e.g- systems such as in
\cite{Masuda2004,Zhan2005,Marson2014}. Microscopically, such a geometry arises in asymmetric semi-solid mono/bi-layers as a result of two competing geometries: the (in-plane) bond-induced geometry, i.e- the spatial arrangement that best satisfies the intermolecular bonds; and the molecular shape geometry. The former depends solely on the number and direction of bonds between different molecules, the latter only on the shape of a single molecule. In this paper, we consider only systems whose bond-geometry is a flat, Euclidean geometry (as is ,in fact, very common- e.g, all defect-free crystalline structures). This is to say that the preferred distances between neighbouring elements are satisfied (everywhere) by setting them in a planar geometry. In contrast, the molecules' shape may prescribe a very different geometry- molecular asymmetry (corresponding to the bi/mono-layer asymmetry ), which may arise due to different sized headgroups, or difference in surface tension, results with a curved geometry (Figure \ref{fig:spherical_chem}), as the intermolecular distance on one sige are larger than those on the other. Unless anisotropy is introduced, the system has the same preferred curvautre, $k_0$ is all directions.
We begin this paper by studying the (mechanical) equilibrium shape of positively curved ribbons (section \ref{s_ch:Euilbrium}), deriving their unique conformational behaviour which includes shape transition at a critical width, abnormal floppiness of wide ribbons and dominance of boundary layer in setting ribbon's rigidity. We then (section \ref{s_ch:StatMech}) show that these unique results manifest themselves as unusual statistical behaviour. We divide the problem into bending dominated and stretching dominated regimes (section \ref{sub:NarrRibb} and \ref{sub:WideRibb}), and study common measures of conformation in elastic ribbons at these regimes. These include the persistence length $\ell_p$, describing the decay tangent-tangent correlations, the Kuhn length $\ell_k$ and the gyration radius $R_g$. We show that these measures on their own fail to wholly capture the phase diagram, and derive new measures to assist in this task. We conclude the paper in \ref{ch:Conc}, where we calculate the phase diagram of such ribbons and discuss the meaning and implications of the results. We suggest that in general, incompatible ribbons will have a phase diagram richer than compatible ones, since in addition to high/low temperature regimes, for such ribbons there are the wide/narrow width regimes. The transitions between them and the resultant abnormal mechanics determine a wider {configurational} phase space.
\begin{figure}[h!]
\centering
\includegraphics[width=0.9\textwidth]{spherical-chem.pdf}
\caption{Illustration of systems with positive spontaneous curvature (as indicated by the "form" wedge - dashed lines). Zigzag lines correspond to carbon chains, different head groups are marked by triangles and circles. a- Asymmetric bola-amphiphile mono-layer; b- Asymmetric bi-layer; c- Other shape asymmetries of the constituent molecules (in this case another carbon "tail"; d- Also chemical differences (in this case different surface tension); e- 3D visualization of the way such systems self assemble into positively curved surfaces. }\label{fig:spherical_chem}
\end{figure}
\section{Elastic Modeling of Positively Curved ribbons}\label{ch:PosRibbons}
Within a continuum mechanics description, a ribbon is a thin, narrow, and long sheet with thickness, width and length which satisfy $t\ll W \ll L$ respectively. We choose coordinates on the ribbon $(x,y)\in \left[0,L\right]\times\left[-\frac{W}{2},\frac{W}{2}\right]$, such that the line $y=0$ parametrizes the ribbon's mid-line. Following \cite{Panyukov2000a} and \cite{Grossman2016} we assume that the ribbon's shape $\vec{r}(x,y)$ is well described by the curvatures at the mid-line. These are the normal curvature ($l(x)$), the twist ($m(x)$) describing the mid-line's shape and transverse curvature ($n(x)$) describing the ribbon's profile (see Fig \ref{fig:lmn}).
\begin{figure}[!h]
\centering
\begin{tikzpicture}[scale=1]
\node[above, right] (ln) at (0,0) {\includegraphics[clip=true, trim={0.5cm 0.1cm 0.4cm .1cm}, width=.2\textwidth]{L_N_frame.pdf}};
\node[above, right] (m) at (3.5,0) {\includegraphics[clip=true, trim={0.1cm 0.1cm 0.1cm 0.1cm}, width=.2\textwidth]{Only_M_w_Frame_2.pdf}};
\node[above, right] (binorm) at (1.2,-1.2) {\large$\hat{v}_1$};
\node[above, right] (norm) at (2.3,-2) {\large$\hat{v}_2$};
\node[above, right] (tang) at (2.4,-.7) {\large$\hat{v}_3$};
\node[above, right] (y0) at (4.85,3.3) {$y=0$};
\node[above, right] (L) at (2.4,0.5) {\huge $l$};
\node[above, right] (N) at (1.55,2.8) {\huge $n$};
\node[above, right] (M) at (5.35,-1) {\huge $m$};
\end{tikzpicture}
\caption{Visualization of the curvatures. Black line is the mid-line ($y=0$). Left- a ribbon with $l$ (bent mid-line) and $n$ (the transverse curvature along the cyan, $x=const$, line) with opposite sign ($m=0$). Arrows correspond to the local frame at two different positions along the ribbon mid-line (Blue for tangent vector- $\hat{v}_3$, Green for bi-normal - $\hat{v}_1$ and Red for the normal vector $\hat{v}_2$). $l$ corresponds to curvature along the tangent, and $n$ along the bi-normal. Right- A ribbon with pure twist, $m$, around the mid-line ($l=n=0$).}
\label{fig:lmn}
\end{figure}
Formally, we assign a Darboux frame at the ribbon's mid-line ($\left\{\hat{v}_1,\hat{v}_2,\hat{v}_3\right\} $) such that $\hat{v}_3(s)$ is tangent to the ribbon's mid-line, $\hat{v}_2(s)$ is normal to the ribbon (pointing outside the ribbon) and the bi-normal $\hat{v}_1 (s)$ pointing along the width (narrow dimension) of the ribbon. This Darboux frame satisfies the generalized Frenet-Serret equations (see e.g \cite{Panyukov2000})
\begin{subequations} \label{eq:Gen_FrenetSerret}
\begin{align}
\hat{v}'_3&= l \, \hat{v}_2 \\ \nonumber
\hat{v}'_2&= -l \, \hat{v}_3 - m \, \hat{v}_1 \\ \nonumber
\hat{v}'_1&= m\, \hat{v}_2. \\ \nonumber
\end{align}
where $X' \equiv \partial_x X $ is the derivative of $X$ along the ribbon. In principle, there is also contribution from the geodesic curvature $k_g$ of the mid-line, however, we assume it is zero. The general solution of these equations, given the frame at some position $x'<x$ along the ribbon is
\begin{align}\label{eq: Solution}
\hat{v}_i(x) & =O_{ij}(x,x') \, \hat{v}_j(x'),
\end{align}
where $O_{ij}$ is an element of the a rotation matrix
\begin{align
\mathbf{O}(x,x') &= T_x\left[e^{-\int_{x'}^x \mathbf{\Omega} \,\text{d} x''}\right] = \lim\limits_{N \rightarrow \infty} e^{-\mathbf{\Omega}(x_{N}) \Delta x}e^{-\mathbf{\Omega}(x_{N-1}) \Delta x}...e^{-\mathbf{\Omega}(x_1) \Delta x},
\end{align}
where $T_x$ is the position ordering operator, $\Delta x = \frac{x-x'}{N}$ and $x\geq x_N > x_{N-1} > \dots >x_1 \geq x'$.
$\mathbf{\Omega}$ is the generator of rotations, given by
\begin{align}
\mathbf{\Omega}(x) &= \left(\begin{array}{ccc}
0 & -m(x) & 0 \\
m(x) & 0 & l(x) \\
0 & -l(x) & 0
\end{array}\right).
\end{align}
The mid-line's configuration is then given by
\begin{align}
\vec{r}(x,0) &= \int_0^x \hat{v}_3(x') \,\text{d} x' ,
\end{align}
and the configuration of ribbon in 3D Euclidean space is
\begin{align}\label{eq:Rib_Shape}
\vec{r}(x,y) &\simeq \vec{r}(x,0) + y \, \hat{v}_1(x) + \frac{1}{2} y^2 n(x) \hat{v}_2
\end{align}
\end{subequations}
where $-\frac{W}{2}\leq y \leq \frac{W}{2}$ along the ribbon's width.
\colorlet{deepg}{green!20!black}
\colorlet{deepb}{blue!50!black}
The shape of a an elastic ribbon is described by its metric ($a$) describing distances between neighboring material elements and curvature tensor ($b$) describing relative angles between neighboring elements. The actual shape is then determined by minimizing the the elastic energy \cite{Efrati2009a} which is schematically given by
\begin{align}
H &= \int {\color{deepb} t \left|a- \bar{a}\right|^2} +{\color{deepg} t^3 \left|b-\bar{b}\right|^2 }\,\text{d} S
\end{align}
Where $\,\text{d} S$ is an area element of the ribbon. This energy is composed of two terms- the first is the stretching energy (dark blue), which is linear in the thickness and penalizes metric ($a$) deviations from the {reference metric ($\bar{a}$)} (describing preferred in-plane distances between material elements), while the second is the bending energy (dark green), cubic in $t$, penalizes deviations from the reference (or spontaneous) curvature \cite{Efrati2009a}. In the reduced 1D model, the energy functional takes the form: (\cite{Grossman2016})
\begin{align}\label{eq:hamiltonian1D}
\nonumber
H =& \frac{Y}{8 \left(1-\nu ^2\right)} \int
\left\{ {\color{deepb} \frac{1}{80} t W^5 \left(\bar{K}- n \,l + m^2\right)^2 }
\right. \\ &
+{\color{deepg} \frac{1}{3} t^3 W \left[2 (1-\nu ) \biggl( \frac{W^2}{12} \left(n'\right)^2 \biggr. \right.} \\ \nonumber
& {\color{deepg} \left. \biggl.
- \left( \bar{l} - l\right) \left(\bar{n} - n \right)+\left(\bar{m} - m \right)^2 \biggr)
\right. }\\ \nonumber
& \left. {\color{deepg} \left.
+ \left( \bar{l} + \bar{n}-n-l \right)^2
+\frac{W^2}{12} \left(m'\right)^2
\right] } \right\} \,\text{d} x,
\end{align}
where $Y$ is Young's modulus, $\nu$ is Poisson's ratio, $\left(\bar{l}(x), \bar{m}(x),\bar{n}(x)\right)$ are the \emph{reference} curvature components;
$\bar{K}(x)$ is the \emph{reference} Gaussian curvature which depends only on $\bar{a}$ and is related to preferred distances between neighbouring material elements. When $\bar{K} = \bar{l}\bar{n}-\bar{m}^2$ (i.e. the reference metric and curvature fulfill Gauss's \emph{Theorema Egregium}), the ribbon is said to be compatible since in this case setting the trivial solution $l=\bar{l},m=\bar{m}, n=\bar{m}$ is the only local and global solution for every width $W$ and thickness $t$. Generally, however, the ribbon is incompatible, $\bar{K} \neq \bar{l}\bar{n}-\bar{m}^2$ and there is no solution that can satisfy simultaneously both the stretching and bending terms, giving rise to residual stresses. It's important to note that the above energy functional accounts for various reference geometries, as well as mechanical limits. For example, in the case of un-stretchable, flat, compatible ribbon, it reduces to the well known Sadowsky functional (\cite{Sadowsky1930,Giomi2010}).
We now turn to the specific case of positive spontaneous curvature and zero reference Gaussian curvature. Specifically $\bar{K}=\bar{m}=0$ and constant $\bar{l}=\bar{n}= k_0$. Such a geometry is evidently incompatible as $\bar{l}\bar{n}-\bar{m}^2= k_0^2 \neq\bar{K}$, and it describes a material which "wants" to have the shape of a sphere on one hand (prescribed by $\bar{l}$ and $\bar{n}$ in the bending term), while keeping it's in-plane distances as in a flat sheet on the other (as prescribed by the stretching term). As mentioned before, this geometry is likely to arise naturally in self-assembled, uneven bi-layers and bola-amphiphile mono-layers in the $L_\beta$ (gel-like) phase. The preferred planar geometry is encompassed by $\bar{K}=0$, while the isotropic difference between the two side is expressed by $\bar{l}=\bar{n}=k_0$, $\bar{m}=0$.
We begin in solving the mechanical equilibrium equations in subsection \ref{s_ch:Euilbrium}, and then move to the statistical behaviour of these ribbons in subsection \ref{s_ch:StatMech}.
\subsection{Mechanical Equilbrium}\label{s_ch:Euilbrium}
We begin by simplifying the Hamiltonian (\ref{eq:hamiltonian1D}) via changing of variables
\begin{subequations}
\begin{align}
l &\equiv h+z\cos(\theta),\\
n &\equiv h-z\cos(\theta),\\
m &\equiv z\sin(\theta).
\end{align}
Thus we defined
\begin{align}
h &\equiv \frac{1}{2}\left(l+n \right),\\
z &\equiv\sqrt{\frac{1}{4}\left(l-n \right)^2+m^2},\\
\theta&\equiv \arctan\left( 2 \frac{m}{l-n}\right).
\end{align}
$h(x)$ is the mean curvature of the mid-line, $z(x)=\sqrt{h^2- K}$ (where $K= n l -m^2$ is the Gaussian curvature at the mid-line) measures the a-sphericity of the ribbon's geometry (in the case of a spherical geometry $z=0$), and $\theta/2$ measures (for $z\neq 0 $ ) the relative angle between the $x$ coordinate (ribbon's long axis) and the principal curvatures axes.
\end{subequations}
Using these variables (and neglecting derivatives) the Hamiltonian of a positively curved ribbon assumes the form
\begin{align}\label{eq:hamiltonian_spherical}
\nonumber
H =& \frac{Y}{8 \left(1-\nu ^2\right)} \int
\left\{ \frac{1}{80} t W^5 \left(h^2 -z^2\right)^2 \right.
\\ & \left.
+ \frac{2}{3} t^3 W \left[(h-k_0)^2(1+\nu)+z^2(1-\nu)
\right] \right\} \,\text{d} x.
\end{align}
Note, that the Hamiltonian, describing the bulk energy of a ribbon, is independent of $\theta$. {This implies that continuous deformations of the ribbon via change of theta (stretching the ribbon spring) cost no bulk energy and the ribbon is expected to be infinitely floppy \cite{Guest2011,Pezzulla2016}}. Such is the case for every "locally spherical" reference geometry (i.e- $\bar{m}=0$, $\bar{l}=\bar{n}=k_0(x)$). The degeneracy is partially lifted by the derivative terms, and by possible boundary layer (see \cite{Efrati2009} and Eq. \ref{eq:boundEner})- at this point both are neglected, we will later see how they affect the result.
By defining the length scale (critical width), and energy scale.
\begin{subequations}
\begin{align}
W^*&= 2^{3/2} \left(\frac{5\left(1-\nu\right)}{3}\right)^{1/4} \sqrt{\frac{t}{(1+\nu)k_0}}\\
E&=\frac{5^{1/4} ~Y t^{7/2} k_0^{1/2}}{ 3^{5/4}\sqrt{2}\left(1-\nu\right)^{3/4}(1+\nu)^{7/2}},
\end{align}
whose significance will be shown immediately, we may transform into dimensionless variables-
\begin{align}
\tilde{h} &= h/k_0\\
\tilde{z} &= z/k_0\\
\tilde{w}&=W/W^*\\
\tilde{x}&= k_0 x,\\
\end{align}
where $\tilde{h}$ is the dimensionless mean curvature of the mid-line, $\tilde{z}$ is the dimensionless a-sphericity, $\tilde{w}$ is dimensionless width and $\tilde{x}$ is dimensionless arc-length along the mid-line.
\end{subequations}
We may then rewrite the Hamiltonian
\begin{align}\label{eq:hamiltonian_spherical_dimless}
\nonumber
H =& E \int
\left\{2(1-\nu) \tilde{w}^5 \left(\tilde{h}^2 -\tilde{z}^2\right)^2 \right.
\\ & \left.
+ \left(1+\nu\right)^2\tilde{w} \left[(\tilde{h}-1)^2(1+\nu)+\tilde{z}^2(1-\nu)
\right] \right\} \,\text{d} \tilde{x} \\ \nonumber
\end{align}
The equilibrium equations obtained by variation of $H$ with respect to the mean curvature $\tilde{h}$, and a-sphericity $\tilde{z}$:
\begin{subequations}
\begin{align}
4\left(1-\nu\right)\tilde{w}^4\tilde{h}\left(\tilde{h}^2-\tilde{z}^2\right)+ \left(1+\nu\right)^3 \left(\tilde{h}-1\right)=0
\end{align}
\begin{align}
2\left(1-\nu\right)\tilde{z}\left[4 \tilde{w}^4\left(\tilde{h}^2-\tilde{z}^2\right)-\left(1+\nu\right)^2\right]=0
\end{align}
\end{subequations}
The solution is given by-
\begin{subequations} \label{eq:equi}
\begin{align}
\tilde{h}_0&= \left\{ \begin{array}{cc}
\frac{1}{2}(1+\nu) \frac{\left(\tilde{\Xi}^2(\tilde{w})-(1-\nu^2) 3^{1/3}\right)}{3^{2/3}(1-\nu)\tilde{\Xi}(\tilde{w})\tilde{w}^2} & \tilde{w}\leq 1 \\
\frac{1}{2} \left(1+\nu\right) & \tilde{w}>1
\end{array}\right.
\\
\tilde{z}_0&= \left\{\begin{array}{cc}
0 & \tilde{w} \leq 1 \\
\frac{1}{2} \left(1+\nu\right) \frac{\sqrt{\tilde{w}^4-1}}{\tilde{w}^2} & \tilde{w} >1
\end{array}\right.,
\end{align}
where
\begin{align*}
\tilde{\Xi}(\tilde{w})&=\left[9\left(1-\nu\right)^2 \tilde{w}^2+\sqrt{81\left(1-\nu\right)^4 \tilde{w}^4+3\left(1-\nu^2\right)^3}\right]^{1/3}.
\end{align*}
\end{subequations}
As can be seen $\tilde{w}=1$ is the critical dimensionless width where a sharp, yet continuous transition between two regimes occurs: bending dominated, narrow regime ($0<\tilde{w}<1$), and stretching dominated, wide regime ($1<\tilde{w}$, Fig. \ref{fig:curvs}). Also note, that these equations are identical to those written in \cite{Grossman2016} for ribbons with negative (saddle) reference curvature,
under the map: $\tilde{h} \leftrightarrow \tilde{m}$, $\tilde{z} \leftrightarrow \tilde{l}$, $\nu \leftrightarrow -\nu$.
\begin{figure}
\centering
\begin{tikzpicture}{scale=1}
\node[above] at (0,0){\includegraphics[width=.7\textwidth]{curvs_sphere.pdf}};
\node[below](a) at (-3.8*0.7,0){\mbox{\Large (a)}};
\node[below](b) at (4.5*0.7,.5*0.7){\mbox{\Large (b)}};
\node[below] at (-4*0.7,-1*0.7){\includegraphics[clip, trim={0cm 0cm 0 5cm },width=0.35\textwidth]{MidAll.jpg}};
\node[below] at (4.5*0.7,-0.3*0.7){\includegraphics[width=0.35\textwidth]{WideAll.jpg}};
\draw[dashed, very thick] (a)--(-3*0.7,1.95*0.7) -- (-3*0.7,12.3*0.7);
\draw[dashed,very thick] (b)--(.51*0.7,1.95*0.7) -- (.51,12.3*0.7);
\end{tikzpicture}
\caption{Plots of the solution given by Eq. \ref{eq:equi} for the $\nu=0$ case. $\tilde{h}$ in Blue and $\tilde{z}$ in Red. Larger/smaller values of $\nu$ result with higher/lower asymptotic values. A narrow ribbon ($\tilde{w}<1$ ) has a ring like shape with radius which depends on the width. A wide ribbon, at any $\tilde{w}>1$ may have a degenerate family of shapes related by a single parameter $\theta$. These shape are visualized at two given widths $\tilde{w}=1.075$ and $\tilde{w}=2$ (markers (a) and (b) respectively). Below the main figure.}\label{fig:curvs}
\end{figure}
When narrow ($\tilde{w}<1$, $\tilde{z}=0$), the ribbon is shaped as a ring taken around the equatorial of a sphere, who's radius grows monotonically with the width (around $\tilde{w}=0$, $R(\delta\tilde{w})-R(0) \propto \delta\tilde{w}^4$ respectively), up until the critical width. Above it (wide regime), the system is locally an ellipsoid (hence non-zero a-sphericity), where the mean curvature remains constant, but the principal curvatures change such that one of them reaches zero at the limit $\tilde{w} \gg 1$ (geometry of a cylinder, see Fig. \ref{fig:curvs}).
The angle $\theta$ has no physical meaning in the bending dominated (narrow) regime, as the system is locally a sphere (as indicated by the zero a-sphericity, $z=0$ ). However, in the stretching dominated (wide) regime it relates to the local shape, the degeneracy in $\theta$ means there is a degenerate continuous family of configurations all related by a local change in $\theta$. Physically, however, this huge degeneracy is partially lifted, as variation along the ribbon's length cost energy ($\Delta E _{der} \propto Y t^3 W^3 (m')^2$ and a similar expression for $n$), thus the ground states are uniform along the ribbon. They correspond to shapes varying continually between a ring and different helices (both left and right handed). For a very wide ribbon, a straight mid-line (infinite pitch) is also a possible solution. In Fig.\ref{fig:curvs} we see some of the possible configurations of a wide ($\tilde{w}>1$) ribbon, close to the transition (a), and far from it (b). For a given width, all the configurations have the same energy according to Eq. \ref{eq:hamiltonian_spherical_dimless}, and are related by different values of $\theta$. The limit $\tilde{w} \gg 1$ recovers the pre-stressed metal ribbons studied by Guest et al.(\cite{Guest2011})
The remaining degeneracy (as depicted in Fig. \ref{fig:curvs}) is lifted by a boundary layer \cite{Efrati2009}, where the local geometry is bending dominated (rather than the bulk which is stretching dominated at $\tilde{w}>1$). In other other words, the local geometry is set by the spontaneous curvatures ($\bar{l},\bar{m}, \bar{n}$) rather than the reference {metric} ($\bar{K}$). This results with a boundary layer along the ribbon's width, such that exactly at the ribbon edge $\tilde{n}=\bar{n}$. The boundary layer's energy for $\tilde{w}\gg1$ and $\nu=0$ is given by (for a general expression for any $\tilde{w}>1$ and $\nu$ see appendix \ref{app:Bound_Layer})
\begin{align}\label{eq:boundEner}
H_{bound}&\propto -Y t^{7/2}k_0^{3/2} \int |\cos(\theta/2)|^3 \,\text{d} x.
\end{align}
Therefore, the boundary layer defines the actual shape of a wide ribbon at mechanical equilibrium (a "ring" configuration ($\theta=0$) is preferential). The resulting shape in such cases is therefore sensitive to the shape and orientation of the edges of the elastic sheet (\cite{Pezzulla2016}). Thus by controlling the shape of the edge one is able to produce many different shape from the same material.
However, as can be seen from Eq.(\ref{eq:boundEner}), the boundary layer's energy (and also size- see \cite{Efrati2009,Levin2016}) is independent of the ribbon's width (in contrast with the bulk energy), and vanishes at the infinitely thin limit. Therefore, in the context of statistical mechanics, we may consider the degeneracy of a global angle $\theta$ only weakly broken. It can be shown (see section \ref{sub:curv_fluct} ) that for wide enough ribbons we may always work at temperatures where the boundary layer's contribution to the statistical behaviour of the ribbon is negligible.
\subsection{Shape Fluctuations}\label{s_ch:StatMech}
In a thermal environment, our ribbons fluctuate, causing their configuration to deviate from their (mechanical) equilibrium one. Characterizing these deviations is of great importance. In principle, both boundary layer and derivatives contribute to the statistical properties of these ribbons. In practice, however, boundary layer contributes only to wide ribbons (no boundary layer for thin ribbons) and is negligible at the thin ribbon limit ($t\rightarrow0$, see the end of section \ref{sub:curv_fluct}). In what follows we assume a thin enough ribbon such that the boundary layer may be neglected. In other words, we assume that the temperatures are high enough so that the system is practically indifferent to the boundary layer.
The derivatives contribution, is somewhat more subtle, as it affects at every temperatures range (there is always small enough scale such that fluctuations on that scale will be suppressed). Nevertheless, their effect (apart of a finite correlation length) is usually quantitative rather than qualitative. Such is the case in every ribbon who's equilibrium shape is extended (in contrast to compact, which is our case), such as in \cite{Panyukov2000a,Giomi2010,Ghafouri2005,Grossman2016}. As will be seen, in positively curved ribbons, the derivatives affect some aspects of the thermal behaviour of the ribbon qualitatively. For simplicity, in what follows we neglect both boundary layer and derivatives from the analysis, unless otherwise mentioned.
\subsubsection{Curvature Fluctuations}\label{sub:curv_fluct}
One way to describe the statistical nature of elastic ribbons is to calculate the fluctuations in their curvatures which are the variables in the Hamiltonian. As such they are a natural choice to describe the statistical behaviour of a ribbon. Calculation is done in a similar manner to the one described in \cite{Grossman2016}, we limit ourselves to the Gaussian approximation, i.e around the solutions in Eqs. \ref{eq:equi}. We start by expanding $H$ to second order about the equilibrium values
\begin{align}
H \simeq H_{0} + H_{(2)}.
\end{align}
In principle, $H_{(2)} =H_{(2)}\left[z(x),h(x),\partial_x z,\partial_x h,\partial_x \theta\right]$, however, as mentioned earlier, we start by neglecting derivatives. The average of a quantity Q is then given by the functional integral
\begin{align}
\langle Q \rangle = \frac{1}{\mathcal{Z}}\int Q e^{-\beta H_{(2)}\left[z(x),h(x)\right]} \prod_x z(x)\,\text{d} z(x) \,\text{d} h(x) \,\text{d} \theta(x).
\end{align}
Where we defined the partition function $\mathcal{Z}= \int e^{-\beta H_{2}\left[z(x),h(x)\right]}\prod_x z(x)\,\text{d} z(x) \,\text{d} h(x) \,\text{d} \theta$.
At the given approximation, integration over the angle is trivial. Neglecting the energetic contribution of derivatives results with no correlation at different positions. i.e-
$ \langle \Delta Q(\tilde{x}) \Delta Q(\tilde{x}')\rangle = \delta(\tilde{x}-\tilde{x}')\langle \Delta Q^2 \rangle$. Hence, the averages and fluctuations of the curvatures are given by
\begin{subequations} \label{eq:avgs}
\begin{align}
\langle \tilde{l} \rangle& = \langle \tilde{h} \rangle+ \langle \tilde{z}\rangle \langle \cos \theta \rangle = \langle \tilde{h} \rangle = \langle \tilde{n} \rangle \equiv \tilde{h}_{eq}\\
\langle \tilde{m} \rangle &=0 \\
\langle \Delta\tilde{l}(\tilde{x}) \Delta \tilde{l}(\tilde{x}')\rangle &=\langle \tilde{l}(\tilde{x}) \tilde{l}(\tilde{x}') \rangle - \langle \tilde{l} \rangle^2 = \delta (\tilde{x}-\tilde{x}') \langle \Delta \tilde{l}^2 \rangle\\ \nonumber
\langle \Delta \tilde{l}^2 \rangle &= \langle\Delta \tilde{h}^2 \rangle + \frac{1}{2} \left( \langle\Delta \tilde{z}^2 \rangle \right) = \langle \Delta \tilde{n}^2 \rangle\\
\langle \Delta \tilde{m}^2 \rangle &= \frac{1}{2} \langle\Delta \tilde{z}^2 \rangle \\ \nonumber
\langle \Delta \tilde{h}^2\rangle&= \langle \left(\tilde{h}- \tilde{h}_{eq} \right)^2 \rangle = \langle \left(\tilde{h}- \tilde{h}_{0} \right)^2 \rangle- \left(\tilde{h}_{eq}-\tilde{h}_0\right)^2 \\ \nonumber
\langle \Delta \tilde{z}^2\rangle&= \langle \left(\tilde{z}- \tilde{z}_{eq} \right)^2 \rangle = \langle \left(\tilde{z}- \tilde{z}_{0} \right)^2 \rangle- \left(\tilde{z}_{eq}-\tilde{z}_0\right)^2,
\end{align}
\end{subequations}
Where $\tilde{h}_0, \tilde{z}_0$ are given in Eq \ref{eq:equi}.
Due to the non trivial measure (a result from $z$ being a non- negative variable), the resulting averages are cumbersome. They are given fully in appendix \ref{app:Specific_Calcs} and depicted in Fig. \ref{fig:moments}. It is clearly seen in Fig. \ref{fig:moments} (a) that the thermal equilibrium values diverge near $\tilde{w}=1$ indicating that the calculation breaks near the critical width. Indeed we perform in appendix \ref{app:Saddle_Point} a more accurate calculation using a saddle point approximation (taking into account the non-trivial measure in the integral) in which these values are finite. At low enough temperatures the treatment shown in Fig. \ref{fig:moments} coincides with the saddle point approximation. In any case, at low enough temperatures (even near $\tilde{w}=1$), the more accurate treatment affect only quantitatively and not qualitatively on any of the following results. We therefore settle for clarity over accuracy in what we show hereafter. For practical use, low enough temperatures and $\tilde{w} \neq 0,1$, we may approximate
\begin{subequations}
\begin{align}
\tilde{h}_{eq} \simeq &\tilde{h}_0 \\
\tilde{z}_{eq} \simeq &\tilde{z}_0 + \left\{ \begin{array}{cc}
\frac{\sqrt{\pi}}{2\sqrt{(1-\nu)\Psi \tilde{w}\left((1+\nu)^2-4 \tilde{w}^4\tilde{h}_{eq}^2\right)}} & \tilde{w}<1 \\
0 & \tilde{w}>1
\end{array} \right. \\
\langle \Delta \tilde{h}^2 \rangle \simeq & \left\{ \begin{array}{c c}
\frac{1}{2 \Psi \tilde{w} \left((1+\nu)^3 + 12 (1-\nu) \tilde{w}^4 \tilde{h}_{eq}^2\right)} & \tilde{w}<1 \\
\frac{1}{4 \Psi \tilde{w} (1+\nu)^2} & \tilde{w} >1
\end{array} \right.\\
\langle \Delta \tilde{z}^2 \rangle \simeq & \left\{ \begin{array}{c c}
\frac{1}{ \Psi (1-\nu) \tilde{w} \left((1+\nu)^2 \tilde{w} - 4 (1-\nu) \tilde{w}^5 \tilde{h}_{eq}^2\right)} & \tilde{w}<1 \\
\frac{1+{(1-\nu)\tilde{w}^4}}{4 \Psi \tilde{w} (1-\nu) (1+\nu)^2 \left(\tilde{w}^4-1\right)} & \tilde{w} >1
\end{array} \right.
\end{align}\label{eq:flucs_simp}
\end{subequations}
Where $\Psi= \frac{E}{k_B T}=\frac{5^{1/4}}{ 3^{5/4}\sqrt{2}\left(1-\nu\right)^{3/4}(1+\nu)^{7/2}} \frac{Y t^{7/2} k_0^{1/2} }{k_B T}$, and we kept deviations of $\tilde{z}_{eq}$ from $\tilde{z}_0$ for the narrow ribbon as $\tilde{z}_0(\tilde{w}<1)=0$ and they are of order $\sqrt{T}$ ( not $T$) and therefore important. It is worth to note that even at high temperatures $\langle \Delta \tilde{h}^2 \rangle\propto 1/\Psi $, $\langle \Delta \tilde{z}^2 \rangle \propto 1/\Psi$, yet with different dependence on $\tilde{w}$.
\begin{figure}
\centering
\begin{tabular}{cc}
\begin{subfigure}{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{moments1.pdf}
\caption{}\label{}
\end{subfigure} &
\begin{subfigure}{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{moments2.pdf}
\caption{}\label{}
\end{subfigure} \\
\end{tabular}
\caption{Moments of $\Delta \tilde{h}$ and $\Delta \tilde{z}$, for the case of $\nu=0, \Psi=\frac{E}{k_B T}=10$. In Red are the moments of the mean curvature $\tilde{h}$, in Blue those of the a-sphericity $\tilde{z}$. Different values of $\Psi$ or $\nu$ results is similar graphs (differences are numerical only).} \label{fig:moments}
\end{figure}
While the statistics of a narrow ($\tilde{w}<1$) ribbon are well captured even without including derivatives, some of the wide ($\tilde{w}>1$) ribbon's statistics are governed by those terms. Therefore, we include here results of the correlations including those terms as they will come in handy soon enough. Expanding the Hamiltonian to 2$^{nd}$ order, and changing into Fourier space, we end up with an angle ($
\theta$) dependent expression. However, to our needs, as we eventually integrate the angle out, we may approximate it without introducing any significant error as (see appendix \ref{app:Theta_avg} for a more detailed analysis)
\begin{align}\label{eq:second_order_fourier}
\Psi H_{(2)}&= \Psi \int \,\text{d} q ~ \frac{1}{2} \left\{ \left[\left(8 \tilde{w}^5\left(1-\nu\right)\left(3 \tilde{h}^2_{eq} -\tilde{z}^2_{eq}\right)+2\tilde{w} \left(1+\nu\right)^3 \right)|\Delta \tilde{h}(q)|^2 \right.\right. \\ \nonumber
&\left. +\left(8 \tilde{w}^5\left(1-\nu\right)\left(3 \tilde{z}^2_{eq} -\tilde{h}^2_{eq}\right)+2\tilde{w} \left(1-\nu\right)\left(1+\nu\right)^2 \right)|\Delta \tilde{z}(q)|^2 - 32 \tilde{w}^5 \left(1-\nu\right)\tilde{h}_{eq} \tilde{z}_{eq}\Re(\Delta \tilde{z}(q) \Delta \tilde{h}^\dagger(q))\right] \\ \nonumber
& \left.+ 16 \sqrt{ \frac{5}{3} \left(1-\nu\right) } \frac{k_0 t}{1+\nu} \left[\frac{(1+\nu)^2\tilde{w}^3 q^2}{48} \left(4\left(1-\nu\right) |\Delta{\tilde{h}}(q)|^2+ \left(3-2\nu\right)\left(\tilde{z}_{eq}^2 |\Delta \theta(q)|^2 + |\Delta{\tilde{z}(q)}|^2\right)\right)\right] \right\}.
\end{align}
The correlations and fluctuations of the mean curvature $\tilde{h}$ and a-sphericity $\tilde{z}$ result with the usual finite correlation lengths $\xi_h(\tilde{w}),~\xi_z(\tilde{w})$, with the slight exception (as in \cite{Grossman2016}, stemming from the fact that the shape transition is function of $\tilde{w}$ and not $T$)
that they are temperature independent. Such that
\begin{subequations}
\begin{align}\label{eq:FiniteCorrsLength}
\langle \Delta \tilde{h}(\tilde{x}) \Delta \tilde{h}(\tilde{x}') \rangle &= \frac{\langle \Delta \tilde{h}^2 \rangle}{2 \xi_h}e^{-\frac{\left|\tilde{x}-\tilde{x}'\right|}{\xi_h}} \\
\langle \Delta \tilde{z}(\tilde{x}) \Delta \tilde{z}(\tilde{x}') \rangle &= \frac{\langle \Delta \tilde{z}^2 \rangle}{2 \xi_z}e^{-\frac{\left|\tilde{x}-\tilde{x}'\right|}{\xi_z}},
\end{align}
where $\langle \Delta \tilde{h}^2 \rangle,~\langle \Delta \tilde{z}^2 \rangle$ are as given above in Eq. \ref{eq:flucs_simp}. This result is true for any finite width, except at $\tilde{w}=1$, where technically $\xi_z\rightarrow \infty$ (and $\langle \Delta \tilde{z}^2 \rangle$).
\end{subequations}
The expressions in Eqs. \ref{eq:avgs} now change
\begin{subequations}
\begin{align}\label{eq:newAvgs}
\langle \Delta\tilde{l}(\tilde{x}) \Delta \tilde{l}(\tilde{x}')\rangle &= \frac{\langle\Delta \tilde{h}^2 \rangle}{2 \xi_h}e^{-\frac{\left|\tilde{x}-\tilde{x}'\right|}{\xi_h}} + \frac{1}{2} e^{-\frac{\left|\tilde{x}-\tilde{x}'\right|}{\xi_\theta}} \tilde{z}_{eq}^2+ \frac{\langle\Delta \tilde{z}^2 \rangle}{4\xi_z} e^{-\frac{\left|\tilde{x}-\tilde{x}'\right|}{\xi_z} }= \langle \Delta{n}(\tilde{x}) \Delta{n}(\tilde{x'})\rangle \\
\langle \Delta{m}(\tilde{x}) \Delta{m}(\tilde{x'})\rangle &=\frac{1}{2} e^{-\frac{\left|\tilde{x}-\tilde{x}'\right|}{\xi_\theta}} \tilde{z}_{eq}^2+ \frac{\langle\Delta \tilde{z}^2 \rangle}{4\xi_z} e^{-\frac{\left|\tilde{x}-\tilde{x}'\right|}{\xi_z} }
\end{align}
\end{subequations}
where $\xi_\theta=16 \Psi \sqrt{ \frac{5}{3} \left(1-\nu\right) } \frac{k_0 t}{1+\nu}\frac{(3-2\nu) (1+\nu)^2\tilde{w}^3 \tilde{z}_{eq}^2}{48}$.
By taking the limit $\xi_z,\xi_h,\xi_\theta \rightarrow0$ we retrieve the previous results. Thus far, inclusion of derivatives has changed the nature of the correlation by adding finite correlation lengths. In the following sections we will see how (and when) these correlation length affect the shape of the ribbon {(see section \ref{sub:WideRibb})}.
Finally, as a wide ribbon also has energetic contribution from a boundary layer, it is worth to see when is the boundary layer important. From Eq. \ref{eq:boundEner}, the boundary layer's contribution scales as $\Psi$, therefore for $\Psi \ll 1$ we may neglect it completely. From Eq. \ref{eq:avgs} it is clear that the low temperature limit is achieved for $\Psi \gg \frac{1}{\tilde{w}}$ , hence in the wide regime we may choose to work in regimes so that $\frac{1}{\tilde{w}} \ll \Psi \ll 1$. From this point on we neglect the boundary layer.
\subsubsection{Shape Fluctuations}
Experimentally the curvatures are hard to measure, and they are not an intuitive tool to understand the ribbon's shape. In polymer science, other measures are used to describe the ribbon's stiffness and shape - most common \cite{Rubinstein2003} are the persistence length $\ell_p$, the gyration radius $R_g$ and the Kuhn length $\ell_k$. The persistence length, $\ell_p$ is the scale over which tangent - tangent correlation decay,
\begin{subequations}
\begin{align}
\ell_p: & \langle \hat{v}_3 (x) \hat{v}_3 (x') \rangle \propto e^{-\frac{\left| x-x' \right|}{\ell_p}}.
\end{align}
The gyration radius $R_g$ approximates the shape enclosing volume of gyrating ribbon in a thermal environment as a sphere and is given by
\begin{align}
R_g^2&= \langle \int_0^L \frac{\,\text{d} x}{L} \left[r(x)^2 - 2\vec{r}(x) \vec{r}_0+ r_0^2 \right] \rangle=- \langle r_0^2 \rangle + \frac{1}{L}\int_0^L \langle r(x)^2\rangle \,\text{d} x
\end{align}
where $\vec{r}(L)$ is the end-to-end distance of the ribbon, and $\vec{r}_0$ is it's center of mass. The Khun length $\ell_k$ is defined as the segment length of a random freely-jointed chain that reproduces same end-to-end length.
\begin{align}
\ell_k = \lim\limits_{L\rightarrow \infty }\frac{1}{L} \langle r^2(L) \rangle.
\end{align}
Another, less common, measure is the torsional correlation length $\ell_\tau$ which is the scale over which bi-normal - bi-normal correlation decay \cite{Giomi2010}.
\begin{align}
\ell_\tau: & \langle \hat{v}_1 (x) \hat{v}_1 (x') \rangle \propto e^{-\frac{\left| x-x' \right|}{\ell_\tau}}.
\end{align}
Nevertheless (as will be seen in the next sections), these quantities fail to encompass the shape of a gyrating ribbon. To this end we should calculate the gyration tensor which approximates the volume in which a ribbon gyrates as an ellipsoid (rather than a sphere).
\begin{align} \label{eq:gyration_tensor}
R_{ij}(L) = \frac{1}{L} \int_{0}^L \left[\langle r_i(x) r_j(x) \rangle - \langle (r_0)_i(r_0)_j \rangle \right]\,\text{d} x.
\end{align}
where $r_i(x)= \vec{r}(x) \cdot \hat{v}_i(0)$. Note that $R_g^2= R_{11}+R_{22}+R_{33}$. As it turns out, $R_{ij}$ is hard to calculate (see appendix \ref{app:Stat_Meas} we therefore add to the list another measure to probe the shape of a gyrating ribbon, the Frame-Origin Correlations
\begin{align}\label{eq:FOCM}
\rho_{ij}(x) &=\langle v^i_3(x)v^j_3(x) \rangle
\end{align}
where $v^i_3(x)=\hat{v}_i(x) \cdot \hat{v}_3(0)$.
\end{subequations}
Using the 2$^{nd}$ order expansion of the Hamiltonian \ref{eq:hamiltonian_spherical_dimless}, we find that for $x>x'$ (see appendix \ref{app:Stat_Meas})
\begin{align} \label{eq:corrmat}
\langle \hat{v}_i(x) \hat{v}_j(x')\rangle &= \langle e^{-\int_{{x'}}^{x} \mathbf{\Omega} \,\text{d} \xi''} \rangle_{ij} =\left[e^{-\mathbf{ \Lambda} (x-x')}\right]_{ij}
\end{align}
where (neglecting derivatives)
\begin{align} \label{eq:Lambda}
\Lambda &= \left( \begin{array}{ccc}
0 & 0 & 0 \\
0 & 0 & \langle l \rangle \\
0 & -\langle l \rangle & 0
\end{array}\right) + \frac{1}{2}\left( \begin{array}{ccc}
\langle \Delta m^2 \rangle & 0 & 0\\
0 & \langle\Delta m^2\rangle + \langle\Delta l^2\rangle & 0 \\
0& 0 & \langle\Delta l^2 \rangle
\end{array}\right).
\end{align}
The averages are given above in Eqs. \ref{eq:avgs}, \ref{eq:flucs_simp}. Retrieving dimensionality is straightforward (note that $\langle\Delta m^2\rangle = k_0 \langle \Delta \tilde{m}^2 \rangle$, and similarly for $\Delta l^2$). The $\Lambda$ matrix is central for calculations of the entities presented above.
As $\Lambda$ encompasses significant information (yet not all) regarding the ribbon's thermal behaviour and structure, it is useful to study some of it's properties. Specifically, it's eigenvalues are
\begin{align} \label{eq:eigen}
\lambda_0 &= \Lambda_{11} =\frac{1}{2} \langle \Delta m^2 \rangle \\ \nonumber
\lambda_{\pm} &= \frac{1}{4} \left(\langle \Delta m^2 \rangle +2 \langle \Delta l^2 \rangle \pm \sqrt{\varpi} \right). \\ \nonumber
\text{\small $\varpi (T,\tilde{w})$} &= \text{\small $ \langle \Delta m^2 \rangle^2 - 16 \langle l \rangle^2$}
\end{align}
At any given width, $\langle \Delta m^2 \rangle$, and $\langle \Delta l^2 \rangle$ roughly scale as $T$ (\cite{Bigtheta}, see also Eq. \ref{eq:flucs_simp} for low temperature approximation). This is in contrast to $\langle l \rangle$ which is a non-zero constant as $T\rightarrow 0$, and (depending on $\tilde{w}$) at hight $T$ scales as $T$. Therefore, at low $T$, $\varpi < 0$ and $\lambda_{\pm}$ are complex conjugates. At some $T^*$, $\varpi(T^*)=0$, and $\lambda_{\pm}$ are equal. Above $T^*$, they are positive reals, signifying that the ribbon lost all structure, and behaves as a random coil. Since for every $\tilde{w}$ there exists only such one (positive) temperature, $T^*(\tilde{w})$ is well defined. It is important to note that experimentally the ribbon will seem to lose structure at lower temperatures. This will happen roughly as the typical fluctuation will be of magnitude similar to the curvature of the ribbon. In other words, when either $\langle \Delta l^2 \rangle \sim \langle l \rangle $ or $\langle \Delta m^2 \rangle \sim \langle l \rangle $. As it turns out, in this case at least, the former happens earlier than the latter. Nevertheless, $T^*$ remains a better defined parameter. Solving for $T^*$ analytically is hard, we therefore plot it in Fig \ref{fig:Tstar} for the case $\nu=0$. We see the critical temperature divides the graph into three regions. It is essentially the phase diagram of the ribbon, which we are yet to characterize. Region I is the high temperature limit- or the Ideal Chain phase, where the ribbons behaves as a random coil. We named regions II and III the Plane Ergodic and the Random Structured phases respectively. In the following subsections we characterize them.
\begin{figure}
\centering
\begin{tikzpicture}
\node[above] at (0,0){\includegraphics[width=0.6\textwidth]{Tstar.pdf}};
\node[above] at (-2.25,5){\fontfamily{arev}\color{darkgray} \huge I};
\node[above] at (-2.25,2){\fontfamily{arev}\color{darkgray}\huge II};
\node[above] at (2,3){\fontfamily{arev}\color{darkgray}\huge III};
\end{tikzpicture}
\caption{$\tilde{T}^*=\frac{1}{\Psi^*}$ , the temperature above which a ribbon behaves like a random coil as a function of ribbon width $\tilde{w~}$ ($\nu=0$). $T^* =\frac{5^{1/4} }{3^{5/4} \sqrt{2} (1-\nu)^{3/4}(1+\nu)^{7/2} } \frac{Y t^{7/2} k_0^{1/2}}{k_B} \tilde{T}^*$. At $\tilde{w}=0,1$, $\tilde{T}^*=0$. Area $I$ corresponds to the Ideal Chain phase, area II is the Plane Ergodic phase and area II is the Random Structured phase (see subsections \ref{sub:NarrRibb}, and \ref{sub:WideRibb} respectively) \label{fig:Tstar}}
\end{figure}
\subsubsection{Narrow Ribbon $\tilde{w}<1$}\label{sub:NarrRibb}
In this section we characterize the thermal dependence of a narrow, cold $T<T^*$ ribbon. In all calculations in this section we omit derivatives as their contribution is negligible in the limit of $T\rightarrow 0$ (as they give rise to finite, temperature independent, correlation lengths).
As in \cite{Giomi2010}, $\ell_\tau$, and $\ell_p$ hold important information about the structure of the ribbon. Specifically they are the scales on which bi-normal - bi-normal and tangent - tangent correlation decay exponentially. From $\Lambda$ (Eq. \ref{eq:Lambda}) we can easily extract $\ell_\tau = 2/\langle\Delta m^2 \rangle$ and $\ell_p=2/\langle \Delta l^2 \rangle$. Asymptotically
\begin{subequations}\label{eq:ell_tau_ell_p_ass_narrow}
\begin{align}
\ell_\tau/\ell_p & \xrightarrow{\tilde{w}\rightarrow 0} \frac{8-\pi(1+\nu)}{(4-\pi)(1+\nu)}.
\end{align}
\end{subequations}
While $\tilde{\ell}_\tau$ and $\tilde{\ell}_p$ have a slight dependence on temperature, their ratio $\ell_\tau/\ell_p$ is independent of it. $\tilde{\ell}_\tau$ and $\ell_\tau/\ell_p$ are depicted in Figs. (\ref{fig:l_tau_vs_l_k} ,\ref{fig:ltauOnlp}) respectively, for the cases $\nu=0,\pm \frac{1}{2}$. The result remain qualitatively the same for different values of $\nu$.
\begin{figure}[h]
\centering
\includegraphics[width=0.6\textwidth]{elltauOnp_narrow.pdf}
\caption{The ratio, $\ell_\tau/\ell_p $ for a narrow ribbon is independent of temperature, but varies with $\tilde{w}$, plotted here for the cases $\nu=0,\pm \frac{1}{2}$. At different values of $\nu$, the narrow limit follows ${\ell_\tau}/{\ell_p}|_{\tilde{w}=0}=\frac{8-\pi(1+\nu)}{(4-\pi)(1+\nu)}$. At $\tilde{w}=1$, ${\ell_\tau}/{\ell_p}|_{\tilde{w}=1}=1$.
\label{fig:ltauOnlp}}
\end{figure}
\begin{figure}[h!]
\hspace*{-1em}\begin{tabular}{c c c}
\begin{subfigure}{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{ellk10_narrow.pdf}
\caption{$\tilde{T}=.1$}\label{fig:KunhLengths_sub10}
\end{subfigure}
&
\begin{subfigure}{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{ellk1_narrow.pdf}
\caption{$\tilde{T}=1$}\label{fig:KunhLengths_sub01}
\end{subfigure}&
\begin{subfigure}{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{ellk01_narrow.pdf}
\caption{$\tilde{T}=10$}\label{fig:KunhLengths_sub1}
\end{subfigure} \\
\multicolumn{3}{ c}{\begin{subfigure}{0.3\textwidth}
\raggedright
\includegraphics[width=\textwidth]{elltau10_narrow.pdf}
\caption{} \vspace*{2em}\label{fig:l_tau_sub10}
\end{subfigure}\hspace*{1em}}
\end{tabular}
\caption{Dimensionless Kuhn length, $\tilde{\ell}_k= \ell_k k_0$ (a-c), and dimensionless torsional correlation length, $\tilde{\ell}_\tau = \ell_\tau k_0/\Psi$ (d), for $\nu=0$ (note the difference in definition). While, $\ell_\tau$ is independent of temperature (up to scaling by $\Psi=\frac{1}{\tilde{T}}$), $\ell_k$ is very sensitive to temperature change. At $\tilde{w}=1$ both $\ell_\tau, \ell_k=0$ as the ribbon is anomalously soft at the critical width.\label{fig:l_tau_vs_l_k} }
\end{figure}
A direct calculation (\ref{app:Stat_Meas}) shows that the Kuhn length is given by $(\Lambda^{-1})_{33}$. The inverse of $\Lambda$ is
\begin{align}
\Lambda^{-1}= \left(\begin{array}{ccc}
\frac{2}{\langle \Delta m^2 \rangle} & 0 & 0 \\
0 & \frac{ 2 \langle \Delta l^2 \rangle}{\varsigma} &\frac{- 4 \langle l \rangle}{\varsigma} \\
0 & \frac{4 \langle l \rangle}{\varsigma} & 2\frac{ \langle \Delta l^2 \rangle+\langle \Delta m^2 \rangle}{\varsigma}
\end{array}\right),&~~~
\text{\small $\varsigma$}= \text{\small $\langle \Delta l^2 \rangle(\langle \Delta l^2 \rangle+\langle \Delta m^2 \rangle)+ 4 \langle l \rangle^2$}\\ \nonumber
\Downarrow &\\
\ell_k = & 2\frac{\langle \Delta l^2 \rangle+\langle \Delta m^2 \rangle}{\varsigma}
\end{align}
$\ell_k$ is plotted at different temperatures is in Fig. (\ref{fig:l_tau_vs_l_k}) (for the case of $\nu=0$, sub figures (a)-(c)), the result stands in stark contrast to $\ell_\tau$ (Fig. \ref{fig:l_tau_sub10}) and $\ell_p$. Not only that $\ell_k$ has a non-monotonic dependence on the ribbon's width, its temperature dependence is also very unique (in contrast with \cite{Grossman2016}) as it converges to $0$ also at low temperatures ($T \rightarrow 0$), not only high ($T \rightarrow \infty$), while it reaches a maximum at some intermediate temperature. This unique dependence of the Kuhn length $\ell_k$ on the temperature, in contrast to that of the persistence and torsional correlation lengths ($\ell_p$ and $\ell_\tau$) directly relates to the fact that $\ell_p$ and $\ell_\tau$ measure arc-distance (\emph{on the ribbon}), while $\ell_k$ measures distance in the embedding space.
At high temperatures ($T\gg T^*$), this result is easily understandable as the ribbon loses all structure, and reverts to the classical behaviour of a freely jointed chain where $\ell_k \sim \ell_p \propto \frac{1}{T}$. This result is merely the "ideal chain" phase that every ribbon of any kind exhibits at high enough temperatures. At low temperatures ($T\ll T^*$), the ribbon mostly retains it's structure (large $\ell_p$, $\ell_\tau$). Since at $T=0$ a narrow ($\tilde{w}<1$) ribbon is ring shaped, the end-to-end distance has a maximal value, $\langle r^2 \rangle \leq 4 R^2$ (where $R$ is the ring's radius). Therefore at the limit we get
\begin{align*}
\lim\limits_{L \rightarrow \infty} \frac{1}{L} \langle r^2(L)\rangle_{T=0} = 0.
\end{align*}
At finite low temperatures, the ribbon retains its shape for a long, but finite, arc-length. This results with $\ell_k (T\ll T^*) \propto T$. The local maxima in Figs. (\ref{fig:KunhLengths_sub10}, \ref{fig:KunhLengths_sub1}), occur roughly around the widths such that $T\sim T^*(\tilde{w})$. Indeed, at $\tilde{T}=4$, a narrow ribbon may be considered "completely melted" (i.e- lost all structure) as at any $0<\tilde{w}<1$, $T^*(\tilde{w})\leq T$. At $\tilde{w}=1$ the ribbon is infinitely soft, indicated by the fact that $\ell_\tau,~ \ell_p,~\ell_k$, and $T^*(\tilde{w}=1)$ are all equal to zero. Counter intuitively, the ribbon gets softer as it widens and approaches the transition. Both $\ell_\tau$, and $\ell_k$ are depicted as a function of $\tilde{T}$, and $\tilde{w}$ in Fig. (\ref{fig:lp_ltau}). In it, also $\ell_\tau(T^*)$ is plotted to visualize $T*$ (red line), it is immediately seen that the apparent "melting" of the ribbon (peaks in $\ell_k$) occur when $T$ is significantly smaller than $T^*$. Nevertheless, $T^*$ remains a better defined parameter.
\begin{figure}
\centering
\includegraphics[width=0.7\textwidth,clip,trim={0 0 0 6.7cm}]{lpltau_nar.jpg}
\caption{$\ell_k$ (coloured surface) and $\ell_\tau$ (black mesh) as a function of $\tilde{w}$ and $\tilde{T}$. The red curve marks $\ell_\tau(\tilde{T}^*)$. The orange peaks in $\ell_k$ occur at lower temperatures, marking an ``apparent melting" of the ribbon.}\label{fig:lp_ltau}
\end{figure}
Following these results, the ribbon's shape comes into question- what should we expect to see when looking into a thermal soup of such ribbons? The fact that $\ell_p \leq \ell_\tau$ at most widths, suggests that the ribbon gyrates non-spherically, at least for some range of lengths. However, $\ell_\tau$ and $\ell_p$ are "internal" measures, in the sense that they tell us how long we should measure the arc-length of a ribbon before we see some change. $\ell_k$ supplies an "external" measure, yet it turns out to be insufficient as there is no randomly jointed chain of finite length rods that can reproduce the statistical behaviour of the end-to-end vector in the cold limit (as indicated by the fact that $\ell_k=0$ at this limit).
As discussed earlier, one way to quantify the shape of a gyrating ribbon is the gyration tensor $R_{ij}$ \ref{eq:gyration_tensor}. In appendix (\ref{app:Stat_Meas}) we develop a formal expression of this tensor, yet even a numerical approximation of it is difficult to calculate as it requires an infinite sum without any immediate cut-off parameter. We therefore decide to use other measures to shed light on the overall shape of the ribbon. First we calculate the gyration radius $R_g^2= \text{Tr}(R_{ij})$, that is essentially a spherical approximation of the ribbon's shape. We also calculate the Frame-Origin Correlation Matrix, $\rho_{ij}$, (Eq. \ref{eq:FOCM}) analytically. Together these two measures enable us explore the overall shape ($R_g$) and any anisotropies ($\rho_{ij}$) of the ribbon, allowing us to estimate the ribbon's (anisotropic) shape.
A direct calculation (see appendix \ref{app:Stat_Meas}) yields
\begin{align}\label{eq:gyration_radius}
R_g^2 = \left[\frac{1}{3}\Lambda^{-1} L - \Lambda^{-2} + \frac{2}{L}\Lambda^{-3}-\frac{2}{L^2}\Lambda^{-4}\left(1-e^{-\Lambda L}\right)\right]_{33}
\end{align}
$\tilde{R}_g^2= k_0^2 R_g^2$ is plotted in Fig, \ref{fig:gyr_rad}, as a function ribbon's length $\tilde{L}=k_0 L$ for different temperatures. At low temperatures, one can describe
\begin{align}\label{eq:Rg_spherical}
R_g^2 \propto \left\{ \begin{array}{cc}
L^2 & 0\leq L < L_{min} \\
const & L_{min}<L <L^*\\
\ell_k L & L^*< L
\end{array}\right.
\end{align}
$L^*(T)\propto \ell_p$ is the scale at which ribbons start to behave as ideal chain, shorter ribbons should be considered as stiff. This scale depends on the temperature and width of the ribbon, but also on it's structure ($k_0,~t$), it exists for ribbons of any type- not just the ones studies here. $L_{min}$ is temperature independent and scales as the ribbon's curvature radius, it marks the scale at which the ribbon's ring-shape manifests itself. Ribbons shorter than $L_{min}$ may be considered, practically at least, as straight and featureless. Since $L^*$ decreases as $T$ increases, there exists some temperature such $L^* \sim L_{min}$ beyond which the ribbon will appear "melted" and structureless. At the chosen width in Fig. \ref{fig:gyr_rad} $T^*=1$, however we already see that the ribbon has an apparent "melting" point around $T\sim T^*/10$ (at which point the typical magnitude of fluctuations is of similar size to the curvature scale, $\langle \Delta m^2 \rangle \sim \langle l^2 \rangle$). It is worth noting that { a similar dependence of $R_g$ should be expected also for non compact ribbons (e.g- helical ribbons). The main difference is the behaviour at the mid range $L_{min} < L< L^*$, where $R_g^2 \propto \sin(\phi) L^2 $, $\phi$ being ribbons' pitch angle}
\begin{figure}[h!]
\centering
\begin{tikzpicture}[scale=1]
\node (pic) at (0,0) {\includegraphics[width=0.7\textwidth]{gyrationRad2.pdf}};
\node (L2) at (-2.1,-.7) {$\sim \tilde{L}^2$};
\node (L1) at (2.2,2.6) {$\sim \tilde{L}$};
\end{tikzpicture}
\caption{The squared gyration radius $\tilde{R}_g^2= k_0^2 R_g^2$ as a function of $\tilde{L}=k_0 L$. At different temperatures (as indicated in the figure), for $\tilde{w}=0.055$, $\nu=0$. Gray dash-dotted lines mark asymptotics of $\tilde{R}_g^2$, as indicated in figure. At this width $\tilde{T}^*=\frac{1}{\Psi^*}=1$, note that the (narrow) ribbon seems to lose structure already at $T=1/10$ (i.e. lose the $R_g^2 \sim const$ region between the $\sim L^2$ and $\sim L $ regions). Also note that at $T^*$ the gyration radius (and $\ell_k$) already start to decrease. The graphs remain qualitatively the same for different values of $\nu$ and $\tilde{w}$. At $T\rightarrow0$, $R_g^2 \xrightarrow{\tilde{L}\geq \tilde{L}_{min}} const.$ }\label{fig:gyr_rad}
\end{figure}
Being a scalar quantity, $R_g^2$ cannot provide information about statistical structural properties of the the ribbon. A direct example to this is the the fact that at $L_{min} < L <L^*$ (especially at low temperatures), the ribbon is shaped like a ring. Such configurations are described by a bounding oblate spheroid, while a spherical description using $R_g $is misleading. To see this we study the frame origin correlation matrix (FOCM) $\rho_{i j}(x)$ (as the gyration tensor $R_{ij}$ is difficult to calculate even in simplet cases). It measures correlations between components of the $i^{th}$ and $j^{th}$ frame vectors at $x$ which are parallel to the ribbon's tangent at the origin $x=0$.
\begin{align}
\rho_{ij}(x) &= \langle {v}_3^i (x) {v}_3^j (x)\rangle= \langle \left(\hat{v}_i(x)\cdot \hat{v}_3(0)\right)\left(\hat{v}_j(x)\cdot \hat{v}_3(0)\right)\rangle
\end{align}
This matrix is plotted in Figure \ref{fig:tangent_corrs}, Using our knowledge of $\ell_\tau$, $\ell_p$ the curvatures and the fact that on long distances $\rho_{ij} \rightarrow \frac{1}{3} \delta_{ij}$ ($\delta_{ij}$ being Kronecker's delta), we may heuristically evaluate $\rho_{ i j}$ (which turns out to be a good approximation)-
\begin{align}
\rho_{ij}(x) &= \left(\begin{array}{ccc}
\frac{1}{3}- \frac{1}{3}e^{-\frac{x}{\ell_\tau/2}} & 0 & 0 \\
0 & -\frac{1}{2}\cos(2 \langle l \rangle x)e^{-\frac{x}{\ell_p/4}} +\left(\frac{1}{2}-\frac{1}{3}\right)e^{-\frac{x}{\ell_\tau/2}}+\frac{1}{3} & \frac{1}{2}\sin(2 \langle l \rangle x)e^{-\frac{x}{\ell_p/4}}\\
0 & \frac{1}{2}\sin(2 \langle l \rangle x)e^{-\frac{x}{\ell_p/4}} & \frac{1}{2}\cos(2 \langle l \rangle x)e^{-\frac{x}{\ell_p/4}} +\left(\frac{1}{2}-\frac{1}{3}\right)e^{-\frac{x}{\ell_\tau/2}}+\frac{1}{3}\
\end{array} \right).
\end{align}
This evaluation, and the actual results (Figure \ref{fig:tangent_corrs}) may be explained as follows:
At low temperatures, and at short distances ($x < \ell_p$), the ribbon behaves as a stiff ring, as indicated by the fact $\rho_{11}\simeq 0$, and by the strong oscillatory contributions in the remaining nonzero elements of $\rho$. At intermediate positions $\ell_p < x< \ell_\tau$, while $\rho_{11}\sim 0$ remains, we see the oscillations subside (depending on ${\ell_\tau}/{\ell_p}$) and we may roughly approximate (especially for large values of this ratio)
\begin{align}
\rho_{ij} &\sim \left(\begin{array}{ccc}
0&0&0 \\
0& \frac{1}{2} & 0\\
0 & 0& \frac{1}{2}
\end{array}\right),
\end{align}
suggesting the ribbon's tangent is randomly directed in 2D. That is, the ribbon occupies an oblate spherical volume with a large aspect ratio. At further positions $x >\ell_\tau $ we find that the gyrating ribbon occupies an evermore spherically shaped volume such that \begin{align}
\rho_{ij}(x\rightarrow \infty) &= \frac{1}{3} \delta_{ij},
\end{align}
corresponding to randomly directed tangent in 3D. The fact that the tangents of the mid-line at distance $x$ from the origin are randomly distributed does not mean that the ribbon lacks any structure (as also indicated by the fact that some of the eigenvalues of $\mathbf{\Lambda}$ are complex). Rather, that there is no long range order (of the Darboux frame) in the system. Knowledge at one point along the ribbon does not tell us anything about parts of the system far away from it.
\begin{figure}[h!]
\begin{tabular}{ccc}
\begin{subfigure}{0.33\textwidth}
\centering
\includegraphics[width=.95\textwidth]{tangentcorrs1.pdf}
\caption{$\Psi=10^{3}$, $\tilde{w}=0.1$}\label{fig:corrs_psi_1000_w_1}
\end{subfigure} &
\begin{subfigure}{0.33\textwidth}
\centering
\includegraphics[width=.95\textwidth]{tangentcorrs2.pdf}
\caption{$\Psi=10^{3}$, $\tilde{w}=0.5$}\label{fig:corrs_psi_1000_w_5}
\end{subfigure} &
\begin{subfigure}{0.33\textwidth}
\centering
\includegraphics[width=.95\textwidth]{tangentcorrs3.pdf}
\caption{$\Psi=10^{3}$, $\tilde{w}=0.9$}\label{fig:corrs_psi_1000_w_9}
\end{subfigure} \\
\begin{subfigure}{0.33\textwidth}
\centering
\includegraphics[width=.95\textwidth]{tangentcorrs4.pdf}
\caption{$\Psi=10^{2}$, $\tilde{w}=0.1$}\label{fig:corrs_psi_100_w_1}
\end{subfigure} &
\begin{subfigure}{0.33\textwidth}
\centering
\includegraphics[width=.95\textwidth]{tangentcorrs5.pdf}
\caption{$\Psi=10$, $\tilde{w}=0.1$}\label{fig:corrs_psi_10_w_1}
\end{subfigure}& \\
\end{tabular}
\caption{Correlation at different temperatures and width. Subfigures (a)-(e): The components of $\rho_{ i j}(x,x)$, semi-logarithmic scale. $\rho_{11}$-red, $\rho_{22}$-blue, $\rho_{33}$-green, $\rho_{23}$-black }\label{fig:tangent_corrs}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.6\textwidth,clip,trim={0cm 0cm 0cm 0cm}]{Blobs.jpg}
\caption{ Visualization mid-line configuration at $\tilde{T}=10^{-3}$ (except Inset III) and $\tilde{w}=0.1$ (all), obtained using MC simulation, all panels are to scale. Colours correspond to different direction of the bi-normal: red- up/down, blue- right/left, green-front/back. Main- full length of a ribbon $\tilde{L}=k_0 L= 10^5$; Inset (I)- a segment of $\tilde{L}=10^4$ of the same ribbon; Inset(II)- a segment of $\tilde{L}=10^3$; Inset(III)- Left- A segment of length $\tilde{L}=10^4$ at $\tilde{T}=10^{-4}$, Right- a similar segment facing sideways (hence coloured blue) to emphasize the high aspect ratio. Uniform colour indicates the ribbon has not lost it's bi-normal correlation, yet the shape clearly shows it has lost any long range tangent-tangent correlation.}\label{fig:MC_psi_103_w_1}
\end{figure}
We used Monte-Carlo (MC) simulations, to visualize a thermalized ribbon at different temperatures. The results are plotted in Figure \ref{fig:MC_psi_103_w_1}, where different colours correspond to different directions of the bi-normal ($\hat{v}_1$). It is immediately seen that at lower temperatures, larger same-colour segments appear, indicating large correlation lengths of the bi-normal. We term this behavior (and hence the phase of the ribbon) "Plane Ergodic" (PE). The existence of this state depends on the ratio of $\ell_\tau/ \ell_p$ and temperature. If $\ell_\tau/ \ell_p \gg 1$, at a given temperature, the "planar" regions of the ribbon are made of longer segments and are more easily observed. If, however $\ell_\tau/ \ell_p = 1$, then this phase does not exist at all. At higher temperatures, this phase is harder to observes as it spans shorter distances, while at temperatures larger than $T^*$ the ribbon has lost structure completely, therefore behaving as an ideal chain and no such phase may be observed.
\subsubsection{Wide Ribbons $\tilde{w}>1$}\label{sub:WideRibb}
We now turn to study the statistics of a wide ribbon. As mentioned in \ref{s_ch:Euilbrium}, at the wide regime one cannot neglect derivatives even at mechanical equilibrium. As they are the only energy scale (regarding $\theta$ fluctuations), they affect the statistics significantly. We therefore repeat the calculations of $\langle \hat{v}_i (x) \hat{v}_j (x') \rangle= \langle e^{-\int_{\tilde{x}'}^{\tilde{x}} \Omega \,\text{d} x} \rangle $ (Eq. \ref{eq:corrmat}), and find that $\log\langle e^{-\int_{\tilde{x}'}^{\tilde{x}} \Omega \,\text{d} x} \rangle = \langle \Omega \rangle \Delta + O(\Delta^2)$, for small $\Delta=|\tilde{x}-\tilde{x}'|$, while $\log\langle e^{-\int_{\tilde{x}'}^{\tilde{x}} \Omega \,\text{d} x} \rangle \simeq \Lambda_\xi \Delta$ for large $\Delta$. Where to leading order
\begin{align}\label{eq:Lambda_xi}
\Lambda_\xi&= \langle \Omega \rangle +\frac{1}{2} \left(\begin{array}{ccc}
\frac{1}{2}\left(\langle \Delta \tilde{z}^2 \rangle +\tilde{z}_{eq}^2 \xi_\theta\right) & 0 & 0 \\
0 & \frac{1}{2} \langle \Delta \tilde{h}^2\rangle + \langle \Delta \tilde{z}^2 \rangle +\tilde{z}_{eq}^2 \xi_\theta & 0 \\
0 & 0 & \frac{1}{2}\left(\langle \Delta \tilde{h}^2\rangle + \langle \Delta \tilde{z}^2 \rangle +\tilde{z}_{eq}^2 \xi_\theta \right)
\end{array}\right),
\end{align}
It is easily verified that $\lim\limits_{\xi_\theta \rightarrow 0} \Lambda_\xi = \Lambda$, thus retrieving Eq. \ref{eq:Lambda}.
At low temperatures ($\xi_\theta \rightarrow \infty$), $\Lambda_\xi$ reduces into
\begin{align}\label{eq:Lambda_xi_low}
\Lambda_\xi= \left(\begin{array}{ccc}
\frac{ \tilde{z}_{eq}^2 \xi_\theta}{4} & 0 &0 \\
0& \frac{\tilde{z}_{eq}^2 \xi_\theta}{2} & 0 \\
0 & 0 & \frac{ \tilde{z}_{eq}^2 \xi_\theta}{4}
\end{array}\right),
\end{align}
where we also omittede the $\langle \Omega \rangle $ term as it is negligible.
This means that on scales much larger than $\xi_\theta$ the ribbon statistically behaves as an ideal chain $\ell_\tau=\ell_p=\ell_k$, while on smaller scales it behaves as rigid. However, unlike an ideal chain, the different Kuhn segments are not structureless. Rather, they have different shapes ranging from rings, through helices, to straight segment depending on the chosen angle and the width of the ribbon (as depicted in Fig \ref{fig:MC_psi_w_3}), we term this unusually soft phase "Random Structured". Due to the non trivial position dependence, even numeric calculation of $R_g$ and $\rho_{ i j}$ is hard. Nevertheless we can compare the above interpretation with MC simulation. Indeed, in Figure \ref{fig:MC_psi_w_3}, we compare the result of a MC simulation (left) to "naive" model consisting of a random ,uniform, angle for segments exactly $\xi_\theta$ long. The similarity between plots is clear, thus our interpretation is supported by the simulation.
As mentioned before, at very low temperatures $\tilde{T} \ll 1$ the boundary layer dominates the ribbon's configuration giving rise to statistical behaviour identical to that of the Plane Ergodic phase. As temperature get higher the ribbons smoothly transitions into the Random Structured phase. Direct calculation shows that the RS phase is describes the ribbon sufficiently at $\tilde{T}\gtrsim 0.1$.
\begin{figure}[!h]
\begin{tabular}{ccc}
\centering
\includegraphics[width=.45\textwidth,clip,trim={1cm .5cm 1cm .5cm}]{WideRibSim2.pdf}
\includegraphics[width=.45\textwidth,clip,trim={1cm .5cm 1cm .5cm}]{WideRib_NO_Sim2.pdf}
\end{tabular}
\caption{ Left- a MC simulation of a possible mid-line configuration at the wide regime with $\tilde{T}=10$, $\tilde{w}=10$ (random structured phase) of total length $\tilde{L} = 10^3$ ; Colours correspond to different values of $\theta$, $\xi_\theta \sim 100$ ; Right- a naive reproduction of a random structured ribbon. Made by assigning a random angle to segment exactly $\xi_\theta =100$ long
}\label{fig:MC_psi_w_3}
\end{figure}
\subsubsection{Ribbon's Phase Diagram}
We may now finally plot the phase diagram of our ribbons. In Figure \ref{fig:Phases} we plotted all three phases we've described so far. The blue curve is $T^*(\tilde{w})$ separating the "structured" phases from the "melted" (ideal chain) phase (light red). This phase is termed Ideal Chain (IC) phase, since at high temperatures the ribbon is well described by the ideal chain model at all scales. Under $T^*$, in a continuous, yet sharp, manner, we find two other phases, separated at the critical width. A narrow ribbon will exhibit the Plain Ergodic (PE) phase, in which at intermediate scales we may find the ribbon coiled into rings which slowly drift away yet contained within a plane, while at large scales the ribbon drifts out of plane. Thus the ribbon may be describe as made of "planar" segments that are randomly connected (both in orientation and position, as seen in Fig. \ref{fig:MC_psi_103_w_1}). Note that in this phase $\ell_p\propto \frac{1}{T}$ yet $\ell_k \propto T$, suggesting that the actual size of the planar segments varies only slightly with temperature. It is important to emphasize, that this phase is geometrical in nature and appears solely due to the ribbon's underlying structure (ring-like).
Finally we have the Random Structured (RS) phase at $\tilde{w}>1,~ T<T^*$. This is a unique phase appearing due to the incompatibility of the ribbon. A ribbon in this phase is soft, and appears almost like an ideal chain on large scales. On intermediate scale one can see that the ribbon has a randomly selected structure- it is made of segments whose shapes are uniform along the segment and may either be right handed, or left handed helical of various pitches as visualized in Fig. (\ref{fig:MC_psi_w_3}). These segments' size scales as $\frac{1}{T}$. It is important to note that on intermediate temperatures ($T<T^*$ but not $T\ll T^*$) the segments shape is not completely uniform and may fluctuate. Nevertheless, they are still distinct, and well defined in their shape, losing it only when we get close to $T^*$. At extremely low temperatures ($\tilde{T} \lesssim 0.1$) this phase exhibits a PE-like behaviour, as the boundary layer becomes important also from a statistical point of view (and not only mechanical).
\begin{figure}[h!]
\centering
\includegraphics[width=1\textwidth]{Phases.jpg}
\caption{Ribbon phases. IC- Ideal Chain, RS- random structured, PE- plane ergodic. The RS phase does not exist in compatible ribbons, the thin green region at the bottom of the RS phase marks the temperature range at which the boundary layer also affects the statistic, exhibiting a similar behaviour to that of PE phase. PE exists even in compatible ribbons, but is different in the details. IC- is the natural limit of every statistical theory of elastic ribbons.}\label{fig:Phases}
\end{figure}
\section{Conclusions}\label{ch:Conc}
Using newly developed formalism to describe incompatible ribbons (\cite{Grossman2016}), we identified and quantified the phase space of incompatible ribbons with positive spontaneous curvature at different temperatures and widths. Such ribbons exhibit configuration transition at a critical width, even at zero temperature. At small widths, the ribbon is bending dominated- i.e its configuration is set by its spontaneous curvature, while at large widths, it is stretching dominated, and its flat (Euclidean) in-plane geometry prescribes a range of developable configurations. We showed that the mechanics of such ribbons is nontrivial and strongly depends on ribbon width. In particular, wide ribbons are abnormally floppy, as their bulk energy is degenerate and their rigidity stems solely from variations in boundary layer energy. At finite temperature, these nontrivial mechanics lead to non trivial statistical properties, which do not exist in compatible ribbons. We calculate explicitly and find that different statistical geometrical measures, such as the persistence length $\ell_p$, the torsional correlation length $\ell_\tau$, and the Kuhn length $\ell_k$ vary non-monotonically with ribbon width and with temperature. The volume occupied by a ribbon varies in temperature and width in a non trivial way. Part of this variation is presented as three different phases (Fig. \ref{fig:Phases}), which we term Ideal Chain (at high temperatures), Plain Ergodic (narrow, cold ribbons), and Random Structured (wide, cold ribbons).
While the first is common to all elastic ribbons, the second is purely geometrical in nature and we would expect all ribbons (including compatible) with some underling structure to have a similar phase. The third phase exists only in positively curved incompatible ribbons and arises purely due to the unique residual stresses in the problem we studied. In this phase, degeneracy in the bulk energy of the ribbon leads to continuous family of (mechanical-) equilibrium configurations. This, at low but finite temperatures, leads the ribbon to look as a random coil on large scales, yet with segments that have a definite and well defined shape, in contrast to compatible ribbons where there is a single equilibrium shape. Though the details of the transitions and the characteristic configurations are unique to positively curved ribbons, we expect a qualitatively similar phase space to appear in other cases of incompatible ribbons. The mechanics and geometry of all such systems are non-trivial and dominated by the competition between the residual bending and stretching energies. In this sense, our results point to a general phase space, that should be expected even in relatively simple self assembled systems.
It should be emphasized that the work on such ribbons is far from closure as such ribbons (especially in Plain Ergodic phase) are densely coiled, and self avoidance effects should be important. Other important corrections may arise at high temperatures, where the Gaussian approximation should fail. It is worth to note that the resulting phase diagram is reminiscent of quantum phase transition, in the sense of the existence of a critical point even in zero temperature. While somewhat similar connections between elastic and quantum systems were made in the past \cite{Moroz1997}, the extent of this similarity in the context of this paper is ill understood and requires further investigation.
\section{Acknowledgment}
This research was supported by the US-Israel Binational Science Foundation \# 2014310. D. G. was also supported by The Harvey M. Kruger Family Center of Nanoscience and Nanotechnology.
\clearpage
\newpage
|
{
"timestamp": "2018-04-17T02:06:37",
"yymm": "1804",
"arxiv_id": "1804.05223",
"language": "en",
"url": "https://arxiv.org/abs/1804.05223"
}
|
\section{Introduction}
Fifth generation (5G) wireless networks are deemed to be a promising technology for the rapid growth of mobile data traffic demand. Millimeter-Wave (mmWave) communications operating in the $28$-$300$ GHz range has been proposed as one of the feasible solutions for 5G networks~\cite{r1}. On one hand, the existence of a large communication
bandwidth at mmWave frequencies represents the potential for
significant throughput gains. On the other hand, the shorter wavelengths at the mmWave band allow for the deployment of a large number of antenna elements in a small area, which enables
mmWave systems to potentially support a higher degree
of beamforming gain and multiplexing~\cite{r1}. However, significant path loss, blockage, and hardware limitations are major obstacles for the deployment of mmWave systems. To address these obstacles, several mmWave systems
have been proposed to date~\cite{r5,el2014spatially,r9,r19,almasi2018new}.
An analog beamforming mmWave system is designed in~\cite{r5} which uses one radio frequency (RF) chain and can support only one data stream. In order to transmit multiple streams, by exploiting several RF chains hybrid beamforming (HB) mmWave systems are designed~\cite{el2014spatially}. In~\cite{r9}, the concept of beamspace multi-input multi-output (MIMO) is introduced where several RF chains are connected to a lens antenna array via switches. Finally, multi-beam lens-based reconfigurable antenna MIMO systems are newly proposed to overcome severe pathloss and shadowing in mmWave frequencies~\cite{r19,almasi2018new}. Beside mmWave communications, non-orthogonal multiple access (NOMA) is another enabling technique for 5G networks. NOMA can be used to augment the spectral efficiency in multi-user scenarios~\cite{higuchi2015non}.
In the aforementioned systems, each beam is used to serve only one mobile user (MU)~\cite{alkhateeb2015limited,almasi2018reconfigurable}. The work in~\cite{alkhateeb2015limited}, shows that exploiting hybrid beamforming in multi-user systems achieves higher spectral efficiency. Also,~\cite{almasi2018reconfigurable} enhances the spectral efficiency by supporting several MUs through multi-beam reconfigurable antenna. Nevertheless, the number of served MUs maybe not sufficient to support the larger user base in 5G networks. To overcome this issue, the integration of NOMA in mmWave systems (mmWave-NOMA) which allows several MUs to share the same beam has been studies in~\cite{chen2017exploiting,ding2017noma,wang2017spectrum,hao2017energy,yang2017noma,wu2017non}. In~\cite{chen2017exploiting,ding2017noma,wang2017spectrum,hao2017energy,yang2017noma} NOMA is evaluated for mmWave systems assuming only baseband precoders/combiners, whereas the work in~\cite{wu2017non} has recently studied NOMA in hybrid beamforming systems. In particular, due to use of hybrid beamforming, the digital precoder of the base station (BS) is not perfectly aligned with the MU's effective channel. While considering this imperfect alignment, a power allocation algorithm that maximizes the sum-rate has been proposed. However, the approach in~\cite{wu2017non} does not study the effect of imperfect beamforming on the sum-rate. Indeed, the approach in~\cite{wu2017non} can be considered a preliminary study applicable to small scale systems, since only two clusters with each of them containing only two MUs are considered.
In this paper, inspired by the hybrid beamforming system in~\cite{alkhateeb2015limited} and the work in~\cite{wu2017non}, we study the impact of using NOMA on the performance of hybrid beamforming systems which we denote as HB-NOMA. To this end, we consider a scenario in which a single BS is equipped with a hybrid beaforming system similar to that of~\cite{alkhateeb2015limited}. The BS transmits several streams through the formed beams and each beam is allowed to be shared with multiple MUs inside a cluster. Due to application of analog components, unlike digital precoders, in hybrid beamforming, the precoders are not able to perfectly align a beam toward the effective channel of all MUs. This causes the angle between the effective channel vectors of the first MU (see Fig. 1) and other MUs located in the same cluster to be non-zero. Therefore, this paper evaluates the effect of the imperfect alignment on the achievable rate of MUs. The contributions of this paper can be summarized as
\begin{enumerate}
\item We propose a generalized hybrid beamforming-based NOMA system that consists of one BS and $N$ clusters, where each cluster has $M$ MUs (see Fig. 1). It is assumed that the BS performs hybrid beamforming, i.e., analog/digital precoder, whereas the MUs only perform analog combining. The sum-rate is formulated for the $m$th MU in the $n$th cluster.
\item A suboptimal algorithm is proposed in three steps to maximize the sum-rate. In the first step, we design the analog precoder/combiner while in the second step, we derive the digital precoder architecture. Finally, we formulate a suboptimal power allocation technique for the proposed setup.
\item We derive a lower bound for the achievable rate of the $m$th MU in the $n$th cluster in the existence of imperfect correlation amongst the channels of MUs in the same cluster. Our analysis shows that under the assumption of imperfect correlation, there can be a significant data-rate loss.
\end{enumerate}
The paper is organized as follows: Section~\ref{sec:system} presents the
system model for the HB-NOMA system. In Section~\ref{sec:problem}, the sum-rate problem for the HB-NOMA system is formulated. In Section~\ref{algorithm}, the proposed algorithm for maximizing the sum-rate is presented. Section~\ref{sec:lower} derives a lower bound for the achievable rate of an HB-NOMA user. In Section~\ref{sec:simulation}, we present simulations investigating the performance of the rate. Section~\ref{sec:conclusion} concludes the paper.
\textbf{Notations:} Hereafter, $j = \sqrt{-1}$, small letters, bold letters and bold capital letters will designate scalars, vectors, and matrices, respectively. Superscripts $(\cdot)^{\dagger}$ and $(\cdot)^{*}$ denote the transpose and transpose-conjugate operators, respectively. Further, $|\cdot|$, and $\norm[]{\cdot}^2$ denote the absolute value, and norm-$2$ of $(\cdot),$ respectively. Finally, the element in $i$th row of $j$th column of matrix $\mathbf{X}$ is denoted by $\mathbf{X}(i,j)$.
\section{System Model}\label{sec:system}
We consider a mmWave downlink system for 5G wireless networks composed of a BS and multiple MUs as shown in Fig.~\ref{fig:system}. It is assumed that the BS is equipped with $N_\text{RF}$ chain and $T_\text{BS}$ antennas, while each MU has one RF chain and $T_\text{MU}$ antennas. Further, we assume that the BS communicates with each MU via only one stream, which is in line with prior work such as~\cite{alkhateeb2015limited}. Note that in traditional hybrid beamforming based multi-user systems it is assumed that the maximum number of MUs that can be simultaneously served by the BS equals to the number of BS RF chains~\cite{alkhateeb2015limited}.
In order to establish better connectivity in a dense area and further improve the sum-rate, this paper utilizes NOMA in hybrid beamforming multi-user systems which is denoted by HB-NOMA. To this end, each beam is allowed to serve more than one MU. For the sake of simplicity, the number of MUs served by each beam is identical and equal to $M$. Also, the MUs are grouped into $N\leq N_\text{RF}$ clusters. Hence, the proposed HB-NOMA system can simultaneously serve $MN \gg N_\text{RF}$ MUs.
On the downlink, the hybrid beamforming is done through two stages. In the first stage, the transmitter applies an $N\times N$ baseband precoder $\mathbf{F}_\text{BB}$ using its $N_\text{RF}$ RF chains. Next, using analog phase shifters a $T_\text{BS} \times N$ RF precoder, $\mathbf{F}_\text{RF}$, is applied. Thus, the transmit signal vector, $\mathbf{x}$, is given by
\begin{equation} \label{eq1}
\mathbf{x} = \mathbf{F}_\text{RF}\mathbf{F}_\text{BB}\mathbf{s},
\end{equation}
where $\mathbf{s} = [s_1, s_2, \cdots, s_N]^{\dagger}$ denotes the information signal vector. Each $s_{n} = \sum_{m = 1}^{M}\sqrt{P_{n,m}^\prime}s_{n,m}$ is the superposition coded signal due to NOMA with $P_{n,m}^\prime$ and $s_{n,m}$ denoting the transmit power and transmitted information signal for the $m$th MU in the $n$th cluster, respectively. Hereafter, MU-$(n,m)$ denotes the $m$th user in the $n$th cluster. Since $\mathbf{F}_\text{RF}$ is implemented by using analog phase shifters, it is assumed that all elements of $\mathbf{F}_\text{RF}$ have equal norm, i.e., $\left|\mathbf{F}_\text{RF}(i,j)\right|^2 = T_\text{BS}^{-1}$ for $i=1,2,\cdots,M$ and $j=1,2,\cdots,N$. Further, the total power of the hybrid transmitter is constrained to $\norm[\big]{\mathbf{F}_\text{RF}\mathbf{F}_\text{BB}}^2_F = N$. The received signal at MU-$(n,m)$, $\mathbf{r}_{n,m}$, is given by
\begin{equation}\label{eq2}
\mathbf{r}_{n,m} = \mathbf{H}_{n,m}\mathbf{F}_\text{RF}\mathbf{F}_\text{BB}\mathbf{s} + \mathbf{n}_{n,m},
\end{equation}
where $\mathbf{H}_{n,m}$ of size $T_\text{MU}\times T_\text{BS}$ denotes the mmWave channel between the BS and MU-(n,m). $\mathbf{n}_{n,m}\sim \mathcal{N}(\mathbf{0},\sigma^2\mathbf{I})$ is the additive white Gaussian noise vector of size $T_\text{MU}\times 1$.
\begin{figure}
\vspace*{-0.7cm}
\hspace*{-0.9cm}
\centering
\includegraphics[scale=1.175]{image/system.pdf}
\vspace*{-1cm}
\caption{Schematic of the HB-NOMA system with one BS and \textit{NM} MUs.}
\label{fig:system}
\end{figure}
At MU-$(n,m)$, the RF combiner, $\mathbf{w}_{n,m}$, is used to process the received vector as
\begin{align}\label{eq3}
y_{n,m} &= \underbrace{\mathbf{w}_{n,m}^*\mathbf{H}_{n,m}\mathbf{F}_\text{RF}\mathbf{f}^n_\text{BB}\sqrt{P_{n,m}^\prime}s_{n,m}}_{\text{desired signal}} \nonumber \\
& \ \ +\underbrace{\mathbf{w}_{n,m}^*\mathbf{H}_{n,m}\mathbf{F}_{RF}\mathbf{f}^n_{BB}\sum_{k \neq m}^M\sqrt{P_{n,k}^\prime}s_{n,k}}_{\text{intra-cluster interference}} \nonumber \\
& \ \ + \underbrace{\mathbf{w}_{n,m}^*\mathbf{H}_{n,m}\sum_{\ell \neq n }^N\mathbf{F}_{RF}\mathbf{f}^\ell_{BB}\sum_{q=1}^M\sqrt{P_{\ell,q}^\prime}s_{\ell,q}}_{\text{inter-cluster interference}} + \underbrace{\mathbf{w}^*_{n,m}\mathbf{n}_{n,m}}_{\text{noise}},
\end{align}
where $\mathbf{w}_{n,m}$ is of size $T_\text{MU}\times 1$. After combining, each MU decodes the intended signal by using successive interference cancellation (SIC)~\cite{higuchi2015non}. More details on SIC will be provided in Section III. For the sake of simplicity, $P_{n,m}$ denotes the normalized transmitted power such that $P_{n,m}=\frac{P_{n,m}^\prime}{\sigma^2}$.
In mmWave communications, the single-path channel between the BS and MU-$(n,m)$ can be expressed as
\begin{equation}\label{eq4}
\mathbf{H}_{n,m} = \sqrt{T_\text{BS}T_\text{MU}} \beta_{n,m}\mathbf{a}_\text{MU}(\vartheta_{n,m})\mathbf{a}_\text{BS}^*(\varphi_{n,m}),
\end{equation}
where $\beta_{n,m} = g_{n,m}D_{n,m}^{\frac{-\nu}{2}}$, $g_{n,m}$ is the complex gain with zero-mean and unit-variance, $D_{n,m}$ is the distance between the BS and MU-$(n,m)$, and $\nu$ is the path loss factor~\cite{yang2017noma}. Also, $\mathbf{a}_\text{BS}(\varphi_{n,m})$ and $\mathbf{a}_\text{MU}(\vartheta_{n,m})$ are the antenna array response vectors of the BS and MU-$(n,m)$, respectively, where $\vartheta_{n,m}$ and $\varphi_{n,m} \in [-1,1]$ are related to the angle of arrival (AoA) $\vartheta \in [-\frac{\pi}{2},\frac{\pi}{2}]$ and angle of departure (AoD) $\phi \in [-\frac{\pi}{2},\frac{\pi}{2}]$ as $\vartheta_{n,m} = \frac{2d\text{sin}(\vartheta)}{\lambda}$ and $\varphi_{n,m} = \frac{2d\text{sin}(\phi)}{\lambda}$, respectively.
In particular, for a uniform linear array (ULA), $\mathbf{a}_\text{BS}(\varphi_{n,m})$ is given by
\begin{equation}\label{eq5}
\mathbf{a}_{\text{BS}}(\varphi_{n,m}) = \frac{1}{\sqrt{T_\text{BS}}}\left[1, e^{-j\pi\varphi_{n,m}},\cdots, e^{-j\pi(T_\text{BS}-1)\varphi_{n,m}}\right]^\dagger,
\end{equation}
with $d$ is the antenna spacing and $\lambda$ denotes wavelength. The antenna array response vector for $\mathbf{a}_\text{MU}(\vartheta_{n,m})$ also has a similar structure to that of \eqref{eq5}~\cite{el2014spatially,alkhateeb2015limited}.
To solely quantify the effect of digital/analog precoding on the sum-rate of HB-NOMA systems, throughout this paper several assumptions are considered:
\begin{itemize}
\item Full CSI of each user, $\mathbf{H}_{n,m}$, $m=1,2,\cdots,M$ and $n=1,2,\cdots,N$, is available at that MU.
\item The BS and all MUs steer the beams with continuous angles. That is, the quantization error is neglected for $\mathbf{F}_\text{RF}$ and $\mathbf{w}_{n,m}$, $m=1,2,\cdots,M$ and $n=1,2,\cdots,N$.
\item The first MU of each cluster feeds complete effective channel back to the BS, i.e., infinite-resolution codebooks are used.
\item The BS knows all MUs' channel gains $\left|\beta_{n,m}\right|$, for $m=1,2,\cdots,M$ and $n=1,2,\cdots,N$.
\item Each MU is capable of preforming error-free SIC.
\end{itemize}
\section{Problem Formulation}\label{sec:problem}
As mentioned, the goal of this work is to quantify the impact of combining hybrid beamforming and NOMA on the sum rate of a dense network. In particular, we study the effect of jointly designing the hybrid precoders and analog combiners for clustered MUs on the sum-rate of the overall system.
In~(\ref{eq3}), after applying superposition coding at the transmitter, each user experiences two forms of interference. The intra-cluster inference which is due to other MUs within the cluster and inter-cluster interference which is due to MUs within other clusters.
Suppressing the intra-cluster interference directly depends on efficient power allocation and deploying SIC. At the receiver side, each user performs SIC to decode the desired signal. To do this, the signal of users that have more power are decided and subtracted from the received signal. This process continues until the intended user decodes its signal. The remainder can be categorized as intra-cluster interference. Here, we assume that each user can perfectly perform SIC decoding. To mitigate the inter-cluster interference, the transmitter needs to design a proper beamforming matrix. In this paper, we adopt zero-forcing beamforming (ZFBF) which achieves a balance between implementation complexity and performance.
Sum-rate has been categorically used to analyze the performance of NOMA. Consequently, we evaluate the sum-rate for the proposed HB-NOMA. The sum-rate is expressed as
\begin{equation}\label{sumrate}
R_\text{sum} = \displaystyle \sum_{n=1}^N\sum_{m=1}^MR_{n,m},
\end{equation}
where $R_{n,m}$ is defined as
\begin{equation}\label{eq6}
R_{n,m} = \text{log}_2\left(1 + \frac{P_{n,m}\left|\mathbf{w}_{n,m}^*\mathbf{H}_{n,m}\mathbf{F}_\text{RF}\mathbf{f}_\text{BB}^n\right|^2}{I_\text{intra}^{n,m} + I_\text{inter}^{n,m} + 1}\right).
\end{equation}
Here, the intra-cluster interference after SIC processing at the $m$th user in the $n$th cluster, $I_\text{intra}^{n,m}$, is given by
\begin{equation}\label{eq61}
I_\text{intra}^{n,m} =
\sum_{k=1}^{m-1}P_{n,k}\left|\mathbf{w}_{n,m}^*\mathbf{H}_{n,m}\mathbf{F}_\text{RF}\mathbf{f}_\text{BB}^n\right|^2.
\end{equation}
Notice that the first user in each cluster is assumed to have the lowest allocated power. Further, the inter-cluster interference at the $m$th user in the $n$th cluster, $I_\text{inter}^{n,m}$, is given by
\begin{equation}\label{eq62}
I_\text{inter}^{n,m} = \sum_{\ell\neq n}^{N}\sum_{q=1}^{M}P_{\ell,q}\left|\mathbf{w}_{n,m}^*\mathbf{H}_{n,m}\mathbf{F}_\text{RF}\mathbf{f}_\text{BB}^\ell\right|^2.
\end{equation}
To improve the achievable rate, we need to obtain the optimal hybrid precoder $\breve{\mathbf{F}}_\text{RF}$, $\breve{\mathbf{f}}^{n}_\text{BB}$, for $n=1 \cdots N$, combiner $\breve{\mathbf{w}}_{n,m}$, for $n=1 \cdots N$, $m=1 \cdots M$, and transmit power $\breve{P}_{n,m}$, for $n=1 \cdots N$, $m=1 \cdots M$. To this end, the following optimization problem must be solved
\begin{IEEEeqnarray*}{rCl}
&\underset{\{\mathbf{F}_\text{RF},\mathbf{f}_\text{BB}^n,\mathbf{w}_{n,m},P_{n,m}\}}{\text{maximize}} \ & \sum_{n=1}^N\sum_{m=1}^M R_{n,m}
\IEEEyesnumber\\
&\text{subject to} & \left|\mathbf{F}_\text{RF}(i,j)\right|^2 = T_\text{BS}^{-1} \IEEEyessubnumber*\\
& &\norm[\big]{\mathbf{F}_\text{RF}[\mathbf{f}_\text{BB}^1, \mathbf{f}_\text{BB}^2,\cdots,\mathbf{f}_\text{BB}^N]}^2_F = N \\
& & \left|\mathbf{w}_{n,m}(i,j)\right|^2 = T_\text{MU}^{-1} \\
& & \sum_{n=1}^N\sum_{m=1}^M P_{n,m}\leq P_\text{t}, \\
& & P_{n,m} > 0
\end{IEEEeqnarray*}
where $i=1,2,\cdots,M$, $j=1,2,\cdots,M$, and $P_\text{t}$ equals to the signal-to-noise ratio (SNR), i.e, $P_\text{t} = \text{SNR}$.
\section{The Maximization Algorithm}\label{algorithm}
Unfortunately, the maximization problem in (7) is non-convex since the objective function has a complicated expression and there is a coupling between the analog and digital precoders. Thus, finding a closed-form solution is non-trivial. Hence, we propose a simple but effective algorithm in three steps.
In the first step the BS and MU-$(n,m)$ solve the following problem
\begin{equation}\label{eq7}
\underset{\{\mathbf{w}_{n,m},\mathbf{f}_\text{RF}^{n,m}\}}{\text{maximize}} \ \left|\mathbf{w}_{n,m}^*\mathbf{H}_{n,m}\mathbf{f}_\text{RF}^{n,m}\right| \qquad \text{subject to (10a) and (10c)}.
\end{equation}
Since the channel $\mathbf{H}_{n,m}$ has only one path, and given the continuous beamsteering capability assumption, the optimal solutions will be $\breve{\mathbf{w}}_{n,m} = \mathbf{a}_\text{MU}(\vartheta_{n,m})$ and $\breve{\mathbf{f}}_\text{RF}^{n,m} = \mathbf{a}_\text{BS}(\varphi_{n,m})$. To design the RF precoder, the BS selects the first user of each cluster. Here, the first MU is selected based on the following criterion,
\begin{equation}\label{eq8}
\left|\beta_{n,1}\right| > \left|\beta_{n,2}\right| > \cdots > \left|\beta_{n,M}\right| \quad \text{for} \quad n = 1, 2, \cdots, N.
\end{equation}
Stacking the the RF precoder vector of the first MUs obtained in~(\ref{eq7}), i.e., $\breve{\mathbf{f}}_\text{RF}^{n,1}$ for $n = 1, 2, \cdots, N$, we can find the RF or analog precoding matrix as
\begin{equation}\label{eq81}
\mathbf{F}_\text{RF} = \left[\breve{\mathbf{f}}^{1,1}_\text{RF}, \breve{\mathbf{f}}^{2,1}_\text{RF},\cdots,\breve{\mathbf{f}}^{N,1}_\text{RF}\right].
\end{equation}
The motivation for steering the beams to the first or closest user to the base station in each cluster is dictated by the structure of NOMA. More detail will be given in the next step.
In the second step, the effective channel for MU-$(n,m)$ is expressed as
\begin{equation}\label{eq9}
\overline{\mathbf{h}}_{n,m}^* = \mathbf{w}_{n,m}^*\mathbf{H}_{n,m}\mathbf{F}_\text{RF} = \sqrt{T_\text{BS}T_\text{MU}}\beta_{n,m}\mathbf{a}_\text{BS}^*(\varphi_{n,m})\mathbf{F}_\text{RF}.
\end{equation}
We define the effective channel matrix as
\begin{equation}\label{eq91}
\overline{\mathbf{H}} = \left[\overline{\mathbf{h}}_{1,1}, \overline{\mathbf{h}}_{2,1}, \cdots, \overline{\mathbf{h}}_{N,1} \right]
\end{equation}
where $\overline{\mathbf{h}}_{n,1}$ denotes the effective channel vector of MU-$(n,1)$. In HB-NOMA, the first MU of each cluster has to decode other MUs' signal in that cluster first before it can decode its own signal. Thus, by steering the beams toward the first MU within each cluster, we ensure that SIC can be performed in an effective fashion. Hence, the design of the RF precoder has to follow that of~(\ref{eq81}) to reduce the intra-cluster interference. Further, the effective channel matrix~(\ref{eq91}) causes the inter-cluster interference on the first MU in each cluster to be eliminated, which is outlined in more detail below.
Recall that designing a proper digital or baseband precoder $\mathbf{F}_\text{BB}$ remarkably reduces the inter-cluster interference. Also, recall that we have selected zero-forcing method to design $\mathbf{F}_\text{BB}$. Thus, designing the baseband precoder equals to solving
\begin{equation}\label{eq10}
\underset{\{\mathbf{f}_\text{BB}^\ell\}_{\ell\neq n}}{\text{minimize}} \ \left|I_\text{inter}^{n,m}\right| \qquad \text{subject to (10b)}.
\end{equation}
Based on ZFBF, the solution for (\ref{eq10}) is obtained as~\cite{alkhateeb2015limited}
\begin{equation}\label{eq11}
\mathbf{F}_\text{BB} = \overline{\mathbf{H}}^*\left(\overline{\mathbf{H}}\overline{\mathbf{H}}^*\right)^{-1}\bf{\Lambda},
\end{equation}
where the diagonal elements of $\mathbf{\Lambda}$ are given by~\cite{alkhateeb2015limited}
\begin{equation}\label{eq12}
\mathbf{\Lambda}_{n,n} = \sqrt{\frac{T_\text{BS}T_\text{MU}}{\left(\mathbf{F}_\text{RF}^*\mathbf{F}_\text{RF}\right)_{n,n}^{-1}}}\left|\beta_{n,1}\right|, \quad \text{for} \quad n = 1, 2, \cdots, N.
\end{equation}
According to ~(\ref{eq11}) and the hybrid beamforming results in~\cite{alkhateeb2015limited}, we can conclude that the inter-cluster interference amongst the first MUs is zero, i.e., $\overline{\mathbf{h}}^*_{n,1}\mathbf{f}^\ell_\text{BB} = 0$ for $n = 1, 2, \cdots, N$ and $\ell \neq n$. That is to say, inter-cluster interference is perfectly eliminated for the first MUs. This completes our justification about the orienting the beams toward the first MUs and choosing their effective channel vector in designing $\mathbf{F}_\text{BB}$.
At the third step, the BS first reorders the MUs, then allocates the power. The reordering process is done based on the effective channel vectors as
\begin{equation}\label{eq121}
\norm[\big]{\overline{\mathbf{h}}_{n,1}} > \norm[\big]{\overline{\mathbf{h}}_{n,2}}>, \cdots,
> \norm[\big]{\overline{\mathbf{h}}_{n,M}} \quad \text{for} \quad n = 1, 2, \cdots, N.
\end{equation}
It is worth mentioning that in~(\ref{eq8}) we aimed to find the first MUs based on the large-scale channel gain. However, in HB-NOMA the power allocation is conducted based on order of the effective channel gains, which takes both the large scale gain and the impact of hybrid beamforming and combining into account. Here, in~\eqref{eq121}, we propose to improve HB-NOMA's sum-rate by carrying out the power allocation based on the effective channel. Accordingly, it is true that while using~(\ref{eq8}) or~(\ref{eq121}) we arrive at the same first MU in each cluster, the order of the remaining MUs might be different when using~(\ref{eq121}) compared to~(\ref{eq8}). This can be intuitively explained by considering that the alignment of the MUs in each cluster with regard to the first MU may play a more important role on the quality of the their channels than their proximity to the base station.
The optimal power allocation in~(10) is non-trivial and iterative procedures are needed to solve for the power of the $m$th user in the $n$th cluster according to
\begin{equation}\label{eq13}
\underset{\{P_{n,m}\}}{\text{maximize}} \ \sum_{n=1}^{N}\sum_{m=1}^M R_{n,m} \qquad \text{subject to (10d) and (10e)}.
\end{equation}
Obtaining the optimal solution for (\ref{eq13}) is beyond the scope of this paper. Hence, we propose a suboptimal solution. Our solution has two stages. First the BS divides power equally amongst the clusters, i.e., $P_\text{c} = P_\text{t}/N$. Then a fixed power allocation~\cite{higuchi2015non} is utilized for the users in each cluster with regard to the constraint $\sum_{m=1}^M P_{n,m} = P_\text{c}$.
\section{The Rate Evaluation for HB-NOMA}\label{sec:lower}
In this section we concentrate on studying the achievable rate of an HB-NOMA MU.
\begin{comment}\subsection{Perfect Correlation}
By perfect correlation we mean that $\mathbf{a}_\text{BS}(\varphi_{n,m})$ is the same for all MUs in the $n$th cluster, i.e., $\mathbf{a}_\text{BS}(\varphi_{n,1}) = \mathbf{a}_\text{BS}(\varphi_{n,2}) = \cdots = \mathbf{a}_\text{BS}(\varphi_{n,M})$ for $n = 1, 2,\cdots,N$. That is, the RF precoder vector is the same for all MUs in each cluster. Although this assumption is ideal and never happens in practical scenarios, the derived lower bound characterizes useful results on the rate performance of HB-NOMA MUs.
\begin{theorem}
Considering the perfect correlation, the lower bound of the rate for HB-NOMA MU-$(n,m)$ is given by
\begin{equation}\label{eq14}
R_{n,m}^\text{P} \geq \text{log}_2\left(1 + \frac{P_{n,m}T_\text{BS}T_\text{MU}\left|\beta_{n,m}\right|^2}{\sum_{k=1}^{m-1}P_{n,k}T_\text{BS}T_\text{MU}\left|\beta_{n,m}\right|^2 + \eta_{n,m} }\right),
\end{equation}
where $\eta_{n,m} = \frac{1}{4}\left(\frac{\lambda_\text{max}(\mathbf{F}_\text{RF}^*\mathbf{F}_\text{RF})}{\lambda_\text{min}(\mathbf{F}_\text{RF}^*\mathbf{F}_\text{RF})} + \frac{\lambda_\text{min}(\mathbf{F}_\text{RF}^*\mathbf{F}_\text{RF})}{\lambda_\text{max}(\mathbf{F}_\text{RF}^*\mathbf{F}_\text{RF})} + 2\right)$.
\end{theorem}
\begin{proof}
Given the perfect correlation assumption and (\ref{eq9}), the effective channel vector for MU-$(n,m)$ becomes $\overline{\mathbf{h}}_{n,m}^* = \sqrt{T_\text{BS}T_\text{MU}}\beta_{n,m}\mathbf{a}_\text{BS}^*(\varphi_{n,m})\mathbf{F}_\text{RF} = \beta_{n,m}\beta_{n,1}^{-1}\overline{\mathbf{h}}^*_{n,1}$.
This gives $\overline{\mathbf{h}}^*_{n,1}\mathbf{f}^n_\text{BB} = \boldsymbol{\Lambda}_{n,n}$ for $n = 1, 2, \cdots, N$ and $\overline{\mathbf{h}}^*_{n,m}\mathbf{f}^\ell_\text{BB} = 0$ for $\ell \neq n$. Therefore, the numerator in~(\ref{eq6}) becomes $P_{n,m}\left|\beta_{n,m}\right|^2\left|\beta_{n,1}\right|^{-2}\mathbf{\Lambda}_{n,n}^2$. Indeed, we get the intra-cluster interference expression in (\ref{eq61}) as $I_\text{intra}^{n,m} = \sum_{k = 1}^{m-1}P_{n,k}\left|\beta_{n,m}\right|^2\left|\beta_{n,1}\right|^{-2}\mathbf{\Lambda}_{n,n}^2$ and the inter-cluster interference term becomes
$I_\text{inter}^{n,m} = 0$. Substituting the numerator, $I_\text{intra}^{n,m}$, and $I_\text{inter}^{n,m}$ in~(\ref{eq6}) gives
\begin{align}\label{eq15}
R_{n,m}^\text{P} & = \text{log}_2\left(1 + \frac{P_{n,m}\left|\beta_{n,m}\right|^2\left|\beta_{n,1}\right|^{-2}\mathbf{\Lambda}_{n,n}^2}{\sum_{k = 1}^{m-1}P_{n,k}\left|\beta_{n,m}\right|^2\left|\beta_{n,1}\right|^{-2}\mathbf{\Lambda}_{n,n}^2 + 1}\right) \nonumber \\
& \overset{(a)}{=} \text{log}_2\left(1 + \frac{P_{n,m}T_\text{BS}T_\text{MU}\left|\beta_{n,m}\right|^2}{\sum_{k = 1}^{m-1}P_{n,k}T_\text{BS}T_\text{MU}\left|\beta_{n,m}\right|^2 + \left(\mathbf{F}_\text{RF}^*\mathbf{F}_\text{RF}\right)^{-1}_{n,n}}\right)\nonumber \\
& \overset{(b)}{\geq} \text{log}_2\left(1 + \frac{P_{n,m}T_\text{BS}T_\text{MU}\left|\beta_{n,m}\right|^2}{\sum_{k = 1}^{m-1}P_{n,k}T_\text{BS}T_\text{MU}\left|\beta_{n,m}\right|^2 + \eta_{n,m}}\right).
\end{align}
We get (a) by plugging~(\ref{eq10}) into the expression in first line of (\ref{eq15}) and using simple manipulations. (b) follows from the Kantorovich inequality~\cite{kantorovich1948functional} and the lemma below.
\begin{lemma}
\cite{alkhateeb2015limited} The element $Q = \left(\mathbf{F}_\text{RF}^*\mathbf{F}_\text{RF}\right)_{n,n}^{-1}$ has an upper bound as
\begin{equation}\label{eq151}
Q \leq \frac{1}{4[\mathbf{F}_\text{RF}^*\mathbf{F}_\text{RF}]_{n,n}}\left(\frac{\lambda_\text{max}(\mathbf{F}_\text{RF}^*\mathbf{F}_\text{RF})}{\lambda_\text{min}(\mathbf{F}_\text{RF}^*\mathbf{F}_\text{RF})} + \frac{\lambda_\text{min}(\mathbf{F}_\text{RF}^*\mathbf{F}_\text{RF})}{\lambda_\text{max}(\mathbf{F}_\text{RF}^*\mathbf{F}_\text{RF})} + 2\right).
\end{equation}
where $\lambda_\text{max}(\mathbf{F}_\text{RF}^*\mathbf{F}_\text{RF})$ and $\lambda_\text{min}(\mathbf{F}_\text{RF}^*\mathbf{F}_\text{RF})$ are the largest and smallest eigenvalues of $\mathbf{F}_\text{RF}^*\mathbf{F}_\text{RF}$, respectively.
\end{lemma}
Now, noting that $[\mathbf{F}_\text{RF}^*\mathbf{F}_\text{RF}]_{n,n} = 1$ for $n = 1, 2, \cdots, N$ and substituting the upper bound in~(\ref{eq151}) into the last line of~(\ref{eq15}) gives the inequality (b).
\end{proof}
Theorem 1 indicates that when the correlation amongest the MUs in each cluster is perfect, still two terms degrades the rate performance of a single HB-NOMA MU. The first term $\sum_{k = 1}^{m-1}P_{n,k}T_\text{BS}T_\text{MU}\left|\beta_{n,m}\right|^2 $ is due to serving MUs with NOMA scheme which leads to an inevitable interference. The second term $\eta_{n,m}$ is due to realizing the beamforming with digital and analog components, i.e., hybrid beamforming. It is worth to mention that in fully-digital beamforming the first term happens but the second is always one. Therefore, even under ideal condition the hybrid beamforming intrinsically imposes interference on the sum-rate. However, when $T_\text{BS}$ approaches the infinity the $\eta_{n,m}$ nears to one~\cite{alkhateeb2015limited}.\end{comment}
In practical scenarios, correlation amongst the users in each cluster is always imperfect which gives $\mathbf{a}_\text{BS}(\varphi_{n,1}) \neq \mathbf{a}_\text{BS}(\varphi_{n,2}) \neq \cdots \neq \mathbf{a}_\text{BS}(\varphi_{n,M})$ for $n = 1, 2, \cdots, N$. The reason is that since $\varphi_{n,k}$ and $\varphi_{n,m}$ for $k=m$ are independent, the probability for the event $\varphi_{n,k} = \varphi_{n,m}$ is zero~\cite{alkhateeb2015limited}. Of interest, we here study the impact of imperfect correlation on the sum-rate in this subsection. Before this, we calculate the norm of the effective channel defined in Eq.~(\ref{eq9}). Defining
\begin{equation}\label{eqFejer}
\left|\mathbf{a}_\text{BS}^*(\varphi_{n,m})\mathbf{a}_\text{BS}(\varphi_{\ell,1})\right|^2 = K_{T_\text{BS}}(\varphi_{\ell,1}-\varphi_{n,m})
\end{equation}
where $K_{T_\text{BS}}$ is Fej$\acute{\text{e}}$r kernel of order $T_\text{BS}$~\cite{strichartz2000way}, we get
\begin{equation}\label{eq1601}
\norm[\big]{\overline{\mathbf{h}}^\text{Im}_{n,m}}^2 = T_\text{BS}T_\text{MU}\left|\beta_{n,m}\right|^2\displaystyle \sum_{\ell = 1}^N K_{T_\text{BS}}\left(\varphi_{\ell,1}-\varphi_{n,m}\right).
\end{equation}
The following lemma describes the relationship between the effective channel vectors of MU-$(n,1)$ and MU-$(n,m)$ for $m=2,3,\cdots,M$ and $n=1,2,\cdots,N$.
\begin{lemma}\label{lemma:2}
The relationship between the imperfect effective channel for MU-$(n,m)$ and MU-$(n,1)$ for $m = 2, 3,\cdots, M$ and $n = 1,2,\cdots,N$ can be modelded as
\begin{equation} \label{eq19}
\widetilde{\mathbf{h}}_{n,m}^\text{Im} \approx \rho_{n,m}\widetilde{\mathbf{h}}_{n,1}^\text{Im} + \sqrt{1 - \rho_{n,m}^2}\boldsymbol{\varpi}_{n,m}
\end{equation}
where $\widetilde{\mathbf{h}}_{n,m}^\text{Im}$ denotes the normalized effective channel, $\rho_{n,m} = \left|\widetilde{\mathbf{h}}_{n,m}^\text{Im*}\widetilde{\mathbf{h}}_{n,1}^\text{Im}\right|$
, and $\boldsymbol{\varpi}_{n,m}$ is a unit-norm vector.
\end{lemma}
\begin{proof}
Assume that the effective channel vectors are fed back by using infinite-resolution codebook. Also, let $\widetilde{\mathbf{h}}_{n,m}^\text{Im}$ denotes the normalized effective channel vector for MU-$(n,m)$ given by
\begin{equation}\label{eq40}
\widetilde{\mathbf{h}}_{n,m}^\text{Im} = \frac{\overline{\mathbf{h}}_{n,m}^\text{Im}}{\norm[\big]{\overline{\mathbf{h}}^\text{Im}_{n,m}}}. \end{equation}
For two complex-valued vectors $\overline{\mathbf{h}}_{n,m}^\text{Im},\overline{\mathbf{h}}_{n,1}^\text{Im} \in V_\mathbb{C}$ the angle between them is obtained as~\cite{scharnhorst2001angles}
\begin{equation}\label{anglevector}
\rho e^{j\omega} = \widetilde{\mathbf{h}}_{n,m}^\text{Im*}\widetilde{\mathbf{h}}_{n,1}^\text{Im},
\end{equation}
where $(\rho\leq 1)$ is equal to~\cite{scharnhorst2001angles}
\begin{equation}\label{rho}
\rho = \text{cos}{\Phi}_\text{H}(\widetilde{\mathbf{h}}_{n,m}^\text{Im},\widetilde{\mathbf{h}}_{n,1}^\text{Im}) = \left|\widetilde{\mathbf{h}}_{n,m}^\text{Im*}\widetilde{\mathbf{h}}_{n,1}^\text{Im}\right|,
\end{equation}
in which $\Phi_\text{H}(\widetilde{\mathbf{h}}_{n,m}^\text{Im},\widetilde{\mathbf{h}}_{n,1}^\text{Im})$, $0\leq \nu_\text{H}\leq\frac{\pi}{2}$, is called the Hermitian angle between two complex-valued vectors $\overline{\mathbf{h}}_{n,m}^\text{Im},\overline{\mathbf{h}}_{n,1}^\text{Im}$ and $\omega$, $-\pi\leq\omega\leq\pi$, is called their pseudo-angle. The factor $\rho$ is related to the angle between two lines in the complex vector space $V_\mathbb{C}$ while the angle $\omega$ is defined in the context of pseudoconformal transformations~\cite{scharnhorst2001angles,goldman1999complex}. Here, we rename the factor $\rho$ as the correlation between two vectors $\overline{\mathbf{h}}_{n,m}^\text{Im},\overline{\mathbf{h}}_{n,1}^\text{Im}$ and disregard the pseudo angle $\omega$. Therefore, the angle between the two vectors is approximated by $\rho$. Then, for some $n$ the two normalized vectors, $\widetilde{\mathbf{h}}_{n,m}^\text{Im}$ for $m = 2, 3, \cdots, M$ and $\widetilde{\mathbf{h}}_{n,1}^\text{Im}$, are related together through the factor $\rho_{n,m}$ and vector $\boldsymbol{\varpi}_{n,m}$ which is a unit-norm in the null-space of $\widetilde{\mathbf{h}}_{n,1}^\text{Im}$.
\end{proof}
Now, we derive a lower bound for the sum-rate of MU-$(n,m)$ in HB-NOMA systems when the correlation is imperfect.
\begin{theorem}\label{theo:2}
Regarding the imperfect correlation, the lower bound of the rate of HB-NOMA MU-$(n,m)$ for $m>1$ cluster is given by
\begin{equation}\label{eq16}
R_{n,m}^\text{Im} \geq \text{log}_2\left(1 +\frac{P_{n,m}\rho_{n,m}^2T_\text{BS}T_\text{MU}\left|\beta_{n,m}\right|^2}{\zeta_\text{intra}^{n,m}+ \zeta_\text{inter}^{n,m} + \zeta_\text{noise}^{n,m}}\right)
\end{equation}
where
\begin{equation}\label{eq161}
\zeta_\text{intra}^{n,m} = \sum_{k = 1}^{m-1}P_{n,k}\rho_{n,m}^2T_\text{BS}T_\text{MU}\left|\beta_{n,m}\right|^2,
\end{equation}
and
\begin{align}\label{eq162}
\zeta_\text{inter}^{n,m} = & P_\text{c}\left(1-\rho_{n,m}^2\right)T_\text{BS}T_\text{MU}\left|\beta_{n,m}\right|^2\lambda_\text{max}\left(\mathbf{S}\right)\eta_nK_{T_\text{BS},\Sigma_1},
\end{align}
$\lambda_\text{max}\left(\mathbf{S}\right)$ showing the maximum eigenvalue of $\mathbf{S} = \mathbf{F}_\text{BB}^{-n}\mathbf{F}_\text{BB}^{-n*}$. $\mathbf{F}_\text{BB}^{-n}$ denotes the $\mathbf{F}_\text{BB}$ after eliminating the $n$th column. Also, for some $m$ we define
\begin{equation}\label{eq163}
K^{-1}_{T_{BS},\Sigma_m} = \left(\displaystyle \sum_{\ell = 1}^N K_{T_\text{BS}}\left(\varphi_{\ell,1}-\varphi_{n,m}\right)\right)^{-1}
\end{equation}
with $K_{T_\text{BS}}\left(\varphi_{\ell,1}-\varphi_{n,m}\right)$ denotes the Fej$\acute{\text{e}}$r kernel in~(\ref{eqFejer}). Finally $\zeta_\text{noise}^{n,m}$ is expressed as
\begin{equation}\label{eq164}
\zeta_\text{noise}^{n,m} = \eta_nK_{T_{BS},\Sigma_1}K^{-1}_{T_{BS},\Sigma_m},
\end{equation}
where $K^{-1}_{T_{BS},\Sigma_m}$ is defined in~(\ref{eq163}) when $\varphi_{n,1}$ is replaced by $\varphi_{n,m}$.
\end{theorem}
\begin{comment}
\begin{proof}
Let denote the normalized effective channel for MU-$(n,m)$ by $\widetilde{\mathbf{h}}_{n,m}^\text{Im} = \frac{\overline{\mathbf{h}}_{n,m}^\text{Im}}{\norm[\big]{\overline{\mathbf{h}}^\text{Im}_{n,m}}}$. Now, $\widetilde{\mathbf{h}}_{n,m}^\text{Im}$ can be modeled as
\begin{equation} \label{eq19}
\widetilde{\mathbf{h}}_{n,m}^\text{Im} = \rho_{n,m}\widetilde{\mathbf{h}}_{n,1}^\text{Im} + \sqrt{1 - \rho_{n,m}^2}\boldsymbol{\varpi}_{n,m}
\end{equation}
where $\rho_{n,m} = \text{Cor}\left(\widetilde{\mathbf{h}}_{n,m}^\text{Im},\widetilde{\mathbf{h}}_{n,1}^\text{Im}\right)$ and $\boldsymbol{\varpi}_{n,m}$ is a unit-norm vector in the null-space of $\widetilde{\mathbf{h}}_{n,1}^\text{Im}$. To get Eq.~(\ref{eq19}), we use the assumption that the effective channel vectors are fed back by using infinite-resolution codebook. Then, the two normalized vectors are related together through the correlation between them $\rho_{n,m}$ an the unit-norm vector $\boldsymbol{\varpi}_{n,m}$. By utilizing Eq.~(\ref{eq19}), we obtain the following important expressions.
\begin{align}\label{eq191}
\left|\overline{\mathbf{h}}^\text{Im*}_{n,m}\mathbf{f}_\text{BB}^n\right|^2 &= \rho_{n,m}^2\norm[\Big]{\overline{\mathbf{h}}^\text{Im}_{n,m}}^2\left|\widetilde{\mathbf{h}}^\text{Im*}_{n,1}\mathbf{f}_\text{BB}^n\right|^2 + \left(1 - \rho_{n,m}^2\right)\left|\boldsymbol{\varpi}^*_{n,m}\mathbf{f}_\text{BB}^n\right|^2 \nonumber \\
& \overset{(a)}{=} \rho_{n,m}^2\norm[\Big]{\overline{\mathbf{h}}^\text{Im}_{n,m}}^2\left|\widetilde{\mathbf{h}}^\text{Im*}_{n,1}\mathbf{f}_\text{BB}^n\right|^2 \nonumber \\
& \overset{(b)}{\approx} \rho_{n,m}^2\norm[\Big]{\overline{\mathbf{h}}^\text{Im}_{n,m}}^2.
\end{align}
We get (a) since $\boldsymbol{\varpi}^*_{n,m}\mathbf{f}_\text{BB}^n = 0$ and (b) follows from the fact that for large $T_\text{BS}$ the term $\left|\widetilde{\mathbf{h}}^\text{Im*}_{n,1}\mathbf{f}_\text{BB}^n\right|^2$ approaches to zero.
\begin{equation}\label{eq192}
\left|\overline{\mathbf{h}}^\text{Im*}_{n,m}\mathbf{f}_\text{BB}^\ell\right|^2 = \left(1 - \rho_{n,m}^2\right)\norm[\Big]{\overline{\mathbf{h}}^\text{Im}_{n,m}}^2\Big|\boldsymbol{\varpi}^*_{n,m}\mathbf{f}_\text{BB}^\ell\Big|^2,\quad \text{for} \quad \ell \neq n,
\end{equation}
as $\widetilde{\mathbf{h}}_{n,1}^\text{Im*}\mathbf{f}_\text{BB}^\ell = 0$.
Now, plugging Eq.~(\ref{eq1601}) into Eq.~(\ref{eq191}), then putting (\ref{eq191}) into Eq.~(\ref{eq61}) gives
\begin{equation}\label{eq30}
I_\text{intra}^{n,m} = \sum_{k = 1}^{m-1}P_{n,k}\rho_{n,m}^2T_\text{BS}T_\text{MU}\left|\beta_{n,m}\right|^2\displaystyle \sum_{\ell = 1}^N K_{T_\text{BS}}\left(\varphi_{\ell,1}-\varphi_{n,m}\right).
\end{equation}
Likewise, substituting (\ref{eq1601}) into~(\ref{eq192}), then putting~(\ref{eq192}) into Eq.~(\ref{eq62}) gives
\begin{align}\label{eq31}
I_\text{inter}^{n,m} = & P_\text{c}\left(1-\rho_{n,m}^2\right)T_\text{BS}T_\text{MU}\left|\beta_{n,m}\right|^2\sum_{\ell \neq n}^N\left|\boldsymbol{\varpi}_{n,m}^*{\mathbf{f}}_\text{BB}^\ell\right|^2 \times \nonumber \\
& \quad \displaystyle \sum_{\ell = 1}^N K_{T_\text{BS}}\left(\varphi_{\ell,1}-\varphi_{n,m}\right).
\end{align}
Further, after substituting~(\ref{eq191}),~(\ref{eq30}) and~(\ref{eq31}) into Eq.~(\ref{eq6}) it becomes
\begin{align}\label{eq32}
R_{n,m}^\text{Im} &= \text{log}_2\left(1+ \frac{P_{n,m}\rho_{n,m}^2T_\text{BS}T_\text{MU}\left|\beta_{n,m}\right|^2\displaystyle \sum_{\ell = 1}^N K_{T_\text{BS}}\left(\varphi_{\ell,1}-\varphi_{n,m}\right)}{I_\text{intra}^{n,m}+ I_\text{inter}^{n,m} + 1}\right)\nonumber \\
&\overset{(a)}{\geq} \text{log}_2\left(1+ \frac{P_{n,m}\rho_{n,m}^2T_\text{BS}T_\text{MU}\left|\beta_{n,m}\right|^2}{\zeta_\text{intra}^{n,m}+ \zeta_\text{inter}^{n,m} + \left(\displaystyle \sum_{\ell = 1}^N K_{T_\text{BS}}\left(\varphi_{\ell,1}-\varphi_{n,m}\right)\right)^{-1}}\right)\nonumber \\
&\overset{(b)}{\geq}\text{log}_2\left(1+ \frac{P_{n,m}\rho_{n,m}^2T_\text{BS}T_\text{MU}\left|\beta_{n,m}\right|^2}{\zeta_\text{intra}^{n,m}+ \zeta_\text{inter}^{n,m} + K^{-1}_{T_\text{BS}}\left(\varphi_{n,1}-\varphi_{n,m}\right)}\right)
\end{align}
The (a) follows taking the common factor $\sum_{\ell = 1}^N K_{T_\text{BS}}\left(\varphi_{\ell,1}-\varphi_{n,m}\right)$ from the numerator and denominator. Also, to replace $I_\text{inter}^{n,m}$ with $\zeta_\text{inter}^{n,m}$, we have the following Lemma.
\begin{lemma}\label{l1}
The upper bound of
$\sum_{\ell \neq n}^N\left|\boldsymbol{\varpi}_{n,m}^*{\mathbf{f}}_\text{BB}^\ell\right|^2$ is the maximum eigenvalue of $\mathbf{F}_\text{BB}^{-n}\mathbf{F}_\text{BB}^{-n*}$, i.e., $\lambda_\text{max}\left(\mathbf{F}_\text{BB}^{-n}\mathbf{F}_\text{RF}^{-n*}\right)$.
\end{lemma}
\begin{proof}
We rewrite $\sum_{\ell \neq n}^N\left|\boldsymbol{\varpi}_{n,m}^*{\mathbf{f}}_\text{BB}^\ell\right|^2 = \norm[\big]{\boldsymbol{\varpi}_{n,m}^*\mathbf{F}_\text{BB}^{-n}}^2$. $\mathbf{F}_\text{BB}^{-n}$ is the baseband precoder in~(\ref{eq11}) after eliminating the $n$th column. Maximizing $\norm[\big]{\boldsymbol{\varpi}_{n,m}^*\mathbf{F}_\text{BB}^{-n}}^2$ given $\norm[]{\boldsymbol{\varpi}_{n,m}} = 1$ is similar to maximizing a beamforming vector for maximum ratio transmission systems~\cite{love2003grassmannian,dighe2003analysis}. Hence, the maximum value of $\boldsymbol{\varpi}_{n,m}$ is the dominant right singular vector of $\mathbf{F}_\text{BB}^{-n}$ given in~(\ref{eq162})~\cite{love2003grassmannian,dighe2003analysis}. Thus, the maximum of $\norm[\big]{\boldsymbol{\varpi}_{n,m}^*\mathbf{F}_\text{BB}^{-n}}^2$ is the maximum eigenvalue of $\mathbf{F}_\text{BB}^{-n}\mathbf{F}_\text{RF}^{-n*}$.
\end{proof}
Lemma~\ref{l1} indicates that $I_\text{inter}^{n,m} \leq \zeta_\text{inter}^{n,m}$. To get (b), we use the inequality $\sum_{\ell = 1}^N K_{T_\text{BS}}\left(\varphi_{\ell,1}-\varphi_{n,m}\right) \geq K_{T_\text{BS}}\left(\varphi_{n,1}-\varphi_{n,m}\right)$ that completes the proof.
\end{proof}
\end{comment}
The proof is eliminated due to lack of enough space. Since for MU-$(n,1)$ the correlation factor $\rho = 1$ even in the case of imperfect correlation, Thoerem 1 is still valid for these users.
Theorem~\ref{theo:2} clearly states that the achievable rate of each MU extremely depends on the correlation between the first and the intended MU such as a weak correlation reduces the power of the effective channel of that MU. Besides, the bound gives an useful insight on the clustering the MUs. That is, the relation between the AoD $\phi_{n,m}$ and the correlation factor $\rho_{n,m}$ is nonlinear. On the other hand, $\phi_{n,m}$ does not explicitly appears in the rate expression whereas $\rho_{n,m}$ does. As a result, the clustering can be done through $\rho_{n,m}$s as a design criterion.
The following two remarks are in order:
\begin{itemize}
\item This paper concentrates only on studying the rate performance of HB-NOMA systems under imperfect effective channel correlation, several ideal assumptions are considered in Section~\ref{sec:system}. Studying the performance of the imperfect effective channel correlation along with practical assumptions can be done in the future works. For instance, in practical scenarios the effective channel at the transmitter is quantized. The impact of the quantization error on the sum-rate of proposed HB-NOMA system with imperfect correlation is required.
\item In the algorithm proposed in Section~\ref{algorithm} the fixed intra-cluster and inter-cluster power allocation strategies are ordered. Adaptive but complex power allocation procedure for MUs inside each cluster can be designed through iterative algorithm~\cite{zhang2016robust}. On the other hand, in practice, each cluster poses different inter-cluster interference. Therefore, the algorithm will be suboptimum in term of inter-cluster power allocation. A useful inter-cluster power allocation regarding the interference caused by each cluster is proposed in~\cite{chen2017exploiting}.
\end{itemize}
\section{Numerical Results}\label{sec:simulation}
In this section we discuss the numerical simulations for the achievable rate of the HB-NOMA users under the imperfect correlation assumption.
\begin{comment}\begin{figure}[!b]
\vspace*{-0.5cm}
\hspace*{-0.3cm}
\centering
\includegraphics[scale=0.4]{image/sumRateVsSNR.pdf}
\caption{The achievable rate vs SNR performance for the perfect correlation between MU-(1,1) and MU-(1,2).}
\label{fig:snr}
\end{figure}
\end{comment}
\begin{figure}
\centering
\includegraphics[scale=0.4]{image/sumRateVsCorr.pdf}
\caption{The achievable rate vs correlation factor $\rho$ performance for the imperfect correlation between MU-(1,1) and MU-(1,2) for SNR 5 and 0 dB.}
\label{fig:corr}
\end{figure}
\begin{figure}
\centering
\includegraphics[scale=0.4]{image/RhoVsAngle}
\caption{Relationship between the correlation and AoD for MU-$(1,2)$ when there are three clusters. The AoDs for MU-$(n,1)$ for $n=1, 2, \text{and}, 3$ are assumed to be respectively $0^\circ$, $-40^\circ$, and $40^\circ$.}
\label{fig:1RhoVsAngle}
\end{figure}
\begin{comment} Fig.~\ref{fig:snr} illustrates the performance of the rate of MU-(1,1) and MU-(1,2) for the hybrid beamforming with the perfect correlation. By increasing the SNR both MUs achieve higher rate. The simulation result shows that the hybrid beamforming can achieve a rate close to the fully-digital beamforming. The small gap between the two curves is due to using RF precoders that causes a residual interference. It is worth to note that in both beamforming methods the inter-cluster interference is zero and intra-cluster interference terms are identical. Indeed, the figure indicates that the derived lower bound in Theorem 1 is tight. This performance is similar to that of the traditional multi-user system proposed in~\cite{alkhateeb2015limited}.\end{comment}
Fig.~\ref{fig:corr} evaluates the effect of the imperfect correlation on the rate performance of MU-(1,2). We consider that the BS is equipped with a $16\times 1$ ULA which serves two clusters each containing two MUs. Each MU has a $4\times 1$ ULA. Also, there is a single path channel from the BS to each MU and full CSI is available at the BS. The elevation AoAs/AoDs have a uniform distribution in $\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$. Further, the proposed algorithm in Section~\ref{algorithm} is used to maximize the rate where The allocated power for the close MU is assumed to be 1/4$P_\text{c}$ and for the far MU it is 3/4$P_\text{c}$ with $P_\text{c} = 1/2P_\text{t}$. Finally, the large-scale fading, i.e, $D^{-\nu}$ is assumed 0 dB for MU-(1,1) and MU-(2,1) and -10 dB for MU-(1,2) and MU-(2,2). It is assume that the SNR is 0 and 5 dB. Also, the AoD, i.e., $\phi_{1,2}$, is increasing from $50^\circ$ to $60^\circ$. As $\phi_{1,2}$ grows up, the correlation factor increases but it has a nonlinear behavior. This means $\rho$ is a nonlinear function of AoDs. Obviously, the small correlation considerably degrades the rate performance, e.g., a channel correlation 0.92 decreases the rate about 1 bits/s/Hz compare to $\rho = 1$. By increasing the correlation factor the rate increases. That is, by increasing $\rho$ the effect of inter-cluster, i.e, $\zeta_\text{inter}$, decreases and angle of the effective channel of the MU approaches to that of MU-(1,1) which leads to a higher value for $K_{T_\text{BS}}$ in Theorem~2. Also, the figure shows that the derived lower bound in Theorem~\ref{theo:2} is accurate and close to the simulation value.
In Fig.~\ref{fig:1RhoVsAngle} we investigate the relationship between the correlation and AoD for $-90^\circ \leq \phi \leq 90^\circ$ in the case of three clusters and two MUs in each of them. First we set AoD of MU-$(n,1)$ for $n=1, 2, 3$ respectively $0^\circ$, $-40^\circ$, and $40^\circ$. As red dash line shows the correlation between MU-$(1,1)$ and MU-$(1,2)$ around $0^\circ$ is 1. It also shows that when AoD of MU-$(1,2)$ is in range $[-7^\circ,7^\circ]$ the correlation remains greater than $0.95$. The lowest correlation happens around $-40^\circ$ and $40^\circ$ which are the AoDs of the second and third clusters.
\section{Conclusion}\label{sec:conclusion}
Hybrid beamforming-based NOMA has been studied. To this end, we formulated an optimization problem for the sum-rate of HB-NOMA system. Then, due to the complicate objective function and constrains an algorithm is proposed in three steps. In order to evaluate the sum-rate, we derived a lower bound for each MU for imperfect correlation between the effective channel of the first MU and other MUs. Our lower bound analysis demonstrates that under the assumption of imperfect correlation the lower bound indicates that an improper correlation can cause a remarkable degradation in the rate performance. The numerical results support our analytical findings.
\appendices
\ifCLASSOPTIONcaptionsoff
\newpage
\fi
\bibliographystyle{IEEEtran}
|
{
"timestamp": "2018-04-17T02:12:12",
"yymm": "1804",
"arxiv_id": "1804.05444",
"language": "en",
"url": "https://arxiv.org/abs/1804.05444"
}
|
\section*{Introduction}
A central feature of K\"ahler geometry is the fact that, once a complex structure is chosen, the Riemannian metric depends locally on only a single real-valued function, the K\"ahler potential. Indeed, in holomorphic coordinates we have the familiar expression for the metric tensor
\[
g_{i\bar\jmath} = \frac{\partial^2 f}{\partial z_i \partial\bar z_j}.
\]
This fundamental observation is crucial to many aspects of K\"ahler geometry, especially to the problem of finding K\"ahler-Einstein metrics and to studying the K\"ahler-Ricci flow; in both cases these problems are reduced to differential equations for a single real-valued function.
The importance of K\"ahler structures in physics derives from the discovery by
Zumino~\cite{ZUMINO1979203} that if a Riemannian manifold is equipped with a K\"ahler structure, then the 2-dimensional sigma model, a field theory whose fields are maps from a Lorentzian 2-manifold to a Riemannian target manifold, may naturally be extended, by introducing additional fields and using the K\"ahler structure on the target, to a field theory with $\mathcal{N}=2$ supersymmetry, an enlargement of the usual Lorentz symmetry of the original model. Zumino also made the key observation that the additional fields of the resulting theory could be interpreted, together with the original field, as the components of a single map called a superfield, but where the domain of the map is modified to be a supermanifold
and the map satisfies a certain constraint.
Zumino showed that in this more geometric formulation of the extended model, the Lagrangian density simplifies dramatically, and is given precisely by the pullback of the K\"ahler potential function $f$ described above.
It was later discovered by Gates, Hull and Ro\v{c}ek~\cite{MR776369} that K\"ahler geometry is not the only structure giving rise to such a supersymmetric extension; they showed that what is required is \emph{generalized K\"ahler geometry}, consisting of a pair $I_+, I_-$ of complex structures compatible with the Riemannian metric and whose associated Hermitian forms $\omega_+, \omega_-$ are not necessarily closed. Instead, they satisfy the conditions
\[
d_{+}^{c}\omega_{+} + d_{-}^{c}\omega_{-} = 0, \qquad dd_{\pm}^{c}\omega_{\pm} = 0.
\]
Gates, Hull and Ro\v{c}ek argued that while a naive application of the $\partial\overline\partial$-lemma is impossible in this case, there should still be a reformulation of the sigma model generalizing the one found by Zumino, for which the Lagrangian density would again be a single real-valued function, which, in turn, determines the metric. The authors established this under the assumption that the complex structures $I_+, I_-$ commute, and in subsequent works~\cite{lindstrom2007potential,lindstrom2007generalized} it was established under the assumption that the commutator $[I_+,I_-]$ has constant rank.
The purpose of this paper is to introduce a new mathematical approach to this problem, and to solve the general case, where the rank of $[I_+,I_-]$ is not locally constant, a common situation in examples. It is important to emphasize that in this paper, we do make the simplifying assumption that $I_++I_-$ is invertible (what we call ``symplectic type''); in the language of superfields, we assume there are chiral and semichiral superfields, but no twisted chiral superfields.
We now give a brief outline of the main ideas and results in the paper.
In Section~\ref{symptyp}, we review the key tools from Dirac geometry and generalized complex geometry which we use, and we define the notion of a generalized K\"ahler structure of symplectic type to which all our main results apply. Most importantly, we explain Hitchin's observation that the commutator $[I_+, I_-]$ defines a pair of holomorphic Poisson structures $\sigma_+, \sigma_-$ relative to the complex structures $I_+, I_-$.
In Section~\ref{Donkahler}, we revisit the classical K\"ahler case and explain an important observation of Donaldson that the K\"ahler potential function has a global interpretation as a Lagrangian submanifold of a holomorphic symplectic affine bundle deforming the cotangent bundle and determined by the K\"ahler class. Our main insight is that it is this viewpoint on K\"ahler geometry which may be profitably generalized.
The space which plays the role of the holomorphic symplectic affine bundle in our more general construction is a \emph{Morita equivalence} bibundle relating the Poisson structures $\sigma_+$ and $\sigma_-$. Morita equivalence is a relation between Poisson manifolds introduced by Weinstein and Xu~\cite{weinstein1987symplectic,xu1991morita}, and involves a symplectic manifold which maps via Poisson maps to the spaces being related; it may be viewed as an invertible generalized morphism. We review the key properties of this equivalence relation on Poisson manifolds in Section~\ref{moreq}.
In Section~\ref{branbi}, we prove our main result, Theorem~\ref{main}, showing that generalized K\"ahler structures of symplectic type are equivalent to holomorphic symplectic Morita equivalences equipped with a bisection which is Lagrangian for the imaginary part of the holomorphic symplectic form (i.e., a Lagrangian brane). We also explain the curious way in which such a Lagrangian brane determines a symmetric tensor---the generalized K\"ahler metric.
Having expressed the geometry in terms of a Lagrangian brane in a holomorphic symplectic manifold, we are able in Section~\ref{secpot} to generalize the classical generating function technique to obtain a local description in terms of a single real-valued function, the generalized K\"ahler potential. We verify that our potential reduces to those found earlier in special cases, and we use it to produce a new example (Proposition~\ref{globex}) of a generalized K\"ahler metric which has a global generalized K\"ahler potential, yet whose Poisson structures are not of constant rank.
In Section~\ref{picsec}, we explore the basic features of the category of holomorphic symplectic Morita equivalences equipped with brane bisection. Real versions of this category or \emph{Picard groupoid} have been investigated recently~\cite{bursztyn2004picard, bursztyn2015picard}, and so we carefully explain the unexpected link to generalized K\"ahler geometry.
Since our main result represents a generalized K\"ahler metric as a Lagrangian submanifold, it is clear that the group of Hamiltonian flows must act on the space of generalized K\"ahler metrics. In Section~\ref{hamflo}, we develop this idea into a method for deforming generalized K\"ahler structures. We also show that this flow method specializes to the existing constructions~\cite{MR1702248,MR2217300,MR2371181,gualtieri2010branes} of generalized K\"ahler metrics. Finally, in Section~\ref{locdef}, we use the above results to prove, in Theorem~\ref{GKlocallyflow}, that any generalized K\"ahler structure of symplectic type may locally be constructed via the flow method applied to a natural ``degenerate'' generalized K\"ahler structure which exists on any holomorphic Poisson manifold. In other words, we have the striking result that, locally, any generalized K\"ahler structure of symplectic type may be obtained by applying the flow construction to a holomorphic Poisson structure.
\vspace{.05in}
\noindent \textbf{Acknowledgements.} We would like to thank Nigel Hitchin, Chris Hull, Ulf Lindstr\"om, Martin Ro\v{c}ek, David Mart\'ines Torres, and Rikard von Unge for discussions on this subject over many years. F.B. is supported by an NSERC CGS Doctoral award, M.G. is supported by an NSERC Discovery Grant, and M.Z. is supported by Vetenskapsr\r{a}det under grant W2014-5517, by the STINT grant and by the grant ``Geometry and Physics'' from the Knut and Alice Wallenberg foundation.
\pagebreak
\section{Generalized K\"ahler structures of symplectic type}\label{symptyp}
We start by reviewing the concept of Generalized K\"ahler (GK) geometry and its many reformulations, as well as the subclass of GK structures of \emph{symplectic type} which will be the main subject of this paper. We begin with the notion of bihermitian geometry, first formulated by Gates, Hull, and Ro\v{c}ek in the context of $N = (2,2)$ supersymmetry \cite{MR776369}. A \emph{bihermitian} structure on a manifold $M$ consists of the data $(g, I_{+}, I_{-})$ of a Riemannian metric $g$ and two complex structures $I_{\pm}$ such that $g$ is Hermitian with respect to both complex structures and such that the following integrability conditions are satisfied:
\[
d_{+}^{c}\omega_{+} + d_{-}^{c}\omega_{-} = 0, \qquad dd_{\pm}^{c}\omega_{\pm} = 0,
\]
where $\omega_{\pm} = gI_{\pm}$ are the associated Hermitian forms and $d_{\pm}^{c} = i(\overline{\partial}_{\pm} - \partial_{\pm})$ are the real operators determined by the two complex structures. The closed $3$-form $H = \pm d_{\pm}^{c}\omega_{\pm}$ is used to define the Wess--Zumino--Witten term in the non-linear sigma model. We assume that it is trivial in cohomology and choose an explicit potential $b$ such that $H = -db$, called the $B$-field. Hence we are concerned with the following data: $(g, I_{+}, I_{-}, b)$. Note that these data admit an action of the additive group of closed $2$-forms $\Omega^{2, cl}(M)$:
\[
B : (g, I_{+}, I_{-}, b) \mapsto (g, I_{+}, I_{-}, b + B),
\]
which we refer to as $B$-field gauge transformations.
As explained in \cite{gualtieri-2010}, this geometry admits a reformulation in terms of commuting \emph{generalized complex structures} on $TM \oplus T^{\ast}M$ known as \emph{generalized K\"ahler geometry}. Let us briefly recall the relevant notions. The bundle $\bb{T} = TM \oplus T^{\ast}M$ admits a natural split-signature non-degenerate symmetric pairing (let $X, Y \in \Gamma(TM)$ and $\xi, \eta \in \Gamma(T^{\ast}M)$):
\[
\langle X + \xi, Y + \eta \rangle = \frac{1}{2}(\xi(Y) + \eta(X)),
\]
as well as an extension of the Lie bracket, known as the \emph{Courant bracket}:
\[
[\![ X + \xi, Y + \eta ]\!] = [X,Y] + \cal{L}_{X}(\eta) - \iota_{Y}(d\xi),
\]
which satisfies the Jacobi identity but is skew-symmetric only up to an exact term involving the above pairing. A generalized complex (GC) structure is defined to be an endomorphism $\bb{J}$ of $\bb{T}$ squaring to $-1$ which is compatible with the symmetric pairing and such that its $+i$ eigenbundle $L$ in the complexification $\bb{T}_\bb{C} = \bb{T}\otimes\bb{C}$ is closed under the (complexified) Courant bracket. A generalized K\"ahler structure is then given by a pair $(\bb{J}_{\cal{A}}, \bb{J}_{\cal{B}})$ of commuting generalized complex structures such that the product $\bb{G} = - \bb{J}_{\cal{A}} \bb{J}_{\cal{B}}$ defines a positive definite metric on $\bb{T}M$, which is to say that for all non-zero $v \in \bb{T}M$:
\[
\langle \bb{G} v, v \rangle > 0.
\]
The bihermitian structure $(g,I_{+},I_{-},b)$ may be encoded in the following Generalized K\"ahler pair:
\begin{align*}
\bb{J}_{\cal{B}} &= \frac{1}{2} e^{b} \begin{pmatrix} I_{+} + I_{-} & -(\omega_{+}^{-1} - \omega_{-}^{-1}) \\ \omega_{+} - \omega_{-} & -(I_{+}^{\ast} + I_{-}^{\ast}) \\ \end{pmatrix} e^{-b}, \\
\bb{J}_{\cal{A}} &= \frac{1}{2} e^{b} \begin{pmatrix} I_{+} - I_{-} & -(\omega_{+}^{-1} + \omega_{-}^{-1}) \\ \omega_{+} + \omega_{-} & -(I_{+}^{\ast} - I_{-}^{\ast}) \\ \end{pmatrix} e^{-b},
\end{align*}
where the conjugation is by the natural orthogonal symmetry determined by $b$:
\[
e^{b} = \begin{pmatrix} 1 & 0 \\ b & 1 \end{pmatrix}.
\]
Conversely, given a GK pair $(\bb{J}_{\cal{A}}, \bb{J}_{\cal{B}})$, it is possible to recover the bihermitian data in the following way: the product $\bb{G}= - \bb{J}_{\cal{A}}\bb{J}_{\cal{B}}$ has eigenvalues $\pm 1$, with corresponding eigenbundles \[
C_{\pm} = \{X + (b\pm g)X \ |\ X\in TM\},
\]
which determine $g$ and $b$. These eigenbundles map isomorphically to $TM$ by the natural projection, and $I_{\pm}$ is given by the restriction of $\bb{J}_{\cal{B}}$ to $C_{\pm}$. Note that $B$-field gauge transformations act on the $GK$ structure by conjugation:
\[
B: (\bb{J}_{\cal{A}}, \bb{J}_{\cal{B}}) \mapsto (e^{B}\bb{J}_{\cal{A}}e^{-B}, e^{B}\bb{J}_{\cal{B}}e^{-B}).
\]
The GC structures are themselves fully specified by their $+i$ eigenbundles $L_{\cal{A}}, L_{\cal{B}} \subset \bb{T}_\mathbb{C}$, which are complex \emph{Dirac structures}: maximal isotropic subbundles of $\bb{T}_\bb{C}$ which are involutive for the Courant bracket. As such, it is possible to completely characterize the conditions for generalized K\"ahler geometry in terms of the Dirac structures $L_{\cal{A}}$ and $L_{\cal{B}}$, leading to yet another reformulation of the geometry. This is what is done in \cite{mgualt-hamdef}, and we reproduce the key statements here. For this, we need two basic observations concerning Dirac structures.
The first is that holomorphic Poisson structures may be viewed as Dirac structures in the following way. Given a complex structure $I$ on the manifold $M$ and a holomorphic Poisson tensor $\sigma$, we decompose it into real and imaginary parts
$
\sigma = -\frac{1}{4}(IQ + iQ),
$
and define the following GC structure:
\[
\bb{J}_{\sigma} = \begin{pmatrix}
-I & Q \\
0 & I^{\ast} \\
\end{pmatrix}.
\]
The $+i$ eigenbundle of $\bb{J}_\sigma$ is the complex Dirac structure
\[
L_{\sigma} = \{ X + \sigma(\zeta) + \zeta \ | \ X \in T^{0,1}M, \ \zeta \in T_{1,0}^{*}M \}.
\]
Note that this Dirac structure encodes both the complex structure $I$ and the Poisson tensor $\sigma$; $L_{\sigma}$ is involutive if and only if $I$ is integrable, $\sigma$ is holomorphic, and $\sigma$ is Poisson. Also, because the intersection with the complexified tangent bundle $T_\mathbb{C} = TM\otimes\mathbb{C}$ is given by $L_{\sigma} \cap T_\mathbb{C} = T^{0,1}$, the Dirac structure $L_\sigma$ satisfies the special property that $T_\bb{C} = (L_{\sigma} \cap T_\bb{C}) \oplus (\overline{L}_{\sigma} \cap T_\bb{C})$. It is shown in \cite{mgualt-hamdef} that this property is in fact enough to guarantee that a complex Dirac structure arises from a holomorphic Poisson structure in the manner described above.
Second, we recall the notion of linear combination of Dirac structures. Given a Dirac structure $L \subset \bb{T}_\bb{C}$, we may scale it by $\lambda \in \bb{C}^*$ to produce a new Dirac structure:
\[
\lambda L = \{ X + \lambda \alpha \ | \ X + \alpha \in L \}.
\]
Similarly, given a pair of Dirac structures $L_{1}$ and $L_{2}$ such that their projections to $T_\bb{C}$ are transverse, we define the sum Dirac structure:
\[
L_{1} + L_{2} = \{ X + \alpha + \beta \ | \ X + \alpha \in L_{1}, X + \beta \in L_{2} \}.
\]
Since it occurs frequently, we use the notation $L_1-L_2$ to denote $L_1 + (-1)L_2$. Any closed 2-form $B$ defines a Dirac structure through its graph $\Gamma_B = \{X + i_XB\ |\ X\in TM\}$; a key property of the Dirac sum is that addition of $\Gamma_B$ coincides with gauge transformation: for any Dirac structure $L$,
\begin{equation}\label{gaugesum}
e^B(L) = L + \Gamma_B.
\end{equation}
\begin{proposition}\label{diffdir}\cite[Proposition 4.3]{mgualt-hamdef}
A pair $L_{\cal{A}}, L_{\cal{B}} \subset \bb{T}M \otimes \bb{C}$ of complex Dirac structures defines a GK structure precisely when the following three properties are satisfied:
\begin{enumerate}
\item Both Dirac structures define GC structures, which is to say that $L_{\cal{A}} \cap \overline{L}_{\cal{A}} = 0$ and $L_{\cal{B}} \cap \overline{L}_{\cal{B}} = 0$.
\item The complex Dirac structures
\[
L_{\sigma_{+}} = \frac{i}{2} (\overline{L}_{\cal{B}} - \overline{L}_{\cal{A}}), \qquad L_{\sigma_{-}} = \frac{i}{2} (\overline{L}_{\cal{B}} - L_{\cal{A}}),
\]
define holomorphic Poisson structures $(I_{+}, \sigma_{+}), (I_{-}, \sigma_{-})$ respectively.
\item For all nonzero $u \in L_{\cal{A}} \cap L_{\cal{B}}$, we have $\langle u, \overline u \rangle > 0$.
\end{enumerate}
\end{proposition}
The complex structures $I_\pm$ arising in the Proposition are indeed those of the original bihermitian data, and the Poisson tensors $\sigma_{\pm}$ are those first observed by Hitchin in \cite{MR2217300}. They share a common imaginary part which can be expressed in terms of the original bihermitian data $(g, I_{+}, I_{-})$. More precisely, the holomorphic Poisson structures are given by $\sigma_{\pm} = -\frac{1}{4}(I_{\pm}Q + iQ)$, where $Q = \frac{1}{2}[I_{-},I_{+}] g^{-1}$.
\subsection{Symplectic type}\label{symptype}
In this paper, we consider only GK structures of symplectic type, defined as follows:
\begin{definition}
A generalized K\"ahler structure $(\mathbb J_\cal{A},\mathbb J_\cal{B})$ is of \emph{symplectic type} if $\mathbb J_\cal{A}$ is gauge equivalent to a symplectic structure. That is, there is a closed 2-form $\beta$ such that
\[
e^{\beta} \mathbb J_\cal{A} e^{-\beta} = \begin{pmatrix}
0 & -F^{-1} \\
F & 0 \\
\end{pmatrix},
\]
where $F$ is a symplectic form.
\end{definition}
The $+i$-eigenbundle of such a $\mathbb J_\cal{A}$ is then
\begin{equation}\label{ella}
L_{\cal{A}} = \Gamma_{(\beta - iF)},
\end{equation}
which, using Proposition~\ref{diffdir}, leads to holomorphic Poisson structures given by
\begin{equation}\label{sumsigma}
L_{\sigma_{+}} =\frac{i}{2}(\overline{L}_{\cal{B}} - \Gamma_{\beta+iF}),
\qquad
L_{\sigma_{-}} =\frac{i}{2}(\overline{L}_{\cal{B}} - \Gamma_{\beta-iF}).
\end{equation}
Since the Dirac sum specializes to a gauge transformation~\eqref{gaugesum}, the above immediately implies that the two Poisson structures are in fact gauge equivalent by the action of the symplectic form:
\[
L_{\sigma_{+}} = e^{F} L_{\sigma_{-}}.
\]
If we unpack what this means in terms of the data $(I_{+}, I_{-}, Q, F)$ of Proposition~\ref{diffdir}, we arrive at the following two equations, first considered in \cite{gualtieri2010branes}:
\begin{align} \label{star1}
&I_{+} - I_{-} = QF, \\
&FI_{+} + I_{-}^{\ast}F = 0. \label{star2}
\end{align}
Due to the fact that Dirac sum by $\Gamma_{\beta\pm iF}$ is invertible, we see from~\eqref{sumsigma} that we may reconstruct the entire GK structure from $(L_{\sigma_{+}}, L_{\sigma_{-}}, F, \beta)$. For instance, the metric $g$ and $B$-field $b$ of the bihermitian geometry are given as follows:
\begin{equation}
g = -\frac{1}{2}F(I_{+} + I_{-}),\qquad \ b = \beta - \frac{1}{2}F(I_{+} - I_{-}), \label{metricF}
\end{equation}
and the Dirac structure $L_{\cal{B}}$ is given by
\begin{equation}
L_{\cal{B}} = e^{\beta + iF}(2i \overline{L}_{\sigma_{-}}). \label{LB}
\end{equation}
On the other hand, it is clear that equations \ref{star1} and \ref{star2} are not sufficient to guarantee that the data $(L_{\sigma_{+}}, L_{\sigma_{-}}, F, \beta)$ defines a GK structure: for example, if $I_{+} + I_{-}$ fails to be invertible, then we can see from equation \ref{metricF} that we will not get a metric. For this reason, we make the following definition:
\begin{definition}
A \emph{degenerate} GK structure of symplectic type consists of the data $(L_{\sigma_{+}}, L_{\sigma_{-}}, F, \beta)$ of two holomorphic Poisson structures $(I_{\pm}, \sigma_{\pm})$ and two closed $2$-forms $F, \beta \in \Omega^{2, cl}(M,\bb{R})$ such that
\[
L_{\sigma_{+}} = e^{F} L_{\sigma_{-}}.
\]
\end{definition}
\begin{example}\label{trivdeg}
Associated to any holomorphic Poisson structure $(I, \sigma)$ there is a canonical degenerate GK structure given by $(L_{\sigma}, L_{\sigma}, 0, 0)$.
\end{example}
Given a degenerate GK structure, we may construct complex Dirac structures $L_{\cal{A}}, L_{\cal{B}}$ according to equations \ref{ella} and \ref{LB} respectively. Then $L_{\cal{A}}$ defines a GC structure if and only if $F$ is non-degenerate (i.e. symplectic) and $L_{\cal{B}}$ defines a GC structure if and only if $I_{+} + I_{-}$ is invertible. In the case that both these conditions hold then we get a GK structure but with a metric that may be indefinite. Therefore, in order to get a genuine GK structure it is necessary to require that the symmetric tensor $g$ defined through equation \ref{metricF} is positive-definite, in which case we say that $F$ is \emph{positive}. This summarizes the content of the following theorem:
\begin{theorem} \cite[Theorem 6.2] {gualtieri2010branes} \label{GK=star}
There is a bijection between Generalized K\"ahler structures of symplectic type $(L_{\cal{A}}, L_{\cal{B}})$ and degenerate GK structures of symplectic type $(L_{\sigma_{+}}, L_{\sigma_{-}}, F, \beta)$ such that the $2$-form $F$ is positive.
\end{theorem}
We will find it useful to study the larger class of degenerate GK structures, applying the condition of positivity of the form $F$ when we want to return to the setting of GK geometry. Also, the action of $B$-field gauge transformations is particularly simple in the case of degenerate GK structures: it acts only (and transitively) on the closed two-form $\beta$:
\[
B : (L_{\sigma_{+}}, L_{\sigma_{-}}, F, \beta) \mapsto (L_{\sigma_{+}}, L_{\sigma_{-}}, F, \beta + B).
\]
In the remainder of this paper we will fix the gauge such that $\beta = 0$. Hence when speaking of degenerate GK structures, we will only specify the data $(L_{\sigma_{+}}, L_{\sigma_{-}}, F)$, or equivalently, $(I_{+}, I_{-}, Q, F)$.
The following is a useful result giving us another means of extracting the metric $g$ and the $B$-field $b$ from the data $(I_{+}, I_{-}, Q, F)$:
\begin{lemma} \label{metric11}
The Hermitian forms $\omega_{\pm} = g I_{\pm}$ associated to the bi-Hermitian data $(g, I_{\pm})$ coincide with the $(1,1)$ components of the symplectic form $F$ relative to the complex structures $I_\pm$:
\[
\omega_{\pm} = F^{(1,1)_{\pm}}.
\]
As a result, we obtain expressions for the metric and $B$-field
\[
g = - F^{(1,1)_{\pm}} I_{\pm},\qquad b = \mp F^{(2,0) + (0,2)_{\pm}} I_{\pm}.
\]
\begin{proof}
We prove the result for the $I_{-}$ complex structure; the other case is completely analogous. Starting with \eqref{metricF} and using~\eqref{star2}, we have
\[
\omega_{-} = gI_{-} = \frac{1}{2}F(1 - I_{+} I_{-}) = \frac{1}{2}(F + I_{-}^{\ast}FI_{-}) = F^{(1,1)_{-}}.
\]
The remaining statement is then another application of~\eqref{metricF}.
\end{proof}
\end{lemma}
\begin{remark}
The above Lemma holds true even in the case that $g$ and $F$ are degenerate.
\end{remark}
\section{Introduction to the problem: the K\"ahler case}\label{Donkahler}
It is fundamental to K\"ahler geometry that, given the underlying complex manifold, one can describe the K\"ahler metric locally by specifying a single real-valued function: the K\"ahler potential. For this reason, we may think of the complex moduli and the potential function as independent degrees of freedom comprising the K\"ahler structure.
The main problem we solve is to determine the analogous degrees of freedom inherent in Generalized K\"ahler geometry. More precisely, we give an answer to the following two questions in the setting of GK structures of symplectic type:
\begin{enumerate}
\item What is the holomorphic structure underlying a Generalized K\"ahler manifold?
\item What additional data is needed to specify the Riemannian metric, and how is it determined locally by a real-valued function?
\end{enumerate}
To understand the methods we use to solve these questions, it is helpful to revisit them in the usual K\"ahler case, where our approach coincides with a reformulation of K\"ahler geometry first considered by Donaldson in his study of the complex Monge-Amp\`{e}re equation \cite{donaldson2002holomorphic}.
Let $X = (M,I)$ be a complex manifold equipped with a K\"ahler metric $g$. Notice that this defines a GK structure of symplectic type where $I_{+} = I_{-} = I$ and $F = \omega=gI$ is the K\"ahler form. The Hitchin Poisson structure in this case vanishes: $Q = \frac{1}{2}[I,I] g^{-1} = 0.$ We begin by explaining how the K\"ahler structure defines a deformation of the holomorphic cotangent bundle $T^\ast X$.
First, cover the manifold $X$ by open sets $U_{i}$ on which a K\"ahler potential $K_{i} \in C^{\infty}(U_{i}, \bb{R})$ is chosen: $\omega = i \partial \overline{\partial} K_{i}$. On the double overlap $U_{i} \cap U_{j}$ we have $\partial \overline{\partial}(K_{i} - K_{j}) = 0$ and so $\mu_{ij} = i\partial(K_i-K_j)$ defines a closed holomorphic $(1,0)$-form. Therefore, we can define a holomorphic affine transformation on the double overlap:
\[
A_{ij} : T^{\ast}U_{i}|_{U_{i} \cap U_{j}} \longrightarrow T^{\ast}U_{j}|_{U_{i} \cap U_{j}},\qquad \alpha_{x} \mapsto \alpha_{x} + \mu_{ij}(x).
\]
In fact, $A_{ij}$ is also a symplectomorphism: if $\Omega_0$ is the canonical symplectic form on $T^\ast X$, then
\[
A_{ij}^{\ast} (\Omega_0) = \Omega_0 + \pi^{\ast} \mu_{ij}^{\ast}(\Omega_0) = \Omega_0 + \pi^{\ast} d\mu_{ij} = \Omega_0,
\]
where $\pi : T^{\ast}X \to X$ is the vector bundle projection. Since $A_{ij}$ defines a cocycle, we may define an affine bundle $Z$ modelled on $T^{\ast}X$ as follows:
\[
Z = (\coprod_{i} T^{\ast}U_{i}) /\sim, \ \ (\alpha_{x})_{i} \sim (A_{ij}(\alpha_{x}))_{j}.
\]
Then $Z$ inherits a symplectic form $\Omega$, making it into a holomorphic symplectic manifold. In fact, more is true: since the symplectic form on $T^{\ast}X$ is compatible with the additive structure, it follows that the form $\Omega$ is compatible with the action of $T^{\ast}X$ on $Z$, in the sense that the graph of the action map $A : T^{\ast}X \times_{X} Z \to Z$ in $(T^{\ast}X, \Omega_0) \times (Z, \Omega) \times (Z, - \Omega)$ is a holomorphic Lagrangian submanifold. This holomorphic symplectic affine bundle $(Z, \Omega)$ is what encodes the underlying holomorphic structure of the K\"ahler manifold as well as the K\"ahler class.
To see where the metric comes in, observe that the potentials $K_{i}$ provide us with a global section of $Z \to X$. More precisely, if we define $\cal{L}_{i} = - i \partial K_{i} : U_{i} \to T^{\ast}U_{i}$, then because
\[
A_{ij} \circ \cal{L}_{i}(x) = -i \partial K_{i} + \mu_{ij} = -i \partial K_{i} + i \partial(K_{i} - K_{j}) = \cal{L}_{j},
\]
we get a global section $\cal{L} : X \to Z$. This section fails to be holomorphic, but it has the interesting property of being Lagrangian with respect to $\Im(\Omega)$ and symplectic with respect to $\Re(\Omega)$. Indeed, computing the pullback we see that
\[
\cal{L}^{\ast}(\Omega) = \cal{L}_{i}^{\ast}(\Omega_0) = d(-i\partial K_{i}) = i \partial \overline{\partial}K_{i} = \omega,
\]
recovering the K\"ahler form and hence the metric $g$. Submanifolds such as the image of $\cal{L}$, which are Lagrangian with respect to $\Im(\Omega)$ and which provide sections of the projection $Z \to X$, are what we will come to call \emph{brane bisections}. They are what encode the extra data needed to recover the metric in GK geometry.
Let us note one more thing about the affine bundle $Z$: using the brane $\cal{L}$ we can define a diffeomorphism between the underlying smooth manifolds
\[
\psi : T^{\ast} X \to Z, \ \ \alpha_{x} \mapsto \alpha_{x} + \cal{L}(x).
\]
Using the above mentioned fact that the symplectic forms are compatible with the affine bundle structure we see that
\[
\psi^{\ast} \Omega = \Omega_0 + \pi^{\ast} \cal{L}^{\ast} \Omega = \Omega_0 + \pi^{\ast} \omega.
\]
In other words, the affine bundle $(Z, \Omega)$ is nothing but a \emph{twisted cotangent bundle} $(T^{\ast}X, \Omega_0 + \omega)$, and the brane bisection $\cal{L}$ corresponds simply to the zero section.
Having separated the K\"ahler degrees of freedom into the holomorphic moduli of $(Z,\Omega)$ and the choice of a smooth brane $\cal{L}$, it is natural to study the deformations of $\cal{L}$ keeping $(Z,\Omega)$ fixed. Any deformation $\cal{L}' : X \to Z$ of $\cal{L}$ is given by the graph of a $(1,0)$-form $\alpha$. Pulling back the symplectic form, we get
\[
\omega' := (\cal{L}')^{\ast} \Omega = \alpha^{\ast} (\Omega_0 + \pi^{\ast} \omega) = \omega + d\alpha.
\]
That $\cal{L}'$ is a brane imposes the condition that $\omega'$ is real; using the $\partial\overline\partial$-lemma, we conclude that $\omega' - \omega = i \partial \overline{\partial} f$, for a real-valued function $f$. In short, varying $\mathcal{L}$ with $(Z,\Omega)$ fixed is equivalent to varying $\omega$ within its K\"ahler class.
\begin{proposition}\cite[Section~2.]{donaldson2002holomorphic}
On a K\"ahler manifold, the cohomology class $[\omega]$ of the K\"ahler form determines a holomorphic symplectic affine bundle $(Z, \Omega)$ modelled on the cotangent bundle, and the metric $g$ determines a smooth section $\cal{L}$ of the bundle $Z$ which is symplectic for $\Re(\Omega)$ and Lagrangian for $\Im(\Omega)$. Conversely, this data $(Z,\Omega,\cal{L})$ uniquely determines the K\"ahler structure. Under this correspondence, deforming $\cal{L}$ is equivalent to varying $\omega$ within the K\"ahler class.
\end{proposition}
As we will see, the situation is much the same in the general case of GK geometry of symplectic type. We will again be able to encode the GK geometry in terms of a brane bisection in a holomorphic symplectic manifold. The crucial difference will be that the symplectic manifold, far from being a twisted cotangent bundle, is determined by the Hitchin Poisson structure according to the theory of symplectic Morita equivalence.
\section{Holomorphic symplectic Morita equivalence}\label{moreq}
In order to access the holomorphic structure underlying GK geometry, we need to begin with the underlying holomorphic Poisson structures $\sigma_{\pm}$. In this section, we recall the necessary tools from Poisson geometry.
A fundamental feature of Poisson geometry is that a Poisson structure $Q$ determines a (singular) foliation of the underlying manifold $M$ by \emph{symplectic leaves}. These leaves are equivalence classes of points related by Hamiltonian flows. Taking the quotient of $M$ by this equivalence relation defines a generalized space, which can be suitably upgraded to a differentiable stack. Viewed in this way, i.e., as stacks, Poisson manifolds inherit a more general notion of morphism than the usual Poisson maps.
Such generalized morphisms between the associated stacks may be conveniently described in terms of spans of Poisson maps, as follows. A generalized morphism between Poisson manifolds $(M_{1}, Q_{1})$ and $(M_{2}, Q_{2})$ is given by a symplectic manifold $(S, \omega)$, together with Poisson maps
\[
\xymatrix{ &(S, \omega)\ar[dl]\ar[dr] & \\(M_{1}, Q_{1}) & &(M_{2}, -Q_{2}) }
\]
satisfying a list of properties (see \cite{MR2166451} for more details). Note that a symplectic manifold $(S, \omega)$ equipped with a Poisson map to $(M, Q)$ is called a \emph{symplectic realization}, a concept introduced by Weinstein in \cite{weinstein1983local}. Poisson manifolds, spans of symplectic realizations, and isomorphisms between spans assemble into a $2$-category. The equivalences in this category are the main objects of interest in our study of GK geometry of symplectic type. These are known as \emph{Morita equivalences} and were first introduced by Xu \cite{xu1991morita} who gave the following definition:
\begin{definition}[Morita equivalence] A Morita equivalence between Poisson manifolds $(M_{1}, Q_{1})$ and $(M_{2}, Q_{2})$ is a symplectic manifold $(S, \omega)$ together with a pair of surjective submersions $\pi_{1} : S \to M_{1}, \pi_{2} : S \to M_{2}$ with connected and simply-connected fibres such that
\begin{enumerate}
\item $\pi_{1}$ is Poisson and $\pi_{2}$ is anti-Poisson;
\item the vertical distributions $\text{ker}(d\pi_{1})$ and $\text{ker}(d\pi_{2})$ are symplectically orthogonal;
\item $\pi_{1},\pi_2$ are \emph{complete} in the sense that the pullback of any complete Hamiltonian vector field is complete.
\end{enumerate}
We view this as a morphism from $(M_{2}, Q_{2})$ to $(M_{1}, Q_{1})$.
\end{definition}
The identity Morita self-equivalence $(\Sigma(M), \Omega)$ of a Poisson manifold $(M, Q)$ comes equipped with an isomorphism $\Sigma(M) \circ \Sigma(M) \cong \Sigma(M)$, reflecting the fact that composition with the identity is trivial up to isomorphism. This endows the space $\Sigma(M)$ with the structure of a Lie groupoid, making it into a \emph{symplectic groupoid}, a notion due to Karas\"ev, Weinstein and Zakrzewski \cite{MR854594,weinstein1987symplectic,MR1081010,MR1081011}.
\begin{definition}[Symplectic groupoid] A symplectic groupoid $(\cal{G} \rightrightarrows M, \Omega)$ is a Lie groupoid $\cal{G}$ equipped with a symplectic structure $\Omega$ such that the graph of the multiplication map is a Lagrangian submanifold of $\cal{G} \times \bar{\cal{G}} \times \bar{\cal{G}} $, where $\bar{\cal{G}}$ denotes $\cal{G}$ with the opposite symplectic form $-\Omega$.
\end{definition}
\begin{remark}
A symplectic form on $\cal{G}$ satisfying the above property is called a \emph{multiplicative form}. Equivalently, $\Omega$ satisfies the following equation on the fibre product $\cal{G} \times_{M} \cal{G}$
\[
m^{\ast} \Omega = p_{1}^{\ast} \Omega + p_{2}^{\ast} \Omega,
\]
where $m : \cal{G} \times_{M} \cal{G} \to \cal{G}$ is the groupoid multiplication, and $p_{i} : \cal{G} \times_{M} \cal{G} \to \cal{G}$ are the two projections.
\end{remark}
Given a Poisson manifold, the existence of an identity Morita self-equivalence is governed by whether a certain Lie algebroid determined by the Poisson structure is integrable to a Lie groupoid.
For this reason we call $(\Sigma(M), \Omega)$ an \emph{integration} of the Poisson manifold, and we call the Poisson manifold \emph{integrable} if such an integration exists. In the category of smooth Lie groupoids this problem was solved by Crainic and Fernandes \cite{crainic2003integrability}, inspired by the work of Cattaneo and Felder \cite{cattaneo2001poisson}. In the holomorphic category, this problem was studied by Laurent-Gengoux, Sti\'{e}non and Xu \cite{laurent2009integration}, who showed that a holomorphic Poisson manifold is integrable if and only if its underlying real or imaginary part is an integrable real Poisson structure. We assume that all our Poisson manifolds are integrable. A given Poisson manifold may have several different integrations, but there is a unique \emph{source simply connected} one, characterized by the fact that the fibers of the source map are connected and simply connected \cite[Proposition
6.8]{MR2012261}; this is the identity Morita self-equivalence $\Sigma(M)$ above, and we refer to it as the \emph{Weinstein groupoid}.
Let $(S, \omega)$ be a Morita equivalence between Poisson manifolds $(M_{1}, Q_{1})$ and $(M_{2}, Q_{2})$. Since composition with the identity Morita equivalence is trivial up to isomorphism, we obtain isomorphisms $\Sigma(M_{1}) \circ S \cong S$ and $S \circ \Sigma(M_{2}) \cong S$. These define actions of the groupoids $\Sigma(M_{i})$ on the space $S$, endowing $S$ with the structure of a \emph{bi-principal groupoid bi-bundle} for the pair $(\Sigma(M_{1}), \Sigma(M_{2}))$. Most importantly for our purposes, these actions are symplectic in the sense that the graphs of the action maps are Lagrangian submanifolds in the product $\Sigma(M_{i}) \times S \times \bar{S}$, where $\bar{S}$ denotes $S$ with the opposite symplectic form. In this way, we make contact with the theory of Morita equivalence for symplectic Lie groupoids. The following theorem of Xu tells us that the notions of Morita equivalence for symplectic groupoids and Poisson manifolds agree.
\begin{theorem}\cite[Theorem 3.2]{xu1991morita}
Two integrable Poisson manifolds $(M_{1}, Q_{1})$ and $(M_{2}, Q_{2})$ are Morita equivalent if and only if their Weinstein groupoids $\Sigma(M_{1}), \Sigma(M_{2})$ are (symplectically) Morita equivalent.
\end{theorem}
\subsection{Examples} We end this section with examples of symplectic groupoids and Morita equivalences.
\begin{example}[Zero Poisson structure]
The Weinstein groupoid integrating the zero poisson structure on a manifold $M$ is given by the cotangent bundle $T^{\ast}M$ with its canonical symplectic form and groupoid multiplication given by fibrewise addition.
\end{example}
\begin{example}[Symplectic manifold] \label{symplectic manifold groupoid}
The Weinstein groupoid of a simply connected symplectic manifold $(M, \omega)$ is given by the \emph{pair groupoid} $Pair(M) = (M \times M, \omega \oplus -\omega)$. The source and target maps are given by the two projections and the multiplication is given by the formula $(a,b)*(b,c) = (a,c)$. If $M$ is not simply connected, the pair groupoid still provides an integration of $(M,\omega)$, but now the Weinstein groupoid is given by the fundamental groupoid $\Pi(M)$, with symplectic form $\Omega = t^{\ast}\omega - s^{\ast}\omega$, where $t$ and $s$ are the target and source maps, respectively.
\end{example}
\begin{example}[Lie-Poisson structure]
Let $\mathfrak{g}$ be a Lie algebra. Its dual $\mathfrak{g}^{\ast}$ has a natural Poisson bracket extending the Lie bracket on $\mathfrak{g}$.
If $G$ is a Lie group integrating $\mathfrak{g}$ then $T^{*}G$ with its canonical symplectic form gives a symplectic groupoid integrating $\mathfrak{g}^{\ast}$. The source and target maps are given by left and right trivialization respectively, and the multiplication is given by the following formula:
\[
\xi_{g_{1}} \star \eta_{g_{2}} := d(R_{g_{2}^{-1}})^{\ast}_{g_{1}g_{2}}(\xi_{g_{1}}) = d(L_{g_{1}^{-1}})^{\ast}_{g_{1}g_{2}}(\eta_{g_{2}}),
\]
where $R_{g}$ and $L_{g}$ are right and left multiplication, respectively. By using left or right trivialization we can see that this groupoid is isomorphic to the action groupoid induced by the coadjoint action of $G$ on $\mathfrak{g}^{\ast}$. See \cite{coste1987groupoides} for further details.
\end{example}
\begin{example}\label{affpois}
As a concrete instance of the previous example, consider the linear Poisson structure on $\bb{C}^{2}$ given by $\Pi = x \partial_{x} \wedge \partial_{y}$. This is a Poisson structure corresponding to the Lie algebra $\frak{g} = \bb{C}x \oplus \bb{C}y$ with bracket $[x,y] = x$. The Lie group integrating this algebra is the group $G$ of affine transformations of $\bb{C}$ (or rather the universal cover of this). Therefore we can obtain the symplectic groupoid integrating our Poisson structure as the action groupoid for the coadjoint action of $G$ on $\frak{g}^{*}$.
However, in the present case a more direct approach to integration is possible: note that the Hamiltonian vector fields corresponding to the coordinate functions are given by $X_{x} = x \partial_{y}$ and $X_{-y} = x \partial_{x}$. The vector field $x \partial_{x}$ generates the multiplicative action of $\bb{C}$: $(a,x) : x \mapsto e^{a}x$, and the vector field $x \partial_{y}$ generates the additive action of $\bb{C}$ rescaled by a factor of $x$: $(b,y): y \mapsto y + xb$. Combining the two actions we get the action groupoid $\cal{G} = \bb{C}^{2} \ltimes \bb{C}^{2}$, with coordinates $(a,b,x,y)$, with the source and target maps given respectively by
\[
s(a,b,x,y) = (x,y) \qquad t(a,b,x,y) = (e^{a}x, y + xb),
\]
and with the multiplication given by
\[
(a_{1}, b_{1}, x_{1}, y_{1}) \star (a_{2}, b_{2}, x_{2}, y_{2}) = (a_{1} + a_{2}, b_{1} e^{a_{2}} + b_{2}, x_{2}, y_{2}).
\]
Note that indeed $\cal{G} = \widetilde{\text{Aff}(\bb{C})} \ltimes \bb{C}^{2}$: the action groupoid for the action of (the universal cover of) the affine group on the dual of its Lie algebra.
To compute the symplectic form on $\cal{G}$ we make use of the following fact: in the case of an invertible Poisson structure with corresponding symplectic form $\omega$, the symplectic form on the integrating groupoid is given by $\Omega = t^{\ast} \omega - s^{\ast} \omega$ (see Example~\ref{symplectic manifold groupoid} above). In the present case the Poisson structure is invertible on a dense subset, corresponding to the meromorphic symplectic form
\[
\omega = \frac{1}{x} dx \wedge dy.
\]
Hence, it follows that the symplectic form on the groupoid $\cal{G}$ is given by the same formula:
\begin{align*}
\Omega &= t^{\ast} \omega - s^{\ast} \omega \\
&= \frac{1}{e^{a}x}d(e^{a}x)\wedge d(y + xb) - \frac{1}{x} dx \wedge dy \\
&= da \wedge d(y + xb) - db \wedge dx.
\end{align*}
See \cite{radko2006picard} for further discussion of this example in the smooth category.
\end{example}
We now consider examples of Morita equivalences.
\begin{example}
Recall from the discussion above that the Weinstein groupoid of a Poisson manifold $(M,Q)$ provides the trivial Morita self-equivalence of $Q$. Therefore all of the above examples of groupoids also give examples of Morita equivalences.
\end{example}
\begin{example}
Let $S_{1}$ and $S_{2}$ be simply connected symplectic manifolds. Then $S_{1} \leftarrow S_{1} \times \bar{S_{2}} \to S_{2}$ defines a Morita equivalence. More generally, two symplectic manifolds are Morita equivalent if and only if they have the same fundamental group. See \cite{xu1991morita} for details.
\end{example}
\begin{example}
Morita equivalences can be composed with Poisson diffeomorphisms. Let $\sigma : M_{2} \to M_{1}$ be a Poisson diffeomorphism and let $M_{2} \xleftarrow{\pi_{2}} S \xrightarrow{\pi_{3}} M_{3}$ be a Morita equivalence. Then we get the following Morita equivalence
\[
\xymatrix{ &S\ar[dl]_-{\sigma \circ \pi_{2}}\ar[dr]^-{\pi_{3}} & \\M_{1} & &M_{3} }
\]
In particular, by composing Poisson diffeomorphisms with the trivial Morita equivalence we get a map from Poisson diffeomorphisms to Morita equivalences.
\end{example}
\begin{example}\label{btrans}
Let $(M,Q)$ be a Poisson manifold. If $B \in \Omega^{2, cl}(M)$ is a closed $2$-form such that $id + B \circ Q: T^{\ast}M \to T^{\ast}M$ is invertible then we can define a new Poisson structure
\[
Q^{B} := Q \circ (id + B \circ Q)^{-1},
\]
which is called the \emph{$B$-field transform} of $Q$. Bursztyn and Radko have shown in \cite{bursztyn2003gauge} that in this case $(M,Q)$ and $(M, Q^{B})$ are Morita equivalent in the following way
\[
\xymatrix{ &(\Sigma(M), \Omega + t^{\ast}B)\ar[dl]_-t\ar[dr]^-s & \\(M, Q^{B}) & &(M, Q) }
\]
where $(\Sigma(M), \Omega)$ is the Weinstein groupoid of $(M,Q)$ and $s$ and $t$ are the source and target maps, respectively. In particular, if we apply this construction to the zero Poisson structure $(M, Q= 0)$ then we get a Morita self-equivalence given by $(T^{\ast}M, \Omega_0 + p^{*}B)$, where $\Omega_0$ is the canonical symplectic form and $p : T^{\ast}M \to M$ is the projection.
\end{example}
\section{Generalized K\"ahler metrics as brane bisections}\label{branbi}
For a GK structure of symplectic type, we have seen that there are two underlying holomorphic Poisson structures $\sigma_\pm$. These Poisson manifolds are not biholomorphically equivalent in general; in fact, the underlying complex manifolds $(M, I_\pm)$ may not even be isomorphic. But, in analogy to Example~\ref{btrans}, it was shown in~\cite{bailey2016integration} that they are actually holomorphically Morita equivalent.
\begin{proposition}\cite[Proposition 6.4]{bailey2016integration} \label{Star->ME}
Let $(I_{+}, I_{-}, Q, F)$ be a degenerate GK structure of symplectic type, and let $\sigma_{\pm} = -\frac{1}{4}(I_{\pm}Q + iQ)$ denote the corresponding holomorphic Poisson structures on the complex manifolds $X_{\pm} := (M, I_{\pm})$. Then $(X_{+}, \sigma_{+})$ and $(X_{-}, \sigma_{-})$ are holomorphically symplectically Morita equivalent in a canonical way.
\end{proposition}
We recall the construction of the canonical Morita equivalence as it will be one of our main objects of study. It can be broken up into two steps as follows:
\begin{enumerate}
\item Integrate $(X_{-}, \sigma_{-})$ to its holomorphic symplectic groupoid $(\Sigma(X_{-}), \Omega_{-})$.
\item Deform $\Sigma(X_-)$ to the holomorphic symplectic manifold
\begin{equation}\label{omegaf}
(Z,\Omega) = (\Sigma(X_-), \Omega_{-} + t^{*}F),
\end{equation}
where $t$ is the target map of $\Sigma(X_-)$.
\end{enumerate}
Note that by modifying the symplectic form, we are also modifying the underlying complex structure. To see this, recall that a $(2,0)$-form can be written as $\Omega = \omega I + i \omega$, where $I$ is the complex structure and $\omega$ is real. Therefore, specifying the form $\Omega$ and insisting that it be holomorphic symplectic is enough to specify a new complex structure. With respect to this deformed holomorphic symplectic structure, the source map $s$ remains holomorphic and anti-Poisson, while the target map $t$ is now holomorphic and Poisson as a map from $(Z,\Omega)$ to $(X_{+}, \sigma_{+})$. We use the notation $\pi_{+} = t$ and $\pi_{-} = s$ for these maps from the modified domain $(Z,\Omega)$. Proposition~\ref{Star->ME} then states that the diagram
\[
\xymatrix{ &(Z,\Omega)\ar[dl]_{\pi_+}\ar[dr]^{\pi_-} & \\(X_{+},\sigma_{+}) & &(X_{-}, \sigma_{-}) }
\]
defines a Morita equivalence between $(X_{-},\sigma_{-})$ and $(X_{+}, \sigma_{+})$.
This result shows that underlying any GK structure of symplectic type is a holomorphic symplectic Morita equivalence between holomorphic Poisson structures. In order to recover the GK structure from this Morita equivalence, we need some real (i.e. non-holomorphic) data. To see how this arises, observe that since $Z$ was obtained by deforming the holomorphic symplectic groupoid $\Sigma(X_-)$, it contains a distinguished submanifold $\cal{L}$ coming from the identity bisection. This submanifold is neither holomorphic nor Lagrangian in $(Z,\Omega)$. It is, however, characterized by the following two properties:
\begin{enumerate}
\item The submanifold $\cal{L}\subset Z$ is a smooth section of both $\pi_{-}$ and $\pi_{+}$; in other words, it is a smooth \emph{bisection}. This defines a diffeomorphism between the underlying smooth manifolds of $X_+$ and $X_-$, which is required because a GK structure involves two complex structures living on the same real manifold.
\item The bisection $\cal{L}$ is Lagrangian with respect to $\omega = \Im(\Omega)$: from the point of view of generalized complex geometry, this is known as a \emph{brane} in $(Z, \Omega)$. Notice that~\eqref{omegaf} implies that $\Omega|_{\cal{L}}=F$, so that we recover the real closed 2-form $F$.
\end{enumerate}
We record the above properties satisfied by $\cal{L}$ in the following definition.
\begin{definition}
A \emph{brane bisection} in the Morita equivalence $(Z,\Omega)$ is a smooth submanifold of $Z$ which is Lagrangian for $\Im(\Omega)$ and which is a section of both $\pi_{-}$ and $\pi_{+}$.
\end{definition}
\begin{theorem} \label{main}
A degenerate GK structure of symplectic type $(I_{+}, I_{-}, Q, F)$ is equivalent to a holomorphic symplectic Morita equivalence with brane bisection $(Z, \Omega, \cal{L})$ between the holomorphic Poisson structures $\sigma_{\pm} = -\frac{1}{4}(I_{\pm}Q + iQ)$.
\begin{proof}
One direction of the theorem follows from Proposition~\ref{Star->ME} and the above remarks. For the converse direction let us start with a holomorphic symplectic Morita equivalence with brane bisection $(Z, \Omega = B + i\omega, \cal{L})$ and construct a degenerate GK structure. Such a Morita equivalence goes between two holomorphic Poisson manifolds $(X_{\pm}, \sigma_{\pm} = -\frac{1}{4}(I_{\pm} Q_{\pm} + iQ_{\pm}))$. Now observe that $(Z, \omega)$ defines a real smooth symplectic Morita equivalence between the real Poisson structures $(X_{+}, Q_{+})$ and $(X_{-}, Q_{-})$. The fact that $\cal{L}$ is a brane bisection means precisely that it defines a Lagrangian bisection in this Morita equivalence, and therefore that it defines a Poisson diffeomorphism between the two Poisson structures. The upshot of this is that when we use the bisection to identify $X_{-} = \cal{L} = X_{+}$, then $Q_{-} = Q_{+}$. We denote this real Poisson structure by $Q$ and the underlying smooth manifold by $M$.
Now that we have made these identifications $(Z, \omega)$ defines a self-Morita equivalence of $(M, Q)$ and $\cal{L}$ is a Lagrangian bisection inducing the identity diffeomorphism on $M$. The symplectic groupoid $(\Sigma(X_{-}), \Omega_{-} = B_{-} + i \omega_{-})$ of $(X_{-}, \sigma_{-})$ acts principally on $(Z, \Omega)$, and so the brane $\cal{L}$ induces the following isomorphism of smooth symplectic Morita equivalences
\[
\phi : (\Sigma(X_{-}), \omega_{-}) \to (Z, \omega), \ g \mapsto \lambda(t(g)) \ast g,
\]
where $\lambda : M \to Z$ is the section of $\pi_{-}$ induced by $\cal{L}$. Using $\phi$ to compare the real parts of $\Omega_{-}$ and $\Omega$ we see that
\begin{equation}
\phi^{\ast}( B) = B_{-} + t^{\ast} (B|_{\cal{L}}). \label{B-eq}
\end{equation}
Let $F := B|_{\cal{L}}$. Then we see that under this isomorphism
\[
\phi^{\ast} (\Omega) = \Omega_{-} + t^{\ast}(F).
\]
On the manifold $M$, we now have the data of the two complex structures $I_{\pm}$ coming from the spaces $X_{\pm}$, the real Poisson structure $Q$, and a closed 2-form $F$. We then show that $(I_{+}, I_{-}, Q, F)$ satisfy equations \ref{star1} and \ref{star2}. For simplicity, we use $\phi$ to identity $(Z, \omega)$ with $(\Sigma(X_{-}), \omega_{-})$. On this space, we have complex structures $I$ and $I_-$ coming from $Z$ and $\Sigma(X_{-})$, respectively. Note that the complex structure $I$ satisfies
\[
t_{*} I = I_{+} t_{*},\qquad s_{*} I = I_{-} s_{*}.
\]
In order to verify equations \ref{star1} and \ref{star2}, we show that they hold for $(I, I_{-}, \omega^{-1}, t^{*}(F))$ on $\Sigma(X_{-})$ and therefore on $M$ by push-forward. First we note that since $B = \omega I$ and $B_{-} = \omega I_{-}$, we have from equation \ref{B-eq} that $\omega (I - I_{-}) = t^{\ast} (F)$, which we can rearrange to equation \ref{star1}:
\[
I - I_{-} = \omega^{-1} t^{\ast} (F).
\]
Applying the target projection we get
\[
(I_{+} - I_{-}) t_{*} = t_{*} \omega^{-1} t^{*} F t_{*} = Q F t_{*},
\]
which implies equation \ref{star1} on $M$ since $t_{*}$ is surjective. Note that the $2$-form $t^{*}(F)$ gives rise to the map $t^{\ast}Ft_{\ast}:T \to T^{\ast}$. Equation \ref{star2} on the groupoid follows from a direct computation:
\[
I^{*} t^{*} (F) + t^{*} (F) I_{-} = I^{*} \omega (I - I_{-}) + \omega (I - I_{-}) I_{-} =0,
\]
where we have used the identities $t^{*}(F) = \omega(I - I_{-})$ and $I^{*} \omega = \omega I$. But this implies the corresponding equation on $M$ since
\[
t^{*} (I_{+}^{*} F + F I_{-} )= I^{*} t^{*}( F) + t^{*}( F) I_{-} = 0,
\]
and $t^{*}$ is injective on differential forms. This establishes equations \ref{star1} and \ref{star2} on $M$. It is clear that the two constructions outlined are inverse to each other, so we get the desired equivalence.
\end{proof}
\end{theorem}
Theorem~\ref{main} answers the main questions we raised in Section~\ref{Donkahler}, identifying the Morita equivalence $(Z,\Omega)$ as the underlying holomorphic data of a GK manifold of symplectic type, and showing how the additional data of a brane bisection $\cal{L}$ specifies the GK metric. In Section~\ref{secpot}, we explain how $\cal{L}$ may be locally given by a real-valued potential function.
\subsection{Induced metric}
We now discuss how the metric and $B$-field of the GK structure arise from the holomorphic symplectic Morita equivalence with brane bisection $(Z, \Omega, \cal{L})$. From the equivalence established in Theorem~\ref{main}, we know that a Morita equivalence with brane bisection gives rise to a degenerate GK structure, and from this we can extract the data of a (possibly degenerate) metric and $B$-field. In this section, we detail how to determine the metric and $B$-field directly from the geometry of the Morita equivalence.
Given a real $2$-form $F$ and a complex structure $I$, the tensor $FI$ decomposes into symmetric and anti-symmetric parts: $FI = S + A$, where
\[
S = F^{(1,1)}I,\qquad A = F^{(2,0) + (0,2)}I.
\]
Given a Morita equivalence with brane bisection $(Z, \Omega, \cal{L})$, we have two induced complex structures $I_{\pm}$ on the brane, as well as the real 2-form $F = \Omega|_\cal{L}$. Therefore the tensors $FI_{\pm}$ give rise to symmetric tensors $S_{\pm} = F^{(1,1)_{\pm}} I_{\pm}$ and anti-symmetric tensors $A_{\pm} = F^{(2,0) + (0,2)_{\pm}}I_{\pm}$.
From the theory of GK structures we know to expect that $S_{+} = S_{-}$ and $A_{+} = - A_{-}$. Indeed, from Lemma~\ref{metric11} we know that $g = - S_{\pm}$ and $b = \mp A_{\pm}$. Here we explain these facts directly.
\begin{proposition}
Let $(Z, \Omega, \cal{L})$ be a holomorphic symplectic Morita equivalence with brane bisection, let $I_{\pm}$ denote the complex structures induced on the brane, and let $F = \Omega|_{\cal{L}}$ denote the real $2$-form on the brane obtained by pulling back the symplectic form. If $S_{\pm} = F^{(1,1)_{\pm}} I_{\pm}$ and $A_{\pm} = F^{(2,0) + (0,2)_{\pm}}I_{\pm}$, then
\[
S_{+} = S_{-} \ \ \text{and } \ A_{+} = - A_{-}.
\]
\begin{proof}
Let $\Omega = B + i \omega $ be the holomorphic symplectic form on $Z$, so that $F = B|_{\cal{L}}$. Writing $A_{\pm}$ and $S_{\pm}$ as the (anti-)symmetrizations of $FI_{\pm}$, we have
\[
2S_{\pm}(u,v) = B(I_{\pm}u, v) + B(I_{\pm}v,u),\qquad 2A_{\pm}(u,v) = B(I_{\pm}u, v) - B(I_{\pm}v,u),
\]
for $u,v \in T\cal{L}$. Using the two direct sum decompositions of $TZ$ along $\cal{L}$ determined by the fibres of the projections $\pi_{\pm}$,
\[
TZ|_{\cal{L}} = T\cal{L} \oplus K_{\pm},\qquad K_{\pm} = \text{ker}(d \pi_{\pm}),
\]
we can write the following expressions for $I_{\pm}$:
\[
I u = I_{\pm}u + k_{\pm}u,\qquad I_{\pm}u \in T\cal{L}, \ k_{\pm}u \in K_{\pm},
\]
for $u \in T\cal{L}$. Then using the fact that $K_{\pm}$ are $\omega$-symplectic orthogonal, $T\cal{L}$ is $\omega$-Langrangian, and $\omega I = I^{*} \omega$, we get the following identity
\begin{align*}
0 &=\omega(k_+u,k_-v)= \omega(Iu - I_{+}u, Iv - I_{-} v)
= \omega(Iv, I_{+}u) - \omega(Iu, I_{-}v),
\end{align*}
for $u, v \in T\cal{L}$. Using the relation $B = \omega I$, this gives the following identity:
\[
B(I_{+}u, v) = B(I_{-}v, u),
\]
which implies the two identities above for $S_{\pm}$ and $A_{\pm}$.
\end{proof}
\end{proposition}
The upshot of the present discussion is as follows: when we pullback the holomorphic $(2,0)$-form $\Omega$ along a brane we get a real $2$-form $F$ which is no longer $(2,0)$ with respect to either of the induced complex structures $I_{\pm}$. As such, it gives rise to symmetric and anti-symmetric tensors on the brane. The fact that $F$ was obtained by pulling back the symplectic form of a holomorphic Morita equivalence implies that we end up with a single symmetric and anti-symmetric tensor; these give rise to the metric and $B$-field of the GK structure. The positivity or non-degeneracy of the metric arises from the configuration of the brane in the Morita equivalence. For example, $g$ is non-degenerate if and only if $I(T\cal{L}) \cap K = 0 = I(T\cal{L}) \cap T\cal{L} $, where $K$ is the average of $K_{\pm}$ with respect to $T\cal{L}$. Indeed, the condition $I(T\cal{L}) \cap T\cal{L} = 0$ is equivalent to invertibility of $F$ and $I(T\cal{L}) \cap K = 0$ is equivalent to invertibility of $I_{+} + I_{-}$. Recall that these two conditions imply that we get a GK structure with non-degenerate but possibly indefinite metric.
\subsection{Examples}
We end this section by providing examples of generalized K\"ahler structures of symplectic type and their corresponding holomorphic symplectic Morita equivalences with brane bisection.
\begin{example}[Holomorphic Poisson structure]\label{integtriv}
Let $(M, I, \sigma)$ be a Poisson manifold, and let $Q = -4 \Im (\sigma)$. As in Example~\ref{trivdeg}, the associated degenerate GK structure is given by the data $(I, I, Q, 0)$. The associated Morita equivalence with brane bisection is given by the holomorphic symplectic groupoid $(\Sigma(X), \Omega)$ integrating $\sigma$, viewed as the trivial Morita self-equivalence, with brane bisection given by the identity bisection $\epsilon$.
\end{example}
\begin{example}[K\"ahler Manifold]\label{kahman}
A K\"ahler manifold $(M, I, g, \omega)$ defines a generalized K\"ahler structure of symplectic type for which $I_{\pm} = I$, $Q = 0$, and $F = \omega$. Therefore the Morita equivalence is given by the holomorphic cotangent bundle of $X = (M, I)$ twisted by $\omega$:
\[
(Z, \Omega) = (T^{\ast}X, \Omega_0 + \pi^{\ast}\omega),
\]
where $\pi: T^{*}X \to X$ is the projection and $\Omega_0$ is the canonical symplectic form. The brane bisection $\cal{L}$ is given by the zero section viewed inside of $Z$. As discussed in Section~\ref{Donkahler}, $(Z, \Omega)$ is an affine bundle modelled on the holomorphic cotangent bundle of $X$, and the brane bisection is both symplectic with respect to $\Re(\Omega)$ and Lagrangian with respect to $\Im(\Omega)$. Conversely, any such affine bundle $\cal{A} \to X$ gives a Morita equivalence and any section $\cal{L}$ of $\pi = \pi_{+} = \pi_{-}$ which is symplectic for $\Re(\Omega)$ and Lagrangian for $\Im(\Omega)$ automatically defines a non-degenerate bilinear form on $M$. This is due to the fact that $K = \text{ker}(d \pi)$ is a complex Lagrangian, which implies that $I(T\cal{L}) \cap \text{ker} (d\pi) = 0$.
\end{example}
\begin{example}[Hyper-K\"ahler Manifold] \label{hyperKahler}
Let $(M, I, J, K, g)$ be a hyper-K\"ahler structure, and let
\[
(\omega_{I}, \omega_J, \omega_K) = (gI, gJ, gK)
\]
be the corresponding triple of K\"ahler forms. This data defines the GK structure of symplectic type
\[
(I_{+}, I_{-}, Q, F) = (I, J, \omega_{K}^{-1}, \omega_{I} + \omega_{J}).
\]
The induced holomorphic Poisson structures are non-degenerate, with corresponding holomorphic symplectic forms
\[
\Omega_{+} = \omega_{J} + i \omega_{K},\qquad \Omega_{-} = -\omega_{I} + i \omega_{K}.
\]
Let $X_{\pm} = (M, I_{\pm})$. Then, assuming that $M$ is simply-connected (or using a non source-simply connected integration), we have a holomorphic Morita equivalence given by
\[
(Z, \Omega) = (X_{+}, \Omega_{+}) \times (X_{-}, -\Omega_{-}),
\]
with $\pi_{+}$ and $\pi_{-}$ given by the projections onto the first and second factors respectively. The brane bisection $\cal{L}$ is then given by the diagonally embedded copy of $M$:
\[
\cal{L} = \{ (m, m) \ | \ m \in M \} \subseteq Z.
\]
\end{example}
\section{The generalized K\"ahler potential}\label{secpot}
In this section, we discuss an application of our main result to the problem of locally describing a generalized K\"ahler structure in terms of a real-valued function analogous to the K\"ahler potential. There is evidence in the physics literature for the existence of such a \emph{generalized K\"ahler potential} (see for example \cite{zabzine2009generalized, lindstrom2007generalized, hull2012generalized, lindstrom2007potential}). However, all previous constructions only work in a neighbourhood where the Poisson structure $Q$ is regular (i.e. has constant rank). As we will see, by viewing a GK structure as a brane in a holomorphic symplectic Morita equivalence, one can describe the geometry in terms of a real-valued function even in a neighbourhood where the Poisson structure changes rank. The fundamental reason for this is that the brane $\cal{L}$ is a Lagrangian submanifold of the real symplectic manifold $(Z, \Im(\Omega))$, and so may be described using a generating function.
Our generalized K\"ahler potential is obtained in the following way. Let $(Z, \Omega, \cal{L})$ be a holomorphic symplectic Morita equivalence with brane bisection. Choose a \emph{Darboux chart} of $(Z, \Omega)$, i.e. a local holomorphic symplectomorphism
\[
(T^{\ast}L, \Omega_0) \to (Z, \Omega),
\]
where $L\subset Z$ is a holomorphic Lagrangian submanifold and $\Omega_0$ is the canonical holomorphic symplectic form on the holomorphic cotangent bundle $T^*L$.
Using the identification of $T^*L$ with the bundle of $(1,0)$-forms, the brane $\cal{L}$ can be viewed as a smooth submanifold of $T^{\ast}_{1,0}L$. We require that the Darboux chart be \emph{generic} in the sense that $\cal{L}$ is the graph of a $(1,0)$-form $\eta \in \Omega^{1,0}(L)$. Then, since
\[
d \eta = \eta^{\ast} \Omega_0 = \Omega|_{\cal{L}} = F
\]
is a real closed 2-form, we have that $d( \Im (\eta)) = 0$, which locally implies that $\Im(\eta) = -\frac{1}{2}dK$, for $K \in C^{\infty}(L,\bb{R})$ a smooth real-valued function on $L$, uniquely determined up to a real constant. And since $\eta$ is a $(1,0)$-form, its real part is determined by its imaginary part, so that $\eta = - i \partial K$ and, finally,
\[
F = i \partial \bar \partial K.
\]
Therefore we call $K$ the \emph{generalized K\"ahler potential} for the GK structure, and we record these observations in the following theorem.
\begin{theorem} \label{GK potential}
Let $(Z, \Omega, \cal{L})$ be a holomorphic symplectic Morita equivalence with brane bisection. Given a generic Darboux chart $(T^{\ast}L, \Omega_0) \to (Z, \Omega)$, there is a smooth real-valued function $K \in C^{\infty}(L, \bb{R})$, unique up to an additive real constant, such that $\cal{L} = Gr(- i \partial K)$. Furthermore, the real $2$-form $F = \Omega|_{\cal{L}}$ is given by $F = i \partial \bar \partial K$. We call $K$ the generalized K\"ahler potential.
\end{theorem}
\subsection{Examples}
In the first example, we see that the generalized K\"ahler potential does indeed specialize to the classical K\"ahler potential.
\begin{example}[K\"ahler potential] \label{Kahlerex} Suppose the Poisson structure vanishes: $Q = 0$. Then a GK structure of symplectic type is automatically K\"ahler, and as we saw in Example~\ref{kahman}, the Morita equivalence is given by a twist of the holomorphic cotangent bundle by the K\"ahler form $\omega$:
\[
(Z, \Omega) = (T^{\ast}X, \Omega_0 + \pi^{\ast}\omega),
\]
and the brane bisection $\cal{L}$ is given by the zero section of $T^{\ast}X$ viewed as a brane in $Z$. Now choose a local holomorphic Lagrangian section $\lambda$ of $\pi: Z \to X$ with domain $U\subset X$ and image $L$. The projection $\pi$ defines a holomorphic isomorphism from $L$ to $U$, and so we can use the natural action of $T^{\ast}X$ on $Z$ to define the following holomorphic Darboux chart:
\[
(T^{\ast}U, \Omega_0) \to (Z, \Omega), \ \alpha_{x} \mapsto \alpha_{x} + \lambda(x),
\]
where $\alpha_{x} \in T^{\ast}_{x}U$. Theorem \ref{GK potential} then tells us that $\cal{L} = Gr(-i \partial K)$ and $\omega = i \partial \bar \partial K$, for a real-valued function $K \in C^{\infty}(U, \bb{R})$, recovering the usual notion of K\"ahler potential.
\end{example}
In the case where the underlying Poisson structure $Q$ is regular, it is possible to choose the Darboux chart in such a way as to recover the GK potentials occurring in the physics literature:
\begin{example}[Non-degenerate Poisson structure] If the Poisson structure $Q = \omega^{-1}$ is non-degenerate, then both of the holomorphic Poisson structures $\sigma_\pm$ are as well, with corresponding holomorphic symplectic forms
\[
\Omega_{\pm} = \omega I_{\pm} + i \omega.
\]
The Morita equivalence between the resulting holomorphic symplectic manifolds is then
\[
(Z, \Omega) = (X_{+}, \Omega_{+}) \times (X_{-}, -\Omega_{-}),
\]
with $\pi_{+}$ and $\pi_{-}$ the projections onto the first and second factors, respectively. The brane bisection $\cal{L}$ is given by the diagonally embedded manifold. We now construct a holomorphic Darboux chart on $Z$ from a pair of holomorphic Darboux charts for $(X_{\pm}, \Omega_{\pm})$: let $(q^{\alpha},p_{\alpha})_{\alpha = 1}^{n}$ define a holomorphic Darboux chart on $X_{+}$ so that $\Omega_{+} = dp_{\alpha} \wedge dq^{\alpha}$, and let $(Q^{\alpha},P_{\alpha})_{\alpha = 1}^{n}$ define a holomorphic Darboux chart on $X_{-}$ so that $\Omega_{-} = dP_{\alpha} \wedge dQ^{\alpha}$. Then $(q^{\alpha},p_{\alpha},Q^{\alpha},P_{\alpha})_{\alpha = 1}^{n}$ defines a holomorphic chart on $Z$ with respect to which the form $\Omega$ has the following expression:
\[
\Omega = dp_{\alpha} \wedge dq^{\alpha} + dQ^{\alpha} \wedge dP_{\alpha}.
\]
A natural choice for the complex Lagrangian in $(Z, \Omega)$ is given by $L = \{ Q^{\alpha} = p_{\alpha} = 0 \}$, so that $ (q^{\alpha},p_{\alpha},Q^{\alpha}, P_{\alpha})_{\alpha = 1}^{n} $ define a holomorphic Darboux chart with $q^{\alpha}$ and $P_{\alpha}$ coordinates along $L$ and $p_{\alpha}$ and $Q^{\alpha}$ the fibre coordinates. Theorem \ref{GK potential} then tells us that in the generic situation, the brane is given in these coordinates by the graph of $-i \partial K$ for a real-valued function $K(q^{\alpha},P_{\alpha}, \bar{q}^{\alpha}, \bar{P}_{\alpha})$:
\[
\cal{L} = \{ (q^{\alpha},p_{\alpha},Q^{\alpha},P_{\alpha})_{\alpha = 1}^{n} \ |\ p_\alpha=- i \frac{\partial K}{\partial q^{\alpha}},\ Q^\alpha= - i \frac{\partial K}{\partial P_{\alpha}}
\},
\]
and the symplectic form is given by $F = i \partial \bar \partial K$.
Using $q^\alpha$ and $P_\alpha$ as coordinates on this brane, we can then express all of the remaining data in terms of the GK potential $K$ as follows. First, the two projection maps $\pi_{\pm}$ have the following coordinate expressions
\[
\pi_{+}|_{\cal{L}}(q^{\alpha},P_{\alpha}) = (q^{\alpha}, -i \frac{\partial K}{\partial q^{\alpha}}), \qquad \pi_{-}|_{\cal{L}}(q^{\alpha}, P_{\alpha}) = (-i \frac{\partial K}{\partial P_{\alpha}}, P_{\alpha}),
\]
and therefore the holomorphic symplectic forms $\Omega_{\pm}$ can be expressed as follows
\begin{align*}
\Omega_{+} &= i \Big( \frac{\partial^{2} K}{\partial \bar{q}^{\beta} \partial q^{\alpha}} dq^{\alpha} \wedge d\bar{q}^{\beta}+ \frac{\partial^{2} K}{\partial P_{\beta} \partial q^{\alpha}} dq^{\alpha} \wedge dP_{\beta} + \frac{\partial^{2} K}{\partial \bar{P}_{\beta} \partial q^{\alpha}} dq^{\alpha} \wedge d\bar{P}_{\beta} \Big) \\
\Omega_{-} &= i \Big( \frac{\partial^{2} K}{\partial \bar{P}_{\beta} \partial P_{\alpha}} d\bar{P}_{\beta} \wedge dP_{\alpha} + \frac{\partial^{2} K}{\partial q^{\beta} \partial P_{\alpha}} dq^{\beta} \wedge dP_{\alpha}+ \frac{\partial^{2} K}{\partial \bar{q}^{\beta} \partial P_{\alpha}} d\bar{q}^{\beta} \wedge dP_{\alpha} \Big).
\end{align*}
Since these forms are holomorphic symplectic, they determine the complex structures $I_{\pm}$. The symplectic form $\omega$ is the common imaginary part of $\Omega_{\pm}$:
\[
\omega = \frac{1}{2}\Big(\frac{\partial^{2} K}{\partial P_{\beta} \partial q^{\alpha}} dq^{\alpha} \wedge dP_{\beta} + \frac{\partial^{2} K}{\partial \bar{P}_{\beta} \partial q^{\alpha}} dq^{\alpha} \wedge d\bar{P}_{\beta} + c.c. \Big).
\]
In this way, we recover the expressions found in~\cite[Section 4.2]{zabzine2009generalized}.
\end{example}
We now use our notion of GK potential to construct new examples of 4-dimensional GK structures for which the Hitchin Poisson structure is \emph{not of constant rank}, dropping from full rank to zero along a codimension 2 submanifold.
\begin{example}
Recall from Example~\ref{affpois} that the Poisson structure $\Pi = x \partial_{x} \wedge \partial_{y}$ on $\bb{C}^{2}$ has symplectic groupoid $\cal{G} = \bb{C}^{4}$, with source and target maps
\[
t(a,b,x,y) = (e^{a}x,y + xb), \qquad s(a,b,x,y) = (x, y),
\]
and symplectic form $\Omega = da \wedge d(y + xb) - db \wedge dx$. Viewing this as the trivial Morita equivalence we can upgrade this to an example of a GK structure by the appropriate choice of a brane bisection. In order to apply Theorem \ref{GK potential} we choose a Darboux chart for the groupoid as follows:
\[
(p_{1}, p_{2}, q_{1}, q_{2}) = (a, - b, y + xb, x).
\]
This puts the symplectic form into the standard form
\[
\Omega = dp_{1} \wedge dq_{1} + dp_{2} \wedge dq_{2}.
\]
We take $q_{1}, q_{2}$ to be coordinates on the Lagrangian and $p_{1}, p_{2}$ the cotangent fibre coordinates. The source and target maps then have the form
\[
t(p_{1}, p_{2}, q_{1}, q_{2}) = (e^{p_{1}}q_{2}, q_{1}),
\qquad
s(p_{1}, p_{2}, q_{1}, q_{2}) = (q_{2}, q_{1} + p_{2}q_{2}).
\]
According to Theorem~\ref{GK potential}, to define a brane it suffices to choose a real-valued function of $q_{1}$ and $q_{2}$. We take
\[
K(q_{1}, q_{2}) = q_{1} \bar{q}_{1} + q_{2} \bar{q}_{2}.
\]
Then the brane is given by the graph of the $(1,0)$-form $-i \partial K = -i(\bar{q}_{1}dq_{1} + \bar{q}_{2}dq_{2})$. That is, the brane is given by
\[
\cal{L} = \{(p_1,p_2,q_1,q_2)\ |\ p_1=-i \bar{q}_{1},\ \ p_2= -i \bar{q}_{2} \}.
\]
Using $q_{1}$ and $q_{2}$ as coordinates on $\cal{L}$, the source and target maps (restricted to $\cal{L}$) are
\[
t(q_{1}, q_{2}) = (e^{-i \bar{q}_{1}}q_{2}, q_{1}), \qquad s(q_{1}, q_{2}) = (q_{2}, q_{1} - i |q_{2}|^{2}).
\]
Because these are diffeomorphisms, $\cal{L}$ is a bisection of $s$ and $t$. Hence the triple $(\cal{G}, \Omega, \cal{L})$ defines a degenerate GK structure of symplectic type. By Theorem~\ref{GK potential}, the 2-form is given by
\[
F = i \partial \bar{\partial} K = i (dq_{1} \wedge d \bar{q}_{1} + dq_{2} \wedge d \bar{q}_{2}).
\]
In these coordinates, the Hitchin Poisson structure is given by
\[
Q = \frac{1}{2i}(q_{2} \partial_{q_{2}} \wedge \partial_{q_{1}} - \bar{q}_{2} \partial_{\bar{q}_{2}} \wedge \partial_{\bar{q}_{1}}),
\]
and the complex structures $I_{\pm}$ can be specified by their holomorphic coordinate functions: the $I_{+}$-holomorphic functions are given by $t^{\ast}x = e^{-i \bar{q}_{1}}q_{2}$ and $t^{\ast}y = q_{1}$, while the $I_{-}$-holomorphic functions are given by $s^{\ast}x = q_{2}$ and $s^{\ast}y = q_{1} - i |q_{2}|^{2}$.
In order to see that this defines a GK structure, we must determine whether, and where, the induced metric $g$ is positive definite. First we determine where $g$ is invertible: by Lemma \ref{metric11} this happens precisely where $\omega_{-} = g I_{-} = F^{(1,1)_{-}}$ is invertible, where $F^{(1,1)_{-}}$ is the $(1,1)$-component of $F$ with respect to the $I_{-}$ complex structure. We have
\[
F^{(1,1)_{-}} = i dq_{1} \wedge d\bar{q}_{1} + i(1 - 2|q_{2}|^{2})dq_{2}\wedge d\bar{q}_{2} + \bar{q}_{2}dq_{2} \wedge dq_{1} + q_{2} d\bar{q}_{2} \wedge d \bar{q}_{1},
\]
so that
\[
F^{(1,1)_{-}} \wedge F^{(1,1)_{-}} = -2(1 - |q_{2}|^{2}) dq_{1} \wedge d\bar{q}_{1} \wedge dq_{2} \wedge d\bar{q}_{2}.
\]
Hence $g$ is invertible whenever $|q_{2}| \neq 1$. Along $q_2=0$, we have $s^{\ast}(dx \wedge dy) = dq_{2} \wedge dq_{1}$, so that $I_{-}$ coincides with the standard complex structure in coordinates $(q_1,q_2)$, and $F^{(1,1)_{-}} = i dq_{1} \wedge d\bar{q}_{1} + idq_{2}\wedge d\bar{q}_{2}$. Since these formulas coincide with the standard K\"ahler structure on $\bb{C}^2$, we know that $g$ is positive definite when $q_{2} = 0$ and therefore in the region $|q_{2}| < 1$. Therefore this defines a generalized K\"ahler structure on $\bb{C} \times D$, where $D = \{ q_{2} \ | \ |q_{2}| < 1 \}$. The metric has the explicit form
\[
g = 2(dq_{1} d\bar{q}_{1} + dq_{2} d\bar{q}_{2} + i \bar{q}_{2} dq_{1} dq_{2} - i q_{2} d\bar{q}_{1} d \bar{q}_{2}).
\]
Note that if we restrict to the disc $\{q_{1} = c\} \times D$ the metric pulls back to $2dq_{2} d\bar{q}_{2}$, and so this disc has finite volume. Therefore the metric is not complete by the Hopf-Rinow theorem.
\end{example}
\begin{example}\label{gkansatz}
We now generalize the previous example by choosing a different generalized K\"ahler potential $K$, of the following form:
\[
K(q_{1}, q_{2}) = a(q_{1},\bar{q}_{1}) + b(q_{2}, \bar{q}_{2}),
\]
for real-valued functions $a$ and $b$. The brane is then given by
\[
\cal{L} = \{(p_1,p_2,q_1,q_2)\ |\ p_1=-i \partial_{q_{1}}a, \ \ p_2 = -i \partial_{q_{2}}b \}.
\]
Again, $t$ and $s$ restrict to $\cal{L}$ to be diffeomorphisms, and hence $\cal{L}$ defines a brane bisection, defining a degenerate GK structure. In terms of the real-valued functions
\[
\alpha(q_{1}, \bar{q}_{1}) = \frac{\partial^2 a}{\partial{q_{1}} \partial{\bar{q}_{1}}}, \qquad \beta(q_{2}, \bar{q}_{2}) = \frac{\partial^2 b}{\partial{q_{2}} \partial{\bar{q}_{2}}},
\]
we have
\[
F = i (\alpha dq_{1} \wedge d \bar{q}_{1} + \beta dq_{2} \wedge d\bar{q}_{2}),
\]
which has $(1,1)$ component with respect to $I_{-}$ given by
\[
F^{(1,1)_{-}} = i \alpha dq_{1} \wedge d\bar{q}_{1} + i \beta(1 - 2 \alpha \beta |q_{2}|^{2})dq_{2}\wedge d\bar{q}_{2} + \alpha \beta \bar{q}_{2}dq_{2} \wedge dq_{1} + \alpha \beta q_{2} d\bar{q}_{2} \wedge d \bar{q}_{1},
\]
so that
\[
F^{(1,1)_{-}} \wedge F^{(1,1)_{-}} = -2 \alpha \beta (1 - \alpha \beta |q_{2}|^{2}) dq_{1} \wedge d\bar{q}_{1} \wedge dq_{2} \wedge d\bar{q}_{2}.
\]
Furthermore, the induced metric is given by
\[
g = 2(\alpha dq_{1} d\bar{q}_{1} + \beta dq_{2} d\bar{q}_{2} + i \alpha \beta \bar{q}_{2} dq_{1} dq_{2} - i \alpha \beta q_{2} d\bar{q}_{1} d \bar{q}_{2}).
\]
We want the metric to be positive definite. Setting $q_{2} = 0$ shows that we must have $\alpha, \beta > 0$. The expression for $F^{(1,1)_{-}} \wedge F^{(1,1)_{-}}$ shows that $g$ will be invertible precisely when
\[
1 \neq \alpha \beta |q_{2}|^{2}.
\]
Setting $q_{2} = 0$ shows that we must therefore require that $\alpha \beta |q_{2}|^{2} < 1$. Finally, because $\alpha$ and $\beta$ depend only on $q_{1},\bar{q}_{1}$ and $q_{2},\bar{q}_{2}$ respectively, $\alpha$ must be bounded by a constant and $\beta$ must be bounded by $\frac{C}{|q_{2}|^{2}}$, for $C$ a positive constant. Under these assumptions, we obtain examples of generalized K\"ahler structures of symplectic type.
\end{example}
\begin{proposition}\label{globex}
In the context of the previous example, choose the generalized K\"ahler potential to be
\[
K(q_{1}, q_{2}) = \frac{|q_{1}|^{2}}{C} - Li_{2}(-|q_{2}|^{2}),
\]
for $C > 1$ a constant, where $Li_{2}(z) = - \int_{0}^{z} \text{log}(1-u)\frac{du}{u}$ is the dilogarithm. Then we get $\alpha = \frac{1}{C}$ and $\beta = \frac{1}{1 + |q_{2}|^2}$, and hence the metric is given by
\[
g = 2(\frac{1}{C} dq_{1} d\bar{q}_{1} + \frac{1}{1+|q_{2}|^2}(dq_{2} d\bar{q}_{2} + \frac{i}{C} \bar{q}_{2} dq_{1} dq_{2} - \frac{i}{C} q_{2} d\bar{q}_{1} d \bar{q}_{2})).
\]
This gives a generalized K\"ahler structure on $\bb{R}^4$ for which the metric is complete.
\begin{proof}
After the argument of Example~\ref{gkansatz}, we need only show the completeness of the metric. Note that the metric is translation invariant in the $q_{1}$ direction. Quotienting by a $\bb{Z}^2$ lattice, we obtain a metric on $S^1 \times S^1 \times \bb{C}$. Completeness of this metric is equivalent to completeness of the original. To show completeness, we need only investigate geodesics escaping to infinity in the $q_{2}$-direction. In coordinates $q_{1} = x + iy$ and $q_{2} = \sinh(s)e^{i \theta}$, the metric has the following form:
\[
g = \frac{2}{C}(dx^2 + dy^2) + 2(ds^2 + \tanh^2(s)d\theta^2 - \frac{2}{C}\tanh^2(s) dx d\theta - \frac{2}{C}\tanh(s)dyds).
\]
For any geodesic $\gamma(t) = (x(t), y(t), s(t), \theta(t))$, we have
\[
\sqrt{g(\dot{\gamma}, \dot{\gamma})} \geq \sqrt{\frac{2 (C-1)}{C}} \dot{s},
\]
from which it follows that the length of the curve over the interval $[t_{0}, t_{1}]$ is bounded below in the following way:
\[
L(\gamma) \geq \sqrt{\frac{2 (C-1)}{C}} (s(t_{1}) - s(t_{0})).
\]
Therefore the length of a curve that escapes to infinity is unbounded and so the metric is complete.
\end{proof}
\end{proposition}
\section{The Picard group}\label{picsec}
In this section we focus on generalized K\"ahler structures where the two holomorphic Poisson structures are isomorphic. In this case, by choosing an isomorphism, we can assume that the two holomorphic Poisson structures coincide and then study the self-Morita equivalences of the given Poisson structure. This leads to the notion of the \emph{Picard group} of a Poisson structure $(X, \sigma)$, first introduced by Weinstein and Bursztyn in the smooth category in \cite{bursztyn2004picard}. Since there is a composition for Morita equivalences, (integrable) holomorphic Poisson manifolds can be viewed as the objects of a groupoid where the morphisms are given by holomorphic symplectic Morita equivalences.
\begin{definition}
The \emph{holomorphic Picard groupoid} $\mathcal{PG}$ is the category whose objects are integrable holomorphic Poisson manifolds and whose morphisms are isomorphism classes of holomorphic symplectic Morita equivalences. The \emph{Picard group} of a holomorphic Poisson manifold $(X, \sigma)$ is the automorphism group of $(X, \sigma)$ in $\mathcal{PG}$:
\[
\Pic(X, \sigma) = \Hom[\mathcal{PG}]{ (X, \sigma), (X, \sigma) }.
\]
\end{definition}
Since Morita equivalences equipped with bisections also compose, it is possible to upgrade the above groupoid so that the morphisms are holomorphic symplectic Morita equivalences with brane bisection.
\begin{definition}
The \emph{Picard groupoid with branes} $\mathcal{PG}^\cal{L}$
is the category whose objects are integrable holomorphic Poisson manifolds and whose morphisms are isomorphism classes of holomorphic symplectic Morita equivalences equipped with brane bisections. The automorphism group of the object $(X, \sigma)$ is denoted by
\[
\mathrm{Pic}^\cal{L}(X, \sigma) = \Hom[\mathcal{PG}^\cal{L}]{ (X, \sigma), (X, \sigma) }.
\]
\end{definition}
\begin{remark}
Morphisms between two fixed Poisson manifolds $\Hom[\mathcal{PG}^\cal{L}]{ (X_{+}, \sigma_{+}), (X_{-}, \sigma_{-})}$ correspond, by Theorem~\ref{main}, to degenerate GK structures of symplectic type.
\end{remark}
There is a natural forgetful functor from $\mathcal{PG}^\cal{L}$ to $\mathcal{PG}$ which drops the data of the brane bisection. This gives rise to a homomorphism
\[
\mathrm{Pic}^\cal{L}(X, \sigma) \to \Pic(X, \sigma),\qquad [(Z, \Omega, \lambda)] \mapsto [(Z, \Omega)].
\]
The kernel of this map consists of objects where the underlying Morita equivalence is trivial, i.e. isomorphic to the Weinstein groupoid $(\Sigma(X), \Omega)$. Therefore, the kernel is given by the image of the following natural homomorphism
\[
\mathrm{Bis}^\cal{L}(\Sigma(X)) \to \mathrm{Pic}^\cal{L}(X, \sigma), \qquad \lambda \mapsto [(\Sigma(X), \Omega, \lambda)],
\]
where $\mathrm{Bis}^\cal{L}(\Sigma(X))$ is the group of brane bisections in $(\Sigma(X), \Omega)$. Note that we are viewing a brane bisection as a map $\lambda: X \to Z$ such that $\pi_{-} \circ \lambda = id_{X}$, the map $\phi_{\lambda} := \pi_{+} \circ \lambda$ is a diffeomorphism, and $\lambda^{\ast} \Im(\Omega) = 0$.
Using Theorem~\ref{main}, we are able to give a concrete description of $\mathrm{Pic}^\cal{L}(X, \sigma)$, as follows. An object $[(Z, \Omega, \lambda)]$ of this group is a Morita self-equivalence with brane bisection of the holomorphic Poisson structure $(X, \sigma)$, where we let $M$ denote the underlying smooth manifold of $X$, $I$ its complex structure, and $Q = -4 \Im(\sigma)$. By Theorem~\ref{main}, this self-equivalence with brane corresponds to a degenerate GK structure of symplectic type, i.e. a solution $(I_{+}, I_{-}, Q, F)$ of equations \ref{star1} and \ref{star2}. Since we are viewing the brane as a section $\lambda$ of $\pi_{-}$ in this correspondence, we obtain
$I_{-} = I$, $F = \lambda^{\ast} \Omega$, and $I_{+} = (\phi_{\lambda}^{-1})_{\ast}(I)$, where $\phi_{\lambda} = \pi_{+} \circ \lambda$. Therefore, given the holomorphic Poisson structure, the remaining data is encoded by the real 2-form $F$ and the diffeomorphism $\phi_{\lambda}$. Using equation \ref{star1} to express $I_{+}$ as $I^{F} := I + QF$, equation \ref{star2} then becomes
\[
FI + I^{*}F + FQF = 0,
\]
and the relation between $\phi_{\lambda}$, $I_{+}$ and $I_{-}$ can then be expressed as
\[
(\phi_{\lambda})_{\ast}(I^{F}) = I.
\]
Based on these observations, we define the following subgroup of $\mathrm{Diff}_Q(M) \ltimes \Omega^{2,\mathrm{cl}}(M)$, the semi-direct product of the group of diffeomorphisms preserving $Q$ with the group of closed 2-forms:
\[
\mathrm{Aut}_\cal{C}({I, \sigma}) = \{ (\phi, F) \in \mathrm{Diff}_Q(M) \times \Omega^{2,\mathrm{cl}}(M) \ | \ FI + I^{\ast} F + FQF = 0\text{ and } \phi_{\ast}(I^{F}) = I \},
\]
with multiplication defined as follows:
\begin{equation} \label{groupproduct}
(\phi_{1}, F_{1}) \ast (\phi_{2}, F_{2}) = (\phi_{1} \circ \phi_{2}, \phi_{2}^{\ast}F_{1} + F_{2}).
\end{equation}
This is the group of \emph{Courant automorphisms of $(I, \sigma)$} when the holomorphic Poisson structure is viewed as a generalized complex structure (See~\cite{gualtieri2010branes} for a study of this group). An upshot of the present discussion is the existence of a map between $\mathrm{Pic}^\cal{L}(X, \sigma)$ and $\mathrm{Aut}_\cal{C}({I, \sigma})$, and a consequence of Theorem~\ref{main} is the fact that these two groups are isomorphic.
\begin{corollary}
There is an isomorphism of groups
\[
\chi : \mathrm{Pic}^\cal{L}(X, \sigma) \to \mathrm{Aut}_\cal{C}({I, \sigma}), \qquad [Z, {\Omega},\lambda] \mapsto (\pi_{+} \circ \lambda, \lambda^{\ast} {\Omega} ).
\]
\begin{proof} The above discussion shows that the map $\chi$ is well-defined. Hence it remains to show that $\chi$ is a homomorphism and a bijection.
Step 1: $\chi$ is a group homormophism. Let $(Z_{1}, \Omega_{1}, \lambda_{1})$ and $(Z_{2}, \Omega_{2}, \lambda_{2})$ be Morita equivalences with brane bisections. The product of the bimodules is given by the quotient
\[
Z_{1} \ast Z_{2} = (Z_{1} \times_{X} Z_{2}) / \Sigma(X),
\]
with symplectic form $\Omega$ defined via symplectic reduction, and the product of the bisections is given by
\[
\lambda_{1} \ast \lambda_{2}(x) = [\lambda_{1} \circ \pi_{+,2} \circ \lambda_{2}(x), \lambda_{2}(x) ].
\]
Therefore $\pi_{+} \circ (\lambda_{1} \ast \lambda_{2}) = (\pi_{+,1} \circ \lambda_{1}) \circ (\pi_{+,2} \circ \lambda_{2})$, and
\begin{align*}
(\lambda_{1} \ast \lambda_{2})^{\ast} \Omega &= (\lambda_{1} \circ \pi_{+,2} \circ \lambda_{2})^{\ast} \Omega_{1} + \lambda_{2}^{\ast} \Omega_{2} \\
&= (\pi_{+,2} \circ \lambda_{2})^{\ast} (\lambda_{1}^{\ast} \Omega_{1}) + \lambda_{2}^{\ast} \Omega_{2},
\end{align*}
which, according to equation \ref{groupproduct}, is the product of $(\pi_{+, 1} \circ \lambda_{1}, \lambda_{1}^{\ast} {\Omega_{1}} )$ and $(\pi_{+, 2} \circ \lambda_{2}, \lambda_{2}^{\ast} {\Omega_{2}} )$.
Step 2: $\chi$ is an isomorphism. We show this by defining an explicit inverse $\zeta : \mathrm{Aut}_\cal{C}({I, \sigma}) \to \mathrm{Pic}^\cal{L}(X,\sigma)$. Given $(\phi, F) \in \mathrm{Aut}_\cal{C}({I, \sigma})$, Proposition~\ref{Star->ME} implies that we get the following element of the Picard
\[
\xymatrix{
&(\Sigma(X),\Omega + t^{\ast}F)\ar[dl]_-{\phi \circ t}\ar[dr]^-{s} & \\
(X,\sigma) & &(X,\sigma)
}
\]
where the brane bisection is given by the identity bisection $\epsilon$. We therefore take this to define $\zeta(\phi, F)$. Theorem \ref{main} then implies that $\zeta$ and $\chi$ are inverse to each other.
\end{proof}
\end{corollary}
As a result of this isomorphism, we immediately see that the kernel of the map $\mathrm{Bis}^\cal{L}(\Sigma(X)) \to \mathrm{Pic}^\cal{L}(X, \sigma)$ is given by the subgroup $\mathrm{IsoLBis}(\Sigma(X))$ of holomorphic Lagrangian bisections of $(\Sigma(X), \Omega)$ which induce the identity diffeomorphism on $M$. Collecting these facts we get the following exact sequence of groups
\begin{equation} \label{exactseq}
1 \to \mathrm{IsoLBis}(\Sigma(X)) \to \mathrm{Bis}^\cal{L}(\Sigma(X)) \to \mathrm{Aut}_\cal{C}({I, \sigma}) \to \Pic(X, \sigma).
\end{equation}
\begin{remark} Bursztyn and Fernandes studied a similar sequence in \cite{bursztyn2015picard} in the setting of real smooth Poisson structures. The above may be seen as a generalization of their results to the case of holomorphic Poisson structures.
\end{remark}
\begin{remark}
A central claim of this paper has been that the holomorphic substructure underlying GK manifolds of symplectic type consists of holomorphic symplectic Morita equivalences, and that the additional real data needed to determine the metric consists of a brane bisection. This is reflected in the sequence \ref{exactseq} by the fact that $\mathrm{Aut}_\cal{C}({I, \sigma})$ is an extension of its image in the holomorphic Picard group by the group of brane bisections.
\end{remark}
\section{Generalized K\"ahler metrics via Hamiltonian flows}\label{hamflo}
An important application of the ideas of Section~\ref{picsec} is to the construction and deformation of generalized K\"ahler metrics. The basic idea is as follows:
Theorem \ref{main} allows us to view GK structures of symplectic type as Morita equivalences with brane bisection, which are morphisms in the groupoid $\mathcal{PG}^\cal{L}$. So, it is possible to compose them. More precisely, if $(Z_{1}, \Omega_{1}, \cal{L}_{1})$ and $(Z_{2}, \Omega_{2}, \cal{L}_{2})$ are degenerate GK structures of symplectic type going between holomorphic Poisson structures $(X_{1}, \sigma_{1}), (X_{2}, \sigma_{2})$ and $(X_{2}, \sigma_{2}), (X_{3}, \sigma_{3})$ respectively, then we may compose them to get a degenerate GK structure going between $(X_{1}, \sigma_{1})$ and $(X_{3}, \sigma_{3})$. In particular, there is an action of the group $\mathrm{Pic}^\cal{L}(X_{+}, \sigma_{+})$ on the space of morphisms $\mathcal{PG}^\cal{L}((X_{+}, \sigma_{+}), (X_{-}, \sigma_{-}))$ and we can use this to \emph{deform} GK structures of symplectic type.
Indeed, all of the constructions of GK metrics contained in~\cite{MR2371181,MR2217300,gualtieri2010branes} are special cases of this.
The idea of the construction in \cite{gualtieri2010branes} is to start with the infinitesimal counterpart of $\mathrm{Pic}^\cal{L}(X,\sigma) \cong \mathrm{Aut}_\cal{C}({I, \sigma})$, i.e. an infinitesimal Courant symmetry, integrate it to a Courant automorphism (a path in $\mathrm{Aut}_\cal{C}({I, \sigma})$), and then use this to deform a given GK structure. We will see how this manifests itself in terms of Morita equivalences with brane bisections. In particular we will see that in certain cases these deformations are obtained by flowing the brane bisections using Hamiltonian vector fields.
The Lie algebra of $\mathrm{Aut}_\cal{C}({I, \sigma})$ is the following subalgebra of $\frak{X}_{Q}(M) \ltimes \Omega^{2, cl}(M)$, the semi-direct product of the Lie algebra of vector fields preserving $Q$ with the group of closed $2$-forms:
\[
\frak{aut}_\cal{C}(I, \sigma) = \{ (V, \omega) \in \frak{X}_{Q}(M) \times \Omega^{2, cl}(M) \ | \ \omega I + I^{\ast} \omega = 0 \text{ and } \cal{L}_{V}I = Q \omega \}.
\]
This is the Lie algebra of \emph{infinitesimal Courant symmetries} of the holomorphic Poisson structure $(I, \sigma)$. Let $(V, \omega) \in \frak{aut}_\cal{C}(I, \sigma)$ be an element of this Lie algebra. This defines an infinitesimal automorphism of the Courant algebroid $T \oplus T^{*}$ and therefore it integrates to the following family of automorphisms of $T\oplus T^{*}$:
\[
(\phi_{t}, F_{t} = \int_{0}^{t} (\phi_{s}^{\ast}\omega) ds),
\]
where $\phi_{t}$ is the flow of the vector field $V$. In \cite[Section 7]{gualtieri2010branes} it is shown that this family lies in $\mathrm{Aut}_\cal{C}({I, \sigma})$ for all $t$ (where it is defined), and therefore this family defines the $1$-parameter subgroup integrating $(V, \omega)$. Hence we have the following result.
\begin{lemma}
The exponential map
\[
\exp : \frak{aut}_\cal{C}(I, \sigma) \to \mathrm{Aut}_\cal{C}({I, \sigma}) \cong \mathrm{Pic}^\cal{L}(X, \sigma)
\]
is given by flowing the above family of automorphisms to $t = 1$:
\[
(V, \omega) \mapsto (\phi_{1}, F_{1} = \int_{0}^{1} (\phi_{s}^{\ast} \omega) ds).
\]
\end{lemma}
The Lie algebra of the group of brane bisections $\mathrm{Bis}^\cal{L}(\Sigma(X))$ is given by the space of closed $1$-forms $\Omega^{1, cl}(M)$ with Lie bracket induced by $Q$ \cite{xu1997flux}. The exponential map $\exp : \Omega^{1, cl}(M) \to \mathrm{Bis}^\cal{L}(\Sigma(X))$ has the following description: given $\alpha \in \Omega^{1, cl}(M)$, we can consider the vector field induced by $t^{\ast}\alpha$ on the Weinstein groupoid with respect to the imaginary part of the symplectic form $\omega = \Im(\Omega)$:
\[
V_{t^{\ast} \alpha} = \omega^{-1}(t^{\ast} \alpha).
\]
This is the right-invariant vector field associated to $\alpha$ when it is viewed as a section of the Lie algebroid $T^{\ast}_{Q}M$ of $(\Sigma(X), \omega)$. It is $t$-related to the vector field $Q(\alpha)$ on $X$. Let $\phi_{t}$ be the flow of $V_{t^{\ast}\alpha}$. Then the $1$-parameter subgroup integrating $\alpha$ is given by the family $\lambda_{t} = \phi_{t} \circ \epsilon$ of bisections, where $\epsilon$ is the identity bisection, so that
\[
\exp(\alpha) = \lambda_{1}.
\]
Mapping the $1$-parameter subgroup $\lambda_{t}$ to $\mathrm{Aut}_\cal{C}({I, \sigma})$ and taking the derivative at $t = 0$ determines the induced Lie algebra morphism $\Omega^{1, cl}(M) \to \frak{aut}_\cal{C}(I, \sigma)$: $\lambda_{t} \in \mathrm{Bis}^\cal{L}(\Sigma(X))$ gets sent to the family $[ \Sigma(X), \Omega, \lambda_{t} ] \in \mathrm{Pic}^\cal{L}(X, \sigma)$ of Morita equivalences with brane bisections, and this corresponds in $\mathrm{Aut}_\cal{C}({I, \sigma})$ to the family of Courant automorphisms given by $(\psi_{t}, \int_{0}^{t} (\psi_{s}^{\ast} d^{c} \alpha) ds )$, where $\psi_{t}$ is the flow of $Q(\alpha)$ and $d^{c} = i (\bar{\partial} - \partial)$. Therefore the map of Lie algebras is given by
\[
\Omega^{1, cl}(M) \to \frak{aut}_\cal{C}(I, \sigma), \qquad \alpha \mapsto (Q(\alpha), d^{c} \alpha).
\]
One upshot of the present discussion is that the exponential map for $\mathrm{Aut}_\cal{C}({I, \sigma})$ has a particularly nice description when it is applied to elements in the image of $\Omega^{1, cl}(M)$.
\begin{proposition} \label{flow1}
Let $\alpha \in \Omega^{1, cl}(M)$ be a real closed $1$-form and let $(Q(\alpha), d^{c} \alpha) \in \frak{aut}_\cal{C}(I, \sigma)$ be the infinitesimal Courant symmetry that it determines. Exponentiating this symmetry yields a family of holomorphic symplectic Morita equivalences with brane bisections. This family has the following simple form:
\[
[ (\Sigma(X), \Omega, \phi_{t} \circ \epsilon )],
\]
where $(\Sigma(X), \Omega)$ is the Weinstein groupoid viewed as a trivial Morita equivalence, $\phi_{t}$ is the flow of the vector field $(\Im \Omega)^{-1}(t^{\ast} \alpha)$ on the groupoid, and $\phi_{t} \circ \epsilon$ is the result of applying this flow to the identity bisection.
\end{proposition}
We now explain how the flow of infinitesimal Courant symmetries can be used to deform a given GK structure of symplectic type to a nearby one. Such deformations are obtained via the action of the group $\mathrm{Pic}^\cal{L}(X_{+}, \sigma_{+})$ on the space of morphisms $\mathcal{PG}^\cal{L}((X_{+}, \sigma_{+}), (X_{-}, \sigma_{-}))$. More precisely, let
\[
\xymatrix{
&(Z,\Omega, \cal{L})\ar[dl]_-{\pi_{+}}\ar[dr]^-{\pi_{-}} & \\
(X_{+},\sigma_{+})& & (X_{-}, \sigma_{-})
}
\]
be a GK structure viewed as a Morita equivalence with brane bisection, and let $\zeta(\phi,F)$
be an element of $\mathrm{Pic}^\cal{L}(X_{+}, \sigma_{+})$ corresponding to the Courant automorphism $(\phi, F) \in \mathrm{Aut}_\cal{C}({I_{+}, \sigma_{+}})$. Composing with the above Morita equivalences, we obtain $\zeta(\phi,F) \ast (Z,\Omega,\cal{L}) = (Z,\Omega + \pi_{+}^{\ast}F, \cal{L})$, again a morphism in $\mathcal{PG}^\cal{L}((X_{+}, \sigma_{+}), (X_{-}, \sigma_{-}))$ as depicted below.
\[
\xymatrix{
&(Z,\Omega + \pi_{+}^{\ast}F, \cal{L})\ar[dl]_-{\phi\circ \pi_{+}}\ar[dr]^-{\pi_{-}} & \\
(X_{+},\sigma_{+})& & (X_{-}, \sigma_{-})
}
\]
Since the positivity of the induced metric is an open condition, this will define a new GK structure if $(\phi, F)$ is `close enough' to the identity. Given an infinitesimal symmetry $(V, \omega) \in \frak{aut}_\cal{C}(I_{+}, \sigma_{+})$ we can exponentiate it to the family $\zeta(\exp(t(V,\omega)))$ in $\mathrm{Pic}^\cal{L}(X_{+}, \sigma_{+})$. Then $\zeta(\exp(t(V,\omega))\ast (Z,\Omega,\cal{L})$
defines a family of degenerate GK structures deforming the initial structure, and the metric will remain positive-definite for sufficiently small $t$. This family of deformations has a particularly nice form for infinitesimal symmetries in the image of $\Omega^{1, cl}(M)$.
\begin{proposition} \label{flow2}
Let $(Z,\Omega, \cal{L})$ be a GK structure of symplectic type, viewed as a holomorphic symplectic Morita equivalence with brane bisection going between the holomorphic Poisson structures $(I_{\pm}, \sigma_{\pm})$. Let $\alpha \in \Omega^{1, cl}(M)$ be a real closed $1$-form, and let $(Q(\alpha), d^{c} \alpha) \in \frak{aut}_\cal{C}(I_{+}, \sigma_{+})$ be the infinitesimal Courant symmetry that it determines. Exponentiating this symmetry, we get a 1-parameter family of Courant automorphisms which act on $(Z,\Omega, \cal{L})$ to produce a family of degenerate GK structures of symplectic type deforming the given one. This family has the following simple form:
\[
[(Z, \Omega, \cal{L}_{t} = \eta_{t}(\cal{L}))],
\]
where $\eta_{t}$ is the flow of the vector field $V_{\pi_{+}^{\ast} \alpha} = (\Im \Omega)^{-1}( \pi_{+}^{\ast} \alpha)$, and $\cal{L}_{t} = \eta_{t}(\cal{L})$ is the result of applying this flow to the brane bisection $\cal{L}$.
\begin{proof}
Let $\lambda_{t} = \phi_{t} \circ \epsilon$ be the 1-parameter subgroup corresponding to $\alpha$ in $\mathrm{Bis}^\cal{L}(\Sigma(X))$, where $\phi_{t}$ is the flow of $V_{t^{\ast} \alpha}$. By Proposition~\ref{flow1}, the family of degenerate GK structures obtained by exponentiating $(Q(\alpha), d^{c} \alpha)$ and acting on the given GK structure is given by the composition of $(\Sigma(X), \Omega, \lambda_{t})$ and $ (Z,\Omega, \cal{L})$. This gives $(Z, \Omega, \cal{L}_{t} = \lambda_{t} \ast \cal{L})$, where
\[
(\lambda_{t} \ast \cal{L})(x) = \lambda_{t}(\pi_{+} \circ \cal{L}(x)) \ast \cal{L}(x),
\]
and where we are using the action $\ast$ of $\Sigma(X)$ on $Z$. Note that we are abusing notation by using $\cal{L}$ to denote both the section of $\pi_{-}$ and its image in $Z$. Now let $V_{\pi_{+}^{\ast} \alpha} = \Im(\Omega)^{-1}( \pi_{+}^{\ast} \alpha)$, a vector field on $Z$, and let $\eta_{t}$ be its flow. We claim that $\cal{L}_{t} = \eta_{t}(\cal{L})$. This can be seen as follows. Using the brane $\cal{L}$ we define a smooth symplectomorphism
\[
\Phi: (\Sigma(X_{+}), \Im(\Omega)) \to (Z, \Im(\Omega)), \qquad g \mapsto g \ast (r \circ s)(g),
\]
where $r : X_{+} \to Z$ is the section of $\pi_{+}$ induced by $\cal{L}$, and $s$ is the source map of the Weinstein groupoid. This map satisfies the relation $\pi_{+} \circ \Phi = t$, and therefore $d\Phi (V_{t^{\ast} \alpha}) = V_{\pi_{+}^{\ast} \alpha}$. This implies that $\Phi$ intertwines the flows $\phi_{t}$ and $\eta_{t}$ of the vector fields $V_{t^{\ast} \alpha}$ and $V_{\pi_{+}^{\ast} \alpha}$, respectively. Then, by the definitions of $\Phi$ and $\cal{L}_{t}$ we see that on the one-hand
\[
\Phi \circ \lambda_{t} \circ \pi_{+}\circ \cal{L}(x) = \lambda_{t}(\pi_{+} \circ \cal{L}(x)) \ast \cal{L}(x) = \cal{L}_{t}(x),
\]
and by the property that $\Phi$ intertwines $\phi_{t}$ and $\eta_{t}$ we see that on the other hand
\[
\Phi \circ \lambda_{t} \circ \pi_{+}\circ \cal{L}(x) = \eta_{t} \circ \Phi(1_{\pi_{+} \circ \cal{L}(x)}) = \eta_{t}(\cal{L}(x)).
\]
\end{proof}
\end{proposition}
\begin{remark}
In Propositions \ref{flow1} and \ref{flow2}, we may specialize to the case of exact $1$-forms $\alpha = dK$, in which case the families we obtain are given by flowing the brane bisections by Hamiltonian vector fields.
\end{remark}
\begin{example}
Recall from Example~\ref{hyperKahler} that the Morita equivalence of a hyper-K\"ahler structure $(M, I, J, K, g)$ is given by
\[
(Z, \Omega) = (X_{+}, \Omega_{+}) \times (X_{-}, -\Omega_{-}),
\]
with brane bisection $\cal{L}$ given by the diagonally embedded copy of $M$. It was observed in~\cite{MR1702248} that such a GK structure may be deformed to a new GK structure which is not hyper-K\"ahler using a real-valued function $f$.
In the present setting we can view this as a deformation obtained by flowing the brane bisection using a Hamiltonian vector field of $f$. More precisely, given a real valued-function $f$, Proposition \ref{flow2} says that the brane is flowed by the Hamiltonian vector field of $\pi_{+}^{\ast} f$, using the imaginary part of the symplectic form on $Z$, namely, $\Im (\Omega_{+}, - \Omega_{-}) = (\omega_{K}, - \omega_{K})$. Therefore, the brane is given by the flow of
\[
(\omega_{K}^{-1}(df), 0),
\]
and so if $\phi_{t}$ is the flow of the Hamiltonian vector field of $f$ with respect to $\omega_{K}$ then the deformed brane is given by
\[
\cal{L}_{t} = \{ (\phi_{t}(m), m) \ | \ m \in M\},
\]
which is the graph of $\phi_{t}$.
\end{example}
\section{Universal local construction via time-dependent flows}\label{locdef}
Let $(M, I, \sigma)$ be a holomorphic Poisson structure, which determines a degenerate GK structure as in Example~\ref{trivdeg}. As explained in Example~\ref{integtriv}, the corresponding Morita equivalence provided by Theorem~\ref{main} is the symplectic groupoid $(\Sigma(X), \Omega, \epsilon)$ integrating $\sigma$, with the identity bisection $\epsilon$ as its brane bisection. Propositions \ref{flow1} and \ref{flow2} tell us that given a real-valued function $f$ we can construct a family of degenerate GK structures deforming the given one by deforming the identity bisection $\epsilon$ using the Hamiltonian vector field of $t^{\ast}f$ on the groupoid: $V_{t^{\ast}df} = \omega^{-1}(dt^{\ast}f)$, where here we decompose $\Omega = B+i\omega$ into its real and imaginary parts. In this section, we show that, locally, all degenerate GK structures of symplectic type arise in this way if we allow the function $f$ to depend on time. This gives a universal local construction (similar to the one using a generalized K\"ahler potential) which places a greater emphasis on the underlying holomorphic Poisson geometry. In effect, it tells us that locally a GK structure of symplectic type is determined by a holomorphic Poisson structure and a single time-dependent real-valued function.
Let $(Z, \Omega, \cal{L})$ be a GK structure viewed as a Morita equivalence with brane bisection, and let $z \in \cal{L}$ be a point on the brane. Choose a local holomorphic Lagrangian bisection $\Lambda$ passing through the point $z$ and let $U := \pi_{-}(\Lambda) \subseteq X_{-}$, and $V := \pi_{+}(\Lambda) \subseteq X_{+}$. Note first that this bisection induces a holomorphic Poisson isomorphism $\phi : (U, \sigma_{-}) \to (V, \sigma_{+})$; this means that the two holomorphic Poisson structures underlying a GK structure of symplectic type are always locally isomorphic, although not in a canonical way. Now consider the local Morita equivalence $\pi_{+}^{-1}(V) \cap \pi_{-}^{-1}(U)$ going between $(U, \sigma_{-})$ and $(V, \sigma_{+})$. Using the Lagrangian $\Lambda$, we can identify this Morita equivalence with the trivial Morita equivalence $t^{-1}(U) \cap s^{-1}(U) \subseteq \Sigma(X_{-})$. That is to say, we have a holomorphic symplectomorphism
\begin{equation}\label{grpcht}
\Phi : t^{-1}(U) \cap s^{-1}(U) \to \pi_{+}^{-1}(V) \cap \pi_{-}^{-1}(U), \qquad g \mapsto \Lambda(t(g)) \ast g,
\end{equation}
satisfying $\pi_{+} \circ \Phi = \phi \circ t$, as well as $\pi_{-} \circ \Phi = s$ and $\Phi \circ \epsilon = \Lambda.$
We view this as a \emph{groupoid chart}, in the sense that it identifies a neighbourhood of $z$ in $Z$ with a neighbourhood of the zero section in $\Sigma(X_-)$. The chart is adapted to the groupoid structure, unlike the Darboux charts considered in section \ref{secpot}. In this chart, the brane $\cal{L}$ intersects the identity bisection at the point $z$. Our goal is to describe $\cal{L}$ as a Hamiltonian flow applied to the zero section: for this, we require a family of brane bisections $\cal{L}_{t}$ interpolating between $\cal{L}$ and $\Lambda$. First we show that such a family exists.
\begin{proposition} \label{bisectionpath}
There is a local family of brane bisections interpolating between a given brane $\cal{L}$ and a holomorphic Lagrangian bisection $\Lambda$.
\end{proposition}
Such a family of brane bisections consists of a family of Lagrangian submanifolds for $\omega = \Im(\Omega)$ which is transverse to both $K_{\pm} = \text{ker} (d \pi_{\pm})$ at all times. Note that since $K_{+}$ and $K_{-}$ are symplectic orthogonal, a Lagrangian $L$ which is transverse to $K_{+}$ is automatically transverse to $K_{-}$:
\[
L \cap K_{-} = L^{\omega} \cap K_{+}^{\omega} = (L + K_{+})^{\omega} = TZ^{\omega} = 0.
\]
The linear version of this problem has an immediate solution: let $(V, \omega)$ be a $2n$-dimensional (real) symplectic vector space and let $K$ be an arbitrary $n$-dimensional subspace. Let $M_{V,K}$ denote the space of Lagrangians in $V$ which are transverse to $K$; this is a connected open subset of the Lagrangian Grassmannian, showing that the linear version of such an interpolation is available.
\begin{proof}[Proof of Proposition \ref{bisectionpath}]
Choose a holomorphic Darboux chart centred at the point $z$ : $(TZ_{z}, \Omega_{z}) \cong (Z, \Omega)$, and let $\Lambda$ be a holomorphic Lagrangian subspace of $TZ_{z}$ which is transverse to $K_{\pm}$. This defines a (local) holomorphic Lagrangian bisection $\Lambda$. The tangent space $L = T_{z} \cal{L}$ defines a Lagrangian subspace of $(TZ_{z}, \omega_{z})$ which is also transverse to $K_{\pm}$.
Since $M_{TZ_{z},K_+}$ is connected, we choose a family $L_{t}$ of Lagrangian subspaces of $(TZ_{z}, \omega_{z})$ which remain transverse to $K_{\pm}$ for all time interpolating between $\Lambda$ and $L$. We can view this as a family of brane bisections going from $\Lambda$ to the brane $L$. Hence it remains to find a path going from $L$ to $\cal{L}$. For this choose a Weinstein neighbourhood of $L$: $(T^{\ast}L, \Omega_0) \cong (Z, \omega)$. In this chart, $\cal{L}$ is given by the graph of a closed $1$-form $\alpha \in \Omega^{1}(L)$. Then $\cal{L}_{t} = Gr(t \alpha)$ defines a family of Lagrangians interpolating between $L$ and $\cal{L}$. Since $T\cal{L}_{z} = TL_{z}$ it follows that $T(\cal{L}_{t})_{z} = TL_{z}$ for all $t$ implying that this family is transverse to $K_{\pm}$ at all times. Combining the two families, we obtain a family of branes interpolating between the holomorphic Lagrangian $\Lambda$ and the brane bisection $\cal{L}$. This family fixes the point $z$, and at all times the tangent space at $z$ is transverse to $K_{\pm}$. Thus we obtain the required interpolation on a (possibly smaller) neighbourhood of $z$.
\end{proof}
Having the interpolating family of branes, we wish to describe it as a Hamiltonian flow. For this, we return to the groupoid chart $t^{-1}(U) \cap s^{-1}(U)$ where we have the following data:
\begin{enumerate}
\item A holomorphic symplectic groupoid $(\Sigma(U), \Omega_{-})$ integrating $(U, \sigma_{-})$, with underlying imaginary part the smooth real symplectic groupoid $(\Sigma(U), \omega)$;
\item Over a fixed neighbourhood of $z$, $W \subseteq U$, we have a family of Lagrangian bisections of $(\Sigma(U), \omega)$, viewed as sections of the source $s$,
\[
\lambda_{t} : W \to (\Sigma(U), \omega),
\]
such that $\lambda_{0}$ is the identity bisection and $\lambda_{1}$ is the given brane bisection $\cal{L}$ viewed in the groupoid chart. Note that $\lambda_{t}(z) = z$ for all time $t$.
\end{enumerate}
Now let $\psi_{t} := t \circ \lambda_{t}$ denote the resulting family of Poisson diffeomorphisms (for the Poisson structure $Q$), and let $W_{t} = \psi_{t}(W)$. Left multiplication by the bisection defines the following family of symplectomorphisms:
\[
\tau_{t} : t^{-1}(W) \to t^{-1}(W_{t}), \ g \mapsto \lambda_{t}(t(g)) \ast g,
\]
which satisfy $s \circ \tau_{t} = s$ and $t \circ \tau_{t} = \psi_{t} \circ t$, as well as $\tau_{t} \circ \epsilon = \lambda_{t}$. This family of symplectomorphisms defines a time-dependent vector field $Y_{t} \in \mathcal{X}^{1}(t^{-1}(W_{t}))$ via the equation
\[
Y_{t}(\tau_{t}(g)) = \frac{d}{dt} \tau_{t}(g).
\]
As explained in \cite{xu1997flux} this family of vector fields is Hamiltonian for a $t$-basic closed $1$-form:
\[
\iota_{Y_{t}}\omega = t^{\ast}(\alpha_{t}),
\]
where $\alpha_{t} \in \Omega^{1}(W_{t})$ is a closed time-dependent 1-form. In fact, the restriction of $Y_{t}$ to the bisection gives the following vector field along $\lambda_{t}$:
\[
X_{t}(x) := Y_{t}(\lambda_{t}(x)) = \frac{d}{dt} \lambda_{t}(x).
\]
Since $\lambda_{t}$ is a section of $s$, $X_{t} \in \text{ker}(ds)$ and hence $\omega(X_{t})$ is in the image of $t^{\ast}$, defining the form $\alpha_{t}$. Note that since $t$ is a Poisson map we have
\[
t_{\ast}(Y_{t}) = Q(\alpha_{t}),
\]
and so $\psi_{t}$ is the flow of this Hamiltonian vector field. The time-dependent form $\alpha_{t} : [0,1] \to \Omega_{cl}^{1}$ is the infinitesimal version of the family of branes $\lambda_{t} : [0,1] \to \mathrm{Bis}^\cal{L}(\Sigma)$.
Now choose a neighbourhood $W'$ of $z$ such that $W' \subseteq W_{t}$ for all time $t$. Restricting to this neighbourhood, we have $\alpha_{t} \in \Omega^{1,cl}(W')$, and if we assume that $W'$ is contractible then $\alpha_{t}$ are exact. Choose a primitive: let $f_{t} \in C^{\infty}(W')$ be a time-dependent function such that $df_{t} = \alpha_{t}$. We conclude that it is now possible to describe the GK structure purely in terms of this function and the holomorphic Poisson structure $(I_{-}, \sigma_{-})$:
\begin{theorem} \label{GKlocallyflow}
Let $(Z, \Omega, \cal{L})$ be a Generalized K\"ahler structure of symplectic type, viewed as a holomorphic symplectic Morita equivalence with brane bisection going between holomorphic Poisson structures $(X_{\pm}, \sigma_{\pm})$ with common imaginary part $-\frac{1}{4}Q$. Let $z \in \cal{L}$ be a chosen point on the brane. Then
\begin{enumerate}
\item It is possible to choose a family of local brane bisections $\cal{L}_{t}$ such that $\cal{L}_{1} = \cal{L}$, $\cal{L}_{0} = \Lambda$, a holomorphic Lagrangian bisection, and such that $z \in \cal{L}_{t}$ for all $t\in[0,1]$.
\item There is a (locally defined) time-dependent real-valued function $f_{t} \in C^{\infty}(X_{-}, \bb{R})$ such that in a neighbourhood of $z$ the family of brane bisections $\cal{L}_{t}$ is given by $\tilde{\tau}_{t}(\Lambda)$, where $\tilde{\tau}_{t}$ is the flow of the Hamiltonian vector field of $(\phi^{-1} \pi_{+})^{*} f_{t}$ with respect to the symplectic form $\Im (\Omega)$.
\item In a neighbourhood of $\pi_{-}(z)$ the GK structure is given by the data $(I_{+}, I_{-}, Q, F)$, where $I_{-}$ is the given complex structure on $X_{-}$, $I_{+} = \psi_{1}^{\ast}(I_{-})$ and
\[
F = \int_{0}^{1} \psi_{t}^{\ast}( d_{-}^{c} d f_{t}) dt,
\]
for $\psi_{t}$ the flow of the Hamiltonian vector field $X_{f_{t}} = Q(df_{t})$, and where $d_{-}^{c} = i (\bar{\partial}_{I_{-}} - \partial_{I_{-}})$.
\end{enumerate
In other words, in a neighbourhood of any point, a GK structure of symplectic type is determined by a holomorphic Poisson structure, together with a time-dependent real-valued function, via the Hamiltonian flow construction of Section~\ref{hamflo}.
\end{theorem}
\begin{proof}
It remains only to prove the formula for $F = \lambda_{1}^{\ast} \Omega_{-}$. Differentiating the pullback $\lambda_{t}^{\ast} \Omega_{-}$ gives
\[
\frac{d}{dt}(\lambda_{t}^{\ast} \Omega_{-}) = \lambda_{t}^{\ast} \cal{L}_{Y_{t}}\Omega_{-} = \lambda_{t}^{\ast} d I^{\ast} \omega(Y_{t}) = \lambda_{t}^{\ast} d I^{\ast} t^{\ast} \alpha_{t} = \psi_{t}^{\ast} d I_{-}^{\ast} \alpha_{t} = \psi_{t}^{\ast} d_{-}^{c} \alpha_{t} = \psi_{t}^{\ast} (d_{-}^{c} d f_{t}).
\]
Therefore $\lambda_{1}^{\ast} \Omega_{-} = \int_{0}^{1} \frac{d}{dt}(\lambda_{t}^{\ast} \Omega_{-}) dt = \int_{0}^{1} \psi_{t}^{\ast}( d_{-}^{c} d f_{t}) dt$.
\end{proof}
\begin{example}
Example \ref{Kahlerex} of a K\"ahler structure showed how the generalized K\"ahler potential of Section~\ref{secpot} generalizes the usual K\"ahler potential. However, another look at this example shows that what we were doing is actually an instance of what is described in the present section. Namely, given a holomorphic Lagrangian $L$, viewed as a section $\lambda$, we used the groupoid structure to induce a local isomorphism of Morita equivalences:
\[
(T^{\ast}X, \Omega_0) \to (Z, \Omega), \qquad \alpha_{x} \mapsto \alpha_{x} + \lambda(x),
\]
so that our Darboux chart in this case is actually a groupoid chart in the sense of~\eqref{grpcht}. Now we view the brane bisection $\cal{L}$ in this chart as the graph of a $(1,0)$-form $\eta = I^{\ast} \alpha + i \alpha$, where $\alpha$ is a closed $1$-form. For our interpolating family of branes, we may choose $\cal{L}_{t} = Gr(t \eta)$. Then the induced family of closed $1$-forms on $X$ is given by the time-independent form $\alpha$. As in example \ref{Kahlerex}, we choose a real-valued function $K$ on $X$ such that $\alpha = - \frac{1}{2} dK$. Then noting that the Hamiltonian vector field $X_{K} = 0$, and appealing to theorem \ref{GKlocallyflow}, we see that the K\"ahler form is given by
\[
\omega = -\frac{1}{2} d^{c} d K = i \partial \bar{\partial} K.
\]
Therefore the construction of Theorem \ref{GKlocallyflow} provides an alternate generalization of the K\"ahler potential which is more closely adapted to the underlying groupoid structure.
\end{example}
\bibliographystyle{hyperamsplain}
|
{
"timestamp": "2018-04-17T02:11:45",
"yymm": "1804",
"arxiv_id": "1804.05412",
"language": "en",
"url": "https://arxiv.org/abs/1804.05412"
}
|
\section{Introduction}
Research on comics have been done independently in several research fields such as document image analysis, multimedia, human-computer interaction, etc. with different sets of values.
We propose to review the research of all of these fields and to organize them in order to understand what is possible to do about comics with the state of the art methods.
We also give some ideas about the future possibility of comics research.
We introduced a brief overview of comics research in computer science~\cite{augereau2017overview} during the second edition of the international workshop on coMics ANalysis, Processing and Understanding (MANPU).
The first edition of MANPU workshop took place during ICPR 2016 (International Conference on Pattern Recognition) and the second one took place during ICDAR 2017 (International Conference on Document Analysis and Recognition).
It shows that comics can interest a large variety of researchers from pattern recognition to document analysis.
We think that the multimedia and interface communities could have some interest too, so we propose to present the research about comics analysis with a broader view.
In the next part of the introduction we will explain the importance of comics and its impact on the society with a brief overview of the open problems.
\subsection{Comics and society}
Comics in the USA, mangas in Japan or bandes dessin\'ees in France and Belgium are graphic novels which have a worldwide audience.
They are respectively an important part of the American, Japanese and Francophone cultures. They are often considered as a soft power of these countries, especially mangas for Japan~\cite{lam2007japan,hall2013struggle}.
In France, bandes dessin\'ee is considered as an art, and is commonly refereed as the ``ninth art''~\cite{Screech2005} (as compared to cinema which is the seventh art).
However, several years ago it was not the case.
Comics was considered as ``children literature'' or ``sub-literature'' as it contains a mixture of images and text.
But more lately comics got a great deal of interest when people recognized it as a complex form of graphic expression that can convey deep ideas and profound aesthetics \cite{Christiansen2000}.
The market of comics is large.
According to a report published in February 2017 by ``The All Japan Magazine and Book Publisher's and Editor's Association'' (AJPEA), the sale of mangas in Japan represents 445.4 billion yens (around 4 billion dollars) in 2016\footnote{\url{http://www.ajpea.or.jp/information/20170224/index.html}}.
In this report, we can see that the market is stable between 2015 and 2014, but a large progression of the digital market can be observed: it almost doubled from 2014 to 2016.
The digital format has several advantages for the readers: it can be displayed on smartphones or tablets and be read anytime, anywhere.
For the editors, the cost of publication and distribution is much lower as compared to the printed version.
However, even if the format changed from paper to screen, no added value has been proposed to the customer.
We think that the democratization of digital format is a good opportunity for the researchers from all computer science fields to propose new services such as augmented comics, recommendation systems, etc.
\begin{figure}[t]
\centering
\includegraphics[width=0.95\linewidth]{figures/map.png}
\caption{We arranged the comics research into three inter-dependent categories: 1) content analysis, 1) content generation, and 3) user interaction. }
\label{map}
\end{figure}
\subsection{Research and open problems}
The research about comics is quite challenging because of the nature of this medium.
Comics contains a mixture of drawings and text.
To fully analyze and understand the content of comics, we need to consider natural language processing to understand the story and the dialogues; and computer vision to understand the line drawings, characters, locations, actions, etc.
A high-level analysis is also necessary to understand events, emotions, storytelling, the relations between the characters, etc.
A lot of related research has been done for covering similar aspect for the case of natural images (i.e. photographic imagery) and videos by classic computer vision.
However, the hight variety of drawings and low availability of labeled dataset make the task harder than natural images.
We organized the research about comics in the three following main categories, as illustrated in Fig.~\ref{map}:
\begin{enumerate}
\item content analysis: getting information about raw images and extracting from high to low-level structured descriptions,
\item content generation: comics can be used as an input or output to generate new contents. Content conversion and augmentation are possible from comics to comics, comics to other media, other media to comics;
\item user interaction: analyzing human reading behavior and internal states (emotions, interests) based on comics contents, and reciprocally, analyzing comics contents based on human behavior and interactions.
\end{enumerate}
Research about comics in computer science has been done covering several aspects but is still an emerging field.
Much research has been done by researchers from the DIA (Document Image Analysis) and AI (Artificial Intelligence) communities and focuses on content analysis, understanding, and segmentation.
Another part of the research is addressed by graphics and multimedia communities and consists in generating new contents or enriching existing contents such as adding colors to black and white pages, creating animation, etc.
The last aspect concerns the interaction between users and comics which is mainly addressed by the HCI (Human-Computer Interaction) researchers.
All these three parts are inter-dependent: segmenting an area of a comic page is important if we want to manipulate and modify it, or if we want to know which area the user is interacting with.
Analyzing the user behavior can be used to drive the content changes or to measure the impact of these changes on the user.
\begin{figure*}[t]
\centering
\includegraphics[width=1\linewidth]{figures/manga2.jpg}
\caption{Example of two double comic pages.
The largest black rectangle encloses two pages which are represented by blue rectangles.
The red rectangles are examples of panel frames. They can be small, large, overlapping each other and have different shapes. Some panels do not have frames and some others can be drawn on more than a page.
The green rectangles are examples of gutters, a white area used for separating two panels.
The yellow ellipses are examples of dialogue balloons. They can have different shapes to represent the feelings of the speaker.
The purple triangles are examples of onomatopoeias, they represent the sound made by people (such as footsteps) or object (water falling, metal blades knocking each other), etc.
Source: images extracted from the Manga109 dataset~\cite{Fujimoto2016}, \textcopyright Sasaki Atsushi. }
\label{manga}
\end{figure*}
In Section 3 We will state in more detail the current state-of-the-art and discuss the open problems.
Large datasets with ground truth information such as layout, characters, speech balloon, text, etc. are not available so using deep learning is hardly possible in such conditions and most of the researchers proposed handcrafted features or knowledge-driven approaches until the very recent years.
The availability of tools and datasets that can be accessed and shared by the research communities is another very important aspect to transform the research about comics, we will talk about the major existing tools and datasets in Section 4.
In the next parts of the paper all the research which are applied to comics, mangas, bande dessin\'ees or any graphics novels will be referred as ``comics'' in order to simplify the reading.
We start the next section of the paper with general information about comics.
\section{What is comics?}
The term comics (as a singular uncountable noun) refers to the comics medium; such as television, radio, etc. comics is a way to transfer information.
We can also refer to a comic (as a countable noun), in this case, we refer to the instance of the medium such as a comic book or a comic page.
As for any art, there are strictly no rules for creating comics.
The authors are free to draw whatever and however they want.
Still, some classic layouts or patterns are usually used by the author as they want to tell a story, transmit feelings and emotions, and drive the attention of the readers~\cite{Jain2012}.
The author needs experience and knowledge to drive smoothly the attention of the readers through the comics~\cite{Cao2014}.
Furthermore, the layout of comics is evolving over time~\cite{pederson2016changing}, moving away from conventional grids to a more decorative and dynamic way.
Usually, comics are printed on books and can be seen as a single or double pages.
When the book is opened, the reader can see both pages so some authors use this physical layout as part of the story: some drawings can be spread in two pages, and when the reader turn one page something might happen in the next page.
Figure~\ref{manga} illustrates a classic comics content.
A page is usually composed of a set of panels defining a specific action or situation.
The panels can be enclosed in a frame and separated by a white space area named gutter.
The reading order of the panels depends on the language.
For example, in Japanese (see Fig.~\ref{order}), the reading order is usually from right to left and top to bottom.
Speech balloons and captions are included in the panel to describe conversations or the narration of the story.
The dialog balloons also have a specified reading order which is usually the same as the reading order of the panels.
Some sound effects or onomatopoeias are often included to give more sensations to the reader such as smell or sound.
Japanese comics often contains ``manpu'' (see Fig.~\ref{manpu}) which are symbols used to visualized feelings and sensations of the characters such as sweating marks on the head of a character to show that he feels uncomfortable even if he is not actually sweating.
The authors are free to draw the characters as they want, so they can be deformed or disproportioned as illustrated in Fig.~\ref{faces}. In some genres such as fantasy, the characters can also be non-human which makes the segmentation and recognition task challenging.
There are also many drawing effects such as speed lines, focusing lines, etc.
For example, in Fig~\ref{manga}, a texture surrounding the female character in the lower-right panel represents her warm atmosphere as contrasted with the cold weather.
\begin{figure}[t]
\centering
\includegraphics[width=0.95\linewidth]{figures/manpu.png}
\caption{Examples of ``manpu'': a mark used to intensify the emotions of the characters such as concentration, anger, surprise, embarrassment, confidence, etc.
The original images are extracted from the Manga109 dataset~\cite{Fujimoto2016}, \textcopyright Yoshi Masako, \textcopyright Kobayashi Yuki, \textcopyright Arai Satoshi, \textcopyright Okuda Momoko, \textcopyright Yagami Ken. }
\label{manpu}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=0.3\linewidth]{figures/order.png}
\caption{Example of the reading order of a Japanese manga. Image under GNU Free Documentation License: \protect\url{ https://commons.wikimedia.org/wiki/File:Manga_reading_direction.svg}}
\label{order}
\end{figure}
Even if more and more digitized versions of the printed version are available few comics are produced digitally and taking advantage of the new technology.
Figure~\ref{protanopia} illustrates an example of digital comics taking advantage of tablet functions: the images are animated continuously and the user can tilt the tablet to control the camera angle.
This comics is created by Andre Bergs\footnote{\url{http://andrebergs.com/protanopia}} and is freely available on App store and Google Play.
We imagine that in the future, it could be possible to create such interactive comics automatically.
\begin{figure}[t]
\centering
\includegraphics[width=0.9\linewidth]{figures/protanopia.jpg}
\caption{The digital comic ``Protanopia" created by Andre Bergs. The reader can control the camera angle by tilting the screen. The panel are animated with continuous loops. Image extracted from the video on: http://andrebergs.com/protanopia}
\label{protanopia}
\end{figure}
\section{Comics research}
We organized the studies done about comics in computer science into three main categories that we will present in this section.
One of the main research fields focuses on analyzing the content of comics images, extracting the text, the characters, segmenting the panels, etc.
Another category is about generating new content from or for comics.
The last category is about analyzing the reader's behavior and interaction with comics.
\subsection{Content analysis}
In order to understand the content of comics and to provide services such as retrieval or recommender systems, it is necessary to extract the content of comics.
The DIA community started to cover this problem with classic approaches.
Images can be analyzed from the low levels such as screentones~\cite{ito2015separation} or text~\cite{Arai2011} to the high level such as style~\cite{Chu2016} or genre~\cite{daiku2017comic} recognition.
Some elements are interdependent; for example finding the text and speech balloons, as one can contain the other.
But also the positions can be relative to each other, as the speech balloon is usually coming from the mouth of a character.
These elements are usually grouped inside a panel, but not necessarily.
As the authors are free to draw whatever and however they want, there is a wide disparity among all comics which make the analysis complex.
For example, some authors exaggerate the facial deformation of the face of a character to make him angrier or more surprised.
We present the related work from the low level to high-level analysis as follow.
\subsubsection*{Textures, screentones, and structural lines}
Black and white textures are often used to enrich the visual experience of non-colored comics.
It is especially used for creating an illusion of shades or colors.
However, the identification and segmentation of the textures is challenging as they can have various forms and are sometimes mixed with the other parts of the drawing.
Ito et al. proposed a method for separating the screentones and line drawings~\cite{ito2015separation}.
More recently, Liu et al.~\cite{Liu2017} proposed a method for segmenting the textures in comics.
Extracting the structural lines of comics is another challenging problem which is related to the analysis of the texture.
The result of such an analysis is displayed in Fig.~\ref{texture}.
The difference between structural lines and arbitrary ones must be considered carefully.
Li et al.~\cite{Li2017} recently proposed a deep network model to handle this problem.
Finding textures and structural lines is an important analysis step to generate colorized and vectorized comics.
\begin{figure}[t]
\centering
\includegraphics[width=1\linewidth]{figures/texture.png}
\caption{Structural line extraction. For each pair of images, the one on the left is the original image, the one on the right is obtained after removing the textures and detecting the structural lines by Li et al. algorithm~\cite{Li2017}.
Downloaded from: \protect\url{http://exhibition.cintec.cuhk.edu.hk/exhibition/project-item/manga-line-extraction/}. }
\label{texture}
\end{figure}
\subsubsection*{Text}
The extraction of text (such as Latin or Chinese) characters has been investigated by several researchers but is still a difficult problem as many authors write the text by hand.
Arai and Tolle~\cite{Arai2011} proposed a method to extract frames, balloon, and text based on connected components and fixed thresholds about their sizes.
This is a simple approach which works well for ``flat'' comics, i.e. conventional comics where each panel is defined by a black rectangle and has no overlapping parts.
Rigaud et al. also proposed a method to recognize the panels and text based on the connected components~\cite{Rigaud2013}.
By adding some other features such as the topological and spatial relations, they successfully increased the performance of~\cite{Arai2011}.
More recently, Aramaki et al. combined connected component and region-based classifications to make a better text detection system~\cite{Aramaki2016}.
A recent method also addresses the problem of speech text recognition~\cite{rigaud2017segmentation}.
In order to simplify the problem, Hiroe and Hotta have proposed to detect and count the number of exclamation marks in order to represent a comic book by its distribution of exclamation marks or to find the scene changes~\cite{hiroe2017histogram}.
\subsubsection*{Faces and pose}
One of the most important elements of comics is the characters (persons) of the story.
However, identifying the characters is challenging because of the posture, occlusions, and other drawing effects.
Also, the characters can be humans, animals, robots or anything with various drawing representations.
Sun et al.~\cite{Sun2013} proposed to locate and identify the characters in comics pages by using local feature matching.
New methods have recently been proposed to recognize the face and characters in comics based on deep neural networks~\cite{Chu2017,qin2017faster,nguyen2017comic}.
\begin{figure}[t]
\centering
\includegraphics[width=1\linewidth]{figures/faces.png}
\caption{Different examples of comics character faces. Some part of the face such as the nose, eyes, or mouth can be deformed to emphasize the emotion of the character.
The original images are extracted from the Manga109 dataset~\cite{Fujimoto2016}, \textcopyright Kurita Riku, \textcopyright Yamada Uduki, \textcopyright Tenya. }
\label{faces}
\end{figure}
Estimating the pose of the character is another challenge.
As we can see in Fig.~\ref{pose}, if the characters have human proportion and are not too deformed, they can be well recognized by a popular approach such as Open Pose~\cite{cao2017realtime}.
Knowing the character poses could lead to activity recognition, but a method such as Open Pose will fail on almost all comics.
\begin{figure}[t]
\centering
\includegraphics[width=1\linewidth]{figures/pose.jpg}
\caption{Example of application of Open Pose on comics~\cite{cao2017realtime}. This model works well for comics as long as the drawings are realistic, so it fail in most cases. Source: images extracted from the Manga109 dataset~\cite{Fujimoto2016}, \textcopyright Yoshi Masako, \textcopyright Kanno Hiroshi.}
\label{pose}
\end{figure}
\subsubsection*{Balloons}
The balloons are an important component of comics where most of the information is conveyed by the discussion between the protagonists.
So one important step is to detect the balloons~\cite{Correia2016} and then to associate the balloons to the speaker ~\cite{Rigaud2015a}.
The shape of the balloon conveys also information about the speaker feelings~\cite{yamanishi2017speech}. For example, a balloon with wavy shape represents anxiety, an explosion shape represents the anger, a cloudy shape represents joy, etc.
\subsubsection*{Panel}
The layout of a comics page is described by Tanaka et al. as a sequence of frames named panels~\cite{Tanaka2007}.
Several methods have been proposed to segment the panels, mainly based on the analysis of connected components~\cite{Arai2010},~\cite{Rigaud2013} or on the page background mask~\cite{Pang2014}.
As these methods based on heuristics rely on white backgrounds and clean gutters, Iyyer et al. recently proposed an approach based on deep learning~\cite{Iyyer2017} to process eighty-year-old American comics.
\begin{figure}[t]
\centering
\includegraphics[width=1\linewidth]{figures/illust2vec.jpg}
\caption{Example application of illustration2vec~\cite{saito2015illustration2vec}. The model recognize several attributes of the character such as her haircut and clothes. The web demo used to generate this image is not online anymore.}
\label{illust2vec}
\end{figure}
\subsubsection*{High level understanding}
Rigaud et al. proposed a knowledge-driven system that understands the content of comics by segmenting all the sub-parts~\cite{Rigaud2015}.
But understanding the narrative structure of comics is much more than simply segmenting its different sub-parts.
Indeed, the reader makes inferences about what is happening from one frame to another by looking at all graphical and textual elements~\cite{McCloud1993}.
Iyyer et al. introduced some methods to explore how readers connect panels into a coherent story~\cite{Iyyer2017}.
They show that both text and images are important to guess what is happening in a panel by knowing the previous ones.
Daiku et al.~\cite{daiku2017comic} proposed to analyze the comics storytelling by analyzing the genre of each page of the comics.
Then the story of a comic book is represented as a sequence of genres such as: ``11 pages of action'', ``5 pages of romance'', ``8 pages of comedy'', etc.
Analyzing the text of the dialogues and stories has not been investigated yet specifically for comics. Similar research as sentiment analysis~\cite{mohammad2016sentiment} could be applied to analyze the psychology of the characters or to analyze and compare the narrative structure of different comics.
From the cognitive point of view, Cohn proposed a theory of ``Narrative Grammar'' based on linguistics and visual language which are leading the understanding process~\cite{Cohn2013}.
A lot of information is inferred by the reader who is constructing a representation of the depicted pictures in his mind. This is how we can recognize that two characters drawn slightly in a different way are the same, or that a character is doing an action by looking at a still image. These concepts must be inferred by the computer too, in order to obtain a high-level representation of comics.
\subsubsection*{Applications}
From these analyses, retrieval systems can be built, and some have already been proposed in the literature such as sketch~\cite{Matsui2016,narita2017sketch} or graphs based~\cite{Le2016} retrieval.
The drawing style has also been studied~\cite{Chu2016}.
The possible applications are artist retrieval, art movement retrieval, and artwork period analysis.
Saito and Matsui proposed a model for building a feature vector for illustrations named illustration2vec~\cite{saito2015illustration2vec}.
As showed on Fig.\ref{illust2vec}, this model can be used to predict the attributes of a character such as its hair or eye color, the size of the hair, the clothes worn by the character, etc. and to research specific illustrations.
Vie et al. proposed a recommender system using the illustration comics covers based on illustration2vec in a cold-start scenario~\cite{vie2017using}.
\subsubsection*{Conclusion (content analysis)}
Segmenting the panels or reading the text of any comics is still challenging because of the complexity of some layouts and the diversity of the content.
Figure~\ref{manga} illustrates the difficulty of segmenting the panels.
Most of the current methods focus on using handcrafted features for the segmentation and analysis and will fail on an unusual layout.
The segmentation of faces and body of the characters is still an open problem and a large amount of labeled data will be necessary to adapt the deep learning approaches.
Even if the text contains very rich information, surprisingly few methods have been proposed to analyze the storyline or the content of comics based on the text.
Also, some parts of comics has not been addressed at all, such as the detection of onomatopoeias.
The future research about high-level information should be more considered as it can be used to represent information that could interest the reader such as the style or genre, the storytelling, etc.
\subsection{Content generation}
The aim of content generation or enrichment is to use comics to generate new content either based on comics or other media.
\subsubsection*{Vectorization}
As most of comics are not created digitally, vectorization is a way to transform scanned comics to a vector representation for real-time rendering with arbitrary resolution~\cite{Yao2017}.
Generating vectorized comics is necessary for visualizing them nicely in digitized environments.
This is also an important step for editing the content of comics and one of the basic step of comics enrichment~\cite{Zhang2009}.
\subsubsection*{Colorization}
Several methods have been proposed for automatic colorization~\cite{Qu2006,Sato2014,cinarelZ17,furusawa2017comicolorization,zhang2017style} and color reconstruction~\cite{Kopf2012}, as comics with colors can be more attractive for some readers.
Colorization is quite a complex problem as the different parts of a character such as his arms, hands, fingers, face, hair, clothes, etc. must be retrieved to color each part in a correct way.
Furthermore, the poses of a character can be very different from each other: some parts can appear, disappear or be deformed.
An example of colorization is displayed in Fig.~\ref{color}.
\begin{figure}[t]
\centering
\includegraphics[width=0.9\linewidth]{figures/color.jpg}
\caption{Example of colorization process based on style2paints. Image downloaded from \protect\url{https://github.com/lllyasviel/style2paints}.}
\label{color}
\end{figure}
Recently, deep learning based colorization approach has been used for creating color version manga books which are distributed by professional companies in Japan\footnote{\url{https://www.preferred-networks.jp/en/news/pr20180206}}.
\subsubsection*{Comics and character generation}
One problem for generating comics is to create the layout and to place the different components such as the characters, text balloons, etc. at a correct position to provide a fluid reading experience.
Cao et al. proposed a method for creating stylistic layout automatically \cite{Cao2012} and then another one for placing and organizing the elements in the panels according to high-level user specification \cite{Cao2014}.
The relation between real-life environment or situations and the one represented in comics can be used to generate or augment comics.
Wu and Aizawa proposed a method to generate a comics image directly from a photograph~\cite{Wu2014}.
At the end of 2017, Jin et al.~\cite{jin2017towards} presented a method to generate automatically comics characters. An example of a generated character by their online demo\footnote{\url{http://make.girls.moe/#/}} is displayed Fig.~\ref{moe}.
The result of the generation is not always visually perfect, but still, this is a powerful tool as an unlimited number of characters can be generated.
\begin{figure}[t]
\centering
\includegraphics[width=1\linewidth]{figures/moe.jpg}
\caption{Example of random character generation based Jin et al. method~\cite{jin2017towards}. In this example, we set some attributes such as green hair color, blue eyes, smile and hat.}
\label{moe}
\end{figure}
\subsubsection*{Animation}
As comics are still images, a way to enhance the visualization of comics is to generate animations.
Recently, some researchers proposed a way for animating still comics images through camera movements~\cite{Cao2017,Jain2016}.
Several animation movies and series have been adapted in comics paper book and vice versa.
Some possible outlook could be to generate an animated movie from a paper comics or a paper comics from an animated movie.
\begin{figure}[t]
\centering
\includegraphics[width=1\linewidth]{figures/smile.png}
\caption{Example of smiling animation in the conceptual space~\cite{white2017generating}. Similar animation could be obtained for comics images. Image source: \protect\url{https://vusd.github.io/toposketch/}}
\label{smile}
\end{figure}
For the natural images, some methods have been proposed to animate the face of people by using latent space interpolations.
As illustrated in Fig.~\ref{smile} the latent vectors can be computed for a neutral and smiling face to generate a smiling animation~\cite{white2017generating}.
Another application is to use extract the facial keypoints and to use another source (text, speech, or face) to animate the mouth of the character.
For example, this has been done for generating photorealistic video of Obama speech based on a text input~\cite{kumar2017obamanet}.
\subsubsection*{Media conversion}
More broadly, we can imagine to convert text, videos, or any content into comics, and vice-versa.
This problem can be seen as media conversion.
For example, Jing et al. proposed a system to convert videos to comics \cite{Jing2015}.
There are many challenges to do a successful conversion: summarizing the videos, stylizing the images, generating the layout of comics and positions of text balloons.
An application which as not been done to comics but to natural videos is to add generated sound to a video~\cite{zhou2017visual}. No application has been done for comics, but we could imagine a similar application to generate sound effects (swords which are banging to each other, a roaring tailpipe, etc.) or atmosphere sounds (village, countryside, crowd, etc.).
Creating a descriptive text based on comics or generating comics based on descriptive text could be possible in the future, as it has been done for the natural images.
Reed et al.~\cite{reed2016generative} proposed a method for automatic synthesis of realistic natural images from text.
We can also imagine changing the content, adding or removing some parts, changing the genre or style depending on the user or author preference.
\subsubsection*{Conclusion (content generation)}
In order to generate contents, some model or labeled data are necessary. In order to generate automatically characters, Jin et al. used around 42000 images.
Deep learning approaches such as Generative Adversarial Networks (GAN)~\cite{goodfellow2014generative} has been widely used for natural image applications such as style transfer~\cite{zhu2017unpaired}, reconstructing 3D models of objects from images~\cite{wu2016learning}, generating images from text~\cite{reed2016generative}, editing pictures~\cite{zhu2016generative}, etc.
These applications could be done for comics too.
Another possibility to enhance comics is to add other modes such as sound, vibrations, etc.
Adding sounds should be easily possible by using the soundtracks from animation movies.
But, in order to be able to produce these effects at a correct timing, information about the user interactions is necessary.
This is possible by using an eye tracker or detecting when the user turns a specific page in real time.
\subsection{User interaction}
Apart from the content analysis and generation, we have identified another category of research based on the interaction between users and comics.
One part consists of analyzing the user himself instead of analyzing comics.
For example, we would like to understand or predict what the user feels or how he behaves while reading comics.
Another part consists in creating new interfaces or interactions between the readers and comics.
Also, new technology can be used to improve the access for impaired people.
\begin{figure}[t]
\centering
\includegraphics[width=0.9\linewidth]{figures/eye.jpg}
\caption{One the left: eye gaze fixations (blue circles) and saccades (segment between circles) of one reader. One the right: heat map accumulated over several readers; the red color corresponds to longer fixation time. }
\label{eye}
\end{figure}
\subsubsection*{Eye gaze and reading behavior}
In order to know where and when a user is looking at some specific parts of a comic, researchers are using eye tracking systems.
By using eye trackers it is possible to detect how long a user spends to read a specific part of a comic page.
Knowing the user reading behavior and interest is an important information that can be used by the author or editors as a feedback.
It also can be used to provide other services to readers such as giving more details about the story of a character that a specific user likes, removing part of battle if he does not likes violence, etc.
Carroll et al. ~\cite{carroll1992visual} showed that the readers tend to look at the artworks before reading the text.
Rigaud et al. found that, in France, the readers spend most of the time at reading the text and looking at the face of the characters~\cite{Rigaud2016}. The same experiment repeated in Japan lead to the same conclusion, as illustrated in Fig.~\ref{eye}.
Another way to analyze how the readers understand the content of comics is to ask them to manually order the panels.
Cohn presented different kinds of layouts with empty panels and showed that various manipulations to the arrangement of panels push readers to navigate panels in alternate routes~\cite{cohn2015navigating}.
Some cognitive tricks can ensure that most of the readers will follow the same reading path.
In order to augment comics with new multimedia contents such as sounds, vibration, etc. it is important to trigger these effects at a good timing.
In this case, detecting when the user turns a page or estimating which position he is looking at will be useful.
\subsubsection*{Emotion}
Comics contains exciting contents. Many different genres of comics exist such as comedy, romance, horror, etc. and trigger different kinds of emotions to the readers.
Much research has been done on emotion detection based on face image and physiological signals such as electroencephalogram (EEG) while watching videos~\cite{koelstra2012deap,soleymani2012multimodal,soleymani2016analysis}.
However such research has not been conducted while reading comics.
We think that analyzing the emotion while reading might be more challenging as movie contain animations and sounds that might stimulate more the emotions of the user.
By recording and analyzing the physiological signals of the readers as illustrated in Fig.~\ref{e4}; Lima Sanches et al. showed that it is possible to estimate if the user is reading a comedy, a romance or a horror comics, based on the emotions felt by the readers~\cite{Sanches2016}. For example, when reading a horror comic book, the user feels stressful and his skin temperature is decreasing.
Emotions are usually represented as two axes: arousal and valence, where the arousal represents the strength of the emotion and the valence relates to a positive or negative emotion.
Matsubara et al. showed that by analyzing the physiological signals of the reader, it is possible to estimate the reader's arousal~\cite{matsubara}.
Both experiments are using the E4 wristband\footnote{\url{https://www.empatica.com/en-eu/research/e4/}} which contains a photoplethysmogram sensor (to analyze the blood volume pulse), an electrodermal activity sensor (to analyze the amount of sweat), an infrared thermopile sensor (to read the peripheral skin temperature), and a 3-axis accelerometer (to captures motion-based activity).
Such device is commonly used for stress detection~\cite{kalimeri2016exploring,greene2016survey}.
\begin{figure}[t]
\centering
\includegraphics[width=0.9\linewidth]{figures/E4.jpg}
\caption{The user wear the E4 wristband, measuring his physiological signals such as heartbeat, skin conductance and skin temperature.}
\label{e4}
\end{figure}
Still, each reader has is own preferences and feels emotions in a different way while reading so these analyses are quite challenging.
Depending on the user state of mind or mood, he might prefer to read content that is eliciting specific kind of emotions.
Emotion detection could be used by author or editors to analyze which content stimulate more the readers.
\subsubsection*{Visualization and interaction}
Comics can be read on books, tablets, smartphones or any other devices.
Visualization and interaction on smartphones can be difficult, especially if the screen is small~\cite{Augereau2016}.
The user needs to zoom and do many operations which can be inconvenient.
Some researchers are also trying to use more interactive devices such as multi-touch tables to attract the users~\cite{Andrews2012}.
Another important challenge is to make comics accessible to impaired people. Rayar~\cite{Rayar} explained that a multidisciplinary collaboration between Human-Computer Interactions, Cognitive Science, and Education Research is necessary to fulfill such a goal.
Up to now, the three main ways to access images for visually impaired people are: audio description, printed Braille description and printed tactile pictures (in relief).
Such way could be generated automatically thanks to new research and technology.
New haptic feedback tablet such as the one proposed by Meyer et al.~\cite{meyer2014dynamics} illustrated in Fig.~\ref{tanvas} could help visually impaired people to access comics.
Others application such as detecting and magnifying the text or moving the comics automatically could be helpful for impaired people.
\begin{figure}[t]
\centering
\includegraphics[width=0.9\linewidth]{figures/tanvas.jpg}
\caption{Tanvas tablet enable the user to feel different textures. This could be used to enhance the interaction with comics. Source: \protect\url{https://youtu.be/ohL_B-6Vy6o?t=19s}}
\label{tanvas}
\end{figure}
\subsubsection*{Education}
It has been proven that the representation of knowledge as comics can be a good way to attract students to read~\cite{Eneh2008} or to learn language~\cite{Sarada2016}.
It could be interesting to measure the impact on the representation of the knowledge.
Comics could be, for some students, a more interesting way to learn, so using comics in education might be a way to augment their attention level and memory if comics are nicely designed.
A challenge related to media conversion is then to transform normal textbooks into comics and to compare the interactions of the students with both books.
\subsubsection*{Conclusion (user interaction)}
The interactions between the user and comics have not been analyzed deeply yet.
Many sensors can be used to analyze the user with respect to brain activity, muscle activity, body movement and posture, heart rate, sweating, breath, pupil dilation, eye movement, etc.
Collecting such information can give more information about the readers and comics.
\section{Available materials}
In this section, we present some tools and datasets which are publicly available for the research on comics.
\subsection{Tools}
Several tools for comics image segmentation and analysis are available on the Internet and can be freely used by anybody, such as:
\begin{itemize}
\item Speech balloon segmentation~\cite{Rigaud2015b},
\item Speech text recognition~\cite{Rigaud2016a},
\item Automatic text extraction cbrTekStraktor\footnote{\url{https://sourceforge.net/projects/cbrtekstraktor/}},
\item Annotation tool to create ground truth label\footnote{\url{http://www.manga109.org/en/tools/}},
\item Semi-Automatic Manga Colorization~\cite{furusawa2017comicolorization}\footnote{\url{https://github.com/DwangoMediaVillage/Comicolorization}},
\item Deep learning library for estimating a set of tags and extracting semantic feature vectors from illustrations~\cite{saito2015illustration2vec},\footnote{\url{https://github.com/rezoo/illustration2vec}}.
\end{itemize}
The speech balloon~\cite{Rigaud2015b} and text segmentation~\cite{Rigaud2016a} algorithms are available on the author's Github\footnote{\url{https://github.com/crigaud}}.
As we can see, even if many papers have been published about comics segmentation and understanding, still few tools are available on the Internet.
To improve the algorithms significantly and being able to compare them, making the code available is an important step for the community.
\subsection{Datasets}
Few dataset has been made publicly available because of copyright issues.
Indeed, it is not possible for researchers to use and share large dataset of copyrighted materials.
So making competition and reproducible research is not easy.
Hopefully, recently, several datasets have been made available.
The Graphic Narrative Corpus (GNC)~\cite{dunst2017graphic} provide metadata information for 207 titles such as the authors, number of pages, illustrators, genres, etc. Unfortunately, the corresponding images are not available because of copyright protections.
So the usefulness of this dataset is very limited.
Still, the authors are willing to share segmentation ground truth and eye gaze data.
However such data has not been released yet.
eBDtheque~\cite{Guerin2013}\footnote{\url{http://ebdtheque.univ-lr.fr/registration/}} contains 100 comic pages, mainly in French language.
The following elements have been labeled on the dataset: 850 panels, 1092 balloons, 1550 characters and 4691 text lines.
Even if the number of images is limited, creating such detailed labeled data is time-consuming and very useful for the community.
\begin{figure}[t]
\centering
\includegraphics[width=1\linewidth]{figures/manga109.png}
\caption{Example of Japanese manga cover page contained in the Manga109 dataset~\cite{Fujimoto2016}.}
\label{manga109}
\end{figure}
Manga109~\cite{Fujimoto2016}\footnote{\url{http://www.manga109.org/index_en.php}} is illustrated in Fig.~\ref{manga109}.
This dataset which contains 109 manga volumes from 93 different authors.
On average, a volume contains 194 pages.
These mangas were published between the 1970's and 2010's and are categorized into 12 different genres such as fantasy, humor, sports, etc.
Only a limited labeled data are available for now such as the text for few volumes. The strong point of this dataset is to provide all pages of one volume which allows analyzing the sequences of pages.
COMICS~\cite{Iyyer2017}\footnote{\url{https://obj.umiacs.umd.edu/comics/index.html}} contains 1,229,664 panels paired with automatic textbox transcriptions from 3,948 American comics books published between 1938 and 1954.
The dataset includes ground truth labeled data such as the rectangular bounding boxes of panels on 500 pages and 1,500 textboxes.
BAM!~\cite{wilber2017bam}\footnote{\url{https://bam-dataset.org}}
contains around 2.5 million artistic images such as: 3D com-
puter graphics, comics, oil painting, pen ink, pencil sketches, vector art, and watercolor. The images contain emotion labels (peaceful, happy,
gloomy, and scary) and object labels (bicycles, birds, buildings, cars, cats,
dogs, flowers, people, and trees).
Figure~\ref{bam} shows a sample of the dataset containing comics.
The dataset is interesting due to the labels and large variety of content and languages. However, the images are just examples provided by the authors and cannot always be understood without the previous or following pages.
BAM!, COMICS, Manga109, and eBDtheques are the four main comics datasets that have been made available with the corresponding images.
Building such datasets is a time and money consuming task, especially for building the ground truth and labeled data.
The main problem to create such dataset comes from the legal and copyright protection which prevent the researchers to make publicly available image datasets.
The content of the dataset is also important depending on the research to proceed. For example, it is interesting to have a variety of comics from different countries, with different languages and genres. It is also interesting to have several continuous pages from the same volumes and several volumes from the same series in order to analyze the evolution of the style of an author, the mentality of the character, or the storyline.
\begin{figure}[t]
\centering
\includegraphics[width=1\linewidth]{figures/bam.jpg}
\caption{Example of comics contained in the BAM! dataset~\cite{wilber2017bam}. From left to right we selected two images containing the following label: bicycles, birds, buildings, cars, dogs, and flowers. }
\label{bam}
\end{figure}
\section{General conclusion}
The research about comics in computer science has been done about several aspects.
We organized the research into three inter-dependent categories: content analysis, content generation, and user interaction.
A mutual analysis of the reader and comics is necessary to understand more about how can we augment comics.
A large part of previous work is focusing on the low-level image analysis by using handcrafted features and knowledge-driven approaches.
Recent research focuses more on deep learning and high-level image understanding.
Still, many applications have been done for natural image and the research about artworks and comics get more attention only very recently~\cite{wilber2017bam}.
A lot of unexplored fields remain, especially, the content generation and augmentation.
Only few companies started to use research for automatic colorization for example, but it is clear that it could be possible to help the authors with content automatic (or semi-automatic) generation of content or animation.
The analysis of the behavior and emotions of the readers have been superficially covered. However, using the opportunity given by new technologies and sensors could be helpful to create the next age of comics. If could be also a way to help the access of comics to impaired people.
For now, few tools and dataset have been made available. Making publicly available copyrighted images is a problem but it would greatly contribute to the improvement of comics research.
\begin{acknowledgements}
The authors would like to thanks the students of the Intelligent Media Processing Group of Osaka Prefecture University who made some of the presented research and illustrations: Yuki Daiku, Mizuki Matsubara, Charles Lima Sanches, Seiichiro Hara, and Yusuke Maeda.
This work is in part supported by JST CREST (JPMJCR16E1), JSPS Grant-in-Aid for Scientific Research (15K12172), JSPS KAKENHI Grant Number (16K16089) and the Key Project
Grant Program of Osaka Prefecture University.
\end{acknowledgements}
\bibliographystyle{spmpsci}
|
{
"timestamp": "2018-04-17T02:13:10",
"yymm": "1804",
"arxiv_id": "1804.05490",
"language": "en",
"url": "https://arxiv.org/abs/1804.05490"
}
|
\section{Applications}
We develop two prototype applications to demonstrate the utility of the recovered shape structure in structure-aware shape editing and processing.
\vspace{-15pt}
\paragraph{Structure-aware image editing.}
In~\cite{zheng_sigg12}, a structure-aware image editing is proposed, where a cuboid structure is
manually created for the object in the image to assist a plausible shape deformation.
With our method, this cuboid structure can be automatically constructed. Moreover,
the part relations are also recovered which can be used to achieve structure-aware editing.
Given an RGB image, we first recover the 3D shape structure of the object of interest using our method.
To align the inferred cuboid structure with the input image,
we train another network to estimate the camera view.
The 3D cuboids are then projected to the image space according to the estimated view.
The object in the image is segmented with a CRF-based method constrained with the cuboid projections~\cite{xu2011photo}.
Each segment is assigned to a 3D cuboid based their image-space overlapping. At this point, the image editing method in~\cite{zheng_sigg12} can be employed to deform the object of interest.
Fig.~\ref{fig:teaser} and~\ref{fig:application_image} show a few examples of structure-aware image editing based on our 3D shape structure recovery.
\input{edit}
\paragraph{Structure-assisted 3D volume refinement.}
A common issue with 3D reconstruction with volumetric shape representation is that the resolution of volume
is greatly limited due to the high computational cost. This usually results in missing parts
and hence broken structure in the reconstructed volume.
Our recovered 3D structures can be used to refine the 3D volumes estimated by existing approaches such as 3D-GAN~\cite{wu2016learning}.
Given a 2D image, a 3D volume is estimated with 3D-GAN and a cuboid structure recovered by our method.
They can be easily aligned with the help of camera view estimation (as have been done above).
Each voxel is assigned to the closest cuboid, leading to a part-based segmentation of the volume.
We then utilize the part symmetry relation in our recovered structure to complete the missing voxels; see results in Fig.~\ref{fig:application_voxel}.
\input{volume}
\section{Conclusion}
We have proposed a deep learning framework that directly recovers 3D shape structures from single 2D images.
Our network joins a structure masking network for decerning the object structure and a structure recovery network for inferring 3D cuboid structure. The recovered 3D structures achieve both fidelity with respect to the input image and plausibility as a 3D shape structure. To the best of our knowledge, our work is the first that recovers detailed 3D shape structures from single 2D images.
Our method fails to recover structures for object categories unseen from the training set. For such cases, it would be interesting to learn an incremental part assembler. Our method currently recovers 3D cuboids only but not the underlying part geometry. A worthy direction is to synthesize detailed part geometry matching the visual appearance of the input image. Another interesting topic is to study the profound correlation between 2D features and 3D structure, so as to achieve a more explainable 3D structure decoding.
\section{Experiments}
We collected a dataset containing $800$ 3D shapes from three categories in ShapeNet: chairs ($500$), tables ($200$), aeroplanes ($100$). The dataset is split into two subsets for training($70\%$) and testing ($30\%$), respectively.
With these 3D shapes, we generate training and testing pairs of image mask and shape structure to train the network and evaluate our method quantitatively. We also evaluate our methods qualitatively with a Google image search challenge. Both quantitative and qualitative evaluations demonstrate the capability of our method in recovering 3D shape structures from single RGB images faithfully and accurately.
\subsection{Training data generation}
\label{train:data}
\paragraph{Image-structure pair generation.}
For each 3D shape, we create $36$ rendered views around the shape for every $30^\circ$ rotation and with $3$ elevations.
Plus another $24$ randomly generated views, we create $60$ rendered RGB images in total for each shape.
The 3D shapes are rendered with randomly selected backgrounds from NYU v2 dataset.
For each RGB image, the ground-truth object mask can be easily extracted using the depth buffer for rendering.
All 3D shapes in our dataset are pre-segmented based on their original mesh components or using the symmetry-aware segmentation proposed in~\cite{wang2011}.
We utilize symmetry hierarchy~\cite{wang2011} to represent the shape structure, which defines how parts in a shape are recursively assembled by connectivity or grouped by symmetry. We adopt the method in~\cite{li2017grass} to infer consistent hierarchy trees for the shapes of each category.
Specifically, we train a unsupervised auto-encoder with the task of self-reconstruction for all shapes. During testing, we use this auto-encoder to perform a greedy search of grouping hierarchy for each shape.
For more details on this process, please refer to the original work.
Consequently, we generate $60$ image-structure pairs for each 3D shape.
\paragraph{Data processing and augmentation.}
To further enhance our dataset and alleviate overfitting, we conduct on each training 3D shape structure-aware deformation~\cite{xu2012fit} based on component-wise controllers~\cite{zheng2011component} to generate a set of structurally plausible variations for the training shape.
Such structure-aware deformation preserves the connection and symmetry relations between shape parts, while maintaining the shape texture for each part. This step is fully automatic and the parameters for each variation generation is randomly set within a given range. In our implementation, we randomly generate $20$ new variations for each 3D shape, thus enlarging our database to $16$K 3D shapes.
For the input images (and the corresponding object masks), we employ the common operations for image data augmentation~\cite{li2017ICCV} such as color perturbation, contrast adjustment, image flip and transformation, etc.
\subsection{Results and evaluation}
\input{result_mask}
We first show in Fig.~\ref{fig:result_mask} some results of object mask prediction by our structure masking network.
As can be seen in the output, the background clutters are successfully filtered out and some detailed structures of the objects are captured.
\paragraph{Google image challenge for structure recovery.}
We first perform a qualitative evaluation on the capability and versatility of our structure recovery.
In order for a more objective study, instead of cherry-picking a few test images, we opt to conduct
a small-scale stress test with a Google image challenge~\cite{xu2011photo}.
During the test, we perform text-based image search on Google using the keywords of ``chair'', ``table''
and ``airplane'', respectively. For each search, we try to recover a 3D cuboid structure for each of the top 8 returned images using our method.
The results are shown in Fig.~\ref{fig:result_internet}.
From the results, we can see that our method is able to recover 3D shape structures from real images in a detailed and accurate way. More importantly, our method can recover the connection and symmetry relations of the shape parts from single view inputs, leading to high quality results with coherent and plausible structure. Examples of symmetry recovery include the reflectional symmetry of chair legs or airplane wings, the rotational symmetry of legs in a swivel chair or a table.
There are some failure cases (marked with red boxes in Fig.~\ref{fig:result_internet}).
The marked chair example is not even composed of multiple parts and hence may not admit a part structure.
When the structure of the object of interest is unseen from our training dataset of 3D shapes, such as the marked table example, our method fails to recover a reasonable structure.
\input{result_shapenet_internet}
\paragraph{Quantitative evaluation.}
We quantitatively evaluate our algorithm with our test dataset.
For the structure masking network, we evaluate the mask accuracy by the overall pixel accuracy
and per-class accuracy against the ground-truth mask (see table~\ref{table:masknetresult}).
We provide a simple ablation study by comparing our method with two baselines:
single-scale (without refinement network) and two-scale (without jump connection).
The results demonstrate the effectiveness of our multi-scale masking network.
\begin{table}[h]\center
\scalebox{0.95}{
\begin{tabular}{| c | c | c | }
\hline
Method & Overall Pixel & Per-Class \\ \hline
single-scale & 0.953 & 0.917 \\ \hline
two-scale (w/o jump) & 0.982 & 0.964 \\ \hline
two-scale (with jump) & 0.988 & 0.983 \\ \hline
\end{tabular}
}
\caption{An ablation study of structure masking network.}
\label{table:masknetresult}
\end{table}
For 3D shape structure recovery, we develop two measures to evaluate the accuracy:
\begin{itemize}
\item \emph{Hausdorff Error}: $ \frac{1}{2T}\sum_{i}^{T} (D(S_i, S^\text{gt}_i) +D(S^\text{gt}_i, S_i))$,
where $S_i$ is a recovered shape structure (represented by a set of boxes) and $S^{\text{gt}}_i$ its corresponding ground-truth. $T$ is the number of models in the test dataset.
$D(S_1, S_2) = \frac{1}{n}\sum_{B^1_j\in{S_1}}\min\limits_{B^2_k\in{S_2}}H(B^1_j, B^2_k)$ measures the
averaged minimum Hausdorff distance from the boxes in structure $S_1$ to those in $S_2$, where
$B^1_j$ and $B^2_k$ represent the boxes in $S_1$ and $S_2$, respectively.
$H(B^1,B^2) = \max\limits_{p\in{B^1}}\min\limits_{q\in B^2} ||p-q||$ is the Hausdorff
distance between two boxes, with $p$ and $q$ being the corner points of box.
Since Hausdorff is asymmetric, the distance is computed for both directions and averaged.
\item \emph{Thresholded Accuracy}: The percentage of boxes $B_{i}$ such that $\delta = H(B_i, B^{*}_i)/L(B^{*}_i) < threshold $, where $B_i$ is the $i$-th box in recovered shape structure $S$ and $B^{*}_i$ its nearest box in the ground-truth $S^\text{gt}$. $H$ is the Hausdorff distance between two boxes as defined above. $L$ is the diagonal length of a box.
\end{itemize}
We consider as our baseline where the structure masking network is simply a vanilla VGG-16 network. In table~\ref{table:comparison}, we compare the accuracy of structure recovery, based on the above two measures, for our method and the baseline. We also compare the two methods where VGG-16 is replaced with VGG-19.
The results demonstrate the significant effect of our structure masking network in helping the structure decoding.
This can also be observed from the reconstruction error plotted in Figure~\ref{fig:Loss} (bottom).
A deeper structure masking network (with VGG-19) also boosts the performance to a certain degree.
\begin{table}[h]\center
\scalebox{0.9}{
\begin{tabular}{| c | c | c | c | }
\hline
\multirow{2}{*}{Method} & \multicolumn{1}{c|}{Hausdorff} & \multicolumn{2}{c|}{Thresholded Acc.} \\ \cline{3-4}
\multicolumn{1}{|r|}{} & \multicolumn{1}{c|}{Error} & $\delta<0.2$ & $\delta<0.1$ \\ \hline
Vanilla VGG-16 & 0.0980 & $ 96.8\% $ & $ 67.8\% $ \\ \hline
Structure masking (VGG-16) & 0.0894 & $ 97.8\% $ & $ 75.3\% $ \\ \hline
Vanilla VGG-19 & 0.0922 & $ 96.4\% $ & $ 72.2\% $ \\ \hline
Structure masking (VGG-19) & 0.0846 & $ 97.6\% $ & $ 78.5\% $ \\ \hline
\end{tabular}
}
\caption{Comparison of structure recovery accuracy over different methods.}
\label{table:comparison}
\end{table}
\paragraph{Comparison.}
In Fig.~\ref{fig:compare_star}, we give a visual comparison of 3D shape reconstruction
from single-view images between our method and two state-of-the-art methods, \cite{huang2015single} and \cite{tulsiani2016learning}.
Both the two alternatives produce part-based representation of 3D shapes, making them comparable to our method.
The method by Huang et al.~\cite{huang2015single} recovers 3D shapes through assembling parts from database
shapes while preserving their symmetry relations.
The method of Tulsiani et al.~\cite{tulsiani2016learning} generates cuboid representation similar to ours,
but does not produce symmetry relations.
As can be seen, our method produces part structures which are more faithful to the input, due to the integration of the structure masking network, and meanwhile structurally more plausible, benefiting from our part relation recovery.
\input{compare_star}\vspace{-10pt}
\section{Introduction}
The last few years have witnessed a continued interest in single-view image-based 3D modeling~\cite{choy20163d,fan2016point,girdhar2016learning}.
The performance of this task has been dramatically boosted, due to the tremendous success
of deep convolutional neural networks (CNN) on image-based learning tasks~\cite{krizhevsky2012imagenet}.
The existing deep models, however,
have so far been mainly targeting the output of volumetric representation of 3D shapes~\cite{choy20163d}.
Such models are essentially learned to map an input 2D image
to a 3D image (voxel occupancy of a 3D shape in a 3D volume).
Some compelling results have been demonstrated.
\input{fig_teaser}
While enjoying the high capacity of deep models in learning the image-to-image mapping,
the 3D volumes reconstructed by these methods lose
an important information of 3D shapes -- shape topology or part structure.
Once a 3D shape is converted into a volumetric representation,
it would be hard to recover its topology and structure, especially when there exist topological
defects in the reconstructed volume.
Shape structure, encompassing part composition and part relations, has been found highly
important to semantic 3D shape understanding and editing~\cite{mitra2013structure}.
Inferring a part segmentation for a 3D shape (surface or volumetric model) is known to be difficult~\cite{kalogerakis20163d}.
Even if a segmentation is given, it is still challenging to reason about part relations
such as connection, symmetry, parallelism, etc.
We advocate learning a deep neural network that directly recovers 3D shape structure
of an object, from a single RGB image.
The extracted structure can be used for enhancing the volumetric reconstruction
obtained by existing methods, facilitating structure-aware editing of the reconstructed 3D shapes,
and even enabling high-level editing of the input images (see Fig.~\ref{fig:teaser}).
However, directly mapping an image to a part structure seems a dunting task.
Tulsiani et al.~\cite{tulsiani2016learning} proposed a deep architecture to map a 3D volume to a set of cuboid primitives. Their method, however, cannot be adapted to our problem setting since the output primitive set
does not possess any structural information (mutual relations between primitives are not recovered).
Our problem involves the reasoning not only about
shape geometry, but also for higher level information of part
composition and relations. It poses several special challenges.
1) Different from shape geometry, part decomposition and relations
do not manifest explicitly in 2D images.
Mapping from pixels to part structure is highly ill-posed, as
compared to pixel-to-voxel mapping studied in many existing
2D-to-3D reconstruction works.
2) Many 3D CAD models of man-made objects contain diverse, fine-grained substructures. A faithful recovery of
those complicated 3D structures goes far beyond shape synthesis modulated by a shape classification.
3) Natural images always contain cluttered background and the imaged objects have large variations
of appearance due to different textures and lighting conditions.
Human brains do well both in shape inference based on low-level visual stimulus
and structural reasoning with the help of prior knowledge about 3D shape compositions.
The strength of human perception is to integrate the two ends of processing and reasoning
to form a capable vision system for high-level 3D shape understanding.
Motivated by this, we propose to learn and integrate two networks,
a \emph{structure masking network} for accentuating multi-scale object structures
in an input 2D image, followed by a \emph{structure recovery network} to
recursively recover a hierarchy of object parts abstracted by cuboids
(see Figure~\ref{fig:overview}).
The structure masking network produces a multi-scale attentional
mask for the object of interest, thereby decerning its shape structures in various forms and scales.
It designed as a multi-scale convolutional neural networks (CNN)
augmented with jump connections to retain shape details while
screening out the structure-irrelevant information such as background and textures in the output mask image.
The structure recovery network fuses
the features extracted in the structure masking network and
the CNN features of the original input image and feed them into
a recursive neural network (RvNN) for 3D structure decoding~\cite{li2017grass}.
The RvNN decoder, which is trained to explicitly model part relations,
expands the fused image features recursively into a tree organization of 3D cuboids
with plausible spatial configuration and reasonable mutual relations.
The two networks are jointly trained, with the training data of image-mask and cuboid-structure pairs.
Such pairs can be generated by rendering 3D CAD models and extracting the box structure based
on the given parts of the shape.
Several mechanisms are devised to avoid overfitting in training this model.
Experiments show that our method is able to faithfully recover diverse and detailed part structures of
3D objects from single 2D natural images. Our paper makes the following contributions:
\begin{itemize}
\item We propose to directly recover 3D shape structures from single RGB images.
The faithful and detailed recovery of 3D structural information of an object, such as part connectivity and symmetries, from 2D images has never been seen before, to our knowledge.
\item We present an architecture to tackle the hard learning task, via integrating a convolutional structure masking network and a recursive structure recovery network.
\item We develop two prototype applications where we use the recovered box structures 1) to refine
the 3D shapes reconstructed from single images by existing methods and 2) to assist structure-aware
editing of 2D images.
\end{itemize}
\section{Method}
We introduce our architecture for learning 3D shape structures from single images. It is an auto-encoder
composed of two sub-networks: a \emph{structure masking network} for decerning the object structures from the input 2D image and a \emph{structure recovery network} for recursive inference of a hierarchy of 3D boxes along with their mutual relations.
\subsection{Network architecture}
Our network is shown in Fig.~\ref{fig:overview}, which is composed of two modules:
a two-scale convolutional structure masking network and a recursive structure recovery network.
The structure masking network is trained to estimate the contour of the object of interest.
This is motivated by the observation that object contours provides strong cues for understanding
shape structures in 2D images~\cite{shotton2005contour,toshev2012shape}.
Instead of utilizing the extracted contour mask,
we feed the feature map of the last layer of the structure masking network into the structure recovery network.
To retain more information in the original image, this feature is fused with the CNN feature of
the input image via concatenation and fully connected layers, resulting in a $80$D feature code.
An RvNN decoder then recursively unfolds the feature code into a hierarchical organization of boxes,
with plausible spatial configuration and mutual relations,
as the recovered structure.
\subsection{Structure Masking Network.}
Our structure masking network is inspired by the recently proposed multi-scale network for detailed depth estimation~\cite{li2017ICCV}.
Given an input RGB image rescaled to $224 \times 224$,
we design a two-scale structure masking network to output a binary contour mask with a quarter of the input resolution ($56 \times 56$).
The first scale captures the information of the whole image while
the second produces a detailed mask map at a quarter of the input resolution.
As our prediction target is a binary mask, we use the SoftMax Loss as our training loss.
We employ VGG-16 to initialize the convolutional layers (up to pool5) of the first scale network, followed by two fully connected layers. The feature maps and outputs of the first scale network are fed into various layers of the second scale one for refined structure decerning. The second scale network, as a refinement block, starts from one $9 \times 9$ convolution and one pooling over the original input image, followed by nine successive $5\times 5$ convolutions without pooling. The feature maps from the pool3, pool4 and the output of last fully connected layer of first scale network are fused into the second, the fourth and the sixth convolutional layer of the second scale network, respectively. All the feature fusions get through a jump connections layer, which has a $5\times5$ convolutional layer and a 2x or 4x up-sampling to match the $56\times 56$ feature map size in the second scale; the jump connection from the fully connected layer is a simple concatenation.
It is shown that jump connections help extracting detailed structures from images effectively~\cite{li2017ICCV}.
\subsection{Structure Recovery Network}
The structure recovery network integrates the features extracted
from the structure masking network and for the input image into a bottleneck feature
and recursively decodes it into a hierarchy of part boxes.
\paragraph{Feature fusion.}
We fuse features from two convolutional channels. One channel takes as input the feature map of the structure masking network (the last feature map before the mask prediction layer), followed by two convolutions and poolings. Another channel is the CNN feature of the original image extracted by a VGG-16. The output feature maps from the two channels are then concatenated with size $7\times7$, and further encoded into a $80$D code after two fully connected layers, capturing the object structure information from the input image.
We found through experiments such fused features not only improve the accuracy of structure recovery, but also attain good domain-adaption from rendered images to real ones.
We believe the reason is that the extracted features for mask prediction task retain shape details through factoring them out of background clutters, texture variations and lighting conditions.
Since it is hard for the masking network to produce perfect mask prediction, the CNN feature of the original image provides complimentary information via retaining more object information.
\paragraph{Structure decoding.} We adopt a recursive neural network (RvNN) as box structure decoder like in~\cite{li2017grass}. Starting from a root feature code, RvNN recursively decodes its into a hierarchy of features until reaching the leaf nodes which each can be further decoded into a vector of box parameters.
There are three types of nodes in our hierarchy: leaf node, adjacency node and symmetry node.
During the decoding, two types of part relations are recovered as the class of internal nodes: \emph{adjacency} and \emph{symmetry}. Thus, each node can be decoded by one of the three decoders below, based on its type (adjacency node, symmetry node or box node):
\begin{description}
\item[Adjacency decoder.] Decoder \mbox{\sc AdjDec}\xspace splits a parent code $p$ into two child codes $c_1$ and $c_2$, using the mapping function:
\[
[c_1 \ c_2] \ = \ \tanh(W_{ad} \cdot p + b_{ad})
\]
where $W_{ad} \in \mathbb{R}^{2n \times n}$ and $b_{ad} \in \mathbb{R}^{2n}$. $n=80$ is the dimension of a non-leaf node.
\item[Symmetry decoder.] Decoder \mbox{\sc SymDec}\xspace recovers a symmetry group in the form of a symmetry generator (a node code $c$) and a vector of symmetry parameters $s$:
\[
[c \ s] \ = \ \tanh(W_{sd} \cdot p + b_{sd})
\]
where $W_{sd} \in \mathbb{R}^{(n + m) \times n}$, and $b_{sd} \in \mathbb{R}^{m + n}$. We use $m = 8$ for symmetry parameters consisting of: symmetry type ($1$D); number of repetitions for rotational and translational symmetries ($1$D); and the reflectional plane for reflective symmetry, rotation axis for rotational symmetry, or position and displacement for translational symmetry ($6$D).
\item[Box decoder.] Decoder \mbox{\sc BoxDec}\xspace converts the code of a leaf node to a $12$D box parameters defining the center, axes and dimensions of a 3D oriented box, similar to~\cite{li2017grass}.
\[
[x] \ = \ \tanh(W_{ld} \cdot p + b_{ld})
\]
where $W_{ld} \in \mathbb{R}^{12 \times n}$, and $b_{ld} \in \mathbb{R}^{12}$.
\end{description}
\input{decoding}
The decoders are recursively applied during decoding.
The key is how to determine the type of a node so that the corresponding decoder can be used at the node.
This is achieved by learning a node classifier based on the training task of structure recovery where the ground-truth
box structure is known for a given training pair of image and shape structure.
The node classifier is jointly trained with the three decoders.
The process of structure decoding is illustrated in Fig.~\ref{fig:decoding}.
In our implementation, the node classifier and the decoders for both adjacency and symmetry are two-layer networks, with the hidden layer and output layer being $200$D and $80$D vectors, respectively.
\subsection{Training details}
There are two stages in the training. First, we train the structure masking network to estimate a binary object mask for the input image. The first and the second scale of the structure masking network are trained jointly.
In the next, we jointly refine the structure masking network and train the structure recovery network, during which a low learning rate for structure masking network is used. The structure recovery loss is computed as the sum of the box reconstruction error and the cross entropy loss for node classification.
The reconstruction error is calculated as the sum of squared differences between the input and output parameters for each box and symmetry node. Prior to training, all 3D shapes are resized into a unit bounding box to make the reconstruction error comparable across different shapes. In Fig.~\ref{fig:Loss} (top), we plot the training and testing losses for box reconstruction, symmetry recovery and node classification, respectively, demonstrating the convergence of our structure recovery network.
\input{loss_plot}
We use the Stochastic Gradient Descent (SGD) to optimize our structure recovery network with back-propagation through structure (BPTT) for the RvNN decoder training. The convolutional layers of VGG-16 are initialized with the parameters pre-trained over ImageNet; all the other convolutional layers, the fully connected layers and the structure recovery network are randomly initialized. The learning rate of the structure masking network is $10^{-4}$ for pre-training and $10^{-5}$ for fine-tuning.
During joint training,
the learning rate is $10^{-3}$ for structure masking network, $0.2$ for RvNN decoder and $0.5$ for RvNN node classifier. These learning rates are decreased by a factor of $10$ for every $50$ epoches.
Our network is implemented with Matlab based on the MatConvNet toolbox~\cite{vedaldi2015matconvnet}.
The details on generating training data is provided in Section~\ref{train:data}.
\section{Related work}
Reconstructing 3D shapes from a single image has been a long-standing pursue in both
vision and graphics fields.
Due to its ill-posedness, many priors and assumptions have been attempted,
until the proliferation of high-capacity deep neural networks.
We will focus only on those deep learning based models and categorize
the fast-growing literature in three different dimensions.
\paragraph{Depth estimation vs. 3D reconstruction.}
Depth estimation is perhaps the most straightforward solution
for recovering 3D information from single images.
Deep learning has been shown to be highly effective for depth estimation~\cite{eigen2015predicting,laina2016deeper}.
Compared to depth estimation, reconstructing a full 3D model is much more challenging due to
the requirement of reasoning about the unseen parts. The latter has to resort to shape or structure priors.
Using deep neural networks to encode shape priors of a specific category has received much attention lately,
under the background of fast growing large 3D shape repositories~\cite{Shapenet}.
Choy et al.~\cite{choy20163d} develop a 3D Recurrent Reconstruction
Neural Network to generate 3D shapes in volumetric representation, given a single image as input.
A point set generation network is proposed for generating from a 2D image a 3D shape in point cloud~\cite{fan2016point}.
We are not aware of any previous works that can generate part-based structures
directly from a single image.
\paragraph{Discriminative vs. Generative.}
For the task of 3D modeling from 2D images, discriminative models are learned to map an input
image directly to the output 3D representation, either by a deep CNN for one-shot generation
or a recurrent model for progressive generation~\cite{choy20163d}.
The advantages of such approach include ease of training and high-quality results.
With the recent development of deep generative models such as variational auto-encoder (VAE)~\cite{kingma2013auto},
generative adversarial nets (GAN)~\cite{goodfellow2014generative} and their variants.
Learning a generative model for 3D shape generation has gained extensive research~\cite{wu2016learning,girdhar2016learning,li2017grass}.
For generative models, the input image can be used to condition the sampling from the predefined parameter space
or learned latent space~\cite{wu2016learning,girdhar2016learning}.
Generative models are known hard to train. For the task of cross-modality mapping,
we opt to train a discriminative model with a moderate size of training data.
\paragraph{Geometry reconstruction vs. Structure recovery.}
The existing works based on deep learning models mostly utilize volumetric 3D shape representation~\cite{wu2016learning,girdhar2016learning}.
Some notable exceptions include generating shapes in point clouds~\cite{fan2016point}, cuboid primitives~\cite{tulsiani2016learning} and manifold surfaces~\cite{lun20173d}.
However, none of these representations contains structural information of parts and part relations.
Interestingly, structure recovery is only studied with non-deep-learning approaches~\cite{xu2011photo,su2014estimating,huang2015single}.
This is largely because of the lack of a structural 3D shape representation suitable for deep neural networks.
Recently, Li et al.~\cite{li2017grass} propose to use recursive neural networks
for structural representation learning, nicely addressing the encoding/decoding of arbitrary number of shape parts
and various types of part relations.
Our method takes the advantage of this and integrate it into a cross-modality
mapping architecture for structure recovery from an RGB image.
\section{Results and Evaluations}
We evaluate the two core components of our pipeline: the co-planarity network and the robust optimization. Since our network is the first trained to predict co-planarity, we evaluate it against several baselines, to justify our design choices. For the optimization, we will show both quantitative and qualitative evaluations on scene reconstruction, especially for the cases which cannot be handled by key-point based methods.
\subsection{Training Set Explanation}
Our training data is generated from both ScanNet~\cite{} and Sun3D~\cite{} datasets. The $1516$ scenes in ScanNet are reconstructed by BundleFusion~\cite{}. $25$ scenes in Sun3D are reconstructed semi-automatically with human-assisted registration. For ScanNet, we adopt the training/testing split provided with the dataset. All $25$ Sun3D scenes and the training set ($1045$ scenes) of ScanNet are used to generate our training triplets. For evaluating our network, we also build a co-planarity benchmark using the $100$ of the testing scenes of ScanNet.
Each training scene contributes $10K$ triplets. About $10M$ triplets in total are generated from all training scenes, among which a half is randomly sampled and the other half is generated with the hard triplet mining heuristics.
\subsection{Co-planarity Benchmark}
We create a benchmark for evaluating co-planarity-based patch matching.
The benchmark dataset contains {\color{blue}[X]~} patch pairs which
are organized according to two aspects of matching hardness: patch size and pair distance.
We create three subsets with average physical patch size ranging from {\color{blue}[X]~} m$^2$, {\color{blue}[X]~} m$^2$, to {\color{blue}[X]~} m$^2$.
In these three subsets, the patch pairs are sampled at random distances.
For pair distance, we build another three subsets with average distance ranging from {\color{blue}[X]~} m, {\color{blue}[X]~} m, to {\color{blue}[X]~} m.
For each subset, the numbers of positive and negative pairs are roughly equal.
For both positive and negative, half of the data are sampled with our heuristic hard example mining.
The benchmark is provided in the supplementary material and will be made public upon acceptance.
\MATTHIAS{Add that table we have found}
\subsection{Network Evaluation}
\subsection{Registration Evaluation}
The evaluation of registration is conducted on the TUM RGB-D dataset by~\cite{sturm12iros},
for which ground-truth camera trajectories are available.
To better demonstrate the advantage of our method in handling sequences with
few frame-to-frame overlap, we generate a challenging test set, by down-sampling frames
from the TUM sequences by every $5$, $10$ and $15$, named TUM-d5, TUM-d10 and TUM-d15, respectively.
\paragraph{Quantitative Results}
To evaluate the robustness of our optimization, we first investigate
how tolerant it is to the precision rate of the co-planarity prediction.
In Figure~\ref{}, we plot the reconstruction error of our method
on the TUM dataset (original), with varying co-planarity precision.
The varying precisions are achieved by randomly disturbing the correct predictions
obtained by our network.
Reconstruction error is measured as the absolute trajectory error (ATE), i.e.,
the root-mean-square error (RMSE) of camera positions along a trajectory.
\kev{From the results, ...}
Normal estimation for planar patches is crucial to our co-planarity-based alignment.
In Figure~\ref{}, we demonstrate how our optimization is able to tolerate
noise in normals.
In reconstructing the TUM dataset, we add random noise (with a standard deviation of $5^\circ$)
to the estimated normals in each iteration, for varying percentages of patches.
\kev{The results show that our method is robust ...}
On the down-sampled TUM sequences, we compare our method
with six state-of-the-art reconstruction methods, including
RGB-D SLAM~\cite{endres2012evaluation},
VoxelHashing~\cite{niessner2013hashing},
ElasticFusion~\cite{Whelan15rss},
Redwood~\cite{choi2015robust},
BundleFusion~\cite{dai2017bundlefusion} and
Fine-to-Coarse~\cite{halber2016fine}.
Table~\ref{} reports the ATE RMSE comparison on TUM-d10. The results for TUM-d5 and TUM-d15
can be found in the supplementary material.
\begin{table}[h]\center
\scalebox{0.9}{
\begin{tabular}{| c | c | c | c | c |}
\hline
& fr1/desk & fr2/xyz & fr3/office & fr3/nst \\ \hline
RGB-D SLAM & 2.3cm & 0.8cm & 3.2cm & 1.7cm \\ \hline
VoxelHashing & 2.3cm & 2.2cm & 2.3cm & 8.7cm \\ \hline
Elastic Fusion & 2.0cm & 1.1cm & 1.7cm & 1.6cm \\ \hline
Redwood (rigid) & 2.7cm & 9.1cm & 3.0cm & 192.9cm \\ \hline
BundleFusion & 1.6cm & 1.1cm & 2.2cm & 1.2cm \\ \hline
Fine-to-Coarse & 1.6cm & 1.1cm & 2.2cm & 1.2cm \\
\hline \hline
Ours & XXX & XXX & XXX & XXX \\ \hline
\end{tabular}
\label{tab:comp_tum}
}
\caption{ATE RMSE (in cm) on downsampled TUM sequences~\protect\cite{sturm12iros}, TUM-d10. Note that unlike the other methods listed, Redwood does not use color information.}
\end{table}
\paragraph{Qualitative Results}
\MATTHIAS{Here, we would pick examples that traditional key-point based methods cannot address; e.g., we where we can get loop closures of two frames where no key point overlaps to begin with}
|
{
"timestamp": "2018-04-17T02:12:46",
"yymm": "1804",
"arxiv_id": "1804.05469",
"language": "en",
"url": "https://arxiv.org/abs/1804.05469"
}
|
\section{Introduction}
A pseudo-Riemannian
manifold $(M,g)$ admits a \textit{Ricci soliton} \cite{Hamilton-1982} if there exists a smooth vector
field $V$ (called the potential vector field) such that
\begin{equation}
\frac{1}{2}\pounds _{V}\,g+S+\lambda g=0, \label{int-1}
\end{equation
where $\pounds _{V}$ denotes the Lie derivative along the vector field $V$ and
\lambda $ is a real constant. By perturbing (\ref{int-1}) with a term $\mu \eta\otimes \eta$, for $\mu$ a real constant and $\eta$ a $1$-form, we obtain the \textit{$\eta$-Ricci soliton}, introduced by Cho and Kimura \cit
{Cho-Kimura} and more general, if we replace the two constants $\lambda$ and $\mu$ by smooth functions, we get \textit{almost $\eta$-Ricci solitons} \cite{bla}.
In the present paper we consider almost $\eta$-Ricci solitons on $\left( \varepsilon \right) $-para Sasakian manifolds which satisfy certain curvature properties. \textit{$\left( \varepsilon \right) $-para Sasakian manifolds} were defined by Tripathi, K\i l\i \c{c}, Y\"{u}ksel Perkt
\c{s}, and Kele\c{s} \cite{Tri-KYK-10} as a conterpart of almost paracontact metric geometry \cite{Sato-76}.
Our interest is also to characterize the geometry of an almost $\eta$-Ricci soliton on $\left( \varepsilon \right) $-para Sasakian manifolds in the case when the potential vector field is of gradient type and explicitly compute the scalar curvature for the case of gradient $\eta$-Ricci solitons. Remark that some properties of $\eta$-Ricci solitons on $\left( \varepsilon \right) $-almost paracontact metric manifolds were studied in \cite{per}.
\section{$\left( \varepsilon \right) $-para Sasakian structures}
Recall that an \textit{almost paracontact structure} on an $n$-dimensional manifold $M$ is a triple $(\varphi ,\xi ,\eta
)$ \cite{Sato-76} consisting of a $(1,1)$-tensor field $\varphi $, a vector field $\xi $ and a $1$-form $\eta $ satisfying:
\begin{equation}
\varphi ^{2}=I-\eta \otimes \xi , \label{eq-phi-eta-xi}
\end{equation
\begin{equation}
\eta (\xi )=1, \label{eq-eta-xi}
\end{equation
\begin{equation}
\varphi \xi =0, \label{eq-phi-xi}
\end{equation
\begin{equation}
\eta \circ \varphi =0. \label{eq-eta-phi}
\end{equation
One can easily check that (\ref{eq-phi-eta-xi}) and one of (\ref{eq-eta-xi
), (\ref{eq-phi-xi}) and (\ref{eq-eta-phi}) imply the other two relations.
Moreover, if $g$ is a pseudo-Riemannian metric such that
\begin{equation}
g\left( \varphi \cdot,\varphi \cdot\right) =g\left( \cdot,\cdot\right) -\varepsilon \eta
\otimes\eta, \label{eq-metric-1}
\end{equation
where $\varepsilon =\pm 1$, then $M$ is called $\left( \varepsilon
\right) ${\it -almost paracontact metric manifold} equipped with an \textit{$\left(
\varepsilon \right) ${\em -}almost paracontact metric structure} $(\varphi
,\xi ,\eta ,g,\varepsilon )$ \cite{Tri-KYK-10}. In particular, if ${\rm inde
}(g)=1$ (that is when $g$ is a Lorentzian metric), then the $(\varepsilon )
-almost paracontact metric manifold is called {\it Lorentzian almost
paracontact manifold}. From (\ref{eq-metric-1}) we obtain:
\begin{equation}
i_{\xi}g=\varepsilon \eta, \label{eq-metric-3}
\end{equation
\begin{equation}
g\left( X,\varphi Y\right) =g\left( \varphi X,Y\right), \label{eq-metric-2}
\end{equation
and from (\ref{eq-metric-3}) it follows that
\begin{equation}
g\left( \xi ,\xi \right) =\varepsilon, \label{eq-g(xi,xi)}
\end{equation
that is, the structure vector field $\xi $ is never lightlike.
\bigskip
Let $(M,\varphi ,\xi ,\eta ,g,\varepsilon )$ be an $(\varepsilon )$-almost
paracontact metric manifold (resp. a Lorentzian almost paracontact
manifold). If $\varepsilon =1$, then $M$ is said to be a spacelike
(\varepsilon )$-almost paracontact metric manifold (resp. a spacelike
Lorentzian almost paracontact manifold) and if $\varepsilon =-\,1$,
then $M$ is said to be a timelike $(\varepsilon )$-almost paracontact
metric manifold (resp. a timelike Lorentzian almost paracontact manifold)
\cite{Tri-KYK-10}.
\bigskip
An $\left( \varepsilon \right) $-almost paracontact metric structure $(\varphi ,\xi ,\eta ,g,\varepsilon )$ is
called $\left( \varepsilon \right) ${\it -para Sasakian structure} if
\begin{equation}
(\nabla _{X}\varphi )Y=-\,g(\varphi X,\varphi Y)\xi -\varepsilon \eta \left(
Y\right) \varphi ^{2}X, \label{para2}
\end{equation
for any $X,Y\in \Gamma (TM)$, where $\nabla $ is the Levi-Civita connection with respect to $g$. A
manifold endowed with an $\left( \varepsilon \right) $-para Sasakian
structure is called $\left( \varepsilon \right) ${\it -para Sasakian
manifold} \cite{Tri-KYK-10}. In an $\left( \varepsilon \right) ${\em -}para
Sasakian manifold, we have
\begin{equation}
\nabla \xi =\varepsilon \varphi \label{para3}
\end{equation
and the Riemann
curvature tensor $R$ and the Ricci tensor $S$ satisfy the following
equations \cite{Tri-KYK-10}
\begin{equation}
R\left( X,Y\right) \xi =\eta \left( X\right) Y-\eta \left( Y\right) X,
\label{eq-eps-PS-R(X,Y)xi}
\end{equation
\begin{equation}
R\left( \xi ,X\right) Y=-\,\varepsilon g\left( X,Y\right) \xi +\eta \left(
Y\right) X, \label{eq-eps-PS-R(xi,X)Y}
\end{equation
\begin{equation}
\eta \left( R\left( X,Y\right) Z\right) =-\,\varepsilon \eta \left( X\right)
g\left( Y,Z\right) +\varepsilon \eta \left( Y\right) g\left( X,Z\right),
\label{eq-eps-PS-eta(R(X,Y),Z)}
\end{equation
\begin{equation}
S(X,\xi )=-(n-1)\eta (X), \label{eq-eps-PS-S(X,xi)}
\end{equation}
for any $X,Y,Z\in \Gamma (TM)$.
\bigskip
Also remark that if $(\varphi,\xi,\eta,g,\varepsilon)$ is an $\left( \varepsilon \right) $-para Sasakian structure on the manifold $(M,g)$ of constant curvature $k$, if $M$ is spacelike (resp. timelike), then $M$ is hyperbolic (resp. elliptic) manifold. Indeed, if $R(X,Y)Z=k[g(Y,Z)X-g(X,Z)Y]$, for any $X$, $Y$, $Z\in \Gamma(TM)$, applying $\eta$ to this relation and using (\ref{eq-eps-PS-eta(R(X,Y),Z)}) we obtain $k=-\varepsilon$.
\bigskip
\begin{example}
\cite{Tri-KYK-10} Let ${\Bbb R}^{5}$\ be the $5$-dimensional real number
space with a coordinate system $\left( x,y,z,t,s\right) $. Defining
\[
\eta =ds-ydx-tdz\ ,\qquad \xi =\frac{\partial }{\partial s}\,
\
\[
\varphi \left( \frac{\partial }{\partial x}\right) =-\,\frac{\partial }
\partial x}-y\frac{\partial }{\partial s}\ ,\qquad \varphi \left( \frac
\partial }{\partial y}\right) =-\,\frac{\partial }{\partial y}\,
\
\[
\varphi \left( \frac{\partial }{\partial z}\right) =-\,\frac{\partial }
\partial z}-t\frac{\partial }{\partial s}\ ,\qquad \varphi \left( \frac
\partial }{\partial t}\right) =-\,\frac{\partial }{\partial t}\ ,\qquad
\varphi \left( \frac{\partial }{\partial s}\right) =0\,
\
\[
g_{1}=\left( dx\right) ^{2}+\left( dy\right) ^{2}+\left( dz\right)
^{2}+\left( dt\right) ^{2}-\eta \otimes \eta \,
\
\begin{eqnarray*}
g_{2} &=&-\,\left( dx\right) ^{2}-\left( dy\right) ^{2}+\left( dz\right)
^{2}+\left( dt\right) ^{2}+\left( ds\right) ^{2} \\
&&-\,t\left( dz\otimes ds+ds\otimes dz\right) -y\left( dx\otimes
ds+ds\otimes dx\right),
\end{eqnarray*
then $(\varphi ,\xi ,\eta ,g_{1})$ is a timelike Lorentzian almost
paracontact structure in ${\Bbb R}^{5}$, while $(\varphi ,\xi ,\eta
,g_{2})$\ is a spacelike $\left( \varepsilon \right) $-almost paracontact
structure.
\end{example}
\section{Almost $\eta$-Ricci solitons in $(M,\varphi ,\xi ,\eta ,g,\varepsilon )$}
Let $(M,g)$ be an $n$-dimensional Riemannian manifold ($n>2$). Consider the equation:
\begin{equation}\label{e8}
\pounds_{\xi}g+2S+2\lambda g+2\mu\eta\otimes \eta=0,
\end{equation}
where $\pounds_{\xi}$ is the Lie derivative operator along the vector field $\xi$, $S$ is the Ricci curvature tensor field of the metric $g$, $\eta$ is a $1$-form and $\lambda$ and $\mu$ are smooth functions on $M$.
The data $(\xi,\lambda,\mu)$ which satisfy the equation (\ref{e8}) is said to be an \textit{almost $\eta$-Ricci soliton} on $(M,g)$ called \textit{steady} if $\lambda=0$, \textit{shrinking} if $\lambda<0$ or \textit{expanding} if $\lambda>0$; in particular, if $\mu=0$, $(\xi,\lambda)$ is an \textit{almost Ricci soliton} \cite{pi}.
\bigskip
\begin{example}
Let $M=\mathbb{R}^3$, $(x,y,z)$ be the standard coordinates in $\mathbb{R}^3$ and $g$ be the Lorentzian metric:
$$g:=e^{-2z}dx\otimes dx+e^{2x-2z}dy\otimes dy-dz\otimes dz.$$
Consider the $(1,1)$-tensor field
$\varphi$, the vector field $\xi$ and the $1$-form $\eta$:
$$\varphi:=-\frac{\partial}{\partial x}\otimes dx-\frac{\partial}{\partial y}\otimes dy, \ \ \xi:=\frac{\partial}{\partial z}, \ \ \eta:=dz.$$
In this case, $(M,\varphi, \xi,\eta,g)$ is a timelike Lorentzian almost paracontact manifold.
Moreover, for the orthonormal vector fields:
$$E_1=e^z\frac{\partial}{\partial x}, \ \ E_2=e^{z-x}\frac{\partial}{\partial y}, \ \ E_3=\frac{\partial}{\partial z},$$
we get:
$$\nabla_{E_1}E_1=-E_3, \ \ \nabla_{E_1}E_2=0, \ \ \nabla_{E_1}E_3=-E_1, \ \ \nabla_{E_2}E_1=e^zE_2, \ \ \nabla_{E_2}E_2=-e^zE_1-E_3,$$$$\nabla_{E_2}E_3=-E_2, \ \ \nabla_{E_3}E_1=0, \ \ \nabla_{E_3}E_2=0, \ \ \nabla_{E_3}E_3=0$$
and the Riemann and the Ricci curvature tensor fields are given by:
$$R(E_1,E_2)E_2=(1-e^{2z})E_1, \ \ R(E_1,E_3)E_3=-E_1, \ \ R(E_2,E_1)E_1=(1-e^{2z})E_2,$$
$$R(E_2,E_3)E_3=-E_2, \ \ R(E_3,E_1)E_1=E_3, \ \ R(E_3,E_2)E_2=E_3,$$
$$S(E_1,E_1)=S(E_2,E_2)=2-e^{2z}, \ \ S(E_3,E_3)=-2.$$
Therefore, the data $(\xi,\lambda,\mu)$ for $\lambda=e^{2z}-1$ and $\mu=e^{2z}+1$ defines an almost $\eta$-Ricci soliton on $(M,g)$.
\end{example}
\bigskip
Writing $\pounds_{\xi}g$ in terms of the Levi-Civita connection $\nabla$, we obtain:
\begin{equation}\label{e9}
S(X,Y)=
-\frac{1}{2}[g(\nabla_X\xi,Y)+g(X,\nabla_Y\xi)]-\lambda g(X,Y)-\mu\eta(X)\eta(Y),
\end{equation}
for any $X$, $Y\in \Gamma(TM)$.
If $(M,\varphi ,\xi ,\eta ,g,\varepsilon )$ is an $\left( \varepsilon \right) $-para Sasakian manifold, then (\ref{e9}) becomes:
\begin{equation}\label{ea9}
S(X,Y)=
-\varepsilon g(\varphi X,Y)-\lambda g(X,Y)-\mu\eta(X)\eta(Y),
\end{equation}
for any $X$, $Y\in \Gamma(TM)$.
\bigskip
Also remark that on an $\left( \varepsilon \right) $-para Sasakian manifold $(M,\varphi ,\xi ,\eta ,g,\varepsilon )$, since the Ricci curvature tensor field $S$ satisfies (\ref{eq-eps-PS-S(X,xi)}), using (\ref{ea9}) we obtain:
\begin{equation}\label{ad}
\varepsilon \lambda+\mu=n-1
\end{equation}
and we can state:
\begin{proposition}
The scalar curvature of an $\left( \varepsilon \right) $-para Sasakian manifold $(M,\varphi ,\xi ,\eta ,g,\varepsilon )$ admitting an almost $\eta$-Ricci soliton $(\xi,\lambda,\mu)$ is:
\begin{equation}\label{c}
scal=-div(\xi)-n\lambda - \varepsilon \mu.
\end{equation}
\end{proposition}
From (\ref{ad}) we also deduce that:
\begin{proposition}
An almost Ricci soliton $(\xi,\lambda)$ on a spacelike (resp. timelike) $\left( \varepsilon \right) $-para Sasakian manifold $(M,\varphi ,\xi ,\eta ,g,\varepsilon )$ is expanding (resp. shrinking).
\end{proposition}
\begin{remark}
If the almost $\eta$-Ricci soliton is steady, then $\mu=n-1$ and the scalar curvature of $(M,\varphi ,\xi ,\eta ,g,\varepsilon )$ is $scal=-div(\xi)-\varepsilon (n-1)$.
\end{remark}
\bigskip
Similar like in the case of $\eta$-Ricci solitons on Lorentzian para-Sasakian manifolds \cite{bl}, the next theorems formulate some results in case of $\left( \varepsilon \right) $-para Sasakian manifold which is Ricci symmetric, has $\eta$-parallel, Codazzi or cyclic $\eta$-recurrent Ricci curvature tensor.
\begin{proposition}\label{t1}
Let $(\varphi ,\xi ,\eta ,g,\varepsilon )$ be an $\left( \varepsilon \right) $-para Sasakian structure on the manifold $M$ and let $(\xi,\lambda,\mu)$ be an almost $\eta$-Ricci soliton on $(M,g)$.
\begin{enumerate}
\item If the manifold $(M,g)$ is Ricci symmetric (i.e. $\nabla S=0$), then $\xi(\mu)=-\varepsilon \xi(\lambda)$.
\item If the Ricci tensor is $\eta$-recurrent (i.e. $\nabla S=\eta\otimes S$), then $\xi(\varepsilon \lambda+\mu)=n-1$.
\item If the Ricci tensor is Codazzi (i.e. $(\nabla_X S)(Y,Z)=(\nabla_Y S)(X,Z)$, for any $X$, $Y$, $Z\in \Gamma(TM)$), then $d(\varepsilon \lambda+\mu)=\xi(\varepsilon \lambda+\mu)\eta$.
\item If the Ricci tensor is $\eta$-parallel (i.e.$(\nabla_X S)(\varphi Y,\varphi Z)=0$), then the scalar function $\lambda$ is locally constant.
\end{enumerate}
\end{proposition}
\begin{proof}
Replacing the expression of $S$ from (\ref{ea9}) in $(\nabla_XS)(Y,Z):=X(S(Y,Z))-S(\nabla_XY,Z)-S(Y,\nabla_XZ)$ we obtain:
\begin{equation}\label{m}
(\nabla_XS)(Y,Z)=\eta(Y)g(X,Z)+\eta(Z)g(X,Y)-2\varepsilon \eta(X)\eta(Y)\eta(Z)-\end{equation}$$-X(\lambda)g(Y,Z)-X(\mu)\eta(Y)\eta(Z)-\mu[\eta(Y)g(\varphi X,Z)+\eta(Z)g(\varphi X,Y)].$$
For the first two assertions, just take $X=Y=Z:=\xi$ in the expression of $\nabla S$ from (\ref{m}) and we obtain the required results. Concerning the case when $S$ is Codazzi, taking $Y=Z:=\xi$ in $(\nabla_X S)(Y,Z)=(\nabla_Y S)(X,Z)$ we obtain $\varepsilon X(\lambda)+X(\mu)=[\varepsilon \xi(\lambda)+\xi(\mu)]\eta(X)$, for any $X\in \Gamma(TM)$, which is equivalent to the stated relation. If $S$ is $\eta$-parallel, then for any $X$, $Y$, $Z\in \Gamma(TM)$, $0=(\nabla_X S)(\varphi Y,\varphi Z)=-X(\lambda)g(\varphi Y,\varphi Z)$ which implies $X(\lambda)=0$, for any $X\in \Gamma(TM)$.
\end{proof}
\bigskip
\begin{remark}
If on the $\left( \varepsilon \right) $-para Sasakian manifold $(M,\varphi ,\xi ,\eta ,g,\varepsilon )$ we consider the almost $\eta$-Ricci soliton $(V, \lambda, \mu)$ with the potential vector field $V$ conformal Killing (i.e. $\frac{1}{2}\pounds_{\xi}g=fg$), for $f$ a smooth function on $M$, then
$$S=-(f+\lambda)g-[n-1-\varepsilon (f+\lambda)]\eta\otimes \eta$$
and the manifold is Einstein if and only if $\lambda=\varepsilon(n-1)-f$; in this case, we have an almost Ricci soliton.
The same conclusion is reached if the potential vector field $V:=\xi$ is torse-forming (i.e. $\nabla \xi =f\varphi ^{2}$ according to \cite{per}).
\end{remark}
\bigskip
When the potential vector field of (\ref{e8}) is of gradient type, i.e. $\xi=grad(f)$, then $(\xi,\lambda,\mu)$ is said to be a \textit{gradient almost $\eta$-Ricci soliton} and the equation satisfied by it becomes:
\begin{equation}\label{e22}
Hess(f)+S+\lambda g+\mu\eta\otimes \eta=0,
\end{equation}
where $Hess(f)$ is the Hessian of $f$ defined by $Hess(f)(X,Y):=g(\nabla_X\xi,Y)$.
\begin{proposition}
Let $(M,\varphi,\xi,\eta,g,\varepsilon)$ be an $\left( \varepsilon \right) $-para Sasakian manifold. If (\ref{e22}) defines a gradient almost $\eta$-Ricci soliton on $(M,g)$
with the potential vector field $\xi:=grad(f)$ and $\eta=df$ is the $g$-dual of $\xi$, then:
\begin{equation}
(\nabla_XQ)Y-(\nabla_YQ)X=(d(f-\lambda)\otimes I-I\otimes d(f-\lambda)+(df\otimes d\mu-d\mu\otimes df)\otimes \xi+\end{equation}$$+\varepsilon \mu(df\otimes \varphi-\varphi\otimes df))(X,Y),
$$
for any $X$, $Y\in\Gamma(TM)$, where $Q$ stands for the Ricci operator defined by $g(QX,Y):=S(X,Y)$.
\end{proposition}
\begin{proof}
Notice that (\ref{e22}) can be written:
\begin{equation}\label{e23}
\nabla\xi+Q+\lambda I+\mu df\otimes \xi=0.
\end{equation}
Then:
\begin{equation}
(\nabla_XQ)Y=-(\nabla_X\nabla_Y\xi-\nabla_{\nabla_XY}\xi)-X(\lambda)Y-X(\mu)df(Y)\xi-\mu[g(Y,\nabla_X\xi)\xi+df(Y)\nabla_X\xi].
\end{equation}
Replacing now $\nabla\xi=\varepsilon \varphi$ in the previous relation, after a long computation, we get the required relation.
\end{proof}
\begin{remark}
i) Remark that since $\xi$ is geodesic vector field, from (\ref{e23}) follows that $\xi$ is an eigenvector of $Q$ corresponding to the eigenvalue $-(\lambda+\mu)$. In particular, if $\lambda=-\mu$, then $\xi\in \ker Q$.
ii) The Ricci operator is $\varphi$-invariant (i.e. $Q\circ \varphi=\varphi \circ Q$).
iii) If $Q$ is Codazzi (i.e. $(\nabla_X Q)Y=(\nabla_Y Q)X$, for any $X$, $Y\in \Gamma(TM)$), then for the gradient $\eta$-Ricci soliton case, $\mu \varphi X=\varepsilon \varphi ^2X$, for any $X\in \Gamma(TM)$ which implies $\mu^2=1$. Therefore, the soliton is expanding if $M$ is spacelike or shrinking if $M$ is timelike and it is given by $(\lambda,\mu)\in \{ (\varepsilon n, -1), (\varepsilon (n-2), 1)\}$.
\end{remark}
\bigskip
A lower and an upper bound of the Ricci curvature tensor's norm \cite{cr} for a gradient almost $\eta$-Ricci soliton on an $\left( \varepsilon \right) $-para Sasakian manifold will be given \cite{bla}.
\begin{theorem}
If (\ref{e22}) defines a gradient almost $\eta$-Ricci soliton on the $n$-dimensional $\left( \varepsilon \right) $-para Sasakian manifold $(M,\varphi,\xi,\eta,g,\varepsilon)$ and
$\eta=df$ is the $g$-dual of the gradient vector field $\xi:=grad(f)$, then:
\begin{equation}\label{e21}
n-1+\mu^2-\frac{(\Delta(f)+\varepsilon\mu)^2}{n}\leq |S|^2\leq n-1+\mu^2+\frac{(scal)^2}{n}.
\end{equation}
\end{theorem}
\begin{remark}
The simultaneous equalities hold for $(scal)^2=-(\Delta(f)+\varepsilon\mu)^2 \ (=0)$ i.e. for steady gradient almost $\eta$-Ricci soliton ($\lambda=0$) with $scal=0$ and $\Delta(f)=-\varepsilon\mu$. In this case, $|S|^2=n-1+\mu^2$.
\end{remark}
\bigskip
Using a Bochner-type formula for the gradient almost $\eta$-Ricci solitons \cite{bla}, in the case of $\left( \varepsilon \right) $-para Sasakian manifold we obtain the condition satisfied by the potential function:
\begin{theorem}
If (\ref{e22}) defines a gradient almost $\eta$-Ricci soliton on the $n$-dimensional $\left( \varepsilon \right) $-para Sasakian manifold $(M,\varphi,\xi,\eta,g,\varepsilon)$ and
$\eta=df$ is the $g$-dual of the gradient vector field $\xi:=grad(f)$, then:
\begin{equation}\label{e1}
\Delta(f)=\frac{1}{2}[\varepsilon(n-1)+\lambda +\varepsilon \mu+\varepsilon (n-2)\xi(\lambda)-\xi(\mu)].
\end{equation}
\end{theorem}
Considering (\ref{c}) and (\ref{e1}) we obtain:
\begin{equation}\label{e}
scal=-\frac{1}{2}[\varepsilon(n-1)+(2n+1)\lambda+3\varepsilon\mu+\varepsilon(n-2)\xi(\lambda)-\xi(\mu)].
\end{equation}
\begin{remark}
i) If (\ref{e22}) defines a gradient $\eta$-Ricci soliton on the $n$-dimensional $\left( \varepsilon \right) $-para Sasakian manifold $(M,\varphi,\xi,\eta,g,\varepsilon)$, then:
$$scal=-\frac{1}{2}[\varepsilon(n-1)+(2n+1)\lambda+3\varepsilon\mu].$$
ii) In this case, if $M$ is connected and has constant scalar curvature, then:
$$\mu=-\frac{2n+1}{3\varepsilon}\lambda+C, \ \ C\in \mathbb{R}.$$
\end{remark}
|
{
"timestamp": "2018-04-17T02:11:26",
"yymm": "1804",
"arxiv_id": "1804.05389",
"language": "en",
"url": "https://arxiv.org/abs/1804.05389"
}
|
\section{Quaternionic projective space}\label{sec:action}
We begin by examining a very natural action of $S^1$ on
quaternionic projective space. It is in some ways analogous
to the natural action of $C_2$ on $\mathbb{C}P^{\infty}$
by complex conjugation.
\begin{construction}\label{cstr:quat-action} Consider the action of
$\mathrm{Sp}(1)$ on the quaternions $\mathbb{H}$
by conjugation. The center $\{\pm 1\}$ acts trivially,
so this produces an action of $\mathrm{Sp}(1)/\{\pm1\}
= \mathrm{SO}(3)$ on $\mathbb{H}$. This produces
an action of $S^1 \subseteq SO(3)$ on $\mathbb{H}$
which can be described in two equivalent ways:
\begin{itemize}
\item $z \in S^1$ acts on $q \in \mathbb{H}$
by $(\sqrt{z}) q (\sqrt{z})^{-1}$;
\item $z \in S^1$ acts on $q = u+vj$
by $u+ (zv)j$, where $u,v \in \mathbb{C}$.
\end{itemize}
From the first description it's clear that $\mathbb{H}$
is an $S^1$-equivariant algebra. From the second
description it's clear that $\mathbb{H} = 2+\lambda$
as a representation.
\end{construction}
\begin{definition}
Let $\mathbb{H}P^{\infty}$ denote the $S^1$-space
obtained from $\mathbb{H}^{\infty}$ by imposing the
relation
\[
[q_0: q_1: \cdots] \sim [q_0h: q_1h: \cdots], \, h \in \mathbb{H}^{\times}
\]
and acting by $S^1$ componentwise as in Construction \ref{cstr:quat-action}.
\end{definition}
We will, without comment, also denote by $\mathbb{H}P^{\infty}$
the $G$-space obtained by restricting the action to a finite subgroup
$G \subseteq S^1$. We will assume that $G$ is a fixed,
\emph{nontrivial} finite subgroup of $S^1$ for the remainder of
this section.
\begin{remark}\label{rmk:quat-fixed} It follows from the definition
that the natural inclusion $\mathbb{C}P^{\infty} \subseteq \mathbb{H}P^{\infty}$
is equivariant for the \emph{trivial} action on complex projective space
and identifies the fixed points
\[
\mathbb{C}P^{\infty} = \left(\mathbb{H}P^{\infty}\right)^{G}.
\]
The reader might compare this to the equivalence
$ \mathbb{R}P^{\infty}=\left(\mathbb{C}P^{\infty}\right)^{C_2}$
where $C_2$ acts by complex conjugation on projective space.
\end{remark}
We now study the loop spaces of $\mathbb{H}P^{\infty}$.
\begin{proposition}
$\Omega \mathbb{HP}^{\infty} \simeq S^{\lambda+1}$.
\end{proposition}
\begin{proof} The usual inclusion $S^{\lambda+2} \simeq \mathbb{H}P^1
\to \mathbb{H}P^{\infty}$ is equivariant for the action above
and induces a map $S^{\lambda+1} \to \Omega \mathbb{H}P^{\infty}$.
This is an underlying equivalence, and on fixed points for nontrivial
subgroups we have the standard map (see
Remark \ref{rmk:quat-fixed})
$S^1 \to \Omega \mathbb{C}P^{\infty}$, which is also an equivalence.
\end{proof}
\begin{remark}\label{rmk:nu-fixed-eta} As a corollary of the proof, we record that
the map $\nu: S^{2\lambda+3} \to S^{\lambda+2}$ becomes
$\eta: S^{3} \to S^2$ upon passage to fixed points.
\end{remark}
\begin{proposition}
If $G=C_2$, then $\Omega^{\sigma} \mathbb{H}P^{\infty} \simeq S^{\rho+1}$.
\end{proposition}
\begin{proof} Again, the usual inclusion
$S^{2\sigma+2} \to \mathbb{H}P^{\infty}$ is equivariant and we get
a map $S^{\sigma+2} = S^{\rho +1} \to \Omega^{\sigma}\mathbb{H}P^{\infty}$
which is an underlying equivalence. To compute the fixed points of
the right hand side, we use the fiber sequence of spaces
\[
\left(\Omega^{\sigma}\mathbb{H}P^{\infty}\right)^{C_2}
\to \mathbb{C}P^{\infty} \to \mathbb{H}P^{\infty}
\]
arising from the cofiber sequence
$C_{2+} \to S^0 \to S^{\sigma}$ of $C_2$-spaces.
But the fiber of $\mathrm{B}S^1 \to \mathrm{B}S^3$ is
$S^3/S^1 \simeq S^2$, by the Hopf fibration, which completes the proof.
\end{proof}
\begin{corollary}
If $G=C_2$, then
$$\Omega^{\lambda} S^{\lambda+1} \simeq \Omega^{\rho} S^{\rho+1}.$$
Moreover, this equivalence respects the inclusion of $S^1$ up to homotopy.
\end{corollary}
\begin{proof}
There is a chain of equivalences
$$\Omega^{\lambda} S^{\lambda+1} \simeq \Omega^{2 \sigma} \Omega \mathbb{H}P^{\infty} \simeq \Omega^{\rho} \Omega^{\sigma} \mathbb{H}P^{\infty} \simeq \Omega^{\rho} S^{\rho+1}.$$
\end{proof}
\begin{remark} If $V$ is a representation of $C_2$, then the free $\mathbb{E}_V$-algebra on any $C_2$-connected, pointed $C_2$-space $X$ is given by $\Omega^{V} \Sigma^{V} X$ \cite[Theorem 1]{rs}. The above discussion therefore implies that the free $\mathbb{E}_{2\sigma}$-space and free $\mathbb{E}_{\rho}$-space on the pointed $C_2$-space $S^1$ coincide. A prototype of this result is that the
free group-like $\mathbb{E}_{\sigma}$-space and free group-like
$\mathbb{E}_{1}$-space on the pointed $C_2$-space
$S^0$ also coincide, i.e.
$\Omega S^1 \simeq \Omega^{\sigma}S^{\sigma} \simeq \mathbb{Z}$
(with trivial action).
This can be proved in much the same way, using the equivalences
\[
\Omega^{\rho}\mathbb{C}P^{\infty} \simeq \mathbb{Z},
\quad \Omega \mathbb{C}P^{\infty} \simeq S^{\sigma},
\quad \text{and }\Omega^{\sigma}\mathbb{C}P^{\infty}
\simeq S^1,
\]
where $C_2$ acts on $\mathbb{C}P^{\infty}$ by complex
conjugation.
\end{remark}
As a consequence of the equivalence $\Omega \mathbb{H}P^{\infty}
\simeq S^{\lambda+1}$, we observe that $\Omega^{\lambda}S^{\lambda+1}$
inherits an $\mathbb{E}_{\lambda+1}$-structure.
This gives the fixed points $\left(\Omega^{\lambda}S^{\lambda+1}\right)^G$
an $\mathbb{E}_1$-algebra structure.
\begin{proposition} As an $\mathbb{E}_1$-space,
\[
\left(\Omega^{\lambda+1}\mathbb{H}P^{\infty}\right)^{G}
\simeq \Omega^2S^3 \times \Omega S^2.
\]
\end{proposition}
\begin{warning} At odd primes, the classifying map
$\mu^G$ is not an $\mathbb{E}_1$-map.
\end{warning}
\begin{proof}[Proof of the proposition]
Define $S^{\lambda/2}$ by the cofiber sequence
\[
G_+ \to S^0 \to S^{\lambda/2},
\]
and notice that we have a cofiber sequence
\[
G_+ \wedge S^1 \to S^{\lambda/2} \to S^{\lambda}.
\]
From the second cofiber sequence, we learn that there is a fiber sequence
\[
\Omega^{\lambda}\mathbb{H}P^{\infty}
\to \mathrm{map}_*(S^{\lambda/2}, \mathbb{H}P^{\infty})
\to
\mathrm{map}_*(G_+ \wedge S^1, \mathbb{H}P^{\infty}).
\]
Taking fixed points, we get
\[
\left(\Omega^{\lambda}\mathbb{H}P^{\infty}\right)^G
\to \mathrm{map}_*(S^{\lambda/2}, \mathbb{H}P^{\infty})^G
\to
\Omega \mathbb{H}P^{\infty} \simeq S^3.
\]
The first cofiber sequence identifies the middle term as the fiber
of the inclusion of fixed points:
\[
\mathrm{map}_*(S^{\lambda/2}, \mathbb{H}P^{\infty})^G
\to \mathbb{C}P^{\infty} \to \mathbb{H}P^{\infty}.
\]
In other words, $\mathrm{map}_*(S^{\lambda/2}, \mathbb{H}P^{\infty})^G
\simeq S^2$ and we can identify our previous fiber sequence with:
\[
\left(\Omega^{\lambda}\mathbb{H}P^{\infty}\right)^G
\to S^2
\to
S^3.
\]
The second map is null, so we learn that there is an equivalence:
\[
\left(\Omega^{\lambda}\mathbb{H}P^{\infty}\right)^G
\simeq \Omega S^3 \times S^2,
\]
and hence an equivalence of loop spaces:
\[
\left(\Omega^{\lambda+1}\mathbb{H}P^{\infty}\right)^G
= \Omega \left(\Omega^{\lambda}\mathbb{H}P^{\infty}\right)^G
\simeq \Omega(\Omega S^3 \times S^2).
\]
\end{proof}
We stress that the above result does not concern
multiplicative structure on the Thom spectrum in question.
This is the subject of the next section at the prime two,
and of the subsequent section at odd primes.
\section{Proof of Theorem \ref{thm:non-eqvt}} \label{sec:appendix}
For convenience, we recall the statement of Theorem \ref{thm:non-eqvt}, which this appendix is devoted to proving. The result is entirely non-equivariant.
\begin{thm*}
Let $S^0_p$ denote the $p$-complete sphere spectrum, and suppose that $p>2$. Then there is no triple loop map
$$X \longrightarrow \mathrm{BGL}_1(S^0_{p}),$$
for any triple loop space $X$, that makes $\mathrm{H}\mathbb{F}_p$ as a Thom spectrum.
\end{thm*}
\begin{remark}
It follows also that $\mathrm{H}\mathbb{F}_p$ is not a triple loop Thom spectrum over the $p$-local sphere spectrum. If it were, then the composition
$$X \longrightarrow \mathrm{BGL}_1(S^0_{(p)}) \longrightarrow \mathrm{BGL}_1(S^0_p)$$
would provide a counterexample to the above.
\end{remark}
\begin{proof}
Suppose, for the sake of contradiction, that such a triple loop map
$$X \longrightarrow \mathrm{BGL}_1(S^0_p)$$
exists. The Thom isomorphism then implies that
$$\mathrm{H}\mathbb{F}_p \smsh \Sigma^{\infty}_+ X \simeq \mathrm{H}\mathbb{F}_p \smsh \mathrm{H}\mathbb{F}_p$$
as $\mathrm{H}\mathbb{F}_p$--$\mathbb{E}_3$-algebras.
In particular, by Theorem \ref{thm:Hop},
$$\mathrm{H}\mathbb{F}_p \smsh \Sigma^{\infty}_+ X \simeq \mathrm{H}\mathbb{F}_p \smsh \Sigma^{\infty}_+ \Omega^2 S^3$$
as $\mathrm{H}\mathbb{F}_p$--$\mathbb{E}_2$-algebras, and the latter object is the free $\mathrm{H}\mathbb{F}_p$--$\mathbb{E}_2$-algebra on a class in degree $1$.
The Hurewicz theorem gives a map $S^1 \rightarrow X$, which extends to a double-loop map $\Omega^2 S^3 \rightarrow X$, and the above discussion implies that this double loop map is a homology isomorphism. Thus, the $p$-completion of $X$ is the $p$-completion of $\Omega^2 S^3$, as a double loop space.
Transporting the $\mathbb{E}_3$-algebra structure on $X$ yields an $\mathbb{E}_3$-algebra structure on the $p$-completion of $\Omega^2 S^3$, extending the usual $\mathbb{E}_2$-algebra structure. The theorems of Dwyer, Miller, and Wilkerson \cite{dwyer-miller-wilkerson} show that there is a unique such $\mathbb{E}_3$-algebra structure, and so the $p$-completion of $\mathrm{B}^3 X$ must be the $p$-completion of $\mathbb{H}P^{\infty}$.
Now, the composite
$$X \longrightarrow \mathrm{BGL}_1(S^0_p) \longrightarrow \mathrm{BGL}_1(\mathrm{H}\mathbb{F}_p)$$
is null, and it follows that there is a factorization through the fiber $F$ of $\mathrm{BGL}_1(S^0_p) \longrightarrow \mathrm{BGL}_1(\mathrm{H}\mathbb{F}_p)$.
The equivalence $\mathbb{Z}_p^{\times} \cong \mu_{p-1} \times \mathbb{Z}_p$ implies that the homotopy groups of $F$ are $p$-complete. Thus, with $\mathbb{H}P^{\infty}_p$ denoting the $p$-completion of $\mathbb{H}P^{\infty}$, there is a commuting diagram
$$
\begin{tikzcd}
& \mathrm{B}^3X \arrow{r} \arrow{d} & B^4GL_1(S^0_p). \\
\mathbb{H}P^{\infty} \arrow{r} & \mathbb{H}P^{\infty}_p \arrow{ru}
\end{tikzcd}
$$
In particular, there is a triple-loop map
$$\Omega^2 S^3 \longrightarrow \Omega^3 \mathbb{H}P^{\infty} \longrightarrow \mathrm{BGL}_1(S^0_p)$$
with Thom spectrum equivalent (at least after $p$-completion) to $\mathrm{H}\mathbb{F}_p$.
The underlying double loop map is determined by a class in $1+p\alpha \in \pi_3(\mathrm{B}^3\mathrm{GL}_1(S^0_p)) \cong \mathbb{Z}^{\times}_p$. Our original assumption, made for the sake of contradiction, is reduced to the assertion that a dashed arrow exists the diagram below:
$$
\begin{tikzcd}[column sep = 5.0em]
S^4 \arrow{d} \arrow{r}{1+p\alpha} & \mathrm{B}^4\mathrm{GL}_1(S^0_p). \\
\mathbb{H}P^{\infty} \arrow[dashed]{ur}
\end{tikzcd}
$$
We will show this to be impossible by proving the non-existence of a solution to the weaker lifting problem
$$
\begin{tikzcd}
S^4 \arrow{d} \arrow{r}{1+p\alpha} & \Sigma^{\infty}
\mathrm{B}^4\mathrm{GL}_1(S^0_p) \arrow{r}{\ell} & \Sigma^4 L_{K(1)} S^0, \\
\Sigma^{\infty} \mathbb{H}P^{\infty} \arrow[dashed]{urr}
\end{tikzcd}
$$
where $L_{K(1)} S^0$ is $K(1)$-local sphere spectrum and $\ell$ is the Rezk logarithm \cite{rezk}. We first calculate the composite
$$S^4 \stackrel{1+p\alpha}{\longrightarrow} \mathrm{B}^4\mathrm{GL}_1(S^0_p) \stackrel{\ell}{\longrightarrow} \Sigma^4 L_{K(1)} S^0,$$
using Rezk's formula \cite[Theorem 1.9]{rezk} for the logarithm at odd primes:
$$\ell(1+p\alpha)=log(1+p\alpha) - \frac{1}{p} log(1+p\alpha).$$
If $\alpha$ were not a $p$-adic unit, then the composite $\Omega^2 S^3 \longrightarrow \mathrm{BGL}_1(S^0) \longrightarrow \mathrm{BGL}_1(\mathrm{H}\mathbb{Z}/p^2)$ would be null as a $2$-fold loop map, providing a ring map $\mathrm{H}\mathbb{F}_p \longrightarrow \mathrm{H}\mathbb{Z}/p^2$. Since this is absurd, $\alpha$ must be a $p$-adic unit, and we learn that $\ell(1+p\alpha)$ is also a $p$-adic unit.
Without loss of generality, then, we are reduced to showing the impossibility of the following lifting problem:
$$
\begin{tikzcd}
S^4 \arrow{r}{1} \arrow{d} & \Sigma^4 L_{K(1)} S^0, \\
\Sigma^{\infty} \mathbb{H}P^{\infty} \arrow[dashed]{ur}
\end{tikzcd}
$$
where $1$ is the unit of the ring spectrum $L_{K(1)} S^0$.
Let $\mathrm{KU}_p$ denote $p$-complete complex $K$-theory. Recall that the composite
$$L_{K(1)} S^0 \longrightarrow \mathrm{KU}_p \stackrel{\psi^q-1}{\longrightarrow} \mathrm{KU}_p$$
is null for any Adams operation $\psi^q$ with $q$ relatively prime to $p$.
Since $p$ is odd, to finish the problem it will suffice for us to show that no element of $\mathrm{KU}_p^4(\mathbb{H}P^{\infty})$ simultaneously:
\begin{enumerate}
\item Restricts to the unit in $\mathrm{KU}_p^4(S^4)$.
\item Is invariant under the action of $\psi^2$.
\end{enumerate}
Now, $$\mathrm{KU}_p^*(\mathbb{H}P^{\infty}) \cong \mathbb{Z}_p\llbracket e\rrbracket[\beta^{\pm}],$$ where $|e|=0$ and $\beta$ is the Bott class in degree $-2$. Of course, $\psi^2(\beta)=2\beta$, and it will be necessary also to understand $\psi^2(e)$.
Remembering that $\mathbb{H}P^{\infty}$ is $\mathrm{BSU}(2)$, we may calculate $\psi^2(e)$ by determining the restriction of $e$ along the inclusion of the maximal torus $\mathrm{B}S^1 \longrightarrow \mathrm{BSU}(2)$. Indeed, $\mathrm{KU}_p^*(\mathrm{B}S^1) \cong
\mathbb{Z}_p\llbracket x\rrbracket [\beta^{\pm 1}]$, where $x=L-1$. On the other hand, $e=V-2$, where $V$ is the standard representation of
$\mathrm{SU}(2)$ on $\mathbb{C}^2$. The restriction of $e$ is thus $L+L^{-1}-2$, where
$$L^{-1}=(x+1)^{-1} = 1 - x + x^2 - x^3 + \cdots.$$
Since
$$\psi^2(L+L^{-1}-2)=L^2+ L^{-2} -2= (x+1)^2 +\frac{1}{(x+1)^2}-2 = \left(x+1+\frac{1}{x+1} -2\right)^2+4 \left(x+1+\frac{1}{x+1} -2\right),$$
we calculate that
$$\psi^2(e)=e^2+4e.$$
An element of $\mathrm{KU}^4(\mathbb{H}P^{\infty})$ is of the form $\beta^{-2} P(e)$, where $P(e)$ is a power series in $\mathbb{Z}_p\llbracket
e\rrbracket$. The lifting problem in question is equivalent to finding a power series $P(e)=e + c_2 e^2 + \cdots$ such that
$$P(e) = 2^{-2} P\left( \psi^2(e) \right).$$
Using the calculations above, this can be rewritten as the relation
$$4 P(e) = P(e^2+4e).$$
The relation
$$4(e+c_2e^2+c_3e^3+\cdots) = (e^2+4e)+c_2(e^2+4e)^2+c_3(e^2+4e)^3 + \cdots$$
inductively determines each $c_i$, given $c_2,\cdots,c_{i-1}$, according to the formula
$$c_i = \frac{2}{(2i)!} \prod_{j=2}^{i} \left(-(j-1)^2\right).$$
In particular, this formula does not yield a $p$-adic integer for $i=\frac{p+1}{2}$, implying that there is no lift through $\mathbb{H}P^{\frac{p+1}{2}}$.
\end{proof}
\begin{remark}
The Adams conjecture provides a map from the connective cover of the $K(1)$-local sphere spectrum into $\mathrm{gl}_1(S^0_{(p)})$.
Using a variant of this due to Bhattacharya and Kitchloo \cite{bhattacharya-kitchloo}, it is possible to construct maps $\mathbb{H}P^{k} \longrightarrow \mathrm{B}^4 \mathrm{GL}_1(S^0_{(p)})$, for small values of $k$. Indeed, \cite{bhattacharya-kitchloo} employs arguments very similar to the ones above in order to produce multiplicative structures on Moore spectra. The authors believe, but have not verified, that it is possible to equip the map $S^3 \stackrel{1-p}{\longrightarrow} \mathrm{B}^3
\mathrm{GL}_1(S^0_{(p)})$ with an $\mathbb{A}_{\frac{p-1}{2}}$-algebra structure in this manner.
\end{remark}
\begin{remark}
It is well-known that the integral Eilenberg--Maclane spectrum $\mathrm{H}\mathbb{Z}_{(p)}$ is the Thom spectrum of a double loop composite
$$\Omega^2 (S^3 \langle 3 \rangle) \longrightarrow \Omega^2 S^3 \longrightarrow \mathrm{BGL}_1(S^0_p).$$
One could attempt to refine this to a triple loop map, using the equivalence $\Omega \mathbb{H}P^{\infty} \langle 4 \rangle \simeq S^3 \langle 3 \rangle$.
The same obstruction as above proves that this strategy cannot work at odd primes, because the map $$\mathbb{H}P^{\infty} \langle 4 \rangle \longrightarrow \mathbb{H}P^{\infty}$$ is a $K(1)$-local equivalence.
\end{remark}
\section{Concluding remarks}\label{sec:epilogue}
\subsubsection*{The integral Eilenberg-MacLane spectrum}
Define $S^{\lambda+1}\langle \lambda+1\rangle$
as the fiber of the unit map
\[
S^{\lambda+1} \to K(\underline{\mathbb{Z}}, \lambda+1)
:= \Omega^{\infty}
\left(\Sigma^{\lambda+1}\mathrm{H}\underline{\mathbb{Z}}\right).
\]
Then we have the following result.
\begin{theorem}\label{thm:Z} There is an equivalence of
$\mathbb{E}_{\lambda}$-algebras
\[
\left(
\Omega^{\lambda}(S^{\lambda+1}\langle \lambda+1\rangle)\right)^{\mu}
\simeq \mathrm{H}\underline{\mathbb{Z}}_{(p)}.
\]
\end{theorem}
\begin{proof} We argue as in
Antol\'{\i}n-Camarena-Barthel
\cite[\S5.2]{omar-toby}, though we need
not develop all the technology present there.
We have a fiber sequence
\[
\Omega^{\lambda}S^{\lambda+1}\langle \lambda+1\rangle
\to \Omega^{\lambda}S^{\lambda+1} \to S^1.
\]
Decomposing $S^1$ into a 0-cell and a 1-cell, and
trivializing the fibration on each cell, produces a decomposition
of the Thom spectrum $\left(\Omega^{\lambda}S^{\lambda+1}\right)^{\mu}$
as a cofiber
\[
\xymatrix{
\left(\Omega^{\lambda}S^{\lambda+1}\langle \lambda+1\rangle
\right)^{\mu}
\ar[r]^{x} &
\left(\Omega^{\lambda}S^{\lambda+1}\langle \lambda+1\rangle
\right)^{\mu}
\ar[r] &
\left(\Omega^{\lambda}S^{\lambda+1}\right)^{\mu}}
\simeq \mathrm{H}\underline{\mathbb{F}}_p.
\]
Each of these Thom spectra came from bundles classified by
$\mathbb{A}_2$-maps, which is enough to ensure
that the map $x$ induces a map
$\pi_*\left(\Omega^{\lambda}S^{\lambda+1}\langle \lambda+1\rangle\right)^{\mu}
\to
\pi_*\left(\Omega^{\lambda}S^{\lambda+1}\langle \lambda+1\rangle
\right)^{\mu}$ of modules over
$\pi_0\left(\Omega^{\lambda}S^{\lambda+1}\langle \lambda+1\rangle
\right)^{\mu}$. In particular, on homotopy the map corresponds
to multiplication by some element $x \in \pi_0\left(\Omega^{\lambda}S^{\lambda+1}\langle \lambda+1\rangle
\right)^{\mu}$.
Arguments similar to those in the proof
of the main theorem show that
\[\underline{\pi}_0
\left(\Omega^{\lambda}S^{\lambda+1}\langle \lambda+1\rangle
\right)^{\mu}\simeq\underline{\mathbb{Z}}_{(p)},\] so we must have
$x = p$. The result follows
from Nakayama's lemma once
one argues that the genuine fixed point spectra
have finitely generated homotopy groups
in each degree. (For example, isotropy separation
reduces us to the corresponding statement on
geometric fixed points, where it follows from the Thom isomorphism.)
\end{proof}
\begin{remark} The map
$S^{\lambda+1} \to K(\underline{\mathbb{Z}}, \lambda+1)$ deloops to a map
$\mathbb{H}P^{\infty} \to K(\underline{\mathbb{Z}},
\lambda+2)$. It follows from Theorem
\ref{thm:main-even} that the equivalence above
is one of $\mathbb{E}_{\lambda+1}$-algebras
when $p=2$.
\end{remark}
\begin{remark} Unlike the classical case,
it is unclear whether the statement globalizes to
a construction of $\mathrm{H}\underline{\mathbb{Z}}$. Our methods do
not construct
$\mathrm{H}\underline{\mathbb{F}}_{\ell}$ as a Thom spectrum
when $\ell$ does not divide the order of $G$.
\end{remark}
\subsubsection*{Questions}
We conclude with a few open-ended questions.
\begin{question}
Is $\mathrm{H}\underline{\mathbb{F}}_p$ a Thom spectrum for any group $G$ that is not cyclic of $p$-power order? It seems plausible that this is so for dihedral groups, as was suggested to the authors by Stefan Schwede. Can obstructions be found for other $G$?
\end{question}
\begin{question} Calculations indicate that the spectrum
$\left(\Omega^{\lambda}S^{\lambda+1}\right)^{\mu}$
is not $\mathrm{H}\underline{\mathbb{F}}_p$ for
the groups $C_n$ when $n$ is not a power of $p$.
What can be said about $\left(\Omega^{\lambda}S^{\lambda+1}\right)^{\mu}$ as an $S^1$ or $O(2)$-equivariant spectrum? At least one expects an interesting $C_{p^{\infty}}$-spectrum.
\end{question}
\begin{question} One of Mahowald's motivations for proving the equivalence
$(\Omega^2S^3)^{\mu} \simeq \mathrm{H}\mathbb{F}_2$ is that
the left hand side carries a natural filtration due to Milgram and May.
This produces a filtration of $\mathrm{H}\mathbb{F}_2$ by
spectra which turn out to be the Brown-Gitler spectra
of \cite{brown-gitler} (see \cite{brown-peterson, cohen, hunter-kuhn}).
The $G$-space $\Omega^{\lambda}S^{\lambda+1}$ also carries
the arity filtration from the $\mathbb{E}_{\lambda}$-operad,
so we could \emph{define} equivariant Brown-Gitler spectra
using this filtration. It would be interesting to know if these spectra are of any use.
In the case $G = C_2$ there are
two different operadic filtrations of $\Omega^{2\sigma}S^{2\sigma+1}
\simeq \Omega^{\rho}S^{\rho+1}$. This leads to two different notions
of Brown-Gitler spectra. How are they related?
\end{question}
\begin{question} What are the Thom spectra obtained
by killing other natural elements in the Burnside ring
in a highly structured manner?
What is the free $\mathbb{E}_{\lambda}$-algebra
in $C_p$-spectra with
$[C_p]=0$?
\end{question}
\begin{question}
Can the $C_{p^n}$-equivariant dual Steenrod algebra $\mathrm{H}\underline{\mathbb{F}}_p \wedge \mathrm{H}\underline{\mathbb{F}}_p$ be profitably studied via its equivalence with $\mathrm{H}\underline{\mathbb{F}}_p \wedge \Sigma^{\infty}_+ \Omega^{\lambda} S^{\lambda+1}$?
\end{question}
\section{An equivariant \texorpdfstring{$H$}{H}-space orientation}\label{sec:exotic}
The purpose of this section is to prove the following theorem.
\begin{theorem}\label{thm:h-orient} There is an $\mathbb{A}_2$-structure on
$S^{\lambda+1}_{(p)}$ such that
\begin{enumerate}[\normalfont (i)]
\item The adjoint to $1-p \in \pi_0(S^0_{(p)})^{\times}$ refines to
an $\mathbb{A}_2$-map
\[
S_{(p)}^{\lambda+1}
\to \mathrm{B}^{\lambda+1}\mathrm{GL}_1(S^0_{(p)}).
\]
\item The composite
\[
S^{\lambda+1}_{(p)} \to \mathrm{B}^{\lambda+1}
\mathrm{GL}_1(S^0_{(p)}) \to
\mathrm{B}^{\lambda+1}
\mathrm{GL}_1(\mathrm{H}\underline{\mathbb{F}}_p)
\]
is nullhomotopic through $\mathbb{A}_2$-maps.
\end{enumerate}
\end{theorem}
Before doing so, we record a corollary which follows from
\cite[X.6.4]{LMS} and the previous theorem:
\begin{corollary} The Thom class
\[
\left(\Omega^{\lambda}S^{\lambda+1}_{(p)}\right)^{\mu}
\longrightarrow
\mathrm{H}\underline{\mathbb{F}}_p
\]
has the structure of a map of $\mathbb{A}_2$-algebras in
$\mathbb{E}_{\lambda}$-algebras.
\end{corollary}
\begin{remark}\label{rmk:a2-str} The $\mathbb{A}_2$-structure we define is necessarily
$p$-local. We only use this structure to produce a multiplication
on the homology of $\left(\Omega^{\lambda}S^{\lambda+1}_{(p)}\right)^{\mu}$
which is preserved by the map induced by the Thom class. The
localization map
\[
\Omega^{\lambda}S^{\lambda+1}\longrightarrow
\Omega^{\lambda}S^{\lambda+1}_{(p)}
\]
induces an isomorphism on mod $p$ homology and using this
one can show that the map
\[
\left(\Omega^{\lambda}S^{\lambda+1}\right)^{\mu}
\longrightarrow
\left(\Omega^{\lambda}S^{\lambda+1}_{(p)}\right)^{\mu}
\]
is a $p$-local equivalence. On the other hand,
the left hand side is automatically $p$-local, being
an (equivariant)
homotopy colimit of $p$-local spectra by construction.
We can transport
the $\mathbb{A}_2$-structure from the target to the source
and get a map
\[
\left(\Omega^{\lambda}S^{\lambda+1}\right)^{\mu} \to
\mathrm{H}\underline{\mathbb{F}}_p
\]
of $\mathbb{A}_2$-algebras
in $\mathbb{E}_{\lambda}$, even though the
$\mathbb{A}_2$-structure does not arise from applying
the Thom construction to an $\mathbb{A}_2$-map of $G$-spaces.
\end{remark}
We will need to recall a few facts about $\mathbb{A}_2$-monoid
objects, mainly to establish notation.
\begin{definition} Let $\mathcal{C}$
be an $\infty$-category which admits products.
For $0\le k \le \infty$,
an \textbf{$\mathbb{A}_k$-monoid} $X$ in $\mathcal{C}$
is a truncated simplicial object
\[
\mathbf{B}_{\le k}X:
\Delta_{\le k}^{op} \longrightarrow \mathcal{C}
\]
such that
\begin{itemize}
\item The object $(\mathbf{B}_{\le k}X)_0$ is final.
\item If $k\ge 1$, then for $1\le j \le k$, the maps
\[
(\mathbf{B}_{\le k}X)_j \to (\mathbf{B}_{\le k}X)_1,
\]
induced by $\{i, i+1\} \to [j]$, exhibit the source
as the $j$-fold product of $(\mathbf{B}_{\le k}X)_1$.
\end{itemize}
In this case we denote $(\mathbf{B}_{\le k}X)_1$ by $X$.
If $\mathcal{C}$ admits
homotopy colimits, we denote the homotopy colimit of the diagram
$\mathbf{B}_{\le k}X$ by $\mathrm{B}_{\le k}X$.
The $\infty$-category of $\mathbb{A}_k$-monoids,
$\mathsf{Mon}_{\mathbb{A}_k}(\mathcal{C})$, is
the full subcategory of $\mathsf{Fun}(\Delta_{\le k}^{op}, \mathcal{C})$
spanned by the $\mathbb{A}_k$-monoids. Note that
restriction defines forgetful functors
\[
\mathsf{Mon}_{\mathbb{A}_k}(\mathcal{C})
\to \mathsf{Mon}_{\mathbb{A}_j}(\mathcal{C})
\]
for $j\le k$, and we have natural maps
\[
\mathrm{B}_{\le j}X \longrightarrow \mathrm{B}_{\le k}X.
\]
\end{definition}
\begin{example}[$k=0$] An $\mathbb{A}_0$-monoid is a final
object of $\mathcal{C}$. The natural maps above provide
each $\mathrm{B}_{\le j}X$ with a basepoint.
\end{example}
\begin{example}[$k=1$] An $\mathbb{A}_1$-monoid in $\mathcal{C}$
is specified by the data of an object $X$ and a map
$\ast \to X$, where $\ast$ is a final object in $\mathcal{C}$.
The object $\mathrm{B}_{\le 1}X$ is computed as the colimit of
\[
\xymatrix{
X \ar@<-.5ex>[r]\ar@<.5ex>[r] & \ast\ar[l]
}
\]
which is the suspension $\Sigma X$.
\end{example}
\begin{example}[$k=2$] Suppose $\mathcal{C}$ admits
limits and colimits.
By \cite[A.2.9.16]{HTT}, extending
a diagram $\mathbf{B}_{\le 1}X$
to a diagram $\mathbf{B}_{\le 2}X$
is equivalent to specifying a factorization
\[
\xymatrix{
&C\ar[dr]^{(d_0, d_1, d_2)}&\\
X \vee X \ar[rr]
\ar[ur] && X \times X \times X
}
\]
where $C \in \mathcal{C}$ is some object and
$X \vee X$ denotes the pushout $X \amalg_{\ast} X$
where $\ast = (\mathbf{B}_{\le 1}X)_0$ is a final object.
In order for this extended diagram to define an $\mathbb{A}_{2}$-monoid,
the maps $d_0$ and $d_2$ must give an equivalence
$C \simeq X \times X$. Under this equivalence, the map
$X \vee X \to X \times X$ is the standard one. The only additional
data is the map $d_1: X \times X \simeq C \to X$.
In summary: an $\mathbb{A}_2$-monoid in $\mathcal{C}$
is precisely the data of a pointed object $X$
together with a map $m: X \times X \to X$ which extends
the fold map $X \vee X \to X$. It follows that the cofiber of
$\mathrm{B}_{\le 1}X \to \mathrm{B}_{\le 2}X$ is
given by $\Sigma^2(X \wedge X)$.
\end{example}
\begin{remark} If we endow $\mathcal{C}$ with the Cartesian monoidal
structure, then we see that an $\mathbb{A}_2$-monoid object in
$\mathcal{C}$ is the same data as an $\mathbb{A}_2$-algebra object
as in Definition \ref{dfn:a2}.
\end{remark}
\begin{example}[Loop spaces]\label{ex:loops} If $\mathcal{C} = \mathsf{Spaces}^G$
and $Y$ is a pointed $G$-space, then $\Omega Y$ has
a natural $\mathbb{A}_{\infty}$-structure.
\end{example}
Now we will restrict attention to the $\infty$-category
$\mathsf{Spaces}^{G}$ of $G$-spaces.
The following is proved just as in the classical case,
for which there are many references. The earliest
appears to be \cite[Proposition 3.5]{stash-ab}.
\begin{lemma} If $X$ is
an $\mathbb{A}_2$-algebra, then
the map $X \to \Omega \mathrm{B}_{\le 2}X$
adjoint to
\[
\Sigma X = \mathrm{B}_{\le 1}X
\to \mathrm{B}_{\le 2} X
\]
extends to an $\mathbb{A}_2$-map.
\end{lemma}
We now return to the case of interest. We begin by establishing
the existence of the $\mathbb{A}_2$-structure we need on
$S^{\lambda+1}$.
\begin{proposition}\label{prop:exotic-h} For $p$ an odd prime,
there is an
$\mathbb{A}_2$-structure on $S_{(p)}^{\lambda+1}$
with the property that the map $\Sigma X \to \mathrm{B}_{\le 2}X$
stably splits.
\end{proposition}
We will deduce this proposition from the following calculation,
which is an equivariant version of a classical result
(see, e.g. \cite{james}). Here we use
\[
E: [X,Y] \to [\Sigma X, \Sigma Y]
\]
to denote the homomorphism on
equivariant homotopy classes of maps
given by suspending by
$S^1$ equipped with the trivial $G$-action.
\begin{proposition}\label{prop:dbl-susp}
Denote by $\nu: S^{2\lambda+3} \to S^{\lambda+2}$
the attaching map for the inclusion
$S^{\lambda+2}\simeq \mathbb{H}P^1 \to
\mathbb{H}P^{2}$.
Then, after localization at $p$,
\[
E(2\nu) \in E^2(\pi_{2\lambda+2}S^{\lambda+1}).
\]
\end{proposition}
\begin{proof}[Proof of Proposition \ref{prop:exotic-h} assuming
Proposition \ref{prop:dbl-susp}]
Recall from
that there is an equivalence $\Omega \mathbb{H}P^{\infty} \simeq
S^{\lambda+1}$ and hence, by Example \ref{ex:loops},
we get an $\mathbb{A}_{\infty}$-structure on
$S^{\lambda+1}$.
This, in turn, determines an $\mathbb{A}_2$-structure on
$S^{\lambda+1}$, and
the attaching map for $\mathrm{B}_{\le 1}S^{\lambda+1}
\to \mathrm{B}_{\le 2}S^{\lambda+1}$ is precisely
\[
\nu: S^{2\lambda+3} \to S^{\lambda+2}.
\]
In general,
if one modifies an $\mathbb{A}_2$-structure on $X$ by an element
$d \in [X \wedge X, X]$, then the attaching map
\[
\Sigma (X \wedge X) \to \Sigma X
\]
for $\mathrm{B}_{\le 2}X$ is altered by $E(d)$.
After inverting 2, Proposition \ref{prop:dbl-susp} implies that
$E(\nu) = E^2(x)$ for some $x \in \pi_{2\lambda+2}S^{\lambda+1}$.
So alter the $\mathbb{A}_2$-structure above by $x$ and the suspension
of the attaching map for $\mathrm{B}_{\le 2}S^{\lambda+1}$ becomes
null, proving the result.
\end{proof}
Now we turn to the proof of Proposition \ref{prop:dbl-susp}.
We will deduce this theorem from a slightly stronger result.
Recall that, given classes $x \in [\Sigma A, X]$ and
$y \in [\Sigma B, X]$, the Whitehead product
${[x,y] \in [\Sigma (A \wedge B), X]}$ is induced by
the commutator of $\pi_Ax$ and $\pi_By$ in the group
$[\Sigma(A\times B), X]$.
\begin{lemma}\label{lem:bracket}
Let $\iota_{\lambda+2} \in \pi_{2\lambda+3}S^{\lambda+2}$
be the fundamental class. Then, after localization at $p$,
\[
[\iota_{\lambda+2}, \iota_{\lambda+2}] \equiv 2\nu
\mod
E(\pi_{\lambda+2}S^{\lambda+1}).
\]
\end{lemma}
\begin{proof}[Proof of Proposition \ref{prop:dbl-susp} assuming
Lemma \ref{lem:bracket}.] Lemma \ref{lem:bracket} states
$2\nu - [\iota_{\lambda+2}, \iota_{\lambda+2}]$ lies in
the image of $E$. Hence $E(2\nu) - E([\iota_{\lambda+2},
\iota_{\lambda+2}])$ lies in the image of $E^2$.
The result now follows from the observation that suspensions
of Whitehead products vanish. Indeed, with notation
as in the definition of the Whitehead product above,
$E([x,y])$ is computed as a commutator in
$[\Sigma^2(A \times B), X]$. But this is an abelian group
by the Eckmann-Hilton argument, so commutators vanish.
\end{proof}
In order to prove Lemma \ref{lem:bracket} we will establish
an exact sequence of the form
\[
\xymatrix{
\pi_{2\lambda+2}S^{\lambda+1}\ar[r]^{E}&
\pi_{2\lambda+3}S^{\lambda+2}\ar[r]^{H} &
\pi_{2\lambda+3}S^{2\lambda+3}
}
\]
and then identify the image of the Whitehead product
in the last group. To that end, we note that the James
splitting
\[
\Sigma \Omega \Sigma X \simeq
\Sigma \left(\bigvee_{k\ge 1} X^{\wedge k}\right)
\]
holds in $\mathsf{Spaces}^G_*$ (see \cite{kron}). This provides
a natural transformation
\[
H: \Omega\Sigma X \to \Omega \Sigma X^{\wedge 2}
\]
which induces a map
\[
H: \pi_{\star+1} \Sigma X \to \pi_{\star+1}\Sigma X^{\wedge 2}
\]
for any $X$.
\begin{lemma} The sequence
\[
\xymatrix{
S^{\lambda+1} \ar[r]^{E} & \Omega S^{\lambda+2}\ar[r]^H &
\Omega S^{2\lambda+3}
}
\]
is a fiber sequence when localized at $p$.
\end{lemma}
\begin{proof} Let $F$ denote the homotopy fiber of
$H$ so that we have a natural map $S^{\lambda+1}\to F$.
We would like to show this is an equivalence.
Since restriction to underlying spaces and fixed points
preserves homotopy limits and colimits, we are reduced to
the nonequivariant statement that
\[
\xymatrix{
S^{2n+1} \ar[r]^E& \Omega S^{2n+2} \ar[r]^H& \Omega S^{4n+3}
}
\]
is a fiber sequence when localized
at $p$ for $n=0, 1$. In fact, it is a classical result of
James \cite{james}
that this is a $p$-local fiber sequence for any $n\ge 0$.
\end{proof}
We will need some control over the last term in this sequence,
which is provided by an equivariant version of the
Brouwer-Hopf degree theorem. For us, the only fact we need
is that the homomorphism
\[
\pi_{2\lambda+3}S^{2\lambda+3} \to \bigoplus_{K \subseteq G} \mathbb{Z},
\]
recording each of the degrees of a map on $K$-fixed points, is
an injection. See, e.g., \cite[8.4.1]{tdieck}. We now prove the only remaining lemma necessary for
producing the exotic $H$-space structure on $S^{\lambda+1}$.
\begin{proof}[Proof of lemma \ref{lem:bracket}] The formation
of Whitehead products commutes with passage to fixed points
and restriction to underlying classes, as does the map $H$.
From the remarks above, it then suffices to check the nonequivariant
formulas:
\[
H([\iota_4, \iota_4]) = 2H(\nu),
\]
\[
H([\iota_2, \iota_2]) = 2H(\nu^K), \quad K\ne\{e\}.
\]
But $\nu$ and $\nu^K = \eta$
(see Remark \ref{rmk:nu-fixed-eta}) have Hopf invariant 1,
while $[\iota_{2n}, \iota_{2n}]$ has Hopf invariant 2 for any $n\ge 1$,
whence the result.
\end{proof}
Since the attaching map in $\mathrm{B}_{\le 2}S^{\lambda+1}_{(p)}$
is stably null, the following lemma is immediate.
\begin{lemma}\label{lem:h-extend-p}
There exists a dotted map making the diagram
below commute up to homotopy in $\mathsf{Sp}^G$:
\[
\xymatrix{
S_{(p)}^{\lambda+2} \ar[r]^-{1-p} \ar[d]&
\Sigma^{\lambda+2}\mathrm{gl}_1S^0_{(p)}\\
\Sigma^{\infty}\mathrm{B}_{\le 2}S_{(p)}^{\lambda+1}\ar@{-->}[ur] &
}
\]
\end{lemma}
We will eventually need to produce a Thom isomorphism
in mod $p$ cohomology which respects our extra structure. For
that we require the next lemma.
\begin{lemma}\label{lem:h-null}
Choose a dotted map $\tilde{f}$ as in the previous
lemma. Then the composite
\[
\xymatrix{
\Sigma^{\infty}\mathrm{B}_{\le 2} S_{(p)}^{\lambda+1}\ar[r]^{\tilde{f}}&
\Sigma^{\lambda+2}\mathrm{gl}_1(S^0_{(p)}) \ar[r] &
\Sigma^{\lambda+2}\mathrm{gl}_1(\mathrm{H}\underline{\mathbb{F}}_p)
}
\]
is null.
\end{lemma}
\begin{proof} The composite
\[
\xymatrix{
S^{\lambda+2}_{(p)} \ar[r] &
\Sigma^{\infty}\mathrm{B}_{\le 2} S_{(p)}^{\lambda+1}\ar[r]^{\tilde{f}}&
\Sigma^{\lambda+2}\mathrm{gl}_1(S^0_{(p)}) \ar[r] &
\Sigma^{\lambda+2}\mathrm{gl}_1(\mathrm{H}\underline{\mathbb{F}}_p)
}
\]
vanishes since
$1-p = 1 \in \pi_0^G(\mathrm{gl}_1(\mathrm{H}\underline{\mathbb{F}}_p)$
is the basepoint component. So the map $\tilde{f}$ factors through
some map
\[
S^{2\lambda+4} \longrightarrow
\Sigma^{\lambda+2}\mathrm{gl}_1(\mathrm{H}\underline{\mathbb{F}}_p).
\]
But
\[
\pi_{\lambda+2}^{G}\mathrm{gl}_1(\mathrm{H}\underline{\mathbb{F}}_p)
\simeq \pi_{\lambda+2}^{G}\mathrm{H}\underline{\mathbb{F}}_p
= 0
\]
since $S^{\lambda+2}$ is $2$-connective, whence the claim.
\end{proof}
Finally, we arrive at the proof of the main theorem of the section.
\begin{proof}[Proof of Theorem \ref{thm:h-orient}]
Choose a dotted map as in Lemma
\ref{lem:h-extend-p} and let $f$ be its adjoint,
\[
f: \mathrm{B}_{\le 2}S^{\lambda+1}_{(p)}
\longrightarrow \mathrm{B}^{\lambda+2}\mathrm{GL}_1(S^0_{(p)}).
\]
Then the map $1-p: S^{\lambda+1}_{(p)} \to
\mathrm{B}^{\lambda+1}\mathrm{GL}_1(S^0_{(p)})$
factors as a composite:
\[
\xymatrix{
S^{\lambda+1}_{(p)}
\ar[r] & \Omega \mathrm{B}_{\le 2}S^{\lambda+1}_{(p)}
\ar[r]^-{\Omega f} &
\Omega \mathrm{B}^{\lambda+2}\mathrm{GL}_1(S^0_{(p)})
\ar[r] & \mathrm{B}^{\lambda+1}\mathrm{GL}_1(S^0_{(p)})
},
\]
each of which is an $H$-map. This proves part (i) of the theorem.
To prove part (ii), consider the diagram:
\[
\xymatrix{
S^{\lambda+1}_{(p)}
\ar[r] & \Omega \mathrm{B}_{\le 2}S^{\lambda+1}_{(p)}
\ar[r]^-{\Omega f} &
\Omega \mathrm{B}^{\lambda+2}\mathrm{GL}_1(S^0_{(p)})
\ar[d]_-{\Omega\mathrm{B}^{\lambda+2}\mathrm{GL}_1(\iota)}
\ar[r] & \mathrm{B}^{\lambda+1}\mathrm{GL}_1(S^0_{(p)})\ar[d]\\
&&
\Omega\mathrm{B}^{\lambda+2}\mathrm{GL}_1(
\mathrm{H}\underline{\mathbb{F}}_p) \ar[r] &
\mathrm{B}^{\lambda+1}
\mathrm{GL}_1(\mathrm{H}\underline{\mathbb{F}}_p)
}
\]
where $\iota: S^0_{(p)} \to \mathrm{H}\underline{\mathbb{F}}_p$
is the unit map.
The composite
\[
\xymatrix{
\mathrm{B}_{\le 2}S^{\lambda+1}_{(p)}
\ar[r]^-{f} & \mathrm{B}^{\lambda+2}\mathrm{GL}_1(S^0_{(p)})
\ar[rr]^-{\mathrm{B}^{\lambda+2}\mathrm{GL}_1(\iota)} &&
\mathrm{B}^{\lambda+2}\mathrm{GL}_1(
\mathrm{H}\underline{\mathbb{F}}_p)
}
\]
is null by Lemma \ref{lem:h-null}. The loop of this composite
is then null through $\mathbb{A}_{\infty}$-maps and the result follows.
\end{proof}
\section{Computing the zeroth homotopy Mackey functor}\label{sec:norm}
In this section, we establish that the zeroth homotopy Mackey functor of our Thom spectrum is as expected. That is to say, we give a proof that
\[
\underline{\pi}_0\left(\Omega^{\lambda}S^{\lambda+1}\right)^{\mu}
=
\underline{\mathbb{F}}_p.
\]
By construction,
$\left(\Omega^{\lambda}S^{\lambda+1}\right)^{\mu}$ receives
a map from the mod $p$ Moore spectrum
$M(p) = (S^1)^{\mu}$. This is enough to guarantee that
$p=0$ in
$\underline{\pi}_0\left(\Omega^{\lambda}S^{\lambda+1}\right)^{\mu}$.
However, $\underline{\pi}_0S^0$ is the Burnside Mackey
functor $\underline{A}$, and $\underline{A}/(p)$ is not
$\underline{\mathbb{F}}_p$. For example, when $G=C_p$,
we have
\[
\underline{A}/(p) = \begin{gathered}
\xymatrix{
\mathbb{F}_p\left\{ [C_p]\right\}\ar@/_/[d]\\
\mathbb{F}_p\ar@/_/[u]
}
\end{gathered}.
\]
We will need to use some extra structure on
$\left(\Omega^{\lambda}S^{\lambda+1}\right)^{\mu}$
to show that $[C_p]$ also vanishes. More generally,
we must show that $[C_{p^n}/C_{p^k}]$ vanishes
in the Hurewicz image for all $k$.
For the remainder of this section we write $G = C_{p^n}$
for a cyclic group of prime power order.
\begin{definition} We say that a $G$-spectrum
$X$ is \textbf{weakly normed} if it is equipped
with a map $S^0 \to X$, and, for each $H \subseteq G$, a map of $H$-spectra
$\mathrm{N}^HX \to X$ such that the diagram
\[
\xymatrix{
\mathrm{N}^H(S^0)\ar[d]\ar@{=}[r] & S^0\ar[d]\\
\mathrm{N}^H(X) \ar[r] & X
}
\]
commutes in the homotopy category.
\end{definition}
\begin{remark} This is the weakest structure necessary to run the
arguments below, but it is perhaps not the most natural definition.
In most examples one at least has compatibility between the norms as
$H$ varies, and the map $S^0 \to X$ acts as a unit for an
underlying multiplication.
\end{remark}
The Thom spectrum $\left(\Omega^{\lambda}S^{\lambda+1}\right)^{\mu}$
is weakly normed, as we now show. This result is well-known; compare,
for example, \cite[Theorem 2.12]{hill-disks}.
\begin{lemma}\label{lem:lambda-has-norms} If $X$ is an $\mathbb{E}_{\lambda}$-algebra
then it is canonically weakly normed.
\end{lemma}
\begin{proof}[Proof of \ref{lem:lambda-has-norms}]
In order to conform with the existing literature
we will present a proof within the point-set model of orthogonal $G$-spectra
as in \cite[\S B]{HHR}. In particular, we will model $X$ by a
positively cofibrant orthogonal $G$-spectrum.
Since the
restriction of an $\mathbb{E}_{\lambda}$-algebra
is still an $\mathbb{E}_{\lambda}$-algebra,
it will suffice to construct the norm
$\mathrm{N}^GX \to X$.
By definition, $X$ comes equipped with a map
\[
\widetilde{\mathrm{Conf}}_{p^n}(\lambda)_+ \wedge_{\Sigma_{p^n}}
X^{\wedge p^n} \longrightarrow X,
\]
where $\widetilde{\mathrm{Conf}}_{p^n}(\lambda)$ denotes
the $G$-space of configurations of $p^n$ ordered points in
$\lambda$. Consider the inclusion
$G \hookrightarrow \Sigma_{p^n}$ which sends
a generator to the standard $p^n$-cycle
$(1, 2, ..., p^n)$ and let
$\Gamma$ denote the graph of this inclusion.
Let $\zeta = e^{2\pi i/p^n}$.
Then the ordered tuple $(1, \zeta, \zeta^2, ..., \zeta^{p^n-1})
\in \widetilde{\mathrm{Conf}}_{p^n}(\lambda)$ produces
a $G \times \Sigma_{p^n}$-equivariant inclusion:
\[
\frac{G \times \Sigma_{p^n}}{\Gamma}
\longrightarrow \widetilde{\mathrm{Conf}}_{p^n}(\lambda).
\]
This, in turn, gives us a map
\[
\left(\frac{G \times \Sigma_{p^n}}{\Gamma}\right)_+ \,\,
\underset{\Sigma_{p^n}}{\wedge} X^{\wedge p^n}
\longrightarrow X.
\]
To complete the proof, we note that (\cite[Prop. 6.2]{blumberg-hill}),
for any $G$-spectrum $Y$,
we have
\[
\left(\frac{G \times \Sigma_{p^n}}{\Gamma}\right)_+\,\,
\underset{\Sigma_{p^n}}{\wedge} Y^{\wedge p^n}
\simeq \mathrm{N}^GY.
\]
\end{proof}
\begin{proposition}\label{prop:norm-and-vanish} Suppose $X$ is
weakly normed.
Suppose further
that $p=0 \in \pi_0^{G}X$. Then
$[H/K] = 0 \in \pi_0^{H}X$ for all
$K \subseteq H \subseteq G$.
\end{proposition}
\begin{corollary}\label{cor:pi-0} We have
\[
\underline{\pi}_0\left(\Omega^{\lambda}S^{\lambda+1}\right)^{\mu}
= \underline{\mathbb{F}}_p.
\]
\end{corollary}
\begin{proof}[Proof of Proposition \ref{prop:norm-and-vanish}]
Recall that $G = C_{p^n}$. If the result is proved for $C_{p^{n-1}}
\subseteq G$, then the classes
\[
p, \mathrm{tr}_{C_{p^{n-1}}}^G([C_{p^{n-1}}])
= [G], \mathrm{tr}_{C_{p^{n-1}}}^G([C_{p^{n-1}}/C_{p}])
= [G/C_p], ..., [G/C_{p^{n-2}}]
\]
all vanish in $\pi_0^GX$. The result now follows from the next lemma.
\end{proof}
\begin{lemma}\label{lem:norm-p-formula} If $X$ is weakly normed, then
\[
\mathrm{N}^G(p) \equiv [G/C_{p^{n-1}}] \mod
(p, [G/K] : K \subsetneq C_{p^{n-1}}).
\]
\end{lemma}
\begin{proof} It suffices to prove this
formula when $X = S^0$, i.e. for the Burnside Mackey
functor $\underline{A}$.
By \cite[Lemma A.36]{HHR}, the
norm of $p$ is the class of the $G$-set
$\mathrm{map}(G, \{1, ..., p\})$. By recording the size of the
image of a map, we get an equality in $A(G)$:
\[
\left[\mathrm{map}(G, \{1, ..., p\})\right]
= \sum_{0<k\le n} \binom{p}{k} [\mathrm{surj}(G, \{1, ..., k\})]
\]
where $\mathrm{surj}(G, \{1, ..., k\})$ denotes the $G$-set of
surjective maps $G \to \{1, ..., k\}$. So we have
\[
\mathrm{N}^G(p) \equiv [\mathrm{surj}(G, \{1, ..., p\})] \mod p.
\]
We are only concerned with the orbits in $\mathrm{surj}(G, \{1, ..., p\})$
with isotropy $C_{p^{n-1}}$ or $G$. There is only one
orbit with isotropy $C_{p^{n-1}}$, namely the orbit
of the quotient map $G \to G/C_{p^{n-1}} \simeq \{1, ..., p\}$.
There are $p$ orbits with isotropy $G$, namely the $p$
constant maps. This completes the proof.
\end{proof}
\section{Outline of the proof of Theorem \ref{thm:MainThom}} \label{sec:outline}
We fix, for the remainder of the paper, a prime $p$ as well as a non-negative integer $n$. We let $G=C_{p^n}$ denote the cyclic group of order $p^n$.
The Thom spectrum of the map
$$S^1 \stackrel{1-p}{\longrightarrow} \mathrm{BGL}_1(S^0_{(p)})$$
is the mod $p$ Moore spectrum $M(p)$, which admits a Thom class
for $\mathrm{H}\underline{\mathbb{F}}_p$. It follows formally
(see the argument for Proposition 5.3 in \cite{behrens-wilson})
that there is a Thom class
\[
\alpha: \left(\Omega^{\lambda}S^{\lambda+1}\right)^{\mu}
\longrightarrow \mathrm{H}\underline{\mathbb{F}}_p.
\]
Our goal is to show that this map is an equivalence of $G$-spectra.
By induction on the order of the group
(the base case being supplied by Hopkins-Mahowald),
it will suffice to prove that the map on geometric fixed
points
\[
\alpha^{\Phi G}:
\left(\left(\Omega^{\lambda}S^{\lambda+1}\right)^{\mu}\right)^{\Phi G}
\longrightarrow \mathrm{H}\underline{\mathbb{F}}^{\Phi G}_p
\]
is an equivalence.
The proof proceeds in several steps.
\begin{itemize}
\item[\underline{Step 1}.] Compute the homotopy groups
of $\left(\left(\Omega^{\lambda}
S^{\lambda+1}\right)^{\mu}\right)^{\Phi G}$.
\item[\underline{Step 2}.] Compute the homotopy groups of
$\mathrm{H}\underline{\mathbb{F}}^{\Phi G}_p$.
\item[\underline{Step 3}.] Show that $\alpha^{\Phi G}$
is a ring map (for some ring structure on the source).
\item[\underline{Step 4}.]
Show that $\alpha^{\Phi G}$ hits algebra generators
in the target.
\end{itemize}
The computation in Step 2 is well-known, and is stated as
Lemma \ref{lem:em-geo-fixed} below. At odd primes, one learns that
\[
\pi_*\mathrm{H}\underline{\mathbb{F}}_p^{\Phi G}
= \mathbb{F}_p[t] \otimes \Lambda(s), \, |t|=2, |s|=1.
\]
Step 1 is more difficult. In \S\ref{sec:action} we show that
$\left(\Omega^{\lambda}S^{\lambda+1}\right)^{G} =
\Omega^2S^3 \times \Omega S^2$, and,
after developing some more properties
of this Thom spectrum,
we verify (Lemma \ref{lem:htpy-thom-geo-fixed}) that the homotopy groups
of
$\left(\left(\Omega^{\lambda}S^{\lambda+1}\right)^{\mu}\right)^{\Phi G}$
are \emph{additively} given by
\[
\pi_*\left(\left(\Omega^{\lambda}S^{\lambda+1}\right)^{\mu}\right)^{\Phi G}
= \mathbb{F}_p[x] \otimes \Lambda(y),\, |x|=2, |y|=1.
\]
There is a serious problem to be addressed in Step 3: by construction,
$\left(\Omega^{\lambda}S^{\lambda+1}\right)^{\mu}$ is an
$\mathbb{E}_{\lambda}$-spectrum and the Thom class is represented
by an $\mathbb{E}_{\lambda}$-map. When we take geometric fixed points,
we are left without any obvious multiplicative structure on either the source or the map. We must do additional work to equip with $\alpha^{\Phi G}$ with additional multiplicative structure. The weakest structure that suffices for our purposes is that of an $\mathbb{A}_2$-algebra; we recall the definition below:
\begin{dfn}\label{dfn:a2}
An $\mathbb{A}_2$-algebra in a symmetric monoidal $\infty$-category $(\mathcal{C},\otimes,\mathbf{1})$ consists of the following data:
\begin{enumerate}
\item An object $X \in \mathcal{C}$.
\item A \emph{multiplication} map $m:X \otimes X \to X$.
\item A \emph{unit} map $u:\mathbf{1} \to X$.
\item Choices of $2$-simplices filling each of the diagrams
$$
\begin{tikzcd}
X \otimes 1 \arrow[swap]{d}{id. \otimes u} \arrow{r}{\simeq} &X &&& 1 \otimes X \arrow[swap]{d}{u \otimes id.} \arrow{r}{\simeq} &X\\
X \otimes X, \arrow[swap]{ur}{m} &&&& X \otimes X. \arrow[swap]{ur}{m}
\end{tikzcd}
$$
\end{enumerate}
\end{dfn}
\begin{rmk}
An $\mathbb{A}_2$-algebra in pointed spaces (with the cartesian symmetric monoidal structure) is an $H$-space.
\end{rmk}
To accomplish Step 3, we show that $\alpha^{\Phi G}$ is an $\mathbb{A}_2$-algebra morphism for a certain $\mathbb{A}_2$-structure on the source.
To do so requires a different argument at the prime 2
than at odd primes.
\begin{itemize}
\item[\underline{Step 3a})] In \S\ref{sec:action} we show that
$S^{\lambda+1}= \Omega \mathbb{H}P^{\infty}$ for
a certain $G$-action on $\mathbb{H}P^{\infty}$.
We then show (\S\ref{sec:prime2}) that, when $p=2$, the map
\[
S^{\lambda+1} \longrightarrow \mathrm{B}^{\lambda}
\mathrm{BGL}_1(S^{0}_{(2)})
\]
deloops once. Thus, $\alpha^{\Phi G}$ is an $\mathbb{A}_{\infty}$-map
in this case. This section contains some material that may be of
independent interest, such as a description of one of the spaces
in the equivariant $K$-theory spectrum in terms of bundles of (twisted)
$G$-$\mathbb{H}$-modules.
\item[\underline{Step 3b})]
In \S\ref{sec:exotic} we produce an exotic $\mathbb{A}_2$-structure
on $S^{\lambda+1}$ at odd primes with respect to which
the map $\mu$ is an $\mathbb{A}_2$-map in
$\mathbb{E}_{\lambda}$-spaces.
Our proof uses a small dose of unstable equivariant homotopy theory,
in particular the EHP sequence.
\end{itemize}
Finally, we come to Step 4. The element
$s \in \pi_1\mathrm{H}\underline{\mathbb{F}}_p^{\Phi G}$
arises as a witness to the fact that the composite
\[
S^0 \stackrel{\nabla}{\longrightarrow}
G_+ \stackrel{1}{\longrightarrow} \mathrm{H}\underline{\mathbb{F}}_p
\]
is null. Said differently, $s$ witnesses the vanishing of the element
$[G] \in \pi_0^{G}S^0$ in the Hurewicz image. Nonequivariantly,
the zeroth homotopy group of $\left(\Omega^2S^3\right)^{\mu}$
is already detected by the map
\[
(S^1)^{\mu} = M(p) \longrightarrow \left(\Omega^2S^3\right)^{\mu}.
\]
Equivariantly, this is no longer true. Recall \cite{segal, tomdieck}
that $\pi_0^{G}S^0 = A(G)$ is the Burnside ring of finite $G$-sets.
While the element $p$ dies in the Moore space $(S^1)^{\mu}$,
the same is not true of the elements $[G/K]$ for $K \subsetneq G$.
The proof that these elements vanish in the Hurewicz image of
$\left(\Omega^{\lambda}S^{\lambda+1}\right)^{\mu}$ is
a consequence of the vanishing of $p$
together with the existence of \emph{norms} supplied by
the $\mathbb{E}_{\lambda}$-structure. This is proved in
\S\ref{sec:norm} and the final pieces of the proof of Step 4
and the main result are then spelled out in \S\ref{sec:proof}.
We end in \S\ref{sec:epilogue} with an explanation of how to produce
$\mathrm{H}\underline{\mathbb{Z}}_{(p)}$ as a Thom spectrum
as well as some unanswered questions.
\section{Extra structure at the prime \texorpdfstring{$2$}{2}}\label{sec:prime2}
Hopkins observed that the map $\mu: \Omega^2S^3 \to \mathrm{BO}$
admits a \emph{triple} delooping as the composite:
\[
\mathbb{H}P^{\infty} = \mathrm{BSp}(1) \longrightarrow
\mathrm{BSp} \simeq \mathrm{B}^5\mathrm{O}
\stackrel{\eta}{\longrightarrow} \mathrm{B}^4\mathrm{O}.
\]
We would like to establish an equivariant version of this result.
The statement requires a few preliminaries.
The first results of this section hold for any finite subgroup $G \subseteq S^1$. We will indicate later when we must restrict attention to $G=C_{2^n}$.
\begin{definition} A \textbf{$G$-$\mathbb{H}$-module}
is a real $G$-representation $V$ equipped with a
$G$-equivariant algebra map $\mathbb{H} \to \mathrm{End}(V)$.
Here $\mathrm{End}(V)$ is the $G$-representation of
all endomorphisms and we use the $G$-action
on $\mathbb{H}$ constructed in (\ref{cstr:quat-action}).
More generally,
a \textbf{$G$-$\mathbb{H}$-bundle} on a
$G$-space $X$ is a $G$-equivariant, real vector bundle
$E \to X$ together with a $G$-equivariant algebra
map $\mathbb{H} \to \mathrm{End}(E)$.
\end{definition}
\begin{construction} For $F = \mathbb{R}$ or $\mathbb{H}$,
let $\mathcal{U}_{F}$ be a \textbf{complete $G$-$F$-universe}.
That is: $\mathcal{U}_F$ is a direct sum of infinitely many
finite dimensional $G$-$F$-modules which contains every
finite dimensional $G$-$F$-module as a summand (up
to isomorphism). Let $\mathrm{Gr}^F(\mathcal{U}_F)$
denote the infinite grassmanian with its induced $G$-action.
Then we define:
\begin{align*}
\mathrm{BO}_G=
\mathrm{BGL}(\mathbb{R})&:= \mathrm{Gr}^{\mathbb{R}}
(\mathcal{U}_{\mathbb{R}}),\\
\mathrm{BGL}(\mathbb{H})&:= \mathrm{Gr}^{\mathbb{H}}
(\mathcal{U}_{\mathbb{H}}).
\end{align*}
\end{construction}
The $G$-space $\mathrm{BO}_G$ is well-known, and there
is an equivalence
\[
\Omega^{\infty}\mathrm{KO}_G = \mathbb{Z}
\times \mathrm{BO}_G.
\]
\begin{warning} Since we have
$\mathrm{BO}_G \simeq \mathrm{BGL}(\mathbb{R})$
and the former is well-known, we will use the notation
$\mathrm{BO}_G$. Beware, however, that
the $G$-space $\mathrm{BGL}(\mathbb{H})$
is \emph{not} the same as the space
$\mathrm{BSp}_G$ associated to equivariant, symplectic $K$-theory.
The latter does not incorporate a nontrivial action of $G$ on
$\mathbb{H}$.
\end{warning}
\begin{warning} Neither $\mathrm{BO}_G$ nor
$\mathrm{BGL}(\mathbb{H})$ are equivariantly connected
when $G$ is nontrivial. For example, $\pi_0^G\mathrm{BO}_G$
is the group of virtual real representations of virtual dimension zero.
\end{warning}
\begin{theorem}[Karoubi]
\[
\Omega^{\infty}\Sigma^{\lambda+2}\mathrm{KO}_G
\simeq \mathbb{Z} \times \mathrm{BGL}(\mathbb{H})
\]
where $\mathrm{BGL}(\mathbb{H})$ is the $G$-space
constructed above.
\end{theorem}
\begin{proof}[Proof sketch] We indicate how to recover this
result from the much more general work of Karoubi.
First, if we endow $\lambda$ with the standard negative
definite quadratic form, then the Clifford algebra
$C\ell(\lambda)$ is $G$-equivariantly isomorphic
to $\mathbb{H}$ as an algebra.
It follows from the `fundamental theorem'
\cite[Theorem 1.1]{karoubi-eqvt} that, when $X$
is compact,
$\mathrm{KO}_G^{2+\lambda}(X)$
is given by Karoubi's (graded)
$K$-theory of the graded Banach
category of $G$-$C\ell(2+\lambda)$-bundles,
in the sense of \cite[Definition 2.1.6]{karoubi-gen}.
From the interpretation of
this $K$-theory group explained on \cite[p.192]{karoubi-eqvt},
we learn that a class in
$\mathrm{KO}_G^{2+\lambda}(X)$ is specified
by
a $G$-$C\ell(2+\lambda)$-bundle $E$ on $X$ together with two extensions
to a $G$-$C\ell(3+\lambda)$-bundle structure on $E$.
Such a triple is declared trivial
if the two extensions give isomorphic bundles, and
two triples are equivalent if they become isomorphic after adding
a trivial triple.
The naturality statement in \cite[Thm. 3.10]{karoubi-book}
produces equivariant isomorphisms:
\begin{align*}
C\ell(2+\lambda) &\simeq \mathrm{M}_2(\mathbb{H})\\
C\ell(3+\lambda) &\simeq \mathrm{M}_2(\mathbb{H})
\times \mathrm{M}_2(\mathbb{H})
\end{align*}
By Morita invariance, we may reinterpret elements in
$\mathrm{KO}_G^{2+\lambda}(X)$ as equivalence classes of
$G$-$\mathbb{H}$-bundles $E$ equipped
with two decompositions
$\eta_1: E \simeq E_0 \oplus E_1$ and $\eta_2: E \simeq E'_0 \oplus E'_1$.
Arguing
as in \cite[Proposition 4.26]{karoubi-book} and
\cite[Proposition 2.4]{segal-K}, one can show that every such
datum $(E, \eta_1, \eta_2)$ is equivalent to one of the
form $(X \times (M_0 \oplus M_1), \mathrm{id}, \eta)$ where
$M_0$ and $M_1$ are $G$-$\mathbb{H}$-modules.
After supplying a metric, we may replace $\eta$ by the data
of a sub-$G$-$\mathbb{H}$-module of $M_0 \oplus M_1$.
For fixed $M_0$ and $M_1$, this data is equivalent
to an equivariant map
$X \to \coprod_{k\ge 0}\mathrm{Gr}^{\mathbb{H}}_k(M_0\oplus M_1)$.
Now the result follows by the definition of a
complete $G$-$\mathbb{H}$-universe and the construction
of $\mathrm{BGL}(\mathbb{H})$.
\end{proof}
At this point we may form the composite
\[
\xymatrix{
\mathbb{H}P^{\infty} \ar[rr]^-{\mathcal{O}(-1)-1}&&
\mathbb{Z} \times \mathrm{BGL}(\mathbb{H})
\simeq \Omega^{\infty}\Sigma^{\lambda+2}\mathrm{KO}_G
\ar[r]^-{\eta} &
\Omega^{\infty}\Sigma^{\lambda+1}\mathrm{KO}_G
}
\]
where
\begin{itemize}
\item $\mathcal{O}(-1)$ is the tautological
$G$-$\mathbb{H}$-bundle on $\mathbb{H}P^{\infty}$.
\item $\eta \in \pi_1^GKO_G$ is the image
of $\eta \in \pi_1S^0 \subseteq \pi_1^GS^0$.
\end{itemize}
To complete the construction, we will need
an equivariant version of the $J$-homomorphism.
The $J$-homomorphism
and equivariant spherical fibrations have been studied previously
(e.g. \cite{segal,mcclure,waner1,waner2}) and
it is shown in \cite{cw2} and \cite{shim} that
the classifying space of equivariant stable spherical fibrations
is an equivariant infinite loop space.
For the reader's convenience, we prove this here, as well as
the corresponding notion and results regarding Picard spectra,
which provides the target for the $J$-homomorphism.
The construction below is natural from the point
of view of \cite{barwick-et-al}, from whom we draw
inspiration.
\begin{construction}[Picard $G$-spectrum]
Bachmann-Hoyois \cite[\S 9.2]{bachmann-hoyois}
refined the Hill-Hopkins-Ravenel
norm construction to a product-preserving functor:
\[
\underline{\mathsf{Sp}}^G:
\mathsf{A}^{\textit{eff}}(G) \longrightarrow \mathsf{CAlg}(\mathsf{Cat}_{\infty}),
G/H \mapsto \mathsf{Sp}^H,
\]
where the left-hand side denotes the (effective) Burnside (2,1)-category
of finite $G$-sets and spans \cite{barwick}.
Given any symmetric monoidal $\infty$-category $\mathcal{C}$,
define $\mathcal{P}\mathrm{ic}(\mathcal{C})$ to be the maximal subgroupoid
of objects which are invertible under the tensor product. This
is a group-like $\mathbb{E}_{\infty}$-space and so deloops
to a spectrum $\mathrm{pic}(S^0)$; moreover
the formation
of Picard spectra
is product preserving \cite[2.2]{akhil-vesna}.
We define $\underline{\mathrm{pic}}(S^0)$
as the composite functor
\[
\underline{\mathrm{pic}}(S^0): \mathsf{A}^{\textit{eff}}(G) \longrightarrow
\mathsf{CAlg}(\mathsf{Cat}_{\infty})
\longrightarrow \mathsf{Sp}.
\]
This is a spectral Mackey functor, and the $\infty$-category
of spectral Mackey functors is equivalent to the
$\infty$-category of genuine $G$-spectra \cite{guillou-may,denis}.
Thus we have produced a $G$-spectrum
which we call the \textbf{Picard $G$-spectrum}
of $S^0$. We denote by $\mathcal{P}\mathrm{ic}(S^0)$ the
$0$th space of this spectrum, which is a group-like
$G$-$\mathbb{E}_{\infty}$-space
(by which we mean, here and below, a $G$-commutative
monoid in the sense of \cite{denis}). We note that this $G$-space,
without extra structure,
may be obtained directly from $\underline{\mathsf{Sp}}^G$
by assigning to the orbit $G/H$ the maximal
subgroupoid of the full subcategory of $\mathsf{Sp}^H$
consisting of invertible objects. If one further restricts to the
full subcategory consisting of objects equivalent to $S^0$,
this defines the $G$-space $\mathrm{BGL}_1(S^0)$. This subcategory
is closed under the formation of norms and smash products and
hence inherits the structure of a group-like $G$-$\mathbb{E}_{\infty}$-space
for which the inclusion
\[
\mathrm{BGL}_1(S^0) \to \mathcal{P}\mathrm{ic}(S^0)
\]
is a map of $G$-$\mathbb{E}_{\infty}$-spaces. In particular,
the $G$-space $\mathrm{BGL}_1(S^0)$ is the zero space of
a $G$-spectrum $\Sigma\mathrm{gl}_1(S^0)$.
More generally, given any virtual $G$-representation $V$,
restricting, for each $H$,
to the full subcategory of objects equivalent to
$S^{\mathrm{res}_H(V)}$ produces a $G$-$\mathbb{E}_{\infty}$-space
canonically equivalent to $\mathrm{BGL}_1(S^0)$.
\end{construction}
\begin{remark} The construction above works without
any change for the category of $p$-local $G$-spectra
to define $G$-infinite loop spaces $\mathcal{P}\mathrm{ic}(S^0_{(p)})$
and $\mathrm{BGL}_1(S^0_{(p)})$.
\end{remark}
\begin{warning} The space $\mathcal{P}\mathrm{ic}(S^0)$
does not decompose into a disjoint union of copies of
$\mathrm{BGL}_1(S^0)$ when $G$ is nontrivial.
For example, take $G=C_2$ and let $Y \subseteq \mathcal{P}\mathrm{ic}(S^0)$
be the $C_2$-space with
\begin{itemize}
\item underlying space the subspace of nonequivariant spectra equivalent to $S^0$,
\item fixed points the subspace of $C_2$-spectra
equivalent to $S^V$ where $V$ has virtual dimension zero.
\end{itemize}
Then $Y$ splits off of $\mathcal{P}\mathrm{ic}(S^0)$, but does not decompose further.
Indeed, $S^{\sigma-1}$ and $S^0$ lie in different components of $Y^{C_2}$
but restrict to the same component on the underlying space. On the other hand,
$\mathrm{BGL}_1(S^0)$ is $G$-connected, whence the claim.
\end{warning}
\begin{construction}[Equivariant $J$-homomorphism]
Let $\mathbf{Vect}_G$ denote the topological category
of finite-dimensional $G$-representations. We use
the same notation for the associated $\infty$-category.
Consider the product preserving functors
\[
\underline{\mathbf{Vect}}_G,
\underline{\mathsf{Spaces}}_*^G:
\mathsf{A}^{\textit{eff}}(G) \longrightarrow
\mathsf{CAlg}({\mathsf{Cat}_{\infty}})
\]
given by:
\begin{itemize}
\item $\underline{\mathbf{Vect}}_G(G/H):= \mathbf{Vect}_H$
with direct sum,
functoriality by
restriction and coinduction;
\item $\underline{\mathsf{Spaces}}_*^G(G/H):=
\mathsf{Spaces}_*^H = \mathsf{Psh}(\mathcal{O}_H, \mathsf{Spaces})$
with $\wedge$ ,
functoriality by restriction and norm defined by
\[
N_H^G(X):= \mathrm{map}_H(G, X)/\{f: * \in f(G)\}.
\]
\end{itemize}
The assignment $V \mapsto S^V$ produces a natural transformation
\[
\underline{\mathbf{Vect}}_G \to \underline{\mathsf{Spaces}}_*^G
\stackrel{\Sigma^{\infty}}{\longrightarrow}
\underline{\mathsf{Sp}}^G.
\]
Restricting to maximal subgroupoids,
and noting that each $S^V$ is invertible,
we get a natural transformation
\[
\underline{\mathbf{Vect}}_G^{\simeq}
\longrightarrow \mathcal{P}\mathrm{ic}(S^0)
\]
which we may regard as a map of $G$-$\mathbb{E}_{\infty}$-spaces.
The target is group-like, so this map factors through the group-completion
of the source. One can identify the underlying space of that
group-completion with $\mathbb{Z} \times \mathrm{BO}_G$,
so we have produced a $G$-$\mathbb{E}_{\infty}$-map
\[
J: \mathbb{Z} \times \mathrm{BO}_G
\longrightarrow \mathcal{P}\mathrm{ic}(S^0).
\]
We also denote by $J$ the restriction to $\{0\} \times \mathrm{BO}_G
= \mathrm{BO}_G$ as well as any deloopings.
\end{construction}
\begin{warning} Unlike the classical case, the restriction to virtual
dimension zero representations
\[
\mathrm{BO}_G \to \mathcal{P}\mathrm{ic}(S^0)
\]
does
\emph{not} factor through $\mathrm{BGL}_1(S^0)$ when
$G$ is nontrivial. Again, an explicit example is given by
the virtual representation $\sigma - 1$ when $G = C_2$.
\end{warning}
\begin{remark} Since $\Omega^{\lambda+1}\mathbb{H}P^{\infty}
=\Omega^{\lambda}S^{\lambda+1}$ is equivariantly connected,
the map $\mathbb{H}P^{\infty} \to \Omega^{\infty}\Sigma^{\lambda+1}
\mathrm{KO}_G$ constructed above
factors through $\mathrm{B}^{\lambda+1}\mathrm{BO}_G$.
\end{remark}
Now we specialize to the case $G = C_{2^n}$.
\begin{proposition} Let $g$ denote the composite
\[
\mathbb{H}P^{\infty} \longrightarrow
\mathrm{B}^{\lambda+1}\mathrm{BO}_G
\longrightarrow
\mathrm{B}^{\lambda+1}\mathcal{P}\mathrm{ic}(S^0)
\]
Then $\Omega^{\lambda+1}g$ factors through $\mathrm{BGL}_1(S^0)$
and is homotopic to $\mu$ under the equivalence
${\Omega^{\lambda+1}\mathbb{H}P^{\infty} \simeq
\Omega^{\lambda}S^{\lambda+1}}$.
\end{proposition}
\begin{proof} Since $\Omega^{\lambda+1}\mathbb{H}P^{\infty}$
is equivariantly connected, $\Omega^{\lambda+1}g$
automatically factors through
$\mathrm{BGL}_1(S^0)$. To complete the proof, we need only
identify the map
\[
S^{\lambda+2} \to \mathbb{H}P^{\infty}
\to \mathrm{B}^{\lambda+1}\mathcal{P}\mathrm{ic}(S^0).
\]
To begin, notice that the map
\[
S^{\lambda+2} \to \mathrm{B}^{\lambda+1}\mathrm{BO}_G
\]
corresponds to an element of $\mathrm{KO}_G^{-1}$. By
\cite[p.17]{atiyah-segal}, this group is $RO(G)/R(G)$.
For $G = C_{2^n}$ we have
\[
RO(G)/R(G) =
\begin{cases}
\mathbb{F}_2\{1, \sigma\} & n=1\\
\mathbb{F}_2 & n>1
\end{cases}
\]
Even when $n=1$, the bundle we started with was restricted
from a bundle defined for $n>1$, so the element in question
is either 0 or 1. But we know that the
\emph{underlying} class is nonzero, so we must be looking
at the nonzero element in $RO(G)/R(G)$. Moreover, this
class corresponds precisely to the M\"obius bundle on $S^1$,
whence the claim.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:main-even}
assuming Theorem \ref{thm:MainThom}]
Combine the above proposition with Theorem \ref{thm:MainThom}.
\end{proof}
\section{Introduction}
Both authors are fond of the following result of Mahowald \cite{mahowald}:
\begin{theorem}[Mahowald] \label{thm:MahE2}
The Thom spectrum of the unique non-trivial double loop map
\begin{equation} \label{eqn:Mah}
\Omega^2 S^3 \longrightarrow \mathrm{BO}
\end{equation}
is $\mathrm{H}\mathbb{F}_2$.
\end{theorem}
This result and several variants have enjoyed many subsequent proofs in the literature. Examples include \cite{mahowald-thom,cohen-may-taylor,priddy,temss,omar-toby,akhil-niko-justin}. In joint work with Mark Behrens \cite{behrens-wilson}, the second author proved a $C_2$-equivariant generalization of Mahowald's Theorem \ref{thm:MahE2}:
\begin{theorem}[Behrens--Wilson] \label{thm:BehrensWilson}
Let $\rho$ denote the real regular representation of the group $C_2$. Then there is a $\Omega^{\rho}$-map
$$\Omega^{\rho} S^{\rho+1} \to \mathrm{BO}_{C_2}$$
with Thom spectrum $\mathrm{H}\underline{\mathbb{F}}_2$, the Eilenberg--Maclane object associated to the constant Mackey functor $\underline{\mathbb{F}}_2$.
\end{theorem}
A less well-known fact is that Mahowald's map (\ref{eqn:Mah}) is in fact a triple loop map, as the authors first learned from Mike Hopkins:
\begin{obs} \label{obs:TripleLoop}
Mahowald's map (\ref{eqn:Mah}) may be obtained by thrice looping the composite
$$\mathbb{H}P^{\infty} \simeq \mathrm{BSp}(1) \longrightarrow \mathrm{BSp}
\simeq \mathrm{B}^5\mathrm{O} \stackrel{\eta}{\longrightarrow} \mathrm{B}^4\mathrm{O}.$$
\end{obs}
The present work arose from the authors' attempt to understand how the above
observation
generalizes to the $C_2$-equivariant setting. We obtain the following theorem:
\begin{thm} \label{thm:main-C2}
Let $\sigma$ denote the sign representation of the group $C_2$. There is a $C_2$-action on the space $\mathbb{H}P^{\infty}$ and a $\Omega^{\rho+\sigma}$-map
$$\Omega^{\rho+\sigma} \mathbb{H}P^{\infty} \to \mathrm{BO}_{C_2},$$
with Thom spectrum $\mathrm{H}\underline{\mathbb{F}}_2$.
\end{thm}
Since $C_2$ is the only group with a $2$-dimensional real regular representation, the statement of Theorem \ref{thm:BehrensWilson} makes it unclear how the constant $G$-Mackey functor $\mathrm{H}\underline{\mathbb{F}}_2$ might be an equivariant Thom spectrum for any larger group $G$. On the other hand, the $C_2$-representation $\rho+\sigma = 2\sigma+1$ is naturally the restriction of a $C_4$-representation. In this paper, we in fact obtain Theorem \ref{thm:main-C2} as a special case of the following more general result:
\begin{thm} \label{thm:main-even}
Fix an integer $n \ge 0$, and let $G=C_{2^n}$ denote the cyclic group of order $2^n$. Let $\lambda$ denote the \textit{standard} representation of $G$ on the complex plane, where the generator acts by $e^{2 \pi i /2^n}$. Then there is a $G$-action on $\mathbb{H}P^{\infty}$, and a $\Omega^{\lambda+1}$-map
$$\Omega^{\lambda+1} \mathbb{H}P^{\infty} \to \mathrm{BO}_{G},$$
with Thom spectrum $\mathrm{H}\underline{\mathbb{F}}_2$.
\end{thm}
\begin{remark}
The group $G=C_2$ is special: only for this group does $\lambda=2\sigma$ split as the sum of two smaller representations. This leads to a non-obvious equivalence of $C_2$-spaces
$$\Omega^{\rho} S^{\rho+1} = \Omega^{\sigma+1} S^{\sigma+2} \simeq \Omega^{2\sigma+1} \mathbb{H}P^{\infty} \simeq \Omega^{2 \sigma} S^{2 \sigma+1} = \Omega^{\lambda} S^{\lambda+1},$$
which implies a $\Omega^{\lambda}$-analog of Theorem \ref{thm:BehrensWilson} (for details, see Section \ref{sec:action}).
A Borel equivariant version of this $\Omega^{\lambda}$-analog previously appeared as Lemma $3.1$ in \cite{klang}, where it is used in an essential manner to compute the factorization homology of Eilenberg--Maclane spectra.
\end{remark}
We next turn to odd primes $p>2$. So long as one is willing to contemplate Thom spectra of $p$-local spherical fibrations (as in, e.g., \cite[\S 3.4]{bcs}), Mahowald's Theorem \ref{thm:MahE2} admits an analog due to Mike Hopkins \cite[Theorem 4.18]{akhil-niko-justin}:
\begin{theorem}[Hopkins] \label{thm:Hop}
Let
\[
S^3 \stackrel{1-p}{\longrightarrow} \mathrm{B}^3\mathrm{GL}_1(S^0_{(p)})
\]
denote the class $1-p \in \pi_0(S_{(p)})^{\times} =
\pi_3(\mathrm{B}^3\mathrm{GL}_1(S_{(p)})).$ Then, applying $\Omega^2$, one obtains a map
\[
\Omega^2 S^3 \longrightarrow \mathrm{BGL}_1(S^0_{(p)})
\]
with Thom spectrum equivalent to $\mathrm{H}\mathbb{F}_p$.
\end{theorem}
In light of the situation at $p=2$, it is natural to wonder if the map
$$1-p:S^3 \longrightarrow \mathrm{B}^3 \mathrm{GL}_1(S^0_{(p)})$$
may be delooped. The authors were surprised to find that it \textit{cannot}, which we record as our only non-equivariant result:
\begin{thm} \label{thm:non-eqvt}
Let $S^0_{(p)}$ denote the $p$-local sphere spectrum, and suppose $p>2$. Then there is no triple loop map
$$X \to \mathrm{BGL}_1(S^0_{(p)}),$$
for any triple loop space $X$, with Thom spectrum $\mathrm{H}\mathbb{F}_p$. The same is true if $S^0_{(p)}$ is replaced by the $p$-completed sphere spectrum.
\end{thm}
\begin{remark}
At the prime $p=3$, the map
$$1-p:S^3 \longrightarrow \mathrm{B}^3 \mathrm{GL}_1(S^0_{(p)})$$
is not even an $H$-space map for the standard $H$-space structure on $S^3=\mathrm{SU}(2)$. Crucially, we will see in Section \ref{sec:exotic} that the map \emph{is} an $H$-space map for a certain exotic $H$-space structure on $S^3$. Determining the maximum amount of structure present on this map remains an interesting question.
\end{remark}
Our equivariant generalization of Theorem \ref{thm:Hop} is as follows:
\begin{thm}[Theorem \ref{thm:main}] \label{thm:MainThom}
Fix an integer $n \ge 0$ and a prime number $p$, and let $G$ denote the cyclic group $C_{p^n}$. Let $\lambda$ denote the standard representation of $G$ on the complex numbers, where a generator acts by $e^{2\pi i / p^n}$, and let $S^0_{(p)}$ denote the $G$-equivariant sphere spectrum. Finally, let
\[
\mu: \Omega^{\lambda}S^{\lambda+1}
\longrightarrow \mathrm{BGL}_1(S^0_{(p)})
\]
denote the $\Omega^{\lambda}$-map obtained by applying $\Omega^{\lambda}$ to the map
\[
S^{\lambda+1} \longrightarrow \mathrm{B^{\lambda+1}GL}_1(S^0_{(p)})
\]
corresponding to $1-p \in (\pi_0^{G}S^0_{(p)})^{\times}$.
Then the Thom spectrum $\left(\Omega^{\lambda}S^{\lambda+1}\right)^\mu$
of $\mu$ is $\mathrm{H}\underline{\mathbb{F}}_p$.
\end{thm}
\begin{remark}
We will include a detailed discussion of Thom spectra of $G$-equivariant, $p$-local spherical fibrations in Section \S \ref{sec:prime2}. This will include in particular a discussion of $\mathrm{BGL}_1(S^0_{(p)})$ as a $G$-infinite loop space, allowing us to form its $\lambda$-delooping $\mathrm{B^{\lambda+1}GL}_1(S^0_{(p)})$. The basic definitions are due to Lewis and May, and found in \cite[Chapter X]{LMS}.
\end{remark}
We also establish a ($p$-local) integral variant of Theorem \ref{thm:MainThom}:
\begin{thm}[Theorem \ref{thm:Z}] \label{thm:introZ} Let $S^{\lambda+1}\langle \lambda+1\rangle$
denote the fiber of the unit
\[
S^{\lambda+1} \longrightarrow
\Omega^{\infty}\left(\Sigma^{\lambda+1}
\mathrm{H}\underline{\mathbb{Z}}\right).
\]
Then there is an equivalence
\[
\left(\Omega^{\lambda}
(S^{\lambda+1}\langle \lambda+1\rangle)\right)^{\mu}
\simeq \mathrm{H}\underline{\mathbb{Z}}_{(p)}.
\]
\end{thm}
\begin{remark}
Collections of little disks in the representation $\lambda$ form an equivariant operad $\mathcal{O}_{\lambda}$, leading to the notion of an $\mathbb{E}_{\lambda}$-algebra \cite{hauschild}. A nice modern discussion of $\mathbb{E}_\lambda$-algebras may be found in \cite{hill-disks}.
If $X$ is any pointed $G$-space, then $\Omega^{\lambda} X$ is naturally an $\mathbb{E}_\lambda$-algebra in $G$-spaces. If $X$ is a $G$-connected, pointed $G$-space, then the free $\mathbb{E}_\lambda$-algebra on $X$ is $\Omega^{\lambda} \Sigma^{\lambda} X$ \cite[Theorem 1]{rs}. Any $G$-$\mathbb{E}_\infty$-ring, such as the Eilenberg--Maclane object $\mathrm{H}\underline{\mathbb{Z}}_{(p)}$
\cite[\S 5.7]{schwede}, is naturally an $\mathbb{E}_\lambda$-algebra object in $G$-Spectra. It was proved by Lewis \cite[Remark X.6.4]{LMS} that the Thom spectrum of a $\Omega^{\lambda}$-map is naturally an $\mathbb{E}_{\lambda}$-algebra in $G$-spectra, and it follows from our proof that the equivalence of Theorem \ref{thm:introZ} is one of $\mathbb{E}_\lambda$-algebras.
\end{remark}
\begin{remark}
Just as Mahowald's original Theorem \ref{thm:MahE2} may be phrased as the fact that $\mathrm{H}\mathbb{F}_2$ is the free $\mathbb{E}_2$-algebra with a nullhomotopy $2 \simeq 0$ \cite{akhil-niko-justin, omar-toby}, our Theorem \ref{thm:MainThom} may be read as the statement that the free $\mathbb{E}_\lambda$-algebra with $p \simeq 0$ is $\mathrm{H}\underline{\mathbb{F}}_p$.
\end{remark}
\subsubsection*{Conventions} We freely use the language
of $\infty$-categories (alias quasicategories,
alias weak Kan complexes) in the form developed
in \cite{HTT} (though we give specific references within \cite{HTT} whenever technical results are applied). We denote by $\mathsf{Spaces}$ the $\infty$-category
of spaces, and append decorations to obtain the
$\infty$-category of pointed spaces or of $G$-spaces
for a finite group $G$,
the latter being defined as the $\infty$-category
of presheaves $\mathsf{Psh}(\mathcal{O}_G)$ on the
1-category of transitive $G$-sets. We denote
by $\mathsf{Sp}^G$ the $\infty$-category of
(genuine) $G$-spectra, and assume the reader
is familiar with the standard notations of equivariant
homotopy theory. We could not improve on the
summary given in
\cite[\S2-3]{HHR}, and recommend it to the reader. In particular,
we will require the Eilenberg-MacLane
spectrum associated to the constant Mackey
functor $\underline{\mathbb{F}}_p$, and the
notion of the geometric fixed points $X^{\Phi G}$
of a $G$-spectrum. Finally, we denote the $G$-space
classifying equivariant stable spherical fibrations
with fiber of type $S^0$ by
$\mathrm{BGL}_1(S^0)$. (See \S\ref{sec:prime2}
for more details).
\subsubsection*{Acknowledgements}
The authors would like to thank Mark Behrens, Jun-Hou Fung, Mike Hopkins, and Inbar Klang for helpful conversations related to this paper. We also thank
Nick Kuhn and Peter May for comments on an earlier draft. Special thanks are due to the anonymous referee, whose painstaking work has led to enormous improvement of the exposition.
\section{Proof of the main theorem}\label{sec:proof}
We are now ready to prove the main theorem, which we recall here
for convenience.
\begin{theorem}\label{thm:main} Let $G = C_{p^n}$
and let $S^1 \to \mathrm{BGL}_1(S^0_{(p)})$
be adjoint to $1-p \in \pi_0^{G}S^0_{(p)}$. Denote
by $\mu: \Omega^{\lambda}S^{\lambda+1}\to \mathrm{BGL}_1(S^0_{(p)})$
the extension of this map over the $\lambda$-loop space.
Then the Thom class
\[
\left(\Omega^{\lambda}S^{\lambda+1}\right)^{\mu}
\longrightarrow \mathrm{H}\underline{\mathbb{F}}_p
\]
is an equivalence of $G$-spectra.
\end{theorem}
Before the proof, we record a well-known computation
(the $p=2$ case is proven in \cite[Proposition 3.18]{HHR}
and the odd primary proof is much the same).
\begin{lemma}\label{lem:em-geo-fixed} For $G = C_{p^n}$ and $p$
odd, we have
\begin{align*}
\pi_*\mathrm{H}\underline{\mathbb{Z}}^{\Phi G}
&= \mathbb{F}_p[t], \, |t|=2,\\
\pi_*\mathrm{H}\underline{\mathbb{F}}_p^{\Phi G}&=
\mathbb{F}_p[t] \otimes \Lambda(s), \, s=\beta t
\end{align*}
When $p=2$, the second computation becomes
\[
\pi_*\mathrm{H}\underline{\mathbb{F}}_2^{\Phi G}
= \mathbb{F}_2[s], \, |s|=1.
\]
\end{lemma}
We also will need the corresponding result about our Thom spectrum.
\begin{lemma}\label{lem:htpy-thom-geo-fixed}
Let $X$ denote the Thom spectrum $(\Omega^\lambda S^\lambda)^{\mu}$. Then the homotopy group $\pi_*(X^{\Phi G})$ of the geometric fixed points is isomorphic to $\mathbb{F}_p$ for each $* \ge 0$.
\end{lemma}
\begin{proof}
As explained in Remark \ref{rmk:a2-str}, we have equipped $X$ with the structure of an $\mathbb{A}_2$-algebra in $\mathbb{E}_\lambda$-algebras and hence the norm map
$$\mathrm{N}^G(X) \longrightarrow X$$
is a map of $\mathbb{A}_2$-algebras.
In particular, $X^{\Phi G}$ is a module over $(\mathrm{N}^G(X))^{\Phi G}
\simeq (\mathrm{N}^G(\mathrm{H}\mathbb{F}_p))^{\Phi G} \simeq
\mathrm{H}\mathbb{F}_p$.
Since $\mathrm{H}\mathbb{F}_p$ is a field spectrum, $X^{\Phi G}$ splits as a wedge of suspensions of $\mathrm{H}\mathbb{F}_p$. The homotopy groups of $X^{\Phi G}$ are then determined by the homology groups of $X^{\Phi G}$.
By the Thom isomorphism, $\mathrm{H}_*(X^{\Phi G}) \simeq
\mathrm{H}_*(\Omega^2 S^3 \times \Omega S^2)$, and the result follows.
\end{proof}
\begin{proof}[Proof of the main theorem] We prove the theorem by induction on $n$. When $n=0$,
this is the non-equivariant result of Hopkins-Mahowald.
For the induction hypothesis we assume that the map
\[
\alpha: X:=\left(\Omega^{\lambda}S^{\lambda+1}\right)^{\mu}
\longrightarrow \mathrm{H}\underline{\mathbb{F}}_p
\]
is an equivalence after restriction to $C_{p^{n-1}}$, and
we assume that $n\ge 1$ from now on.
We need only show
that the map on geometric fixed point
spectra
\[
\alpha^{\Phi G}:
X^{\Phi G}=\left(\left(\Omega^{\lambda}S^{\lambda+1}\right)^{\mu}\right)^{\Phi G}
\longrightarrow \mathrm{H}\underline{\mathbb{F}}^{\Phi G}_p
\]
is an equivalence. By Lemma \ref{lem:htpy-thom-geo-fixed}, we know that
(additively)
\[
\pi_*X^{\Phi G} = \mathbb{F}_p[x] \otimes \Lambda(y),
\, |x| = 2, |y|=1.
\]
Thus, by Lemma \ref{lem:em-geo-fixed}, both the source and target
have the same homotopy groups, additively.
By Theorem \ref{thm:h-orient}, when $p$ is odd
we can put an $\mathbb{A}_2$-structure on $X$ such that
$\alpha$, and hence $\alpha^{\Phi G}$, is an $\mathbb{A}_2$-map.
When $p=2$, the Thom class is already an $\mathbb{E}_{\lambda+1}$-map.
In either case, $\alpha^{\Phi G}$ induces a ring map on homotopy.
It therefore suffices
to show that $s$ and $t$ lie in the image of $\alpha^{\Phi G}$.
This follows somewhat formally
from the inductive hypothesis and the
computation of (\ref{cor:pi-0}), as we now explain.
First we show that $s$ lies
in the image. Define a $G$-spectrum $S^{1-\lambda/2}$ as the cofiber
of the transfer map
\[
S^0 \to G_+\wedge S^0 \to
S^{1-\lambda/2}.
\]
By Corollary \ref{cor:pi-0},
we know $\underline{\pi}_0X =\underline{\mathbb{F}}_p$,
which has zero transfer map. From the diagram of exact
sequences
\begin{equation}
\xymatrix{
\pi_1X^G \ar[r]\ar[d]&
[S^{1-\lambda/2}, X] \ar[r]\ar[d] & \pi_0^uX \ar[r]^0
\ar[d]& \pi_0X^G\ar[d]\\
0\ar[r]&[S^{1-\lambda/2}, \mathrm{H}\underline{\mathbb{F}}_p]
\ar[r] & \mathbb{F}_p
\ar[r]^0 & \mathbb{F}_p
}\label{eqn:diagram}
\end{equation}
we learn that
\[
[S^{1-\lambda/2}, X]
\to
[S^{1-\lambda/2}, \mathrm{H}\underline{\mathbb{F}}_p]
\]
is surjective. On the other hand,
passage to geometric fixed points produces an
isomorphism
\[
[S^{1-\lambda/2}, \mathrm{H}\underline{\mathbb{F}}_p]
\to
\pi_1\mathrm{H}\underline{\mathbb{F}}_p^{\Phi G}.
\]
It follows that $s$ lies in the image of $\alpha^G$.
Next, we argue that $\underline{\pi}_1X = 0$.
Indeed, let $\mathcal{P}$ denote the family of proper subgroups
of $G$ and let $X_{h\mathcal{P}}$
denote the spectrum $(X \wedge \mathrm{E}\mathcal{P}_+)^G$
(see, for example, \cite[2.5.2]{HHR}). Then there is a diagram
of cofiber sequences of spectra
\[
\xymatrix{
X_{h\mathcal{P}} \ar[r]\ar[d]& X^G \ar[r]\ar[d] &X^{\Phi G}\ar[d]\\
\left(\mathrm{H}\underline{\mathbb{F}}_p\right)_{h\mathcal{P}}
\ar[r] & \mathrm{H}\mathbb{F}_p \ar[r] &
\mathrm{H}\underline{\mathbb{F}}_p^{\Phi G}
}
\]
By the induction hypothesis, the left vertical map is an equivalence.
Since we have shown that $s$ is hit, the right vertical
map is an isomorphism on both $\pi_0$ and $\pi_1$. It follows
that the middle vertical map is injective on $\pi_1$, and hence
that $\pi_1X^G = 0$, proving the claim.
Returning to the diagram (\ref{eqn:diagram}) we learn that
$[S^{1-\lambda/2}, X] \to
[S^{1-\lambda/2}, \mathrm{H}\underline{\mathbb{F}}_p]$
is in fact an isomorphism.
Finally, we turn to the following cofiber sequence:
\[
S^{1-\lambda/2} \to G_+\wedge S^0 \to S^{2-\lambda}.
\]
This leads to the diagram:
\[
\xymatrix{
0 \ar[r] & \pi_{2-\lambda}X \ar[r] \ar[d]& \pi_0^uX\ar[d]^{\simeq}
\ar[r] & [S^{1-\lambda/2}, X] \ar[d]^{\simeq}\\
0\ar[r] & \pi_{2-\lambda}\mathrm{H}\underline{\mathbb{F}}_p
\ar[r] & \pi_0^u\mathrm{H}\underline{\mathbb{F}}_p
\ar[r] & [S^{1-\lambda/2}, \mathrm{H}\underline{\mathbb{F}}_p]
}
\]
Thus $\pi_{2-\lambda}X\to\pi_{2-\lambda}\mathrm{H}\underline{\mathbb{F}}_p$
is an isomorphism. Passage to geometric
fixed points gives an isomorphism
$\pi_{2-\lambda}\mathrm{H}\underline{\mathbb{F}}_p
\to \pi_2\mathrm{H}\underline{\mathbb{F}}_p^{\Phi G}$,
and this completes the proof.
\end{proof}
|
{
"timestamp": "2019-06-12T02:02:10",
"yymm": "1804",
"arxiv_id": "1804.05292",
"language": "en",
"url": "https://arxiv.org/abs/1804.05292"
}
|
\section{INTRODUCTION}
Mobile robotic systems that move autonomously in complex environments are becoming more prevalent. However, the perceptual input available to a mobile robot, for example from computer vision, is uncertain. Therefore it is {\em not possible to certify that a path is collision-free}. When such robots perform complex manoeuvres among obstacles, absolute safety therefore cannot be guaranteed~\cite{r19}. But instead, robots can operate within a controlled level of risk of collision~\cite{r1}.
Typically, the environment is perceived through sensors such as stereo vision or LIDAR. Uncertainty arises directly from sensor noise, and then indirectly through perception algorithms that detect discrete
obstacles~\cite{r22, r23} or impassable terrain. It is therefore realistic to expect each perception module to output a {\em probability distribution}~\cite{r2} over pose, for each detected object. Then, given these random variables from detector outputs, candidate paths can be assessed for the risk of collision.
\begin{figure}[htbp]
\centering
\vspace{3mm}\includegraphics[width=0.45\textwidth]{figures/fig-trajectory-risk.png}
\caption{\small{\bf Robot motion problem.} Data taken from an aerial view of a car-park. The obstacles here are 35 cars with shapes given by the green bounding boxes, and uncertainties in location visualised as a green halo. The robot vehicle (blue) has a set of candidate paths, evaluated here as high risk (red) to low risk (blue) according to the scale shown. Bounds on collision risk are represented by different colours as on the logarithmic scale shown. Some higher risk paths involve squeezing through a narrow gap.} \label{f:trajectory-risk}
\end{figure}
The FPR algorithm introduced here efficiently computes a bound $F_{\rm D}$ on risk of collision. Note that a bound on risk would not be useful for determining an optimal path. However the problem in this paper is different: to select paths whose risk fall below a certain threshold. For that a bound is entirely usable.
\subsection{Specifying the Problem}
It is assumed that a static freespace in the plane is defined deterministically, and that a robot of known shape $A$ translates and rotates in that freespace. In addition, discrete ``tethered'' obstacles $k=1\ldots,K$ of known shape and uncertain location are perceived by the robot's sensor systems. Of course there is a great deal of work in Robotics addressing uncertain estimation of robot location. This has been so successful (e.g.\cite{Cremers13}) that here we assume that uncertainty in robot location is negligible compared with uncertainty in the locations of perceived obstacles.
The problem is then, over a (short) time-interval $t\in\,[0,\ldots,T]$ to:
\begin{enumerate}
\item{\bf Generate} $N$ candidate paths in configuration space $\Re^2 \times S$ for the robot. Each such path then sweeps out a shape $A$ in the plane $\Re^2$.
\item{\bf Bound} the risk of collision: a bound $F_{\rm D}$ on risk is computed for each candidate path.
\end{enumerate}
The main contribution of the paper relates not to 1. above, for which off-the-shelf methods are used, but to 2. where we introduce a novel, fast computation of a bound on the risk of collision for a given path. Its computational complexity is $O(N+K)$, compared with the naive $O(NK)$. Once the bound $F_{\rm D}$ has been computed for the first path, the cost of computing the bound for subsequent paths is {\em independent} of the {\em number of obstacles} $K$.
The following inputs to the risk computation are assumed:
\begin{enumerate}
\item{\bf Freespace}: it is assumed that freespace $F$ is initially defined as a subset of $\Re^2$. Then, within $F$, further, tethered obstacles are defined stochastically as below.
\item{\bf Robot}: assumed to have a deterministic spatial extent and to be manoeuvrable in translation and rotation. Over the interval $t\in\,[0,\ldots,T]$, it sweeps out the set $A$ in the plane.
\item{\bf Obstacle shape}: the $k^{\rm th}$ obstacle is assumed to have deterministic shape and mean orientation represented by the set $B_k$.
\item{\bf Tethered obstacle}: tethering here means specifying a probability distribution $p_k({\bf r})$ for the location of the $k^{\rm th}$ obstacle, where ${\bf r}=(x, y)$ are coordinates in the plane. This takes into account: i) variability arising from any possible motion over the time-interval $[0,\ldots,T]$; and ii) modelled uncertainty in object detector output. Obstacle locations are assumed to be mutually independent --- arising from independent sensor observation/detection.
\item{\bf Obstacle rotation:} treated as part of its shape so, to the extent that an obstacle may rotate over the time interval $[0,\ldots,T]$, that rotation must be absorbed in a deterministic expansion of the set $B_k$.
\end{enumerate}
This treatment of obstacle rotation is a limitation of our framework which, however, is reasonable if the time interval $[0,\ldots,T]$ is short, so rotation is limited. For longer time intervals it may be necessary to consider the full $xyt$-space, rather than more simply the $xy$-plane as in this paper. However our treatment is fully general in the rotation of the robot.
\subsection{Existing approaches}
We review some prominent approaches to computing the risk of collision under uncertainty. There are a number of recent approaches to estimating the risk of collision under uncertain robot dynamics \cite{ud1,ud2,ud3}. In our problem it is the environment that is uncertain rather than the dynamics. One approach to this problem involves casting ``shadows'' around obstacles~\cite{r13} but that does not facilitate the resolution of uncertainty from multiple different sources. Probability density functions can usefully model robot and obstacle uncertainty, as in~\cite{ab2}, which however requires Monte-Carlo computation and has $O(NK)$ complexity. Bevilacqua et al.~\cite{ab3} model obstacles stochastically, but deal with just one obstacle, and do not allow for sensor or perceptual uncertainty. Empirical probability distributions can also be useful~\cite{ab4} in the case of a single obstacle. Alternatively Althoff et al.~\cite{ab5} elegantly avoid Monte Carlo computation by compiling stochastic reachability of moving obstacles down to finite Markov Chains, but the risk computation remains $O(NK)$.
Probabilistic Occupancy Grids are an established mechanism for dealing with spatial uncertainty probabilistically~\cite{r3,r10a} and can be used to find paths. However, to calculate the risk of collision along a path with numerous obstacles, a single grid is not enough. Laugier and collaborators~\cite{r10b,r10} show that certain ``Laugier integrals'' (our term) over multiple grids, one grid per point-obstacle, can be combined nonlinearly to compute the total risk of collision. We build on this approach.
An important question is then whether the combination of the Laugier integrals can be simplified somehow, despite the nonlinearity. For example, if the set of obstacles could somehow be replaced by the union of obstacles, that could live on a single grid, it would simplify computation. However, given that obstacles are each defined here not just by their shape but also by the uncertainty in their location, it turns out that constructing a composite obstacle as a trivial union of obstacle shapes is not valid. Therefore, in the FPR algorithm, we derive and justify a non-trivial combination of shape properties and location distributions, onto just 2 grids. This ultimately leads to the qualitative improvement in computation time of the FPR algorithm.
\subsection{Main contributions}
Note that this paper is not about path-planning {\it per se}. It claims no new contribution whatsoever to the extensive science of path-planning~\cite{Latombe12}. Its novel contribution is entirely directed at the efficient computation of the risk of collision.
Our principal contributions are as follows.
\begin{enumerate}
\item A linearisation of the Laugier integrals scheme gives a close approximation and a bound on the risk, and allows the entire computation to be done over just two grids, regardless of the number of obstacles. That reduces computational complexity from $O(NK)$ to $O(N+K)$, for $K$ point obstacles and $N$ paths. So far, this applies only to point obstacles, not obstacles of finite size.
\item The Laugier integrals can however be extended by means of Minkowski sums to apply to obstacles of finite size.
That, together with a new ``convolution trick'', leads to the FPR algorithm for computing a bound on the risk of collision with finite, tethered obstacles. The computational complexity of the FPR algorithm is $O(N+K)$, as desired.
\item Simulations quantify the difference between the FPR bound on risk and the true risk, under various circumstances.
\item Simulations with simulated and real data show that the $O(N+K)$ computational complexity does indeed lead to substantial reductions in practical computation times.
\end{enumerate}
\section{EFFICIENT COMPUTATION OF BOUNDS ON COLLISION PROBABILITIES}
Given the shapes and uncertain location of obstacles in an environment, the problem is to estimate the risk of collision for a set of candidate paths. This risk computation uses the probability distributions for encroachment by obstacles on the path swept by a moving robot, during a given time interval. Our starting point is the work by Laugier and collaborators~\cite{r10b,r10} who propose a probabilistic framework of this sort. This section extends the framework and develops an efficient algorithm for computing risk.
\subsection{Probabilistic Obstacle Framework}
Consider an environment consisting of a set of $K$ point obstacles. The obstacles are typically detected by perception modules whose outputs are uncertain (by design), so the position of each obstacle is a random variable given by the density function $p_k({\bf r})$ where ${\bf r}=(x,y)\in\Re^2$.
Then the probability of collision between the robot and the $k^{\rm th}$ point obstacle can be written~\cite{r10} as
\begin{equation}
\label{e:Laugier0}
P_{\rm D}(k) = \int_A p_k({\bf r})
\end{equation}
where $A$ is the swept area of the robot along a path $\pi$ and over a time interval $t \in [0,T]$.
Now the total probability of collision $P_{\rm D}$ is computed~\cite{r10} as
\begin{equation}
\label{e:PD}
P_{\rm D} = 1 - \prod_{k=1}^K (1-P_{\rm D}(k)),
\end{equation}
which must be recomputed for {\textit{each}} swept path $A$ of $N$ candidate paths. However, we propose instead a bound $P_{\rm D}\leq \bar{P}_{\rm D}$ that can be computed as
\begin{equation}
\bar{P}_{\rm D} = \sum_{k=1}^K P_{\rm D}(k) .
\label{e:PDbar}
\end{equation}
Moreover, when $P_{\rm D}\ll 1$, as we expect in practical, relatively safe situations, the bound $\bar{P}_{\rm D}$ is tight.
The bound can be computed efficiently, exploiting the linearity of (\ref{e:PDbar}) cf. (\ref{e:PD}), to calculate
\begin{equation}
\bar{P}_{\rm D} = \int_A G({\bf r}) \mbox{ where } G = \sum_{k=1}^K p_k ({\bf r}) .
\label{e:FDG}
\end{equation}
The computation is then trivially $O(N+K)$ not $O(NK)$, since $G$ in (\ref{e:FDG}) can be precomputed and re-used for all candidate paths $A$. However, with obstacles of finite size (as opposed to point obstacles) achieving $O(N+K)$ complexity is no longer trivial, as we see next.
\subsection{Finite obstacles}
Generally for obstacles that are not just points but have finite area, (\ref{e:Laugier0}) has been extended~\cite{r10b} for the case of circular obstacles by ``adding on'' the obstacle radius to the robot $A$.
More generally, for an obstacle shape $B_k \subset \Re^2$, situated at the origin, Minkowski sum can be used to expand the robot shape $A$. At a general position ${\bf r}$, the displaced obstacle is
\begin{equation}
B_k({\bf r}) = B_k + {\bf r} = \{ {\bf r} + {\bf r}' : {\bf r}' \in B_k \}.
\end{equation}
So the probability of collision with the obstacle can be rewritten as
\begin{equation}
P_{\rm D}(k) = \int_{A_k} p_k({\bf r}) ,
\label{Laugier_int}
\end{equation}
where $A_k = A \oplus B_k$, the Minkowski sum of the robot shape and the obstacle shape, as in figure~\ref{fig2}. We term this equation (\ref{Laugier_int}) the {\em Laugier Integral}.
In a search-based algorithm, the Minkowski sums for $A_k$ must then be recomputed for each of $N$ candidate swept paths $A$, and for every obstacle $B_k$. We would therefore like to find a way to replace this naive $O(NK)$ computation, by an efficient $O(N+K)$ computation --- as was done above for point obstacles, but now in the finite obstacle case.
\begin{figure}[htbp]
\centering
\vspace{3mm}\includegraphics[width=0.45\textwidth]{figures/fig-expanded-swept-area.png}
\caption{\small {\bf Minkowski Sum.} Path $\pi$ with swept area $A$, dilated by Minkowski sum with obstacle $B_k$, to give the expanded swept area $A_k$.}\label{fig2}
\end{figure}
\subsection{Minkowski Sum and the ``Convolution Trick''}
Note that the integral in (\ref{Laugier_int}) can be rewritten as the mathematical convolution of two functions, evaluated at the origin:
\begin{equation}
\label{e:LaugierConvol}
P_{\rm D}(k) = [{\rm I}_{A_k} \ast \tilde{p}_k({\bf r})]({\bf 0}) ,
\end{equation}
where ${\rm I}_S$ denotes the indicator function of the set $S$, $\tilde{f}$ denotes the reflection of a function, i.e., $\tilde{f}({\bf r}) = f(-{\bf r})$, and the notation $[\ldots]({\bf r})$ means that the function defined in square brackets is evaluated at the location ${\bf r}$. (So in this instance (\ref{e:LaugierConvol}), the function in square brackets is a convolution of two functions, which is then evaluated at the origin ${\bf r}= {\bf 0}$.)
There is also a known connection~\cite{ab1} (see also~\cite{r20b, r20}) between the convolution of the indicator functions of two sets, and the Minkowski sum of the two sets, as follows:
\begin{equation}
X \oplus Y = \mbox{supp}({\rm I}_X \ast \tilde{\rm I}_Y) ,
\end{equation}
where $\mbox{supp}(f)$ is the support of the function $f$. In particular,
\begin{equation}
A_k = \mbox{supp}({\rm I}_A \ast \tilde{\rm I}_{B_k}) .
\end{equation}
It is not generally the case that the indicator of a Minkowski sum is simply equal to the (normalised) convolution of the two indicator functions (see figure~\ref{fig3}). Nonetheless, over a restricted portion of the domain, corresponding to the case when the obstacle $B_k$ lies {\textit{inside}} the robot path $A$, equality does hold:
\begin{equation}
{\rm I}_{A_k}({\bf r}) = \lambda_k ~ [{\rm I}_A \ast \tilde{\rm I}_{B_k}]({\bf r}) ~\mbox{when}~ B_k({\bf r}) \subset A ,
\label{conv_eq_1}
\end{equation}
where $\lambda_k = \frac{1}{area(B_k)}$. The expression on the right of this equation is everywhere positive as it is a convolution of indicator functions which are positive.
This gives us a formula for ${\rm I}_{A_k}$ when $B_k({\bf r}) \subset A$. Next, we need the corresponding formula for the complementary case when $B_k({\bf r}) \not\subset A$.
\begin{figure}[htbp]
\centering
\vspace{3mm}\includegraphics[width=0.45\textwidth]{figures/fig-Minkowski-convolution.png}
\caption{\small {\bf Convolution and the Minkowski sum.} Illustration in 1D of the fact that the indicator of a Minkowski sum is not generally equal to the convolution of the two indicators, but they do share the same support.}\label{fig3}
\end{figure}
\subsection{Contour convolution}
For the complementary component of the Minkowski sum, where $B_k({\bf r}) \not\subset A$, then ${\rm I}_{A_k}$ can be bounded using a convolution of the bounding contours of obstacles $\partial{B_k}$, and of the robot $\partial{A_k}$. This leads to an upper bound on any integral of the form $\int_{A_k} f({\bf r})$, and in particular on the collision probability (\ref{Laugier_int}).
Given the set $A$, we define the {\em delta function ridge} around its boundary $\partial{A}$ as:
\begin{equation}
\partial{A_\sigma}({\bf r}) = |\nabla g_\sigma({\bf r}) \ast {\rm I}_A ({\bf r})|
\end{equation}
where $g_\sigma({\bf r})$ is a normalised, isotropic, $2$D Gaussian function with a (small) diameter $\sigma$.
Similarly, we define $\partial{B}_{k,\sigma}({\bf r})$ as the delta function ridge around $\partial{B}_{k}$.
Now, we claim that the indicator function for the Minkowski sum is bounded in the complementary condition, and in the limit that $\sigma \rightarrow 0$, by the convolution of these two delta function ridge functions, as follows:
\begin{equation}
{\rm I}_{A_k}({\bf r}) \le \frac{1}{2} [\partial{A_\sigma} \ast \tilde{\partial{B}}_{k, \sigma}]({\bf r}) ~\mbox{when}~ B_k({\bf r}) \not\subset A
\label{conv_eq_2}
\end{equation}
This is illustrated in figure~\ref{fig4}, and proved later.
\begin{figure}[htbp]
\centering
\vspace{3mm}\includegraphics[width=0.45\textwidth]{figures/fig-contour-convolution.png}
\caption{\small {\bf Contour convolution.} Approximating the indicator function of the Minkowski sum of sets $A$ and $B$ via contour convolution.}
\label{fig4}\label{f:contour-convolution}
\end{figure}
\subsection{Combined bound for the FPR algorithm}
As with (\ref{conv_eq_1}), the right hand side of the inequality (\ref{conv_eq_2}) is everywhere positive. So, now the mutually complementary expressions (\ref{conv_eq_1}) and (\ref{conv_eq_2}) can be combined into a single bound on the indicator function of the Minkowski sum:
\begin{equation}
{\rm I}_{A_k}({\bf r}) \le \frac{1}{2} \left[\partial{A_\sigma} \ast \tilde{\partial{B}}_{k, \sigma} \right]({\bf r}) + \lambda_k ~\left[{\rm I}_A \ast \tilde{\rm I}_{B_k}\right]({\bf r}) .
\label{conv_eq_3}
\end{equation}
As this bound holds everywhere, we can simply write
\begin{equation}
{\rm I}_{A_k} \le \frac{1}{2} \partial{A_\sigma} \ast \tilde{\partial{B}}_{k, \sigma} + \lambda_k ~{\rm I}_A \ast \tilde{\rm I}_{B_k} .
\label{conv_eq_4}
\end{equation}
Returning to the earlier expression~(\ref{e:LaugierConvol}) for the collision probability, we have
\begin{equation}
P_{\rm D}(k) \le \left[ \left( \frac{1}{2} \partial{A_\sigma} \ast \tilde{\partial{B}}_{k, \sigma} + \lambda_k {\rm I}_A \ast \tilde{\rm I}_{B_k}\right) \ast \tilde{p_k} \right] ({\bf 0}) .
\label{conv_eq_5}
\end{equation}
Now using the associativity of the convolution operator, this can be rewritten as
\begin{equation}
P_{\rm D}(k) \le \frac{1}{2} \left[ \partial{A_\sigma} \ast \tilde{\partial{B}}_{k, \sigma} \ast \tilde{p_k} \right] ({\bf{0}}) + \lambda_k \left[ {\rm I}_A \ast \tilde{\rm I}_{B_k} \ast \tilde{p_k}\right]({\bf 0}) .
\label{conv_eq_6}
\end{equation}
This can equivalently be written as
\begin{equation}
P_{\rm D}(k) \le \int \partial{A_\sigma} \frac{1}{2} \left(\partial{B_{k,\sigma}} \ast {p_k}\right) + \int {\rm I}_A \left( \lambda_k {I}_{B_k} \ast {p_k} \right) .
\label{bound_eq_1}
\end{equation}
Finally, summing up over obstacles as in (\ref{e:PDbar}), the bound $F_{\rm D}$ on the number of collisions is given by:
\begin{equation}
\bar{P}_{\rm D} \leq F_{\rm D} = \int \partial{A_\sigma}({\bf r}) \partial G_\sigma({\bf r}) + \int {\rm I}_A({\bf r}) G({\bf r})
\label{bound_eq_2}
\end{equation}
where $\partial G_\sigma$ and $G$ are:
\begin{align}
\label{Gsigma_eq} \partial G_\sigma &= \frac{1}{2} \sum_k \partial{B_{k,\sigma}} \ast p_k \\
\label{G_eq} G &= \sum_k \lambda_k {\rm I}_{B_k} \ast p_k .
\end{align}
Note that $G$ and $\partial G_\sigma$ are independent of $A$ and {\textit{do not}} need to be {\em recomputed} every time $A$ changes. So the repeated computation of the bound (\ref{bound_eq_2}), for $N$ different swept paths $A$, would indeed have complexity $O(N+K)$.
This combined ``convolution trick'' gives the FPR method to calculate the bound on the number of collisions which is summarised in Algorithm~1.
The two equations (\ref{Gsigma_eq}) and (\ref{G_eq}) combine the full set of obstacles (together with their location distributions) onto 2 grids or planes. This non-trivial combination of $K$ obstacle shapes and distributions onto just 2 grids is what gives the FPR algorithm its increased efficiency. However, it is important to note that this is {\em not simply a union of obstacles}. It is a complex combination of shapes, outlines and location distributions, which is by no means obvious, but is derived and justified in this paper by means of the convolution trick.
\subsection{Proof of the Contour Convolution Formula}
We show that the indicator function for the Minkowski sum is indeed bounded, in the limit $\sigma \rightarrow 0$, as in inequality (\ref{conv_eq_2}).
For values of ${\bf r}$ such that $B_k({\bf r}) \cap A = \phi$, both sides of the inequality in (\ref{conv_eq_2}) are $0$, in the limit. Elsewhere $B_k({\bf r}) \not\subset A$, so the contours $\partial A$ and $\partial B_k$ must intersect at least twice. In that case, the convolution
\begin{equation}
[\partial{A}_\sigma \ast \tilde{\partial{B}}_{k,\sigma}]({\bf r}) =
\int_{{\bf r}'} \partial{A_\sigma}({\bf r}') \partial{B}_{k,\sigma} ({\bf r}' -{\bf r})
\label{proof_eq_2}
\end{equation}
integrates across two or more contour intersections. The integral at each intersection of two smooth contours (crossing at an angle $\theta$) has the general form
\begin{equation}
J = \int\int g_\sigma(x) g_\sigma(x \cos\theta + y \sin\theta) \,{\rm d}x \,{\rm d}y .
\label{proof_eq_3}
\end{equation}
Now as $g_\sigma$ is a normalised Gaussian, $\int_x g_\sigma(x)\,{\rm d}x = 1$, and applying this above in $y$ (with a straightforward substitution), and in $x$, yields
\begin{equation}
J = \frac{1}{\sin\theta} \ge 1 ,
\label{proof_eq_4}
\end{equation}
so the integral (\ref{proof_eq_2}) accumulates a value of at least 1 for each contour intersection. Therefore, with 2 or more intersections, the right hand side of inequality~(\ref{conv_eq_2}) is at least $\frac{1}{2}\times 2=1$, compared with the value of 1 for the indicator function ${\rm I}_{A_k}({\bf r})$ on the left hand side, so the inequality does indeed hold.
\RestyleAlgo{ruled}
\begin{algorithm}
\KwData{$N$~instances~of~path~$A$, $B_{1:K}$, $p_{1:K}({\bf r})$, $\sigma$}
\KwResult{$F_{\rm D}$}\;
\textit{// Compute $\partial G_\sigma$}\;
\For{k in 1 to K} {
$\partial{B_{k, \sigma}}({\bf r}) = |\nabla g_\sigma({\bf r}) \ast {\rm I}_{B_k} ({\bf r})|$
}
\;
$\partial G_\sigma({\bf r}) = \frac{1}{2}\sum_{k=1}^{K}{\partial{B_{k, \sigma}}({\bf r}) \ast p_{k}({\bf r})}$\;
\;
\textit{// Compute $G$}\;
$G({\bf r}) = \sum_{k=1}^{K}{\frac{1}{\text{area}(B_k)}{\rm I}_{B_k}({\bf r}) \ast p_{k}({\bf r})}$\;
\;
\For{{\rm each}~A} {
\textit{// Compute $\partial A_\sigma$}\;
$\partial{A_\sigma}({\bf r}) = |\nabla g_\sigma({\bf r}) \ast {\rm I}_A ({\bf r})|$\;
\;
\textit{// Integrate over a box around A with margins $4\sigma$}\;
$F_{\rm D} = \int_{\mathbb{R}^2}{\partial{A_\sigma}({\bf r})\partial G_\sigma({\bf r}) + {\rm I}_A({\bf r})G({\bf r})}$
}
\;
\caption{\small {\bf FPR algorithm} for the bound on collision risk given $N$ paths, and $K$ obstacles.}
\end{algorithm}
\section{Results}
In this section we demonstrate that randomly generated trajectories of an SE(2) robot can be efficiently labelled by the FPR algorithm, according to the bound $F_{\rm D}$ on the risk of collision for each path. The FPR algorithm is agnostic as to the motion planner used to synthesise the candidate trajectories and is generally compatible with state of the art methods for motion planning \cite{katrakazas2015real}. We use an off-the-shelf Closed Loop variant CL-RRT~\cite{kuwata2009real} of the RRT algorithm~\cite{lavalle1998rapidly}
to generate candidate paths in the environment, drawn from the kinodynamic model for a particular robot. This has the advantage of generating typically smooth paths, that are plausible paths for that robot. Then the risk bound is calculated for each generated path.
First our results demonstrate the FPR algorithm for a simulated environment, then for a real environment taken from an aerial view, and finally a substantial dataset of 7481 birds-eye views each with several goals and multiple trajectories for each goal --- 240,498 trajectories in all. In each case, higher risk paths take tighter lines around obstacles, as would be expected. We show: i) how close the bound on risk is to the true risk; and ii) that the use in FPR of the convolution trick, which improves computational complexity from $O(NK)$ to $O(N+K)$ as explained earlier, leads to substantial reductions in practical computation times.
\subsection{Simulated Environments}
We first use a $2$D simulation in which a rectangular SE(2) robot of size $2{\rm m} \times 4{\rm m}$ navigates along continuous paths, defined as the progression of $(x,y,\theta)$ pose over time (though our visualisations only depict the centroid). A simulation scenario is defined as a collection of obstacles within the environment, each specified as a shape (a subset of $\Re^2$), and pose, together with positional uncertainty, as well as start and goal poses for the ego vehicle.
The simulated scenario shown here in figure~\ref{f:sim-results} resembles sections of a car park; 400 paths are generated at random by CL-RRT. The uncertainty over each obstacle's position is modelled as a two-dimensional Gaussian distribution with standard deviation $0.3$m, which is 15\% of each obstacle's width.
In figure~\ref{f:sim-results}, paths with lower $F_{\rm D}$ are seen to maintain a greater clearance from the obstacles, as expected.
\begin{figure}[htbp]
\centering
\vspace{3mm}\includegraphics[width=0.45\textwidth]{figures/fig-sim-results.png}
\caption{\small {\bf Visualisation of paths.} The figure shows paths for a simulated environment. A set of candidate paths are generated with fixed start and end poses. The dark objects are obstacles whose position is known with an uncertainty visualised here as a shaded halo. Risk bounds $F_{\rm D}$ on each path are represented by different colours, on the logarithmic scale shown. Safer paths maintain greater clearance around obstacles as expected. Some risky paths (bottom) involve squeezing through a narrow gap. }\label{f:sim-results}
\end{figure}
In figure~\ref{f:trajectory-risk} an aerial view of a car park is shown, with a set of candidate paths generated by CL-RRT, between fixed start and end points. Obstacle vehicle shapes are represented as bounding rectangles. Error in estimated position of the obstacle-cars is Gaussian with standard deviation of 0.3m.
The candidate paths are coloured according to the computed value of the bound on collision risk. This turns out to include safer paths with collision risk down to $10^{-5}$ and below, and riskier paths, above $10^{-2}$ risk of collision, that involve squeezing through a narrow gap.
Finally, for the sole purpose of researching the behaviour of the FPR algorithm, a larger dataset derived from the birds-eye view KITTI collection~\cite{KITTI13} is used. Each scene contains a number of vehicles, obstacles $B_k$ that are represented as rectangles, with positions labelled and assumed here to have Gaussian error with standard deviation of 0.7m. One example view, from the total of 7481 birdseye views, is illustrated in figure~\ref{f:KittiTrajectories}.
\begin{figure}[htbp]
\centering
\vspace{3mm}\includegraphics[width=0.48\textwidth]{figures/fig-KittiTrajectories.png}
\caption{\small {\bf KITTI birdseye data.} Example view from KITTI dataset of 7,481 birdseye views of traffic scenes. Obstacles shown in blue, each have simulated Gaussian uncertainty with standard deviation 0.7m. A goal is chosen automatically. Possible paths for an SE(2) robot (red) are shown. FPR bound on collision risk is displayed on the colour scale shown.
}\label{f:KittiTrajectories}
\end{figure}
In each scene, an SE(2) robot is given an initial position, and several goals are chosen, automatically. Then up to 10 paths per goal are generated --- a total of 240,498 paths. Shapes of obstacles are assumed known, and in practice this could be achieved by recognition of known objects such as vehicles.
\subsection{Performance Evaluation}
Simulation results given here illustrate the computational benefits of the FPR approach for evaluating bounds on risk. In figure~\ref{f:timings}, we present empirical data regarding the computational efficiency of our method compared to the exact computation of the integral in (\ref{Laugier_int}). Computing the FPR bound is, on average, significantly faster than exact computation. Even for the first path, FPR is more than 3 times as efficient on average, thanks to the use of efficiently implemented convolution, in place of Minkowsi sum. For subsequent paths FPR is on average two orders of magnitude more efficient, at 10ms per path. This is consistent with $O(N+K)$ complexity c.f $O(NK)$ complexity (for $N$ evaluated paths), as expected from theory.
\begin{figure}[htbp]
\centering
\vspace{3mm}\includegraphics[width=0.45\textwidth]{figures/fig-KittiTimings.png}
\caption{\small {\bf Performance of FPR bound computation} over the KITTI birdseye data, compared with exact (\ref{Laugier_int}) computation of risk.
}\label{f:timings}
\end{figure}
\subsection{How tight is the bound on collision risk?}
It is reasonable to ask how close the FPR bound is, in practice, to the exact risk $P_{\rm D}$. The ratio of the bound to the exact risk is evaluated over all the 240,498 paths derived from the KITTI data. The ratio is about 2.7 on average, with a distribution largely between 1 and 10 (93\% of examples), as in figure~\ref{f:KittiSlack}.
\begin{figure}[htbp]
\centering
\vspace{3mm}\includegraphics[width=0.45\textwidth]{figures/fig-KittiSlackFactor.png}
\caption{\small {\bf Tightness of bounds -- KITTI data.} Bounds $F_{\rm D}$ on collision risk are computed for all 240,498 trajectories and compared with exact risk. The risk bounds have an average ratio of 2.72 times the exact risk, and that ratio is distributed as shown in the histogram.}
\label{f:KittiSlack}
\end{figure}
The effect of this ratio is that the bound will lead to conservative decisions. For example if the ratio is $3$, then to achieve a desired collision risk of say $10^{-3}$ or better, setting the FPR risk to $10^{-3}$ will actually achieve a lower risk of $\frac{1}{3} 10^{-3}$. As a result, selected paths may leave more clearance from obstacles than strictly necessary. Occasionally, it is possible that no path in a certain set may have an FPR bound within the acceptable level of risk, even when a path with an acceptable level of true risk does actually exist.
\subsection{Implementation details}
We use an augmented version of the CL-RRT planner, with probabilistic sampling, similar to other approaches for heuristically biasing RRT growth~\cite{urmson2003approaches}, and choosing tree nodes for expansion according to their scores. It discretises steering and accelerator input ranges when expanding the random tree, to generate realisable trajectories, and in order to restrict abrupt steering or velocity changes. Nodes in the RRT are scored based on their proximity to the goal, and similarity to its orientation and velocity. We treat each
tethered obstacle as deterministic, just for the purposes of CL-RRT, taking the shape $B_k$ at the mean location over $p_k$.
For all simulations, space is discretized on a grid with a resolution of 5 cm/px. All convolutions
in our implementation use a Gaussian or gradient of Gaussian kernel and so we exploit the separability property in order to perform convolutions efficiently. Additionally, we approximate the final integration step in Alg. 1 with a computationally efficient Riemann sum over the discretised
grid. (If it were desired to use non-Gaussian $p_k({\bf r})$ then the convolutions with $p_k({\bf r})$ could be done by FFT or morphologically~\cite{Samaniego19}.)
The constant for Gaussian convolution, $\sigma=2$ grid squares, is just big enough for good numerical behaviour.
All of the numerical computations are implemented using the GPU-enabled Python module for linear algebra CuPy~\cite{cupy_learningsys2017}. The goal and trajectory data, used with KITTI data in simulations, will be made available on the web.
\section{CONCLUSIONS}
Our FPR algorithm bounds the risk of collision for candidate paths in a given environment.
It builds on a probabilistic framework for calculating collision risk, using the convolution trick to render these computations in linear time in $N$ and $K$. Amongst trajectories deemed safe enough, there would then be freedom to optimise for other criteria such as rider comfort and travel time.
Other sources of uncertainty would of course also need to be taken into account in an end-to-end implementation, such as missed detections of obstacles. The current state of the art~\cite{r23} suggests risk of the order of $10^{-3}$ for missed detections. It is to be hoped however that this improves with future advances in temporal and cross-modal fusion.
The effect of computing risk in $xy$-space, as opposed to the full $xyt$-space, is further to approximate (in fact bound) the computed risk. The bounding effect preserves safety, but in some circumstances is overly conservative and may overestimate risk, which could lead to `frozen robots'~\cite{r9}.
Future work looks at extending FPR from tethered obstacles to fully dynamic obstacles whose position evolves stochastically. Then $p_k({\bf r})$ in (\ref{Laugier_int}) would be extended to a spatio-temporal (stochastic) process as in~\cite{ab2,ab3,ab5}. Risk computation would be in the full $xyt$-space. The question is, to what extent could full Monte-Carlo computation of risk be avoided, and linear time-complexity in $N$ and $K$ be retained?
\addtolength{\textheight}{-13cm}
|
{
"timestamp": "2019-09-18T02:13:05",
"yymm": "1804",
"arxiv_id": "1804.05384",
"language": "en",
"url": "https://arxiv.org/abs/1804.05384"
}
|
\section{Introduction}
A well-known probability puzzle asks how to simulate a fair coin toss given a coin with unknown bias. A solution, due to Von Neumann \cite{V}, is to flip the coin twice and if the flips disagree use the first, otherwise flip twice again and repeat. We leave it to the reader to verify that this algorithm does in fact solve the problem. The purpose of extractors in theoretical science is to address a similar problem: Given the output of a random variable/source with unknown biases, subject to only some weak assumptions, use these outputs to simulate a random variable with much stronger (more uniform) randomness properties.
We say that a real-valued random variable $X:\Omega \rightarrow R$ has min-entropy $k$ if $\sup_{r \in R}\mathbb{P}[X = r] \leq 2^{-k}.$ Given random variables $X$ and $Y$ we define the statistical distance between them as
$$SD(X,Y) := \frac{1}{2} \sum_{r\in R} \left| \mathbb{P}[X=r] - \mathbb{P}[Y=r] \right|.$$
We call a family of maps $Ext_N : [N] \times [N] \rightarrow \{0,1\}$ an extractor family with min-entropy $\rho$ if there exists a fixed $\delta >0$ such that for any independent random variables $X,Y: \Omega \rightarrow [N]$ with min-entropy rate $\rho \log_2 N$ one has
$$ SD \left(Ext_N(X,Y), U \right) \lesssim \log^{-\delta} N.$$
where $U$ is the uniform distribution on $\{0,1\}$. To ease notation, we will say that $Ext_N$ is an extractor family with min-entropy rate \textit{near} $\rho$ if it extracts from all sources with min-entropy rate $\rho' > \rho$. We will say that $Ext_N$ has exponentially small error if one has the stronger estimate
$$ SD \left(Ext_N(X,Y), U \right) \lesssim N^{-\delta}$$
for some fixed $\delta >0$.
\begin{remark}The computer science literature typically considers maps from $\{0,1\}^\ell \times \{0,1\}^\ell \rightarrow \{0,1\}$ when defining extractors. One can convert to this regime by taking $\ell = \left \lceil{2 \log p}\right \rceil$.
\end{remark}
There are a number of elementary constructions of extractor families with min-entropy rate greater than $1/2$. In 2005, Bourgain \cite{B} gave the first explicit constructions of an extractor with min-entropy rate less than $1/2$. His proof depended on the finite field Szemer\'di-Trotter theorem \cite{BKT} and gave an unspecified min-entropy rate. An optimization of the key ingredient, the Szemer\'edi-Trotter incidence theorem, by Helfgott and Rudnev \cite{HR} in 2010 gave an exponent $\frac{3}{2} - \frac{1}{10678}$ in that result. Combined with Bourgain's method in the form below, this gives a min-entropy rate estimate of $\frac{16017}{32036} = .49996\ldots$. The incidence estimate was further refined by Jones in 2011 who gave an incidence estimate with exponent of $\frac{3}{2}- \frac{1}{622} = \frac{496}{331}$ which gives a min-entropy rate of around $\frac{2647}{5296}=.4998\ldots$. More recently, Stevens and de Zeeuw \cite{Sd} introduced several new ideas which allowed them to improve the exponent in the incidence theorem to $\frac{3}{2}- \frac{1}{30} = \frac{22}{15}$. This yields an extractor with min-entropy rate near $\frac{45}{92}=.489\ldots$. Lastly we note that an optimal Szemer\'edi-Trotter theorem in finite fields with exponent $4/3$ would imply Bourgain's extractor has min-entropy rate near $\frac{9}{20}=.45$, following the initial approach of \cite{B}. See also \cite{HH} for generalizations of Bourgain's work.
In a 2015 breakthrough paper, Chattopadhyay and Zuckerman \cite{CZ} gave an alternative approach which produced explicit extractors which extract from sources with arbitrarily small (in fact poly-logarithmic) min-entropy rate. One advantage to Bourgain's Fourier analytic constructions is that these extractors have exponentially small error, where Chattopadhyay and Zuckerman's only have poly-logarithmically small error. Finding explicit constructions with both an arbitrarily small min-entropy rate and exponentially small error remains an important open problem in the field. Our interest here is to show how one can push Bourgain's Fourier analytic approach further using recent advances in finite field incidence geometry. This allows us to produce explicit constructions with exponentially small error and a lower min-entropy rate than available in the prior literature.
We start by recalling one of Bourgain's construction. Let $F$ be a prime order finite field. Given an element $x \in F$, we let $\sigma(x) \in [0,\frac{p-1}{p} ]$ denote the real numbers in the interval $[0,1]$ given by considering the integer representation of $x$ between $0$ and $p-1$ and dividing that number by $p$. Moreover we let $\rho : F \rightarrow \{0,1\}$ be defined by $\rho(x) := \sign \sin \left( 2 \pi \sigma(x) \right)$ where we adopt the convention that $\sign(0) = 1$. We take $N=p^2$ and consider the maps $Ext_{p^2}$ from $F^2 \times F^2 \rightarrow \{0,1\}$ given by
$$(x,y) \rightarrow \rho (x\cdot y + x \cdot x \times y \cdot y).$$
If $p$ is chosen so that there does not exists an element $i \in F$ such that $i^2=-1$, then the map just defined is a Bourgain extractor.
Our first result is a refined analysis of one of Bourgain's extractor:
\begin{theorem}\label{thm:3d}Bourgain's extractor (for $F$ a prime order finite field in which $-1$ is not a square) defined above extracts from two independent sources with min-entropy rate near $\frac{21}{44} =.477\ldots$ and has exponentially small error.
\end{theorem}
Next we show that one can construct extractors with lower min-entropy rate by replacing $F^2$ in Bourgain's construction with $F^3$. More precisely:
\begin{theorem}\label{thm:4d}Let $F$ be finite field of order $p$ and identify $[N] = F^3$. Then the maps $Ext_{p^3} (x,y) \rightarrow \rho(x\cdot y + x \cdot x \times y \cdot y)$ from $F^3 \times F^3 \rightarrow \{0,1\}$ is a two-source extractor with min-entropy rate near $\frac{4}{9} =.444\ldots$ and has exponentially small error.
\end{theorem}
We note that Bourgain's extractor requires one to restrict to prime order fields in which $-1$ is not a square, while this hypothesis on the prime order field in Theorem \ref{thm:4d}. In Bourgain's case if $F$ is such that there exists an $i \in F$ such that $i^2=-1$ then the the zero set of $z:=\{x \in F^2 : \sigma(x)=0\}$ contains the line $\{(t, it) : t \in F \} \subset F^2$ and the required additive energy will not hold for certain subsets of this line. In the second case, the analogous level set will contain a line regardless of the field. However to prove Theorem \ref{thm:4d} one only needs control of the additive energy for sets substantially larger than that of a line. This phenomenon is explained by the theory of quadratic forms in finite fields, and is discussed in detail in \cite{LewkoKakeya} and also is connected to the recent observation that one can obtain sharp $L^2$ restriction theorems for the finite field paraboloid in high dimensions while the analogous problem remains open in $3$ dimensions. See \cite{IKL} and \cite{RS}.
The proofs proceed by observing that Bourgain's argument can be extended to a rather general statement relating extraction properties of certain maps to the additive energy of certain subsets in finite fields. Lemma \ref{lem:reduct} and \ref{lem:expSum} below articulates this generalization. The second step is to observe that these statements reduce the maps above to additive energy estimates for subsets of the paraboloid obtained by Rudnev and Shkredov \cite{RS} in their recent work on the finite field restriction problem. See \cite{IKL}, \cite{LewkoEnd}, \cite{LewkoKakeya}, \cite{LewkoImp}, and \cite{MT} for a discussion of this problem.
Finally we remark that inserting an optimal finite field Szemer\'edi-Trotter theorem into the Rudnev-Shkredov \cite{RS} machinery yields an additive energy estimate which when inserted into the machinery below shows that Bourgain's extractor defined above extracts from sources with min-entropy rate near $3/8 = .375.$
\textbf{Acknowledgment} We thank David Zuckerman for comments on an earlier draft of this note.
\section{Notation and Preliminaries}
We will use $R$ and $C$ to denote the fields of real and complex numbers, respectively. We will use $F$ to denote a finite field, which will always be of prime order. We denote the non-zero elements of $F$ by $F_*$. As usual we write the additive character on $F$ as $e(x):=e^{2\pi i x /p}$. Given $x,y \in F^{n}$ we write $x\cdot y := x_1 y_1 + \ldots+ x_n y_n$. Given a subset $A \in F^{n}$ we define the additive energy $\Lambda(A):= \sum_{\substack{a+b=c+d \\ a,b,c,d \in A}} 1 = \sum_{x \in F^n} \left( \sum_{\substack{a+b=x \\ a,b \in A}} 1 \right)^{2}.$ Parsavel's identify is the equality
$$ \sum_{x \in F^n} \left| \sum_{\xi \in F^n} f(\xi) e(x \cdot \xi) \right|^2 = |F| \sum_{\xi \in F^n} |f(\xi)|^2.$$
If $f$ is a real or complex valued function on a domain $D$ we define $|| f ||_{\ell^\infty} := \sup_{x\in D} |f(x)|$. We will write $X \sim Y$ to indicate that $Y/2 \leq X \leq 2Y$. Typically we use this to select a level set of a function. For example $D_{\lambda } := \{x \in D : f(x) \sim \lambda\}$ would denote the elements of the domain, $D$, of $f$ where $\lambda/2 \ \leq f(x) \leq 2 \lambda $. We will also use the notation $a \lesssim b$ to indicate that the inequality $a \leq c b$ holds with some universal constant $c$. Let $F^d$ denote the $d$ dimensional vector space over $F$. We define the $d$-dimensional paraboloid $P_d \subset{F}^d$ by $P_d:=\{ (\underline{x},\underline{x}\cdot\underline{x}), \underline{x} : F^{d-1}\}$.
\section{Estimates}
We start by showing how Fourier analytic estimates imply extractor-type properties. Recall that the Fourier coefficients in the expansion $sign \sin(x) = \sum_{\xi \in F} c(\xi) e(\xi x)$ satisfy
\begin{equation}\label{eq:ssCo} \sum_{\xi \in F} |c(\xi)| \lesssim \log |F|.
\end{equation}
This can be seen from a simple and direct computation with the Dirichlet Kernel, see Remark 3.3 in \cite{B}.
\begin{lemma}\label{lem:reduct}Let $f:F^n \times F^n \rightarrow F$ be a family of maps indexed by $F$, such that the following holds. For some fixed $\delta >0$, one has for all functions $a,b: F^n \rightarrow F$ with $||a||_{\ell^\infty},||b||_{\ell^\infty} \leq 1$ with respective supports $A,B \in F^n$ satisfying $|A|,|B| \geq |F|^{n \rho}$ that
$$\max_{\lambda \in F_*} |\sum_{x\in A, y\in B} a(x) b(y) e(\lambda f(x,y)) | \lesssim |F|^{-\delta} |A| |B|$$ then the family of maps $ \rho (f(x,y))$ from $F^n \times F^n \rightarrow \{0,1\}$ is a two-source extractor family with min-entropy rate near $\rho$.
\end{lemma}
\begin{proof}Let $U$ denote the uniform distribution on $\{0,1\}$. Our goal is to show $SD( \rho (f(x,y)), U) \leq |F|^{-\delta}$ for some $\delta >0$. Let $X$ and $Y$ denote independent random variables with min-entropy $k$ and probability mass functions $A$ and $B$, respectively. Let $J':F \times F \rightarrow \{0,1\}$ denote the probability mass function of $\rho (f(x,y))$, that is $J'(x):= \mathbb{P}[ \rho (f(x,y)) =x]$ and $J(x) = J'(x) - 2^{-1}$. We then have
$$ SD( \rho (f(x,y)), U) = \frac{1}{2} \sum_{x\in \{0,1\}} \left| J(x) \right|.$$
Expanding in a Fourier series $\sign \sin x = \sum_{\lambda \in F} c(\lambda) e(\lambda x)$, and applying the estimate \eqref{eq:ssCo} we have that the above is
$$\lesssim \left| \mathbb{E} \rho (f(x,y)) \right| = \left| \sum_{x \in F^n}\sum_{y \in F^n}\sum_{\lambda \in F} A(x) B(y) c(n) e(\lambda f(x,y) ) \right|$$
$$\lesssim \log |F| \max_{\lambda \in F_*}\left| \sum_{y \in F^n}\sum_{x \in F} A(x) B(y) e(\lambda f(x,y) ) \right|.$$
The min-entropy assumption implies that $A(x),B(x) \leq |F|^{-\eta}$. Let $S_\theta := \{x \in F^n : A(x) \sim \theta \}$ and similarly $W_\theta := \{x \in F^n : B(x) \sim \theta \}$. Now we split $A(x)$ and $B(y)$ into $O(\log |F|)$ level sets such that $A_\ell(x) = A(x) \sim 2^{-\ell}$ and $B_\ell(x) = B(x) \sim 2^{-\ell}$ on their support which is at most $2^{\ell}$. From the hypotheses we may assume that $\ell \geq k$ and have
$$ \max_{\lambda \in F_{*}} |\sum_{x\in A, y\in B} A(x)B(y) e(\lambda f(x,y))| \leq \sum_{i,j=k}^{\log |F| } \max_{\lambda \in F_{*}} |A_i(x) B_j(y) e(\lambda f(x,y))|$$
$$ \lesssim |F|^{-\delta} \sum_{i,j=k}^{\log |F| } 2^{-i-j}|\text{supp}(A_i)| |\text{supp}(B_j)| \lesssim |F|^{-\delta} \log |F| $$
This completes the proof.
\end{proof}
\begin{remark}The above argument is a special case of what is called Vazirani's XOR lemma in the computer science literature. See Lemma 4.1 in \cite{Rao}. This lemma, which considers the more general case where the function $\rho$ is replaced by a map into other finite abelian groups larger than $\{0,1\}$, allows one to use the methods presented here to obtain extractors that output more than one bits. We refer the reader to \cite{Rao} and omit the routine details.
\end{remark}
Next we show how to obtain estimates on exponential sums of the form appearing in the statement of Lemma \ref{lem:reduct}.$•$
\begin{lemma}\label{lem:expSum}Let $A,B \subseteq F^n$ and $a,b$ functions on $A$ and $B$, respectively, such that $||a||_{\ell^\infty}, ||b||_{\ell^\infty} \leq 1$. Then
$$ \max_{\lambda \in F_*}\left|\sum_{x \in A, b \in B} a(x) b(y) e(\lambda x\cdot y) \right| \leq |A|^{1/2} |B|^{1/2 } |F|^{n/8} \left( \Lambda(A) \Lambda(B) \right)^{1/8}. $$
\end{lemma}
\begin{proof}By Cauchy-Schwarz
$$ \max_{\lambda \in F_*} \left|\sum_{x \in A, y \in B} a(x) b(y) e(\lambda x\cdot y) \right| \leq |A|^{1/2} \max_{\lambda \in F_*}\left( \sum_{x\in A } \left|\sum_{y \in B} a(x) b(y) e(\lambda x\cdot y) \right|^2 \right)^{1/2}$$
$$ = |A|^{1/2} \max_{\lambda \in F_*} \left( \sum_{x\in A } \sum_{y_1, y_2 \in B} a(x) b(y_1) b(y_2) e\left(\lambda x\cdot (y_1 - y_2) \right) \right)^{1/2}$$
$$ \leq |A|^{1/2}|B|^{1/2} \max_{\lambda \in F_*} \left( \sum_{x_1, x_2 \in A } \sum_{y_1, y_2 \in B} a(x_1)a(x_2) b(y_1) b(y_2)e\left( \lambda (x_1 - x_2) \cdot (y_1 - y_2) \right) \right)^{1/4} $$
Now for $\xi, \eta \in F^n$ let
$$\mathcal{A}_{\lambda}(\xi) = \sum_{\substack{ x_1, x_2 \in A \\ x_1- x_2 = \lambda^{-1} \xi}} a(x_1)a(x_2),$$
$$\mathcal{B}(\eta) = \sum_{\substack{ y_1, y_2 \in B \\ y_1- y_2 = \eta}} a(x_1)a(x_2)$$
where we let $\mathcal{A}(\xi):=\mathcal{A}_{1}(\xi)$. With this notation, we may rewrite the above as
$$ \leq |A|^{1/2}|B|^{1/2} \max_{\lambda} \left( \sum_{ \xi, \eta \in F^n} \mathcal{A}_\lambda (\xi) \mathcal{B}(\eta) e\left( \xi \cdot \eta \right) \right)^{1/4} $$
$$\leq |A|^{1/2}|B|^{1/2} \left( \left(\sum_{\xi \in F^n} |\mathcal{A}(\xi)|^2 \right)^{1/2} \left( \sum_{\xi \in F^n } | \sum_{ \eta \in F^n} \mathcal{B}(\eta) e\left( \xi \cdot \eta \right)|^{2} \right)^{1/2} \right)^{1/4}.$$
Applying Parsavel and noting that $\sum_{\xi \in F^n}|\mathcal{A}(\xi)|^{2} \leq \Lambda(A)$ gives
$$\leq |A|^{1/2} |B|^{1/2 } |F|^{n/8} \left( \Lambda(A) \Lambda(B) \right)^{1/8}. $$
This completes the proof.
\end{proof}
\begin{remark}The referee has pointed out that nearly the same lemma appears in \cite{BG}. \end{remark}
Combining Lemma \ref{lem:reduct} and Lemma \ref{lem:expSum} gives us the following:
\begin{proposition}\label{prop:energyToExtract}Let $n > d$ and $M: F^d \rightarrow F^n$. Let $\eta >0 $ and assume that for every subset $A \subseteq F^d$ with $|A| \sim |F|^{\frac{n}{(8-2\alpha )}}$ one has the energy estimate $\Lambda( M(A)) \lesssim |A|^{\alpha}$. Then the map $(x,y) \rightarrow \rho \left( M(x)\times M(y) \right) $ is an extractor with min-entropy rate near $\frac{n}{d(8-d\alpha )}.$
\end{proposition}
At this point we state the following additive energy estimates of Rudnev and Shkredov \cite{RS} for subsets of the $3$-dimensional and $4$-dimensional paraboloids, $P_3$ and $P_4$.
\begin{theorem}We have the following additive energy estimates. Let $F$ be a prime order field in which $-1$ is not a square. $A \subset P_{3}$ with $|A| \leq |F|^{\frac{26}{21}}$, then
$$\Lambda(A) \lesssim |A|^{\frac{17}{7}}.$$
Let $F$ be an arbitrary prime order finite field. Let $B \subset P_4$ with $p^{4/3} \leq |B| \leq p^{2}$, then
$$\Lambda(B) \lesssim |B|^{\frac{5}{2}}.$$
\end{theorem}Inserting these estimates into Proposition \ref{prop:energyToExtract} proves Theorem \ref{thm:3d} and Theorem \ref{thm:4d}. \textbf{Note Added:} Recently the author has slightly improved the exponent in the first energy estimate above from $17/7$ to $99/41+\epsilon$ for any $\epsilon >0$. This improves the min-entropy rate of the respective extractor to near $123/260$ from $21/41$. See \cite{LewkoRect}.
|
{
"timestamp": "2019-03-19T01:24:29",
"yymm": "1804",
"arxiv_id": "1804.05451",
"language": "en",
"url": "https://arxiv.org/abs/1804.05451"
}
|
\section{Introduction}
Ultimate scientific exploitation of astronomical observations is made recurring to post-processing techniques - commonly called data-reduction. The current way to process scientific images relies commonly on standard packages, such as StarFinder~\citep{Diolaiti2000}, SeXTRACTOR~\citep{Bertin1996}, or DAOPHOT~\citep{Stetson1987}. Nonetheless, past studies~\citep{Fritz2010,Yelda2010} have quantified an astrometry error breakdown on the Galactic Center~(GC), that pointed out the Point spread function~(PSF) model accounts at 50-60~\% level on the astrometry accuracy.
Photometry measurements are impacted by PSF mis-knowledge as well~\citep{Shodel2010,Sheehy2006}, but a recent analysis~\citep{Ascenso2015} has unveiled that photometry measurements accuracy is improved from 0.2 down to 0.02 mag by providing StarFinder with the exact PSF.
Besides, ground-based astronomy is improved thanks to Adaptive optics~(AO) that allows to restore the angular resolution to nearly the diffraction limit. However AO produces a PSF that varies across space and time and does not match standard parametric models, such as Gaussian or Moffat functions. Retrieving the AO PSF shape from focal-plane observations might be also compromised when observing crowded fields suffering from large source confusion~\citep{Turri2017,Shodel2010,Lu2013,Ghez2008}, or deep cosmological fields where no PSF is observed~\citep{Falomo2008,Schramm2013}, calling for alternative approach as PSF reconstruction~(PSF-R)~\citep{Veran1997,Gilles2012}.
PSF-R is a data processing approach that delivers the PSF from AO control loop data, without any priors on its shape. PSF-R reliability has been demonstrated on-sky multiple times~\citep{Veran1997,Flicker2008,Jolissaint2015,Martin2016JATIS}. PSF is reconstructed as a convolution of independent patterns that characterize different physical limitations of AO systems~\citep{Veran1997,Gilles2012}. In this paper, we focus on the specific step of PSF extrapolation that relies on anisoplanatism~\citep{Fried1982} and accounts for PSF spatial variations across the field. The AO system measures the incoming distorted wave-front in the particular direction of the guide star; wave-fronts that propagate along another direction do not cross the exact same turbulence are partially compensated. This introduces an anisoplanatism error that grows with the angular separation from the guide star and the seeing of altitude layers.
AO systems can also guide on Laser guide stars~(LGS)~\citep{Foy1985} which are focused in a range from 80 up to 100~km. When the LGS beam propagates as a spherical wave downwards to the pupil, it crosses only a portion of the turbulence above the telescope. The phase aberrations in the science direction are not fully corrected; it includes an additional term in the residual phase that stands as the cone effect or focal anisoplanatism.
On top of that, probing the wave-fronts using LGSs does not grant access to the wave-front angle of arrival because of the round trip of the light from telescope to the sodium (Na) layer. The PSF position in the focal plane is stabilized by measuring the wave-front angle of arrival using an Natural guide star~(NGS) whose location may be different from the scientific target. This angular separation yields an anisoplanatism effect on tip-tilt modes only, named as tip-tilt anisoplanatism or anisokinetism~\citep{Winick1988,Fried1996,Ellerbroek2001,Flicker2003,Correia2011}.
Reconstructing the off-axis PSF for LGS-based systems requires a model of the angular, focal and tilt anisoplanatism plus the relevant input parameters the model depends on which are the vertical distribution of turbulence known as the $C_n^2(h)$ profile. Such models exist in the literature, but are either not laser-compliant~\citep{Tyler1994,Rigaut1998,Jolissaint2010}, or not numerically efficient~\citep{Fusco2000,Britton2006,Flicker2008}, particularly for future AO systems on next 40~m class telescopes with large number of degrees of freedom. The $C_n^2(h)$ profile is accessible either from internal methods handling AO telemetry of multiple guide-stars~\citep{Ono2017,Martin2016L3S,Guesalaga2016}, or from external profilers~\citep{Wilson2002,Tokovinin2007,Osborn2013,Osborn2015} or more recently from meso-scale models~\citep{Masciadri2017}.
For single guide-star AO systems, the $C_n^2(h)$ is not measurable from AO telemetry, calling for external profilers for modelling the off-axis PSF. However, studies~\citep{Ono2017} have highlighted discrepancies on the retrieved profile by comparing outputs of external to internal profiling techniques. Consequently, using an external $C_n^2(h)$ profile along a likely different line-of-sight from the observations might degrade the accuracy of PSF modeling.
Besides, predicting performance of future AO-based instrument relies on a $C_n^2(h)$ profile as well; we must constrain the $C_n^2(h)$ accuracy required for retrieving faithfully AO-corrected PSFs.
We propose in this paper an investigation of how the PSF is degraded regarding the accuracy on the $C_n^2(h)$. Yet, such an analysis is compromised if relies on on-sky observations, in a way the real $C_n^2(h)$ is not be perfectly known; downstream results would be contaminated by anisoplanatism model errors, such as the discrete number of layers or the exact LGS height that is not perfectly identified for instance.
As an alternative we have applied this approach on the HeNOS Multi-conjugated AO~(MCAO) testbed designed to be the demonstrator for NFIRAOS~\citep{Conan2010}. We lay in this paper all the theoretical background our near-future work will be based on, that will be particularly focused on PSF characterisation application to crowded fields observation, such as the Galactic center.
This paper is organized as follows. In Sect.~\ref{S:PSFR}, we derive a generalized anisoplanatism model that combines both focal and tip-tilt anisoplanatism recurring to an accurate and fast implementation. In Sect.~\ref{S:comparison} it is compared to physical-optics simulations on a 10~m telescope. We consider morphological PSF metrics, as Strehl ratio, FWHM and fraction of variance unexplained, as science metrics~(photometry and astrometry for tight binaries) as well. In Sect.~\ref{S:sensitivity}, we present how those criteria are sensitive to $C_n^2(h)$ accuracy, including number of layers and weight/height precision. We validate the meaning of this approach using observations on the HeNOS bench in Sect.~\ref{S:HENOS}. We conclude in Sect.~\ref{S:conclusions}.
\section{Spatial PSF extrapolation}
\label{S:PSFR}
\subsection{Anisoplanatism transfer function}
We define $\phi_\varepsilon(\rvec,t)$ as the residual wave-front delivered by the AO system. At the focal plane downstream the AO system, the long-exposure Optical Transfer Function~(OTF) in the science direction 1 is~\citep{Roddier1981}
\begin{equation}
\begin{aligned}
\otf{1}(\rhovec/\lambda) =\dfrac{1}{S} \iint_{\mathcal{\mathcal{P}}} \mathcal{P}(\boldsymbol{r})\mathcal{P}(\boldsymbol{r}+\rhovec)\cdot
\exp\para{-\dfrac{1}{2}D_{\phi_\varepsilon}(\boldsymbol{r},\rhovec)}\boldsymbol{dr},
\end{aligned}
\label{E:otfdef}
\end{equation}
where $\mathcal{P}$ is the pupil function, $\boldsymbol{r}$ and $\boldsymbol{\rho}$ the location and separation vectors in the pupil and $S$ is the area of the telescope aperture which normalizes the PSF to unit energy and $D_{\phi_\varepsilon}(\boldsymbol{r},\rhovec)$ is the residual phase structure function defined by
\begin{equation}
\label{E:dphidef}
D_{\phi_\varepsilon}(\boldsymbol{r},\rhovec) = \aver{\module{\phi_\varepsilon(\boldsymbol{r},t) - \phi_\varepsilon(\boldsymbol{r}+\rhovec,t)}^2}_t,
\end{equation}
where $\aver{x(t)}_t$ denotes the temporal average of process $x$. Under the assumption that the anisoplanatism term is not correlated to other AO error terms, such as servo-lag or aliasing, $D_{\phi_\varepsilon}(\boldsymbol{r},\rhovec)$ can be split as follows
\begin{equation}
D_{\phi_\varepsilon}(\boldsymbol{r},\rhovec) = D_{0}(\boldsymbol{r},\rhovec) + D_{\Delta}(\boldsymbol{r},\rhovec),
\label{E:dphisum}
\end{equation}
where $D_{0}(\boldsymbol{r},\rhovec)$ characterizes the AO residual phase structure function in the AO guide star direction~(NGS or LGS as well), while $D_{\Delta}(\boldsymbol{r},\rhovec)$ is the structure function of the anisoplanatic phase $\phi_\Delta(\boldsymbol{r},t)$ defined as
\begin{equation}
\phi_\Delta(\boldsymbol{r},t) = \phi_1(\boldsymbol{r},t) - \phi_0(\boldsymbol{r},t),
\label{E:phiDelta}
\end{equation}
where $\phi_1$ and $\phi_0$ refer to the atmospheric phase in direction 1~(science) and direction 0~(guide star). As explicitly mentioned in Eq.~\ref{E:otfdef}, the phase structure is a function of both position and separation in the pupil. For on-axis PSF reconstruction, we handle the residual phase as a stationary process~\citep{Veran1997}; we assume $D_{0}(\boldsymbol{r},\rhovec)$ is a function of separation only, that allows to average $D_{0}$ over the pupil location as
\begin{equation}
\bar{D}_{\text{0}}(\rhovec) = \dfrac{\iint_{\mathcal{P}} \mathcal{P}(\boldsymbol{r})\mathcal{P}(\boldsymbol{r}+\rhovec) D_{0}(\boldsymbol{r},\boldsymbol{\rho}) \boldsymbol{dr}}{\iint_{\mathcal{P}}\mathcal{P}(\boldsymbol{r})\mathcal{P}(\boldsymbol{r}+\rhovec)\boldsymbol{dr} },
\end{equation}
that allows to pull the exponential term in Eq.~\ref{E:otfdef} out of the integral, which makes for an easier and more convenient numerical implementation.\\
For the LGS case, \citep{Flicker2008} has pointed out the stationarity hypothesis is no longer accurate and degrades the PSF model because the cone effect. We have confirmed on our side that this assumption degrades PSF metrics~(introduced in Sect.~\ref{S:metrics}) at the level of 5\%, while it is maintained to less than 1\% in the NGS case. Because we have the numerical ability to derive the full calculation $D_{\Delta}(\boldsymbol{r},\rhovec)$, the formalism we present here does not rely on the stationarity hypothesis. We include into $D_0$ all AO-residual in the guide star direction and separate any focal and angular anisoplanatism effect into $D_{\Delta}$, in a manner we can still apply the pupil-averaged process on $D_0$.
The OTF reduces to the expression
\begin{equation}
\begin{aligned}
\otf{1}(\rhovec/\lambda) =\dfrac{1}{S} \exp\para{-\dfrac{1}{2}\bar{D}_{0}(\rhovec)}\cdot
\iint_{\mathcal{P}} \mathcal{P}(\boldsymbol{r})\mathcal{P}(\boldsymbol{r}+\rhovec)\exp\para{-\dfrac{1}{2}D_{\Delta}(\boldsymbol{r},\rhovec)} \boldsymbol{dr},
\end{aligned}
\label{E:otfsci}
\end{equation}
We introduce $\otf{\text{DL}}(\rhovec/\lambda)$ as the diffraction-limit OTF that characterizes the angular frequencies distribution imposed by the pupil shape
\begin{equation}
\otf{\text{DL}}(\rhovec/\lambda) = 1/S\iint_{\mathcal{P}} \mathcal{P}(\boldsymbol{r})\mathcal{P}(\boldsymbol{r}+\rhovec)\boldsymbol{dr}
\label{E:otfDL}
\end{equation}
that allows to write equation~\ref{E:otfsci} as
\begin{equation}
\otf{1}(\rhovec/\lambda) = \otf{0}(\rhovec/\lambda) \cdot \atf{\Delta}(\rhovec/\lambda),
\label{E:gs2sci}
\end{equation}
where
\begin{equation}
\otf{0}(\rhovec/\lambda) = \otf{\text{DL}}(\rhovec/\lambda)\cdot\exp\para{-\dfrac{1}{2}\bar{D}_{0}(\rhovec)},
\label{E:otf0}
\end{equation}
and
\begin{equation}
\atf{\Delta}(\rhovec/\lambda) = \dfrac{\iint_{\mathcal{P}} \mathcal{P}(\boldsymbol{r})\mathcal{P}(\boldsymbol{r}+\rhovec)\cdot\exp\para{-\dfrac{1}{2}D_{\Delta}(\boldsymbol{r},\rhovec)}\boldsymbol{dr} }{\otf{\text{DL}}(\rhovec/\lambda)}
\label{E:atf}
\end{equation}
is the Anisoplanatism Transfer Function~(ATF) as introduced by~\citep{Fusco2000} which is a convenient formulation to derive the residual OTF anywhere in the field from the knowledge of the on-axis OTF. A practical result to extrapolate the PSF anywhere in the field consists of computing
$\otf{2}$. From Eqs.~\ref{E:gs2sci} and~\ref{E:atf} comes
\begin{equation}
\otf{2}(\rhovec/\lambda) = \otf{1}(\rhovec/\lambda) \cdot \dfrac{\otf{\Delta_2}(\rhovec/\lambda)}{\otf{\Delta_1}(\rhovec/\lambda)},
\label{E:otf2t1}
\end{equation}
which introduces an "OTF ratio" that is the key quantity for deriving the OTF from an direction to any other. The PSF is seamlessly computed from the Fourier transform of the OTF.
In summary, the capability to model a PSF anywhere in a field requires, firstly to have access to a PSF in a direction 1 using either sources extraction method, parametric modelling or PSF-reconstruction, secondly be capable of computing accurately an OTF ratio. This latter is the main focus of following sections.
\subsection{Modelling anisoplanatism}
We now focus on the implementation of the phase structure function $D_{\Delta}(\boldsymbol{r},\rhovec)$ for computing the ATF in Eq.~\ref{E:atf}. Results in the remainder of this paper rely on the calculation of the anisoplanatic covariance $\mathcal{C}_\Delta(\boldsymbol{r},\rhovec)$ which writes
\begin{equation}
\mathcal{C}_\Delta(\boldsymbol{r},\rhovec) = \aver{\phi_\Delta(\boldsymbol{r},t)\phi^t_\Delta(\boldsymbol{r}+\rhovec,t)}
\label{E:cphi0}
\end{equation}
from which $D_\Delta(\boldsymbol{r},\rhovec)$ is
\begin{equation}
D_\Delta(\boldsymbol{r},\rhovec) = 2\times\para{\mathcal{C}_\Delta(\rvec,0) - \mathcal{C}_\Delta(\boldsymbol{r},\rhovec)}.
\end{equation}
The problem of modeling anisoplanatism is now a matter of describing how the atmospheric phase is spatially correlated. In general terms, phase is described as a linear combination of $n_m$ (orthonormal) $\mathcal{M}$ modes, yielding
\begin{equation}
\begin{aligned}
&\phi_1(\boldsymbol{r},t) = \sum_{i=1}^{n_m} a_i(t)\times\mathcal{M}_i(\boldsymbol{r})\\
&\phi_0(\boldsymbol{r},t) = \sum_{i=1}^{n_m} b_i(t)\times\mathcal{M}_i(\boldsymbol{r}),
\end{aligned}
\end{equation}
where $\boldsymbol{a}(t)$ and $\boldsymbol{b}(t)$ are respectively modal coefficients in the science and guide star directions. The anisoplanatic phase $\phi_\Delta(\boldsymbol{r},t)$ becomes
\begin{equation} \label{E:phiDelta_modal}
\phi_\Delta(\boldsymbol{r},t) = \sum_{i=1}^{n_m} \Delta_i(t)\times\mathcal{M}_i(\boldsymbol{r}),
\end{equation}
with $\Delta_i(t) = (a_i(t) - b_i(t))$.
Combining Eqs.~\ref{E:cphi0} and~\ref{E:phiDelta_modal} gives the following expression
\begin{equation}
\begin{aligned}
\cov{\Delta}(\boldsymbol{r},\rhovec) = \sum_{i=1,j}^{n_m} \aver{\Delta_i\Delta_j^t}\times\mathcal{M}_i(\boldsymbol{r})\mathcal{M}^t_j(\boldsymbol{r} + \rhovec),
\end{aligned}
\label{E:covDelta_modal}
\end{equation}
where
\begin{equation}
\Delta_i\Delta_j^t = {a_i(t)a^t_j(t)} + {b_i(t)b^t_j(t)} - {a_i(t)b^t_j(t)} - b_j(t)a^t_i(t).
\end{equation}
Eq.~\ref{E:covDelta_modal} reminds the $U_{ij}$ formalism introduced by~\citep{Veran1997} for the computation of the residual phase structure function; it simplifies to the multiplication $\mathcal{M}_i(\boldsymbol{r})\mathcal{M}_j(\boldsymbol{r} + \rhovec)$ when deriving the covariance.
On top of that, anisoplanatism modeling requires the calculation of modal coefficients correlation $\aver{\Delta_i\Delta_j^t}$. \citep{Fusco2000} has derived the ATF using a Zernike expansion of the phase, based on Zernike coefficients spatial correlation provided in~\citep{Chassat1989}. Although this study has delivered successful results on a 8~m telescope, its application on a 40~m class telescope is computationally-demanding because of the $ \mathcal{M}_i(\boldsymbol{r})\mathcal{M}_j(\boldsymbol{r} + \rhovec)$ derivations. An alternative computation approach was proposed by~\citep{Gendron2006} to tackle the numerical complexity of the $U_{ij}$ technique. However, although focal anisoplanatism could be covered in using covariance terms introduced by~\citep{Molodij1997}, it would not be as accurate as the NGS case because the intrinsic stationarity assumption of the Zernike expansion.
To decrease the numerical complexity whilst maintaining model accuracy, we may resort to a spatial frequency basis~\citep{Rigaut1998,Jolissaint2010}. Although such a technique is efficient from the numerical computation point of view, statistical independence of Fourier modes assume underlying stationarity; the long-exposure OTF is derived from the AO residual phase Power Spectrum Density~(PSD), although such a description is only accurate for linear and space-invariant systems. In order to generalise it to the LGS case, correlations between the residual phase errors at different frequencies must be included~\citep{VanDam2006,Flicker2008} due to the cone stretching factor. Besides, Fourier expansion does not comply with an accurate tip-tilt filtering~\citep{Sasiela1994} and assume an infinite pupil that degrade the PSF model. Such considerations call for alternative approaches such as the point-wise method that is our formulation baseline described below.
\subsection{Point-wise calculation}
To maintain the accuracy of the anisoplanatism model whilst reducing the computational complexity, we focus on a point-wise approach.
We follow the technique proposed by~\citep{Gilles2012}~: the phase is discretised over a grid of $N\times N$ pixels in the real domain, i.e. $\mathcal{M}_i(\boldsymbol{r})$ functions are turned into Dirac distributions $\delta_i(\boldsymbol{r}) = \delta(\boldsymbol{r} - \boldsymbol{r}_i)$ that are centered at the $i^\text{th}$ pixel location. The anisoplanatic covariance takes the following form
\begin{equation} \label{E:covDelta_zonal}
\begin{aligned}
\cov{\Delta}(\boldsymbol{r},\rhovec)
= \sum_{i,j=1}^{N}\delta(\boldsymbol{r}-\boldsymbol{r}_i)\delta(\boldsymbol{r} + \rhovec - \boldsymbol{r}_i - \rhovec_j)\times\aver{\phi_\Delta(\boldsymbol{r},t)\phi^t_\Delta(\boldsymbol{r}+\boldsymbol{\rho},t)},
\end{aligned}
\end{equation}
where $\delta(\boldsymbol{r}-\boldsymbol{r}_i)\delta(\boldsymbol{r} + \rhovec - \boldsymbol{r}_i - \rhovec_j)$ is 1 only for the couple of pixels located at $\rvec_i$ and separated by $\rhovec_j$. Computation of $\cov{\Delta}(\boldsymbol{r},\rhovec)$ is reduced to the determination of phase covariance at specific separations in the bi-dimensional plane. To include the dependency on both pupil location and separation, we compute the covariance of any two samples leading consequently $N^4$ values. All of these are concatenated into a $N^2\times N^2$ matrix which defines $\cov{\Delta}$. For a given separation, the latter is well described for the Von-K\'arm\'an spectrum of turbulence as
\begin{equation}\label{E:cov_fcn_zonal}
\begin{aligned}
C_\phi(\rho) = \left(\frac{L_{0}}{r_{0}}\right)^{5/3}\times
\frac{\Gamma(11/6)}{2^{5/6}
\pi^{8/3}}\times\left[\frac{24}{5}\Gamma\left(\frac{6}{5}\right)\right]\times
\left(\frac{2\pi \rho}{L_{0}}\right)^{5/6}\times K_{5/6}\left(\frac{2 \pi \rho}{L_{0}}\right)
\end{aligned}
\end{equation}
with $L_{0}$ and $r_{0}$ respectively the outer scale and Fried's parameter, $\Gamma$ the 'gamma' function and finally $K_{5/6}$ a modified Bessel function of the third order.
Derivation of $\cov{\Delta}$ relies on the numerical implementation of Eq.~\ref{E:cov_fcn_zonal} that is fed with vector of separations to estimate the covariance terms in Eq.~\ref{E:covDelta_zonal}. The main challenge in the anisoplanatic covariance calculation lies in the proper definition of separations.
Consider the phase covariance along two different directions $\theta_1$ and $\theta_2$ at a single layer located at altitude $h_l$. Define $z_1$ and $z_2$ the source heights in direction 1 and 2 respectively. Finally, let $\boldsymbol{r}_1$ and $\boldsymbol{r}_2$ be the global pixel location vectors in the pupil projected from the atmosphere along straight paths. Fig.~\ref{F:SACovMat} provides a schematic view of this sketch.
\begin{figure}[h!]
\centering
\includegraphics[width=9cm]{covarianceSketch.pdf}
\caption{Sketch representing how the separation vector $\rhovec_l$ is defined between two resolution elements at a given turbulent layer $l$. Vectors $\boldsymbol{r}_1$ and $\boldsymbol{r}_2$ are the coordinates back-projected on the pupil.}
\label{F:SACovMat}
\end{figure}
From Fig.~\ref{F:SACovMat}, we note the separation of two phase samples has two components. The first is related to the pupil plane coordinates that are stretched down with respect to cone-related squeeze factor $1-h_l/z_1$ and $1-h_l/z_2$. On top of that, there is a pupil shift in altitude created by the angular separation that leads to $h_l\times(\theta_1 - \theta_2)$. The separation vector $\rho_{l}(i,j)$, between aperture points raster index $i$ in direction $\thetavec_1$ and index $j$ in direction $\thetavec_2$ at altitude $h_l$ is
\begin{equation}
\rho_{l}(i,j) = r_1(i)\times(1 - h_l/z_1) - r_2(j)\times(1 - h_l/z_2) + h_l\times(\theta_1 - \theta_2).
\label{E:rhodef}
\end{equation}
We assume the turbulence is discretised along $n_l$ statistically independent layers. From the last equation the element $(i,j)$ of $\cov{\Delta}$ takes the following expression
\begin{equation}
\begin{aligned}
\cov{\Delta}(i,j) &= \sum_{l=1}^{n_l} f_l\times \left(\cov{\phi}((1 - h_l/z_1)r_1(i) - (1 - h_l/z_2)r_2(j)+ h_l\theta_1)\right.
+\cov{\phi}((1 - h_l/z_1)r_1(i) - (1 - h_l/z_2)r_2(j) + h_l\theta_2)\\
&- \cov{\phi}((1 - h_l/z_1)r_1(i) - (1 - h_l/z_2)r_2(j) + h_l\Delta\theta)
\left.- \cov{\phi}((1 - h_l/z_1)r_1(j) - (1 - h_l/z_2)r_2(i) + h_l\Delta\theta)\right),
\end{aligned}
\end{equation}
where $\Delta \theta = \theta_1 - \theta_2$, $f_l$ is the fractional power of the $l^\text{th}$ layer that is defined from the $C_n^2(h_l)$ value as
\begin{equation}
f_l = 0.06\lambda^2r_0^{-5/3}/C_n^2(h_l),
\end{equation}
where $C_n^2(h_l)$ is the $C_n^2(h)$ profile at altitude $h_l$. This implementation of the ATF is freely made available with the simulator OOMAO~\citep{Conan2014OOMAO}.
Other point-wise approaches have been developed~\citep{Tyler1994,Britton2006,Flicker2008}, but only the last reference is laser-compliant. For sanity check purpose, we have also coded the Flicker's method as an alternative point-wise calculation. Our formulation of anisoplanatism is strictly equivalent from the Flicker's one in the mathematical point of view; both of them relies on a non-stationarity calculation and include focal anisoplanatism. However we propose a direct derivation of the covariance function~(instead of the structure function as done by Flicker) that relies on optimal routines implementation within the OOMAO framework, that are 40~m class telescopes-compliant in terms of memory and cpu usage.
\subsection{Generalization to laser-based systems}
\label{SS:lgsaniso}
In this section we provide the full expression of anisoplanatic covariance for laser-based systems which includes angular, focal and tilt terms.
We define $\phi_\text{s},\phi_\text{l}$ and $\phi_\text{n}$ as the atmospheric phase in respectively the science, LGS and NGS directions. The anisoplanatic phase $\phi_\Delta$ results from the subtraction of the high-order modes measured from the LGS and the tip-tilt measured on the NGS from the phase in the science direction as
\begin{equation} \label{E:totaniso}
\phi_\Delta = \phi_\text{s} - \boldsymbol{\text{P}}_\text{TTR}.\phi_\text{l} - \boldsymbol{\text{P}}_\text{TT}.\phi_\text{n}
\end{equation}
where $\boldsymbol{\text{P}}_\text{TTR}(\boldsymbol{r})$ is a spatial filter that removes the angle of arrival from LGS measurements. Conversely, $\boldsymbol{\text{P}}_\text{TT}(\boldsymbol{r})$ is the filter that conserves only this angle of arrival. Modal spatial filters are of the kind
\begin{equation}
\boldsymbol{\text{P}}_\text{TTR} = \mathcal{I}_d - \boldsymbol{\text{P}}_\text{TT}
\end{equation}
with $\boldsymbol{\text{P}}_\text{TT} = \boldsymbol{T}\boldsymbol{T}^\dagger$ and $T$ the vector projecting the phase onto tip or tilt modes. Based on properties of $\boldsymbol{\text{P}}_\text{TT}$ and $\boldsymbol{\text{P}}_\text{TTR}$, $\phi_\Delta$ conforms to
\begin{equation}
\begin{aligned}
\phi_\Delta &= \boldsymbol{\text{P}}_\text{TTR}\para{\phi_\text{s} - \phi_\text{l}} + \boldsymbol{\text{P}}_\text{TT}\para{\phi_\text{s}-\phi_\text{n}}\\
& = \boldsymbol{\text{P}}_\text{TTR}\phi_{\Delta_\text{sl}} + \boldsymbol{\text{P}}_\text{TT}\phi_{\Delta_\text{sn}}
\end{aligned}
\label{E:phiDelta_lgs}
\end{equation}
where a first term $\boldsymbol{\text{P}}_\text{TTR}\phi_{\Delta_\text{sl}}$ relates to the anisoplanatism, both angular and focal whereas the second term $\boldsymbol{\text{P}}_\text{TT}\phi_{\Delta_\text{sn}}$ represents tilt anisoplanatism. From Eq.~\ref{E:phiDelta_lgs}, the anisoplanatic covariance $\cov{\Delta}$ in Eq.~\ref{E:cphi0} becomes
\begin{equation}
\begin{aligned}
C_{\Delta}(\boldsymbol{r},\rhovec) & = \boldsymbol{\text{P}}_\text{TTR}\aver{\phi_{\Delta_\text{sl}}(\boldsymbol{r})\phi^t_{\Delta_\text{sl}}(\boldsymbol{r}+\rhovec)}\boldsymbol{\text{P}}_\text{TTR}^t\\
& + \boldsymbol{\text{P}}_\text{TT}\aver{\phi_{\Delta_\text{sn}}(\boldsymbol{r})\phi^t_{\Delta_\text{sn}}(\boldsymbol{r}+\rhovec)}\boldsymbol{\text{P}}_\text{TT}^t\\
& + \boldsymbol{\text{P}}_\text{TT}\aver{\phi_{\Delta_\text{sn}}(\boldsymbol{r})\phi^t_{\Delta_\text{sl}}(\boldsymbol{r}+\rhovec)}\boldsymbol{\text{P}}_\text{TTR}^t\\
& + \boldsymbol{\text{P}}_\text{TTR}\aver{\phi_{\Delta_\text{sl}}(\boldsymbol{r})\phi^t_{\Delta_\text{sn}}(\boldsymbol{r}+\rhovec)}\boldsymbol{\text{P}}_\text{TT},
\end{aligned}
\label{E:cov2}
\end{equation}
which turns into
\begin{equation}
\begin{aligned}
C_{\Delta}(\boldsymbol{r},\rhovec) & = \boldsymbol{\text{P}}_\text{TTR}\cov{\Delta_\text{sl}}(\boldsymbol{r},\rhovec)\boldsymbol{\text{P}}_\text{TTR}^t\\
&+ \boldsymbol{\text{P}}_\text{TT}\cov{\Delta_\text{sn}}(\boldsymbol{r},\rhovec)\boldsymbol{\text{P}}_\text{TT}^t \\
& + \cov{\text{cross}}(\boldsymbol{r},\rhovec) + \cov{\text{cross}}^t(\boldsymbol{r},\rhovec),
\end{aligned}
\label{E:cov3}
\end{equation}
where covariance terms $\cov{\Delta_\text{sl}}$ and $\cov{\Delta_\text{sn}}$ are respectively focal-angular and tip-tilt anisoplanatism that are derived independently using the formalism introduced in the previous section. Eq.~\ref{E:cov3} introduces cross-correlation on high-order and tip-tilt modes. From Eqs.~\ref{E:cov2} and~\ref{E:phiDelta_lgs}, this cross term writes
\begin{equation}
\begin{aligned}
\cov{\text{cross}}(\boldsymbol{r},\rhovec) &= \boldsymbol{\text{P}}_\text{TT}\aver{(\phi_\text{s}(\boldsymbol{r}) - \phi_\text{n}(\boldsymbol{r}))(\phi_\text{s}(\boldsymbol{r}+\rhovec)-\phi_\text{l}(\boldsymbol{r}+\rhovec))^t}\boldsymbol{\text{P}}_\text{TTR}^t\\
& = \boldsymbol{\text{P}}_\text{TT}\left(\aver{\phi_\text{s}(\boldsymbol{r})\phi_\text{s}(\boldsymbol{r}+\rhovec)^t} + \aver{\phi_\text{n}(\boldsymbol{r})\phi_\text{l}^t(\boldsymbol{r}+\rhovec)}\right.\\
& - \left.\aver{\phi_\text{s}(\boldsymbol{r})\phi_\text{l}^t(\boldsymbol{r}+\rhovec)} + \aver{\phi_\text{n}(\boldsymbol{r})\phi_\text{s}^t(\boldsymbol{r}+\rhovec)}\right).\boldsymbol{\text{P}}_\text{TTR}^t,
\end{aligned}
\end{equation}
and includes the tip-tilt/high order modes correlation. These cross-terms are generally neglected in the literature on account of its small amplitude when comparing with other terms; we propose to confirm this assumption using full physical-optics simulations in Sect.~\ref{SS:lgscase}. We now introduce the terminology \emph{total anisoplanatism} when including all these cross-terms as in Eq.~\ref{E:cov3}, compared to the {split anisoplanatism} that considers $\cov{\text{cross}} = 0$ in Eq.~\ref{E:cov3}.
\subsection{Spatial filtering}
The number of phase samples $N$ must be chosen wisely; large values will make the calculation time-consuming, while small values will make us loose in accuracy. In terms of spatial frequencies, $N$ phase samples allows to derive the PSF within a $N/2\times \lambda/D$-wide 2D region. Besides, the AO correction band breaks at $n_\text{act}/2\times \lambda/D$, with $n_\text{act}$ the linear number of DM actuators, where the anisoplanatism occurs actually. As a consequence, it is enough to define the $\cov{\Delta}$ as a $n_\text{act}\times n_\text{act}$ matrix to represent accurately the anisoplanatism in the AO correction band. However numerical simulations require to sample the phase with a resolution given by at least 2-3 points per $r_0$, which potentially makes $N > n_\text{act}$. We solve this issue by multiplying $\cov{\Delta}$ with a filter matrix $\mathcal{P}_\text{DMR}$ that is designed to filter out uncontrolled DM modes; it sets to zero all spatial frequencies greater than $n_\text{act}/2D$
\begin{equation} \label{E:atf_filt}
\begin{aligned}
\atf{\Delta}(\rhovec/\lambda) = \dfrac{1}{\otf{\text{DL}}(\rhovec/\lambda)}\times\iint_{\mathcal{P}} \mathcal{P}(\boldsymbol{r})\mathcal{P}(\boldsymbol{r}+\rhovec)\times
\exp\para{\mathcal{P}_\text{DMR}\para{\cov{\Delta}(\boldsymbol{r},\rhovec) - \cov{\Delta}(\boldsymbol{r},0)}\mathcal{P}_\text{DMR}^t} \boldsymbol{dr}
\end{aligned}
\end{equation}
where $\mathcal{P}_\text{DMR}$ is a zonal DM filter of size $N\times N$, calculated from the DM influence functions $\boldsymbol{h}$ of size $N^2\times n_\text{act}$ as follows
\begin{equation}
\mathcal{P}_\text{DMR} = \mathcal{I}_d - \boldsymbol{h}\boldsymbol{h}^\dagger
\end{equation}
where $\mathcal{I}_d$ is the $N\times N$ identity matrix. This filter removes spatial frequencies above the DM cut-off frequency that are beyond the correction space spanned by the latter. The removed high spatial frequencies contribute to the fitting error; it manifests itself as the PSF halo and is a function of the atmospheric seeing only.
\section{Analytic models versus physical-optics simulations}
\label{S:comparison}
In this section we compare existing anisoplanatism models in the literature to simulations. In a first step, we aim at checking the point-wise derivation of anisoplanatism using our general formalism, refereed as OOMAO in next results, complies with existing models, as Zernike~\citep{Fusco2000}, Fourier~\citep{VanDam2006} and~\citep{Flicker2008}.
\subsection{Metrics}
\label{S:metrics}
Results presented in the next sections evaluate PSF model deviations to simulation. First insights on PSF are given by morphological scalar values, as the Strehl-ratio and the FWHM. Both these parameters mostly refer to AO performance; we might prefer having a metric that encompasses the entire structure of the PSF, including especially the PSF halo outside the AO-correction band, as the Fraction of Variance Unexplained~(FVU). If $X$ is 2-D image and $\widehat{X}$ the estimation of this, the FVU is defined as\citep{King1983}
\begin{equation}
\text{FVU}_X = \dfrac{\sum_{i,j} \para{X(i,j) - \widehat{X}(i,j)}^2}{\sum_{i,j} \para{X(i,j) - \sum_{i,j}X(i,j)}^2},
\label{E:FVU}
\end{equation}
where $(i,j)$ are respectively the $\xth{i}$ and $\xth{j}$ pixel of the image. The great advantage of such a metric is to yield an overall error on all angular frequencies; all the PSF patterns, as the PSF core and wings, are included into this calculation.
Besides PSF-related metrics, characterising off-axis PSFs is motivated by science exploitation, calling for science-based metrics, such as photometry and astrometry. Contrary to PSF-metrics, science ones must refer to a specific observed object and image processing tools~(deconvolution or model-fitting for instance). We focus on a particular science case of imaging a binary system to measure relative fluxes and astrometry for deriving the binary’s orbit. In this case, the field typically lacks independent PSF stars and the ability to characterize the PSF is of tremendous value. We define a binary model from a reference off-axis PSF and a set of parameters as
\begin{equation}
\begin{aligned}
\mathcal{B}\para{\text{PSF},\Delta F,\Delta \alpha_x,\Delta \alpha_y} =\Delta F\times\para{\text{PSF}\para{\alpha_x,\alpha_y}\ + \text{PSF}\para{\alpha_x + \Delta \alpha_x ,\alpha_y+\Delta \alpha_y}}
\end{aligned}
\end{equation}
where $(\alpha_x,\alpha_y)$ represents the angular separations in the focal plane, $\Delta F$ the relative stars flux and ($\Delta \alpha_y$,$\Delta \alpha_y$) the differential angular offsets. We did not include any source of noise and AO residual in the focal plane to really extract the real impact of anisoplanatism characterization onto our metrics.
We define a reference binary model from the simulated off-axis PSF, sampled at $\lambda/4D$, with $\Delta F^0 = 1$ and a star separation set to $\Delta\alpha_y^0 = \lambda/D$. Accuracy on photometry and astrometry is evaluated by minimizing the following criterion
\begin{equation} \label{E:cost}
\begin{aligned}
\varepsilon^2(\Delta F,\Delta\alpha_x,\Delta\alpha_y) =\norme{\mathcal{B}\para{\text{PSF}_\varepsilon,\Delta F^0,\Delta\alpha_x^0,\Delta\alpha_y^0} - \mathcal{B}\para{\text{P}\widehat{\text{S}}\text{F}_\varepsilon,\Delta F,\Delta \alpha_x,\Delta \alpha_y}}^2_2
\end{aligned}
\end{equation}
where $\norme{\boldsymbol{x}}_2^2$ is the $\mathcal{L}_2$ norm of the vector $\boldsymbol{x}$. We retrieve photometry and astrometry by fitting a synthetic PSF-based binary on the reference model. Because we know the exact binary parameters, we can estimate how we deviate from those regarding the PSF model. Particularly the photometry error is given by
\begin{equation}
\Delta \text{mag} = -2.5\times\log_\text{10}\para{\dfrac{\widehat{\Delta}F}{\Delta F^0}}
\end{equation}
while the astrometry error results from
\begin{equation}
\Delta \alpha = \sqrt{\para{\widehat{\Delta}\alpha_x - \Delta\alpha_x^0}^2 + \para{\widehat{\Delta}\alpha_y -\Delta\alpha_x^0}^2}.
\end{equation}
Photometry and astrometry, as defined in last equations, may be derived differently regarding the science case and the image-processing method. However, gathering science and PSF-related metrics in our analysis will enhance the overall evaluation of anisoplanatism characterization; we will point out the consequence of PSF errors on science images exploitation, that is the real information that matters in the end. Finally, we estimate astrometry accuracy with an infinite signal-to-noise ratio; we consider only PSF morphology impact into this derivation.
\subsection{NGS case}
We have simulated H-band atmospheric phase screens using the simulator OOMAO~\citep{Conan2014OOMAO} to compare all anisoplanatism models described in the previous section to simulations. These latter include only anisoplanatism patterns into the PSF; we do not account for AO-residual in the guide star direction, static patterns ans science camera parasites such as noise. Our goal is strictly dedicated to anisoplanatism determination and impact on the PSF.
We have considered the median $C_n^2(h)$ profile~\citep{Sarazin2013} at Paranal degraded to 7-layers with $r_0 = 16$ cm. See next section for explanation about the choice of a 7-layers based profile. Fig.~\ref{F:atf0} provides a qualitative comparison and illustrates how each approach achieves an accurate model of the ATF within a percent of the full physical-optics simulation. We notice the Zernike approach becomes less accurate when putting the NGS farther away in the field. We may believe that the stationarity assumption causes this effect, as invoked by~\citep{Fusco2000}, but in such a case, we would observe a similar degradation compared to the Fourier approach that relies on the same assumption as well. The main issue here is that the DM spatial filtering is translated into a modal truncation at the radial order $n$ given by $0.3\times(n+1)/D$~\citep{Conan1994}. Zernike modes do not match exactly the DM frequency behaviour; such an approximation in the DM cut-off frequency may introduce slight errors on the ATF, especially when the anisoplanatism level is stronger. The Fourier method is the most accurate for the NGS case in the FVU sense despite the approximation made of infinite pupil translating into potential edge effects.
\begin{figure}
\centering
\includegraphics[height=9cm]{modelComparison_residualAnisoSimuvZonal_NGS40_60p}
\caption{\small Relative errors on ATF azimuthal average deduced from simulation and analytical calculations difference, for a NGS 40"-off. Negative values on residuals corresponds to overestimation of analytical calculation.}
\label{F:atf0}
\end{figure}
Tab.~\ref{T:model} provides quantitative assessment of ATF compared to simulations and highlights each method leads to very similar results. What we see is consistent with Fig.~\ref{F:atf0}: all models match very well to end-to-end simulations, within 0.2\% of FVU on OTF. We also verified that FVU on PSF are systematically within one or two order of magnitude lower than FVU on ATF. It is explained by the PSF derivation that results from the Fourier-transform of the ATF, that is preliminary multiplied by the diffraction-limit OTF, given by Eq.~\ref{E:otfDL}, that filtered out high-angular frequencies. Strehl ratio is estimated within few percent, which is definitely below errors bars obtained on images acquired on-sky, while FWHM is perfectly well estimated whatever the approach. Science metrics reach a level of milli-mag~(0.1\% on photometry) and micro-arcsec~(1\% of pixel-size), that is definitely at several order of magnitude lower than usual estimations~\citep{Turri2017,Ascenso2015}.
\begin{table*}
\centering
\captionof{table}{\small Relative errors obtained on considered PSF metrics by comparing in simulated anisoplanatic PSF to modeled ones. Models refer respectively to OOMAO, Flicker, Zernike and Fourier. Photometric errors are given in H-band milli-mag and astrometry in $\mu$as. Zero values corresponds to machine precision. Strehl-ratio and FWHM given on table header are values extracted out the simulation.}
\small
\begin{tabular}{|c|c|c|c|c||c|c|c|c||c|c|c|c|}
\hline
NGS location & \multicolumn{4}{c|}{{10"}} & \multicolumn{4}{c|}{{20"} } & \multicolumn{4}{c|}{{40"}} \\
\hline
Strehl-ratio [\%] & \multicolumn{4}{c|}{54} & \multicolumn{4}{c|}{19} & \multicolumn{4}{c|}{5}\\
\hline
FWHM [mas] &\multicolumn{4}{c|}{80} & \multicolumn{4}{c|}{90} & \multicolumn{4}{c|}{260} \\
\hline
& \multicolumn{12}{c|}{Relative residual errors [\%]} \\
\hline
Model & O & F & Ze & Fo & O & F & Ze& Fo& O& F& Ze&Fo \\
\hline
SR &3.8&4.0&1.1 &2.3 & 3.4& 3.9&1.7 &1.6 & 2.7& 3.3& 3.5&0.6 \\
\hline
FWHM & 0& 0& 0.1 &0.1 & 0.04& 0.04 & 0.2& 0.6& 0.4& 0.06& 1.0&1.5 \\
\hline
FVU$_\text{OTF}$ & 0.21 &0.23 & 0.02 & 0.08 & 0.15&0.18 & 0.04& 0.05& 0.08& 0.1& 0.08& 0.03\\
\hline
& \multicolumn{12}{c|}{Science estimates} \\
\hline
$\Delta$mag & 25 &29 &2 &13 & 22& 28 &4 & 6& 15&20 &20 &5 \\
\hline
$\Delta \alpha $ & 2 & 9 & 37 &28 &13 &49 &56 &50 & 50& 10& 14&70 \\
\hline
\end{tabular}
\label{T:model}
\end{table*}
As a conclusion, our point-wise method matches accurately existing angular anisoplanatism models and simulations in the literature within 1\%-level differences.
\subsection{LGS case}
\label{SS:lgscase}
We now focus on anisoplanatism on LGS-based systems; we have simulated the total anisoplanatism from Eq.~\ref{E:totaniso}, with one LGS and one NGS distributed along a L-shaped asterism while keeping the science on-axis. We have estimated the anisoplanatic covariance given in Eq.~\ref{E:cov2} and the resulting ATF using Eq.~\ref{E:atf}. Furthermore, we have also derived independently focal and tilt anisoplanatic covariance matrices $\cov{\Delta_{sl}}$ and $\cov{\Delta_{sn}}$ in Eq.~\ref{E:cov3}, using both simulations and analytic calculations. The goal of the analysis is two-fold: firstly demonstrate the generalized analytic model matches simulations including all anisoplanatism terms, secondly confirming high order/tip-tilt cross terms that appear in Eq.~\ref{E:cov2} are not determinant in the anisoplanatism characterization.
Fig~\ref{F:atfs} provides a comparison of ATFs maps for a 20"-off LGS and 40"-off NGS from on-axis in perpendicular directions. It highlights clearly that analytic calculations fit very well simulations results to within two percent points of accuracy. On top of that, residual errors are mostly located at the map border that corresponds to pupil edges. As previously, FVU on PSF reach two order of magnitude lower level compared to FVU on OTF. When extrapolating the on-axis OTF, we will multiply these angular frequencies above $D/\lambda$ to zero. Fig~\ref{F:splitVtotal} illustrate cuts of ATF and residuals along the elongated direction. We observe that analytic calculations reproduce very well simulations within 2\% in maximal range and 0.1\% in FVU when looking at Tab.~\ref{T:simuvzonal}.
\begin{figure}
\centering
\includegraphics[height=9cm]{atfMapComparison_LGS20_NGS40_fullp}
\captionof{figure}{\small Comparison of ATF maps derived either from simulation and analytical calculations. Figure distinguishes total and split anisoplanatism that respectively do and do not include tip-tilt/high order modes cross-correlation as discussed in Sect.~\ref{SS:lgsaniso}. ATF maps are derived for a LGS and NGS 40"-off respectively to the north and east. }
\label{F:atfs}
\end{figure}
\begin{figure}
\centering
\includegraphics[height=9cm]{residualAnisoSimuvZonal_LGS20_NGS40_fullp}
\caption{\small Relative errors on ATF azimuthal average deduced from simulation and analytical calculations difference for an LGS at 20" and NGS at 40" separation respectively to the north and east.}
\label{F:splitVtotal}
\end{figure}
\begin{table}
\centering
\captionof{table}{\small Relative errors obtained on considered PSF metrics by comparing in simulated PSFs to analytical ones considering split anisoplanatism. Sources were distributed along a L-shape constellation with the science target on-axis. Photometric errors are given in H-band milli-mag and astrometry in $\mu$as. Zero values corresponds to machine precision. Strehl-ratio and FWHM given on table header are values extracted out the simulation.}
\small
\begin{tabular}{|c||c|c|c||c|c|c|}
\hline
LGS location & \multicolumn{3}{c||}{0"} & \multicolumn{3}{c|}{20"} \\
\hline
NGS location & 0" & 20" & 40" & 0" & 20" & 40" \\
\hline
Strehl-ratio [\%] & 64 &56 & 41 & 22 & 19 & 15\\
\hline
FWHM [mas] & 79 &84 & 97 & 86 &92 & 108\\
\hline
& \multicolumn{6}{c|}{Relative residual errors [\%]} \\
\hline
SR& 2.2 & 2.0 & 2.0 & 2.5 & 2.03 & 2.2 \\
\hline
FWHM & 0 & 0.31&0.35& 0 & 0.23 & 1.3 \\
\hline
FVU$_\text{OTF}$ & 0.07 & 0.06 & 0.06 & 0.10 &0.08 & 0.08 \\
\hline
& \multicolumn{6}{c|}{Science estimates} \\
\hline
$\Delta$mag & 6.3 & 3.1 & 4.8 & 8.6 & 1.8 & 6.0 \\
\hline
$\Delta \alpha $ & 7.7 & 41.4 & 28.2 & 67.6 & 60.5 & 77.6 \\
\hline
\end{tabular}
\label{T:simuvzonal}
\end{table}
We went through systematic comparisons over different asterism geometries to investigate whether cross terms are always negligible. Tab.~\ref{T:totvsplit} highlights that cross-terms independence assumption is affecting the PSF within extremely low differences, lower than model deviation from simulations. As a sub-product of our analysis, we confirm the LGS-based anisoplanatism is accurately represented by two independent terms, as angular+focal terms and tilt anisoplanatism.
\begin{table}
\centering
\captionof{table}{\small Relative errors obtained on considered PSF metrics by comparing in simulation PSFs produced by the total and split anisoplanatism. Sources were distributed along a L-shape constellation with the science target on-axis. Photometric errors are given in H-band milli-mag and astrometry in $\mu$as. Zero values corresponds to machine precision.}
\small
\begin{tabular}{|c||c|c|c||c|c|c|}
\hline
LGS location & \multicolumn{3}{c||}{0"} & \multicolumn{3}{c|}{20"} \\
\hline
NGS location & 0" & 20" & 40" & 0"& 20"& 40" \\
\hline
& \multicolumn{6}{c|}{Relative residual errors [\%]} \\
\hline
SR& 0 & 3e-2 & 3e-2 & 0 & 2e-2 & 1e-2 \\
\hline
FWHM & 0 & 0.35& 0.39& 0.22 & 0.39 & 1.39 \\
\hline
FVU$_\text{OTF}$ & 0 & 1e-4& 5e-4& 0 &2e-3 & 4e-3 \\
\hline
& \multicolumn{6}{c|}{Science estimates} \\
\hline
$\Delta$mag &0&0.30 &0.21 &2e-3 & 1.05 & 1.5 \\
\hline
$\Delta \alpha $ & 0& 2.0 & 16.7& 0.05&8.1 &36.0 \\
\hline
\end{tabular}
\label{T:totvsplit}
\end{table}
\section{Off-axis PSF sensitivity to $\boldsymbol{C_n^2(h)}$ profile accuracy}
\label{S:sensitivity}
In this section we quantify the impact of $C_n^2(h)$ profile knowledge on PSF morphological and science metrics.
We split our study into two complementary analyses. On a first step, we introduce a bias on the $C_n^2(h)$ estimation by binning down the number of layers. On a second step, we keep the same number of bins, but we introduce random variations on both heights and weights.
\subsection{Impact of number of bins}
We have initially considered a 35-layers $C_n^2(h)$ profile~\citep{Sarazin2013} as our reference to compute the PSF at different separations ($\theta_0/2$, $\theta_0$, $1.5\times\theta_0$ and $2\times\theta_0$) with $\theta_0$ = 24.5" in H-band.
We have opted for the mean-weighted compression~\citep{Robert2010} method for reducing the problem from 15 down to 2 layers. At each iteration, we retrieve the anisoplanatic PSF to compare to the reference full profile PSF. Other binning options could be considered instead~\citep{Saxenhuber2017}. Keeping the angular coherence angle constant across profiles seems to us a sensible choice as we look particularly into anisoplanatic effects.
We report on Figs.~\ref{F:fvuVnlayer},~\ref{F:srfwhmVnLayer} and~\ref{F:photoastroVnLayer} the FVU and accuracy on Strehl-ratio, FWHM, photometry and astrometry as function of the number of modelled layers for a median profile at Paranal and different off-axis position in the field~($0.5\times\theta_0$,$\theta_0$,$1.5\times\theta_0$ and $2\times\theta_0$). Curves envelopes are deduced from quartiles profiles~\citep{Sarazin2013}.
An immediate observation is that 15 layers instead of 35 can be used with little impact on the PSFs. If 1\% errors are allowed, at least 7 layers are required. For such a profile, photometry errors are at the level of 3\%-level while astrometry is given at 5\% of pixel size level in the worst case. External profilers commonly deliver profiles with such a number of layers; these results suggest that model-dependent errors based on that input are of the percent level.
This is an important take home message for anisoplanatism characterization and PSF-R on 10~m class telescopes: for a field roughly given by $\theta_0$, we only need a representation of the $C_n^2(h)$ over 7-layers to model anisoplanatism signature onto PSF at 1\%-level.
\begin{figure}
\centering
\includegraphics[height=9cm]{fvuVnLayer_Hband}
\captionof{figure}{\small Fraction of variance unexplained as function of the number of reconstructed layer.}
\label{F:fvuVnlayer}
\end{figure}
\begin{figure}
\centering
\includegraphics[height=8cm]{SRaccuracyVnLayer_Hband}
\includegraphics[height=8cm]{FWHMaccuracyVnLayer_Hband}
\captionof{figure}{\small \textbf{Left~:} H-band long-exposure Strehl ratio \textbf{Right~:} FWHM accuracy versus the number of reconstructed layer.}
\label{F:srfwhmVnLayer}
\end{figure}
\begin{figure}
\centering
\includegraphics[height=8cm]{PhotoaccuracyVnLayer_Hband}
\includegraphics[height=8.25cm]{AstroaccuracyVnLayer_Hband}
\captionof{figure}{\small \textbf{Left~:} H-band photometry \textbf{Right~:} Astrometry accuracy versus the number of reconstructed layer.}
\label{F:photoastroVnLayer}
\end{figure}
\subsection{Impact of heights and weights precision}
We consider the 7-layer equivalent profile at Paranal~\citep{Sarazin2013} as the new reference for this section; our purpose is to evaluate how much the precision on both weights and heights of turbulent layers impact the PSF. We denote $h_l^0$ and $w_l^0$ respectively to present the $l^\text{th}$ layer heights and weights values of reference.
We apply a random variation on each layer, either on its weight $w_l$ or height $h_l$. We define for each of those layers a zero-mean Gaussian statistical variable - $\eta_h$ and $\eta_w$ -, by setting its standard-deviation to unity. We then define $\sigma_h$ and $\sigma_w$ as $p$-size vectors such as
\begin{equation}
\begin{aligned}
&h_l(p,k) = h_l^0 + \sigma_h(p)\times\eta_h(l,k)\\
& w_l(p,k) = w_l^0\times\para{1 + \sigma_w\times\eta_w(l,k)}
\end{aligned}
\end{equation}
where $k$ refers to the random selection and $p$ to the value of deviation introduced. For height sensitivity, we allow up to 1km of variability, while weights are de-tuned by up to 30\%. For each iteration $k$, we define a new 7-layer atmosphere and get the PSF at any separation starting from the PSF on-axis as described earlier. We finally compute metrics as a function of $\sigma_h$ and $\sigma_w$, the separation $\theta$ and the random selection $k$. Metrics are averaged out over 1000 realizations providing error bars as well. The choice of 1000 iterations is a compromise between the computation time and relevance of results.
Figs.~\ref{F:srfwhmVweight},~\ref{F:photoastroVweight},~\ref{F:srfwhmVheight} and~\ref{F:photoastroVheight} provide curves of mean values at multiple separations versus weights and heights precision given by $\sigma_p$. Also Fig.~\ref{F:fvuVweight} illustrates FVU errors regarding accuracy on height and weight. Curves envelopes are not represented, but range as similarly as they do in Figs.~\ref{F:fvuVnlayer},~\ref{F:srfwhmVnLayer} and~\ref{F:photoastroVnLayer} for a 7-layers profile.
Globally, PSF metrics deviate monotonically with respect to inputs precision level, with a speed that grows with the angular separation from the guide star. It is an expected results: the PSF model differs more largely when introducing more errors on inputs and for stronger anisoplanatism cases.
Strehl ratio is known to follow a $\exp(\theta_0^{-5/3})$ law while FWHM is proportional to $\theta_0^{-5/3}$. This latter is given by $r0.(\sum(h^{5/3}.C_n^2(h))^{3/5}$ that makes FWHM proportional to $r_0^{-5/3}.\sum(h^{5/3}.C_n^2(h))$ and SR proportional to $1+ (r0^{-5/3}.\sum(h^{5/3}.C_n^2(h)))$ for small amount of variations on inputs. Weights as introduced in Fig.~\ref{F:fvuVweight} scale with $r0^{-5/3}$; we directly see why FWHM/SR are supposed to be linear regarding the weight. About the altitude, both FWHM and SR should not be linear in h, but because the exponential term involved in the Strehl expression, the non-linear regime on the Strehl appears after 1km of altitude precision, while we do not have this mitigation effect of the non-linear component on the FWHM that follows directly a 5/3 power law on altitude.
Strehl and FWHM are estimated within 1\% accuracy as long as we ensure a precision of 10\% on weights and roughly 200~m on heights. This latter means that heights of the retrieved 7-layers profile must be measured within 200~m precision at 1 $\sigma$. Astrometry is more critical regarding the weight precision that must be ensured to a 7\%-level to get an astrometry of 10\% of the pixel scale. Also, photometry is only impacted by height precision that also must provided within 200~m to get 1\% level of photometry. Such a level of accuracy on $C_n^2(h)$ weights is accessible from external profilers~\citep{Butterley2006}, but altitude resolution reaches generally 500~m up to 1~km, leading to an accuracy of 5 up to 10\% on PSF estimates, which can be still acceptable for some science cases that are noise-limited for instance.
Fig.~\ref{F:srfwhmVweight} highlights the error introduced on weights does not degrade the PSF linearly with the separation. Errors at $2\times\theta_0$ are lower compared to $1.5\times\theta_0$. There is a physical explanation: at such a separation, the phase is largely decorrelated. Eq.\ref{E:covDelta_zonal} involves both the phase auto-covariance and cross-covariance terms. The latter converge towards zero when the separation goes to infinity. However these are the terms that carry the sensitivity to the fractional weights of the $C_n^2(h)$ profile, which means the off-axis PSF is less and less sensitive to weights precision for increasing separations.
How do our results translate to ELT-scales? Anisoplanatism variance degrades with $(\theta/\theta_0)^{5/3}$, where $\theta_0$ is only an atmosphere-dependent variable; it therefore seems to us the findings ought to be conserved at an ELT scale, in a way we would need 10ish layers to describe the anisoplanatism. However our analysis is focused on the anisoplanatism only and does not include any tomography or multi-laser configuration. On top of that, others metrics may be more relevant regarding the science case. As a conclusion, we have an order of magnitude for the number of layers of about 10 for anisoplanatism characterisation only.
\begin{figure}
\centering
\includegraphics[height=9cm]{FVUaccuracyVdH_Hband}
\includegraphics[height=9cm]{FVUaccuracyVdW_Hband}
\captionof{figure}{\small FVU as function of \textbf{Top:} height accuracy \textbf{Bottom:} weight accuracy.}
\label{F:fvuVweight}
\end{figure}
\begin{figure}
\centering
\includegraphics[height=8cm]{SRaccuracyVdW_Hband}
\includegraphics[height=8cm]{FWHMaccuracyVdW_Hband}
\captionof{figure}{\small \textbf{Top:} Strehl ratio \textbf{Bottom:} FWHM accuracy versus accuracy on weights estimation in H-band.}
\label{F:srfwhmVweight}
\end{figure}
\begin{figure}
\centering
\includegraphics[height=8cm]{SRaccuracyVdH_Hband}
\includegraphics[height=8cm]{FWHMaccuracyVdH_Hband}
\captionof{figure}{\small \textbf{Top:} Strehl ratio \textbf{Bottom:} FWHM accuracy versus precision on heights estimation in H-band.}
\label{F:srfwhmVheight}
\end{figure}
\begin{figure}
\centering
\includegraphics[height=7.75cm]{PhotoaccuracyVdW_Hband}
\includegraphics[height=8cm]{AstroaccuracyVdW_Hband}
\captionof{figure}{\small \textbf{Top:} Photometry \textbf{Bottom:} astrometry accuracy versus precision on weights estimation in H-band.}
\label{F:photoastroVweight}
\end{figure}
\begin{figure}
\centering
\includegraphics[height=8.75cm]{PhotoaccuracyVdH_Hband}
\includegraphics[height=9cm]{AstroaccuracyVdH_Hband}
\captionof{figure}{\small \textbf{Top:} Photometry \textbf{Bottom:} astrometry accuracy versus precision on heights estimation in H-band.}
\label{F:photoastroVheight}
\end{figure}
\subsection{Metrics correlation}
We have gathered all metrics values from previous analyses and compared to each other in order to track correlations. We ended with
the following regression relationship:
\begin{equation}
\begin{aligned}
&\Delta \text{mag [mag]} = 0.0067 \pm 0.00052 \times \Delta \text{SR [\%]}\\
&\Delta \text{ast [mas]} = 0.25 \pm 0.024 \times \Delta \text{FWHM [\%]}
\end{aligned}
\end{equation}
that allows to match accurately observations as illustrated in Fig.~\ref{F:correl}.
\begin{figure}
\centering
\includegraphics[height=8.75cm]{dmagVsdSR.pdf}
\includegraphics[height=8.75cm]{dastVsdFWHM.pdf}
\caption{\textbf{Up:} Photometry accuracy versus Strehl ratio accuracy \textbf{Down:} Astrometry accuracy versus FWHM accuracy}
\label{F:correl}
\end{figure}
We have noticed a quadratic dependency of the FVU with the photometry~(and so Strehl-ratio as well) accuracy; for accurate photometric measurements~($\Delta$mag< 5\%), we have FVU$~=5.4\pm 0.6\times\Delta$mag,while for less accurate measurements, we get FVU$~=24\pm 3\times\Delta$mag. We also observed a clear correlation trend between FVU and astrometry accuracy values, given by FVU$~=1.2\pm 0.2\times\Delta$ast, although we observed a large discrepancy of samples around the linear regression. In summary, FVU is an efficient metric to characterize the PSF; it defines a comprehensive scale value that depends on both PSF-related parameters and key science observables.
\section{Towards ELTs~: sensitivity analysis validated using HeNOS}
\label{S:HENOS}
HeNOS (Herzberg NFIRAOS Optical Simulator) is a MCAO test bench designed to be a scaled down version of NFIRAOS,
the first light adaptive optics system for the Thirty Meter Telescope~\citep{Mieda2018,Rosensteiner2016}. We used HeNOS in single-conjugated mode with in closing the loop on ones of the 4 LGSs distributed over a 4.5"-side length square constellation, while atmosphere is created using three phase screens. Summary of main parameters is given in Tab.~\ref{T:setup}. To simulate the expected PSF degradation across the field on NFIRAOS at TMT, all altitudes are stretched up by a factor 11. Moreover, at the time we acquired HeNOS data, science camera was conjugated at the LGSs altitude; LGSs beams are propagating along a cone but are arriving in-focus at the science camera entrance.
\begin{table}
\centering
\begin{tabular}{l|l}
Asterism side length & 4.5"\\
Sources wavelength & 670 nm \\
$r_0$ (670 nm) & 0.751 \\
$\theta_0$ (670 nm) & 0.854" \\
fractional $r_0$ & 74.3\% ,17.4\%,8.2\% \\
altitude layer & (0.6, 5.2, 16.3) km\\
source height & 98.5 km \\
Telescope diameter & 8.13~m\\
DM actuator pitch & 0.813~m\\
\end{tabular}
\caption{HeNOS set up summary.}
\label{T:setup}
\end{table}
We did acquire closed-loop data when guiding on a LGS in single-conjugated mode in July 2017. Two data sets have been acquired respectively with and without phase screens in the LGSs beams. This second measurement allows to measure the best Non common path aberrations~(NCPA)-limited PSFs over all directions. See~\citep{Lamb2016} about phase diversity/focal plane sharpening methods deployed on HeNOS to calibrate NCPA.
Our purpose is to demonstrate the anisoplanatism model we developed allows for a good representation of PSF characteristics within the expected accuracy given by the sensitivity analysis. We derive off-axis PSF from the PSF in the guide-star direction by the use of Eq.~\ref{E:otf2t1}. Although off-axis PSFs are largely dominated by anisoplanatism, static aberrations that vary across the field must be taken into account to improve the PSF modeling.
We have compensated for on-axis NCPA and put back off-axis static aberrations for each individual direction by convolving PSFs with best NCPA-limited PSF. These PSF includes both NCPA calibration and field static aberrations residual in off-axis direction. OTF in the off-axis direction $\theta$ is thus yielded by the following calculation
\begin{equation} \label{E:otfHenos}
\otf{}(\theta) = \otf{}(0) \times \dfrac{\otf{\text{stat}}(\theta)}{\otf{\text{stat}}(0)} \times\text{ATF}_\Delta(\theta)
\end{equation}
where $\otf{\text{stat}}(\theta)$ is derived from observations without phase screens in direction $\theta$ while $\otf{}(0)$ is the on-axis OTF during AO operation on phase screens.
Because all sources are focused at the same height, we only need to consider angular anisoplanatism to extrapolate the on-axis PSF to any other direction. Fig.~\ref{F:henosPSFs} provides focal-plane images acquired in visible~(670 nm) whilst closing the AO loop on LGS 1 (top-left PSF). Anisoplanatism on LGS 2 (bottom-right), 3 (bottom left) and 4 (top right) is clearly visible and produces a strong PSF elongation in the guide star direction. We also illustrate off-axis PSFs modelled derived from Eq.~\ref{E:otfHenos}. At a glance we are capable of reproducing the good shape and elongation of off-axis PSFs. We report in Tab.~\ref{T:henosStats} a quantification of PSF characteristics measured versus estimated, that shows that we get 10\%-level of accuracy on PSF metrics and reach 5\% of fraction of variance unexplained on all PSFs.
\begin{figure*}
\centering
\includegraphics[width=12cm]{henosPSFs}\\
\includegraphics[width=12cm]{henosPSFs_predicted}\\
\hspace{.1cm}\includegraphics[width=12cm]{henosPSFs_residual}
\captionof{figure}{\textbf{Up~:} 670 nm HeNOS PSFs while running the AO loop on LGS 1 \textbf{Middle~:} PSFs predicted from on-axis PSF and anisoplanatism \textbf{Down~:} Residual on the prediction.}
\label{F:henosPSFs}
\end{figure*}
\begin{table}
\centering
\small
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
& \multicolumn{3}{c|}{Bench} &\multicolumn{3}{c|}{Predicted}\\
\hline
\# LGS & 2 & 3 & 4 &2 &3 &4\\
\hline
SR[\%] & 4.1& 6.0&5.0 &3.2&4.2 &4.2\\
\hline
FWHM [mas] &117 & 90 &100 & 105&84 &82\\
\hline
Aspect ratio & 1.41& 1.27 & 1.42& 1.41&1.34 &1.33\\
\hline
EE [\%] & 53& 56 & 64&65 &67 &64\\
\hline
FVU$_\text{PSF}$ [\%] & x & x & x & 5.8 & 4.1 & 6.0\\
\hline
\end{tabular}
\caption{PSF characteristics measured on laboratory PSF compared to predict values from on-axis PSF and anisoplanatism. EE is taken at $10\lambda/D$ and FVU values are estimated using Eq.~\ref{E:FVU} is derived using 60 pixels.}
\label{T:henosStats}
\end{table}
Residuals are mostly composed by a combination of high spatial frequencies patterns and a large structure oriented towards the guide star direction. The first component may be introduced by residual speckles and static aberrations, while the second feature suggests the anisoplanatism effect is not perfectly well characterized. It echoes the previous discussion on sensitivity: the parameters in Tab.~\ref{T:setup} we have considered as inputs of our model have been measured with a certain accuracy. Weights are estimated within 10\% while heights are retrieved every 1~km.
To confirm whether such a level of accuracy may explain bias on PSF estimates, we have deliberately introduced a random error on $C_n^2(h)$ following the methodology presented in Sect.~\ref{S:sensitivity}. We illustrate in Fig.~\ref{F:FWHMhenosvdH} how the off-axis PSF FWHM vary regarding the layer height precision. At this range of separations~(4.5" compared to 0.854" for $\theta_0$), the anisoplanatism characterization does not depend on the weight precision as discussed earlier and it has been confirmed for HeNOS. In Fig.~\ref{F:FWHMhenosvdH}, we highlight that the relative error on FWHM, averaged out the three LGS, reaches zero for an altitude precision of 750~m. From~\citep{Rosensteiner2016}, altitude have been measured at every 1~km, which comply with our results. It does not mean we will get a better PSF characterization by shifting all layers by this quantity; instead it confirms the precision on inputs parameters translate into accuracy on PSF-metrics. In next works, we will particularly investigate for inverting the problem, in order to retrieve the $C_n^2(h)$ profile from a collection of observed off-axis PSFs.
\begin{figure}
\centering
\includegraphics[height=8cm]{FWHMvsdeltaH}
\caption{\small PSF FWHM predicted versus the absolute precision in altitude layers. Plots represent FWHM variations on each individual off-axis PSF as the overall mean FWHM over PSFs as well.}
\label{F:FWHMhenosvdH}
\end{figure}
\section{Conclusions}
\label{S:conclusions}
We carried out developments to characterise anisoplanatism in LGS (and NGS) systems in order to estimate off-axis PSFs using a point-wise method that is accurate, numerically efficient and can potentially be used with 40~m class telescopes featuring a large number of degrees of freedom. For a 7-layers median profile at Paranal, we have demonstrated our modelling complies with physical-optics simulations to within 0.1\% on the fraction of variance unexplained, together with 1\% on PSF-related metrics, as Strehl ratio and FWHM, with a LGS and a NGS at respectively 20" and 40" off-axis. Stars flux and position are conserved respectively at a mill-mag and $\mu$arcsec-level. Finally, the total anisoplanatism with LGS is accurately split into a focal+angular and a tilt component.
With the purpose of determining how $C_n^2(h)$ knowledge constrains off-axis PSFs, we have investigated how input parameter errors on the number of layers, its heights and weights translate into PSF errors. Firstly, we have highlighted for 10~m class telescopes that percent-level accuracy on SR, FWHM and FVU can be met with seven turbulent layers. On tight binaries, photometry can be retrieved at 3\% while astrometry stays within 5\% of pixel size. If we may admit 10\% of accuracy on PSF metrics, $C_n^2(h)$ must be provided within 500 m accuracy on altitude height. Those results can be extended to ELT scale, with a conclusion that we need roughly ten layers to characterize at 1\%-level a PSF on a 40~m class telescope.
We have finally validated this methodology on the HeNOS testbed where $C_n^2(h)$ values and its accuracy are known; By introducing random errors on anisoplanatism model input parameters, we have improved the off-axis PSF FWHM estimation. It demonstrates the precision in those parameters measurements by explaining the bias we have observed on PSF off-axis PSF modelling.
Next works will aim at applying our analytical formulation to real on-sky observations of crowded fields at Keck for improving the PSF characterisation and the estimation of key science parameters. Moreover we are also focusing in inverting the problem in order to retrieve the $C_n^2(h)$ profile from a collection of observed off-axis PSFs. Such a technique is investigated for providing an accurate reconstructed PSF for single-conjugated AO systems for which $C_n^2(h)$ profile is not measurable from the AO control loop data.
\section*{Acknowledgments}
This work was supported by the A*MIDEX project (no. ANR-11-IDEX-0001-02) funded by the "Investissements d'Avenir" French Government programme, managed by the French National Research Agency (ANR).
\bibliographystyle{plain}
|
{
"timestamp": "2018-04-17T02:05:38",
"yymm": "1804",
"arxiv_id": "1804.05198",
"language": "en",
"url": "https://arxiv.org/abs/1804.05198"
}
|
\subsection{Deep learning for Person Re-ID}
Deep learning based method has dominated in Re-ID community \cite{zheng2016person}. Yi~\cite{yi2014deep} first employed deep neural network to determine if a pair of input images belong to the same ID. In general, two types of models are used for person re-identification: verification and identification model.
For the verification model, they adopt siamese neural network or triplet loss to pull the pair of images with same identity and push away that with different identity~\cite{ahmed2015improved,hermans2017defense,ding2015deep,chen2017multi,li2014deepreid}. In~\cite{hermans2017defense}, Hermans~\emph{et al.}. proposed a variant of triplet loss to perform end-to-end deep metric learning, which can outperforms many other published methods by a large margin. However, generally, this kind of model may have a compromised efficiency on large gallery. This is because it does not make full use of Re-ID annotations.
For the identification model~\cite{xiao2016learning,zheng2017discriminatively,sun2017beyond}, it tries to learn a discriminative representation of given input image and it always yields superior accuracy compared with verification model. Xiao~\emph{et al.}~\cite{xiao2016learning} propose a novel dropout strategy to train a classification model with multiple datasets jointly. In~\cite{zheng2017discriminatively}, the verification and classification losses are combined together to learn a discriminative embedding and a similarity measurement at the same time. In~\cite{sun2017beyond}, a Part-based Convolutional network is proposed to learn discriminative part-informed features.
\subsection{Part-based Model}
Recently, many works generate deep representation from local parts for fine-grained discriminative features of person. This kind of part-based model can be divided into three categories. First one is based on some prior knowledge like pose estimation and landmark detection~\cite{zheng2017pose,su2017pose,wei2017glad}. These methods share a same drawback that is the gap lying between datasets for pose estimation and person retrieval. Second, several other works abandon the semantic cues for partition~\cite{yao2017deep,liu2017hydraplus,Zhao2017DeeplyLearnedPR,li2018harmonious}. For example, Yao~\emph{et al.}~\cite{yao2017deep} employed the Part Loss Networks which enforces the deep network to learn representations for different parts and gain the discriminative power on unseen persons. Third, the partition is cropped into pre-defined patches~\cite{sun2017beyond,li2017learning}. Sun\emph{et al.}~\cite{sun2017beyond} proposed Part-based Convolutional Baseline (PCB) to learn discriminative partition features. However, the PCB may suffer some outliers, which make the inconsistency in each partition, thus they proposed Refined Part Pooling (RPP) to enhance within-part consistency.
\subsection{Spatial Pyramid Pooling}
Since convolutional neural networks with the fully connectedly layer always require the fixed input size. In order to remove this constrain, He~\emph{et al.}~\cite{he2014spatial} proposed the Spatial Pyramid Pooling network, which is able to generate a fixed length output regardless of the input size and maintain spatial information by pooling in local spatial bins. Multi-level spatial pooling has also shown to be robust to object deformations. It can improve the performance of classification and object detection tasks. Similarly pyramid pooling module is also used in~\cite{zhao2017pyramid}, the pyramid level pooling separates the feature map into different sub-regions and forms pooled representation for different locations.
\begin{figure*}[t]
\centering
\includegraphics[width=1\textwidth]{fig/framework.pdf}
\caption{Overview of the proposed Horizontal Pyramid Matching (HPM) approach. The input image firstly goes through a convolutional neural network to extract its feature maps. Then, the horizontal pyramid pooling is leveraged to producing feature representation of each part using both global average pooling and global max pooling. Finally, prediction of each part is fed into the classifier to conduct partial-level person Re-ID. During the testing stage, we concatenate features of parts at different pyramid scales to form the final feature representation of each image.}
\label{fig:framework}
\vspace{-5mm}
\end{figure*}
\subsection{Horizontal Pyramid Matching}
\noindent {\bf Backbone Network} The HPM can take various network architecture like VGG~\cite{simonyan2014very}, Resnet~\cite{he2016deep} and Google Inception~\cite{szegedy2016rethinking} as the backbone. Our paper choose the Resnet50 as backbone network with some modifications following the previous state-of-the-art~\cite{sun2017beyond}. First, the average pooling layer and the fully connected layer are removed. Also, the stride of the conv4\_{1} is set to 1. As a result, the size extracted feature maps will be $\frac{1}{16}$ of the input image size.
\noindent {\bf Horizontal Pyramid Pooling (HPP) module}
HPP is inspired by Special Pyramid Pooling (SPP)~\cite{he2014spatial}, which is proposed to eliminate uncertain length of feature vectors caused by different input sizes of images. The differences between our HPP module and SPP mainly include two aspects: 1) {\bf motivation: } HPP is designed to learn to enhance the discriminative information of partial person body at various scales, while SPP is to address the issue of inconsistent length of image feature vectors. 2) {\bf operation: } Since the distribution of distinguish partitions of a person is from head to foot, HPP slices the feature maps into multiple scrips in a horizontal manner, which is different from SPP using a 2-D spatial manner. With HPP, we can obtain vectors of fixed length for person parts at different horizontal pyramid scales. These vectors are further fed into one convolutional layer and one fully-connected layer for learning classification. In this way, the discriminative ability of person parts can be captured from global to local, from coarse to fine.
Formally, denote the feature maps extracted by the backbone network as $F$. We adopt 4 pyramid scales within the HPP module and $F$ is sliced into several spatial bins horizontally and equally according to different scales. Specifically,
assume each spatial bin as $F_{i, j}$. $i, j$ stand for the index of scale and the index of bins in each scale. For instance, $F_{3, 4}$ means the fourth bin in third pooling scale. Then, we pool each spatial bin $F_{i, j}$ by global average and max pooling to generate column feature vector, $G_{i, j}$. $$G_{i, j} = avgpool(F_{i, j}) + maxpool(F_{i, j})$$ After that, each $G_{i, j}$ is fed into a convolutional layer to reduce the dimensions from 2048 to 256, denote as $H_{i, j}$. These $H_{i, j}$ with the same i can be considered as a description of the person. This kind of description covers more detailed partial features with the increasing pyramid scales.
\subsection{Loss Function}
We leverage the classification-based model to tackle the person Re-ID task. Therefore, the target is to predict the ID of each person, thus person-specific feature representations can be then learned by the optimized classification model. We use a branch of fully connected layer as the classifier, each feature column vector $H_{i,j}$ is fed into a corresponding classifier $FC_{i,j}$ and following a softmax function to predict its ID. During training, the output of given image $I$ is a set of predictions $\hat{y}_{i, j}$. Each $\hat{y}_{i, j}$ can be formulated as
$$\hat{y}_{i, j} = \argmax_{c\in P} \frac{exp((W_{i, j}^{c})^{T} H_{i, j}(I))}{\sum_{p=1}^{P}exp((W_{i, j}^{p})^{T} H_{i, j}(I))}$$
where the P is the total number of person ID, $W_{i,j}$ is the learnt weights of $FC_{i,j}$, $y$ is the ground truth ID of input image $I$. The loss function is sum of Cross Entropy loss of each output $\hat{y}_{i,j}$.
$$Loss = \sum_{n=1}^{N}\sum_{i,j} CE(\hat{y}_{i, j}^{n}, y^{n})$$
where N is the size of mini-batch, CE is the Cross Entropy loss.
\subsection{Variant of HPM}
HPM may have some variants different from the basic framework describe above, ~\emph{e.g. } different pyramid scales and pooling strategies.
{\bf Number of pyramid scales}
The HPM can have several different number of scales. Instead of the 4 scales, it can be any number up to the $log_{2}(h)$, where $h$ is the height of feature map. The HPM structure with different pyramid scales is shown in Table~\ref{method:t1}.The model will focus on more detailed and fine partitions of the given person with the increasing of pyramid scales. Since our loss function is a linear combination of each pyramid scales, if there are too many pyramid scales, the global information of the person may be underestimated. On the other hand, if too few pyramid scales, the features of local discriminative partition may be more difficult to extract. Thus, choosing a proper pyramid scales that can balance the global and local features is vital for the performance of HPM.
\begin{table}[ht]\setlength{\tabcolsep}{5pt}
\centering
\begin{tabular}{c|l|l}
\hline
\# Pyramid Scale & \# Spatial Bins & Size of Spatial Bins \\ \hline
1 & 1 & 24x8\\
2 & 1, 2 & 24x8, 12 x 8\\
3 & 1, 2, 4 & 24 x 8, 12x8, 6x8 \\
4 & 1, 2, 4, 8 & 24 x 8, 12x8, 6x8, 3x8 \\
\hline
\end{tabular}
\caption{HPM Structure with different pyramid scales.}
\label{method:t1}
\end{table}
{\bf Pooling strategies}
The HPM uses both average pooling and max pooling. The global average pooling is a traditional operation in many classification framework, because it enforces a corresponding relation between feature maps and categories. However, the global average pooling can lose some very discriminative information by the average operation. For example, if one partition of the person is very discriminative but surrounded by background, in this case, the global average pooling will obtain the average of the discriminative part and the background region, which may lead to a low response and miss it. To deal with this problem, we use both average pooling and max pooling, which can maintain the global relation with the identification and preserve the discriminative part.
We will provide extreme ablation experiments in the following section to validate the effectiveness of our settings.
\subsection{Dataset and Evaluation Protocol}
{\bf Market1501}~\cite{zheng2015scalable} contains 32,668 images of 1,501 labeled persons of six camera views. There are 19,732 gallery images and 12,936 training images detected by DPM~\cite{felzenszwalb2010object}, including 751 identities in the training set and 750 identities in the testing set. It also contains 500,000 images as some distractors, which may has a considerable influence on the retrieval accuracy.
{\bf DukeMTMC-ReID}~\cite{ristani2016performance,zheng2017unlabeled} is a subset of the DukeMTMC dataset. It contains 1,812 identities captured by 8 cameras. There are 2,228 query images, 16,522 training images and 17,661 gallery images, with 1,404 identities appear in more than two cameras.Also, similar with the Market1501, the rest 408 IDs are considered as distractor images. DukeMTMC-ReID is one of the most challenging re-ID datasets up to now with so many images from 8 multi-cameras.
{\bf CUHK03}~\cite{li2014deepreid} consists of 14,097 cropped images from 1,467 identities. For each identity, images are captured from two cameras and there are about 5 images for each view. There are two ways to obtain the annotations: human labeled and detected by DPM. Our evaluation is based on the detected label image.
{\bf Evaluation Protocol}
In our experiment, we use Cumulative Matching Characteristic (CMC) curve and the mean average precision (mAP) to evaluate our approach. CMC represents the accuracy of the person retrieval, it is accurate when each query only has one ground truth. However, when multiple ground truths exist in the gallery, the goal is to return all right match to user. In this case, CMC may not have enough discriminative ability, but the mAP could reflect the recall. For Market-1501 and DukeMTMC-ReID. We use the evaluation packages provided by~\cite{zheng2015scalable} and~\cite{zheng2017unlabeled}, respectively. And for CUHK03, we adopt the new training/testing protocol proposed in~\cite{zhong2017re}. Moreover, for simplicity, all results reported in this paper is under the single-query setting and does not use the re-ranking proposed in~\cite{zhong2017re}.
\subsection{Implementation Details}
In order to obtain enough information from person image and proper size of feature map for horizontal pyramid pooling, we resize all the image to 384x128. For the backbone network, we use Resnet50 that initialized with the weights pretrained on ImageNet~\cite{deng2009imagenet}. We remove the last fully connected layer and average pooling layer and set the stride of last resent conv4\_{1} from 2 to 1. The training images are augmented with horizontal flipping and normalization. The batch size is set to 64 and we train model for 60 epoch. The base learning rate is set to 0.1 and decay to 0.01 after 40 epochs. Notice that learning rate for all pretrained Resnet layer is set to 0.1 x base learning rate. The stochastic gradient descent (SGD) with 0.9 momentum is implemented in each mini-batch to update the parameters. During evaluation, we concatenate the feature vectors after the $1x1$ convolution operation together to generate feature representation of query image. The feature from original image and horizontal flipped image are added up and normalized feature for retrieval evaluation. Our model is implemented on Pytorch platform and train with two NVIDIA TITAN X GPUs. All datasets share the same experiments setting as above.
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{fig/samples.pdf}
\caption{Qualitative Results: (a) Queries and the corresponding discriminative heatmaps learned by the proposed HPM. (b) Comparisons of R5 of w/ HPM and w/o HPM schemes.}
\label{fig:samples}
\vspace{-5mm}
\end{figure}
\begin{table*}[t]\setlength{\tabcolsep}{8pt}
\centering
\begin{tabular}{l|c|c|c|c|c|c|c|c}
\hline
\multirow{2}{*}{Model} & \multicolumn{4}{c}{Market1501} & \multicolumn{2}{|c}{DukeMTMC-ReID} & \multicolumn{2}{|c}{CUHK03} \\ \cline{2-9}
& R1 & R5 & R10 & mAP & R1 & mAP & R1 & mAP \\ \hline
BoW+kissme \cite{zheng2015scalable} & 44.4 & 63.9 & 72.2 & 20.8 & 25.1 & 12.2 & 6.4 & 6.4\\
WARCA \cite{jose2016scalable} & 45.2 & 68.1 & 76.0 & -- & -- & -- & -- & --\\
SVDNet \cite{sun2017svdnet} & 82.3 & 92.3 & 95.2 & 62.1 & 76.7 & 56.8 & 41.5 & 37.3\\
PAN \cite{zheng2017pedestrian} & 82.8 & -- & -- & 63.4 & 71.6 & 51.5 & 36.3 & 34.0\\
PAR \cite{Zhao2017DeeplyLearnedPR} & 81.0 & 92.0 & 94.7 & 63.4 & -- & -- & -- & --\\
MultiLoss \cite{Li2017PersonRB} & 83.9 & -- & -- & 64.4 & -- & -- & -- & --\\
TripletLoss \cite{hermans2017defense} & 84.9 & 94.2 & -- &69.1 & -- & -- & --\\
MultiScale \cite{Chen2017PersonRB} & 88.9 & -- & -- & 73.1 & 79.2 & 60.6 & 40.7 & 37.0 \\
MLFN \cite{chang2018multi} & 90.0 & -- & -- & 74.3 & 81.0 & 62.8 & 54.7 & 49.2\\
HA-CNN \cite{li2018harmonious} & 91.2 & -- & -- & 75.7 & 80.5 & 63.8 & 41.7 & 38.6\\
AlignedReID \cite{zhang2017alignedreid} & 91.0& 96.3& -- & 79.4 & -- & -- & -- & -- \\
Deep-Person \cite{bai2017deep} & 92.3 & -- & -- & 79.5 & 80.9 & 64.8 & -- & --\\
PCB \cite{sun2017beyond} & 92.4 & 97.0 & 97.9 & 77.3 & 81.8 & 66.1 & 61.3 & 54.2\\
PCB + RPP\cite{sun2017beyond} & 93.8 & 97.5 & 98.5 & 81.6 & 83.3 & 69.2 & 63.7 & 57.5 \\ \hline
HPM(ours) & {\bf 94.2} & 97.5 & {\bf 98.5} & {\bf 82.7} & {\bf 86.6} & {\bf 74.3} & {\bf 63.9} & 57.5 \\ \hline
\end{tabular}
\caption{Comparison of the proposed method with the state-of-art on Market-1501, DukeMTMC-ReID and CUHK03 with new protocol. HPM is implemented with four pyramid scales and combine both average pooling and max pooling described in Fig~\ref{fig:framework}.}
\vspace{-2mm}
\label{exp:soa}
\end{table*}
\begin{table}\setlength{\tabcolsep}{2pt}
\centering
\begin{tabular}{l|c|c|c}
\hline
\multirow{2}{*}{Model} & \multicolumn{3}{c}{Market1501} \\ \cline{2-4}
& R1 & R5 & mAP \\ \hline
No Pyramid Structure & 88.1 & 94.6 & 71.2 \\ \hline
HPM & {\bf 94.2}(+6.1) & {\bf 97.5}(+2.9) & {\bf 82.7}(+11.5) \\ \hline
\end{tabular}
\caption{Evaluation of effectiveness of Pyramid Structure, HPM uses four pyramid scales and combine both average pooling and max pooling, Non pyramid structure split the feature into same partitions as the last scale in HPM but without pyramid structure}
\label{exp:ab2}
\end{table}
\begin{table*}[t]\setlength{\tabcolsep}{3pt}
\centering
\begin{tabular}{l|c|c|c|c|c|c|c|c|c|c|c|c|c}
\hline
\multirow{2}{*}{Model} & \multirow{2}{*}{Feature Dim} & \multicolumn{4}{c}{Market1501} & \multicolumn{4}{|c}{DukeMTMC-ReID} & \multicolumn{4}{|c}{CUHK03} \\ \cline{3-14}
& & R1 & R5 & R10 & mAP & R1 & R5 &R10 & mAP & R1 & R5 & R10 & mAP \\ \hline
HPM + \#PS 1 + Avg pool & 256 &88.1 & 94.6 & 96.4 & 71.2 & 79.3 & 89.7 & 91.9 & 61.0 & 39.2 & 61.1 & 71.6 & 37.3 \\ \hline
HPM + \#PS 2 + Avg pool & 256x(1+2) & 92.0 & 96.9 & 97.9 & 78.3 & 83.1 & 91.9 & 93.4 & 68.9 & 53.2 & 73.2 & 79.6 & 48.9 \\ \hline
HPM + \#PS 3 + Avg pool & 256x(1+2+4) &92.3 & 97.2 & 97.9 & 79.3 & 84.5 & 92.4 & 94.1 & 70.8& 58.2 & 76.7 & 83.1 & 52.8 \\ \hline
HPM + \#PS 4 + Avg pool &256x(1+2+4+8)& 93.2 & 97.3 & 98.1 & 79.5 & 84.8 & 92.5 & 94.1 & 72.1 & 58.6 & 76.8 & 83.8 & 53.4 \\ \hline
HPM + \#PS 4 + Max pool &256x(1+2+4+8)& 93.6 & {\bf 97.7} & 98.3 & 81.6 & 86.2 & {\bf 93.2} & 94.8 & 74.1 & 62.4 & 78.9 & {\bf 86.3} & 57.4 \\ \hline
HPM + \#PS 4 + Max+Avg pool & 256x(1+2+4+8) & {\bf 94.2} & 97.5 & {\bf 98.5} & {\bf 82.7} & {\bf 86.6} & 93.0 & {\bf 95.1} & {\bf 74.3} & {\bf 63.9} & {\bf 79.7} & 86.1 & {\bf 57.5} \\ \hline
\end{tabular}
\caption{Performance comparison of the proposed method with different pyramid scales and different pooling strategies as described in Section3.4. PS is the abbreviation of Pyramid Scales.}
\vspace{-5mm}
\label{exp:ab1}
\end{table*}
\subsection{Comparison with the State-of-the-arts}
{\bf Results on Market1501} Comparisons between HPM and state-of-art approaches on Market1501 are shown in Table~\ref{exp:soa}. The results show that our HPM achieves the mAP of 83.1\% and Rank 1 accuracy 94.2\%, which both surpass all existing works more than 1.5\% and 0.4\%, respectively. It should be noted that we do not conduct any post-processing operation (~\emph{e.g. } the re-rank algorithm given by~\cite{zhong2017re}), which can further bring a considerably improvement in terms of mAP. PCB~\cite{sun2017beyond} is closest competitor, which also leverages partial-based leaning for person Re-ID. However, there are mainly two disadvantages of PCB,~\emph{i.e. } 1) it splits the features maps into pre-defined patches (6 in PCB) with the assumption that most persons in the given images are well aligned, which not make scene and resist to some outliers; 2) its state-of-the-art results are benefited from a powerful post-processing approach called RPP, which enable the optimized model cannot be trained in an end-to-end manner. In contrast, our HPM splits the feature maps according to various pyramid scales, which is more robust compared with PCB in addressing the outliers that are not well aligned. In addition, our HPM can be end-to-end learned and we believe that any post-processing operation can make a continued improvements upon the current results. From Table~\ref{exp:soa}, we can observe that our HPM makes a 5.4\% improvements compared with PCB in mAP. Even without post-processing, our HPM is still better than PCB+RPP (~\emph{i.e. } 83.1\% \emph{vs.} 81.6\%). Beyond PCB, the best model aims to deal with different size of person MultiScale~\cite{chen2017multi} yields the mAP of 73.1\% and Rank 1 accuracy of 88.9\%. Our HPM model outperforms it by 5.3\% and 10.0\% on Rank 1 and mAP, respectively.
{\bf Results on DukeMTMC-ReID} Person Re-ID results on Duke MTMC-ReID are given in Table~\ref{exp:soa}. This dataset is challenging because it has 8 different camera and the person bounding box size varies drastically across different camera views, however, our HPM achieves even better performance on this dataset. Without any post-processing, it still achieves 74.8\% on mAP and 86.6\% on Rank 1 accuracy, which is better than all other state-of-the-art methods by a large margin, 5.3\% and 3.3\%. Note that our HPM is the first model that can achieve above 80\% on mAP, which surpass all state-of-the-art methods by more than 5.0\%
{\bf Results on CUHK03} Table~\ref{exp:soa} shows results on CUHK03 when detected person bounding boxes are used for both training and testing. HPM achieves the best result of mAP, 59.7\% under this setting. Although the Rank 1 accuracy of HPM is a little lower than PCB+RPP, there's a clear gap, more than 2\%, between HPM and other methods, including the PCB+RPP, on mAP. And We believe that, as a end-to-end and part-based model, the RPP can also boost the performance HPM further.
{\bf Qualitative Result} We visualize some examples in Figure~\ref{fig:samples}. Concretely, Figure~\ref{fig:samples} (a) shows the queries and the corresponding heatmaps\footnote{We normalize each feature map of the last convolutional feature maps and sum them together to obtain the heatmap.} of the last convolutional feature maps. We observe that the discriminative abilities of multiple person parts are enhanced with our HPM. Figure~\ref{fig:samples} (b) compares the Re-ID results of w/ HPM and w/o HPM schemes. It can be seen that our HPM is very effective in guaranteeing accurate Re-ID results.
\subsection{Ablation Study}
To verify the effectiveness of each component and setting of HPM, we design several ablation study with different settings on Market-1501, DukeMTMC-ReID and CUHK03, including different number of pyramid scales, w/ and w/o using pyramid structure, different pooling strategies. Note that all unrelated settings are the same as HPM implementation detailed in Section 4.2
{\bf Effectiveness of Pyramid Structure}
Previous analysis shows that the HMP reaches the best performance with four pyramid scales, which has up to 8 partial bins on the feature map. In order to verify the effectiveness of pyramid structure, we remove other branches and just preserve the branch with 8 partial bins. From Table~\ref{exp:ab2}, we can observe that Rank 1, Rank 5 and mAP drop from 94.2\%, 97.5\% and 82.7\% to 92.0\%, 96.3\% and 76.4\%, respectively. Such an operation is similar to PCB, which leverage 6 partial bins. The reason is that many persons are not well aligned in the images, and naively split the feature maps into a pre-defined number of bins cannot well resist to unaligned outliers. In addition, discriminative information may hardly be learned for some parts if we apply too dense division. In contrast, with our pyramid structure, we can formulate partial features from coarse to fine, which can finally form into a more robust feature representation for person images.
{\bf Number of Pyramid Scales}
Table~\ref{exp:ab1} shows the Re-ID results of HPM with different pyramid scales, ~\emph{e.g. } 1, 2, 4, 8. From these results, we can find that HPM reaches the best performance with four pyramid scales. Intuitively, the number of pyramid $p$ determines the granularity of the partition feature. When $p=1$, it is equivalent to global pooling. With the increasing the number of $p$, Rank 1 accuracy and mAP are significant improved from 88.1\% and 71.2\% to 93.2\% and 79.5\%, as illustrated in Fig~\ref{exp:compare}. The reason why the HPM does not drops dramatically at some point as introduced in~\cite{sun2017beyond} is that the pyramid structure can combine both global and local features, which may increase the discriminative ability of very small partition. Since the last convolutional feature maps are with 24 horizontal units, we also try more dense pyramid scales, such as 12 and 24. However, more pyramid scales will bring additional computational cost and there is no obvious improvement can be observed. Therefore, we finally adopt 4 pyramid scales in this work.
\begin{figure}
\begin{center}
\bgroup
\def\arraystretch{0.2}
\setlength\tabcolsep{1.0pt}
\begin{tabular}{cc}
\includegraphics[width=0.5\hsize]{fig/compare_r1.png} &
\includegraphics[width=0.5\hsize]{fig/compare_mAP.png}
\end{tabular} \egroup
\end{center}
\caption{Impact of pyramid scales. Rank-1 accuracy and mAP are compared.}
\vspace{-5mm}
\label{exp:compare}
\end{figure}
{\bf Pooling Strategies}
Row4 and Row 5 in Table~\ref{exp:ab1} shows the performance of HPM with different pooling strategies. It can be observed that max pooling performs better than average pooling in the most cases. The reason is that average pooling will take all locations of a particular parts into account and all the locations contribute equally to the final partial representation. Thus, the discriminative ability of the representation produced by average pooling can be easily influenced by the unrelated background patterns. In contrast, the global max pooling only preserve the largest response values for a local view. We consider these two pooling strategies are complementary in producing feature representations from global and local vies. Therefore, we integrate them into a unified model to take advantages from these two strategies. Experimental results in Table~\ref{exp:ab1} demonstrate that mixing the two pooling strategies achieves better results compared with using either of them.
\section{Introduction}
\input{TEX/1_intro.tex}
\section{Related Work}
\input{TEX/2_related.tex}
\section{Proposed Method}
\input{TEX/3_method.tex}
\section{Experiments}
\input{TEX/4_experiments.tex}
\section{Conclusion}
\input{TEX/5_conclusion.tex}
|
{
"timestamp": "2018-11-13T02:04:57",
"yymm": "1804",
"arxiv_id": "1804.05275",
"language": "en",
"url": "https://arxiv.org/abs/1804.05275"
}
|
\section{Introduction}
\label{sec:1}
The phase of matter can be characterized by symmetry and topology \cite{Hasan2010,Haldane2017}. For certain effects, symmetry provides the basic condition to occur while topology can add exceptional robustness. A text-book example is the Hall effect enabled by the broken time-reversal symmetry ($\cal{T}$) in the applied magnetic field. The exact discrete resistance values in the quantum Hall effect (QHE), unperturbed by disorder, are then a consequence of the topological Landau level form of the electronic structure in a strong quantizing magnetic field \cite{Hansson2017}. In this chapter, we show how the fundamental concepts of symmetry and topology apply to antiferromagnetic spintronics \cite{Jungwirth2016}.
We start our chapter by briefly illustrating the symmetry and topology principles on three key functionalities of spintronic memory devices, namely the retention, reading, and writing of magnetic information \cite{Chappert2007,Kent2015}. Side by side we compare in this introductory section how the principles apply when considering the more conventional ferromagnetic and the emerging antiferromagnetic spintronic devices.
Ferromagnetism can (and often does) lower the symmetry of the crystal, depending on the direction of magnetic moments. For example, a rotation along a certain crystal axis remains a symmetry operation when moments are aligned with the axis but the symmetry is broken when the moments are perpendicular to the rotation axis. It implies that reorientation of ferromagnetic moments can, in the presence of spin-orbit coupling (SOC), change the electronic structure and by this the total energy. This is the origin of the magnetocrystalline anisotropy energy (MAE) barrier that supports the non-volatile storage in spintronic memories \cite{Chappert2007}. The same symmetry principle and corresponding magnetic storage functionality apply equally to antiferromagnets (AFs) \cite{Jungwirth2016}. On top of that, the lack of a net magnetic moment and suppressed dipolar fields make the storage in AFs less sensitive to magnetic fields and allow for denser integration of memory bits than in ferromagnets. Apart from binary storage, AFs naturally host series of different stable multi-domain reconfigurations which is suitable for integrated memory-logic or neuromorphic computation devices \cite{Wadley2016,Kriegner2016,Olejnik2017,Borders2017}.
\begin{figure}[h]
\sidecaption
\includegraphics[width=0.9\linewidth]{Fig1.pdf}
\caption{\textbf{Discovery of the manipulation of the antiferromagnetic order via electric currents.} (a) The crystal structure of the CuMnAs AF with marked non-equilibrium spin polarisations $\delta\textbf{s}$ generated by the current $\textbf{J}$ applied along the [010] direction. (b) The reorientation of the moments is observed in the microbars of the CuMnAs. The writing current is applied along one of two orthogonal directions and generates a spin-orbit field which reorients the moments along the perpendicular direction to the applied current. In the bottom panel, we show the anisotropic magnetoresistance signal corresponding to the reorientation of the N\'{e}el order parameter. Panel (b) adapted from Ref. \cite{Olejnik2017a}, and Ref. \cite{Wadley2016}.}
\label{fig:1}
\end{figure}
Anisotropic resistance, i.e. the sensitivity of electronic transport to the current direction, requires broken cubic symmetry. Ferromagnetism where spins align with a specific crystal axis always breaks cubic symmetry. This implies that ferromagnets can have the anisotropic resistance. Here typically the leading dependence of the resistance on current direction is when measured with respect to the magnetization axis. The effect called (spontaneous) anisotropic magnetoresistance (AMR) is known for more than 150 years \cite{Thomson1856} and provides arguably the most straightforward means for electrically detecting different directions of ferromagnetic moments. AMR was used, e.g., in the first generation of magnetoresistive field sensors for hard-disk read-heads or for electrical readout in the first generation of magnetic random access memories (MRAMs) \cite{Daughton1992}. The same symmetry argument applies to the AMR in AFs where it has been used to demonstrate electrical readout in experimental memory devices, as shown in Fig. \ref{fig:1} \cite{Park2011b,Marti2014,Wadley2016,Kriegner2016,Olejnik2017}. On the other hand, AMR in AFs is not suitable for external magnetic field sensing because of the lack of the net magnetic moment.
\begin{figure}[h]
\sidecaption
\includegraphics[width=0.9\linewidth]{Fig2.pdf}
\caption{\textbf{Observation of the anomalous Hall effect in noncollinear AFs.} (a) The noncollinear magnetic order on the kagome lattice in the Mn$_{\text{3}}$Sn breaks the time reversal symmetry and allows for the net magnetization in the [100] direction. Although the net moment is almost perfectly compensated, a strong emergent magnetic field, e.g. Berry curvature $\textbf{b}$, along the [100] direction generates a large anomalous Hall effect in the perpendicular, (001), plane. (b) The measured Hall resistivity $\rho_{H}^{AF}$ obtained by subtracting the signal from the small net moment and ordinary Hall effect. The panel (b) adapted from Ref. \cite{Nakatsuji2015}.}
\label{fig:2}
\end{figure}
Another transport effect that can be used to internally detect magnetic moments of a conductor is the anomalous Hall effect (AHE). It can occur in crystals with magnetic space groups whose symmetry allows for the presence of a net magnetic moment. Remarkably, this symmetry argument holds independently of whether the system indeed has a ferromagnetic moment or is in a fully compensated antiferromagnetic state. While the AHE in ferromagnets was discovered more than 100 years ago \cite{Hall1881}, its experimental demonstration in AFs, shown in Fig. \ref{fig:2}, is one of the most recent developments in spintronics \cite{Nakatsuji2015}.
Unlike time-reversal, the symmetry operation of space-inversion ($\cal{P}$) does not rotate the axial magnetic moment vector which implies that ferromagnets cannot have a combined $\cal{PT}$-symmetry. This removes the Kramers $\cal{PT}$-symmetry protection of the spin-up/spin-down degeneracy of electronic bands. As a result, electrons moving in the unequal spin-up and spin-down bands have different resistivities. In ferromagnetic bilayers this leads to different resistance states for parallel and antiparallel alignments of moments in the two layers and the corresponding giant/tunnelling magnetoresistance (GMR/TMR) effects \cite{Chappert2007}. These phenomena, that tend be stronger than the AMR, are used in modern hard disk read-heads and MRAMs.
Different resistivities in spin-up and spin-down transport channels in a ferromagnet can be also used to filter an unpolarised current passing through the ferromagnetic layer by suppressing one spin-component of the electrical current. The resulting spin-polarized current filtered through such a ferromagnetic polarizer can exert a spin transfer torque (STT) on the adjacent ferromagnetic layer and switch its magnetic moment \cite{Ralph2008}. This reversible electrical writing method is used in the latest generation of MRAMs.
There is no equivalent counterpart of the two-spin-channel GMR/TMR or STT phenomena in AFs with equal spin-up and spin-down bands. On the other hand, AFs can have the combined $\cal{PT}$-symmetry which opens a possibility of the Dirac crossing of two doubly-degenerate bands, as shown in Fig. 3(a) \cite{Tang2016,Smejkal2016}. The topological protection of these Dirac points can be turned on and off by changing other symmetries of the antiferromagnetic crystal, e.g., via changing the direction of the antiferromagnetic N\'eel vector. This could enable very large topological AMR effects in AFs and remedy the absence of the two-spin-channel GMR/TMR.
In time-reversal symmetric paramagnets, a broken space-inversion symmetry leads to the Kramers degeneracy of states with opposite spins and opposite crystal momenta, while the states at a given crystal momentum can be spin split. As a result, the crystal can develop a net spin polarization in a non-equilibrium, current-carrying state. When these spin Hall or Edelstein effects occur at an inversion-asymmetric interface between a paramagnet and a ferromagnet, or inside a ferromagnetic crystal that lacks inversion symmetry, they can induce a spin-orbit torque (SOT) in the ferromagnet \cite{Sinova2015,Hellman2016}. The charge to spin conversion efficiency driving the SOT can outperform that of the STT and is explored as a prospect writing mechanism for future fast MRAMs. A particularly large charge to spin conversion efficiency is expected to occur in time-reversal symmetric topological insulators (TIs) whose 2D surfaces host Dirac cones with the spins locked perpendicular to the 2D momenta \cite{Mellnik2014,Fan2014a,Fan2016}. The ultimate charge to spin conversion efficiency would then occur in 1D surface states of 2D TIs, the so called quantum spin Hall states \cite{Roth2009}, with opposite electron spins locked to the opposite crystal momenta at a given 1D edge.
\begin{figure}[h]
\sidecaption
\includegraphics[width=0.9\linewidth]{Fig3.pdf}
\caption{\textbf{Concepts of topological antiferromagnetic structures in crystal momentum space.} (a) The antiferromagnetic effective time reversal symmetry $\mathcal{TP}$ combining time reversal and spatial inversion can protect the Dirac semimetal state. The exemplar crystal structure of orthorhombic CuMnAs is shown in the inset. (b) The antiferromagnetic effective time reversal symmetry $\mathcal{T}T_{\frac{1}{2}}$ combining time reversal and half-unit cell translation can protect the topological insulator. The GdPtBi candidate is drawn in the inset.
}
\label{fig:3}
\end{figure}
A direct counterpart of the SOT is observed in AFs that break the $\cal{T}$ symmetry and the $\cal{P}$ symmetry but have the combined $\cal{PT}$ symmetry. In these antiferromagnetic crystals, a global electrical current induces a staggered non-equilibrium spin polarization that is commensurate with the staggered equilibrium N\'eel order \cite{Zelezny2014}. The phenomenon was used to demonstrate, in combination with AMR, experimental antiferromagnetic memory devices with electrical writing and readout \cite{Wadley2016}. Because of the THz-scale antiferromagnetic resonance, compared to the GHz-scale ferromagnetic resonance, SOT switching in antiferromagnetic memory devices was demonstrated with writing current pulses as short as 1~ps \cite{Olejnik2017a}. Moreover, since it is again the $\cal{PT}$-symmetry that enables the antiferromagnetic SOT, it is potentially compatible with the large topological AMR \cite{Smejkal2016}.
Another class of AFs, with no symmetry counterparts in ferromagnets, has the combined ${\cal{T}}T_{\frac{1}{2}}$-symmetry, where $T_{\frac{1}{2}}$ is the translation by a half of the magnetic unit cell. This symmetry allows in principle for TIs with spin-momentum locked surface states, despite the breaking of the $\cal{T}$ symmetry by the magnetic order, as illustrated in Fig. \ref{fig:3}(b) \cite{Mong2010}. Overall, an antiferromagnetic order can occur in 1421 magnetic space groups out of which only 275 allow also for the ferromagnetic order. Similarly, the 122 magnetic point groups are all compatible with the antiferromagnetic order out of which only 31 also support ferromagnetic states. This not only underlines why antiferromagnetism is more common than ferromagnetism and spans the whole range of materials from insulators to superconductors but also highlights how much is the symmetry and topology playground enlarged by including AFs. We are just beginning to unravel what new spintronics phenomena and functionalities this may offer. In Section 2 we give a brief overview of symmetry and topology concepts in condensed matter systems. In Section 3 we discuss in more detail antiferromagnetic topological semimetals, Chern insulators, and TIs. Topological spintronic phenomena in AFs are reviewed in Section 4 and Section 5 gives a brief summary of the Chapter.
\section{Antiferromagnets: symmetry and topology}
The phases of matter can be classified by Landau symmetry breaking mechanism. In an AF, N\'{e}el vector breaks the rotational symmetry present in the paramagnetic state. The order parameter is the N\'{e}el vector and SOC determines the particular direction(s) of the magnetic moments in crystal, the so-called easy axes, corresponding to the lowest MAE. While certain AFs tends to be to a very good approximation isotropic Heisenberg magnets, e.g. Mn$_{\text{2}}$Au AF can have in-plane MAE at the level of 10\,$\mu$eV per formula unit \cite{Shick2010,Bodnar2017}, noncollinear Mn$_{3}$Sn AF was reported to have MAE of 0.1\,eV per formula unit \cite{Sandratskii1996,Duan2015}, and a giant MAE of 10\,meV per formula unit was predicted in noncollinear IrMn$_{\text{3}}$ \cite{Szunyogh2009}.
The discoveries in the 1980s in the theory of superfluid vortices, QHE, or dislocation defects revealed an additional label of the phases different from the symmetry, based on topology \cite{david1998topological}. The phases can be characterized by an integer topological index which does not change upon continuous transformations of the Hamiltonian and thus supports the relative robustness of a topological phase. For instance, the topological invariants in TIs are the $Z2$ indices, which in simple centrosymmetric non-magnetic TIs are related to counting the number of parities at time-reversal invariant crystal momenta \cite{Hasan2010}. Dirac semimetals (DSMs), such as graphene, are protected by the vorticity around the Dirac point \cite{bernevig2013topological}. The DSMs protected by crystalline symmetry can be assigned a topological index by subtracting this crystalline symmetry eigenvalues of the conduction and valence band along the line in Brillouin zone (BZ) invariant under this symmetry. Red and blue lines in Fig. \ref{fig:3}(a) illustrate the symmetry eigenvalues with an opposite sign.
We note that TIs and semimetals represent a symmetry protected topological order \cite{Hermanns2017}. For instance, AF DSMs or TIs require the presence of the $\mathcal{PT}$ or $\mathcal{T}T_{1/2}$ symmetry as illustrated in Fig. \ref{fig:3}(a,b). In contrast, Weyl semimetals (WSM) and Chern insulators are protected by Chern numbers. WSM can be realized in system with broken $\mathcal{PT}$ symmetry, while Chern insulators materialize in systems with broken $\mathcal{T}$ symmetry as we explain further. The slices of constant wavevector component $k_{z}$ between two Weyl points in simple model WSM \cite{Vafek2013} (with $\mathcal{PT}$ symmetry broken by breaking $\mathcal{T}$ symmetry, Fig. \ref{fig:8}(a)) can be thought of as Chern insulators, whose examples are QHE or quantum anomalous Hall effect (QAHE). If we use the Bloch ansatz, the Chern number of a $k_{z}$=const. plane reads,
\begin{equation}
\mathcal{C}= \frac{1}{2\pi} \int dk_{x}dk_{y}b_{z}(\textbf{k}),
\end{equation}
where the Berry curvature has a meaning of an emergent magnetic field $\textbf{b}$ in the crystal momentum space and quantifies the underlying topology of the wavefunction \cite{bernevig2013topological}:
\begin{equation}
\textbf{b}(\textbf{k})=-\text{Im} \langle\partial_{\textbf{k}}u(\textbf{k}) \vert \times \vert \partial_{\textbf{k}}u(\textbf{k}) \rangle.
\label{Eq_Berry}
\end{equation}
The topological index of a Weyl point can be defined as a Chern number of a closed surface surrounding the Weyl point, which can be calculated due to the Gauss theorem as a difference between two Chern numbers along the line connecting the Weyl points:
\begin{equation}
\mathcal{Q} = \mathcal{C}(k_{z,W}+\delta)-\mathcal{C}(k_{z,W}+\delta)=\frac{1}{2\pi}\int_{\delta S}d^{2}k \; \textbf{n}\cdot \textbf{b}(\textbf{k}).
\label{WP}
\end{equation}
Here $\delta S$ is a small sphere surrounding the Weyl point at $k_{z,W}$, $\textbf{n}$ is the surface normal vector, and $\mathcal{C}$ is the Chern number of the plane slightly below and above the Weyl point $k_{z,W}\pm\delta$. Thus the Chern number is nonzero along the $k_{z}$ between the two Weyl points and zero outside as marked in Fig. \ref{fig:8}(a).
The Berry curvature acts as a source and sink at the Weyl points as we illustrate in Fig. \ref{fig:8}(a) and consequently the two Weyl points along $k_{z}$ have $\cal{Q}$ $=+1$, and $-1$.
\begin{figure}[h]
\sidecaption
\includegraphics[width=1\linewidth]{Fig4.pdf}
\caption{\textbf{Topological edge states and transport.} (a) Fermi arcs in Weyl semimetal. (b) The realization of quantum Hall effect as seen in quantized transversal resistivity in EuMnBi$_{\text{2}}$ AF \cite{Masuda2016}. (c) The edge states in Chern insulator. (d) Spin polarised edge states in quantum spin Hall effect. Panel (b) adapted from \cite{Masuda2016}.}
\label{fig:8}
\end{figure}
Topological phases in the crystal momentum space are very often accompanied by quantized or almost quantized and low dissipation transport properties and nontrivial surface states \cite{Hasan2010,Wang2017b}. In Fig. \ref{fig:8}(b) we show the quantized Hall plateaus in EuMnBi$_{\text{2}}$ AF \cite{Masuda2016}. The Chern insulator exhibits quantized Hall conductivity:
$
\sigma_{xy}=\frac{e^{2}}{h}\mathcal{C}.
$
On the other hand, quantum spin Hall effect (QSHE) in the 2D TI shows a quantized spin Hall conductivity
$
\sigma_{xy}^{S}=2\frac{e^{2}}{h}.
$
In a Chern insulator, chiral edge states arise (single spin polarized electrons), which are in fact 1D Weyl fermions of a given chirality \cite{Armitage2017}, as we show in Fig. \ref{fig:8}(c).
In 2D TIs, helical edge states are observed (two counter-propagating perfectly polarized currents with opposite spins) as we illustrate in Fig. \ref{fig:8}(d). In contrast, a WSM exhibits surface BZ Fermi arcs for the constant $k_{z}$ in-between Weyl points as we show in Fig. \ref{fig:8}(a). The stacked Fermi arcs from chiral edge states of QAHE subsystems yield almost quantized Hall conductivity:
$
\sigma_{xy}=\frac{e^{2}}{h}\frac{\delta k_{W}}{\pi},
$
where $\delta k_{W}$ is the distance between the Weyl points in the BZ. In Section 3 we will discuss in detail possible realizations of topological semimetals, Chern insulators, and TIs in AFs.
\subsection{Magnetic symmetry and spintronics effects}
The potential presence of long-range ordered antiferromagnetic textures and spintronics effects in AFs is determined by magnetic symmetries.
Magnetic point groups (MPGs) \cite{bradley2010mathematical} are obtained from the ordinary point groups by adding an additional antiunitary operation $\mathcal{T}$ whose application reverses the direction of magnetic moments. Antiferromagnetic order can be in principle found in all 122 MPGs. The colorless MPGs, the so called category I \cite{bradley2010mathematical} (see an example of a MPG I in Fig. \ref{fig:5}), are those which do not contain the operation $\mathcal{T}$ at all and there are 32 of them which is the same number as the number of nonmagnetic classical point groups. The grey magnetic point groups (category II \cite{bradley2010mathematical}) contains $\mathcal{T}$ as an element of the magnetic symmetry group. There is also 32 of them obtained from the category I by adding the $\mathcal{T}$ operation. Antiferromagnetic order may appear in this category since the point groups can be obtained from the magnetic space groups by removing all nontrivial unit cell translations. Thus antiferromagnetic sublattices connected by a combination of nontrivial translation $T$ and time reversal fall into this category and we show in Fig. \ref{fig:5}(MPG II) exemplar FeSe AF structure with the $\mathcal{T}T_{\frac{1}{2}}$ operation. Finally, the category III black and white MPGs contain $\mathcal{T}$ only in a combination with another point group symmetry (mirror, or rotation). There are 58 of them and we show in Fig. \ref{fig:5}(MPG III) three antiferromagnetic examples (from left): A DSM AF crystal\cite{Smejkal2016}, a WSM AF crystal\cite{Wan2011} and a WSM AF crystal with a nonzero AHE \cite{Yang2016c}.
\begin{figure}[h]
\sidecaption
\includegraphics[width=1\linewidth]{Fig5.pdf}
\caption{\textbf{Classical magnetic point groups (MPGs) and exemplar AFs.} (I) Colorless MPG example of a layered AF MnTe. (II) Grey MPG and zig-zag antiferromagnet FeSe. (III) Black and white MPG. Three different types (from left): $\mathcal{PT}$ AF CuMnAs, centrosymmetric AF with 3Q magnetic order and prohibited anomalous Hall effect (AHE) - IrMn\cite{Kohn2013} or certain pyrochlore AF\cite{Wan2011}, and centrosymmetric AF with nonzero AHE, Mn$_{\text{3}}$Ge. The three colors represent overlap of the antiferromagnetic symmetries allowing for Dirac quasiparticles, superconductivity, and AHE.}
\label{fig:5}
\end{figure}
The form of spintronics linear response tensors is obtained by the application of magnetic symmetries. The Neumann principle states that \textit{any physical observable of a system must exhibit symmetry of the point group of the system} \cite{bradley2010mathematical}. A special role is played by $\mathcal{T}$ and $\mathcal{P}$ symmetries which define the basic transformation properties of tensors as shown in Tab. \ref{tab1}.
\begin{table}[htbp]
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|l|l|}
\hline
Tensor rank & \multicolumn{4}{c|}{Even (scalar, matrix)} & \multicolumn{4}{c|}{Odd (vector, 3rd rank)} \\ \hline
Time reversal $\mathcal{T}$ & \multicolumn{2}{c|}{+} & \multicolumn{2}{c|}{-} & \multicolumn{2}{c|}{+} & \multicolumn{2}{c|}{-} \\ \hline
Spatial inversion $\mathcal{P}$ & + (polar) & - (axial) & + (polar) & - (axial) & - (polar) & + (axial) & - (polar) & + (axial) \\ \hline
$\mathcal{PT}$ & + & - & - & + & - & + & + & - \\ \hline \hline
Exemplar tensor & AMR & & $\sigma_{ij}^{AHE}$ & & & $\sigma_{ij}^{S}$ & & $\textbf{b}$\\ \hline
\end{tabular}
\end{center}
\caption{Spatial inversion and time reversal transformations of tensors.}
\label{tab1}
\end{table}
For the conductivity $\sigma_{ij}$ and spin Hall conductivity $\sigma_{ij}^{S}$ analysis, it is sufficient to use the magnetic Laue group \cite{Seemann2015}. $\sigma_{ij}$ and $\sigma_{ij}^{S}$ do not change sign under spatial inversion and thus this symmetry can be omitted leading to only 32 magnetic Laue groups to investigate \cite{Kleiner1966}. In contrast, the SOT torkance tensor $\tau _{ij}$ changes sign under spatial inversion and thus non-centrosymmetric lattice sites are required and all 122 MPGs have to be considered \cite{Wimmer2016a,Zelezny2017}. This procedure leads to the conclusion that in the MPG from category I, and III there are in total 31 MPGs which allow for uncompensated moments, ferrimagnetism, ferromagnetism and also a nonzero AHE conductivity $\sigma_{ij}^{AHE}$. In Fig. \ref{fig:5}(MPG III) - right panel we show a corresponding example of the non-collinear AF structure of Mn$_{\text{3}}$Ge.
\begin{figure}[h]
\sidecaption
\includegraphics[width=\linewidth]{Fig6.pdf}
\caption{\textbf{The symmetry of spin-orbit fields.}
(a) All possible nonequilibrium spin polarization at noncentrosymmetric positions in crystals can be decomposed into combinations of Rashba, Dresselhaus, or Weyl spin texture. (b) Example of realistic calculation of the spin-orbit fields induced by electric current in tetragonal CuMnAs. The symmetry of the fields was confirmed in experiment \cite{Wadley2016}. Panel (b) adapted from \cite{Wadley2016}.}
\label{fig:6}
\end{figure}
The torkance tensor is defined by $\textbf{t}=\tau \textbf{E}$ \cite{Freimuth2014a}, where $\textbf{t}$ is a torque generated by an applied electric field $\textbf{E}$. The Onsager reciprocal for SOT is the inverse SOT obtained by interchanging the perturbation and response and both effects can be desribed by the torkance tensor \cite{Freimuth2015}. SOT can be decomposed into an even part $\textbf{t}_{FL}$ and an odd part $\textbf{t}_{AL}$: $\textbf{t}=\textbf{t}_{FL}+\textbf{t}_{AL}$. There are 21 non-centrosymmetric point groups which allow for a global or local current-induced nonequilibrium spin polarisation, and thus for SOT, in the lowest order that is independent of magnetization \cite{Ciccarelli2016}. Spin-orbit fields generating the SOT can be decomposed into a combination of Rashba, Dresselhaus, and Weyl symmetry as shown in Fig. \ref{fig:6}(a). The AF variants of SOT can be found in AFs with non-centrosymmetric magnetic sublattices connected by crystalline symmetries as we illustrate on the example of the CuMnAs AF in Fig. \ref{fig:6}(b) \cite{Zelezny2014,Wadley2016,Smejkal2016,Bodnar2017}. Here the dominating SOT is driven by a staggered effective current-induced field of Rashba symmetry and the magnetic sublattices are connected via $\mathcal{PT}$ symmetry \cite{Smejkal2016}. Wimmer et al. \cite{Wimmer2016a} list forms of torkance tensors for all 122 MPGs.
There are also 21 MPGs which contain combined $\mathcal{PT}$ symmetry and can host Dirac quasiparticles as we explain in Section 3.
Other effects can be treated analogically. For instance, the magneto-optical Kerr effect \cite{Feng2015} and anomalous Nernst effect \cite{Ikhlas2017} were predicted and observed in non-collinear AFs as well. Also, this scheme can be applied to the layer-resolved quantities in heterostructures, e.g. the layer-resolved conductivity \cite{Seemann2015}.
Finally, AF order can coexist with superconductivity as we illustrate in Fig. \ref{fig:5} by the grey shaded ellipse and examples are the iron-based superconductors \cite{Wang2016e}. More complicated magnetic structures such as spin spirals, spin density waves, and skyrmions may not allow to be described completely in the classical MPG framework. Gopalan and Litvin \cite{Gopalan2011} have shown the possibility of new hidden symmetries, an example being local roto-inversion. This operation does not rotate the whole crystal but just a finite subset while unchanging its MPG. The novel magnetic counterpart symmetries might be also relevant for the complete description of antiferromagnetic structures.
\subsection{Electronic structure and band touchings}
Antiferromagnetic exchange interactions arise as a result of the complex interplay among the electrons. Different types of exchange interactions are possible, e.g., direct, indirect, superexchange, or itinerant exchange. The electronic structure of AFs is thus very often complicated and requires the inclusion of correlation and many particle effects. A realistic insight into the electronic structure and the existence of the antiferromagnetic phase can be determined by the density functional theory (DFT). In DFT, the interacting many-particle problem is mapped onto non-interacting electrons in an effective Kohn-Sham potential \cite{Kohn1965}. Hohenberg and Kohn have shown that the ground-state properties of the effective electronic gas are uniquely determined by the electronic density \cite{Hohenberg1964}. The reformulation of the problem as a variational one tremendously decreases the macroscopic $\sim$10$
^{23}$ degrees of freedom to just 3 - the spatial coordinates of the electronic density. In a magnetic system with a strong relativistic SOC, the magnetic relativistic "spin-only" (neglected diamagnetic effects \cite{strange1998relativistic}) Kohn-Sham-Dirac Hamiltonian reads \cite{strange1998relativistic}:
\begin{equation}
H_{\text{KSD}}=c\alpha \cdot \textbf{p}+\beta mc^{2}+V^{\text{eff}}(\textbf{r})-\textbf{m}(\textbf{r})\cdot\textbf{B}^{\text{eff}}(\textbf{r}),
\label{KSD}
\end{equation}
where $\textbf{p}=-i\hbar \nabla$ is a momentum, $V^{\text{eff}}$ and $\textbf{B}^{\text{eff}}$ is the spin independent part of the potential and the exchange-correlation magnetic field, $\alpha, \beta$ are 4x4 Dirac matrices, $\textbf{m}$ is spin density, and the electronic density is obtained from $n(\textbf{r})=\sum \Psi_{i}^{\dagger}(\textbf{r})\Psi_{i}(\textbf{r})$ \cite{strange1998relativistic}.
The Kohn-Sham potential is not known exactly and has to be approximated, e.g., by the local density approximation (LDA) or generalized gradient approximation (GGA) \cite{prasad2013electronic}.
The set of equations for the electronic wavefunctions and Kohn-Sham potential is solved iteratively. The procedure yields ground state wavefunctions and Hamiltonian from which other quantities, e.g., linear response coefficients can be calculated. For instance in Fig. \ref{fig:9}(b) we show electronic bands of the orthorhombic AF CuMnAs as calculated within GGA \cite{Smejkal2016}.
AF systems are often correlated and disordered. Electronic correlations can be treated within DFT+Hubbard U, or DFT+dynamical mean field theory (DMFT). We show the generalized band structure of an AF BaFe$_{\text{2}}$As$_{\text{2}}$ calculated by DFT+DMFT in Fig. \ref{fig:FeSe}(b). The effects of disorder can be captured by the supercell technique \cite{prasad2013electronic} or coherent potential approximation \cite{prasad2013electronic} as was demonstrated, for instance, for disordered Mn$_{\text{2}}$Au AF \cite{Bodnar2017}.
Symmetries impose constraints on the electronic spectrum, including the existence and protection of band touchings. The palette of quasiparticles in solids is more rich than the three types of high energy physics excitations: Weyl, Dirac and Majorana fermions \cite{Vafek2013}. This is because of the more complex crystalline symmetries that are not present in the high energy vacuum \cite{Wieder2016,Bradlyn2016}. In the next Section we illustrate how the effective Hamiltonian arises for Dirac and Weyl quasiparticles in an AF.
\section{Topological antiferromagnetic phases}
Topological magnetic phases can be found in heterostructures with antiferromagnetic elements as well as in bulk AFs. In Tab. \ref{tab2}, we list promising topological AFs for spintronics together with the status of theoretical predictions and experimental observations of topological state or spintronics effects.
\begin{table}
\caption{Topological antiferromagnets}
\label{tab:1}
\begin{tabular}{p{1.5cm}p{3.5cm}p{1.5cm}p{1.5cm}p{5.5cm}}
\hline\noalign{\smallskip}
AF & Phase & T$_{N}$ & Space group & Representative effect \\ \hline
\noalign{\smallskip}\svhline\noalign{\smallskip}
FeSe & QSHE \cite{Wang2016e} & & P4/nmm & Superconductivity \\ \hline
GdPtBi & \textit{TI}\cite{Mong2010,Li2011,Li2015}/Weyl \cite{Hirschberger2016} & 9 \cite{Suzuki2016} & F$\overline{4}$3m & Large thermopower \cite{Hirschberger2016}, AHE \cite{Suzuki2016} \\ \hline
\noalign{\smallskip}\svhline\noalign{\smallskip}
SrMnBi$_{2}$ & Dirac metal\cite{Park2011a} & 290 \cite{Park2011a} & I4/mmm & Angular dependent magnetoresistance \cite{Wang2011e} \\ \hline
CaMnBi$_{2}$ & Dirac metal & 300 \cite{Guo2014} & P4/nmm & Dirac fermions coupled to magnetism \cite{Guo2014} \\ \hline
EuMnBi$_{2}$ & Dirac metal & 22* \cite{Masuda2016} & I4/mmm & QHE controlled by magnetism \cite{Masuda2016} \\ \hline
BaFe$_{2}$As$_{2}$ & 2D Dirac metal\cite{Chen2017} & 143 \cite{Huang2008} & I4/mmm & Superconductivity \\ \hline
CuMnAs & \textit{Dirac semimetal}\cite{Maca2012,Tang2016,Smejkal2016} & $\sim$400 \cite{Maca2012} & Pnma & \textit{TopoMIT, TopoAMR} \cite{Smejkal2016} \\ \hline \hline
X$_{2}$ Ir$_{2}$ O$_{7}$ & Weyl semimetal\cite{Wan2011,Sushkov2015} & & Fd$\overline{3}$m & TopoMIT \cite{Tian2015}, \textit{wealth of topo. phases} \cite{Wan2011,Kondo2015} \\ \hline
Mn$_{3}$Sn & Weyl (semi)metal \cite{Yang2016c,Kuroda2017} & 430 \cite{Nakatsuji2015} & P63/mmc & AHE controlled by magnetic field\cite{Nakatsuji2015} \\ \hline
\end{tabular}
$^a$ Italic font marks theoretical prediction. Normal font marks existing experimental signatures.
\label{tab2}
\end{table}
\subsection{Low dimensional Dirac antiferromagnets and superconductors}
Introducing magnetism into TIs is known to modify the spin texture of the Dirac quasiparticles \cite{Xu2012}. Magnetism can couple to TIs either by creating the magnetically doped TI (MTI) \cite{Xu2012}, or by proximity coupling between the TI and magnetic order in heterostructures \cite{Katmis2016}. AF order was shown to increase the critical temperature of the adjacent MTIs in the MTI/AF CrSb/MTI heterostructure \cite{He2016}. Improved performance of the SOT in terms of larger spin Hall angles and lower critical currents was achieved in the TI/ferrimagnetic CoTb alloys \cite{Han2017} with the AF coupled \cite{Finley2016} Co and Tb sublattices.
\begin{figure}[h]
\sidecaption
\includegraphics[width=1\linewidth]{Fig7r.pdf}
\caption{\textbf{Dirac quasiparticles in iron-based AF superconductors.} (a) Realization of QSHE in a monolayer of FeSe AF. Angle resolved photoemission spectroscopy data overlayed with \textit{ab initio} bands \cite{Wang2016e}.
(b) Topological quasi-2D Dirac quasiparticles in BaFe$_{\text{2}}$As$_{\text{2}}$ AF. (c) Single Dirac cone at M point in FeSe monolayer AF with a stripe order. Panel (a) adapted from \cite{Wang2016e}, panel (b) from \cite{Chen2017}, and panel (c) from \cite{Young2016}.}
\label{fig:FeSe}
\end{figure}
The first TI AF was predicted in systems with combined $\mathcal{T}T_{1/2}$ symmetry \cite{Mong2010}. GdPtBi AF was suggested as a possible candidate, as we show in Fig. \ref{fig:3}(b). GdPtBi was up to date not confirmed as a TI presumably due to the low resolution of data obtained by the angle resolved photoemission spectroscopy (ARPES) \cite{Liu2011f}. However, signatures of the coexistence of a 2D TI (Fig. \ref{fig:8}(c)), and a superconducting state in hole-doped and electron-doped antiferromagnetic monolayers of
FeSe were demonstrated \cite{Wang2016e}. FeSe belongs to the metallic building block of the iron-based high-T$_{\text{C}}$ superconductors. Remarkably, the combined effects of SOC, substrate strain, and electronic correlations can induce band inversion and QSHE edge states, as we show in Fig. \ref{fig:FeSe}(a) \cite{Wang2016e}. Creating a p-n junction across FeSe and attaching two ferromagnetic electrodes can generate Majorana zero modes at the interfaces \cite{Tsai2016}. Majorana states are considered for a possible use in quantum computing \cite{Beenakker2016}.
One of the first systems explored for observing Dirac quasiparticles in condensed matter beyond graphene were the SrMnBi$_{\text{2}}$ type AFs. The electronic structure of these systems is governed by the quasi-2D square Bi planes. The Bi states create close to the Fermi level massive Dirac quasiparticles. High mobilities, and Fermi velocities, and pseudospin structure of wavefunctions are reminiscent of graphene properties. In contrast to graphene, however, the quasiparticles are highly anisotropic with anisotropy factor of $\sim$ 8 \cite{Wang2011e}. Several of these types of AFs were reported in recent years including SrMnBi$_{\text{2}}$ \cite{Wang2011e}, or CaMnBi$_{\text{2}}$ \cite{Guo2014}. The systems belong to the 112-type pnictides where the antiferromagnetism and Dirac quasiparticles might coexist also with superconductivity. Related systems, e.g. YbMnBi$_{\text{2}}$, were inconclusively \cite{Armitage2017} reported to be either WSM \cite{Borisenko2015,Chinotti2016} or DSM \cite{Wang2016k,Chaudhuri2017}. Most likely the collinear AF order cants in, e.g., the (Sr,Yb)MnBi$_{\text{2}}$ alloy \cite{Liu2017} where the double band degeneracy breaks and Weyl points might emerge. Despite many recent studies, more accurate and detailed measurements are needed to reveal the detailed nature of Dirac quasiparticles in these systems. Finally, in the sister compound, EuMnBi$_{\text{2}}$ AF, the half-integer QHE was reported controllable by the strength of an external magnetic field, as we showed in Fig. \ref{fig:8}(b). EuMnBi$_{\text{2}}$ contains at very low temperatures two antiferromagnetic sublattices. The presence of QHE was linked to the confinement of the massive Dirac quasiparticles by the spin-flop at the Eu AF sites \cite{Masuda2016}.
Recently, the high temperature superconducting 122-type pnictide XFe$_{\text{2}}$As$_{\text{2}}$ (X=Ba, or Sr) AFs showed signatures of topological Dirac quasiparticles in the infrared spectra in high magnetic fields. In Fig. \ref{fig:FeSe}(b) we show the state-of-the-art DFT+DMFT calculation of quasi-2D Dirac cones close to the Fermi level which are consistent with the observed Landau level spectra and density of states \cite{Chen2017}.
When the Fermi states are dominated by Dirac quasiparticles, the topological semimetal is achieved. Electron filling enforced semimetals with a single Dirac cone were predicted theoretically in 2D model AFs \cite{Young2016}. In Fig. \ref{fig:FeSe}(c) we show the single Dirac cone at the $M$ point in the BZ of the monolayer of the FeSe AF with a stripe order. The quasi-low dimensional systems and heterostructures, however, suffer from fragile magnetism and low critical temperatures. In two following subsection, we describe possible room temperature 3D Dirac and Weyl semimetal AFs.
\subsection{3D Dirac semimetal antiferromagnets}
\begin{figure}[h]
\sidecaption
\includegraphics[width=1\linewidth]{Fig8.pdf}
\caption{\textbf{Antiferromagnetic Dirac semimetals, spin-orbit torques, and metal-insulator transition.} (a) Unit cell of orthorhombic CuMnAs AF with marked $\mathcal{PT}$ symmetry center (black sphere) and nonequilibrium spin polarizations $\delta\textbf{s}$ for current applied along the [010]. (b) Band structure of the CuMnAs AF calculated \textit{ab initio} without SOC. (c) The density of states. (d) Detail of Dirac quasiparticles in CuMnAs as calculated \textit{ab initio} with SOC switched on. Panel (a) adapted from Ref.\cite{Smejkal2017a}, and panels (b-d) adapted from Ref.\cite{Smejkal2016}.}
\label{fig:9}
\end{figure}
Dirac quasiparticles are allowed in doubly-degenerate bands \cite{bernevig2013topological} realized in systems invariant under $\mathcal{PT}$ symmetry.
\begin{svgraybox}
The low energy Hamiltonian might maintain an effective Dirac form \cite{Vafek2013,Yang2014a,Smejkal2017}, corresponding to the Dirac Hamiltonian \eqref{KSD} \cite{Burkov2016,Smejkal2017,Armitage2017}:
\begin{align}
\mathcal{H}_{D}(\textbf{k}) =
\left(\begin{matrix}
\hbar v_{F}\textbf{k}\cdot \boldsymbol\sigma & m \\
m & -\hbar v_{F}\textbf{k}\cdot \boldsymbol\sigma
\end{matrix}\right).
\label{MDirac}
\end{align}
Here $v_{F}$ is the Fermi velocity, $\textbf{k}=\textbf{q}-\textbf{q}_{0}$ is the crystal momentum measured from the Dirac point at $\textbf{q}_{0}$, $m$ is the mass (in units of energy), and $\boldsymbol\sigma$ is the vector of Pauli matrices. The energy dispersion gives massive Dirac cones,
\begin{equation}
E(\textbf{k})=\pm \hbar v_{\text{F}}\sqrt{k_{x}^{2}+k_{y}^{2}+k_{z}^{2}+\left(\frac{m}{\hbar v_{F}}\right)^2}.
\end{equation}
\end{svgraybox}
The mass term can be removed by the presence of an additional crystalline symmetry,
and in this case $\mathcal{H}_{D}(\textbf{k})$ describes the
four-fold degenerate band touching \cite{Burkov2016,Smejkal2017} of a
3D DSM as we show in Fig. \ref{fig:9}(b) on the band dispersion of an antiferromagnetic DSM orthorhombic CuMnAs with a high N\'{e}el temperature of $\sim$ 400\,K \cite{Maca2012}.
The 3D DSM state cannot occur in ferromagnets because $\mathcal{T}$-symmetry breaking prevents double band degeneracy. However, a topological crystalline 3D DSM was predicted in an AF, namely in the orthorhombic phase of CuMnAs \cite{Tang2016,Smejkal2016}. The unit cell of the orthorhombic CuMnAs contains four Mn sublattices that are connected in pairs by the $\mathcal{PT}$ symmetry \cite{Tang2016,Smejkal2016} as we show in Fig. \ref{fig:9}(a). Although individually the $\mathcal{P}$ and $\mathcal{T}$ symmetries are broken, the preserved combined $\mathcal{PT}$ symmetry ensures the double band degeneracy over the whole BZ. In the calculation with a switched-off SOC (see Fig. \ref{fig:9}(b)), we observe three Dirac points at the Fermi level along the $\Gamma−X$, $X−U$, and $Z−X$ lines which are part of the nodal line protected by the $\mathcal{PT}$ symmetry. The Dirac quasiparticles are 3D as can be seen from the quadratically vanishing DOS at the Fermi level/Dirac point as we show in Fig. \ref{fig:9}(c). In the presence of SOC and for the N\'{e}el order along the [001] axis, the nodal lines become gapped except for the two Dirac points along the $UXU$ line, as we show in Fig. \ref{fig:9}(d). The Dirac points are protected by the non-symmorphic screw axis symmetry $\mathcal{S}ß_{z}=\left\lbrace 2_{z} \vert (\frac{1}{2},0,\frac{1}{2}) \right\rbrace$ \cite{Tang2016,Smejkal2016} and are connected via nontrivial surface states \cite{Tang2016}. The topological invariants and surface states can be linked to the crystalline symmetry protecting the degeneracy and even in the non-magnetic DSM, the surface states are in general less stable than in WSM and strongly depend on the crystalline orientation at the surface termination \cite{Yang2014a,Kargarian2016}. The easy axis in orthorhombic CuMnAs tends to be along [100] according to \textit{ab initio} calculation \cite{Smejkal2016}, however, we will discuss in the next section the possibility of reorienting the N\'{e}el vector.
The orthorhombic CuMnAs AF is an attractive {\it hydrogen atom} for magnetic DSMs induced by band inversion since only a single pair of Dirac points appears near the Fermi level of the {\em ab initio} band structure. However, presumably, the correlation and disorder effects prevented the observation of Dirac quasiparticles in non-stoichiometric CuMnAs to date \cite{Emmanouilidou2017,Zhang2017e}.
\subsection{Weyl semimetal antiferromagnets}
In solids, quite often at least one of the $\mathcal{P}$ or $\mathcal{T}$ symmetries (and also $\mathcal{PT}$) is broken and thus the double band degeneracy is lifted. When the two non-degenerate bands are touching close to the Fermi level a 3D WSM can be formed.
\begin{svgraybox}
WSM is described by the generalized two-band Weyl Hamiltonian \cite{Burkov2016,Smejkal2017}:
\begin{equation}
\mathcal{H}_{W}(\textbf{k})=\epsilon_{0}\pm\hbar v_{\text{F}}\left( \textbf{q} - \textbf{q}_{0} \right)\cdot \boldsymbol{\sigma},
\label{Weyl}
\end{equation}
where the first term corresponds to the tilting of the Weyl cone and $\textbf{k}=\textbf{q} - \textbf{q}_{0} $. Weyl points always come in pairs with opposite topological charges. Dimensionality is important here. Because the Weyl points are 3D objects in the BZ, the effective Hamiltonian uses all three Pauli matrices. Thus any small perturbation expressed without loss of generality as a linear combination of these three Pauli matrices just shifts but not gaps the Weyl point. We illustrate this on the dispersion around the Weyl points for a perturbation of a form $m\sigma_{z}$,
\begin{equation}
E(\textbf{k})=\epsilon_{0}\pm \hbar v_{\text{F}}\sqrt{k_{x}^{2}+k_{y}^{2}+\left(k_{z}+\frac{m}{\hbar v_{\text{F}}}\right)^{2}}.
\label{WSMdisp}
\end{equation}
\end{svgraybox}
WSM states can be found in nonmagnetic, ferromagnetic \cite{Xu2011,Wang2016c} or antiferromagnetic solids, where the $\mathcal{PT}$ symmetry is broken. A WSM state was observed in non-centrosymmetric non-magnetic mono-pnictides of the TaAs type \cite{Xu2015b,Lv2015,Yang2015d}. TaAs is well described by the DFT single quasiparticle picture. Despite numerous predictions, the true magnetic WSM remained for a long time experimentally elusive \cite{Kuroda2017}. The reason is that the magnetic system is very often also strongly correlated, disordered, and the symmetry breaking is provided by the complex collective phenomenon - magnetism. Antiferromagnetic candidates include pyrochlore irridates \cite{Wan2011} like the Eu$_{2}$Ir$_{2}$O$_{7}$ \cite{Sushkov2015}, or YbMnBi$_{2}$ AF which were suggested to be either the WSM \cite{Borisenko2015,Chinotti2016} or the DSM \cite{Wang2016k,Chaudhuri2017}.
\begin{figure}[h]
\sidecaption
\includegraphics[width=1\linewidth]{Fig9.pdf}
\caption{\textbf{Antiferromagnetic Weyl semimetal and surface Fermi arcs.}
(a) \textit{Ab initio} calculation of Fermi arcs in Mn$_{3}$Sn AF. (b) ARPES of Weyl points in Mn$_{3}$Sn close to the Fermi level overlayed with \textit{ab initio} band structure. Panel (a) adapted from Ref. \cite{Yang2016c} and panel (b) from \cite{Kuroda2017}.}
\label{fig:10}
\end{figure}
Weyl fermions were proposed also in non-collinear AFs Mn$_{3}$Ge and Mn$_{3}$Sn \cite{Kubler2014,Yang2016c} (see Fig. \ref{fig:2}(a)) \cite{Yang2016c}.
These AFs are potentially appealing for spintronics due to the measured large AHE, established magnetic structure, and N\'{e}el temperatures reaching 365-420 K.
The structure of Mn$_{3}$Ge and Mn$_{3}$Sn crystals is built from stacked kagome planes along the [001] axis, as we show in the right panel in Fig. \ref{fig:5}(MPG III).
These AFs have a relatively weak magnetic anisotropy, reaching 0.1meV per formula unit for Mn$_{3}$Sn \cite{Sandratskii1996,Duan2015} due to the vanishing second and fourth order MAE of the inverted triangular AF structure on the kagome lattice \cite{Nagamiya1982,Tomiyoshi1982}. The magnetic order can be thus reoriented by low external magnetic fields. Ref. \cite{Kuroda2017} reports reorientation fields of $\sim$ 200 Gauss. The net magnetic moment reaches 0.005 $\mu$B per unit cell \cite{Tomiyoshi1982}. In spite of the weak anisotropy of the inverted chiral structure, the materials show relatively high stability against thermal fluctuations. Also, a possibility to influence the in-plane chiral AF magnetic structure by a spin-filtering effect was reported \cite{Fujita2017}.
Mn$_{3}$Ge and Mn$_{3}$Sn were predicted to exhibit several different types of Weyl points in their metallic bandstructure coexisting with trivial bands close to the Fermi level \cite{Yang2016c}. The Weyl points found by tracking the Berry curvature in the BZ are tilted, thus of the so-called type-II \cite{Soluyanov2015}. The Fermi arc surface states - the hallmark of a WSM - were predicted by first-principles calculations of the local density of states (LDOS) as we show in Fig. \ref{fig:10}(a) \cite{Yang2016c}. The tilting of the Weyl points does not influence the Berry curvature, however, the electron and hole pockets due to the tilting influence the transport effects, particularly they can renormalize the almost perfectly quantized AHE \cite{Zyuzin2016a}.
Signatures of these band crossings were reported recently in an ARPES study of Mn$_{3}$Sn AF \cite{Kuroda2017}, as we show in Fig. \ref{fig:10}(b). Weyl band crossings were found along $MK$ or $M'K$ lines presumably depending on the orientation of the triangular magnetic texture stabilized by an external magnetic field \cite{Kuroda2017}. A comparison between DFT and ARPES points towards strong electron correlations in the Mn 3d bands as seen in the strong renormalization of the bands in ARPES in Fig. \ref{fig:10}(b). In the next Section, we describe in detail the AHE in Mn$_{3}$Ge and Mn$_{3}$Sn and its relation to Weyl points.
\section{Topological antiferromagnetic spintronics effects}
Topological variants of common magnetotransport effects and their novel cousins can potentially offer large signal to noise ratios important for reading signals in spintronics nanodevices. Again, we will demonstrate that very often the unique antiferromagnetic symmetries are of vital importance for certain topological spintronics effects to occur, such as topological AMR or AHE.
\subsection{Large magnetoresistance and chiral anomaly}
\begin{figure}[h]
\sidecaption
\includegraphics[width=1\linewidth]{Fig10.pdf}
\caption{\textbf{Possible observation of a chiral anomaly in magnetically induced Weyl semimetal and antiferromagnetic Weyl semimetal.} (a) Cartoon of the chiral anomaly principle. (b) A magnetically induced Weyl semimetal in GdPtBi. (c) Observed negative magnetoresistance in GdPtBi. (d) Observed positive magnetoconductance in the Mn$_{3}$Sn AF. Panels (a,b) adapted from \cite{Yan2016a}, panel (c) from \cite{Hirschberger2016}, and panel (d) from \cite{Kuroda2017}.}
\label{fig:11}
\end{figure}
Application of a magnetic field perpendicular to the current flow results in a positive magnetoresistance, defined as
\begin{equation}
\text{MR}=\frac{\rho(B)-\rho(0)}{\rho(0)},
\end{equation}
where $\rho(B)$ is the resistivity in a magnetic field $B$, as commonly observed in metals, semimetals, and semiconductors. However, in a topological semimetal with Weyl quasiparticles, a negative magnetoresistance might occur, attributed to a chiral anomaly. In the original proposal for a consensed matter realization of the chiral anomaly, Nielsen and Ninomiya \cite{Nielsen1983} considered a chiral Weyl linear dispersion of the zero-th Landau levels. Application of an electric field parallel to the magnetic field generates an imbalance between the zero Landau levels of opposite chiralities. This axial current leads to a positive magnetoconductivity \cite{Armitage2017}:
\begin{equation}
\sigma(B)=\sigma+\frac{e^{4}B^{2}}{4\pi^{4}g(E_{F})},
\end{equation}
where $g(E_{F})$ is the density of states at the Fermi level. Remarkably, this expression can be derived both in the quantum limit or in the semi-classical framework without introducing Landau levels \cite{Armitage2017}. We illustrate the chiral anomaly with Weyl fermions in Fig. \ref{fig:11}(a).
The negative magnetoresistance became accepted as the signature of the presence of linearly dispersing topological quasiparticles, and was possibly observed for instance in GdPtBi \cite{Hirschberger2016}, which is a quadratic gapless semiconductor. In the magnetic field, the fourfold degenerate band-touching splits and pairs of Weyl points are created as we illustrate in schematics in Fig. \ref{fig:11}(b). Rotating the external magnetic field from out-of-plane to in-plane (see Fig. \ref{fig:11}(c)) changes the magnetoresistance from positive to negative. Positive non-saturating magnetoconductance was observed recently also in the Mn$_{3}$Sn AF \cite{Kuroda2017}. Alternative sources of positive magnetoconductance such as current jetting and weak localization were carefully ruled out in this study. The positive magnetoconductance in Mn$_{3}$Sn is linear in a magnetic field which was attributed to the type-II Weyl fermions \cite{Kuroda2017} in contrast to the quadratic magnetoconductance observed in type-I (non-tilted) Weyl semimetals \cite{Zyuzin2017}. The detailed role of topological quasiparticles in the negative and large non-saturating magnetoresistance remains to be clarified \cite{Ali2014,Pletikosic2014,Soluyanov2015,Khouri2016}. Here the first step was made by \textit{ab initio} \cite{Kim2017e} and transport studies \cite{Zhang2017f} of Weyl points in strong magnetic fields signaling the importance of linear dispersion for the observation of negative magnetoresistance. In strong magnetic fields, the negative magnetoresistance disappears which was attributed to the gapping of Weyl points by the Zeeman splitting.
\subsection{Topological phase transitions and anisotropic magnetoresistance in antiferromagnetic systems}
Topological phase transitions were experimentally demonstrated in heterostructures with TIs and AFs \cite{He2016,He2016a} or systems with an artificially engineered AF coupling \cite{Mogi2017}. AF CrSb/TI (Bi,Sb)$_{\text{2}}$Te$_{\text{3}}$ /AF CrSb heterostructure shows spikes in the magnetoresistance which were attributed to a topological phase transition of Dirac quasiparticles at the interfaces \cite{He2016a}. A MTI/TI/MTI heterostrucutre was reported for a presumed topological phase transition between QAHE state and axion insulator (quantized topological magnetoelectric effect) by switching the magnetic order in the MTI from ferromagnetic to antiferromagnetic by an external magnetic field or electric gating \cite{Mogi2017,Tokura2017}. Albeit at mK temperatures, the phase transition yields very large magnetoresistance or electroresitance changes corresponding to switching on and off the quantized conductivity plateous $h/e^{2} \sim 25,8 k\Omega$ \cite{Mogi2017,Tokura2017}. The QAHE effect was to date observed only at mK temperatures \cite{Wang2017b}. Searching for novel mechanism and material candidates with a more robust, controllable and room temperature QAHE states represents an important direction of future research in topological spintronics. Here for instance, QAHE induced by electrical gating was predicted in Sr$_{\text{2}}$FeOsO$_{\text{6}}$ AF \cite{Dong2016}.
\begin{figure}[h]
\sidecaption
\includegraphics[width=1\linewidth]{Fig11r.pdf}
\caption{\textbf{Extreme anisotropies in magnetoresistance effects in topological semimetals.} (a) Large topological anisotropic magnetoresistance in a tetragonal AF Dirac semimetal model. (b) Band structure of the model Dirac AF. (c) Angular dependence of magnetoresistance in a magnetically induced Weyl semimetal GdPtBi. (d) Angular dependence of magnetoresistance in AF Weyl semimetal Mn$_{3}$Sn. Panel (b) adapted from Ref\cite{Smejkal2016}, panel (c) from \cite{Hirschberger2016}, and panel (d) from \cite{Kuroda2017}.}
\label{fig:12}
\end{figure}
The prediction of tuning the N\'{e}el order parameter by the N\'{e}el SOT or gating in antiferromagnetic DSMs opens the possibility of using the topological metal-insulator transition (TopoMIT) in a bulk AF \cite{Smejkal2016,Smejkal2017a}. The origin of these effects is in the sensitivity of Dirac crossing hybridations on the orientation of the N\'{e}el vector in the orthorhombic magnetic crystalline symmetry, as shown in Fig. \ref{fig:9}(d). In the presence of SOC, only for the N\'{e}el vector along the [001] axis, protected Dirac fermions emerge. All the other N\'{e}el vector orientations lead to a gapped spectrum at the Fermi level \cite{Smejkal2016}. The transport counterpart of the topoMIT was predicted to be the topological anisotropic magnetoresistance (topoAMR) \cite{Smejkal2016}. The topoAMR can be extremely large and is understood as a limiting case of the crystalline AMR. In Fig. \ref{fig:12}(a) we show the tetragonal lattice of a minimal model of the DSM AF \cite{Smejkal2016}. The corresponding Hamiltonian reads $H_{\textbf{k}}=-2t\tau_{x}\cos\frac{k_{x}}{2}\cos\frac{k_{y}}{2}-t'\left(\cos k_{x}+\cos k_{y}\right) +
\lambda \tau_{z}\left( \sigma_{y} \sin k_{x} - \sigma_{x} \sin k_{y} \right) + \tau_{z}J_n\boldsymbol\sigma\cdot\textbf{n}$, where first two terms are first and second neighbor hoppings, third term is a staggered SOC, last term is AF s-d type exchange and $\tau$, and $\sigma$ are Pauli matrices corresponding to orbital and spin-degree of freedom, respectively. The longitudinal conductivity is calculated from the Boltzmann formula in the limit of a small spectral broadening $\Gamma$:
%
\begin{equation}
\sigma_{xx}(\phi)=\frac{e^{2}}{\hbar 4\Gamma L^{2}}\sum_{\textbf{k}n}\frac{\partial E_{\textbf{k}n}}{\partial k_{x}}\frac{\partial E_{\textbf{k}n}}{\partial k_{x}}\delta(E-E_{F}),
\end{equation}
where $L$ is the size of the system, and $\phi$ is the angle between [100] axis and magnetization. The AMR is defined as:
\begin{equation}
\text{AMR}=-\frac{\sigma_{xx}(\phi)-\sigma}{\sigma},
\end{equation}
where $\sigma$ is the average conductivity within the plane. In Fig.\ref{fig:12}(a) we show the angular dependence of the AMR in this model \cite{YamamotoSmejkal2017}. In Fig. \ref{fig:12}(b) we show the band structure of the model. For the N\'{e}el vector orientations [100] and [010], preserving the glide mirror planes of the system \cite{Smejkal2016}, the Dirac points are gapless and conduct. Once the N\'{e}el vector is rotated away from these high symmetry axes, the crystalline symmetries are broken, Dirac bands hybridize and gap opens. Consequently the conductivity decreases exponentially. The sharp peaks in the angular dependence are very different when comparing to the standard harmonic AMR dependence in ferromagnetic alloys. Also the origin is very distinct. The topoAMR originates in Fermi surface topology changes instead of scattering effects responsible for the standard AMR.
The difference in conductivity between the [100] and [010] direction originates in the anisotropy of the Dirac cones. The orthorhombic CuMnAs was predicted as the realistic material candidate based on \textit{ab initio} calculations \cite{Smejkal2016}. The interplay of the Dirac points and topoAMR with disorder, interaction effects, and nonequilibrium currents needs to be carefully addressed to potentially make the effect relevant for real spintronics device applications. Foreseen applications include topological transistors or memories \cite{Smejkal2017a}.
Since the control of the N\'{e}el vector can be achieved either by the N\'{e}el SOT due to the applied current or due to the tuning of the MAE by electric gating, the effect is presumably more favorable for spintronics than the MIT manipulated by external magnetic field in pyrochlores \cite{Tian2015} or AF topological semimetal candidate NdSb \cite{Wakeham2016}. Although this topoAMR due to the MIT was not experimentally discovered yet, analogical effects controlled by external magnetic field were observed in WSMs.
In GdPtBi, Weyl point positions are sensitive to the orientation of the applied magnetic field \cite{Hirschberger2016}. This leads in turn to a pronounced angular dependence of the magnetoconductance, as we show in Fig. \ref{fig:12}(c).
The changes in magnetoconductance are attributed to the varying angle between the crystalline axis and the Zeeman field what is in contrast to the behavior predicted for the AF DSM CuMnAs. The spikes in magnetoconductance have been measured also in the correlated WSM AF Mn$_{3}$Sn as we illustrate in Fig. \ref{fig:12}(d) \cite{Kuroda2017}. Here the magnetic order is controlled by the relatively weak external magnetic field. The reorientation of the moments changes the local symmetry and possibly redistributes the Weyl points close to the Fermi level \cite{Kuroda2017}. Importantly, the spikes in magnetoconductance persist to temperatures $\sim$100\,K, despite the WSM is highly correlated and disordered. These temperatures are much higher temperatures than the reported QAHE critical temperatures of $\sim$10\,mK.
\subsection{Anomalous Hall effect in noncollinear antiferromagnets}
AHE refers to the transversal electric current generation in the magnet subjected to an applied longitudinal electric field. The anomalous Hall conductivity is for the magnetization along $z$ axis the antisymmetric part of the conductivity tensor:
\begin{equation}
\sigma_{\text{AHE}}=\frac{\sigma_{xy}-\sigma_{yx}}{2}.
\end{equation}
For a long time, the AHE was considered to scale with the magnetization: $\rho_{H}=R_{0}H_{z}+R_{S}M_{z}$, where the first part corresponds to the ordinary Hall effect due to the external magnetic field $H_{z}$, and the second term is the AHE due to the $\cal{T}$ symmetry breaking due to the magnetization $M_{z}$, and $R_{0}$, and $R_{S}$ are ordinary and (spontaneous) anomalous Hall coefficients, respectively. AHE was traditionally attributed to the simultaneous presence of $\cal{T}$ symmetry breaking by the ferromagnetism and SOC. Thus, naively, one would expect that the AHE must vanish in AFs because of the compensation moments of the opposite sublattices. Indeed, this picture is valid in simple AFs where the combination of $\cal{T}$ symmetry with another crystalline symmetry forces the AHE to vanish. Typical examples include collinear AFs with $\mathcal{PT}$ or $\mathcal{T}T_{1/2}$ symmetries which we discussed in the context of the Dirac quasiparticles and TIs.
Remarkably, the AHE was observed in systems with a negligible net magnetization and without the necessity for SOC.
Interestingly, already Haldane pointed out in 1988 \cite{Haldane1988} the possibility of the quantized AHE in honeycomb lattice with complex hoppings with a staggered potential and Shindou et al. \cite{Shindou2001} later demonstrated the nonzero AHE in a model calculation in AHE induced by distorting the FCC lattice with 3Q AF order. More recently Hua Chen et al. \cite{Chen2014} and K\"{u}bler and Felser \cite{Kubler2014} predicted a large AHE in noncollinear AFs with a negligible net moment. Presumably, the large MAE in Mn$_{3}$Ir AF makes it impossible to orient the magnetic domains and thus prevents the experimental detection of the AHE in this compound.
However, Mn$_{3}$Ge and Mn$_{3}$Sn have a much smaller MAE. AHE from the AF texture, $\rho_{H}^{AF}$, was indeed observed in these compounds by carefully substracting the Hall effect originating from the external field, $R_{0}H_{z}$, and from the small ferromagnetic moment, $R_{S}M$ \cite{Nakatsuji2015,Nayak2016}:
\begin{equation}
\rho_{H}=R_{0}H_{z}+R_{S}M+\rho_{H}^{AF},
\label{hall_exp_af}
\end{equation}
where $M$ is the net magnetization. The experimental value of the AHE in Mn$_{3}$Ge is $\sigma_{xz}$ $\approx$ 380\,$\Omega^{-1}\text{cm}^{-1}$ \cite{Kiyohara2015} while the \textit{ab initio} calculation from Berry curvature gives $\sigma_{xz}\approx$ 330\,$\Omega^{-1}\text{cm}^{-1}$ \cite{Zhang2016d}.
The noncollinear AF order on the kagome lattice breaks the time-reversal symmetry as we show in Fig. \ref{fig:13}(a) and \ref{fig:13}(b). For the AF structure in Fig.~\ref{fig:13}(a) \cite{Tomiyoshi1982,Kiyohara2015} there is an effective time reversal symmetry $\mathcal{T}\mathcal{M}_{x}$ ($\mathcal{M}_{x}$ is the mirror (100) plane symmetry) which gives $\sigma_{yz}=0$, and the emergent magnetic field lies along $\textbf{B}\parallel$[010], and only $\sigma_{xz}\neq 0$. In constrast, for the chiral texture in Fig.~\ref{fig:13}~(b) \cite{Tomiyoshi1982,Kiyohara2015}, the effective $\mathcal{T}\mathcal{G}_{y}$ symmetry ($\mathcal{G}_{y}$ is the glide mirror plane $\left\lbrace \mathcal{M}_{y} \vert (0,0,\frac{1}{2}) \right\rbrace$) implies $\sigma_{xz}=0$, the emergent magnetic field points along $\textbf{B}\parallel$[100], and only $\sigma_{zy}\neq 0$. Furthermore, independent on the in-plane orientation, there is an effective time reversal symmetry $ \mathcal{T}\mathcal{M}_{z}$ ($\mathcal{M}_{z}$ is the mirror (001) plane symmetry) making the component
$\sigma_{xy}=0$. The symmetry analysis is consistent with the experimental data measured on Mn$_{3}$Sn and Mn$_{3}$Ge and presented in Fig. \ref{fig:2}(b) and \ref{fig:13}(c). In conclusion, the spin-orbit entangled bands generate a large fictitious magnetic field in the crystal momentum space parallel to the direction of the field stabilizing the triangular order, and the AHE takes place in the plane perpendicular to the field.
\begin{figure}[h]
\sidecaption
\includegraphics[width=1\linewidth]{Fig12.pdf}
\caption{\textbf{Anomalous and topological Hall effects in chiral AFs.} Two chiral inverted AF structure stabilized by an external magnetic field along (a) [010], and (b) [100] direction. (c) Measured temperature dependence of AHE in Mn$_{3}$Ge originating from AF texture. (d) Spin chirality and non-coplanar magnetic moments of an antiferromagnetic candidate for the quantum topological Hall effect. (e) Topological Hall effect observed in the spin liquid state on a pyrochlore lattice - fragment shown in the inset. Panels (a,b) adapted from \cite{Smejkal2017}, panel (c) from \cite{Kiyohara2015}, and panel (e) from \cite{Machida2010}.}
\label{fig:13}
\end{figure}
The emergent magnetic field was estimated to be very large, of the order of $\sim$ 200\,T in Mn$_{3}$Ge \cite{Kiyohara2015}. Although the Mn$_{3}$Ge and Mn$_{3}$Sn were predicted to host Weyl points close to the Fermi level, the \textit{ab initio} calculation of the AHE shows that the largest contributions come from BZ regions not related to any identified Weyl points, but rather from spin-orbit entangled avoided crossings \cite{Zhang2016d}. A recent study by K\"{u}bler and Felser \cite{Felser2017}, however, demonstrates the possibility of propagation of Fermi sea Weyl points in Mn$_{3}$Ge and Mn$_{3}$Sn to the Fermi surface-quasiparticle transport \cite{Haldane2004,Gos2015}.
\subsection{Topological Hall effects and antiferromagnetic skyrmions}
In the topological Hall effect, the role of SOC is overtaken by the spin chirality. We show in Fig.\ref{fig:13}(d) the spin chirality generating a nonzero Berry curvature. Spin chirality is nonzero in non-coplanar spins, in contrast, it vanishes in coplanar non-collinear antiferromagnetic structures of, e.g., Mn$_{3}$Ge and Mn$_{3}$Sn. The spin chirality generates a fictitious magnetic field (see red arrow in Fig. \ref{fig:13}(d)), $\hat{\textbf{m}}\cdot \left(\partial_{x}\hat{\textbf{m}} \times \partial_{y}\hat{\textbf{m}}\right)$. This field acting on the Bloch electrons generates a Hall response. The topological Hall effect and the AHE can be possible to experimentally disentangled by analyzing the disorder dependence \cite{Kanazawa2011}. The topological Hall effect was initially reported in antiferromagnetic pyrochlore iridates (see Fig.\ref{fig:13}(e))
\cite{Machida2010} and later in MnSi chiral antiferromagnetic alloys \cite{Surgers2014,Surgers2016}. We note that the effect does not imply in this case a correspondence to a topological invariant, in sharp contrast to the topological Hall effect in skyrmions. However, in the quantum topological Hall effect proposed for the non-coplanar AF K$_{0.5}$RhO$_{2}$ \cite{Zhou2016}, with its magnetic sublattices shown schematically in Fig.\ref{fig:13}(d), the topological charge occurs in the momentum space as in the QAHE.
The topological Hall effect from a skyrmion spin texture is associated with a topological winding number of the skyrmion \cite{Tokura2017}. It is important to distinguish it from the skyrmion Hall effect which refers to the deflection of skyrmion center due to the Magnus force. The Magnus force is according to micromagnetic calculations not present in AF skyrmions, which implies that AF skyrmions might move in straight lines \cite{Barker2016}. This is favorable when considering skyrmions for storing information in racetracks.
Finally, the topological Hall effect in AF skyrmions with a compensating sublattices vanishes, while the topological spin Hall effect can be still sizable and can be used to detect an AF skyrmion \cite{Buhl2017,Gobel2017}.
\section{Summary}
Antiferromagnetic spintronics has been recently established as a new branch of magnetism \cite{MacDonald2011,Jungwirth2016,Smejkal2017,Sander2017a}. In parallel, last few years have seen progress in coupling magnetism with topological states of matter, giving rise to a new spin-off: topological spintronics \cite{Fan2016b}. We have shown that AF order might play an important role in topological spintronics due to the unique AF symmetries \cite{Smejkal2017a}. While signatures of correlated AF WSM were already observed \cite{Kuroda2017}, other topological AF phases remain to be discovered. The large signal to noise ratio was reported in magnetoresistance signals of topological semimetals \cite{Smejkal2016,Kuroda2017}. Further theoretical and experimental progress will possibly lead to topological spintronics effects improving the reading and writing signals in AFs \cite{Smejkal2017a}. The progress in writing efficiency due to the nontrivial topologies is in its infancy, although an increase of the spin Hall angle in TI/FM\cite{Mellnik2014}, TI/MTI\cite{Fan2014a}, or TI/ferrimagnetic systems \cite{Han2017} was already reported and novel mechanisms for dissipationless SOT were suggested \cite{Hanke2017a}. Here we focused on the state-of-the-art effects which were predicted, and some of them already experimentally confirmed, in antiferromagnetic systems. The unique AF symmetries and the abundancy of AF allow for other research directions to emerge such as topological superconductivity in AFs and spintronics based on antiferromagnetic skyrmions \cite{Smejkal2017a}.
\begin{acknowledgement}
We acknowledge support from the Ministry of Education of the Czech Republic Grants LM2015087 and LNSM-LNSpin, the Grant Agency of the Czech Republic Grant No. 14-37427, and the EU FET Open RIA Grant No. 766566.
\end{acknowledgement}
\bibliographystyle{spphys}
|
{
"timestamp": "2018-04-17T02:16:18",
"yymm": "1804",
"arxiv_id": "1804.05628",
"language": "en",
"url": "https://arxiv.org/abs/1804.05628"
}
|
\section{Introduction}
Recent successes in machine learning, especially deep learning, rely
greatly on the training processes that operate on hundreds, if not
thousands, of labeled training instances for each class. However,
in practice, it might be extremely expensive or even infeasible to
obtain many labelled samples, \textit{e.g.} for rare objects or objects
that may be hard to observe. In contrast, humans can easily learn
to recognize a novel object category after seeing only few training
examples \cite{Thrun96learningto}. Inspired by this ability, few-shot
learning aims to build classifiers from few, or even a single, examples.
The major obstacle of learning good classifiers in a few-shot learning
setting is the lack of training data. Thus a natural recipe for few-shot
learning is to first augment the data in some way. A number of approaches
for data augmentation have been explored. The dominant approach, adopted
by the previous work, is to obtain more images \cite{KrizhevskySH12}
for each category and use them as training data. These additional
augmented training images could be borrowed from unlabeled data \cite{transductiveEmbeddingJournal}
or other relevant categories \cite{yuxiong2016eccv,yuxiong2016nips,zhizhong2016eccv,lim2011nips}
in an unsupervised or semi-supervised fashion. However{, the augmented
data that comes from related classes is often semantically noisy and
can result in the \emph{negative transfer} which leads to reduced
(instead of improved) performance. On the other hand, synthetic images
rendered from virtual examples \cite{Movshovitz2015phd,Park2015cvpr,attias2015cvpr,Dosovitskiy2015cvpr,zhu2015ijcv,opelt2006alphabet}
are semantically correct but require careful domain adaptation to
transfer the knowledge and features to the real image domain. To avoid
the difficulty of generating the synthesized images directly, it is
thus desirable to augment the samples in the feature space itself.
For example, the state-of-the-art deep Convolutional Neural Networks
(CNNs) stack multiple feature layers in a hierarchical structure;
{we hypothesize that feature augmentation can, in this case, be done
in feature spaces produced by CNN layers.
Despite clear conceptual benefits, feature augmentation techniques
have been relatively little explored. The few examples include \cite{zhu2015ijcv,opelt2006alphabet,AGA_2017}.
Notably, \cite{zhu2015ijcv} and \cite{opelt2006alphabet} employed
the feature patches (\emph{e.g.} HOG) of the object parts and combined
them to synthesize new feature representations. Dixit {\em et al.}
\cite{AGA_2017}, for the first time, considered attributes-guided
augmentation to synthesize sample features. Their work, however, utilizes
and relies on a set of pre-defined semantic attributes.
A straightforward approach to augment the image feature representation
is to add random (vector) noise to a representation of each single
training image. However, such simple augmentation procedure may not
substantially inform/improve the decision boundary. Human learning
inspires us to search for related information in the concept space.
Our key idea is to leverage additional semantic knowledge, \emph{e.g.}
encapsulated by the semantic space pre-trained using the linguistic
model such as Google's word2vec \cite{distributedword2vec2013NIPS}.
In such semantic manifold similar concepts tend to have similar semantic
feature representations. The overall space demonstrates semantic continuity,
which makes it ideal for feature augmentation.
To leverage such semantic space, we propose a dual TriNet architecture
($g\left(\mathbf{x}\right)=g_{Dec}\circ g_{Enc}\left(\mathbf{x}\right)$)
to learn the transformation between the {image} features {at}
multiple layers and the semantic {space}. The dual TriNet is {paired}
with the 18-layer residual net (ResNet-18) \cite{he2015deep}; it
has encoder TriNet ($g_{Enc}(\mathbf{x})$) and the decoder TriNet
($g_{Dec}(\mathbf{x})$). Specifically, given one training instance,
we can use the ResNet-18 to extract the features at different layers.
The $g_{Enc}(\mathbf{x})$ efficiently maps these features into the
semantic {space}. In the semantic space, the projected instance
features can be corrupted by adding Gaussian noise, or replaced by
its nearest semantic word vectors. {We assume that} slight changes
of feature values {in the semantic space will allow us to maintain
semantic information while spanning the potential class variability.
The decoder TriNet ($g_{Dec}(\mathbf{x})$) is {then} adapted to
map the perturbed semantic instance features back to {multi-layer
(ResNet-18)} feature space. {It is worth noting that Gaussian augmentations/perturbations
in the semantic space ultimately result in highly non-Gaussian augmentations
in the original feature space. This is the core benefit of the semantic
space augmentation. Using three classical supervised classifiers,
we show that the augmented features can boost the performance in few-shot
classification.
\noindent \textbf{Contributions}. Our contributions are in several
fold. First, we propose a simple and yet elegant deep learning architecture:
ResNet-18+dual TriNet with an efficient end-to-end training for few-shot
classification. Second, we illustrate that the proposed dual TriNet
can effectively augment visual features produced by multiple layers
of ResNet-18. Third, and interestingly, we show that we can utilize
semantic spaces of various types, including semantic attribute space,
semantic word vector space, or even subspace defined by the semantic
relationship of classes. Finally, extensive experiments on four datasets
validate the efficacy of the proposed approach in addressing the few-shot
image recognition task.
\section{Related work}
\subsection{Few-Shot Learning}
\noindent Few-shot learning is inspired by human ability to learn
new concepts from very few examples \cite{Jankowski,compositional_1shot}.
Being able to recognize and generalize to new classes with only one,
or few, examples \cite{bart2005cross_gen} is beyond the capabilities
of typical machine learning algorithms, which often rely on hundreds
or thousands of training examples. Broadly speaking there are two
categories of approaches for addressing such challenges:
\vspace{0.05in}
\noindent \textbf{Direct supervised learning-based approaches,} directly learn
a one-shot classifier via instance-based learning (such as K-nearest
neighbor), non-parametric methods \cite{feifei2003unsup_1s_objcat_learn,feifei2006one_shot,tommasi2009transfercat},
deep generative models \cite{generative_1shot,deep_1shot_recent},
or Bayesian auto-encoders \cite{kingama2014iclr}. Compared with our
work, these methods employ a rich class of generative models to explain
the observed data, rather than directly augmenting instance features
as proposed.
\vspace{0.05in}
\noindent \textbf{Transfer learning-based approaches,} are explored
via the paradigm of learning to learn \cite{Thrun96learningto} or
meta-learning \cite{JVilalta2002AIR}. Specifically, these approaches
employ the knowledge from auxiliary data to recognize new categories
with few examples by either sharing features \cite{bart2005cross_gen,hertz2016icml,Fleuret2005nips,amit2007icml,wolfc2005cvpr,torralba2005pami},
semantic attributes \cite{lampert13AwAPAMI,transferlearningNIPS,rohrbach2010semantic_transfer},
or contextual information \cite{one_shot_TL_contexutal}. Recently,
the ideas of learning metric spaces from source data to support one-shot
learning were quite extensively explored. Examples include matching
networks \cite{matchingnet_1shot} and prototypical networks \cite{prototype_network}.
Generally, these approaches can be roughly categorized as either meta-learning
algorithms (including MAML {\cite{MAML}, Meta-SGD {\cite{meta-sgd},
DEML+Meta-SGD \cite{DEML+Meta-SGD}, META-LEARN LSTM {\cite{Sachin2017},
Meta-Net \cite{MetaNetwork}, R2-D2\cite{closedform}, Reptile\cite{DBLP:journals/corr/abs-1803-02999},
WRN~\cite{predictFromActivation}) and metric-learning algorithms (including{
}Matching Nets {\cite{matchingnet_1shot}, }PROTO-NET {\cite{prototype_network}},
RELATION NET \cite{relation_net}, MACO {\cite{2018arXiv180204376H},
and \textcolor{black}{Cos \& Att. }\textcolor{black}{\cite{dym}}). In
\cite{zhongwen2016,memorymatching}, they maintained external memory for
continuous learning. MAML \cite{pmlr-v70-finn17a} can learn good
initial neural network weights which can be easily fine-tuned for
unseen categories. The \cite{2017arXiv171104043G} used graph neural
network to perform message-passing inference from support images to
test images. TPN~\cite{TPN} proposed a framework for transductive
inference thus to solve the data-starved problem.\textcolor{black}{{}
Multi-Attention~\cite{multiAttention} utilized semantic information
to generate attention map to help one-shot recognition, whereas we
directly augment samples in the semantic space and then map them back
to the visual space.} With respect to these works, our framework is
orthogonal but potentially useful -- it is useful to augment instance
features of novel classes before applying such methods.
\subsection{Augmenting training instances\label{subsec:One-shot-learning-by}}
The standard augmentation techniques are often directly applied in
the image domain, such as flipping, rotating, adding noise and randomly
cropping images \cite{KrizhevskySH12,returnDevil2014BMVC,visualizing_network}.
Recently, more advanced data augmentation techniques have been studied
to train supervised classifiers. In particular, {augmented} training
data can also be employed to alleviate the problem of instances {scarcity}
and thus avoid overfitting in one-shot/few-shot learning settings.
{P}revious approaches {can be categorized into six classes of methods}:
(1) Learning one-shot models by utilizing the manifold information
of a large amount of unlabelled data in a semi-supervised or transductive
{setting} \cite{transductiveEmbeddingJournal}; (2) Adaptively learning
the one-shot classifiers from off-shelf trained models \cite{yuxiong2016eccv,yuxiong2016nips,zhizhong2016eccv};
(3) Borrowing examples from relevant categories \cite{lim2011nips,Delta-encoder}
or semantic vocabularies \cite{ssvoc_2016_CVPR,deep_0shot} to augment
the training set; (4) Synthesizing additional labelled training instances
by rendering virtual examples \cite{Movshovitz2015phd,Park2015cvpr,attias2015cvpr,Dosovitskiy2015cvpr,renderCNN}
or composing synthesized representations \cite{zhu2015ijcv,opelt2006alphabet,gait_augmentation,mocap_guide,detector_3d,2017ICCVaug}
or distorting existing training examples \cite{KrizhevskySH12}; (5)
Generating new examples {using} Generative Adversarial Networks
(GANs) \cite{gan_cycle,zhu2015eccv,gan2014,reed2016generative,dcgan,LSgan,ishan2017iclr,xun2017cvpr,imaginaryData};
\textcolor{black}{(6) Attribute-guided augmentation (AGA) and Feature
Space Transfer~\cite{AGA_2017,FSTransfer} to synthesize sample{s}
at desired values, poses or strength.}
{Despite the breadth of research,} previous methods may suffer from
several problems: (1) semi-supervised algorithms {rely on} the manifold
assumption, which, however, cannot {be} effectively validated {in
practice}. (2) {transfer learning} may suffer from the \emph{negative
transfer }when the off-shell models or relevant categories are very
different from one-shot classes; (3) rendering, composing or distorting
{existing} training examples {may require domain expertise;} (4)
GAN-based approaches mostly focuse on learning good generators to
synthesize ``realistic'' images to ``cheat'' the discriminators.
{S}ynthesized images may not necessarily preserve the discriminative
information. This is in contrast to our network structure, where the
discriminative instances are directly synthesized in visual feature
domain. The AGA~\cite{AGA_2017} mainly employed the attributes of
3D depth or pose information for augmentation; in contrast, our methods
can additionally utilize semantic {information} to augment data.
Additionally, the proposed dual TriNet networks can effectively augment
multi-layer features.
\subsection{Embedding Network structures}
Learning {of} visual-semantic embedding{s} has been explored in
various ways, including {with} neural networks, \emph{e.g.}, Siamese
network \cite{Bromley1993ijcai,siamese_1shot}, discriminative methods
(\emph{e.g.}, Support Vector Regressors (SVR) \cite{lampert13AwAPAMI,farhadi2009attrib_describe,Kienzle2006icml}),
metric learning methods \cite{matchingnet_1shot,quattoni2008sparse_transfer,fink2005nips},
or kernel embedding methods \cite{hertz2016icml,wolf2009iccv}. One
of the most common embedding approaches is to project visual features
and semantic entities into a common {\em new} space. However, when
dealing with the feature space of different layers in CNNs, previous
methods have to learn an individual visual semantic embedding for
each layer. In contrast, the proposed Dual TriNet can effectively
learn a single visual-semantic embedding for multi-layer feature spaces.
Ladder Networks \cite{ladderNet} utilize the lateral connections
as auto-encoders for semi-supervised learning tasks. In \cite{deeplyfused}, the authours
fused different intermediate layers of different networks to improve
the image classification performance. Deep Layer Aggregation \cite{layer_aggregation}
aggregated the layers and blocks across a network to better fuse the
information across layers. Rather than learn a specific aggregation
node to merge different layers, our dual TriNet directly transforms,
rescales and concatenates the features of different layers in an encoder-decoder
structure.
\section{Dual TriNet Network for Semantic Data Augmentation}
\begin{figure*}
\centering{}\includegraphics[scale=0.45]{trinet.png}\caption{\label{fig:Overview-of-our}\textbf{Overview of our framework.} We
extract image features by ResNet-18 and augment features by dual TriNet.
Encoder TriNet projects features to the semantic space. After augmenting
data in semantic space, we use the decoder TriNet to obtain the corresponding
augmented features. Both real and augmented data are used to train
the classification model. Note that: (1) the small green arrow indicates
the max pooling with $2\times2$, and following by a ``conv'' layer
which is the sequence Conv-BN-ReLU.}
\end{figure*}
\subsection{Problem setup\label{subsec:Problem-setup}}
In one-shot learning, we are given the base categories $C_{base}$,
and novel categories $C_{novel}(C_{base}\bigcap C_{novel}=\emptyset)$
with the total class label set $\mathcal{C}=\mathcal{C}_{base}\cup\mathcal{C}_{novel}$.
The base categories $C_{base}$ have sufficient labeled image data
and we assume the base dataset $D_{base}=\left\{ \mathbf{I}_{i}^{base},z_{i}^{base},\mathbf{u}_{z_{i}}^{base}\right\} _{i=1}^{N_{base}}$
of $N_{base}$ samples. $\mathbf{I}_{i}^{base}$ indicates the raw
image $i$; $z_{i}^{base}\in\mathcal{C}_{base}$ is a class label
from the base class set; $\mathbf{u}_{z_{i}}^{base}$ is the semantic
vector of the instance $i$ in terms of its class label. The semantic
vector $\mathbf{u}_{z_{i}}^{base}$ can be either semantic attribute
\cite{lampert13AwAPAMI}, semantic word vector \cite{distributedword2vec2013NIPS}
or any representation obtained in the subspace constructed or learned
from semantic relationship of classes.
For novel categories $C_{novel}$, we consider the another dataset
$D_{novel}=\left\{ \mathbf{I}_{i}^{novel},z_{i}^{novel},\mathbf{u}_{z_{i}}^{novel}\right\} $
and each class $z_{i}^{novel}\in\mathcal{C}_{novel}$ . For the novel
dataset, we have a support set and test set. Support set $D_{support}=\left\{ \mathbf{I}_{i}^{support},z_{i}^{support},\mathbf{u}_{z_{i}}^{support}\right\} $
($D_{support}\in D_{novel}$) is composed of a small number of training
instances of each novel class. The test set $D_{test}=\left\{ \mathbf{I}_{i}^{test},z_{i}^{test},\mathbf{u}_{z_{i}}^{test}\right\} $
($D_{test}\in D_{novel},D_{support}\bigcap D_{test}=\emptyset$) is
not available for training, but is used for testing. In general, we
only train on $D_{base}$ and $D_{support}$ ,which contain adequate
instances of base classes and a small number of instances of novel
classes respectively. Then we evaluate our model on $D_{test}$, which
only consists of novel classes. We target learning a model that can
generalize well to the novel categories,
by using only a small support set $D_{support}$.
\subsection{Overview\label{subsec:Overview}}
\noindent \textbf{Objective.} We seek to directly augment the features
of the training instances of each target class. Given one training
instance $\mathbf{I}_{i}^{support}$ from the novel classes, the feature
extractor network can output the instance feature $\left\{ f_{l}\left(\mathbf{I}_{i}^{support}\right)\right\} $
($l=1,\cdots,L$); and the augmentation network $g\left(\mathbf{x}\right)$
can generate a set of synthesized features $g\left(\left\{ f_{l}\left(\mathbf{I}_{i}^{support}\right)\right\} \right)$.
Such synthesized features are used as additional training instances
for one-shot learning. As illustrated in Fig. \ref{fig:Overview-of-our},
we use the ResNet-18 \cite{he2015deep} and propose a Dual TriNet
network as the feature extractor network and the augmentation network
respectively. The whole architecture is trained in an end-to-end manner
by combining the loss functions of both networks,
\begin{equation}
\left\{ \Omega,\Theta\right\} =\underset{\Omega,\Theta}{\mathrm{argmin}}J_{1}\left(\Omega\right)+\lambda\cdot J_{2}\left(\Theta\right)\label{eq:jointly_loss}
\end{equation}
where $J_{1}\left(\Omega\right)$ and $J_{2}\left(\Theta\right)$
are the loss functions for ResNet-18 \cite{he2015deep} and dual TriNet
network respectively; $\Omega$ and $\Theta$ represent corresponding
parameters. The cross entropy loss is used for $J_{1}\left(\Omega\right)$
as in~\cite{he2015deep}. Eq.~(\ref{eq:jointly_loss}) is optimized
using base dataset $D_{base}$.
\vspace{0.05in}
\noindent \textbf{Feature extractor network.} We train
ResNet-18~\cite{he2015deep} to convert the raw images into feature
vectors. ResNet-18 has 4 sequential residual layers, \emph{i.e.},
layer1, layer2, layer3 and layer4 as illustrated in Fig.~\ref{fig:Overview-of-our}.
Each residual layer outputs a corresponding feature map $f_{l}\left(\mathbf{I}_{i}\right),\quad l=1,\dots4$.
If we consider each feature map a different image representation,
ResNet-18 actually learns a Multi-level Image Feature (M-IF) encoding.
Generally, different layer features may be used for various one-shot
learning tasks. For example, as in \cite{KrizhevskySH12}, the features
of fully connected layers can be used for one-shot image classification;
and the output features of fully convolutional layers may be preferred
for one-shot image segmentation tasks \cite{fully_conv_seg,one_shot_seg17BMVC,one-shot-video17cvpr}.
By combining features from multiple levels, our method can be applied
to a variety of different visual tasks.
\vspace{0.05in}
\noindent \textbf{Augmentation network.} We propose an encoder-decoder
architecture -- dual TriNet ($g\left(\mathbf{x}\right)=g_{Dec}\circ g_{Enc}\left(\mathbf{x}\right)$).
As illustrated in Fig.~\ref{fig:Overview-of-our}, our dual TriNet
can be divided into encoder-TriNet $g_{Enc}\left(\mathbf{x}\right)$
and decoder-TriNet sub-network $g_{Dec}\left(\mathbf{x}\right)$.
The encoder-TriNet maps visual feature space to a semantic space.
This is where augmentation takes place. The decoder-TriNet projects
the augmented semantic space representation back to the feature space.
Since ResNet-18 has four layers, the visual feature spaces produced
by different layers can use the same encoder-decoder TriNet for data
augmentation.
\subsection{Dual TriNet Network}
The dual TriNet is paired with ResNet-18. Feature representations
obtained from different layers of such a deep CNN architecture, are
hierarchical, going from \emph{general} (bottom layers) to more \emph{specific}
(top layers) \cite{transferable_deep_feat}. For instance, the features
produced by the first few layers are similar to Gabor filters \cite{visualizing_network}
and thus agnostic to the tasks; in contrast, the high-level layers
are specific to a particular task, \emph{e.g}., image classification.
The feature representations produced by layers of ResNet-18 have different
levels of abstract semantic information. Thus a natural question is
whether we can augment features at different layers? Directly learning
an encoder-decoder for each layer will not fully exploit the relationship
of different layers, and thus may not effectively learn the mapping
between feature spaces and the semantic space. To this end, we propose
the dual TriNet network.
Dual TriNet learns the mapping between the Multi-level Image Feature
(M-IF) encoding and the Semantic space. The semantic space can be
either semantic attribute space, or semantic word vector space introduced
in Sec. \ref{subsec:Problem-setup}. Semantic attributes can be pre-defined
by human experts \cite{AGA_2017}. Semantic word vector $\mathbf{u}_{z_{i}}^{base}$
is the projection of each vocabulary entity $w_{i}\in\mathcal{W}$,
where vocabulary $\mathcal{W}$ is learned by word2vec \cite{distributedword2vec2013NIPS}
on a large-scale corpus. Furthermore, the subspace $\mathbf{u}_{z_{i}}^{base}$
can be spanned by Singular Value Decomposition (SVD) of the semantic
relationship of classes. Specifically, we can use $\left\{ \mathbf{u}_{z_{i}}^{base};\mathbf{u}_{z_{j}}^{novel}\right\} _{z_{i}\in\mathcal{C}_{base},z_{j}\in\mathcal{C}_{novel}}$
to compute the semantic relationship $\mathbf{M}$ of classes using
cosine similarity. We decompose $\mathbf{M}=\mathbf{U}\mathbf{\Sigma}\mathbf{V}$
by SVD algorithm. The $\mathbf{U}$ is a unitary matrix and defines
a new semantic space. Each row of $\mathbf{U}$ is taken as a new
semantic vector of one class.
Encoder TriNet is composed of four layers corresponding to each layer
of ResNet-18. It aims to learn the function $\hat{\mathbf{u}}_{z_{i}}=g_{Enc}\left(\left\{ f_{l}\left(\mathbf{I}_{i}\right)\right\} \right)$
to map all layer features $\left\{ f_{l}\left(\mathbf{I}_{i}\right)\right\} $
of instance $i$ as close to the semantic vector $\mathbf{u}_{z_{i}}$
of instance $i$ as possible. The structure of subnetwork is inspired
by the tower of Hanoi as shown in Fig.~\ref{fig:Overview-of-our}.
Such a structure can efficiently exploit the differences and complementarity
of information encoded in multiple layers. The encoder TriNet is trained
to match the four layers of ResNet-18 by merging and combining the
outputs of different layers. The decoder TriNet has inverse architecture
to project the features $\hat{\mathbf{u}}_{z_{i}}$ from semantic
space to the feature space $\hat{f_{l}}\left(\mathbf{I}_{i}\right)=g_{Dec}\left(g_{Enc}\left(\left\{ f_{l}\left(\mathbf{I}_{i}\right)\right\} \right)\right)$.
We learn TriNet by optimizing the following loss:
\begin{equation}
\;J_{2}\left(\Theta\right)=\mathbb{E}\left[\sum_{l=1}^{4}\left(f_{l}\left(\mathbf{I}_{i}\right)-\hat{f_{l}}\left(\mathbf{I}_{i}\right)\right)^{2}+\left(\hat{\mathbf{u}}_{z_{i}}-\mathbf{u}_{z_{i}}\right)^{2}\right]+\lambda P\left(\Theta\right)\label{eq:auto_encoder}
\end{equation}
\noindent \textcolor{black}{where $\mathbf{I}_{i}\in D_{base}$ and
$\Theta$ indicates the parameter set of dual TriNet network }and
$P\left(\cdot\right)$ is the $L_{2}-$regularization term. The dual
TriNet is trained on $D_{base}$ and used to synthesize instances
in the form of $l$-th layer feature perturbations with respect to
a given training instance from $D_{support}$.
\subsection{Feature Augmentation by Dual TriNet \label{subsec:Data-Augmentation-by}}
With the learned dual TriNet, we have two ways to augment the features
of training instances. Note that the augmentation method is only used
to extend $D_{support}$.
\vspace{0.05in}
\noindent \textbf{Semantic Gaussian (SG).} A natural way to augment
features is by sampling instances from a Gaussian distribution. Specifically,
for the feature set $\left\{ f_{l}\left(\mathbf{I}_{i}^{support}\right)\right\} $($l=1,\cdots,L$)
extracted by ResNet-18, the encoder TriNet can project the $\left\{ f_{l}\left(\mathbf{I}_{i}^{support}\right)\right\} $
into the semantic space, $g_{Enc}\left(\left\{ f_{l}\left(I_{i}^{support}\right)\right\} \right)$.
In such a space, we assume that vectors
can be corrupted by a random Gaussian noise without changing a semantic
label.
This can be used to augment the data. Specifically, we sample the
$k-th$ semantic vector $\mathbf{v}_{i}^{G_{k}}$ from $I_{i}^{support}$
using semantic Gaussian as follows,
\begin{equation}
\mathbf{v}_{i}^{G_{k}}\sim\mathcal{N}\left(g_{Enc}\left(\left\{ f_{l}\left(\mathbf{I}_{i}^{support}\right)\right\} \right),\sigma\mathbf{E}\right)\label{eq:semantic_gaussian}
\end{equation}
where $\sigma\in\mathbb{R}$ is the variance of each dimension and
$\mathbf{E}$ is the identity matrix; $\sigma$ controls the standard
deviation of the noise being added. To make the augmented semantic
vector $\mathbf{v}_{i}^{G_{k}}$ still be representative of the class
of $z_{i}^{support}$, we empirically set $\sigma$ to $15\%$ of
the distance between $u_{z_{i}}^{support}$ and its nearest other
class instance $\mathbf{u}_{z_{j}}^{support}$ ($z_{i}^{support}\neq z_{j}^{support}$)
as this gives the best performance. The decoder TriNet generates the
virtual synthesized sample $g_{Dec}\left(\mathbf{v}_{i}^{G_{k}}\right)$
which shares the same class label $z_{i}^{support}$ with the original
instance. By slightly corrupting the values of some dimensions of
semantic vectors, we expect the sampled vectors $\mathbf{\mathbf{v}}_{i}^{G_{k}}$
to still have the same semantic meaning.
\vspace{0.05in}
\noindent \textbf{Semantic Neighborhood (SN).} Inspired by the recent
work on vocabulary-informed learning \cite{ssvoc_2016_CVPR}, the
large amount of vocabulary in the semantic word vector space (\textit{e.g.},
word2vec \cite{distributedword2vec2013NIPS}) can also be used for
augmentation. The distribution of such vocabulary reflects the general
semantic relationships in the linguistic corpora. For example, in
word vector space, the vector of ``truck'' is closer to the vector
of ``car'' than to the vector of ``dog''. Given the features $\left\{ f_{l}\left(\mathbf{I}_{i}^{support}\right)\right\} $
of training instance $i$, the $k$-th augmented data $\mathbf{v}_{i}^{N_{k}}$
can be sampled from the neighborhood of\emph{ }$g_{Enc}\left(\left\{ f_{l}\left(\mathbf{I}_{i}^{support}\right)\right\} \right)$,
\emph{i.e.},
\begin{equation}
\mathbf{v}_{i}^{N_{k}}\in Neigh\left(g_{Enc}\left(\left\{ f_{l}\left(\mathbf{I}_{i}^{support}\right)\right\} \right)\right)\label{eq:semantic_neighbourhood}
\end{equation}
$Neigh\left(g_{Enc}\left(\left\{ f_{l}\left(\mathbf{I}_{i}^{support}\right)\right\} \right)\right)\subseteq\mathcal{W}$
indicates the nearest neighborhood vocabulary set of $g_{Enc}\left(\left\{ f_{l}\left(\mathbf{I}_{i}^{t}\right)\right\} \right)$
and $\mathcal{W}$ indicate vocabulary set learned by word2vec \cite{distributedword2vec2013NIPS}
on a large-scale corpus. These neighbors correspond to the most semantically
similar examples to our training instance. The features of synthesized
samples can again be decoded by $g_{Dec}\left(\mathbf{v}_{i}^{N_{k}}\right)$.
There are several points we want to highlight. (1) For one training
instance $\mathbf{I}_{i}^{support}$, we use as the Gaussian mean
in Eq (\ref{eq:semantic_gaussian}) or neighborhood center in Eq (\ref{eq:semantic_neighbourhood}),
the $g_{Enc}\left(\left\{ f_{l}\left(\mathbf{I}_{i}^{support}\right)\right\} \right)$
rather than its ground-truth word vector $\mathbf{u}_{z_{i}}^{support}$.
This is due to the fact that $\mathbf{u}_{z_{i}}^{support}$ only
represents the semantic center of class $z_{i}^{support}$, not the
center for the instance $i$. Experimentally, on \emph{mini}ImageNet
dataset, augmenting features using $\mathbf{u}_{z_{i}}^{support}$,
rather than $g_{Enc}\left(\left\{ f_{l}\left(\mathbf{I}_{i}^{t}\right)\right\} \right)$,
leads to $3\sim5\%$ performance drop (on average) in 1-shot/5-shot
classification. (2) Semantic space Gaussian noise added in Eq~(\ref{eq:semantic_gaussian})
or semantic neighborhood used in Eq (\ref{eq:semantic_neighbourhood})
result in the synthesized training features that are highly nonlinear
(non-Gaussian) for each class. This is the result of non-linear decoding
provided by TriNet $g_{Dec}\left(\mathbf{x}\right)$ and ResNet-18
($\left\{ f_{l}\left(\mathbf{I}_{i}^{t}\right)\right\} $). (3) Directly
adding Gaussian noise to $\left\{ f_{l}\left(\mathbf{I}_{i}^{t}\right)\right\} $
is another naive way to augment features. However, in \emph{mini}ImageNet
dataset, such a strategy does not give any significant improvement
in one-shot classification.
\subsection{One-shot Classification\label{subsec:One-shot-Classification-and}}
Having trained feature extractor network and dual TriNet on base dataset
$D_{base}$, we now discuss conducting one-shot classification on
the target dataset $D_{novel}$. For the instance $i$ in $D_{novel}$
we can extract the M-IF representation $f_{l}\left(\mathbf{I}_{i}^{novel}\right)$
($l=1,2,...,L$) using the feature extractor network. We then use
the encoder part of TriNet to map all layer features $\left\{ f_{l}\left(\mathbf{I}_{i}\right)\right\} $
of instance $i$ to semantic vector $g_{Enc}\left(\left\{ f_{l}\left(\mathbf{I}_{i}^{support}\right)\right\} \right)$.
Our framework can augment the instance, producing multiple synthetic
instances in addition to the original one $\left\{ \mathbf{v}_{i}^{G_{k}}\right\} \cup\left\{ \mathbf{v}_{i}^{N_{k}}\right\} $
using semantic Gaussian and/or semantic neighborhood approaches discussed.
For each new semantic vector $\mathbf{v}_{i}^{k}\in\left\{ \mathbf{v}_{i}^{G_{k}}\right\} \cup\left\{ \mathbf{v}_{i}^{N_{k}}\right\} $,
we use decoder TriNet to map them from semantic space to all layer
features $\left\{ x_{l}^{augment_{i}}\right\} =g_{Dec}(v_{i}^{k})$
($l=1,2,...,L$). The features that are not at the final $L$-th layer
are feed through from $l+1$-th layer to $L$-th layer of feature
extractor network to obtain $\left\{ \hat{x}_{l}^{augment}\right\} $.
Technically, one new semantic vector $\mathbf{v}_{i}^{k}$ can generate
$L$ instances: one from each of the $L$ augmented layers. Consistent
with previous work~\cite{KrizhevskySH12,he2015deep}, the features
produced by the final layer are utilized for one-shot classification
tasks. Hence the newly synthesized $L$-th layer features $\left\{ \hat{x}_{l}^{augment_{i}}\right\} $
obtained from the instance $i$ and original $L$-th layer feature
$f_{L}\left(\mathbf{I}_{i}^{support}\right)$ are used to train the
one-shot classifier $g_{one-shot}(x)$ in a supervised manner. Note
that all augmented feature vectors obtained from instance $i$ are
assumed to have the same class label as the original instance $i$.
In this work, we show that the augmented features can benefit various
supervised classifiers. To this end three classical classifiers, \emph{i.e.},
the K-nearest neighbors (KNN), Support Vector Machine (SVM) and Logistic
Regression (LR), are utilized as one-shot classifier $g_{one-shot}(x)$.
In particular, we use $g_{one-shot}(x)$ to classify the $L$-th layer
feature $f_{l}\left(\mathbf{I}_{i}^{test}\right)$ of test sample
$\mathbf{I}_{i}^{test}$ at the test time.
\section{Experiments}
\subsection{Datasets}
We conduct experiments on four datasets. Note that (1) on all datasets,
ResNet-18 is only trained on the training set (equivalent to base
dataset) in the specified splits of previous works. (2) The same networks
and parameter settings (including the size of input images) are used
for all the datasets; hence all images are resized to $224\times224$.
\vspace{0.05in}
\noindent \textbf{\emph{mini}}\textbf{ImageNet.} Originally proposed
in \cite{matchingnet_1shot}, this dataset has 60,000 images from
100 classes; each class has around 600 examples. To make our results
comparable to previous works, we use the splits in \cite{Sachin2017}
by utilizing 64, 16 and 20 classes for training, validation and testing
respectively.
\vspace{0.05in}
\noindent \textbf{Cifar-100.} Cifar-100 contains 60,000 images from
100 fine-grained and 20 coarse-level categories \cite{Krizhevsky2009LearningML}.
We use the same data split as in \cite{2018arXiv180203596Z} to enable
the comparison with previous methods. In particular, 64, 16 and 20
classes are used for training, validation and testing respectively.
\vspace{0.05in}
\noindent \textbf{Caltech-UCSD Birds 200-2011 (CUB-200).} CUB-200
is a fine-grained dataset consisting of a total of 11,788 images from
200 categories of birds \cite{WahCUB_200_2011}. As the split in \cite{2018arXiv180204376H},
we use 100, 50 and 50 classes for training, validation and testing.
This dataset also provides 312 dimensional semantic attribute vectors
on a per-class level.
\vspace{0.05in}
\noindent \textbf{Caltech-256.} Caltech-256 has 30,607 images from
256 classes \cite{griffin2007caltech}. As in \cite{2018arXiv180203596Z},
we split the dataset into 150, 56 and 50 classes for training, validation
and testing respectively.
\begin{table*}
\centering{}%
\begin{tabular}{c|c|c|c|c}
\hline
\multirow{2}{*}{Methods} & \multicolumn{2}{l|}{\emph{mini}ImageNet{} ($\%$)} & \multicolumn{2}{l}{{}CUB-200($\%$)}\tabularnewline
\cline{2-5}
& {}1-shot & {}5-shot & {}1-shot & {}5-shot \tabularnewline
\hline
\hline
{}META-LEARN LSTM \cite{Sachin2017} & {}43.44\textpm 0.77 & {}60.60\textpm 0.71 & {}40.43 & {}49.65 \tabularnewline
\hline
{}MAML \cite{MAML} & {}48.70\textpm 1.84 & {}63.11\textpm 0.92 & {}38.43 & {}59.15 \tabularnewline
\hline
{}Meta-Net \cite{MetaNetwork} & {}49.21\textpm 0.96 & - & {}- & {}- \tabularnewline
\hline
\textcolor{black}{Reptile\cite{DBLP:journals/corr/abs-1803-02999}} & \textcolor{black}{49.97} & \textcolor{black}{65.99} & \textcolor{black}{-} & \textcolor{black}{-}\tabularnewline
\hline
MAML{*} \cite{MAML} & 52.23\textpm 1.24 & 61.24\textpm 0.77 & - & -\tabularnewline
\hline
Meta-SGD{*} \cite{meta-sgd} & 52.31\textpm 1.14 & 64.66\textpm 0.89 & - & -\tabularnewline
\hline
{}DEML+Meta-SGD \cite{DEML+Meta-SGD} & \textbf{{}}58.49{}\textpm 0.91\textbf{{}} & {}71.28\textpm 0.69 & {}- & {}- \tabularnewline
\hline
\hline
{}MACO \cite{2018arXiv180204376H} & {}41.09\textpm 0.32 & {}58.32\textpm 0.21 & {}60.76 & {}74.96 \tabularnewline
\hline
Matching Nets{*} \cite{matchingnet_1shot} & 47.89\textpm 0.86 & 60.12\textpm 0.68 & - & -\tabularnewline
\hline
{}PROTO-NET \cite{prototype_network} & {}49.42\textpm 0.78 & {}68.20\textpm 0.66 & {}45.27 & {}56.35 \tabularnewline
\hline
\textcolor{black}{GNN~\cite{2017arXiv171104043G}} & \textcolor{black}{50.33\textpm 0.36} & \textcolor{black}{66.41\textpm 0.63} & \textcolor{black}{-} & \textcolor{black}{-}\tabularnewline
\hline
\textcolor{black}{R2-D2~\cite{closedform}} & \textcolor{black}{51.5\textpm 0.2 } & \textcolor{black}{68.8\textpm 0.1 } & \textcolor{black}{-} & \textcolor{black}{-}\tabularnewline
\hline
\textcolor{black}{MM-Net~\cite{memorymatching}} & \textcolor{black}{53.37\textpm 0.48} & \textcolor{black}{66.97\textpm 0.35} & \textcolor{black}{-} & \textcolor{black}{-}\tabularnewline
\hline
\textcolor{black}{Cos \& Att.~\cite{dym}} & \textcolor{black}{55.45\textpm 0.89} & \textcolor{black}{70.13 \textpm 0.68} & \textcolor{black}{-} & \textcolor{black}{-}\tabularnewline
\hline
\textcolor{black}{TPN~\cite{TPN}} & \textcolor{black}{55.51} & \textcolor{black}{69.86} & \textcolor{black}{-} & \textcolor{black}{-}\tabularnewline
\hline
SNAIL \cite{SNAIL} & 55.71\textpm 0.99 & 68.88\textpm 0.92 & - & -\tabularnewline
\hline
{}RELATION NET \cite{relation_net} & {}57.02\textpm 0.92 & {}71.07\textpm 0.69 & {}- & {}- \tabularnewline
\hline
\textcolor{black}{Delta-encoder~\cite{Delta-encoder}} & \textcolor{black}{58.7} & \textcolor{black}{73.6} & \textcolor{black}{-} & \textcolor{black}{-}\tabularnewline
\hline
\textcolor{black}{WRN~\cite{predictFromActivation}} & \textbf{\textcolor{black}{59.60}}\textcolor{black}{\textpm 0.41 } & \textcolor{black}{73.74\textpm 0.19 } & \textcolor{black}{-} & \textcolor{black}{-}\tabularnewline
\hline
\hline
{}ResNet-18 & {}52.73\textpm 1.44 & {}73.31\textpm 0.81 & {}66.54\textpm 0.53 & {}82.38\textpm 0.43\tabularnewline
\hline
ResNet-18+Gaussian Noise & 52.14\textpm 1.51 & 71.78\textpm 0.89 & 65.02\textpm 0.60 & 80.79\textpm 0.49\tabularnewline
\hline
{}Ours: ResNet-18+Dual TriNet & {}58.12\textpm 1.37 & \textbf{{}76.92}{}\textpm 0.69 & \textbf{{}69.61}{}\textpm 0.46 & \textbf{{}84.10}{}\textpm 0.35\tabularnewline
\hline
\end{tabular}\caption{\label{tab:miniimagenet}Results on \emph{mini}ImageNet and CUB-200.
The ``\textpm '' indicates $95\%$ confidence intervals over tasks.{*}:
indicates the corresponding baselines that are using ResNet-18. Note
that ``\textpm '' is not reported on CUB-200 in previous works.}
\end{table*}
\subsection{Network structures and Settings}
The same ResNet-18 and dual TriNet are used for all four datasets
and experiments.
\vspace{0.05in}
\noindent \textbf{Parameters.} The dropout rate and learning rate
of the auto-encoder network are set to 0.5 and $1e^{-3}$ respectively
to prevent overfitting. The learning rate is divided by 2 every 10
epochs. The batch size is set to 64. The network is trained using
Adam optimizer and usually converges in 100 epochs. To prevent randomness
due to the small training set size, all experiments are repeated multiple
times.
\textcolor{black}{The Top-1 accuracies are reported with $95\%$ confidence
interval and are averaged over multiple test episodes, the same as
previous work \cite{Sachin2017}. }
\vspace{0.05in}
\noindent \textbf{Settings. }We use the 100-dimensional semantic word
vectors extracted from the vocabulary dictionary released by \cite{ssvoc_2016_CVPR}.
The class name is projected into the semantic space as a vector $\mathbf{u}_{z_{i}}^{base}$
or $\mathbf{u}_{z_{i}}^{novel}$. The semantic attribute space is
pre-defined by experts \cite{lampert13AwAPAMI,WahCUB_200_2011}. In
all experiments, given one training instance
the dual TriNet will generate 4 augmented instances in the semantic
space. Thus we have 4 synthesized instances of each layer which results
in
16 synthesized instances in the form of $4-th$ layer features. So
one training instance becomes 17 training instances at the end.
\subsection{Competitors and Classification models}
\noindent \textbf{Competitors}. The previous methods we compare to
are run using the same source/target and training/testing splits as
used by our method. We compare to Matching Nets {\cite{matchingnet_1shot}},
MAML {\cite{MAML}}, Meta-SGD {\cite{meta-sgd},} DEML+Meta-SGD
{\cite{DEML+Meta-SGD}}, PROTO-NET {\cite{prototype_network}},
RELATION NET {\cite{relation_net}}, META-LEARN LSTM {\cite{Sachin2017}},
Meta-Net {\cite{MetaNetwork}}, SNAIL \cite{SNAIL} , MACO \cite{2018arXiv180204376H},
\textcolor{black}{GNN \cite{2017arXiv171104043G}, MM-Net\cite{memorymatching},
Reptile \cite{DBLP:journals/corr/abs-1803-02999}, TPN \cite{TPN},
WRN \cite{predictFromActivation}, Cos \& Att. \cite{dym}, Delta-encoder \cite{Delta-encoder}
and R2-D2\cite{closedform}. To make} a fair comparison, we implement
some of the methods and use ResNet-18 as a commmon backbone architecture.
\vspace{0.05in}
\noindent \textbf{Classification model.} KNN, SVM, and LR are used
as the classification models to validate the effectiveness of our
augmentation technique. \textcolor{black}{The hyperparameters of classification
models are selected using cross-validation on a validation set.}
\subsection{Experimental results on \emph{mini}ImageNet and {CUB-200}}
\begin{table*}
\begin{centering}
\begin{tabular}{c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c}
\hline
& \multicolumn{5}{c|}{Semantic Neighbourhood} & \multicolumn{5}{c|}{Semantic Gaussian} & \multicolumn{5}{c}{Attribute Gaussian}\tabularnewline
\hline
\hline
& 0 & 2 & 4 & 10 & 50 & 0 & 2 & 4 & 10 & 50 & 0 & 2 & 4 & 10 & 50\tabularnewline
\hline
L1 & 66.5 & 67.1 & 67.2 & \textbf{67.3} & 67.2 & 66.5 & 67.1 & \textbf{67.2} & 67.1 & \textbf{67.2} & 66.5 & \textbf{67.1} & \textbf{67.1} & \textbf{67.1} & \textbf{67.1}\tabularnewline
\hline
L2 & 66.5 & 67.0 & \textbf{67.1} & \textbf{67.1} & 67.0 & 66.5 & \textbf{67.1} & \textbf{67.1} & \textbf{67.1} & 67.0 & 66.5 & 67.1 & \textbf{67.3} & \textbf{67.3} & 67.2\tabularnewline
\hline
L3 & 66.5 & 67.1 & \textbf{67.3} & \textbf{67.3 } & \textbf{67.3} & 66.5 & 67.1 & \textbf{67.3} & \textbf{67.3} & 67.2 & 66.5 & 67.0 & 67.4 & \textbf{67.5} & \textbf{67.5}\tabularnewline
\hline
L4 & 66.5 & 67.4 & \textbf{67.5} & 67.4 & 67.4 & 66.5 & 67.1 & \textbf{67.3} & 67.2 & \textbf{67.3} & 66.5 & 67.1 & \textbf{67.6} & \textbf{67.6} & 67.5\tabularnewline
\hline
M-L & 66.5 & 68.0 & \textbf{68.1} & \textbf{68.1} & \textbf{68.1} & 66.5 & 67.9 & \textbf{68.0} & \textbf{68.0} & \textbf{68.0} & 66.5 & 68.2 & 68.3 & \textbf{68.4} & 68.3\tabularnewline
\hline
\end{tabular}
\par\end{centering}
\caption{\label{tab:augnum}Ablation study of the number of augmented samples
in semantic space on CUB. We report 5-way 1-shot accuracy. L1, L2,
L3, and L4 indicate that we only use the augmented features of Layer
1, Layer 2, Layer 3 and Layer 4 respectively. M-L indicates that we
use all augmented features from four layers; }
\end{table*}
\noindent \textbf{Settings.}{ For }\emph{mini}{ImageNet dataset
we only have a semantic word space. Give one training instance, we
can generate 16 augmented instances for Semantic Gaussian (SG) and
Semantic Neighborhood (SN) each. On CUB-200 dataset, we use both the
semantic word vector and semantic attribute spaces. Hence for one
training instance, we generate 16 augmented instances for SG and SN
each in semantic word vector space; and additionally, we generate
16 virtual instances (in all four layers) for Semantic Gaussian (SG)
in semantic attribute space, which we denote Attribute Gaussian (AG).}
\vspace{0.05in}
\noindent \textbf{Variants of the number of augmented samples. }Varying the
number of augmented samples does not significantly affect our performance.
To show this, we provide 1-shot accuracy on CUB dataset with the different
numbers of augmented samples (Table~\ref{tab:augnum}). As shown,
the improvements from increasing the number of augmented samples saturate
at a certain point.
\begin{figure*}
\begin{centering}
\includegraphics[scale=0.5]{classifier.png}
\par\end{centering}
\caption{\label{fig:aug-layer} One-shot Results of feature augmentation by
different layers/classifiers on CUB-200 and \emph{mini}ImageNet. ``NoAug'',
``Layer1'', ``Layer2'', ``Layer3'', ``Layer4'' indicate the
one-shot learning results without any augmentation, with the feature
augmentation by using layer 1, layer 2, layer 3, layer 4 of ResNet-18.
``Multi-L'' denotes the performance of using all augmented instances
of one-shot learning. The X-axis represents the different supervised
classifiers. }
\end{figure*}
\vspace{0.05in}
\noindent \textbf{Results. }{As shown in Tab. \ref{tab:miniimagenet},
the competitors can be divided into two categories: Meta-learning
algorithms (including MAML, Meta-SGD, {}DEML+Meta-SGD, META-LEARN
LSTM and Meta-Net)} and Metric-learning algorithms (including Matching
Nets, PROTO-NET, RELATION NET ,SNAIL and MACO). We also report the
results of ResNet-18 (without data augmentation).
The accuracy of our framework (ResNet-18+Dual TriNet) is also reported.
The Dual TriNet synthesizes each layer features of ResNet-18 as described
in Sec. \ref{subsec:Data-Augmentation-by}. ResNet18+Gaussian Noise
is a simple baseline that synthesizes 16 samples of each test example
by adding Gaussian noise to the $4-th$ layer features. We use SVM
classifiers for ResNet-18 , ResNet18+Gaussian Noise and ResNet-18+Dual
TriNet in Tab. \ref{tab:miniimagenet}. In particular, we found that,
\vspace{0.05in}
\noindent \textbf{(1) Our baseline (ResNet-18) nearly beats all the
other baselines.} Greatly benefiting from learning the residuals,
Resnet-18 is a very good feature extractor for one-shot learning tasks.
Previous works \cite{MAML}\cite{meta-sgd}\cite{matchingnet_1shot}
designed their own network architectures with fewer parameters and
used different objective functions. As can be seen from Tab. \ref{tab:miniimagenet},
after replacing their backbone architecture with ResNet-18, they still
behave worse than our baseline (Resnet-18). We argue that this is
because ResNet-18 is more adaptable to classification task and it
can generate more discriminative space using Cross Entropy Loss than
other objective functions used in metric learning. However, this topic
is beyond our discussion. We want to clarify that since our augmentation
method is capable of being combined with arbitrary approaches, we
choose the strongest baseline to the best of our knowledge. This baseline
can be enhanced by our approach, illustrate the universality of our
augmentation.
\vspace{0.05in}
\noindent \textbf{(2) Our framework can achieve the best performance.}
As shown in Tab. \ref{tab:miniimagenet}, the results of our framework,
\emph{i.e.}, ResNet-18+Dual TriNet can achieve the best performance
and we can show a clear improvements over all the other baselines
on both datasets. This validates the effectiveness of our framework
in solving the one-shot learning task. Note, DEML+Meta-SGD~\cite{DEML+Meta-SGD}
uses the ResNet-50 as the baseline model and hence has better one-shot
learning results than our ResNet-18. Nevertheless, with the augmented
data produced by Dual TriNet we can observe a clear improvement over
ResNet-18.
\begin{figure*}
\begin{centering}
\includegraphics[scale=0.5]{space.png}
\par\end{centering}
\caption{\label{fig:aug-layer-1} One-shot results of feature augmentation
by different types of semantic spaces on CUB-200 and \emph{mini}ImageNet.
``Single Layer'' indicates the best one-shot performance augmented
by using only single layer. ``Multi-layer'' represents the results
of using synthesized instances from all layers.}
\end{figure*}
\vspace{0.05in}
\noindent \textbf{(3) Our framework can effectively augment multiple
layer features. }We analyze the effectiveness of augmented features
in each layer as shown in Fig. \ref{fig:aug-layer}. On CUB-200 and
\emph{mini}ImageNet, we report the results in 1-shot learning cases.
We have several conclusions: (1) Only using the augmented features
from one single layer (\emph{e.g.}, Layer1 -- Layer 4 in Fig. \ref{fig:aug-layer})
can also help improve the performance of one-shot learning results.
This validates the effectiveness of our dual TriNet of synthesizing
features of different layers in a single framework. (2) The results
of using synthesized instances from all layers (Multi-L) are even
better than those of individual layers. This indicates that the augmented
features at different layers are intrinsically complementary to each
other.
\vspace{0.05in}
\noindent \textbf{(4) Augmented features can boost the performance
of different supervised classifiers. }Our augmented features are not
designed for any one supervised classifier. To show this point and
as illustrated in Fig. \ref{fig:aug-layer}, three classical supervised
classifiers (\emph{i.e.}, KNN, SVM and LR) are tested along the X-axis
of Fig. \ref{fig:aug-layer}. Results show that our augmented features
can boost the performance of three supervised classifiers on one-shot
classification cases. This further validates the effectiveness of
our augmentation framework.
\vspace{0.05in}
\noindent \textbf{(5) The augmented features by SG, SN and AG can
also improve few-shot learning results.} We compare different types
of feature augmentation methods of various semantic spaces in Fig.
\ref{fig:aug-layer-1}. Specifically, we compare the SG and SN in
semantic word vector space; and AG in semantic attribute space. On
CUB-200 dataset, the augmented results by SG, SN and AG are better
than those without augmentation. The accuracy of combining the synthesized
instance features generated by any two of SG, SN, and AG can be further
improved over those of SG, SN or AG only. This means that the augmented
feature instances of SG, SN and AG are complementary to each other.
Finally, we observe that by combining augmented instances from all
methods (SG, SN and AG), the accuracy of one-shot learning is the
highest one.
\begin{table*}
\begin{centering}
\begin{tabular}{c|c|c|c|c}
\hline
\multirow{2}{*}{Methods } & \multicolumn{2}{l|}{{}Caltech-256 ($\%$)} & \multicolumn{2}{l}{{}CIFAR-100 ($\%$)}\tabularnewline
\cline{2-5}
& {}1-shot & {}5-shot & {}1-shot & {}5-shot \tabularnewline
\hline
\hline
{}MAML \cite{MAML} & {}45.59\textpm 0.77 & {}54.61\textpm 0.73 & {}49.28\textpm 0.90 & {}58.30\textpm 0.80 \tabularnewline
\hline
{}Meta-SGD \cite{meta-sgd} & {}48.65\textpm 0.82 & {}64.74\textpm 0.75 & {}53.83\textpm 0.89 & {}70.40\textpm 0.74 \tabularnewline
\hline
{}DEML+Meta-SGD \cite{DEML+Meta-SGD} & {}62.25\textpm 1.00 & {}79.52\textpm 0.63 & {}61.62\textpm 1.01 & {}77.94\textpm 0.74 \tabularnewline
\hline
\hline
{}Matching Nets \cite{matchingnet_1shot} & {}48.09\textpm 0.83 & {}57.45\textpm 0.74 & {}50.53\textpm 0.87 & {}60.30\textpm 0.82 \tabularnewline
\hline
\hline
{}ResNet-18 & {}60.13\textpm 0.71 & {}78.79\textpm 0.54 & {}59.65\textpm 0.78 & {}76.75\textpm 0.73\tabularnewline
\hline
{}ResNet-18+Dual TriNet & \textbf{{}63.77}{}\textpm 0.62 & \textbf{{}80.53}{}\textpm 0.46 & \textbf{{}63.41}{}\textpm 0.64 & \textbf{{}78.43}{}\textpm 0.62\tabularnewline
\hline
\end{tabular}
\par\end{centering}
\caption{\label{tab:cifar100}Results on Caltech-256 and CIFAR-100 datasets.
The ``{\small{}{}{}{}\textpm '' indicates $95\%$ confidence
intervals over tasks. }}
\end{table*}
\vspace{0.05in}
\noindent \textbf{(6) Even the semantic space inferred from the semantic
relationships of classes can also work well with our framework. }To
show this point, we again compare the results in Fig. \ref{fig:aug-layer-1}.
Particularly, we compute the similarity matrix of classes in \emph{mini}ImageNet
obtained using semantic word vectors. The SVD is employed to decompose
the similarity matrix and the left singular vectors of SVD are assumed
to span a new semantic space. Such a new space is hence utilized in
learning the dual TriNet. We employ the Semantic Gaussian (SG) to
augment the instance feature in the newly spanned space for one-shot
classification. The results are denoted ``SVD-G''. We report the
results of SVD-G augmentation in \emph{mini}ImageNet dataset in Fig.
\ref{fig:aug-layer-1}. We highlight several interesting observations.
(1) The results of SVD-G feature augmentation are still better than
those without any augmentation. (2) The accuracy of SVD-G is actually
slightly worse than that of SG, since the new spanned space is derived
from the original semantic word space. (3) There is almost no complementary
information in the augmented features between SVD-G and SG, still
partly due to the new space spanned from the semantic and the word
space. (4) The augmented features produced by SVD-G are also very
complementary to those from SN as shown in the results of Fig. \ref{fig:aug-layer-1}.
This is due to the fact that additional neighborhood vocabulary information
is not used in deriving the new semantic space. We have a similar
experimental conclusion on CUB-200 as shown in Tab. \ref{tab:CUB200}.
\begin{table*}
\centering{}%
\begin{tabular}{c|c|c|c|c|c|c|c|c|c|c}
\hline
\multirow{2}{*}{{\small{}Method} } & \multirow{2}{*}{{\small{}Shots }} & \multirow{2}{*}{{\small{}R-18 }} & \multirow{2}{*}{{\small{}Layer} } & \multicolumn{7}{c}{{\small{}{}Data Augmentation}}\tabularnewline
\cline{5-11}
& & & & \multicolumn{1}{c|}{{\small{}{}{}SN}} & \multicolumn{1}{c|}{{\small{}{}{}SG}} & \multicolumn{1}{c|}{{\small{}{}{}SD}} & \multicolumn{1}{c|}{{\small{}{}{}SN+SG}} & \multicolumn{1}{c|}{{\small{}{}{}SG+SD}} & \multicolumn{1}{c|}{{\small{}{}{}SN+SD}} & {\small{}{}{}SN+SG+SD }\tabularnewline
\hline
\multirow{4}{*}{{\small{}KNN }} & \multirow{2}{*}{1 } & \multirow{2}{*}{{\small{}{}64.30} } & {\small{}{}{}S.} & \multicolumn{1}{c|}{{\small{}{}{}65.12}} & \multicolumn{1}{c|}{{\small{}{}{}65.21}} & \multicolumn{1}{c|}{{\small{}{}{}65.38}} & \multicolumn{1}{c|}{{\small{}{}{}66.82}} & \multicolumn{1}{c|}{{\small{}{}{}65.50}} & \multicolumn{1}{c|}{{\small{}{}{}67.21}} & {\small{}{}{}67.23 }\tabularnewline
& & & {\small{}{}{}M.} & \multicolumn{1}{c|}{{\small{}{}{}65.58}} & \multicolumn{1}{c|}{{\small{}{}{}65.61}} & \multicolumn{1}{c|}{{\small{}{}{}65.78}} & \multicolumn{1}{c|}{{\small{}{}{}67.29}} & \multicolumn{1}{c|}{{\small{}{}{}65.77}} & \multicolumn{1}{c|}{{\small{}{}{}67.82}} & {\small{}{}{}67.91 }\tabularnewline
\cline{2-11}
& \multirow{2}{*}{5} & \multirow{2}{*}{{\small{}{}77.66} } & {\small{}{}{}S.} & \multicolumn{1}{c|}{{\small{}{}{}78.34}} & \multicolumn{1}{c|}{{\small{}{}{}78.42}} & \multicolumn{1}{c|}{{\small{}{}{}78.62}} & \multicolumn{1}{c|}{{\small{}{}{}79.01}} & \multicolumn{1}{c|}{{\small{}{}{}78.66}} & \multicolumn{1}{c|}{{\small{}{}{}79.12}} & {\small{}{}{}79.36 }\tabularnewline
& & & {\small{}{}{}M.} & \multicolumn{1}{c|}{{\small{}{}{}79.01}} & \multicolumn{1}{c|}{{\small{}{}{}78.96}} & \multicolumn{1}{c|}{{\small{}{}{}79.04}} & \multicolumn{1}{c|}{{\small{}{}{}79.51}} & \multicolumn{1}{c|}{{\small{}{}{}79.09}} & \multicolumn{1}{c|}{{\small{}{}{}79.56}} & {\small{}{}{}79.71 }\tabularnewline
\hline
\multirow{4}{*}{{\small{}SVR }} & \multirow{2}{*}{1} & \multirow{2}{*}{{\small{}{}66.54} } & {\small{}{}{}S.} & \multicolumn{1}{c|}{{\small{}{}{}67.63}} & \multicolumn{1}{c|}{{\small{}{}{}67.49}} & \multicolumn{1}{c|}{{\small{}{}{}67.69}} & \multicolumn{1}{c|}{{\small{}{}{}68.23}} & \multicolumn{1}{c|}{{\small{}{}{}67.60}} & \multicolumn{1}{c|}{{\small{}{}{}68.41}} & {\small{}{}{}68.56 }\tabularnewline
& & & {\small{}{}{}M.} & \multicolumn{1}{c|}{{\small{}{}{}68.10}} & \multicolumn{1}{c|}{{\small{}{}{}68.03}} & \multicolumn{1}{c|}{{\small{}{}{}68.22}} & \multicolumn{1}{c|}{{\small{}{}{}68.71}} & \multicolumn{1}{c|}{{\small{}{}{}67.98}} & \multicolumn{1}{c|}{{\small{}{}{}68.89}} & {\small{}{}{}69.01 }\tabularnewline
\cline{2-11}
& \multirow{2}{*}{5 } & \multirow{2}{*}{{\small{}{}82.38 }} & {\small{}{}{}S.} & \multicolumn{1}{c|}{{\small{}{}{}83.01}} & \multicolumn{1}{c|}{{\small{}{}{}83.07}} & \multicolumn{1}{c|}{{\small{}{}{}83.02}} & \multicolumn{1}{c|}{{\small{}{}{}83.59}} & \multicolumn{1}{c|}{{\small{}{}{}83.11}} & \multicolumn{1}{c|}{{\small{}{}{}83.42}} & {\small{}{}{}83.44 }\tabularnewline
& & & {\small{}{}{}M.} & \multicolumn{1}{c|}{{\small{}{}{}83.47}} & \multicolumn{1}{c|}{{\small{}{}{}83.51}} & \multicolumn{1}{c|}{{\small{}{}{}83.60}} & \multicolumn{1}{c|}{{\small{}{}{}83.82}} & \multicolumn{1}{c|}{{\small{}{}{}83.49}} & \multicolumn{1}{c|}{{\small{}{}{}83.99}} & {\small{}{}{}84.10 }\tabularnewline
\hline
\multirow{4}{*}{{\small{}LR }} & \multirow{2}{*}{1} & \multirow{2}{*}{{\small{}{}64.22} } & {\small{}{}{}S.} & \multicolumn{1}{c|}{{\small{}{}{}65.29}} & \multicolumn{1}{c|}{{\small{}{}{}65.33}} & \multicolumn{1}{c|}{{\small{}{}{}65.43}} & \multicolumn{1}{c|}{{\small{}{}{}66.59}} & \multicolumn{1}{c|}{{\small{}{}{}65.46}} & \multicolumn{1}{c|}{{\small{}{}{}66.89}} & {\small{}{}{}67.01 }\tabularnewline
& & & {\small{}{}{}M.} & \multicolumn{1}{c|}{{\small{}{}{}65.71}} & \multicolumn{1}{c|}{{\small{}{}{}65.92}} & \multicolumn{1}{c|}{{\small{}{}{}65.89}} & \multicolumn{1}{c|}{{\small{}{}{}67.12}} & \multicolumn{1}{c|}{{\small{}{}{}65.74}} & \multicolumn{1}{c|}{{\small{}{}{}67.63}} & {\small{}{}{}67.55 }\tabularnewline
\cline{2-11}
& \multirow{2}{*}{{\small{}{}5}} & \multirow{2}{*}{{\small{}{}82.51} } & {\small{}{}{}S.} & \multicolumn{1}{c|}{{\small{}{}{}83.37}} & \multicolumn{1}{c|}{{\small{}{}{}83.31}} & \multicolumn{1}{c|}{{\small{}{}{}83.60}} & \multicolumn{1}{c|}{{\small{}{}{}83.61}} & \multicolumn{1}{c|}{{\small{}{}{}83.59}} & \multicolumn{1}{c|}{{\small{}{}{}83.62}} & {\small{}{}{}83.69 }\tabularnewline
& & & {\small{}{}{}M.} & \multicolumn{1}{c|}{{\small{}{}{}}\textbf{\small{}{}83.82}} & \multicolumn{1}{c|}{{\small{}{}{}}\textbf{\small{}{}83.83}} & \multicolumn{1}{c|}{{\small{}{}{}}\textbf{\small{}{}83.90}} & \multicolumn{1}{c|}{{\small{}{}{}}\textbf{\small{}{}84.21}} & \multicolumn{1}{c|}{{\small{}{}{}}\textbf{\small{}{}83.71}} & \multicolumn{1}{c|}{{\small{}{}{}}\textbf{\small{}{}84.23}} & {\small{}{}{}}\textbf{\small{}{}84.17 }\tabularnewline
\hline
\end{tabular}{\tiny{}{}\caption{\label{tab:CUB200} \textbf{The classification accuracy of one-shot
learning on Caltech-UCSD Birds in 5-way. Note that: ``S.'' and ``M.''
indicates the single and multiple layers respectively. ``SD'' is
short for ``SVD-G''. ``R-18'' is short for ``ResNet-18''.}}
}
\end{table*}
\subsection{Experimental results on Caltech-256 and CIFAR-100}
\begin{figure*}
\begin{centering}
\includegraphics[scale=0.3]{visualize}
\par\end{centering}
\caption{\label{fig:Visualization-of-the}Visualization of the original and
augmented features.}
\vspace{-0.15in}
\end{figure*}
\noindent \textbf{Settings. }{On {}Caltech-256 and CIFAR-100 dataset
we also use the semantic word vector space. For one training instance,
we synthesize }16 sugmented features for SG and SN individually from
all four layers of ResNet-18. On these two datasets, the results of
competitors are implemented and reported in {\cite{DEML+Meta-SGD}}.
Our reported results are produced by using the augmented feature instances
of all layers, both by SG and SN. The SVM classifier is used as the
classification model.
\vspace{0.05in}
\noindent \textbf{Results. }The results on Caltech-256 and CIFAR-100
are illustrated in Tab. \ref{tab:cifar100}. We found that (1) our
method can still achieve the best performance as compared to the state-of-the-art
algorithms, thanks to the augmented feature instances obtained using
the proposed framework. (2) The ResNet-18 is still a very strong baseline;
and it can beat almost all the other baselines, except the DEML+Meta-SGD
which uses ResNet-50 as the baseline structure. (3) There is a clear
margin of improvement from using our augmented instance features over
using ResNet-18 only. This further validates the efficacy of the proposed
framework.
\section{Further analysis}
\begin{table*}
\centering{}%
\begin{tabular}{c|l|l|l|l|l|l|l|l}
\hline
\multirow{2}{*}{Methods} & \multicolumn{2}{l|}{{}{}MiniImagenet} & \multicolumn{2}{l|}{{}{}CUB-200} & \multicolumn{2}{l|}{{}{}Caltech-256} & \multicolumn{2}{l}{{}{}CIFAR-100}\tabularnewline
\cline{2-9}
& {}{}1-shot & {}{}5-shot & {}{}1-shot & {}{}5-shot & {}{}1-shot & {}{}5-shot & {}{}1-shot & {}{}5-shot \tabularnewline
\hline
\hline
{}{}ResNet-18 & {}{}52.73 & {}{}73.31 & {}{}66.54 & {}{}82.38 & {}{}60.13 & {}{}78.79 & {}{}59.65 & {}{}76.75 \tabularnewline
\hline
{}{}ResNet-18+U-net & {}{}56.41 & {}{}75.67 & {}{}68.32 & {}{}83.24 & {}{}61.54 & {}{}79.88 & {}{}62.32 & {}{}77.87 \tabularnewline
\hline
{}{}ResNet-18+Auto-encoder & {}{}56.80 & {}{}75.27 & {}{}68.56 & {}{}83.24 & {}{}62.41 & {}{}79.77 & {}{}61.76 & {}{}76.98 \tabularnewline
\hline
{}Ours without encoder & {}50.69 & {}70.79 & {}64.15 & {}80.06 & {}58.78 & {}76.45 & {}57.46 & {}75.12\tabularnewline
\hline
{}Ours without decoder & {}48.75 & {}65.12 & {}62.04 & {}78.16 & {}58.67 & {}76.45 & {}53.19 & {}68.74\tabularnewline
\hline
\hline
{}{}Ours & \textbf{{}{}58.12}{}{} & \textbf{{}{}76.92}{} & \textbf{{}{}69.61}{} & \textbf{{}{}84.10}{}{} & \textbf{{}{}63.77}{} & \textbf{{}{}80.53}{} & \textbf{{}{}63.41}{} & \textbf{{}{}78.43 }\tabularnewline
\hline
\end{tabular}\caption{\label{tab:Alternative-augmentation-network}Results of using alternative
augmentation networks.Ours indicates ResNet-18+Dual-TriNet.}
\end{table*}
\subsection{Comparison with standard augmentation methods}
Besides our feature augmentation method, we also compare the standard
augmentation methods \cite{KrizhevskySH12} in one-shot learning setting.
These methods include cropping, rotation, flipping, and color transformations
of training images of one-shot classes. Furthermore, we also try the
methods of adding the Gaussian noise to the ResNet-18 features of
training instances of one-shot classes as shown in Tab. \ref{tab:miniimagenet}.
However, none of these methods can improve the classification accuracy
in one-shot learning. This is reasonable since the one-shot classes
have only very few training examples. This is somewhat expected: such
naive augmentation methods intrinsically just add noise/variance,
but do not introduce extra information to help one-shot classification.
\subsection{Dual TriNet structure}
We propose the dual TriNet structure which intrinsically is derived
from the encoder-decoder architecture. Thus we further analyze the
other alternative network structures for feature augmentation. In
particular, the alternative choices of augmentation network can be
the auto-encoder \cite{hinton2006science} of each layer or U-net
\cite{unet2015}. The results are compared in Tab. \ref{tab:Alternative-augmentation-network}.
We show that our dual TriNet can best explore the complementary information
of different layers, and hence our results are better than those without
augmentation (ResNet-18), with U-net augmentation ({ResNet-18+U-net)
and with auto-encoder augmentation (ResNet-18+Auto-encoder). This
validates that our dual TriNet can efficiently merge and exploit the
information of multiple layers for feature augmentation.} In addition,
we conduct experiments to prove that the encoder part and decoder
part is necessary. If we simply used the semantic vector of true label
$\mathbf{u}_{z_{i}}^{base}$instead of using encoder $g_{Enc}\left(\left\{ f_{l}\left(\mathbf{I}_{i}^{support}\right)\right\} \right)$,
the augmented samples actually hurt the performance.
In the case where we do the classification in the semantic space,
effectively disabling the decoder, the performance drops by over 5\%.
This is because of the loss of information during the mapping from
visual space to semantic space, but in our approach, we keep original
information and have additional information from semantic space.
\subsection{Visualization }
Using the technique in \cite{2014arXiv1412.0035M}, we can visualize
the image that can generate the augmented features $\hat{f_{l}}\left(\mathbf{I}_{i}\right)=g\left(f_{l}\left(\mathbf{I}_{i}\right)\right)$
in ResNet-18. We first randomly generate an image $\mathbf{I}_{i_{0}}$.
Then we optimize $\mathbf{I}_{i_{0}}$ by reducing the distance betwen
$f_{l}\left(\mathbf{I}_{i_{0}}\right)$ and $\hat{f_{l}}\left(\mathbf{I}_{i}\right)$
(both are the output of ResNet-18):
\begin{equation}
\mathbf{I}_{i_{0}}=\mathrm{\underset{\mathbf{I}_{i_{0}}}{\mathrm{argmin}}}\frac{1}{2}\left\Vert f_{l}\left(\mathbf{I}_{i_{0}}\right)-\hat{f_{l}}\left(\mathbf{I}_{i}\right)\right\Vert _{2}^{2}+\lambda\cdot R\left(\mathbf{I}_{i_{0}}\right)\label{eq:visualization}
\end{equation}
where $R\left(\cdot\right)$ is the Total Variation Regularizer for
image smoothness; $\lambda=1e-2$. When the difference is small enough,
$\mathbf{I}_{i_{0}}$ should be representation of the image that can
generate the corresponding augmented feature.
By using SN and the visualization algorithm above, we visualize the
original and augmented features in Fig. \ref{fig:Visualization-of-the}.
The top row shows the input images of two birds, one roof, and one
dog. The blue circles and red circles indicate the visualization of
original and augmented features of Layer 1 -- Layer 4 respectively.
The visualization of augmented features is similar, and yet different
from that of original image. For example, the first two columns show
that the visualization of augmented features actually slightly change
the head pose of the bird. In the last two columns, the augmented
features clearly visualize a dog which is similar have a different
appearance from the input image. This intuitively shows why our framework
works.
\section{Conclusions}
This work purposes an end-to-end framework for feature augmentation.
The proposed dual TriNet structure can efficiently and directly augment
multi-layer visual features to boost the few-shot classification.
We demonstrate our framework can efficiently solve the few-shot classification
on four datasets. We mainly evaluate on classification tasks; it is
also interesting and future work to extend augmented features to other
related tasks, such as one-shot image/video segmentation \cite{one_shot_seg17BMVC,one-shot-video17cvpr}.
Additionally, though dual TriNet is paired with ResNet-18 here, we
can easily extend it for other feature extractor networks, such as
ResNet-50.
\bibliographystyle{splncs}
|
{
"timestamp": "2019-03-18T01:14:32",
"yymm": "1804",
"arxiv_id": "1804.05298",
"language": "en",
"url": "https://arxiv.org/abs/1804.05298"
}
|
\section{Introduction}\label{sec1}
Imagine a system ${\sf H}$ of hyperplanes in Euclidean space ${\mathbb R}^d$ ($d\ge 2$) that induces a tessellation $T_{\sf H}$ of ${\mathbb R}^d$. This means that any bounded subset of ${\mathbb R}^d$ meets only finitely many hyperplanes of ${\sf H}$ and that the components of ${\mathbb R}^d\setminus \bigcup_{H\in {\sf H}} H$ are bounded. The closures of these components are then convex polytopes which cover ${\mathbb R}^d$ and have pairwise no common interior points. The set of these polytopes is denoted by $T_{\sf H}$. We impose the additional assumption that the hyperplanes of ${\sf H}$ are in general position; then each polytope of $T_{\sf H}$ is simple, that is, each of its vertices is contained in precisely $d$ facets. The polytopes appearing in $T_{\sf H}$ may be rather boring; they could, for example, all be parallelepipeds. However, if the hyperplanes of ${\sf H}$ have sufficiently many different directions, one can imagine that quite different shapes of polytopes appear in $T_{\sf H}$. Is it possible that every combinatorial type of a simple $d$-polytope is realized in $T_{\sf H}$? This can be achieved in a much stronger sense.
In fact, suppose that $\widehat X$ is a stationary and isotropic Poisson hyperplane process in ${\mathbb R}^d$ (explanations are found in \cite{SW08}, for example). Its hyperplanes are almost surely in general position and induce a random tessellation of ${\mathbb R}^d$, denoted by $X$. The general character of the polytopes in $X$ was recently investigated in \cite{RS16}. For example, it was shown there that almost surely (a.s.) the translates of the polytopes in $X$ are dense in the space of convex bodies in ${\mathbb R}^d$ (with the Hausdorff metric). Another result was that a.s. the polytopes of $X$ realize every combinatorial type of a simple $d$-polytope infinitely often. In the following, we improve the latter result considerably, replacing `infinitely often' by `with positive density'. In the subsequent definition, $B_n$ is the ball in ${\mathbb R}^d$ with center at the origin and radius $n\in{\mathbb N}$, and $\lambda_d$ denotes Lebesgue measure in ${\mathbb R}^d$. Further, ${\mathbbm 1}_A$ is the indicator function of $A$.
\begin{definition}
Let $T$ be a tessellation of ${\mathbb R}^d$, and let $A$ be a translation invariant set of polytopes in ${\mathbb R}^d$. We say that $A$ {\bf appears in $T$ with density $\delta$} if
$$ \liminf_{n\to \infty} \frac{1}{\lambda_d(B_n)} \sum_{P\in T,\,P\subset B_n} {\mathbbm 1}_A(P)=\delta.$$
\end{definition}
With this definition, we prove below that in a Poisson hyperplane tessellation in ${\mathbb R}^d$ which is stationary and isotropic (that is, has a motion invariant distribution), almost surely every combinatorial type of a simple $d$-polytope appears with positive density. The actual result will, in fact, be more general: it is sufficient that the Poisson hyperplane tessellation is stationary and that its directional distribution, a measure on the unit sphere, is not zero on any nonempty open set and is zero on any great subsphere. The precise theorem is formulated in the next section.
\section{Explanations}\label{sec2}
We work in the $d$-dimensional Euclidean space ${\mathbb R}^d$ ($d\ge 2$) with its usual scalar product $\langle\cdot\,,\cdot\rangle$. By $\lambda_d$ we denote its Lebesgue measure, by $o$ its origin, by $B^d$ its unit ball (with $nB^d=:B_n$), and by ${\mathbb S}^{d-1}$ its unit sphere. The space of hyperplanes in ${\mathbb R}^d$, with its usual topology, is denoted by $\mathcal{H}$, and $\mathcal{B}(\mathcal{H})$ is the $\sigma$-algebra of Borel sets in $\mathcal{H}$. Hyperplanes in ${\mathbb R}^d$ are often written in the form
$$ H(u,\tau)= \{x\in {\mathbb R}^d: \langle x,u\rangle \le\tau\}$$
with $u\in{\mathbb S}^{d-1}$ and $\tau\in {\mathbb R}$.
We assume that $\widehat X$ is a stationary Poisson hyperplane process in ${\mathbb R}^d$, thus, a Poisson point process in the space $\mathcal{H}$ of hyperplanes, with the property that its distribution is invariant under translations (we refer, e.g., to \cite{SW08} for more details). The {\em intensity measure} $\widehat\Theta$ of $\widehat X$ is defined by
$$ \widehat\Theta(A)={\mathbb E}\, \widehat X(A)\quad\mbox{for }A\in\mathcal{B}(\mathcal{H}).$$
Here ${\mathbb E}\,$ denotes expectation, and we write $(\Omega,{\mathcal A},{\mathbb P})$ for the underlying probability space. It is assumed that $\widehat\Theta$ is locally finite and not identically zero. That $\widehat X$ is a Poisson process includes that
$$ {\mathbb P}(\widehat X(A)=k)= e^{-\widehat \Theta(A)}\frac{\widehat \Theta(A)^k}{k!} \quad \mbox{for }k\in {\mathbb N}_0,$$
for any $A\in \mathcal{B}(\mathcal{H})$ with $\widehat\Theta(A)<\infty$.
Since $\widehat X$ is stationary, the measure $\widehat\Theta$ has a decomposition
$$ \widehat\Theta(A) =\widehat \gamma \int_{{\mathbb S}^{d-1}} \int_{-\infty}^\infty {\mathbbm 1}_A(H(u,\tau))\,{\rm d}\tau\,\varphi({\rm d} u)$$
for $A\in\mathcal{B}(\mathcal{H})$ (see \cite{SW08}, Theorem 4.4.2 and (4.33)). The number $\widehat\gamma>0$ is the {\em intensity} of $\widehat X$, and $\varphi$ is a finite, even Borel measure on the unit sphere. It is called the {\em spherical directional distribution} of $\widehat X$. For any such measure $\varphi$ and any number $\widehat\gamma>0$, there exists a stationary Poisson hyperplane process in ${\mathbb R}^d$ with these data, and it is unique up to stochastic equivalence.
The hyperplane process $\widehat X$ induces a random tessellation of ${\mathbb R}^d$, which we denote by $X$. As usual, a random tessellation is formalized as a particle process; we refer again to \cite{SW08}.
Since we are considering only simple processes, it is convenient to identify such a process, which by definition is a counting measure, with its support, which is a locally finite set. In particular, a realization of $\widehat X$ is also considered as a set of hyperplanes, and a realization of $X$ is considered as a set of polytopes. The notations $\widehat X(\{H\})=1$ and $H\in \widehat X$ for a hyperplane $H$, for example, are therefore used synonymously.
The combinatorial type of a polytope $P$ in ${\mathbb R}^d$ is the set of all polytopes in ${\mathbb R}^d$ that are combinatorially isomorphic to $P$. Now we can formulate our result.
\begin{theorem}\label{T1}
Let $X$ be a tessellation of ${\mathbb R}^d$ that is induced by a stationary Poisson hyperplane process $\widehat X$ with spherical directional distribution $\varphi$. Suppose that the support of $\varphi$ is the whole unit sphere ${\mathbb S}^{d-1}$ and that $\varphi$ assigns measure zero to each great subsphere of ${\mathbb S}^{d-1}$. Then, with probability one, each combinatorial type of a simple $d$-polytope appears with positive density in $X$.
\end{theorem}
Theorem \ref{T1} implies, trivially, that under its assumptions almost surely each combinatorial type of a simple $d$-polytope appears infinitely often in $X$. When the latter fact was proved, among other results, in \cite{RS16}, a tool was a strengthened version of the Borel--Cantelli lemma, due to Erd\"os and R\'{e}nyi \cite{ER59} (see also \cite[p. 327]{Ren66}). When the note \cite{RS16} was submitted, an anonymous referee wrote ``that the use of ergodicity of the mosaic could lead to a possibly shorter alternative proof'', and he/she briefly indicated a possible approach. After thorough consideration, we preferred the more elementary Borel--Cantelli lemma. However, reconsideration revealed that ergodicity, applied in a different way, might lead to a stronger result, as far as the occurrence of combinatorial types is concerned. This is carried out in the following.
\section{Proof}\label{sec3}
Let $X$ satisfy the assumptions of Theorem \ref{T1}. Under the only assumption that the spherical directional distribution of the stationary Poisson hyperplane tessellation $X$ is zero on every great subsphere, it was shown in \cite[Thm. 10.5.3]{SW08} that $X$ is mixing and hence ergodic. This requires a few explanations. To model $X$ as a point process, we consider the space ${\mathcal K}$ of convex bodies (nonempty, compact, convex subsets) in ${\mathbb R}^d$ with the Hausdorff metric. By $\mathcal{B}({\mathcal K})$ we denote the $\sigma$-algebra of Borel sets in ${\mathcal K}$. Let ${\sf N}_s({\mathcal K})$ be the set of simple, locally finite counting measures on $\mathcal{B}({\mathcal K})$ and ${\mathcal N}_s({\mathcal K})$ its usual $\sigma$-algebra (for details see, e.g., \cite[Sect. 3.1]{SW08}). As underlying probability space $(\Omega,{\mathcal A},{\mathbb P})$, on which $X$ is defined, we can use $({\sf N}_s({\mathcal K}),{\mathcal N}_s({\mathcal K}),{\mathbb P}_X)$, where ${\mathbb P}_X$ is the distribution of $X$. For $t\in{\mathbb R}^d$, a bijective map ${\sf T}_t:\eta\mapsto {\sf T}_t\eta$ of ${\sf N}_s({\mathcal K})$ onto itself is defined by
$$ ({\sf T}_t\eta)(B):= \eta(B-t),\quad B\in \mathcal{B}({\mathcal K}),\,\eta\in {\sf N}_s({\mathcal K}).$$
Since $X$ is stationary, we have
$$ {\mathbb P}_X({\sf T}_tA)={\mathbb P}_X(A)\quad\mbox{for }A\in{\mathcal N}_s({\mathcal K}),$$
thus ${\sf T}_t$ induces a measure preserving map of ${\mathcal N}_s({\mathcal K})$ into itself. Let ${\mathcal T}:=\{ {\sf T}_t:t\in{\mathbb R}^d\}$. As shown in \cite[Thm. 10.5.3]{SW08}, the dynamical system $({\sf N}_s({\mathcal K}),{\mathcal N}_s({\mathcal K}),{\mathbb P}_X,{\mathcal T})$ is mixing, that is,
$$ \lim_{\|t\|\to\infty} {\mathbb P}_X(A\cap {\sf T}_tB)={\mathbb P}_X(A){\mathbb P}_X(B)$$
holds for all $A,B\in{\mathcal N}_s({\mathcal K})$. It follows that the system is ergodic, which means that ${\mathbb P}_X(A)\in\{0,1\}$ for all $A\in {\bf T}:= \{A\in{\mathcal N}_s({\mathcal K}):{\sf T}_tA=A\mbox{ for all }t\in{\mathbb R}^d\}$. Therefore, the `Individual Ergodic Theorem for $d$-dimensional Shifts' yields the following.
\begin{proposition}\label{P1}
Let $f$ be an integrable random variable on $({\sf N}_s({\mathcal K}),{\mathcal N}_s({\mathcal K}),{\mathbb P}_X)$. Then
$$ \lim_{n\to\infty} \frac{1}{\lambda_d(B_n)} \int_{B_n} f({\sf T}_t\,\omega)\,\lambda({\rm d} t) = {\mathbb E}\, f$$
holds for ${\mathbb P}_X$-almost all $\omega\in {\sf N}_s({\mathcal K})$.
\end{proposition}
We refer to Daley and Vere--Jones \cite[Proposition 12.2.II]{DV08} for a more general formulation (with hints to proofs of more general results in Tempel'man \cite{Tem72}). However, we have already incorpated into our Proposition \ref{P1} the information that in our case $({\sf N}_s({\mathcal K}),{\mathcal N}_s({\mathcal K}),{\mathbb P}_X,{\mathcal T})$ is ergodic, which yields that the limit is equal to the expectation of $f$.
We apply this Proposition in the following way. First we choose a center function $c$ on ${\mathcal K}$; for example, let $c(K)$ denote the circumcenter of $K\in {\mathcal K}$, which is the center of the smallest ball containing $K$. Let $A\in \mathcal{B}({\mathcal K})$ be a translation invariant Borel set of convex bodies. Given any bounded Borel set $B\in\mathcal{B}({\mathbb R}^d)$, we define
$$ f(B,\omega):= \sum_{K\in X(\omega),\,c(K)\in B} {\mathbbm 1}_A(K)$$
for $\omega\in\Omega$, where we use $(\Omega,{\mathcal A},{\mathbb P})=({\sf N}_s({\mathcal K}),{\mathcal N}_s({\mathcal K}),{\mathbb P}_X)$ as the underlying probability space. Then $f(B,\cdot)$ is measurable, and $f(B+t,\omega)= f(B,{\sf T}_{-t}\,\omega)$ for $t\in{\mathbb R}^d$. The following generalizes an approach of Cowan \cite{Cow80} in the plane (``Tricks with small disks''). Assuming that $n>1$, we have
\begin{eqnarray*}
&& \int_{B_{n-1}} f(B_1+t,\omega)\,\lambda_d({\rm d} t)\\
&&= \sum_{K\in X(\omega)} \int_{{\mathbb R}^d} {\mathbbm 1}\{t\in B_{n-1}\}{\mathbbm 1}\{K\in A\}{\mathbbm 1}\{c(K)\in B_1+t\}\,\lambda_d({\rm d} t).
\end{eqnarray*}
Since
$$ {\mathbbm 1}\{t\in B_{n-1}\}{\mathbbm 1}\{c(K)\in B_1+t\}\le {\mathbbm 1}\{t\in -B_1+c(K)\}{\mathbbm 1}\{ c(K)\in B_n\},$$we get
\begin{eqnarray*}
&& \int_{B_{n-1}} f(B_1+t,\omega)\,\lambda_d({\rm d} t)\\
&&\le \sum_{K\in X(\omega)} \int_{{\mathbb R}^d} {\mathbbm 1}\{t\in -B_1+c(K)\}{\mathbbm 1}\{K\in A\}{\mathbbm 1}\{c(K)\in B_n\}\,\lambda_d({\rm d} t)\\
&& = \lambda_d(B_1)f(B_n,\omega).
\end{eqnarray*}
Similarly,
\begin{eqnarray*}
&& \int_{B_{n+1}} f(B_1+t,\omega)\,\lambda_d({\rm d} t)\\
&& \ge \sum_{K\in X(\omega)} \int_{{\mathbb R}^d} {\mathbbm 1} \{t\in -B_1+c(K)\} {\mathbbm 1} \{K\in A\}{\mathbbm 1} \{c(K)\in B_n\}\,\lambda_d({\rm d} t)\\
&& = \lambda_d(B_1)f(B_n,\omega).
\end{eqnarray*}
We conclude that
\begin{eqnarray*}
&& \frac{\lambda_d(B_{n-1})} {\lambda_d(B_n)} \frac{1}{\lambda_d(B_{n-1})} \int_{B_{n-1}} f(B_1,{\sf T}_{-t}\,\omega)\,\lambda_d({\rm d} t)\\
&& \le \frac{\lambda_d(B_1)}{\lambda_d(B_n)}f(B_n,\omega)\\
&& \le \frac{\lambda_d(B_{n+1})}{\lambda_d(B_n)} \frac{1}{\lambda_d(B_{n+1})} \int_{B_{n+1}} f(B,{\sf T}_{-t}\,\omega)\,\lambda_d({\rm d} t).
\end{eqnarray*}
By the Proposition, the lower and the upper bound converge, for $n\to\infty$, almost surely to ${\mathbb E}\, f(B_1,\cdot)$, hence a.s.
\begin{equation}\label{3.1}
\lim_{n\to\infty} \frac{1}{\lambda_d(B_n)}f(B_n,\cdot) = \frac{{\mathbb E}\, f(B_1,\cdot)}{\lambda_d(B_1)}.
\end{equation}
Now we assume in addition that there is a constant $D>0$ such that all convex bodies $K\in A$ satisfy ${\rm diam}\,K\le D$, where ${\rm diam}$ denotes the diameter. The center function $c$ satisfies $c(K)\in K$, hence if $c(K)\in B_{n-D}$ (with $n>D$) and ${\rm diam}\,K\le D$, then $K\subset B_n$. It follows that, for $n>D$,
\begin{eqnarray*}
&& \frac{\lambda_d(B_{n-D})} {\lambda_d(B_n)} \frac{1}{\lambda_d(B_{n-D})} \sum_{K\in X} {\mathbbm 1}_A(K) {\mathbbm 1}\{c(K)\in B_{n-D}) \\
&& \le\frac{1}{\lambda_d(B_n)} \sum_{K\in X,\,K\subset B_n} {\mathbbm 1}_A(K) \\
&&\le \frac{1}{\lambda_d(B_n)} \sum_{K\in X} {\mathbbm 1}_A(K\in A){\mathbbm 1}\{c(K)\in B_n\}.
\end{eqnarray*}
As $n\to\infty$, the lower and the upper bound converge a.s. to the right side of (\ref{3.1}), hence a.s. we have
\begin{equation}\label{3.1a}
\delta(X,A):=\lim_{n\to\infty} \frac{1}{\lambda_d(B_n)} \sum_{K\in X,\,K\subset B_n} {\mathbbm 1}_A(K) = \frac{1}{\lambda_d(B^d)} \,{\mathbb E}\, \sum_{K\in X,\,c(K)\in B^d} {\mathbbm 1}_A(K).
\end{equation}
Now we consider the special case where $A_D$ is the set of polytopes that are combinatorially isomorphic to a given simple $d$-polytope $P$ and have diameter at most $D$, for some fixed number $D>0$. We remark that (\ref{3.1a}) shows that
\begin{equation}\label{3.1b}
\delta(X,A_D) = \frac{1}{\lambda_d(B^d)} \,{\mathbb E}\, \sum_{K\in X,\,c(K)\in B^d} {\mathbbm 1}\{K\in A_D\},
\end{equation}
It remains to show that
\begin{equation}\label{3.2}
{\mathbb E}\, \sum_{K\in X,\,c(K)\in B^d} {\mathbbm 1}\{K\in A_D\}>0.
\end{equation}
For this, we use an argument from \cite{RS16}, which we recall for completeness.
Without loss of generality, we can assume that $c(P) =o$. Let $F_1,\dots,F_m$ be the facets of $P$. We denote by $B(x,\varepsilon)$ the ball with center $x$ and radius $\varepsilon>0$, set $[B(x,\varepsilon)]_\mathcal{H}:= \{H\in\mathcal{H}:H\cap B(x,\varepsilon)\not=\emptyset\}$, and define
$$ A_j(P,\varepsilon) := \bigcap_{v\in{\rm vert}F_j} [B(v,\varepsilon)]_\mathcal{H},\quad j=1,\dots,m,$$
where ${\rm vert}$ denotes the set of vertices. Each hyperplane from $A_j(P,\varepsilon)$ is said to be {\em $\varepsilon$-close} to $F_j$. A polytope $Q$ is said to be {\em $\varepsilon$-close} to $P$ if it has $m$ facets $G_1,\dots,G_m$ and, after suitable renumbering, the affine hull of $G_j$ is $\varepsilon$-close to $F_j$, for $j=1,\dots,m$. Since $P$ is simple and $c(P)=o$, we can choose numbers $D,\varepsilon_0>0$ such that for $0<\varepsilon\le \varepsilon_0$, the following is true:\\[1mm]
$\bullet$ the sets $A_1(P,\varepsilon),\dots,A_m(P,\varepsilon)$ are pairwise disjoint, and any hyperplanes $H_j\in A_j(P,\varepsilon)$, $j=1,\dots,m$, are the facet hyperplanes of a polytope $Q$ that is $\varepsilon$-close to $P$.\\[1mm]
$\bullet$ Any polytope $Q$ that is $\varepsilon$-close to $P$ satisfies the following:\\[1mm]
$\bullet$ $Q$ is combinatorially isomorphic to $P$,\\[1mm]
$\bullet$ $Q\subset P+B^d$,\\[1mm]
$\bullet$ ${\rm diam}\,Q\le D$,\\[1mm]
$\bullet$ $c(Q)\in B^d$.\\[1mm]
That this can be achieved by suitable choices of $D$ and $\varepsilon_0$, follows from easy continuity considerations and the fact that $P$ is simple.
Now we define
$$ C(P,\varepsilon):= \{H \in \mathcal{H}: H\cap (P+B^d)\not=\emptyset,\; H\notin A_j(P,\varepsilon) \mbox{ for }j=1,\dots,m\}$$
and consider the event $E(P,\varepsilon)$ defined by
$$ \widehat X(A_j(P,\varepsilon))=1 \mbox{ for }j=1,\dots,m \enspace \mbox{and} \enspace\widehat X(C(P,\varepsilon))=0.$$
Let $0<\varepsilon\le\varepsilon_0$. The following was proved in \cite{RS16}:\\[1mm]
$\bullet$ If the event $E(P,\varepsilon)$ occurs, then some polytope $Q$ of the tessellation $X$ is $\varepsilon$-close to $P$ and hence satisfies $Q\in A_D$ and $c(Q)\in B^d$,\\[1mm]
$\bullet$ The event ${\mathbb P}(E(P,\varepsilon))$ has positive probability.
Now it follows that
$$ {\mathbb E}\, \sum_{K\in X,\,c(K)\in B^d} {\mathbbm 1}\{K\in A_D\} \ge {\mathbb P}(E(P,\varepsilon))>0,$$
which proves (\ref{3.2}).
The result is that $\delta(X,A_D)>0$ a.s. This implies, in particular, that with probability one the polytopes of the combinatorial type of $P$ appear in $X$ with positive density. Since there are only countably many combinatorial types, it also holds with probability one that each combinatorial type of a simple $d$-polytope appears in $X$ with positive density.
|
{
"timestamp": "2018-04-17T02:16:09",
"yymm": "1804",
"arxiv_id": "1804.05622",
"language": "en",
"url": "https://arxiv.org/abs/1804.05622"
}
|
\section*{Energy of a Gaussian-localised particle}
In our protocol we exchange Gaussian-localised single-particle excitations between the labs. If we use the Hamiltonian operator, we can show that these single-particle excitations have finite energy provided they are not strictly localised. The Hamiltonian is,
\begin{equation}
H = \sum_{j}\int \mathrm{d}k \frac{|k|}{(2\pi)^{1/2}} a_{k,j}^\dagger a_{k,j}.
\end{equation}
We find that the expectation,
\begin{align}
&\Braket{1,j|H|1,j} = \int \frac{\mathrm{d}k}{2\pi \sigma} |k| e^{-\frac{(k-k_0)^2}{2\sigma^2}}\\
&\stackrel{k_0\gg \sigma}{\approx}\frac{\sigma}{2\pi} \rbracket{\sqrt{2\pi} \frac{k_0}{\sigma} + e^{-\frac{k_0^2}{\sigma^2}} \frac{2\sigma^2}{k_0^2} + \mathcal{O}\sbracket{\rbracket{\frac{\sigma}{k_0}}^3}}
\end{align}
is finite in energy for $\sigma <\infty$ (i.e. not strictly localised).
\section*{The Mode selective mirror}
In the main text we modelled the mode selective mirror as a projective measurement onto the lab mode. Here we present a more detailed model of the mirror. Let us consider Alice's lab. Fig. 1 represents the mode selective mirror. A complete set of orthonormal modes, $\cbracket{a_i}$, impinges from the outside. This basis set is chosen such that Alice's lab mode, $a_0$, is a member of the set (this can always be done \cite{rohde_spectral_2007}. A complementary \footnote{Complementary in the sense that one mode becomes the other if their propagation direction is reflected by $90^\circ$} and orthogonal set of modes, $\cbracket{b_i}$, impinges from the inside. An incoming mode, $c_\text{in}$ from the outside can then be decomposed as
\begin{align}
c_\text{in} = \sqrt{\eta} ~a_0 + \sum_{i \ne 0} f_i ~a_i,
\end{align}
where $\sqrt{\eta} = \sbracket{c_\text{in}, a_0^{\dagger}}$ is given by the overlap of $c_\text{in}$ and $a_0$. Also note that $\eta +\sum_{i \ne 0} |f_i|^2 =1$. Alice's mode selective mirror can then be modelled by the direct product of unitaries
\begin{align}
U = \prod_{i \ne 0} e^{i \frac{\pi}{2} \rbracket{a_i b_i^\dagger + a_i^\dagger b_i}}, \label{eq:unitary}
\end{align}
which reflects all $a_i$ with $i \ne 0$, but transmit $a_0$.
\begin{figure}[h]
\centering
\includegraphics[width=0.35\textwidth]{beamsplitter3.pdf}
\caption{A setup of the mode selective mirror. The beamsplitter is given by the unitary in \cref{eq:unitary}. The state from bob enters from the left and the lab is to the right of the beamsplitter.}
\end{figure}
So a single photon state from Bob, $\rbracket{\Ket{\Psi} = c_\text{in}^\dagger \Ket{0}}$, going through the mirror becomes
\begin{equation}
U \Ket{\Psi} = U c_\text{in}^\dagger U^{\dagger} U \Ket{0} = \rbracket{\sqrt{\eta}~ a_0^\dagger + i\sum_{i \ne 0} f_i~ b_{i}^\dagger}\Ket{0},
\end{equation}
where we have used that $U^{\dagger} U$ is the identity and $U \Ket{0} = \Ket{0}$. If we trace over the reflected outside modes $b_i$, the reduced density operator of the state in mode $a_0$ inside the lab is,
\begin{equation}
\rho = \eta ~ a_0^\dagger \Ket{0}\Bra{0} a_0 + (1-\eta) \Ket{0}\Bra{0}.
\end{equation}
All other modes are in the vacuum state. Any operation carried out in the lab will have the maximum probability ($\eta$) of interacting with the photon if it is carried out on the lab mode, $a_0$.
A physical implementation of the mode-selective mirror requires an active interaction such as the pulse gate introduced by \textcite{eckstein_quantum_2011}.
\section*{Measurements with different timing precision than the mode}
Let us suppose that Alice sends out a mode with a width $\sigma_A$ and Bob tries to measure a mode with a width $\sigma_B$, then we find that
\begin{align}
P_\text{Bob's mirror} &=\left| \Braket{0|a(t_B,x_B,\sigma_B) a^\dagger(t_A,x_A,\sigma_A)|0}\right|^2 \\
& = \frac{2\sigma_A \sigma_B e^{ -\frac{2 (\Delta t +\tau)^2 \sigma_A^2 \sigma_B^2}{\sigma_A^2 + \sigma_B^2}}}{\sigma_A^2 + \sigma_B^2}.
\end{align}
In the case of maximum probability, this gives
\begin{align}
P_\text{Bob's mirror, max} & = \frac{2\sigma_A \sigma_B }{\sigma_A^2 + \sigma_B^2}\\
&=\frac{2 \frac{\sigma_A}{\sigma_B}}{1+ \frac{\sigma_A^2}{\sigma_B^2}}.
\end{align}
Which is strictly $<1$ for $\frac{\sigma_A}{\sigma_B} \neq 1$. The generalised probability of success is therefore
\begin{equation}
P_\text{succ} = \frac{1}{4} \rbracket{2 + \frac{2\sigma_A \sigma_B e^{ -\frac{2 (\Delta t +\tau)^2 \sigma_A^2 \sigma_B^2}{\sigma_A^2 + \sigma_B^2}}}{\sigma_A^2 + \sigma_B^2}+\frac{2\sigma_A \sigma_B e^{ -\frac{2 (-\Delta t +\tau)^2 \sigma_A^2 \sigma_B^2}{\sigma_A^2 + \sigma_B^2}}}{\sigma_A^2 + \sigma_B^2} }.
\end{equation}
So we see that anything other than $\sigma_A = \sigma_B$ would cause a decrease in the violation of the causal inequality. In particular, the violation would be reduced if Bob tries to measure a mode with greater timing precision (i.e. $\sigma_B>\sigma_A$) than the mode that Alice actually sent.
\section{Operations in the lab}
We allow all physical operations to be carried out in the labs. However, as previously noted, efficient coupling to any incoming photon will only occur by addressing the lab mode. Similarly, efficient coupling to a photon that will successfully leave the lab via the mode-selective output mirror will only occur by addressing the lab mode. Thus any unitary operations should act specifically on the lab mode or on ancillary states in complementary modes.
There is some subtlety in this, as the physical unitaries doing the operations are localised in space while the mode itself is delocalised. This means that the unitaries are delocalised in time. Such unitaries have a causal order in terms of their central time, or equivalently in terms of their spatial ordering within the lab, but their temporal spread means their operations overlap in time.
However, if all of the above conditions are fulfilled, we could perform any unitary within the lab. This would include measurement and preparing the output state. This is indicated in \cref{fig:setup} a). In particular, the output state can be prepared conditional on the measurement outcome of the input state, thus justifying the view that---from the laboratory perspective---the measurement causally precedes the preparation. This would not be possible in a protocol where causal inequalities are violated thanks to ``open laboratories'', where a party performs the preparation first and the measurement later, after the system has gone through the other party's lab.
The violation of the causal inequality indicates that signals can be sent efficiently both from Alice to Bob and from Bob to Alice. As we have commented, preparations of outputs conditional on inputs is allowed by our formalism. One might then worry that this somehow leads to inconsistent behaviour such as Alice sending a message to her own input telling her not to send a message. Of course, our formalism is based on quantum field theory so we expect consistent solutions. The situation we have described is in fact a quantum feedback loop
\cite{yanagisawa_self-consistent_2010}. Whilst in general this problem is very difficult to solve there exists solutions for zero-time feedback loops \cite{combes_slh_2017}. In the next section we investigate a non-trivial loop in the limit of zero-time feedback, where $\tau \sigma \ll 1$ such that the time of travel is much smaller than the temporal spread in the wave packet (i.e. an extreme case of scenario c in the main text).
\section{Feedback loop for a CNOT gate}
Let us consider a CNOT gate implemented with a cross Kerr non-linearity and dual rail encoding. The CNOT gate is depicted in \cref{fig:CNOT}.
\begin{figure}[H]
\centering
\includegraphics[width=0.35\columnwidth]{Cnotkerr.pdf}
\caption{CNOT gate with a cross Kerr non-linearity. The beamsplitters are 50:50. A control qubit is encoded as $\Ket{0_c} = a^\dagger \Ket{0}$ and $\Ket{1_c} = {a'}^\dagger\Ket{0}$. The target qubit is encoded as $\Ket{0_t} = b^\dagger \Ket{0}$ and $\Ket{1_c} = {b'}^\dagger\Ket{0}$.}
\label{fig:CNOT}
\end{figure}
The cross Kerr non-linearity is given by a unitary,
\begin{equation}
U = e^{i \pi c^\dagger c {a'}^\dagger a'}
\end{equation}
The output of this circuit is,
\begin{align}
a'_\text{out} &= e^{-i \pi c^\dagger c} a'\\
d_\pm &= \frac{1}{2}\sbracket{\rbracket{e^{-i\pi {a'}^\dagger a' } \pm 1 } b + \rbracket{e^{-i\pi {a'}^\dagger a' } \mp 1 } b'}
\end{align}
Now if we feed the output $a$ \& $a'$ to the input $b$ \& $b'$, then we have the circuit in \cref{fig:feedback}. Notice that nominally this assignment can be inconsistent. For example if we prepare the $a$ modes in the state $\Ket{+} = 1/\sqrt{2}\rbracket{\Ket{01}+\Ket{10}}$ and the $b$ modes in the state $\Ket{+}$, then the $a_\text{out}$ modes are in the state $\Ket{-} = 1/\sqrt{2}\rbracket{\Ket{01}-\Ket{10}}$, so the $a_\text{out}$ and $b$ modes seems inconsistent. If we try to fix this by making the $b$ modes in the state $\Ket{-}$ then the $a_\text{out}$ modes switch to $\Ket{+}$ -- seemingly inconsistent again. However, we will see that the actual solution is consistent.
\begin{figure}[H]
\centering
\includegraphics[width=0.35\columnwidth]{cnotfeedback.pdf}
\caption{CNOT gate with zero-time feedback.}
\label{fig:feedback}
\end{figure}
By equating $b = a$ and $b' = a'_\text{out}$ we are assuming the loop is short and the feedback is effectively instantaneous. Notice that we have reduced the Hilbert space of the problem down to 2 dimensions from the previous 4.
The output is now given by,
\begin{equation}
d_\pm = \frac{1}{2}\sbracket{\rbracket{e^{-i\pi {a'}^\dagger a' } \pm 1 }a + \rbracket{e^{-i\pi {a'}^\dagger a' } \mp 1 } e^{-i \pi c^\dagger c} a'}
\end{equation}
While we have a self-recursive expression for $c = \frac{1}{\sqrt{2}} \rbracket{a + e^{-i \pi c^\dagger c} a'}$ we will see that we don't need an explicit expression. We can now calculate what this circuit does to logical 0s and 1s.
\begin{align*}
d_\pm a^\dagger \Ket{0} &= \frac{1}{2} \rbracket{1\pm 1} \Ket{0}\\
\implies &\Braket{0| a d_\pm^\dagger d_\pm a^\dagger|0} = \begin{cases}
1 & \text{for} \quad d_+\\
0 & \text{for} \quad d_-
\end{cases}\\
d_\pm {a'}^\dagger \Ket{0} &= \rbracket{e^{-i\pi {a'}^\dagger a' } \mp 1 } e^{-i \pi c^\dagger c} a' {a'}^\dagger \Ket{0}\\
&=\frac{1}{2} \rbracket{1\mp 1} \Ket{0}\\
\implies &\Braket{0| a' d_\pm^\dagger d_\pm {a'}^\dagger|0} = \begin{cases}
0 & \text{for} \quad d_+\\
1 & \text{for} \quad d_-
\end{cases}
\end{align*}
We see that although we do not know the expression for $c$, $e^{-i \pi c^\dagger c}$ acts on the vacuum. For arbitrary input states we find
\begin{align}
&\Braket{0| (\alpha^* a + \beta^* a') d_+^\dagger d_+ (\alpha a^\dagger + \beta a'^\dagger)|0} = |\alpha|^2 \\
&\Braket{0| (\alpha^* a + \beta^* a') d_-^\dagger d_- (\alpha a^\dagger + \beta a'^\dagger)|0} = |\beta|^2
\end{align}
So we see that the zero-time feedback for a CNOT gate (up to a phase rotation) is actually just the identity.
Let us now consider modes extended in time. For the case in \cref{fig:CNOT}, it is clear how to proceed, we simply specify that the unitary and mirrors are mode matched to modes $a,a', b, b'$. However, when there is a finite-time feedback loop, the modes rentering are shifted in time. If we continue using the mode matched unitary as before, then in the case of scenario a) where the temporal spread of the modes is small compared to the distance between labs, we expect that the unitary would not be matched to the mode by the time most of it propagates back. Therefore in scenario a), we expect that the unitary is also the identity.
\end{document}
|
{
"timestamp": "2019-02-11T02:08:17",
"yymm": "1804",
"arxiv_id": "1804.05498",
"language": "en",
"url": "https://arxiv.org/abs/1804.05498"
}
|
\section{Introduction}
Data Center Networks (DCNs) are core infrastructures for various online services and cloud applications such as social networking and cloud computing.
Those services and applications are engendering an exponential traffic growth, which places a significant demand on network bandwidth.
To meet this demand, all-optical DCNs arise as promising architectures because they offer extremely high bandwidth by adopting Wavelength Division Multiplexing (WDM) \cite{all_optical_scheduling, wave_Cube_ToN, OSA_optical_switching}. Besides, optical DCNs are reported to consume much less power compared with electronic DCNs \cite{Helios,OSA_optical_switching}.
In a large-scale datacenter deployment, traditional hierarchical tree topologies face issues such as link oversubscription and network bisection-bandwidth bottlenecks.
To address these issues, researchers have proposed various scalable topology solutions such as fat-trees \cite{al2008scalable}, BCube \cite{guo2009bcube}, and ExCCC \cite{ExCCC-DCN}.
A fat-tree is a folded version of a Clos network which was first designed in mid-1950s \cite{clos1953study}.
BCube, as a modified version of Hypercube, was proposed recently by Guo \textit{et al.} \cite{guo2009bcube} for building modular datacenters.
Both fat-trees and BCube achieve linear relationships between the network bisection bandwidth and the network size.
However, BCube is reported to be more cost-effective than fat-trees.
In addition, a BCube network can push the routing and scheduling functionalities to end-servers, which helps alleviate the routing burden on intermediate switches.
For simplicity, we refer to servers or end-servers as hosts.
Note that all-optical DCNs are promising and BCube is highly scalable and economic for building modular datacenters.
In this paper, we make the first attempt to study the fundamental problem of Routing and Wavelength Assignment (RWA) in an all-optical BCube network considering a \textit{host-to-host} traffic, where we assign every Source-Destination (S-D) host pair with a nonblocking \textit{lightpath} --- consists of a single physical path and a single wavelength --- such that all host pairs can communicate simultaneously.
We describe lightpaths are \textit{nonblocking} if lightpaths that share a common link have different wavelengths.
Since wavelength is a limited resource, the goal of the RWA problem is to minimize wavelength usage \cite{routing_W, RWA_opt}.
Although a host-to-host traffic may not arise frequently in practice, it evaluates the maximum communication capacity of a network and also locates a reference point for further communication analysis.
To simplify the analysis, we divide the RWA problem into two parts: path allocation and wavelength assignment.
In the part of path allocation, we aim to find a set of dipaths that minimizes the maximum link load;
in the part of wavelength assignment, we aim to minimize the usage of wavelengths.
Specifically, the minimum of the maximum link load over all possible routings is referred to as the \textit{forwarding index} when link load is measured by the number of paths passing through it \cite{optical_fowarding, forward_optical_paper}.
We refer to the minimum number of wavelengths, used to support simultaneous host-to-host communication, as the \textit{optical index}.
It has been shown that the optical index is naturally lower bounded by the forwarding index \cite{forward_optical_paper, optical_fowarding}.
It is NP-hard to derive either the optical index or the forwarding index in a general network since these problems are shown to be more complicated than a vertex coloring problem \cite{full_connections}.
Therefore, there have been numerous attempts to study the optical and forwarding index in various interconnection networks such as fat-trees \cite{global_packing}, $4$-regular circulant networks \cite{forward_optical_paper}, and some Cartesian product of chains or cycles \cite{full_connections, all-to-alloptical}.
In particular, Lo \textit{et al.} \cite{global_packing} derived the optical index in an all-optical fat-tree network through explicit construction of an RWA scheme; Beauquier \cite{full_connections} derived the forward and optical indices for some Cartesian product of simple graphs, such as cycles, chains and complete graphs.
In this paper, we report three results shown as follows.
First, we derive the value of the \textit{ fowarding index} in a BCube network.
Second, we propose an \textit{oblivious} RWA scheme; the term oblivious signifies that the RWA assigns a lightpath to an S-D pair based only on its source and destination addresses.
Third, we derive an upper bound and a lower bound of the optical index in a BCube network.
The derived results can provide insights into optimal lightpath allocation, and serve as a baseline for future research in more sophisticated RWA schemes in BCube networks.
The rest of the paper is organized as follows.
Section 2 introduces some preliminaries.
Section 3 introduces the BCube topology.
Section 4 presents analysis on host-to-host communication in a BCube network.
We conclude the paper in section 5.
\section{Preliminaries}
In this paper, we consider a full-duplex network, where each node can send and receive messages at the same time.
Hence, we can model an all-optical network by a \textit{symmetric digraph} --- a directed graph $G $ with vertex set $V(G)$ and arc set $A(G)$ such that if $\alpha_{x, y} \in A(G)$ then $\alpha_{y, x}\in A(G)$.
Here $\alpha_{x, y}$ represents an arc directed from node $x$ to node $y$.
Let $P_{s, d}$ denote a directed path (dipath) from source node $s$ to destination node $d$.
A set of dipaths is called routing. For a given routing $R$ of $G$, let
$\pi(G, R, \alpha_{x, y})$ denote the load of arc $\alpha_{x,y}$ with respect to $R$.
which is measured by the number of dipaths in $R$ that pass through $\alpha_{x,y}$.
The maximum link load is then denoted by $\pi(G, R) := \max_{\alpha_{x, y} \in A(G)} \pi(G, R, \alpha_{x, y})$.
Let $\mathcal{R}$ denote the collection of all possible routings.
Then the \textit{forwarding index} of a graph $G$, denoted by $\pi(G)$, is defined as the minimum of the maximum link load over all routings, i.e.,
\begin{equation}
\pi(G) :=\min_{R\in \mathcal{R}} \pi(G, R).
\end{equation}
To study the RWA problem, we represent wavelengths by different colors.
In an optimal wavelength assignment, the number of colors required is minimal.
Let $\omega(G, R)$ denote the minimum number of colors required to color dipaths of $R$ such that dipaths are assigned with different colors if they share a common arc.
The \textit{optical index} in a graph of $G$, denoted by $\omega(G)$, is defined as
\begin{equation}
\omega(G) = \min_{R \in \mathcal{R}} \omega(G, R).
\end{equation}
Since dipaths sharing a common arc should be assigned with different colors, we have
\begin{equation}
\label{inequality_1}
\omega(G)\ge \pi(G). \quad \quad \quad
\end{equation}
It is difficult to investigate whether the equality in (\ref{inequality_1}) holds for a general topology.
However, researchers have shown that the equality holds for some specific topologies such as cycles \cite{full_connections}, Hypercubes \cite{full_connections}, trees of cycles \cite{all-to-all3}, some Cartesian product of paths or cycles with equal length \cite{full_connections, all-to-alloptical}, and some circulant graphs \cite{forward_optical_paper}.
In this paper, we evaluate the forwarding and optical indices in a BCube network by considering host-to-host routings.
More precisely, by denoting $V_h$ the set of hosts in a BCube, a host-to-host routing is given by
$R = \{P_{s, d}: s, d\in V_h(G), s\neq d\}$.
The structure of BCubes will be given in next section.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width = 3.2 in]{figures/product}
\caption{ $\mathcal{H}(3, 3) = K_3 \times K_3$}
\label{product}
\end{center}
\end{figure}
BCube is closely related to the generalized Hypercube. Towards a better understanding of BCube, we first briefly explain some properties of Hypercube.
Let $K_{n}$ denote a complete graph with $n$ nodes
indexed by integers in $\mathbb{Z}_{n}$, where $\mathbb{Z}_{n} := \{0, 1, ..., n-1 \}$.
Any two nodes in $K_{n}$ are adjacent to each other.
Since a generalized $\ell$-dimensional Hypercube $\mathcal{H}(n_1, ..., n_\ell)$ is the Cartesian product of complete graphs $K_{n_i}, i= 1, 2, ..., \ell$, we have
\begin{displaymath}
\mathcal{H}(n_1, ..., n_\ell) : = K_{n_1} \times ... \times K_{n_\ell}.
\end{displaymath}
The node set $V(\mathcal{H}(n_1, ..., n_\ell))$ is $\{(v_1,\ldots,v_n): \ v_i \in V(K_{n_i})\}$, where each node of $\mathcal{H}(n_1, ..., n_\ell)$ can be expressed by an $\ell$-dimensional vector, $\textbf{h} =h_1 ... h_\ell \in \mathbb{Z}_{n_1}\times ... \times \mathbb{Z}_{n_\ell}$.
Two nodes in $\mathcal{H}(n_1, ..., n_\ell)$ are adjacent if their vectors differ only in one component.
Fig. \ref{product} illustrates $\mathcal{H}(3, 3)$, which is the Cartesian product of $K_3$ and $K_3$.
Readers can refer to \cite{bhuyan1984generalized} for more details on Cartesian product of graphs and Hypercube. It has been shown in \cite{full_connections} that $\omega(\mathcal{H}(n_1, ..., n_\ell) )= d^{\ell}-d^{\ell-1}$ when $n_i= d $ for any $i\in \{1, 2, ..., \ell\}$.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width = 3.2 in]{figures/hypercube_to_BCube}
\caption{An intermediate state of transforming $\mathcal{H}(3, 3)$ into $\mathcal{B}(2, 3)$}
\label{hypercube-to-BCube}
\end{center}
\end{figure}
The main difference between Hypercube and BCube lies in how adjacent nodes are connected to each other.
In Hypercube adjacent nodes are connected directly by edges; however, in BCube they are connected via common switches.
Such a difference contributes to a reduction in wiring complexity for building large-scale networks.
Fig. \ref{hypercube-to-BCube} shows an intermediate process of transforming Hypercube into BCube, where each torus is replaced by a switch.
\section{ The BCube Topology}
We use $\mathcal{B}(\ell, d)$ to denote a BCube network which has one host layer and $\ell$ switch layers; this network is constructed by optical $d$-port switches.
We index switch layers from $1$ to $\ell$ from bottom to top, and index ports in a switch from $0$ to $d-1$ from left to right. Since we consider full-duplex networks, we assume these ports are bidirectional.
Similar to Hypercube, we denote hosts in $\mathcal{B}(\ell, d)$ by $\ell$-dimensional vectors, $\textbf{h} = h_1... h_\ell \in \mathbb{Z}_d^\ell$, and we denote switches in $\mathcal{B}(\ell, d)$ by $(\ell-1)$-dimensional vectors, $\textbf{s}^k = s^k_1 ... s^k_{\ell-1} \in \mathbb{Z}^{\ell-1}_d$. Here $k$ indicates a switch at the $k$-th switch layer. Hereinafter, we simply use layer to refer to switch layer.
Fig. \ref{fig.0} and Fig. \ref{fig.1} illustrate the structures of $\mathcal{B}(2, 3)$ and $\mathcal{B}(3, 3)$, respectively.
Since $\mathcal{B}(\ell, d)$ consists of switches and hosts, we have $V(\mathcal{B}(\ell, d)) = V_s(\mathcal{B}(\ell, d)) \cup V_h(\mathcal{B}(\ell, d))$, where $V_s(\mathcal{B}(\ell, d))$ is the switch set and $V_h(\mathcal{B}(\ell, d))$ is the host set.
By letting $N_h$, $N_s$ and $N_\alpha$ be the number of hosts, switches and arcs, we have $N_h = d^\ell$ and $N_\alpha = 2\ell N_h$ since each host has $\ell$ bidirectional links.
\begin{figure}[ht]
\centering
\includegraphics[width = 3.2 in]{figures/B2}
\caption[Recursive Construction:]
{The bottom layer in purple is the host layer; the remaining layers are switch layers. This BCube has three built-in $\mathcal{B}(1, 3)$, each of which is in a blue dashed rectangle.
}
\label{fig.0}
\end{figure}
\noindent \textbf{Recursive construction:}
$\mathcal{B}(1, d)$ is constructed by $d$ hosts and one switch, where these hosts are all connected directly to this switch.
For $ \ell > 1$, $\mathcal{B}(\ell, d)$ is constructed by $d$ $\mathcal{B}(\ell-1, d)$, where hosts in different $\mathcal{B}(\ell-1, d)$ are connected by switches at the $\ell$-th layer.
In particular, we refer to these $\mathcal{B}(\ell-1, d)$ as \textit{built-in} BCubes of $\mathcal{B}(\ell, d)$ since they are inside $\mathcal{B}(\ell, d)$.
For example, the three $\mathcal{B}(2, 3)$ in Fig. \ref{fig.1} are built-in BCubes of $\mathcal{B}(3, 3)$.
Accordingly, we can split a host vector in $\mathcal{B}(\ell, d)$ into two parts, i.e., $\textbf{h} = h_{1:\ell-1} h_{\ell}$, where $h_{1:\ell-1} = h_1...h_{\ell-1}$ is equal to a host vector in $\mathcal{B}(\ell-1, d)$, and $h_{\ell}$ can identify different built-in $\mathcal{B}(\ell-1, d)$.
In particular, $h_\ell$ is the same for all hosts belonging to the same build-in BCube.
For simplicity, we let $h_{\ell}$ be the index of a built-in $\mathcal{B}(\ell-1, d)$.
For example, the indices of three built-in $\mathcal{B}(2,3)$ from left to right in Fig. \ref{fig.1} are $0$, $1$, and $2$, respectively.
In $\mathcal{B}(\ell, d)$, a link exists only between a host and a switch; a host is physically connected to $\ell$ switches at different layers; and a switch is physically connected to $d$ hosts.
We describe two hosts are \textit{neighbors} if their vectors differ in only one component.
If two neighbor hosts differ in the $k$-th components, the two hosts are connected directly to a common switch at the $k$-th layer.
Mathematically, a physical link exits between a host $\textbf{h}$ and a switch $\textbf{s}^k$ if and only if the following equation is satisfied.
\begin{equation}
\label{connection_def}
s^k_1... s^k_{\ell -1} = h_1 ... h_{k-1} h_{k+1} ... h_\ell.
\end{equation}
If a host is connected to the $h_k$-th port of switch $\textbf{s}^k$ via a link, we can infer the vector of this host according to (\ref{connection_def}).
For example, in Fig. \ref{fig.1}, switch $\textbf{s}^3 = 20$ is physically connected to hosts $2 0 \underline{0}$, $2 0 \underline{1}$ and $2 0 \underline{2}$ via its ports $0$, $1$, and $2$, respectively; the underlined numbers are determined by the corresponding port indices.
Moreover, switch $\textbf{s}^2 =2 0$ is directly connected to hosts $2 \underline{0} 0$, $2 \underline{1} 0$, and $2 \underline{2} 0$ via its ports $0, 1$, and $ 2$, respectively; switch $\textbf{s}^1= 2 0$ at the 1st layer is directly connected to hosts $\underline{0} 2 0$, $\underline{1} 2 0$ and $\underline{2} 2 0$ via its ports $0$, $1$, and $2$, respectively.
A directed link is referred to as an \textit{uplink} if its direction is from a host to a switch and is referred to as a \textit{downlink} if its direction is from a switch to a host.
We use $\textbf{u}^k = u^k_1 ... u^k_\ell$ and $\textbf{d}^k=d^k_1 ... d^k_\ell$ to denote an uplink and a downlink, respectively. Here $k$ indicates that the corresponding directed link is connected to a switch at layer $k$. Note that the value of $\textbf{u}^k $ (or $\textbf{d}^k $) is determined by the vector of its connected host. Hence, we have $\textbf{u}^k= \textbf{h}$ ($\textbf{d}^k = \textbf{h}$) if uplink $\textbf{u}^k$ (downlink $\textbf{d}^k$) is connected to host $\textbf{h}$.
\begin{figure*}[ht]
\centering
\includegraphics[width= 7.2 in]{figures/B3}
\caption[Topology of $ \mathcal{B}(3, 3)$]
{$\mathcal{B}(3, 3)$ is constructed by three $\mathcal{B}(2, 3)$. Purple and blue rectangles represent hosts and switches, respectively.}
\label{fig.1}
\end{figure*}
\noindent \textbf{Routing in BCube:}
For an S-D host pair ($\textbf{h}^s, \textbf{h}^d$), we use the Hamming distance $\| \textbf{h}^s - \textbf{h}^d\|_0 $ to measure the distance between $\textbf{h}^s$ and $\textbf{h}^d$.
If the Hamming distance of pair $(\textbf{h}^s, \textbf{h}^d)$ is $m$, any of its shortest dipaths has $m$ \textit{hops}.
A hop is defined here as from a host to a switch, and then back to a host.
Note that neighbor hosts in a BCube network can be reached through one hop.
Let $\textbf{h}\to \textbf{h}^\prime$ denote the hop from host $\textbf{h}$ to host $\textbf{h}^\prime$.
Then, we can represent a shortest dipath of $m$ hops from host $\textbf{h}^s$ to host $\textbf{h}^d$ as follows.
\begin{displaymath}
\textbf{h}^s \to \textbf{h}^1 \to ... \to \textbf{h}^{m-1} \to \textbf{h}^d.
\end{displaymath}
Each hop in a shortest dipath corresponds to one different component between $\textbf{h}^s$ and $ \textbf{h}^d$.
In particular, if $\textbf{h}^s$ and $ \textbf{h}^d$ differs in the $k$-th component, there exits a hop that shall traverse a switch at layer $k$; thus, we describe the hop fixes the $k$-th component.
We define \textit{component-fixing order} as the order of fixing different components by a sequence of hops.
For an S-D pair of distance $m$, there are $m!$ different component-fixing order, where each order uniquely determines a shortest path.
In view of this, we conclude that BCube maintains a high degree of path diversity.
For example, in Fig. \ref{fig.1}, S-D pair $(0 0 0, 1 2 2 )$ has the following six shortest dipaths with each corresponding to an unique component-fixing order.
\begin{equation}
\nonumber
\begin{cases}
0 0 0 \to 1 0 0 \to 1 2 0 \to 1 2 2,\\
0 0 0 \to 1 0 0 \to 1 0 2 \to 1 2 2, \\
0 0 0 \to 0 2 0 \to 1 2 0 \to 1 2 2, \\
0 0 0 \to 0 2 0 \to 0 2 2 \to 1 2 2, \\
0 0 0 \to 0 0 2 \to 1 0 2 \to 1 2 2, \\
0 0 0 \to 0 0 2 \to 0 2 2 \to 1 2 2,
\end{cases}
\end{equation}
In particular, dipaths that follow the descending component-fixing order are called \textit{descending} dipaths.
For example, the descending dipath of $(0 0 0, 1 2 2 )$ is $0 0 0 \to 0 0 2 \to 0 2 2 \to 1 2 2$.
Without loss of generality, we conduct the analysis using descending dipaths in the remaining of this paper.
\begin{defn}
For any positive integers $\ell$ and $d$, let $R^{*}(\ell,d)$ denote the host-to-host routing in $\mathcal{B}(\ell, d)$ where all involved dipaths are descending.
\end{defn}
\section{Forwarding and Optical Indices}
In this section, we first derive the exact value of $\pi(\mathcal{B}(\ell, d))$.
Then we propose an oblivious RWA for a $\mathcal{B}(\ell, d))$.
Finally, we derive the upper and lower bound of $\omega(\mathcal{B}(\ell, d))$.
We divide the analysis of the host-to-host communication into two parts: path allocation and wavelength assignment.
In the part of path allocation, we aim to find a set of dipaths that minimizes the maximum link load;
in the part of wavelength assignment, we aim to minimize the usage of wavelengths.
\subsection{Forwarding Index}
To begin with, we first present an existing result on the forwarding index of a graph (see e.g., \cite{forwarding_bound}).
\begin{lem}
\label{lem1}
For a given $G= (V, A)$, we have
\begin{displaymath}
\pi(G) \ge \frac{N_v ( N_v -1) \bar{d}(G)}{N_\alpha},
\end{displaymath}
where
\begin{displaymath}
\bar{d}(G) = \frac{1}{N_v(N_v-1)}\sum_{x, y\in V, x\neq y}d_{x, y}.
\end{displaymath}
Here $N_v$ and $N_\alpha$ refer to the number of nodes and arcs, respectively, and $d_{x, y}$ denotes the distance between node $x$ and node $y$. Note that $\bar{d}(G)$ denotes the average distance over all nodes.
\end{lem}
Recall that nodes in BCube have two types: switches and hosts, and we only assign dipaths to host pairs.
In order to derive a lower bound of $\pi(\mathcal{B}(\ell, d))$ based on Lemma \ref{lem1}, we replace $N_v$ with $N_h$ which is the number of hosts in $\mathcal{B}(\ell, d)$.
Besides, we should use the average distance only over host nodes (average host distance for short).
Note that the average host distance is given by
\begin{equation}
\label{avg_distance}
\bar{d}(\mathcal{B}(\ell, d)) = 2\ell \frac{d-1}{d}\frac{N_h}{N_h-1},
\end{equation}
according to Theorem 4 of \cite{guo2009bcube}.
We then have
\begin{equation}
\label{forwarding_lower}
\pi(\mathcal{B}(\ell, d)) \ge \frac{N_h(N_h-1) \bar{d}(\mathcal{B}(\ell, d))}{N_\alpha} = d^{\ell} - d^{\ell-1}.
\end{equation}
In what follows, we shall show $d^\ell-d^{\ell-1}$ is also an upper bound of $\pi(\mathcal{B}(\ell, d))$.
\begin{thm}
\label{thm1}
For any positive integers $\ell$ and $d$, one has
\begin{displaymath}
\pi (\mathcal{B}(\ell, d)) = d^{\ell} - d^{\ell-1}.
\end{displaymath}
\end{thm}
\begin{proof}
We proceed by induction on $\ell$. The steps are as follows.
\begin{enumerate}[(1)]
\item We first show $\pi(\mathcal{B}(1, d)) = d-1$;
\item we \textit{assume} $\pi(\mathcal{B}(k, d )) = d^k - d^{k-1}$;
\item we \textit{prove} $\pi(\mathcal{B}(k+1, d)) =d^{k+1} - d^{k}$.
\end{enumerate}
Note that $\mathcal{B}(k+1, d)$ contains $d$ built-in $\mathcal{B}(k,d)$.
In $\mathcal{B}(1,d)$, there are $d$ hosts and only one switch. Since every host plays as the role of source and destination exactly $d-1$ times, and the shortest dipath for an S-D pair is unique, we clearly have $\pi(\mathcal(1,d)) \leq d-1$.
Together with (\ref{forwarding_lower}), we have $\pi(\mathcal{B}(1, d))= d-1$.
Next, we assume the result holds for $\ell=k$ and then prove the result for $\ell=k+1$.
We first show that the maximum link load in layer (k+1) is equal to $d^{k+1} -d^k$; we then show that the maximum link load in each built-in BCube is equal to $d^{k+1} -d^k $.
Consider an arbitrary uplink $\textbf{u}^{k+1}$.
If this uplink is traversed by the dipath of an S-D pair $(\textbf{h}^s, \textbf{h}^d)$ in $R^*(k+1, d)$, we have
\begin{equation}
\label{uplink_traverse_condition}
\textbf{h}^s = \textbf{u}^{k+1}\ \textup{and} \ h^d_{k+1}\neq h^s_{k+1}.
\end{equation}
Furthermore, the number of satisfied S-D pairs in $R^*(k+1, d)$ is $d^k(d-1) = d^{k+1} -d^k$ S-D pairs due to the following facts.
(1) Uplink $\textbf{u}^{k+1}$ determines the source of these pairs. (2) Any of the first $k$ components of $\textbf{h}^d$ has $d$ choices, whereas the last component has $d-1$ choices.
In other words, the load of uplink $\textbf{u}^{k+1}$ is $d^{k+1} -d^k$.
Similarly, consider an arbitrary downlink $\textbf{d}^{k+1}$.
If downlink $\textbf{d}^{k+1}$ is traversed by the dipath of an S-D pair $(\textbf{h}^s, \textbf{h}^d)$ in $R^*(k+1, d)$, we have
\begin{equation}
\label{downlink_traverse_condition}
h^s_1... h^s_k h^d_{k+1} = \textbf{d}^{k+1}\ \textup{and} \ h^d_{k+1} \neq h^s_{k+1}.
\end{equation}
Furthermore, the number of satisified S-D pairs in $R^*(k+1, d)$ is $(d-1)d^k = d^{k+1} -d^k$ due to the following facts. (1) Downlink $\textbf{d}^{k+1}$ determines the first $k$ components of $\textbf{h}^s$, whereas the last component of $\textbf{h}^s$ has $d-1$ choices. (2) Downlink $\textbf{d}^{k+1}$ determines the last component of $\textbf{h}^d$, whereas any of the first $k$ components of $\textbf{h}^d$ has $d$ choices.
In other words, the load of downlink $\textbf{d}^{k+1}$ is $d^{k+1} -d^k$.
Together with (\ref{forwarding_lower}), we infer that the maximum link load in layer $k+1$ is $d^{k+1} -d^k$.
Next, we show that the maximum link load in each built-in BCube is also $d^{k+1}-d^{k} $.
First, we consider the dipath of an S-D pair $( h^s_1 ... h^s_k, h^d_1 ... h^d_k )$ in $R^*(k, d)$.
The same dipath becomes a part of the dipath in $R^*(k+1, d)$ whose source and destination are given by
\begin{equation}
\small
\label{dorSet}
(h^s_1 \cdots h^s_k x, h^d_1 \cdots h^d_k y)
\end{equation}
where $x$ is the index of the built-in BCube in $\mathcal{B}(k+1, d)$ that the source belongs to, and $y$ is the index of the built-in BCube that the destination belongs to.
The above statement applies to any dipath in $R^*(k, d)$.
Fix $y$, i.e., a built-in BCube.
Since $x$ has a range $\{0, ..., d-1\}$, we learn that the link load in a built-in BCube is $d^{k+1} -d^k$, which is $d$ times the link load in $\mathcal{B}(k, d)$.
Together with (\ref{forwarding_lower}), we infer that the maximum link load in a built-in BCube is $d^{k+1} -d^k$.
This completes the proof.
\end{proof}
\subsection{The Proposed RWA scheme }
To analyze the RWA problem in BCube, we introduce a specific pattern of permutation routing called \textit{Cyclic Permutation Routing} (CPR).
We first show that BCube provides link-disjoint dipaths for a CPR.
We then propose an oblivious RWA scheme and derive upper bounds of the optical index.
A permutation here is referred to as a set of S-D pairs wherein each host is a source and a destination of exactly one S-D pair.
\begin{defn}
For a given $\ell$-dimensional vector $p_1 ... p_\ell \in \mathbb{Z}^\ell_d$, we define a permutation, denoted by $P(p_1 ... p_{\ell})$, as follows.
\begin{equation}
\label{permutation}
\begin{split}
P(p_1 ... p_\ell ) :=& \{(\textbf{h}^s, \textbf{h}^d): \\
& \textbf{h}^s\in \mathbb{Z}^{\ell}_d, h^d_i = (h^s_i +p_i)_d , i =1, 2, ..., \ell \},
\end{split}
\end{equation}
where $(x)_d := x \mod d $, and $ h^d_i = (h^s_i +p_i)_d $ also implies that $p_i = (h^d_i -h^s_i)_d$ since $h^d_i, h^s_i, \textup{and} \ p_i \in \mathbb{Z}_d$.
\end{defn}
In particular, we call $P(0 ... 0)$ the \textit{zero} permutation where $p_i=0$ for all $i$.
\begin{lem}
\label{communication decomposition}
All host pairs in $\mathcal{B}(\ell, d)$ can be classified into $d^\ell$ CPRs with $p_1...p_\ell$ ranging over $\mathbb{Z}^\ell_d$.
\end{lem}
\begin{proof}
Given an arbitrary S-D pair $(\textbf{h}^s, \textbf{h}^d)$, it must belong to some $ P(p_1...p_\ell) $ whose $p_1, ..., p_\ell$ is equal to
\begin{displaymath}
p_i = (h^d_i -h^s_i)_d, \ i=1, 2, ..., \ell.
\end{displaymath}
Thus, this lemma follows.
\end{proof}
\begin{defn}
For a given $P(p_1 ... p_\ell)$ in $\mathcal{B}(\ell, d)$, we define a CPR, denoted by $R(p_1 ... p_\ell )$, as follows.
\begin{displaymath}
R{(p_1 ... p_\ell )} := \{P_{\textbf{h}^s, \textbf{h}^d}: (\textbf{h}^s, \textbf{h}^d)\in P(p_1 ... p_\ell ) \},
\end{displaymath}
where $P_{\textbf{h}^s, \textbf{h}^d}$ is a descending path.
\end{defn}
In particular, dipaths in $R(p_1 ... p_{\ell-1} 0)$, where none of layer-$\ell$ links is involved, can be classified into $d$ different sets such that dipaths in each set consist of links that only belong to some built-in BCube.
Furthermore, each of such sets is isomorphic to $R(p_1...p_{\ell-1})$, a dipath set in $\mathcal{B}(\ell-1, d)$.
Next, we show dipaths in $R{(p_1...p_\ell )}$ are link disjoint.
\begin{lem}
\label{link-disjoint permutation routes}
Given $\mathcal{B}(\ell, d)$ and $P(p_1 ... p_\ell )$, we have dipaths in $R{(p_1 ... p_\ell )}$ are link disjoint.
\end{lem}
\begin{proof}
We proceed by induction on $\ell$. The steps are as follows.
\begin{enumerate}[(1)]
\item We first prove the result holds in $\mathcal{B}(k, d)$ when $k=1$;
\item we assume the result holds in $\mathcal{B}(k, d)$ for some $k$ in $\{1, 2, ...., \ell-1\}$;
\item we prove the result holds in $\mathcal{B}(k+1, d)$.
\end{enumerate}
If $\ell=1$, we have each dipath in $R(p_1)$, $p_1\neq 0$, consists of only one uplink and only one downlink; the uplink connects a source and the downlink connects a destination.
Since a host is a source and a destination of exactly one pair in a CPR, each directed link is traversed by only one dipath in $R(p_1)$. In other words, dipaths in $R(p_1)$ are link disjoint.
Next, we show this result holds for $\ell = k+1$ based on the assumption for $\ell =k$.
Consider an uplink $\textbf{u}^{k+1}$ in $\mathcal{B}(k+1, d)$.
If the dipath of an S-D pair $(\textbf{h}^s, \textbf{h}^d)$ traverses this uplink, we have
\begin{equation}
\textbf{u}^{k+1} = \textbf{h}^s, h^s_{k+1} \neq h^d_{k+1}.
\end{equation}
Here $h^s_{k+1} \neq h^d_{k+1}$ implies that $p_{k+1} \neq 0$.
Consider a downlink $\textbf{d}^{k+1}$.
If the dipath of an S-D pair $(\textbf{h}^s, \textbf{h}^d)$ traverses this downlink, we have
\begin{equation}
h^s_1 ... h^s_{k} h^d_{k+1} =\textbf{d}^{k+1} \ \textup{and} \ h^s_{k+1} \neq h^d_{k+1}.
\end{equation}
According to $h^s_{k+1} = (h^d_{k+1} -p_{k+1})_d =(d^{k+1}_{k+1} - p_{k+1})_d $ in $P(p_1 ... p_{k+1})$, we learn that uplink $\textbf{u}^{k+1}$ (or downlink $\textbf{d}^{k+1}$) uniquely determines the source of a pair.
We thus infer by the CPR definition that dipaths of $R(p_1 ... p_{k+1})$ collide on neither uplinks nor downlinks in layer $k+1$. If $p_{k+1} =0$, the above statement holds naturally because dipaths in $R(p_1 ... p_{k} 0)$ do not traverse any links in layer $k+1$.
Next, we further show that dipaths in $R(p_1 ... p_{k+1})$ do not collide inside each built-in BCube.
Consider an S-D pair $(\textbf{h}^s, \textbf{h}^d)$ in $P(p_1 ... p_k p_{k+1})$ and a host $\textbf{h}^I$ with $\textbf{h}^I = h^s_1 ... h^s_k h^d_{k+1}$.
If $p_{k+1} \neq 0$, the descending dipath $P_{\textbf{h}^s, \textbf{h}^d}$ arrives at host $\textbf{h}^I$ after its first hop; otherwise, $\textbf{h}^I$ is its source node.
One can check that each $P_{\textbf{h}^s, \textbf{h}^d}$ in $R(p_1 ... p_k p_{k+1})$ has a distinct $\textbf{h}^I$, and each descending dipath $P_{\textbf{h}^I, \textbf{h}^d}$ belongs to $R(p_1 ... p_k 0)$.
On the other hand, $R(p_1 ... p_k 0)$ can be divided into $d$ dipath sets, where each set is isomorphic to $R(p_1 ... p_k )$. Recall that we have assumed dipaths in $R(p_1 ... p_k )$ are link disjoint.
We can infer that dipaths in $R(p_1 ... p_k p_{k+1})$ do not collide in each built-in BCube. Thus we finish the proof.
\end{proof}
On the basis of Lemma \ref{link-disjoint permutation routes}, we divide RWA into two parts: path allocation and wavelength assignment.
In the part of path allocation, we use descending dipaths only.
In the part of wavelength assignment, we first indicate wavelengths by $\ell$-dimensional vectors $w_1... w_\ell$ in $\mathbb{Z}^\ell_d$,
and then assign all dipaths in $R(p_1 ... p_\ell)$ with a single wavelength whose vector is given by
\begin{equation}
\label{RWA_1}
w_1... w_\ell = p_1...p_\ell
\end{equation}
Algorithm 1 illustrates more details on the proposed wavelength assignment.
It is easy to see the wavelength assignment scheme in (\ref{RWA_1}), besides its simplicity, guarantees nonblocking lightpaths for a host-to-host traffic.
\begin{algorithm}[!t]
\caption{An oblivious RWA}
\KwIn{$\textbf{h}^\textup{s}, \textbf{h}^\textup{d}$}
\KwOut{ $w_1 .... w_\ell$}
\For{$i=1; i\le \ell; $ ++$i$}
{ $w_i = (h^d_i - h^s_i)_d$\;
}
return $w_1 .... w_\ell$\;
\end{algorithm}
Since the host-to-host communication is composed of $d^\ell-1$ non-zero permutations, the scheme in (\ref{RWA_1}) uses at most $d^\ell-1$ wavelengths.
In other words, we have $\omega(\mathcal{B}(\ell, d)) \le d^\ell-1$.
Recall that we have $ \pi(\mathcal{B}(\ell, d)) = d^\ell -d^{\ell-1} \le \omega(\mathcal{B}(\ell, d)) $.
Combining the two results together, we get
\begin{equation}
\label{loose_bound}
d^{\ell} - d^{\ell-1} \le \omega(\mathcal{B}(\ell, d)) \le d^\ell - 1.
\end{equation}
However, the proposed RWA does not use wavelengths in an optimal way.
For example, dipath sets $R(1 0 0)$, $R(0 2 0)$, and $R(0 0 2)$ in $\mathcal{B}(3, 3)$ can be assigned with a same wavelength since their dipaths use directed links of different layers.
Towards a better understanding on the minimum usage of wavelengths, we conduct a deeper investigation on the upper bound of $\omega(\mathcal{B}(\ell, d))$ in the next section.
\subsection{Bounds of the Optical Index}
To derive a tigher bound of $\omega(\mathcal{B}(\ell, d))$, we transform this problem into a vertex coloring problem.
Then, we derive an upper bound of $\omega(\mathcal{B}(\ell, d))$ using existing results on the chromatic number in Graph Theory.
To begin with, we bring the following property of CPRs, which motivates the problem transformation.
\begin{lem}
\label{collision_analysis}
Consider $R(x_1 ... x_\ell)$ and $R(y_1 ... y_\ell)$.
Dipaths in $R(x_1 ... x_\ell)$ and $R(y_1 ... y_\ell)$ must collide at links of layer $i$ if $x_i\neq 0$ and $y_i\neq 0$.
\end{lem}
\begin{proof}
If $x_i\neq 0$, a dipath in $R(x_1 ... x_\ell)$ must traverse an uplink and a downlink of layer $i$. Moreover, according to the link-disjoint property in Lemma \ref{link-disjoint permutation routes}, different dipaths of $R(x_1 ... x_\ell)$ use different directed links at layer $i$.
Since the number of diapths in $R(x_1 ... x_\ell)$ is equal to that of uplinks (downlinks) at layer $i$, we infer that each uplink (downlink) of layer $i$ is traversed by exactly one dipath in $R(x_1 ... x_\ell)$.
The above result also applies to $R(y_1 ... y_\ell)$ with $y_i\neq 0$.
The result follows.
\end{proof}
We continue to follow the idea of assigning a single wavelength to $R(p_1 ... p_\ell)$.
To achieve nonblocking lightpaths, we assign different wavelengths to $R(x_1...x_\ell)$ and $R(y_1 ... y_\ell)$ if there exists some $i$ such that $x_i\neq 0$ and $y_i \neq 0$.
We refer to the above constraint as Wavelength Assignment Constraint (WAC), based on which, we draw a graph, denoted by $G=(R, E)$.
Each node in $R$ indicates a $R(p_1 ... p_\ell)$; two nodes are adjacent if they satisfy WAC.
We then use colors to represent wavelengths, and color adjacent nodes in $(R, E)$ with different colors.
Clearly, the goal of vertex coloring is to minimize the usage of colors, which is known as a vertex coloring problem.
In a vertex coloring problem, the chromatic number of a graph $\chi(G)$, is defined as the minimum number of colors in order to color adjacent nodes with different colors.
Let $\Delta(G)$ be the maximum degree of graph $G$.
Brooks's theorem \cite{Brooks} proved that
\begin{equation}
\label{vertex_coloring}
\chi(G)\le \Delta(G),
\end{equation}
where $G$ is a connected simple graph that is neither a complete graph nor an odd cycle.
Besides, it has been shown that complete graphs have
\begin{equation}
\label{vertex_coloring_2}
\chi(G)= \Delta(G)+1.
\end{equation}
Since $(R, E)$ is derived under the restriction of assigning a single wavelength to a CPR, we further have
\begin{equation}
\label{chromatic_upper_bounds}
\omega(\mathcal{B}(\ell, d)) \le \chi(G).
\end{equation}
We then prove upper bounds of $\omega(\mathcal{B}(\ell, d))$ as follows.
\begin{thm}
\label{specific case}
Let $\ell$ and $d$ be positive integers.
Then $\omega(\mathcal{B}(1, d)) = d-1$, $\omega(\mathcal{B}(2, d)) = d^2-d$, and for $\ell \ge 3$ we have
\begin{displaymath}
d^{\ell} - d^{\ell-1} \le \omega(\mathcal{B}(\ell, d)) \le d^\ell - d^{\lfloor \frac{\ell}{2}\rfloor}- (\lfloor \frac{\ell}{2}\rfloor-1).
\end{displaymath}
\end{thm}
\begin{proof}
We only prove the upper bound here since we have validated the lower bound in the last subsection.
We consider respectively three cases, $\ell=1$, $\ell=2$, and $\ell >2$, and prove each case, separately.
For the case $\ell =1$, the result holds naturally according to (\ref{loose_bound}).
For the case $\ell =2$, we assign dipaths in $R(p_1 p_2)$ with a wavelength $w_1 w_2$, whose value is given by
\begin{equation}
\begin{cases}
w_1 = p_1, w_2 = p_2, & \textup{if}\ p_2 \neq 0,\\
w_1 = 0, w_2 = p_1, & \textup{if}\ p_2 = 0.
\end{cases}
\end{equation}
In other words, wavelength $0 x$ is used by dipaths in $R(x 0)$ and $R(0x)$ for any $x\in \{1, ..., d-1\}$.
Since dipaths in $R(x 0)$ and $R(0 x)$ use links of different layers, they can share a common wavelength.
Note that we do not include the zero permutation in host-to-host communication.
Therefore, we always have $w_2\neq 0$, which implies that the number of involved wavelengths is $d(d-1) =d^2-d$.
In other words, we have $\omega(\mathcal{B}(2, d)) \le d^2-d$.
Together with (\ref{loose_bound}), we get $\omega(\mathcal{B}(2, d)) = d^2-d$.
The problem becomes much more complicated when $\ell \ge 3$, where we fail to derive an exact number of the optical index.
Instead, we derive a tighter upper bound than the one in (\ref{loose_bound}). The steps are as follows.
First, we classify all CPRs into $\ell$ classes, $\mathcal{C}_1, ..., \mathcal{C}_\ell$, according to the number of non-zero components in $p_1...p_\ell$.
That is, if $P(p_1...p_\ell)$ has $k$ nonzero components in $p_1...p_\ell$, then $P(p_1 ...p_\ell)\in \mathcal{C}_k$.
Second, we analyze the upper bound of wavelengths used by each class, separately.
By summing up all these upper bounds together, we finally achieve an upper bound of $\omega(\mathcal{B}(\ell, d))$ for $\ell\ge 3$.
To begin with, we draw a graph, denoted by $G_k$, for each $\mathcal{C}_k$, respectively. The nodes of $G_k$ are CPRs in $\mathcal{C}_k$, and the edges of $G_k$ are added according to WAC.
The total number of nodes in $G_k$ is $\dbinom{\ell}{k}(d-1)^k$ since each CPR has $k$ nonzero components and each nonzero component has $d-1$ choices.
Let $\dbinom{\ell-k}{k} = 0 \ \textup{if} \ k>\frac{\ell}{2}$.
Consider any node in $G_k$ as a target node.
According to WAC, we learn that there are $ \dbinom{\ell-k}{k}(d-1)^k$ nodes that are not adjacent to the target node in $G_k$.
In other words, the degree of this target node is given by
\begin{equation}
\left ( \dbinom{\ell}{k} - \dbinom{\ell-k}{k}\right)(d-1)^k -1.
\end{equation}
Besides, we can infer by symmetry that $G_k$ is a regular graph.
Therefore, we have
\begin{equation}
\label{degree_G_k}
\Delta(G_k) = \left( \dbinom{\ell}{k} - \dbinom{\ell-k}{k}\right)(d-1)^k-1.
\end{equation}
Note that $G_k$ with $k=1$ consists of $\ell$ independent complete graphs.
According to (\ref{vertex_coloring_2}), the chromatic number of each independent complete graph is $d-1$.
In particular, we have $\chi(G_1) = d-1$ since these independent complete graphs can share a common set of colors.
If $k>\lfloor \frac{\ell}{2}\rfloor$, we also have $\chi(G_k) = \Delta(G_k) +1$ since $G_k$ is a complete graph.
For $k\in [2,\lfloor \frac{\ell}{2}\rfloor ]$, according to (\ref{vertex_coloring}), we have $\chi(G_k) \le \Delta(G_k)$.
In conclusion, we have
\begin{equation}
\sum_{k=1}^\ell \chi(G_k) \le d-1+ \sum_{k=2}^{\lfloor \frac{\ell}{2}\rfloor}\Delta(G_k) + \sum_{k=1+\lfloor \frac{\ell}{2}\rfloor}^{\ell} (\Delta(G_k)+1)
\end{equation}
In particular, let
\begin{equation}
\label{total number}
N_w = \sum_{k=1}^{\ell}(\Delta(G_k)+1),
\end{equation}
we have
\begin{equation}
\sum_{k=1}^\ell \chi(G_k) \le N_w -(\lfloor \frac{\ell}{2}\rfloor-1).
\end{equation}
By substituting (\ref{degree_G_k}) into (\ref{total number}), we get
\begin{equation}
\begin{split}
N_w &= \sum_{k=1}^{\ell} \dbinom{\ell}{k}(d-1)^k - \sum_{k=1}^{\lfloor\frac{\ell}{2}\rfloor}\dbinom{\ell-k}{k} (d-1)^k\\
& \le \sum_{k=1}^{\ell} \dbinom{\ell}{k}(d-1)^k -\sum_{k=1}^{\lfloor\frac{\ell}{2}\rfloor} \dbinom{\lfloor\frac{\ell}{2}\rfloor}{k} (d-1)^k\\
& = d^\ell -1 -(d^{\lfloor\frac{\ell}{2}\rfloor} -1) = d^\ell -d^{\lfloor\frac{\ell}{2}\rfloor}
\end{split}
\end{equation}
Therefore, we have
\begin{equation}
\sum_{k=1}^\ell \chi(G_k) \le d^\ell -d^{\lfloor\frac{\ell}{2}\rfloor} -(\lfloor \frac{\ell}{2}\rfloor-1).
\end{equation}
Thus, according to (\ref{chromatic_upper_bounds}), we finish the proof.
\end{proof}
\section{Conclusion}
In this paper, we study host-to-host routing in an all-optical BCube network, where we mainly focus on the forwarding and optical indices.
Specifically, we succeed in deriving the exact values of the forwarding indices in all BCube networks and the exact values of the optical indices in BCube networks that have only one or two switch layers.
For BCube networks with more than two switch layers, we derive tight upper bounds of the optical indices after formulating the problem as a vertex coloring problem on the basis of CPRs.
Besides, we also propose an oblivious RWA scheme which can assign a lightpath to every S-D host pair based only on its source and destination addresses.
Although we have shown that the proposed RWA is not optimal in wavelength usage, it has the advantage in low implementation complexity.
\bibliographystyle{ieeetr}
|
{
"timestamp": "2018-04-17T02:10:31",
"yymm": "1804",
"arxiv_id": "1804.05358",
"language": "en",
"url": "https://arxiv.org/abs/1804.05358"
}
|
\section{Introduction}
\label{sec:intro}
The recent breakthroughs of Bourgain, Demeter and Guth in \cite{BDG}
and Wooley in \cite{Woo,WooNest}
has led to a full proof of the main conjecture in Vinogradov's Mean
Value Theorem (VMVT for short).
As one consequence among many, new estimates for Weyl sums are available.
With a standard approach, in this article, we show
that these already lead to strong estimates for moments of Weyl sums
(see Theorem~\ref{Satz2}).
In this context we record an observation that for moments of Weyl
sums, a small extra-improvement can be made using Montgomery's
so-called alternative derivation from \cite[\S 4]{Mont} incorporating
VMVT (see Theorem~\ref{Satz1}).
This additional gain can be exploited in a certain range
for the approximating denominator (see \eqref{eq:impr_range})
assuming the length of summation is large enough.
We formulate a conjecture stating where this gain might lead to, if
further refinements were available (see Conjecture~\ref{conj:c}).
Then, in Section~\ref{sec:mvexposums}, the Weyl sum moment estimates
are used to prove $k$-th derivative tests
for discrete moments of exponential sums with smooth functions
(Theorems~\ref{th:Thb} and \ref{th:Thb2}, the mentioned
extra-improvement is incorporated in Theorem~\ref{th:Thb}).
The achieved bounds for moments of Weyl sums and exponential sums with
smooth functions lead to improvements in some number-theoretic
applications, and we present two such applications.
The first application, discussed in Section~\ref{sec:appl1},
is the problem of counting integer points close to smooth curves.
For this, we use a new approach involving exponential sums such
that strong bounds for the counting quantity $\mathcal{R}(f,N,\delta)$, see
Definition~\ref{defR}, can be obtained.
The novelty is to perform an efficient Weyl shift step over a
set of indices known to be $H'$-spaced so that
the cluster structure of the indices is respected. This allows a
saving of an extra factor $H^{-1}$ in the proof of
Theorem~\ref{th:thR}, see the arguments before \eqref{eq:eqsuma}.
Compared to some existing bounds the resulting
bound in Theorem~\ref{th:thR}
is stronger, but is valid only for certain appropriate
functions. This is discussed at the end of the section.
The second application, discussed in Section~\ref{sec:polyLSI}, concerns
the polynomial large sieve inequality from \cite{KH,KH2}.
In the one-dimensional case we obtain a new improvement
of the bound. That new bound comes from the extra-improvement in
Theorem~\ref{Satz1}.
\subsection{Notations and conventions}
\label{ssec:notations}
Let $k$ denote a fixed positive integer and
let $\varepsilon$ be an arbitrary small positive real number that may change
its value during calculations. By $s,s_{0},s_{1}\geq 1$
we denote integers that depend on $k$.
In this article,
we suppress the dependence of the implicit constants
on $k$, $s$ or $\varepsilon$ in our notation, simply writing
$\ll$ for $\ll_{k,s,\varepsilon}$.
Moreover, we write $f\lll g$ if $f(x)=o(g(x))$, that is if
$f(x)/g(x)\to 0$ for $x\to \infty$.
For $\alpha\in\mathbb{R}$ we write
$\mathrm{e}(\alpha):=\exp(2\pi i \alpha)$ for
the complex exponential function and $\|\alpha\|$ denotes the distance
from $\alpha$ to the nearest integer.
For integers $k\geq 1$, $s\geq 1$ and a real number $x>0$ we use the
notation $J_{k}(x,s)$ for Vinogradov's integral, that is
the number of solutions to Vinogradov's system
\begin{align*}
m_{1}+\dots+m_{s}&= n_{1}+\dots+n_{s} \\
m_{1}^{2}+\dots+m_{s}^{2}&= n_{1}^{2}+\dots+n_{s}^{2} \\
&\cdots\\
m_{1}^{k}+\dots+m_{s}^{k}&= n_{1}^{k}+\dots+n_{s}^{k}
\end{align*}
with $1\leq m_{1},\dots,m_{s},n_{1},\dots,n_{s}\leq x$.
In this work, although no use of the integral representation of
$J_{k}(x,s)$ is made, the $\ell_{2}$-norm of the counting
function $r_{s}(\boldsymbol{\lambda})$ is used in the proof of
Theorem \ref{Satz1}.
Given a positive integer $n$ we write $\tau(n)$ for the number of
divisors of $n$, and $\tau_{3}(n)$ denotes the number of ways one can
write $n$ as a product of $3$ factors. We will use the well-known
estimates $\tau(n)\ll n^{\varepsilon}$ and $\tau_{3}(n)\ll n^{\varepsilon}$.
The set of real functions with continuous derivatives of
order up to $k$ on an interval $I$ is denoted by $C^{k}(I)$.
\subsection{Auxiliaries}
\label{ssec:auxlemmas}
We collect some auxiliary results needed as tools in
this article.
The following is the well-known sum lemma,
see e.g.~\cite[Lemma 4C]{Schm} for a proof.
\begin{lemma}[Sum lemma]
\label{lem:sumlemma}
For $\alpha\in\mathbb{R}$ let $u,q$ be integers with $(u,q)=1$, $0\leq u\leq
q-1$ and $|\alpha-u/q|<q^{-2}$. Let $\beta\in\mathbb{R}$. Then
\[
\sum_{Z\leq h\leq Y} \min(X,\|\alpha h +\beta\|^{-1}) \ll
(X+q\log(q))((Y-Z)q^{-1}+1).
\]
\end{lemma}
For further improvements, we will also make use of the following
result from \cite[Lemma 9C]{Schm}.
\begin{lemma}[Variant of sum lemma]
\label{lem:varsumlemma}
For $\alpha\in\mathbb{R}$ let $u,q$ be integers with $(u,q)=1$,
$0\leq u\leq q-1$ and $|\alpha-u/q|<q^{-1}X^{-1}$. Let $\beta\in\mathbb{R}$. Then
\[
\sum_{1\leq j\leq q} \min(X,\|\alpha j +\beta\|^{-1}) \ll
\min(X,q\|\beta q\|^{-1}) + q\log q.
\]
\end{lemma}
Next, we need the following simple bound
for the number of curve points close to integer points.
This is Lemma 2 in \cite{HB-V}, see also \cite[Thm. 5.6]{Bor}, where
a proof is provided.
\begin{lemma}[Curve points close to integer points]
\label{lem:smf}
Let $N$ be a positive integer, and suppose that $g(x) : [0, N] \to \mathbb{R}$ has
a continuous derivative on $(0, N )$. Suppose further that
$0 < \lambda \leq g'(x) \leq A \lambda$ for all $x \in (0, N)$.
Then
\[
\#\{n \leq N : \|g(n)\| \leq \delta\}
\ll (1 + A \lambda N )(1 + \delta/\lambda).
\]
\end{lemma}
We also need the following simple assertion.
\begin{lemma}
\label{lem:sri}
Consider positive real functions $S,f$ and $A$.
Assume that $S(x)\ll f(x)S(x) + A(x)$. If $f(x)$ tends to zero for
$x\to \infty$, then $S(x)\ll A(x)$.
\end{lemma}
\begin{proof}
If $C$ denotes the implicit constant, then $S(x)(1-Cf(x))\leq C
A(x)$. With $f(x)\leq C/2$ for all large $x$ we deduce $S(x)\leq 2C A(x)$.
\end{proof}
Another important ingredient is
Vinogradov's Mean Value Theorem. The theorem is elementary for $k=1$
and $k=2$. For the highly nontrivial cases $k\geq 3$ it has been
proved in \cite{BDG} by Bourgain, Demeter and Guth for $k\geq 4$
and in \cite{Woo} by Wooley for $k=3$, and again in \cite{WooNest} by Wooley
for $k\geq 3$. In our analysis, we will make use of this deep estimate.
\begin{theorem}[VMVT]
\label{th:VMVT}
Let $s\geq 1$, $k\geq 1$ be integers and
$\varepsilon>0$. Then
$J_{k}(x,s)\ll (x^{s} + x^{2s-k(k+1)/2})x^{\varepsilon}$.
\end{theorem}
\section{Discrete moments of Weyl sums}
\label{sec:WeylVMVT}
It is known that VMVT has the following impact on
Weyl sum estimates:
\begin{theorem}[Weyl sum estimate]
\label{th:weyloriginal}
Let $P\in\mathbb{R}[X]$ be a polynomial of degree $k\geq 2$, and for
the leading coefficient $\alpha_{k}$ of $P$ let $u,q$ be integers with
$(u,q)=1$, $q\geq 1$ and $|\alpha_{k}-u/q|<q^{-2}$.
Then we have
\[
\sum_{n\leq x} \mathrm{e}(P(n)) \ll x \Big(\frac{1}{q}+\frac{1}{x}
+\frac{q}{x^{k}}\Big)^{1/k(k-1)} x^{\varepsilon}.
\]
\end{theorem}
A proof can easily be found using Montgomery's
exposition \cite[\S 4]{Mont}. Our analysis of this proof yields a
generalization of this estimate for discrete moments of Weyl sums.
By changing a small aspect, it comes
with an extra-improvement, stated below as Theorem~\ref{Satz1}.
Compared to this, Theorem
\ref{Satz2} below is just a straight forward generalization of Theorem
\ref{th:weyloriginal} that stems from Montgomery's original approach
presented in \cite[\S 4]{Mont}.
\bigskip
We give the definition of the discrete moments we look at.
\begin{definition}
\label{def1}
Let $k\geq 2$ be a fixed integer and
consider a fixed polynomial $P_{\boldsymbol{\alpha}}\in\mathbb{R}[X]$ of degree $k$ with
$P_{\boldsymbol{\alpha}}(0)=0$, say
\[
P_{\boldsymbol{\alpha}}(X)=\alpha_{k}X^{k}+\alpha_{k-1}X^{k-1}+\dots+\alpha_{1}X
\]
with $\alpha_{1},\dots,\alpha_{k}\in \mathbb{R}$.
Let $x>1$ be a sufficiently large real number
and for $a\in\mathbb{N}$ let
\[
S_{a}(\boldsymbol{\alpha}):=\sum_{m\leq x} \mathrm{e}(aP_{\boldsymbol{\alpha}}(m))
\]
be a corresponding Weyl sum of $P_{\boldsymbol{\alpha}}$. The twist with $a$ allows us
to consider discrete moments of the form
\begin{equation}
\label{eq:DMWeylsum}
\sum_{a\leq T}|S_{az}(\boldsymbol{\alpha})|^{2s}
\end{equation}
with large real $T>1$ and with
fixed numbers $z,s\in\mathbb{N}$.
\end{definition}
The role of $z$ is to control a possible dependence of a further
factor in the argument of the exponential. We might think of a small
$z$, or even $z=1$.
Sums of the shape \eqref{eq:DMWeylsum} occur in numerous applications,
like in Dirichlet's divisor problem, counting integer points close to curves,
or, as we will see below, in the polynomial large sieve
inequality (for one variable polynomials). We will restrict on
presenting just the latter two applications which work well.
Our first goal is to give good estimates for the expression in
\eqref{eq:DMWeylsum} depending on $x$ and $T$.
Note that bounds for other moments can then easily be derived by H\"older's
inequality
\begin{equation}
\label{eq:Hoelder}
\sum_{a\leq T}|S_{az}(\boldsymbol{\alpha})|^{\ell}\leq T^{1-\ell/2s}
\Big(\sum_{a\leq T} |S_{az}(\boldsymbol{\alpha})|^{2s}\Big)^{\ell/2s},\ 0<\ell< 2s.
\end{equation}
Since our results use different values of $s$, it is convenient
to state bounds for the first moment, which makes the statements
easy to compare. Therefore, the results Theorem \ref{Satz1} and
Theorem \ref{Satz2} below are stated for the first moment.
\section{Improved moment estimate}
\label{sec:WeylImpr}
We are following the estimate of Weyl sums along the
lines of Montgomery's so-called alternative derivation in \cite[\S 4.4]{Mont}
and carry it over to the situation of discrete moments.
This approach yields
the following result for Weyl sums as given in Definition \ref{def1}.
We call it the improved moment estimate.
The direct approach leading to Theorem\ \ref{Satz2}
yields a bound that is weaker in certain ranges. This is discussed in
Subsection \ref{ssec:compconj}.
\begin{theorem}[Improved moment estimate]
\label{Satz1}
Let $k\geq 3$, $s_{0}=(k-1)(k-2)/2+1$ and $u,q$ be integers with $q\geq 1$,
$(u,q)=1$ and $|\alpha_{k}-u/q|<q^{-2}$. Then
\begin{equation*}
\sum_{a\leq T} |S_{az}(\boldsymbol{\alpha})| \ll Tx
\Big(\frac{zx^{k-1}}{q}+\frac{zx^{k-1}\log(q)}{T}
+\frac{1}{x}+\frac{q\log(q)}{Tx}\Big)^{1/2s_{0}}x^{\varepsilon}.
\end{equation*}
\end{theorem}
In the bound, we ordered the factors on the right hand side:
it starts with
the trivial estimate $Tx$, then we give the improvement factor and
then a small additional factor $x^{\varepsilon}$.
\begin{proof}
We need to introduce some of the notations from \cite[\S 4.4]{Mont},
but writing $x$ instead of $N$.
Thus for $j\in\mathbb{N}$, the $j$-th power sum of a tuple
$\mathbf{m}}%{\underline{j}=(m_{1},\dots,m_{s})\in\mathbb{N}^{s}$ is written as
$s_{j}(\mathbf{m}}%{\underline{j}):=m_{1}^{j}+\dots+m_{s}^{j}$,
and the difference of two power sums as
$d_{j}=d_{j}(\mathbf{u}}%{\underline{j},\mathbf{v}}%{\underline{j}):=s_{j}(\mathbf{u}}%{\underline{j})-s_{j}(\mathbf{v}}%{\underline{j})$, where
$u_{i},v_{i}\in\{-x,\dots,x\}$.
Multiplying $S_{az}(\boldsymbol{\alpha})^{s}$
out, sorting the summands according
to the value of the power sums with power $j=1,\dots,k-2$,
and an application of Cauchy--Schwarz's inequality yields
\begin{equation*}
|S_{az}(\boldsymbol{\alpha})|^{2s} \leq s^{k-2} x^{(k-1)(k-2)/2} \mathcal{T}(a)
\end{equation*}
with
\begin{align*}
\mathcal{T}(a)&:=\sum_{\substack{\mathbf{m}}%{\underline{j},\mathbf{n}}%{\underline{j}
\\ s_{j}(\mathbf{m}}%{\underline{j})=s_{j}(\mathbf{n}}%{\underline{j}),\ j=1,\dots,k-2}}
\mathrm{e}(azP(m_{1}))
\cdots\mathrm{e}(azP(m_{s}))\\ &\hspace{5cm}\cdot
\mathrm{e}(-azP(n_{1}))\cdots\mathrm{e}(-azP(n_{s})) \\
&=\sum_{\substack{\mathbf{m}}%{\underline{j},\mathbf{n}}%{\underline{j}
\\ s_{j}(\mathbf{m}}%{\underline{j})=s_{j}(\mathbf{n}}%{\underline{j}),\ j=1,\dots,k-2}}
\mathrm{e}\big( (s_{k}(\mathbf{m}}%{\underline{j})-s_{k}(\mathbf{n}}%{\underline{j})) az\alpha_{k} \\ &\hspace{5cm}+
(s_{k-1}(\mathbf{m}}%{\underline{j}) - s_{k-1}(\mathbf{n}}%{\underline{j}))az\alpha_{k-1}\big) \\
&= \sum_{\substack{\mathbf{u}}%{\underline{j},\mathbf{v}}%{\underline{j}
\\ d_{j}(\mathbf{u}}%{\underline{j},\mathbf{v}}%{\underline{j})=0,\ j=1,\dots,k-2}}
\mathrm{e}\big( d_{k} az\alpha_{k} + d_{k-1}az\alpha_{k-1}\big)
\sum_{m\in I} \mathrm{e}\big(kd_{k-1}maz\alpha_{k}\big),
\end{align*}
compare \cite[Eq.\ (34)--(38) in \S 4.4]{Mont}.
Here, $m$ runs through an interval
$I=I(\mathbf{u}}%{\underline{j},\mathbf{v}}%{\underline{j},x)$ that contains at most $x$ many successive integers,
and we have put $m=m_{1}$, $m_{i}=m+u_{i}$ for $2\leq i\leq s$,
and $n_{i}=m+v_{i}$ for $1\leq i\leq s$. Note that the vector $\mathbf{u}}%{\underline{j}$
consists of one variable less, it has $s-1$ components.
In this step, all variables $m_{i}, n_{i}$ in the Vinogradov System
$s_{j}(\mathbf{m}}%{\underline{j})=s_{j}(\mathbf{n}}%{\underline{j})$, $j=1,\dots,k-2$, have been translated by
$m=m_{1}$, thus we make use of the translation invariance of the
Vinogradov system.
Now let $h=d_{k-1}\leq 2skx^{k-1}$ and sort the tuples $\mathbf{u}}%{\underline{j},\mathbf{v}}%{\underline{j}$
by their value for $d_{k-1}=h$.
Then the summation of $\mathcal{T}(a)$ over $a\leq T$ yields
\begin{multline}
\label{eq:TSumme}
\sum_{a\leq T}\mathcal{T}(a) = \sum_{h\ll x^{k-1}}
\sum_{\substack{\mathbf{u}}%{\underline{j},\mathbf{v}}%{\underline{j}\\d_{k-1}=h\\d_{j}=0,\ j=1,\dots,k-2}}
\sum_{m\in I} \sum_{a\leq T}
\mathrm{e}(az(\alpha_{k}d_{k}+\alpha_{k}khm+h\alpha_{k-1})),
\end{multline}
where the last geometric sum can be estimated by
\[
\ll \min(T,\|\alpha_{k}zd_{k}+\alpha_{k}zkhm+zh\alpha_{k-1}\|^{-1}).
\]
In the following, the notation $\sum'$ at the sum over $\mathbf{u}}%{\underline{j},\mathbf{v}}%{\underline{j}$
abbreviates the condition that $d_{j}=0$ holds for $j=1,\dots,k-2$.
Using this, we obtain
\begin{align*}
\sum_{a\leq T} \mathcal{T}(a) &\ll \sum_{h\ll x^{k-1}}
\sideset{}{'}\sum_{\substack{\mathbf{u}}%{\underline{j},\mathbf{v}}%{\underline{j} \\ d_{k-1}=h}}
\sum_{m\in I} \min(T, \| \alpha_{k}zd_{k}+\alpha_{k}zkhm
+zh\alpha_{k-1}\|^{-1}) \\
&\ll \sum_{h\ll x^{k-1}} \sum_{m\in I'}
\sideset{}{'}\sum_{\substack{\mathbf{u}}%{\underline{j},\mathbf{v}}%{\underline{j} \\ d_{k-1}=h}}
\min(T, \| \alpha_{k}zd_{k}+\alpha_{k}zkhm
+zh\alpha_{k-1}\|^{-1}),
\end{align*}
where we extended the interval $I$ of length at most $x$ to an
interval $I'$ of length at most $3x$, in order to remove the
dependence on the variables $\mathbf{u}}%{\underline{j},\mathbf{v}}%{\underline{j}$ except on $h$, which makes the
separation of summation possible.
We continue with
\begin{multline*}
\sum_{a\leq T} \mathcal{T}(a)
\ll
\sum_{d\ll x^{k}} \sum_{h\ll x^{k-1}} \sum_{m\in I'}
\min(T, \|\alpha_{k}zd+\alpha_{k}zkhm+zh\alpha_{k-1}\|^{-1}) \\
\cdot \sideset{}{'}\sum_{\substack{\mathbf{u}}%{\underline{j},\mathbf{v}}%{\underline{j} \\ d_{k-1}=h,\ d_{k}=d}} 1,
\end{multline*}
and changing $hm$ to $w$ yields
\begin{align}
\sum_{a\leq T} \mathcal{T}(a)
&\ll\sum_{d\ll x^{k}} \sum_{w\ll x^{k}}
\sum_{\substack{h\mid w\\ h\ll x^{k-1}}}
\min(T, \| \alpha_{k}z(d+kw)+zh\alpha_{k-1}\|^{-1})\notag\\ &\hspace{7cm}
\cdot \sideset{}{'}\sum_{\substack{\mathbf{u}}%{\underline{j},\mathbf{v}}%{\underline{j} \\ d_{k-1}=h,\ d_{k}=d}} 1\notag \\
&\ll\sum_{j\ll zx^{k}} \sum_{\substack{d\ll x^{k} \\ dz\equiv j (zk)}}
\sum_{\substack{h\ll x^{k-1}\\ h\mid (j-zd)/zk}}
\min(T, \| \alpha_{k}j+zh\alpha_{k-1}\|^{-1})
\sideset{}{'}\sum_{\substack{\mathbf{u}}%{\underline{j},\mathbf{v}}%{\underline{j} \\ d_{k-1}=h,\ d_{k}=d}} 1 \notag\\
&\ll \sum_{d\ll x^{k}} \sum_{h\ll x^{k-1}} \Big( \sum_{j\ll zx^{k}}
\min(T, \| \alpha_{k}j+zh\alpha_{k-1} \|^{-1}) \Big)
\sideset{}{'}\sum_{\substack{\mathbf{u}}%{\underline{j},\mathbf{v}}%{\underline{j} \\ d_{k-1}=h,\ d_{k}=d}} 1, \label{eq:cest}
\end{align}
and an application of Lemma \ref{lem:sumlemma} to the sum in large
brackets yields
\begin{align*}
\sum_{a\leq T} \mathcal{T}(a)
&\ll \sum_{d\ll x^{k}} \sum_{h\ll x^{k-1}} (T+q\log(q))(zx^{k}/q+1)
\sideset{}{'}\sum_{\substack{\mathbf{u}}%{\underline{j},\mathbf{v}}%{\underline{j} \\ d_{k-1}=h,\ d_{k}=d}} 1 \\
&= (T+q\log(q))(zx^{k}/q+1)\sideset{}{'}\sum_{\mathbf{u}}%{\underline{j},\mathbf{v}}%{\underline{j}} 1,
\end{align*}
assuming that the integer $q\geq 1$ is such that there exists an integer $u$
with $(u,q)=1$ and $|\alpha_{k}-u/q|<q^{-2}$.
Now we shall give an estimate for the last sum.
For $\boldsymbol{\lambda}\in\mathbb{Z}^{k-2}$ let
\[
r_{s-1}(\boldsymbol{\lambda}):=\#\{\mathbf{u}}%{\underline{j};\ u_{2}+\dots+u_{s}=\lambda_{1},\dots,\
u_{2}^{k-2}+\dots+u_{s}^{k-2}=\lambda_{k-2}\}
\]
and similarly
\[
r_{s}(\boldsymbol{\lambda}):=\#\{\mathbf{v}}%{\underline{j};\ v_{1}+\dots+v_{s}=\lambda_{1},\dots,\
v_{1}^{k-2}+\dots+v_{s}^{k-2}=\lambda_{k-2}\}.
\]
Then the Cauchy--Schwarz inequality yields
\begin{align*}
\sideset{}{'}\sum_{\mathbf{u}}%{\underline{j},\mathbf{v}}%{\underline{j}} 1 &= \sum_{\substack{\mathbf{u}}%{\underline{j},\mathbf{v}}%{\underline{j} \\
d_{j}=0,\ j=1,\dots,k-2}} 1 =
\sum_{\boldsymbol{\lambda}\in\mathbb{Z}^{k-2}}r_{s-1}(\boldsymbol{\lambda})r_{s}(\boldsymbol{\lambda}) \\
&\leq \Big(\sum_{\boldsymbol{\lambda}}r_{s-1}(\boldsymbol{\lambda})^{2}\Big)^{1/2}
\Big(\sum_{\boldsymbol{\lambda}}r_{s}(\boldsymbol{\lambda})^{2}\Big)^{1/2}=(J_{k-2}(x,s-1)J_{k-2}(x,s))^{1/2},
\end{align*}
and so we obtain
\begin{align*}
\sum_{a\leq T} \mathcal{T}(a)
&\ll (J_{k-2}(x,s-1)J_{k-2}(x,s))^{1/2}
\Big(\frac{Tzx^{k}}{q}+zx^{k}\log(q)+T+q\log(q)\Big) \\
&=Tx^{k}(J_{k-2}(x,s-1)J_{k-2}(x,s))^{1/2}
\Big(\frac{z}{q}+\frac{z\log(q)}{T}
+\frac{1}{x^{k}}+\frac{q\log(q)}{Tx^{k}}\Big).
\end{align*}
For the desired moment of Weyl sums this yields
\begin{align*}
\sum_{a\leq T} &|S_{az}(\boldsymbol{\alpha})|^{2s} \\
&\ll Tx^{(k-1)(k-2)/2+k}
(J_{k-2}(x,s-1)J_{k-2}(x,s))^{1/2} \\ &\hspace{6cm}
\cdot\Big(\frac{z}{q}+\frac{z\log(q)}{T}
+\frac{1}{x^{k}}+\frac{q\log(q)}{Tx^{k}}\Big) \\
&= Tx^{2s}
\Big(\frac{J_{k-2}(x,s-1)J_{k-2}(x,s)}{x^{4s-(k-1)(k-2)-2k}}
\Big)^{1/2}\\ &\hspace{6cm}
\cdot \Big(\frac{z}{q}+\frac{z\log(q)}{T}
+\frac{1}{x^{k}}+\frac{q\log(q)}{Tx^{k}}\Big).
\end{align*}
Now that we have VMVT, Theorem \ref{th:VMVT},
at hand, we can apply the best possible
bound for the term in big brackets that includes the Vinogradov integrals.
Choosing $s=s_{0}$ with $s_{0}=(k-1)(k-2)/2+1$, we have
\begin{multline*}
\frac{J_{k-2}(x,s-1)J_{k-2}(x,s)}{x^{4s-(k-1)(k-2)-2k}} \\ \ll
x^{s-1}x^{2s-(k-1)(k-2)/2}x^{-4s+(k-1)(k-2)+2k}x^{\varepsilon}=x^{2k-2+\varepsilon}
\end{multline*}
for this value of $s$. (Note that we have $\ll x^{2k-1+\varepsilon}$
when choosing $(k-1)(k-2)/2$ for $s$ instead,
so the choice $s=s_{0}$ is optimal.)
We arrive at the following estimate.
\begin{equation}
\label{eq:S1}
\sum_{a\leq T} |S_{az}(\boldsymbol{\alpha})|^{2s_{0}} \ll Tx^{2s_{0}} x^{k-1}
\Big(\frac{z}{q}+\frac{z\log(q)}{T}
+\frac{1}{x^{k}}+\frac{q\log(q)}{Tx^{k}}\Big) x^{\varepsilon}.
\end{equation}
\bigskip
Using H\"older's inequality \eqref{eq:Hoelder}, we obtain an estimate
for the first moment. In this way, we obtain
the asserted bound from equation \eqref{eq:S1}.
\end{proof}
\bigskip
Theorem \ref{Satz1} has to be compared with the result obtained by the
straight-forward approach, that is, the following bound.
\begin{theorem}[Standard approach estimate]
\label{Satz2}
Let $k\geq 2$, $s_{1}=k(k-1)/2$ and $u,q$
be integers with $q\geq 1$, $(u,q)=1$ and
$|\alpha_{k}-u/q|<q^{-2}$. Then
\begin{equation*}
\sum_{a\leq T} |S_{az}(\boldsymbol{\alpha})| \ll
Tx \Big(\frac{z}{q}+\frac{z}{x}+\frac{q}{Tx^{k}}\Big)^{1/2s_{1}}(Txz)^{\varepsilon}.
\end{equation*}
\end{theorem}
Note that with $T=1$ and $z=1$, we get back Theorem
\ref{th:weyloriginal} above as a special case.
\begin{proof}
Montgomery's original approach in \cite[\S 4.4,p.81,l.15]{Mont} yields
\begin{multline*}
\sum_{a\leq T} |S_{az}(\boldsymbol{\alpha})|^{2s} \\ \ll x^{(k-1)(k-2)/2}
x^{-1}J_{k-1}(3x,s)\sum_{a\leq T} \sum_{0\leq h\leq 2sx^{k-1}}
\min(x,\| akhz\alpha_{k}\|^{-1}),
\end{multline*}
where the last double sum can be estimated using Lemma \ref{lem:sumlemma}.
Together with the substitution $w=akhz$ this yields
\[
\ll \sum_{w\leq 2sTzkx^{k-1}} \tau_{3}(w)\min(x,\|w\alpha_{k}\|^{-1})
\ll \Big(\frac{zTx^{k}}{q} + zTx^{k-1}+q\Big)(Txz)^{\varepsilon},
\]
where there exist integers $u,q$ with $q\geq 1$, $(u,q)=1$ and
$|\alpha_{k}-u/q|<q^{-2}$. We proceed with
\[
\sum_{a\leq T} |S_{az}(\boldsymbol{\alpha})|^{2s} \ll x^{2s}T
\Big(\frac{J_{k-1}(3x,s)}{x^{2s-k(k-1)/2}}\Big)
\Big(\frac{z}{q}+\frac{z}{x}+\frac{q}{Tx^{k}}\Big)(Txz)^{\varepsilon}.
\]
Next, using VMVT (Theorem \ref{th:VMVT}) with the optimal
$s=s_{1}=k(k-1)/2$ leads to the estimate
\begin{equation}
\label{eq:S2}
\sum_{a\leq T} |S_{az}(\boldsymbol{\alpha})|^{2s_{1}} \ll
x^{2s_{1}}T\Big(\frac{z}{q}+\frac{z}{x}+\frac{q}{Tx^{k}}\Big)(Txz)^{\varepsilon}.
\end{equation}
Applying H\"older's inequality \eqref{eq:Hoelder}
to equation \eqref{eq:S2} yields the desired
first moment as given in the assertion of the theorem. \end{proof}
\subsection{Comparison and conjectural considerations}
\label{ssec:compconj}
The expressions in large brackets in Theorems \ref{Satz1} and
\ref{Satz2} show the improvements compared
to the trivial estimate $Tx$.
They lead to a nontrivial assertion
if $zx^{k-1}\lll q\lll Tx$ in Theorem \ref{Satz1}
respectively if $z\lll q\lll Tx^{k}$ in Theorem \ref{Satz2}.
Let $s_{0}=(k-1)(k-2)/2+1$ and $s_{1}=k(k-1)/2$.
We compare these improvement expressions
(supposing $z$ is small in this comparison) and obtain
the following assertions.
\begin{enumerate}[1.)]
\item
In these expressions, we compare the typical dominant
terms, $x^{-1/2s_{0}}$ (for $zx^{k}\ll q \ll T$)
with $(z/x)^{1/2s_{1}}$ (for $x\ll q\ll zTx^{k-1}$),
we immediately see that Theorem
\ref{Satz1} yields a sharper estimate in the intersection range
$zx^{k}\ll q\ll T$.
\item
The dominant term $(zx^{k-1}/q)^{1/2s_{0}}$ for $q\ll \min(zx^{k},T)$
in Theorem \ref{Satz1} is sharper than $(z/x)^{1/2s_{1}}$ for $x\ll q\ll T$,
if $q\gg z^{\sigma}x^{k-\sigma}$ with
\begin{equation}
\label{eq:sigma}
\sigma=\sigma_{k}:=1-s_{0}/s_{1}=2/k-2/k(k-1).
\end{equation}
To summarize, with Theorem \ref{Satz1} we obtain an improvement in the range
$z^{\sigma}x^{k-\sigma}\ll q\ll \min(zx^{k},T)$.
\item
The dominant term $(zx^{k-1}/T)^{1/2s_{0}}$ for $T\ll q\ll zx^{k}$
is sharper than $(z/x)^{1/2s_{1}}$ for $x\ll q\ll zTx^{k-1}$ if $T\gg
z^{\sigma}x^{k-\sigma}$. Thus in the intersection range $T\ll q
\ll zx^{k}$ we obtain an improvement.
\item
The dominant term $(q/Tx)^{1/2s_{0}}$ for $q\gg \max(T,zx^{k})$
is sharper than $(z/x)^{1/2s_{1}}$ for $x\ll q\ll zTx^{k-1}$ if $q\ll
Tx^{\sigma}z^{1-\sigma}$. Thus in the intersection range
$\max(T,zx^{k})\ll q\ll Tx^{\sigma}z^{1-\sigma}$, for which $T\gg
z^{\sigma}x^{k-\sigma}$ has to hold necessarily, we obtain an improvement.
\end{enumerate}
To summarize, Theorem \ref{Satz1} yields an improvement
only if $T\gg z^{\sigma}x^{k-\sigma}$, so
this term $z^{\sigma}x^{k-\sigma}$ turns out to be a critical value
for $T$ from which on we obtain improvements.
Moreover, above conditions on $q$ have to hold, that is the range
\begin{equation}
\label{eq:impr_range}
z^{\sigma}x^{k-\sigma}\ll q\ll Tx^{\sigma}z^{1-\sigma}.
\end{equation}
For any other $q$, Theorem \ref{Satz2} gives a sharper bound.
\bigskip
An observation is that in \eqref{eq:cest} we made a very coarse
estimate. Heuristically, one would expect that it could be doable
with the mean value over $h$. This would provide a gain of an extra
factor $x^{k-1}$ in the estimate. In this way,
we would save it also in \eqref{eq:S1} and
arrive at the following conjectural bound.
\begin{conjecture}
\label{conj:c}
For $k\geq 3$, $s_{0}=(k-1)(k-2)/2+1$, and $|\alpha_{k}-a/q|<q^{-2}$ for
$(a,q)=1$, the estimate
\begin{equation*}
\sum_{a\leq T} |S_{az}(\boldsymbol{\alpha})| \ll Tx
\Big(\frac{z}{q}+\frac{z\log(q)}{T}
+\frac{1}{x^{k}}+\frac{q\log(q)}{Tx^{k}}\Big)^{1/2s_{0}}(Txz)^{\varepsilon}
\end{equation*}
is conjectured to hold true.
\end{conjecture}
Compared to Theorem \ref{Satz1}, this would lead
to an improvement factor
$x^{-k/2s_{0}}$ (around $x^{-1/k}$) instead of $x^{-1/2s_{0}}$, provided that the secondary
terms do not matter.
It is interesting to see that we can indeed improve further towards
Conjecture~\ref{conj:c} if we assume suitable
rational approximations to $\alpha_{k}$ and $\alpha_{k-1}$ as follows.
\begin{theorem}[Second improved moment estimate]
\label{Satz3}
Let $k\geq 2$, $s_{2}=k(k-1)/2+1$ and $u,q$ be integers with $q\geq 1$,
$(u,q)=1$ and $|\alpha_{k}-u/q|<q^{-1}T^{-1}$. Further, let $v,w$ be
integers with $1\leq w\leq x^{k-1}z$ and $|zq\alpha_{k-1}-v/w|<w^{-2}$.
Then
\begin{equation*}
\sum_{a\leq T} |S_{az}(\boldsymbol{\alpha})| \ll Tx
\Big(\frac{x^{k-1}}{qw} +
\frac{x^{k-1}}{T}+ \frac{1}{xw}+\frac{q}{Tx}\Big)^{1/2s_{2}}x^{\varepsilon}.
\end{equation*}
\end{theorem}
\begin{proof}
We start as in the proof of Theorem~\ref{Satz1}, but continue \eqref{eq:cest}
with
\begin{align}
\label{eq:eq4neu}
\sum_{a\leq T} \mathcal{T}(a)
&\ll \sum_{d\ll x^{k}} \sum_{h\ll x^{k-1}} \Big( \sum_{j\ll zx^{k}}
\min(T, \| \alpha_{k}j+hz\alpha_{k-1} \|^{-1}) \Big) \notag
\sideset{}{'}\sum_{\substack{\mathbf{u}}%{\underline{j},\mathbf{v}}%{\underline{j} \\ d_{k-1}=h,\ d_{k}=d}} 1 \\
&\ll \Big(\sum_{h\ll x^{k-1}} \sum_{j\ll zx^{k}}
\min(T, \| \alpha_{k}j+zh\alpha_{k-1} \|^{-1}) \Big)
\max_{h_{0}}\sideset{}{'} \sum_{\substack{\mathbf{u}}%{\underline{j},\mathbf{v}}%{\underline{j} \\ d_{k-1}=h_{0}}} 1.
\end{align}
To handle the last sum, let
\begin{multline*}
\tilde{r}_{s}(\boldsymbol{\lambda},h_{0}):=\#\{\mathbf{v}}%{\underline{j};\ v_{1}+\dots+v_{s}=\lambda_{1},\dots,\
v_{1}^{k-2}+\dots+v_{s}^{k-2}=\lambda_{k-2},\\
v_{1}^{k-2}+\dots+v_{s}^{k-2} =\lambda_{k-2}+h_{0}\},
\end{multline*}
we obtain
\begin{align*}
\sideset{}{'} \sum_{\substack{\mathbf{u}}%{\underline{j},\mathbf{v}}%{\underline{j} \\ d_{k-1}=h_{0}}} 1
&= \sum_{\boldsymbol{\lambda}\in\mathbb{Z}^{k-1}}
\tilde{r}_{s-1}(\boldsymbol{\lambda},0)\tilde{r}_{s}(\boldsymbol{\lambda},h_{0}) \\ &\leq
\Big(\sum_{\boldsymbol{\lambda}}\tilde{r}_{s-1}(\boldsymbol{\lambda},0)^{2}\Big)^{1/2}
\Big(\sum_{\boldsymbol{\lambda}}\tilde{r}_{s}(\boldsymbol{\lambda},h_{0})^{2}\Big)^{1/2}\\
&=\Big(J_{k-1}(x,s-1)J_{k-1}(x,s)\Big)^{1/2},
\end{align*}
uniformly in $h_{0}$. Now we turn to the sum over $j,h$ in
\eqref{eq:eq4neu}.
For each block $B=[1+bq,\dots,q-1+bq]$ of consecutive positive
integers with $b\geq 1$, we have
\[
\sum_{j\in B}\min(T,\|j\alpha_{k}+hz\alpha_{k-1}\|^{-1})
\ll \sum_{1\leq j\leq q} \min(T,\|j\alpha_{k} +hz\alpha_{k-1}\|^{-1}),
\]
since $\min(T,\|j\alpha_{k}+q\alpha_{k}+hz\alpha_{k-1}\|^{-1})\ll
\min(T, \|j\alpha_{k} +hz\alpha_{k-1}\|^{-1})$ holds true
under the assumption $\|q\alpha_{k}\|<T^{-1}$.
Therefore
\begin{align*}
\sum_{h\ll x^{k-1}} &\sum_{j\ll zx^{k}}
\min(T, \| \alpha_{k}j+hz\alpha_{k-1} \|^{-1}) \\
&\ll
(zx^{k}/q+1) \sum_{h\ll x^{k-1}} \sum_{1\leq j\leq q}
\min(T, \|\alpha_{k}j+hz\alpha_{k-1} \|^{-1}) \\
&\ll
(zx^{k}/q+1) \sum_{h\ll x^{k-1}} (\min(T,q\|hzq\alpha_{k-1}\|^{-1}) +
q\log q),
\end{align*}
where we applied Lemma~\ref{lem:varsumlemma} in the last step.
Writing $m=hz$ and noting that the number of divisors of $m$ is
$\ll x^{\varepsilon}$, we continue using
Lemma~\ref{lem:sumlemma} by
\begin{multline*}
\ll (zx^{k}/q+1) x^{\varepsilon} q(T/q + w\log w)(x^{k-1}z/w+1)
+ zx^{2k-1+\varepsilon}+x^{k-1+\varepsilon}q\\
\ll x^{\varepsilon}z^{2} (x^{k}+q) (T/q+w)x^{k-1}/w +
zx^{2k-1+\varepsilon}+x^{k-1+\varepsilon}q \\ \ll
x^{\varepsilon}z^{2} Tx^{k}
\Big(\frac{x^{k-1}}{qw}+\frac{x^{k-1}}{T}+\frac{1}{xw}+\frac{q}{Tx}\Big),
\end{multline*}
supposing $w\leq x^{k-1}z$.
Then together with \eqref{eq:eq4neu}, we arrive at
\begin{multline*}
\sum_{a\leq T} \mathcal{T}(a) \ll
x^{\varepsilon}x^{k}T\Big(\frac{x^{k-1}}{qw} +
\frac{x^{k-1}}{T}+ \frac{1}{xw}+\frac{q}{Tx}\Big)
(J_{k-1}(x,s-1)J_{k-1}(x,s))^{1/2}.
\end{multline*}
For the desired moment of Weyl sums this yields now
\begin{align*}
\sum_{a\leq T} &|S_{az}(\boldsymbol{\alpha})|^{2s} \\
&\ll Tx^{(k-1)(k-2)/2+k+\varepsilon}
(J_{k-1}(x,s-1)J_{k-1}(x,s))^{1/2} \\ &\hspace{1cm}
\cdot \Big(\frac{x^{k-1}}{qw} +
\frac{x^{k-1}}{T}+ \frac{1}{xw}+\frac{q}{Tx}\Big) \\
&= Tx^{2s+\varepsilon}
\Big(\frac{J_{k-1}(x,s-1)J_{k-1}(x,s)}{x^{4s-(k-1)(k-2)-2k}}
\Big)^{1/2}
\cdot \Big(\frac{x^{k-1}}{qw} +
\frac{x^{k-1}}{T}+ \frac{1}{xw} +\frac{q}{Tx}\Big).
\end{align*}
Again, we need to choose the optimal parameter $s$ which fits best
with the Vinogradov integrals. This is provided by
the choice $s_{2}=k(k-1)/2+1$, an application of VMVT (Theorem \ref{th:VMVT})
yields
\[
\frac{J_{k-1}(x,s-1)J_{k-1}(x,s)}{x^{4s-(k-1)(k-2)-2k}}\ll
x^{s-1}x^{2s-k(k-1)/2}x^{-4s+(k-1)(k-2)+2k}\ll 1.
\]
Thus applying H\"older's inequality, we arrive at the assertion
\[
\sum_{a\leq T} |S_{az}(\boldsymbol{\alpha})| \ll Tx
\Big(\frac{x^{k-1}}{qw} +
\frac{x^{k-1}}{T}+ \frac{1}{xw}+\frac{q}{Tx}\Big)^{1/2s_{2}}.
\]
\end{proof}
We see that in the setting of Theorem~\ref{Satz3} we improved the term
$1/x$ by $1/xw$, where $w$ may be taken as large as $x^{k-1}$.
This would allow a saving of up to $x^{-1/k}$ in the estimate, namely
\[
x^{-k/2s_{2}}=x^{-k/(k(k-1)+2)}=x^{-1/(k-1+2/k))}\ll x^{-1/k},
\]
assuming best parameter choices for $q$ and $T$ (say $x^{k}\leq
q$ and $T\geq qw$).
Like this, we come close to Conjecture~\ref{conj:c}, but
the assumptions on $\alpha_{k}$ and $\alpha_{k-1}$ are more restrictive.
Theorem~\ref{Satz3} also suggests that there might be limitations to
Conjecture~\ref{conj:c}, such as if $\alpha_{k-1}$ is close to
$0$ mod $1$ or has good approximation to a rational $v/w$ with small
denominator $w$. In cases like these it seems that we may not estimate the sum
in \eqref{eq:cest} much better than the way we proceed.
\section{Discrete moments of exponential sums}
\label{sec:mvexposums}
We turn now to discrete moments of general exponential sums
with smooth functions $f$. The main idea is to approximate $f$ with a
polynomial using Taylor's theorem and apply the bounds of the previous
sections.
We proceed similar as in Bordell\`es' book \cite[\S 6.6.7]{Bor},
or in Heath-Brown's recent article \cite{HB-V}.
The first result is as follows.
\begin{theorem}
\label{th:Thb}
Let $N$ be a large positive integer,
and let $f\in\mathbb{C}^{k}((0,3N))$, $k\geq 3$.
Suppose that there exists real numbers
$\lambda,A$ such that $0<\lambda\leq f^{(k)}(x)\leq A\lambda$ holds for
all $x\in (0,3N)$. Let $\rho=1/((k-2)(k-3)+2)$ and
$\mu=1+A\lambda N$.
Let $z$ be a positive integer that is considered
to be small, and let $T$ be a real number with
$N^{-k}(zA\lambda)^{-1}\leq T\leq (zA\lambda)^{-1}$.
Then
\begin{multline}
\label{eq:Thb}
\sum_{a\leq T} \Big|\sum_{N<m<2N} \mathrm{e}(azf(m))\Big|
\ll NT(zA\lambda T)^{\rho/k+\varepsilon} + T(zA\lambda T)^{-1/k} \\
+T\mu z^{2}(zA\lambda T)^{2/k-2}
+ \mu z (zA\lambda T)^{1/k-1}\lambda^{-1}.
\end{multline}
\end{theorem}
We note that $\lambda$ as well as $A$ and $z$ may depend on $N$
and $T$. In the case if $A$ and $z$ is depending on $k$ only, we may hide $A$
and $z$ in the implicit constant leading to a slightly easier expression.
Additionally assuming $\mu=1$, the upper bound simplifies to
\begin{equation}
\label{eq:Thbstr}
NT(\lambda T)^{\rho/k}+ T(\lambda T)^{-1/k}
+ T(\lambda T)^{2/k-2}
+ (\lambda T)^{1/k-1}\lambda^{-1}.
\end{equation}
The proof uses an adapted circle method.
The first term in the bound \eqref{eq:Thb} respectively \eqref{eq:Thbstr}
comes from the the minor arc contribution, the
second gives a trivial contribution from a Weyl-shift,
and the last two terms come from the major arc contribution.
\begin{proof}
Let $\mathcal{L}_{f}$ denote the left hand side of \eqref{eq:Thb}.
We start with a Weyl-shift with $1\leq H\leq N$. For this,
let $\beta_{m}=\mathrm{e}(azf(m))$ if $N<m<2N$, and $\beta_{m}=0$
otherwise. Then for each $h'\leq H$,
\begin{multline*}
\sum_{N<m<2N} \mathrm{e}(azf(m)) = \sum_{m\in\mathbb{Z}} \beta_{m+h'} =
\frac{1}{H}\sum_{h\leq H}\ \sum_{m\in\mathbb{Z}} \beta_{m+h} \\
= \frac{1}{H}\sum_{m=N-H+1}^{2N-1}\ \sum_{h\leq H} \beta_{m+h}
= \frac{1}{H}\sum_{m=N+1}^{2N-1}\ \sum_{h\leq H} \mathrm{e}(azf(m+h)) +O(H).
\end{multline*}
We obtain
\begin{equation}
\label{eq:Lfglg}
\mathcal{L}_{f} =\frac{1}{H} \sum_{a\leq T}\ \sum_{N<m<2N}\Big|\sum_{h\leq H}
\mathrm{e}(azf(m+h))\Big| + O(TH).
\end{equation}
An application of Taylor's theorem provides the expansion
$f(m+h)=Q_{m}(h)+u_{m}(h)$ with
\[
Q_{m}(h)=hf'(m)+h^{2}f''(m)/2!+\dots+h^{k-1}f^{(k-1)}(m)/(k-1)!.
\]
Note that $f^{(k-1)}(m)/(k-1)!$ is the leading
coefficient of this polynomial of degree $k-1$ in $h$, and that
\[
u_{m}(h)=f(m)+\frac{1}{(k-1)!}\int_{0}^{h} (h-v)^{k-1}f^{(k)}(m+v)dv,
\]
so that $\mathrm{e}(azf(m+h))=\mathrm{e}(azQ_{m}(h))\mathrm{e}(azu_{m}(h))$.
We separate the exponential expressions containing $Q_{m}$ and $u_{m}$
by partial summation, this yields
\[
\mathcal{L}_{f} \leq \mathcal{S}_{1}+\mathcal{S}_{2}+O(TH)
\]
with
\begin{align*}
\mathcal{S}_{1} &\leq \frac{1}{H} \sum_{N<m<2N} \sum_{a\leq T}
\Big|\sum_{h\leq H}\mathrm{e}(azQ_{m}(h))\Big| \\
\end{align*}
and
\begin{align*}
\mathcal{S}_{2} &\leq \sum_{a\leq T} \frac{2\pi az}{H}\sum_{N<m<2N}
\int_{1}^{H} \Big|\sum_{h\leq x} \mathrm{e}(azQ_{m}(h))\Big| \cdot |u'_{m}(x)|\,\mathrm{d}x \\
&\ll z H^{k-2} \sum_{N<m<2N}\int_{1}^{H}\sum_{a\leq T}a
\Big|\sum_{h\leq x} \mathrm{e}(azQ_{m}(h))\Big| \,\mathrm{d}x \cdot
\sup_{v\in (0,H)} |f^{(k)}(m+v)| \\
&\ll zA\lambda H^{k-2} \int_{1}^{H} \sum_{N<m<2N}
\sum_{a\leq T}a
\Big|\sum_{h\leq x} \mathrm{e}(azQ_{m}(h))\Big| \,\mathrm{d}x.
\end{align*}
Next, we abbreviate
\[
S_{a,m}(x):= \sum_{h\leq x} \mathrm{e}(azQ_{m}(h)),
\]
summarize the bounds for $\mathcal{S}_{1}$ and $\mathcal{S}_{2}$ and arrive at
\begin{multline}
\label{eq:Lfbound}
\mathcal{L}_{f} \ll TH + \frac{1}{H} \sum_{N<m<2N} \sum_{a\leq T}
|S_{a,m}(H)| \\ + zA\lambda H^{k-2}T
\int_{1}^{H} \sum_{N<m<2N}\sum_{a\leq T} |S_{a,m}(x)| \,\mathrm{d}x.
\end{multline}
For the next argument,
fix $x$ with $x\leq H\leq N$
and let $\Delta_0:=z^{-1}T^{-1}H^{1-k}$.
Consider $m\in(N,2N)\cap\mathbb{Z}$ and let
\[
\mathcal{A}_{m}:=\Big\{\alpha\in[0,1];\ \Big\|
\frac{f^{(k-1)}(m)}{(k-1)!}-\alpha \Big\|\leq \Delta_0 \Big\}.
\]
Fix an $\alpha\in \mathcal{A}_{m}$.
We replace the leading coefficient in $Q_{m}(h)$ by $b_{k-1}\in\mathbb{R}$ such
that $ \frac{f^{(k-1)}(m)}{(k-1)!}-b_{k-1}\in\mathbb{Z}$, so that
$|b_{k-1}-\alpha|\leq \Delta_0$. Like this, we look at
\[
f_{m}^{*}(h):=hf'(m)+\dots+h^{k-2}f^{(k-2)}(m)/(k-2)!+b_{k-1}h^{k-1}.
\]
Let $S_{a,m}^{*}(x):=\sum_{h\leq x} \mathrm{e}(azf_{m}^{*}(h))$, so that
$|S_{a,m}^{*}(x)|=|S_{a,m}(x)|$ and we are able to work with $S_{a,m}^{*}(x)$
instead of $S_{a,m}(x)$ in \eqref{eq:Lfbound}. Moreover, let
\[
\tilde{f}_{m,\alpha}(h):=hf'(m)+\dots+h^{k-2}f^{(k-2)}(m)/(k-2)!+\alpha
h^{k-1}
\]
and
\[
\tilde{S}_{a,m}(\alpha,x):=\sum_{h\leq x} \mathrm{e}(az\tilde{f}_{m,\alpha}(h)).
\]
Then we have
\[
\frac{d}{dx} (f_{m}^{*}(x)-\tilde{f}_{m,\alpha}(x)) \ll
|b_{k-1}-\alpha|x^{k-2} \ll \Delta_0 x^{k-2},
\]
and we conclude by a partial summation, that
\begin{equation}
\label{eq:Sabsch}
S_{a,m}^{*}(x) \ll |\tilde{S}_{a,m}(\alpha,x)| +
az\int_{1}^{x} y^{k-2}\Delta_0 |\tilde{S}_{a,m}(\alpha,y)|\,\mathrm{d}y.
\end{equation}
Our task is reduced to prove good upper bounds for the term
\[
\mathcal{T}_{x}:=\sum_{N<m<2N} \sum_{a\leq T}
|\tilde{S}_{a,m}(\alpha,x)|
\]
with $x\leq H$. For each $m$ in the sum there is a chosen
$\alpha\in\mathcal{A}_{m}$.
We intend to apply Theorem \ref{Satz1}. We expect a good
result if we assume
$T$ to be much bigger than $zx^{k-2}$. (Note that $\deg
\tilde{S}_{a,m}(\alpha,x)=k-1$.)
For this purpose, introduce appropriate major and minor arcs.
Let
\[
\mathfrak{M}=\bigcup_{q\leq zx^{k-1}} \bigcup_{(u,q)=1}
\Big[\frac{u}{q}-\frac{1}{qT},\frac{u}{q}+\frac{1}{qT}\Big]
\]
denote the set of major arcs, and $\mathfrak{m}=[0,1]\setminus \mathfrak{M}$.
Now we distinguish two cases:
Say case ($\mathfrak{m}$) occurs if $m$ is such that
there exists a real number $\alpha\in\mathcal{A}_{m}\cap\mathfrak{m}$.
We choose then such an $\alpha$ for each such $m$.
By Dirichlet's approximation theorem, there exists
coprime integers $u$ and $q$
with $1\leq q\leq T$ such that
\[
\Big|\alpha-\frac{u}{q}\Big|\leq \frac{1}{qT}.
\]
Since $\alpha$ is contained in $\mathfrak{m}$, we conclude that even $q\geq
zx^{k-1}$ holds true.
A closer look at the improvement expression in Theorem
\ref{Satz1} yields
\[
\Big(\frac{zx^{k-2}}{q}+\frac{zx^{k-2}}{T}+\frac{1}{x}
+\frac{q}{Tx}\Big)^{\rho} \ll x^{-\rho}
\]
with $\rho=1/((k-2)(k-3)+2)$.
Therefore by Theorem \ref{Satz1},
\[
\sum_{a\leq T} |\tilde{S}_{a,m}(\alpha, x)|\ll Tx^{1-\rho+\varepsilon},
\]
hence, summing up over these $m$,
\begin{equation}
\label{eq:minor}
\mathcal{T}_{x,\text{($\mathfrak{m}$)}}:= \sideset{}{^{(\mathfrak{m})}}\sum_{N<m<2N}
\ \sum_{a\leq T} |\tilde{S}_{a,m}(\alpha, x)| \ll NTx^{1-\rho+\varepsilon}.
\end{equation}
In the major arc case,
$\mathcal{A}_{m}$ is contained completely in a major arc interval.
Then we conclude for $m$ with $N<m<2N$ in case ($\mathfrak{M}$),
that there exist $q\leq zx^{k-1}$ and
$(u,q)=1$ such that $\|f^{(k-1)}(m)/(k-1)!-u/q\|<1/qT$.
Summing up over these $m$, we obtain
\begin{multline*}
\mathcal{T}_{x,\text{($\mathfrak{M}$)}}:=\sideset{}{^{(\mathfrak{M})}}\sum_{N<m<2N}
\ \sum_{a\leq T}
|\tilde{S}_{a,m}(\alpha, x)|\\ \ll Tx\sum_{q\leq zx^{k-1}}
\sum_{(u,q)=1} \#\{m\in(N,2N);\ \\
\hspace{5cm}\|f^{(k-1)}(m)/(k-1)!-u/q\|<1/qT\} \\
\ll Tx \sum_{q\leq zx^{k-1}} \varphi(q) (1+A\lambda N)(1+1/\lambda qT),
\end{multline*}
where in the last step we applied Lemma \ref{lem:smf} with
$g(x):=f^{(k-1)}(x)/(k-1)!-u/q$.
From now on we make use of the abbreviation $\mu=1+A\lambda N$.
This yields the bound
\[
\mathcal{T}_{x,\text{($\mathfrak{M}$)}} \ll Tx (\mu z^{2}x^{2k-2} + \mu zx^{k-1}\lambda^{-1} T^{-1}),
\]
hence, in the major arc case,
\begin{equation}
\label{eq:major}
\mathcal{T}_{x,\text{($\mathfrak{M}$)}} \ll T\mu z^{2}x^{2k-1} + \mu zx^{k}\lambda^{-1}.
\end{equation}
Then joining the estimates \eqref{eq:minor} and \eqref{eq:major}
together yields
\begin{equation}
\label{eq:sxab}
\mathcal{T}_{x}\ll \mathcal{T}_{x,\text{($\mathfrak{m}$)}} + \mathcal{T}_{x,\text{($\mathfrak{M}$)}} \ll
NTx^{1-\rho+\varepsilon} + T\mu z^{2}x^{2k-1} + \mu zx^{k}\lambda^{-1}.
\end{equation}
Next, from estimate \eqref{eq:Sabsch} together with \eqref{eq:sxab}
we obtain
\begin{multline*}
\sum_{N<m<2N} \sum_{a\leq T} |S_{a,m}^{*}(x)| \ll
\mathcal{T}_{x} + Tz\Delta_{0}\int_{1}^{x}y^{k-2}\mathcal{T}_{y}\,\mathrm{d}y \\
\ll NTx^{1-\rho+\varepsilon} +Tz\Delta_{0}\int_{1}^{x} y^{k-2} NTy^{1-\rho+\varepsilon}\,\mathrm{d}y
\\
+ T\mu z^{2}x^{2k-1} + Tz\Delta_{0}\int_{1}^{x} y^{k-2} T\mu
z^{2}y^{2k-1} \,\mathrm{d}y \\
+ \mu zx^{k}\lambda^{-1} + Tz\Delta_{0}\int_{1}^{x} y^{k-2}
\mu zy^{k}\lambda^{-1}\,\mathrm{d}y \\
\ll NTx^{1-\rho+\varepsilon} + T\mu z^{2}x^{2k-1}+\mu zx^{k}\lambda^{-1},
\end{multline*}
where only in the last step we used $\Delta_0=z^{-1}T^{-1}H^{1-k}$
and $x\leq H$.
Therefore, by \eqref{eq:Lfbound}, we arrive at
\begin{align}
\mathcal{L}_{f} &\ll TH+H^{-1}\sum_{m}\sum_{a\leq T} |S_{a,m}^{*}(H)| \notag\\ &\qquad
+zA\lambda H^{k-2}T \int_{1}^{H} \sum_{m} \sum_{a\leq T}
|S_{a,m}^{*}(x)|\,\mathrm{d}x \notag\\ &\ll TH+
H^{-1}NTH^{1-\rho+\varepsilon}+zA\lambda H^{k-2}T\int_{1}^{H}NTx^{1-\rho+\varepsilon}\,\mathrm{d}x
\notag\\
&+ H^{-1} T\mu z^{2}H^{2k-1} + zA\lambda H^{k-2}T\int_{1}^{H} T\mu
z^{2}x^{2k-1} \,\mathrm{d}x \notag \\ &+ H^{-1} \mu zH^{k}\lambda^{-1}
+ zA\lambda H^{k-2}T\int_{1}^{H} \mu zx^{k-1}\lambda^{-1}\,\mathrm{d}x \notag\\
&\ll NTH^{-\rho+\varepsilon} + TH + T\mu z^{2}H^{2k-2} + \mu zH^{k-1}\lambda^{-1},
\label{eq:LF}
\end{align}
where we have chosen $H=[(zA\lambda T)^{-1/k}]$ in \eqref{eq:LF}.
This gives the bound
\begin{multline*}
\mathcal{L}_{f}\ll NT(zA\lambda T)^{\rho/k+\varepsilon} +
T(zA\lambda T)^{-1/k} \\
+T\mu z^{2}(zA\lambda T)^{2/k-2}
+ \mu z (zA\lambda T)^{1/k-1}\lambda^{-1}.
\end{multline*}
As necessary constraint for $T$ we get $N^{-k}\leq zA\lambda T \leq
1$, since we need $1\leq H=[(zA\lambda T)^{-1/k}]\leq N$.
\end{proof}
\textbf{Remark.}
We have to discuss in which range for $T$ Theorem \ref{th:Thb}
provides a nontrivial upper bound for $\mathcal{L}_{F}$.
The first two terms of the bound \ref{eq:Thb}
clearly give a nontrivial upper bound $\lll TN$,
and also the third term is $\lll TN$ provided that $T\mu
z^{2}H^{2k-2}\lll TN$ which means
\begin{equation}
\label{eq:t1}
T\ggg \mu^{k/(2k-2)}z^{1/(k-1)}(A\lambda)^{-1}N^{k/(2-2k)}.
\end{equation}
And also the fourth term is $\lll TN$ provided that
\begin{equation}
\label{eq:t2}
T\ggg\mu^{k/(2k-1)}z^{1/(2k-1)}A^{(1-k)/(2k-1)} \lambda^{-1} N^{k/(1-2k)}.
\end{equation}
Note that this means $T\ggg \mu (\lambda N)^{-1}zH^{k-1}$, which is
stronger than just $T\ggg zH^{k-1}$ which was expected in the proof to
lead to nontrivial results.
A short calculation shows that
these lower bounds \eqref{eq:t1} and \eqref{eq:t2}
for $T$ are admissible with the constraint $T\leq
(zA\lambda)^{-1}$ provided that $z^{2}\mu \ll N$ and $z^{2}\mu A\ll
N^{k}$.
We conclude that then, for small $z$, there exists a range for $T$ where
a nontrivial bound is achieved.
The lower bounds \eqref{eq:t1} and \eqref{eq:t2} for $T$ are quite
restrictive, but realize
the advantage of Theorem \ref{Satz1} compared to Theorem \ref{Satz2}.
Using Theorem \ref{Satz2} in the proof instead will lead to the
following slightly weaker bound \eqref{eq:Thb2} since $\tau<\rho$,
but provides a larger range for $T$.
\begin{theorem}
\label{th:Thb2}
Let $N$ be a large positive integer and let
$f\in\mathbb{C}^{k}((0,3N))$, $k\geq 3$.
Suppose that there exists real numbers
$\lambda,A$ such that $0<\lambda\leq f^{(k)}(x)\leq A\lambda$ holds for
all $x\in (0,3N)$. Let $\tau=1/(k-1)(k-2)$ and
$\mu=1+A\lambda N$.
Let $z$ be a positive integer that is considered to be small
and let $T$ be a positive real number with
$N^{-k}(zA\lambda)^{-1}\leq T\leq (zA\lambda)^{-1}$.
Then
\begin{multline}
\label{eq:Thb2}
\sum_{a\leq T} \Big|\sum_{N<m<2N} \mathrm{e}(azf(m))\Big|
\ll NT(zA\lambda T)^{\tau/k+\varepsilon} +T(zA\lambda T)^{-1/k} \\
+A\mu T (zA\lambda T)^{-2/k}.
\end{multline}
\end{theorem}
\begin{proof}
We proceed as before in Theorem \ref{th:Thb}, but choose now the
major arc set to be
\[
\mathfrak{M}=\bigcup_{q\leq x} \bigcup_{(u,q)=1}
\Big[\frac{u}{q}-\frac{1}{qTzx^{k-1}},
\frac{u}{q}+\frac{1}{qTzx^{k-1}}\Big].
\]
In the minor arc case,
we treat $m$ with $x\leq q\leq zx^{k-1}T$
and we are in the situation
to use Theorem \ref{Satz2} instead, leading to the slightly weaker
estimate
\[
\mathcal{T}_{x,\text{($\mathfrak{m}$)}} \ll NTx^{1-\tau+\varepsilon},
\]
since $\tau<\rho$, where $\tau=1/(k-1)(k-2)$.
Like this, we estimate the major arc contribution in a better way, namely
\begin{multline*}
\mathcal{T}_{x,\text{($\mathfrak{M}$)}}:= \sideset{}{^{(\mathfrak{M})}}\sum_{N-H<m<2N}
\ \sum_{a\leq T} |\tilde{S}_{a,m}(\alpha, x)|\\ \ll Tx\sum_{q\leq x}
\sum_{(u,q)=1} \#\{m\in(N-H,2N);\ \\
\hspace{5cm}\|f^{(k-1)}(m)/(k-1)!-u/q\|<1/qTzx^{k-1}\} \\
\ll Tx \sum_{q\leq x} \varphi(q) (1+A\lambda N)(1+1/\lambda qTzx^{k-1}) \\
\ll Tx (\mu x^{2} + \mu x /\lambda Tzx^{k-1})
\ll \mu Tx^{3}+\mu (\lambda z)^{-1}x^{3-k},
\end{multline*}
with $\mu=1+A\lambda N$, again by using Lemma \ref{lem:smf}.
We similarly arrive at
\[
\mathcal{L}_{f}\ll NTH^{-\tau+\varepsilon} + TH
+ \mu TH^{2} + \mu(\lambda z)^{-1}H^{2-k},
\]
if we choose $H=[(zA\lambda T)^{-1/k}]$ again.
Since
\[
\max\{1, (T\lambda z)^{-1}H^{-k}\} = \max\{1, A\}=A,
\]
the last two terms are $\ll A\mu TH^{2}$.
Again noting that $1\leq H\leq N$ provides the assertion
of Theorem \ref{th:Thb2}.
\end{proof}
\textbf{Remark.} Again, we give the range for $T$ where
Theorem \ref{th:Thb2} provides a nontrivial bound for $\mathcal{L}_{f}$.
We need to inspect the
third term in this bound, it is $\lll TN$
provided that $A\mu TH^{2}\lll TN$ which means
\begin{equation}
\label{eq:tt1}
T\ggg \mu^{k/2}A^{k/2-1}(z\lambda)^{-1}N^{-k/2}.
\end{equation}
A short calculations shows that this lower bound \eqref{eq:tt1}
for $T$ is admissible with the constraint $T\leq (zA\lambda)^{-1}$
provided that $\mu A\ll N$.
Compared to \eqref{eq:t1} and \eqref{eq:t2}, the range for $T$
due to \eqref{eq:tt1} will be much bigger in most cases.
\medskip
We compare our theorems with the direct application of the
following recent result of Heath-Brown in \cite[Thm.~1]{HB-V}.
\begin{theorem}[Heath-Brown]
\label{th:HB-V}
Let $k\geq 3$, let $f:[0,N]\to\mathbb{R}$ denote a function in
$C^{k}((0,N))$, and suppose that $0<\lambda\leq
f^{(k)}(x)\leq A\lambda$ for all $x\in(0,N)$. Then
\[
\sum_{n\leq N} \mathrm{e}(f(n)) \ll_{A,k,\varepsilon} N^{1+\varepsilon} (\lambda^{1/k(k-1)} +
N^{-1/k(k-1)} + N^{-2/k(k-1)}\lambda^{-2/k^{2}(k-1)}).
\]
\end{theorem}
In principle,
$\mathcal{L}_{f}$ can be estimated by using Theorem~\ref{th:HB-V},
but one needs then the dependence of the implicit constant
on $A$ explicitly since the term
$az$ occurs in the argument of the complex exponential
function, so that $A$ in Theorem~\ref{th:HB-V}
contains this factor $az\leq zT$.
Writing down the dependence on $A$ from the proof
in \cite[Thm.~1]{HB-V} explicitly,
we will have a factor $A^{4}$ occurring in the quantity $\mathcal{N}$ there.
The resulting bound for $\mathcal{L}_{f}$ will then
contain the factor $A^{4/2s}=A^{4/k(k-1)}$. Thus the
main term from this method will provide the extra factor
$T^{4/k(k-1)}$ which is much larger than the factors $T^{\rho/k}$ or
$T^{\tau/k}$ from our Theorems here.
So compared to this, Theorems \ref{th:Thb} and \ref{th:Thb2} give
sharper estimates for long Weyl sum averages. When no or short
averages are considered, Heath-Brown's bound is sharper.
Note that the potential improvements depend also on the type
of functions considered. For example, if $f(n)=t\log n$ and $k$ is
large with $t=N^{k/2}$, then the minor arc contribution from
Theorem~\ref{th:Thb} is around $tN^{1-1/k(k-1)(k-2)}$,
whereas Heath-Brown's bound is around $tN^{1-1/k^{2}}$. It may be that
Theorem~\ref{th:Thb}, as it stands,
can \emph{not} be improved if uniformity for \emph{all} functions is
desired, but further improvements
may be feasible if a suitable type of function is assumed.
A careful combination of the methods from
Theorem~\ref{Satz1}
and Theorem~\ref{th:Thb} together with the idea from the proof of
Theorem~\ref{Satz3} could lead to further improvements, which is
not discussed in this article.
\section{First application: Integer points close to smooth curves}
\label{sec:appl1}
In this application, we will use Theorem \ref{th:Thb2}.
We introduce the following quantity.
\begin{definition}\label{defR}
Let $N$ be a large positive integer, let
$f\in\mathbb{C}^{k}((0,3N))$, $k\geq 3$ and $0<\delta<1/4$.
Define
\[
\mathcal{R}(f,N,\delta):=\#\{n\in[N,2N]\cap\mathbb{Z};\ \|f(n)\|<\delta \}.
\]
\end{definition}
Like this, we count lattice points in $\mathbb{Z}^{2}$ close to the
graph of $f$.
Bordell\`es gives in \cite[Ch.~5]{Bor} an overview of
several known nontrivial bounds for this quantity and their
applications.
In Lemma \ref{lem:smf} we gave already a bound for $\mathcal{R}(f,N,\delta)$
in the case $k=1$, it is also known as the first derivative test
for $\mathcal{R}(f,N,\delta)$.
In what follows, we use a property of the set which is
counted by $\mathcal{R}(f,N,\delta)$. It is proved in \cite[Thm.~5.11]{Bor}.
\begin{lemma}
\label{lem:space}
Suppose that there exists real numbers
$\lambda, A>0$
such that $\lambda\leq f^{(k)}(x)\leq A\lambda\leq 1/4$ holds for
all $x\in(0,3N)$.
In the set $\{n\in[N,2N]\cap\mathbb{Z};\ \|f(n)\|<\delta \}$ there exists a
$H'$-spaced subset $\mathcal{R}$ such that $\mathcal{R}(f,N,\delta)\leq (k+1)(1+\#\mathcal{R})$,
where $H'=(A\lambda)^{-2/k(k+1)}$.
\end{lemma}
A set is called $H'$-spaced
if any two elements differ by more than $H'>0$.
Theorem \ref{th:Thb2} of Section \ref{sec:mvexposums}
allows us to prove a strong bound for $\mathcal{R}(f,N,\delta)$
which is the following.
\begin{theorem}
\label{th:thR}
Suppose that there exists real numbers
$\lambda, A>0$
such that $\lambda\leq f^{(k)}(x)\leq A\lambda\leq c_{0}$ holds for
all $x\in(0,3N)$ and some small constant $c_{0}<1/4$,
and assume $(A\lambda)^{-2/k(k+1)}\leq N$.
Let $\lambda_{1}>0$ be such that $|f'(x)|\leq \lambda_{1}$ holds for all
$x\in(0,3N)$.
Assume that $\delta>0$ is such that
\begin{equation}
\label{eq:condthR}
A\lambda\ll \delta+ \lambda_{1}\ll (A\lambda)^{1-2(k-1)/k(k+1)},
\end{equation}
then we have the bound
\begin{equation}
\label{eq:Rb}
\mathcal{R}(f,N,\delta)\ll
1+ A(1+A\lambda N)((\delta+\lambda_{1})/A\lambda)^{1/(k-1)}.
\end{equation}
\end{theorem}
\begin{proof}
We begin the proof as indicated in \cite[Ex.\ 6.7.4]{Bor}.
Let $m\in\mathbb{Z}$ with $N\leq m\leq 2N$ and $\|f(m)\|<\delta$,
then
\begin{multline}
\label{eq:Tqu}
\Big|\sum_{a=0}^{T-1}\mathrm{e}(af(m))\Big|^{2}=\sum_{a_{1},a_{2}}\mathrm{e}((a_{1}-a_{2})f(m))
\\ = \sum_{a_{1},a_{2}}\Re(\mathrm{e}((a_{1}-a_{2})f(m))) \geq T^{2}/2,
\end{multline}
since we have $\Re(\mathrm{e}(af(m)))\geq \sqrt{2}/2$ for all $a\in\mathbb{Z}$ with $|a|<T$,
provided that $1\leq T\leq [1/8\delta]+1$.
From Lemma \ref{lem:space} we know that there is a $H'$-spaced subset $\mathcal{R}$ of
$\{m;\ N\leq m\leq 2N, \|f(m)\|<\delta\}$ with $\mathcal{R}(f,N,\delta)\leq
(k+1)(1+\#\mathcal{R})$, where $H'= (A\lambda)^{-2/k(k+1)}$.
Hence
\[
\mathcal{R}(f,N,\delta)\ll 1+\sum_{m\in \mathcal{R}} 1\leq 1+
\frac{2}{T^{2}} \sum_{m\in\mathcal{R}} \Big|\sum_{a=0}^{T-1}\mathrm{e}(af(m))\Big|^{2}.
\]
Now opening the square and separating the summand for $a=0$ shows
\begin{multline}
\label{eq:amdef}
\Big|\sum_{a=0}^{T-1}\mathrm{e}(af(m))\Big|^{2} = \sum_{|a|\leq T-1}
(T-|a|)\mathrm{e}(af(m))
\\ = T + \sum_{0<|a|\leq T-1}(T-|a|)\mathrm{e}(af(m))=:T+a_{m}
\end{multline}
say. Clearly $a_{m}>0$ for large $T$ and for $m\in\mathcal{R}$ due to \eqref{eq:Tqu}.
We proceed with
\begin{equation}
\label{eq:Rexposum}
\mathcal{R}(f,N,\delta)\ll 1+ \mathcal{R}(f,N,\delta)T^{-1} +
T^{-2}\sum_{m\in\mathcal{R}}a_{m} \ll 1+T^{-2}\sum_{m\in\mathcal{R}}a_{m},
\end{equation}
by an application of Lemma \ref{lem:sri} if we take $T\gg 1$ (in fact
$T\geq 4(k+1)$ suffices).
Let $m\in\mathcal{R}$,
then there exists an integer $n$ with $|f(m)-n|<\delta$. Thus,
by an application of the mean value theorem,
$|f(m-h)-f(m)|=|f'(t)|h\leq \lambda_{1}H$
for some $t\in (-h,0)$.
We conclude $|f(m-h)-n|\leq|f(m-h)-f(m)|+|f(m)-n|\leq\lambda_{1}H+\delta$, so
$\|f(m-h)\|\leq \lambda_{1}H+\delta$ for all $h\leq H$.
This argument shows that for $m\in\mathcal{R}$
we have $a_{m-h}\geq T^{2}/2-T\gg T^{2}$ by \eqref{eq:Tqu}
assuming
\begin{equation}
\label{eq:Tbound}
T\leq \frac{1}{8(\lambda_{1}H+\delta)}.
\end{equation}
Since $T^{2}\geq a_{m}$ we conclude that
\[
a_{m}\ll \frac{1}{H}\sum_{h\leq H}a_{m-h}.
\]
This implies
\[
\sum_{m\in\mathcal{R}}a_{m}\ll \sum_{m\in\mathcal{R}}\frac{1}{H}\sum_{h\leq
H}a_{m-h}.
\]
Now we have
\[\sum_{m\in\mathcal{R}}\frac{1}{H}\sum_{h\leq
H}a_{m-h}
= \frac{1}{H}\sum_{\substack{n\in\mathbb{Z}\\ \exists h\leq H:
n+h\in\mathcal{R}}} a_{n},
\]
since for each $m\in\mathcal{R}$, the summands $a_{m-1},\dots,a_{m-h}$ occur
in the sum on the right hand side. Assuming $H\leq H'$, the elements
of $\mathcal{R}$ are $H$-spaced, so each such summand occurs exactly once on
both sides of this equality.
This improves upon the Weyl step in the proofs above, namely
\begin{multline*}
\sum_{m\in\mathcal{R}} a_{m} \ll
H^{-1} \sum_{\substack{n\in\mathbb{Z}\\\exists h'\leq H: n+h'\in\mathcal{R}}}
a_{n}
= H^{-2}\sum_{h\leq H} \sum_{\substack{n\in\mathbb{Z}\\\exists h'\leq H:
n+h+h'\in\mathcal{R}}} a_{n+h} \\
= H^{-2} \sum_{\substack{n\in\mathbb{Z}\\\exists h_{0}\in[1,2H]:
n+h_{0}\in\mathcal{R}}}\ \sum_{\substack{h\leq H \\
h\in[h_{0}-H,h_{0}-1] }} a_{n+h}\\
= H^{-2} \sum_{\substack{n\in\mathbb{Z}\cap[N-2H,2N]\\\exists h_{0}\in[1,2H]:
n+h_{0}\in\mathcal{R}}}\ \sum_{\substack{h\leq H \\
h\in[h_{0}-H,h_{0}-1] }} a_{n+h}.
\end{multline*}
Note that the number of $n\in\mathbb{Z}$ in this sum is $\ll
\min\{H\#\mathcal{R},N\}=:R$, which gives an improvement for a sparse set
$\mathcal{R}$ since then $H\#\mathcal{R}\ll N$.
Further, $h$ lies in the intersection $\mathcal{H}:=[1,H]\cap[h_{0}-H, h_{0}-1]$
which is an interval
of length at most $H$. The extra factor
$H^{-1}$ due to the improved Weyl step will be an advantage in the
minor arc analysis.
With above definition \eqref{eq:amdef} of $a_{m}$,
we derive
\begin{equation}
\label{eq:eqsuma}
\sum_{m\in\mathcal{R}} a_{m}\ll TH^{-2} \sum_{a\leq T} \sum_{\substack{m\in[N-2H,2N]\\
\exists h_{0}\leq 2H:m+h_{0}\in\mathcal{R}}}
\Big|\sum_{h\in\mathcal{H}} \mathrm{e}(af(m+h))\Big|.
\end{equation}
The right hand side in \eqref{eq:eqsuma} can now
be handled like $\mathcal{L}_{f}$ in the proofs of the Theorems
\ref{th:Thb} and \ref{th:Thb2} above.
For this, let us denote in analogy to $\mathcal{L}_{f}$,
\[
\mathcal{L}_{f}':= H^{-1} \sum_{a\leq T} \sum_{\substack{m\in[N-2H,2N]\\
\exists h_{0}\leq 2H:m+h_{0}\in\mathcal{R}}} \Big|\sum_{h\in\mathcal{H}} \mathrm{e}(af(m+h))\Big|.
\]
Thus
\begin{equation}
\label{eq:rr}
\mathcal{R}(f,N,\delta)\ll 1+ T^{-1}H^{-1}\mathcal{L}_{f}'
\end{equation}
by \eqref{eq:Rexposum}
and \eqref{eq:eqsuma}. Note that $\mathcal{L}_{f}$ differs from $\mathcal{L}_{f}'$
only by the summation over $m$ and that $h$ runs through an interval
$\mathcal{H}$ of length
at most $H$, which boundary points depend on $m$.
The estimation of $\mathcal{L}_{f}'$ follows now that of $\mathcal{L}_{f}$.
The only small change in above proof of Theorem
\ref{th:Thb2} lies in the minor arc estimate
\eqref{eq:minor}, where we are able to replace $N$ by $R$ so that
\[
\mathcal{T}'_{x,(\mathfrak{m})} \ll RTx^{1-\tau+\varepsilon}.
\]
To verify this, note that
\[
\mathcal{T}_{x}':=\sum_{\substack{N-2H<m<2N\\ \exists h_{0}\leq 2H:m+h_{0}\in\mathcal{R}}} \sum_{a\leq T}
|\tilde{S}_{a,m}'(\alpha,x)|
\]
involves a restriction of $h$ in the sum
\[
\tilde{S}_{a,m}'(\alpha,x)=\sum_{h\in\mathcal{H},h\leq x} \mathrm{e}(a\tilde{f}_{m,\alpha}(h))
\]
to $h\in\mathcal{H}$.
Still Theorem \ref{Satz2} can be applied since the condition $h\in \mathcal{H}$
restricts the summation down to a set which is an interval.
Since this interval has an upper bound of at most $x$,
this provides the stated bound for $\mathcal{T}'_{x,(\mathfrak{m})}$.
Next, for appropriate $H$, choose $T=[\frac{1}{H^{k}A\lambda}]$.
Especially, if $H^{k}A\lambda -8\lambda_{1}H\geq 8\delta$
then \eqref{eq:Tbound} is true.
We obtain in the same way as in
Theorem \ref{th:Thb2} that
\[
\mathcal{L}_{f}'\ll RTH^{-\tau+\varepsilon} + A\mu TH^{2}.
\]
We put this bound inside \eqref{eq:rr} above and obtain
\begin{multline*}
\mathcal{R}(f,N,\delta) \ll 1+ T^{-1}H^{-1}(RTH^{-\tau+\varepsilon} + A\mu TH^{2})
\\ \ll
1+\mathcal{R}(f,N,\delta) H^{-\tau+\varepsilon} + A\mu H.
\end{multline*}
By an application of Lemma \ref{lem:sri} we leave out the term on
the right hand side containing $\mathcal{R}(f,N,\delta)$ assuming that $H$ is
large enough in terms of the implicit constant.
This works since $H^{-\tau+\varepsilon}$ gets
arbitrary small if $H$ increases.
We arrive at the bound
\[
\mathcal{R}(f,N,\delta)\ll 1+ A(1+A\lambda N)H.
\]
Now we collect all the assumptions made on $H$.
Due to Lemma \ref{lem:space} we need $H\leq (A\lambda)^{-2/k(k+1)}
\leq N$.
Moreover, due to \eqref{eq:Tbound} we need
$H^{k}A\lambda-8\lambda_{1}H\geq 8\delta$,
that is if
\begin{equation}
\label{eq:deltab}
8\delta \leq H(H^{k-1}A\lambda-8\lambda_{1}),
\end{equation}
for which necessarily $H^{k-1}A\lambda>8\lambda_{1}$ has to be true.
Let
$H=((8\delta+8\lambda_{1})/A\lambda)^{1/(k-1)}$,
so that \eqref{eq:deltab} holds true.
Now if
\[
((\delta+\lambda_{1})/A\lambda)^{1/(k-1)} \ll (A\lambda)^{-2/k(k+1)},
\]
that is, if
\begin{equation}
\label{eq:cond}
\delta+ \lambda_{1}\ll (A\lambda)^{1-2(k-1)/k(k+1)}
\end{equation}
holds true,
we conclude that $H$ is appropriate
for the asserted bound in the theorem to be valid
for all small $\delta$ such that \eqref{eq:cond} holds.
We further need to assume that $H$ is bigger
than some constant which makes above step with Lemma \ref{lem:sri}
work, so we shall assume also $A\lambda \ll \delta+\lambda_{1}$.
This yields the assertion.
\end{proof}
\medskip
We shall compare the bound in Theorem \ref{th:thR}
with the well-known theorem of Huxley and Sargos from
\cite[Thm.~1]{HuxSar}, it states the following bound for $\mathcal{R}(f,N,\delta)$.
The given version here is explicit in $A$ and has been taken from
\cite[Thm.~5.12]{Bor} where a proof is provided. The known proofs
are geometric and do not depend on any exponential sum technique.
\begin{theorem}[Huxley and Sargos, explicit in $A$]
\label{th:HS}
Let $k\geq 3$ be an integer and $f\in C^{k}([N,2N])$ be such that there
exist $\lambda,A>0$ with $\lambda \leq |f^{(k)}(x)|\leq A\lambda$
for all $x\in[N,2N]$. Let $0<\delta<1/4$. Then
\begin{equation*}
\mathcal{R}(f,N,\delta)\ll
N(A\lambda)^{2/k(k+1)} +
N(A\delta)^{2/k(k-1)} + (\delta/\lambda)^{1/k}+1.
\end{equation*}
\end{theorem}
In Theorem \ref{th:HS}, the first term
$N(A\lambda)^{2/k(k+1)}$ dominates if
$\delta\leq\lambda^{1-2/(k+1)}$
$A^{-2/(k+1)}$
and $\delta\ll N^{k}A^{2/(k+1)}\lambda^{1+2/(k+1)}$,
especially when examining very small $\delta$.
The main term $N(A\lambda)^{2/k(k+1)}$
is commonly called the smoothness term
and being regarded as very difficult to improve,
compare also \cite[p.275]{Bor}.
A comparison of the term $NA^{2}\lambda((\delta+\lambda_{1})/A\lambda)^{1/(k-1)}$
in the bound of Theorem \ref{th:thR} with the smoothness term
shows that under the stated assumptions,
Theorem \ref{th:thR} gives a sharper bound.
Also note that if we let $\delta\to 0$
in Theorem \ref{th:thR}, we obtain the improved bound
$\mathcal{R}(f,N,0)\ll 1+ A(1+A\lambda N)(\lambda_{1}/A\lambda)^{1/(k-1)}$,
compared to Theorem \ref{th:thR} which yields
the smoothness term as upper bound for $\mathcal{R}(f,N,0)$.
The bound of Theorem \ref{th:thR} is quite satisfying,
the only obstruction lies in the restrictive assumption
\begin{equation}
\label{eq:obstr}
\lambda_{1}\ll (A\lambda)^{1-2(k-1)/k(k+1)}
\end{equation}
needed for $\lambda_{1}$ and $A\lambda$.
This shows that Theorem \ref{th:thR}
is in fact of limited use in applications since this condition
is true only for certain appropriate functions. Unfortunately,
functions like $f(x)=B/x^{r}$ or $f(x)=(B/x)^{1/r}$ for
integers $r\geq 2$, which are occurring in interesting applications,
are not of this kind when $A$ is taken as constant.
And if $A$ is so big such that \eqref{eq:obstr} holds, the bound of Theorem
\ref{th:thR} can be quite weak.
In this context, we state the following theorem of Gorny \cite{Gor}.
\begin{theorem}[Gorny]
\label{th:gor}
Let $k\geq 2$, $f\in C^{k}([1,1+3N])$. Let $M,A,\lambda\in\mathbb{R}$ such that
$|f(x)|\leq M$ and $|f^{(k)}(x)|\leq A\lambda$ holds for all $x\in [1,1+3N]$.
Then for all $x\in [1,1+3N]$,
\[
|f'(x)| \ll MN^{-1} + M^{1-1/k}(A\lambda)^{1/k},
\]
with implicit constant that depends on $k$ only.
\end{theorem}
By this theorem, we conclude that \eqref{eq:obstr} is true for all
sufficiently large $N$ provided that $M\leq (A\lambda)^{1-2/(k+1)}$.
Therefore, functions on $[1,1+3N]$ that are much smaller in absolute value
compared to the maximum of the absolute value of the
$k$-th derivative are admissible for Theorem \ref{th:thR}.
This gives a nice criterion for Theorem \ref{th:thR} to hold, but it
seems to be hard to find easy examples.
\textbf{Remark.}
Applying Theorem \ref{th:Thb} instead of Theorem \ref{th:Thb2} in
the proof of Theorem \ref{th:thR} would
also lead to a vanishing of the minor arc contribution in the bound,
but the resulting bound for $\mathcal{R}(f,N,\delta)$
would be weaker due to the bigger major arc contribution.
Instead, the presented vanishing trick may be used with even
bigger minor arc contributions probably leading to further refinements.
\section{Second application:
The polynomial large sieve inequality
(LSI) in the one-dimensional case}
\label{sec:polyLSI}
In this section we present an application of Theorem \ref{Satz1},
namely to the polynomial large sieve inequality. This generalization of the
classical large sieve inequality to sparse moduli sets, usually given as values
of some fixed polynomial, has been studied intensely in past research and has
already influenced some other areas of Number Theory, especially the
version with $k=2$ from \cite{BaiZhao}.
To name a few such topics, it has been found useful to find
variants of Bombieri--Vinogradov's theorem \cite{Bak,KHBVvar},
new results on
primes of polynomial shape \cite{FooZhao} or primes in APs to spaced
moduli \cite{Bak},
divisibility questions with Fermat quotients \cite{BFKS},
and mean value estimates for character sums with
applications \cite{BaiYou,BaiZhao2,BloGoLa, LamYou}. Furthermore, the
multidimensional polynomial LSI can be used
for sieving with high powers like seen in \cite{KH2} and reaches questions
related to the abc-conjecture.
We start by giving the setting and basic assumptions in the polynomial
LSI in the one-dimensional case.
\textbf{Setting.} Let $P\in\mathbb{R}[x]$ be a fixed monic polynomial of
degree $k\geq 2$ with $P(0)=0$.
Assume that $P$ has only positive values in $[Q,2Q]$ for each real $Q\geq 1$
and let $M_{Q}:=\max \{P(q);\ q\in [Q,2Q]\}$ be the maximal value for
integers $q\in[Q,2Q]$. Clearly $M_{Q}\ll Q^{k}$,
assume also that $P(q)\gg Q^{k}$ holds true for all
integers $q\in[Q,2Q]$ and some implicit constant that may depend only on $k$.
Let $N,M$ be integers and $(v_{n})_{n\in\mathbb{N}}$ be a sequence of complex
numbers.
In the theory of the large polynomial LSI, see
\cite{BaiZhao,KH,Zhao},
we aim to give upper bounds for the quantity
\[
\Sigma_{P}:=\sum_{q\leq Q} \sum_{\substack{1\leq a\leq
P(q)\\(a,P(q))=1}} \Big| \sum_{M<n\leq M+N}
v_{n}\mathrm{e}\Big(\frac{an}{P(q)}\Big) \Big|^{2}.
\]
When we put the current form for Weyl's inequality,
Theorem~\ref{th:weyloriginal}, in the machinery of \cite{KH,KH2},
the bound $\Sigma_{P}\ll Q^{\varepsilon} \|v\|^{2}
(Q^{k+1}+A_{k}(Q,N))$ is easily derived with
\[
A_{k}(Q,N):= NQ^{1-1/k(k-1)}+N^{1-1/k(k-1)}Q^{1+1/(k-1)}.
\]
The interesting range for $N$ is $Q^{k}\ll N\ll Q^{2k}$, since outside
that, it is already known by an application of the standard large
sieve inequality that the sharp bound
$\Sigma_{P}\ll Q^{\varepsilon} \|v\|^{2} (Q^{k+1}+N)$ holds true. So we
assume without loss of generality that $N$ lies in this range.
This result already offers an improvement compared to \cite{BaiZhao2}
in the range $Q^{k}\ll N \ll Q^{2k-2+2/k(k-1)}$ when $k\geq 3$. We
will now improve on that.
\subsection{The connection of the Polynomial LSI with Weyl sums}
\label{sub:integral}
From \cite[Lemma~1]{KH}, we know that
\begin{equation}\label{eq:Fint}
\Sigma_{P} \ll Q^{\varepsilon}\|v\|^{2}\bigg(\sum_{Q<q\leq 2Q} P(q) +
\max_{Q<r\leq 2Q} \max_{\substack{1\leq
b<P(r)\\\gcd(b,P(r))=1}}
\int_{1/N}^{1/2} \#\mathcal{F}_{b,P(r)}(x) \frac{dx}{x^{2}}\bigg).
\end{equation}
The first term in large brackets
can be estimated as $\ll QM_{Q}$ and is admissible.
It is not necessary to repeat the definition of $\#\mathcal{F}_{b,P(r)}(x)$
since we will just make use of the upper bound
\[
\#\mathcal{F}_{b,P(r)}(x) \ll B^{-1} Q + B^{-1}\sum_{1\leq a\leq B}|S_{a}|
\]
with $B^{-1}=2M_{Q}x$ and
\[
S_{a}:=\sum_{q\leq 2Q} \mathrm{e}\Big(\frac{ab}{P(r)}P(q)\Big),
\]
which has been shown in the deduction of \cite[(8)]{KH}.
To consider the integral expression in \eqref{eq:Fint},
fix a pair $b,r$ with $r\in[Q,2Q]$, $1\leq b<P(r)$ and
$\gcd(b,P(r))=1$.
We substitute $B^{-1}=2M_{Q}x$
and estimate as follows.
\begin{multline*}
\begin{aligned}
\int_{1/N}^{1/2} &\#\mathcal{F}_{b,P(r)}(x) \frac{dx}{x^{2}}\\
&\ll \int_{1/4M_{Q}}^{N/2M_{Q}} \bigg(B^{-1}Q +
B^{-1}\sum_{a\leq B} |S_{a}|\bigg) M_{Q} \,dB \\
&\ll QM_{Q}\log N + M_{Q} \int_{1/4M_{Q}}^{N/2M_{Q}}
B^{-1}\sum_{a\leq B} |S_{a}| \,dB \\
&\ll QM_{Q}\log N + M_{Q} \sum_{a\leq N/2M_{Q}} |S_{a}|
\int_{a}^{N/2M_{Q}} B^{-1} \,dB \\
&\ll QM_{Q}\log N + M_{Q} \log N \sum_{a\leq N/2M_{Q}}|S_{a}|.
\end{aligned}
\end{multline*}
Here the last sum is a discrete moment of a Weyl sum
with the polynomial $bP(x)/P(r)$ and leading term $b/P(r)$ since $P$
is monic.
We are able to apply Theorem \ref{Satz1} directly with $P(r)$ as approximating
denominator (when $k\geq 3$). By this, we have
\[
\sum_{a\leq N/2M_{Q}}|S_{a}| \ll \frac{N}{M_{Q}}Q^{1+\varepsilon}
\Big(\frac{Q^{k-1}}{P(r)} + \frac{Q^{k-1}}{N/M_{Q}} + \frac{1}{Q}
+ \frac{P(r)}{QN/M_{Q}}\Big)^{1/2s_{0}}
\]
with
\[s_{0}=(k-1)(k-2)/2+1, \text{ so that }2s_{0}=k(k-1)-2k+4.
\]
Let $\omega:=1/2s_{0}$.
In the big bracket expression, the last summand $P(r)M_{Q}/NQ$
dominates since $P(r)\gg Q^{k}$ and $1/Q\ll P(r)M_{Q}/NQ$ for $N\ll
Q^{2k}$.
So we continue with
\begin{align*}
\int_{1/N}^{1/2} &\#\mathcal{F}_{b,P(r)}(x) \frac{dx}{x^{2}}\\
&\ll QM_{Q}\log Q + N^{1-\omega}M_{Q}^{\omega}
Q^{1-\omega+\varepsilon} P(r)^{\omega} \\
&\ll QM_{Q}\log Q + N^{1-\omega} M_{Q}^{2\omega}
Q^{1-\omega+\varepsilon} \\
&\ll Q^{k+1}\log Q + N^{1-\omega} Q^{1+2k\omega-\omega+\varepsilon},
\end{align*}
where we used $M_{Q}\ll Q^{k}$ in the last step.
Compared to the dominating term $NQ^{1-1/k(k-1)}$ in the former bound
$A_{k}(Q,N)$,
we get an advantage if $N^{1-\omega}Q^{1+(2k-1)\omega}\leq
NQ^{1-1/k(k-1)}$,
which is the case if
$N\geq Q^{2k-2/(k-1)+4/k(k-1)}$, so when $N$ is close to $Q^{2k}$,
but still in the interesting range $Q^{k}\ll N\ll Q^{2k}$.
We have therefore shown the following new improved bound
for the polynomial LSI.
\begin{theorem}
\label{th:newPLSI}
In the setting of Section~\ref{sec:polyLSI} when $k\geq 3$,
\begin{equation}
\label{eq:newPLSI}
\Sigma_{P}\ll Q^{\varepsilon} \|v\|^{2} (Q^{k+1} +
\min\{A_{k}(Q,N), N^{1-\omega} Q^{1+(2k-1)\omega}\})
\end{equation}
with $\omega=1/((k-1)(k-2)+2)$.
\end{theorem}
Theorem~\ref{th:newPLSI} offers an improvement compared to known previous
results when $k\geq 4$. This is since for $k=3$, the additional bound from
\cite{BaiZhao2} is still stronger in this case.
It is interesting what we would obtain having Conjecture~\ref{conj:c}.
In this case, we would be able to
gain a factor $Q^{(1-k)\omega}$.
Then, we would arrive at the following result.
\begin{conjecture}
\label{conj1} In the setting of Section~\ref{sec:polyLSI} when $k\geq 3$,
\[
\Sigma_{P}\ll Q^{\varepsilon} \|v\|^{2} (Q^{k+1} + \min\{A_{k}(Q,N),
N^{1-\omega} Q^{1+k\omega}\})
\]
with $\omega=1/((k-1)(k-2)+2)$.
\end{conjecture}
Note that if we could take even $1/k(k-1)$ at the place of $\omega$,
the expression $N^{1-1/k(k-1)} Q^{1+1/(k-1)}$ coincides with the
second summand in $A_{k}(Q,N)$.
Still, these conjectural bounds are far from Zhao's conjecture
in \cite{Zhao} stating
\[
\Sigma_{P}\ll Q^{\varepsilon} \|v\|^{2} (Q^{k+1} + N).
\]
Conjecture \ref{conj1} might be rather within reach
of further refinements of the methods
presented in this article.
We remark that an attempt to use Theorem~\ref{Satz3} will not give
more if more on the coefficient of $q^{k-1}$ in $P(q)$, say
$\alpha_{k-1}$, is known. This is since the coefficient of $q^{k-1}$
in the polynomial $bP(q)/P(r)$ is then $\alpha_{k-1}b/P(r)$. By
Theorem~\ref{Satz3}, one needs to look then at the rational approximations
to $\alpha_{k-1}b$. Since $b$ is supposed to be \emph{any} coprime
residue mod $P(r)$, there will always be one with small denominator
which offers no advantage in Theorem~\ref{Satz3}.
\section{Acknowledgements}
The author thanks the organizers of the Workshop
on Efficient Congruencing in March 2017
at the Fields institute in Toronto during the Thematic Program
on Unlikely Intersections, Heights, and Efficient Congruencing
for an inspiring stay. The author also
thanks the referee for many useful suggestions and comments on the
manuscript which led to an improvement of some of the
material.
|
{
"timestamp": "2019-10-01T02:32:28",
"yymm": "1804",
"arxiv_id": "1804.05587",
"language": "en",
"url": "https://arxiv.org/abs/1804.05587"
}
|
\section*{Results}
\subsection*{Growth of 30\textdegree-tBLG}
The 30\textdegree-tBLG sample was grown on Pt(111) substrate by carbon segregation from the bulk substrate \cite{SutterPRB, YW2015prb}. Figure \ref{fig2a} shows the low energy electron diffraction (LEED) of bottom monolayer graphene at 30\textdegree~azimuthal orientation (blue arrow) relative to the substrate. Our previous work shows that distinguished from other graphene/Pt with commensurate (e.g. 2$\times$2, 3$\times$3) Moir\'{e} superstructures \cite{SutterPRB}, the interface between the 30\textdegree~monolayer graphene and the Pt(111) substrate is incommensurate without forming any Moir\'{e} pattern, leading to nearly free-standing monolayer graphene \cite{YW2015prb}. Further increasing the annealing temperature and time can lead to thicker graphene sample with a new set of diffraction peaks emerging at 0\textdegree~orientation (red arrow in Fig.~\ref{fig2b}) coexisting with those at 30\textdegree, and such graphene sample is the focus of current work. The lattice constants extracted from LEED show that there is negligible strain (<0.2$\%$, see Fig.~S1 in SI Appendix for details) between the 30\textdegree~and 0\textdegree~graphene layers, and the absence of additional reconstructed diffraction spots in the LEED pattern further supports that they do not form commensurate superlattice. If these diffraction peaks come from the same graphene domains, this would imply that 0\textdegree~graphene is stacked on top of the bottom 30\textdegree~graphene layer, namely, a 30\textdegree-tBLG is formed. In the following, we will provide direct experimental evidence for the conjectured incommensurate 30\textdegree-tBLG from Raman spectroscopy and reveal its electronic structure from ARPES measurements.
\subsection*{Intervalley double-resonance Raman mode}
Raman spectroscopy is a powerful tool for characterizing the vibrational mode in graphene \cite{FerrariRaman} and can provide direct information about the sample thickness and stacking. Figure \ref{fig2g} shows a typical optical image of the as-grown sample. The optical image shows strong intensity contrast, with a darker region of $\approx$ 10 $\mu$m overlapping with the brighter region. The Raman spectra in Fig.~\ref{fig2c} show stronger intensity for the darker region (red curve) than the brighter region (blue curve), suggesting that the darker region is thicker. The spectra for both regions show characteristic features of monolayer graphene \cite{FerrariRaman}, in which the 2D mode shows a single Lorentzian peak with stronger intensity than the G mode. This suggests that the top and bottom flakes are both monolayer graphene, and the thicker region is not a Bernal (AB stacking) bilayer graphene, but a bilayer graphene with a large twisting angle instead \cite{twistRaman}. More importantly, when zooming in the spectra between 1300 to 1430 cm$^{-1}$, the bilayer region shows two peaks centered at 1353 cm$^{-1}$ and 1383 cm$^{-1}$ respectively (Fig.~\ref{fig2d}). The peak at 1353 cm$^{-1}$ is the D mode of graphene which is caused by the limited size or defects \cite{FerrariRaman}. The Raman mapping for this peak (Fig.~\ref{fig2h}) shows that it only appears at the edges of the bilayer region or defect centers, consistent with the nature of the D peak. In contrast, the peak at 1383 cm$^{-1}$ (``R'' peak) is observed in the whole bilayer graphene region (Fig.~\ref{fig2i}), indicating that it is intrinsic to the twisted bilayer graphene.
Since the energy of the R mode is close to that of the D mode, the R mode likely has similar origin as the D mode: intervalley double-resonance (DR) Raman process \cite{DoubleResonance,CarozoRaman}. The key process for the intervalley DR Raman process involves a special scattering process (by defect, phonon or Moir\'{e} pattern etc.) in which photoexcited electrons at one valley are scattered to another valley by a non-zero momentum transfer $\bm{q}$ and subsequently decay back to the original valley by emitting a phonon with wave vector $\bm{Q_{\rm phonon}}$=$\bm{q}$ (see Fig.~\ref{fig2f}). While D mode is induced by defect scattering between two neighboring Dirac cones, instead, we claim that the R mode in our case is caused by scattering between two opposite Dirac cones which are connected by one reciprocal lattice vector of the bottom graphene layer with $\bm{q}$=$\bm{G_{\rm bottom}}$ (see Fig.~\ref{fig2e} and \ref{fig2f}). This is verified by observing a phonon mode with matching momentum and energy in the transverse optical (TO) phonon spectrum along $\Gamma$-K-M-K$^\prime$-$\Gamma$ direction (see Fig.~\ref{fig2j}). The phonon momentum of R mode is not a Moir\'{e} reciprocal vector of any commensurate superlattice, suggesting that the bilayer region is an incommensurate 30\textdegree-tBLG. It also reveals an unreported scattering process involving the reciprocal lattice vector of the bottom layer in this interesting 30\textdegree-tBLG region. This novel scattering process for the observed R mode is closely related to the interlayer interaction to be discussed below and is supported by the exotic electronic structure of the 30\textdegree-tBLG which we will investigate next.
\begin{figure*}
\subfloat{\label{fig2a}}
\subfloat{\label{fig2b}}
\subfloat{\label{fig2c}}
\subfloat{\label{fig2d}}
\subfloat{\label{fig2e}}
\subfloat{\label{fig2f}}
\subfloat{\label{fig2g}}
\subfloat{\label{fig2h}}
\subfloat{\label{fig2i}}
\subfloat{\label{fig2j}}
\centering
\includegraphics[width=17.4 cm] {fig2.png}
\caption{\textbf{Observation of double-resonance Raman mode in 30\textdegree-tBLG.} \textbf{(A)}, The LEED pattern for the graphene sample after annealing at 1200 \textcelsius. \textbf{(B)}, The LEED pattern for the graphene sample at higher annealing temperature of 1600 \textcelsius. \textbf{(C)}, Measured Raman spectrum for monolayer and bilayer region respectively, covering the range of G peak and the 2D peak. \textbf{(D)}, Zoom in of (C) for range from 1300 to 1430 cm$^{-1}$. \textbf{(E)}, The geometry for electron momentum transferring in DR Raman process of the observed R mode. \textbf{(F)}, The schematic drawing for the DR process of R mode in graphene band structure (not in scale). Electron at one Dirac cone from the top layer is photo-excited to the conduction band, scattered by one reciprocal lattice vector of the bottom layer $\bm{q}=\bm{G}_{\rm bottom}$, and subsequently scattered back by phonon with vector $\bm{Q}_{\rm phonon}$. \textbf{(G)}, The optical image for the measured area. The red and blue dots indicate the measuring positions of spectra in (C) and (D). \textbf{(H)}, The Raman map for the D peak, integrating intensity from 1343 to 1363 cm$^{-1}$. \textbf{(I)}, The Raman map for the R peak, integrating intensity from 1373 to 1393 cm$^{-1}$. \textbf{(J)}, The phonon spectrum of the transversal optical (TO) mode along high symmetric direction, taken from Ref.~\cite{CarozoRaman}. The arrow indicates the momentum of transferring R mode phonon. The inset shows the high-symmetry path in $k$-space for the phonon spectrum. The left hexagon is the first Brillouin zone.}
\end{figure*}
\begin{figure}
\subfloat{\label{fig3a}}
\subfloat{\label{fig3b}}
\subfloat{\label{fig3c}}
\subfloat{\label{fig3d}}
\subfloat{\label{fig3e}}
\centering
\includegraphics[width=7.3 cm] {fig3.png}
\caption{\textbf{Observation of mirrored Dirac cone in 30\textdegree-tBLG.} \textbf{(A)}, Intensity map at -0.8 eV. The BZs of 0\textdegree-graphene and 30\textdegree-graphene are indicated by the dashed lines. \textbf{(B)}, The same map as (A) but with logarithmic scale. Green arrows indicate the traces of replica Dirac cones. \textbf{(C)}, Schematic drawing of the Brillouin zone (left) and the 2D curvature image of constant energy maps for the region marked by rectangle at energies from E$_F$ to -0.8 eV (right). The BZ boundary of 0\textdegree, 30\textdegree~and replica bands are shown as red, blue and green lines. \textbf{(D)}, The measured Dirac-type dispersion near the K point of the 0\textdegree~orientation. The 3 vertical arrows indicate the K point of 0\textdegree-graphene, the M point of 30\textdegree-graphene and the K point of the replica BZ. The gap is indicated by the horizontal arrows. \textbf{(E)}, Schematic illustration showing the mirrored relation between the original and the emerging replica Dirac cone.}
\end{figure}
\subsection*{Observation of mirrored Dirac cone}
The electronic structure of the sample was first measured by conventional ARPES with a beam size of $\sim$ 100 $\mu$m. Figure~\ref{fig3a} shows the ARPES intensity map at -0.8 eV. Characteristic conical contours appear at the K points of the graphene Brillouin zone (BZ) for both 0\textdegree~and 30\textdegree~layers (labeled as K$_{0^\circ}$ and K$_{30^\circ}$), in agreement with the coexistence of 0\textdegree- and 30\textdegree-graphene layers observed in the LEED pattern. Additional contours with weaker intensity are detected inside the first BZ near K$_{0^\circ}$ (green arrows in Fig.~\ref{fig3b}) when enhancing the weak features using logarithmic scale. The curvature images (Fig.~\ref{fig3c}) at different energies with higher visibility show that these contours expand from low to high binding energy similar to the Dirac cones at K$_{0^\circ}$, and they occur at the reflected position of K$_{0^\circ}$ (indicated by green arrow and labeled as K$_{\rm R}$) with respect to the 30\textdegree~BZ edges (blue lines). The dispersion near the K$_{\rm R}$ also shows linear behavior, which is the reflected image of the Dirac cone at K$_{0^\circ}$ (Fig.~\ref{fig3d}). The mirrored Dirac cone shows similar intensity asymmetry caused by the dipole matrix element \cite{Shirley}, with stronger intensity between K$_{0^\circ}$ (K$_{\rm R}$) and the M point of the 30\textdegree~graphene layer M$_{30^\circ}$, suggesting that it comes from the original Dirac cone at the opposite momentum valley and is strongly related to the transfer mechanism discussed above for the double-resonance Raman mode (more details are shown in the Discussion section). Moreover, the suppression of intensity at the crossing point M$_{30^\circ}$ between the original and mirrored Dirac cones indicates a gap opening (pointed by black arrows in Fig.~\ref{fig3d}). Such mirrored Dirac cones and gap opening are not observed in the bottom monolayer 30\textdegree-graphene (see SI Appendix for details). Furthermore, the gap happens to appear at BZ boundary of bottom layer and is away from the substrate bands (see SI Appendix for details), suggesting that they are not caused by the graphene-substrate interaction, but intrinsic properties of the 30\textdegree-tBLG. Therefore our ARPES results confirm that the 30\textdegree~graphene layer spatially overlap with 0\textdegree~graphene layer and they interact with each other.
\begin{figure*}
\subfloat{\label{fig4a}}
\subfloat{\label{fig4b}}
\subfloat{\label{fig4c}}
\subfloat{\label{fig4d}}
\centering
\includegraphics[width=13 cm] {fig4.png}
\caption{\textbf{Confirmation of 30\textdegree-tBLG and the electronic structure by NanoARPES measurements.} \textbf{(A)}, Spatial intensity map obtained by integrating the energy from -1.5 eV to E$_F$. The structures for regions B and C are schematically shown on the right. \textbf{(B), (C)}, Dispersion cuts taken at regions marked by \textbf{B} and \textbf{C} in (A). \textbf{(D)}, The energy distribution curves (EDCs) taken in the momentum range marked by the arrows in (C). The EDC through the cross point of the original and mirrored bands is shown by the red color. Blue and green circles mark the peak positions.}
\end{figure*}
\subsection*{The band structures with spatial resolution}
The electronic structure of 30\textdegree-tBLG is further confirmed by NanoARPES \cite{NanoARPES}, which is capable of resolving the electronic band structure of coexisting graphene structures with spatial resolution at $\approx$ 120 nanometer scale. A spatially resolved intensity map is shown in Fig.~\ref{fig4a}, where we can distinguish different regions and map out their distinct electronic structure. Region labeled by \textbf{b} shows dispersion along the $\Gamma$-M direction of the 30\textdegree~bottom graphene layer (Fig.~\ref{fig4b}), while region \textbf{c} shows dispersions from the 30\textdegree-tBLG, with dispersion along the $\Gamma$-K direction of 0\textdegree~top graphene layer coexisting with that along the $\Gamma$-M direction of the bottom 30\textdegree~graphene layer (Fig.~\ref{fig4c}). Therefore, dispersions measured by NanoARPES with spatial resolution provides definitive experimental evidence for the 30\textdegree-tBLG, and the mirrored Dirac cones indeed originate from this newly discovered tBLG structure. Moreover, by focusing only in the 30\textdegree-tBLG region, sharper dispersions can be obtained and the gap at the M point becomes more obvious. Analysis from the energy distribution curves (EDCs) for 30\textdegree-tBLG (Fig.~\ref{fig4d}) shows a gap at the M$_{30^\circ}$ point, suggesting the Dirac cone at K$_{0^\circ}$ and the mirrored Dirac cone are hybridized.
\begin{figure*}
\subfloat{\label{fig5a}}
\subfloat{\label{fig5b}}
\subfloat{\label{fig5c}}
\subfloat{\label{fig5d}}
\subfloat{\label{fig5e}}
\centering
\includegraphics[width=15.5 cm] {fig5.png}
\caption{\textbf{Evolution of gap size and gap position in \textit{k}-space.} \textbf{(A)}, 5 different cuts locating at different positions in the $k$-space shown in (B). The EDCs crossing the gap are shown on the right of the spectrum as red symbols. The fitting curves are plotted as blue lines and green curves are the two Lorentzian peaks from fitting results. Black markers show the positions of the two peaks. \textbf{(B)}, Schematic BZ showing the positions of the 5 cuts in (A). \textbf{(C)}, The gap size along the $k_y$ direction. \textbf{(D)}, The gap position along the $k_y$ direction. The green line is the fitting curve by the tight-binding model. \textbf{(E)}, Cartoon diagram for the two Dirac cones and their intersecting line with a gap. The color indicates the gap size.}
\label{fig4}
\end{figure*}
\subsection*{Band gap in \textit{k}-space}
To track the evolution of the gap at the crossing points between the original Dirac cone at K$_{0^\circ}$ and the mirrored Dirac cone at K$_{\rm R}$, we show in Fig.~\ref{fig5a} dispersions measured from cut 1 to 5 (labeled in Fig.~\ref{fig5b}). The gap size is quantified by the peak separation of the EDCs at the crossing points and the extracted value is plotted in Fig.~\ref{fig5c}. It is clear that the gap size decreases from the maximum value of $\sim$ 280 meV at the M$_{30^\circ}$ point to $\sim$ 200 meV when deviating from the M$_{30^\circ}$ point. In addition, we can define the gap position as the average of the two peak positions. The gap position curve in Fig.~\ref{fig5d} shows that the gap is located at the intersecting line of the two Dirac cones, hence it is possible to fit the gap positions by the energy dispersion obtained from a tight-binding model (see SI Appendix for details). The fitting curve (green curve in Fig.~\ref{fig5d}) is in good agreement with the experimental data, giving a nearest-neighbor hopping parameter $\gamma_0$ to be 3.10 eV which is closed to the value derived from theoretical calculations \cite{Wallace1947}. The Fermi velocity $\nu_F=\sqrt{3}a\gamma_0/2$ ($a\sim$ 2.46 \AA) is extracted to be 1.003$\pm$0.002$\times10^6$ m/s. A schematic summary of the gap between the original and mirrored Dirac cones is shown in Fig.~\ref{fig5e}.
\begin{figure}
\subfloat{\label{fig6a}}
\subfloat{\label{fig6b}}
\subfloat{\label{fig6c}}
\centering
\includegraphics[width=8.5 cm] {fig6.png}
\caption{\textbf{Electronic structure of 30\textdegree-tBLG from first-principles calculations.} \textbf{(A)}, Band structure of a $(5\times5)/(3\sqrt{3}\times3\sqrt{3})$R30\textdegree~tBLG in the extended zone scheme. The energy gap of $\pi$-band at M point of the other layer is indicated. \textbf{(B)}, Real-space projection of the Bloch state at $\Gamma$ point near Fermi level, showing a clear bonding state between $p_z$-orbital of C (brown balls) and $d$-orbital of Pt (white balls). \textbf{(C)}, Schematic illustration for the band structure of 30\textdegree-tBLG, showing the emergence of mirrored Dirac cones.}
\label{fig6}
\end{figure}
\subsection*{Theoretical calculation}
\textit{Ab initio} calculations are performed to reveal the electronic structure of 30\textdegree-tBLG and investigate the stability. Since Bloch theorem does not apply to the non-periodic structure and an infinitely large cell is beyond the calculation capability, a finite supercell of $(5\times5)/(3\sqrt{3}\times3\sqrt{3})$R30\textdegree~ is constructed to mimic the non-periodic 30\textdegree-tBLG system. The calculated electronic structure in Fig.~\ref{fig6a} shows band gap opening at the M point of the counterpart layer, with gap value close to the experimental measurements. An interesting observation regards the stability of 30\textdegree-tBLG when the Pt substrate is included in the calculation. While the formation energy of free-standing 30\textdegree-tBLG is 1.6 meV/(C atom) higher than the more common AB-stacking BLG, inclusion of the substrate leads to formation energy of 4.1 meV/(C atom) lower than AB-stacking one, which explains why 30\textdegree-tBLG emerges in our sample and suggests an important role of the Pt substrate in stabilizing the 30\textdegree-tBLG. By investigating real-space projection of the Bloch states, a significant coupling between the carbon p orbital and platinum d orbital (shown in Fig.~\ref{fig6b}) is revealed near Fermi level. Since such p-d coupling occurs at the $\Gamma$ point and phase matching between graphene and Pt substrate is less critical (due to zero $k$-vector). Therefore it could thus exist in systems with weak and incoherent sample-substrate interaction such as our sample, contributing positively to the stability of grown 30\textdegree-tBLG on Pt substrate. Extrapolating to larger supercell, like 5\texttimes5, 7\texttimes7 and 16\texttimes16, the stability of 30\textdegree-tBLG is further enhanced (See SI Appendix for details) and the electronic structure remains similar, suggesting that the calculation conclusion mentioned above still holds at the limit of infinitely large cell or the incommensurate structure. Our calculation, although not perfect, still provides some theoretical insights into the physics and stability of the 30\textdegree-tBLG.
\section*{Discussion}
Unlike the satellite Dirac cones appearing in commensurate graphene system with long-ranged Moir\'{e} pattern \cite{PletikosicPRL,Eli2011PRB,WEY2016A}, the emergence of mirrored Dirac cones in incommensurate 30\textdegree-tBLG indicates an unusual scattering mechanism in this novel structure. Based on a generalized Umklapp scattering process \cite{KoshinoNJP}, we build up a tight-binding model with second-order perturbation further included (See SI Appendix for details) and reveal a general coupling condition for two Bloch states in one layer of twisted bilayer system:
\begin{equation}
\bm{k}_1^u = \bm{k}_2^u - \bm{G}^u + \bm{G}^d
\label{eq2}
\end{equation}
where $\bm{k}_1^u$ and $\bm{k}_2^u$ are wave vectors of the two Bloch states in one layer, $\bm{G}^u$ and $\bm{G}^d$ are the reciprocal lattice vectors of the same layer and its counterpart respectively. Applying this condition, we propose a scattering mechanism for electrons in 30\textdegree-tBLG: Dirac cone at the K$^\prime$ point of one graphene layer K$_{0^\circ}^\prime$ is scattered by one reciprocal lattice vector of the other layer $\bm{G}_{30^\circ}$, forming a mirrored Dirac cone K$_R$ near the opposite momentum valley at K$_{0^\circ}$ as schematically shown in Fig.~\ref{fig6c}. This corresponds to the simplest case in \eqref{eq2} for $\bm{G}^u=\bm{0}$ and $\bm{G}^d=\bm{G}_{30^{\circ}}$. Such a mechanism is confirmed by three experimental observations. First of all, the momentum of the mirrored Dirac point $\bm{K}_{\rm R}$ is connected to the other graphene valley $\bm{K}_{0^{\circ}}^\prime$ by just one reciprocal lattice vector of the other 30\textdegree~layer $\bm{G}_{30^\circ}$. Secondly, the asymmetry of the intensity contour for the mirrored Dirac cone is identical to the one at K$^\prime$ (see Fig.~\ref{fig3d} or Fig.~\ref{fig4c}), suggesting that the mirrored Dirac cone is scattered from K$^\prime$ point. Thirdly, such a scattering mechanism is fully consistent with the observed R mode in Raman spectrum. Not only for the incommensurate structure, this scattering mechanism is also able to account for the appearance of replica Dirac cones in those commensurate graphene systems following the same analysis \cite{KoshinoNJP}. Thus, this mechanism can be applied to other van der Waals heterostructure beyond tBLG.
\section*{Conclusions}
In summary, we have successfully grown 30\textdegree-tBLG, a typical example for incommensurate superlattice with quasicrytalline order. The realization of 30\textdegree-tBLG provides new opportunities for investigating the intriguing physics of quasicrystalline superlattice. Moreover, by revealing the mirrored Dirac cones in a 30\textdegree-tBLG, we provide direct experimental evidence for the strong interlayer coupling through a coherent scattering process in such an incommensurate superlattice. Such scattering mechanism can be applied to engineer the band structure of both commensurate and incommensurate tBLG as well as other van der Waals heterostructures.
\matmethods{
\subsection*{Sample Growth}
The graphene sample was obtained by annealing one Pt(111) substrate to about 1600 \textcelsius. At this high temperature, the carbon impurities would segregate from the bulk to surface forming graphene film.
\subsection*{ARPES and NanoARPES}
Conventional ARPES measurements were performed at the home laboratory with a Helium discharge lamp and beamline 10.0.1 of Advanced Light Source (ALS) with synchrotron radiation source. The NanoARPES measurements were performed at ANTARES endstation of Synchrotron SOLEIL, France, with a lateral spatial resolution of 120 nm. The sample temperature was kept at 20 K for measurements at ALS and 40 K for those at SOLEIL.
\subsection*{Calculations}
The calculations were based on VASP code \cite{VASP1,VASP2} with plane wave basis set \cite{PAW1,PAW2}. We adopted opt86b-vdW functional \cite{vdW1,vdW2} to include the van der Waals interactions. A periodic (5\texttimes5) on $(3\sqrt{3}$\texttimes$3\sqrt{3})$R30\textdegree~structure was created to simulate the 30\textdegree-tBLG because it is impossible to directly simulate the very large non-periodic bilayer. The lower layer is slightly compressed about 1.8\% to fit the Pt lattice and the upper layer is slightly stretched about 2\% to partially compensate the compression from the lower layer. Extensive convergence tests have been performed in terms of vacuum, film thickness, and k-point sampling. Although our calculations were based on one periodic structure, such effects would be extrapolated to non-periodic structures of 30\textdegree-tBLG.
}
\showmatmethods
\acknow{This work is supported by the Ministry of Science and Technology of China (Grant No. 2016YFA0303004 and 2015CB921001), National Natural Science Foundation of China (Grant No. 11334006 and 11725418), Science Challenge Project (No. 20164500122) and Beijing Advanced Innovation Center for Future Chip (ICFC).
The Synchrotron SOLEIL is supported by the Centre National de la Recherche Scientifique (CNRS) and the Commissariat \`{a} l$^\prime$Energie Atomique et aux Energies Alternatives (CEA), France. This work is supported by a public grant overseen by the French National Research Agency (ANR) as part of the ``Investissements d$^\prime$Avenir'' program (Labex NanoSaclay, reference: ANR-10-LABX-0035), as well as the the French Ministre des affaires trangres et europennes (MAEE), the Centre National de la Recherche Scientifique (CNRS) through the ICT-ASIA programme grant 3226/DGM/ATT/RECH.}
\showacknow
|
{
"timestamp": "2018-06-20T02:13:17",
"yymm": "1804",
"arxiv_id": "1804.05372",
"language": "en",
"url": "https://arxiv.org/abs/1804.05372"
}
|
\section{Introduction}
During planet formation, ice plays a key role in multiple ways. Water ice enhances the solid mass reservoir available for planet formation, it improves the sticking properties of small dust grains for coagulation into larger bodies \citep[e.g.][]{Wada2007, Wettlaufer2010, Wada2013, Gundlach2015}. More indirectly, ices are an extra opacity source thus affecting the disk thermal structure. In addition, they could play an important role in providing the reservoir to make larger organic molecules \citep[see review by][]{MunozCaro2013}.
In Herbig disks, water ice was first detected by \citet{Malfait1999} in the ISO/SWS and LWS spectra of HD\,142527, a detection followed up later by Herschel/PACS \citep{Min2016b}. The disk contains an outer reservoir of highly crystalline ice (beyond $\sim\!130$~au), which comprises a major fraction of the available oxygen (\mbox{$\sim\!80$\%}). \citet{Chiang2001} report the detection of the 45~$\mu$m water ice feature in ISO/LWS spectra of CQ~Tau and AA~Tau; the spectra unfortunately lack full wavelength coverage of this feature. Later Herschel/PACS spectra of a small sample of T Tauri stars also suffer from partial coverage of the ice features and baseline stability issues. However, \citet{McClure2012} were able to report a detection of the 63~$\mu$m crystalline ice feature in GQ~Lup and \citet{McClure2015} in Haro~6-13 and DO~Tau.
The history and evolution of the ices in disks remains unclear. \citet{Visser2009} show from cloud collapse simulations that water ice is incorporated into the outer reservoir of a protoplanetary disk without desorption (never heated above the sublimation temperature). Recent observations of the comet 67P/Churyumov-Gerasimenko with the Rosetta mission show high abundances of the noble gas argon suggesting that the ice of that comet has never been heated above $\sim\!20$~K \citep{Balsiger2015}. On the other hand, the high fraction of cold crystalline ice seen in the disk around HD\,142527 requires some heating and/or mixing processes within the disk.
Photodesorption processes can remove ices from grains in the surface layers of cold outer disks \citep{Dominik2005, Woitke2009b}, thereby creating a cold water vapour reservoir. This cold water vapour was subsequently detected by Herschel/HIFI in the disks around TW~Hya and DG~Tau \citep{Hogerheijde2011, Podio2013}. After establishing the relevance of non-thermal desorption, \citet{Min2016b} model for the first time the water ice reservoir self consistently in the disk around a Herbig star with the 2D thermal disk structure and photodesorption. They use an ice line description that is calibrated on thermo-chemical disk models.
In this work, we expand the disk modeling to T Tauri disks. We use the reference T Tauri disk model presented in \citet{Woitke2016} (Sect.~\ref{sec:models}). By varying key disk parameters such as the disk size, dust mass, grain sizes, dust settling, and ice thickness, we exploit the diagnostic value of the far-IR water ice features at 45 and 63~$\mu$m (Sect.~\ref{sec:results}). In addition, we explore whether we can observationally distinguish the three scenarios of warm-up, in-situ formation, and cool-down for water ice. In Sect.~\ref{sec:future}, we discuss our results in the context of the approved SOFIA/HIRMES instrument and the proposed SPICA/SAFARI instrument.
\section{The disk models}
\label{sec:models}
\begin{table}[th]
\caption{Basic model parameters for the reference T Tauri disk.}
\begin{tabular}{lll}
\hline
\hline\\[-3mm]
Quantity & Symbol & Value \\
\hline\\[-3mm]
Stellar mass & $M_\ast$ & 0.7~M$_\odot$\\
Effective temperature & $T_{\rm eff}$ & 4000~K\\
Stellar luminosity & $L_\ast$ & 1.0~L$_\odot$\\
FUV excess & $f_{\rm UV}$ & 0.01\\
& $p_{\rm UV}$ & 1.3\\
\hline\\[-3mm]
Disk dust mass$^1$ & $M_{\rm dust}$ & $3.3\,10^{-4}$~M$_\odot$\\
Inner disk radius & $R_{\rm in}$ & 0.07~au\\
Outer disk radius$^2$ & $R_{\rm out}$ & 700~au\\
Tapered edge radius & $R_{\rm taper}$ & 100~au\\
Column density power index & $\epsilon$ & 1.0\\
Reference radius & $R_0$ & 100~au \\
Scale height at $R_{0}$ & $H_0$ & 10.0~au \\
Disk flaring power index & $\beta$ & 1.15 \\
\hline\\[-3mm]
Minimum dust particle radius & $a_{\rm min}$ & 0.05 $\mu$m\\
Maximum dust particle radius & $a_{\rm max}$ & 3000.0 $\mu$m \\
Dust size dist.\ power index & $a_{\rm pow}$ & 3.5\\
Turbulent mixing parameter & $\alpha_{\rm settle}$ & $10^{-2}$\\
Max.\ hollow volume ratio & $V_{\rm hollow}^{\rm max}$ & 80\% \\
Dust composition & Mg$_{0.7}$Fe$_{0.3}$SiO$_3$\hspace*{-2mm} & 60\% \\
(volume fractions) & amorph.\ carbon\hspace*{-2mm} & 15\% \\
& porosity & 25\% \\
Grain material density & $\rho_{\rm grain}$ & 2.076~g~cm$^{-3}$ \\
\hline
\end{tabular}
\label{tab:diskparameters}
\tablefoot{(1) The disk mass is a factor 3.3 higher than in the original \citet{Woitke2016} model. (2) The outer radius is defined as the radius where the surface density column drops to $N_{\rm \langle H\rangle, ver}=10^{20}$~cm$^{-2}$.}
\end{table}
All disk models are calculated using the 3D Monte Carlo radiative transfer code MCMax \citep{Min2009}. The code uses a grain size distribution where the fraction of ices is a free parameter. We use the DIANA\footnote{DIANA is the EU FP7 project ``Disc Analysis'' (PI: P.\ Woitke) that developed tools for the interpretation of multi-wavelengths observations of protoplanetary disks and applied them to a large number of protoplanetary disks using a consistent approach.} grain opacities \citep{Min2016a} which were constructed to fit simultaneously observed thermal and scattered light properties in protoplanetary disks. The ice opacities are taken from \citep{Smith1994} and for amorphous ice from \citet{Li1998}; the code can use either opacities at a fixed temperature or temperature dependent opacities. In the following we use 2D disk models and solve the continuum radiative transfer iteratively together with the location of the water ice reservoir. For the latter, we use the formalism developed in \citet{Min2016b}, where the snow line depends on the temperature, pressure and UV radiation field (thermal and non-thermal desorption processes); the oxygen abundance is constrained by setting the total ice fraction in the disk $f_{\rm ice}$. Typically, three iterations are needed for the disk structure and spectral energy distribution (SED) to converge.
Table~\ref{tab:diskparameters} summarizes the key parameters of the reference T Tauri disk model from \citet{Woitke2016}. We focus here on passive irradiated disks where viscous heating plays a negligible role in the midplane close to the star. Since we investigate in this work the far-IR water ice features typically originating from beyond 10~au, the precise radial position of the midplane snowline is not expected to affect our results.
The fraction of ice is fixed to 0.8 for the reference model and the disk parameter series including the thermal history of ice series. However, we include one model series where we explore the impact of the water ice fraction on the SED and the strength of the water ice features. The total dust mass is spread over silicates, carbon and vacuum (60, 15 and 25\% respectively). The fraction of ice is defined as
\begin{equation}
M_{\rm ice}=f_{\rm ice} M_{\rm dust}\,\,\, .
\end{equation}
In this way, changing the ice fraction does not change the underlying bare grain opacities. Having a non-zero ice fraction adds a mantle around each grain and the total mass of extra ice is spread using the underlying bare grain size distribution. The values of the first, second and third moment of the bare grain size distribution are $\langle a \rangle \!=\! 8.33\,10^{-2}~{\rm \mu m}$, $\langle a^2 \rangle \!= \! 1.245\,10^{-2}~{\rm \mu m^2}$, $\langle a^3 \rangle \!= \! 1.525\,10^{-1}~{\rm \mu m^3}$. The ice fractions chosen here (see Table~\ref{tab:variedparameters}) explore a reasonable range since thermo-chemical disk models suggest that the grains grow almost a factor two in size due to the ice mantles \citep{Chaparro2016}. The fractions of 0.4, 0.8, 1.2, 1.6 and 2.0 correspond to a grain size increase of a factor 1.9, 2.2, 2.4, 2.5 and 2.6 respectively assuming an average water ice density of 1~g~cm$^{-3}$.
\subsection{Model series}
In order to isolate disk effects on the SED and the 45 and $63~\mu$m water ice features, the base choice for opacities is crystalline ice at 140~K unless stated otherwise. For comparison, the reference disk model is also calculated using amorphous ice opacities and without taking photodesorption into account. For all model series, only one of the parameters of the disk is altered at a time. This allows us again to isolate the effect of several properties of the disk on the water ice features. Table~\ref{tab:variedparameters} shows all the parameters that were varied; the values of the reference disk model are found in Table~\ref{tab:diskparameters}.
\begin{table}[th]
\caption{Varied parameters for the disk models.}
\begin{tabular}{lll}
\hline
\hline\\[-3mm]
Quantity & Symbol & Values \\
\hline\\[-3mm]
Inner radius & $R_{\rm in}$ & 0.07, 0.2, 1, 10, 30, 50 au\\
Tapering-off radius & $R_{\rm taper}$ & 50, 100, 200 au\\
Disk dust mass & $M_{\rm dust}$ & $10^{-3}$, $5\,10^{-4}$, $3.3\,10^{-4}$, \\
& & $2\,10^{-4}$, $10^{-4}$, $10^{-5}$, $10^{-6}$, \\
& & $10^{-7}$~M$_\odot$ \\
Minimum grain size & $a_{\rm min}$ & 0.05, 0.5, 2, 30, 50 $\mu$m\\
Turbulent mixing & $\alpha_{\rm settle}$ & $10^{-2}$, $10^{-3}$, $10^{-4}$\\
fraction water ice & $f_{\rm ice}$ & 0.4, 0.8, 1.2, 1.6, 2.0\\
\hline
\end{tabular}
\label{tab:variedparameters}
\end{table}
An additional series was done where the composition of the water ice is varied, based on its thermal history. We used the same series that are described by \citet{Smith1994}, which consist of\\[-5mm]
\begin{itemize}
\item Series 1: Warmup from 10 to 150~K\\[-3mm]
\item Series 2: Direct deposit between 10 and 150~K\\[-3mm]
\item Series 3: Cooldown from 140 to 10~K\\[-5mm]
\end{itemize}
The warmup series refers to a scenario where the water ice is formed in cold, optically thick regions. After its formation it moves to warmer areas in the disk. We mimick this by using for each temperature interval in the disk the corresponding ice opacity from the warmup series of \citet{Smith1994}. The water ice in the direct deposit scenario is formed locally and stays in the same thermal region. Also here, we change the ice opacity as a function of disk temperature using this time the direct deposity series from Smith's work. In the cooldown scenario, the water ice is formed in warm areas close to the midplane snowline ($\sim\!0.5$~au, around 140~K), then moves outwards to colder areas in the disk. This is simulated using the same methodology as the other two series, but now with the cooldown opacity series.
For each disk model, we calculate the disk thermal structure, extent of the ice reservoir and spectral energy distribution self-consistently. In the following, we describe the analysis of the strength of the water ice features.
\subsection{Analysis of water ice features}
\label{Sect:Analysis}
The trend that we are seeing in the shape of the water ice features between our different disk models is quantified by using a polynomial best fit to the continuum of the SED. Given the strong changes in SED shape related to disk geometry, the placement of the continuum can be very uncertain. Also, observers have no possibility to choose a more refined method since the underlying continuum could not be estimated independently like we could do in the models. To avoid these problems, we chose the simplest solution, a linear fit between $40.0$ and $57.5~\mu$m that covers the $45~\mu$m water ice feature. The $63~\mu$m feature cannot be analyzed this way because of the broad and weak nature of the feature and the uncertainty of the shape of the underlying continuum.
\begin{figure}[thb]
\includegraphics[width=9.5cm]{ZoomedSED_reference-photodesorptionoff-amorphous.png}
\caption{SED of the reference disk model using crystalline ice opacities (140~K) with photodesorption switched on (solid line) and off (dashed line) and amorphous ice (photodesorption on, dotted line).}
\label{fig:ZoomedSED_reference-photodesoffl}
\end{figure}
We define the height of the $45~\mu$m water ice features $S_{\rm \lambda,peak}$ as
\begin{equation}\label{fluxSED-fluxContinuum}
\hspace{2.7cm} S_{\rm \lambda,peak} = \frac{F_{\rm \lambda,peak} - F_{\rm \lambda,cont}}{F_{\rm \lambda,cont}}
\end{equation}
where $F_{\rm \lambda,cont}$ and $F_{\rm \lambda,peak}$ are the continuum and peak flux of the SED respectively. Since the height of the features is sensitive to the definition of the continuum, trends are only considered real if they are significantly larger than the error bars. We estimate the errors, by also using higher degree polynomials to fit the continuum and derive a peak strength. Due to the specific shape of the continuum over the 30-60~$\mu$m wavelength range (mostly concave), the polynomials systematically provide higher peak values (typically $10-25$\%) compared to the linear fit. The uncertainty within the polynomials is smaller, typically only 10\%.
\section{Results}
\label{sec:results}
\subsection{Ice opacities and photodesorption}
Photodesorption is extremely important for the strength of the water ice features. Taking this effect not into account will overestimate their strengths by a factor 4-5 (Fig.~\ref{fig:ZoomedSED_reference-photodesoffl}). Also, amorphous water ice produces features that are a factor $\sim\!7$ weaker than those of crystalline ice, to the extent that the peak strength for the $45~\mu$m feature becomes only $0.02$. Hence, in all subsequent parameter studies except for the `thermal history' series, we adapt crystalline ice opacities (140~K). Note that this does not imply that we think water ice in disks has a temperature of 140~K, but it merely ensure a straightforward and clean analysis of the features. The more realistic `thermal history' water ice scenarios will be discussed in Sect.~\ref{sect:dustice-changefeature}. It is also important to note that the underlying shape of the continuum as well as its absolute value changes depending on the ice opacity choice. This is due to the self-consistent solution of the dust radiative transfer.
\subsection{Location of the water ice reservoir}
We focus in the following on the reference disk model to explain the location of the water ice reservoir and to illustrate where the 45 and 63~$\mu$m ice features originate with respect to the continuum optical depth at those wavelengths.
\begin{figure}[thb]
\vspace*{-5mm}
\includegraphics[width=10cm]{2Dtemp_reference.png}
\caption{The 2D temperature disk structure for the reference disk model. The thick white dashed contour outlines the water ice reservoir. The black dashed lines are the 160 and 100~K temperature contours. The thick red and yellow contours denote the vertical optical depth $\tau\!=\!1$ at 45 and 63~$\mu$m. The green and orange contours denote the radial optical depth $\tau\!=\!1$ at $0.55$ and 3~$\mu$m respectively.}
\label{fig:2Dtemp_referencemodel}
\end{figure}
Fig. \ref{fig:2Dtemp_referencemodel} shows the 2D temperature disk structure of the reference disk model. The thick white, dashed contour outlines the water ice reservoir of the disk (snowline). Note that at the outer radius of the disk, even though these are very cold regions, there is no water ice present. This is because of the interstellar radiation field, which causes photodesorption of ice mantles on dust grains in low density regions. The snowline is defined by the equation derived by \citet{Min2016b}, an equilibrium between the vapor pressure and thermal/non-thermal desorption processes. The black lines show the region where the temperature is between 100 and 160~K. This is roughly the area where the crystallization of water ice is happening \citep{Smith1994}. The red and yellow lines show the location where the photons can escape the disk without further absorption and re-emission. Photons emitted below the vertical optical depth $\tau\!=\!1$ line at 45 and $63~\mu$m will get absorbed again, and observers can therefore not obtain information about the structure of the disk below this region. The green and orange lines show where stellar photons with a wavelength of 0.55 and $3~\mu$m, respectively, first interact with the disk. The stellar emission peaks at roughly $0.55~\mu$m. Stellar photons with a wavelength of $3~\mu$m are not directly absorbed by the ice reservoir, if the region has a hard cut-off and therefore the $3~\mu$m water ice feature may not show in observations of protoplanetary disks with simple standard geometries (full disk, no gaps, holes, no strong settling).
The 45 and $63~\mu$m water ice features in the SED of protoplanetary disks, originate between the snowline and the vertical optical depth $\tau\!=\!1$ lines at 45 and $63~\mu$m. The following sections will use various model series to understand in more detail the radial range over which the ice feature originate.
\subsection{The peak strength of the $45~\mu$m feature}
Table \ref{tab:heightfeatures} shows the height of the 45~$\mu$m water ice feature $S_{\rm45, peak}$ for the different disk model series. It is important to recall that the peak strength carries an error of typically less than $10$\% (from use of different polynomial methods).
In the following sections, we discuss in detail the different parameter series in two groups: Parameters that do not change the water ice features and those which do. This enables us to assess the key science that can be learned from studying these features in the future.
\begin{table}[h]
\caption{Peak strength of the $45~\mu$m water ice feature in the reference model and the various T Tauri disk model series.}
\begin{tabular}{lll}
\hline
\hline\\[-3mm]
Model series & Value & $S_{\rm45, peak}$ \\
\hline\\[-3mm]
Reference (crystalline) & - & 0.134\\
Reference (amorphous) & - & 0.020\\
\hline\\[-3mm]
$R_{\rm in}$ [au] & 0.07 & 0.133\\
& 0.2 & 0.138\\
& 1 & 0.108\\
& 10 & * \\
& 30 & * \\
& 50 & * \\
\hline\\[-3mm]
$R_{\rm taper}$ [au] & 50 & 0.141\\
& 100 & 0.130\\
& 200 & 0.128\\
\hline\\[-3mm]
$a_{\rm min}$ [$\mu$m] & 0.05 & 0.131 \\
& 0.5 & 0.132\\
& 2 & 0.111\\
& 10 & 0.043\\
& 30 & $8\,10^{-4}$\\
& 50 & 0.0\\
\hline\\[-3mm]
$M_{\rm dust}$ [M$_\odot$] & $10^{-3}$ & 0.138\\
& $5\,10^{-4}$ & 0.141 \\
& $3.3\,10^{-4}$ & 0.132\\
& $2\,10^{-4}$ & 0.127\\
& $10^{-4}$ & 0.127\\
& $10^{-5}$ & 0.107\\
& $10^{-6}$ & 0.050\\
& $10^{-7}$ & 0.0\\
\hline\\[-3mm]
$\alpha_{\rm settle}$ & $10^{-2}$ & 0.132 \\
& $10^{-3}$ & 0.146 \\
& $10^{-4}$ & 0.157 \\
\hline\\[-3mm]
$f_{\rm ice}$ & 0.4 & 0.087 \\
& 0.8 & 0.128 \\
& 1.2 & 0.149 \\
& 1.6 & 0.170 \\
& 2.0 & 0.176 \\
\hline
\end{tabular}
\label{tab:heightfeatures}
\tablefoot{* indicates that we cannot measure the strength of the ice feature accurately due to the peak of the SED falling into the range of the water ice emission features.}
\end{table}
\subsection{Disk properties that do not change the ice features}
The disk inner and outer radius, within the limits we probe here, do not affect the water ice features. However, they are very instructive in narrowing down the region where the ice features predominantly form. The peak height of the $45~\mu$m feature is very constant between $0.11$ and $0.14$ throughout this series, that is they stay within $\pm\!17$\%, which is close to the error margin on the continuum placement (10\%).
Fig.~\ref{fig:ZoomedSED_rin} shows that while the continuum changes strongly with increasing disk inner radius from 0.07 to 50~au, the water ice features remain present. This indicates that the dominant contribution to the water ice features comes from beyond 50~au. This is also illustrated by Fig.~\ref{fig:2Dtemp_rin}: The ice reservoir (outlined by dashed white thick contour) stays unchanged for radii beyond $\sim\!50$~au.
\begin{figure}[htb]
\includegraphics[width=9.5cm]{ZoomedSED_rin.png}
\caption{SED for the disk models with disk inner radius varying between 0.07 and 50~au.}
\label{fig:ZoomedSED_rin}
\end{figure}
Fig.~\ref{fig:ZoomedSED_rtap} shows that while the continuum changes strongly with increasing disk tapering off radius (50-200~au), the water ice features remain again very stable. Increasing the tapering off radius spreads the disk material to larger radii; however, this material further out is not substantially increasing the strength of the ice feature. This indicates that the dominant contribution to the water ice features comes from within $100$~au. The dust temperature within the ice reservoir beyond 100~au drops well below 50~K and hence the icy dust grains are not at the peak of their emissivity. Fig.~\ref{fig:2Dtemp_rtap} also shows that the ice reservoir touches the outer disk radius for large tapering off radii. However, the ice column density from those outer regions is likely too small to significantly contribute to the ice feature.
\begin{figure}[htb]
\includegraphics[width=9.5cm]{ZoomedSED_rtap.png}
\caption{SED for the disk models with disk tapering off radius varying between 50 and 200~au.}
\label{fig:ZoomedSED_rtap}
\end{figure}
We use the turbulent $\alpha$ to parametrize dust settling \citep{Dubrulle1995}. The lower the turbulence, the stronger the dust settling in the outer disk. In the settled case, mostly the large grains settle towards the midplane, while the smaller micron-sized grains are less affected. These are the carriers of the far-IR ice features as demonstrated in the minimum grain size parameter series (see Sect.~\ref{sect:dustice-changefeature}). The water ice features change weakly with the level of turbulence. The ice feature becomes $\sim\!15$\% stronger for the lowest turbulence value compared with the highest one. This is largely due to the change in local continuum opacity pushing the $\tau\!\sim\!1$ lines at 45 and 63~$\mu$m closer to the disk midplane (see Fig.~\ref{fig:2Dtemp_alphaturb}).
Of course this result is limited by the intrinsic assumption of settling being an equilibrium process and ice evaporation/re-condensation occurring on shorter timescales compared to possible vertical mixing processes.
\begin{figure}[htb]
\includegraphics[width=9.5cm]{ZoomedSED_alphaturb.png}
\caption{SED for the disk models with different levels of turbulence $\alpha$ (see legend).}
\label{fig:ZoomedSED_alphaturb}
\end{figure}
\subsection{Dust/ice properties that change the ice features}
\label{sect:dustice-changefeature}
The 45 and $63~\mu$m water ice features are affected by the minimum grain size $a_{\rm min}$, the disk dust mass, the mass fraction of ice-to-refractory (so the thickness of the ice mantle) and also the thermal history of the water ice. In the following, we summarize the key results from those four model series.
Fig.~\ref{fig:ZoomedSED_amin} shows that while the absolute level of the continuum changes strongly with increasing minimum grain size, the water ice features remain very stable up to $a_{\rm min}\!\sim\!10~\mu$m. Hence, we conclude that the dominant carrier of the far-IR water ice features are grains up to sizes of a few $\mu$m. This also explains the result in the previous section that dust settling is not affecting the strength of the 45 and $63~\mu$m water ice features. For a grain size distribution, where small grains are lacking, so $a_{\rm min}\!=\!10~\mu$m, the $45~\mu$m feature becomes a factor of three smaller than the reference model ($a_{\rm min}\!=\!0.05~\mu$m).
\begin{figure}[htb]
\includegraphics[width=9.5cm]{ZoomedSED_amin.png}
\caption{SED for the disk models with minimum grain size varying between 0.05 and $50~\mu$m (see legend).}
\label{fig:ZoomedSED_amin}
\end{figure}
The strength of the ice features correlates with dust mass until the disk reaches a mass of $\sim\!10^{-4}$~M$_\odot$. For higher dust masses, the ice features saturate and the feature strength levels off around 0.13 (Table~\ref{tab:heightfeatures}). Typical dust masses of class\,{\sc ii} disks around stars with less than 1~M$_\odot$ are below $2\,10^{-4}$~M$_\odot$ \citep[e.g.][]{Pascucci2016} and hence it should be possible to estimate the total ice mass in many young star forming regions.
\begin{figure}[htb]
\includegraphics[width=9.5cm]{ZoomedSED_Mdust.png}
\caption{SED for the disk models with disk dust masses varying between $10^{-3}$ and $10^{-7}$~M$_\odot$ (see legend).}
\label{fig:ZoomedSED_Mdust}
\end{figure}
The ice fraction produces also a strong effect: The thicker the ice mantle, the larger the peak strength. However, at high ice fractions of $>\!1.6$, the peak strength saturates. Adding more ice to the mantle does not contribute anymore to the emission; the ice becomes `optically thick'. Overall, an increase in the thickness of the ice mantle by a factor of $1.3$ causes an increase in peak strength by a factor of approximately two ($f_{\rm ice}\!=\!0.4$ with respect to $1.6$).
We would like to note here that we assume in the above model series the ice to be in crystalline shape (i.e.\ formed at temperatures above 100~K), causing prominent features. This assumption is plausible given the results from \citet{Min2016b}. However, if the ice in the disk is formed and still located at temperatures below 100~K, a significant fraction could be in amorphous form, which is more difficult to observe.
\begin{figure}[htb]
\includegraphics[width=9.5cm]{ZoomedSED_fraction.png}
\caption{SED for the disk models with different water ice fraction $f_{\rm ice}$ on the grains ($f_{\rm ice}\!=\!2$ means that the models contains two times the dust mass added in ice mantles). }
\label{fig:ZoomedSED_ice-fraction}
\end{figure}
Figure~\ref{fig:ZoomedSED_ice-scenarios} shows that the different thermal histories of water ice will lead to clearly different feature strength and shape. Table \ref{tab:locationfeatures} shows the peak wavelength of both water ice features for these series. The $63~\mu$m feature shows no clearly defined peak for the warmup and direct deposit scenarios.
Most interestingly, in the cooldown series, we see a significant shift of the peak of the $45~\mu$m ice feature to shorter wavelengths ($43.55~\mu$m) with respect to the reference series with opacities at 140~K ($44.96~\mu$m). This indicates that a large fraction of the ice feature originates rather from $\sim\!70$~K ice \citep[see Table~1 of][]{Smith1994}. The $63~\mu$m feature also shows a shift corresponding to $\sim\!90$~K ($62.6~\mu$m). In the reference model, temperatures of 70-90~K correspond to temperatures at the snow line surface at a distance of 10-20~au.
In the direct deposit and warm-up series, the water ice features are more indicative of amorphous ice. Also, their peaks shift to longer wavelengths ($45.93~\mu$m and $46.42~\mu$m) with respect to the reference model, again indicating on average low ice temperatures in the emitting region ($< \! 110$~K) in agreement with their amorphous nature and an origin beyond 6~au.
\begin{table}[htb]
\caption{Peak wavelengths position of the water ice features for the different thermal history scenarios.}
\begin{tabular}{lll}
\hline
\hline\\[-3mm]
& 45~$\mu$m & 63~$\mu$m \\
\hline\\[-3mm]
Reference & 44.96 & 63.91 \\
Warmup (thin) & 45.93 & - \\
Warmup (thick) & 46.42 & - \\
Direct deposit & 44.96 & - \\
Cooldown & 43.55 & 62.56 \\
\hline
\end{tabular}
\label{tab:locationfeatures}
\end{table}
This model series is the only one where we include proper temperature dependent ice opacities. It shows that our approach to find the spatial origin of these water ice features using the model series with varying inner and tapering-off radii might be too simple. For example, the emitting region of the ice features could change in the model series with increasing $R_{\rm in}$, and an increasing surface area could compensate for the emission becoming weaker, thus leading to the apparent `constancy' of the far-IR water ice features over a wide range of disk geometries. The disk model series with water ice histories may provide here a more reliable estimate of the main emitting region through the measurement of the $45~\mu$m ice feature peak.
\begin{figure}[htb]
\includegraphics[width=9.5cm]{ZoomedSED_coolingdown-series.png}
\caption{SED for the disk models with different water ice `thermal histories'.}
\label{fig:ZoomedSED_ice-scenarios}
\end{figure}
\section{Future observations}
\label{sec:future}
\subsection{SOFIA/HIRMES}
HIRMES was selected as a third generation instrument for the SOFIA
Observatory. The SOFIA Observatory consists of a 2.5~m telescope mounted
inside a modified Boeing~747SP operating at altitudes from 12-14~km.
HIRMES is an infrared spectrograph operating between 25 and 122~$\mu$m
with a spectral resolving power of 350-650 in low resolution mode and
$5\,10^4-10^5$ in high resolution mode. HIRMES will be commissioned on
SOFIA in early to mid 2019. For the low resolution mode, rebinning to a resolution of 100,
the expected sensitivity is $\sim\!0.04$~Jy (1$\sigma$ in 1~hr; Klaus Pontoppidan, private communication)).
\begin{figure}[htb]
\includegraphics[width=9.cm]{SimulateHIRMES_all.pdf}
\caption{Simulated HIRMES spectra for several ice model series: crystalline ice and amorphous ice, different water ice `thermal histories' shifted upwards by 0.7~Jy ($+0.7$) and different ice thicknesses shifted upwards by 1.4~Jy ($+1.4$). The exposure time is 1~hr and the resolution is $R\!=\!100$.}
\label{fig:HIRMESice-composite}
\end{figure}
Figure~\ref{fig:HIRMESice-composite} shows the noise level that HIRMES will achieve in a 1 hr exposure for various of our model series using $R\!=\!100$. Even though it could provide detections in the case of brighter sources and/or for particularly strong features (e.g. very high ice fraction, crystalline ice at 140~K), for most of the model simulations of this paper detailed studies of the ice properties and thermal history will be challenging with a resolution of $R\!=\!100$. Given the current sensitivities and our model predictions, ice studies of T Tauri disks with HIRMES will require long exposure times, for typical T Tauri continuum fluxes (1~Jy and lower) even beyond 1 hour.
\subsection{SPICA/SAFARI}
The SPace Infrared telescope for Cosmology and Astrophysics (SPICA) is a joint Japanese/European 2.5~m cooled telescope proposed for ESAs M5 call. The SAFARI instrument on board will allow spectroscopy between 34 and 230~$\mu$m at a resolution of 300-11000. In low resolution mode, it will reach 0.31, 0.45 and 0.72~mJy (5$\sigma$ in 1~hr) in the SW (34-56~$\mu$m), MW (54-89~$\mu$m) and LW (87-143~$\mu$m) band respectively.
\begin{figure}[htb]
\includegraphics[width=9.cm]{SimulateSAFARI_all.pdf}
\caption{Simulated SAFARI spectra for several ice model series: crystalline ice and amorphous ice, different water ice `thermal histories' shifted upwards by 0.7~Jy ($+0.7$) and different ice thicknesses shifted upwards by 1.4~Jy ($+1.4$). The exposure time is 10~min and the resolution is $R\!=\!300$.}
\label{fig:SAFARIice-composite}
\end{figure}
Figure~\ref{fig:SAFARIice-composite} shows the noise level that SAFARI will achieve in a short 10~min exposure for various of our model series using $R\!=\!300$. The \mbox{SAFARI} instrument with its high sensitivity will be able to distinguish between cooldown and direct deposit/warmup ice histories, but also ice thicknesses, i.e.\ ice fractions. The short exposure times (10~min) will enable ice surveys of hundreds of T~Tauri disks with the SPICA mission. The most important limitation will be the baseline stability and calibration of the instrument. For detailed ice characterization studies, we require a stability of better than a few \% across the 35-100~$\mu$m spectral range.
Long exposures (1~hr) can provide more reliable ice fraction measurements. However, the distinction between direct deposit and warmup scenarios, and the detection of amorphous ice will remain difficult and hinges mostly on a reliable continuum placement due to the broadness of the weak features.
\section{Discussion}
\label{sec:discussion}
The constant nature of the far-IR water ice features against physical disk parameters such as disk sizes and settling makes these features a robust probe of the grain and ice history in disks.
A closely related water ice feature study is \citet{McClure2015}. One of their Herschel observations, DO~Tau, shows far-IR continuum fluxes very close to our reference model. The peak strength of the detected feature at $63~\mu$m is $\sim\!0.2$, at least a factor of two stronger than the feature we typically see using crystalline water ice in our reference model. The other water ice detections they report are even stronger. On the other hand, the weak features predicted from our `thermal history' modeling series, would have been undetectable given the typical PACS noise level. The only way in our modeling approach to enhance the strength of the $63~\mu$m feature to a level that would have been detectable by PACS, would be to neglect the photodesorption and have the water ice reservoir extend high up in the disk. It is however important to keep in mind that the continuum placement with PACS was very difficult due to the pointing stability and the data was obtained in range spectroscopy mode (51-73~$\mu$m) thus only covering the central part of the feature without much neighboring continuum.
The `thermal history' modeling series illustrates the power of having access to the peak position of the $45~\mu$m ice feature, the strength of the peak as well as the shape of the $45$ and $63~\mu$m feature complex. If the ice is purely amorphous, the $63~\mu$m feature will be hard to distinguish from the continuum due to its very broad nature. However, the shorter wavelength feature remains in that case an important diagnostic and the combination of peak wavelength and strength has the potential to discriminate between in-situ formation and a scenario where the ice has been formed cold either in the precursor molecular cloud or in the outer disk and later distributed more widely through radial migration processes. The opposite process, a formation close to the midplane snow line and later migration outwards leaves the ice in a crystalline state, which is clearly detectable in both far-IR water ice features.
The next step should be to couple detailed grain growth/migration models with the self-consistent snow line and dust radiative transfer approach. The optical depth of the disk will likely change if the grain size distribution varies with position in the disk. This could affect the appearance and strength of the water ice features. A step in this direction is the dust settling series in which the height of the far-IR continuum of the disk changes substantially. Similar changes can be expected for a varying radial distribution driven by dust migration. Once the smaller dust grains change their spatial distribution, also the UV/optical $\tau\!\sim\!1$ line will change and this can also change the location of the snow line in the disk surface. Dust drift models suggest a high efficiency in depleting small $\mu$m-sized grains from a wide region around $\sim\!80$~au \citep[region\,{\sc iv} in Fig.~2 of][]{Birnstiel2015}. This is the region we identify here as the main contributor to the water ice thermal emission feature. Leaving this region devoid of particle sizes that can produce the thermal ice emission features (grains $\lesssim\!10~\mu$m) would have an effect on the strength of the feature.
It is interesting to note that the absence of a water ice feature in the spectrum does not imply the absence of an extended water ice reservoir. This can be caused by the absence of grains smaller than $10~\mu$m in the outer parts of the disk ($\gtrsim\!30$~au). The disk model with $a_{\min}\!=\!30~\mu$m shows no features in the SED, but still shows a significant ice reservoir.
\section{Conclusions}
\label{sec:conclusion}
This paper studies for the first time the far-IR water ice features in a self-consistent modeling approach to set the location of the water snowline together with consistent dust/ice opacities and dust disk thermal structure. We demonstrate that photodesorption of water ice is key in the quantitative interpretation of the peak strengths of the 45 and $63~\mu$m features. The water ice features originate from grains smaller than $\sim\!10~\mu$m.
The water ice features originate predominantly at the surface snow line of the disk between 10 and 100~au. As such they are very robust against several disk parameters such as inner and outer radii, within limits the grain size distribution and dust settling. However, the feature strength scales with disk dust mass in the range of typical T Tauri disks. The features also
change substantially with the thickness of the ice mantle, although the strength levels off for mantles much thicker than 2.5 times the original grain sizes.
Most interestingly, the features do carry an imprint of the thermal history of the ice and thus can distinguish between a warmup scenario for the water ice (water ice is formed in cold, optically thick regions and then moves to warmer areas in the disk) and a cooldown scenario (water ice is formed in warm areas close to the midplane snowline and then moves outwards to colder areas in the disk). The direct deposit scenario (ice stays where it was formed locally in the disk) resembles qualitatively the warmup scenario and while the peak strength change of the 45~$\mu$m feature may not allow a discrimination, the shift of the peak wavelengths to shorter values might be a discriminator.
Given the results of our ice model series, simulations using the sensitivities for the planned SOFIA/HIRMES instrument show that the low resolution as well as the sensitivity will severely limit detailed ice studies --- beyond simple detection --- in T Tauri disks at distance of typical star forming regions. The $45~\mu$m ice feature is strong enough to be detected if it originates from warm grains with thick ice mantles that are crystalline in nature. Similar simulations for the SPICA/SAFARI instrument show that both the differences in peak strengths as well as the wavelength shifts in the peak position can be easily measured for samples of several hundred T Tauri disks (exposure times of 10~min). At low spectral resolution ($R\!=\!300$) weak flat ice features at $45~\mu$m (e.g.\ amorphous, direct deposit, warmup of ice) require a good continuum characterization in deep surveys.
\begin{acknowledgements}
IK and MM acknowledge funding from the EU FP7-2011 under Grant Agreement no. 284405. LK is supported by a grant from the Netherlands Research School for Astronomy (NOVA). We thank the anonymous referee for comments that improved the clarity of the paper and the suggestion to include the dust mass series.
\end{acknowledgements}
\newpage
\clearpage
\begin{appendix}
\section{SEDs and disk structures}
\label{App:modelseries}
\subsection{Disk inner radius and tapering off radius}
\label{App:sect:rin:rtap}
Fig.~\ref{fig:2Dtemp_rin} and Fig.~\ref{fig:2Dtemp_rtap} illustrate the temperature distribution in the disk models with varying inner radius and tapering off radius. To assess where the water ice features originate, we also show the snow line and the vertical optical depth $\tau\!=\!1$ contours for 45 and $63~\mu$m and the radial optical depth $\tau\!=\!1$ at $0.55~\mu$m.
\begin{figure*}[tbh]
\includegraphics[width=6.cm]{2Dtemp_rin=0_07.png}
\includegraphics[width=6.cm]{2Dtemp_rin=0_2.png}
\includegraphics[width=6.cm]{2Dtemp_rin=1.png}
\includegraphics[width=6.cm]{2Dtemp_rin=10.png}
\includegraphics[width=6.cm]{2Dtemp_rin=30.png}
\includegraphics[width=6.cm]{2Dtemp_rin=50.png}
\caption{The 2D dust temperature distribution in the model series where the disk inner radius changes from 0.07 to 50~au. The contour lines have the same meaning as in Fig.~\ref{fig:2Dtemp_referencemodel}.}
\label{fig:2Dtemp_rin}
\end{figure*}
\begin{figure*}[!htb]
\includegraphics[width=6.cm]{2Dtemp_rtap=50.png}
\includegraphics[width=6.cm]{2Dtemp_rtap=100.png}
\includegraphics[width=6.cm]{2Dtemp_rtap=200.png}
\caption{The 2D dust temperature distribution in the model series where the disk tapering off radius changes from 50 to 200~au. The contour lines have the same meaning as in Fig.~\ref{fig:2Dtemp_referencemodel}.}
\label{fig:2Dtemp_rtap}
\end{figure*}
\subsection{Dust settling}
\label{App:sect:alphaturb}
Fig.~\ref{fig:2Dtemp_alphaturb} shows the 2D temperature distribution for disk models with various degrees of settling ($\alpha_{\rm turb}\!=10^{-2}$ to $10^{-4}$). The change of opacities is strongest in the outer disk and hence temperature changes there are largest. Lower values for $\alpha_{\rm turb}$ lead to strong dust settling towards the midplane, thus reducing the opacities in the upper/outer layers. This shifts the snow line to lower heights (from $\sim\!0.3$ at 100 au for $\alpha_{\rm turb}\!=\!10^{-2}$ to $\sim\!0.25$ for $\alpha_{\rm turb}\!=\!10^{-4}$) and produces a larger water vapor reservoir in the outer disk.
\begin{figure*}[htb]
\includegraphics[width=6cm]{2Dtemp_alphaturb=1e-02.png}
\includegraphics[width=6cm]{2Dtemp_alphaturb=1e-03.png}
\includegraphics[width=6cm]{2Dtemp_alphaturb=1e-04.png}
\caption{The 2D dust temperature distribution in the model series where the turbulence decreases from 0.01 to $10^{-4}$. The contour lines have the same meaning as in Fig.~\ref{fig:2Dtemp_referencemodel}.}
\label{fig:2Dtemp_alphaturb}
\end{figure*}
\subsection{Minimum grain size}
\label{App:sect:amin}
When we change the minimum grain size from 0.05 to $50~\mu$m, the optical depth in the disk changes dramatically as illustrated in Fig.~\ref{fig:2Dtemp_amin}. Removing the smaller grains lowers the opacity and the largest impact is seen in the surface layers, especially in the change of height at which the disk becomes optically thick (green contour, $\tau\!=\!1$ at $0.55~\mu$m). Subsequently, also the snow line moves closer to the midplane ($\sim\!0.3$ at 100~au for $a_{\rm min}\!=\!0.05~\mu$m grains to $\sim\!0.2$ for $a_{\rm min}\!=\!50~\mu$m grains).
If the disk surfaces are strongly depleted in small grains, there is an increasing chance of observing the ice feature also in scattered light; the green radial optical depth line touches the snow line, i.e.\ optical stellar light can be scattered off icy grains even at large distances from the star (out to several 10~au).
\begin{figure*}[htb]
\includegraphics[width=6cm]{2Dtemp_amin=0_05.png}
\includegraphics[width=6cm]{2Dtemp_amin=0_5.png}
\includegraphics[width=6cm]{2Dtemp_amin=10.png}
\includegraphics[width=6cm]{2Dtemp_amin=30.png}
\includegraphics[width=6cm]{2Dtemp_amin=50.png}
\caption{The 2D dust temperature distribution in the model series where the minimum grain size increases from 0.05 to $50~\mu$m. The contour lines have the same meaning as in Fig.~\ref{fig:2Dtemp_referencemodel}.}
\label{fig:2Dtemp_amin}
\end{figure*}
\subsection{Disk dust mass}
\label{App:sect:Mdust}
The disks become increasingly transparent at mid-IR wavelength when the dust mass decreases from $10^{-3}$~M$_\odot$ to $10^{-7}$~M$_\odot$ (see red and yellow contour in Fig.~\ref{fig:2Dtemp_Mdust}). The full ice column density will be visible for dust masses below $10^{-4}$~M$_\odot$ beyond 100~au and for dust masses below $10^{-6}$~M$_\odot$ beyond 2~au. At the same time, the disk ice reservoir (white dashed contours) shrinks because the disk becomes warmer with decreasing dust mass and photodesorption can act at lower disk height due to an overall lower optical depth to UV photons. Especially in the lowest disk mass models, the change in opacity due to the presence of ice is now clearly visible in the optical depth contours.
\begin{figure*}[htb]
\includegraphics[width=6cm]{2Dtemp_Mdust=1e-03.png}
\includegraphics[width=6cm]{2Dtemp_Mdust=5e-04.png}
\includegraphics[width=6cm]{2Dtemp_Mdust=3-3e-04.png}
\includegraphics[width=6cm]{2Dtemp_Mdust=2e-04.png}
\includegraphics[width=6cm]{2Dtemp_Mdust=1e-04.png}
\includegraphics[width=6cm]{2Dtemp_Mdust=1e-05.png}
\includegraphics[width=6cm]{2Dtemp_Mdust=1e-06.png}
\includegraphics[width=6cm]{2Dtemp_Mdust=1e-07.png}
\caption{The 2D dust temperature distribution in the model series where the disk dust mass decreases from $10^{-3}$~M$_\odot$ to $10^{-7}$~M$_\odot$. The contour lines have the same meaning as in Fig.~\ref{fig:2Dtemp_referencemodel}.}
\label{fig:2Dtemp_Mdust}
\end{figure*}
\subsection{Ice fraction}
\label{App:sect:ice-fraction}
Figure~\ref{fig:2Dtemp_ice-fraction} clearly shows that the water ice fraction has a negligible impact on the disk thermal structure. It contributes very little to the overall grain energy balance. The explanation lies in the implementation, where the ice fraction determines an amount of ice to be added on top of the bare grain opacities. In that sense, changing the ice fraction does not change the underlying bare grain opacities. The grains become effectively thicker by growing more ice on the surface.
\begin{figure*}[htb]
\includegraphics[width=9cm]{2Dtemp_fraction=0_4.png}
\includegraphics[width=9cm]{2Dtemp_fraction=2_0.png}
\caption{The 2D dust temperature distribution in the model series with different water ice fraction of 0.1 and 2. The contour lines have the same meaning as in Fig.~\ref{fig:2Dtemp_referencemodel}.}
\label{fig:2Dtemp_ice-fraction}
\end{figure*}
\subsection{Warmup versus cooldown scenario}
\label{App:sect:ice-scenarios}
Figure~\ref{fig:2Dtemp_ice-scenarios} shows the difference between using fixed temperature 140~K crystalline ice opacities in the reference model and the full temperature dependent ice opacities from the cooldown series. While the temperature changes in the disk model are very small, there is a shift in the optical depth at $45~\mu$m to lower depth in the cooldown model. It is clear that the crystalline fixed 140~K ice opacity of the reference disk model systematically overestimates the ice opacity.
\begin{figure*}[htb]
\includegraphics[width=9cm]{2Dtemp_reference.png}
\includegraphics[width=9cm]{2Dtemp_series3cooldown.png}
\caption{The 2D dust temperature distribution in the model series with different water ice `thermal histories'; for comparison, the first panel shows the reference disk model. The contour lines have the same meaning as in Fig.~\ref{fig:2Dtemp_referencemodel}.}
\label{fig:2Dtemp_ice-scenarios}
\end{figure*}
\end{appendix}
\bibliographystyle{aa
|
{
"timestamp": "2018-04-17T02:09:51",
"yymm": "1804",
"arxiv_id": "1804.05324",
"language": "en",
"url": "https://arxiv.org/abs/1804.05324"
}
|
\section{Introduction}
Massive Goldstone (MG) modes, often referred to as Higgs amplitude modes, and Nambu-Goldstone (NG) modes are ubiquitous in systems that involve spontaneous breaking of continuous symmetries \cite{goldstone-61,goldstone-62,higgs-64,pekker-15}. In the simplest U(1) symmetry breaking, the former induce amplitude oscillation of a complex order parameter \cite{higgsdef} and the latter induce phase oscillation.
Whereas NG modes have been studied in various condensed matter systems, MG modes have evaded observations until recently with only a few exceptions \cite{sooryakumar-81,giannetta-80,demsar-99}.
\par
Despite the increasing number of observations, for example, in superconductors \cite{sooryakumar-81,matsunaga-13,matsunaga-14,measson-14,sherman-15,katsumi-18}, quantum spin systems \cite{ruegg-08,jain-17,hong-17,souliou-17}, charge-density-wave materials \cite{demsar-99,yusupov-10}, and ultracold atomic gases \cite{bissbort-11,endres-12,leonard-17,behrle-18}, and theoretical studies \cite{littlewood-81,engelbrecht-97,tsuchiya-13,gazit-13,cea-15,nakayama-15,bjerlin-16, krull-16,moor-17,liberto-18}, fundamental aspects of MG modes in condensed matter systems have not been fully understood, in contrast to NG modes; spontaneous breaking of a continuous symmetry does not guarantee emergence of MG modes, while that of NG modes is ensured by the Goldstone theorem \cite{goldstone-62}.
For instance, whereas a MG mode appears in a Bardeen-Cooper-Schrieffer (BCS) superconductor \cite{sooryakumar-81,littlewood-81}, it does not exist in a Bose-Einstein condensate (BEC) \cite{varma-02}, despite the fact that both of the systems involve U(1) symmetry breaking and furthermore one evolves continuously to the other through the BCS-BEC crossover \cite{leggett-80,nozieres-85,sademelo-93,ohashi-02,regal-04}. Varma pointed out that the approximate particle-hole (p-h) symmetry, i.e., the linearly approximated fermionic dispersion $\xi_{\bm k}\simeq v_F(k-k_F)$ ($v_F$ is the Fermi velocity and $k_F$ is the Fermi wave number), results in the effective Lorentz invariance of the time-dependent Ginzburg-Landau equation in the weak-coupling BCS limit, which yields the decoupled amplitude and phase modes \cite{varma-02}. A pure amplitude MG mode also appears in lattice systems if the energy bands exhibit the rigorous p-h symmetry \cite{tsuchiya-13,cea-15}.
\par
It has been thus recognized in the previous studies that a pure amplitude MG mode emerges in superconductors if the dispersion of fermions $\xi_{\bm k}$ exhibits the p-h symmetry \cite{littlewood-81,engelbrecht-97,varma-02,tsuchiya-13,cea-15}. However, the p-h symmetry in the context of the previous works refers to the characteristic feature of the fermionic dispersion $\xi_{\bm k}$ that should be distinguished from the symmetry of the Hamiltonian. Meanwhile, clear understanding of the relation between {\it the symmetry of the Hamiltonian and MG modes} has not been reached.
\par
In this paper, we reveal the fundamental connection between {\it the discrete symmetry of the Hamiltonian and the emergence of pure amplitude MG modes}. We introduce three discrete operations for general non-relativistic systems of fermions, which we refer to ``charge-conjugation" ($\mathcal C$), ``parity" ($\mathcal P$), and ``time-reversal" ($\mathcal T$). The product of $\mathcal{CPT}$ (or its permutations) represents an exact symmetry analogous to the CPT theorem in the relativistic field theory \cite{lee-81}. We show that the standard BCS Hamiltonian with a p-h symmetric dispersion is invariant under $\mathcal C$, $\mathcal P$, $\mathcal T$, and $\mathcal{CPT}$ in addition to the global U(1) gauge invariance. If the U(1) symmetry is spontaneously broken in the superconducting ground state, the symmetries under $\mathcal P$ and $\mathcal T$ are simultaneously broken while the symmetry under $\mathcal C$ is unbroken. We rigorously show that amplitude and phase fluctuations of the gap function are uncoupled due to the unbroken $\mathcal C$. The MG mode thus induces pure amplitude oscillations of the gap function in a p-h symmetric system. It is also shown that the MG and NG modes reduce to the degenerate states in the normal phase due to the U(1) symmetry and they have opposite parity under $\mathcal T$. Therefore, the lifting of the degeneracy in the superconducting phase and the resulting emergence of the pure amplitude MG mode can be identified as a consequence of the the spontaneous breaking of $\mathcal T$ symmetry but not of $\mathcal P$ or U(1). Thus, the breaking of $\mathcal T$ proves to be responsible for the emergence of the pure amplitude MG mode.
\par
This paper is organized as follows: In Sec.~II, we present the model and introduce the pseudospin representation. In Sec.~III, we define the three discrete operations $\mathcal C$, $\mathcal T$, and $\mathcal P$ to discuss the symmetries of the Hamiltonian under the operations of $\mathcal C$, $\mathcal T$, $\mathcal P$, and $\mathcal{CPT}$. In Sec.~IV, we study the symmetry of the superconducting ground state. In Sec.~V, we discuss collective modes within the classical spin analysis. In Sec.~VI, we give a rigorous proof of the uncoupled amplitude and phase fluctuations of the gap function in a p-h symmetric system due to the unbroken $\mathcal C$. In Sec.~VII, we give a direct demonstration of the relation between the emergence of the pure amplitude MG mode and the spontaneously broken $\mathcal T$ symmetry. In Sec.~VIII, we summarize. We set $\hbar=k_{\rm B}=1$ throughout the paper.
\section{Pseudospin representation}
We study for simplicity the reduced BCS Hamiltonian \cite{fullH}
\begin{eqnarray}
\mathcal H&=&\sum_{\bm k,s}\xi_{\bm k}c_{\bm k s}^\dagger
c_{\bm k s} -g \sum_{\bm k,\bm k'} c_{\bm k\uparrow}^\dagger c_{-\bm
k\downarrow}^\dagger c_{-\bm k'\downarrow}c_{\bm k'\uparrow},
\label{eq.HBCS}
\end{eqnarray}
where $c_{\bm k s}^\dagger$ ($c_{\bm k s}$) is the creation (annihilation) operator of a fermion with momentum $\bm k$ and spin $s$ $(=\uparrow,\downarrow)$, $g(>0)$ denotes the attractive interaction between fermions, and $\xi_{\bm k}=\varepsilon_{\bm k}-\mu$ is the kinetic energy of a fermion measured from the chemical potential $\mu$. For example, $\varepsilon_{\bm k}=k^2/2m$ in a continuous system ($m$ is the mass of a fermion). We do not specify the form of $\varepsilon_{\bm k}$ for generality of argument.
\par
To discuss the symmetries of the Hamiltonian (\ref{eq.HBCS}), it is convenient to introduce the pseudospin representation \cite{anderson-58}: $S_{\mu\bm k}=\frac{1}{2}\Psi_{\bm k}^\dagger\tau_{\mu}\Psi_{\bm k}$ ($\mu=x,y,z$),
where $\bm\tau=(\tau_x,\tau_y,\tau_z)$ are Pauli matrices and $\Psi_{\bm k}=(c_{\bm k\uparrow},c_{-\bm k\downarrow}^\dagger)^t$ is the Nambu spinor \cite{nambu-60}.
Note that $S_{z\bm k}$ is related to the fermion number operator $n_{\bm k s}=c_{\bm k s}^\dagger c_{\bm k s}$ by $S_{z\bm k}=\frac{1}{2}(n_{\bm k\uparrow}+n_{\bm k\downarrow}-1)$. In the pseudospin language, the fermion vacuum is the spin-down state ($|0\rangle_{\bm k}=|\!\downarrow\rangle_{\bm k}$) and the fully occupied state is the spin-up state ($c_{\bm k\uparrow}^\dagger c_{-\bm k\downarrow}^\dagger|0\rangle_{\bm k}=|\!\uparrow\rangle_{\bm k}$).
\par
The pseudospin representation of the Hamiltonian (\ref{eq.HBCS}) is given by
\begin{eqnarray}
\mathcal H=\sum_{\bm k}2\xi_{\bm k} S_{z\bm k}-g\sum_{\bm k,\bm k'}\bm S_{\perp\bm k}\cdot \bm S_{\perp\bm k'},
\label{eq.HBCS_spin}
\end{eqnarray}
where $\bm S_{\perp\bm k}=(S_{x\bm k},S_{y\bm k})$. The kinetic energy (interaction) term is translated into the Zeeman (ferromagnetic XY exchange) term in the pseudospin language. The rotational symmetry of the Hamiltonian (\ref{eq.HBCS_spin}) in the $xy$-plane represents the U(1) symmetry of Eq.~(\ref{eq.HBCS}) with respect to the transformation $\Psi_{\bm k}\to e^{i\tau_z\alpha}\Psi_{\bm k}$.
\section{Hidden discrete symmetries}
In this section, we define three discrete transformations for fermions and discuss the symmetry of the Hamiltonian (\ref{eq.HBCS_spin}) under those operations.
\subsection{Charge-conjugation}
Let us consider a unitary transformation for the Nambu spinor \cite{serene-83}:
\begin{eqnarray}
{\mathcal C} \Psi_{\bm k} {\mathcal C}=\tau_x\Psi_{\underline{\bm k}},\quad {\mathcal C} \Psi_{\bm k}^\dagger {\mathcal C}=\Psi_{\underline{\bm k}}^\dagger\tau_x .
\end{eqnarray}
Here, $\underline{\bm k}$ is the mirror reflected wave vector of $\bm k$ with respect to the Fermi surface, i.e., $\bm k$ and $\underline{\bm k}$ are on the opposite side of the Fermi surface and away from it with the same minimum distance (see Figs.~\ref{fig.phsymmetry} (a)-(c)). For example, $\underline{\bm k} =(2k_F-k)\bm k/|\bm k|$ in a continuous system. Note that $\underline{\bm k}=\bm k$ if $\bm k$ is on the Fermi surface.
\begin{figure}
\centering
\includegraphics[width=6cm]{phsymmetry.eps}
\caption{Illustration of the wave vector $\underline{\bm k}$ and the dispersion $-\xi_{\underline{\bm k}}$ in (a) a continuous system and (b) the 1D lattice at half-filling ($\mu=0$). (c) $\underline{\bm k}$ for the half-filled energy band in the square lattice.}
\label{fig.phsymmetry}
\end{figure}
Since $\mathcal C$ transforms a particle ($c^\dagger$) into a hole ($c$) and {\it vice versa}, it can be referred to as a ``charge conjugation" operation.
$\mathcal C$ is specifically given by
\begin{eqnarray}
\mathcal C={\mathcal F}\prod_{\bm k}\sigma_{x\bm k},\quad \mathcal F=\prod_{\xi_{\bm k}>0} f_{\bm k,\underline{\bm k}},\label{eq.operatorC}
\end{eqnarray}
where $\sigma_{\mu\bm k}=2S_{\mu\bm k}$. The operator $f_{\bm k,\underline{\bm k}}$ swaps the state of $\bm k$ and that of $\underline{\bm k}$: $f_{\bm k,\underline{\bm k}}|\psi\rangle_{\bm k}|\phi\rangle_{\underline{\bm k}}= |\phi\rangle_{\bm k}|\psi\rangle_{\underline{\bm k}}$. One can show $\mathcal C^\dagger=\mathcal C$ and ${\mathcal C}^2=1$ from Eq.~(\ref{eq.operatorC}).
\par
The pseudospin operators are transformed by $\mathcal C$ as
\begin{eqnarray}
{\mathcal C}S_{\mu\bm k}{\mathcal C}=(-1)^{\delta_{\mu,x}+1}S_{\mu\underline{\bm k}},\quad{\mathcal C}S_{\mu}{\mathcal C}=(-1)^{\delta_{\mu,x}+1}S_{\mu},\label{eq.CSkC}
\end{eqnarray}
where $S_\mu=\sum_{\bm k}S_{\mu\bm k}$ is the total spin.
Equation~(\ref{eq.CSkC}) shows that $\mathcal C$ consists of the $\pi$ rotation of pseudospins about the $x$-axis and the swapping of $\bm k$ and $\underline{\bm k}$.
\par
Transforming Eq.~(\ref{eq.HBCS_spin}) by $\mathcal C$, we obtain
\begin{eqnarray}
{\mathcal C}\mathcal H{\mathcal C}=\sum_{\bm k}2(-\xi_{\underline {\bm k}})S_{z\bm k}-g\sum_{\bm k,\bm k'}{\bm S}_{\perp\bm k}\cdot{\bm S}_{\perp\bm k'}.\label{eq.CHC}
\end{eqnarray}
Hence, ${\mathcal C}{\mathcal H}{\mathcal C}={\mathcal H}$ and equivalently $[{\mathcal H},\mathcal C]=0$, if the fermion dispersion satisfies the condition
\begin{equation}
-\xi_{\underline{\bm k}}=\xi_{\bm k}.\label{eq.phcondition}
\end{equation}
Equation~(\ref{eq.phcondition}) indicates the invariance of the dispersion $\xi_{\bm k}$ under the successive mirror reflections with respect to $\xi=0$ and $k=k_F$ (see Figs.~\ref{fig.phsymmetry} (a) and \ref{fig.phsymmetry} (b)), which we refer to {\it particle-hole (p-h) symmetry} in view of the fact that the density of states $N(\xi)=\sum_{\bm k}\delta(\xi-\xi_{\bm k})$ is even if Eq.~(\ref{eq.phcondition}) holds.
\par
Figure~\ref{fig.phsymmetry}(a) shows that, whereas $\xi_{\bm k}=k^2/2m-\mu$ is not p-h symmetric, the linearized dispersion $\xi_{\bm k}\simeq v_F(k-k_F)$ is p-h symmetric. Therefore, a continuous system has an approximate p-h symmetry if the interaction is weak enough. On the other hand, Fig.~\ref{fig.phsymmetry} (b) illustrates that the tight-binding energy band in the $d$-dimensional cubic lattice $\xi_{\bm k}=-2t\sum_{i=1}^d\cos(k_i)$ ($t$ is the hopping matrix element) exhibits a rigorous p-h symmetry at half-filling ($\mu=0$).
\subsection{Time-reversal}
The ``time-reversal" operation of the Nambu spinor and the pseudospin operators are defined to be
\begin{eqnarray}
&&{\mathcal T}\Psi_{\bm k}{\mathcal T}^{-1}=\tau_y\Psi_{\underline{\bm k}},\quad {\mathcal T}\Psi_{\bm k}^\dagger {\mathcal T}^{-1}=\Psi_{\underline{\bm k}}^\dagger\tau_y,\\
&&{\mathcal T}S_{\mu\bm k} {\mathcal T}^{-1}=-S_{\mu\underline{\bm k}},\quad \mathcal T{S_\mu}{\mathcal T}^{-1}=-S_\mu.\label{eq.TST}
\end{eqnarray}
The time-reversal $\mathcal T$ can be written in the form
\begin{eqnarray}
\mathcal T =U_T\mathcal K,\quad U_T={\mathcal F}\prod_{\bm k}(-i\sigma_{y\bm k}),\label{eq.operatorT}
\end{eqnarray}
where $\mathcal K$ is the complex conjugation operator and $U_T$ is the unitary operator that rotates pseudospins $\pi$ about the $y$-axis and swaps $\bm k$ and $\underline{\bm k}$.
From Eq.~(\ref{eq.TST}), the p-h symmetric Hamiltonian that satisfies Eq.~(\ref{eq.phcondition}) is time-reversal invariant ${\mathcal T}\mathcal H{\mathcal T}^{-1}
=\mathcal H$. $\mathcal T$ reverses the time in the Heisenberg representation as ${\mathcal T}S_{\mu}(t){\mathcal T}^{-1}=-S_{\mu}(-t)$.
\par
It is important to note that $\mathcal T$ represents ``time-reversal" in the pseudospin space, which is different from the usual time-reversal operation discussed, for example, in Ref.~\onlinecite{sigrist-91}. Although the usual time-reversal symmetry is not broken in $s$-wave superconductors \cite{sigrist-91}, $\mathcal T$ is spontaneously broken simultaneously with the U(1) symmetry breaking as we shall show later.
\subsection{Parity}
The ``parity" operation, denoted by $\mathcal P$, is defined to be the inversion of pseudospins in the $xy$-plane. It is equivalent to the $\pi$ rotation about the $z$-axis and therefore can be represented as
\begin{equation}
\mathcal P=\prod_{\bm k}\sigma_{z\bm k}.\label{eq.operatorP}
\end{equation}
It satisfies $\mathcal P^\dagger=\mathcal P$ and ${\mathcal P}^2=1$. The transformation by $\mathcal P$ is given as
\begin{eqnarray}
\mathcal P\Psi_{\bm k}{\mathcal P}=\tau_z\Psi_{\bm k}, \quad {\mathcal P}\Psi_{\bm k}^\dagger {\mathcal P}=\Psi_{\bm k}^\dagger\tau_z,\\
{\mathcal P}S_{\mu\bm k}{\mathcal P}=(-1)^{\delta_{\mu,z}+1}S_{\mu\bm k},\ {\mathcal P}S_{\mu}{\mathcal P}=(-1)^{\delta_{\mu,z}+1}S_{\mu}.
\end{eqnarray}
The Hamiltonian (\ref{eq.HBCS_spin}) is invariant by $\mathcal P$: $\mathcal P\mathcal H\mathcal P={\mathcal H}$. Since the $\pi$ rotation in the $xy$-plane is an element of U(1), $\mathcal P$ is trivially broken in the U(1) broken ground state.
\subsection{CPT invariance}
The transformation by the product $\Theta={\mathcal C}{\mathcal P}{\mathcal T}$ is given as
\begin{eqnarray}
\Theta\Psi_{\bm k}\Theta^{-1}=i\Psi_{\bm k}, \quad \Theta\Psi_{\bm k}^\dagger\Theta^{-1}=-i\Psi_{\bm k}^\dagger,\\
{\Theta}S_{\mu\bm k}{\Theta}=(-1)^{\delta_{\mu,y}+1}S_{\mu\bm k},\ {\Theta}S_{\mu}{\Theta}=(-1)^{\delta_{\mu,y}+1}S_{\mu}.
\end{eqnarray}
Using Eqs.~(\ref{eq.operatorC}), (\ref{eq.operatorT}), and (\ref{eq.operatorP}), we obtain $\Theta=\prod_{\bm k}(-1)\cdot\mathcal K$ and thus $\Theta \mathcal H\Theta^{-1}=\mathcal H$.
Since the Lagrangian
$\mathcal L=\sum_{\bm k}i\Psi_{\bm k}^\dagger\frac{\partial }{\partial t}\Psi_{\bm k}-{\mathcal H}$
is transformed as
$\Theta\mathcal L(t)\Theta^{-1}=\mathcal L(-t)$, the action is invariant and therefore $\mathcal C\mathcal P\mathcal T$ and all other permutations of $\mathcal C$, $\mathcal P$, and $\mathcal T$ are exact symmetries analogous to the CPT invariance in relativistic systems \cite{lee-81}.
\section{Symmetry of the ground state}
We study the symmetries of the superconducting ground state focusing on that of a p-h symmetric system. It is reasonable to expect that all the symmetries of the true ground state are realized in the BCS wave function $|\Psi\rangle=\prod_{\bm k}(u_{\bm k}|\downarrow\rangle_{\bm k}+v_{\bm k}|\uparrow\rangle_{\bm k})$. Here, $u_{\bm k}=\sqrt{(1+\xi_{\bm k}/E_{\bm k})/2}$ and $v_{\bm k}=\sqrt{(1-\xi_{\bm k}/E_{\bm k})/2}$. The gap function is set positive real in the ground state without loss of generality $\Delta_0=g\sum_{\bm k}\langle c_{-\bm k\downarrow} c_{\bm k\uparrow} \rangle=g\langle S_{x}\rangle>0$. $E_{\bm k}=\sqrt{\xi_{\bm k}^2+\Delta_0^2}$ is the dispersion of single-particle excitations (bogolons).
$|\Psi\rangle$ represents the ground state of the mean-field (MF) Hamiltonian ${\mathcal H}_{\rm MF}=-\sum_{\bm k}\bm H_{\bm k}^0\cdot\bm S_{\bm k}$, where ${\bm S}_{\bm k}=(S_{x\bm k},S_{y\bm k}, S_{z\bm k})$.
The effective magnetic field $\bm H_{\bm k}^0=(2\Delta_0,0,-2\xi_{\bm k})$ lies in the $xz$-plane with the polar angle $\varphi_{\bm k}$ (see Fig.~\ref{fig.spinconfig} (b)), where $\sin\varphi_{\bm k}=\Delta_0/E_{\bm k}$ and $\cos\varphi_{\bm k}=-\xi_{\bm k}/E_{\bm k}$. Note that $\varphi_{\underline{\bm k}}=\pi-\varphi_{\bm k}$, if Eq.~(\ref{eq.phcondition}) holds.
The requirement that the average spin $\bm S_{\bm k}^0=\langle \bm S_{\bm k}\rangle$ is in parallel with $\bm H_{\bm k}^0$ leads to the MF gap equation
$1=g\sum_{\bm k}\frac{1}{2E_{\bm k}}$ \cite{anderson-58}.
\par
Figures~\ref{fig.spinconfig}(a) and \ref{fig.spinconfig}(b) show the pseudospin configuration of the superconducting ground state described by $|\Psi\rangle$. The pseudospins smoothly rotate sidewise in the $xz$-plane from up to down towards the positive $x$-direction as $k$ increases \cite{anderson-58}. The spontaneous U(1) symmetry breaking with respect to the phase of the gap function sets the direction of rotating spins projected in the $xy$-plane. In a p-h symmetric system, ${\bm S}_{\underline{\bm k}}$ is the mirror reflected image of $\bm S_{\bm k}$ with respect to the $xy$-plane.
\begin{figure}
\centering
\includegraphics[width=8.5cm]{spinconfig.eps}
\caption{Schematic illustration of the pseudospin distribution $\bm S_{\bm k}$ described by the BCS wave function $|\Psi\rangle$ for a positive real gap function (a) as a function of $k$ \cite{anderson-58} and (b) on the Bloch sphere. (a) Spins rotate in the $xz$-plane from up to down towards the positive $x$-direction as $k$ increases from below to above $k_F$. (b) In a p-h symmetric system, ${\bm S}_{\underline{\bm k}}$ is the mirror reflected image of $\bm S_{\bm k}$ with respect to the $xy$-plane.}
\label{fig.spinconfig}
\end{figure}
\par
The symmetry under $\mathcal C$ is unbroken in the ground state of a p-h symmetric system. In fact, using $u_{\underline{\bm k}}=v_{\bm k}$ and $v_{\underline{\bm k}}=u_{\bm k}$, the BCS wave function is shown to be parity even ($\mathcal C|\Psi\rangle=|\Psi\rangle$) that reflects the invariance of the MF Hamiltonian ($\mathcal C\mathcal H_{\rm MF}\mathcal C=\mathcal H_{\rm MF}$). As shown in Fig.~\ref{fig.spinconfig}, the pseudospin configuration is indeed invariant under the $\pi$ rotation of spins about the $x$-axis followed by the swapping of $\bm k$ and $\underline{\bm k}$.
In contrast, the symmetries under $\mathcal T$ and $\mathcal P$ are spontaneously broken accompanied with the U(1) symmetry breaking. The operation of either $\mathcal T$ or $\mathcal P$ flips the sign of the gap as
\begin{eqnarray}
{\mathcal T}{\mathcal H}_{\rm MF}{\mathcal T}^{-1}={\mathcal P}{\mathcal H_{\rm MF}}{\mathcal P}=-\sum_{\bm k}\bar{\bm H}^{0}_{\bm k}\cdot\bm S_{\bm k}=\bar{\mathcal H}_{\rm MF},\\
\mathcal T|\Psi\rangle=|\bar{\Psi}\rangle,\quad\mathcal P|\Psi\rangle=\left\{\prod_{\bm k}(-1)\right\}\cdot|\bar\Psi\rangle.
\end{eqnarray}
\begin{figure}
\centering
\includegraphics[width=5cm]{doublewell.eps}
\caption{Schematic illustration of the double-well potential for a real gap function and the spontaneous breaking of the symmetries under $\mathcal T$ and $\mathcal P$. The operation of either $\mathcal T$ or $\mathcal P$ flips the sign of the gap function and transforms $|\Psi\rangle$ to $|\bar\Psi\rangle$.}
\label{fig.doublewell}
\end{figure}
Figure~\ref{fig.doublewell} schematically illustrates the spontaneous breaking of the symmetry under $\mathcal T$ and $\mathcal P$ and their operations on $|\Psi\rangle$. Hereafter, the overline represents the replacement $\Delta_0\to -\Delta_0$, e.g., $\bar{\bm H}^{0}_{\bm k}=(-2\Delta_0,0,-2\xi_{\bm k})$ and $|\bar\Psi\rangle=\prod_{\bm k} (u_{\bm k}|\downarrow\rangle_{\bm k}-v_{\bm k}|\uparrow\rangle_{\bm k})$.
\par
The symmetries of the Hamiltonian and the ground state are compared between p-h symmetric and non-symmetric systems in Table~1. It shows that the broken symmetry of $\mathcal T$ and unbroken symmetry of $\mathcal C$ are characteristic to a p-h symmetric system. Given the fact that a pure amplitude MG mode arises only in a p-h symmetric system as shown later, Table~1 implies that it results from the broken $\mathcal T$ and $\mathcal C$, which we reveal in the following.
\begin{table}[tb]
\begin{tabular}{|c||cc|cc|} \hline
& \multicolumn{2}{c|}{p-h symmetric} & \multicolumn{2}{c|}{p-h non-symmetric} \\ \cline{2-5}
Symmetry & $\mathcal H$ & $|\Psi\rangle$ & $\mathcal H$ & $|\Psi\rangle$ \\ \hline
$\mathcal C$ & \checkmark & \checkmark & $\times$ & $\times$ \\
$\mathcal T$ & \checkmark & $\times$ & $\times$ & $\times$ \\\hline
$\mathcal P$ & \checkmark & $\times$ & \checkmark & $\times$ \\
$\Theta=\mathcal{CPT}$ & \checkmark& \checkmark & \checkmark & \checkmark \\
U(1) & \checkmark & $\times$ & \checkmark & $\times$\\ \hline
\end{tabular}
\caption{Symmetry of the Hamiltonian $\mathcal H$ and the ground state wave function $|\Psi\rangle$ for a p-h symmetric system ($\xi_{\bm k}=-\xi_{\underline{\bm k}}$) and a p-h non-symmetric system ($\xi_{\bm k}\neq-\xi_{\underline{\bm k}}$). \checkmark and $\times$ mean presence and absence of the symmetry, respectively.}
\end{table}
\section{Collective modes}
We first discuss collective modes within the classical spin analysis~\cite{anderson-58} (Details are given in Appendix B). We study dynamics of the pseudospins based on the MF Hamiltonian
$\mathcal H_{\rm MF}'=-\sum_{\bm k}{\bm H}_{\bm k}\cdot{\bm S}_{\bm k}$.
Here, the magnetic field $\bm H_{\bm k}=(2{\rm Re}\Delta,-2{\rm Im}\Delta,-2\xi_{\bm k})$ is self-consistently determined by the gap function $\Delta=g\sum_{\bm k}\langle c_{-\bm k\downarrow}c_{\bm k\uparrow}\rangle=g(\langle S_x\rangle-i\langle S_y\rangle)$, which is allowed to take complex values. The time evolution of $\bm S_{\bm k}(t)$, which is treated as a classical spin, is described by the equation of motion
\begin{equation}
\frac{d\bm S_{\bm k}}{dt}={\bm S}_{\bm k}\times\bm H_{\bm k}.\label{eq.Skt}
\end{equation}
\par
Introducing amplitude and phase fluctuations from the ground state $\Delta=(\Delta_0+\delta \Delta)e^{i\delta\theta}$, one finds that spin fluctuations in the $x$-direction induce amplitude fluctuations $\delta \Delta =g\delta S_x$ and those in the $y$-direction induce phase fluctuations $\delta\theta=-g\delta S_y/\Delta_0$, where $\delta\bm S_{\bm k}=\bm S_{\bm k}(t)-{\bm S}_{\bm k}^0$.
Linearizing Eq.~(\ref{eq.Skt}) by fluctuations $\delta\Delta,\delta\theta\propto e^{-i\omega t}$, we obtain
\begin{eqnarray}
\left(1-2g\chi_{xx}(\omega)\right)\delta \Delta-2g\chi_{xy}(\omega)\Delta_0\delta\theta=0,\label{eq.deltaSx}\\
2g\chi_{yx}(\omega)\delta\Delta-\left(1-2g\chi_{yy}(\omega)\right)\Delta_0\delta\theta=0,\label{eq.deltaSy}
\end{eqnarray}
where $\chi_{\mu\nu}(\omega)$ are the dynamical spin susceptibilities defined as
$\chi_{\mu\nu}(\omega)=-i\int_0^\infty \langle[S_\nu,S_\mu(t)]\rangle e^{-i\omega t}dt$ ($S_\mu(t)$ is the Heisenberg representation and $\langle\dots\rangle$ denotes the average). For example, $\chi_{xy}$ represents the coupling of amplitude and phase, while $\chi_{zx}$ represents that of density and amplitude. The susceptibilities are calculated as
\begin{eqnarray}
&&\chi_{xx}=\sum_{\bm k}\frac{\xi_{\bm k}^2}{E_{\bm k}(4E_{\bm k}^2-\omega^2)},\!\chi_{yy}=\sum_{\bm k}\frac{E_{\bm k}}{4E_{\bm k}^2-\omega^2},\label{eq.suscept_xxyy}\\
&&\chi_{xy}=-\chi_{yx}=\frac{i\omega}{2}\sum_{\bm k}\frac{\xi_{\bm k}}{E_{\bm k}(4E_{\bm k}^2-\omega^2)}.\label{eq.suscept_xy}
\end{eqnarray}
\par
Using the MF gap equation, one finds that Eqs.~(\ref{eq.deltaSx}) and (\ref{eq.deltaSy}) have the NG mode solution ($\delta\theta\neq0$, $\delta\Delta=0$) with $\omega=0$. They also have a solution for a pure amplitude mode ($\delta\Delta\neq 0$ and $\delta \theta=0$) with $\omega=2\Delta_0$, if phase and amplitude are uncoupled $\chi_{xy}(2\Delta_0)=\chi_{yx}(2\Delta_0)=0$. From Eq.~(\ref{eq.suscept_xy}), this leads to the condition
\begin{eqnarray}
\sum_{\bm k}\frac{1}{E_{\bm k}\xi_{\bm k}}=\int d\xi\frac{N(\xi)}{\xi\sqrt{\xi^2+\Delta_0^2}}=0.\label{eq.p-h}
\end{eqnarray}
Equation~(\ref{eq.p-h}) is satisfied if $N(\xi)$ is even. Thus, MG mode arises as a {\it pure amplitude mode} in a p-h symmetric system \cite{engelbrecht-97}.
\par
The p-h symmetry also ensures fermion number conservation ($\delta S_z=0$) \cite{deltaSz0}. $\delta S_z$ is represented as
\begin{eqnarray}
\delta S_z=2\chi_{zx}(\omega)\delta\Delta+2\chi_{zy}(\omega)\Delta_0\delta\theta,\label{eq.deltaSz}
\end{eqnarray}
where $\chi$s are given by
\begin{eqnarray}
\chi_{zx}=\sum_{\bm k}\frac{\Delta_0\xi_{\bm k}}{E_{\bm k}(4E_{\bm k}^2-\omega^2)},\!
\chi_{zy}=\sum_{\bm k}\frac{i\omega \Delta_0/2}{E_{\bm k}(4E_{\bm k}^2-\omega^2)}\label{eq.suscept_zxy}.
\end{eqnarray}
The MG mode solution ($\delta\Delta\neq 0$, $\delta\theta=0$, and $\omega=2\Delta_0$) satisfies $\delta S_z=0$, if $\chi_{zx}(2\Delta_0)=0$, which reduces to Eq.~(\ref{eq.p-h}). Hence, the MG mode does not induce density fluctuation and indeed conserves total fermion number $N=2S_z+\sum_{\bm k}1$.
\par
If the p-h symmetry is absent, due to $\chi_{zx}(\omega)\neq 0$ and $\chi_{zy}(\omega)\neq 0$, Eq.~(\ref{eq.deltaSz}) indicates that $\delta \Delta$ and $\delta \theta$ must be finite in order to satisfy $\delta S_z=0$. As a result, $\delta\Delta$ is inevitably coupled with $\delta \theta$ and therefore the MG mode induces both amplitude and phase fluctuations. The energy of the MG mode becomes greater than $2\Delta_0$ \cite{tsuchiya-13,cea-15}.
\section{Rigorous proof of $\chi_{xy}=\chi_{zx}=0$}
The arguments in the last section are based on the MF approximation restricted to zero temperature ($T=0$). We rigorously show that amplitude is decoupled from phase and density in a p-h symmetric system at any temperature.
We focus on $\chi_{zx}(\omega)$ and evaluate $\langle [S_x,S_z(t)] \rangle\propto\sum_n e^{-E_n/T}\langle n|[S_x,S_z(t)]|n\rangle$. Here, $|n\rangle$ denotes an exact eigenstate of $\mathcal H$ with energy $E_n$. Since $\mathcal C$ is not broken, $|n\rangle$ is parity either even or odd under $\mathcal C$. Using the fact that $S_x$ and $S_z$ have opposite parity under $\mathcal C$, we obtain
\begin{eqnarray}
\langle n|S_xS_z(t)|n\rangle&=&(\langle n|{\mathcal C})({\mathcal C}S_x{\mathcal C})({\mathcal C}S_z(t){\mathcal C})({\mathcal C}|n\rangle)\nonumber\\
&=&-\langle n|S_xS_z(t)|n\rangle=0.
\end{eqnarray}
One can analogously show $\langle n|S_z(t)S_x| n\rangle=\langle n|S_zS_x(t)|n\rangle=\langle n|S_x(t)S_z|n\rangle=0$ and therefore $\chi_{zx}(\omega)=\chi_{xz}(\omega)=0$.
$\chi_{xy}(\omega)=\chi_{yx}(\omega)=0$ can be shown analogously using the opposite parity of $S_x$ and $S_y$. Thus, the unbroken symmetry under $\mathcal C$ is essential for the pure amplitude character of the MG mode.
\section{Emergence of the MG mode by the broken $\mathcal T$ symmetry}
We show that the spontaneous breaking of $\mathcal T$ is responsible for the emergence of the MG mode.
The creation operator of the MG mode $\beta_{\rm H}^\dagger$ and that of the NG mode $\beta_{\rm NG}^\dagger$ derived by the Holstein-Primakoff theory are given by (see Appendix C for details)
\begin{eqnarray}
&&\beta_{\rm H}^\dagger=A\sum_{\bm k}\frac{\xi_{\bm k}}{E_{\bm k}}\left(\frac{S'^+_{\bm k}}{2|\Delta_0|-2E_{\bm k}}+\frac{S'^-_{\bm k}}{2|\Delta_0|+2E_{\bm k}}\right),\label{eq.betaH}\\
&&\beta_{\rm NG}^\dagger=A'\sum_{\bm k}\frac{1}{E_{\bm k}}(S'^+_{\bm k}+S'^-_{\bm k}).\label{eq.betaNG}
\end{eqnarray}
Here, $S'^{\pm}_{\bm k}=S'_{x\bm k}\pm iS'_{y\bm k}$, which creates and annihilates a pair of bogolons, are the raising and lowering operators of the pseudospins for bogolons $\bm S'_{\bm k}=(S'_{x\bm k},S'_{y\bm k},S'_{z\bm k})$. $S_{\bm k}'^{\pm}$ are transformed as (see Appendix A)
\begin{eqnarray}
{\mathcal C}S_{\bm k}'^{\pm}{\mathcal C}=-S_{\underline{\bm k}}'^{\pm},\mathcal PS_{\bm k}'^\pm\mathcal P=- \bar{S}_{\bm k}'^\pm,\mathcal TS_{\bm k}'^\pm\mathcal T^{-1}=\bar{S}_{\underline{\bm k}}'^{\pm}.\label{eq.TSpT}
\end{eqnarray}
Using Eq.~(\ref{eq.TSpT}), one can show that the MG mode is even and the NG mode is odd under $\mathcal C$:
\begin{equation}
{\mathcal C}\beta_{\rm H}^\dagger{\mathcal C}=\beta_{\rm H}^\dagger,\quad{\mathcal C}\beta_{\rm NG}^\dagger {\mathcal C}=-\beta_{\rm NG}^\dagger.\label{eq.CbetaC}
\end{equation}
Their opposite parity under $\mathcal C$ is consistent with the uncoupled phase and amplitude. A single MG mode is thus prohibited to decay into odd number of NG modes by the selection rule. Moreover, since the excited states of energy $2\Delta_0$ with a pair of bogolons are odd under $\mathcal C$ (see Appendix A), a MG mode with energy $2\Delta_0$ is stable against decay into independent bogolons.
\par
The MG and NG modes thus have definite parity under $\mathcal C$ due to the unbroken $\mathcal C$, while the discrete symmetries under $\mathcal T$ and $\mathcal P$ are broken. From Eq.~(\ref{eq.TSpT}), we obtain
\begin{eqnarray}
&&{\mathcal T}\beta_{\rm H}^\dagger {\mathcal T}^{-1}={\mathcal P}\beta_{\rm H}^\dagger {\mathcal P}=-\bar\beta_{\rm H}^\dagger,\label{eq.TbetaHT}\\
&&{\mathcal T}\beta_{\rm NG}^\dagger {\mathcal T}^{-1}=\bar\beta_{\rm NG}^\dagger,\quad{\mathcal P}\beta_{\rm NG}^\dagger {\mathcal P}=-\bar\beta_{\rm NG}^\dagger\label{eq.TbetaNGT},
\end{eqnarray}
where $\beta_{\rm H}^\dagger\to\bar\beta_{\rm H}^\dagger$ and $\beta_{\rm NG}^\dagger\to\bar\beta_{\rm NG}^\dagger$ by the replacement $\Delta_0\to-\Delta_0$.
Note that using Eqs.~(\ref{eq.CbetaC}), (\ref{eq.TbetaHT}), and (\ref{eq.TbetaNGT}), $\Theta\beta_{\rm H}^\dagger\Theta^{-1}=\beta_{\rm H}^\dagger$ and $\Theta\beta_{\rm NG}^\dagger\Theta^{-1}=\beta_{\rm NG}^\dagger$ are indeed satisfied.
\par
Denoting the vacuum state for $\beta_{\rm H}$ and $\beta_{\rm NG}$ ($\bar\beta_{\rm H}$ and $\bar\beta_{\rm NG}$) as $|{\rm vac}\rangle$ ($|\overline{\rm vac}\rangle$), we have the relation $\mathcal T|{\rm vac}\rangle=\mathcal P|{\rm vac}\rangle=|\overline{\rm vac}\rangle$, since either $\mathcal T$ or $\mathcal P$ flips the sign of the gap function \cite{Pvac}.
From Eqs.~(\ref{eq.CbetaC}), (\ref{eq.TbetaHT}) and (\ref{eq.TbetaNGT}), one obtains
\begin{eqnarray}
&&\mathcal C(\beta_{\rm H}^\dagger|{\rm vac}\rangle)=\beta_{\rm H}^\dagger|{\rm vac}\rangle,\label{eq.CbetaH}\\
&&\mathcal T(\beta_{\rm H}^\dagger|{\rm vac}\rangle)=\mathcal P(\beta_{\rm H}^\dagger|{\rm vac}\rangle)=-\bar\beta_{\rm H}^\dagger|\overline{\rm vac}\rangle,\label{eq.TbetaH}\\
&&\mathcal C(\beta_{\rm NG}^\dagger|{\rm vac}\rangle)=-\beta_{\rm NG}^\dagger|{\rm vac}\rangle,\label{eq.CbetaNG}\\
&&\mathcal T(\beta_{\rm NG}^\dagger|{\rm vac}\rangle)=\bar\beta_{\rm NG}^\dagger|\overline{\rm vac}\rangle,\label{eq.TbetaNG}\\
&&\mathcal P(\beta_{\rm NG}^\dagger|{\rm vac}\rangle)=-\bar\beta_{\rm NG}^\dagger|\overline{\rm vac}\rangle.\label{eq.PbetaNG}
\end{eqnarray}
\par
In the normal phase, setting $\Delta_0=0$, $\beta_{\rm H}^\dagger|\rm vac\rangle$ and $\bar\beta_{\rm H}^\dagger|\overline{\rm vac}\rangle$ trivially reduce to the same state $\beta_{\rm H0}^\dagger|\rm FS\rangle\equiv|\phi_{\rm H}\rangle$, while $\beta_{\rm NG}^\dagger|\rm vac\rangle$ and $\bar\beta_{\rm NG}^\dagger|\overline{\rm vac}\rangle$ reduce to $\beta_{\rm NG0}^\dagger|{\rm FS}\rangle\equiv|\phi_{\rm NG}\rangle$.
Here, $|{\rm FS}\rangle$ denotes the vacuum in the normal phase. $\beta_{\rm H0}^\dagger$ and $\beta_{\rm NG0}^\dagger$ are given by
\begin{eqnarray}
\beta_{\rm H0}^\dagger\equiv\left.\beta_{\rm H}^\dagger\right|_{\Delta_0=0}\propto\sum_{\bm k}\frac{1}{\xi_{\bm k}}S_{y\bm k}\label{eq.betaH0},\\
\beta_{\rm NG0}^\dagger\equiv\left.\beta_{\rm NG}^\dagger\right|_{\Delta_0=0}\propto\sum_{\bm k}\frac{1}{\xi_{\bm k}}S_{x\bm k}\label{eq.betaNG0}.
\end{eqnarray}
Since $\beta_{\rm NG0}^\dagger$ can be transformed to $\beta_{\rm H0}^\dagger$ by the $\pi/2$ rotation about the $z$-axis in the pseudospin space, Eqs.~(\ref{eq.betaH0}) and (\ref{eq.betaNG0}) indicate that the $|\phi_{\rm H}\rangle$ and $|\phi_{\rm NG}\rangle$ states are degenerate in the normal phase before breaking the U(1) symmetry \cite{cooperon,degeneracy}.
Setting $\Delta_0=0$ in Eqs.~(\ref{eq.CbetaH}), (\ref{eq.TbetaH}), (\ref{eq.CbetaNG}), (\ref{eq.TbetaNG}), and (\ref{eq.PbetaNG}), we obtain \cite{CPTcommutation}
\begin{eqnarray}
\mathcal C|\phi_{\rm H}\rangle=|\phi_{\rm H}\rangle,\ \mathcal T|\phi_{\rm H}\rangle=\mathcal P|\phi_{\rm H}\rangle=-|\phi_{\rm H}\rangle,\\
\mathcal T|\phi_{\rm NG}\rangle=|\phi_{\rm NG}\rangle,\ \mathcal C|\phi_{\rm NG}\rangle=\mathcal P|\phi_{\rm NG}\rangle=-|\phi_{\rm NG}\rangle.
\end{eqnarray}
The above equations show that $|\phi_{\rm H}\rangle$ is odd and $|\phi_{\rm NG}\rangle$ is even under $\mathcal T$. On the other hand, both $|\phi_{\rm H}\rangle$ and $|\phi_{\rm NG}\rangle$ are odd under $\mathcal P$. From these facts, we can conclude that the lifting of the degeneracy of $|\phi_{\rm H}\rangle$ and $|\phi_{\rm NG}\rangle$ in the superconducting phase
should be induced by the spontaneous breaking of $\mathcal T$ symmetry, not by the breaking of $\mathcal P$ or U(1) symmetry. Consequently, the breaking of $\mathcal T$ proves to be responsible for the emergence of the pure amplitude MG mode.
The spontaneously induced magnetic field that breaks the $\mathcal T$ symmetry is given by $H_{x\bm k}^0=2\Delta_0$. Therefore, the energy splitting between the MG and NG modes should be of the order of $|H_{x\bm k}^0|=2\Delta_0$. This is consistent with the fact that the energy gap of the MG mode is $2\Delta_0$.
\section{Conclusions}
Extending the previous understanding of the emergence of the MG mode in the presence of the p-h symmetric fermionic dispersion, we have revealed the fundamental connection between the emergence of the pure amplitude MG mode and the discrete symmetry of the Hamiltonian in superconductors, which has not been clarified in the previous works. We have shown that a non-relativistic Hamiltonian for fermions with a p-h symmetric dispersion exhibits nontrivial discrete symmetries under $\mathcal C$, $\mathcal P$, $\mathcal T$, and $\mathcal{CPT}$. In the U(1) broken superconducting ground state of such a p-h symmetric system, $\mathcal T$ and $\mathcal P$ are spontaneously broken, while $\mathcal C$ is unbroken. We have shown that the spontaneous breaking of the discrete $\mathcal T$ symmetry leads to the emergence of the MG mode that induces pure amplitude oscillation of the gap function due to the unbroken $\mathcal C$. It may be possible to show a similar relation between the discrete symmetry of the Hamiltonian and the emergence of the MG modes in other non-relativistic systems, such as ultracold bosons in optical lattices \cite{endres-12,liberto-18} and quantum spin systems \cite{jain-17,hong-17}.
\section*{Acknowledgments}
ST is grateful to C. A. R. S\'a de Melo, T. Nikuni, and N. Tsuji for inspiring discussions. DY thanks the support of CREST, JST No. JPMJCR1673, and of JSPS Grant-in-Aid for Scientific Research (KAKENHI Grant No. 18K03525). The work of RY and MN is supported by the Ministry of Education, Culture, Sports, Science (MEXT)-Supported Program for the Strategic Research Foundation at Private Universities ``Topological Science'' (Grant No. S1511006). The work of MN is also supported in part by
JSPS Grant-in-Aid for Scientific Research (KAKENHI Grant No.~16H03984 and 18H01217), and by a Grant-in-Aid for Scientific Research on Innovative Areas ``Topological Materials Science'' (KAKENHI Grant No.~15H05855) from the MEXT of Japan.
|
{
"timestamp": "2018-11-07T02:08:46",
"yymm": "1804",
"arxiv_id": "1804.05577",
"language": "en",
"url": "https://arxiv.org/abs/1804.05577"
}
|
\section{Introduction} \label{sec:intro}
Molecules are a unique and powerful tool in astronomy.
They provide an excellent diagnosis of the physical conditions and processes in the regions where they reside.
Progress in this field called astrochemistry is mainly driven by observations from various single-dish and interferometers at millimeter/sub-millimeter wavelengths along with space telescopes at mid- and far-infrared wavelengths \citep[e.g.,][]{2017arXiv171005940V}.
Unsaturated carbon-chain molecules tend to be abundant in young low-mass starless cores and deficient in low-mass star-forming cores \citep{1992apj...392...551, 2009apj...699...585}, because they are mainly formed from ionized carbon (C$^{+}$) and atomic carbon (C) via ion-molecule reactions in the early stage of molecular clouds and destroyed mainly by reaction with oxygen atoms and depleted onto dust grains in the late stage.
Carbon-chain molecules have been found to be abundant around two low-mass protostars; IRAS 04368+2557 in L1527 \citep{2008ApJ...672..371S} and IRAS 15398$-$3359 in Lupus \citep{2009ApJ...697..769S}.
In these protostars, carbon-chain molecules are newly formed from CH$_{4}$ evaporated from dust grains in the lukewarm ($T \sim 20-30$ K) gas \citep{2008ApJ...672..371S}.
Such a carbon-chain chemistry around the low-mass protostars was named warm carbon chain chemistry \citep[WCCC,][]{2008ApJ...672..371S}.
The difference between hot corino chemistry and WCCC is considered to be brought by the different timescale of the starless core phase; the long and short starless core phases lead to hot corino and WCCC sources, respectively \citep{2008ApJ...672..371S}.
Recently, sources possessing both hot corino and WCCC characteristics have been found and high-spatial-resolution observations showed that the spatial distributions of carbon-chain molecules and COMs are different with each other \citep[e.g.,][]{2016ApJ...830L..37I}.
Saturated complex organic molecules (COMs), organic species consisting of more than six atoms and being rich in hydrogen \citep{2009ARA&A..47..427H}, are classically known to be abundant in the dense and hot ($n > 10^{6}$ cm$^{-3}$, $T \geq 100$ K) gas around young stellar objects (YSOs).
Besides, COMs also have been found in the gas phase before ice thermally evaporates at temperatures above 100 K \citep{2014ApJ...795L...2V}.
At these low temperatures, COMs can be desorbed from icy grain mantles via different types of non-thermal desorption process such as: (1) the cosmic-ray desorption mechanism \citep{2014MNRAS.440.3557R}, (2) the chemical desorption mechanism \citep[desorption due to the exothermicity of surface reactions;][]{2007A&A...467.1103G}, and (3) the photo-desorption \citep{2016MNRAS.459.3756R}.
Role of barrier-less gas phase reactions to form COMs was also proposed recently by \citet{2015MNRAS.449L..16B}.
Recent observations show the presence of some COMs, methylformate (HCOOCH$_{3}$), dimethyl ether (CH$_{3}$OCH$_{3}$), and methyl cyanide (CH$_{3}$CN), and the complex radical methoxy (CH$_{3}$O) in regions where the dust temperature is less than 30 K; pre-stellar cores \citep{2012A&A...541L..12B, 2014ApJ...795L...2V, 2016A&A...594A.117P} and cold envelopes of low-mass protostars \citep{2010ApJ...716..825O, 2012ApJ...759L..43C, 2014ApJ...791...29J}.
Grain-surface chemistry certainly plays a role, for example in forming hydrogenated species during the prestellar phase \citep{1982A&A...114..245T, 2012A&ARv..20...56C}, but not necessarily in the formation of all COMs.
Not only in the low-mass star-forming regions, new questions arise in the chemistry around massive young stellar objects (MYSOs).
\citet{2015A&A...576A..45F} compared the chemistry between organic-poor MYSOs and organic-rich MYSOs, namely hot cores.
They suggested that hot cores are not required to form COMs and temperature and initial ice composition possibly affect complex organic distributions around MYSOs.
\citet{2014MNRAS...443...2252} detected HC$_{5}$N, the second shortest cyanopolyynes (HC$_{2n+1}$N, $n=1,2,3,...$), in 35 hot cores associated with the 6.7 GHz methanol masers, which give us the exact positions of MYSOs \citep{2013MNRAS.431.1752U}.
However, there remained the possibility that the emission of HC$_{5}$N comes from the outer cold molecular clouds or other molecular clouds in the large single-dish beam ($\sim 0.95$\arcmin).
\citet{2017ApJ...844...68T} carried out observations toward four MYSOs where \citet{2014MNRAS...443...2252} detected HC$_{5}$N, using the Green Bank 100-m telescope (GBT) and the Nobeyama 45-m radio telescope.
The four target sources were selected adding three criteria mentioned in Section \ref{sec:obs} and we chose three sources showing the highest HC$_{5}$N peak intensities (G12.89+0.49, G16.86$-$2.16, and G28.28$-$0.36) and a source showing the low HC$_{5}$N peak intensity with the high CH$_{3}$CN peak intensities (G10.30$-$0.15).
They detected the high-excitation-energy ($E_{\rm {u}}/k \sim 100$ K) lines of HC$_{5}$N, which cannot be detected if HC$_{5}$N exists in the cold dark clouds, in the three sources (G12.89+0.49, G16.86$-$2.16, and G28.28$-$0.36) and confirmed that HC$_{5}$N exists in the warm gas around MYSOs.
Therefore, carbon-chain molecules seem to be formed in the lukewarm gas around MYSOs, as well as WCCC sources in the low-mass star-forming regions.
At the present stage, we do not know the relationships between carbon-chain molecules and COMs around MYSOs.
In this paper, we report the observational results in the 42$-$46, 82$-$103, 338.2$-$339.2 and 348.45$-$349.45 GHz bands obtained with the Nobeyama 45-m radio telescope and the Atacama Submillimeter Telescope Experiment (ASTE) 10-m telescope toward three MYSOs, G12.89+0.49, G16.86$-$2.16, and G28.28$-$0.36.
We derive the rotational temperatures and beam-averaged column densities of HC$_{3}$N, CH$_{3}$OH, and CH$_{3}$CCH (Section \ref{sec:ana}).
We compare the spectra and the chemical composition among the three sources, combining with previous HC$_{5}$N data \citep{2017ApJ...844...68T}, in order to investigate the relationship between carbon-chain species and COMs in high-mass star-forming regions (Section \ref{sec:discuss}).
\section{Observations} \label{sec:obs}
The observations presented in this paper were conducted in the Nobeyama 45-m radio telescope and the ASTE joint observation program (Proposal ID: JO161001, PI: Kotomi Taniguchi, 2016-2017 season).
The observing parameters of each frequency band are summarized in Table \ref{tab:obs}.
The Nobeyama 45-m telescope data have been partly published in \citet{2017ApJ...844...68T}.
The source selection criteria were described in \citet{2017ApJ...844...68T} as follows:
\begin{enumerate}
\item The source declination is above $-21$\arcdeg,
\item the distance ($D$) is within 3 kpc, and
\item CH$_{3}$CN was detected \citep[$\int T_{\rm {mb}}dv > 0.5$ K km s$^{-1}$ for $J_{K}=5_{0}-4_{0}$ line,][]{2006MNRAS...367...553}.
\end{enumerate}
Eight sources in the HC$_{5}$N-detected source list of \citet{2014MNRAS...443...2252} meet the above three criteria.
We chose three sources among the eight selected sources which show the highest peak intensities of HC$_{5}$N ($T_{\rm {mb}} > 120$ mK).
Table \ref{tab:source} summarizes the properties of our three target sources.
The observed positions correspond to the 6.7 GHz methanol maser positions, which show exact positions of the MYSOs \citep{2013MNRAS.431.1752U}.
\floattable
\begin{deluxetable}{ccccccc}
\tablecaption{Observing parameters\label{tab:obs}}
\tablewidth{0pt}
\tablehead{
\colhead{Frequency} & \colhead{Telescope} & \colhead{Beam size} & \colhead{$\eta_{\rm {mb}}$} & \colhead{$T_{\rm {sys}}$} & \colhead{$\Delta \nu$} & \colhead{$T_{\rm {rms}}$\tablenotemark{a}} \\
\colhead{(GHz)} & \colhead{} & \colhead{($^{\prime\prime}$)} & \colhead{(\%)} & \colhead{(K)} & \colhead{(kHz)} & \colhead{(mK)}
}
\startdata
42$-$46 & Nobeyama & 37 & 71 & 120$-$150 & 122.07 & 6$-$14 \\
82$-$103 & Nobeyama & 18\tablenotemark{b} & 54\tablenotemark{b} & 120$-$200 & 244.14 & 3$-$6 \\
338$-$349.4 & ASTE & 22 & 60 & 300$-$700 & 1000 &16$-$24 \\
\enddata
\tablenotetext{a}{In $T_{\rm {A}}^{*}$ scale.}
\tablenotetext{b}{The values are at the 86 GHz.}
\end{deluxetable}
\floattable
\begin{deluxetable}{ccccccc}
\tablecaption{Properties of our target sources \label{tab:source}}
\tablewidth{0pt}
\tablehead{
\colhead{Source} & \colhead{R.A.\tablenotemark{a}} & \colhead{Decl.\tablenotemark{a}} &\colhead{$D$} & \colhead{$V_{\rm {LSR}}$\tablenotemark{a}} & \multicolumn{2}{c}{Other Association\tablenotemark{b}} \\
\cline{6-7}
\colhead{} & \colhead{(J2000)} & \colhead{(J2000)} & \colhead{(kpc)} & \colhead{(km s$^{-1}$)} & \colhead{UC\ion{H}{2}\tablenotemark{a}} & \colhead{outflow}
}
\startdata
G12.89+0.49 & 18$^{\rm h}$11$^{\rm m}$51\fs4 & -17\arcdeg31\arcmin30\arcsec & 2.50\tablenotemark{c} & 33.3 & N & Y\tablenotemark{d} \\
G16.86$-$2.16 & 18$^{\rm h}$29$^{\rm m}$24\fs4 & -15\arcdeg16\arcmin04\arcsec & 1.67\tablenotemark{d} & 17.8 & N & Y\tablenotemark{d} \\
G28.28$-$0.36 & 18$^{\rm h}$44$^{\rm m}$13\fs3 & -04\arcdeg18\arcmin03\arcsec & 3.0\tablenotemark{e} & 48.9 & Y & Y\tablenotemark{f} \\
\enddata
\tablenotetext{a}{\citet{2006MNRAS...367...553}}
\tablenotetext{b}{The symbols of ``Y" and ``N" represent detection and non-detection, respectively.
``UC\ion{H}{2}" indicates an ultracompact \ion{H}{2} region lying within a radius of $19\arcsec$.
The 6.7 GHz methanol masers are associated with all of the four sources.}
\tablenotetext{c}{\citet{2014apj...783...130}}
\tablenotetext{d}{\citet{2016aj...152...92L}}
\tablenotetext{e}{\citet{2014MNRAS...443...2252}}
\tablenotetext{f}{\citet{2008AJ...136...2391}}
\end{deluxetable}
\subsection{Observations with the Nobeyama 45-m Radio Telescope} \label{sec:obsNRO}
We carried out observations with the Nobeyama 45-m radio telescope from 2017 January to March.
We employed the position-switching mode.
The integration time was 20 seconds per on-source and off-source positions.
The on-source positions are summarized in Table \ref{tab:source} and the off-source positions were set to be $+15\arcmin$ away in declination.
The total integration time is $\sim 1$ hr and 2$-$4.5 hr in the 42$-$46 GHz and 82$-$103 GHz band observations, respectively.
The Z45 receiver \citep{2015PASJ...67..117N} and the TZ receiver \citep{2013PASP..125..252N} were used in the observations at 42$-$46 GHz and 82$-$103 GHz, respectively.
The main beam efficiency ($\eta_{\rm {mb}}$) and the beam size (HPBW) at 43GHz were 71\% and $37\arcsec$, respectively.
The main beam efficiency and the beam size at 86 GHz were 54\% and $18\arcsec$, respectively.
The system temperatures were 120$-$150 K and 120$-$200 K during the observations at 42$-$46 GHz and 82$-$103 GHz, respectively.
We used the SAM45 FX-type digital correlator \citep{2012PASJ...64...29K} in frequency settings whose bandwidths and resolutions are 500 MHz and 122.07 kHz for the Z45 observations, and 1000 MHz and 244.14 kHz for the TZ observations, respectively.
The frequency resolutions correspond to the velocity resolution of $\sim 0.85$ km s$^{-1}$.
The telescope pointing was checked every 1.5 hr by observing the SiO maser line ($J=1-0$; 43.12203 GHz) from OH39.7+1.5 at ($\alpha_{2000}$, $\delta_{2000}$) = (18$^{\rm h}$56$^{\rm m}$03\fs88, +06\arcdeg38\arcmin49\farcs8).
We used the Z45 receiver for the pointing check during the 42$-$46 GHz band observations and the H40 receiver for the pointing check during the 82$-$103 GHz band observations.
The pointing error was less than $3\arcsec$.
The rms noises are 6$-$14 and 3$-$6 mK in $T_{\rm {A}}^{*}$ scale in the 42$-$46 GHz and 82$-$103 GHz bands, respectively.
The baseline was fitted with a linear function.
The absolute flux calibration error is approximately 10\%.
\subsection{Observations with the ASTE 10-m Telescope} \label{sec:obsASTE}
The observations with the ASTE 10-m telescope were conducted in 2016 September and October.
The DASH345 receiver and the WHSF FX-type digital spectrometer \citep{2008PASJ...60..857I, 2008PASJ...60..315O} were used.
The observed frequency ranges are 338.2$-$339.2 and 348.45$-$349.45 GHz.
The main beam efficiency and beam size were 60\% and $22\arcsec$, respectively.
The system temperatures were between 300 and 700 K, depending on the elevation and weather conditions.
The frequency setting was 2048 MHz bandwidth and 1 MHz frequency resolution.
The frequency resolution corresponds to 0.86 km s$^{-1}$, which is almost equal to that of observations with the Nobeyama 45-m telescope.
The total integration time is approximately 1-- 2.5 hr, which is different among sources.
We checked the telescope pointing every 2 hr by observing the $^{12}$CO ($J = 3-2$) line from W Aql at ($\alpha_{2000}$, $\delta_{2000}$) = (19$^{\rm h}$15$^{\rm m}$23\fs35, -07\arcdeg02\arcmin50\farcs3).
The pointing error was less than $2\arcsec$.
The rms noise levels in the line-free regions are 23$-$24, 19$-$20, and 16$-$17 mK in $T_{\rm {A}}^{*}$ scale in G12.89+0.49, G16.86$-$2.16, and G28.28$-$0.36, respectively.
Some scans were excluded due to bad baselines.
A linear fit was applied for baseline subtraction.
The absolute flux calibration error is approximately 10\%.
\section{Results}
We conducted data reduction using the Java Newstar software, an astronomical data analyzing system of the Nobeyama 45-m telescope and the ASTE 10-m telescope.
\subsection{Observational Results with the Nobeyama 45-m Radio Telescope} \label{sec:resNRO}
Figure \ref{fig:f1} shows the spectra of the main isotopologue HC$_{3}$N and its $^{13}$C and D isotopologues in the three sources obtained with the Nobeyama 45-m radio telescope.
We fitted the spectra with a Gaussian function and obtained the spectral line parameters as summarized in Table \ref{tab:resHC3N}.
Two rotational transitions, $J=5-4$ and $10-9$, of the main isotopologue are in the observed frequency bands, and they were detected from all of the three sources.
Its three $^{13}$C isotopologues have been detected from G28.28$-$0.36 with a signal-to-noise (S/N) ratio above 4.
HC$^{13}$CCN was not detected in G12.89+0.49, whereas H$^{13}$CCCN was not detected in G16.86$-$2.16.
DC$_{3}$N was detected in G28.28$-$0.36 with a S/N ratio of 3 and in G16.86$-$2.16 with a S/N ratio above 4.
The $V_{\rm {LSR}}$ values agree with the systemic velocities of each source (Table \ref{tab:source}).
The line profiles of the main isotopologue show wing emission, suggesting that HC$_{3}$N also exists in the molecular outflow \citep[e.g.,][]{2015ApJS..221...31S, 2018ApJ...854..133T}.
Such wing emission is most prominent in G16.86$-$2.16, where the blue and red components are clearly detected.
The red and blue components are prominent in G12.89+0.49 and G28.28$-$0.36, respectively.
These features of wing emission of HC$_{3}$N are similar to those of CH$_{3}$OH (Figure \ref{fig:f3}), as mentioned later.
Hence, the origin of molecular outflows is plausible.
\begin{figure}
\figurenum{1}
\plotone{HC3N.eps}
\caption{Spectra of the main isotopologue HC$_{3}$N and its $^{13}$C and D isotopologues in the three sources. The red lines show the Gaussian fitting results and gray vertical lines show the systemic velocity for each source.\label{fig:f1}}
\end{figure}
\floattable
\rotate
\begin{deluxetable}{lllcccccccccccccc}
\tabletypesize{\scriptsize}
\tablecaption{Spectral line parameters of the main isotopologue HC$_{3}$N and its $^{13}$C and D isotopologues in the three sources with the Nobeyama 45-m telescope \label{tab:resHC3N}}
\tablewidth{0pt}
\tablehead{
\colhead{} & \colhead{} & \colhead{} & \multicolumn{4}{c}{G12.89+0.49} & \colhead{} & \multicolumn{4}{c}{G16.86$-$2.16} & \colhead{} & \multicolumn{4}{c}{G28.28$-$0.36} \\
\cline{4-7}\cline{9-12}\cline{14-17}
\colhead{Species} & \colhead{Frequency\tablenotemark{a}} & \colhead{$E_{\rm {u}}/k$} & \colhead{$T_{\rm {mb}}$} & \colhead{FWHM} & \colhead{$\int T_{\mathrm {mb}}dv$} & \colhead{$V_{\rm{LSR}}$\tablenotemark{b}} & \colhead{} & \colhead{$T_{\rm {mb}}$} & \colhead{FWHM} & \colhead{$\int T_{\mathrm {mb}}dv$} & \colhead{$V_{\rm{LSR}}$\tablenotemark{b}} & \colhead{} & \colhead{$T_{\rm {mb}}$} & \colhead{FWHM} & \colhead{$\int T_{\mathrm {mb}}dv$} & \colhead{$V_{\rm{LSR}}$\tablenotemark{b}} \\
\colhead{Transition} & \colhead{(GHz)} & \colhead{(K)} & \colhead{(K)} & \colhead{(km s$^{-1}$)} & \colhead{(K km s$^{-1}$)} & \colhead{(km s$^{-1}$)} & \colhead{} & \colhead{(K)} & \colhead{(km s$^{-1}$)} & \colhead{(K km s$^{-1}$)} & \colhead{(km s$^{-1}$)} & \colhead{} & \colhead{(K)} & \colhead{(km s$^{-1}$)} & \colhead{(K km s$^{-1}$)} & \colhead{(km s$^{-1}$)}
}
\startdata
HC$_{3}$N & & & & & & & & & & & & & & & & \\
$J=5-4$ & 45.490314 & 6.5 & 1.64 (4) & 3.35 (9) & 5.8 (2) & 32.8 & & 1.76 (5) & 3.50 (12) & 6.6 (3) & 18.7 & & 1.62 (6) & 2.22 (10) & 3.8 (2) & 49.0 \\
$J=10-9$ & 90.979023 & 24.0 & 2.58 (12) & 4.1 (2) & 11.3 (8) & 32.5 & & 3.15 (7) & 3.22 (8) & 10.8 (4) & 17.7 & & 1.64 (3) & 2.39 (5) & 4.16 (11) & 49.5 \\
H$^{13}$CCCN & & & & & & & & & & & & & & & & \\
$J=5-4$ & 44.084172 & 6.4 & 0.037 (7) & 3.0 (7) & 0.12 (4) & 33.0 & & $< 0.036$ & ... & ... & ... & & 0.036 (6) & 2.7 (5) & 0.10 (3) & 49.8 \\
HC$^{13}$CCN & & & & & & & & & & & & & & & & \\
$J=5-4$ & 45.2973345 & 6.5 & $< 0.027$ & ... & ... & ... & & 0.050 (12) & 2.8 (8) & 0.15 (6) & 18.9 & & 0.051 (8) & 3.0 (6) & 0.16 (4) & 48.7 \\
HCC$^{13}$CN & & & & & & & & & & & & & & & & \\
$J=5-4$ & 45.3017069 & 6.5 & 0.042 (7) & 3.4 (7) & 0.15 (4) & 32.5 & & 0.090 (13) & 3.7 (6) & 0.36 (8) & 19.3 & & 0.073 (13) & 1.7 (4) & 0.13 (4) & 48.8 \\
DC$_{3}$N & & & & & & & & & & & & & & & & \\
$J=12-11$ & 101.314818 & 31.6 & $< 0.013$ & ... & ... & ... & & 0.034 (6) & 4.3 (9) & 0.16 (4) & 18.2 & & 0.014 (3) & 3.2 (7) & 0.049 (13) & 49.2 \\
\enddata
\tablecomments{Numbers in the parentheses are the standard deviation of the Gaussian fit, expressed in units of the last significant digits. For example, 1.64 (4) means $1.64 \pm 0.04$. The upper limits correspond to the $3 \sigma$ limits.}
\tablenotetext{a}{Taken from the Cologne Database for Molecular Spectroscopy \citep[CDMS;][]{2005JMoSt...742...215}.}
\tablenotetext{b}{The errors are 0.85 km s$^{-1}$, which corresponds to the velocity resolution (Section \ref{sec:obsNRO}).}
\end{deluxetable}
Figure \ref{fig:f2} shows the spectra of CH$_{3}$CCH obtained with the Nobeyama 45-m radio telescope.
Its $J=5-4$ and $6-5$ $K$-ladder lines ($K=0-0$, $1-1$, $2-2$, and $3-3$) were detected from the three sources with a S/N ratio above 4.
We fitted the spectra with four-component Gaussian profiles.
We fixed the centroid velocities to be the systemic velocities for each source (Table \ref{tab:source}).
The spectral line parameters are summarized in Table \ref{tab:resCH3CCH}.
There is no presence of wing emission in the CH$_{3}$CCH spectra and the lines are well fitted with the Gaussian profile.
\begin{figure}
\figurenum{2}
\plotone{CH3CCH.eps}
\caption{Spectra of CH$_{3}$CCH in the three sources. The red lines show the Gaussian fitting results.\label{fig:f2}}
\end{figure}
\floattable
\rotate
\begin{deluxetable}{lllccccccccccc}
\tabletypesize{\scriptsize}
\tablecaption{Spectral line parameters of CH$_{3}$CCH in the three sources with the Nobeyama 45-m telescope \label{tab:resCH3CCH}}
\tablewidth{0pt}
\tablehead{
\colhead{} & \colhead{} & \colhead{} & \multicolumn{3}{c}{G12.89+0.49} & \colhead{} & \multicolumn{3}{c}{G16.86$-$2.16} & \colhead{} & \multicolumn{3}{c}{G28.28$-$0.36} \\
\cline{4-6}\cline{8-10}\cline{12-14}
\colhead{Transition} & \colhead{Frequency\tablenotemark{a}} & \colhead{$E_{\rm {u}}/k$} & \colhead{$T_{\rm {mb}}$} & \colhead{FWHM} & \colhead{$\int T_{\mathrm {mb}}dv$} & \colhead{} & \colhead{$T_{\rm {mb}}$} & \colhead{FWHM} & \colhead{$\int T_{\mathrm {mb}}dv$} & \colhead{} & \colhead{$T_{\rm {mb}}$} & \colhead{FWHM} & \colhead{$\int T_{\mathrm {mb}}dv$} \\
\colhead{} & \colhead{(GHz)} & \colhead{(K)} & \colhead{(K)} & \colhead{(km s$^{-1}$)} & \colhead{(K km s$^{-1}$)} & \colhead{} & \colhead{(K)} & \colhead{(km s$^{-1}$)} & \colhead{(K km s$^{-1}$)} & \colhead{} & \colhead{(K)} & \colhead{(km s$^{-1}$)} & \colhead{(K km s$^{-1}$)}
}
\startdata
$J=5-4$ & & & & & & & & & & & & & \\
$K=0-0$ & 85.4573003 & 12.3 & 0.958 (14) & 3.03 (5) & 3.09 (6) & & 0.616 (18) & 3.18 (9) & 2.08 (8) & & 0.638 (14) & 2.19 (5) & 1.49 (4) \\
$K=1-1$ & 85.4556667 & 19.5 & 0.785 (14) & 3.20 (6) & 2.67 (7) & & 0.496 (18) & 3.02 (11) & 1.59 (8) & & 0.477 (13) & 2.46 (7) & 1.25 (5) \\
$K=2-2$ & 85.4507663 & 41.2 & 0.324 (14) & 3.08 (13) & 1.06 (6) & & 0.190 (19) & 2.8 (3) & 0.57 (8) & & 0.160 (13) & 2.44 (19) & 0.42 (5) \\
$K=3-3$ & 85.4426012 & 77.3 & 0.168 (14) & 3.2 (3) & 0.56 (7) & & 0.065 (16) & 3.9 (9) & 0.27 (9) & & 0.044 (13) & 2.6 (7) & 0.12 (5) \\
$J=6-5$ & & & & & & & & & & & & & \\
$K=0-0$ & 102.5479844 & 17.2 & 1.741 (13) & 3.51 (3) & 6.50 (8) & & 1.079 (14) & 3.07 (5) & 3.53 (7) & & 0.791 (12) & 2.16 (4) & 1.82 (4) \\
$K=1-1$ & 102.5460242 & 24.5 & 1.412 (13) & 3.81 (4) & 5.72 (8) & & 0.845 (13) & 3.26 (6) & 2.93 (7) & & 0.676 (12) & 2.09 (4) & 1.50 (4) \\
$K=2-2$ & 102.5401446 & 46.1 & 0.637 (13) & 3.75 (9) & 2.54 (8) & & 0.385 (13) & 3.19 (13) & 1.31 (7) & & 0.239 (12) & 2.09 (12) & 0.53 (4) \\
$K=3-3$ & 102.5303476 & 82.3 & 0.374 (13) & 3.61 (14) & 1.44 (8) & & 0.192 (14) & 3.0 (2) & 0.61 (7) & & 0.086 (13) & 1.8 (3) & 0.16 (4) \\
\enddata
\tablecomments{Numbers in the parentheses are the standard deviation of the Gaussian fit, expressed in units of the last significant digits.}
\tablenotetext{a}{Taken from the Cologne Database for Molecular Spectroscopy \citep[CDMS;][]{2005JMoSt...742...215}.}
\end{deluxetable}
Nine thermal CH$_{3}$OH lines were detected from G12.89+0.49, and eight lines, except for $6_{-2, 5}-7_{-1, 7}$ $E$ transition, were detected from the other two sources with a S/N ratio above 4 as shown in Figure \ref{fig:f3}.
We fitted the spectra with Gaussian profiles and the spectral line parameters are summarized in Table \ref{tab:resCH3OH}.
For the four lines shown in the top panels of Figure \ref{fig:f3}, we applied the four-component Gaussian fitting.
In the same way as for CH$_{3}$CCH, the centroid velocities were fixed to be the systemic velocities of each source (Table \ref{tab:source}).
The wing emission for red and blue velocity components are seen in G16.86$-$2.16, while only the red component is found in G12.89+0.49.
In G28.28$-$0.36, no wing emission is seen, which may be due to their lower line intensities.
\begin{figure}
\figurenum{3}
\plotone{CH3OH.eps}
\caption{Spectra of CH$_{3}$OH in the three sources. The red lines show the Gaussian fitting results and gray vertical lines show the systemic velocity for each source.\label{fig:f3}}
\end{figure}
\floattable
\rotate
\begin{deluxetable}{lllcccccccccccccc}
\tabletypesize{\scriptsize}
\tablecaption{Spectral line parameters of CH$_{3}$OH in the three sources with the Nobeyama 45-m telescope \label{tab:resCH3OH}}
\tablewidth{0pt}
\tablehead{
\colhead{} & \colhead{} & \colhead{} & \multicolumn{4}{c}{G12.89+0.49} & \colhead{} & \multicolumn{4}{c}{G16.86$-$2.16} & \colhead{} & \multicolumn{4}{c}{G28.28$-$0.36} \\
\cline{4-7}\cline{9-12}\cline{14-17}
\colhead{Transition} & \colhead{Frequency\tablenotemark{a}} & \colhead{$E_{\rm {u}}/k$} & \colhead{$T_{\rm {mb}}$} & \colhead{FWHM} & \colhead{$\int T_{\mathrm {mb}}dv$} & \colhead{$V_{\rm{LSR}}$\tablenotemark{b}} & \colhead{} & \colhead{$T_{\rm {mb}}$} & \colhead{FWHM} & \colhead{$\int T_{\mathrm {mb}}dv$} & \colhead{$V_{\rm{LSR}}$\tablenotemark{b}} & \colhead{} & \colhead{$T_{\rm {mb}}$} & \colhead{FWHM} & \colhead{$\int T_{\mathrm {mb}}dv$} & \colhead{$V_{\rm{LSR}}$\tablenotemark{b}} \\
\colhead{} & \colhead{(GHz)} & \colhead{(K)} & \colhead{(K)} & \colhead{(km s$^{-1}$)} & \colhead{(K km s$^{-1}$)} & \colhead{(km s$^{-1}$)} & \colhead{} & \colhead{(K)} & \colhead{(km s$^{-1}$)} & \colhead{(K km s$^{-1}$)} & \colhead{(km s$^{-1}$)} & \colhead{} & \colhead{(K)} & \colhead{(km s$^{-1}$)} & \colhead{(K km s$^{-1}$)} & \colhead{(km s$^{-1}$)}
}
\startdata
$5_{-1,5}-4_{0,4}$ $E$ & 84.521169 & 40.4 & 1.97 (8) & 3.44 (14) & 7.2 (4) & 32.5 & & 1.20 (6) & 4.9 (2) & 6.2 (4) & 17.9 & & 0.222 (8) & 2.79 (10) & 0.66 (3) & 48.9 \\
$6_{-2, 5}-7_{-1, 7}$ $E$ & 85.568084 & 74.7 & 0.206 (7) & 5.06 (17) & 1.11 (5) & 32.9 & & $<0.018$ & ... & ... & ... & & $<0.011$ & ... & ... & ... \\
$8_{0, 8}-7_{1, 7}$ $A^{+}$ & 95.169463 & 83.5 & 1.82 (7) & 3.70 (15) & 7.2 (4) & 32.4 & & 0.96 (4) & 2.92 (13) & 2.99 (18) & 17.7 & & 0.045 (4) & 3.0 (3) & 0.14 (2) & 49.3 \\
$2_{1, 2}-1_{1, 1}$ $A^{+}$ & 95.914309 & 21.4 & 0.664 (13) & 4.39 (10) & 3.10 (9) & 32.7 & & 0.141 (4) & 5.55 (19) & 0.83 (4) & 18.0 & & 0.050 (4) & 2.8 (3) & 0.149 (19) & 48.9 \\
$2_{-1, 2}-1_{-1, 1}$ $E$ & 96.739362 & 12.5 & 1.81 (2) & 4.09 (6) & 7.87 (15) & $-$ & & 1.63 (2) & 5.31 (9) & 9.2 (2) & $-$ & & 1.06 (3) & 2.80 (10) & 3.16 (14) & $-$ \\
$2_{0, 2}-1_{0, 1}$ $A^{+}$ & 96.741375 & 7.0 & 2.39 (2) & 3.60 (4) & 9.14 (14) & $-$ & & 2.05 (2) & 4.50 (7) & 9.81 (19) & $-$ & & 1.42 (3) & 3.02 (7) & 4.57 (15) & $-$ \\
$2_{0 , 2}-1_{0 , 1}$ $E$ & 96.744550 & 20.1 & 0.99 (2) & 4.40 (11) & 4.66 (15) & $-$ & & 0.485 (18) & 7.8 (4) & 4.0 (2) & $-$ & & 0.23 (3) & 3.5 (5) & 0.87 (16) & $-$ \\
$2_{1 , 1}-1_{1 , 0}$ $E$ & 96.755511 & 28.0 & 0.64 (2) & 4.47 (17) & 3.06 (15) & $-$ & & 0.16 (2) & 5.2 (8) & 0.87 (18) & $-$ & & 0.08 (3) & 2.6 (12) & 0.22 (14) & $-$ \\
$2_{1, 1}-1_{1, 0}$ $A^{-}$ & 97.582804 & 21.6 & 0.702 (12) & 4.18 (8) & 3.12 (8) & 32.7 & & 0.172 (7) & 5.3 (2) & 0.97 (6) & 17.6 & & 0.065 (6) & 2.7 (3) & 0.18 (2) & 48.8 \\
\enddata
\tablecomments{Numbers in the parentheses are the standard deviation of the Gaussian fit, expressed in units of the last significant digits. The upper limits correspond to the $3 \sigma$ limits.}
\tablenotetext{a}{Taken from the Cologne Database for Molecular Spectroscopy \citep[CDMS;][]{2005JMoSt...742...215}.}
\tablenotetext{b}{The errors are 0.85 km s$^{-1}$, which corresponds to the velocity resolution (Section \ref{sec:obsNRO}).}
\end{deluxetable}
Figures \ref{fig:f6} and \ref{fig:f7} show the spectra in the frequency bands covered with the TZ receiver.
Several lines from COMs have been detected with a S/N ratio above 4.
An analysis of the COMs requires the simultaneous fitting of multiple species to avoid blending effects with other lines.
In our case, detected COMs have similar excitation-energy and such fitting cannot degenerate excitation temperatures and column densities.
Therefore, we cannot derive their rotational temperatures and column densities accurately and we will not discuss COMs in the rest of this paper.
We summarize the detection and non-detection of COMs in each source in Table \ref{tab:resCOMs}.
G12.89+0.49 is the most line-rich source with strong peak intensities, and both nitrogen-bearing COMs and oxygen-bearing COMs have been detected.
On the other hand, only CH$_{3}$OH and CH$_{3}$CHO were detected with weak peak intensities in G28.28$-$0.36.
\begin{figure}
\figurenum{4}
\plotone{COM85.eps}
\caption{Spectra of the complex organic molecules in the 84.5$-$84.7 and 89.3$-$89.7 GHz bands in the three sources obtained with the Nobeyama 45-m telescope. \label{fig:f6}}
\end{figure}
\begin{figure}
\figurenum{5}
\plotone{COM90.eps}
\caption{Spectra of the complex organic molecules in the 90.0$-$90.2, 94.9$-$95.1, and 95.9$-$96.1 GHz bands in the three sources obtained with the Nobeyama 45-m telescope. \label{fig:f7}}
\end{figure}
\floattable
\begin{deluxetable}{lccc}
\tablecaption{Summary of detection of complex organic molecules in the three sources with the Nobeyama 45-m telescope \label{tab:resCOMs}}
\tablewidth{0pt}
\tablehead{
\colhead{Species} & \colhead{G12.89+0.49} & \colhead{G16.86$-$2.16} & \colhead{G28.28$-$0.36}
}
\startdata
CH$_{3}$OCHO & Y & Y & N \\
CH$_{3}$CH$_{2}$OH & Y & N & N \\
CH$_{3}$CHO & Y & Y & Y \\
CH$_{3}$OCH$_{3}$ & Y & N & N \\
NH$_{2}$CHO & Y & N & N \\
CH$_{2}$CHCN & Y & N & N \\
CH$_{3}$CH$_{2}$CN & Y & N & N \\
\enddata
\tablecomments{`Y' and `N' represent detection and non-detection with a S/N ratio above 4, respectively.}
\end{deluxetable}
\subsection{Observational Results with the ASTE 10-m Telescope} \label{sec:resASTE}
Figure \ref{fig:f5} shows the spectra in the 338.2$-$339.2 and 348.45$-$349.45 GHz bands obtained with the ASTE 10-m telescope.
Table \ref{tab:resASTE} summarizes the spectral line parameters obtained from the Gaussian fitting.
The detection limit was set at a S/N ratio above 4.
The $V_{\rm {LSR}}$ values are consistent with the systemic velocities of each source (Table \ref{tab:source})\footnote{The $V_{\rm {LSR}}$ values of molecular emission lines are shifted by $\sim 0.2$ km s$^{-1}$ due to bug in ASTE Newstar. Results and discussions in this paper are not affected by this bug.}.
Several high-excitation energy ($E_{\rm {u}}/k > 200$ K) lines from CH$_{3}$OH and CH$_{3}$CN were detected from G12.89+0.49.
On the other hand, only two lower-excitation energy ($E_{\rm {u}}/k < 75$ K) lines of CH$_{3}$OH and CCH were detected from G28.28$-$0.36.
The line density in G16.86$-$2.16 is between those in G12.89+0.49 and G28.28$-$0.36.
The relatively high-excitation energy line of CH$_{3}$OH ($7_{4, 3}- 6_{4, 2}$ $A^{+}$ transition; $E_{\rm {u}}/k = 145.3$ K) was detected in G16.86$-$2.16.
The results suggest that hot gas exists in G12.89+0.49 and G16.86$-$2.16, which is supported by the detection of the metastable inversion transition NH$_{3}$ lines with the very-high-excitation energies, ($J, K$) = (8, 8) at 26.51898 GHz ($E_{\rm {u}}/k = 686 $ K) and ($J, K$) = (9, 9) at 27.47794 GHz ($E_{\rm {u}}/k = 852 $ K), with the GBT \citep{2017ApJ...844...68T}.
In G12.89+0.49 and G16.86$-$2.16, the line profiles of CH$_{3}$OH show wing emission suggestive of molecular outflow origins.
In addition, the line widths (FWHM) of CH$_{3}$OH in these two sources ($\sim 5-6$ km s$^{-1}$) are larger than those in G28.28$-$0.36 ($\sim 2-3$ km s$^{-1}$).
This may suggest that CH$_{3}$OH in G28.28$-$0.36 exists in less turbulent regions, as it is discussed in more detail in Section \ref{sec:discuss1}.
\begin{figure}
\figurenum{6}
\plotone{ASTE.eps}
\caption{Spectra in the 338.2$-$339.2 and 348.45$-$349.45 GHz bands toward the three sources with the ASTE 10-m telescope.\label{fig:f5}}
\end{figure}
\floattable
\rotate
\begin{deluxetable}{lllcccccccccccccc}
\tabletypesize{\scriptsize}
\tablecaption{Spectral line parameters in the three sources with the ASTE 10-m telescope \label{tab:resASTE}}
\tablewidth{0pt}
\tablehead{
\colhead{} & \colhead{} & \colhead{} & \multicolumn{4}{c}{G12.89+0.49} & \colhead{} & \multicolumn{4}{c}{G16.86$-$2.16} & \colhead{} & \multicolumn{4}{c}{G28.28$-$0.36} \\
\cline{4-7}\cline{9-12}\cline{14-17}
\colhead{Species \&} & \colhead{Frequency\tablenotemark{a}} & \colhead{$E_{\rm {u}}/k$} & \colhead{$T_{\rm {mb}}$} & \colhead{FWHM} & \colhead{$\int T_{\mathrm {mb}}dv$\tablenotemark{b}} & \colhead{$V_{\rm{LSR}}$\tablenotemark{c}} & \colhead{} & \colhead{$T_{\rm {mb}}$} & \colhead{FWHM} & \colhead{$\int T_{\mathrm {mb}}dv$\tablenotemark{b}} & \colhead{$V_{\rm{LSR}}$\tablenotemark{c}} & \colhead{} & \colhead{$T_{\rm {mb}}$} & \colhead{FWHM} & \colhead{$\int T_{\mathrm {mb}}dv$\tablenotemark{b}} & \colhead{$V_{\rm{LSR}}$\tablenotemark{c}} \\
\colhead{Transition} & \colhead{(GHz)} & \colhead{(K)} & \colhead{(K)} & \colhead{(km s$^{-1}$)} & \colhead{(K km s$^{-1}$)} & \colhead{(km s$^{-1}$)} & \colhead{} & \colhead{(K)} & \colhead{(km s$^{-1}$)} & \colhead{(K km s$^{-1}$)} & \colhead{(km s$^{-1}$)} & \colhead{} & \colhead{(K)} & \colhead{(km s$^{-1}$)} & \colhead{(K km s$^{-1}$)} & \colhead{(km s$^{-1}$)}
}
\startdata
SO$_{2}$ & \multicolumn{16}{c}{} \\
$18_{4,14}-18_{3,15}$ & 338.305993 & 196.8 & 0.13 (4)\tablenotemark{d} & 5.9 (9) & 0.81 (9) & 33.1 & & $<0.09$ & ... & ... & ... & & $<0.09$ & ... & ...& ... \\
CH$_{3}$OH & \multicolumn{16}{c}{} \\
$7_{-1, 7}- 6_{-1, 6}$ $E$ & 338.344628 & 70.5 & 1.23 (4) & 4.6 (9) & 5.99 (10) & 32.9 & & 0.82 (3) & 4.5 (9) & 3.94 (8) & 17.3 & & 0.17 (3) & 2.2 (8) & 0.40 (6) & 48.9 \\
$7_{0, 7}- 6_{0, 6}$ $A^{+}$ & 338.408681 & 65.0 & 1.50 (4) & 5.1 (9) & 8.06 (11) & 33.0 & & 1.07 (3) & 4.8 (9) & 5.42 (7) & 17.2 & & 0.19 (3) & 2.5 (8) & 0.49 (5) & 49.2 \\
$7_{-6, 1}- 6_{-6, 0}$ $E$ & 338.430933 & 253.9 & 0.23 (4) & 5.8 (9) & 1.40 (9) & 32.9 & & $<0.09$ & ... & ... & ... & & $<0.09$ & ... & ...& ... \\
$7_{6, 1}- 6_{6, 0}$ $A^{+}$ & 338.442344 & 258.7 & 0.26 (4) & 4.5 (9) & 1.22 (9) & 33.0 & & $<0.09$ & ... & ... & ... & & $<0.09$ & ... & ...& ... \\
$7_{-5, 2}- 6_{-5, 1}$ $E$ & 338.456499 & 189.0 & 0.26 (4) & 5.0 (9) & 1.38 (11) & 32.9 & & $<0.09$ & ... & ... & ... & & $<0.09$ & ... & ...& ... \\
$7_{5 , 3}- 6_{5 , 2}$ $E$ & 338.475290 & 201.1 & 0.25 (4) & 5.2 (9) & 1.42 (11) & 32.7 & & $<0.09$ & ... & ... & ... & & $<0.09$ & ... & ...& ... \\
$7_{5, 3}- 6_{5, 2}$ $A^{+}$ & 338.486337 & 202.9 & 0.25 (4) & 5.9 (9) & 1.55 (11) & 33.2 & & $<0.09$ & ... & ... & ... & & $<0.09$ & ... & ...& ... \\
$7_{-4, 4}- 6_{-4, 3}$ $E$ & 338.504099 & 152.9 & 0.30 (4) & 6.0 (9) & 1.91 (10) & 32.8 & & $<0.09$ & ... & ... & ... & & $<0.09$ & ... & ...& ... \\
$7_{4, 3}- 6_{4, 2}$ $A^{+}$ & 338.512627 & 145.3 & 0.51 (4) & 4.7 (9) & 2.56 (11) & 32.7 & & 0.14 (3) & 6.1 (9) & 0.92 (10) & 17.7 & & $<0.09$ & ... & ...& ... \\
$7_{4 , 3}- 6_{4 , 2}$ $E$ & 338.530249 & 161.0 & 0.27 (4) & 6.0 (9) & 1.72 (9) & 32.8 & & $<0.09$ & ... & ... & ... & & $<0.09$ & ... & ...& ... \\
$7_{3, 5}- 6_{3, 4}$ $A^{+}$ & 338.540795 & 114.8 & 0.56 (4) & 6.1 (9) & 3.64 (12) & 32.0 & & 0.15 (3) & 6.7 (9) & 1.10 (10) & 16.1 & & $<0.09$ & ... & ...& ... \\
$7_{-3, 5}- 6_{-3, 4}$ $E$ & 338.559928 & 127.7 & 0.36 (4) & 4.5 (9) & 1.73 (10) & 32.8 & & $<0.09$ & ... & ... & ... & & $<0.09$ & ... & ...& ... \\
$7_{3 , 4}- 6_{3 , 3}$ $E$ & 338.583195 & 112.7 & 0.36 (4) & 5.2 (9) & 2.00 (10) & 33.0 & & $<0.09$ & ... & ... & ... & & $<0.09$ & ... & ...& ... \\
$7_{1 , 6}- 6_{1 , 5}$ $E$ & 338.614999 & 86.1 & 0.73 (4) & 5.5 (9) & 4.28 (11) & 33.3 & & 0.26 (3) & 4.3 (9) & 1.20 (8) & 17.3 & & $<0.09$ & ... & ...& ... \\
$7_{2, 5}- 6_{2, 4}$ $A^{+}$ & 338.639939 & 102.7 & 0.45 (4) & 4.9 (9) & 2.36 (11) & 33.2 & & 0.12 (3) & 5.8 (9) & 0.72 (9) & 17.9 & & $<0.09$ & ... & ...& ... \\
$7_{-2, 6}- 6_{-2, 5}$ $E$ & 338.72294 & 90.9 & 0.89 (4) & 4.9 (9) & 4.64 (11) & 32.3 & & 0.51 (3) & 5.8 (9) & 3.10 (7) & 18.2 & & $<0.09$ & ... & ...& ... \\
$14_{1, 13}- 14_{0, 14}$ $A^{- +}$ & 349.10702 & 260.2 & 0.36 (4) & 6.0 (9) & 2.29 (12) & 33.2 & & $<0.09$ & ... & ... & ... & & $<0.09$ & ... & ...& ... \\
$^{13}$CH$_{3}$OH & \multicolumn{16}{c}{} \\
$13_{0, 13}- 12_{1, 12}$ $A^{+}$ & 338.759948 & 205.9 & 0.17 (4) & 4.5 (9) & 0.80 (6) & 33.0 & & $<0.09$ & ... & ... & ... & & $<0.09$ & ... & ...& ... \\
H$_{2}$CS & \multicolumn{16}{c}{} \\
$10_{1, 9}- 9_{1, 8}$ & 348.5343647 & 105.2 & 0.55 (4) & 4.5 (9) & 2.64 (11) & 32.9 & & 0.10 (3) & 6.2 (9) & 0.64 (6) & 17.4 & & $<0.09$ & ... & ...& ... \\
CCH & \multicolumn{16}{c}{} \\
$N= 4- 3, J=9/2-7/2$ & 349.3374558 & 41.9 & 1.17 (4) & 4.2 (9) & 5.22 (11) & 33.7 & & 1.12 (3) & 3.2 (9) & 3.82 (8) & 16.8 & & 0.34 (3) & 3.4 (9) & 1.25 (7) & 48.1 \\
$N= 4- 3, J=7/2-5/2$ & 349.3992738 & 41.9 & 0.80 (4)\tablenotemark{e} & 3.8 (9) & 3.27 (11) & 32.1 & & 0.86 (3) & 3.3 (9) & 3.04 (8) & 17.1 & & 0.25 (3) & 3.4 (9) & 0.92 (7) & 48.3 \\
CH$_{3}$CN & \multicolumn{16}{c}{} \\
$J_{K}=19_{6}-18_{6}$ & 349.2123106 & 424.7 & 0.14 (4) & 5.4 (9) & 0.81 (12) & 33.1 & & $<0.09$ & ... & ... & ... & & $<0.09$ & ... & ...& ... \\
$J_{K}=19_{5}-18_{5}$ & 349.2860057 & 346.2 & 0.16 (4) & 5.8 (9) & 0.98 (12) & 33.9 & & $<0.09$ & ... & ... & ... & & $<0.09$ & ... & ...& ... \\
$J_{K}=19_{4}-18_{4}$ & 349.3463428 & 282.0 & 0.20 (4)\tablenotemark{e} & 5.1 (9) & 1.07 (11) & 33.3 & & $<0.09$ & ... & ... & ... & & $<0.09$ & ... & ...& ... \\
$J_{K}=19_{2}-18_{2}$ & 349.4268497 & 196.3 & 0.23 (4) & 5.3 (9) & 1.32 (11) & 33.0 & & $<0.09$ & ... & ... & ... & & $<0.09$ & ... & ...& ... \\
\enddata
\tablecomments{Numbers in the parentheses are the standard deviation, expressed in units of the last significant digits. The upper limits correspond to the $3 \sigma$ limits.}
\tablenotetext{a}{Taken from the Cologne Database for Molecular Spectroscopy \citep[CDMS;][]{2005JMoSt...742...215}.}
\tablenotetext{b}{The errors were derived using the following formula; $\Delta T_{\rm {mb}} \sqrt{n} \Delta v$, where $\Delta T_{\rm {mb}}$, $n$, $\Delta v$ are the rms noise in the emission-free regions, the number of channels, and velocity resolution per channel, respectively.}
\tablenotetext{c}{The errors are 0.86 km s$^{-1}$, which corresponds to the velocity resolution (Section \ref{sec:obsASTE}).}
\tablenotetext{d}{Tentative detection with the signal-to-noise ratio above 3.}
\tablenotetext{e}{These lines are contaminated and fitting results are tentative.}
\end{deluxetable}
\section{Analyses} \label{sec:ana}
\subsection{Rotational Diagram Analysis} \label{sec:rotational}
We derived the rotational temperatures and beam-averaged column densities of HC$_{3}$N, CH$_{3}$CCH, and CH$_{3}$OH with the beam size of $18\arcsec$ in the three sources from the rotational diagram analysis, using the following formula \citep{1999ApJ...517..209G};
\begin{equation} \label{rd}
{\rm {ln}} \frac{3k \int T_{\mathrm {mb}}dv}{8\pi ^3 \nu S \mu ^2} = {\rm {ln}} \frac{N}{Q(T_{\rm {rot}})} - \frac{E_{\rm {u}}}{kT_{\rm {rot}}},
\end{equation}
where $k$ is the Boltzmann constant, $S$ is the line strength, $\mu$ is the permanent electric dipole moment, $N$ is the column density, and $Q(T_{\rm {rot}})$ is the partition function.
The permanent electric dipole moments are 3.7312, 0.784, and 0.899 D for HC$_{3}$N, CH$_{3}$CCH, and CH$_{3}$OH, respectively.
We used $\int T_{\mathrm {mb}}dv$ values summarized in Tables \ref{tab:resHC3N}, \ref{tab:resCH3CCH}, and \ref{tab:resCH3OH}.
Figure \ref{fig:f4} shows the fitting results of HC$_{3}$N, CH$_{3}$CCH, and CH$_{3}$OH in the three sources.
The errors include the Gaussian fitting errors, the uncertainties from the main beam efficiency of 10\%, the chopper-wheel method of 10\%, and pointing calibration error of 30\%.
The derived rotational temperatures and column densities are summarized in Table \ref{tab:rot}.
In the case of HC$_{3}$N, we added data of the $J=3-2$ transition obtained with the GBT \citep[Table 3 in][]{2017ApJ...844...68T}.
The beam sizes are different among the three transitions, $27\arcsec$, $37\arcsec$, and $18\arcsec$ for the $J=3-2$, $5-4$, and $10-9$, respectively.
Assuming a small beam filling factor, we multiplied the integrated intensities of the GBT data by ($\frac{27\arcsec}{18\arcsec}$)$^{2}$ and the 45 GHz band data by ($\frac{37\arcsec}{18\arcsec}$)$^{2}$ for the correction of the different beam sizes (filled circles in Figure \ref{fig:f4}).
The red lines are the best fitting results for $J=5-4$ without beam size correction and $J=10-9$ transitions, and the green lines show the fitting results for the three transitions with beam size correction.
In the case of the fitting for the $J=5-4$ and $10-9$ transitions, the $1\sigma$ fluctuation in the integrated intensity causes unlikely large errors in the derived rotational temperature and column density (e.g., $T_{\rm {rot}} \sim 600$ K), and we do not include the errors in Table \ref{tab:rot}.
All of the three transitions are better fitted with beam size correction.
The fitting results imply that the spatial distribution of HC$_{3}$N is smaller than 18\arcsec, corresponding to $\sim 0.07-0.1$ pc radii at the source distances (Table \ref{tab:source}), like in the case for HC$_{5}$N \citep{2017ApJ...844...68T}.
\citet{2017ApJ...844...68T} derived the rotational temperatures of HC$_{5}$N with beam size correction to be $\sim 13-20$ K in the three sources.
The different rotational temperatures between HC$_{3}$N and HC$_{5}$N probably arise from the fact that we observed only lower excitation-energy lines ($E_{\rm {u}}/k < 24.0$ K), compared to HC$_{5}$N \citep[$E_{\rm {u}}/k \leq 100$ K;][]{2017ApJ...844...68T}, and the presence of complex excitation conditions.
In the case of CH$_{3}$CCH, we multiplied the integrated intensities of the $J=6-5$ transition lines by ($\frac{85.4 {\rm GHz}}{102.5 {\rm GHz}}$)$^{2}$ in order to correct the values for the $18\arcsec$ beam size at the 85 GHz band in G12.89+0.49 and G16.86$-$2.16, because data of the $J=6-5$ transition are systematically higher than those of the $J=5-4$ transition.
In Figure \ref{fig:f4}, open and filled circles represent the integrated intensities without and with correction of frequency dependence of the beam size, respectively.
The black lines show the fitting results with correction of frequency dependence.
All of the data can be fitted simultaneously, which means that the spatial distribution of CH$_{3}$CCH is smaller than $18\arcsec$.
This is the same case as for HC$_{3}$N and HC$_{5}$N.
The derived rotational temperatures are $33^{+20}_{-9}$, $29^{+15}_{-8}$, and $23^{+9}_{-6}$ K in G12.89+0.40, G16.86$-$2.16, and G28.28$-$0.36, respectively.
In the case of CH$_{3}$OH, we fitted the data using all the lines in the 85$-$98 GHz band summarized in Table \ref{tab:resCH3OH}, and the fitting results are shown by the black lines in Figure \ref{fig:f4}.
As is the case of CH$_{3}$CCH, we derived beam-averaged column densities a beam size of $18\arcsec$, correcting the frequency dependence of the beam size by multiplying the integrated intensities of the lines in the 95$-$98 GHz by ($\frac{85 {\rm GHz}}{\nu {\rm GHz}}$)$^{2}$, where $\nu$ is the frequency of each line.
The rotational temperatures are $42^{+33}_{-13}$, $36^{+18}_{-9}$, and $18^{+5}_{-3}$ K in G12.89+0.49, G16.86$-$2.16, and G28.28$-$0.36, respectively (Table \ref{tab:rot}).
\citet{2009MNRAS.394..323P} derived the rotational temperatures of CH$_{3}$OH to be $5\pm 0.8$, $6 \pm 0.1$, and $5 \pm 0.5$ K in G12.89+0.49, G16.86$-$2.16, and G28.28$-$0.36, respectively, from fitting only three or four line transitions, which was interpreted as CH$_{3}$OH being sub-thermally excited.
This does not seem to be the case when considering more CH$_{3}$OH transitions in the derivation of the rotational temperature.
\begin{figure}
\figurenum{7}
\plotone{rot_rev4.eps}
\caption{Rotational diagram of HC$_{3}$N, CH$_{3}$CCH, and CH$_{3}$OH in the three sources. HC$_{3}$N (upper panels): The filled and open circles represent with and without beam size correction, respectively. The red line is the fitting results for the $J=5-4$ data without beam size correction and $J=10-9$ data. The green line shows the fitting results for all of the lines with the beam size correction. CH$_{3}$CCH (middle panels): The filled and open circles represent with and without correction of frequency dependence, respectively. The black lines show the fitting results with correction of frequency dependence. CH$_{3}$OH (lower panels): The black lines show the fitting results for all of the observed lines. \label{fig:f4}}
\end{figure}
\floattable
\begin{deluxetable}{ccccccccc}
\tablecaption{The rotational temperatures and beam-averaged column densities of HC$_{3}$N, CH$_{3}$CCH, and CH$_{3}$OH \label{tab:rot}}
\tablewidth{0pt}
\tablehead{
\colhead{} & \multicolumn{2}{c}{G12.89+0.49} & \colhead{} & \multicolumn{2}{c}{G16.86$-$2.16} & \colhead{} & \multicolumn{2}{c}{G28.28$-$0.36} \\
\cline{2-3}\cline{5-6}\cline{8-9}
\colhead{Species} & \colhead{$T_{\rm {rot}}$} & \colhead{$N$} & \colhead{} & \colhead{$T_{\rm {rot}}$} & \colhead{$N$} & \colhead{} & \colhead{$T_{\rm {rot}}$} & \colhead{$N$} \\
\colhead{} & \colhead{(K)} & \colhead{(cm$^{-2}$)} & \colhead{} & \colhead{(K)} & \colhead{(cm$^{-2}$)} & \colhead{} & \colhead{(K)} & \colhead{(cm$^{-2}$)}
}
\startdata
HC$_{3}$N & & & & & & & & \\
$J=5-4$ \& $10-9$ & $24$ & $4.4 \times 10^{13}$ & & $20$ & $4.3 \times 10^{13}$ & & $13.4$ & $2.0 \times 10^{13}$ \\
All & $7.7^{+1.6}_{-2.8}$ & ($1.4 \pm 0.3$)$\times 10^{14}$ & & $7.2^{+1.4}_{-2.4}$ & ($1.3 \pm 0.3$)$\times 10^{14}$ & & $6.2^{+1.1}_{-1.7}$ & ($7.6 \pm 2.1$)$\times 10^{13}$ \\
\multicolumn{9}{c}{} \\
CH$_{3}$CCH & $33^{+20}_{-9}$ & $1.0^{+0.11}_{-0.02} \times 10^{15}$ & & $29^{+15}_{-8}$ & $5.4^{+0.7}_{-0.8} \times 10^{14}$ & & $23^{+9}_{-6}$ & $3.7^{+0.5}_{-0.9} \times 10^{14}$ \\
\multicolumn{9}{c}{} \\
CH$_{3}$OH & $42^{+33}_{-13}$ & $2.9^{+1.9}_{-1.3} \times 10^{15}$ & & $36^{+18}_{-9}$ & $1.5^{+0.9}_{-0.7} \times 10^{15}$ & & $18^{+5}_{-3}$ & $2.3^{+1.6}_{-1.1} \times 10^{14}$ \\
\enddata
\tablecomments{The errors represent the standard deviation.}
\end{deluxetable}
\subsection{Column Densities of the D and $^{13}$C Isotopologues of HC$_{3}$N}
We derived the column densities of the D and $^{13}$C isotopologues of HC$_{3}$N in the three sources assuming the LTE condition \citep{1999ApJ...517..209G}.
We used the following formulae;
\begin{equation} \label{tau}
\tau = - {\mathrm {ln}} \left[1- \frac{T_{\rm {mb}} }{J(T_{\rm {ex}}) - J(T_{\rm {bg}})} \right]
\end{equation}
where
\begin{equation} \label{tem}
J(T) = \frac{h\nu}{k}\Bigl\{\exp\Bigl(\frac{h\nu}{kT}\Bigr) -1\Bigr\} ^{-1},
\end{equation}
and
\begin{equation} \label{col}
N = \tau \frac{3h\Delta v}{8\pi ^3}\sqrt{\frac{\pi}{4\mathrm {ln}2}}Q\frac{1}{\mu ^2}\frac{1}{J_{\rm {lower}}+1}\exp\Bigl(\frac{E_{\rm {lower}}}{kT_{\rm {ex}}}\Bigr)\Bigl\{1-\exp\Bigl(-\frac{h\nu }{kT_{\rm {ex}}}\Bigr)\Bigr\} ^{-1}.
\end{equation}
In Equation (\ref{tau}), $\tau$ denotes the optical depth, and $T_{\rm {mb}}$ the peak intensities summarized in Table \ref{tab:resHC3N}.
$T_{\rm{ex}}$ and $T_{\rm {bg}}$ are the excitation temperature and the cosmic microwave background temperature ($\simeq 2.73$ K).
We used the rotational temperatures of HC$_{3}$N summarized in Table \ref{tab:rot} as the excitation temperatures.
We calculated two cases of excitation temperatures (Table \ref{tab:rot}) for each source.
$J$($T$) in Equation (\ref{tem}) is the effective temperature equivalent to that in the Rayleigh-Jeans law.
In Equation (\ref{col}), {\it N} is the column density, $\Delta v$ is the line width (FWHM, Table \ref{tab:resHC3N}), $Q$ is the partition function, $\mu$ is the permanent electric dipole moment, and $E_{\rm {lower}}$ is the energy of the lower rotational energy level.
The electric dipole moments are 3.7408 D and 3.73172 D for DC$_{3}$N and the three $^{13}$C isotopologues, respectively.
The derived column densities and the D/H and $^{12}$C/$^{13}$C ratios are summarized in Table \ref{tab:LTE}.
\floattable
\begin{deluxetable}{ccccccccc}
\tablecaption{The column densities of the D and $^{13}$C isotopologues of HC$_{3}$N \label{tab:LTE}}
\tablewidth{0pt}
\tablehead{
\colhead{} & \multicolumn{2}{c}{G12.89+0.49} & \colhead{} & \multicolumn{2}{c}{G16.86$-$2.16} & \colhead{} & \multicolumn{2}{c}{G28.28$-$0.36} \\
\cline{2-3}\cline{5-6}\cline{8-9}
\colhead{Species} & \colhead{$N$} & \colhead{D/H,} & \colhead{} & \colhead{$N$} & \colhead{D/H,} & \colhead{} & \colhead{$N$} & \colhead{D/H,} \\
\colhead{} & \colhead{(cm$^{-2}$)} & \colhead{$^{12}$C/$^{13}$C} & \colhead{} & \colhead{(cm$^{-2}$)} & \colhead{$^{12}$C/$^{13}$C} & \colhead{} & \colhead{(cm$^{-2}$)} & \colhead{$^{12}$C/$^{13}$C}
}
\startdata
& \multicolumn{2}{c}{$T_{\rm {ex}} = 24$ K (fixed)} & & \multicolumn{2}{c}{$T_{\rm {ex}} = 20$ K (fixed)} & & \multicolumn{2}{c}{$T_{\rm {ex}} = 13.4$ K (fixed)} \\
\cline{2-3}\cline{5-6}\cline{8-9}
DC$_{3}$N & $< 6.2 \times 10^{10}$ & $< 0.0014$ & & ($5.5 \pm 1.5$)$\times 10^{11}$ & $0.013 \pm 0.003$ & & ($3.7 \pm 1.0$)$\times 10^{11}$ & $0.018 \pm 0.005$ \\
H$^{13}$CCCN & ($5.8 \pm 1.7$)$\times 10^{11}$ & $76 \pm 23$ & & $<2.6 \times 10^{11}$ & $<165$ & & ($6.8 \pm 1.8$)$\times 10^{11}$ & $30 \pm 8$ \\
HC$^{13}$CCN & $<1.9 \times 10^{11}$ & $< 237$ & & ($7.6 \pm 2.8$)$\times 10^{11}$ & $57 \pm 21$ & & ($1.0 \pm 0.3$)$\times 10^{12}$ & $20 \pm 5$ \\
HCC$^{13}$CN & ($7.1 \pm 1.9$)$\times 10^{11}$ & $63 \pm 16$ & & ($1.8 \pm 0.4$)$\times 10^{12}$ & $24 \pm 5$ & & ($8.2 \pm 2.3$)$\times 10^{11}$ & $25 \pm 7$ \\
\multicolumn{9}{c}{} \\
& \multicolumn{2}{c}{$T_{\rm {ex}} = 7.7$ K (fixed)} & & \multicolumn{2}{c}{$T_{\rm {ex}} = 7.2$ K (fixed)} & & \multicolumn{2}{c}{$T_{\rm {ex}} = 6.2$ K (fixed)} \\
\cline{2-3}\cline{5-6}\cline{8-9}
DC$_{3}$N & $< 5.4 \times 10^{11}$ & $< 0.0046$ & & ($4.9 \pm 1.4$) $\times 10^{12}$ & $0.037 \pm 0.010$ & & ($3.3 \pm 0.9$)$\times 10^{12}$ & $0.043 \pm 0.012$ \\
H$^{13}$CCCN & ($2.5 \pm 0.8$)$\times 10^{12}$ & $46 \pm 14$ & & $<1.2 \times 10^{12}$ & $<114$ & & ($3.1 \pm 0.8$)$\times 10^{12}$ & $24 \pm 7$ \\
HC$^{13}$CCN & $<8.2 \times 10^{11}$ & $<142$ & & ($3.4 \pm 1.3$)$\times 10^{12}$ & $38 \pm 14$ & & ($4.7 \pm 1.2$)$\times 10^{12}$ & $16 \pm 4$ \\
HCC$^{13}$CN & ($3.1 \pm 0.8$)$\times 10^{12}$ & $37 \pm 10$ & & ($6.0 \pm 1.4$)$\times 10^{12}$ & $22 \pm 5$ & & ($3.9 \pm 1.1$)$\times 10^{12}$ & $19 \pm 5$ \\
\enddata
\tablecomments{The errors represent the standard deviation. The upper limits were derived using the $1\sigma$ noise level.}
\end{deluxetable}
\section{Discussion} \label{sec:discuss}
In this section, we will discuss comparisons of the chemical composition among MYSOs using CH$_{3}$OH, CH$_{3}$CCH, and HC$_{5}$N, all of which are considered to be in the lukewarm envelopes (Table \ref{tab:rot}).
In the following discussion, we assume that the spatial distributions of CH$_{3}$OH, CH$_{3}$CCH, and HC$_{5}$N are similar to each other.
This assumption is based on the following results;
the source sizes of both HC$_{5}$N and CH$_{3}$CCH seem to be smaller than $18\arcsec$ (see \citet{2017ApJ...844...68T} and Section \ref{sec:rotational}), and the rotational temperatures of CH$_{3}$OH are comparable with those of CH$_{3}$CCH in our three sources.
Nitrogen-bearing COMs are usually originated from hot cores \citep[e.g.,][]{2007A&A...465..913B}, and thus we do not include them in discussion.
We will also discuss the relationship between the line width and excitation energy of rotational lines of CH$_{3}$OH.
\subsection{Comparisons of Fractional Abundances in the High-Mass Star-Forming Regions} \label{sec:discuss2}
We derived the fractional abundances of HC$_{3}$N, CH$_{3}$OH, and CH$_{3}$CCH, $X$($a$) = $N$($a$)/$N$(H$_{2}$), in the three sources.
The H$_{2}$ column densities, $N$(H$_{2}$), in the three sources were derived by \citet{2017ApJ...844...68T}.
The $N$(H$_{2}$) values and the fractional abundances of HC$_{3}$N, CH$_{3}$OH, CH$_{3}$CCH, and HC$_{5}$N derived by \citet{2017ApJ...844...68T} are summarized in Table \ref{tab:frac}.
The derived $X$(CH$_{3}$OH) values are $1.2^{+1.8}_{-0.9} \times 10^{-7}$, $8.8^{+11}_{-5.8} \times 10^{-8}$, and $4.6^{+7.0}_{-3.4} \times 10^{-8}$ in G12.89+0.49, G16.86$-$2.16, and G28.28$-$0.36, respectively.
The $X$(CH$_{3}$CCH) values are derived to be $4.2^{+2.8}_{-2.2} \times 10^{-8}$, $3.2^{+1.9}_{-1.5} \times 10^{-8}$, and $7.6^{+5.2}_{-4.4} \times 10^{-8}$ in G12.89+0.49, G16.86$-$2.16, and G28.28$-$0.36, respectively.
We derive the $X$(HC$_{3}$N) values for two cases, fitting results of two transitions and all transitions (Table \ref{tab:rot}).
Figure \ref{fig:f9} shows their fractional abundances in the three sources.
All of the column densities including the molecular hydrogen were derived as beam-averaged values for a beam size of $18\arcsec$.
More accurate column densities and fractional abundances can be obtained in the future via interferometric observations that resolve the size of the sources.
Although our sample is small, there seems to be an anti-correlation between $X$(CH$_{3}$OH) and $X$(HC$_{5}$N).
G12.89+0.49 shows the highest $X$(CH$_{3}$OH) value and the lowest $X$(HC$_{5}$N) value among the three sources, while G28.28$-$0.36 shows the lowest $X$(CH$_{3}$OH) value and the highest $X$(HC$_{5}$N) value.
We further discuss the relationship between HC$_{5}$N and CH$_{3}$OH in Section \ref{sec:discuss3}.
The $X$(HC$_{3}$N) has a positive correlation with $X$(HC$_{5}$N), while a negative correlation with $X$(CH$_{3}$OH).
The $X$(CH$_{3}$CCH) values in the three sources are consistent within their $1 \sigma$ errors, and we cannot find any clear correlations between HC$_{5}$N and CH$_{3}$CCH.
According to the gas-grain-bulk three-phase chemical network simulation \citep{2016MNRAS.458.1859M, 2017MNRAS.467.3525M, 2017MNRAS.466.4470M} results assuming that the temperatures are 20 and 30 K and the density is $10^{5}$ cm$^{-3}$, CH$_{3}$CCH is formed on the grains and desorbed non-thermally \citep{1993MNRAS.261...83H, 2007A&A...467.1103G}.
HC$_{5}$N is formed in the gas phase mainly by the neutral-neutral reaction of CN + C$_{4}$H$_{2}$ and the electron recombination reaction of H$_{2}$C$_{5}$N$^{+}$.
The neutral-neutral reaction is also a main formation pathway to HC$_{5}$N in the model by \citet{2009MNRAS...394...221}.
\citet{2009MNRAS...394...221} suggested that C$_{4}$H$_{2}$ is formed by the reaction of C$_{2}$H$_{2}$ + CCH.
C$_{2}$H$_{2}$ can be formed in the gas phase from CH$_{4}$ \citep{2008ApJ...681...1385} and can be directly evaporated from grain mantles \citep{2000A&A...355..699L}.
Hence, the productions of CH$_{3}$CCH and HC$_{5}$N seem to be related to the grain-surface reactions even in the lukewarm envelopes ($T \sim 20-30$ K).
No correlation between CH$_{3}$CCH and HC$_{5}$N may imply that there is no direct relationship between them, which is also indicated in the chemical network simulation.
\floattable
\begin{deluxetable}{lcccccc}
\tablecaption{The $N$(H$_{2}$) and fractional abundances of CH$_{3}$OH, CH$_{3}$CCH, HC$_{3}$N, and HC$_{5}$N in the three sources \label{tab:frac}}
\tablewidth{0pt}
\tablehead{
\colhead{Source} & \colhead{$N$(H$_{2}$)\tablenotemark{a}} & \colhead{$X$(CH$_{3}$OH)} & \colhead{$X$(CH$_{3}$CCH)} & \colhead{$X$(HC$_{3}$N)\tablenotemark{b}} & \colhead{$X$(HC$_{3}$N)\tablenotemark{c}} & \colhead{$X$(HC$_{5}$N)\tablenotemark{a}}\\
\colhead{} & \colhead{($\times 10^{22}$ cm$^{-2}$)} & \colhead{($\times 10^{-8}$)} & \colhead{($\times 10^{-8}$)} & \colhead{($\times 10^{-9}$)} & \colhead{($\times 10^{-9}$)} & \colhead{($\times 10^{-10}$)}
}
\startdata
G12.89+0.49 & $2.4^{+2.5}_{-0.8}$ & $12^{+18}_{-9}$ & $4.2^{+2.8}_{-2.2}$ & $1.8 \pm 0.9$ & $5.8^{+4.8}_{-3.6}$ & $9.9^{+6.1}_{-5.4}$ \\
G16.86$-$2.16 & $1.7^{+1.0}_{-0.5}$ & $8.8^{+11}_{-5.8}$ & $3.2^{+1.9}_{-1.5}$ & $2.5^{+1.1}_{-0.9}$ & $7.7^{+5.7}_{-3.9}$ & $16 \pm 7$ \\
G28.28$-$0.36 & $0.49^{+0.42}_{-0.16}$ & $4.6^{+7.0}_{-3.4}$ & $7.6^{+5.2}_{-4.4}$ & $4.1^{+2.0}_{-1.9}$ & $16^{+14}_{-9.5}$ & $42^{+26}_{-20}$ \\
\enddata
\tablecomments{The errors represent the standard deviation. These values are averaged ones with the beam size of $18\arcsec$.}
\tablenotetext{a}{\citet{2017ApJ...844...68T}}
\tablenotetext{b}{The HC$_{3}$N column densities were derived from $J=5-4$ and $10-9$.}
\tablenotetext{c}{The HC$_{3}$N column densities were derived from three rotational transitions.}
\end{deluxetable}
\begin{figure}
\figurenum{8}
\plotone{frac_rev4.eps}
\caption{Fractional abundances of CH$_{3}$OH, CH$_{3}$CCH, HC$_{3}$N, and HC$_{5}$N in the high-mass star-forming regions. The HC$_{5}$N values are taken from \citet{2017ApJ...844...68T}. \label{fig:f9}}
\end{figure}
\subsection{A Variety of the $N$(HC$_{5}$N)/$N$(CH$_{3}$OH) Ratio} \label{sec:discuss3}
\citet{2017ApJ...844...68T} found that the $N$(HC$_{5}$N)/$W$(CH$_{3}$OH) ratio, where $W$ represents the integrated intensity, varies by more than one order of magnitude among the three sources, and suggested the possibility of the chemical differentiation.
In this paper, we derived the CH$_{3}$OH column densities in the three sources, and then we calculated the $N$(HC$_{5}$N)/$N$(CH$_{3}$OH) ratios in the observed three sources, as shown in Figure \ref{fig:f10}.
We also added the data toward the high-mass protostellar object NGC2264 CMM3 \citep{2015ApJ...809..162W}.
\citet{2015ApJ...809..162W} derived $N$(HC$_{5}$N) and $N$(CH$_{3}$OH) to be ($2.5 \pm 0.9$)$\times 10^{13}$ cm$^{-2}$ and ($1.8 \pm 0.2$)$\times 10^{15}$ cm$^{-2}$, respectively.
We took the cold component value for CH$_{3}$OH, because its rotational temperature ($24.3 \pm 2.6$ K) is comparable with that of HC$_{5}$N ($25.8 \pm 4.6$ K) in NGC2264 CMM3 \citep{2015ApJ...809..162W}.
The $N$(HC$_{5}$N)/$N$(CH$_{3}$OH) ratio in G28.28$-$0.36 ($0.091^{+0.109}_{-0.039}$) is higher than that in G12.89+0.49 ($0.008^{+0.008}_{-0.004}$) by one order of magnitude, and those in G16.86$-$2.16 ($0.019^{+0.017}_{-0.008}$) and NGC2264 CMM3 ($0.014^{+0.007}_{-0.006}$) by a factor of $\sim 5$.
As \citet{2017ApJ...844...68T} discussed, the HC$_{5}$N molecules can be formed from CH$_{4}$ \citep[WCCC,][]{2008ApJ...674...993, 2008ApJ...681...1385} and/or C$_{2}$H$_{2}$ \citep{2009MNRAS...394...221}, which could be evaporated from grain mantles.
CH$_{3}$OH is mainly formed by the successive hydrogenation reaction of CO molecules on dust grains, while CH$_{4}$, may be C$_{2}$H$_{2}$ \citep{2000A&A...355..699L} as well, are formed by the hydrogenation reaction of C atoms on dust grains.
Therefore, the chemical differentiation in the lukewarm envelope may reflect the different ice chemical composition among the MYSOs.
Our source sample was chosen from the HC$_{5}$N-detected source list by \citet{2014MNRAS...443...2252}.
HC$_{5}$N was not detected in more than half of the sources where \citet{2014MNRAS...443...2252} carried out observations.
In addition, \citet{2018ApJ...854..133T} detected HC$_{5}$N in half of high-mass protostellar objects in their source list.
The $N$(HC$_{5}$N)/$N$(CH$_{3}$OH) ratio should cover a broader range of values than the presented here, when considering the HC$_{5}$N-undetected sources into consideration.
\begin{figure}
\figurenum{9}
\plotone{carbon_com_rev4.eps}
\caption{Comparison of the $N$(HC$_{5}$N)/$N$(CH$_{3}$OH) ratio among the three high-mass star-forming regions. The data for NGC2264 CMM3 were taken from \citet{2015ApJ...809..162W}. \label{fig:f10}}
\end{figure}
\subsection{Relationship between $N$(HC$_{5}$N)/$N$(CH$_{3}$OH) and $N$(CH$_{3}$CCH)/$N$(CH$_{3}$OH)} \label{sec:discuss4}
In this subsection, we follow the analysis by \citet{2015A&A...576A..45F} who compared the chemical composition normalized by CH$_{3}$OH among MYSOs.
We compare the chemical composition in the lukewarm envelope using the $N$(HC$_{5}$N)/$N$(CH$_{3}$OH) and $N$(CH$_{3}$CCH)/$N$(CH$_{3}$OH) ratios.
Figure \ref{fig:f11} shows the relationship between the $N$(HC$_{5}$N)/$N$(CH$_{3}$OH) ratio and $N$(CH$_{3}$CCH)/$N$(CH$_{3}$OH) ratio in the four MYSOs, the observed three sources and NCG2264 CMM3 \citep{2015ApJ...809..162W}.
The $N$(HC$_{5}$N)/$N$(CH$_{3}$OH) ratio correlates with the $N$(CH$_{3}$CCH)/$N$(CH$_{3}$OH) ratio.
This is caused by the fraction of the lukewarm envelope in the single-dish beams, rather than the chemical differentiation.
The chemical composition, based on HC$_{5}$N and CH$_{3}$CCH, of the lukewarm gas in G28.28$-$0.36 is similar to those in G16.86$-$2.16 and NGC2264 CMM3 (Ratio $\sim 20$).
On the other hand, the $N$(CH$_{3}$CCH)/$N$(CH$_{3}$OH) ratio compared to the $N$(HC$_{5}$N)/$N$(CH$_{3}$OH) ratio in G12.89+0.49 is higher (Ratio $\approx 40$) than those in the other three sources.
These results suggest the chemical diversity in the lukewarm envelope.
\begin{figure}
\figurenum{10}
\plotone{envelope_rev4.eps}
\caption{Plot of $N$(CH$_{3}$CCH)/$N$(CH$_{3}$OH) versus $N$(HC$_{5}$N)/$N$(CH$_{3}$OH). The data for NGC2264 CMM3 were taken from \citet{2015ApJ...809..162W}. The labels of G12, G16, G28, and NGC2264 indicate the plots for G12.89+0.49, G16.86$-$2.16, G28.28$-$0.36, and NGC2264 CMM3, respectively. \label{fig:f11}}
\end{figure}
\subsection{Relationship of the Chemical Composition between Central Core and Envelope} \label{sec:discuss5}
\citet{2012ApJS..202....1H} carried out 1-mm spectral line survey observations toward 89 GLIMPSE Extended Green Objects \citep[EGOs;][]{2008AJ...136...2391}.
From their survey observations, there are largely two types of EGO clouds; line-rich and line-poor sources.
It is unclear why some sources show line-poor spectra, and others show many lines of COMs.
\citet{2014MNRAS.445.1170G} suggested that the EGO cloud cores are possibly in the short onset phase of the hot core stage, when CH$_{3}$OH ice is quickly evaporating from grain surfaces at a gas temperature of $\sim 100$ K.
Hence, the difference between the line-poor and line-rich EGO cloud cores seems to originate from the chemical differentiation rather than the chemical evolution.
Besides, \citet{2015A&A...576A..45F} compared the chemistry between organic-poor MYSOs and organic-rich MYSOs, namely hot cores, but the relationship between the central core and the envelope was not clear.
In this subsection, we discuss a possibility of the relationship of the chemical composition between the core and the envelope.
There are clear differences in the spectra among the three sources (Figures \ref{fig:f6} $-$ \ref{fig:f5}).
As summarized in Table \ref{tab:resCOMs}, the detected COMs are different among the three observed sources.
CH$_{3}$CHO was detected toward all of the three sources, and it is considered to exist in the envelope \citep{2015A&A...576A..45F}.
In G12.89+0.49, the largest number of COMs have been detected, while G28.28$-$0.36 is a line-poor source.
The results in G28.28$-$0.36, a line-poor source, are consistent with those of \citet{2012ApJS..202....1H}, showing that only H$^{13}$CO$^{+}$ was detected in G28.28$-$0.36.
As discussed in Section \ref{sec:discuss3}, the $N$(HC$_{5}$N)/$N$(CH$_{3}$OH) ratio in G28.28$-$0.36 is higher than that in G12.89+0.49 by an order of magnitude.
In the case of G16.86$-$2.16, the $N$(HC$_{5}$N)/$N$(CH$_{3}$OH) ratio shows a value between G12.89+0.49 and G28.28$-$0.36, and the line density and the COM's line intensities are also between the others.
From these results, the line-poor MYSOs are likely to be surrounded by the carbon-chain-rich envelope, while the line-rich MYSOs, namely hot cores, appear to be surrounded by the CH$_{3}$OH-rich envelope.
\subsection{Isotopic Fractionation of HC$_{3}$N} \label{sec:discuss6}
The $^{12}$C/$^{13}$C ratio shows a gradient with distance from the Galactic center \citep[$D_{\rm {GC}}$; e.g.,][]{2002ApJ...578...211, 2005ApJ...634...1126}.
Recent observations derived the following relationship \citep{2017ApJ...845..158H};
\begin{equation} \label{equ:carbon}
^{12}{\rm {C}}/^{13}{\rm {C}} = 5.21 (\pm 0.52) D_{\rm {GC}} + 22.6 (\pm 3.3),
\end{equation}
where $D_{\rm {GC}}$ is in units of kpc.
We estimated the $D_{\rm {GC}}$ values of each source using trigonometry.
The $D_{\rm {GC}}$ values are estimated at 5.8 and 8.9 kpc for G12.89+0.49 and G16.86$-$2.16, respectively.
The $D_{\rm {GC}}$ value of G28.28$-$0.36 was derived to be 5.4 kpc by \citet{2016ApJ...830..106T}.
From Equation (\ref{equ:carbon}), the $^{12}$C/$^{13}$C ratios are calculated at $53 \pm 6$, $69 \pm 8$, and $51 \pm 6$ in G12.89+0.49, G16.86$-$2.16, and G28.28$-$0.36, respectively.
The $^{12}$C/$^{13}$C ratios derived from observations (Table \ref{tab:LTE}) are generally lower than or consistent with the values calculated from Equation (\ref{equ:carbon}) within the errors, except for non-detection species.
These results mean that the $^{13}$C species of HC$_{3}$N are not heavily diluted.
The $^{12}$C/$^{13}$C ratios of cyanopolyynes in dark clouds \citep[e.g., TMC-1, L1521B, and L134N;][]{2016ApJ...817..147T, 2017ApJ...846...46T} are generally lower than those of other carbon-chain species (e.g., CCH, CCS, C$_{3}$S, C$_{4}$H), which means that the $^{13}$C species of cyanopolyynes are not heavily diluted.
Similar results with the $^{12}$C/$^{13}$C ratios of HC$_{3}$N being not significantly high were also found in the warm gas around the protostar L1527 \citep{2016ApJ...830..106T}.
Our results show the similar tendency to the local low-mass star-forming regions.
\citet{2016A&A...587A..91B} reported a tentative detection of DC$_{3}$N toward the Sgr B2 (N2) hot core and derived the D/H ratio to be 0.09\%.
A tentative detection of DC$_{3}$N was also achieved toward the Compact Ridge and the hot core of Orion KL \citep{2013A&A...559A..51E}, and the deuterium fractionation was estimated at $1.5 \pm 0.9$\%.
The deuterium fractionation in the high-mass protostellar object NGC2264 CMM3 was calculated at $1.8 \pm 1.5$\% from a tentative detection of DC$_{3}$N \citep{2015ApJ...809..162W}.
The higher D/H values were reported toward cold dense cores \citep[5\%$-$10\%,][]{1994MNRAS.267...59H} and toward the L1527 protostar, which is one of the warm carbon chain chemistry sources \citep[$\sim 3$\%,][]{2009ApJ...702.1025S}.
The D/H values of HC$_{3}$N in G16.86$-$2.16 and G28.28$-$0.36 ($\sim 1 - 5$\%) is higher than that in Sgr B2, lower than cold dense cores, and comparable to Orion KL hot core and L1527.
On the other hand, DC$_{3}$N was not detected in G12.89+0.49 and we only derived the upper limit for its D/H ratio, which is significantly lower than those in G16.86$-$2.16 and G28.28$-$0.36.
In general, the D/H ratio becomes lower in the higher temperature regions \citep[e.g.,][]{2012A&ARv..20...56C}.
Hence, the results may imply that the emission of HC$_{3}$N comes from the G12.89+0.49 central hot core position, and the lukewarm envelope component is less than those in the other two sources.
In the observations presented here, the lukewarm envelopes and the central cores were covered by the single-dish beam, and we cannot distinguish between them.
The D/H ratio significantly depends on the temperature, and we need the high-spatial resolution observations using interferometers such as ALMA to derive the temperature dependence of the D/H ratio.
\subsection{A Comparison of Line Width of CH$_{3}$OH} \label{sec:discuss1}
Line width is a key tool to characterize the environments where the molecules exist.
In general, high-excitation energy lines are expected to trace the hotter gas and show broad line widths, whereas low-excitation energy lines come mainly from cold molecular clouds and show narrow line widths.
We detected several CH$_{3}$OH emission lines with different excitation energies.
We will investigate the relationship between the excitation energy and the line width of CH$_{3}$OH in this subsection.
Figure \ref{fig:f8} shows a comparison between the line widths of CH$_{3}$OH and the excitation energy of each line.
We derive the line width ($\Delta v$) using the following formula;
\begin{equation}
\Delta v = \sqrt{\Delta v_{\rm {obs}}^2 - \Delta v_{\rm {inst}}^2},
\end{equation}
where $\Delta v_{\rm {obs}}$ and $\Delta v_{\rm {inst}}$ are the observed line widths (Tables \ref{tab:resCH3OH} and \ref{tab:resASTE}) and the instrumental velocity width (0.85 km s$^{-1}$ $-$ 0.86 km s$^{-1}$, Section \ref{sec:obs}), respectively.
We conducted the Kendall's $\tau$ correlation coefficient test.
The probability ($p$) that there is no correlation between the line width and the excitation energy is calculated at 0.03\%, and the $\tau$ value is $+0.51$ in G12.89+0.49.
This suggests the weak positive correlation between the line width and the excitation energy.
The $p$ and $\tau$ values are derived to be 69\% and $+0.08$ in G16.86$-$2.16, respectively.
In G28.28$-$0.36, the $p$ and $\tau$ values are 72\% and $-0.45$, respectively.
Hence, there is no clear correlation between the line width and the excitation energy in G16.86$-$2.16 and G28.28$-$0.36.
This may be caused by the non-detection of the very-high-excitation energy lines in the two sources.
We also compared the distributions of the CH$_{3}$OH line widths among the three sources using the two-sample Kolmogorov-Smirnov test (K-S test).
We compared the distributions for all of the combinations; (a) G12.89+0.49 and G28.28$-$0.36, (b) G16.86$-$2.16 and G28.28$-$0.36, and (c) G12.89+0.49 and G16.86$-$2.16.
The probabilities that the distributions of the line widths in the selected two sources are the same are derived to be $3.2 \times 10^{-4}$\%, $5.8 \times 10^{-3}$\%, and 39\% for case (a), (b), and (c), respectively.
These results mean that the distribution of the CH$_{3}$OH line width in G28.28$-$0.36 is clearly different from those in the other two sources, and we cannot exclude the possibility that the distributions in G12.89+0.49 and G16.86$-$2.16 are different.
The average line widths of CH$_{3}$OH are $4.8 \pm 0.6$, $5.2 \pm 0.6$, and $2.6 \pm 0.5$ km s$^{-1}$ in G12.89+0.49, G16.86$-$2.16, and G28.28$-$0.36, respectively.
Although the average line widths in G12.89+0.49 and G16.86$-$2.16 are consistent with each other within the errors, G16.86$-$2.16 shows the highest value; nevertheless the very-high-excitation energy lines were not detected in G16.86$-$2.16.
These imply that CH$_{3}$OH exists in turbulent gas such as a molecular outflow in G16.86$-$2.16 rather than the hot gas.
This is supported by the strong wing emission in the CH$_{3}$OH spectra in G16.86$-$2.16.
In G12.89+0.49, CH$_{3}$OH seems to exist in the hot gas as well as in molecular outflow suggested by the wing emission.
On the other hand, the line width in G28.28$-$0.36 is much narrower than those in the other two sources.
This suggests that CH$_{3}$OH exists mainly in the relatively quiescent region, i.e. envelope for this source.
The non-detection of the very-high-excitation energy lines is consistent with the envelope origin.
There is a possibility of the molecular outflow origin, but a low S/N ratio prevents us from confirmation.
\begin{figure}
\figurenum{11}
\plotone{FWHM.eps}
\caption{Relationship between excitation energy and line width of CH$_{3}$OH. \label{fig:f8}}
\end{figure}
\section{Conclusions}
We carried out observations in the 42$-$46 and 82$-$103 GHz bands with the Nobeyama 45-m radio telescope, and in the 338.2$-$339.2 and 348.45$-$349.45 GHz bands with the ASTE 10-m telescope toward the three high-mass star-forming regions associated with the 6.7 GHz CH$_{3}$OH masers, G12.89+0.49, G16.86$-$2.16, and G28.28$-$0.36.
The rotational temperatures and the beam-averaged column densities of HC$_{3}$N, CH$_{3}$CCH, and CH$_{3}$OH in the three sources are derived.
The $N$(HC$_{5}$N)/$N$(CH$_{3}$OH) ratio in G28.28$-$0.36 is higher than that in G12.89+0.49 by one order of magnitude and that in G16.86$-$2.16 by a factor of 5.
Moreover, the relationships between the $N$(HC$_{5}$N)/$N$(CH$_{3}$OH) ratio and the $N$(CH$_{3}$CCH)/$N$(CH$_{3}$OH) ratio in G28.28$-$0.36 and G16.86$-$2.16 are similar to each other, while HC$_{5}$N is deficient when compared to CH$_{3}$CCH in G12.89+0.49.
These results may imply the chemical diversity of the lukewarm envelope.
The line density in G28.28$-$0.36 is significantly low and a few COMs have been detected, while oxygen-/nitrogen-bearing COMs and high-excitation-energy lines have been detected from G12.89+0.49.
These results seem to imply that the organic-poor MYSOs (G28.28-0.36) are surrounded by the carbon-chain-rich lukewarm envelope, whereas the organic-rich MYSOs (G12.89+0.49 and G16.86-2.16), hot cores, are surrounded by the CH$_{3}$OH-rich lukewarm envelope.
The results presented in this paper were led based on observations with single-dish telescopes without information about the spatial distributions of each molecular emission and only a few molecular species.
Further observations are required to confirm the trends reported in this work.
\acknowledgments
We thank the anonymous referee who gave us valuable comments which helped us improve the quality of this paper.
We would like to express our great thanks to the members of the Nobeyama Radio Observatory.
The Nobeyama Radio Observatory is a branch of the National Astronomical Observatory of Japan (NAOJ), National Institutes of Natural Sciences.
The Z45 receiver is supported in part by a Granting-Aid for Science Research of Japan (24244017).
We thank to the operation staff members of the ASTE.
The ASTE telescope is operated by the NAOJ.
KT appreciates support from a Granting-Aid for Science Research of Japan (17J03516).
LM also acknowledges support from the NASA postdoctoral program.
A portion of this research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration.
\vspace{5mm}
\facilities{Nobeyama 45-m radio telescope, Atacama Submillimeter Telescope Experiment (ASTE)}
\software{Java NEWSTAR (https://www.nro.nao.ac.jp/~nro45mrt/html/obs/newstar/index.html)}
|
{
"timestamp": "2018-09-03T02:11:51",
"yymm": "1804",
"arxiv_id": "1804.05205",
"language": "en",
"url": "https://arxiv.org/abs/1804.05205"
}
|
\section{Introduction}
\IEEEPARstart{I}{n} recent years, the amount of network traffic has increased significantly because of the large number of users connecting to the network. Moreover, the boom in the Internet of Things is expected to increase network traffic dramatically. To address this soaring traffic demand, next-generation wireless technologies such as 5G are required to provide advantages, such as better spectral efficiency, massive connectivity, and faster response time \cite{1}.
\par
Non-orthogonal multiple access (NOMA) is a promising candidate for 5G to achieve better capacity gains because of its high spectral efficiency \cite{2,3,4}. In \cite{2}, the author classified NOMA as code domain and power domain. In this letter, we aimed for power domain NOMA (hereinafter referred to as NOMA). In NOMA, multiple users are served in each orthogonal resource block, e.g., a time slot, frequency channel, or spreading code, by exploiting the power domain. The signals of multiplexed users are allocated different power levels by the base station (BS), superimposed with each other, and transmitted. To recover their signals, cell center users perform successive interference cancellation (SIC) \cite{5}. However, cell-edge users do not perform SIC, and experience a decrease in spectral efficiency due to degraded signals. Recently, to enhance the spectral efficiency, NOMA with spatial modulation (SM) was investigated in \cite{wang2017achievable}, \cite{wang2017spectral}. In those studies, the authors aimed to analyze the spectral efficiency of SM-NOMA from the point of view of mutual information.
\par
To solve the problem of cell-edge users, NOMA-SSK has been suggested, in which the cell-edge user is multiplexed in the spatial domain to improve the spectral efficiency of the system by using NOMA and space shift keying (SSK) \cite{6}. SSK is a multiple-input multiple-output (MIMO) technique, which transmits information using an antenna index, contrary of the conventional modulation schemes \cite{7}. Moreover, the application of SSK can efficiently reduce transmitter overhead and receiver complexity by using the antenna index instead of any modulation scheme. However, because of the characteristics of SSK, the number of transmit antennas must be a power of two.
\par
In this letter, to further improve the spectral efficiency of cell-edge users and to overcome the limitation on the number of transmit antennas of NOMA-SSK, we propose a novel transmission scheme by combining NOMA and generalized space shift keying (GSSK), called NOMA-GSSK. GSSK is a generalized form of SSK and uses multiple transmit antennas, unlike SSK \cite{8}. The proposed scheme achieves higher spectral and energy efficiency, and lower bit error rate (BER), compared to MIMO-NOMA and NOMA-SSK, because the users are multiplexed in both the power and spatial domains, by using a set of transmit antennas, whereas in MIMO-NOMA, all antennas are used to transmit the NOMA signal. In addition, because cell-edge users are multiplexed in the spatial domain, the complexity of the system is also decreased, because the SIC steps are reduced.
\par
\section{System Model}
\begin{figure}[b]
\centering
\includegraphics[width=0.27\textwidth]{NOMA-GSSK_transmit_model.jpg}
\caption{NOMA-GSSK Transmitter model.}
\label{fig:Transmit_Model}
\end{figure}
Assume $N+K$ users to be uniformly distributed in a cell,
$N$ users are multiplexed using NOMA, and $K$ users exploit the spatial domain. The channel gains of users are in the order $|h_1| \geq \cdots \geq |h_{N}| \geq |h_{N+1}| \geq \cdots \geq |h_{N+K}|$. The users with low channel gain are regarded as cell-edge users, and are multiplexed by GSSK. Fig.\ref{fig:Transmit_Model} shows the downlink transmitter model for $K$ users. Fractional transmit power allocation (FTPA) is used to allocate ower to the $N$ NOMA users. The transmitted signal is
\begin{equation}
X=\sum_{i=1}^{N}\sqrt{\alpha_i P}x_i,
\end{equation}
where $\alpha_i$ is the $i_{th}$ user's power allocation factor, such that $\alpha_1<\alpha_2<\cdots<\alpha_i<\cdots<\alpha_{N}$ and $\sum_{i=1}^{N}\alpha_i=1$, $P$ is the total transmit power, and $x_i$ is the symbol of the $i$-th user.
\par
As shown in Fig.\ref{fig:Transmit_Model}, $N$ users are transmitted using specific antennas from the entire set of transmit antennas, $M_t$. The data symbols of cell-edge users, $UE_{K}$, are transmitted by using the selected specific antenna set on the basis of the antenna index information. Symbols for the cell-edge users are transmitted by the antenna allocation on the basis of the GSSK mapping rule. For the example of $M_{a}=2$ active transmit antennas, transmitted signal $X_i$ is expressed as
\begin{equation}
X_i \triangleq [\dfrac{X}{\sqrt{M_a}}\;\;\;\cdots\;\;\;0\;\;\;\dfrac{X}{\sqrt{M_a}}\;\;\;\cdots\;\;\;0]^T,
\end{equation}
where $M_a$ is number of active transmit antennas.
\par
The received signal of the $i$-th user can be expressed as
\begin{equation}
y_i=h_{i,j}X_i+n_{i,j},
\end{equation}
where $h_{i,j}$ is the channel gain of the $i$-th user using the $j$-th antenna set, and $n_{i,j}$ is additive white Gaussian noise (AWGN).
\begin{figure}[t]
\centering
\begin{subfigure}[b]{0.27\textwidth}
\centering
\includegraphics[width=1\textwidth]{NOMA_freq_band_fix.jpg}
\caption{NOMA}
\end{subfigure}%
~
\begin{subfigure}[b]{0.23\textwidth}
\centering
\includegraphics[width=1\textwidth]{NOMA-GSSK_frequency_band.jpg}
\caption{NOMA-GSSK}
\end{subfigure}
\caption{Frequency Distribution of NOMA and NOMA-GSSK in 4 users case.}
\label{fig:Frequency_Band_NOMA_GSSK}
\end{figure}
\par
In the GSSK case, each active transmit antenna sends only a constant signal $1/{\sqrt{M_a}}$, because it transmits only antenna index information based on the set of transmit antennas. However, NOMA-GSSK transmits $X/\sqrt{M_a}$ symbols , similar to generalized spatial modulation (GSM) \cite{9}. By transmitting the superposed signal $X_i$ and antenna index information together, spectral efficiency is improved.
\par
$N$ NOMA users detect transmitted signals in the same way as NOMA. However, because of the characteristics of NOMA-GSSK, the detection method of the received data is different from that of NOMA. NOMA users are detected by SIC, and $K$ users multiplexed in the spatial domain are detected by a maximum likelihood (ML) detector. Moreover, because only $N$ users are multiplexed in the power domain, the complexity of the system can be decreased by reducing the use of SIC, which requires high complexity.
\par
For example, in the case of $4$ users ($N+K=4$,), Fig. \ref{fig:Frequency_Band_NOMA_GSSK} shows the comparison of frequency distribution between NOMA and NOMA-GSSK. In the example, we assume that N=2 users can be multiplexed by NOMA.
Unlike NOMA, in NOMA-GSSK, $UE_3$ and $UE_4$ are multiplexed in the spatial domain, and the remaining users in the power domain, i.e., NOMA. Assuming that one channel bandwidth is $15$~kHz (LTE's sub-band channel bandwidth), NOMA requires $30$~kHz for $4$ users, considering two users per channel. NOMA-GSSK can support all $4$ users with only $15$~kHz ($1$ sub-channel). This shows that NOMA-GSSK has naturally better spectral efficiency than NOMA. There are $N_H$ possible index sets $j$ representing the active antennas, i.e., $b_H$ = $log2(N_H)$ bits can be conveyed by the particular choice of index set j. If K$>$1 cell-edge users are supposed to be supported, they have to share these $b_H$ bits, i.e., each user will receive $b_H/K$ bits-per-channel-use (bpcu).
\subsection{Cell-edge user detector}
In the proposed NOMA-GSSK, GSSK is used for the symbol transmission of cell-edge users, and symbol detection is performed by determining which set of transmit antennas is actively transmitting. Cell-edge users receive NOMA symbols, and evaluate the transmit antenna index used at the BS.
Received signals are demodulated by using an ML detector. For each $i_{th}$ cell-edge user, ML detection can be expressed as
\begin{equation}
\hat{l}=\underset{\textbf{j}}{\arg\min}\Vert y_i-\sqrt{\rho'}h_{j,eff} \Vert ^2,
\end{equation}
where $\hat{l}$ is the detected transmit antenna set, $\rho'$ is signal-to-noise ratio (SNR), $h_{j,eff}$ is the effective channel gain of the $j_{th}$ antenna set ($h_{j,eff}=h_{j(1)}+h_{j(2)}+\cdots+h_{j(M_a)}$, \textbf{j}($\cdot$) = $j \in \{1,2,...,M_t\}$). It is to be noted that, as the cell-edge users information is modulated using antenna set, they are only concerned about the transmit antenna set detection.
Error performance of the ML detector can be derived as
\setlength{\arraycolsep}{0.0em}
\begin{eqnarray}
P_e \leq \dfrac{1}{N_H log_2 (N_H)} \sum_{j}^{N_H} \sum_{k,k \neq j}^{N_H} N_{b_{j,k}} Q \left( A \right),\\
A = \sqrt{\dfrac{\overline{\gamma}}{M_a} \vert \sum_{l=1}^{M_a}[h_{j(l)}-h_{k(l)}] \vert^2 },
\end{eqnarray}
where $N_{b_{j,k}}$ is the number of error bits between the $j$-th and $k$-th constellation points that follow the GSSK mapping rule, $N_H$ the possible constellation with size of a power of 2 ($j=1,2,...,N_H$), $Q(\cdot)=1/\sqrt{2\pi}\int_{0}^{\infty}exp(-u^2/2)du$, $\overline{\gamma}$ is the SNR (average SNR), and $h_{x(l)}$ is the $x$-th constellation point \cite{8}.
\par
The sum-rate of all cell-edge users ($UE_{N+1}$,$\cdots$,$UE_{N+K}$) is expressed as
\begin{equation}
R_{K}=(1-P_e) \lfloor \log_2(_{M_t}C_{M_a}) \rfloor,
\end{equation}
where $_{M_t}C_{M_a}$ is the binomial coefficient of ($M_t$, $M_a$)
\section{Performance Analysis}
\subsection{Capacity Analysis}
For a total of $N+K$ users in a MIMO-NOMA system\cite{4}, the capacity is given by
\begin{equation}
R_{MIMO-NOMA}=\log_2(\rho \log_2(\log_2{(N+K)})).
\end{equation}
\par
The capacity of NOMA-SSK and NOMA-GSSK can be calculated as the sum of the capacity of $N$ NOMA users and the capacity of cell-edge users using the spatial domain. NOMA-SSK has an average capacity given as
\begin{equation}
R_{NOMA-{SSK}}=\log_2(\rho \log_2(\log_2 (N)))+(1-P_e)\lfloor \log_2(M_t) \rfloor.
\end{equation}
\par
The capacity of NOMA-GSSK is given by
\begin{equation}
R_{NOMA-{GSSK}} = \log_2(\rho \log_2(\log_2 (N)))+R_K.
\end{equation}
\par
\iffalse
\begin{figure}[t!]
\centering
\includegraphics[width=0.30\textwidth]{SE_diff.eps}
\caption{Capacity of MIMO-NOMA, NOMA-SSK, NOMA-GSSK according to the number of transmit antennas (NOMA-GSSK, $M_a$ = 2).}
\label{fig:SE_diff}
\end{figure}
\fi
Fig. \ref{fig:SE_diff_1} shows the capacity comparison with respect to the number of transmit antennas. When the number of transmit antennas is less than $4$, the capacity of NOMA-SSK is equal to that of NOMA-GSSK. However, when the number of transmit antennas is more than $4$, NOMA-GSSK has a higher capacity, because GSSK can have a plurality of active transmit antenna sets rather than one active transmit antenna.
\subsection{Energy Efficiency Analysis}
Generally, the energy efficiency is expressed as
\begin{equation}
\eta = \dfrac{R}{P_T},
\end{equation}
where $R$ denotes the capacity, and $P_T$ is the total transmit power.
\par
The cell-edge user is multiplexed in the spatial domain and the information is transmitted using the antenna index set. Because the cell-edge user of NOMA-GSSK does not use power allocation, NOMA-GSSK has superior energy efficiency compared to MIMO-NOMA.
\par
In MIMO-NOMA, where the total power is allocated to the entire number of users $N+K$, the total power of MIMO-NOMA can be expressed as $P_{T(MIMO-NOMA)} = \sum_{i=1}^{N+K}\alpha_i P$. NOMA-GSSK assigns the total power to users other than the cell-edge users like NOMA-SSK does. Therefore, $P_{T(NOMA-SSK)} = P_{T(NOMA-GSSK)} = \sum_{i=1}^{N}\alpha_i P$ and the energy efficiency of MIMO-NOMA, NOMA-SSK, and NOMA-GSSK are given by
\begin{equation}
\eta_{MIMO-NOMA} = \dfrac{R_{MIMO-NOMA}}{\sum_{i=1}^{N+K}\alpha_i P},
\end{equation}
\begin{equation}
\eta_{NOMA-SSK} = \dfrac{R_{NOMA-SSK}}{\sum_{i=1}^{N}\alpha_i P},
\end{equation}
\begin{equation}
\eta_{NOMA-GSSK} = \dfrac{R_{NOMA-GSSK}}{\sum_{i=1}^{N}\alpha_i P}.
\end{equation}
\par
Eqs. (12)-(14) clearly show that NOMA-GSSK has improved energy efficiency compared to conventional schemes.
\subsection{Complexity Analysis}
The complexity of SIC can be divided into two parts: decoding and subtraction. In this system, because an ML detector is used, the complexity of MIMO-NOMA in UE$_j$ can be obtained as
\begin{equation}
O_{MIMO-NOMA} = (4M_{r}M_{t}M + 2M_{r}M^{M_{t}})(N+K-j+1),
\end{equation}
where $M_r$ is the number of receive antennas, $M$ is the modulation order, and $j$ is the ordering for UE from the nearest UE $(1 \leq j \leq N+K)$. In (15), $4M_{r}M_{t}M + 2M_{r}M^{M_{t}}$ is the decoding part based on the ML detector \cite{10}, $(N+K-j+1)$ is the subtraction part, and the unit of complexity is the number of add-compare operations.
\par
Indeed, the subtraction step of UE$_j$ is $N+K-1$, because the last user of NOMA does not perform SIC. However, NOMA users should decode their own signals after subtraction.
\par
From (15), the complexity for all users can be obtained as
\begin{equation}
O_{MIMO-NOMA,total} = \dfrac{(N+K)(2+(N+K-1))}{2}(4M_{r}M_{t}M + 2M_{r}M^{M_{t}}).
\end{equation}
\par
The complexity of NOMA-SSK and NOMA-GSSK can be calculated using the same approach. However, SSK and GSSK decoding complexity is different from that of the MIMO-ML detector. For this reason, we applied the decoding complexity equation from \cite{7}:
\begin{equation}
O_{NOMA-SSK,total} = \dfrac{N(2+(N-1))}{2}(4M_{r}M_{t}M + 2M_{r}M^{M_{t}})N+(KN_{r}M).
\end{equation}
\par
In the NOMA-GSSK case, the number of transmit antennas is less than in the MIMO-NOMA and NOMA-SSK cases for the same number of accommodated users, ($N+K$). Therefore, it can achieve lower complexity than the other schemes.
\begin{equation}
O_{NOMA-GSSK,total} = \dfrac{N(2+(N-1))}{2}(4M_{r}M_{t}M + 2M_{r}M^{M_{t}})N + (KM_{a}M_{r}log_2(_{M_{t}}C_{M_{a}})).
\end{equation}
\par
In Table \ref{complex_table}, with some numerical examples, we show that the complexity of NOMA-GSSK is lower than that of MIMO-NOMA and NOMA-SSK. The low complexity of NOMA-GSSK is one of the advantages that makes it easier to implement than other schemes (e.g., MIMO-NOMA, NOMA-SSK).
\begin{table*}[]
\centering
\caption{Comparison of the complexity of MIMO-NOMA, NOMA-SSK and NOMA-GSSK.}
\label{complex_table}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
\hline
\multirow{2}{*}{N+K} & \multirow{2}{*}{K} & \multirow{2}{*}{$M_r$} & \multirow{2}{*}{$M_t$ (MIMO-NOMA, NOMA-SSK)} & \multirow{2}{*}{$M_t$ (GSSK)} & \multirow{2}{*}{$M_a$} & \multirow{2}{*}{M} & \multicolumn{3}{c|}{Complexity} \\ \cline{8-10}
& & & & & & & MIMO-NOMA & NOMA-SSK & NOMA-GSSK \\ \hline
5 & 2 & 4 & 4 & 4 & 1 & 2 & 3,840 & 1,024 & 1,024 \\ \hline
5 & 2 & 4 & 8 & 5 & 2 & 3 & 793,080 & 317,256 & 5,072 \\ \hline
\end{tabular}
\end{table*}
\begin{figure*}[t!]
\centering
\begin{subfigure}[t]{0.22\textwidth}
\centering
\includegraphics[height=100pt,width=\textwidth]{SE_diff.eps}
\caption{Data Rate (NOMA-GSSK, $M_a$ = 2).}
\label{fig:SE_diff_1}
\end{subfigure}
\begin{subfigure}[t]{0.22\textwidth}
\centering
\includegraphics[height=100pt,width=\textwidth]{SE.eps}
\caption{Spectral Efficiency}\label{fig:SE}
\end{subfigure}
\begin{subfigure}[t]{0.22\textwidth}
\centering
\includegraphics[height=100pt,width=\textwidth]{EE.eps}
\caption{Energy Efficiency}\label{fig:EE}
\end{subfigure}
\begin{subfigure}[t]{0.22\textwidth}
\centering
\includegraphics[height=100pt,width=\textwidth]{BER_rev.eps}
\caption{BER of cell-edge user}\label{fig:BER}
\end{subfigure}
\caption{Performance comparison of MIMO-NOMA, NOMA-SSK, and NOMA-GSSK.}\label{fig:total}
\end{figure*}
\section{Numerical Results}
This section discusses the evaluation of the performance of the proposed scheme by simulation. We assumed that the receiver perfectly knows the channel state information (CSI) of the flat Rayleigh fading channel with AWGN. We let $M_r=4$, $N=2$, $K=1$ in all simulations (MIMO-NOMA, $N=3$, $K=0$). In Fig.~\ref{fig:SE_diff_1} and \ref{fig:SE}, $M_t=2$ for MIMO-NOMA. User channel gains are in the range $0 \leq |h_i| \leq 1$. For cell-center users, $|h_i| \geq 0.6$, whereas for cell-edge users, $|h_i| \leq 0.4.$
\par
Fig.~\ref{fig:SE} compares the spectral efficiency of MIMO-NOMA, NOMA-SSK, and NOMA-GSSK. NOMA-SSK and NOMA-GSSK show better spectral efficiency than MIMO-NOMA because of the gain of exploiting the spatial domain. In addition, NOMA-GSSK has better spectral efficiency than NOMA-SSK, if the same number of transmit antennas is used, depending on the characteristics of GSSK. This clearly shows the comparison between NOMA-SSK, with $M_t=8$, and NOMA-GSSK, with $M_t=8, M_a=4$.
\par
Fig.~\ref{fig:EE} shows the energy efficiency comparison of MIMO-NOMA, NOMA-SSK, and NOMA-GSSK. In Fig. \ref{fig:EE}, we can see that the energy efficiency of NOMA-SSK with $M_t = 4$ and NOMA-GSSK with $M_t = 4, M_a = 2$ is the same, because the same energy is assigned, and the data rate is equal to 2 bps. However, when the number of transmit antennas is 8, the efficiency of NOMA-GSSK is better than that of NOMA-SSK, because the achievable data rate of NOMA-GSSK with $M_t=8, M_a=3$ is 4 bps. The achievable data rate of NOMA-SSK with $M_t=8$ is 3 bps. In this case, NOMA-GSSK can transmit one more bit at the one-channel bandwidth using the same amount of transmit energy.
\par
Fig.~\ref{fig:BER} shows the BER comparison of cell-edge users in MIMO-NOMA, NOMA-SSK, and NOMA-GSSK for the same number of bpcu case. The total number of users was 3. Therefore, three users were multiplexed in the power domain in MIMO-NOMA, and two users were multiplexed in the power domain in NOMA-SSK and NOMA-GSSK. Because NOMA-SSK and NOMA-GSSK utilize the spatial domain, interference caused by power allocation does not occur. As a result, their BER performance is better than that of MIMO-NOMA.
\section{Conclusion}
In this letter, we propose NOMA-GSSK using multiple active transmit antennas for performance enhancement. In NOMA-GSSK, the spatial domain was assigned to cell-edge users for transmitting symbol information using only the antenna index without SIC. In addition, computational complexity was reduced. Both analytical and simulation results show that the proposed scheme achieves significant spectral and energy efficiency gain, and BER and complexity reduction, compared to MIMO-NOMA.
\bibliographystyle{IEEEtran}
|
{
"timestamp": "2018-04-17T02:15:56",
"yymm": "1804",
"arxiv_id": "1804.05611",
"language": "en",
"url": "https://arxiv.org/abs/1804.05611"
}
|
\section{Introduction}
ACL Anthology\footnote{https://aclweb.org/anthology/} is one of the popular initiatives of the Association for Computational Linguistics (ACL) to curate all publications related to computational linguistics and natural language processing at one common place. At present, it hosts more than 44,000 papers and is actively updated and maintained by Min Yen Kan. Since its inception, ACL Anthology functions as a repository with the collection of papers from ACL and related organizations in computational linguistics. However, it does not provide any additional statistics about authors, papers, venues, and topics. Also, it lacks advance search features such as article ranking by factoring in popularity or relevance, natural language query support, author profiles, topical search etc.
\subsection{Previous systems built on ACL anthology}
Owing to above limitations, ACL anthology remained an archival repository for quite a long time. \citet{bird2008acl} developed the \textit{ACL Anthology Reference Corpus (ACL ARC)} as a collaborative attempt to provide a standardized testbed reference corpus based on the ACL Anthology. Later, \citet{radev2009acl} have invested humongous manual efforts to construct \textit{The {ACL} Anthology Network Corpus (AAN)}. AAN consists of a manually curated database of citations, collaborations, and summaries and statistics about the network. They have utilized two OCR processing tools PDFBox\footnote{https://pdfbox.apache.org/} and ParsCit~\cite{councill2008parscit} for curation. AAN was continuously updated till 2013~\cite{radev2013}. Recently, this project has been moved to Yale University as part of the new LILY group\footnote{http://tangra.cs.yale.edu/newaan/}.
\subsection{The computational linguistic knowledge graph}
As a similar initiative, in this paper, we demonstrate the development of \textit{CL Scholar} which automatically mines ACL anthology and constructs computational linguistic knowledge graph (hereafter \textit{`CLKG'}). The current framework automatically crawls new articles, processes, indexes, constructs knowledge graph and generates searchable statistics without involving tedious manual annotations. We leverage state-of-the-art scientific article processing tool OCR++~\cite{singh-EtAl:2016:COLING2} for robust and automatic information extraction from scientific articles. OCR++ is an open-source framework that can extract from scholarly articles the metadata, the structure and the bibliography.
\begin{figure*}[!tbh]
\centering
\includegraphics[width=0.9\hsize]{fig/data_processing}
\vspace{-0.7cm}
\caption{Data processing flow diagram.}
\label{fig:knowledge_graph}
\vspace{-0.5cm}
\end{figure*}
The constructed $CLKG$ is modeled as a heterogeneous graph~\cite{sun2009ranking} consisting of four entities: author, paper, venue, and field. We utilize metapaths~\cite{sun2012mining} to implement the query retrieval framework.
\subsection{Natural language queries}
In the first-of-its-kind initiative, we extend the functionalities of \textit{CL Scholar} to answer natural language queries (hereafter `\textit{NLQ}') along with standard keyword-based queries. Currently, it answers \textit{binary}, \textit{statistical} and \textit{list} based $NLQ$. Overall, we handle more than 1200 variations of $NLQ$.
\noindent\textbf{Outline:} The rest of the paper is organized as follows. Section~\ref{sec:dataset} describes the ACL Anthology dataset. Section~\ref{sec:preprocess} details step by step extraction procedure for $CLKG$ construction. In section \ref{sec:knowledge_graph}, we describe $CLKG$. We describe our framework in section~\ref{sec:framework}. We conclude in section~\ref{sec:end} and identify future work.
\vspace{-0.2cm}
\begin{table}[!tbh]
\centering
\caption{General statistics about the ACL Anthology dataset.}
\label{tab:dataset}
\begin{tabular}{lc} \toprule
Number of papers &42,069\\
Year range& 1965--2017 \\
Total authors& 37,752\\
Total unique authors& 33,372\\
Total unified venues& 33\\
\bottomrule \hline
\end{tabular}
\vspace{-0.2cm}
\end{table}
\section{Dataset}
\label{sec:dataset}
\textit{CL Scholar} uses metadata and full-text PDF research articles crawled from ACL Anthology. ACL Anthology consists of more than 40,000 research articles published in more than 33 computational linguistic events (venues) including conferences, workshops, and journals. Table~\ref{tab:dataset} presents general statistics of the crawled dataset.
We crawl both metadata information (unique article identifier, article title, authors' names, and venue) as well as full-text PDF articles. Next, we describe in detail several pre-processing steps and knowledge graph construction methodology.
\section{Pre-processing and knowledge graph construction}
\label{sec:preprocess}
We process full-text PDFs using state-of-the-art extraction tool \textit{OCR++}~\cite{singh-EtAl:2016:COLING2}. We extract references, citation contexts, author affiliations and URLs from full-text. \textit{OCR++} also provides reference to citation contexts mapping. Raw information with several variations like author names, venue names and affiliations are assigned unique identifiers using standard indexing approaches. We only consider those reference papers that are present in ACL anthology. This rich textual, as well as citation relationship information, is utilized in the construction of $CLKG$. Figure~\ref{fig:knowledge_graph} presents the $CLKG$ construction from metadata and full-text PDF files crawled from ACL anthology.
\section{Computational linguistic knowledge graph}
\label{sec:knowledge_graph}
Computational linguistic knowledge graph ($CLKG$) is a \textit{heterogeneous graph}~\cite{sun2009ranking} consisting of four entities: author ($A$), paper ($P$), venue ($V$) and field ($F$) as nodes. Each entity is associated with few properties, for example, properties of $P$ are publication year, title, abstract, etc. Similarly, properties of $A$ are name, publication trend, affiliation etc. We utilize \textit{metapaths}~\cite{sun2012mining} between entities to express semantic relations. For example, simple metapaths like $A$$\rightarrow$$P$ and $V$$\rightarrow$$P$ represent ``author of'' and ``published at'' relations respectively, whereas complex metapaths like $V$$\rightarrow$$A$$\rightarrow$$P$ and $F$$\rightarrow$$A$$\rightarrow$$P$ represent ``authors of papers published at'' and ``authors of papers in'' relations respectively. We leverage metapaths to develop \textit{CL Scholar} (described in the next section).
\vspace{-0.5cm}
\begin{figure}[!tbh]
\centering
\includegraphics[trim=0 0 0 15,clip,width=0.9\hsize]{fig/framework}
\vspace{-1cm}
\caption{CL Scholar framework.}
\label{fig:framwork}
\vspace{-0.5cm}
\end{figure}
\section{CL Scholar}
\label{sec:framework}
\textit{CL scholar} fetches information from $CLKG$ as per the input query from the user. The current framework is divided into two modules -- 1) natural language based query retrieval, and 2) entity specific query retrieval. Figure~\ref{fig:framwork} shows \textit{CL Scholar} framework.
\begin{table*}[!tbh]
\centering
\caption{Representative queries from three natural language query classes. $A$ represents author, $P$ represents paper, $V$ represents venue and $F$ represents field. The list of supported queries is available online at \textit{CL Scholar} portal.} \label{tab:NL_query}
\resizebox{0.95\textwidth}{!}{\begin{minipage}{\textwidth}
\begin{tabular}{clll} \toprule
&\multicolumn{1}{c}{\textbf{Binary queries}} & \multicolumn{1}{c}{\textbf{Statistical queries}} &\multicolumn{1}{c}{ \textbf{List queries}}\\
\midrule
1.&Is $V$ accepting papers from $F$&
\parbox[t]{4.5cm}{How many $F$ papers are published in $V$ over the years}&\parbox[t]{4.5cm}{List the papers from $F$ accepted in $V$}\\
2.&Has $A$ published any paper&
\parbox[t]{4.5cm}{How many papers are published by $A$}&\parbox[t]{4.5cm}{List the papers published by $A$}\\
3.&Does $A$ publish papers on $F$&
\parbox[t]{4.5cm}{How many papers are published by $A$ in $F$}&\parbox[t]{4.5cm}{List the papers published by $A$ on $F$}\\
4.&\parbox[t]{4.5cm}{Are there any papers published by $A_1$ and $A_2$ together}&
\parbox[t]{4.5cm}{How many papers are published by $A_1$ and $A_2$ together}&\parbox[t]{4.5cm}{List the papers published by $A1$ and $A2$ together}\\
5.& \parbox[t]{4.5cm}{Are there any papers of $A$ with positive sentiment}&
\parbox[t]{4.5cm}{How many papers are there of $A$ with positive sentiment}&\parbox[t]{4.5cm}{List of papers with positive sentiment of $A$}\\
\bottomrule
\hline
\end{tabular}
\end{minipage}}
\vspace{-0.4cm}
\end{table*}
\subsection{Natural language query retrieval}
\label{sec:nlp_query}
The first module answers natural language queries ($NLQ$). It consists of two sub-modules, 1) the query classifier, and 2) the NL query processor. \textit{Query classifier} classifies user input into one of the three basic types of $NLQ$ using regular expression patterns. \textit{NL query processor} processes query based on its type determined by query classifier. Given an input natural language query, we utilize longest subsequence match to identify entity instances. The three types of $NLQ$ are:
\begin{enumerate}[noitemsep,nolistsep]
\item \textbf{Binary queries:} These represent a set of queries for which user demands a `yes' or `no' type answer. Table~\ref{tab:NL_query} lists few interesting binary queries.
\item \textbf{Statistical queries:} These represent set of queries which the knowledge base returns with some statistics. Currently, we support three types of statistics -- 1) temporal, 2) cumulative, and 3) comparison. Temporal represents year-wise statistics, cumulative represents overall statistics and comparison represents comparative statistics between two or more instances of the same entity type. Table~\ref{tab:NL_query} lists few representative statistical queries.
\item \textbf{List queries:} These represent set of queries for which the knowledge base returns a list of papers, authors or venues. Table~\ref{tab:NL_query} also enumerates few representative list queries.
\end{enumerate}
\subsection{Entity specific query retrieval}
\textit{CL scholar} also supports entity specific retrieval. As described in section~\ref{sec:knowledge_graph}, $CLKG$ consists of four entities: paper, author, venue, and field. Currently, our system supports three\footnote{The fourth sub-module is still under development.} entity specific retrieval schemes handled by three sub-modules:
\begin{enumerate}[noitemsep,nolistsep]
\item \textbf{Paper specific:} This sub-module returns paper specific information. Currently, we retrieve and display author names and affiliations, abstract, publication year and venue, cumulative and year-wise citations, list of references, citer papers, co-cited papers present in ACL anthology and list of URLs present in the paper text. We also show average sentiment score received by the queried paper by utilizing incoming citation contexts. Table~\ref{tab:entity_specific_query} shows three representative paper specific queries.
\item \textbf{Author specific:} This sub-module handles author specific queries. Given an author name, the system shows its cumulative and year-wise publication and citation count, collaborator list with an average number of collaborations, current and temporal H-index and temporal topic distribution. We also list author's publications in ACL anthology. Table~\ref{tab:entity_specific_query} lists three author specific queries with first name, last name and full name respectively.
\item \textbf{Venue specific:} We also answer venue specific queries. For each venue specific query, the system shows cumulative and year-wise publication and citation count, 2-year impact factor, recently held year and list of collaborating venues. Table~\ref{tab:entity_specific_query} shows three representative venue specific queries.
\end{enumerate}
\vspace{-0.3cm}
\begin{table}[!tbh]
\centering
\caption{Representative entity specific queries.} \label{tab:entity_specific_query}
\begin{tabular}{ccc} \toprule
\parbox[t]{1.5cm}{\centering \textbf{Paper specific}} & \parbox[t]{1.5cm}{\centering \textbf{Author specific}}&\parbox[t]{1.5cm}{\centering \textbf{Venue specific}}\\
\midrule
OCR&Chris&NAACL\\
Deep learning&Singh&SIGDAT\\
Word embeddings&Aravind Joshi&ACL
\\
\bottomrule
\hline
\end{tabular}
\vspace{-0.3cm}
\end{table}
\subsection{Additional insights}
\label{sec:insights}
We provide two additional insights by analyzing incoming citation contexts. First, we present a summary generated from incoming the citation contexts~\cite{Qazvinian:2008:SPS:1599081.1599168}. Currently, we show five summary sentences against each paper. Second, we also compute sentiment score of each citation context by leveraging a standard sentiment analyzer~\cite{Athar:2012:CCS:2382029.2382125}. We aggregate by averaging over the sentiment score of all the incoming citation contexts.
\subsection{Ranking}
Currently, we employ popularity based ranking of retrieved results. We utilize current citation count as a measure of popularity. In future, we plan to deploy other ranking schemes like recency, impact, sentiment, relevance, etc.
\subsection{Deployment}
\textit{CL Scholar} is developed using ReactJS framework. The system also supports REST API requests which are powered by a NodeJS server with data being served using MongoDB. It is currently accessible at our research group page\footnote{http://cnerg.iitkgp.ac.in/aclakg}. More information about API usage is available at API support page\footnote{http://cnerg.iitkgp.ac.in/aclakg/api}. In addition, the entire knowledge graph can also be easily downloaded in a plain text format. Figure~\ref{fig:portal} shows a snapshot of the \textit{CL Scholar} landing page.
\begin{figure}[!tbh]
\centering
\includegraphics[width=1\hsize]{fig/portal}
\vspace{-0.7cm}
\caption{Snapshot of CL Scholar landing page.}
\label{fig:portal}
\vspace{-0.3cm}
\end{figure}
The current system is still under development. Currently, we assume that spellings are correct for NLQ. We do not support instant query search. We also do not support query recommendations.
\section{Conclusion}
\label{sec:end}
In this paper, we propose a fully automatic approach for the development of computational linguistic knowledge graph from full-text PDF articles available in ACL Anthology. We also develop first-of-its-kind academic natural language query retrieval system. Currently, our system can answer three different types of natural language queries. In future, we plan to extend the query set. We also plan to append structural information within knowledge graphs such as section labeling of citations, figure and table captions etc. We also plan to conduct extensive evaluation to compare \textit{CL Scholar} with state-of-the-art systems.
\newpage
\bibliographystyle{acl_natbib}
|
{
"timestamp": "2018-04-17T02:13:51",
"yymm": "1804",
"arxiv_id": "1804.05514",
"language": "en",
"url": "https://arxiv.org/abs/1804.05514"
}
|
\section{Introduction}
\subsection{Non-malleable Codes}
Non-malleable codes were introduced by Dziembowski, Pietrzak, and Wichs \cite{DPW10} as an elegant relaxation and generalization of standard error correcting codes, where the motivation is to handle much larger classes of tampering functions on the codeword. Traditionally, error correcting codes only provide meaningful guarantees (e.g., unique decoding or list-decoding) when \emph{part} of the codeword is modified (i.e., the modified codeword is close in Hamming distance to an actual codeword), whereas in practice an adversary can possibly use much more complicated functions to modify the entire codeword. In the latter case, it is easy to see that error correction or even error detection becomes generally impossible, for example an adversary can simply change all codewords into a fixed string. On the other hand, non-malleable codes can still provide useful guarantees here, and thus partially bridge this gap. Informally, a non-malleable code guarantees that after tampering, the decoding either correctly gives the original message or gives a message that is completely unrelated and independent of the original message. This captures the notion of non-malleability: that an adversary cannot modify the codeword in a way such that the tampered codeword decodes back to a related but different message.
The original intended application of non-malleable codes is in tamper-resilient cryptography \cite{DPW10}, where they can be used generally to prevent an adversary from learning secret information by observing the input/output behavior of modified ciphertexts. Subsequently, non-malleable codes have found applications in non-malleable commitments \cite{GPR16}, public-key encryptions \cite{CMTV15}, non-malleable secret sharing schemes \cite{GK18a,GK18b}, and privacy amplification protocols \cite{CKOS18}. Furthermore, interesting connections were found to non-malleable extractors \cite{CG14b}, and very recently to spectral expanders \cite{RS18}. Along the way, the constructions of non-malleable codes used various components and sophisticated ideas from additive combinatorics \cite{ADL13,CZ14} and randomness extraction \cite{CGL15}, and some of these techniques have also found applications in constructing extractors for independent sources \cite{Li16}. Until today, non-malleable codes have become fundamental objects at the intersection of coding theory and cryptography. They are well deserved to be studied in more depth in their own right, as well as to find more connections to other well studied objects in theoretical computer science.
We first introduce some notation before formally defining non-malleable codes.
\begin{define}
For any function $f:S \rightarrow S$, $f$ has a fixed point at $s \in S$ if $f(s)=s$. We say $f$ has no fixed points in $T \subseteq S$, if $f(t) \neq t$ for all $t \in T$. $f$ has no fixed points if $f(s) \neq s$ for all $s \in S$.
\end{define}
\begin{define}[Tampering functions]For any $n>0$, let $\mathcal{F}_n$ denote the set of all functions $f: \{ 0,1\}^n \rightarrow \{0,1\}^n$. Any subset of $\mathcal{F}_n$ is a family of tampering functions.
\end{define}
We use the statistical distance to measure the distance between distributions.
\begin{define}
The statistical distance between two distributions $\mathbf{\mathcal{D}}_1$ and $\mathbf{\mathcal{D}}_2$ over some universal set $\Omega$ is defined as $|\mathbf{\mathcal{D}}_1-\mathbf{\mathcal{D}}_2| = \frac{1}{2}\sum_{d \in \Omega}|\mathbf{Pr}[\mathbf{\mathcal{D}}_1=d]- \mathbf{Pr}[\mathbf{\mathcal{D}}_2=d]|$. We say $\mathbf{\mathcal{D}}_1$ is $\epsilon$-close to $\mathbf{\mathcal{D}}_2$ if $|\mathbf{\mathcal{D}}_1-\mathbf{\mathcal{D}}_2| \le \epsilon$ and denote it by $\mathbf{\mathcal{D}}_1 \approx_{\epsilon} \mathbf{\mathcal{D}}_2$.
\end{define}
To handle fixed points, we need to define the following function.
\[
\textnormal{copy}(x,y) =
\begin{cases}
x & \text{if } x \neq \textnormal{$same^{\star}$} \\
y & \text{if } x = \textnormal{$same^{\star}$}
\end{cases}
\]
Following the treatment in \cite{DPW10}, we first define coding schemes.
\begin{define}[Coding schemes] Let $\textnormal{Enc}:\{0,1\}^k \rightarrow \{0,1\}^n$ and $\textnormal{Dec}:\{0,1\}^n \rightarrow \{0,1\}^k \cup \{ \perp \}$ be functions such that $\textnormal{Enc}$ is a randomized function (i.e.,\ it has access to private randomness) and $\textnormal{Dec}$ is a deterministic function. We say that $(\textnormal{Enc},\textnormal{Dec})$ is a coding scheme with block length~$n$ and message length $k$ if for all $s \in \{0,1\}^k $, $\Pr[\textnormal{Dec}(Enc(s))=s]=1$, where the probability is taken over the randomness in $\textnormal{Enc}$.
\end{define}
We can now define non-malleable codes.\
\begin{define}[Non-malleable codes]\label{nm_def} A coding scheme $\mathcal{C} = (\textnormal{Enc},\textnormal{Dec})$ with block length $n$ and message length $k$ is a non-malleable code with respect to a family of tampering functions $\mathcal{F} \subset \mathcal{F}_n$ and error~$\epsilon$ if for every $f \in \mathcal{F}$ there exists a random variable $D_f$ on $\{ 0,1\}^k \cup \{ \textnormal{$same^{\star}$} \}$ which is independent of the randomness in $\textnormal{Enc}$ such that for all messages $s \in \{0,1\}^k$, it holds that $$ |\textnormal{Dec}(f(\textnormal{Enc}(s))) - \textnormal{copy}(D_f,s)| \le \epsilon. $$
We say the code is explicit if both the encoding and decoding can be done in polynomial time.\ The rate of $\mathcal{C}$ is given by $k/n$.
\end{define}
\paragraph{Relevant prior work on non-malleable codes.}
There has been a lot of exciting research on non-malleable codes, and we do not even attempt to provide a comprehensive survey of them. Instead we focus on relevant explicit constructions in the information theoretic setting, which is also the focus of this paper. One of the most studied classes of tampering functions is the so called \emph{split-state} tampering, where the codeword is divided into (at least two) disjoint intervals and the adversary can tamper with each interval arbitrarily but independently. This model arises naturally in situations where the codeword may be stored in different parts of memory or different devices. Following a very successful line of work \cite{DKO13,ADL13,CG14b,CZ14,ADKO14,CGL15,Li16,Li18}, we now have explicit constructions of non-malleable codes in the $2$-split state model with constant rate and constant error, or rate $\Omega(\log \log n/\log n)$ with exponentially small error \cite{Li18}. For larger number of states, recent work of Kanukurthi, Obbattu, and Sruthi \cite{kan4}, and that of Gupta, Maji and Wang \cite{GMW17} gave explicit constructions in the $4$-split-state model and $3$-split-state model respectively, with constant rate and negligible error.
The split state model is a ``compartmentalized" model, where the codeword is partitioned \emph{a prior} into disjoint intervals for tampering. Recently, there has been progress towards handling non-compartmentalized tampering functions. A work of Agrawal, Gupta, Maji, Pandey and Prabhakaran \cite{AGMPP15} gave explicit constructions of non-malleable codes with respect to tampering functions that permute or flip the bits of the codeword. Ball, Dachman-Soled, Kulkarni and Malkin \cite{BDKM16} gave explicit constructions of non-malleable codes against $t$-local functions for $t \le n^{1-\epsilon}$. However in all these models, each bit of the tampering function only depends on part of the codeword.\ A recent work of Chattopadhyay and Li \cite{CL17} gave the first explicit constructions of non-malleable codes where each bit of the tampering function may depend on all bits of the codeword.\ Specifically, they gave constructions for the classes of linear functions and small-depth (unbounded fain-in) circuits.\ The rate of the non-malleable code with respect to small-depth circuits was exponentially improved by a subsequent work of Ball, Dachman-Soled, Guo, Malkin, and Tan \cite{Liyang18}.
Given all these exciting results, a major goal of the research on non-malleable codes remains to give explicit constructions for broader classes of tampering functions, as one can use the probabilistic method to show the existence of non-malleable codes with rate close to $1-\delta$ for any class $\mathcal{F}$ of tampering functions with $|\mathcal{F}| \leq 2^{2^{\delta n}}$ \cite{CG14a}.
\paragraph{Our results.}
We continue the line of investigation on explicit constructions of non-malleable codes, and give explicit constructions for several new classes of non-compartmentalized tampering functions, where in some classes each bit of the tampering function can depend on all the bits of the codeword. The new classes strictly generalize several previous studied classes of tampering functions. In particular, we consider the following three classes.
\begin{enumerate}
\item \emph{Interleaved $2$-split-state tampering}, where the adversary can divide the codeword into two arbitrary disjoint intervals and tamper with each interval arbitrarily but independently. This model generalizes the split-state model and captures the situation where the codeword is partitioned into two halves in an unknown way by the adversary before applying a $2$-split-state tampering function. Constructing non-malleable codes for this class of tampering functions was left as an open problem by Cheraghchi and Guruswami \cite{CG14b}. \item \emph{Composition of tampering}, where the adversary takes two tampering functions and compose them together to get a new tampering function.\ We note that function composition is a natural strategy for an adversary to achieve more powerful tampering, and it has been studied widely in other fields (e.g., computational complexity and communication complexity).\ Thus we believe that studying non-malleable codes for the composition of known classes of tampering functions is also a natural and important direction. \item \emph{Bounded communication $2$-split-state tampering}, where the two tampering functions in a $2$-split state model are allowed to have some bounded communication.
\end{enumerate}
We now formally define these classes and some related classes below. We use the notation that for any permutation $\pi:[n] \rightarrow [n]$ and any string $x \in [r]^{n}$, $y=x_{\pi}$ denotes the length $n$ string such that $y_{\pi(i)}=x_{i}$
\begin{itemize}
\item The family of $2$-split-state functions $2\sss \subset \mathcal{F}_{2n}$: Any $f \in 2\sss$ comprises of two functions $f_1: \{0, 1\}^n \rightarrow \{0, 1\}^n$ and $f_2: \{0, 1\}^n \rightarrow \{0, 1\}^n$, and for any $x, y \in \{0, 1\}^n$, $f(x,y) = (f_1(x), f_2(x))$. This family of tampering functions has been extensively studied, with a long line of work achieving near optimal explicit constructions of non-malleable codes.
\item The family of linear functions $\lin \subset \mathcal{F}_{n}$: Any $f \in \lin$ is a linear function from $\{0, 1\}^n$ to $\{0, 1\}^n$ (viewing $\{0, 1\}^n$ as $\mathbb{F}_2^n)$.
\item The family of interleaved $2$-split-state functions $2\iss \subset \mathcal{F}_{2n}$: Any $f \in 2\iss$ comprises of two functions $f_1: \{0, 1\}^n \rightarrow \{0, 1\}^n$, $f_2: \{0, 1\}^n \rightarrow \{0, 1\}^n$, and a permutation $\pi: [2n] \rightarrow [2n]$. For any $z = (x \circ y)_{\pi} \in \{0, 1\}^{2n}$, where $x,y \in \{0, 1\}^n$, let $f(z) = (f_1(x) \circ f_2(y))_{\pi}$ (where $\circ$ denotes the string concatenation operation).
\item The family of bounded communication $2$-split-state functions $(2,t)-\css$:\ Consider the following natural extension of the $2$-split-state model.\ Let $c= (x,y)$ be a codeword in $\{0, 1\}^{2n}$, where $x$ is the first $n$ bits of $c$ and $y$ is the remaining $n$ bits of $c$. Let Alice and Bob be two tampering adversaries, where Alice has access to $x$ and Bob has access to $y$. Alice and Bob run a (deterministic) communication protocol based on $x$ and $y$ respectively, which can last for an arbitrary number of rounds but each party sends at most $t$ bits in total. Finally, based on the transcript and $x$ Alice outputs $x' \in \{0, 1\}^n$, similarly based on the transcript and $y$ Bob outputs $y' \in \{0, 1\}^n$. The tampered codeword is $c'=(x',y')$.
\item For any tampering function families $\mathcal{F}, \mathcal{G} \subset \mathcal{F}_{n}$, define the family $\mathcal{F} \circ \mathcal{G} \subset \mathcal{F}_n$ to be the set of all functions of the form $f \circ g$, where $f \in \mathcal{F}$, $g \in \mathcal{G}$ and $\circ$ denotes function composition.
\end{itemize}
We now formally state our results.\ Our main result is an explicit non-malleable code with respect to the tampering class of $\lin \circ 2\iss$, i.e, linear function composed with interleaved $2$-split-state tampering. Specifically, we have the following theorem.
\begin{THM}
There exists a constant $\delta>0$ such that for all integers $n>0$ there exists an explicit non-malleable code with respect to $\lin \circ 2\iss$ with rate $1/n^{\delta}$ and error $2^{-n^{\delta}}$.
\end{THM}
This immediately gives the following corollaries, which give explicit non-malleable codes for interleaved $2$-split-state tampering, and linear function composed with $2$-split-state tampering.
\begin{COR}
\label{cor:intro_int_nm_code}
There exists a constant $\delta>0$ such that for all integers $n>0$ there exists an explicit non-malleable code with respect to $2\iss$ with rate $1/n^{\delta}$ and error $2^{-n^{\delta}}$.
\end{COR}
\begin{COR}
\label{cor:intro_comp_nm_code}
There exists a constant $\delta>0$ such that for all integers $n>0$ there exists an explicit non-malleable code with respect to $\lin \circ 2\sss$ with rate $1/n^{\delta}$ and error $2^{-n^{\delta}}$.
\end{COR}
Next we give an explicit non-malleable code with respect to bounded communication $2$-split-state tampering.
\begin{THM}
\label{thm:intro_com_nm_code}
There exists a constant $\delta>0$ such that for all integers $n, t>0$ with $t \le \delta n$, there exists an explicit non-malleable code with respect to $(2,t)-\css$ with rate $\Omega(\log \log n/\log n)$ and error $2^{-\Omega(n \log \log n/\log n)}$.
\end{THM}
\noindent Prior to our work, no explicit non-malleable code of any rate was known for these tampering classes.
\subsection{Seedless non-malleable extractors}
Our results on non-malleable codes are based on new constructions of seedless non-malleable extractors, which we believe are of independent interest. Before defining seedless non-malleable extractors formally, we first recall some basic notation from the area of randomness extraction.
Randomness extraction is motivated by the problem of purifying imperfect (or defective) sources of randomness. The concern stems from the fact that natural random sources often have poor quality, while most applications require high quality (e.g., uniform) random bits.\ We use the standard notion of min-entropy to measure the amount of randomness in a distribution.
\begin{define} The min-entropy $H_{\infty}(\mathbf{X})$ of a probability distribution $\mathbf{X}$ is defined to be \newline $\min_{x}(-\log(\Pr[\mathbf{X}=x]))$. We say a probability distribution $\mathbf{X}$ on $\{ 0,1\}^n$ is an $(n, H_{\infty}(\mathbf{X}))$-source and the min-entropy rate is $H_{\infty}(\mathbf{X})/n$.
\end{define}
It turns out that it is impossible to extract from a single general weak random source even for min-entropy $n-1$. There are two possible ways to bypass this barrier. The first one is to relax the extractor to be a \emph{seeded extractor}, which takes an additional independent short random seed to extract from a weak random source. The second one is to construct deterministic extractors for special classes of weak random sources.
Both kinds of extractors have been studied extensively. Recently, they have also been generalized to stronger notions where the inputs to the extractor can be tampered with by an adversary. Specifically, Dodis and Wichs \cite{DW09} introduced the notion of \emph{seeded non-malleable extractor} in the context of privacy amplification against an active adversary.\ Informally, such an extractor satisfies the stronger property that the output of the extractor is independent of the output of the extractor on a tampered seed. Similarly, and more relevant to this paper, a seedless variant of non-malleable extractors was introduced by Cheraghchi and Guruswami \cite{CG14b} as a way to construct non-malleable codes. Apart from their original applications, both kinds of non-malleable extractors are of independent interest. They are also related to each other and have applications in constructions of extractors for independent sources \cite{Li16}.
We now define seedless non-malleable extractors. For simplicity, the definition here assumes that the tampering function has no fixed points. See Section $\ref{section:prelims}$ for a more formal definition.
\begin{define}[Seedless non-malleable extractors] Let $\mathcal{F} \subset \mathcal{F}_n$ be a family of tampering functions such that no function in $\mathcal{F}$ has any fixed points. A function $\textnormal{nmExt} : \{0,1\}^{n} \rightarrow \{0,1\}^{m}$ is a seedless $(n,m,\epsilon)$-non-malleable extractor with respect to $\mathcal{F}$ and a class of sources $\mathcal{X}$ if for every distribution $\mathbf{X} \in \mathcal{X}$ and every tampering function $f \in \mathcal{F}$, $$|\textnormal{nmExt}(\mathbf{X}), \textnormal{nmExt}(f(\mathbf{X})) - \mathbf{U}_m, \textnormal{nmExt}(f(\mathbf{X})) | \le \epsilon. $$ Further, we say that $\textnormal{nmExt}$ is $\epsilon'$-invertible, if there exists a polynomial time sampling algorithm $\mathcal{A}$ that takes as input $y \in \{0, 1\}^m$, and outputs a sample from a distribution that is $\epsilon'$-close to the uniform distribution on the set $\textnormal{nmExt}^{-1}(y)$.
\end{define}
In the above definition, when the class of sources $\mathcal{X}$ is the distribution $\mathbf{U}_n$, we simply say that $\textnormal{nmExt}$ is a seedless $(n,m,\epsilon)$-non-malleable extractor with respect to $\mathcal{F}$.
\iffalse
\begin{define}[$(r,n,k)$-Interleaved Sources] Let $\mathbf{X}_1,\ldots,\mathbf{X}_r$ be independent $(n,k)$-sources and let $\pi:[rn] \rightarrow [rn]$ be any permutation. Then, $Z=(\mathbf{X}_1\circ \ldots \circ \mathbf{X}_r)_{\pi}$ is an $(r,n,k)$-interleaved source.
\end{define}
If the parameters $n$ and $k$ are clear from the context, we sometimes drop them and talk about $r$-interleaved sources.
We now specialize the definition of seedless non-malleable extractors to the case of interleaved tampering function family.
\begin{define} A function $\textnormal{nmExt}: \{0, 1\}^{r n}\rightarrow \{0, 1\}^{m}$ is a $(r,n,k,\epsilon)$-interleaved non-malleable extractors if for any $(r,n,k)$-interleaved source $\mathbf{Z}$ and any $(r,n)$-interleaved tampering function $f:\{0, 1\}^{rn} \rightarrow \{0, 1\}^{rn}$, $$ \textnormal{nmExt}(\mathbf{Z}), \textnormal{nmExt}(f(\mathbf{Z})) \approx_{\epsilon} \mathbf{U}_m, \textnormal{nmExt}(f(\mathbf{Z})).$$
\end{define}
\fi
\paragraph{Relevant prior work on seedless non-malleable extractors.}
The first construction of seedless non-malleable extractors was given by Chattopadhyay and Zuckerman \cite{CZ14} with respect to the class of $10$-split-state tampering. Subsequently, a series of works starting with the work of Chattopadhyay, Goyal and Li \cite{CGL15} gave explicit seedless non-malleable extractors for $2$-split-state tampering. The only known construction with respect to a class of tampering functions different from split state tampering is the work of Chattopadhyay and Li \cite{CL17}, which gave explicit seedless non-malleable extractors with respect to the tampering class $\lin$ and small depth circuits. We note that constructing explicit seedless non-malleable extractors with respect to $2\iss$ was also posed as an open problem in \cite{CG14b}.
\paragraph{Our results.}
We give the first explicit constructions of seedless non-malleable extractors with respect to the tampering classes $\lin \circ 2\iss$ and $(2,t)-\css$.\ Note that the first construction also directly implies non-malleable extractors with respect to the classes $2\iss$ and $\lin \circ 2\sss$.\ The non-malleable extractors with respect to $\lin \circ 2\iss$ is a fundamentally new construction. The non-malleable extractor with respect to $(2,t)-\css$ is obtained by showing a reduction to seedless non-malleable extractors for $2\sss$, where excellent constructions are known (e.g., a recent construction of Li \cite{Li18}).
We now formally state our main results.
\begin{THM}
For all $n>0$ there exists an efficiently computable seedless $(n,n^{\Omega(1)},2^{-n^{\Omega(1)}})$-non-malleable extractor with respect to $\lin \circ 2\iss$, that is $2^{-n^{\Omega(1)}}$-invertible.
\end{THM}
This immediately gives the following two corollaries.
\begin{COR}
\label{cor:intro_int_ext}
For all $n>0$ there exists an efficiently computable seedless $(n,n^{\Omega(1)},2^{-n^{\Omega(1)}})$-non-malleable extractor with respect to $2\iss$, that is $2^{-n^{\Omega(1)}}$-invertible.
\end{COR}
\begin{COR}
\label{cor:intro_comp_ext}
For all $n>0$ there exists an efficiently computable seedless $(n,n^{\Omega(1)},2^{-n^{\Omega(1)}})$-non-malleable extractor with respect to $\lin \circ 2\sss$, that is $2^{-n^{\Omega(1)}}$-invertible.
\end{COR}
Next we give the non-malleable extractor with respect to $(2,t)-\css$.
\begin{THM}
\label{thm:intro_com_ext}
There exists a constant $\delta>0$ such for all integers $n,t>0$ with $t \le \delta n$, there exists an efficiently computable seedless $(n,\Omega\l(\frac{ n \cdot (\log \log n) }{\log n}\r),2^{-\Omega(n \log \log n/\log n)})$-non-malleable extractor with respect to $(2,t)-\css$, that is $2^{-\Omega(n \log \log n/\log n)}$-invertible.
\end{THM}
We derive our results on non-malleable codes using the above explicit constructions of non-malleable extractors. In particular we use the following theorem proved by Cheraghchi and Guruswami \cite{CG14b} that connects non-malleable extractors and codes.
\begin{thm}[\cite{CG14b}]\label{thm:connection} Let $\textnormal{nmExt}: \{0,1\}^{n} \rightarrow \{0,1\}^{m}$ be an efficient seedless $(n,m,\epsilon)$-non-malleable extractor with respect to a class of tampering functions $\mathcal{F}$ acting on $\{0, 1\}^n$. Further suppose $\textnormal{nmExt}$ is $\epsilon'$-invertible.
Then there exists an efficient construction of a non-malleable code with respect to the tampering family $\mathcal{F}$ with block length $=n$, relative rate $\frac{m}{n}$ and error $2^{m}\epsilon+\epsilon'$.
\end{thm}
\subsection{Extractors for interleaved sources}\label{intro_sec_ilext}
Our techniques also yield improved explicit constructions of extractors for interleaved sources, which generalize extractors for independent sources in the following way: the inputs to the extractor are samples from a few independent sources mixed (interleaved) in an unknown (but fixed) way. Raz and Yehudayoff \cite{RY08} showed that such extractors have applications in communication complexity and proving lower bounds for arithmetic circuits. In a subsequent work, Chattopadhyay and Zuckerman \cite{CZ15a} showed that such extractors can also be used to construct extractors for certain samplable sources, extending a line of work initiated by Trevisan and Vadhan \cite{TV00}. We now define interleaved sources formally.
\begin{define}[Interleaved Sources] Let $\mathbf{X}_1,\ldots,\mathbf{X}_r$ be arbitrary independent sources on $\{0, 1\}^n$ and let $\pi : [rn] \rightarrow [rn] $ be any permutation. Then $Z = (\mathbf{X}_1 \circ \ldots \circ \mathbf{X}_r)_{\pi}$ is an $r$-interleaved source.
\end{define}
\paragraph{Relevant prior work on interleaved extractors.} Raz and Yehudayoff \cite{RY08} gave explicit extractors for $2$-interleaved sources when both the sources have min-entropy at least $(1-\delta)n$ for a tiny constant $\delta>0$. Their construction is based on techniques from additive combinatorics and can output $\Omega(n)$ bits with exponentially small error. Subsequently, Chattopadhyay and Zuckerman \cite{CZ15a} constructed extractors for $2$-interleaved sources where one source has entropy $(1-\gamma)n$ for a small constant $\gamma>0$ and the other source has entropy $\Omega(\log n)$.\ They achieve output length $O(\log n)$ bits with error $n^{-\Omega(1)}$.
A much better result (in terms of the min-entropy) is known if the extractor has access to an interleaving of more sources. For a large enough constant $C$, Chattopadhyay and Li \cite{CL15} gave an explicit extractor for $C$-interleaved sources where each source has entropy $k \ge {\rm poly}(\log n)$. They achieve output length $k^{\Omega(1)}$ and error $n^{-\Omega(1)}$.
\paragraph{Our results.}
Our main result is an explicit extractor for $2$-interleaved sources where each source has min-entropy at least $2n/3$. The extractor outputs $\Omega(n)$ bits with error $2^{-n^{\Omega(1)}}$.
\begin{THM}
\label{thm:intro_il_ext}
For any constant $\delta>0$ and all integers $n>0$, there exists an efficiently computable function $\textnormal{i$\ell$Ext}: \{0, 1\}^{2n} \rightarrow \{0, 1\}^{m}$, $m = \Omega(n)$, such that for any two independent sources $\mathbf{X}$ and $\mathbf{Y}$, each on $n$ bits with min-entropy at least $(2/3 + \delta)n$, and any permutation $\pi:[2n] \rightarrow [2n]$, $$|\textnormal{i$\ell$Ext}((\mathbf{X} \circ \mathbf{Y})_{\pi}) - \mathbf{U}_m| \le 2^{-n^{\Omega(1)}}.$$
\end{THM}
\subsection{Open questions}
\paragraph{Non-malleable codes for composition of function classes} We gave efficient constructions of non-malleable codes for the tampering class $\lin \circ 2\sss $ (and more generally $\lin \circ \iss$). Many natural questions remain to be answered. For instance, one open problem is to efficiently construct non-malleable codes for the tampering class $2\sss \circ \lin$. It looks like one needs substantially new ideas to give such constructions. More generally, for what other interesting classes of functins $\mathcal{F}$ and $\mcl{G}$ can we construct non-malleable codes for the composed class $\mathcal{F} \circ \mcl{G}$? Is it possible to efficiently construct non-malleable codes for the tampering class $\mathcal{F} \circ \mcl{G}$ if we have efficient non-malleable codes for the classes $\mathcal{F}$ and $\mcl{G}$?
\paragraph{Other applications for seedless non-malleable extractors} The explicit seedless non-malleable extractors that we construct satisfy strong pseudorandom properties. A natural question is to find more applications of these non-malleable extractors in explicit constructions of other interesting objects.
\paragraph{Improved seedless extractors} We construct an extractor for $2$-interleaved sources that works for min-entropy rate $2/3$. It is easy to verify that there exists extractors for sources with min-entropy as low as $ C \log n$, and a natural question here is to come up with such explicit constructions. Given the success in constructing $2$-source extractors for low min-entropy \cite{CZ15,Li18}, we are hopeful that more progress can be made on this problem.
\subsection{Organization}
The rest of the paper is organized as follows. We use Section $\ref{sec:overview}$ to present an overview of our results and techniques. We use Section~$\ref{section:prelims}$ to introduce some background and notation. We present our seedless non-malleable extractor construction with respect to $\lin \circ 2\iss$ in Section~$\ref{section:composed_split_state}$. We use Section $\ref{section:communication_split_state}$ to present our non-malleable extractor construction with respect to $(2,t)-\css$. We present efficient sampling algorithms for our seedless non-malleable extractor constructions in Section $\ref{sec:sampling}$. We use Section $\ref{sec:ilext}$ to present an explicit construction of an extractor for interleaved sources.
\section{Overview of constructions and techniques}
\label{sec:overview}
Our results on non-malleable codes are derived from explicit constructions of invertible seedless non-malleable extractors (see Theorem $\ref{thm:connection}$). In this section, we focus on explicit constructions of seedless non-malleable extractors with respect to the relevant classes of tampering functions, and explicit extractors for interleaved sources.
\paragraph{Seedless non-malleable extractors with respect to $\lin \circ 2\iss$.}
We construct a seedless non-malleable extractor $\textnormal{nmExt}:\{0, 1\}^{n} \times \{0, 1\}^n \rightarrow \{0, 1\}^{m}$, $m=n^{\Omega(1)}$ such that the following hold: Let $\mathbf{X}$ and $\mathbf{Y}$ be two independent uniform sources, each on $n$ bits. Let $h:\{0, 1\}^{2n} \rightarrow \{0, 1\}^{2n}$ be an arbitrary linear function, $f:\{0, 1\}^n \rightarrow \{0, 1\}^{n}$, $g:\{0, 1\}^{n} \rightarrow \{0, 1\}^n$ be two arbitrary functions, and $\pi:[2n] \rightarrow [2n]$ be an arbitrary permutation. Then, $$ \textnormal{nmExt}((\mathbf{X},\mathbf{Y})_{\pi}), \textnormal{nmExt}(h((f(\mathbf{X}) \circ g(\mathbf{Y}))_{\pi})) \approx_{\epsilon} \mathbf{U}_m, \textnormal{nmExt}(h((f(\mathbf{X}) \circ g(\mathbf{Y}))_{\pi})),$$ where $\epsilon=2^{-n^{\Omega(1)}}$. Notice that such an extractor is not possible to construct in general, for example when all $f, g, h$ are the identify function. However, such an extractor exists when the composed function does not have fixed points. For simplicity, we ignore this issue related to fixed points in the proof sketch, and just mention that we have a reduction from the problem of constructing non-malleable codes to the problem of constructing a non-malleable extractor with no fixed points. The argument is similar to the argument in \cite{CG14b} but more complicated since here we are dealing with more powerful adversaries. We refer the reader to Section $\ref{section:composed_split_state}$ for more details.
Our first step is to reduce the problem of constructing non-malleable codes with respect to $\lin \circ 2\iss$ to constructing non-malleable extractors with the following guarantee. For strings $x, y \in \{0, 1\}^{n}$, we use $x+y$ (or equivalently $x-y$) to denote the bit-wise xor of the two strings. Let $\mathbf{X}$ and $\mathbf{Y}$ to be two independent $(n,n-n^{\delta})$-sources and $f_1,f_2,g_1,g_2 \in \mathcal{F}_n$ be four functions that satisfy the following condition:
\begin{itemize}
\item $\forall x \in support(\mathbf{X})$ and $y \in support(\mathbf{Y})$, $f_1(x) + g_1(y) \neq x$ or
\item $\forall x \in support(\mathbf{X})$ and $ y \in support(\mathbf{Y})$, $f_2(x) + g_2(y) \neq y$.
\end{itemize} Then,
\begin{align*}
|\textnormal{nmExt}((\mathbf{X},\mathbf{Y})_{\pi}), \textnormal{nmExt}((f_1(\mathbf{X}) + g_1(\mathbf{Y}),f_2(\mathbf{X}) + g_2(\mathbf{Y}))_{\pi}) - \\ \mathbf{U}_{m}, \textnormal{nmExt}((f_1(\mathbf{X}) + g_1(\mathbf{Y}),f_2(\mathbf{X}) + g_2(\mathbf{Y}))_{\pi})| &\le 2^{-n^{\Omega(1)}}.
\end{align*}
The reduction can be seen in the following way: Define $\ol{x}=(x, 0^n)_{\pi}$ and $\ol{y}=(0^n, y)_{\pi}$. Similarly define $\ol{f(x)}= h((f(x), 0^n)_{\pi})$ and $\ol{g(y)} = h((0^n, g(y))_{\pi})$.\ Thus, $(x, y)_{\pi} = \ol{x} + \ol{y}$ and $h((f(x), g(y))_{\pi}) = \ol{f(x)} + \ol{g(y)}$. Define functions $h_1:\{0, 1\}^{2n} \rightarrow \{0, 1\}^n$ and $h_2:\{0, 1\}^{2n} \rightarrow \{0, 1\}^n$ such that $h((f(x) , g(y))_{\pi})= (h_1(x,y) , h_2(x,y))_{\pi}$. Since $h((f(x), g(y))_{\pi}) = \ol{f(x)} + \ol{g(y)}$, it follows that there exists functions $f_1,g_1, f_2, g_2 \in \mathcal{F}_n$ such that for all $x,y \in \{0, 1\}^n$, the following hold:
\begin{itemize}
\item $h_1(x,y) = f_1(x) + g_1(y)$, and
\item $h_2(x,y) = f_2(x) + g_2(y)$.
\end{itemize}
Thus, $h((f(x) , g(y))_{\pi}) = ((f_1(x) + g_1(y)) , (f_2(x)+ g_2(y)))_{\pi}$.
The loss of entropy in $\mathbf{X}$ and $\mathbf{Y}$ in the reduction (from $n$ to $n-n^{\delta}$) is due to the fact that we have to handle issues related to fixed points of the tampering functions, and we ignore it for the proof sketch here.
The idea now is to use the framework of advice generators and correlation breakers with advice to construct the non-malleable extractor \cite{C15,CGL15}. We informally define these objects below as we describe our explicit constructions.
We start with the construction of the advice generator $\textnormal{advGen}:\{0, 1\}^{2n} \rightarrow \{0, 1\}^{a}$. Informally, $\textnormal{advGen}$ is a weaker object than a non-malleable extractor, and we only need that $\textnormal{advGen}((\mathbf{X} , \mathbf{Y})_{\pi}) \neq \textnormal{advGen}((f_1(\mathbf{X}) + g_1(\mathbf{Y}),f_2(\mathbf{X}) + g_2(\mathbf{Y}))_{\pi})$ (with high probability). Further, it is crucial that $a \ll n$, and in particular think of $a=n^{\gamma}$ for a small constant $\gamma>0$.
Without loss of generality, suppose that $\forall x \in support(\mathbf{X})$ and $y \in support(\mathbf{Y})$, $f_1(x) + g_1(y) \neq x$.
Let $\mathbf{Z}=(\mathbf{X} , \mathbf{Y})_{\pi}$. Let $n_0=n^{\delta'}$ for some small constant $\delta'>0$. We take two slices from $\mathbf{Z}$, say $\mathbf{Z}_1$ and $\mathbf{Z}_2$ of lengths $n_1 = n_0^{c_0}$ and $n_2= 10 n_0$, for some constant $c_0>1$. Next, we use a good linear error correcting code (let the encoder of this code be E) to encode $\mathbf{Z}$ and sample $n^{\gamma}$ coordinates (let $\mathbf{S}$ denote this set) from this encoding using $\mathbf{Z}_1$ (the sampler is based on seeded extractors \cite{Z97}). Let $\mathbf{W}_1 = E(\mathbf{Z})_{\mathbf{S}}$. Next, using $\mathbf{Z}_2$, we sample a random set of indices $\mathbf{T} \subset [2n]$, and let $\mathbf{Z}_3=\mathbf{Z}_{\mathbf{T}}$. We now use an extractor for interleaved sources, i.e., an extractor that takes as input an unknown interleaving of two independent sources and outputs uniform bits (see Section $\ref{intro_sec_ilext}$). Let $\textnormal{i$\ell$Ext}$ be this extractor (say from Theorem $\ref{thm:intro_il_ext}$), and we apply it to $\mathbf{Z}_3$ to get $\mathbf{R} = \textnormal{i$\ell$Ext}(\mathbf{Z}_3)$. Finally, let $\mathbf{W}_2$ be the output of a linear seeded extractor\footnote{A linear seeded extractor is a seeded extractor where for any fixing of the seed, the output is a linear function of the source.} $\textnormal{LExt}$ on $\mathbf{Z}$ with $\mathbf{R}$ as the seed. The output of the advice generator is $\mathbf{Z}_1 \circ \mathbf{Z}_2 \circ \mathbf{Z}_3 \circ \mathbf{W}_1 \circ \mathbf{W}_2$.
The intuition that this works is as follows. We use the notation that if $\mathbf{W} = h((\mathbf{X} , \mathbf{Y})_{\pi})$ (for some function $h$), then $\mathbf{W}'$ or $(\mathbf{W})'$ stands for the corresponding random variable after tampering, i.e., $h(((f_1(\mathbf{X}) + g_1(\mathbf{Y})) , (f_2(\mathbf{X}) + g_2(\mathbf{Y})))_{\pi})$. Further, let $\mathbf{X}_i$ be the bits of $\mathbf{X}$ in $\mathbf{Z}_i$ for $i=1,2,3$ and $\mathbf{X}_4$ be the remaining bits of $\mathbf{X}$. Similarly define $\mathbf{Y}_i$'s, $i=1,2,3,4$.
Without loss of generality suppose that $|\mathbf{X}_1| \ge |\mathbf{Y}_1|$, (where $|\alpha|$ denotes the length of the string $\alpha$).
The correctness of $\textnormal{advGen}$ is direct if $\mathbf{Z}_i \neq \mathbf{Z}_i'$ for some $i \in\{1,2,3\}$. Thus, assume $\mathbf{Z}_i = \mathbf{Z}_i'$ for $i=1,2,3$. It follows that hence $\mathbf{S}=\mathbf{S}',\mathbf{T} = \mathbf{T}'$ and $\mathbf{R} = \mathbf{R}'$. Recall that $(\mathbf{X}, \mathbf{Y})_{\pi} = \ol{\mathbf{X}} + \ol{\mathbf{Y}}$ and $h((f(\mathbf{X}), g(\mathbf{Y}))_{\pi}) = \ol{f(\mathbf{X})} + \ol{g(\mathbf{Y})}$. Since $E$ is a linear code and $\textnormal{LExt}$ is a linear seeded extractor, the following hold:
\begin{align*}
\mathbf{W}_{1} - \mathbf{W}_{1}' = (E(\ol{\mathbf{X}} + \ol{\mathbf{Y}} - \ol{f(\mathbf{X})} - \ol{g(\mathbf{Y})} ))_{\mathbf{S}}, \\
\mathbf{W}_{2} - \mathbf{W}_{2}' = \textnormal{LExt}(\ol{\mathbf{X}} + \ol{\mathbf{Y}} - \ol{f(\mathbf{X})} - \ol{g(\mathbf{Y})},\mathbf{R} ).
\end{align*}
Now the idea is the following: Either (i) we can fix $\ol{\mathbf{X}}-\ol{f(\mathbf{X})}$ and claim that $\mathbf{X}_1$ still has enough min-entropy, or (ii) we can claim that $\ol{\mathbf{X}}-\ol{f(\mathbf{X})}$ has enough min-entropy conditioned on the fixing of $(\mathbf{X}_2, \mathbf{X}_3)$. Let us first discuss why this is enough. Suppose we are in the first case. Then, we can fix $\ol{\mathbf{X}}-\ol{f(\mathbf{X})}$ and $\mathbf{Y}$ and argue that $\mathbf{Z}_1$ is a deterministic function of $\mathbf{X}$ and contains enough entropy. Note that $\ol{\mathbf{X}} + \ol{\mathbf{Y}} - \ol{f(\mathbf{X})} - \ol{g(\mathbf{Y})}$ is now fixed, and in fact it is fixed to a non-zero string (using the assumption that $f_1(x) + g_1(y) \neq x$). Thus, $E(\ol{\mathbf{X}} + \ol{\mathbf{Y}} - \ol{f(\mathbf{X})} - \ol{g(\mathbf{Y})} )$ is a string with a constant fraction of the coordinates set to $1$ (since $E$ is an encoder of a linear error correcting code with constant relative distance), and it follows that with high probability $(E(\ol{\mathbf{X}} + \ol{\mathbf{Y}} - \ol{f(\mathbf{X})} - \ol{g(\mathbf{Y})} ))_{\mathbf{S}}$ contains a non-zero entry (using the fact that $\mathbf{S}$ is sampled using $\mathbf{Z}_1$, which has enough entropy). This finishes the proof in this case since it implies $\mathbf{W}_{1} \neq \mathbf{W}_{1}' $ with high probability.
Now suppose we are in case $(ii)$. We use the fact that $\mathbf{Z}_2$ contains entropy to conclude that the sampled bits $\mathbf{Z}_3$ contain almost equal number of bits from $\mathbf{X}$ and $\mathbf{Y}$ (with high probability over $\mathbf{Z}_2$). Now we can fix $\mathbf{Z}_2$ without loosing too much entropy from $\mathbf{Z}_3$ (by making the size of $\mathbf{Z}_3$ to be significantly larger than $\mathbf{Z}_2$). Next, we observe that $\mathbf{Z}_3$ is an interleaved source, and hence $\mathbf{R}$ is close to uniform. We now fix $\mathbf{X}_3$, and argue that $\mathbf{R}$ continues to be uniform. This follows roughly from the fact that any $2$-source extractor is strong \cite{rao2007exposition}, which easily extends to extractors for $2$ interleaved sources. Thus, $\mathbf{R}$ now becomes a deterministic function of $\mathbf{Y}$ while at the same time, $\ol{\mathbf{X}}-\ol{f(\mathbf{X})}$ still has enough min-entropy. Hence, $\textnormal{LExt}(\ol{\mathbf{X}}- \ol{f(\mathbf{X})},\mathbf{R} )$ is close to uniform even conditioned on $\mathbf{R}$. We can now fix $\mathbf{R}$ and $\textnormal{LExt}(\ol{\mathbf{Y}} - \ol{g(\mathbf{Y})},\mathbf{R} )$ without affecting the distribution $\textnormal{LExt}(\ol{\mathbf{X}}- \ol{f(\mathbf{X})},\mathbf{R} )$, since $\textnormal{LExt}(\ol{\mathbf{Y}} - \ol{g(\mathbf{Y})},\mathbf{R} )$ is a deterministic function of $\mathbf{Y}$ while $\textnormal{LExt}(\ol{\mathbf{X}}- \ol{f(\mathbf{X})},\mathbf{R} )$ is a deterministic function of $\mathbf{X}$ conditioned on the previous fixing of $\mathbf{R}$. It follows that after these fixings, $\mathbf{W}_{2} - \mathbf{W}_{2}' $ is close to a uniform string and hence $\mathbf{W}_{2} - \mathbf{W}_{2}' \neq 0$ with probability $1- 2^{-n^{\Omega(1)}}$, which completes the proof.
The fact that we can only consider case $(i)$ and case $(ii)$ relies on a careful convex combination argument, which is in turn based on the pre-image size of the function $\tau:\{0, 1\}^{n} \rightarrow \{0, 1\}^{2n}$ defined as $\tau(x) = (x , 0^n)_{\pi} - h((f(x) , 0^n)_{\pi})=\ol{x}-\ol{f(x)}$. The intuition is as follows. If conditioned on the fixing of $\tau(\mathbf{X})=\ol{\mathbf{X}}-\ol{f(\mathbf{X})}$ we have that $\mathbf{X}$ still has very high min-entropy, then we can take the slice $\mathbf{Z}_1$ such that $\mathbf{X}_1$ still has enough entropy conditioned on the fixing of $\tau(\mathbf{X})$. On the other hand, if conditioned on the fixing of $\tau(\mathbf{X})$ we have that $\mathbf{X}$ does not have high min-entropy, then $\tau(\mathbf{X})$ itself must have a large support size (or relatively high entropy). Therefore we can take the slice $\mathbf{Z}_2$ and sample a short string $\mathbf{Z}_3$ such that conditioned on the fixing of $(\mathbf{X}_2, \mathbf{X}_3)$, $\tau(\mathbf{X})$ still has enough min-entropy. To make the whole argument work, we need to carefully choose the sizes of the three slices $\mathbf{Z}_1, \mathbf{Z}_2, \mathbf{Z}_3$. In particular, we need to ensure that the size of $(\mathbf{Z}_2, \mathbf{Z}_3)$ is much smaller than that of $\mathbf{Z}_1$.
We now discuss the other crucial component in the construction, the advice correlation breaker $\textnormal{ACB}:\{0, 1\}^{2n} \times \{0, 1\}^{a} \rightarrow \{0, 1\}^{m}$. Informally, $\textnormal{ACB}$ takes $2$ inputs, a source $\mathbf{Z}$ (that contains some min-entropy) and an advice string $s \in \{0, 1\}^a$, and outputs a distribution on $\{0, 1\}^m$ with the following guarantee. If $\mathbf{Z}'$ is the distribution of $\mathbf{Z}$ after tampering, and $s' \in \{0, 1\}^a$ is another advice such that $s\neq s'$, then $\textnormal{ACB}(\mathbf{Z},s), \textnormal{ACB}(\mathbf{Z}',s') \approx \mathbf{U}_m, \textnormal{ACB}(\mathbf{Z}',s')$. Typically, we also assume some structures in $\mathbf{Z}$ (e.g., it consists of two independent sources or an interleaving of two independent sources). Our main result is an advice correlation breaker that satisfies
\begin{align*}
\textnormal{ACB}(\overline{\mathbf{X}} + \ol{\mathbf{Y}},s), \textnormal{ACB}(\ol{f(\mathbf{X})} + \ol{g(\mathbf{Y})},s') \approx_{\epsilon} \mathbf{U}_m, \textnormal{ACB}(\ol{f(\mathbf{X})} + \ol{g(\mathbf{Y})},s'),
\end{align*}
for any fixed strings $s,s' \in \{0, 1\}^a$ where $s \neq s'$. We note that correlation breakers have found important applications in explicit constructions of seedless extractors \cite{C15,li2016improved,Li18}, thus we believe the above correlation breaker can be of independent interest and potentially find other applications.\ By composing the advice generator $\textnormal{advGen}$ and the correlation breaker $\textnormal{ACB}$ in the natural way, we get the non-malleable extractor.\ Here we only briefly mention that the above advice correlation breaker crucially exploits the ``sum-structure" of the source and the tampering function, the fact that extractors are samplers \cite{Z97}, and previous constructions of correlation breakers using linear seeded extractors \cite{CL17}. We refer the reader to Section $\ref{sec:new_ext_composed}$ for more details.
Finally, it is far from obvious how to efficiently invert the extractor, or in other words, sample from the pre-image of this non-malleable extractor.\ This is important since the encoder of the corresponding non-malleable code is doing exactly the sampling and thus we need it to be efficient. We use Section \ref{sec:sampling} to suitably modify our extractor to support efficient sampling. Here we briefly sketch some high level ideas involved. Recall $\mathbf{Z}=(\mathbf{X} \circ \mathbf{Y})_{\pi}$. The first modification is that in all applications of seeded extractors in our construction, we specifically use linear seeded extractors. This allows us to argue that the pre-image we are trying to sample from is in fact a convex combination of distributions supported on subspaces. The next crucial observation is the fact that we can use smaller disjoint slices of $\mathbf{Z}$ to carry out various steps outlined in the construction. This is to ensure that the dimensions of the subspaces that we need to sample from, do not depend on the values of random variables that we fix. For the steps where we use the entire source $\mathbf{Z}$ (in the construction of the advice correlation breaker), we replace $\mathbf{Z}$ by a large enough slice of $\mathbf{Z}$. However this is problematic if we choose the slice deterministically, since in an arbitrary interleaving of two sources, a slice of length less than $n$ might have bits only from one source. We get around this by pseudorandomly sampling enough coordinates from $\mathbf{Z}$ (by first taking a small slice of $\mathbf{Z}$ and using a sampler that works for weak sources \cite{Z97}).
We now use an elegant trick introduced by Li \cite{Li16} where the output of the non-malleable extractor described above (with the modifications that we have specified) is now used as a seed to a linear seeded extractor applied on an even larger pseudorandom slice of $\mathbf{Z}$. The linear seeded extractor that we use has the property that for any fixing of the seed, the rank of the linear map corresponding to the extractor is the same, and furthermore one can efficiently sample from the pre-image of any output of the extractor.\ The final modification needed is a careful choice of the error correcting code used in the advice generator.\ For this we use a dual BCH code, which allows us to argue that we can discard some output bits of the advice generator without affecting its correctness (based on the dual distance of the code). This is crucial in order to argue that the rank of the linear restriction imposed on the free variables of $\mathbf{Z}$ does not depend on the values of the bits fixed so far. We refer the reader to Section $\ref{sec:sampling}$ for more details.
\paragraph{Non-malleable extractors for $(2,t)-\css$.}
We show that any $2$-source non-malleable extractor that works for min-entropy $n-2\delta n$ can be used as non-malleable extractor with respect to $(2,t)-\css$ for $t \le \delta n$. The tampering function $h$ that is based on the communication protocol can be rephrased in terms of functions in the following way. Suppose the protocol lasts for $\ell$ rounds, there exist deterministic functions $f_{i}$ and $g_{i}$ for $i =1,\ldots,\ell$, and $f:\{0, 1\}^{n} \times \{0, 1\}^{2 t} \rightarrow \{0, 1\}^n$ and $g:\{0, 1\}^{n} \times \{0, 1\}^{2 t} \rightarrow \{0, 1\}^n$ such that the communication protocol between Alice and Bob corresponds to computing the following random variables: $\mathbf{S}_1 = f_1(\mathbf{X}), \mathbf{R}_1 = g_1(\mathbf{Y},\mathbf{S}_1),\mathbf{S}_2 = f_2(\mathbf{X},\mathbf{S}_1,\mathbf{R}_1),\ldots, \mathbf{S}_i = f_i(\mathbf{X},\mathbf{S}_1,\ldots,\mathbf{S}_{i-1}, \mathbf{R}_{1},\ldots,\mathbf{R}_{i-1}), \mathbf{R}_i = g_i(\mathbf{Y},\mathbf{S}_1,\ldots,\mathbf{S}_{i},\mathbf{R}_{i,},\ldots,\mathbf{R}_{i-1}),\ldots,\mathbf{R}_{\ell} = g_{\ell}(\mathbf{Y},\mathbf{S}_1,\ldots,\mathbf{S}_{\ell},\mathbf{R}_{1},\ldots,\mathbf{R}_{\ell-1})$.
Finally, $\mathbf{X}' = f(\mathbf{X},\mathbf{R}_1,\ldots,\mathbf{R}_{\ell},\mathbf{S}_1,\ldots,\mathbf{S}_{\ell})$ and $\mathbf{Y}' = g(\mathbf{Y},\mathbf{R}_1,\ldots,\mathbf{R}_{\ell},\mathbf{S}_1,\ldots,\mathbf{S}_{\ell})$ correspond to the output of Alice and the output of Bob respectively. Thus, $h(\mathbf{X},\mathbf{Y}) = (\mathbf{X}',\mathbf{Y}')$.
Similar to the way we argue about alternating extraction protocols, we fix random variables in the following order: Fix $\mathbf{S}_1$, and it follows that $\mathbf{R}_1$ is now a deterministic function of $\mathbf{Y}$. We fix $\mathbf{R}_1$, and thus $\mathbf{S}_2$ is now a deterministic function of $\mathbf{X}$. Thus, continuing in this way, we can fix all the random variables $\mathbf{S}_1,\ldots,\mathbf{S}_{\ell}$ and $\mathbf{R}_1,\ldots,\mathbf{R}_{\ell}$ while maintaining that $\mathbf{X}$ and $\mathbf{Y}$ are independent. Further, invoking Lemma $\ref{lemma:entropy_loss_1}$, with probability at least $1-2^{-\Omega(n)}$, both $\mathbf{X}$ and $\mathbf{Y}$ have min-entropy at least $n-t - \delta n \ge n - 2 \delta n$ since both parties send at most $t$ bits.
Note that now, $\mathbf{X}'$ is a deterministic function of $X$ and $\mathbf{Y}'$ is a deterministic function of $Y$. Thus, any invertible $2$-source non-malleable extractor for min-entropy $n-2\delta n$ can be used. Our result follows by using such a construction from a recent work of Li \cite{Li18}.
\paragraph{Extractors for interleaved sources.}
We construct an explicit extractor $\textnormal{i$\ell$Ext}:\{0, 1\}^{2n} \rightarrow \{0, 1\}^{m}$, $m=\Omega(n)$ that satisfies the following: Let $\mathbf{X}$ and $\mathbf{Y}$ be independent $(n,k)$-sources with $k\ge (2/3 + \delta)n$, for any constant $\delta>0$. Let $\pi:[2n] \rightarrow [2n]$ be any permutation. Then, $$ |\textnormal{i$\ell$Ext}((\mathbf{X} \circ \mathbf{Y})_{\pi}) - \mathbf{U}_m| \le \epsilon.$$
We present our construction and also explain the proof along the way, as this gives more intuition to the different steps of the construction. Let $\mathbf{Z} = (\mathbf{X} \circ \mathbf{Y})_{\pi}$. We start by taking a large enough slice $\mathbf{Z}_1$ from $\mathbf{Z}$ (say, of length $(2/3 + \delta/2)n$). Let $\mathbf{X}$ have more bits in this slice than $\mathbf{Y}$. Let $\mathbf{X}_1$ be the bits of $\mathbf{X}$ in $\mathbf{Z}_1$ and $\mathbf{X}_2$ be the remaining bits of $\mathbf{X}$. Similarly define $\mathbf{Y}_1$ and $\mathbf{Y}_2$. Notice that $\mathbf{X}_1$ has linear entropy and also that $\mathbf{X}_2$ has linear entropy conditioned on $\mathbf{X}_1$. We fix $\mathbf{Y}_1$ and use a condenser (from works of Barak et al. \cite{BRSW12} and Zuckerman \cite{Zuck07}) to condense $\mathbf{Z}_1$ into a matrix with a constant number such that at least one of the row has entropy rate at least $0.9$. Notice that this matrix is a deterministic function of $\mathbf{X}$. The next step is to use $\mathbf{Z}$ and each row of the matrix as a seed to a linear seeded extractor get longer rows. This requires some care for the choice of the linear seeded extractor since the seed has some deficiency in entropy. After this step, we use the advice correlation breaker from \cite{CL15} on $\mathbf{Z}$ and each row of the somewhere random source with the row number as the advice (similar to as done before in the construction of seedless non-malleable extractors for $2\iss$), and compute the bit-wise XOR of the different outputs that we produce. Let $\mathbf{V}$ denote this random variable. Finally, to output $\Omega(n)$ bits we use a linear seeded extractor on $\mathbf{Z}$ with $\mathbf{V}$ as the seed. The correctness of various steps in the proof exploit the fact that $\mathbf{Z}$ can be written as the bit-wise sum of two independent sources, and the fact that we use linear seeded extractors. We refer the reader to Section $\ref{sec:ilext}$ for more details.
\section{Background and notation} \label{section:prelims}
We use $\mathbf{U}_m$ to denote the uniform distribution on $\{0,1 \}^m$. \newline For any integer $t>0$, $[t]$ denotes the set $\{1,\ldots,t \}$.\newline For a string $y$ of length $n$, and any subset $S \subseteq [n]$, we use $y_S$ to denote the projection of $y$ to the coordinates indexed by $S$. \newline We use bold capital letters for random variables and samples as the corresponding small letter, e.g., $\mathbf{X}$ is a random variable, with $x$ being a sample of $\mathbf{X}$. \newline For strings $x, y \in \{0, 1\}^{n}$, we use $x+y$ (or equivalently $x-y$) to denote the bit-wise xor of the two strings.
\subsection{A probability lemma}
The following result on min-entropy was proved by Maurer and Wolf \cite{MW07}.
\begin{lemma}\label{lemma:entropy_loss_1} Let $\mathbf{X},\mathbf{Y}$ be random variables such that the random variable $\mathbf{Y}$ takes at $\ell$ values. Then
\begin{align*}
\mathbf{Pr}_{y \sim \mathbf{Y}}[ H_{\infty}(\mathbf{X}| \mathbf{Y} = y) \ge H_{\infty}(\mathbf{X}) - \log \ell -\log(1/\epsilon)] > 1-\epsilon.
\end{align*}
\end{lemma}
\subsection{Conditional min-entropy}
\begin{define} The average conditional min-entropy of a source $\mathbf{X}$ given a random variable $\mathbf{W}$ is defined as $$ \widetilde{H}_{\infty}(\mathbf{X}|\mathbf{W}) = -\log \l( \E_{w \sim W}\l[\max_{x} \Pr[\mathbf{X}=x | \mathbf{W}=w] \r] \r) = - \log \l(\E\l[ 2^{-H_{\infty}(\mathbf{X}|\mathbf{W}=w)} \r]\r).$$
\end{define}
We recall some results on conditional min-entropy from the work of Dodis et al.\ \cite{DORS08}.
\begin{lemma}[\cite{DORS08}]
\label{lem:avg_worst_min}
For any $\epsilon>0$, $$\mathbf{Pr}_{w \sim \mathbf{W}}\l[H_{\infty}(\mathbf{X}|\mathbf{W}=w) \ge \widetilde{H}_{\infty}(\mathbf{X}|\mathbf{W})-\log(1/\epsilon)\r] \ge 1- \epsilon.$$
\end{lemma}
\begin{lemma}[\cite{DORS08}]\label{lem:entropy_loss} If a random variable $\mathbf{Y}$ has support of size $2^\ell$, then $\widetilde{H}_{\infty}(\mathbf{X}|\mathbf{Y}) \ge H_{\infty}(\mathbf{X}) - \ell$.
\end{lemma}
\begin{define}A function $\textnormal{Ext}:\{0,1\}^{n} \times \{ 0,1\}^d \rightarrow \{ 0,1\}^m$ is a $(k,\epsilon)$-seeded extractor if for any source $\mathbf{X}$ of min-entropy $k$, $|\textnormal{Ext}(\mathbf{X},\mathbf{U}_d) - \mathbf{U}_m| \le \epsilon$. $\textnormal{Ext}$ is called a strong seeded extractor if $|(\textnormal{Ext}(\mathbf{X},\mathbf{U}_d), \mathbf{U}_d) - (\mathbf{U}_m,\mathbf{U}_d) | \le \epsilon$, where $\mathbf{U}_m$ and $\mathbf{U}_d$ are independent.
Further, if for each $s\in \mathbf{U}_d$, $\textnormal{Ext}(\cdot,s):\{ 0,1\}^n\rightarrow \{ 0,1\}^m$ is a linear function, then $\textnormal{Ext}$ is called a linear seeded extractor.
\end{define}
We require extractors that can extract uniform bits when the source only has sufficient conditional min-entropy.
\begin{define} A $(k,\epsilon)$-seeded average case seeded extractor $\textnormal{Ext}:\{ 0,1\}^n \times \{ 0,1\}^d \rightarrow \{ 0,1\}^m$ for min-entropy $k$ and error $\epsilon$ satisfies the following property: For any source $\mathbf{X}$ and any arbitrary random variable $\mathbf{Z}$ with $\tilde{H}_{\infty}(\mathbf{X}|\mathbf{Z})\ge k$, $$\textnormal{Ext}(\mathbf{X},\mathbf{U}_d),\mathbf{Z} \approx_{\epsilon} \mathbf{U}_m, \mathbf{Z}.$$
\end{define}
It was shown in \cite{DORS08} that any seeded extractor is also an average case extractor.
\begin{lemma}[\cite{DORS08}]\label{lem:cond_ext} For any $\delta>0$, if $\textnormal{Ext}$ is a $(k,\epsilon)$-seeded extractor, then it is also a $(k+\log(1/\delta),\epsilon+\delta)$-seeded average case extractor.
\end{lemma}
\subsection{Samplers and extractors}\label{sec:samp_weak}
Zuckerman \cite{Z97} showed that seeded extractors can be used as samplers given access to weak sources. This connection is best presented by a graph theoretic representation of seeded extractors. A seeded extractor $\textnormal{Ext}:\{0, 1\}^n \times \{0, 1\}^d \rightarrow \{0, 1\}^m$ can be viewed as an unbalanced bipartite graph $G_{\textnormal{Ext}}$ with $2^n$ left vertices (each of degree $2^d$) and $2^m$ right vertices. Let $\mathcal{N}(x)$ denote the set of neighbors of $x$ in $G_{\textnormal{Ext}}$.
\begin{thm}[\cite{Z97}]\label{bad_set}Let $\textnormal{Ext}:\{0,1\}^{n} \times \{ 0,1\}^{d} \rightarrow \{ 0,1\}^{m}$ be a seeded extractor for min-entropy $k$ and error $\epsilon$. Let $D=2^d$. Then for any set $R \subseteq \{0,1\}^{m}$, $$ |\{x \in \{ 0,1\}^n : | |\mathcal{N}(x) \cap R| - \mu_R D| > \epsilon D \}| < 2^k,$$ where $\mu_R = |R|/2^{m}$.
\end{thm}
\begin{thm}[\cite{Z97}]\label{thm:seed_samp}Let $\textnormal{Ext}:\{0,1\}^{n} \times \{ 0,1\}^{d} \rightarrow \{ 0,1\}^{m}$ be a seeded extractor for min-entropy $k$ and error $\epsilon$. Let $\{ 0,1\}^d=\{r_1,\ldots,r_D\}$, $D=2^d$. Define $\textnormal{Samp}(x) = \{\textnormal{Ext}(x,r_1),\ldots,\textnormal{Ext}(x,r_D)\}$. Let $\mathbf{X}$ be an $(n,2k)$-source. Then for any set $R \subseteq \{0,1\}^{m}$, $$\mathbf{Pr}_{\mathbf{x} \sim \mathbf{X}}[||\textnormal{Samp}(\mathbf{x}) \cap R | - \mu_RD| > \epsilon D] < 2^{-k},$$ where $\mu_R = |R|/2^{m}$.
\end{thm}
\subsection{Explicit extractors from prior work}
We recall an optimal construction of strong-seeded extractors.
\begin{thm}[\cite{GUV09}]\label{guv} For any constant $\alpha>0$, and all integers $n,k>0$ there exists a polynomial time computable strong-seeded extractor $\textnormal{Ext}: \{ 0,1\}^n \times \{ 0,1\}^d \rightarrow \{ 0,1\}^m$ with $d = O(\log n + \log (1/\epsilon))$ and $m = (1-\alpha)k$.
\end{thm}
The following are explicit constructions of linear seeded extractors.
\begin{thm}[\cite{Tr01,RRV02}]\label{trev_ext} For every $n,k,m \in \mathbb{N}$ and $\epsilon>0$, with $m \le k \le n$, there exists an explicit strong linear seeded extractor $\textnormal{LExt}:\{ 0,1\}^n \times \{ 0,1\}^d \rightarrow \{ 0,1\}^m$ for min-entropy $k$ and error~$\epsilon$, where $d = O\l(\log^2(n/\epsilon)/\log(k/m)\r)$.
\end{thm}
A drawback of the above construction is that the seeded length is $\omega(\log n)$ for sub-linear min-entropy. A construction of Li \cite{Li:affine} achieves $O(\log n)$ seed length for even polylogarithmic min-entropy.
\begin{thm}[\cite{Li:affine}]\label{li_ext} There exists a constant $c>1$ such that for every $n,k \in \mathbb{N}$ with $c\log^8 n \le k \le n$ and any $\epsilon \ge 1/n^2$, there exists a polynomial time computable linear seeded extractor $\textnormal{LExt}: \{ 0,1\}^n \times \{ 0,1\}^d \rightarrow \{0, 1\}^m$ for min-entropy $k$ and error~$\epsilon$, where $d= O(\log n)$ and $m \le \sqrt{k}$.
\end{thm}
A different construction achieves seed length $O(\log(n/\epsilon))$ for high entropy sources.
\begin{thm}[\cite{CGL15,Li16}]
\label{thm:low_error_inv_lin}
For all $\delta>0$ there exist $\alpha, \gamma>0$ such that for all integers $n>0$, $\epsilon \ge 2^{-\gamma n}$, there exists an efficiently computable linear strong seeded extractor $\textnormal{LExt}:\{0, 1\}^n \times \{0, 1\}^d \rightarrow \{0, 1\}^{\alpha d}$, $d = O(\log (n/\epsilon))$ for min-entropy $\delta n$. Further, for any $ y \in \{0, 1\}^{d}$, the linear map $\textnormal{LExt}(\cdot,y)$ has rank $\alpha d$.
\end{thm}
The above theorem is stated in \cite{Li16} for $\delta = 0.9$, but it is straightforward to see that the proof extends for any constant $\delta>0$.
We use a property of linear seeded extractors proved by Rao \cite{Rao09}.
\begin{lemma}[\cite{Rao09}]\label{aff_error} Let $\textnormal{Ext}:\{ 0,1\}^{n} \times \{ 0,1\}^{d} \rightarrow \{ 0,1\}^{m}$ be a linear seeded extractor for min-entropy $k$ with error $\epsilon<\frac{1}{2}$. Let $X$ be an affine $(n,k)$-source. Then $$\Pr_{u \sim U_{d}}[|\textnormal{Ext}(X,u) - U_{m}|>0] \le 2\epsilon. $$
\end{lemma}
We recall a two-source extractor construction for high entropy sources based on the inner product function.
\begin{thm}[\cite{CG88} ]
\label{thm:strong_ip} For all $m,r>0$, with $q=2^{m}, n =rm$, let $\mathbf{X},\mathbf{Y}$ be independent sources on $\bb{F}_q^r$ with min-entropy $k_1,k_2$ respectively. Let $\textnormal{IP}$ be the inner product function over the field $\bb{F}_q$. Then, we have: $$|\textnormal{IP}(\mathbf{X},\mathbf{Y}), \mathbf{X} - \mathbf{U}_m, \mathbf{X}| \le \epsilon, \hspace{0.5cm} |\textnormal{IP}(\mathbf{X},\mathbf{Y}), \mathbf{Y} - \mathbf{U}_m, \mathbf{Y}| \le \epsilon$$ where $\epsilon =2^{-(k_1+k_2-n-m)/2}$.
\end{thm}
\subsection{Advice correlation breakers}
\label{sec:acb}
We use a primitive called `correlation breaker' in our construction. Consider a situation where we have arbitrarily correlated random variables $\mathbf{Y}^1,\ldots,\mathbf{Y}^r$, where each $\mathbf{Y}^i$ is on $\ell$ bits. Further suppose $\mathbf{Y}^1$ is a `good' random variable (typically, we assume $\mathbf{Y}^1$ is uniform or has almost full min-entropy). A correlation breaker $\textnormal{CB}$ is an explicit function that takes some additional resource $\mathbf{X}$, where $\mathbf{X}$ is typically additional randomness (an $(n,k)$-source) that is independent of $\{\mathbf{Y}^1,\ldots,\mathbf{Y}^r\}$. Thus using $\mathbf{X}$, the task is to break the correlation between $\mathbf{Y}^1$ and the random variables $\mathbf{Y}^2,\ldots,\mathbf{Y}^r$, i.e., $\textnormal{CB}(\mathbf{Y}^1,\mathbf{X})$ is independent of $\{\textnormal{CB}(\mathbf{Y}^2,\mathbf{X}),\ldots,\textnormal{CB}(\mathbf{Y}^r,\mathbf{X})\}$. A weaker notion is that of an advice correlation breaker that takes in some advice for each of the $\mathbf{Y}^i$'s as an additional resource in breaking the correlations. This primitive was implicitly constructed in \cite{CGL15} and used in explicit constructions of non-malleable extractors, and has subsequently found many applications in explicit constructions of extractors for independent sources and non-malleable extractors.
We recall an explicit advice correlation breaker constructed in \cite{CL15}. This correlation breaker works even with the weaker guarantee that the `helper source' $\mathbf{X}$ is now allowed to be correlated to the sources random variables $\mathbf{Y}^1,\ldots,\mathbf{Y}^r$ in a structured way. Concretely, we assume the source to be of the form $\mathbf{X} + \mathbf{Z}$, where $\mathbf{X}$ is assumed to be an $(n,k)$-source that is uncorrelated with $\mathbf{Y}^1,\ldots,\mathbf{Y}^r, \mathbf{Z}$. We now state the result more precisely.
\begin{thm}[\cite{CL15}]\label{thm:acb} For all integers $n,n_1,n_2, k, k_1,k_2,t,d,h,{\lambda}$ and any $\epsilon>0$, such that $d=O(\log^2(n/\epsilon))$, $k_1 \ge 2d+ 8tdh + \log(1/\epsilon)$, $n_1\ge2d + 10tdh + (4h t +1)n_2^2+\log(1/\epsilon)$, and $n_2 \ge 2d +3td+\log(1/\epsilon)$, let
\begin{itemize}
\item $\mathbf{X}$ be an $(n,k_1)$-source, $\mathbf{X}'$ a r.v on n bits, $\mathbf{Y}^1$ be an $(n_1,n_1-{\lambda})$-source, $\mathbf{Z},\mathbf{Z}'$ are r.v's on $n$ bits, and $\mathbf{Y}^{2},\ldots,\mathbf{Y}^{t}$ be r.v's on $n_1$ bits each, such that $\{\mathbf{X},\mathbf{X}'\}$ is independent of $\{\mathbf{Z},\mathbf{Z}',\mathbf{Y}^{1},\ldots,\mathbf{Y}^{t}\}$,
\item $id^1,\ldots,id^{t}$ be bit-strings of length $h$ such that for each $i\in \{2,t\}$, $id^1 \neq id^{i}$.
\end{itemize}
Then there exists an efficient algorithm $\textnormal{ACB}:\{0, 1\}^{n_1} \times \{0, 1\}^{n} \times \{0, 1\}^{h} \rightarrow \{0, 1\}^{n_2}$ which satisfies the following: let
\begin{itemize}
\item $\mathbf{Y}^{1}_{h}=\textnormal{ACB}(\mathbf{Y}^1,\mathbf{X}+\mathbf{Z},id^1)$,
\item $\mathbf{Y}^{i}_{h}=\textnormal{ACB}(\mathbf{Y}^i,\mathbf{X}'+\mathbf{Z}',id^i)$, $i\in [2,t]$
\end{itemize}
Then, $$\mathbf{Y}^{1}_{h},\mathbf{Y}^{2}_{h},\ldots,\mathbf{Y}^{t}_{h}, \mathbf{X}, \mathbf{X}' \approx_{O((h+2^{{\lambda}}) \epsilon)} \mathbf{U}_{n_2}, \mathbf{Y}^{2}_{h},\ldots,\mathbf{Y}^{t}_{h}, \mathbf{X}, \mathbf{X}'.$$
\end{thm}
\section{NM extractors for linear composed with interleaved split-state adversaries}
\label{section:composed_split_state}
The main result of this section is an explicit non-malleable extractor against the tampering family $\lin \circ 2\iss \subset \mathcal{F}_{2n}$.
\begin{thm}
\label{theorem:ext_lin_composed_ss_1}
For all integers $n>0$ there exists an explicit function $\textnormal{nmExt}: \{0, 1\}^{2n} \rightarrow \{0, 1\}^{m}$, $m=n^{\Omega(1)}$, such that the following holds: For any linear function $h : \{0, 1\}^{2n} \rightarrow \{0, 1\}^{2n}$, arbitrary tampering functions $f,g \in \mathcal{F}_n$, any permutation $\pi:[2n] \rightarrow [2n]$ and independent uniform sources $\mathbf{X}$ and $\mathbf{Y}$ each on $n$ bits, there exists a distribution $\mathbf{\mathcal{D}}_{h,f,g,\pi}$ on $\{0, 1\}^m \cup \{ \textnormal{$same^{\star}$}\}$, such that
$$ |\textnormal{nmExt}((\mathbf{X} \circ \mathbf{Y})_{\pi}), \textnormal{nmExt}(h((f(\mathbf{X}) \circ g(\mathbf{Y}))_{\pi})) - \mathbf{U}_{m},\textnormal{copy}(\mathbf{\mathcal{D}}_{h,f,g,\pi},\mathbf{U}_m) | \le 2^{-n^{\Omega(1)}}.$$
\end{thm}
Our first step is to show that in order to prove Theorem $\ref{theorem:ext_lin_composed_ss_1}$ it is enough to construct a non-malleable extractor satisfying Theorem $\ref{theorem:ext_lin_composed_ss_2}$.
\begin{thm}
\label{theorem:ext_lin_composed_ss_2}
There exists a $\delta>0$ such that for all integers $n,k>0$ with $n \ge k \ge n - n^{\delta}$, there exists an explicit function $\textnormal{nmExt}: \{0, 1\}^{2n} \rightarrow \{0, 1\}^{m}$, $m=n^{\Omega(1)}$, such that the following holds: Let $\mathbf{X}$ and $\mathbf{Y}$ be independent $(n,n-n^{\delta})$-sources, $\pi:[2n] \rightarrow [2n]$ any arbitrary permutation and arbitrary tampering functions $f_1,f_2,g_1,g_2 \in \mathcal{F}_n$ that satisfy the following condition:
\begin{itemize}
\item $\forall x \in support(\mathbf{X})$ and $y \in support(\mathbf{Y})$, $f_1(x) + g_1(y) \neq x$ or
\item $\forall x \in support(\mathbf{X})$ and $ y \in support(\mathbf{Y})$, $f_2(x) + g_2(y) \neq y$.
\end{itemize} Then,
\begin{align*}
|\textnormal{nmExt}((\mathbf{X} \circ \mathbf{Y})_{\pi}), \textnormal{nmExt}(((f_1(\mathbf{X}) + g_1(\mathbf{Y})) \circ (f_2(\mathbf{X}) + g_2(\mathbf{Y})))_{\pi}) - \\ \mathbf{U}_{m}, \textnormal{nmExt}(((f_1(\mathbf{X}) + g_1(\mathbf{Y})) \circ (f_2(\mathbf{X}) + g_2(\mathbf{Y})))_{\pi})| &\le 2^{-n^{\Omega(1)}}.
\end{align*}
\end{thm}
\begin{proof}[Proof of Theorem $\ref{theorem:ext_lin_composed_ss_1}$ assuming Theorem $\ref{theorem:ext_lin_composed_ss_2}$] Define $\ol{f(x)}= h((f(x) \circ 0^n)_{\pi})$ and $\ol{g(y)} = h((0^n \circ y)_{\pi})$. Thus, $h((f(x) \circ g(y))_{\pi}) = \ol{f(x)} + \ol{g(y)}$. Define functions $h_1:\{0, 1\}^{2n} \rightarrow \{0, 1\}^n$ and $h_2:\{0, 1\}^{2n} \rightarrow \{0, 1\}^n$ such that $h((f(x) \circ g(y))_{\pi})= (h_1(x,y) \circ h_2(x,y))_{\pi}$. Since $h(f(x), g(y)) = \ol{f(x)} + \ol{g(y)}$, it follows that there exists functions $f_1,g_1, f_2, g_2 \in \mathcal{F}_n$ such that for all $x,y \in \{0, 1\}^n$, the following hold:
\begin{itemize}
\item $h_1(x,y) = f_1(x) + g_1(y)$, and
\item $h_2(x,y) = f_2(x) + g_2(y)$.
\end{itemize}
Thus, $h((f(x) \circ g(y))_{\pi}) = ((f_1(x) + g_1(y)) \circ (f_2(x)+ g_2(y)))_{\pi}$.
Now, the idea is to show that $((\mathbf{X} \circ \mathbf{Y})_{\pi}, ((f_1(\mathbf{X})+g_1(\mathbf{Y})) \circ(f_2(\mathbf{X}) + g_2(\mathbf{Y})))_{\pi})$ is $2^{-n^{\Omega(1)}}$-close to a convex combination of $((\mathbf{X} \circ \mathbf{Y})_{\pi}, (\mathbf{X} \circ \mathbf{Y})_{\pi})$ and distributions of the form $((\mathbf{X}' \circ \mathbf{Y}')_{\pi}, ((\eta_1(\mathbf{X})+\nu_1(\mathbf{Y})) \circ (\eta_2(\mathbf{X}) + \nu_2(\mathbf{Y})))_{\pi})$, where $\mathbf{X}'$ and $\mathbf{Y}'$ are independent $(n,n-n^{\delta})$-sources and $\eta_1,\eta_2,\nu_1,\nu_2$ are deterministic functions in $\mathcal{F}_n$ satisfying the conditions that:
\begin{itemize}
\item $\forall x \in support(\mathbf{X}')$ and $y \in support(\mathbf{Y}')$, $\eta_1(x) + \nu_1(y) \neq x$ or
\item $\forall x \in support(\mathbf{X}')$ and $ y \in support(\mathbf{Y}')$, $\eta_2(x) + \nu_2(y) \neq y$.
\end{itemize}
Theorem $\ref{theorem:ext_lin_composed_ss_1}$ is then direct from from Theorem $\ref{theorem:ext_lin_composed_ss_2}$.
Let $n_0 = n^{\delta}$. For any $y \in \{0, 1\}^{n}$ and any function $\eta: \{0, 1\}^{n} \rightarrow \{0, 1\}^{n}$, let $\eta^{-1}(y)$ denote the set $\{ z \in \{0, 1\}^{n}: \eta(z) = y\}$. We partition $\{0, 1\}^n$ into the following two sets: $$\Gamma_1 = \{ y \in \{0, 1\}^{n}: |g_1^{-1}(g_1(y))| \ge 2^{n-n_0}\}, \hspace{1cm}\Gamma_2 = \{0, 1\}^{n} \setminus \Gamma_1.$$
Let $\mathbf{Y}_1$ be uniform on $\Gamma_1$ and $\mathbf{Y}_2$ be uniform on $\Gamma_2$. Clearly, $\mathbf{Y}$ is a convex combination of $\mathbf{Y}_1$ and $\mathbf{Y}_2$ with weights $w_i = |\Gamma_1|/2^n$, $i =1,2$. If $w_{i} \le 2^{-n_0/2}$, we ignore the corresponding source and add an error of $2^{-n_0/2}$ to the extractor. Thus, suppose $w_i \ge 2^{-n_0/2}$ for $i=1,2$. Thus, $\mathbf{Y}_1$ and $\mathbf{Y}_2$ each have min-entropy at least $n- n_0/2$.
We claim that $g_1(\mathbf{Y}_2)$ has min-entropy at least $n_0/2$. This can be seen in the following way. For any $y \in \Gamma_2$, $|g_1^{-1}(g_1(y))| \le 2^{n-n_0}$, and hence it follows $g_1(\mathbf{Y}_2)$ has min-entropy at least $(n-n_0/2)-(n-n_0) = n_0/2$. Thus, clearly for any $x \in \{0, 1\}^n$, $x + g_1(\mathbf{Y}_2) \neq x$ with probability at least $1-2^{-n_0/2}$. We add a term of $2^{-n^{\Omega(1)}}$ to the error and assume that $\mathbf{X}+g_1(\mathbf{Y}_2) \neq \mathbf{X}$. Thus, $(\mathbf{X} \circ \mathbf{Y}_2)_{\pi},((f_1(\mathbf{X})+g_1(\mathbf{Y}_2)) \circ (f_1(\mathbf{X})+g_1(\mathbf{Y}_2)))_{\pi}$ is indeed $2^{-n^{\Omega(1)}}$close to a convex combination of distributions of the required form.
Next, we claim that for any fixing of $g_1(\mathbf{Y}_1)$, the random variable $\mathbf{Y}_1$ has min-entropy at least $n-n_0$. This is direct from the fact that for any $y \in \Gamma_2$, $|g_1^{-1}(g_1(y))| > 2^{n-n_0}$. We fix $g_1(\mathbf{Y}_1)=g$, and let $f_{1,g}(x) = f_1(x) + g$. Thus, $f_{1,g}(\mathbf{X}) = f_1(\mathbf{X}) + g_1(\mathbf{Y}_1)$. We now partition $\{0, 1\}^n$ according to the fixed points of $f_{1,g}$. Let $$\Delta_1 = \{x: f_1'(x) = x \}, \hspace{1cm}\Delta_2 = \{0, 1\}^n \setminus \Delta_1.$$
Let $\mathbf{X}_{1}$ be a flat distribution on $\Delta_1$ and $\mathbf{X}_2$ be a flat distribution on $\Delta_2$. If $|\Delta_1| < 2^{n-n_0/2}$, we ignore the distribution $\mathbf{X}_1$ and add an error of $2^{n-n_0/2}$ to the analyis of the non-malleable extractor. Further, it is direct from definition that $f_1(\mathbf{X}_2) + g \neq \mathbf{X}_2$. We now handle to case when $\Delta_1 > 2^{n-n_0/2}$. Note that in this case, $H_1(\mathbf{X}_1) \ge n- n_0/2$. The idea is now to partition $\Delta_1$ into two sets based on the pre-image size of $f_2$ similar to the way we partioned the support of $\mathbf{Y}$ based on the pre-image size of $g_1$. Define the sets $$\Delta_{11} = \{ x \in \Delta_1: |f_2^{-1}(f_2(x)) \cap \Delta_1 | \ge 2^{n-n_0}\}, \hspace{1cm}\Delta_{12} = \Delta_1 \setminus \Delta_{11}.$$
Let $\mathbf{X}_{11}$ be flat on $\Delta_{11}$ and $\mathbf{X}_{12}$ be flat on $\Delta_{12}$. Clearly, $\mathbf{X}_1$ is a convex combination of the sources $\mathbf{X}_{11}$ and $\mathbf{X}_{12}$. If $\Delta_{11}$ or $\Delta_{12}$ is smaller than $2^{n-3n_0/4}$, we ignore the corresponding distribution and add an error of $2^{-n_0/4}$ to the error analysis of the non-malleable extractor. Thus suppose $\Delta_{1i}\ge 2^{n-3n_0/4}$ for $i=1,2$. Thus, $\mathbf{X}_{11}$ and $\mathbf{X}_{12}$ both have min-entropy at least $n-3n_0/4$.
We claim that $f_2(\mathbf{X}_{12})$ has min-entropy at least $n_0/4$. This can be seen in the following way. For any $x \in \Delta_{12}$, $|f_2^{-1}(f_2(x)) \cap \Delta_1| \le 2^{n-n_0}$, and hence it follows $f_2(\mathbf{X}_{12})$ has min-entropy at least $(n-3n_0/4)-(n-n_0) = n_0/4$. Thus, clearly $f_2(\mathbf{X}_{12}) + g_2(\mathbf{Y}_1) \neq \mathbf{Y}_1$ with probability at least $1-2^{-n_0/4}$. As before, we add an error of $2^{-n^{\Omega(1)}}$ to the error, and assume that $f_2(\mathbf{X}_{12}) + g_2(\mathbf{Y}_1) \neq \mathbf{Y}_1$. Thus, $(\mathbf{X}_{12} \circ \mathbf{Y}_1)_{\pi},((f_1(\mathbf{X}_{12})+g_1(\mathbf{Y}_2)) \circ (f_1(\mathbf{X}_{12})+g_1(\mathbf{Y}_2)))_{\pi}$ is indeed $2^{-n^{\Omega(1)}}$-close to a convex combination of distributions of the required form.
Next, we claim that for any fixing of $f_2(\mathbf{X}_{11})$, the random variable $\mathbf{X}_{11}$ has min-entropy at least $n-n_0$. This is direct from the fact that for any $x \in \Delta_1$, $|f_2^{-1}(f_1(x)) \cap \Delta_1| > 2^{n-n_0}$. We fix $f_2(\mathbf{X}_{11})={\lambda}$, and let $g_{2,{\lambda}}(y) = {\lambda} + g_2(y)$. Thus, $g_{2,{\lambda}}(\mathbf{Y}) = f_1(\mathbf{X}) + g_1(\mathbf{Y}_1)$. We now partition $\Gamma_1$ according to the fixed points of $f_{1,g}$. Let $$\Gamma_{11} = \{y: g_{2,{\lambda}}(y) = y \}, \hspace{1cm}\Gamma_{12} = \{0, 1\}^n \setminus \Gamma_{11}.$$
Let $\mathbf{Y}_{11}$ be a flat distribution on $\Gamma_{11}$ and $\mathbf{Y}_{12}$ be a flat distribution on $\Gamma_{12}$. It follows from definition that $(f_1(\mathbf{X}_{11}) + g_1(\mathbf{Y}_{11}), f_2(\mathbf{X}_{11}) + g_2(\mathbf{Y}_{11})) = (\mathbf{X}_{11}, \mathbf{Y}_{11})$. Further,
$f_2(\mathbf{X}_{11}) + g_2(\mathbf{Y}_{12}) \neq \mathbf{Y}_{12}$, and hence $(\mathbf{X}_{11} \circ \mathbf{Y}_{12})_{\pi}, ((f_1(\mathbf{X}_{11})+g_1(\mathbf{Y}_{12})) \circ (f_1(\mathbf{X}_{11})+g_1(\mathbf{Y}_{12})))_{\pi}$ is $2^{-n^{\Omega(1)}}$-close to a convex combination of distributions of the required form. This completes the proof.
\end{proof}
In the rest of the section, we prove Theorem $\ref{theorem:ext_lin_composed_ss_2}$. We assume the setup given from Theorem $\ref{theorem:ext_lin_composed_ss_2}$. Thus, $\mathbf{X}$ and $\mathbf{Y}$ are independent $(n,n-n^{\delta})$-sources, $\pi:[2n] \rightarrow [2n]$ is an arbitrary permutation and $f_1,f_2,g_1,g_2 \in \mathcal{F}_n$ satisfy the following conditions:
\begin{itemize}
\item $\forall x \in support(\mathbf{X})$ and $y \in support(\mathbf{Y})$, $f_1(x) + g_1(y) \neq x$ or
\item $\forall x \in support(\mathbf{X})$ and $ y \in support(\mathbf{Y})$, $f_2(x) + g_2(y) \neq y$.
\end{itemize}
We use the following notation: if $\mathbf{W} = h((\mathbf{X} \circ \mathbf{Y})_{\pi})$ (for some function $h$), then we use to $\mathbf{W}'$ or $(\mathbf{W})'$ to denote the random variable $h(((f_1(\mathbf{X}) + g_1(\mathbf{Y})) \circ (f_2(\mathbf{X}) + g_2(\mathbf{Y})))_{\pi})$. Further, define $\overline{\mathbf{X}}= (\mathbf{X} \circ 0^n)_{\pi}$, $\overline{\mathbf{Y}} = (0^n \circ \mathbf{Y} )_{\pi}$, $\overline{f_1(\mathbf{X})}=(f_1(\mathbf{X}) \circ 0^n)_{\pi}$, $\overline{f_2(\mathbf{X})}=(0^n \circ f_2(\mathbf{X}))_{\pi}$, $\overline{g_1(\mathbf{Y})}= (g_1(\mathbf{Y}) \circ 0^n)_{\pi}$ and $\overline{g_2(\mathbf{Y})}= (0^n \circ g_2(\mathbf{Y}))_{\pi}$. It follows that $\mathbf{Z}= \overline{\mathbf{X}} + \overline{\mathbf{Y}}$ and $\mathbf{Z}' = \overline{f_1(\mathbf{X})} + \overline{g_1(\mathbf{Y})} + \overline{f_2(\mathbf{X})} + \overline{g_2(\mathbf{Y})}$.
We use Section $\ref{sec:new_adv_1}$ to construct an advice generator and Section~$\ref{sec:new_ext_composed}$ to construct an advice correlation breaker. Finally, we present the non-malleable extractor construction in Section~$\ref{sec:new_ext_composede}$.
\subsection{An advice generator}
\label{sec:new_adv_1}
\begin{lemma}\label{lem:new_adv_1}There exists an efficiently computable function $\textnormal{advGen}:\{0, 1\}^{n} \times \{0, 1\}^{n} \rightarrow \{0, 1\}^{n_4}$, $n_4= n^{\delta}$, such that with probability at least $1-2^{-n^{\Omega(1)}}$ over the fixing of the random variables $\{\textnormal{advGen}((\mathbf{X} \circ \mathbf{Y})_{\pi}), \textnormal{advGen}(((f_1(\mathbf{X}) + g_1(\mathbf{Y})) \circ (f_2(\mathbf{X})+g_2(\mathbf{Y})))_{\pi})\}$, the following hold:
\begin{itemize}
\item $\{\textnormal{advGen}((\mathbf{X} \circ \mathbf{Y})_{\pi}) \neq \textnormal{advGen}(((f_1(\mathbf{X}) + g_1(\mathbf{Y})) \circ (f_2(\mathbf{X})+g_2(\mathbf{Y})))_{\pi})\}$,
\item $\mathbf{X}$ and $\mathbf{Y}$ are independent,
\item $H_{\infty}(\mathbf{X}) \ge k -2 n^{\delta} $, $H_{\infty}(\mathbf{Y}) \ge k - 2n^{\delta}$.
\end{itemize}
\end{lemma}
We prove the above lemma in the rest of this subsection. We claim that the function $\textnormal{advGen}$ computed by Algorithm $\ref{alg:advice_new}$ satisfies the above lemma. We first set up some parameters and ingredients.
\begin{itemize}
\item Let $C$ be a large enough constant and $\delta'=\delta/C$.
\item Let $n_0 = n^{\delta'}, n_1= n_0^{c_0}, n_2= 10n_0$, for some constant $c_0$ that we set below.
\item Let $E:\{0, 1\}^{2n} \rightarrow \{0, 1\}^{n_3}$ be the encoding function of a linear error correcting code $\mathcal{C}$ with constant rate $\alpha$ and constant distance $\beta$.
\item Let $\textnormal{Ext}_1: \{0, 1\}^{n_1} \times \{0, 1\}^{d_1} \rightarrow \{0, 1\}^{\log(n_3)}$ be a $(n_1/20,\beta/10)$-seeded extractor instantiated using Theorem $\ref{guv}$. Thus $d_1= c_1 \log n_1$, for some constant $c_1$. Let $D_1=2^{d_1}=n_1^{c_1}$.
\item Let $\textnormal{Samp}_1:\{0, 1\}^{n_1} \rightarrow [n_3]^{D_1}$ be the sampler obtained from Theorem $\ref{thm:seed_samp}$ using $\textnormal{Ext}_1$.
\item Let $\textnormal{Ext}_2: \{0, 1\}^{n_2} \times \{0, 1\}^{d_2} \rightarrow \{0, 1\}^{\log(2n)}$ be a $(n_2/20,1/n_0)$-seeded extractor instantiated using Theorem $\ref{guv}$. Thus $d_2= c_2 \log n_2$, for some constant $c_2$. Let $D_2=2^d_2$. Thus $D_2= 2^{d_2}= n_2^{c_2}$.
\item Let $\textnormal{Samp}_2:\{0, 1\}^{n_2} \rightarrow [2n]^{D_2}$ be the sampler obtained from Theorem $\ref{thm:seed_samp}$ using $\textnormal{Ext}_2$.
\item Set $c_0 = 2c_2$.
\item Let $\textnormal{i$\ell$Ext}:\{0, 1\}^{D_2} \rightarrow \{0, 1\}^{n_0}$ be the extractor from Theorem $\ref{thm:il_ext}$.
\item Let $\textnormal{LExt}: \{0, 1\}^{2n} \times \{0, 1\}^{n_0} \rightarrow \{0, 1\}^{n_0}$ be a linear seeded extractor instantiated from Theorem $\ref{thm:strong_ip}$ set to extract from min-entropy $n_1/100$ and error $2^{-\Omega(\sqrt{n_0})}$ .
\end{itemize}
\RestyleAlgo{boxruled}
\LinesNumbered
\begin{algorithm}[ht]\label{alg:advice_new}
\caption{$\textnormal{advGen}(z)$ \vspace{0.1cm}\newline \textbf{Input:}
Bit-string $z= (x \circ y)_{\pi}$ of length $2n$, where $x$ and $y$ are each $n$ bit-strings and $\pi: [2n] \rightarrow [2n]$ is a permutation. \newline \textbf{Output:} Bit string $v$ of length $n_4$.
}
Let $z_1 = \textnormal{Slice}(z,n_1), z_2= \textnormal{Slice}(z,n_2)$.
Let $S= \textnormal{Samp}_1(z_1)$.
Let $T = \textnormal{Samp}_2(z_2)$ and $z_3 = z_{T}$.
Let $r = \textnormal{i$\ell$Ext}(z_3)$.
Let $w_{1} = (E(z))_S$.
Let $w_{2} = \textnormal{LExt}(z,r)$.
Output $v = z_1 \circ z_2 \circ z_3 \circ w_1 \circ w_2$.
\end{algorithm}
\begin{lemma}\label{lem:advice_new_2} With probability at least $1-2^{-n^{\Omega(1)}}$, $ \mathbf{V} \neq \mathbf{V}'$.
\end{lemma}
\begin{proof} We prove the lemma assuming $f_1(\mathbf{X}) + g_1(\mathbf{Y}) \neq \mathbf{X}$. The proof in the other case (i.e., $f_2(\mathbf{X}) + g_2(\mathbf{Y}) \neq \mathbf{Y}$) is similar and we skip it.
First observe that the lemma is direct if $\mathbf{Z}_1 \neq \mathbf{Z}_1'$ or $\mathbf{Z}_2 \neq \mathbf{Z}_2'$ or $\mathbf{Z}_3 \neq \mathbf{Z}_3'$. Thus, we can assume $\mathbf{Z}_i = \mathbf{Z}_i'$ for $i=1,2,3$. It is easy to see that $\mathbf{S} = \mathbf{S}', \mathbf{T}= \mathbf{T}'$, and $\mathbf{Z}_4 = \mathbf{Z}_4'$.
Now observe that
\begin{align*}
\mathbf{Z} - \mathbf{Z}' = \overline{\mathbf{X}} + \overline{\mathbf{Y}} - \overline{f_1(\mathbf{X})} - \overline{g_1(\mathbf{Y})} - \overline{f_2(\mathbf{X})} - \overline{g_2(\mathbf{Y})}.
\end{align*}
Note that $\mathbf{Z} - \mathbf{Z}' \neq 0$ which follows from our assumption that $f_1(\mathbf{X}) + g_1(\mathbf{Y}) \neq \mathbf{X}$.
Now define the function $h_1: \{0, 1\}^{2n} \rightarrow \{0, 1\}^{2n}$ as $h_1(z) = z - f_1(z) - f_2(z)$ and $h_2: \{0, 1\}^{2n} \rightarrow \{0, 1\}^{2n}$ as $h_2(z) = z - g_1(z) - g_2(z)$.
Thus,
\begin{align*}
\mathbf{Z} - \mathbf{Z}' = h_1(\overline{\mathbf{X}}) + h_2(\overline{\mathbf{Y}}).
\end{align*}
Let $\mathbf{X}_i$ be the bits of $\mathbf{X}$ in $\mathbf{Z}_i$ for $i=1,2,3$ and $\mathbf{X}_4$ be the remaining bit of $\mathbf{X}$. Similarly define $\mathbf{Y}_i$'s, $i=1,2,3,4$.
Without loss of generality suppose that $|\mathbf{X}_1| \ge |\mathbf{Y}_1|$, (where $|\alpha|$ denotes the length of the string $\alpha$).
Let $\Gamma \subset \{0, 1\}^{2n}$ denote the support of the source $\overline{\mathbf{X}}$. We partition $\Gamma$ into two sets $\Gamma_a$ and $\Gamma_b$ according to the pre-image size of the function $h_1$ in the following way. For any $z \in \{0, 1\}^{2n}$, let $h_1^{-1}(z)$ denote the set $\{ y \in \{0, 1\}^{2n}: h_1(y) = z\}$.
Let $n_{p} = n_1/50$.
Define $$\Gamma_a = \{ z \in \Gamma: |h_1^{-1}(h_1(z)) \cap \Gamma| \ge 2^{n-n_p} \}, \hspace{0.5cm}\Gamma_b = \Gamma \setminus \Gamma_1.$$
Let $p_a = \Pr[\overline{\mathbf{X}} \in \Gamma_a]$ and $p_b = \Pr[\overline{\mathbf{X}} \in \Gamma_b]$.
Let $\overline{\mathbf{X}}_a$ be the source supported on $\Gamma_a$ with the probability law $\Pr[\overline{\mathbf{X}}_a=z] = \frac{1}{p_a} \cdot \Pr[\overline{\mathbf{X}}=z]$. Also define $\overline{\mathbf{X}}_b$ supported on $\Gamma_b$ with the probability law $\Pr[\overline{\mathbf{X}}_a=z] = \frac{1}{p_b} \cdot \Pr[\overline{\mathbf{X}}=z]$.
Clearly $\overline{\mathbf{X}}$ is a convex combination of the distributions $\overline{\mathbf{X}}_a$ and $\overline{\mathbf{X}}_b$, with weights $p_a$ and $p_b$ respectively. If any of $p_a$ or $p_b$ is less that $2^{-n_0}$, we ignore the corresponding source and add it to the error. Thus suppose both $p_a$ and $p_b$ are at least $2^{-n_0}$. This implies that both $\overline{\mathbf{X}}_a$ and $\overline{\mathbf{X}}_b$ have min-entropy at least $n-2n_0$. We record the following two bounds that are direct from the above definitions.
\begin{itemize}
\item For any fixing of $h_1(\overline{\mathbf{X}}_{a})=x_a$, $\overline{\mathbf{X}}_a$ has min-entropy at least $n-n_p$.
\item The distribution $h_1(\mathbf{X}_b)$ has min-entropy at least $n_p-2n_0$.
\end{itemize}
We introduce some notation. For any random variable $\nu = \eta(\overline{\mathbf{X}},\overline{\mathbf{Y}})$ (where $\eta$ is an arbitrary deterministic function), we add an extra $a$ or $b$ to the subscript and use $\nu_a$ to denote the random variable $\eta(\overline{\mathbf{X}}_a,\overline{\mathbf{Y}})$ and $\nu_b$ to denote the random variables $\eta(\overline{\mathbf{X}}_b,\overline{\mathbf{Y}})$ respectively. For example, $\mathbf{Z}_{1,a}' = \overline{f_1(\mathbf{X}_a)} + \overline{g_1(\mathbf{Y})} + \overline{f_2(\mathbf{X}_a)} + \overline{g_2(\mathbf{Y})}$. Further we use $\mathbf{X}_a$ to denote the distribution on $n$ bits such that $\overline{\mathbf{X}}_a = (\mathbf{X}_a \circ 0^n)_{\pi}$. We similarly define the distribution $\mathbf{X}_b$.
We prove the following two statements:
\begin{enumerate}
\item $\mathbf{W}_{1,a} - \mathbf{W}_{1,a}' \neq 0 $ with probability $1-2^{-n^{\Omega(1)}}$.
\item $\mathbf{W}_{2,b} - \mathbf{W}_{2,b}' \neq 0 $ with probability $1-2^{-n^{\Omega(1)}}$.
\end{enumerate}
It is direct that the lemma follows from the above two inequalities.
We begin with the proof of $(1)$.
Since $E$ is a linear code, we have
\begin{align*}
\mathbf{W}_{1,a} - \mathbf{W}_{1,a}' &= (E(\mathbf{Z}_a- \mathbf{Z}_a'))_{\mathbf{S}_a}.\\
&=(E(h_1(\overline{\mathbf{X}_a}) + h_2(\overline{\mathbf{Y}})))_{\mathbf{S}_a}.
\end{align*}
Now fix the random $h_1(\mathbf{X}_a)$, and it follows that $\mathbf{X}_a$ has min-entropy at least $n-n_p$. Recall that we assumed $|\mathbf{X}_1| \ge |\mathbf{Y}_1|$. Thus, $\mathbf{X}_{1,a}$ has min-entropy at least $n_1/2 - n_p-n_0>n_1/10$ with probability at least $1-2^{-n_0}$. Further fix $\mathbf{Y}$, and note that this does not affect the distribution of $\mathbf{X}_{1,a}$. This fixes $E(\mathbf{Z}_a- \mathbf{Z}_a')$. Further $\mathbf{Z}_a \neq \mathbf{Z}_a'$, the $E(\mathbf{Z}_a- \mathbf{Z}_a')$ contains $1$'s at least $\beta$ fraction of its coordinates. Recalling that $\mathbf{S}_a = \textnormal{Samp}_1(\mathbf{Z}_{1,a})$, it now follows from Theorem $\ref{thm:seed_samp}$ that with probability at least $1-2^{-n^{\Omega(1)}}$, $(E(\mathbf{Z}_a- \mathbf{Z}_a'))_{\mathbf{S}}$ is a non-zero string (and hence $\mathbf{W}_{1,a} - \mathbf{W}_{1,a}' \neq 0 $). This completes the proof of this case.
We now proceed to prove $(2)$. Using the fact that $\textnormal{LExt}$ is a linear seeded extractor, it follows that
\begin{align*}
\mathbf{W}_{2,b} - \mathbf{W}_{2,b}' &= \textnormal{LExt}(\mathbf{Z}_b-\mathbf{Z}_b',\mathbf{R}_b) \\
& = \textnormal{LExt}(h_1(\mathbf{X}_b),\mathbf{R}_b) + \textnormal{LExt}(h_2(\mathbf{Y}_b),\mathbf{R}_b).
\end{align*}
Without loss of generality, suppose $\mathbf{X}$ has more bits in $\mathbf{Z}_2$ (the argument is identical in the other case). Since $\mathbf{X}_{2,b}$ has min-entropy at least $n-2n_0$, it follows that $\mathbf{X}_{2,b}$ has min-entropy at least $\frac{n_2}{2} - 3n_0>\frac{n_2}{10}$ with probability at least $1-2^{-n_0}$. Fix the bits of $\mathbf{Y}$ in $\mathbf{Z}_2$, and thus $\mathbf{Z}_{2,b}$ is a deterministic function of $\mathbf{X}_{2,b}$. Recall that $\mathbf{T}_b = \textnormal{Samp}_2(\mathbf{Z}_2)$. It is now straightforward to see that with probability $1-2^{-n^{\Omega(1)}}$ over the fixing of $\mathbf{X}_{2,b}$, $|\mathbf{T}_b|\cdot (1/2-o(1))\le |\mathbf{T}_b \cap \pi([n])| \le |\mathbf{T}_b| \cdot (1/2+o(1)$. Recall $|T_b| = D_2$. We fix $\mathbf{X}_{2,b}$ such that $(1/2-o(1))D_2 \le |\mathbf{T}_b \cap \pi([n])| \le (1/2+o(1))D_2$. Thus, $\mathbf{Z}_3$ contains at least $(1/2-o(1))D_2$ bits from both $\mathbf{X}_b$ and $\mathbf{Y}$. It follows that both $\mathbf{X}_{3,b}$ and $\mathbf{Y}$ both have min-entropy at least $(1/2 - o(1))D_2 - 2n_0 - n_2 = (1/2 - o(1))D_2$ (even with the conditionings so far), and hence $\mathbf{R}_b$ is $2^{-n^{\Omega(1)}}$-close to uniform. We argue this this hold even conditioned on $\mathbf{X}_{3,b}$. This follows roughly from the fact that any $2$-source extractor is strong \cite{rao2007exposition} which easily extends to interleaved extractors. We fix $\mathbf{X}_{3,b}$, and thus $\mathbf{R}_b$ is now a deterministic function of $\mathbf{Y}$.
Next, we note that $h_1(\mathbf{X}_b)$ has min-entropy at least $(n-2n_0) - (n-n_p) -n_2 - D_2-n_0=n_p - 3n_0 - D_2 - n_2 > n_p/2$ (with probability $1-2^{-n^{\Omega(1)}}$). Thus, $\textnormal{LExt}(h_1(\mathbf{X}_b),\mathbf{R}_b)$ is $2^{-n^{\Omega(1)}}$-close to uniform. We fix $\mathbf{R}_b$ and $\textnormal{LExt}(h_1(\mathbf{X}_b),\mathbf{R}_b)$ continues to be close to uniform using the fact that $\textnormal{LExt}$ is a strong-seeded extractor. Further, $\textnormal{LExt}(h_1(\mathbf{X}_b),\mathbf{R}_b)$ is now a deterministic function of $\mathbf{X}_b$ and we can fix $\textnormal{LExt}(h_2(\mathbf{Y}_b),\mathbf{R}_b)$ which is a deterministic function of $\mathbf{Y}$. It thus follows that $\mathbf{W}_{2,b} - \mathbf{W}_{2,b}' \neq 0$ with probability $1-2^{-n^{\Omega(1)}}$ using the fact that $\textnormal{LExt}(h_1(\mathbf{X}_b),\mathbf{R}_b)$ is close to uniform. This completes the proof of $(2)$. The fact that $\mathbf{V}$ and $\mathbf{V}'$ can be fixed such that $\mathbf{X}$ and $\mathbf{Y}$ remain independent with min-entropy at least $k-2n^{\delta}$ (with probability $1-2^{-n^{\Omega(1)}})$ is easy to verify from the construction. This completes the proof of Lemma $\ref{lem:advice_new_2}$.
\end{proof}
\iffalse
Consider any fixing of $g_1(\mathbf{Y}_a)=g$. By definition of $\mathbf{Y}_a$, it follows that there are at least $2^{n-n_p}$ strings in the support of $\mathbf{Y}_a$ such that $g_1$ maps each of these strings to $g$. Thus, it follows that after this conditioning, $\mathbf{Y}_a$ has min-entropy at least $n - n_p$. Let $\mathbf{Y}_{a} = \mathbf{Y}_{1,a} \circ \ol{\mathbf{Y}_{1,a}}$, i.e., $\ol{\mathbf{Y}_{1,a}}$ is the remaining bits of $\mathbf{Y}_{a}$ after slicing off $\mathbf{Y}_{1,a}$. Since the length of $\ol{\mathbf{Y}_{1,a}}$ is $n-n_1$, it follows that $\mathbf{Y}_{1,a}$ has min-entropy at least $(n-n_p) - (n-n_1) = n_1 - n_p = 35n_0 = 7 n_1/10$. Further, $\mathbf{X}_1$ has min-entropy at least $n-n_0- (n-n_1) = n_1 - n_0 = 49 n _1/50$. It follows by Theorem~$\ref{thm:strong_ip}$ that $\mathbf{R}_a= \textnormal{IP}(\mathbf{X}_1, \mathbf{Y}_{1,a})$ is $2^{-n^{\Omega(1)}}$-close to uniform even conditioned on $\mathbf{X}_1$. We fix $\mathbf{X}_1$ and $\mathbf{X} - f_1(\mathbf{X})$. It follows that $\mathbf{X} - f_1(\mathbf{X}) - g_1{\mathbf{Y}_a}$ is now a fixed non-zero string, and hence $E(\mathbf{X} - f_1(\mathbf{X}) - g_1{\mathbf{Y}_a})$ has $1$'s in at least $\beta$ fraction of its coordinates. Since $\mathbf{R}_a$ is uniform, it follows that with probability at least $1- 2^{-n^{\Omega(1)}}$, $(\mathbf{W}_{1,x,a} - \mathbf{W}_{1,x,a})_{\mathbf{T}_a}$ is not the all zero string. Thus, $\mathbf{W}_{1,x,a} - \mathbf{W}_{1,x,a}' \neq 0 $ with probability $1-2^{-n^{\Omega(1)}}$.
We now prove $(2)$. Note that by definition, for any $ y_b \in \mathbf{Y}_b $, $|g_1^{-1}(g_1(Y_b))| \le 2^{n-n_p}$. Since $\mathbf{Y}_b$ has min-entropy at least $n - 2n_0$, it follows that $g_1(\mathbf{Y}_b)$ has min-entropy at least $n-2n_0-(n-n_p)= n_p - 2n_0 = 13n_0$. Next, note that $\mathbf{Y}_{2,b}$ has min-entropy at least $(n-2n_0) - (n-n_2)= n_2 - 2 n_0= 3n_2/5$ and $\mathbf{X}_{2,b}$ has min-entropy at least $(n-n_0)-(n-n_2) = 4 n_2/5$. Fix $\mathbf{Y}_{2,b}$, and it follows by Theorem $\ref{thm:strong_ip}$ that $\mathbf{R}_{2,b}$ is $2^{-n^{\Omega(1)}}$-close to uniform and is a deterministic function of $\mathbf{X}$. Now, $g_1(\mathbf{Y}_b)$ has min-entropy at least $13n_0-n_2 - n_0= 7n_0>n_2$ with probability at least $1-2^{-n^{\Omega(1)}}$. It follows by our choice of parameters that $\textnormal{LExt}(g_1(\mathbf{Y}_b),\mathbf{R}_{2,b})$ is $2^{-n^{\Omega(1)}}$-close to uniform. We fix $\mathbf{R}_{2,b}$, and thus $\textnormal{LExt}(g_1(\mathbf{Y}_b),\mathbf{R}_{2,b})$ is now a deterministic function of $\mathbf{Y}$. Further, $\textnormal{LExt}(\mathbf{X}-f_1(\mathbf{X}),\mathbf{R}_{2,b})$ is now a deterministic function of $\mathbf{X}$, and we fix it. Note that this does not affect the distribution of $\textnormal{LExt}(g_1(\mathbf{Y}_b),\mathbf{R}_{2,b})$. It follows that $\mathbf{W}_{2,x,b} - \mathbf{W}_{2,x,b}= \textnormal{LExt}(g_1(\mathbf{Y}_b),\mathbf{R}_{2,b}) + \textnormal{LExt}(\mathbf{X}-f_1(\mathbf{X}),\mathbf{R}_{2,b})$ is close to uniform, and hence $\mathbf{W}_{2,x,b} - \mathbf{W}_{2,x,b}' \neq 0 $ with probability $1-2^{-n^{\Omega(1)}}$. This completes the proof.
The proof of Lemma $\ref{lem:new_adv_1}$ is now direct from the construction in the following way: Fix $\mathbf{X}_1,\mathbf{Y}_1, \textnormal{Slice}(f_1(\mathbf{X}),n_1),\textnormal{Slice}(f_2(\mathbf{X}),n_1), \textnormal{Slice}(g_1(\mathbf{Y}),n_1), \textnormal{Slice}(g_2(\mathbf{Y}),n_1)$. Note that this fixes $\mathbf{T},\mathbf{T}',\mathbf{R}_2,\mathbf{R}_2'$. Further fix the random variables $E(\mathbf{X})_T, E(\mathbf{Y})_T, E(f_1(\mathbf{X}))_{T'}, E(f_2(\mathbf{X}))_{T'}, E(g_1(\mathbf{Y}))_{T'}, E(f_2(\mathbf{Y}))_{T'}$. This clearly fixes $\mathbf{V},\mathbf{V}'$ and we have maintained that $\mathbf{X}$ and $\mathbf{Y}$ are still independent sources. Further, it can be verified that with probability at least $1-2^{-n^{\Omega(1)}}$, $\mathbf{X}$ and $\mathbf{Y}$ each have min-entropy at least $n- 200n_0$.
\fi
\subsection{An Advice Correlation Breaker}
\label{sec:new_ext_composed}
We recall the setup of Theorem $\ref{theorem:ext_lin_composed_ss_2}$. $\mathbf{X}$ and $\mathbf{Y}$ are independent $(n,k)$-sources, $k \ge n-n^{\delta}$, $\pi:[2n] \rightarrow [2n]$ is an arbitrary permutation and $f_1,f_2,g_1,g_2 \in \mathcal{F}_n$ satisfy the following conditions:
\begin{itemize}
\item $\forall x \in support(\mathbf{X})$ and $y \in support(\mathbf{Y})$, $f_1(x) + g_1(y) \neq x$ or
\item $\forall x \in support(\mathbf{X})$ and $ y \in support(\mathbf{Y})$, $f_2(x) + g_2(y) \neq y$.
\end{itemize}
Further, we defined the following: $\overline{\mathbf{X}}= (\mathbf{X} \circ 0^n)_{\pi}$, $\overline{\mathbf{Y}} = ( 0^n\circ \mathbf{Y})_{\pi}$, $\overline{f_1(\mathbf{X})}=(f_1(\mathbf{X}) \circ 0^n)_{\pi}$, $\overline{f_2(\mathbf{X})}=(0^n \circ f_2(\mathbf{X}))_{\pi}$, $\overline{g_1(\mathbf{Y})}= (g_1(\mathbf{Y}) \circ 0^n)_{\pi}$ and $\overline{g_2(\mathbf{Y})}= (0^n \circ g_2(\mathbf{Y}))_{\pi}$. It follows that $\mathbf{Z}= \overline{\mathbf{X}} + \overline{\mathbf{Y}}$ and $\mathbf{Z}' = \overline{f_1(\mathbf{X})} + \overline{g_1(\mathbf{Y})} + \overline{f_2(\mathbf{X})} + \overline{g_2(\mathbf{Y})}$. Thus, for some functions $f,g \in \mathcal{F}_{2n}$, $\mathbf{Z}'=f(\overline{\mathbf{X}}) + g(\overline{\mathbf{Y}})$. Let $\overline{\mathbf{X}'} = f(\overline{\mathbf{X}})$ and $\overline{\mathbf{Y}'} = g(\overline{\mathbf{Y}})$.
The following is the main result of this section. Assume that we have some random variables such that $\mathbf{X}$ and $\mathbf{Y}$ continue to be independent, and $H_{\infty}(\mathbf{X}), H_{\infty}(\mathbf{Y}) \ge k- 2n^{\delta}$.
\begin{lemma}\label{lem:new_acb} There exists an efficiently computable function $\textnormal{ACB}:\{0, 1\}^{2n} \times\{0, 1\}^{n_1} \rightarrow \{0, 1\}^{m}$, $n_1= n^{\delta}$ and $m=n^{\Omega(1)}$, such that
$$
\textnormal{ACB}(\overline{\mathbf{X}} + \ol{\mathbf{Y}},w), \textnormal{ACB}(\ol{f(\mathbf{X})} + \ol{g(\mathbf{Y})},w') \approx_{\epsilon} \mathbf{U}_m, \textnormal{ACB}(\ol{f(\mathbf{X})} + \ol{g(\mathbf{Y})},w'),
$$
for any fixed strings $w,w' \in \{0, 1\}^{n_1}$ with $w \neq w'$.
\end{lemma}
We use the rest of the section to prove the above lemma. In particular, we prove that the function $\textnormal{ACB}$ computed by Algorithm $\ref{alg:ilnm}$ satisfies the conclusion of Lemma $\ref{lem:new_acb}$.
We start by setting up some ingredients and parameters.
\begin{itemize}
\item Let $\delta>0$ be a small enough constant.
\item Let $n_2=n^{\delta_1}$, where $\delta_1 = 2 \delta$.
\item Let $\textnormal{LExt}_1: \{0, 1\}^{n_2} \times \{0, 1\}^{d} \rightarrow \{0, 1\}^{d_1}$, $d_1= \sqrt{n_2}$, be a linear-seeded extractor instantiated from Theorem $\ref{trev_ext}$ set to extract from entropy $k_1=n_2/10$ with error $\epsilon_1=1/10$. Thus $d= C_1\log n_2$, for some constant $C_1$. Let $D=2^{d}=n^{\delta_2}$, $\delta_2=2C_1 \delta$.
\item Set $\delta' = 20 C_1 \delta$.
\item Let $\textnormal{LExt}_2: \{0, 1\}^{2n} \times \{0, 1\}^{d_1} \rightarrow \{0, 1\}^{n_4}$, $n_4=n^{8\delta_3}$ be a linear-seeded extractor instantiated from Theorem $\ref{trev_ext}$ set to extract from entropy $k_2=0.9 k$ with error $\epsilon_2=2^{-\Omega(\sqrt{d_1})}=2^{-n^{\Omega(1)}}$, such that the seed length of the extractor $\textnormal{LExt}_2$ (by Theorem $\ref{trev_ext}$) is $d_1$.
\item Let $\textnormal{ACB}':\{0, 1\}^{n_{1,acb'}} \times \{0, 1\}^{n_{acb'}} \times \{0, 1\}^{h_{acb'}} \rightarrow \{0, 1\}^{n_{2,acb'}}$, be the advice correlation breaker from Theorem $\ref{thm:acb}$ set with the following parameters: $n_{acb'}=2n, n_{1,acb'}=n_4,n_{2,acb'} =m=O(n^{2\delta_2}), t_{acb'} = 2D, h_{acb'}=n_1+d, \epsilon_{acb'}= 2^{-n^{\delta}}$, $d_{acb'}=O(\log^2(n/\epsilon_{acb'})), {\lambda}_{acb'}=0$. It can be checked that by our choice of parameters, the conditions required for Theorem $\ref{thm:acb}$ indeed hold for $k_{1,acb'} \ge n^{2\delta_2}$.
\end{itemize}
\RestyleAlgo{boxruled}
\LinesNumbered
\begin{algorithm}[ht]\label{alg:ilnm}
\caption{$\textnormal{ACB}(z)$ \vspace{0.1cm}\newline \textbf{Input:}
Bit-strings $z=(x \circ y)_{\pi}$ of length $2n$ and bit string $w$ of length $n_1$, where $x$ and $y$ are each $n$ bit-strings and $\pi:[2n] \rightarrow [2n]$ is a permutation. \newline \textbf{Output:} Bit string of length $m$.
}
Let $z_1= \textnormal{Slice}(z,n_2)$.
Let $v$ be a $D \times n_3$ matrix, with its $i$'th row $v_i = \textnormal{LExt}_1(z_1,i)$.
Let $r$ be a $D \times n_4$ matrix, with its $i$'th row $r_i = \textnormal{LExt}_2(z,v_i)$.
Let $s$ be a $D \times m$ matrix, with its $i$'th row $s_i = \textnormal{ACB}'(r_i,z, w \circ i)$.
Output $\oplus_{i=1}^D s_i$.
\end{algorithm}
Let $\mathbf{X}_1$ be the bits of $\mathbf{X}$ in $\mathbf{Z}_1$ and $\mathbf{X}_2$ be the remaining bit of $\mathbf{X}$. Define $\mathbf{Y}_1$ and $\mathbf{Y}_2$ similarly. Without loss of generality suppose that $|\mathbf{X}_1| \ge |\mathbf{Y}_1|$. Let $\overline{\mathbf{X}}_1= \textnormal{Slice}(\overline{\mathbf{X}},n_2)$ and $\overline{\mathbf{Y}}_1= \textnormal{Slice}(\overline{\mathbf{Y}},n_2)$. Define $\overline{\mathbf{X}}_1' = \textnormal{Slice}(f(\overline{\mathbf{X}}),n_2)$ and $\overline{\mathbf{Y}_1}' = \textnormal{Slice}(g(\overline{\mathbf{Y}}),n_2)$. It follows that $\mathbf{Z}_1 = \overline{\mathbf{X}}_1 + \overline{\mathbf{Y}}_1$ and $\mathbf{Z}_1' = \overline{\mathbf{X}}_1' + \overline{\mathbf{Y}}_1'$.
\begin{claim}\label{cl:ilnm1} Conditioned on the random variables $\mathbf{Y}_1,\overline{\mathbf{Y}}_1'$, $\{ \textnormal{LExt}_2(\overline{\mathbf{X}},\textnormal{LExt}_1(\overline{\mathbf{X}}_1+\overline{\mathbf{Y}}_1,i))\}_{i=1}^{D}$, $\{ \textnormal{LExt}_2(\overline{\mathbf{X}}',\textnormal{LExt}_1(\overline{\mathbf{X}}'_1+ \overline{\mathbf{Y}}'_1,i))\}_{i \in [D]}$, $\mathbf{X}_1$ and $\overline{\mathbf{X}}_1'$, the following hold:
\begin{itemize}
\item the matrix $\mathbf{R}$ is $2^{-n^{\Omega(1)}}$-close to a somewhere random source,
\item $\mathbf{R}$ and $\mathbf{R}'$ are deterministic functions of $\mathbf{Y}$,
\item $H_{\infty}(\mathbf{X}) \ge n-n^{\delta'}$, $H_{\infty}(\mathbf{Y}) \ge n- n^{\delta'}$.
\end{itemize}
\end{claim}
\begin{proof}
By construction, we have that for any $j \in [D]$,
\begin{align*}
\mathbf{R}_j &= \textnormal{LExt}_2(\mathbf{Z},\textnormal{LExt}_1(\mathbf{Z}_1,j)) \\
&=\textnormal{LExt}_2(\overline{\mathbf{X}}+ \overline{\mathbf{Y}}, \textnormal{LExt}_1(\overline{\mathbf{X}_1}+\overline{\mathbf{Y}_1},j)) \\
&= \textnormal{LExt}_2(\overline{\mathbf{X}}, \textnormal{LExt}_1(\overline{\mathbf{X}}_1+\overline{\mathbf{Y}}_1,j)) + \textnormal{LExt}_2(\overline{\mathbf{Y}}, \textnormal{LExt}_1(\overline{\mathbf{X}}_1+\overline{\mathbf{Y}}_1,j))
\end{align*}
Similarly,
\begin{align*}
\mathbf{R}_j' = \textnormal{LExt}_2(\overline{\mathbf{X}}', \textnormal{LExt}_1(\overline{\mathbf{X}}_1'+\overline{\mathbf{Y}}_1',j)) + \textnormal{LExt}_2(\overline{\mathbf{Y}}', \textnormal{LExt}_1(\overline{\mathbf{X}}_1'+\overline{\mathbf{Y}}_1',j)).
\end{align*}
Fix the random variables $\mathbf{Y}_1,\overline{\mathbf{Y}}_1'$. Note that after these fixings, $\overline{\mathbf{Y}}$ has min-entropy at least $k- 2n_1 - n_2 >0.9k$. Now, since $\textnormal{LExt}_2$ is a strong seeded extractor for entropy $0.9k$, it follows that there exists a set $T \subset \{0, 1\}^{d_1}$, $|T| \ge (1-\sqrt{\epsilon_2})2^{d_1}$, such that for any $j \in [T]$, $| \textnormal{LExt}_2(\overline{\mathbf{Y}},j) - \mathbf{U}_{n_4}| \le \sqrt{\epsilon_2}$.
Now viewing $\textnormal{LExt}_1$ as a sampler (see Section $\ref{sec:samp_weak}$) using the weak source $\overline{\mathbf{X}}_{1,y_1}=\overline{\mathbf{X}}_1 + \overline{y_1}$, it follows by Theorem $\ref{thm:seed_samp}$ that
$$ \Pr[ |\{\textnormal{LExt}_1(\overline{\mathbf{X}}_{1,y_1},i):i \in \{0, 1\}^{d}\} \cap T| >(1-\sqrt{\epsilon_2} - \epsilon_1) D ] \ge 1-2^{0.2 n_2} = 1-2^{-n^{\Omega(1)}}. $$
We fix $\overline{\mathbf{X}}_1$, and it follows that with probability at least $1-2^{-n^{\Omega(1)}}$, $\{\textnormal{LExt}_1(\overline{\mathbf{X}}_{1,y_1},i):i \in \{0, 1\}^{d}\} \cap T \neq \emptyset$, and thus there exists a $j \in [D]$ such that $ \textnormal{LExt}_2(\overline{\mathbf{Y}}, \textnormal{LExt}_1(\overline{\mathbf{X}_1}+\overline{\mathbf{Y}_1},j))$ is $2^{-n^{\Omega(1)}}$-close to $\mathbf{U}_{n_2}$ and is a deterministic function of $\mathbf{Y}$.
We now fix the random variables $\overline{\mathbf{X}}_1'$, $\{\textnormal{LExt}_2(\overline{\mathbf{X}}, \textnormal{LExt}_1(\overline{\mathbf{X}_1} + \overline{\mathbf{Y}_1},i))\}_{i=1}^D$, $\{\textnormal{LExt}_2(\overline{\mathbf{X}}', \textnormal{LExt}_1(\overline{\mathbf{X}_1}' +\overline{\mathbf{Y}_1}',i))\}_{i=1}^D$, and note that $ \textnormal{LExt}_2(\overline{\mathbf{Y}}, \textnormal{LExt}_1(\overline{\mathbf{X}_1}+\overline{\mathbf{Y}_1},j))$ continues to be $2^{-n^{\Omega(1)}}$-close to $\mathbf{U}_{n_2}$. It follows that $\mathbf{R}_j$ is $2^{-n^{\Omega(1)}}$-close to $\mathbf{U}_{n_2}$. Further, for any $i \in [D]$, the random variables $\mathbf{R}_i$ and $\mathbf{R}_i'$ are deterministic functions of $\mathbf{Y}$. Finally, note that $\mathbf{X}$ and $\mathbf{Y}$ remain independent after these conditionings, and $H_{\infty}(\mathbf{X}) \ge n-3n_1-2n_2-2Dn_4 \ge n-n^{10\delta_2} $ and $H_{\infty}(\mathbf{Y}) \ge n-3n_1-n_2>n-n^{\delta_2}$.
\end{proof}
Lemma $\ref{lem:new_acb}$ is now direct from the next claim.
\begin{claim}\label{cl:ilnm2} There exists $j \in [D]$ such that
$$ \mathbf{S}_j, \{ \mathbf{S}_i \}_{i \in [D]\setminus j} \approx_{2^{-n^{\Omega(1)}}} \mathbf{U}_{m}, \{ \mathbf{S}_i \}_{i \in [D]\setminus j}.$$
\end{claim}
\begin{proof} Fix the random variables: $\mathbf{W},\mathbf{W}',\mathbf{Y}_1,\overline{\mathbf{Y}}_1'$, $\{ \textnormal{LExt}_2(\overline{\mathbf{X}},\textnormal{LExt}_1(\overline{\mathbf{X}}_1+\overline{\mathbf{Y}}_1,i))\}_{i=1}^{D}$, $\{ \textnormal{LExt}_2(\overline{\mathbf{X}}',\textnormal{LExt}_1(\overline{\mathbf{X}}'_1+ \overline{\mathbf{Y}}'_1,i))\}_{i \in [D]}$, $\mathbf{X}_1$ and $\overline{\mathbf{X}}_1'$. By Lemma $\ref{lem:new_adv_1}$, we have that with probability at least $1-2^{-n^{\Omega(1)}}$, $\mathbf{W} \neq \mathbf{W}'$. Further, by Claim $\ref{cl:ilnm1}$ we have that $\mathbf{R}$ and $\mathbf{R}'$ are deterministic functions of $\mathbf{Y}$, and with probability at least $1-2^{-n^{\Omega(1)}}$, there exists $j \in [D]$ such that $\mathbf{R}_j$ is $2^{-n^{\Omega(1)}}$-close to uniform, and $H_{\infty}(\overline{\mathbf{X}}) \ge \frac{1}{2}n_{acb} - n^{\delta'}>n^{2\delta_2}$. Recall that $\mathbf{Z}=\overline{\mathbf{X}} + \overline{\mathbf{Y}}$ and $\mathbf{Z}' =\overline{\mathbf{X}}' + \overline{\mathbf{Y}}'$. It now follows by Theorem $\ref{thm:acb}$ that
\begin{align*}
\textnormal{ACB}'(\mathbf{R}_{j}, \mathbf{Z}, \mathbf{W} \circ j), \{ \textnormal{ACB}'(\mathbf{R}_{i},\overline{\mathbf{X}}+\overline{\mathbf{Y}}, \mathbf{W} \circ i) \}_{i \in [D] \setminus j}, \{ \textnormal{ACB}'(\mathbf{R}_{i}',\overline{\mathbf{X}}'+\overline{\mathbf{Y}}', \mathbf{W}' \circ i) \}_{i \in [D]} \approx_{2^{-n^{\Omega(1)}}} \\ \mathbf{U}_{m}, \{ \textnormal{ACB}'(\mathbf{R}_{i},\overline{\mathbf{X}}+\overline{\mathbf{Y}}, \mathbf{W} \circ i) \}_{i \in [D] \setminus j}, \{ \textnormal{ACB}'(\mathbf{R}_{i}',\overline{\mathbf{X}}'+\overline{\mathbf{Y}}', \mathbf{W}' \circ i) \}_{i \in [D]}
\end{align*}
This completes the proof of the claim.
\end{proof}
\subsection{The non-malleable extractor}
\label{sec:new_ext_composede}
We are now ready to present the construction of $\textnormal{i$\ell$NM}$ that satisfies the requirements of Theorem~$\ref{theorem:ext_lin_composed_ss_2}$.
\begin{itemize}
\item Let $\delta>0$ be a small enough constant, $n_1= n^{\delta}$ and $m=n^{\Omega(1)}$.
\item Let $\textnormal{advGen}:\{0, 1\}^{2n} \rightarrow \{0, 1\}^{n_1}$, $n_1=n^{\delta}$, be the advice generator from Lemma $\ref{lem:new_adv_1}$.
\item Let $\textnormal{ACB}:\{0, 1\}^{2n} \times\{0, 1\}^{n_1} \rightarrow \{0, 1\}^{m}$ be the advice correlation breaker from Lemma $\ref{lem:new_acb}$.
\end{itemize}
\RestyleAlgo{boxruled}
\LinesNumbered
\begin{algorithm}[ht]\label{alg:ilnme}
\caption{$\textnormal{i$\ell$NM}(z)$ \vspace{0.1cm}\newline \textbf{Input:}
Bit-string $z=(x \circ y)_{\pi}$ of length $2n$, where $x$ and $y$ are each $n$ bit-strings, and $\pi:[2n] \rightarrow [2n]$ is a permutation. \newline \textbf{Output:} Bit string of length $m$.
}
Let $w = \textnormal{advGen}(z)$.
Output $\textnormal{ACB}(z,w)$
\end{algorithm}
We prove that the function $\textnormal{i$\ell$NM}$ computed by Algorithm $\ref{alg:ilnme}$ satisfies the conclusion of Theorem $\ref{theorem:ext_lin_composed_ss_2}$ as follows.
Fix the random variables $\mathbf{W}, \mathbf{W}'$. By Lemma $\ref{lem:new_adv_1}$, it follows that $ \mathbf{X}$ remains independent of $ \mathbf{Y}$, and with probability at least $1-2^{-n^{\Omega(1)}}$, $H_{\infty}(\mathbf{X}) \ge k-2n_1$ and $H_{\infty}(\mathbf{Y}) \ge k-2n_1$ (recall $k \ge n-n^{\delta}$). Theorem $\ref{theorem:ext_lin_composed_ss_2}$ is now direct using Lemma $\ref{lem:new_acb}$.
\section{Non-malleable extractors for split-state adversaries with bounded communication}
\label{section:communication_split_state}
Let $\mathcal{F}_{n,t} \subset \mathcal{F}_{2n}$ be the set of all functions that can be computed in the following way. Let $c= (x,y)$ be the input in $\{0, 1\}^{2n}$, where $x$ is the first $n$ bits of $c$ and $y$ is the remaining $n$ bits of $c$. Let Alice and Bob be two tampering adversaries, where Alice has access to $x$ and Bob has access to $y$. Alice and Bob run a (deterministic) communication protocol based on $x$ and $y$ respectively, which can last for an arbitrary number of rounds but each party sends at most $t$ bits. Finally, based on the transcript and $x$ Alice outputs $x' \in \{0, 1\}^n$, similarly based on the transcript and $y$ Bob outputs $y' \in \{0, 1\}^n$. The function outputs $c'=(x',y')$. The following is our main result.
\begin{thm}
\label{thm:comm_nm_ext} There exists a constant $\delta>0$ such that for all integers $n,t>0$ with $t \le \delta n$, there exists an efficiently computable function $\textnormal{nmExt}:\{0, 1\}^{n} \times \{0, 1\}^{n} \rightarrow \{0, 1\}^{m}$, $m = \Omega(n)$, such that the following holds: let $\mathbf{X}$ and $\mathbf{Y}$ be uniform independent sources each on $n$ bits, and let $h$ be an arbitrary tampering function in $\mathcal{F}_{n,t}$. Then, there exists a distribution $\mathbf{\mathcal{D}}_{h}$ on $\{0, 1\}^{m} \cup \{ \textnormal{$same^{\star}$}\}$ that is independent of $\mathbf{X}$ and $\mathbf{Y}$ such that $$|\textnormal{nmExt}(\mathbf{X},\mathbf{Y}), \textnormal{nmExt}(h(\mathbf{X},\mathbf{Y})) - \mathbf{U}_m, \textnormal{copy}(\mathbf{\mathcal{D}}_{h},\mathbf{U}_m)| \le 2^{-\Omega(n\log \log n/\log n)}.$$
Further, $\textnormal{nmExt}$ is $2^{-\Omega(n \log \log n/\log n)}$-invertible.
\end{thm}
\begin{proof}
We show that any $2$-source non-malleable extractor that works for min-entropy $n-2\delta n$ can be used as the required non-malleable extractor in the above theorem. The tampering function $h$ that is based on the communication protocol can be rephrased in terms of functions in the following way. Suppose the protocol lasts for $\ell$ rounds, there exist deterministic functions $f_{i}$ and $g_{i}$ for $i =1,\ldots,\ell$, and $f:\{0, 1\}^{n} \times \{0, 1\}^{2 t} \rightarrow \{0, 1\}^n$ and $g:\{0, 1\}^{n} \times \{0, 1\}^{2 t} \rightarrow \{0, 1\}^n$ such that the communication protocol between Alice and Bob corresponds to computing the following random variables: $\mathbf{S}_1 = f_1(\mathbf{X}), \mathbf{R}_1 = g_1(\mathbf{Y},\mathbf{S}_1),\mathbf{S}_2 = f_2(\mathbf{X},\mathbf{S}_1,\mathbf{R}_1),\ldots, \mathbf{S}_i = f_i(\mathbf{X},\mathbf{S}_1,\ldots,\mathbf{S}_{i-1}, \mathbf{R}_{1},\ldots,\mathbf{R}_{i-1}), \mathbf{R}_i = g_i(\mathbf{Y},\mathbf{S}_1,\ldots,\mathbf{S}_{i},\mathbf{R}_{i,},\ldots,\mathbf{R}_{i-1}),\ldots,\mathbf{R}_{\ell} = g_{\ell}(\mathbf{Y},\mathbf{S}_1,\ldots,\mathbf{S}_{\ell},\mathbf{R}_{1},\ldots,\mathbf{R}_{\ell-1})$.
Finally, $\mathbf{X}' = f(\mathbf{X},\mathbf{R}_1,\ldots,\mathbf{R}_{\ell},\mathbf{S}_1,\ldots,\mathbf{S}_{\ell})$ and $\mathbf{Y}' = g(\mathbf{Y},\mathbf{R}_1,\ldots,\mathbf{R}_{\ell},\mathbf{S}_1,\ldots,\mathbf{S}_{\ell})$ correspond to the output of Alice and the output of Bob respectively. Thus, $h(\mathbf{X},\mathbf{Y}) = (\mathbf{X}',\mathbf{Y}')$.
Similar to the way we argue about alternating extraction protocols, we fix random variables in the following order: Fix $\mathbf{S}_1$, and it follows that $\mathbf{R}_1$ is now a deterministic function of $\mathbf{Y}$. We fix $\mathbf{R}_1$, and thus $\mathbf{S}_2$ is now a deterministic function of $\mathbf{X}$. Thus, continuing in this way, we can fix all the random variables $\mathbf{S}_1,\ldots,\mathbf{S}_{\ell}$ and $\mathbf{R}_1,\ldots,\mathbf{R}_{\ell}$ while maintaining that $\mathbf{X}$ and $\mathbf{Y}$ are independent. Further, invoking Lemma $\ref{lemma:entropy_loss_1}$, with probability at least $1-2^{-\Omega(n)}$, both $\mathbf{X}$ and $\mathbf{Y}$ have min-entropy at least $n-t - \delta n \ge n - 2 \delta n$ since both parties send at most $t$ bits.
Note that now, $\mathbf{X}' = \eta(\mathbf{X})$ for some deterministic function $\eta$ and $\mathbf{Y}' =\nu(\mathbf{X})$ for some deterministic function $\nu$. Thus, for any $2$-source non-malleable extractor $\textnormal{nmExt}$ that works for min-entropy $n-2\delta n$ with error $\epsilon$, we have that there exists a distribution $\mathbf{\mathcal{D}}_{\eta,\nu}$ over $\{0, 1\}^{m} \cup \{\textnormal{$same^{\star}$}\}$ that is independent of $\mathbf{X}$ and $\mathbf{Y}$ such that $$|\textnormal{nmExt}(\mathbf{X},\mathbf{Y}), \textnormal{nmExt}(\eta(\mathbf{X}), \nu(\mathbf{Y})) - \mathbf{U}_m, \textnormal{copy}(\mathbf{\mathcal{D}}_{\eta,\nu},\mathbf{U}_m)| \le \epsilon+.2^{-\Omega(n)}$$
The theorem now follows by plugging in such a construction from a recent work of Li (\cite{Li18}, Theorem $1.12$). We note the non-malleable extractor in \cite{Li18} is indeed $2^{-\Omega(n \log \log n/\log n)}$-invertible.
\end{proof}
\section{Efficient sampling algorithms}
\label{sec:sampling}
In this section, we provide efficient sampling algorithms for the seedless non-malleable extractor construction presented in Section $\ref{section:composed_split_state}$. This is crucial to get efficient encoding algorithms for the corresponding non-malleable codes. We do not know how to invert the non-malleable extractor constructions in Theorem $\ref{theorem:ext_lin_composed_ss_1}$, but we show that the constructions can suitably modified in a way that admits efficient sampling from the pre-image of the extractor.
\subsection{An invertible non-malleable extractor with respect to linear composed with interleaved adversaries}
The main idea is to ensure that on fixing appropriate random variables that are generated in computing the non-malleable extractor, the source is now restricted onto a known subspace of fixed dimension (i.e., the dimension does not depend on value of the fixed random variables). Once we can ensure this, sampling from the pre-image can simply be done by first uniformly sampling the fixed random variables, and then sampling the other variables uniformly from the known subspace. To carry this out, we need an efficient construction of a linear seeded extractor that has the property that for any fixing of the seed the linear map corresponding linear seeded extractor has the same rank. Such a linear seeded extractor was constructed in prior works \cite{CGL15,Li16} (see Theorem $\ref{thm:low_error_inv_lin}$).
One additional care we need to take is the choice of the error correcting code we use in the advice generator construction. We ensure that the linear constraints imposed by fixing the advice string does not depend on the value of the advice string. This is subtle since the advice generator comprises of a sample from an error correction of the sources as well as the output of a linear seeded extractor on the source. The basic idea is to remove a few sampled coordinates of the error corrected sources and show that this suffices to remove any linear dependencies.
We use the following notation: For any linear map $L:\{0, 1\}^{r} \rightarrow \{0, 1\}^s$ given by $L(\alpha)=M \alpha$ for some matrix $M$, we use $con_{L}$ to denote a maximal set of linearly independent rows of $M$.
We now set up some parameters and ingredients for our construction of an invertible non-malleable extractor.
\begin{itemize}
\item Let $\delta>0$ be a small enough constant and $C$ a large constant.
\item Let $\delta'=\delta/C$.
\item Let $\mathcal{C}$ be a $\textnormal{BCH}$ code with parameters: $[n_{b},n_{b}-t_{b} \log n_{b},2t_{b}]_2$, $t_{b}= \sqrt{n_b}/100$, where we fix $n_{b}$ in the following way. Let $\textnormal{dBCH}$ be the dual code. From standard literature, it follows that $\textnormal{dBCH}$ is a $[n_{b}, t_{b} \log n_{b}, \frac{n
_b}{2}- t_{b}\sqrt{n_{b}}]_2$-code. Set $n_{b}$ such that $t_{b} \cdot \log n_{b}= \sqrt{n_b} \log n_b = 2n$. Let $E$ be the encoder of $\textnormal{dBCH}$. Note that by our choice of parameters, the relative minimum distance of $\textnormal{dBCH}$ is at least $1/3$.
\item Let $n_0 = n^{\delta'}, n_1= n_0^{c_0}, n_2= 10n_0$, for some constant $c_0$ that we set below.
\item Let $n_3= n^{C \delta}, n_4 = n^{C^2 \delta}/5, n_5 =n^{C^{3} \delta}, n_6 = n - \sum_{i=1}^5n_i $. We ensure that $n_6 = n(2-o(1))$.
\item Let $\textnormal{Ext}_1: \{0, 1\}^{n_1} \times \{0, 1\}^{d_1} \rightarrow \{0, 1\}^{\log(n_b)}$ be a $(n_1/20,1/10)$-seeded extractor instantiated using Theorem $\ref{guv}$. Thus $d_1= c_1 \log n_1$, for some constant $c_1$. Let $D_1=2^{d_1}=n_1^{c_1}$.
\item Let $\textnormal{Samp}_1:\{0, 1\}^{n_1} \rightarrow [n_b]^{D_1}$ be the sampler obtained from Theorem $\ref{thm:seed_samp}$ using $\textnormal{Ext}_1$.
\item Let $\textnormal{Ext}_2: \{0, 1\}^{n_2} \times \{0, 1\}^{d_2} \rightarrow \{0, 1\}^{\log(n_6)}$ be a $(n_2/20,1/n_0)$-seeded extractor instantiated using Theorem $\ref{guv}$. Thus $d_2= c_2 \log n_2$, for some constant $c_2$. Let $D_2=2^d_2$. Thus $D_2= 2^{d_2}= n_2^{c_2}$.
\item Let $\textnormal{Samp}_2:\{0, 1\}^{n_2} \rightarrow [n_6]^{D_2}$ be the sampler obtained from Theorem $\ref{thm:seed_samp}$ using $\textnormal{Ext}_2$.
\item Set $c_0 = 2c_2$.
\item Let $\textnormal{i$\ell$Ext}:\{0, 1\}^{D_2} \rightarrow \{0, 1\}^{n_0}$ be the extractor from Theorem $\ref{thm:il_ext}$.
\item Let $\textnormal{LExt}_0: \{0, 1\}^{2n} \times \{0, 1\}^{n_0} \rightarrow \{0, 1\}^{\sqrt{n_0}}$ be a linear seeded extractor instantiated from Theorem $\ref{thm:strong_ip}$ set to extract from min-entropy $n_1/100$ and error $2^{-\Omega(\sqrt{n_0})}$.
\item Let $\textnormal{Ext}_3: \{0, 1\}^{n_3} \times \{0, 1\}^{d_3} \rightarrow \{0, 1\}^{\log (n_6-D_2)}$ be a $(n_3/8,1/100)$-seeded extractor instantiated using Theorem $\ref{guv}$. Thus $d_3= C_1 \log n_3$, for some constant $C_1$.
\item Let $\textnormal{Samp}_3:\{0, 1\}^{n_3} \rightarrow [n_6-D_2]^{n_7}$ be the sampler obtained from Theorem $\ref{thm:seed_samp}$ using $\textnormal{Ext}_3$. Thus $n_7= 2^{d_3}= n_3^{C_1}$.
\item Let $\textnormal{Ext}_4: \{0, 1\}^{n_4} \times \{0, 1\}^{d_4} \rightarrow \{0, 1\}^{n_6 - n_7-D_2}$ be a $(n_4/8,1/100)$-seeded extractor instantiated using Theorem $\ref{guv}$. Thus $d_3= C_1 \log n_4$.
\item Let $\textnormal{Samp}_4:\{0, 1\}^{n_4} \rightarrow [n_5-n_7-D_2]^{n_{8}}$ be the sampler obtained from Theorem $\ref{thm:seed_samp}$ using $\textnormal{Ext}_4$. Thus $n_{8}= 2^{d_3}= n_4^{C_1 }$.
\item Let $\textnormal{LExt}_1: \{0, 1\}^{n_5} \times \{0, 1\}^{d} \rightarrow \{0, 1\}^{d_5}$, $d_5= \sqrt{n_5}$, be a linear-seeded extractor instantiated from Theorem $\ref{trev_ext}$ set to extract from entropy $k_1=n_2/10$ with error $\epsilon_1=1/10$. Thus $d= C_2\log n_5$, for some constant $C_2$. Let $D=2^{d}$.
\item Let $\textnormal{LExt}_2: \{0, 1\}^{n_7} \times \{0, 1\}^{d_5} \rightarrow \{0, 1\}^{m_1}$, $m_1=\sqrt{n_7}$ be a linear-seeded extractor instantiated from Theorem $\ref{trev_ext}$ set to extract from entropy $k_2= n_7/100$ with error $\epsilon_2=2^{-\Omega(\sqrt{d_4})}=2^{-n^{\Omega(1)}}$, such that the seed length of the extractor $\textnormal{LExt}_2$ (by Theorem $\ref{trev_ext}$) is $d_5$.
\item Let $\textnormal{ACB}:\{0, 1\}^{n_{1,acb}} \times \{0, 1\}^{n_{acb}} \times \{0, 1\}^{h_{acb}} \rightarrow \{0, 1\}^{n_{2,acb}}$, be the advice correlation breaker from Theorem $\ref{thm:acb}$ set with the following parameters: $n_{acb}=n_7, n_{1,acb}=m_1,n_{2,acb} =n_{9}=D^2, t_{acb} = 2D, h_{acb}=n^{\delta}+d, \epsilon_{acb}= 2^{-n^{\delta'}}$, $d_{acb}=O(\log^2(n/\epsilon_{acb})), {\lambda}_{acb}=0$. It can be checked that by our choice of parameters, the conditions required for Theorem $\ref{thm:acb}$ indeed hold for $k_{1,acb} \ge n^{C\delta}$.
\item Let $\textnormal{LExt}_3: \{0, 1\}^{n_{8}} \times \{0, 1\}^{n_{9}} \rightarrow \{0, 1\}^{m}$ be the linear seeded extractor from Theorem $\ref{thm:low_error_inv_lin}$ set to extract from min-entropy rate $1/10$ and error $\epsilon= 2^{-n^{\Omega(1)}}$ (such that the seed-length is indeed $n_9$). Thus, $m = \alpha n_{9}$, for some small contant $\alpha$ that arises out of Theorem $\ref{thm:low_error_inv_lin}$.
\end{itemize}
\RestyleAlgo{boxruled}
\LinesNumbered
\begin{algorithm}[ht]\label{alg:inv_ilnm}
\caption{$\textnormal{i$\ell$NM}(z)$ \vspace{0.1cm}\newline \textbf{Input:}
Bit-string $z=(x \circ y)_{\pi}$ of length $2n$, where $x$ and $y$ are each $n$ bit-strings, and $\pi:[2n] \rightarrow [2n]$ is a permutation. \newline \textbf{Output:} Bit string of length $m$.
}
Let $z_i = z_1 \circ z_2 \circ z_3 \circ z_4 \circ z_5 \circ z_6$, where $z_i$ is of length $n_i$.
Let $T_i= \textnormal{Samp}_i(z_i)$, $i=1,2,3,4$.
Let $\overline{z}_2=(z_6)_{T_2}$.
Let $z_2'=\textnormal{i$\ell$Ext}(\ol{z}_2)$.
Let $z_2''=\textnormal{LExt}_0(z,z_2')$.
For any set $Q \subseteq [2n]$, define the linear function $E: \{0, 1\}^{2n} \rightarrow \{0, 1\}^{|Q|}$ as $E_Q(x) = (E(x))_Q$.
Pick a subset $\ol{T_1} \subset T_1$ of size $D_1-\sqrt{n_0}$ such that $con_{E_{\ol{T_1}}}$ is linearly independent of $con_{LExt_0(\cdot,z_2')}$. If there is no such set $\ol{T_1}$, then output $0^m$.
Let $w= z_1 \circ z_2 \circ \ol{z}_2 \circ (E( z))_{\ol{T_1}} \circ z_2''$.
Let $v$ be a $D \times d_4$ matrix, with its $i$'th row $v_i = \textnormal{LExt}_1(z_5,i)$.
Let $z_6'$ be the bits in $z_6$ outside $T_2$. Let $\ol{z_6}=(z_6')_{T_3}$.
Let $r$ be a $D \times n_4$ matrix, with its $i$'th row $r_i = \textnormal{LExt}_2(\ol{z_6},v_i)$.
Let $s$ be a $D \times m$ matrix, with its $i$'th row $s_i = \textnormal{ACB}(r_i,\ol{z_6}, w \circ i)$.
Let $\tilde{s} = \oplus_{i=1}^D s_i$.
Let $z_7$ be the bits in $z_6$ outside the coordinates $T_2 \cup T_3$.
Let $\ol{z_7} = (z_7)_{T_4}$. Let $z_8$ be the bits in $z_{6}$ outside the coordinates $T_2 \cup T_3 \cup T_4$.
Output $g = \textnormal{LExt}_3(\ol{z_7},\tilde{s})$.
\end{algorithm}
\begin{thm}
For all integers $n>0$ there exists an explicit function $\textnormal{nmExt}: \{0, 1\}^{2n} \rightarrow \{0, 1\}^{m}$, $m=n^{\Omega(1)}$, such that the following holds: For any linear function $h : \{0, 1\}^{2n} \rightarrow \{0, 1\}^{2n}$, arbitrary tampering functions $f,g \in \mathcal{F}_n$, any permutation $\pi:[2n] \rightarrow [2n]$ and independent uniform sources $\mathbf{X}$ and $\mathbf{Y}$ each on $n$ bits, there exists a distribution $\mathbf{\mathcal{D}}_{h,f,g,\pi}$ on $\{0, 1\}^m \cup \{ \textnormal{$same^{\star}$}\}$, such that
$$ |\textnormal{nmExt}((\mathbf{X} \circ \mathbf{Y})_{\pi}), \textnormal{nmExt}(h((f(\mathbf{X}) \circ g(\mathbf{Y}))_{\pi})) - \mathbf{U}_{m},\textnormal{copy}(\mathbf{\mathcal{D}}_{h,f,g,\pi},\mathbf{U}_m) | \le 2^{-n^{\Omega(1)}}.$$
\end{thm}
The proof that $\textnormal{i$\ell$NM}$ computed by Algorithm $\ref{alg:inv_ilnm}$ satisfies Theorem $\ref{theorem:ext_lin_composed_ss_1}$ is very similar, and we omit the details. We include a discussion of the key differences and subtleties that arise from the modifications done in the above construction as compared to Algorithm $\ref{alg:ilnm}$.
The first key difference is Step $7$, where we discard some bits from the advice generator's output. The existence of the subset $\ol{T}_1$ is guaranteed by the fact that $E$ has dual distance $t_b = \Omega(n/\log n)$. Thus, for any $T$, it must be that $\textnormal{Con}_{E_{T_1}}$ is a set of size $|T_1| = D_1$. Further, $con_{LExt_0(\cdot,z_2')}$ is a set with cardinality at most $\sqrt{n_0}$. Thus, indeed there exists such a set $\ol{T}_1$. An important detail to notice is that $|T_1 \setminus \ol{T_1}| = o(D_1)$ and the distance of the code computed by $E$ is $\Omega(1)$. Thus, the fact that we discard the bits indexed by the set $T_1 \setminus \ol{T_1}$ from the string $E(\mathbf{Z})_{T_1}$ (and thus from the output of the advice generator) does not affect the correctness of the advice generator.
Another difference is that in the steps where we transform the somewhere random matrix $v$ into a matrix with longer rows, and the subsequent step where the advice correlation breaker is applied is now done using a pseudorandomly sampled subset of coordinates from $\mathbf{Z}$ (as opposed to the entire $\mathbf{Z}$ which we did before). It is not hard to prove that this does not make a difference as long as we sample enough bits. Finally, another difference is the final step where we use a linear seeded extractor, with $\ol{\mathbf{Z}_6}$ as the seed. As done many times in the paper, we use the sum structure of $\ol{\mathbf{Z}_7}$ (into a source that depends on $\mathbf{X}$ and a source that depends on $\mathbf{Y}$) along with the fact that $\textnormal{LExt}_3$ is linear seeded to show that the output is close to uniform.
We now focus on the problem of efficiently sampling from the pre-image of this extractor. The following lemma almost immediately implies a simple sampling algorithm.
\begin{lemma}
\label{lem:sampling_interleaved}
With probability $1-2^{-n^{\Omega(1)}}$ over the fixing of the variables $z_1,z_2,\ol{z_2},z_2'',z_3,z_4,z_5,\ol{z_6}, w$, and any $g \in \{0, 1\}^m$, the set $\textnormal{i$\ell$NM}^{-1}(g)$ is a linear subspace of fixed dimension.
\end{lemma}
\begin{proof}
Consider any fixing of $z_1,z_2,z_3,z_4$. Clearly, these fix the sets $T_i$, $i=1,2,3,4$. Next, note that given $\overline{z}_2$, we have the value of $z_2'$. We note that by Lemma $\ref{aff_error}$ that with probability $1-2^{-n^{\Omega(1)}}$, the linear map $\textnormal{LExt}_0(,z_2')$ has full rank. Using Algorithm $\ref{alg:ilnm}$, determine the set $\ol{T_1}$ (if it exists). Fix $E(z)_{\overline{T}_1}$ and $z_2''$, noting that the value of $w$ is now determined. Now given $z_5,\ol{z_6}$, we can compute $r,s,\tilde{s}$. Next observe that given $g$ and $\tilde{s}$, Theorem $\ref{thm:low_error_inv_lin}$ implies the value of $\ol{z_7}$ belongs to a subspace whose dimension does not depend on the values of $g$ and $\tilde{s}$. Finally, we are left to see how to compute $z_8$. Note that the constraints on $z_8$ are imposed by the fixings of $z_2''$ and $E(C)_{\ol{T_1}}$. However, by construction (Step $7$ of our algorithm), the number of independent linear constraints on $z_8$ is exactly equal to $D_1$ as long as $\textnormal{LExt}_0(,z_2')$ has full rank (which as noted before occurs with probability at least $1-2^{-n^{\Omega(1)}}$). This completes the proof.
\end{proof}
Given Lemma $\ref{lem:sampling_interleaved}$, the sampling algorithm is now straightforward:
Input $g \in \{0, 1\}^{m}$; Output $z$ that is uniform on the set $\textnormal{i$\ell$NM}^{-1}(g)$.
\begin{enumerate}
\item Sample $z_i$, $i=1,2,3,4,5$ uniformly at random. Compute $T_1,T_2,T_3,T_4$ following Algorithm~$\ref{alg:ilnm}$.
\item Sample $\ol{z_2}$ uniformly, and compute $z_2'$. Further, sample $z_2''$ uniformly.
\item Compute $\ol{T_1}$, and sample $(E(z))_{\ol{T_1}}$ uniformly at random.
\item Compute $w,v,r,s,\tilde{s}$ using Algorithm $\ref{alg:ilnm}$.
\item Sample $\ol{z_7}$ from $(\textnormal{LExt}_3(\cdot,\tilde{s}))^{-1}(g)$ efficiently using Theorem $\ref{thm:low_error_inv_lin}$.
\item Sample $z_8$ as described in Lemma $\ref{lem:sampling_interleaved}$. Compute the string $z_6$.
\item Output $z = z_1 \circ z_2 \circ z_3 \circ z_4 \circ z_5 \circ z_6$.
\end{enumerate}
\section{Extractors for interleaved sources}
\label{sec:ilext}
Our techniques yield improved explicit constructions of extractors for interleaved sources. Our extractor works when both sources have entropy at least $2n/3$, and outputs $\Omega(n)$ bits that are $2^{-n^{\Omega(1)}}$-close to uniform.
The following is our main result.
\begin{thm}
\label{thm:il_ext}
For any constant $\delta>0$ and all integers $n>0$, there exists an efficiently computable function $\textnormal{i$\ell$Ext}: \{0, 1\}^{2n} \rightarrow \{0, 1\}^{m}$, $m = \Omega(n)$, such that for any two independent sources $\mathbf{X}$ and $\mathbf{Y}$, each on $n$ bits with min-entropy at least $(2/3 + \delta)n$, and any permutation $\pi:[2n] \rightarrow [2n]$, we have $$|\textnormal{i$\ell$Ext}((\mathbf{X} \circ \mathbf{Y})_{\pi}) - \mathbf{U}_m| \le 2^{-n^{\Omega(1)}}.$$
\end{thm}
We use the rest of the section to prove Theorem $\ref{thm:il_ext}$. An important ingredient in our construction is an explicit somewhere condenser for high-entropy sources constructed in the works of Barak et al.\ \cite{BRSW12} and Zuckerman \cite{Zuck07}.
\begin{thm}
\label{thm:condense}
For all constants $\beta, \delta$ and all integers $n>0$, there exists an efficiently computable function $\textnormal{Con}:\{0, 1\}^{n} \times \{0, 1\}^{d} \rightarrow \{0, 1\}^{\ell}$, $d = 0(1)$ and $\ell = \Omega(n)$ such that the following holds: for any $(n,\delta n)$-source $\mathbf{X}$ there exists a $y \in \{0, 1\}^{d}$ such that $\textnormal{Con}(\mathbf{X},y)$ is $2^{-\Omega(n)}$-close to a source with min-entropy $(1- \beta)\ell$. \\We call such a function $\textnormal{Con}$ to be a $(\delta,1-\beta)$-condenser.
\end{thm}
We prove that Algorithm $\ref{alg:il_ext}$ computes the required extractor. We begin by setting up some ingredients and parameters.
\begin{itemize}
\item Let $\kappa>0$ be a small enough constant.
\item Let $n_1 = (2/3 + \delta/2)n$ and $n_2 = n^{5\kappa}$.
\item Let $\beta$ be a parameter which we fix later. Let $\textnormal{Con}: \{0, 1\}^{n_1} \times \{0, 1\}^{d} \rightarrow \{0, 1\}^{\ell}$ be a $(\delta/4,1-\beta)$-condenser instantiated from Theorem $\ref{thm:condense}$. Thus $\ell = n/C'$, for some constant $C'$ that depends on $\delta, \beta$. Let $D= 2^d$. Note that $D= O(1)$.
\item Let $\textnormal{LExt}_1: \{0, 1\}^{2n} \times \{0, 1\}^{\ell} \rightarrow \{0, 1\}^{n_2}$ be the linear seeded extractor from Theorem $\ref{thm:low_error_inv_lin}$ set to extract from min-entropy rate $1/12$ and error $\epsilon_1= 2^{-2 \beta \ell}$. The seed-length is at most $3 C\beta \ell$, some constant $C$ that arises out of Theorem $\ref{thm:low_error_inv_lin}$. We choose $\beta = min\{1/3C,\gamma\}$, where $\gamma$ is the constant in Theorem $\ref{thm:low_error_inv_lin}$. Note that the seed-length of $\textnormal{LExt}_1$ is indeed at most $\ell$.
\item Let $\textnormal{ACB}:\{0, 1\}^{n_{1,acb}} \times \{0, 1\}^{n_{acb}} \times \{0, 1\}^{h_{acb}} \rightarrow \{0, 1\}^{n_{2,acb}}$, be the advice correlation breaker from Theorem $\ref{thm:acb}$ set with the following parameters: $n_{acb}=2n, n_{1,acb}=n_2, n_{2,acb} =n_3= n^{2\kappa}, t_{acb} = D, h_{acb}=d, \epsilon_{acb}= 2^{-n^{\kappa}}, d_{acb}= O(\log^2(n/\epsilon_{acb})), {\lambda}_{acb}=0$. It can be checked that by our choice of parameters, the conditions required for Theorem $\ref{thm:acb}$ indeed hold for $k_{1,acb} \ge n^{2 \kappa}$.
\item Let $\textnormal{LExt}_2: \{0, 1\}^{2n} \times \{0, 1\}^{n_3} \rightarrow \{0, 1\}^{m}$, $m=\Omega(n)$, be a linear-seeded extractor instantiated from Theorem $\ref{trev_ext}$ set to extract from entropy $k_1=n/10$ with error $\epsilon_1=2^{-\alpha\sqrt{n_3}}$, for an appropriately picked small constant $\alpha$.
\end{itemize}
\RestyleAlgo{boxruled}
\LinesNumbered
\begin{algorithm}[ht]\label{alg:il_ext}
\caption{$\textnormal{i$\ell$Ext}(z)$ \vspace{0.1cm}\newline \textbf{Input:}
Bit-string $z=(x \circ y)_{\pi}$ of length $2n$, where $x$ and $y$ are each $n$ bit-strings, and $\pi:[2n] \rightarrow [2n]$ is a permutation. \newline \textbf{Output:} Bit string of length $m$.
}
Let $z_1= \textnormal{Slice}(z,n_1)$.
Let $v$ be a $D \times n_2$ matrix, with its $i$'th row $v_i = \textnormal{Con}(z_1,i)$.
Let $r$ be a $D \times n_3$ matrix, with its $i$'th row $r_i = \textnormal{LExt}_1(z,v_i)$.
Let $s$ be a $D \times m$ matrix, with its $i$'th row $s_i = \textnormal{ACB}(r_i,z, i)$.
Let $\tilde{s} = \oplus_{i=1}^D s_i$.
Output $\textnormal{LExt}_2(z,\tilde{s})$.
\end{algorithm}
We use the following notation: Let $\mathbf{X}_1$ be the bits of $\mathbf{X}$ in $\mathbf{Z}_1$ and $\mathbf{X}_2$ be the remaining bit of $\mathbf{X}$. Let $\mathbf{Y}_1$ be the bits of $\mathbf{Y}$ in $\mathbf{Z}_1$ and $\mathbf{Y}_2$ be the remaining bits of $\mathbf{Y}$. Without loss of generality assume $|\mathbf{X}_1| \ge |\mathbf{Y}_1|$. Define $\overline{\mathbf{X}}= (\mathbf{X} \circ 0^n)_{\pi}$ and $\overline{\mathbf{Y}} = (\mathbf{Y} \circ 0^n)_{\pi}$. Further, let $\overline{\mathbf{X}}_1= \textnormal{Slice}(\overline{\mathbf{X}},n_1)$ and $\overline{\mathbf{Y}}_1= \textnormal{Slice}(\overline{\mathbf{Y}},n_1)$. It follows that $\mathbf{Z}= \overline{\mathbf{X}} + \overline{\mathbf{Y}}$, and $\mathbf{Z}_1 = \overline{\mathbf{X}_1} + \overline{\mathbf{Y}_1}$. Further, let $k_x =k_y = (2/3 + \delta)n$.
We begin by proving the following claim.
\begin{claim}\label{cl:il1} Conditioned on the random variables $\mathbf{X}_1,\mathbf{Y}_1, \{\textnormal{LExt}_1(\overline{\mathbf{X}}, \textnormal{Con}(\overline{\mathbf{X}_1} + \overline{\mathbf{Y}_1},i))\}_{i=1}^D$, the following hold:
\begin{itemize}
\item the matrix $\mathbf{R}$ is $2^{-\Omega(n)}$-close to a somewhere random source,
\item $\mathbf{R}$ is a deterministic functions of $\mathbf{Y}$,
\item $H_{\infty}(\mathbf{X}) \ge \delta n/4 $, $H_{\infty}(\mathbf{Y}) \ge n/6$.
\end{itemize}
\end{claim}
\begin{proof}
By construction, we have that for any $j \in [D]$,
\begin{align*}
\mathbf{R}_j &= \textnormal{LExt}_1(\mathbf{Z},\textnormal{Con}(\mathbf{Z}_1,j)) \\
&=\textnormal{LExt}_1(\overline{\mathbf{X}}+ \overline{\mathbf{Y}}, \textnormal{Con}(\overline{\mathbf{X}_1}+\overline{\mathbf{Y}_1},j)) \\
&= \textnormal{LExt}_2(\overline{\mathbf{X}}, \textnormal{Con}(\overline{\mathbf{X}_1}+\overline{\mathbf{Y}_1},j)) + \textnormal{LExt}_2(\overline{\mathbf{Y}}, \textnormal{Con}(\overline{\mathbf{X}_1}+\overline{\mathbf{Y}_1},j))
\end{align*}
Fix the random variables $\mathbf{Y}_1$, and $\overline{\mathbf{Y}}$ has min-entropy at least $k_y- n_1/2\ge n/6 + 3 \delta n/4$. Further, note that $\ol{\mathbf{X}_1}$ has min-entropy at least $ n_1/2 - (n- k_x) \ge \delta n/4$. Now, by Theorem $\ref{thm:condense}$, we know that there exists a $j \in [D]$ such that $\textnormal{Con}(\ol{\mathbf{X}}_1 + \ol{\mathbf{Y}}_1,j)$ is $2^{-\Omega(n)}$-close to a source with min-entropy at least $(1 - \beta)\ell$. Further, note that $\mathbf{V}$ is a deterministic function of $\mathbf{X}$.
Now, since $\textnormal{LExt}_1$ is a strong seeded extractor set to extract from min-entropy $n/6$, it follows that $$| \textnormal{LExt}_1(\overline{\mathbf{Y}},\textnormal{Con}( \ol{\mathbf{X}}_1 + \ol{\mathbf{Y}}_1, j)) - \mathbf{U}_{n_2}| \le 2^{\beta \ell}\epsilon_1 + 2^{-\Omega(n)} \le 2^{-\beta \ell + 1}.$$
We now fix the random variables $\ol{\mathbf{X}}_1$ and note that $ \textnormal{LExt}_1(\overline{\mathbf{Y}},\textnormal{Con}( \ol{\mathbf{X}}_1 + \ol{\mathbf{Y}}_1, j))$ continues to be $2^{-\Omega(\ell)}$-close to $\mathbf{U}_{n_2}$. This follows from the fact that $\textnormal{LExt}_1$ is a strong seeded extractor. Note that the random variables $\{\textnormal{Con}(\ol{\mathbf{X}}_1 + \ol{\mathbf{Y}}_1, i)): i \in [D]\}$ are now fixed. Next, fix the random variables $\{\textnormal{LExt}_1(\overline{\mathbf{X}}, \textnormal{Con}(\overline{\mathbf{X}_1} + \overline{\mathbf{Y}_1},i))\}_{i=1}^D$ noting that they are deterministic functions of $\mathbf{X}$. Thus $\mathbf{R}_j$ is $2^{-\Omega(n)}$-close to $\mathbf{U}_{n_2}$ and for any $i \in [D]$, the random variables $\mathbf{R}_i$ are deterministic functions of $\mathbf{Y}$. Finally, note that $\mathbf{X}$ and $\mathbf{Y}$ remain independent after these conditionings, and $H_{\infty}(\mathbf{X}) \ge k_x -n_1 -D n_2 $ and $H_{\infty}(\mathbf{Y}) \ge k_y - n_1/2$.
\end{proof}
The next claim almost gets us to Theorem $\ref{thm:il_ext}$.
\begin{claim}\label{cl:il2} There exists $j \in [D]$ such that
$$ \mathbf{S}_j, \{ \mathbf{S}_i \}_{i \in [D]\setminus j}, \mathbf{X} \approx_{2^{-n^{\Omega(1)}}} \mathbf{U}_{n_3}, \{ \mathbf{S}_i \}_{i \in [D]\setminus j}, \mathbf{X}.$$
\end{claim}
\begin{proof} Fix the random variables: $\mathbf{X}_1,\mathbf{Y}_1, \{\textnormal{LExt}_1(\overline{\mathbf{X}}, \textnormal{Con}(\overline{\mathbf{X}_1} + \overline{\mathbf{Y}_1},i))\}_{i=1}^D$. By Claim $\ref{cl:il1}$ we have that $\mathbf{R}$ is a deterministic function of $\mathbf{Y}$, and with probability at least $1-2^{-\Omega(n)}$, there exists $j \in [D]$ such that $\mathbf{R}_j$ is $2^{-n^{\Omega(1)}}$-close to uniform, and $H_{\infty}(\overline{\mathbf{X}}) \ge \delta n/4 $. Recall that $\mathbf{Z}=\overline{\mathbf{X}} + \overline{\mathbf{Y}}$. It now follows by Theorem $\ref{thm:acb}$ that
\begin{align*}
\textnormal{ACB}(\mathbf{R}_{j}, \mathbf{Z}, \mathbf{W} \circ j), \{ \textnormal{ACB}(\mathbf{R}_{i},\overline{\mathbf{X}}+\overline{\mathbf{Y}}, \mathbf{W} \circ i) \}_{i \in [D] \setminus j}, \mathbf{X} \approx_{2^{-n^{\Omega(1)}}} \\ \mathbf{U}_{n_3}, \{ \textnormal{ACB}(\mathbf{R}_{i},\overline{\mathbf{X}}+\overline{\mathbf{Y}}, \mathbf{W} \circ i) \}_{i \in [D] \setminus j}, \mathbf{X}.
\end{align*}
\end{proof}
It follows by Claim $\ref{cl:il2}$ that $\widetilde{\mathbf{S}}$ is $2^{-n^{\Omega(1)}}$-close to uniform even conditioned on $\mathbf{X}$. Thus, noting that $\textnormal{LExt}_2(\mathbf{Z},\widetilde{\mathbf{S}}) = \textnormal{LExt}_2(\ol{\mathbf{X}},\widetilde{\mathbf{S}}) + \textnormal{LExt}_2(\ol{\mathbf{Y}},\widetilde{\mathbf{S}})$, it follows that we can fix $\widetilde{\mathbf{S}}$ and $\textnormal{LExt}_2(\ol{\mathbf{X}},\widetilde{\mathbf{S}})$ remains $2^{-n^{\Omega(1)}}$-close to uniform and is a deterministic function of $\mathbf{X}$. Next, we fix $\textnormal{LExt}_2(\ol{\mathbf{Y}},\widetilde{\mathbf{S}})$ without affecting the distribution of $\textnormal{LExt}_2(\ol{\mathbf{X}},\widetilde{\mathbf{S}})$. It follows that $\textnormal{LExt}_2(\mathbf{Z},\widetilde{\mathbf{S}})$ is $2^{-n^{\Omega(1)}}$-close to uniform. This completes the proof of Theorem $\ref{thm:il_ext}$.
\bibliographystyle{alpha}
|
{
"timestamp": "2018-11-05T02:17:01",
"yymm": "1804",
"arxiv_id": "1804.05228",
"language": "en",
"url": "https://arxiv.org/abs/1804.05228"
}
|
\section{Introduction and preliminaries}
Graph labeling is a vivid area of combinatorics which started in the middle of
1960's. Much of the area is based on results of Rosa \cite{rosa1967} and of
Graham and Sloane \cite{graham1980additive}. Since then, over 200 different
labelings were introduced. We refer to Gallian's survey \cite{dynamic}, citing
over 2500 papers, gathering most of the results in the area. Applications of
labeling are both theoretical (Rosa introduced so-called \emph{graceful}
labelings to attack Ringel's conjecture on certain graph decompositions) and
practical (for example the frequency assignment problem
\cite{hale1980,vangraph,jha2000}).
We study \emph{edge-sum distinguishing} (abbreviated as ESD) labeling,
introduced by Tuza \cite{tuza2017graph} in 2017. Tuza's primarily concern was
to study several combinatorial games connected to this labeling. Our main
objective is to study structural properties of this labeling on its own.
However, as our secondary objective, we also give some results on game
variants of edge-sum distinguishing labeling.
\paragraph{Structure of the paper.} In the rest of this section we review
basic definitions and show a broader context of ESD labeling to other existing
notions in combinatorics. The second section deals with structural properties
of ESD labeling. For various well-known classes of graphs we show if they have
a canonical ESD labeling or not. In the third section we are concerned with
game variants, the original motivation of Tuza. Finally, in the last section
we summarize our results and propose some open problems.
\paragraph{Notation.}
We use the notation of West \cite{west2001introduction}. All graphs in the
paper are finite, undirected, connected and without multiple edges, unless
we say otherwise.
\subsection{Basic definitions}
We need to formally define what graph labeling is. We will need vertex
labelings only.
\begin{definition}
Let $G = (V,E)$ be a graph and let $L \subseteq \mathbb{N}$ be a set of labels. Then a mapping $\phi: V \rightarrow L$ is called a \emph{vertex labeling}.
We further say that vertex labeling is \emph{canonical} if $|V| = |L|$.
\end{definition}
We will often refer to edge-weights, induced by a vertex labeling.
\begin{definition}
Let $G=(V,E)$ be a graph and $\phi$ a vertex labeling on $G$. The \emph{edge-weight}
of an edge $xy$ is defined as $w_\phi(uv):=\phi(u) + \phi(v)$.
\end{definition}
Now we can finally introduce a definition of \emph{edge-sum distinguishing
labeling}.
\begin{definition}
Let $G=(V,E)$ be a graph and $L = \{1, \ldots, l\}$, $l \in \mathbb N$. A vertex labeling $\phi: V \rightarrow L $ is called \emph{edge-sum distinguishing labeling}
(\emph{ESD labeling}) if $\phi$ is injective and if
$$\forall e, f \in E: e \neq f \rightarrow w_\phi(e) \neq w_\phi(f).$$
\end{definition}
We note that no ESD labeling exists in case $|L| < |V|$. We call a special case
when $|L| = |V|$ a \emph{canonical ESD labeling}.
\begin{example}
Consider a path $P_n$ and denote its vertices consecutively $v_1,\ldots,v_n$.
Choose a labeling $\phi(v_i) = i$. Clearly, this is an ESD labeling and even a canonical ESD labeling.
\end{example}
\subsection{Connections to existing notions}
\paragraph{Edge-antimagic vertex labeling.}
Following the usual terminology in the area of graph labelings, one could name
canonical ESD labelings also as \emph{edge-antimagic vertex labelings}. To
illustrate this, let us recall that an \emph{antimagic labeling} of a graph
with $m$ edges and $n$ vertices is a bijection from the set of edges to the
integers $1,\ldots,m$ such that all $n$ vertex sums are pairwise distinct,
where a vertex sum is the sum of labels of all edges incident with the same
vertex. Antimagic labeling were introduced as a natural generalization of
magic labelings. We refer the reader to
\cite{bavca2007edge,simanjuntak2000two} for more information on antimagic
labelings and to \cite{kotzig1970magic,kotzig1972magic,wallis2001magic} for a
literature on magic labelings.
To our best knowledge, edge-antimagic vertex labelings were not studied yet.
\paragraph{Super edge-magic total labelings.}
A \emph{super edge-magic total labeling} is an injection $f: V \cup E \to
\{1, 2, \ldots , |V| + |E|\}$ such that the weight of every edge $xy$ defined
as $w(xy) = f (x) + f (y) + f (xy)$ is equal to the same magic constant $m$
and the vertex labels are the numbers $1, 2, \ldots , |V|$. One can observe
that such labeling implies an edge-sum distinguishing labeling in a natural
way. If we remove the labels of edges, the edge-weights now form an arithmetic
progression. We can say about the resulting labeling even more; it is an
$(a,1)$-edge antimagic vertex labeling.
An \emph{$(a,d)$-edge antimagic vertex labeling} is a one-to-one mapping
$f$ from $V(G)$ onto $\{1,2,\ldots,|V| \}$ with the property that
for every edge $xy \in E(G)$, the edge-weight set is equal to
$$\{f(x)+f(y): x,y\in V(G) \} = \{a,a+d,a+2d,\ldots,a+(|E|+1)d \},$$
for some $a>0, d\geq 0$. This definition comes from \cite{sugeng2013construction}.
\paragraph{Sidon sequences.} The Sidon sequences were introduced by Simon Sidon
in 1932 \cite{sidon1932satz}. We refer the reader to a dynamically
updated survey of O’Bryant \cite{o2004complete}.
The formulation of the following definition comes from the survey.
\begin{definition}
A \emph{Sidon sequence} is a sequence of integers $a_1 < a_2 < \ldots$ with
the property that sums $a_i+a_j \,\, (i \leq j),$ are distinct.
\end{definition}
ESD labeling can be reformulated in a similar fashion.
\begin{definition}
An \emph{ESD labeling} of a graph $G=(V,E)$, where $V=\{1,\ldots,n\}$, is a sequence of
integers $a_1 < a_2 < \ldots$ with the property that
sums $a_i+a_j, \, i \leq j,\, (i,j) \in E,$ are distinct and $a_1 = 1$.
\end{definition}
With this new definition in hand, we see that ESD labeling
is in some sense a generalization of the Sidon sequence.
The difference that $a_1 = 1$ in the definition of ESD labeling could be
easily dropped (but it is convenient for this paper). Also, one can observe
that without this condition, the original Sidon sequences are ESD labelings
of a sufficiently large complete graph with loops added to each vertex.
However, again for our convenience, we consider only loopless graphs in
this paper.
\paragraph{Harmonious labeling.}
\emph{Harmonious labeling} was introduced by Graham and Sloane \cite{graham1980additive}.
We say that graph $G$ with $k$ edges is \emph{harmonious} if its vertices
can be labeled injectively with integers modulo $k$ so that the sum of the labels
of its endpoints modulo $k$ is unique.
The difference between harmonious and ESD labeling is that we do not take
vertex labels and edge labels modulo number of edges in ESD case. In fact, ESD labelings
and harmonious labelings behave differently. For example, it is
conjectured that trees are harmonious and it is known that not all cycles are
harmonious \cite{guy2013unsolved}. For comparison, we show that
all trees and cycles have a canonical ESD labeling.
\section{Structural results}
\subsection{Necessary condition}
\begin{theorem} \label{thm:nec}
If a graph $G=(V,E)$ such that $|V| > 1$ has a canonical ESD labeling, then the inequality $|E| \leq 2|V|-3$ holds.
\end{theorem}
\begin{proof}
We claim that every canonical ESD labeling of an $n$-vertex graph has at most
$2n-3$ different edge-weights.
To prove this, observe that the smallest possible edge-weight in such
labeling is 3 and
the largest possible is $2n-1$. Also, the edge-weights of $G$ form a subset of the
set $\{3,\ldots,2n-1 \}$ which is of the size $2n-3$. This proves the claim.
Now if a graph $G$ has more than $2|V|-3$ edges we can use our claim
and by the pigeonhole principle, we have two edges with the same
weight, a contradiction.
\qed
\end{proof}
Now we will show that this bound is tight.
\begin{theorem}
For every $n \in \mathbb{N}, n > 1,$ there exist an $n$-vertex graph $G_n$ with $|E(G_n)| = 2n - 3$ which has a canonical ESD labeling.
\end{theorem}
\begin{proof}
For $G_2$ take $K_2$ and for $G_3$ take $K_3$. These cases are trivial.
For $n>3$, take a complete bipartite graph $K_{2,n-2}$ and add
an edge between the two vertices of the part of size 2. See Figure \ref{fig:tight}
for an example.
\begin{figure}
\centering
\includegraphics[scale=0.8]{obrazky/2n3.pdf}
\caption{An example of an ESD graph with $2n-3$ edges.}
\label{fig:tight}
\end{figure}
We will show that this graph has a canonical ESD labeling. We will denote
$x_1,x_2$ the vertices of the part of size 2 and $y_1,\ldots,y_{n-2}$ the vertices
of the other part.
Now we define a labeling $\phi$ in the following way.
\begin{itemize}
\item Let $\phi(x_1) = 1$ and $\phi(x_2) = n$.
\item Let $\phi(y_i) = i+1$ for $1 \leq i \leq n-2$.
\end{itemize}
Observe that the edges incident with $x_1$ have edge-weights from $3$ to $n+1$.
Furthermore, the edges incident with $x_2$, except for the edge $x_1x_2$,
have edge-weights ranging from $n+2$ to $2n-1$. All these weights appear exactly once
and thus we are done.
\qed
\end{proof}
\subsection{Fan graphs}
In the previous part we showed a necessary condition for graph to have a
canonical ESD labeling. The point of this part is to show that this condition
is not sufficient in general by proving that \emph{fan graphs}, which have
$2n-3$ edges, do not have a canonical ESD labeling if their order is bigger
than 8.
\begin{definition}
A \emph{fan graph} $F_n$ is a path $P_{n-1}$ and one other vertex $v$ (we call
it the \emph{central vertex}) joined by an edge with every vertex of the path.
See Figure \ref{fig:vejar} for an example.
\end{definition}
\begin{figure}
\centering
\includegraphics[scale=0.8]{obrazky/vejar.pdf}
\caption{A fan graph $F_6$.}
\label{fig:vejar}
\end{figure}
\begin{theorem} \label{thm:fan}
A fan graph $F_n$ does not have a canonical ESD labeling if and only if $n \ge 8$.
\end{theorem}
\begin{proof}
Note that $F_n$ for $n$ up to 7 has a canonical ESD labeling, as we can see on Figure \ref{fig:vejar_male}. It is obvious that $F_2$ and $F_3$ have canonical ESD labelings.
From Theorem \ref{thm:nec} we know that we have at most $2n-3$ different
edge-weights. Since a fan graph of order $n$ has exactly $2n-3$ edges we need
to use every possible edge-weight from the set $\{3,\ldots,2n-1 \}$ exactly once.
The edge-weights 3 and 4 can be obtained in exactly one possible way. In the first
case on an edge with endpoints labeled 1 and 2, in the second case on an edge
with endpoints 1 and 3. The edge-weight 5 can be obtained in two ways. Either
as the weight of an edge with endpoints $2$ and $3$ or as the weight
on an edge with endpoints $1$ and $4$. We get two possible subgraphs $S_1$ and
$S_2$.
By a similar analysis, one can get the labeled subgraphs $S_3$ and $S_4$.
Hence, exactly one of the labeled subgraphs $S_1$ or $S_2$ has to be in $F_n$
and, analogously, one of the $S_3$ and $S_4$ as well.
However, in all graphs $S_i, i \in \{1,\ldots,4 \}$, one of its vertices has
to be the central vertex. Since $n \ge 8$, we see that the minimum possible
label in $S_3$ and $S_4$ is $5$. Also, the maximum label on $S_1$ and $S_2$ is 4 . Therefore, we cannot properly label the central
vertex and the theorem follows.
\qed
\end{proof}
\begin{figure}
\centering
\includegraphics[scale=1.1]{obrazky/vejar_sit.pdf}
\caption{The subgraphs from the proof of Theorem \ref{thm:fan}.}
\label{fig:vejar-sit}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[scale=1]{obrazky/vejar_male.pdf}
\caption{Canonical ESD labelings for $F_4, F_5, F_6$, and $F_7$.}
\label{fig:vejar_male}
\end{figure}
\subsection{Complete bipartite graphs}
We need to introduce a notion of isomorphism for vertex labelings.
\begin{definition}
Vertex labelings $\phi_1$ and $\phi_2$ on $G$ are isomorphic if there
exists an automorphism $f$ of $G$ such that $\phi_1(v) = \phi_2(f(v))$ for
every $v \in V(G)$.
\end{definition}
We will prove the following theorem, covering all cases for complete
bipartite graphs.
\begin{theorem} \label{thm:kqr}
Let $K_{p,q}$ be a complete bipartite graph on $n=p+q$ vertices, $p \leq q$, then the following holds.
\begin{enumerate}
\item For $p, q > 2$ there is no ESD labeling on $K_{p,q}$.
\item If $p = 2$, then there exists exactly one possible ESD labeling
up to isomorphism.
\item If $p = 1$, then every canonical labeling is an
ESD labeling.
\end{enumerate}
\end{theorem}
\begin{proof}
\begin{enumerate}
\item Suppose for a contradiction that we have some
canonical ESD labeling $\phi$. Denote the parts of $K_{p,q}$ by $P$ and $Q$.
We will divide the proof into two cases.
\begin{itemize}
\item There exist two vertices $v_1, v_2,$ in $P$
and two vertices $u_1,u_2$ in $Q$ such that
$\phi(v_2) = \phi(v_1) +1$ and $\phi(u_2) = \phi(u_1)+ 1.$
Then $w_\phi(v_1u_2) = w_\phi(v_2u_1)$, and we get a contradiction.
\item There exists a part (without loss of generality $P$) such that $\phi(v_1) \neq \phi(v_2)+1$ for every $v_1,v_2 \in P$.
Since $P$ is of size at least 3, there exist two vertices $v'_1, v'_2 \in P$ with labels smaller than $n$. Thus there exists a vertex $u_1 \in Q$ with label $\phi(v'_1)+1$ and $u_2 \in Q$ with label $\phi(v'_2)+1$.
Then $w_\phi(v'_1u_2) = w_\phi(v'_2u_1)$, a contradiction.
\end{itemize}
\item We denote the vertices of the part of the size 2 as $v_1,v_2$. The vertices of the other part will be $u_1,\ldots,u_q$. Let $\psi$ be
a vertex labeling of $K_{2,q}$ defined as follows:
\begin{itemize}
\item $\psi(v_1) = 1$,
\item $\psi(v_2) = n$,
\item $\psi(u_i) = i+1$ for $i \in \{1,\ldots,q \}$.
\end{itemize}
Observe that $\psi$ is indeed a canonical ESD labeling.
For $q = 2$, one can easily check that this is the only
canonical ESD labeling up to isomorphism.
Now, for a contradiction, assume that a canonical ESD
labeling $\psi'$, nonisomorphic to $\psi$, exists.
Furthermore, $n > 4$, and we can assume that $\psi'(v_1) < \psi'(v_2)$.
Either $\psi'(v_1) \neq 1$ or $\psi'(v_2) \neq n$.
We distinguish two cases.
\begin{enumerate}
\item It holds that $\psi'(v_2) = \psi'(v_1) + 1$.
Since $n > 4$, we can find two vertices $a_1,a_2$ in the other part
such that $\psi'(a_2) = \psi'(a_1) + 1$. Similarly as in case (1)
of this theorem, $w_{\psi'}(v_1a_2) = w_{\psi'}(v_2a_1)$ and we get a contradiction.
\item It holds that $\psi'(v_2) \neq \psi'(v_1) + 1$.
Then there exist two distinct vertices $u_j,u_k \in \{u_1,\ldots,u_q \}$
such that one of the following holds.
Either $\psi'(u_j) = \psi'(v_1) + 1$ and $\psi'(u_k) = \psi'(v_2) + 1$, or
$\psi'(u_j) = \psi'(v_1) - 1$ and $\psi'(u_k) = \psi'(v_2) - 1$.
In both cases $w_{\psi'}(v_1u_k) = w_{\psi'}(v_2u_j)$ and we are done.
\end{enumerate}
We conclude that no such $\psi'$ exists.
\item Every edge in a canonical labeling of $K_{1,q}$ has a unique sum since every edge is incident to the central vertex of degree $q$.
\end{enumerate}
\qed \end{proof}
We note that the first part of Theorem \ref{thm:kqr} can be proved
by using Theorem \ref{thm:nec} but we think that our proof is more clear.
\subsection{Trees}
We already showed that paths and stars are ESD graphs. The following theorem
solves the general case of trees.
\begin{theorem} \label{thm:trees}
Every tree has a canonical ESD labeling.
\end{theorem}
\begin{proof}
Let $T$ be an $n$-vertex tree with root in $v_1 \in V(T)$.
We will denote by $v_1,\ldots,v_n$ an ordering of vertices visited
in a breadth-first search on $T$, starting in $v_1$.
We define a labeling $\phi$ as $\phi(v_k) := k, \forall v_k \in V(T)$. We
want to show that $\phi$ is a canonical ESD labeling.
Consider some vertex $v_i, i>1,$ and its parent $v_j$. Denote by $T'$ the
tree induced by vertices $v_1,\ldots,v_{i-1}$. See Figure \ref{fig:tree}
for an illustration. We claim that the following
holds:
$$w_\phi(v_iv_j) > w_\phi(v_av_b),\, \forall v_av_b\in E(T').$$
By the level of a vertex we mean its distance to root vertex $v_1$.
Without loss of generality, assume that $a < b$. We distinguish these cases.
\begin{itemize}
\item The edge $v_av_b$ has both endpoints on a level lower or equal
to the level of $v_j$. Then $a < j$ and $b < i$ and from that $a+b < i+j$.
\item If $v_a = v_j$, then $v_j$ is the common parent of $v_b$ and $v_i$.
Thus $b < i$ and from that $b+j < i+j$.
\item The vertex $v_a$ is on the same level as $v_j$ and $v_a \neq v_j$.
Then $a < j$ and $b < i$, implying that $a+b < i+j$.
\end{itemize}
We proved the claim and the theorem follows.
\qed
\end{proof}
\begin{figure}
\centering
\includegraphics[scale=0.7]{obrazky/strom.pdf}
\caption{An illustration of situation in Theorem \ref{thm:trees}.}
\label{fig:tree}
\end{figure}
\subsection{Cycles}
\begin{theorem} \label{thm:cycles}
Every cycle graph $C_n$ is an ESD graph.
\end{theorem}
\begin{proof}
Let us denote the vertices of $C_n$ as $v_1,\ldots,v_n$ in a circular order. We
distinguish two cases:
\begin{enumerate}
\item If $n$ is even, then we assign labels as follows:
\begin{itemize}
\item $\phi(v_i) = i$ for all $i \in \{1,\ldots,n-2\}$,
\item $\phi(v_{n-1}) = n$,
\item $\phi(v_n) = n - 1$.
\end{itemize}
Weights of the edges $v_iv_{i+1}$ for $i \in \{1,\ldots,n-3\}$
are odd integers $3,5,\ldots,2n-5$. The weight of the edge
$v_{n-1}v_n$ is $2n-1$ and therefore is odd as well. The remaining
edges will be even; $w_\phi(v_nv_1) = n$ and $w_\phi(v_{n-2}v_{n-1}) = 2n-2$.
We conclude that the edge-weights are unique.
\item If $n$ is odd we assign labels as follows:
\begin{itemize}
\item $\phi(v_i) = i$ for all $i \in \{1,\ldots,n\}$.
\end{itemize}
The weights of the edges between $v_iv_{i+1}$ for $i \in \{1,\ldots,n-1\}$
will be odd integers $3,5,\ldots,2n-1$. The weight of the edge
$v_1v_n$ is equal to $n+1$ and therefore it is even. Again, all edge-weights
are unique and we get a canonical ESD labeling.
\end{enumerate}
\qed
\end{proof}
\subsection{Generalized sunlet graphs}
We recall that a graph is \emph{unicyclic} if it contains exactly one cycle.
\begin{definition}
A \emph{generalized sunlet graph} $S_k^p$ is a unicyclic graph obtained by taking a cycle graph
$C_k$, with $V(C_k) = \{c_1,\ldots,c_k \}$, and joining path graphs $R_i, i \in \{1,\ldots,k \}$ of order $p$ to this cycle so that
one of the endpoints of $R_i$ is identified with~$c_i$.
\end{definition}
\begin{theorem}
Let $S_k^p$ be a generalized sunlet graph.
If $k$ is odd and $p$ is even, then $S_k^p$ has a canonical ESD labeling.
\end{theorem}
\begin{proof}
We denote the vertices of $S_k^p$ in the following way.
\begin{itemize}
\item Vertices on the cycle are $v_1, v_{p + 1}, v_{2p+1}, \ldots, v_{(k-1)p+1}$.
\item Vertices on the path joined to the vertex $v_{ip+1}$ are
consecutively\\ $v_{ip+1}, \ldots, v_{(i+1)p}$, for $1 \leq i \leq k$.
\end{itemize}
We define a vertex labeling $\phi$ as $ \phi(v_i) := i$. We claim that $\phi$
is a canonical ESD labeling. All edge-weights on attached paths are odd,
because we get them as a sum of two consecutive numbers. Furthermore, all
edge-weights on a path joined to vertex $v_{ip+1}$ are smaller than
edge-weights on a path joined to vertex $v_{(i+1)p+1}$. Thus all edge-weights on
paths are distinct. All edge-weights on the cycle expect for the edge
$v_1v_{(k-1)p+1}$ are in the form $k'p+2$ for some $k' \in \mathbb{N}$. Thus they are all even
and distinct.
It remains to show that the edge $ v_1v_{(k-1)p+1}$ has an edge-weight
different from all others. For a contradiction we assume that the edge-weight
$(k-1)p+2$ was already used. It is even, so it can be only used on the cycle.
Thus, $k-1$ must be a sum of two distinct consecutive natural numbers. That
gives a contradiction, because $k-1$ is even.
\qed
\end{proof}
For the other parity conditions we were not able to prove that there is always
an ESD labeling. Thus we leave as an open problem to determine if all
generalized sunlet graphs have a canonical ESD labeling. Small examples
suggest that it might be true.
\begin{theorem}
Let $S_k^p$ be a generalized sunlet graph.
If $k$ and $p$ are odd or $k$ is even and $p$ is odd or even, then $S_k^p$ has an ESD labeling with label set $L$ of size $(p+1)k - 2$.
\end{theorem}
\begin{proof}
In both cases of parity of $k$, the unique cycle in $S_k^p$ will be labeled in
the same way as in Theorem \ref{thm:cycles}. Observe that the greatest edge-
weight on the edges of cycle is $2k-1$.
The rest of the vertices is labeled by the following procedure. Start with
label $i := 2k-1$ and label by $i$ an unlabeled vertex which is adjacent to
the vertex with the minimum label. Increment $i$ by one and repeat the step.
We see that in every step we get one new edge-weight. Furthermore, this
edge-weight is always greater than any previous edge-weight created during this
procedure and all these edge-weights are greater than any edge-weight on an
edge in the cycle. Thus, the resulting labeling is ESD. Furthermore, we labeled
the cycle with $k$ labels with $1,\ldots,k$ and then the remaining $p(k-1)$ vertices with labels
$2k-1,\ldots, (p-1)k - 2$. This implies that the set of labels $L$ is of size $(p-1)k-2$.
\qed
\end{proof}
\subsection{Grids}
\begin{definition}
A $k\times l$ grid graph $G_{k,l}$ is the Cartesian product of path graphs $P_k$ and $P_l$.
\end{definition}
\begin{theorem}
Let $G_{k,l}$ be a grid graph. If $k$ or $l$ is even then $G_{k,l}$ has a canonical ESD labeling.
\end{theorem}
\begin{proof}
Without loss of generality assume that $k$, the number of
columns, is even. Let us denote the vertices in the $i$-th row by $v_{(i-1)k+1},
\ldots, v_{ik}$ for every $i \in \{1, \ldots, l\}$. We define a canonical vertex
labeling $\phi$ as $\phi(v_i) := i$. We want to show that $\phi$ is an
ESD labeling on $G_{k,l}$.
The graph $G_{k,l}$ with labeling $\phi$ has the following edge-weights:
\begin{itemize}
\item $2(i-1)k + 3, \ldots, 2ik-1$ in the $i$-th row for every $1 \leq i \leq l$,
\item $2ik + 3, \ldots, 2(i+1)k-1$ in the $(i+1)$-th row for every $0 \leq i \leq l-1$,
\item $(2i-1)k+2, \ldots, (2i+1)k$ on edges between the $i$-th and the $(i+1)$-th row $1 \leq i \leq l-1$.
\end{itemize}
All edge-weights on rows are odd and all edge-weights in the $i$-th row are smaller than all edge-weights in the $(i+1)$-th row. A similar argument holds for all edge-weights in columns. This concludes the proof.
\qed
\end{proof}
\iffalse
\subsection{Friendship graphs}
\begin{definition} \cite{dynamic}
A friendship graph $F_r$ is a graph obtained by taking $r$ copies of the cycle graph $C_3$ with a vertex in common.
\end{definition}
\begin{theorem}
Let $F_r$ be a friendship graph and $r \geq 8$. Then $F_r$ does not have any canonical ESD labeling.
\end{theorem}
\begin{proof} \todo{}
We will denote the central vertex (the vertex of degree $2r$) $v$. For a contradiction we assume that $F_r$ has a canonical ESD labeling $\phi$.
We define $s := \phi(v)$. The edge-weights on the edges incident with $s$ are $\{s+1, \ldots, s+n\} \setminus \{2s\}$.
We will distinguish two cases. In the first case, let $s \geq \lceil n/2 \rceil$.
\todo{Uvažme, s čím amôžu susediť niektoré značky tak, aby sme sa vyhli použitým váham na okolí s.}
Label $s-i$ can be incident only with labels $1,\ldots, i$ and $s+i$ for $i \in \{1, \ldots 5\}$, if they exist. If we choose the labels
that will be incident with $s-1, \ldots, s-4$, then in every case we do not have a label which can be incident with $s-5$.
This is a contradiction because we assume that $\phi$ is an ESD labeling.
In the other case, if $s \leq \lfloor n/2 \rfloor$, we can use similar argument for labels $s+1, \ldots, s+5$ and we are done.
\qed
\end{proof}
\fi
\subsection{Complete graphs}
From Theorem \ref{thm:nec} it is clear that complete graphs $K_n$ for $n > 3$
do not have a canonical ESD labeling. However, the following theorem provides
a simple way how to find an ESD labeling. We recall that \emph{Fibonacci sequence}
is defined as $F_0 := 0, F_1 := 1$, and $F_n = F_{n-1} + F_{n-2}$ for $n > 1$.
We note that the following theorem implies that for any $n$-vertex graph,
$F_{n+1}$ labels suffice to construct an ESD labeling.
\begin{theorem}
There exists an ESD labeling with $F_{n+1}$ labels for every complete graph $K_n$.
\end{theorem}
\begin{proof}
For given $K_n$ we label its vertices $\{v_1, \ldots, v_n\}$ by function
$\phi_n$, defined as $\phi_n(v_i) := F_{i+1}$.
We show that this is an ESD labeling by induction. We see that for $K_1$
and $K_2$, $\phi_1$ and $\phi_2$ are clearly ESD labelings. Now we want to
prove that $\phi_n$ is an ESD labeling. We see that $v_1,\ldots,v_{n-1}$
are labeled as in $\phi_{n-1}$. The largest possible sum on an edge in
$\phi_{n-1}$ is $F_n+F_{n-1}$. The only new label in $\phi_n$ is $F_{n+1}$
and the minimum possible sum on an edge incident with $v_n$ is
$F_{n+1}+F_2 = F_n+F_{n-1}+1$. Thus, assuming that $\phi_{n-1}$ is an ESD
labeling, $\phi_n$ is an ESD labeling as well.
\qed
\end{proof}
\iffalse
We note that the problem of finding $l$ sufficiently large so that we have a
ESD labeling with $l$ labels on complete graphs is connected to finding
precise bounds on the shortest Sidon sequence problem.
The survey of O’Bryant
\cite{o2004complete} features a table, reproduced here in Figure 5, giving upper bounds on
how big $l$ must be for a graph $K_n$, with $n$ up to 13. This is captured
in Table \ref{tab:sidon}.
\begin{table}
\centering
\setlength{\tabcolsep}{4pt}
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|l|l|}
\hline
$n$ & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13\\ \hline
min. $l$& 1 & 3 & $\leq$6 & $\leq$11 & $\leq$17 & $\leq$25 & $\leq$34 & $\leq$44 & $\leq$55 & $\leq$72 & $\leq$85 & $\leq$106 \\ \hline
\end{tabular}
\vspace{0.2cm}
\caption{A table showing bounds on the minimum $l$ such that $K_n$ has an ESD labeling
with $l$ labels.}
\label{tab:sidon}
\end{table}
\fi
\section{Games with ESD labeling}
Tuza in his paper \cite{tuza2017graph} emphasized that only few papers on graph labeling games exist. He defined a new game from ESD labeling.
\begin{definition}
We call a vertex of graph \emph{free} if it is not labeled yet.
\end{definition}
\begin{definition}
Let $G=(V,E)$ be a graph and $L=\{1,\ldots,l \}$ its set of labels. Alice and Bob are two players
who alternate after every move. Alice starts. In each move, player chooses a free vertex of $G$ and assigns
an unused label to it. The move is \emph{legal} if the resulting edge-weights are unique.
The game ends if there is no legal move possible or an ESD labeling is created. Alice wins if
an ESD labeling is created, otherwise Bob wins.
We say that an ESD labeling game is \emph{canonical} on $G$ if $|L| = |V(G)|$.
\end{definition}
One can also define other variants of this game. For example, Bob can be the starting one.
Also, our definition of game is a Maker-Breaker type of game, but it is possible to define
Achievement and Avoidance type of this game as well.
\begin{proposition} \label{thm:bob}
If a graph $G$ does not have a canonical ESD labeling then Bob has a winning strategy in
the canonical game on $G$.
\end{proposition}
\begin{proof}
If a graph $G$ does not have a canonical ESD labeling then Alice can not make any canonical ESD labeling and Bob eventually wins.
\qed
\end{proof}
\begin{theorem}
Alice wins every canonical game on a star $S_n$.
\end{theorem}
\begin{proof}
We already proved in Theorem \ref{thm:kqr} that every canonical vertex labeling on a star graph is edge-sum distinguishing. Thus Alice wins every game regardless on the course of the game.
\qed
\end{proof}
\begin{theorem}
Bob wins every canonical game on
a complete bipartite graph $K_{p,q}$, $p \leq q$, where $p = 2$.
\end{theorem}
\begin{proof}
We recall Theorem \ref{thm:kqr}. The graph $K_{p,q}$, $p \leq q$, where $p = 2$, needs
to have labels $1$ and $p+q$ on the smaller part. Thus a winning strategy for Bob is
to assign a label $w$, such that $1 < w < p+q$, on a free vertex of the smaller part in
his first move. Now it is not possible to build a canonical ESD labeling and Bob wins.
\qed
\end{proof}
Tuza also asked \cite[Problem 3.1]{tuza2017graph} the following question:
Given $G=(V, E)$, for which values of $l$ can Alice win the edge-sum
distinguishing labeling game? We partially answer this question by the
following theorem.
\begin{theorem} \label{thm:odhad}
Let $G$ be a graph, $\Delta$ its maximum degree, and $L$ its set of labels. If $|L| \geq (\Delta^2+1)n + \Delta {n-1 \choose 2}$, then Alice has a wining
strategy.
\end{theorem}
\begin{proof}
For each vertex $v$ of $G$, define a set $S_v$ as the set of labels available for $v$.
In the beginning of every game, $S_v = L$ for every $v \in V(G)$.
Our goal is to build a winning strategy for Alice. In $k$-th move, a player
assigns to a free vertex $v$ some label $\phi(v) \in S_v$. We update
the set of labels in the following three steps right after the player's choice.
\begin{enumerate}
\item We delete $\phi(v)$ from $S_u$ for every $u \in V(G)$. This label
cannot be used twice, since ESD labeling is an injective mapping.
\item For every free vertex $y$ incident to $v$ we delete all labels
$l_{y,e}$ such that $l_{y,e} + \phi(v) = w_\phi(e)$ for some
edge $e$ with both endpoint vertices labeled and incident with $v$. In this process, we delete at most ${k-1 \choose 2}$ labels
from $S_y$.
\item For every free vertex $z$ and for every vertex $z' \in N(z)$,
such that $z'$ is already labeled, we delete from $S_z$
all labels $l'$ such that
$$l' + \phi(z') = w_\phi(vv'),\,\,\forall v' \in N(v).$$
Within these steps, we delete at most $\Delta^2$ labels from label set of
every free vertex.
\end{enumerate}
If the label set for every free vertex is nonempty before every move, Alice
wins. Let us count how many labels are deleted in course
of the game for every free vertex.
\begin{itemize}
\item We delete at most $n-1$ labels through all first steps.
\item We delete at most $\Delta {n-1 \choose 2}$ labels through all second
steps.
\item We delete at most $\Delta^2n$ labels in third steps.
\end{itemize}
Summarized, we delete at most $ (\Delta^2+1)n + \Delta {n-1 \choose 2} - 1$ labels.
If we have one extra label available, we can always find a label for a free vertex
and our bound is proved. Note an important fact that it does not matter how Bob
plays and the resulting labeling is ESD.
\qed
\end{proof}
Observe that this theorem also gives us a bound on the size of label set for general
graphs. This follows by taking Proposition \ref{thm:bob} into account.
Also, by a similar analysis, one can obtain the following theorem for path graphs.
\begin{theorem}
Let $P_n$ be a path graph on $n$ vertices. If $|L| \geq 5n$, then Alice
wins every game on $P_n$.
\end{theorem}
\section{Concluding remarks}
We studied a new type of graph labeling, introduced by Tuza, which is similar
to magic (and antimagic) labelings, harmonious labelings and has a relation to
the Sidon sequences. We would like to highlight our main results.
\begin{itemize}
\item We proved that trees, cycles and complete bipartite graphs with one part of size 2 have a canonical ESD labeling.
\item We proved that in some cases grid graphs and generalized sunlet graphs
do have a canonical ESD labeling.
\item We showed that
fan graphs and complete bipartite graphs with both parts of size at least 3 do not have a canonical ESD labeling.
\item We studied a Maker-Breaker type of game, applied our previous results and
derived a general bound on number of labels such that Maker wins the game.
\end{itemize}
\paragraph{Open problems.} Aside from Tuza's original game-oriented
problems proposed in \cite{tuza2017graph}, we emphasize the following question,
arising from the results in this paper.
\begin{problem}
What is the maximum possible number of edges for $n$-vertex connected graphs
so that every graph with such number of edges has a canonical ESD labeling?
\end{problem}
From Theorem \ref{thm:trees} we see that to answer this question one needs to
resolve the case of unicyclic graphs which is now only partially solved.
\section*{Acknowledgment}
Both authors were supported by the grant SVV–2017–260452. Jan Bok was
supported by the Center of Excellence - ITI (P202/12/G061 of GA\v{C}R). Nikola
Jedličková was supported by the Student Faculty Grant of the Faculty of
Mathematics and Physics, Charles University.
Both authors would like to thank Robert Šámal for his feedback and suggestions
regarding the paper. We would also like to thank the anonymous referee for
valuable advices that led to substantial improvements of the paper.
\bibliographystyle{plain}
|
{
"timestamp": "2019-01-16T02:02:48",
"yymm": "1804",
"arxiv_id": "1804.05411",
"language": "en",
"url": "https://arxiv.org/abs/1804.05411"
}
|
\section{Introduction}
Gradient-based optimization algorithms usually have the following update form,
\begin{align}
\bm{\theta} \leftarrow \bm{\theta} - \eta \mathbf{G}^{-1} \nabla_{\bm{\theta}}f(\bm{\theta})
\label{gradient_descent}
\end{align}
where $\bm{\theta} \in \mathbb{R}^d$ is the model parameters, $\eta$ is a learning rate, $\mathbf{G}$ is a $d\times d$ non-singular matrix and $f(\cdot)$ is the loss function. If $\mathbf{G}$ is the identity matrix, (\ref{gradient_descent}) is the gradient descent method.
In practice, gradient descent often converges slowly and its performance depends critically on how $f(\bm{\theta})$ is parameterized. That is,
if one chooses $\bm{\theta} = g(\bm{\xi})$,
then the behavior of $\bm{\theta} \leftarrow \bm{\theta} - \eta \nabla_{\bm{\theta}}f(\bm{\theta})$ can be significantly different from that of $\bm{\xi} \leftarrow \bm{\xi} - \eta \nabla_{\bm{\xi}}f(g(\bm{\xi}))$. (see an example in~\textsection~\ref{sec:parameter_dependency}).
\newpage
To accelerate gradient descent and deal better with the parameterization issue, one can resort to second order optimization methods,
in which the second order derivative (Hessian) of $f(\cdot)$ or second order statistics of gradients is incorporated in $\mathbf{G}$.
In the Newton method, $\mathbf{G}$ is chosen as the Hessian matrix of $f(\cdot)$ according to
\begin{align}
\mathbf{G}_{ij} = \frac{\partial^2 f(\bm{\theta})}{\partial \theta_i \partial \theta_j}\;.
\end{align}
The Newton method approximates the loss function locally by a quadratic
function, in which the Hessian matrix measures the curvature of the loss function. This in turn yields better descent directions than the ones obtained by solely considering the gradient directions. In fact, under mild conditions, the Newton method converges to a local minimum at a quadratic rate while gradient descent has a linear convergence
rate~\cite{nocedal2006numerical}. Besides, the Newton method is invariant to affine re-parameterization (see the derivation in Appendix).
The concept of natural gradient~\cite{amari1998natural} provides another perspective by conditioning gradient step on the KL-divergence variations induced by the model output distribution. That is, any given step in the natural gradient method produces an equal amount of variation in terms of the KL-divergence.
It is shown that the method of natural gradient is invariant to the parameterization of the model~\cite{pascanu2013revisiting}.
Approximating the KL-divergence variations by its second order Taylor series, one arrives at a form that looks like the Newton method, albeit with
$\mathbf{G}$ becoming the Fisher information matrix
\begin{align}
\label{eqn:fisher_information}
\mathbf{G} = \mathbb{E}_{\mathbf{x}\sim p(\mathbf{x}|\bm{\theta})} \left[ \frac{\partial \log p(\mathbf{x}|\bm{\theta})}{\partial \bm{\theta}}\frac{\partial \log p(\mathbf{x}|\bm{\theta})}{\partial \bm{\theta}}^\intercal \right]
\end{align}
where $p(\mathbf{x}|\bm{\theta})$ is the probabilistic model we try to optimize.
It has been shown that the Fisher information matrix can be viewed as an approximated Hessian matrix~\cite{martens2014new}.
Various studies suggest that natural gradient can have a better convergence rate than that of the gradient descent method (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, blind signal separation~\cite{amari1996new}, reinforcement learning~\cite{peters2008natural} and variational inference \cite{honkela2010approximate}).
\newpage
For stochastic optimization, the AdaGrad algorithm~\cite{duchi2011adaptive} incorporates the previously computed gradients to guide its current descent direction. In AdaGrad, $\mathbf{G}$ is the matrix square root of an outer product matrix of gradient vectors
\begin{align}
\mathbf{G} = \left(\sum_{\tau=1}^t \mathbf{g}_{\tau} \mathbf{g}_{\tau}^\intercal\right)^{\frac{1}{2}}
\end{align}
where $\mathbf{g}_{\tau} = \nabla_{\bm{\theta}}f_{\tau}(\bm{\theta}_{\tau})$ is the gradient estimated from a mini-batch of data at step $\tau$ and $t$ is the current step. The relationship between the matrix $\sum_{\tau=1}^t \mathbf{g}_{\tau} \mathbf{g}_{\tau}^\intercal$
and the Hessian matrix is discussed in \cite{hazan2007logarithmic}.
Despite their intriguing properties and fast convergence rates, the aforementioned second order optimization methods become overwhelming for high-dimensional $\bm{\theta}$.
This is because construction and inversion of $\mathbf{G}$ have a time complexity of $O(d^3)$.
As such, various studies resort to approximation techniques when it comes to high-dimensional problems.
A simple, yet effective approximation is the diagonal approximation, where one only keeps the diagonal elements of $\mathbf{G}$. A classic method is the Levenberg-Marquardt
algorithm~\cite{levenberg1944method,marquardt1963algorithm} which uses the diagonal approximation of the Hessian matrix. Its stochastic version for training neural networks is proposed in~\cite{becker1988improving}. The diagonal approximation of AdaGrad and its variants such as AdaDelta~\cite{zeiler2012adadelta}, RMSprop~\cite{hinton2012rmsprop} and Adam~\cite{kingma2014adam}
have seen increasing popularity in training neural networks recently.
The diagonal approximation of $\mathbf{G}$ amounts to setting an individual learning rate for each parameter. This balances the scale of parameters and often accelerates the convergence.
Nevertheless, one wonders whether disregarding the correlation between the parameters, as captured by the off-diagonal elements of
$\mathbf{G}$, has negative effects on the convergence speed. We will show that the answer to this question is a firm yes. A simple example is given in \textsection \ref{sec:parameter_dependency}. Experiments in \textsection \ref{sec:exp} also show the benefits of capturing off-diagonal elements of
$\mathbf{G}$.
As such, in this paper we propose a new matrix approximation technique. The main idea is to split $\mathbf{G}$ into smaller blocks and approximate each block by one or two numbers. This, in return, will enable us
to approximate the inverse of $\mathbf{G}$ by inverting small matrices, drastically reducing the time complexity while benefiting from the off-diagonal elements of $\mathbf{G}$. We incorporate our method into AdaGrad for training deep neural networks and empirically observe that the resulting algorithm outperforms AdaGrad with diagonal approximation in terms of the convergence speed.
\newpage
\section{Parameter Dependency}
\label{sec:parameter_dependency}
\begin{figure}[t!]
\centering
\subfloat[Original]{
\includegraphics[width=0.45\linewidth]{fig/contour_qp1.pdf}}
\subfloat[Transformed]{
\includegraphics[width=0.45\linewidth]{fig/contour_qp2.pdf}}
\caption{The effect of parametrization in convergence behavior of gradient descent (with line search). (a) Gradient descent moves in a zig-zag fashion
as it approaches to the minimum. (b) By reparameterization the same problem, gradient descent finds the minimum in only one step (see the text for more details).}
\label{fig:gd}
\end{figure}
To show how the dependency of parameters affects the speed of gradient descent, consider a simple quadratic minimization problem
\begin{align}
\min_{\bm{\theta}}f(\bm{\theta}) = \frac{1}{2}\bm{\theta}^\intercal\mathbf{Q}\bm{\theta}\;
\end{align}
where
\begin{align}
\bm{\theta} = [\theta_1, \theta_2]^\intercal, \quad \mathbf{Q} =
\begin{bmatrix}
1.0 & -0.8 \\
-0.8 & 1.0
\end{bmatrix}.
\end{align} The eigenvalues of $\mathbf{Q}$ are $\lambda_1 = 1.8$ and $\lambda_2 = 0.2$.
The minimum is obtained at $\bm{\theta} = \mathbf{0}$.
We note that the scale of $\theta_1$ equals that of $\theta_2$. However, the gradient of $\theta_1$ depends on the value of $\theta_2$ and vice versa as
\begin{align}
\nabla_{\bm{\theta}} f(\bm{\theta}) = \mathbf{Q}\bm{\theta}\;.
\end{align}
The gradient descent method, even with the optimal step size for each iteration, converges with a rate of
$(\frac{\lambda_1 - \lambda_2}{\lambda_1 + \lambda_2})^2$ with such a parameterization.
However, with eigendecomposition $\mathbf{U}\mathbf{\Lambda}\mathbf{U}^\intercal = \mathbf{Q}$ and reparameterization with $\bm{\xi} = [\xi_1, \xi_2]^\intercal = \mathbf{\Lambda}^{\frac{1}{2}}\mathbf{U}\bm{\theta}$, then the optimization problem becomes
\begin{align}
\min_{\bm{\xi}}f(\bm{\xi}) = \frac{1}{2} \bm{\xi}^\intercal\bm{\xi}\;.
\end{align}
The gradient of $\xi_1$ is not dependent on $\xi_2$ anymore and vice versa. As a result, gradient descent with the optimal step size converges to the minimum solution in only one iteration (see Figure~\ref{fig:gd} for an illustration).
The take home message from this textbook example is that \emph{algorithms that exploit the dependency between the parameters can prevail
over the ones that totally ignore such information}.
\newpage
\section{Block Mean Approximation}\label{sec:BMA}
We need the following definitions before introducing our proposed block mean approximation (BMA).
\begin{notation*}
$\mathbf{M}^{ij}$ denotes the ($i,j$)-th block of the matrix $\mathbf{M}$. $\mathbf{M}^{ij}_{mn}$ denotes the ($m,n$)-th element of the matrix $\mathbf{M}^{ij}$.
\end{notation*}
\begin{defn}
For a diagonal matrix
\begin{align}
\mathbf{\Lambda} =
\begin{bmatrix}
\lambda_{1} & 0 & ... & 0 \\
0 & \lambda_{2} & ... & 0 \\
... & ... & ... & ... \\
0 & 0 & ... & \lambda_{L}
\end{bmatrix}\;,
\end{align}
its diagonal expansion matrix with partition vector $\mathbf{s} = (s_1,...,s_L)$ is
\begin{align}
\bar{\mathbf{\Lambda}} =
\begin{bmatrix}
\bar{\mathbf{\Lambda}}^{11} & 0 & ... & 0 \\
0 & \bar{\mathbf{\Lambda}}^{22} & ... & 0 \\
... & ... & ... & ... \\
0 & 0 & ... & \bar{\mathbf{\Lambda}}^{LL}
\end{bmatrix}
\end{align}
where each $\bar{\mathbf{\Lambda}}^{ii}$ is a diagonal matrix of size $s_i \times s_i$ with fixed diagonal elements of $\lambda_i$.
\end{defn}
\begin{defn}
For a matrix
\begin{align}
\mathbf{B} =
\begin{bmatrix}
b_{11} & b_{12} & ... & b_{1L} \\
b_{21} & b_{22} & ... & b_{2L} \\
... & ... & ... & ... \\
b_{L1} & b_{L2} & ... & b_{LL}
\end{bmatrix}\;,
\end{align}
its full expansion matrix with partition vector $\mathbf{s} = (s_1,...,s_L)$ is
\begin{align}
\bar{\mathbf{B}} =
\begin{bmatrix}
\bar{\mathbf{B}}^{11} & \bar{\mathbf{B}}^{12} & ... & \bar{\mathbf{B}}^{1L} \\
\bar{\mathbf{B}}^{21} & \bar{\mathbf{B}}^{22} & ... & \bar{\mathbf{B}}^{2L} \\
... & ... & ... & ... \\
\bar{\mathbf{B}}^{L1} & \bar{\mathbf{B}}^{L2} & ... & \bar{\mathbf{B}}^{LL}
\end{bmatrix}
\end{align}
where each block $\bar{\mathbf{B}}^{ij}$ is a matrix of constant value $b_{ij}$ and has $s_i$ number of rows.
\end{defn}
The above definitions are illustrated in Figure \ref{matrices}.
\begin{figure}[h!]
\centering
\subfloat[$\mathbf{\Lambda}$]{
\input{fig/v.tex}}
\hspace{0.1in}
\subfloat[$\bar{\mathbf{\Lambda}}$]{
\input{fig/V.tex}}
\hspace{0.1in}
\subfloat[$\mathbf{B}$]{
\input{fig/b.tex}}
\hspace{0.1in}
\subfloat[$\bar{\mathbf{B}}$]{
\input{fig/B.tex}}
\caption{Expansion matrices. (a) Diagonal matrix $\mathbf{\Lambda}$. (b) Diagonal expansion of
$\mathbf{\Lambda}$. (c) Full matrix $\mathbf{B}$. (d) Full expansion of $\mathbf{B}$. The partition vector in both cases
is $\mathbf{s}=(2,5,3)$.}
\label{matrices}
\end{figure}
\begin{defn}
The block mean approximation of a matrix $\mathbf{M}$ with the partition vector $\mathbf{s}$ is
\begin{align}
\widehat{\mathbf{M}} = \bar{\mathbf{\Lambda}} + \bar{\mathbf{B}} \approx \mathbf{M}
\end{align}
where $\bar{\mathbf{\Lambda}}$ and $\bar{\mathbf{B}}$ are the diagonal and full expansion matrices with partition vector $\mathbf{s}$, respectively.
\end{defn}
\begin{figure}[t!]
\centering
\subfloat[Original]{
\frame{
\includegraphics[width=0.35\linewidth]{fig/M.png}}}
\hspace{0.2in}
\subfloat[Approximate]{
\frame{
\includegraphics[width=0.35\linewidth]{fig/G.png}}}
\caption{Block mean approximation of a matrix.}
\label{fig:bma}
\end{figure}
The block mean approximation, as illustrated in Figure \ref{fig:bma}, allows one to efficiently store a big matrix as only $\mathbf{\Lambda}$, $\mathbf{B}$ and a partition vector $\mathbf{s}$ need to be kept.
Given a square matrix, its optimal block mean approximation under Frobenius norm is given by the following.
\begin{prop}
The optimal block mean approximation of $\mathbf{M}$ with the partition vector $\mathbf{s}$ according to the Frobenius norm
\begin{align}
\min_{\bar{\mathbf{\Lambda}},\bar{\mathbf{B}}} \| \bar{\mathbf{\Lambda}}+\bar{\mathbf{B}} - \mathbf{M} \|_F^2\;
\end{align}
is given by
\begin{align}
b_{ij} &=
\begin{cases}
0, &i = j, s_i = 1, \\
\frac{\sum_{mn} \mathbf{M}^{ii}_{mn} - \sum_{m} \mathbf{M}^{ii}_{mm}}{s_{i}(s_{i}-1)}, &i = j, s_i \neq 1, \\
\frac{\sum_{mn}\mathbf{M}^{ij}_{mn}}{s_{i}s_{j}}, &i \neq j,
\end{cases} \label{b_ij} \\
\lambda_{i} &= \frac{1}{s_{i}}\sum_m \mathbf{M}^{ii}_{mm} - b_{ii}\;. \label{lambda_i}
\end{align}
\end{prop}
Proposition 1 can be understood as follows. According to (\ref{b_ij}) and (\ref{lambda_i}), the non-diagonal block $\mathbf{M}^{ij}$ is approximated by $\widehat{\mathbf{M}}^{ij} = \mathbf{B}^{ij}$, whose value is the mean value of $\mathbf{M}^{ij}$, which minimizes the Frobenius form. The diagonal block $\mathbf{M}^{ii}$ is approximated by $\widehat{\mathbf{M}}^{ii} = \mathbf{\Lambda}^{ii} + \mathbf{B}^{ii}$, whose diagonal values are equal to the mean diagonal values of $\mathbf{M}^{ii}$ and its off-diagonal values are equal to the mean off-diagonal values of $\mathbf{M}^{ii}$, which again minimizes the Frobenius form.
The power of block mean approximation lies in the ease of computing its inverse and inverse square root matrices, as shown by the following theorems. All the proofs are relegated to the Appendix.
\newpage
\begin{theorem}
For the positive definite matrix $\bar{\mathbf{\Lambda}}+\bar{\mathbf{B}}$, where $\bar{\mathbf{\Lambda}}$ and $\bar{\mathbf{B}}$ are the diagonal and full expansion of $\mathbf{\Lambda}$ and $\mathbf{B}$ with respect to the partition vector $\mathbf{s}$,
\begin{align}
(\bar{\mathbf{\Lambda}}+\bar{\mathbf{B}})^{-1} = \bar{\mathbf{\Lambda}}^{-1} + \bar{\mathbf{D}}
\end{align}
where $\bar{\mathbf{D}}$ is the full expansion matrix with partition vector $\mathbf{s}$ of
\begin{align}
\mathbf{D} = (\mathbf{\Lambda} \mathbf{S} + \mathbf{S}\mathbf{B}\mathbf{S} )^{-1} - (\mathbf{\Lambda}\mathbf{S})^{-1}
\end{align}
where $\mathbf{S}= \text{\normalfont diag}(\mathbf{s})$.
\end{theorem}
\begin{theorem}
For the positive definite matrix $\bar{\mathbf{\Lambda}}+\bar{\mathbf{B}}$, where $\bar{\mathbf{\Lambda}}$ and $\bar{\mathbf{B}}$ are the diagonal and full expansion of $\mathbf{\Lambda}$ and $\mathbf{B}$ with respect to the partition vector $\mathbf{s}$,
\begin{align}
(\bar{\mathbf{\Lambda}}+\bar{\mathbf{B}})^{-\frac{1}{2}} = \bar{\mathbf{\Lambda}}^{-\frac{1}{2}} + \bar{\mathbf{D}}
\end{align}
where $\bar{\mathbf{D}}$ is the full expansion matrix with partition vector $\mathbf{s}$ of
\begin{align}
\mathbf{D} = \mathbf{S}^{-\frac{1}{2}}\left[(\mathbf{\Lambda} + \mathbf{S}^{\frac{1}{2}}\mathbf{B}\mathbf{S}^{\frac{1}{2}} )^{-\frac{1}{2}} - \mathbf{\Lambda}^{-\frac{1}{2}}\right]\mathbf{S}^{-\frac{1}{2}}
\end{align}
where $\mathbf{S}= \text{\normalfont diag}(\mathbf{s})$.
\end{theorem}
The importance of the above theorems can be understood by noting that inverting $\mathbf{G} \in \mathbb{R}^{d \times d}$ by splitting it into $L\times L$ blocks
has a complexity of $O(L^3)$, which can be significantly faster than $O(d^3)$ flops required to obtain $\mathbf{G}^{-1}$ or $\mathbf{G}^{-\frac{1}{2}}$.
\section{BMA for Neural Networks}
As block mean approximation divides a matrix into $L\times L$ blocks, a natural question to ask is \emph{how much prior knowledge is needed to
determine such a block structure?}
In training neural networks, we recommend to group the parameters in each layer (or even each unit) into a block such that $\mathbf{G}$ with block mean approximation represents the scale and dependency between layers (or units). This comes naturally as the output and gradient of each layer often depends on one another.
There are several work that use heuristics to set an individual learning rate for layers of a deep network~\cite{singh2015layer,you2017scaling,abu2017proportionate}. Nevertheless, even with such heuristics, the dependency between parameters is ignored.
In contrast, the BMA gives a principled way to set learning rates per layers in a deep network while capturing the dependency between layers.
In Figure~\ref{eFIM}, we compute the empirical Fisher information matrix $\mathbb{E}_{\mathbf{x}\sim p_{\text{data}}}[ \frac{\partial \log p(\mathbf{x}|\bm{\theta})}{\partial \bm{\theta}}\frac{\partial \log p(\mathbf{x}|\bm{\theta})}{\partial \bm{\theta}}^\intercal ]$ for a convolutional neural network (described in Table 1) and compare BMA with different block partitions. Even for the finest approximation in Figure~\ref{eFIM}, only $45\times 45$ matrices need to be constructed and inverted.
\newpage
\begin{figure}
\centering
\subfloat[Original]{\includegraphics[width=0.5\linewidth]{fig/eFIM.png}}
\subfloat[$10\times 10$ blocks]{\includegraphics[width=0.5\linewidth]{fig/eFIM_bma_layer.png}}
\subfloat[$27\times 27$ blocks]{\includegraphics[width=0.5\linewidth]{fig/eFIM_bma_unit.png}}
\subfloat[$45\times 45$ blocks]{\includegraphics[width=0.5\linewidth]{fig/eFIM_bma_filter.png}}
\caption{Block mean approximations of the empirical Fisher information matrix of a neural network model. The values are normalized for better visualization. (a) The original matrix has a size of $322\times 322$. (b) Each block represents weights or bias in a layer. MSE (mean square error) = 0.1036. (c) Each block represents weights or bias in a unit. MSE=0.1024. (d) Each block represents a group of weights or bias in a unit. MSE=0.0998.}
\label{eFIM}
\end{figure}
\section{Related Work}
There are several generic matrix approximation techniques which have been applied to second order optimization methods. Aside from the diagonal approximation,
block diagonal approximation has been applied to AdaGrad~\cite{duchi2011adaptive} and the Gauss-Newton method~\cite{botev2017practical}.
Low rank approximation has been applied to natural gradient~\cite{roux2008topmoumoute} and AdaGrad~\cite{krummenacher2016scalable}.
Kronecker approximation \cite{martens2015optimizing,grosse2016kronecker}, sparse approximation \cite{grosse2015scaling} and quasi-diagonal approximation \cite{ollivier2015riemannian} have been applied to natural gradient.
We stress that the block mean approximation takes into account all the elements of $\mathbf{G}$,
while the diagonal and block diagonal approximations neglect most of the elements of $\mathbf{G}$.
Furthermore, the block mean approximation does not force any low-rank assumption, which cannot be guaranteed in general. For example, when $\mathbf{G} = \sigma \mathbf{I}$ for $\sigma > 0$, $\mathbf{G}$ has the singular values all equal to $\sigma$ and thus does not have a low-rank structure. The other advantage of the BMA is its ease of implementation. The approximate matrix $\widehat{\mathbf{G}}$ does not need to be constructed explicitly in general.
This is shown in \textsection~\ref{sec:adagrad_bma}.
\section{AdaGrad with BMA}\label{sec:adagrad_bma}
For unconstrained stochastic optimization problems, the full version of AdaGrad \cite{duchi2011adaptive} has the following update rule,
\begin{align}
\bm{\theta}_{t+1} = \bm{\theta}_{t} - \eta \mathbf{G}_t^{-1} \mathbf{g}_{t}
\label{ada_grad}
\end{align}
where $\mathbf{G}_t = \mathbf{H}_t^{\frac{1}{2}}$ and $\mathbf{H}_t = \sum_{\tau=1}^t \mathbf{g}_{\tau} \mathbf{g}_{\tau}^\intercal$.
We approximate the gradient outer product matrix in the following form
\begin{align}
\widehat{\mathbf{H}}_t = \mathbf{Z}_t\mathbf{F}_t\mathbf{Z}_t \approx \mathbf{H}_t
\label{H_t}
\end{align}
where $\mathbf{Z}_t = \text{diag}(\mathbf{z}_t)$ is a diagonal matrix and $\mathbf{F}_t$ is a positive definite matrix. As AdaGrad requires the computation of $\mathbf{H}_t^{-\frac{1}{2}}$, one needs to choose $\mathbf{F}_t$ such that $\mathbf{F}_t^{-\frac{1}{2}}$ is easy to obtain.
To derive the algorithm, we first define the following notations.
\begin{notation*}
We divide vector $\mathbf{g}$ into $L$ blocks such that $\mathbf{g} = (\mathbf{g}^1,...,\mathbf{g}^L)$. $\mathbf{g}^i$ denotes the $i$-th block of vector $\mathbf{g}$. $g^i_m$ denotes the $m$-th element of vector $\mathbf{g}^i$. $({g}^i_m)^2$ denotes the square value of ${g}^i_m$. $\mathbf{g}_t$ denote a vector at step $t$. $\mathbf{g}_t^i$ denote the $i$-th block of $\mathbf{g}_t$. $g_{t,m}^i$ denote the $m$-th element of $\mathbf{g}_t^i$. $\mathbf{g}^2$ denote elementwise square of $\mathbf{g}$. $\sqrt{\mathbf{g}}$ denotes elementwise square root of $\mathbf{g}$. $\mathbf{g} + a$ denotes each element of $\mathbf{g}$ is added by scalar $a$.
\end{notation*}
\subsection{Diagonal Approximation}
If we set $\mathbf{z}_t$ with elements
\begin{align}
z_{t,i} = \sqrt{\sum_{\tau=1}^t (g_{\tau,i})^2}
\label{Z}
\end{align}
and $\mathbf{F}_t = \mathbf{I}$, $\widehat{\mathbf{H}}_t$ is reduced to diagonal approximation.
\subsection{Block Mean Approximation}
In order to capture the off-diagonal elements of $\mathbf{H}$, we approximate $\mathbf{F}_t$ with the block mean approximation proposed in \textsection \ref{sec:BMA}.
We set $\mathbf{z}_t$ as in (\ref{Z})
and seek
\begin{align}
\mathbf{F}_t = \bar{\mathbf{\Lambda}} + \bar{\mathbf{B}} \approx \mathbf{Z}_t^{-1}(\sum_{\tau=1}^t\mathbf{g}_{\tau}\mathbf{g}_{\tau}^\intercal)\mathbf{Z}_t^{-1}.
\label{F_t}
\end{align}
With the block mean approximation, (\ref{H_t}) can be interpreted as follows: $\mathbf{Z}_t$ gives an individual learning rate of each parameter and $\mathbf{F}_t$ captures the dependency between the groups of parameters.
To realize block mean approximation, we partition the parameters into $L$ groups.
\newpage
Let $\mathbf{g}_t = (\mathbf{g}_t^1, \mathbf{g}_t^2,...,\mathbf{g}_t^L)$.
Define $\mathbf{u}_t$ and $\mathbf{v}_t$ with
\begin{align}
u_{t,i} =\sum_{m} g_{t,m}^i, \quad
v_{t,i} = \sum_{m} z_{t,m}^i,
\end{align}
respectively. Let $\mathbf{U}_t = \sum_{\tau=1}^t \mathbf{u}_{\tau}\mathbf{u}_{\tau}^\intercal$.
To approximate $\mathbf{Z}_t^{-1}(\sum_{\tau=1}^t\mathbf{g}_{\tau}\mathbf{g}_{\tau}^\intercal)\mathbf{Z}_t^{-1}$, we choose $\bar{\mathbf{\Lambda}}$ and $\bar{\mathbf{B}}$ to be the expansion matrices of $\mathbf{\Lambda}$ and $\mathbf{B}$
\begin{align}
\mathbf{\Lambda} &= \mathbf{I}, \quad \mathbf{B} = \mathbf{S}^{-\frac{1}{2}}\frac{\mathbf{U}_t - \text{diag}(\mathbf{U}_t)}{\mathbf{v}_t\mathbf{v}_t^\intercal} \mathbf{S}^{-\frac{1}{2}}
\end{align}
where the division is elementwise. Based on Theorem 2, the inverse square root is $\mathbf{I} + \bar{\mathbf{D}}$ where $\bar{\mathbf{D}}$ is the expansion matrix of
\begin{align}
\mathbf{D} = \mathbf{S}^{-\frac{1}{2}}\left[\left(\mathbf{I} + \frac{\mathbf{U}_t - \text{diag}(\mathbf{U}_t)}{\mathbf{v}_t\mathbf{v}_t^\intercal}\right)^{-\frac{1}{2}} - \mathbf{I}\right]\mathbf{S}^{-\frac{1}{2}}. \label{D}
\end{align}
The inverse square root in (\ref{D}) can be computed as follows.
Let $\mathbf{RVR}^\intercal$ be the eigendecomposition of a matrix, then its inverse square root is $\mathbf{R}\mathbf{V}^{-\frac{1}{2}}\mathbf{R}^\intercal$. In case where the eigenvalues are zeros or too small, we clamp the eigenvalues to have a minimal value before computing $\mathbf{V}^{-\frac{1}{2}}$.
We call the above algorithm AdaGrad-BMA, which is summarized in Algorithm 1.
The eigendecomposition has time complexity $O(L^3)$. Therefore, for parameters of dimension $d$ and partitioned into $L$ blocks, AdaGrad-BMA has time complexity $O(L^3 + d)$ per iteration.
\begin{algorithm}[t]
\caption{AdaGrad-BMA}
\label{alg:example}
\begin{algorithmic}[1]
\STATE {\bfseries Input:} Objective function $f(\bm{\theta})$ with parameters $\bm{\theta}$
\STATE {\bfseries Input:} A partition of parameters $\bm{\theta}=\{\bm{\theta}^1,...,\bm{\theta}^L\}$
\STATE {\bfseries Input:} Partition vector $\mathbf{s}$ that $s_i=$ size($\bm{\theta}^i$)
\STATE {\bfseries Input:} Hyperparameters $\eta$ and $\epsilon$
\STATE Initialize $\mathbf{U} = \mathbf{0}$, $\mathbf{v} = \mathbf{0}$, $\mathbf{r} = \mathbf{0}$, $\mathbf{S}=\text{diag}(\mathbf{s})$
\FOR{$t=1$ {\bfseries to} $T$}
\STATE $\mathbf{g} \leftarrow \nabla_{\bm{\theta}}f_t(\bm{\theta}_t)$
\STATE $\mathbf{r} \leftarrow \mathbf{r} + \mathbf{g}^2$
\STATE $\mathbf{z} \leftarrow \sqrt{\mathbf{r}+\epsilon}$
\FOR{$i=1$ {\bfseries to} $L$}
\STATE $u_i \leftarrow \sum_m {g}^i_{m} $
\STATE $v_i \leftarrow \sum_m {z}^i_{m} $
\ENDFOR
\STATE $\mathbf{U} \leftarrow \mathbf{U} + \mathbf{u}\mathbf{u}^\intercal$
\STATE Compute $\mathbf{D}$ according to (\ref{D})
\STATE $\mathbf{g} \leftarrow \mathbf{Z}^{-\frac{1}{2}} \mathbf{g}$
\FOR{$i=1$ {\bfseries to} $L$}
\STATE $\mathbf{g}^i \leftarrow \mathbf{g}^i + \sum_j\mathbf{D}_{ij}{u}_j$
\ENDFOR
\STATE $\mathbf{g} \leftarrow \mathbf{Z}^{-\frac{1}{2}} \mathbf{g}$
\STATE $\bm{\theta} \leftarrow \bm{\theta} - \eta \mathbf{g} $
\ENDFOR
\end{algorithmic}
\end{algorithm}
\newpage
\section{Experiments}\label{sec:exp}
We evaluate AdaGrad-BMA in training convolutional neural networks, against the full version of AdaGrad (AdaGrad-full) and AdaGrad with diagonal approximation (AdaGrad-diag). For AdaGrad-BMA, we group the weights and the bias parameters separately for each layer. For a model of $l$ convolution or fully connected layers, we partition $\mathbf{G}$ into $L\times L$ blocks with BMA where $L = 2l$.
The experiments are done on two standard datasets MNIST and CIFAR-10. We use two simple models: small and large, as described in Table 1 and 2.
We choose the architecture of the small model to ensure that AdaGrad-full is applicable. For the large model, AdaGrad-full is too computationally expensive to use.
Each convolution layer has kernel size $3\times 3$, stride 1 and zero-padding 1. Each convolution layer is followed by a hyperbolic tangent activation function.
The number of parameters of each model is listed in Table 3. As MNIST and CIFAR-10 have different input image size, the fully connected layer in each model has different size of inputs, therefore resulting in different number of parameters.
For each algorithm on each dataset, we tried learning rates $\eta \in \{1,10^{-1},10^{-2},10^{-3},10^{-4}\}$ and report the best performance achieved by the algorithm on the dataset.
The results are shown in Figure \ref{fig:results}, from which we can see AdaGrad-BMA outperforms AdaGrad-diag and achieves similar speed of convergence to AdaGrad-full.
The comparison of runtime of each iteration on MNIST is shown in Table \ref{runtime}.
Although AdaGrad-BMA has longer runtime than AdaGrad-diag for each iteration, in practice one can update $\mathbf{G}^{-1}$ for every several steps to amortize the cost.
The code for the experiments is included in the supplementary materials.
\begin{table}[h!]
\centering
\begin{scriptsize}
\parbox{.45\linewidth}{
\centering
\caption{Small model}
\vspace{0.1in}
\begin{tabular}{|c|}
\hline
Conv 3x3, 3 \\
Max Pooling 2x2 \\
Conv 3x3, 3 \\
Max Pooling 2x2 \\
Conv 3x3, 3 \\
Max Pooling 2x2 \\
Conv 3x3, 3 \\
Max Pooling 2x2 \\
Fully Connected, 10 \\
Softmax, 10 \\
\hline
\end{tabular}}
\parbox{.45\linewidth}{
\centering
\caption{Large model}
\vspace{0.1in}
\begin{tabular}{|c|}
\hline
Conv 3x3, 32 \\
Conv 3x3, 32 \\
Conv 3x3, 32 \\
Conv 3x3, 32 \\
Max Pooling 2x2 \\
Conv 3x3, 32 \\
Conv 3x3, 32 \\
Conv 3x3, 32 \\
Conv 3x3, 32 \\
Max Pooling 2x2 \\
Conv 3x3, 32 \\
Conv 3x3, 32 \\
Conv 3x3, 32 \\
Conv 3x3, 32 \\
Max Pooling 2x2 \\
Conv 3x3, 32 \\
Conv 3x3, 32 \\
Conv 3x3, 32 \\
Conv 3x3, 32 \\
Max Pooling 2x2 \\
Fully Connected, 10 \\
Softmax, 10 \\
\hline
\end{tabular}}
\end{scriptsize}
\end{table}
\begin{table}[h!]
\centering
\caption{Model parameters}
\vspace{0.1in}
\begin{scriptsize}
\begin{tabular}{|c|c|c|c|}
\hline
& MNIST & CIFAR-10 \\
\hline
Small model & 322 & 466 \\
Large model & 139370 & 140906 \\
\hline
\end{tabular}
\end{scriptsize}
\end{table}
\begin{table}[h!]
\centering
\caption{Runtime comparison (ms/iteration)}
\vspace{0.1in}
\begin{scriptsize}
\begin{tabular}{|c|c|c|c|}
\hline
& Small model & Large model \\
\hline
AdaGrad-full & 16.85 & - \\
AdaGrad-diag & 5.70 & 7.55 \\
AdaGrad-BMA & 10.07 & 19.12 \\
\hline
\end{tabular}
\end{scriptsize}
\label{runtime}
\end{table}
\begin{figure}[h!]
\centering
\subfloat[MNIST, small model]{\includegraphics[width=0.45\linewidth]{fig/mnist_train_small.eps}}
\subfloat[MNIST, small model]{\includegraphics[width=0.45\linewidth]{fig/mnist_test_small.eps}}
\subfloat[MNIST, large model]{\includegraphics[width=0.45\linewidth]{fig/mnist_train_large.eps}}
\subfloat[MNIST, large model]{\includegraphics[width=0.45\linewidth]{fig/mnist_test_large.eps}}
\subfloat[CIFAR-10, small model]{\includegraphics[width=0.45\linewidth]{fig/cifar_train_small.eps}}
\subfloat[CIFAR-10, small model]{\includegraphics[width=0.45\linewidth]{fig/cifar_test_small.eps}}
\subfloat[CIFAR-10, large model]{\includegraphics[width=0.45\linewidth]{fig/cifar_train_large.eps}}
\subfloat[CIFAR-10, large model]{\includegraphics[width=0.45\linewidth]{fig/cifar_test_large.eps}}
\caption{Performance of AdaGrad and its approximations on two standard datasets.}
\label{fig:results}
\end{figure}
\clearpage
\section{Discussions}
In this paper, we propose a new matrix approximation method which allows efficient storage and computation of matrix inverse and inverse square root.
The method is applied to AdaGrad and achieves promising results.
In the numerical linear algebra literature, there are two work relevant but different from ours.
\cite{chow1997approximate} proposed an approximate inverse technique which generates sparse solutions for block partitioned matrices. However, in our method the exact inverse of the block mean approximation matrix can be computed, as proved in Theorem 1 and 2. Our method does not assume sparse solutions either. \cite{guillaume2003block} proposed an approximate inverse technique which incorporates block constant structure in the preconditioning matrix. This is different from ours as in our method the block constant structure is incorporated the approximated matrix and its inverse and the analytic solution of the inverse is explicitly given.
Second order optimization methods for training neural networks have many theoretical appeals, as discussed in this paper and reviewed in \cite{shepherd2012second,martens2016second}. We hope our method makes one more step towards their practical implementation.
\newpage
\section*{Appendix}
\setcounter{prop}{0}
\setcounter{theorem}{0}
To show how Newton method is invariant of affine re-parameterization, let $\nabla^2$ be the Hessian operator, $\bm{\xi}=\mathbf{A}\bm{\theta}$ and $g(\bm{\theta}) = f(\mathbf{A}\bm{\theta})$ where $\mathbf{A}$ is an invertible square matrix.
\begin{align}
\bm{\theta}_{t+1} &= \bm{\theta}_{t} - \eta (\nabla^2_{\bm{\theta}}g(\bm{\theta}))^{-1}\nabla_{\bm{\theta}}g(\bm{\theta}_{t}) \\
&=\bm{\theta}_{t} - \eta (\mathbf{A}^\intercal\nabla^2_{\bm{\xi}}f(\mathbf{A}\bm{\theta}_{t})\mathbf{A})^{-1}\mathbf{A}^\intercal\nabla_{\bm{\xi}}f(\mathbf{A}\bm{\theta}_{t}) \\
&=\bm{\theta}_{t} - \eta \mathbf{A}^{-1}(\nabla^2_{\bm{\xi}}f(\mathbf{A}\bm{\theta}_{t}))^{-1}\nabla_{\bm{\xi}}f(\mathbf{A}\bm{\theta}_{t})
\end{align}
we obtain
\begin{align}
\mathbf{A}\bm{\theta}_{t+1} &= \mathbf{A}\bm{\theta}_{t} - \eta (\nabla_{\bm{\xi}}^2f(\mathbf{A}\bm{\theta}_{t}))^{-1}\nabla_{\bm{\xi}}f(\mathbf{A}\bm{\theta}_{t})
\end{align}
which is equivalent to
\begin{align}
\bm{\xi}_{t+1} &= \bm{\xi}_{t} - \eta (\nabla_{\bm{\xi}}^2f(\bm{\xi}_{t}))^{-1}\nabla_{\bm{\xi}}f(\bm{\xi}_{t}).
\end{align}
\begin{prop}
The optimal block mean approximation of $\mathbf{M}$ with the partition vector $\mathbf{s}$ according to the Frobenius norm
\begin{align}
\min_{\bar{\mathbf{\Lambda}},\bar{\mathbf{B}}} \| \bar{\mathbf{\Lambda}}+\bar{\mathbf{B}} - \mathbf{M} \|_F^2\;
\end{align}
is given by
\begin{align}
b_{ij} &=
\begin{cases}
0, &i = j, s_i = 1, \\
\frac{\sum_{mn} \mathbf{M}^{ii}_{mn} - \sum_{m} \mathbf{M}^{ii}_{mm}}{s_{i}(s_{i}-1)}, &i = j, s_i \neq 1, \\
\frac{\sum_{mn}\mathbf{M}^{ij}_{mn}}{s_{i}s_{j}}, &i \neq j,
\end{cases} \label{b_ij} \\
\lambda_{i} &= \frac{1}{s_{i}}\sum_m \mathbf{M}^{ii}_{mm} - b_{ii}\;. \label{lambda_i}
\end{align}
\end{prop}
\begin{proof}
For $i\neq j$, $\bar{\mathbf{\Lambda}}^{ij} = 0$ by construction. Hence, $b_{ij}=\frac{\sum_{mn}\mathbf{M}^{ij}_{mn}}{s_{i}s_{j}}$ is the minimum solution for $\| \bar{\mathbf{B}}^{ij} - \mathbf{M}^{ij} \|_F$. For $i=j$, since $b_{ii} = \frac{\sum_{mn} \mathbf{M}^{ii}_{mn} - \sum_{m} \mathbf{M}^{ii}_{mm}}{s_{i}(s_{i}-1)}$, the off-diagonal elements of $\mathbf{M}^{ii}$ are minimized under the Frobenius norm. And since $\lambda_{i} + b_{ii} = \frac{1}{s_{i}}\sum_m \mathbf{M}^{ii}_{mm}$, the diagonal elements of $\mathbf{M}^{ii}$ are also minimized.
\end{proof}
\begin{theorem}
For positive definite matrix $\bar{\mathbf{\Lambda}}+\bar{\mathbf{B}}$, where $\bar{\mathbf{\Lambda}}$ is the diagonal expansion matrix of $\mathbf{\Lambda}$ and $\bar{\mathbf{B}}$ is the full expansion matrix of $\mathbf{B}$, both of which have the same partition vector $\mathbf{s}$,
\begin{align}
(\bar{\mathbf{\Lambda}}+\bar{\mathbf{B}})^{-1} = \bar{\mathbf{\Lambda}}^{-1} + \bar{\mathbf{D}}
\end{align}
where $\bar{\mathbf{D}}$ is the full expansion matrix with partition vector $\mathbf{s}$ of
\begin{align}
\mathbf{D} = (\mathbf{\Lambda} \mathbf{S} + \mathbf{S}\mathbf{B}\mathbf{S} )^{-1} - (\mathbf{\Lambda}\mathbf{S})^{-1} \label{D_theorem1}
\end{align}
where $\mathbf{S}= \text{\normalfont diag}(\mathbf{s})$.
\end{theorem}
\begin{proof}
First we prove $(\bar{\mathbf{\Lambda}}+\bar{\mathbf{B}})(\bar{\mathbf{\Lambda}}^{-1} + \bar{\mathbf{D}}) = \mathbf{I}$.
\begin{align}
\mathbf{D} &= (\mathbf{\Lambda}\mathbf{S} + \mathbf{S}\mathbf{B}\mathbf{S})^{-1} - (\mathbf{\Lambda}\mathbf{S})^{-1} \\
&= (\mathbf{\Lambda}\mathbf{S})^{-1} \nonumber \\
&- (\mathbf{\Lambda}\mathbf{S})^{-1} \mathbf{S}(\mathbf{I}+\mathbf{BS}(\mathbf{\Lambda}\mathbf{S})^{-1}\mathbf{S})^{-1}\mathbf{BS}(\mathbf{\Lambda}\mathbf{S})^{-1} \nonumber \\
&- (\mathbf{\Lambda}\mathbf{S})^{-1} \label{woodbury} \\
&= - \mathbf{\Lambda}^{-1}(\mathbf{I}+\mathbf{B}\mathbf{\Lambda}^{-1}\mathbf{S})^{-1}\mathbf{B}\mathbf{\Lambda}^{-1} \label{commutativity} \\
&= -(\mathbf{\Lambda}+\mathbf{B}\mathbf{S})^{-1}\mathbf{B}\mathbf{\Lambda}^{-1} \label{D_last}
\end{align}
(\ref{woodbury}) follows from the Kailath variant of Woodbury identity. (\ref{commutativity}) follows from $\mathbf{\Lambda}\mathbf{S} = \mathbf{S}\mathbf{\Lambda}$ since $\mathbf{S}$ and $\mathbf{\Lambda}$ are both diagonal. Multiply both sides of (\ref{D_last}) by $\mathbf{\Lambda}+\mathbf{B}\mathbf{S}$, after some manipulation, we have
\begin{align}
\mathbf{B}\mathbf{\Lambda}^{-1} + \mathbf{\Lambda}\mathbf{D} + \mathbf{B}\mathbf{S}\mathbf{D} = 0
\end{align}
Since $\bar{\mathbf{\Lambda}}$, $\bar{\mathbf{B}}$ and $\bar{\mathbf{D}}$ are the expansion matrices with partition vector $\mathbf{s}$ of $\mathbf{\Lambda}$, $\mathbf{B}$ and $\mathbf{D}$, respectively, we have equivalently
\begin{align}
\bar{\mathbf{B}}\bar{\mathbf{\Lambda}}^{-1} + \bar{\mathbf{\Lambda}}\bar{\mathbf{D}} + \bar{\mathbf{B}}\bar{\mathbf{D}} &= 0 \\
\mathbf{I} + \bar{\mathbf{B}}\bar{\mathbf{\Lambda}}^{-1} + \bar{\mathbf{\Lambda}}\bar{\mathbf{D}} + \bar{\mathbf{B}}\bar{\mathbf{D}} &= \mathbf{I} \\
(\bar{\mathbf{\Lambda}}+\bar{\mathbf{B}})(\bar{\mathbf{\Lambda}}^{-1} + \bar{\mathbf{D}}) &= \mathbf{I}
\end{align}
Second, we prove $(\bar{\mathbf{\Lambda}}^{-1} + \bar{\mathbf{D}})(\bar{\mathbf{\Lambda}}+\bar{\mathbf{B}}) = \mathbf{I}$.
\begin{align}
\mathbf{D} &= (\mathbf{\Lambda}\mathbf{S} + \mathbf{S}\mathbf{B}\mathbf{S})^{-1} - (\mathbf{\Lambda}\mathbf{S})^{-1} \\
&= (\mathbf{\Lambda}\mathbf{S})^{-1} \\
&- (\mathbf{\Lambda}\mathbf{S})^{-1} \mathbf{SB}(\mathbf{I}+\mathbf{S}(\mathbf{\Lambda}\mathbf{S})^{-1}\mathbf{SB})^{-1}\mathbf{S}(\mathbf{\Lambda}\mathbf{S})^{-1} \\
&- (\mathbf{\Lambda}\mathbf{S})^{-1} \\
&= - \mathbf{\Lambda}^{-1}\mathbf{B}(\mathbf{I}+\mathbf{\Lambda}^{-1}\mathbf{S}\mathbf{B})^{-1}\mathbf{\Lambda}^{-1} \\
&= - \mathbf{\Lambda}^{-1}\mathbf{B}(\mathbf{\Lambda}+\mathbf{S}\mathbf{B})^{-1}
\end{align}
Therefore
\begin{align}
\mathbf{\Lambda}^{-1}\mathbf{B} + \mathbf{D}\mathbf{\Lambda} + \mathbf{D}\mathbf{S}\mathbf{B} = 0 \\
\bar{\mathbf{\Lambda}}^{-1}\bar{\mathbf{B}} + \bar{\mathbf{D}}\bar{\mathbf{\Lambda}} + \bar{\mathbf{D}}\bar{\mathbf{B}} = 0 \\
\mathbf{I} + \bar{\mathbf{\Lambda}}^{-1}\bar{\mathbf{B}} + \bar{\mathbf{D}}\bar{\mathbf{\Lambda}} + \bar{\mathbf{D}}\bar{\mathbf{B}} = \mathbf{I} \\
(\bar{\mathbf{\Lambda}}^{-1} + \bar{\mathbf{D}})(\bar{\mathbf{\Lambda}}+\bar{\mathbf{B}}) = \mathbf{I}
\end{align}
\end{proof}
\begin{lemma}
For positive definite matrix $\bar{\mathbf{\Lambda}}+\bar{\mathbf{B}}$, where $\bar{\mathbf{\Lambda}}$ is the diagonal expansion matrix of $\mathbf{\Lambda}$ and $\bar{\mathbf{B}}$ is the full expansion matrix of $\mathbf{B}$, both of which have the same partition vector $\mathbf{s}$,
\begin{align}
(\bar{\mathbf{\Lambda}}+\bar{\mathbf{B}})^{\frac{1}{2}} = \bar{\mathbf{\Lambda}}^{\frac{1}{2}} + \bar{\mathbf{D}}
\end{align}
where $\bar{\mathbf{D}}$ is the full expansion matrix with partition vector $\mathbf{s}$ of
\begin{align}
\mathbf{D} = \mathbf{S}^{-\frac{1}{2}}\left[(\mathbf{\Lambda} + \mathbf{S}^{\frac{1}{2}}\mathbf{B}\mathbf{S}^{\frac{1}{2}} )^{\frac{1}{2}} - \mathbf{\Lambda}^{\frac{1}{2}}\right]\mathbf{S}^{-\frac{1}{2}} \label{D_lemma1}
\end{align}
where $\mathbf{S}= \text{\normalfont diag}(\mathbf{s})$.
\end{lemma}
\begin{proof}
\begin{align}
\mathbf{D} &= \mathbf{S}^{-\frac{1}{2}}\left[(\mathbf{\Lambda} + \mathbf{S}^{\frac{1}{2}}\mathbf{B}\mathbf{S}^{\frac{1}{2}} )^{\frac{1}{2}} - \mathbf{\Lambda}^{\frac{1}{2}}\right]\mathbf{S}^{-\frac{1}{2}}
\end{align}
Left and right multiply both side by $\mathbf{S}^{\frac{1}{2}}$,
\begin{align}
\mathbf{S}^{\frac{1}{2}} \mathbf{D} \mathbf{S}^{\frac{1}{2}} &= (\mathbf{\Lambda} + \mathbf{S}^{\frac{1}{2}}\mathbf{B}\mathbf{S}^{\frac{1}{2}} )^{\frac{1}{2}} - \mathbf{\Lambda}^{\frac{1}{2}} \\
\mathbf{S}^{\frac{1}{2}} \mathbf{D} \mathbf{S}^{\frac{1}{2}}+ \mathbf{\Lambda}^{\frac{1}{2}} &= (\mathbf{\Lambda} + \mathbf{S}^{\frac{1}{2}}\mathbf{B}\mathbf{S}^{\frac{1}{2}} )^{\frac{1}{2}} \\
(\mathbf{S}^{\frac{1}{2}} \mathbf{D} \mathbf{S}^{\frac{1}{2}}+ \mathbf{\Lambda}^{\frac{1}{2}} )^{2} &= (\mathbf{\Lambda} + \mathbf{S}^{\frac{1}{2}}\mathbf{B}\mathbf{S}^{\frac{1}{2}} )
\end{align}
Expanding the square,
\begin{align}
&\mathbf{S}^{\frac{1}{2}} \mathbf{D} \mathbf{S} \mathbf{D} \mathbf{S}^{\frac{1}{2}} + \mathbf{S}^{\frac{1}{2}}\mathbf{D}\mathbf{S}^{\frac{1}{2}}\mathbf{\Lambda}^{\frac{1}{2}}
+ \mathbf{\Lambda}^{\frac{1}{2}}\mathbf{S}^{\frac{1}{2}}\mathbf{D}\mathbf{S}^{\frac{1}{2}}+\mathbf{\Lambda} \\
&= \mathbf{\Lambda} + \mathbf{S}^{\frac{1}{2}}\mathbf{B}\mathbf{S}^{\frac{1}{2}}
\end{align}
Left and right multiply both side by $\mathbf{S}^{-\frac{1}{2}}$,
\begin{align}
\mathbf{D} \mathbf{S} \mathbf{D} + \mathbf{D}\mathbf{\Lambda}^{\frac{1}{2}}
+ \mathbf{\Lambda}^{\frac{1}{2}}\mathbf{D}
= \mathbf{B}
\end{align}
Since $\bar{\mathbf{\Lambda}}$, $\bar{\mathbf{B}}$ and $\bar{\mathbf{D}}$ are the expansion matrices with partition vector $\mathbf{s}$ of $\mathbf{\Lambda}$, $\mathbf{B}$ and $\mathbf{D}$, respectively, we have equivalently
\begin{align}
\bar{\mathbf{D}} \bar{\mathbf{D}} + \bar{\mathbf{D}}\bar{\mathbf{\Lambda}}^{\frac{1}{2}}
+ \bar{\mathbf{\Lambda}}^{\frac{1}{2}}\bar{\mathbf{D}}
&= \bar{\mathbf{B}} \\
\bar{\mathbf{\Lambda}} + \bar{\mathbf{D}} \bar{\mathbf{D}} + \bar{\mathbf{D}}\bar{\mathbf{\Lambda}}^{\frac{1}{2}}
+ \bar{\mathbf{\Lambda}}^{\frac{1}{2}}\bar{\mathbf{D}}
&= \bar{\mathbf{\Lambda}} + \bar{\mathbf{B}} \\
(\bar{\mathbf{\Lambda}}^{\frac{1}{2}}+\bar{\mathbf{D}})^2
&= \bar{\mathbf{\Lambda}} + \bar{\mathbf{B}} \\
\bar{\mathbf{\Lambda}}^{\frac{1}{2}}+\bar{\mathbf{D}}
&= (\bar{\mathbf{\Lambda}} + \bar{\mathbf{B}})^{\frac{1}{2}}
\end{align}
\end{proof}
\begin{theorem}
For positive definite matrix $\bar{\mathbf{\Lambda}}+\bar{\mathbf{B}}$, where $\bar{\mathbf{\Lambda}}$ is the diagonal expansion matrix of $\mathbf{\Lambda}$ and $\bar{\mathbf{B}}$ is the full expansion matrix of $\mathbf{B}$, both of which have the same partition vector $\mathbf{s}$,
\begin{align}
(\bar{\mathbf{\Lambda}}+\bar{\mathbf{B}})^{-\frac{1}{2}} = \bar{\mathbf{\Lambda}}^{-\frac{1}{2}} + \bar{\mathbf{D}}
\end{align}
where $\bar{\mathbf{D}}$ is the full expansion matrix with partition vector $\mathbf{s}$ of
\begin{align}
\mathbf{D} = \mathbf{S}^{-\frac{1}{2}}\left[(\mathbf{\Lambda} + \mathbf{S}^{\frac{1}{2}}\mathbf{B}\mathbf{S}^{\frac{1}{2}} )^{-\frac{1}{2}} - \mathbf{\Lambda}^{-\frac{1}{2}}\right]\mathbf{S}^{-\frac{1}{2}}
\label{D_theorem2}
\end{align}
where $\mathbf{S}= \text{\normalfont diag}(\mathbf{s})$.
\end{theorem}
\begin{proof}
The theorem can be proved by combining results of Theorem 1 and Lemma 1.
Substituting $\mathbf{B}$ in (\ref{D_lemma1}) with $\mathbf{D}$ in (\ref{D_theorem1}) and substituting $\mathbf{\Lambda}$ in (\ref{D_lemma1}) with $\mathbf{\Lambda}^{-1}$, we get $\mathbf{D}$
\begin{align}
= &\mathbf{S}^{-\frac{1}{2}}[(\mathbf{\Lambda} + \mathbf{S}^{\frac{1}{2}} ((\mathbf{\Lambda} \mathbf{S} + \mathbf{S}\mathbf{B}\mathbf{S} )^{-1} \\
-&(\mathbf{\Lambda}\mathbf{S})^{-1})\mathbf{S}^{\frac{1}{2}} )^{\frac{1}{2}} - \mathbf{\Lambda}^{-\frac{1}{2}}]\mathbf{S}^{-\frac{1}{2}} \\
=&\mathbf{S}^{-\frac{1}{2}}[\mathbf{S}^{\frac{1}{2}} (\mathbf{\Lambda} \mathbf{S} + \mathbf{S}\mathbf{B}\mathbf{S} )^{-1} \mathbf{S}^{\frac{1}{2}} )^{\frac{1}{2}}- \mathbf{\Lambda}^{-\frac{1}{2}}]\mathbf{S}^{-\frac{1}{2}} \\
=&\mathbf{S}^{-\frac{1}{2}}[\mathbf{S}^{\frac{1}{2}} \mathbf{S}^{-\frac{1}{2}}(\mathbf{\Lambda} + \mathbf{S}^{\frac{1}{2}}\mathbf{B}\mathbf{S}^{\frac{1}{2}} )^{-1}\mathbf{S}^{-\frac{1}{2}} \mathbf{S}^{\frac{1}{2}} )^{\frac{1}{2}}- \mathbf{\Lambda}^{-\frac{1}{2}}]\mathbf{S}^{-\frac{1}{2}} \\
=& \mathbf{S}^{-\frac{1}{2}}\left[(\mathbf{\Lambda} + \mathbf{S}^{\frac{1}{2}}\mathbf{B}\mathbf{S}^{\frac{1}{2}} )^{-\frac{1}{2}} - \mathbf{\Lambda}^{-\frac{1}{2}}\right]\mathbf{S}^{-\frac{1}{2}}
\end{align}
\end{proof}
\clearpage
|
{
"timestamp": "2018-08-31T02:01:14",
"yymm": "1804",
"arxiv_id": "1804.05484",
"language": "en",
"url": "https://arxiv.org/abs/1804.05484"
}
|
\section{}
Members of young associations are currently incomplete in the low-mass star regime ($\lesssim$\,0.6\,$M_{\odot}$). Two recent data sets (RECONS and URAT-South; \citealp{2015AJ....149....5W,2018AJ....155..176F}) published trigonometric distances for 2664 low-mass stars, making it possible to identify new candidate members just before data release 2 of the \emph{Gaia} mission \citep{2016AA...595A...1G}. We used the Bayesian classification algorithm BANYAN~$\Sigma$ \citep{2018ApJ...856...23G} to identify new candidate members in 27 young associations from this sample based on their sky position, proper motion, trigonometric distance. We complemented our sample with 2MASS $J$-band magnitudes, \emph{Gaia} $G$-band magnitudes, and literature radial velocities when available. All objects were cross-matched with known bona fide members \citep{2018ApJ...856...23G} and new candidate members from \emph{Gaia}--TGAS \citep{Gagne:2018un}, and the 11 matches were excluded. We selected all targets with a Bayesian young association membership probability above 90\%, and an optimal $UVW$ distance from the BANYAN~$\Sigma$ kinematic models below 5\,\hbox{km\,s$^{-1}$}\ and 5$\sigma$.
All resulting 146 stars (listed in Table~\ref{table}) were cross-matched with literature catalogs to gather spectral types, signs of youth (e.g., X-ray, UV, Li absorption, infrared excess), and to identify the previously known members or candidate members. Most are M dwarfs, and 131 are newly identified candidate members of young associations. Three of the new candidates (G~13--39 and LHS~2935 in the Carina-Near moving group, and G~152--1 in the AB~Doradus moving group, or ABDMG hereafter) have complete kinematics, but further observations will be needed to confirm their young age determine if they are new bona fide members. We also identified the helium-atmosphere white dwarf EGGR~344 as a candidate member of the 130--200\,Myr-old ABDMG \citep{2015MNRAS.454..593B}, but its effective temperature of $\sim$\,6950\,K \citep{2015ApJS..219...19L} corresponds to a cooling age of $\sim$\,1.3\,Gyr that is incompatible with this hypothesis, even with a conservatively low mass of 0.5\,$M_{\odot}$\ \citep{2001PASP..113..409F}. We therefore reject EGGR~344 as a candidate member of ABDMG.
This catalog of new potentially young M dwarfs, which will soon be improved with \emph{Gaia}~DR2, will be especially useful to characterize the initial mass function of young associations, calculate lithium depletion boundary ages, study their angular momentum evolution, and search for exoplanet companions by direct imaging.
\begin{longrotatetable}
\global\pdfpageattr\expandafter{\the\pdfpageattr/Rotate 90}
\begin{deluxetable*}{lllllr@{\;$\pm$\;}lr@{\;$\pm$\;}lr@{\;$\pm$\;}lcclll}
\tablecolumns{16}
\tablecaption{New candidate members. \label{table}}
\tablehead{\colhead{Name} & \colhead{Assoc.\tablenotemark{a}} & \colhead{Spectral} & \colhead{R.A.} & \colhead{Decl.} & \multicolumn{2}{c}{$\mu_\alpha\cos\delta$} & \multicolumn{2}{c}{$\mu_\delta$} & \multicolumn{2}{c}{Distance} & \colhead{HR1} & \colhead{$NUV$} & \colhead{IR$_{\rm ex}$} & \colhead{Known} & \colhead{Youth\tablenotemark{c}}\\
\colhead{} & \colhead{} & \colhead{Type\tablenotemark{b}} & \colhead{(hh:mm:ss)} & \colhead{(dd:mm:ss)} & \multicolumn{2}{c}{($\mathrm{mas}\,\mathrm{yr}^{-1}$)} & \multicolumn{2}{c}{($\mathrm{mas}\,\mathrm{yr}^{-1}$)} & \multicolumn{2}{c}{(pc)} & \colhead{} & \colhead{(mag)} & \colhead{$N\sigma$} & \colhead{Group} & \colhead{}
}
\startdata
LP~524--13 & ABDMG & M3 & 00:02:04.017 & +04:08:09.95 & $183.3$ & $3.3$ & $-201.4$ & $3.2$ & $22.3$ & $3.0$ & $\cdots$ & $\cdots$ & $\cdots$ & ABDMG & $\cdots$\\
G~171--61 & ABDMG & M1.5 & 00:32:00.621 & +43:56:33.02 & $296.4$ & $3.9$ & $-268.1$ & $3.9$ & $21.5$ & $2.5$ & $\cdots$ & $\cdots$ & $\cdots$ & $\cdots$ & $\cdots$\\
LP~295--636 & ABDMG & M3 & 01:12:55.574 & +27:44:51.43 & $152.3$ & $4.1$ & $-214.2$ & $4.1$ & $23.7$ & $2.9$ & $\cdots$ & $\cdots$ & $\cdots$ & $\cdots$ & $\cdots$\\
G~244--32 & ABDMG & (M6) & 01:42:27.090 & +61:02:29.49 & $244.8$ & $3.1$ & $-213.2$ & $3.1$ & $23.2$ & $2.2$ & $-0.47$ & $\cdots$ & $\cdots$ & $\cdots$ & $\cdots$\\
UCAC4~485--002908 & ABDMG & (M3) & 02:03:25.984 & +06:47:59.15 & $89.2$ & $1.2$ & $-120.5$ & $1.2$ & $23.5$ & $2.8$ & $\cdots$ & $\cdots$ & $\cdots$ & ABDMG & $\cdots$\\
\enddata
\tablenotetext{a}{Young association. See \cite{2018ApJ...856...23G} for more detail.}
\tablenotetext{b}{Spectral types in parentheses were estimated using the $G - J$ color, similarly to \citep{Gagne:2018un}.}
\tablenotetext{c}{Signs of youth compiled from the literature, see \cite{Gagne:2018un} for more detail.}
\tablecomments{This is a preview of the full electronic-only table.}
\end{deluxetable*}
\end{longrotatetable}
\global\pdfpageattr\expandafter{\the\pdfpageattr/Rotate 0}
\acknowledgments
This research made use of: data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by NASA and the National Science Foundation, and data from the ESA mission {\it Gaia}, processed by the {\it Gaia} Data Processing and Analysis Consortium whose funding has been provided by national institutions, in particular the institutions participating in the {\it Gaia} Multilateral Agreement.
\software{BANYAN~$\Sigma$ \citep{2018ApJ...856...23G}.}
\bibliographystyle{apj}
|
{
"timestamp": "2018-04-17T02:07:27",
"yymm": "1804",
"arxiv_id": "1804.05248",
"language": "en",
"url": "https://arxiv.org/abs/1804.05248"
}
|
\section{Introduction}
\IEEEPARstart{T}{his} demo file is intended to serve as a ``starter file''
for IEEE Communications Society journal papers produced under \LaTeX\ using
IEEEtran.cls version 1.8b and later.
I wish you the best of success.
\hfill mds
\hfill August 26, 2015
\subsection{Subsection Heading Here}
Subsection text here.
\subsubsection{Subsubsection Heading Here}
Subsubsection text here.
\section{Conclusion}
The conclusion goes here.
\appendices
\section{Proof of the First Zonklar Equation}
Appendix one text goes here.
\section{}
Appendix two text goes here.
\section*{Acknowledgment}
The authors would like to thank...
\ifCLASSOPTIONcaptionsoff
\newpage
\fi
\section{Introduction}
\IEEEPARstart{T}{his} demo file is intended to serve as a ``starter file''
for IEEE journal papers produced under \LaTeX\ using
IEEEtran.cls version 1.8b and later.
I wish you the best of success.
\hfill mds
\hfill August 26, 2015
\subsection{Subsection Heading Here}
Subsection text here.
\subsubsection{Subsubsection Heading Here}
Subsubsection text here.
\section{Conclusion}
The conclusion goes here.
\appendices
\section{Proof of the First Zonklar Equation}
Appendix one text goes here.
\section{}
Appendix two text goes here.
\section*{Acknowledgment}
The authors would like to thank...
\ifCLASSOPTIONcaptionsoff
\newpage
\fi
\section{Introduction}
This demo file is intended to serve as a ``starter file''
for IEEE Computer Society conference papers produced under \LaTeX\ using
IEEEtran.cls version 1.8b and later.
I wish you the best of success.
\hfill mds
\hfill August 26, 2015
\subsection{Subsection Heading Here}
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\subsubsection{Subsubsection Heading Here}
Subsubsection text here.
\section{Conclusion}
The conclusion goes here.
\ifCLASSOPTIONcompsoc
\section*{Acknowledgments}
\else
\section*{Acknowledgment}
\fi
The authors would like to thank...
\section{Introduction}
\IEEEPARstart{T}{his} demo file is intended to serve as a ``starter file''
for IEEE \textsc{Transactions on Magnetics} journal papers produced under \LaTeX\ using
IEEEtran.cls version 1.8b and later.
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\hfill August 26, 2015
\subsection{Subsection Heading Here}
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\subsubsection{Subsubsection Heading Here}
Subsubsection text here.
\section{Conclusion}
The conclusion goes here.
\appendices
\section{Proof of the First Zonklar Equation}
Appendix one text goes here.
\section{}
Appendix two text goes here.
\section*{Acknowledgment}
The authors would like to thank...
\ifCLASSOPTIONcaptionsoff
\newpage
\fi
\section{Introduction}\label{sec:introduction}}
\else
\section{Introduction}
\label{sec:introduction}
\fi
\IEEEPARstart{T}{his} demo file is intended to serve as a ``starter file''
for IEEE Computer Society journal papers produced under \LaTeX\ using
IEEEtran.cls version 1.8b and later.
I wish you the best of success.
\hfill mds
\hfill August 26, 2015
\subsection{Subsection Heading Here}
Subsection text here.
\subsubsection{Subsubsection Heading Here}
Subsubsection text here.
\section{Conclusion}
The conclusion goes here.
\appendices
\section{Proof of the First Zonklar Equation}
Appendix one text goes here.
\section{}
Appendix two text goes here.
\ifCLASSOPTIONcompsoc
\section*{Acknowledgments}
\else
\section*{Acknowledgment}
\fi
The authors would like to thank...
\ifCLASSOPTIONcaptionsoff
\newpage
\fi
\section{Introduction}
This document is a model and instructions for \LaTeX.
Please observe the conference page limits.
\section{Ease of Use}
\subsection{Maintaining the Integrity of the Specifications}
The IEEEtran class file is used to format your paper and style the text. All margins,
column widths, line spaces, and text fonts are prescribed; please do not
alter them. You may note peculiarities. For example, the head margin
measures proportionately more than is customary. This measurement
and others are deliberate, using specifications that anticipate your paper
as one part of the entire proceedings, and not as an independent document.
Please do not revise any of the current designations.
\section{Prepare Your Paper Before Styling}
Before you begin to format your paper, first write and save the content as a
separate text file. Complete all content and organizational editing before
formatting. Please note sections \ref{AA}--\ref{SCM} below for more information on
proofreading, spelling and grammar.
Keep your text and graphic files separate until after the text has been
formatted and styled. Do not number text heads---{\LaTeX} will do that
for you.
\subsection{Abbreviations and Acronyms}\label{AA}
Define abbreviations and acronyms the first time they are used in the text,
even after they have been defined in the abstract. Abbreviations such as
IEEE, SI, MKS, CGS, ac, dc, and rms do not have to be defined. Do not use
abbreviations in the title or heads unless they are unavoidable.
\subsection{Units}
\begin{itemize}
\item Use either SI (MKS) or CGS as primary units. (SI units are encouraged.) English units may be used as secondary units (in parentheses). An exception would be the use of English units as identifiers in trade, such as ``3.5-inch disk drive''.
\item Avoid combining SI and CGS units, such as current in amperes and magnetic field in oersteds. This often leads to confusion because equations do not balance dimensionally. If you must use mixed units, clearly state the units for each quantity that you use in an equation.
\item Do not mix complete spellings and abbreviations of units: ``Wb/m\textsuperscript{2}'' or ``webers per square meter'', not ``webers/m\textsuperscript{2}''. Spell out units when they appear in text: ``. . . a few henries'', not ``. . . a few H''.
\item Use a zero before decimal points: ``0.25'', not ``.25''. Use ``cm\textsuperscript{3}'', not ``cc''.)
\end{itemize}
\subsection{Equations}
Number equations consecutively. To make your
equations more compact, you may use the solidus (~/~), the exp function, or
appropriate exponents. Italicize Roman symbols for quantities and variables,
but not Greek symbols. Use a long dash rather than a hyphen for a minus
sign. Punctuate equations with commas or periods when they are part of a
sentence, as in:
\begin{equation}
a+b=\gamma\label{eq}
\end{equation}
Be sure that the
symbols in your equation have been defined before or immediately following
the equation. Use ``\eqref{eq}'', not ``Eq.~\eqref{eq}'' or ``equation \eqref{eq}'', except at
the beginning of a sentence: ``Equation \eqref{eq} is . . .''
\subsection{\LaTeX-Specific Advice}
Please use ``soft'' (e.g., \verb|\eqref{Eq}|) cross references instead
of ``hard'' references (e.g., \verb|(1)|). That will make it possible
to combine sections, add equations, or change the order of figures or
citations without having to go through the file line by line.
Please don't use the \verb|{eqnarray}| equation environment. Use
\verb|{align}| or \verb|{IEEEeqnarray}| instead. The \verb|{eqnarray}|
environment leaves unsightly spaces around relation symbols.
Please note that the \verb|{subequations}| environment in {\LaTeX}
will increment the main equation counter even when there are no
equation numbers displayed. If you forget that, you might write an
article in which the equation numbers skip from (17) to (20), causing
the copy editors to wonder if you've discovered a new method of
counting.
{\BibTeX} does not work by magic. It doesn't get the bibliographic
data from thin air but from .bib files. If you use {\BibTeX} to produce a
bibliography you must send the .bib files.
{\LaTeX} can't read your mind. If you assign the same label to a
subsubsection and a table, you might find that Table I has been cross
referenced as Table IV-B3.
{\LaTeX} does not have precognitive abilities. If you put a
\verb|\label| command before the command that updates the counter it's
supposed to be using, the label will pick up the last counter to be
cross referenced instead. In particular, a \verb|\label| command
should not go before the caption of a figure or a table.
Do not use \verb|\nonumber| inside the \verb|{array}| environment. It
will not stop equation numbers inside \verb|{array}| (there won't be
any anyway) and it might stop a wanted equation number in the
surrounding equation.
\subsection{Some Common Mistakes}\label{SCM}
\begin{itemize}
\item The word ``data'' is plural, not singular.
\item The subscript for the permeability of vacuum $\mu_{0}$, and other common scientific constants, is zero with subscript formatting, not a lowercase letter ``o''.
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\item Be aware of the different meanings of the homophones ``affect'' and ``effect'', ``complement'' and ``compliment'', ``discreet'' and ``discrete'', ``principal'' and ``principle''.
\item Do not confuse ``imply'' and ``infer''.
\item The prefix ``non'' is not a word; it should be joined to the word it modifies, usually without a hyphen.
\item There is no period after the ``et'' in the Latin abbreviation ``et al.''.
\item The abbreviation ``i.e.'' means ``that is'', and the abbreviation ``e.g.'' means ``for example''.
\end{itemize}
An excellent style manual for science writers is \cite{b7}.
\subsection{Authors and Affiliations}
\textbf{The class file is designed for, but not limited to, six authors.} A
minimum of one author is required for all conference articles. Author names
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affiliation. Please keep your affiliations as succinct as possible (for
example, do not differentiate among departments of the same organization).
\subsection{Identify the Headings}
Headings, or heads, are organizational devices that guide the reader through
your paper. There are two types: component heads and text heads.
Component heads identify the different components of your paper and are not
topically subordinate to each other. Examples include Acknowledgments and
References and, for these, the correct style to use is ``Heading 5''. Use
``figure caption'' for your Figure captions, and ``table head'' for your
table title. Run-in heads, such as ``Abstract'', will require you to apply a
style (in this case, italic) in addition to the style provided by the drop
down menu to differentiate the head from the text.
Text heads organize the topics on a relational, hierarchical basis. For
example, the paper title is the primary text head because all subsequent
material relates and elaborates on this one topic. If there are two or more
sub-topics, the next level head (uppercase Roman numerals) should be used
and, conversely, if there are not at least two sub-topics, then no subheads
should be introduced.
\subsection{Figures and Tables}
\paragraph{Positioning Figures and Tables} Place figures and tables at the top and
bottom of columns. Avoid placing them in the middle of columns. Large
figures and tables may span across both columns. Figure captions should be
below the figures; table heads should appear above the tables. Insert
figures and tables after they are cited in the text. Use the abbreviation
``Fig.~\ref{fig}'', even at the beginning of a sentence.
\begin{table}[htbp]
\caption{Table Type Styles}
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
\textbf{Table}&\multicolumn{3}{|c|}{\textbf{Table Column Head}} \\
\cline{2-4}
\textbf{Head} & \textbf{\textit{Table column subhead}}& \textbf{\textit{Subhead}}& \textbf{\textit{Subhead}} \\
\hline
copy& More table copy$^{\mathrm{a}}$& & \\
\hline
\multicolumn{4}{l}{$^{\mathrm{a}}$Sample of a Table footnote.}
\end{tabular}
\label{tab1}
\end{center}
\end{table}
\begin{figure}[htbp]
\centerline{\includegraphics{fig1.png}}
\caption{Example of a figure caption.}
\label{fig}
\end{figure}
Figure Labels: Use 8 point Times New Roman for Figure labels. Use words
rather than symbols or abbreviations when writing Figure axis labels to
avoid confusing the reader. As an example, write the quantity
``Magnetization'', or ``Magnetization, M'', not just ``M''. If including
units in the label, present them within parentheses. Do not label axes only
with units. In the example, write ``Magnetization (A/m)'' or ``Magnetization
\{A[m(1)]\}'', not just ``A/m''. Do not label axes with a ratio of
quantities and units. For example, write ``Temperature (K)'', not
``Temperature/K''.
\section*{Acknowledgment}
The preferred spelling of the word ``acknowledgment'' in America is without
an ``e'' after the ``g''. Avoid the stilted expression ``one of us (R. B.
G.) thanks $\ldots$''. Instead, try ``R. B. G. thanks$\ldots$''. Put sponsor
acknowledgments in the unnumbered footnote on the first page.
\section*{References}
Please number citations consecutively within brackets \cite{b1}. The
sentence punctuation follows the bracket \cite{b2}. Refer simply to the reference
number, as in \cite{b3}---do not use ``Ref. \cite{b3}'' or ``reference \cite{b3}'' except at
the beginning of a sentence: ``Reference \cite{b3} was the first $\ldots$''
Number footnotes separately in superscripts. Place the actual footnote at
the bottom of the column in which it was cited. Do not put footnotes in the
abstract or reference list. Use letters for table footnotes.
Unless there are six authors or more give all authors' names; do not use
``et al.''. Papers that have not been published, even if they have been
submitted for publication, should be cited as ``unpublished'' \cite{b4}. Papers
that have been accepted for publication should be cited as ``in press'' \cite{b5}.
Capitalize only the first word in a paper title, except for proper nouns and
element symbols.
For papers published in translation journals, please give the English
citation first, followed by the original foreign-language citation \cite{b6}.
\section{Introduction}
This demo file is intended to serve as a ``starter file''
for IEEE conference papers produced under \LaTeX\ using
IEEEtran.cls version 1.8b and later.
I wish you the best of success.
\hfill mds
\hfill August 26, 2015
\subsection{Subsection Heading Here}
Subsection text here.
\subsubsection{Subsubsection Heading Here}
Subsubsection text here.
\section{Conclusion}
The conclusion goes here.
\section*{Acknowledgment}
The authors would like to thank...
\section{Introduction}
Named Data Networking (NDN)~\cite{zhang2010named} is a network layer protocol that is being actively researched with the hope of serving as a replacement for the IP protocol. nTorrent~\cite{mastorakis2017ntorrent} is an NDN peer-to-peer file sharing application. The current implementation runs with a few modifications to the base ns-3 network simulator in order to compile and run successfully. The idea behind this paper is to extend the functionality of nTorrent and make it run on top of ndnSIM~\cite{mastorakis2017evolution, mastorakis2016ndnsim, mastorakis2015ndnsim} that features full integration with the NDN Forwarding Daemon (NFD)~\cite{nfd-dev} for simulations.
Our code is available at \url{https://github.com/akshayraman/scenario-ntorrent}.
\section*{Acknowledgments}
We would like to thank Spyridon Mastorakis for providing us with all the help needed to pursue this project.
\bibliographystyle{plain}
\section{Related Work}
nTorrent has been designed to have a hierarchical file structure. At the top of the hierarchy, the .torrent file (Figure~\ref{Figure:torrent-file}) contains the name, size, type of the torrent file, and the signature of the original publisher. It also includes the names of the file manifests that make up the torrent file. Each file manifest (Figure~\ref{Figure:file-manifest}) contains its name, signature of the original publisher, and a list of names of the packets that make up that particular file. Using these names, Interest packets can be sent out to request for the corresponding files or packets.
The fetching strategy currently implemented starts with requesting for the first packet of the first file to the last packet of the last file. Each name also follows a certain naming convention that can help easily identify the name of the torrent file, the file names, and the individual packets in a file. This name is also used by the routers to verify the integrity of the the packet.
\begin{figure}[h]
\centering
\includegraphics[width=\columnwidth]{figures/torrent-file}
\caption{\small Structure of a torrent-file}
\label{Figure:torrent-file}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=\columnwidth]{figures/manifest}
\caption{\small Structure of a file manifest}
\label{Figure:file-manifest}
\end{figure}
|
{
"timestamp": "2018-04-17T02:07:43",
"yymm": "1804",
"arxiv_id": "1804.05250",
"language": "en",
"url": "https://arxiv.org/abs/1804.05250"
}
|
\section{Introduction}\label{Sec:preface}
State surfaces are spanning surfaces of links that are obtained from link diagrams. Their construction is guided by the combinatorics underlying Kauffman's construction of the Jones link polynomial
via \emph{state models}. Geometric properties of state surfaces are often dictated by simple link diagrammatic criteria, and the surfaces themselves carry important information about geometric structures of link
complements. On the other hand, certain state surfaces carry spines (state graphs) that can be used to compute the Jones polynomial of links. From this point of view,
state surfaces provide a tool for establishing relations between Jones polynomials and topological link invariants, such as the crosscap number or invariants coming from geometric structures on link complements (e.g. hyperbolic volume). In this article
we survey the construction of state surfaces of links and some of their recent applications.
\section{Definitions and examples}\label{Sec:Intro}
For a link $K$ in $S^3$, $D = D(K)$ will denote a link diagram, in the equatorial
$2$--sphere of $S^3$. We will often abuse by referring to the projection
$2$--sphere using the common term projection plane. In
particular, $D(K)$ cuts the projection ``plane'' into compact regions each of which is a polygon with vertices at the crossings of $D$.
Given a crossing on a link diagram $D(K)$ there are two ways to resolve
it; the $A$- resolution and the $B$-resolution as shown in Figure \ref{resolve}. The figure is borrowed from \cite{KaLee}. Note that if the link $K$ is oriented, only one of the two resolutions at each crossing
will respect the orientation of $K$.
A Kauffman state $\sigma$ on $D(K)$ is a choice of one of these two resolutions at each crossing of $D(K)$ \cite{states}.
For each state $\sigma$ of a link diagram the \emph {state graph} ${{\mathbb{G}}}_{\sigma}$ is constructed as follows: The result of applying $\sigma$
to $D(K)$
is a collection $v_{\sigma}(D)$ of non-intersecting
circles in the plane, called \emph{state circles}, together with embedded arcs recording the crossing
splice.
Next we obtain the \emph{state surface} $S_{\sigma}$, as follows: Each circle of $v_{\sigma}(D)$ bounds a disk in $S^3$. This collection of disks can be disjointly embedded in the ball below the projection plane.
At each crossing of $D(K)$, we connect the pair of neighboring disks by a half-twisted band to construct a surface $S_{\sigma} \subset S^3$ whose boundary is $K$.
\begin{figure}[ht]
\includegraphics[scale=.6]{surA.eps}
\hspace{2cm}
\includegraphics[scale=.6]{surB.eps}
\label{resolve}
\caption{The $A$-resolution (left), the $B$-resolution (right) of a crossing and their contribution to state surfaces.
}
\end{figure}
\begin{example} \label{AB}{\rm{Given an oriented link diagram $D=D(K)$, the Seifert state, denoted by $s(D)$, is the one that assigns to each crossing of $D$ the resolution that is consistent with the orientation of $D$.
The corresponding state surface $S_s=S_s(D)$ is oriented (a.k.a. a Seifert surface). The process of constructing $S_s$ is known a {\emph{Seifert's algorithm}} \cite{Lickbook}.
By applying the
$A$--resolution to each crossing of $D$, we obtain a
crossing--free diagram $s_A(D)$.
Its state graph, denoted by
${\mathbb{G}_A}={\mathbb{G}_A}(D)$, is called the all--$A$ state graph and the corresponding state surface is denoted by $S_A=S_A(D)$. An example is shown in Figure \ref{fig:statesurface},
which is borrowed from \cite{GutsBook}. Similarly, for the all--$B$ state the crossing--free resulting diagram is denoted by
$s_B(D)$, the state
graph is denoted ${\mathbb{G}_B}$, and the state surface by $S_B$.}}\end{example}
\begin{figure}
\includegraphics[scale=.8]{state-surface.eps}
\caption{Left to right: A diagram, the all-$A$ state graph ${\mathbb{G}_A}$ and the corresponding state
surface $S_A$.}
\label{fig:statesurface}
\end{figure}
By construction, $\mathbb{G}_\sigma$ has one vertex for every circle of
$v_\sigma$ (i.e. for every disk in $S_\sigma$), and one edge for every
half--twisted band in $S_\sigma$. This gives a natural embedding of
$\mathbb{G}_\sigma $ into the surface, where vertices are embedded
into the corresponding disks, and edges run through the
corresponding half-twisted bands. Hence, $\mathbb{G}_\sigma$ is a spine for $S_\sigma$.
\begin{lemma}\label{lemma:ga-spine}
The surface $S_\sigma$ is orientable if and only if
$\mathbb{G}_\sigma$ is a bipartite graph.
\end{lemma}
\begin{proof}
Recall that a graph is bipartite if and only if all cycles (i.e. paths from any vertex to itself)
contain an even number of edges.
If $\mathbb{G}_\sigma$ is
bipartite, we may assign an orientation on
$S_\sigma$, as follows: Pick a normal direction to one disk,
corresponding to a vertex of $\mathbb{G}_\sigma$, extend
over half--twisted bands to orient every adjacent disk, and continue
inductively. This inductive process
$S_\sigma$ will not run into a contradiction since
every cycle in $\mathbb{G}_\sigma$ has even number of edges. Thus $S_\sigma$ is
a two--sided surface in $S^3$, hence orientable. This is the case with the example of Figure \ref{fig:statesurface}.
Conversely, suppose $\mathbb{G}_\sigma$ is not
bipartite, hence contains a cycle with an odd number of edges. By embedding
$\mathbb{G}_\sigma$ as a spine of $S_\sigma$, as above, we see that this cycle is an
orientation--reversing loop in $S_\sigma$.\qed
\end{proof}
\section{Genus and crosscap number of alternating links }\label{Sec:genus}
The genus of an orientable surface $S$ with with $k$ boundary components is defined to be $1-(\chi(S)+k)/2$,
where $\chi(S)$ is the Euler characteristic of $S$ and the \emph{crosscap number} of a non-orientable surface with $k$ boundary components is defined to be
$2-\chi(S)-k$.
\begin{definition}Every link in $S^3$ bounds both orientable and non-orientable surfaces.
The \emph{genus} of an oriented link $K$, denoted by $g(K)$, is the minimum genus over all orientable surfaces $S$
bounded by $K$. That is we have $\partial S=K$.
The \emph{crosscap number} (a.k.a. non-orientable genus) of a link $K$, denoted by $C(K)$, is the minimum crosscap number over all non-orientable surfaces
spanned by $K$.
\end{definition}
For \emph{ alternating links} the genus and the crosscap number
can be computed using state surfaces of alternating link diagrams. For the orientable case,
we recall the following classical result due to Crowell \cite{crowell} (see also \cite{Lickbook}).
\begin{theorem} \cite{crowell} Suppose that $D$ is a connected alternating diagram of a $k$-component link $K$. Then the state surface $S_s(D)$ corresponding to the Seifert state of $D$ realizes the
genus of $K$. That is we have $g(K)=1-{(\chi(S_s(D))+k)} /2$.
\end{theorem}
In \cite{Adamsstate}, Adams and Kindred used state surfaces to give an algorithm for computing crosscap numbers of alternating links.
To summarize their algorithm and state their result, consider a connected alternating diagram $D(K)$ as 4-valent a graph on $S^2$.
Each region in the complement of the graph is an $m$-gon
with vertices at the vertices of the graph.
\begin{lemma} \label{triangle} Suppose that $D(K)$ is a connected alternating link diagram whose complement has no
bigons or 1-gons. Then at least one region must be a triangle.
\end{lemma}
\begin{proof}
Let $V$, $E$, $F$
denote the number of vertices, edges and complimentary regions of $D(K)$, respectively.
Then,
$V - E + F = 2$ and
$E = 2V$, which implies that $F > V$. Suppose that none of the $F$ regions is a triangle.
Then, $F < 4V / 4 = V$ since each region has at least four vertices and each vertex can only be on at most 4 distinct regions. This is a contradiction. \qed \end{proof}
Observe that the Euler characteristic of a surface,
corresponding to a state $\sigma$, is $\chi(S_{\sigma})=v_{\sigma}-c$, where $c$ is the number of crossings on $D(K)$.
Thus to maximize $\chi(S_{\sigma})$ we must maximize the number of state circles $v_{\sigma}$.
Now we outline the algorithm from \cite{Adamsstate} that finds a surface of maximal Euler characteristic (and thus of minimum genus) over all surfaces (orientable and non-orientable) spanned by an alternating link.
\vskip 0.08in
\noindent { {\bf{Adams-Kindred algorithm:}}} \ Let $D(K)$ be a connected, alternating diagram.
\begin{enumerate}
\item Find the smallest $m$ for which the complement of the projection $D(K)$ contains an $m$-gon.
\item If $m=1$, then we resolve the corresponding crossing so that the $1$-gon becomes a state circle.
Suppose that $m=2$. Then some regions of $D(K)$ are bigons.
Create one branch of the algorithm for each bigon on $D(K)$.
Resolve the two crossings corresponding to the vertices of the bigon
so that the bigon is bounded by a state circle. See Figures 1.4 and \ref{fstep2} below. \label{step2a}
\vskip 0.02in
\item Suppose $m>2$. Then by Lemma \ref{triangle}, we have $m = 3$. Pick a triangle region on $D(K)$.
Now the process has two branches: For one branch
we resolve each crossing on the triangle's boundary so that the triangle becomes a state circle. For the other branch, we resolve each of the crossings the opposite way.
\begin{figure}[ht]
\centering
\includegraphics[scale=.15]{branch.eps}
\hspace{1cm}
\includegraphics[scale=.15]{branch1.eps}
\hspace{1cm}
\includegraphics[scale=.15]{branch2.eps}
\caption{The two branch of the algorithm for triangle regions. The is figure borrowed from
\cite{KaLee}.
}
\label{3-gon}
\end{figure}
\item Repeat Steps 1 and 2 until each branch reaches a projection without crossings. Each branch corresponds
to a Kauffman state of $D(K)$ for which there is a corresponding state surface.
Of all the branches
involved in the process choose one that has the largest number of state circles. The surface $S$ corresponding to this
state has maximal Euler characteristic over all the states corresponding to $D(K)$.
Note that, {\emph{a priori}}, more than one branches of the algorithm may lead to surfaces of maximal Euler characteristic.
\end{enumerate}
\begin{theorem}{\cite{Adamsstate}}\label{AK}
Let $S$ be any maximal Euler characteristic surface obtained via above algorithm from an alternating diagram of $k$-component link $K$.
Then,
\begin{enumerate}
\item If there is a surface $S$ as above that is non-orientable then $C(K)=2-\chi(S)-k$.
\vspace{0.1in}
\item If all the surfaces $S$ as above are orientable, we have $C(K)=3-\chi(S)-k$.
Furthermore, $S$ is a minimal genus Seifert surface of $K$ and $C(K)=2g(K)+1$.
\end{enumerate}
\end{theorem}
\smallskip
\begin{example}\label{Fig8} {\rm{Different choices of branches as well as the order in resolving bigon regions following the algorithm above, may result in different state surfaces. In particular at the end of the algorithm we may
have both orientable and non-orientable surfaces that share the same Euler characteristic:
\begin{figure}[ht]
\centering
\includegraphics[scale=.3]{fig8.eps}
\hspace{2cm}
\includegraphics[scale=.3]{fig8b1.eps}
\label{fstep0}
\caption{A diagram of $4_1$ with bigon regions 1 and 2 and the result of applying step \ref{step2a} of the algorithm to bigon 1.}
\end{figure}
Suppose that we choose the bigon labeled by 1 in the left hand side picture of Figure 1.4.
Then, for the next step of the algorithm, we have three choices of bigon regions to resolve, labeled by 1 and 2 and 3 of the figure.
\begin{figure}[ht]
\includegraphics[scale=.3]{fig8ch.eps}
\hspace{2cm}
\includegraphics[scale=.3]{fig8s.eps}
\caption{Two algorithm branches corresponding to different bigons.}
\label{fstep2}
\end{figure}
The choice of bigon 1 leads
to a non-orientable surface, shown in the left panel of Figure \ref{fstep2}, realizing the crosscap number of $4_1$, which is two.
The choice of bigon 2 leads to an orientable surface, shown in the right panel of Figure \ref{fstep2}, realizing the genus of the knot which is one.
Both surfaces realize the maximal Euler characteristic of -1.}}
\end{example}
\section{Jones polynomial and state graphs}\label{Sec:Jones}
A connected link diagram $D$ defines a 4--valent planar graph
$\Gamma \subset S^2$,
which leads to the construction of the \emph{Turaev surface} $F(D)$ as follows \cite{dasbach-futer...}:
Thicken the projection plane to $S^2 \times
[- 1, 1]$, so that $\Gamma$ lies in $S^2 \times \{0\}$. Outside a
neighborhood of the vertices (crossings) the surface intersect
$S^2 \times
[- 1, 1]$, in $\Gamma \times [- 1, 1]$. In the neighborhood of
each vertex, we insert a saddle, positioned so that the boundary
circles on $S^2 \times \{1\}$ are the
components
of the $A$--resolution $s_A(D)$, and the boundary circles on $S^2
\times \{- 1\}$ are the components of $s_B(D)$.
When $D$ is an alternating diagram, each circle of
$s_A(D)$ or $s_B(D)$ follows the boundary of a region in the
projection plane. Thus, for alternating diagrams, the surface $F(D)$
is the projection sphere $S^2$. For general diagrams, the diagram $D$ still is alternating on
$F(D)$.
The surface $F(D)$ has a natural
cellulation: the $1$--skeleton is the graph $\Gamma$ and the
$2$--cells correspond to circles of $s_A(D)$ or $s_B(D)$, hence to
vertices of ${\mathbb{G}_A}$ or ${\mathbb{G}_B}$. These $2$--cells admit a
checkerboard coloring, in which the regions corresponding to the
vertices of ${\mathbb{G}_A}$ are white and the regions corresponding to ${\mathbb{G}_B}$ are
shaded. The graph ${\mathbb{G}_A}$ (resp. ${\mathbb{G}_B}$) can
be embedded in $F(D)$ as the
adjacency graph of white (resp. shaded) regions.
The \emph{faces} of ${\mathbb{G}_A}$ (that is, regions in the complement of ${\mathbb{G}_A}$) correspond to vertices of ${\mathbb{G}_B}$, and vice versa. Hence the graphs are dual to one another on $F(D)$.
Graphs, together with such embeddings into an orientable surface, called \emph{ribbon graphs} have been studied in the literature \cite{BR}. Building on this point of view,
Dasbach, Futer, Kalfagianni, Lin and Stoltzfus
\cite{dasbach-futer...} showed that the ribbon graph embedding of ${\mathbb{G}_A}$ into the Turaev surface $F(D)$ carries at least as much information as the Jones polynomial $J_K(t)$.
To state the relevant result from \cite{dasbach-futer...},
recall that
a \emph{spanning} subgraph of ${\mathbb{G}_A}$ is a subgraph that contains all
the vertices of ${\mathbb{G}_A}$. Given a spanning subgraph ${\mathbb{G}}$ of ${\mathbb{G}_A}$ we
will use $v({\mathbb{G}})$, $e({\mathbb{G}})$ and $f({\mathbb{G}})$ to denote the number of
vertices, edges and faces of ${\mathbb{G}}$ respectively.
\begin{theorem}\cite{dasbach-futer...}\label{JPgraph} For a connected link diagram $D$, the Kauffman bracket $\langle D \rangle\in
{\mathbb{Z}}[A, A^{-1}]$ is expressed as
$$\langle D \rangle = \sum_{{\mathbb{G}}\subset
{\mathbb{G}_A}}^{\phantom{a}}\ A^{e({\mathbb{G}_A})-2e({\mathbb{G}})} (-A^2-A^{-2})^{f({\mathbb{G}})-1},$$
where ${\mathbb{G}}$ ranges over all the spanning subgraphs of ${\mathbb{G}_A}$.
\end{theorem}
Given a diagram $D=D(K)$, the {\emph{Jones polynomial}} of $K$, denoted by $J_K(t)$, is
obtained from $\langle D \rangle$ as follows: Multiply $\langle D
\rangle$ by $(-A)^{-3w(D)}$, where $w(D)$ is the \emph{writhe} of
$D$, and then substitute $A = t^{-1/4}$ \cite{states, Lickbook}.
Theorem \ref{JPgraph} leads to formulae for the coefficients of $J_K(t)$
in terms of topological quantities
of the state graphs ${\mathbb{G}_A}$ , ${\mathbb{G}_B}$ corresponding to any diagram of $K$
\cite{dasbach-futer..., dfkls:determinant}.
These formulae become
particularly effective if ${\mathbb{G}_A}, {\mathbb{G}_B}$ contain no 1-edges loops. In particular, this is the case when ${\mathbb{G}_A}, {\mathbb{G}_B}$ correspond to an alternating diagram that is \emph{reduced} (i.e. contains no redundant crossings).
\begin{corollary} \cite{dfkls:determinant} \label{coeffs} Let $D(K)$ be a reduced alternating diagram and let $\beta_K$ and $\beta'_K$ denote the second and penultimate coefficient of $J_K(t)$, respectively . Let ${\mathbb{G}'_A}$ and ${\mathbb{G}'_B}$ denote the \emph{simple} graphs obtained by removing all duplicate edges between pairs of vertices of
${\mathbb{G}}_A(D)$ and ${\mathbb{G}}_B(D)$. Then,
$$\abs{\beta_K}=1-\chi({\mathbb{G}'_B}), \ \ {\rm and} \ \ \abs{\beta'_K}=1-\chi({\mathbb{G}'_A}).$$
\end{corollary}
\section{Geometric Connections}\label{Sec:Geometry}
To a link $K$ in $S^3$ corresponds a compact 3-manifold with boundary; namely $M_K=S^3\setminus N(K)$, where $N(K)$ is an open tube around $K$.
The interior of $M_K$ is homeomorphic to the link complement $S^3\setminus K$.
In the 80's, Thurston \cite{thurston:notes} proved that link complements decompose canonically into pieces that admit locally homogeneous geometric structures.
A very common and interesting case is when the entire $S^3\setminus K$ has a hyperbolic structure, that is a metric of constant curvature $-1$ of finite volume.
By Mostow rigidity, this hyperbolic structure is unique up to isometry, hence invariants of the metric of $S^3\setminus K$ give topological invariants of $K$.
State surfaces obtained from link diagrams $D(K)$ give rise to properly embedded surfaces in $M_K$.
Many geometric properties of state surfaces can be checked through combinatorial and link diagrammatic criteria.
For instance, Ozawa \cite{ozawa} showed that
the all -$A$ surface $S_A(D)$ is $\pi_1$--injective in $M_K$ if the state graph ${\mathbb{G}_A}(D)$ contains no 1-edge loops. Futer, Kalfagianni and Purcell \cite{GutsBook} gave a different proof of Ozawa's result
and also
showed that $M_K$ is a fiber bundle over the circle with fiber
$S_A(D)$, if and only if the simple state graph ${\mathbb{G}'_A}(D)$ is a \emph{tree}.
State
surfaces have been used to obtain relations between combinatorial or Jones type link invariants and geometric invariants
of link complements.
Below we give a couple of sample of such relations. For additional applications the reader is referred to
to \cite{AdamsCuspSizeBounds, BuKa, GutsBook, fkp:TAMS, lackenby:alt-volume, LPalte} and references therein.
The first result, proven combining \cite{Adamsstate} with hyperbolic geometry techniques, relates the crosscap number and the Jones polynomial of alternating links.
It was used to determine the crosscap numbers of 283 alternating knots of \emph{knot tables} that were previously unknown \cite{Knotinfo}.
\begin{theorem}\label{thm:cupjonesknots}\cite{KaLee} Given an an alternating, non-torus knot $K$, with crosscap number $C(K)$,
we have
$$ \left\lceil \frac{ T_K}{3}\right\rceil + 1\; \leq \; C(K) \; \leq \; {\rm {min}}{ \left\{ T_K + 1, \
\left\lfloor{ \frac{s_K}{2}}\,\right\rfloor \right\}}
$$
\noindent where $T_K:= \abs{\beta_K} + \abs{\beta'_K},$
$\beta_K$,
$\beta'_K$ are
second and penultimate coefficients of $J_K(t)$ and
$s_K$ is the degree span of $J_K(t)$.
Furthermore, both bounds are sharp.
\end{theorem}
\begin{example} {\rm{For $K=4_1$ we have $J_K(t)=t^{-2}-t^{-1}+ 1-t+ t^2$. Thus $T_K=1$ and $s_k=4$ and Theorem \ref{thm:cupjonesknots} gives $C(K)=2$.}}
\end{example}
The next result gives a strong connection of the Jones polynomial
to hyperbolic geometry as it estimates volume of hyperbolic alternating links in terms of coefficients of their Jones polynomials.
The result follows by work of Dasbach and Lin \cite{dfkls:determinant} and work
of Lackenby
\cite{lackenby:alt-volume}.
\begin{theorem} \label{volume}Let $K$ be an alternating link whose exterior admits a hyperbolic structure with volume ${\rm vol}(S^3 \setminus K)$.
Then we have
\begin{equation*}
\frac{{v_{\rm oct}}}{2} (T_K-2) \leq {\rm vol}(S^3 \setminus K) \leq 10{v_{\rm tet}} (T_K - 1),
\end{equation*}
\noindent where ${v_{\rm oct}}=3.6638$ and ${v_{\rm tet}}= 1.0149$.
\end{theorem}
To establish the lower bound of Theorem \ref{volume} one looks at the state surfaces $S_A$, $S_B$ corresponding to a reduced alternating diagram $D(K)$:
Use $M_K {\backslash \backslash} S_A$ to denote the complement in $M_K$ of a collar neighborhood of $S_A$.
Jaco-Shalen-Johannson theory \cite{jaco-shalen} implies that there is a canonical way to decompose $M_K {\backslash \backslash} S_A$ along certain annuli
into three types of pieces: (i) $I$--bundles over subsurfaces of $S_A$; (ii) solid tori; and (iii) the
remaining pieces, denoted by ${\rm guts}(M,S)$. On one hand, by work Agol, Storm, and Thurston \cite{AST:guts}, the quantity
$\abs{ \chi({\rm guts}(M_K,S_A))}$ gives a lower bound
for the volume $ {\rm vol}(S^3 \setminus K)$. On the other hand, \cite{lackenby:alt-volume} shows that this quantity is equal to $1-\chi({\mathbb{G}'_A})$, which by Corollary \ref{coeffs}
is $\abs{\beta'_K}$. A similar consideration applies to the surface $S_B$ giving the lower bound of Theorem \ref{volume}. The approach was developed and generalized to non-alternating links in \cite{GutsBook}.
\bibliographystyle{plain}
|
{
"timestamp": "2018-04-17T02:08:51",
"yymm": "1804",
"arxiv_id": "1804.05281",
"language": "en",
"url": "https://arxiv.org/abs/1804.05281"
}
|
\section{Introduction}
\label{sec:introduction}
The era of precision cosmology will enable us to distinguish between competing gravitational models. While the concordance cosmology of \lcdm{} has performed remarkably well over the past two decades \cite{2016PDU....12...56B}, there exist theoretical problems\footnote{Examples include fine-tuning of $\Lambda$, the lack of direct detection of CDM, possible fine-tuning of inflation \cite{Clifton:2011jh}.} and tensions between observations measured at early and late times.\footnote{Compare measurements with Planck 2015 \cite{2016A&A...594A..13P} to the values of $\sigma_8$ from weak lensing and cluster number counts, $H_0$ from Lyman-$\alpha$ measurements of BAO, $f\sigma_8$ from redshift-space distortions \cite{2016PDU....12...56B}.} Hence a plethora of MG theories (\eg{} \cite{Clifton:2011jh}), and the surge in interest to probe a range of MG models which may alleviate these issues.
The HMF, i.e. the number density of haloes within a given mass range, is the simplest (one point) statistic one can use to study structure formation
and evolution in the Universe. It also possesses the remarkable property that, given GR, it is (nearly-)universal, in the sense that its functional form does
not depend on redshift or cosmology \cite{Zentner:2006vw,2017MNRAS.467.3454B}. These features make the HMF the ideal candidate to investigate
MG theories. Can we apply GR HMFs (and specifically \lcdm{}) to these other theories of gravity? In other words is the
HMF also universal across (\ie{} insensitive to) different gravity theories?
The simplest theoretical model to understand the origin and evolution of the HMF is called the Press-Schechter ansatz \cite{1974ApJ...187..425P} which derived the fraction of haloes above a given mass for the first time. Excursion set theory was then proposed in \cite{1991ApJ...379..440B} to set the Press-Schechter result
on a more solid theoretical footing. This approach relies upon two key quantities. The \lq\lq{}collapse density\rq\rq{} $\delta_{c}$ is the solution to the boundary value problem of spherical collapse of an overdensity to form a halo. The \lq\lq{}resolution\rq\rq{}\footnote{We retain the term used by Bond et al. in their excursion set paper \cite{1991ApJ...379..440B}: other authors use various different names for this quantity.} $S$ is the variance in the (linear) overdensity field $\delta$ for a given radius. Then one can map $S$ from a radius to a halo mass $M$ via a window function. The fundamental idea behind the excursion set formalism is the realisation that the fraction of haloes above a certain mass can be obtained by solving a diffusion problem in the phase space $(\delta, S)$ with an absorbing barrier given by the collapse density $\delta_{c}$. This model has already been applied to a number of MG scenarios \cite{2012MNRAS.425..730L,2012MNRAS.421.1431L,2013JCAP...11..056B,2014PhRvD..89b3523T,2011PhRvD..83f3511P}.
A pre-requisite for using the HMF as a probe of MG is a firm understanding of how it is typically derived in GR. Superficially this may be obvious: the analytical HMF of Press-Schechter has long been supplanted by a range of fitting functions (summarised in \eg{} \cite{2013A&C.....3...23M}) whose free parameters are calibrated using $N$-body simulations. However, this calibration only performed for certain values of the cosmological parameters and a certain range in mass and redshift. It is the fitting function which is deemed universal ---in the sense that it can be applied to a range of redshifts and cosmologies \cite{2008ApJ...688..709T}--- rather than the HMF itself. There has been disagreement over whether \cite{2014MNRAS.439.3156J,Zentner:2006vw,2007ApJ...671.1160L} or not \cite{2007MNRAS.374....2R,2006ApJ...646..881W,2003MNRAS.346..565R} this universality holds --- and these papers only test a handful of the competing fitting functions. Despite these shortcomings, it is widely believed that once various obscuring factors are addressed, the deviations from non-universality in the calibrated HMFs are of a few percent \cite{2008ApJ...688..709T}. In particular, if one uses the correct measurement of halo over-density \cite{Despali:2015yla} and includes $z$-dependence in both the over-density and the resolution, it is thought that the best-fit parameter values provide an HMF which applies over several decades of mass and up to $z = 30$,\cite{2007MNRAS.374....2R}, at least for a fixed cosmology \cite{2011MNRAS.410.1911C}.
Unfortunately, due to the additional computation time and complexity required to run $N$-body simulations in MG, one cannot take the GR approach described in the previous paragraph. Instead, one typically resorts to using either the Press-Schechter or the Sheth-Tormen functions (although \cite{2017JCAP...03..012V} uses the Peacock function). However, there appears to be little consensus on how the fifth-force degree of freedom affects the suitability of the fitting functions. Some authors use the Sheth-Tormen fit on the grounds that this is considered satisfactory in the GR simulations (symmetron \cite{2014PhRvD..89b3523T}, $\fr$ \cite{Lombriser:2013wta}, DGP \cite{2010PhRvD..81f3005S}). A counter-example is the Galileon HMF: the authors of \cite{2013JCAP...11..056B} reason that the assumptions used in the Sheth-Tormen model are unlikely to hold, given the Galileon fifth-force modification to GR, and so use the Press-Schechter fit.
These examples illustrate the additional theoretical and practical complications when deriving an appropriate HMF in MG.
This paper thus seeks to address these issues. Given the profuse and rapidly evolving landscape of MG models \footnote{The recent neutron star merger detection GW170817 (and its electromagnetic counterpart GRB 170817A) seemingly ruled out complex Horndeski gravity models (but not simpler models such as $f(\R)$ because these latter do not change the speed at which gravitational waves propagate) \cite{Baker2017}}, we choose to restrict ourselves to \HS{}-$f(\R)$, Symmetron and DGP gravity. This selection is intended to represent a range of MG families and screening mechanisms, while ensuring that each theory is sufficiently related to take a unified approach. First we outline the various screened models of MG used in \cref{ch:screened_mg_theories}. More detail is provided on: $f(\R)$ in \cref{sub:fr_gravity}, Symmetron in \cref{sub:symmetron_gravity} and DGP in \cref{sub:dgp_gravity}. Then \cref{sec:the_halo_mass_function} describes the theoretical approach to the HMF. \cref{sub:spherical_collapse_in_mg} describes how to extend the spherical collapse model from GR to MG; \cref{sec:excursion_set_theory} outlines the excursion set theorem, which is still used to derive the HMF in MG; \cref{sub:fitting_functions} summarises the fitting functions used in this paper and why they are universal. Some technical details are passed over in this section and described in \cref{app:technical_aspects_of_spherical_collapse_in_mg}. Next we describe the numerical work. \cref{sub:n_body_simulations_haloes} outlines the simulation parameters and algorithms used for the various $N$-body simulations and the extraction of the HMF from the simulations; \cref{sub:bayesian_inference} outlines the theory used to calculate the best-fit values for the free parameters in the HMFs. (We calibrate the \lq\lq{}nested sampling\rq\rq{} algorithm and describe the data processing effects in \cref{app:calibration_of_multinest}). The results section discusses the questions raised at the beginning of the introduction: \cref{sub:assuming_concordance_cosmology} shows how MG manifests itself in the free parameter values even when assuming a \lcdm{} HMF; \cref{sub:recalibrating_best_fit_parameters} recalibrates the fitting functions using the same gravity model in both the $N$-body simulations and the HMFs. Finally, we summarise our method and key results in \cref{sub:summary} and discuss avenues for further work in \cref{sub:further_work}.
The conventions used throughout this paper are:
\begin{itemize}
\item Units: $c = 1$, Einstein constant $\kappa = 8\pi G/c^{2}$, Planck mass $M_{Pl} = \kappa^{-1/2}$
\item Gauge choice is conformal Newtonian with metric signature $(-+++)$
\begin{equation*}
\dx s^{2} = a^{2}(\tau) \left[ -(1+2\Psi) \dx \tau^2 + (1 - 2\Phi) \delta_{ij} \dx \four<i>{x} \dx \four<j>{x} \right]
\end{equation*}
\item The full spacetime metric has Greek indices ranging from $0$ to $3$ while flat spatial hypersurfaces have Roman indices ranging from $1$ to $3$.
\end{itemize}
This accounts for a variety of sign changes in some formulae compared to the equations in the citations.
\section{Screened gravity theories}
\label{ch:screened_mg_theories}
The MG theories studied in this paper share a common trait to ensure that the successes of GR are unaffected by any modifications. The \emph{screening mechanism} is a technique whereby fifth-force modifications to GR are \lq\lq{}screened away\rq\rq{} in regions where the theory must mimic GR in order to be experimentally viable. A range of such techniques exists, so we limit ourselves to three popularly-held categories of screening---\emph{chameleon} \cite{Khoury:2003rn}, \emph{symmetron} \cite{2010PhRvL.104w1301H,2011PhRvD..84j3521H} and \emph{Vainshtein} \cite{Vainshtein:1972sx}---by selecting one theory from each family, namely: \HS{} $\fr$ (\cite{0705.1158}; chameleon), Hinterbichler-Khoury symmetron \cite{2010PhRvL.104w1301H} and \DGP{} (\cite{2000PhLB..485..208D}; Vainshtein). Our specific MG models and parameter values are detailed in \cref{app:screened_gravity_theories}.
The background evolution of these models is always close to \lcdm{} for viable values of the free parameters. Accordingly, we only employ the MG modification in computing the spherical collapse of the haloes within a \lcdm{} background.
By applying the quasi-static approximation in the weak-field limit, we can express the effect of the scalar field in the halo model without evolving the full scalar field equations of motion. Does this produces a reasonable approximation to the perturbed field equations and the equation of motion of the scalar field? In model-specific contexts, \cite{PhysRevD.80.043001} asserts that the QSA is appropriate on sub-horizon scales for DGP, \cite{PhysRevD.89.084023} finds $< 1\%$ changes in the local power spectrum for Symmetron models and \cite{Noller:2013wca} conclude that errors in the matter overdensity are $< 5\%$ for Hu-Sawicki $\fr$ (albeit including super-horizon scales where one expects the QSA to break down). In general, the slow-rolling nature of the scalar field which produces \lcdm{}-like behaviour also causes the QSA matter overdensity to be accurate at the percent level on sub-horizon scales \cite{Noller:2013wca}.
The resulting equations relate (after copious algebra: see \cite{2011PhLB..706..123D} for details) the metric perturbations $\Phi$, $\Psi$ to the perturbed field $\chi$ and gauge-invariant matter density $\delta$. More algebraic manipulation (for details see \cite{2011PhLB..706..123D}) allows us to recast this as a Poisson equation. Following \cite{Lombriser:2013wta,2012MNRAS.421.1431L} one can use the excursion set paradigm to transform this into an equation for the evolution of the critical overdensity for halo collapse $\delta_{c}$, which we call the \lq\lq{}barrier density\rq\rq{} in this paper. Thus we obtain our effective multiplication of $G_{\mathrm{eff}} = (1 + F_{\mathrm{eff}}) G_{N}$ in the excursion set ODE which ultimately determines the value of $\delta_{c}$ in the theoretical HMF. The exact expressions for $F_{\mathrm{eff}}$ are lengthy: we present them in \cref{app:screened_gravity_theories} for each MG model considered here.
Let us characterise the various types of screening used in this paper. Screening mechanisms add new terms to the GR action, which are designed to suppress the non-GR modifications under certain conditions. By construction, this suppression of the fifth-force modification happens on non-linear regimes. The conditions required for screening divide the mechanism into three groups: chameleon, symmetron and Vainshtein. The different screening methods correspond to different behaviours of the Lagrangian, which we can classify by expanding about some value $\phi_0$ by an infinitesimal value $\left( \delta \phi \right)$ \cite{2015CQGra..32x3001B}:
\begin{equation} \label{eq:Lagrangian_screening}
\mathcal{L} = \frac{1}{2} \R M^{2}_{\mathrm{Pl}}
+ \four[\mu]{\partial} \left( \delta\phi \right) \four[\nu]{\partial} \left( \delta\phi \right) \four<\mu\nu>{Z} \left( \phi_0 \right)
+ \left( \delta\phi \right) m^{2} \left( \phi_0 \right)
+ \frac{\rho_{m}}{M_{\mathrm{Pl}}} \beta \left( \phi_0 \right)
\end{equation}
In low density regions the scalar field takes a value $\phi_0 = \phi_{\mathrm{low}}$, for which:@
\begin{equation} \label{eq:gamma_defn}
\gamma \equiv \abs{ \frac{\vec{F}_{\phi}}{\vec{F}_{N}} }
\propto \beta^{2} \left( \phi_{\mathrm{low}} \right) \sim 0
\end{equation}
and we see that the contribution of the fifth force $\lvert \vec{F}_{\phi} \rvert$ is non-negligible compared to the Newtonian value $\lvert \vec{F}_{N} \rvert$. Compare this to the high-density value $\phi_0 = \phi_{\mathrm{high}}$, for which:
\begin{subequations} \label{eq:gamma}
\begin{align}
\gamma &\ll 1 &&\text{by definition}
\intertext{which can only be produced by:}
\beta \left( \phi_{\mathrm{high}} \right) &\ll \beta \left( \phi_{\mathrm{low}} \right) &&\text{matter coupling suppressed} &&\text{(symmetron)} \\
m \left( \phi_{\mathrm{high}} \right) &\gg m \left( \phi_{\mathrm{low}} \right) &&\text{large local mass} &&\text{(chameleon)} \\
\four<\mu\nu>{Z} \left( \phi_{\mathrm{high}} \right) &\gg \four<\mu\nu>{Z}\left( \phi_{\mathrm{low}} \right) &&\text{weakened matter source} &&\text{(Vainshtein)}
\end{align}
\end{subequations}
This permits us to classify a screened MG theory not by abstract considerations (\textit{i.e.} which conditions are relaxed in Lovelock's Theorem) but rather by the practicalities of the mechanism by which it evades local tests of gravity.
\cref{app:screened_gravity_theories} provides more detail on the fifth-force modification caused by various MG theories used in this paper. We have already demonstrated how to use the effective fifth-force contribution to $G$ to find the density required for collapse of a spherical top-hat in \cite{2017JCAP...03..012V}. We now turn our attention to the halo mass functions in the next section.
\section{Generalising the halo mass function from GR to MG}
\label{sec:the_halo_mass_function}
The halo mass function $n(M)$ is defined to be the number density of dark matter haloes in a given mass interval at a certain redshift. This is closely related to the first-crossing distribution $f(S)$ from excursion-set theory. This is the probability for a given random walk in the excursion-set phase space $\left( \delta , S \right)$ to be absorbed at resolution $S$ when the over-density $\delta$ reaches the collapse density $\delta_c$. In turn, $f(S)$ can be expressed in terms of a \lq\lq{}universal\rq\rq{} fitting function $F(\nu)$, which is invariant under changes in redshift and cosmological parameters. The aim of this next section is to \lq\lq{}unpack\rq\rq{} the details of this process, including precisely defining the key quantities $\delta, S, \delta_{c}, \nu$ and the functions $f(S)$ and $F(\nu)$. This enables us to extend the existing formalism to calculate the HMF from GR to the context of screened MG.
In \cref{sub:spherical_collapse_in_mg} we show how the critical density $\delta_{c}$ forms a drifting-and-diffusing barrier which absorbs the trajectories in the excursion-set formalism. We show how to determine this \lq\lq{}barrier density\rq\rq{} via the solution to an ODE which can be tailored to each theory of gravity using an effective Newton's constant. This equation encapsulates the modifications to non-linear collapse from MG.
In \cref{sec:excursion_set_theory} we build upon this result by deriving the first-crossing distribution $f(S)$ from the barrier density. We generalise the theoretical approach provided by excursion set theory from GR to MG.
In \cref{sub:fitting_functions} we apply this method to a variety of empirical fitting functions. These are derived from and calibrated using $N$-body simulations, and we discuss the additional complexities which this presents in MG. In particular we focus on the various ways to account for environment dependence---a problem peculiar to chameleon MG theories---and in \cref{sub:including_denv_barrier_in_mg} we analyse the accuracy of both existing methods and the novel methods which we propose in this paper.
\subsection{Spherical collapse in MG}
\label{sub:spherical_collapse_in_mg}
\begin{figure}
\captionsetup{skip=-1pt}
\begin{center}
\subfloat[{\lcdm{}, DGP1: $r_{c} = 1.2 c/H_{0}$, DGP2: $r_{c} = 5.6 c/H_{0}$, SymA: $a_{\mathrm{SSB}} = 0.5$, SymB: $a_{\mathrm{SSB}} = 0.33$}]{\label{subfig:deltac_lnM_all}\includegraphics[keepaspectratio,width=0.95\textwidth]{deltac_lnM_all}} \\
\subfloat[{F6: $\fr[6]$}]{\label{subfig:deltac_lnM_fr6}\includegraphics[keepaspectratio,width=0.49\textwidth]{plot_deltac_lnM_F6}}
\subfloat[{F6: $\fr[6]$}]{\label{subfig:deltac_denv_fr6}\includegraphics[keepaspectratio,width=0.49\textwidth]{plot_deltac_denv_F6}} \\
\subfloat[{F5: $\fr[5]$}]{\label{subfig:deltac_lnM_fr5}\includegraphics[keepaspectratio,width=0.49\textwidth]{plot_deltac_lnM_F5}}
\subfloat[{F5: $\fr[5]$}]{\label{subfig:deltac_denv_fr5}\includegraphics[keepaspectratio,width=0.49\textwidth]{plot_deltac_denv_F5}} \\
\end{center}
\caption{The barrier density for each of the MG models in this paper. In \cref{subfig:deltac_lnM_all} the barrier is constant for \lcdm{} and only a function of mass for DGP and Symmetron models. In \cref{subfig:deltac_denv_fr5,subfig:deltac_denv_fr6,subfig:deltac_lnM_fr5,subfig:deltac_lnM_fr6} we show cross-sections of the $\fr$ barrier \lq\lq{}surface\rq\rq{} at constant environment density (\cref{subfig:deltac_lnM_fr5,subfig:deltac_lnM_fr6}) and mass (\cref{subfig:deltac_denv_fr5,subfig:deltac_denv_fr6})
}
\label{fig:deltac_all}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[keepaspectratio,width=0.75\textwidth]{plot_density_average}
\end{center}
\caption{The collapse density for \lcdm{}, $\fr[5]$ and $\fr[6]$ averaged over the Eulerian environment distribution \cref{eq:pdf_denv_E} converts the drifting-and-diffusing barrier density to a drifting $\delta_{c}(S)$. }
\label{fig:plot_density_average}
\end{figure}
In this section we quantify how screening affects the critical density $\delta_c$ required for halo collapse in the excursion set formalism. We present a formula for $\delta_c$ which is sufficiently general for all of our screened MG theories. The main difference in modified gravity is that $\delta_c$ depends both on mass and environment density as well as depending on redshift as in $\Lambda$CDM. Throughout this paper we use the terms \lq\lq{}critical density\rq\rq{}, \lq\lq{}barrier density\rq\rq{} and \lq\lq{}drifting-and-diffusing barrier\rq\rq{} synonymously, to indicate $\delta_{c}(S, \Senv, \denv)$. This is they key result of this section: the solution to \cref{eq:sph_collapse_all} (coupled to \cref{eq:sph_collapse_lcdm} if necessary), subject to the initial conditions \cref{eq:sph_collapse_linear,eq:mass_conservation} and the boundary value problem \cref{eq:delta_c_defn}.
The density field $\delta$ is the linearly-extrapolated over-density at the present epoch, convolved with a window function.\footnote{Originally \cite{1991ApJ...379..440B} denoted this by $F$ to avoid confusion with $\delta_{c}$ and to emphasise that it is not purely $\rho / \bar{\rho}$, the usual cosmological definition of over-density compared to the background density. However, present convention dictates that $\delta$ be used instead.} Assuming that the density field exhibits Gaussian fluctuations on all scales, we scale the smoothed density field in units of the variance $\delta = \nu\sigma(R)$ for any positive constant $\nu$ and $\sigma = \sqrt{S}$ defines the resolution $S$ over the scale $R$:
\begin{equation} \label{eq:resolution}
S \equiv \sigma^{2}
= \frac{1}{\left( 2 \pi{} \right)^{3}} \int \dx<3>{k} \left< \abs{\delta \left( \vec{k}; R \right)} \right>
= \frac{1}{2 \pi{}} \int_{0}^{\infty} \dx{\ln k} \; P(k,z) W(k;R)^{2} k^{3}
\end{equation}
where $P(k,z)$ is the power spectrum which completely describes the original Gaussian density field $\delta$ at redshift $z$.
Now we require a condition for the redshift evolution of the density field: we choose a linear evolution $\delta(k,z;R) = D(z) \delta(k;R)$ where $D(z)$ is the growing mode normalised to unity at the present epoch. Since all of our MG models have background evolution close to \lcdm{}, we utilise the usual growing mode \textsc{ode}:
\begin{subequations} \label{eq:d_linear}
\begin{align}
0 &= \frac{\dx<2>{D}}{\dx \ln a^{2}}
+ \left(2 - \frac{3}{2} \Omega_{m}(\ln a) \right) \frac{\dx{D}}{\dx \ln a}
- \left( \frac{3}{2} \Omega_{m}(\ln a) \right) D(\ln a) \\
\Omega_{m}(\ln a) &= \frac {\Omega_{m0} \exp{(-3 \ln a)}} {\Omega_{m0} \exp{(-3 \ln a)} + \Omega_{\Lambda 0}} \\
\Omega_{\Lambda}(\ln a) &= \frac {\Omega_{\Lambda 0}} {\Omega_{m0} \exp{(-3 \ln a)} + \Omega_{\Lambda 0}}
\end{align}
\end{subequations}
Let us view the present-day $\delta$ as a fixed density field, with a critical over-density for collapse $\delta_{c0}$: where $\delta > \delta_{c0} / D(z)$, we expect these over-densities to have already collapsed at redshift $z$; conversely where $\delta < \delta_{c0}$, these parts of the density field have yet to collapse. Our aim is to calculate $\delta_{c0}$ for both GR and our screened MG theories.
The collapse of the environment surrounding the halo is equivalent to collapse in general relativity. Let us assume that the initial over-density is a spherical top-hat in Eulerian space. We can utilise the resulting axisymmetry to simplify the gravitational collapse equation to:
\begin{equation} \label{eq:sph_collapse_lcdm}
0 = \frac{d^2{y}}{d \ln a^{2}}
+ \left(2 - \frac{3}{2} \Omega_{m}(a) \right) \frac{d{y}}{d \ln a}
+ \frac{1}{2} \Omega_{m}(a) \left( \frac{1}{y^{3}} - 1 \right) y
\end{equation}
where $y(a)$ is the ratio of the physical radius of the halo $R_{\rm TH}(a)$ to the physical radius of the filter $a(t)R$ \cite{2012MNRAS.421.1431L}. Now we have an expression for the halo density in general relativity and the environment density in modified gravity.
Since all of our MG theories obey a modified Poisson equation in the weak-field limit, we can use the standard equations for collapse of a spherical top-hat over-density. We need only replace $G_{N}$ by $G_{N}(1 + \Feff)$ in the Poisson equation. Following the same steps as for the environment collapse, we obtain:
\begin{equation} \label{eq:sph_collapse_all}
0 = \frac{\dx<2> y}{\dx \ln a^{2}}
+ \left(2 - \frac{3}{2} \Omega_{m}(a) \right) \frac{\dx y}{\dx \ln a}
+ \frac{1}{2} \left( 1 + \Feff \right) \Omega_{m}(a) \left( \frac{1}{y^{3}} - 1 \right) y
\end{equation}
The enhancement factors $\Feff$ are derived in \cref{app:screened_gravity_theories}. Specifically, they are \cref{eq:feff_fr} for $\fr$, \cref{eq:symm_feff} for Symmetron and \cref{eq:dgp_feff} for DGP. For Symmetron gravity $\Feff$ depends upon the halo over-density in \cref{eq:symm_feff}; while in \DGP{} models $\Feff$ depends upon the halo mass and radius in \cref{eq:dgp_feff}. Therefore for Symmetron and Vainshtein screening we need only solve \cref{eq:sph_collapse_all}. In contrast, the chameleon-screened $\fr$ theory with $\Feff$ in \cref{eq:feff_fr} depends both upon the halo density and the environment density: in this case \cref{eq:sph_collapse_all} (for the halo collapse) must be coupled to \cref{eq:sph_collapse_lcdm} (for the collapse of the surrounding environment). This is the general form for modified gravity collapse of a spherical top-hat.
Finally we set the initial conditions. Since $\Feff \ll 1$ at early times, we can use the same initial conditions in MG and GR. Mass conservation determines $y_{i}$ via equating the physical radii of the physical halo and the top-hat window function at the initial time $a_{i}$:
\begin{equation} \label{eq:mass_conservation}
M = \frac{4}{3} \pi \rho_{m0} a_{i}^{3} R_{\mathrm{TH}}^{3} = \frac{4}{3} \pi \rho_{m0} \left( 1 + \delta_{i} \right) r_{i}^{3}
\implies y(a_{i}) = 1 - \frac{\delta(a_i)}{3}
\end{equation}
At early times we expect the ODE \cref{eq:sph_collapse_all,eq:sph_collapse_lcdm} to be well-approximated by its linearised equivalent. Without loss of generality, we can set the initial redshift of the ODE (we used $z = 500$) to before the modification to GR, hence $F_{\mathrm {eff}} = 0$. Substituting the result for $y_{i}$ gives the corresponding first derivative:
\begin{equation} \label{eq:sph_collapse_linear}
\frac{\dx y(a_{i})}{\dx \ln a} = - \frac{\delta(a_{i})}{3}
\end{equation}
An important corollary of \cref{eq:mass_conservation} is that $\delta = y^{-3} - 1$, which contributes the nonlinearity in the final term of \cref{eq:sph_collapse_all,eq:sph_collapse_lcdm}.
Despite the fact that our (coupled) ODEs are in terms of $y(a)$, our aim is to calculate $\delta_{c}(z_{c})$. We define this via the boundary-value problem \cite{Kopp:2013lea}:
\begin{equation} \label{eq:delta_c_defn}
\delta_{c} \left( z_{c} \right) = \left. \delta_{i} \left( z_{i} \right) \frac{D \left( z_{c} \right)}{D \left( z_{i} \right)} \given y \left( z_{c} \right) = 0 \right.
\end{equation}
This condition specifies that the value of $\delta_{i}$ which causes $y_{c}$ to vanish at the collapse redshift $z_{c}$ is the initial over-density which can be linearly-extrapolated to find $\delta_{c}(z_{c})$. We now have all the requirements to compute the present-day collapse density (in fact, the barrier density required for collapse at any redshift).
The collapse densities using our screening parameters are shown in \cref{fig:deltac_all}. The DGP models produce a flat result with similar values to \lcdm{}. This suggests that these results may be indistinguishable from a \lcdm{} model with non-standard cosmological parameters. In contrast, the two Symmetron models produce a collapse density which is a function of mass. While it asymptotes to \lcdm{} density at high masses, it is substantially lower for small masses and appears to tend to a constant value on sub-cluster scales. As expected, the MG model parameters are correlated with the collapse density divergence from \lcdm{}: DGP models recover the \lcdm{} collapse density as the cross-over radius tends to infinity, whereas symmetron models do so as the symmetry-breaking scale factor tends to unity.
\begin{figure}
\begin{center}
\subfloat[{$\fr[5]$}]{\label{subfig:surf_deltac_linear_F5}\includegraphics[keepaspectratio,width=0.9\textwidth]{surf_delta_linear_F5}} \\[-1ex]
\subfloat[{$\fr[6]$}]{\label{subfig:surf_deltac_linear_F6}\includegraphics[keepaspectratio,width=0.9\textwidth]{surf_delta_linear_F6}}
\end{center}
\caption{The full barrier (surface) and linear approximation (grid) for $\fr$ models with $\fr[5]$ and $\fr[6]$. Both the surface and the grid are coloured according to the value of $\delta_{c}$, so any areas in which the grid is visible indicates a discrepancy between the linear estimate and the true value. This is particularly pronounced for $\fr[5]$ compared to $\fr[6]$. }
\label{fig:surf_deltac_linear}
\end{figure}
The $\fr$ collapse density exhibits a more complex behaviour. We shows its dependence on mass in \cref{subfig:deltac_lnM_fr6,subfig:deltac_lnM_fr5} and environment in \cref{subfig:deltac_denv_fr6,subfig:deltac_denv_fr5}. The collapse density is no longer flat: $\delta_{c}(M, \denv)$ is a monotonically-increasing function of $M$, and the peak-background split $\delta_{c} - \denv$ is a monotonically-decreasing function of $\denv$. Compared to $\Lambda$CDM, we expect haloes to form at higher masses and to have more in low-density regions. \cref{subfig:deltac_denv_fr6,subfig:deltac_denv_fr5} demonstrate that the collapse density is always bounded from below by the environment density and from above by the \lcdm{} result. We have marginalised over the environment distribution according to \cref{eq:pdf_denv_L,eq:pdf_denv_E}. The resulting environment-averaged collapse density is shown in \cref{fig:plot_density_average}. It is also possible to linearise the collapse density in both $M$ (or $S$) and $\denv$, as shown in \cref{fig:surf_deltac_linear}. We will use both of these approximations in \cref{sec:the_halo_mass_function}. This completes our analysis of spherical collapse in MG.
\subsection{Excursion set theory}
\label{sec:excursion_set_theory}
In this section we relate the halo mass function $n(M)$ to the \fcd{} $f(S)$ via excursion-set theory, including \lq\lq{}nuisance parameters\rq\rq{} other than the halo mass $M$. We outline the excursion set formalism in general relativity and then discuss the modifications induced by modified gravity.
The \textit{Ansatz} of the excursion set formalism is the relation of the collapse of the over-density to a halo of mass $M$ in real space to the absorption of trajectories in the density-resolution space by a barrier density at resolution $S$. As shown in \cite{1991ApJ...379..440B,1993MNRAS.262..627L}, the fraction of random walks in the plane $(\delta,S)$ that are absorbed by the collapse over-density $\delta_c$ at resolutions earlier than $S$ is equivalent to the cumulative fraction $F(>M)$ of mass contained in haloes above mass $M$.
The ingredients of the excursion set formalism are:
\begin{enumerate}
\item The window function $W(kR)$ used to smooth the density field. We utilise a Gaussian filter and a top-hat filter in both real and Fourier space.
\item The over-density field $\delta$ corresponding to the fractional, linearly-evolved over-density smoothed over a scale $R$ defined by the aforementioned window function.
\item The resolution $S$, which is related to the linear matter power spectrum $P(k)$ and the halo mass $M$.
\item The collapse density $\delta_{c}$ which acts as a drifting-and-diffusing barrier in the excursion-set parameter space.
\end{enumerate}
Following \cite{1991ApJ...379..440B}, we obtain a diffusion equation for the probability density function $\Pi(S,\delta)$ that a Markovian trajectory which moves randomly in the linearly-extrapolated density field $\delta$ and moving linearly forwards in $S$ from the origin will first exceed the barrier density $\delta_{c}(S)$ at resolution $S$. In GR the barrier is flat, so we obtain an analytical solution for $\Pi$ provided that we use a top-hat filter in Fourier space for $S$. This is given by the Press-Schechter mass function \cite{1974ApJ...187..425P}:
\begin{equation} \label{eq:ps_erfc_integral}
F(>M)
= 2 \int_{\delta_{c}}^{\infty} \dx{} \delta \; \Pi (\delta(S))
= \int_{0}^{S} \dx{} S^{\prime} f (S^{\prime})
= \textrm{erfc} \left(\frac{\nu_h}{\sqrt{2}} \right)
\end{equation}
where $\nu_h\equiv\delta_c/\sqrt{S}$. (A more detailed derivation of this result is in \cite{1991ApJ...379..440B}.)
However, our aim is to express this solely in terms of $S$---the resulting function $f(S)$ is known as the \emph{\fcd{}}---and as the name suggests is the probability of first up-crossing the barrier density at $S$.
By assuming that trajectories in $(\delta , S)$ are uncorrelated, the diffusion equation admits the solution \cite{2011PhRvD..83f3511P}:
\begin{subequations} \label{eq:volterra_all}
\begin{align}
f(S|\Senv , \denv)
&= g(S) + \int_{\Senv}^{S} \! \dx x \; k(S,x) \, f \left( x \given \Senv , \denv \right) \label{eq:volterra_fcd} \\
k(S,x) &= \left[ \frac{\delta_{c}(S) - \delta_{c}(x)}{S - x} - 2\frac{\dx \delta_{c} (S)}{\dx S} \right]
\frac{1}{\sqrt{2\pi (S-x)}} \exp\left\{ -\frac{(\delta_{c}(S) - \delta_{c}(x))^{2}}{2 (S-x)} \right\} \\
g(S) & = \left[ \frac{\delta_{c}(S) - \denv }{S - \Senv } - 2\frac{\dx \delta_{c} (S)}{\dx S} \right]
\frac{1}{\sqrt{2\pi (S - \Senv)}} \exp \left\{ -\frac{(\delta_{c}(S) - \denv)^{2}}{2(S - \Senv)} \right\}
\end{align}
\end{subequations}
This is valid for theories with a non-flat barrier density $\delta_{c}(S,\Senv,\denv)$ which depends upon the starting point of the random walk $(\Senv,\denv)$ and the variance $S$ at which the random walk crosses the barrier.
It is straightforward to show that this has an analytical solution for a linear barrier density \cite{Zhang:2005ar,2011PhRvD..83f3511P}, including the constant case $\delta_{c0}^{\Lambda} \approx 1.676$ which is the \lcdm{} solution:
\begin{equation} \label{eq:ps_conditional}
f(S|\Senv , \denv) = \frac{\delta_{c}^{\Lambda} - \denv}{\sqrt{2 \pi (S - \Senv)^3}} \exp \left[ - \frac{1}{2} \frac{ \left( \delta_{c}^{\Lambda} - \denv \right)^{2} }{S - \Senv} \right]
\end{equation}
We shall refer to this as the (environment-)conditional HMF from now on. This generalises directly from GR to MG: we merely replace the \lcdm{} value $\delta_{c}^{\Lambda}$ with the appropriate density from \cref{fig:deltac_all}. This holds for flat barriers, \lq\lq{}drifting\rq\rq{} barriers $\delta_{c}(S)$ and the full \lq\lq{}drifting-and-diffusing\rq\rq{} barrier $\delta_{c}(S,\Senv,\denv)$.
Finally we must marginalise over the environment. We require a probability distribution for $\denv$, described in \cref{sec:choice_of_environment_density_function}. Then the unconditional mass functions is simply the conditional one marginalised over the nuisance parameter \cite{2012MNRAS.421.1431L,2012MNRAS.425..730L}:
\begin{equation} \label{eq:fcd_ave_integral}
f \left( S \right)
= \langle f \left( S \given \Senv , \denv \right) \rangle_{\mathrm{env}} = \int_{-\infty}^{\delta_{\Lambda}} \! \dx{\denv} \;
p \left( \denv \given \Senv \right) \; f \left( S \given S_{\mathrm{env}} , \delta_{\mathrm{env}} \right)
\end{equation}
In \lcdm{} we obtain an analytic solution which is precisely the \PS{} function:
\begin{equation} \label{eq:ps_unconditional}
f \left( S \right)
= \frac{\delta_{c}^{\Lambda} }{\sqrt{2 \pi S^3}} \exp \left[ - \frac{1}{2} \frac{ \left( \delta_{c}^{\Lambda} \right)^{2} }{S} \right]
\end{equation}
The unconditional HMF \cref{eq:ps_unconditional} has the same functional form as the conditional one \cref{eq:ps_conditional}, albeit with the substitutions $\delta_{c} \rightarrow (\delta_{c} - \denv)$ and $S \rightarrow S - \Senv$. In \cref{sec:the_halo_mass_function} we shall assume that this applies to other fitting functions beyond \PS{}.
So far we have neglected the additional complications that the resolution $S$ is sensitive to the MG parameters (\eg{} $\fr$) via the power spectrum and that $S(R)$ should be based upon a scale $R$ which is Eulerian rather than Lagrangian.
We have chosen to use the linear $P(k)$ computed in the $\Lambda$CDM model in \cref{eq:resolution} when computing the $f(\R)$ gravity predictions. This is in agreement with \cite{2016JCAP...12..024C,2017JCAP...03..012V}. \cref{sec:justification_for_using_the_lcdm_power_spectrum} summarises our reasoning.
Throughout this paper we set $\Senv = S(10\; \mathrm{Mpc}/h) \approx 0.1$, \ie{} an environment radius of $\approx 14 \, \mathrm{Mpc}$. This is the Eulerian size of the environment: see \cref{sub:the_scale_of_the_environment} for details of the Eulerian-Lagrangian distinction. Using a Gaussian window function limits the maximum halo mass to $\lesssim 10^{16} M_{\odot}$. We confirmed that this is above the maximum halo mass found in our $N$-body simulations, so that we do not exclude any data by imposing the excursion-set condition $S < \Senv$.
\subsection{Generalising HMF fitting functions}
\label{sub:fitting_functions}
This section summarises the various halo mass functions used in this paper. In GR a variety of empirical fits to {\it N}-body simulations have been proposed in a \lq\lq{}universal\rq\rq{} form $f(\nu)$. The aim of these fits is to find a fitting function over a broad range of masses which is independent of redshift and cosmology, and the extent to which each fitting function exhibits this universality is a controversial one even in GR. Nonetheless, we show how to generalise the HMFs proposed in GR to three types of screened MG. Furthermore, we show a variety of methods to include environment dependence in the HMF, as required by chameleon-screened MG. A summary of the functions used in this paper is in \cref{tab:hmf_functions}.
Two of the fitting functions used here (\textit{viz.} Press-Schechter and Sheth-Mo-Tormen) can be derived analytically. The Press-Schechter function results from assuming spherical symmetry and a flat (scale-independent) barrier density. That Press-Schechter can be derived analytically in screened MG (\eg{} for a chameleon model \cite{2012MNRAS.421.1431L}) via a simple change of variables is the very reason that we can extend the other mass functions in this paper to MG as well. The Sheth-Mo-Tormen function in \cite{2001MNRAS.323....1S,Sheth:2001dp} was originally a fit to data with the substitution $\delta_{c} \rightarrow \sqrt{a} \delta_{c}$ made to produce a better fit than \PS{}. Later \cite{Maggiore:2009rx} showed that this is equivalent to using excursion set theory with the moving barriers which result from ellipsoidal collapse. To our knowledge, no-one has shown that the assumptions used to construct the Sheth-Tormen function in GR also hold in MG. Nonetheless, the analyticity in GR makes these appealing functions for use in MG.
The remaining fitting functions are purely empirical fits, derived from \lcdm{} N-body simulations.
Having extracted the discrete approximation to the HMF $n(M)$, this can be converted to a discrete \fcd{} and a continuous \lq\lq{}best-fit\rq\rq{} approximation found---either in terms of $\sigma$ or in terms of $\nu$---which holds for a given redshift (range), mass range and cosmology (or family of cosmologies). The resulting HMF also depends upon the halo finder used to extract the halo masses, as well as any subsequent calibration or corrections. These factors all contribute to the final parameter values adopted by a particular function. This is particularly notable for the Sheth-Mo-Tormen fit, for which several authors\footnote{This is why Jenkins appears twice in \cref{tab:hmf_functions}: one instance is their calibration for the SMT function (as with Courtin) and the other is their new fitting function.} \cite{2001MNRAS.323....1S,2001MNRAS.321..372J,2011MNRAS.410.1911C,2006ApJ...646..881W} have proposed their own \lq\lq{}improved\rq\rq{} values for the best-fit parameters of this function. This emphasises that while the same functional form can provide a good fit to data in different background cosmologies, the same $N$-body data and the same fitting function will produce different best-fit parameters and varying degrees of invariance depending upon the halo extraction tehcniques and the theoretical assumptions of the authors. For this reason, we have separated the HMFs into families which share the same functional form. We summarise the HMFs $f(\nu)$ in \cref{tab:hmf_functions}.
There are four steps involved in generalising the \lcdm{} HMFs to MG:
\begin{enumerate}
\item \label{item:hmf_deltac} Calculating the appropriate barrier density;
\item \label{item:hmf_pk} Selecting the appropriate linearised power spectrum;
\item \label{item:hmf_sigma_to_nu} Converting from $\sigma$ (or $S$) to $\nu$ as the dependent variable;
\item \label{item:hmf_params} Rescaling the free parameters accordingly.
\end{enumerate}
We discussed \cref{item:hmf_deltac} in \cref{sub:spherical_collapse_in_mg}. The barrier densities for the models in this paper are in \cref{fig:deltac_all}.
In \cref{item:hmf_pk} we must decide whether to keep the \lcdm{} power spectrum or to adjust it according to the MG theory. We keep the linear $P(k)$ from \lcdm{}, in agreement with \cite{2017JCAP...03..012V,2016JCAP...12..024C}. We discuss the reasons for this in \cref{sec:justification_for_using_the_lcdm_power_spectrum}.
\cref{item:hmf_sigma_to_nu} is concerned with making explicit the dependence on the collapse density. In GR, there are two opinions on the \lq\lq{}correct\rq\rq{} independent variable for the HMF. Those arguing for $\nu$ (\textit{inter alia} \cite{Sheth:2001dp,2001MNRAS.323....1S,Peacock2007,Zentner:2006vw}) assert that according to excursion set theory the (albeit weak) cosmological and redshift dependence of both $\delta_{c}$ and $\sigma$ cancel when the \fcd{} is expressed in terms of $\nu \equiv \delta_{c} / \sigma$ in GR cosmologies. Those arguing for $\sigma$ (or $S$, or $\sigma^{-1}$, or $\ln \sigma$) (\textit{inter alia} \cite{2001MNRAS.321..372J,2007MNRAS.374....2R,2003MNRAS.346..565R,2008ApJ...688..709T,2010MNRAS.403.1353C,2006ApJ...646..881W}) assert that $\delta_{c}$ is a sufficiently weak function of $\Omega_{m0}$ and $z$ that this dependence can be ignored; in fact \cite{2001MNRAS.321..372J} go so far as to say that \lq\lq{}taking $\delta_{c} = 1.686$ in all cosmologies leads to excellent agreement with our numerical data if halos are defined at fixed over-density\rq\rq{}. In GR cosmologies there is an invertible mapping between these two options because $\delta_{c}$ is assumed to be a constant.
This is clearly not the case in our extended gravity theories. Again, there is more than one opinion on this topic. Do we change only the resolution $S(M)$ \cite{2011PhRvD..84h4033L} or only the barrier density $\delta_{c}$ \cite{2017JCAP...03..012V} or both \cite{Lombriser:2013wta}, or do we need to evolve the full scalar field equations \cite{Kopp:2013lea}? The only choice which fulfills the three objectives:
\begin{enumerate}
\item A consistent model applicable in all our MG theories
\item Incorporating the mass- and $\denv$-dependence for $\delta_{c}$ in $\fr$
\item Our choice of the \lcdm{} $P(k)$ to calculate $S(M)$, so that we have the same map between mass and resolution for all haloes
\end{enumerate}
is to use the universal parameter $\nu$ and change only the barrier density from $\delta_{c}^{\Lambda}$ to the full barrier density appropriate for a given MG theory.
\cref{item:hmf_params} is the consequence of \cref{item:hmf_sigma_to_nu}. All the free parameters in the HMFs\footnote{The \lq\lq{}Reed 2007\rq\rq{} fit is the 2003-like fit from \cite{2007MNRAS.374....2R}, rather than the one with $n_{\mathrm{eff}}(\sigma)$ dependence, which creates ambiguity about which terms to convert to $\nu$. In the fit we use the $\sigma$-dependence is clearly only caused by treating $\delta_{c}$ as a constant.} need to be converted from $\ln \left( \sigma^{-1} \right) $ to $\nu$, by absorbing factors of $\delta_{c}$. This requires paying particular attention to whether the SCDM or \lcdm{} collapse density is used in the original papers, as this has implicitly been absorbed into the best-fit parameters.\footnote{If we were concerned with redshift evolution, we would have to decide whether to absorb $\delta_{c0}$ or $\delta_{c}(z)$. Some authors fix $\delta_{c}$ and allow the free parameters to vary with redshift; others do the opposite.}
This is sufficient for MG theories which only have a drifting barrier $\delta_{c}(S)$, but not for the drifting-and-diffusing barrier $\delta_{c}(S,\denv,\Senv)$. We treat this additional generalisation in the next few paragraphs.
The drifting-and-diffusing barrier can be accounted for using a wide variety of methods:
\begin{enumerate}
\item \label{item:scaling_Volterra} Scaling using the Volterra integral solution in \cref{eq:volterra_fcd}
\item \label{item:cosmic_web} Averaging over the cosmic web (described in \cite{2017JCAP...03..012V})
\item \label{item:deltac_ave} Calculating a $\denv$-averaged collapse density, \ie{} converting to a drifting barrier
\item \label{item:barrier_approx} A flat or linear-barrier approximation to the full excursion-set problem
\end{enumerate}
We address each of these in turn.
Technique~\ref{item:scaling_Volterra} utilises the fact that we already know how to account for the extra barrier complexity in the excursion set approach: whereas the flat barrier produces the \PS{} distribution, the drifting-and-diffusing barrier leads to the solution \cref{eq:volterra_fcd}. So we may calculate the unconditional HMF, i.e. assuming $(\denv = 0, \, \Senv = 0)$, then accounting for the effects of the drifting and diffusing barrier using excursion set theory. This results in a rescaling of the unconditional HMF:
\begin{equation} \label{eq:scaling_Volterra}
f(S)_{\mathrm{MG}} = f(S | \Senv = 0, \denv = 0 )_{\mathrm{MG}} \quad \frac{\text{Volterra solution}}{\text{Press-Schechter}}
\end{equation}
This is efficient, as we only need to solve the Volterra equation once for each drifting-and-diffusing barrier density, rather than re-calculating for each HMF as well.
Technique~\ref{item:cosmic_web} uses the $\nueff$ prescription of \cite{2017JCAP...03..012V}. This is substituted directly into the HMF:
\begin{equation}
\nu_{\mathrm{eff}} = \mathrm{max} \left\{ 0, \frac{ \nu_{h} \left( S, \denv \right) - \epsilon^{2} \left( S, \Senv \right) \nu_{\text{env}}
\left( \Senv, \denv \right) }{ \sqrt{ 1 - \epsilon^{2} \left( S, \Senv \right) }} \right\}
\end{equation}
Since we are not interested in restricting ourselves to a particular environment, we can simplify the relevant equations in \cite{2017JCAP...03..012V} to:
\begin{equation} \label{eq:cosmic_web}
f \left( S \right) =
\int_{-\infty}^{\infty} \dx \nu_{\text{env}} \; f \left( \nu_{\mathrm{eff}} \right)
\int_{0}^{\infty} \dx \rho \int_{-\rho}^{\rho} \dx \theta \; p(\rho, \theta, \nu_{\text{env}})
\end{equation}
We have rearranged the order of integration to highlight that the conditional mass function $f \left( \nu_{\mathrm{eff}} \right)$ is independent of $\rho$ and $\theta$.
Technique~\ref{item:deltac_ave} approximates the drifting-and-diffusing barrier by a drifting-only barrier. Effectively we integrate over the environment density at the stage of calculating the collapse density:
\begin{equation} \label{eq:deltac_ave}
\nueff = \frac{\langle \delta_{c} \rangle_{\mathrm{env}} }{S}
\quad \text{where} \quad
\langle \delta_{c} \rangle_{\mathrm{env}} =
\int_{-\infty}^{\delta_{\Lambda}} \! \dx{\denv} \;
p \left( \denv \given \Senv \right) \; \delta_{c} \left( S \given S_{\mathrm{env}} , \delta_{\mathrm{env}} \right)
\end{equation}
This effective-$\nu$ is substituted directly into the unconditional HMF. In contrast to the full barrier solution---where the random walk must up-cross $\delta_c$ at $S$ having started at $(\denv,\Senv)$---here there is no accounting for the environment-dependent absorption of the Markovian trajectories in $(\delta , S)$ caused by the drifting-and-diffusing barrier.
Technique~\ref{item:barrier_approx} approximates the full solution to the Volterra equation in \cref{eq:volterra_fcd} by a linear barrier. This is motivated by the fact that the Volterra equation with a linear barrier $\delta_{c}(S) = \omega - \beta S $ is calculable analytically \cite{2011PhRvD..83f3511P}:
\begin{equation} \label{eq:flat_barrier}
f \left( S, \delta_{c}(S) \given \Senv, \denv \right) = \frac{ \delta_{c} - \denv }{ \sqrt{2 \pi \left( S - \Senv \right)^{3}} } \exp \left[ -\frac{1}{2} \frac{\left( \delta_{c} - \denv - \beta \left( S - \Senv \right) \right)^{2} }{S - \Senv} \right]
\end{equation}
so we can substitute the effective arguments into the fitting function, in the same way as for the cosmic web. However, \cref{fig:surf_deltac_linear} shows that the $\fr$ barriers are not linear in $S$ (except when $\denv \rightarrow \delta_{c}^{\Lambda}$). The linear approximation is always an overestimate and is particularly poor at approximating the sharp rise at $\denv \approx \delta_{c}^{\Lambda} $. Therefore we discard this approach due to its poor approximation.
A \textit{caveat} for all of these methods is that there is no way to eliminate dependence on $\Senv$. The excursion set condition $\Senv < S$ prevents us from marginalising directly. The environment distributions (whether cosmic web or the PDFs for the environment over-density) in the limit $\Senv \rightarrow 0$ do not have a finite limit. Instead we take a sufficiently large environment radius (\eg{} $R_{\text{env}} = 10,\, 20 \text{Mpc}/h$) that $S > \Senv$ is always obeyed (but not too large, otherwise this is no longer a sufficiently local description of the halo surrounds \cite{2012MNRAS.425..730L}).
The main result of \cref{sub:including_denv_barrier_in_mg} is to compare these methods for generating the conditional HMFs for $\fr[5]$ and $\fr[6]$. For the other MG models, the barriers do not have an environment-dependent component, so their calculation is straightforward.
The process described in this section updates the $\lcdm$-calibrated fits to a format which is compatible both with GR and MG. Our fitting functions are shown in \cref{tab:hmf_functions}. The independent parameter is $\nu$ and the other variables are free parameters. It is these free parameters which we vary, in order to optimise the fit between the fitting functions and the HMF derived from $N$-body simulations. In \cref{sub:assuming_concordance_cosmology} we assume \lcdm{} values for $\nu$ in our fitting functions, whereas in \cref{sub:recalibrating_best_fit_parameters} we use the full MG values of $\nu$.
\input{./table_cosmologies}
\input{./table_hmf_functions}
\clearpage
\section{Data processing}
\label{sec:algorithms}
This section summarises the N-body simulations and the process of extracting the halo catalogues which form our data. This includes our choice of halo finder, corrections and mass cuts to the binned halo counts and quantification of the uncertainties in the resulting discrete HMF.
\subsection{$N$-body simulations}
\label{sub:n_body_simulations_haloes}
The {\it N}-body simulations were run using the \texttt{ISIS} and \texttt{ECOSMOG} code \cite{2014A&A...562A..78L, 2012JCAP...01..051L}, which is a modified gravity modification of the high-resolution {\it N}-body code \texttt{RAMSES} \cite{2002A&A...385..337T}. For details about the implementation and for a comparison of these codes see the modified gravity {\it N}-body code comparison project \cite{2015MNRAS.454.4208W}.
We ran two sets of simulations with different mass resolution. Simulation set 1 has $N = 512^3$ particles of mass $8.75 \times 10^{9} M_{\odot}/h$ in a box of $B = 250~\text{Mpc}/h$. The background cosmology is a flat $\Lambda$CDM model with $\Omega_m = 0.269$, $\Omega_\Lambda = 0.732$, $h = 0.704$, $n_s = 0.966$ and $\sigma_8 = 0.8$ . These simulations were presented in \cite{2015MNRAS.454.4208W}. Simulation set 2 has $N = 256^3$ particles of mass $3.531 \times 10^{10} M_{\odot}/h$ in a box of $B = 200~\text{Mpc}/h$. The background cosmology is a flat $\Lambda$CDM model with $\Omega_m = 0.267$, $\Omega_\Lambda = 0.733$, $h = 0.719 $, $n_s = 1.0$ and $\sigma_8 = 0.8$. These simulations were presented in \cite{2012JCAP...10..002B}.
Dark matter {\it N}-body simulations are performed by evolving two equations. The first one is the Poisson equation which gives us the gravitational potential $\Phi$ in terms of the particle positions (which determines the density field $\rho_m$)
\begin{align}\label{eq:poisson}
\nabla_x^2\Phi = 4\pi G(\rho_m - \overline{\rho}_m) a^{2}
\end{align}
and the second is the geodesic equation
\begin{align}\label{eq:geodesic}
\ddot{{\bf x}} + 2H\dot{{\bf x}} = - \frac{\nabla_x\Phi}{a^{2}}
\end{align}
which determines the evolution of the particles. For the modified gravity simulations we consider here the only change is that we have a fifth force $-\nabla\varphi$ that contributes to the right hand side of \cref{eq:geodesic} and we have to solve a field equation similar to \cref{eq:poisson}, but highly non-linear, to get the fifth-force potential $\varphi$. More details about the implementation for the models considered in this paper are contained in \cite{2014A&A...562A..78L,2015MNRAS.454.4208W,2012JCAP...10..002B}.
We utilised two different halo finders of differing complexity:
\begin{enumerate}
\item The friend-of-friend halo-finder \texttt{MatchMaker}\footnote{MatchMaker can be found at https://github.com/damonge/MatchMaker} with linking-length $b=0.2$.
\item The 6D phase-space friend-of-friend halo-finder \texttt{RockStar} \cite{2013ApJ...762..109B}.
\end{enumerate}
Both halo finders use the Friends-of-Friends (FoF) algorithm developed in \cite{1985ApJ...292..371D}, albeit with different distance measures. Particles are formed into connected graphs by drawing an edge between vertex particles if the distance between them is less than some fraction $b$ of the mean inter-particle distance. Each connected graph is defined to be a halo if it is not a subgraph of a larger halo.
The \texttt{MatchMaker} finder is a parallel 3d-FoF finder. We used the canonical linking length $b = 0.2$. The distance between particles is the usual 3d Euclidean distance.
The \texttt{RockStar} finder uses the normal 3d FoF (albeit with $b = 0.28$) to identify groups of particles, within which it uses FoF in the 6d phase space to identify subgroups. After conversion from subgroups to subhaloes, any unbinding of particles from haloes is performed using the halo potentials. (This algorithm is summarised in Figure 1 of \cite{2013ApJ...762..109B}.)
Subsequently halo properties are extracted. The halo property with which we are concerned in this paper is the halo mass of the parent haloes only (\ie{} we ignore subhaloes because we are only interested in the largest mass ranges). This is defined to be $M_{200}$, the total mass of all particles within the over-density satisfying $\rho \geq 200 \rho_{\mathrm{crit}}$, where $\rho_{\mathrm{crit}}$ is the background critical density (not the matter density) \cite{2013ApJ...762..109B}.
A comparative analysis of halo finder performance in \lcdm{} can be found in \cite{2011MNRAS.415.2293K}.
\subsection{Simulation corrections}
\label{sub:simulations_corrections}
Having obtained our halo catalogues we now approximate the continuous HMF using a histogram.
We do not make any corrections to the data. Some authors propose adjusting the values of $\sigma (M)$ in the simulation data. The aim is to correct for the finite box size, which precludes modes with $k \leq 2\pi / L_{\mathrm{box}}$ from contributing to the over-density fluctuations in the halo. This effect can be approximated by the extended Press-Schechter approach (amongst other methods: see \cite{2007ApJ...671.1160L} for details). We avoid corrections for the mass variance due to our large box size, for which corrections are negligible.
It is necessary to remove simulation artifacts from the low-mass end. We truncate the mass function at a lower bound of $100$ particles, where one particle is the mass resolution of the $N$-body simulations. (This is independent of the minimum number of particles required in the halo identification process.) Compared to the cuts of \cite{2007ApJ...671.1160L} this is a conservative cut: faced by a relatively small box size, we wish to retain as much of the HMF as possible. However, the cut is sufficient to remove the low-mass \lq\lq{}tail\rq\rq{} where the mass function---which should be monotonically-decreasing with mass---actually increases with mass. Such a phenomenon arises from the finite (mass) resolution of the simulations. At the lowest masses, there is insufficient resolution to identify all of the bound objects with few particles, so the number density is increasingly suppressed at masses below a characteristic turnover mass. This limitation cannot be alleviated without sub-sampling the simulation box at finer resolution. Slightly higher, at haloes with tens of particles, the uncertainty on the mass values is a significant fraction of the total halo mass. Consequently, the loss or addition of one particle can move the halo between bins. There are two possible ways of accounting for this: either incorporating a mass uncertainty in the likelihood function, or minimising the effect by judicious bin optimisation. We opt for the latter. This low-mass effect is well-known and we do not discuss it further.
At the high mass end our cutoff is artifically imposed by the finite box size. The finite box size curtails the number of large mass haloes found in a finite sub-volume of the horizon (this underestimation is quantified for \lcdm{} in \cite{2013A&C.....3...23M}). In addition, our excursion set technique prevents us from calculating the HMF for masses of $S < \Senv$ (for the excursion-set method) or $\nueff < 0$ (for the cosmic web). Using a Gaussian window function with a radius of $10 \, \mathrm{Mpc}/h$ this corresponds to a mass of $M \approx 10^{16} M_{\odot}$. We have confirmed that this does not remove any haloes from our data.
The remaining factor in our simulated HMF is the bin width of the histogram. For simplicity we adopted constant bin widths, as is common across the literature (although we discuss this and other options in more depth in \cref{sub:number_of_bins}). We verified that the results we obtain are independent of the bin width in \cref{sub:number_of_bins}. Consequently we adopted a value of $N = 30$ bins as an average value across all our data sets.
\subsection{Uncertainty in the data}
\label{sub:data_errors}
Finally we quantify the uncertainty in the HMF.
The uncertainty in the bin occupation is assumed to be Poissonian. The usual uncertainty on bin occupations is taken to be the well-known result\footnote{Eq. (11) in \cite{2011MNRAS.410.1911C} is not true, but is related to the standard deviation as a \emph{fraction} of bin occupation: $\sigma/N = 1 / \sqrt{N} \mathrel{{\ooalign{\hidewidth$\not\phantom{=}$\hidewidth\cr$\implies$}}} \sigma = 1/N$.} that the Poisson standard deviation on a bin containing $N$ haloes is $\sigma = \sqrt{N}$. Various HMF papers (\eg{} \cite{2007ApJ...671.1160L}) use an \lq\lq{}improved\rq\rq{} Poisson error defined to be:
\begin{equation} \label{eq:poisson_err}
\sigma_{\pm} = \sqrt{N + \frac{1}{4}} \pm \frac{1}{2}
\end{equation}
This asymmetric error asymptotes to the usual one for large $N$ but is better-behaved for small $N$, particularly for empty bins. While this does not affect the Poisson-based likelihood, it does enter the Gaussian-based likelihood. The reason is straightforward: the number of counts per bin is known precisely, so there is no uncertainty. The Poisson \lq\lq{}noise\rq\rq{} expresses the uncertainty in the mean of the underlying Poisson \pdf{}, which is equal to the variance term which does enter into the Gaussian \pdf{}. Since the error does not enter the likelihood function, we make no cuts when using constant-width bins (explained below). For variable-width bins, we tolerate an error of $10\%$ or less, which is in line with the choice from other papers (\eg{} \cite{2011MNRAS.410.1911C}).
We must pay attention to combining the asymmetric errors via the method of \cite{2004physics...6120B}. The combined variance is then:
\begin{equation} \label{eq:asymmetric_error}
V \left( x \given \hat{x} \right) = V_{0} + V_{1}(x - \hat{x})
\quad \text{where} \quad V_{0} = \sigma_{+}\sigma_{-}
\quad \text{and} \quad
V_{1} = \sigma_{+} + \sigma_{-}
\end{equation}
This method tightens the uncertainty on the low-occupation bins, weighting the Gaussian likelihood more favourably towards the high-mass end than with symmetric error bars.
\section{Bayesian inference}
\label{sub:bayesian_inference}
Our aim is to find the posterior probability distribution $p \left( H \given D,I \right)$ for a given hypothesis $H$ when we take into account our prior information $I$ and the data $D$. We have already examined $D$, the counts per mass bin from our simulation data in \cref{sub:simulations_corrections}. In this section we define our priors, the appropriate likelihood function and characterise different measures of the high-probability regions in our probability density functions for the HMF free parameters. The effects of the bin width, choice of likelihood function and nested sampling settings are discussed in \cref{app:calibration_of_multinest}.
The priors $I$ contain our assumptions about the prior distribution for the parameters of the hypothesis. We set uniform priors, \ie{} we have no reason to favour any regions of parameter space over another. The lower limit for the priors is zero, because we must avoid an HMF which is negative (if the scaling parameter is $A < 0$), complex (fractional powers of $a\nu$ where $a < 0$), or non--monotonically-decreasing (since $\nu$ is an increasing function of mass, $\nu^{p}$ must have $p > 0$). The upper limit is arbitrary. Given that most published values for the parameters lie in $[0,2]$ (the exception being Jenkins $a = 3.8$), we used an upper limit of $10$ to ensure that the credible regions were well-contained within the prior region.
Now we require an expression for the likelihood function $p \left( D \given I,H \right)$. We apply the Poisson likelihood:
\begin{equation} \label{eq:likelihood_Poisson}
\ln \mathcal{L} = - \sum_{i} \left( \mu_{i} - n_{i} + n_{i} \ln \frac{n_{i}}{\mu_{i}} \right)
\end{equation}
where $\mu_{i}(\mathbf{q})$ is the number of counts given by the parameter set $\mathbf{q}$ and $n_{i}$ is that given by the data in the $i$-th bin. The last term is zero when $n_{i}$ is zero: otherwise $n$ is always a positive integer, whereas $\mu$ is a positive real number.
The likelihood $\mathcal{L}(H)$ is the probability that the hypothesis $H$ produces the data $D$. However, we are interested in the probability that the data are consistent with the hypothesis. Via Bayes\rq{} theorem, we obtain:
\begin{equation}
\label{eq:posterior}
p \left( H \given D,I \right) = \frac{ p \left( H \given I \right) \mathcal{L} \left( D \given H \right)}{ p\left( D \given I \right) }
\end{equation}
This is the complete posterior PDF for our hypothesis $H$ in light of the data $D$ given prior information $I$.
Having chosen a particular HMF, we are interested in the universality of the HMF across different MG theories. One way in which to measure this is to see how the best-fit parameter values change depending on the MG model. \texttt{MultiNest} gives three different options to define \lq\lq{}best-fit\rq\rq{}: the maximum-likelihood (ML), maximum-a-posteriori (MAP) and posterior mean (PM) values.
The ML value for a given parameter $\theta$ is defined to be that which maximises $p \left( D \given \theta,I \right)$, \ie{} the most probable value for the model to give the observed data. In contrast the MAP value maximises the probability $p \left( \theta \given D,I \right)$. Since we have chosen uniform priors, the MAP value is equal to the ML value via \cref{eq:posterior}. Since the ML value is the fastest to converge to the true value in MultiNest \cite{2009MNRAS.398.1601F}, we used this value for our best-fit parameters. We are also interested in the PM value
\begin{equation} \label{eq:posterior_mean}
\langle \theta \rangle = \int \dx \bar\theta \; p \left( \bar\theta \given D, I \right) \bar\theta
\end{equation}
because the $1\sigma$ credible regions given by MultiNest are only provided for the posterior mean.
If the full PDF is not well-described by a single value (\eg{} the three values above are very different, or the posterior is very flat \etc{}), then we are better off examining the credible region $R$ of credibility $C$, which is the set:
\begin{equation} \label{eq:credible_region}
R = \left\{ \four{\theta} : \int_{p \left( \bar{\four{\theta}} \given D, I \right) > c } d \bar{\four{\theta}} \;
p \left( \bar{\four{\theta}} \given D, I \right) = C \right\}
\end{equation}
where $c$ is the level set forming the boundary $\partial R$, inside which the probability is greater than $c$ and outside which it is less than $c$. We refer to the $1\sigma$ and $2\sigma$ credible regions for $C = 0.68$ and $C = 0.95$ respectively.
\section{Results and discussion}
\label{sec:results_and_discussion}
This section addresses the questions posed in the introduction. \cref{sub:including_denv_barrier_in_mg} present our new results for incorporating MG into \lcdm{}-calibrated HMFs. In \cref{sub:assuming_concordance_cosmology} we discuss whether the presence of screened gravity can be mistaken for a change in best-fit parameter values for the \lcdm{} HMF. In \cref{sub:recalibrating_best_fit_parameters} we apply the procedure used in \lcdm{} to calibrate the MG HMF using the full excursion set approach. We compare to existing values from the literature and assess the deviation. In particular we address the universality of the halo mass function, \ie{} the invariance of the best fit free parameters to changes in the underlying gravity model. \cref{app:calibration_of_multinest} confirms that our parameter estimation technique is reliable for the problem at hand and describes the settings we used in the nested sampling algorithm.
\subsection{Accounting for the drifting-and-diffusing barrier in MG}
\label{sub:including_denv_barrier_in_mg}
\cref{sub:including_denv_barrier_in_mg} presents our new results for incorporating MG into \lcdm{}-calibrated HMFs.
In \cref{sec:the_halo_mass_function} we suggested a variety of different methods to implement the full effects of environment dependence in MG for a fitting function. We compare the data from our $\fr$ simulations with $\fr[5]$ and $\fr[6]$ to the theoretical HMF using each method.
\begin{figure
\begin{center}
\subfloat[\lcdm{}]{\label{subfig:plots_danddbarrier_plot_fcd_lenm_30_LCDM_Reed-07}\includegraphics[keepaspectratio,height=0.3\textheight]{danddbarrier/plot_fcd_lenm_30_LCDM_Reed-07}} \\[-1ex]
\subfloat[{$\fr[5]$}]{\label{subfig:plots_danddbarrier_plot_fcd_lenm_30_F5_Reed-07}\includegraphics[keepaspectratio,height=0.3\textheight]{danddbarrier/plot_fcd_lenm_30_F5_Reed-07}} \\[-1ex]
\subfloat[{$\fr[6]$}]{\label{subfig:plots_danddbarrier_plot_fcd_lenm_30_F6_Reed-07}\includegraphics[keepaspectratio,height=0.3\textheight]{danddbarrier/plot_fcd_lenm_30_F6_Reed-07}}
\end{center}
\caption{The first-crossing distribution for a drifting-and-diffusing barrier using the variety of methods explored in this paper are shown as lines. We also plot the data from the $N$-body simulations using the two halo finders.}
\label{fig:plot_fcd_lenm_30_Reed-07_cont1}
\end{figure}
In the order given by \cref{fig:plot_fcd_lenm_30_Reed-07_cont1} we have:
\begin{enumerate}
\item The unconditional HMF using $\langle \delta_{c}\left( S \given \denv, \Senv \right) \rangle_{\mathrm{env}}$, the environment-averaged collapse density \cref{eq:deltac_ave}
\item The conditional HMF marginalised over the Eulerian $\denv$-distribution \cref{eq:pdf_denv_E}
\item The conditional HMF marginalised over the Lagrangian $\denv$-distribution \cref{eq:pdf_denv_L}
\item The unconditional HMF, \ie{} assuming both $\denv$ and $\Senv$ vanish
\item The Volterra solution to \cref{eq:volterra_all}
\item The unconditional HMF scaled with the Volterra solution using \cref{eq:scaling_Volterra}
\item The cosmic web HMF marginalised over the tidal tensor distribution \cref{eq:cosmic_web}
\end{enumerate}
\cref{fig:plot_fcd_lenm_30_Reed-07_cont1} shows these methods for the Reed-07 fitting function\footnote{We might have used any of the fitting functions, because our aim is to compare the behaviour of each technique for extending the mass function to MG. The Reed fit produced the closest fit to the data given the default parameters.} (lines) and the $N$-body HMFs calculated using the \texttt{MatchMaker} and \texttt{RockStar} halo finders (points). These simulations and the workings of the two halo finders are described in \cref{sub:n_body_simulations_haloes}. We examine the \lcdm{} results before discussing each method over the next few paragraphs.
The \lcdm{} behaviour in \cref{subfig:plots_danddbarrier_plot_fcd_lenm_30_LCDM_Reed-07} is consistent with the excursion-set framework of \cite{1991ApJ...379..440B}. The environment-averaged collapse density is $\delta_{c}^{\Lambda}$ (because the excursion-set barrier is flat) and the Volterra solution reduces to \PS{}. Therefore both of these methods produce the same result as the unconditional HMF. The other four methods all differ. While both the Lagrangian and cosmic web methods do equal the unconditional HMF using \PS{}, this is due specifically to the design of this function from excursion set theory. It is the only integrand for which the solution to the integral equation \cref{eq:volterra_fcd} is merely a rescaling between the conditional and unconditional forms of $\nu$. We have now ensured that the different methods behave as expected in GR, before applying them to $\fr$.
The first option avoids using excursion set theory altogether, by pre-emptively converting the drifting-and-diffusing barrier to an average density. The drifting barrier $\delta_{c}(S)$ can be incorporated straightforwardly, just as in the non-chameleon MG models, into the unconditional HMF. Surprisingly, this gives a very good fit, superior to the purely unconditional HMF. Although the peak of \cref{eq:pdf_denv_L} is at zero, this result illustrates that we need to use the entire PDF, rather than only using the peak to approximate the average. In this way we can account for the peak-background split, whereby it is easier for haloes to form as $\denv \rightarrow \delta_{c}$ in dense regions and more difficult in under-dense regions. This method has the advantage that we can compute the barrier density once, rather than re-computing the conditional function at every stage of the MCMC process. We have managed to produce a good fit by considering only the barrier density, rather than accounting for the complex excursion set behaviour of the full drifting-and-diffusing barrier.
The conditional HMFs marginalised over the Lagrangian (\cref{eq:pdf_denv_L}) and Eulerian (\cref{eq:pdf_denv_E}) theoretical distributions $p(\denv)$ have appeared in the literature before\footnote{\cite{Lombriser:2013wta} only plotted the relative enhancement $n_{\mathrm{MG}}(M) - n_{\mathrm{GR}}(M) / n_{\mathrm{GR}}(M)$, rather than the HMF proper $n(M)$, so we cannot readily compare our results to theirs.} \cite{Lombriser:2013wta}, where they were applied to the Sheth-Tormen HMF. The authors suggested discarding the Lagrangian (density) distribution in favour of the Eulerian, on the basis that a density distribution which better reflects the physical formation of over-densities, would correspondingly produce a more accurate HMF. This is supported by the findings of \cite{2012MNRAS.425..730L}. While our results agree using the \texttt{MatchMaker} halo finder, the \texttt{RockStar} halo finder predicts a systematically lower distribution of haloes, better suited to the Lagrangian model.
The question of which PDF to use is somewhat moot considering that neither fit performs particularly well. This is probably due to generating the conditional from the unconditional HMF. The rescaling at the end of \lcdm{} excursion set theory which is used in \cite{2012MNRAS.421.1431L,Lombriser:2013wta} implicitly assumes that a linear function of $\delta_{c}(S)$ is a good approximation for the actual barrier demsity. We have already seen in \cref{fig:surf_deltac_linear} that this is not the case. Therefore we cannot use methods which work for a flat barrier density in \lcdm{} to good effect in $\fr$.
The solution to the integral equation is the MG-equivalent of \PS{}. For this reason, we do not expect it to be a good fit to the data. Indeed the $\fr[5]$ anf $\fr[6]$ plots in \cref{subfig:plots_danddbarrier_plot_fcd_lenm_30_F5_Reed-07,subfig:plots_danddbarrier_plot_fcd_lenm_30_F6_Reed-07} share the recognised flaws of the \PS{} fit in \lcdm{} in \cref{subfig:plots_danddbarrier_plot_fcd_lenm_30_LCDM_Reed-07}, namely that it underpredicts at the high-mass end (low-$S$) and overpredicts at the low-mass end (high-$S$). Nonetheless, it reproduces the general behaviour of the \fcd{}, which is remarkable for such a simple model.
Moreover, we can improve the result from the unconditional HMF by scaling by the ratio of the Volterra solution to \PS{}. The two poor results combine to form a decent approximation. At the low-$S$ end, the Volterra solution forms too few haloes becuase the random walks in excursion set theory are not absorbed early enough by the barrier density, so too many trajectories survive to produce haloes at high-$S$. In contrast, the unconditional HMF assumes $\denv = 0$ (and $\Senv = 0$) so at small values of $S$ the value of $\nu = \delta_{c}(S)/\sqrt{S}$ is large and vice-versa at high-$S$. Since the fitting function $f(\nu)$ is montonically increasing with $\nu$, we have too many haloes at small $S$ and too few at low $S$. These two behaviours counteract one another to reduce the overall discrepancy of the fit. This is because we have deliberately designed a method to combine different strengths of the analytical and empirical approaches. The unconditional fitting function is designed to produce a good approximation to the \lcdm{} data. The Volterra solution captures the excursion set behaviour of the barrier density, which incorporates the main effect of $\fr$ compared to \lcdm{} from a theoretical viewpoint. Thus, we can combine two simple mechanisms to produce a relatively good solution, despite their individual predictions being ineffectual.
The performance of the cosmic web method is discussed in more detail in our previous paper \cite{2017JCAP...03..012V}. In general it underpredicts the HMF at the high-mass end, whereas it works well at the low-mass end. There are two contributing factors, namely the $\nueff$ approximation substituted into the fitting function and the $p(\rho, \theta, \nu_{\text{env}})$ distribution related to the tidal tensor in the cosmic web. The former is not particularly successful even in \lcdm{} (\cite{Alonso:2014zfa}) so we ought not expect any better performance in $\fr$ where the spherical collapse in more complicated due to the fifth force. The latter is the equivalent of the Lagrangian distribution for $\denv$ applied to all three eigenvalues of the tidal tensor rather than its trace. We have already seen that the Lagrangian-$\denv$ fit does not produce an accurate fit. This technique is useful for calculating the HMF in individual structures of the cosmic web (\eg{} voids, sheets \etc{}), wherein we have no other analytical treatment, but not the overall HMF.
Finally we comment on the distinction between the cosmic web results (\cref{eq:cosmic_web}) and those of the Lagrangian PDF. Given that the conditional HMF in the cosmic web doesn't depend upon $(\rho,\theta)$, and that the distributions for $p(\nu_{\text{env}})$ and $p(\denv)$ have the same dependence on $\denv$, one may expect that these two methods should produce the same results. Clearly \cref{fig:plot_fcd_lenm_30_Reed-07_cont1} does not agree. This is due to the effect of the window functions in calculating $S(M)$ and $\Senv(R_{\text{env}})$. In cosmic web we use a Gaussian window function for the environment and a (real-space) top-hat for the halo, whereas in the excursion set we used a top-hat window function for both halo and environment. This affects the value of $\Senv$ obtained from the same environment radius. Additionally, the Lagrangian (and Eulerian) result(s) both assume that the halo-environment correlation is $\epsilon = \Senv / S$. This is due to the assumption in excursion set theory that our trajectories in $(\delta, S)$ are uncorrelated at successive values of $S$, for which one requires a sharp-$k$ window function, whence comes this expression for $\epsilon$. In the cosmic web method we calculate $\epsilon$ numerically for a Gaussian and top-hat window function in the environment and halo respectively. Therefore the same values of $\delta_{c}(S | \denv, \Senv)$ map differently onto the argument of the conditional HMF. Once this is established, it does not matter that the marginalisation over the environment is the same in both methods.
Given the performance of the various technique for extending the fitting functions to chameleon MG, and the performance factor involved in re-calculating a conditional HMF (and marginalising over it) at every stage of the MCMC procedure, we shall use the Volterra-ratio method to calculate the $\fr$ HMF in \cref{sub:recalibrating_best_fit_parameters}. Nevertheless, we have found a broad spectrum of possible methods by means of which we can incorporate MG into fitting functions originally designed for \lcdm{} alone. Moreover, we have found that some are more suited to certain applications (\eg{} the cosmic web approach) or halo finders (\eg{} the two density-marginalised methods) than others. This demonstrates the additional complexity which environment dependence produces in chameleon screening compared to symmetron- and Vainshtein-screened theories. We cannot neglect this and simply substitute the unconditional HMF if we wish to produce a useful empirical function to use in lieu of deriving one from $N$-body simulations.
\begin{figure}
\begin{center}
\subfloat[\lcdm{}: MatchMaker halofinder, 30 bins]{\label{chains_new_lcdm/Jenkins_DGP1_lenm_30_nlive_500_MM_}\includegraphics[keepaspectratio,height=0.45\textheight]{chains_new_lcdm/Jenkins_DGP1_lenm_30_nlive_500_MM_}} \\
\subfloat[\lcdm{}: RockStar halofinder, 30 bins]{\label{chains_new_lcdm/Jenkins_DGP1_lenm_30_nlive_500_RS_}\includegraphics[keepaspectratio,height=0.45\textheight]{chains_new_lcdm/Jenkins_DGP1_lenm_30_nlive_500_RS_}}
\end{center}
\caption{Triangle plot showing the 1-$\sigma$ (dark) and 2-$\sigma$ (light) posterior credible regions for the Jenkins HMF assuming \lcdm{} fitting functions and using $N$-body data from GR and MG (coloured regions). The black dashed lines show the values proposed by \cite{2001MNRAS.321..372J}.}
\label{fig:.plots/chains_new_lcdm/Jenkins_DGP1_lenm_30_nlive_100}
\end{figure}
\begin{figure}
\begin{center}
\subfloat[\lcdm{}: MatchMaker halofinder, 30 bins]{\label{chains_new_lcdm/Peacock_DGP1_lenm_30_nlive_500_MM_}\includegraphics[keepaspectratio,height=0.45\textheight]{chains_new_lcdm/Peacock_DGP1_lenm_30_nlive_500_MM_}} \\
\subfloat[\lcdm{}: RockStar halofinder, 30 bins]{\label{chains_new_lcdm/Peacock_DGP1_lenm_30_nlive_500_RS_}\includegraphics[keepaspectratio,height=0.45\textheight]{chains_new_lcdm/Peacock_DGP1_lenm_30_nlive_500_RS_}}
\end{center}
\caption{Triangle plot showing the 1-$\sigma$ (dark) and 2-$\sigma$ (light) posterior credible regions for the Peacock HMF assuming \lcdm{} fitting functions and using $N$-body data from GR and MG (coloured regions). The black dashed lines show the values proposed by \cite{Peacock2007}.}
\label{fig:.plots/chains_new_lcdm/Peacock_DGP1_lenm_30_nlive_100}
\end{figure}
\begin{figure}
\begin{center}
\subfloat[\lcdm{}: MatchMaker halofinder, 30 bins]{\label{chains_new_lcdm/SMT-Courtin_DGP1_lenm_30_nlive_500_MM_}\includegraphics[keepaspectratio,height=0.45\textheight]{chains_new_lcdm/SMT-Courtin_DGP1_lenm_30_nlive_500_MM_}} \\
\subfloat[\lcdm{}: RockStar halofinder, 30 bins]{\label{chains_new_lcdm/SMT-Courtin_DGP1_lenm_30_nlive_500_RS_}\includegraphics[keepaspectratio,height=0.45\textheight]{chains_new_lcdm/SMT-Courtin_DGP1_lenm_30_nlive_500_RS_}}
\end{center}
\caption{Triangle plot showing the 1-$\sigma$ (dark) and 2-$\sigma$ (light) posterior credible regions for the SMT-Courtin HMF assuming \lcdm{} fitting functions and using $N$-body data from GR and MG (coloured regions). The black dashed lines show the values proposed by \cite{2011MNRAS.410.1911C}.}
\label{fig:.plots/chains_new_lcdm/SMT-Courtin_DGP1_lenm_30_nlive_100}
\end{figure}
\begin{figure}
\begin{center}
\subfloat[\lcdm{}: MatchMaker halofinder, 30 bins]{\label{chains_new_lcdm/Warren-Crocce_DGP1_lenm_30_nlive_500_MM_}\includegraphics[keepaspectratio,height=0.45\textheight]{chains_new_lcdm/Warren-Crocce_DGP1_lenm_30_nlive_500_MM_}} \\
\subfloat[\lcdm{}: RockStar halofinder, 30 bins]{\label{chains_new_lcdm/Warren-Crocce_DGP1_lenm_30_nlive_500_RS_}\includegraphics[keepaspectratio,height=0.45\textheight]{chains_new_lcdm/Warren-Crocce_DGP1_lenm_30_nlive_500_RS_}}
\end{center}
\caption{Triangle plot showing the 1-$\sigma$ (dark) and 2-$\sigma$ (light) posterior credible regions for the Warren-Crocce HMF assuming \lcdm{} fitting functions and using $N$-body data from GR and MG (coloured regions). The black dashed lines show the values proposed by \cite{2006ApJ...646..881W}.}
\label{fig:.plots/chains_new_lcdm/Warren-Crocce_DGP1_lenm_30_nlive_100}
\end{figure}
\subsection{Assuming concordance cosmology}
\label{sub:assuming_concordance_cosmology}
The first problem is perhaps so obvious as to be invisible: if we do not set out to look for modified gravity, we may be unable to detect it. In other words, having assumed a \lcdm{} cosmology \textit{a priori}, does modified gravity present itself \textit{a posteriori} as merely a different (\ie{}, non-standard) set of best-fit parameters in a \lcdm{} halo mass function? In this subsection, we fit the N-body HMFs for each modified gravity model using the \lcdm{} mass functions. We used $10^{2}$ and $10^{3}$ live points to ensure that our results had converged, \ie{} that our estimate of the posteriors PDFs was accurate. We also examined the effects of both the \texttt{MatchMaker} and \texttt{RockStar} halo finders, which we expect to differ due to their varying criteria for finding bound objects, which translates into a different halo mass function.
Throughout this subsection, we will refer to \cref{fig:.plots/chains_new_lcdm/Warren-Crocce_DGP1_lenm_30_nlive_100,fig:.plots/chains_new_lcdm/SMT-Courtin_DGP1_lenm_30_nlive_100,fig:.plots/chains_new_lcdm/Peacock_DGP1_lenm_30_nlive_100,fig:.plots/chains_new_lcdm/Jenkins_DGP1_lenm_30_nlive_100}. These figures show the 1,2-$\sigma$ credible regions for the free parameters in each of the fitting functions which passed our tests in \cref{app:calibration_of_multinest}. The main diagonal shows the 1-$d$ posteriors marginalised over all other parameters, while the off-diagonal plots show correlations between pairs of parameters via the 2-$d$ credible regions at $1,2$-$\sigma$. The black dashed lines show the published values for each HMF, \ie{} those proposed by their original authors. (Where it is not visible, the value is outside the range of the plot provided by \texttt{GetDist}.) The free parameters are shown in \cref{tab:hmf_functions} for each of the four families of HMF fitting function.
First we define appropriate criteria for a fitting function to produce a well-behaved fit to the data. An ill-behaved fitting function, \eg{} with poorly constrained posteriors, suggests that this fitting function is sensitive to MG because our (incorrect) \lcdm{} hypothesis does not fit the data, no matter how far we vary the values of the free parameters. A well-behaved fitting function is one for which the probability of the model given the data is well-constrained, \ie{} our hypothesis of an \lcdm{} cosmology \textit{with new parameter values} is likely \textit{despite} the fact that we know that our data emerge from a non-\lcdm{} simulation. In this scenario, we would fail to detect the influence of MG if we attributed the shift in MAP-values to a factor such as differing cosmology or halo finders from the paper(s) in which the fitting function was first proposed.
We define a fit to be \lq\lq{}successful\rq\rq{} if the credible regions are continuous, smooth and peaked, tending to zero well within the prior parameter range. We do not exclude a fit from being well-behaved if zero is within its $1\sigma$ credible region: instead we examine the proximity of the maximum-a-posteriori value, excluding the fit if the probability at zero is close to that of the MAP. (A more quantitative option would be to exclude a fit if zero lies within the full-width-half-maximum of the Gaussian posterior, but we cannot guarantee that our posteriors are Gaussian.) The upper limit is set by the priors, so our main criterion is the breadth of the posterior. If it does not have a clear peak (or peaks), but a flat posterior, then there is too much uncertainty for the fit to be useful: it is not constrained by the data.
The majority of the mass functions are well-behaved for all MG models and both halo finders: Peacock (\cref{fig:.plots/chains_new_lcdm/Peacock_DGP1_lenm_30_nlive_100}), Jenkins (\cref{fig:.plots/chains_new_lcdm/Jenkins_DGP1_lenm_30_nlive_100}). Initially, SMT-Courtin (\cref{fig:.plots/chains_new_lcdm/SMT-Courtin_DGP1_lenm_30_nlive_100}) appears to not be successful because both halo finders and all MG models have a peak as $p$ approaches zero. (All the curves in the bottom right plot in the triangle in \cref{fig:.plots/chains_new_lcdm/SMT-Courtin_DGP1_lenm_30_nlive_100} increase as $p \rightarrow 0$; \cf{} the peaks on the 1-$d$ posteriors for $A$ and $p$.) However, this lower bound is set by the physical requirement for the parameters to be positive, rather than an indication that we have not explored enough of the parameter space. Moreover, considering that the prior volume of $p$ is in $[0,10]$, the motion of the maximum-likelihood from $0.1$ (Courtin \etal{}'s original value) to $0.0$ is only a $1\%$ shift relative to the size of the parameter space. The Warren-Crocce fitting function exhibits slightly different behaviour in \cref{fig:.plots/chains_new_lcdm/Warren-Crocce_DGP1_lenm_30_nlive_100} for $b$. Using the \texttt{MatchMaker} finder, only LCDM, DGP1 and F5 peak clearly away from $b \approx 0$; whereas using \texttt{RockStar} only SymB peaks at $b \approx 0$. The same logic applies as with SMT-Courtin, except that we can be more reassured here because the majority of MG models produce a ML value at the published value (the black dashed line). However, the posteriors are broader and more complicated in shape than the simple peaks for the other HMFs. This is probably due to the corrections made to the mass of each halo when Warren \etal{} derived the function: a halo of $N$ particles was corrected to $N(1 - N^{-0.6})$, producing a non-linear correction to the resulting $n(M)$. In this way we have found that all of our fitting functions can exhibit a degeneracy between a change in cosmology and a change in the underlying gravity theory.
We can now determine the sensitivity of these fitting functions to the halo data produced by the underlying theory of gravity. We are interested in the overlap between credible regions as we change the MG $N$-body \lq\lq{}data\rq\rq{}. Overlap indicates that these parameter pairs are insensitive to deviations from GR. Conversely, should the values of (or degeneracies between) parameters change sufficiently between $N$-body simulation \lq\lq{}data\rq\rq{}, then we have an HMF in which deviations from \lcdm{} values may reliably indicate the underlying deviation from an \lcdm{} cosmology. Over the next few paragraphs we examine each fitting function.
In \cref{fig:.plots/chains_new_lcdm/Jenkins_DGP1_lenm_30_nlive_100}, the Jenkins HMF shows a high degree of overlap in parameters $a$ and $b$ (the bottom centre 2-$d$ posterior), so we must rely on $A$ to separate the various theories. Both the posterior for $\left\{ A, a \right\}$ (centre-left) and for $\left\{ A, b \right\}$ (bottom-left), the MG theories clump into several groups. Regardless of halo finder, DGP and LCDM are nearly indistinguishable, whereas F5 is distinct. The overlap between F6 and the two symmetron models depends upon the halo finder: using \texttt{MatchMaker} (top triangle plot) F6 and SymA overlap and SymB is distinct; whereas using \texttt{RockStar} (bottom triangle plot) all three overlap. Thus we can clearly identify whether a result is in one of the DGP-LCDM or Sym-FR \lq\lq{}families\rq\rq{} but not confidently be more specific.
The Peacock model shows similar behaviour in \cref{fig:.plots/chains_new_lcdm/Peacock_DGP1_lenm_30_nlive_100}. Here the overlap is more severe. (Consider the posteriors for $\left\{ a, c \right\}$ (bottom-left) and for $\left\{ b, c \right\}$ (bottom-centre).) Thus only $A$ shows any spread in values, with bot DGP models overlapping \lcdm{} and the rest depending upon the halo finder. Again for \texttt{RockStar} there is more overlap between the three \lq\lq{}families\rq\rq{}, whereas for \texttt{MatchMaker} SymB is distinct from the indistinguishable F6, F5 and SymA.
The situation is even more problematic for the SMT-Courtin fit in \cref{fig:.plots/chains_new_lcdm/SMT-Courtin_DGP1_lenm_30_nlive_100}. In the 1-$d$ posteriors, not only does $a$ (centre) have significant overlap, but $p$ (bottom) peaks to the same value for all MG models. Once again we find the DGP and LCDM models (themselves inseparable) not distinguishable from the $\fr$ and Symmetron ones. This time, considering $\left\{ a, b \right\}$ (centre-left) \texttt{MatchMaker} (top triangle) separates SymB from the remaining three models, whereas \texttt{RockStar} (bottom triangle) separates F5 from the other three. This demonstrates the impact of the halo finder on the question of universality, \ie{} invariance to MG.
The Warren-Crocce model has too much overlap to determine the underlying MG model. \cref{fig:.plots/chains_new_lcdm/Warren-Crocce_DGP1_lenm_30_nlive_100} shows overlap for all of the 2-$d$ posteriors, to the extent that not all of the credible regions are visible. The 1-$d$ posteriors (main diagonal) confirm that the parameters overlap, particularly $b$. Although the MAP values do not overlap exactly for every MG model, they are sufficiently close that it would be difficult to distinguish between MG theories in this case. We would not see the shift in parameter MAP-values in MG (coloured curves) compared to \lcdm{} (grey curves) which would signal a problem with our \lcdm{} hypothesis, using MG halo data.
Therefore we find that is it possible for MG to be mis-interpreted as a \lcdm{} result for all of the fitting functions which survived \cref{app:calibration_of_multinest}, bar Warren-Crocce. However, we cannot distinguish between the underlying mechanisms for the deviation from \lcdm{}. The same trends occur: the separation of LCDM and the two DGP models into one group and the $\fr$ and Symmetron ones into one or more others. This latter group has behaviour which is halo-finder--dependent. Once may extrapolate that this is due to the flat barrier of DGP, compared to the mass-dependent barriers of the other MG models (\cf{} \cref{fig:deltac_all}). In practice, we cannot readily use this to test for varying theories of MG because a given point in parameter space can be occupied by the credible regions of multiple MG theories. Nonetheless, it is remarkable that MG theories can be well-approximated by an HMF assuming an \lcdm{} cosmology. The extra fifth-force interactions governing the halo collapse in the $N$-body data can be ignored for the purposes of excursion set theory. Only a change in the best-fit parameters is required for the simple \lcdm{} excursion set model to match MG data.
\begin{figure}
\begin{center}
\subfloat[MG: MatchMaker halofinder, 30 bins]{\label{chains_new_mg/Jenkins_DGP1_lenm_30_nlive_500_MM_}
\includegraphics[keepaspectratio,height=0.45\textheight]{chains_new_mg/Jenkins_DGP1_lenm_30_nlive_500_MM_}} \\
\subfloat[MG: RockStar halofinder, 30 bins]{\label{chains_new_mg/Jenkins_DGP1_lenm_30_nlive_500_RS_}
\includegraphics[keepaspectratio,height=0.45\textheight]{chains_new_mg/Jenkins_DGP1_lenm_30_nlive_500_RS_}}
\end{center}
\caption{Triangle plot showing the 1-$\sigma$ (dark) and 2-$\sigma$ (light) posterior credible regions for the Jenkins HMF assuming the same MG fitting functions as the model in the $N$-body data (coloured regions). The black dashed lines show the values proposed by \cite{2001MNRAS.321..372J}.}
\label{fig:.plots/chains_new_mg/Jenkins_DGP1_lenm_30_nlive_100}
\end{figure}
\begin{figure}
\begin{center}
\subfloat[MG: MatchMaker halofinder, 30 bins]{\label{chains_new_mg/Peacock_DGP1_lenm_30_nlive_500_MM_}\includegraphics[keepaspectratio,height=0.45\textheight]{chains_new_mg/Peacock_DGP1_lenm_30_nlive_500_MM_}} \\
\subfloat[MG: RockStar halofinder, 30 bins]{\label{chains_new_mg/Peacock_DGP1_lenm_30_nlive_500_RS_}\includegraphics[keepaspectratio,height=0.45\textheight]{chains_new_mg/Peacock_DGP1_lenm_30_nlive_500_RS_}}
\end{center}
\caption{Triangle plot showing the 1-$\sigma$ (dark) and 2-$\sigma$ (light) posterior credible regions for the Peacock HMF assuming the same MG fitting functions as the model in the $N$-body data (coloured regions). The black dashed lines show the values proposed by \cite{Peacock2007}.}
\label{fig:.plots/chains_new_mg/Peacock_DGP1_lenm_30_nlive_100}
\end{figure}
\begin{figure}
\begin{center}
\subfloat[MG: MatchMaker halofinder, 100 bins]{\label{chains_new_mg/SMT-Courtin_DGP1_lenm_30_nlive_500_MM_}\includegraphics[keepaspectratio,height=0.45\textheight]{chains_new_mg/SMT-Courtin_DGP1_lenm_30_nlive_500_MM_}} \\
\subfloat[MG: RockStar halofinder, 100 bins]{\label{chains_new_mg/SMT-Courtin_DGP1_lenm_30_nlive_500_RS_}\includegraphics[keepaspectratio,height=0.45\textheight]{chains_new_mg/SMT-Courtin_DGP1_lenm_30_nlive_500_RS_}}
\end{center}
\caption{Triangle plot showing the 1-$\sigma$ (dark) and 2-$\sigma$ (light) posterior credible regions for the SMT-Courtin HMF assuming the same MG fitting functions as the model in the $N$-body data (coloured regions). The black dashed lines show the values proposed by \cite{2011MNRAS.410.1911C}.}
\label{fig:.plots/chains_new_mg/SMT-Courtin_DGP1_lenm_30_nlive_100}
\end{figure}
\begin{figure}
\begin{center}
\subfloat[MG: MatchMaker halofinder, 30 bins]{\label{chains_new_mg/Warren-Crocce_DGP1_lenm_30_nlive_500_MM_}\includegraphics[keepaspectratio,height=0.45\textheight]{chains_new_mg/Warren-Crocce_DGP1_lenm_30_nlive_500_MM_}} \\
\subfloat[MG: RockStar halofinder, 30 bins]{\label{chains_new_mg/Warren-Crocce_DGP1_lenm_30_nlive_500_RS_}\includegraphics[keepaspectratio,height=0.45\textheight]{chains_new_mg/Warren-Crocce_DGP1_lenm_30_nlive_500_RS_}}
\end{center}
\caption{Triangle plot showing the 1-$\sigma$ (dark) and 2-$\sigma$ (light) posterior credible regions for the Warren-Crocce HMF assuming the same MG fitting functions as the model in the $N$-body data (coloured regions). The black dashed lines show the values proposed by \cite{2006ApJ...646..881W}.}
\label{fig:.plots/chains_new_mg/Warren-Crocce_DGP1_lenm_30_nlive_100}
\end{figure}
\subsection{Recalibrating best-fit parameters}
\label{sub:recalibrating_best_fit_parameters}
Now we explore the opposite question to the previous one, namely: how universal is the halo mass function? In this subsection, we use the same extended gravity model for both the $N$-body data and the collapse density for the halo mass function. First we define what we mean by \lq\lq{}universality\rq\rq{} in the context of MG HMFs. Then we explain why our \lcdm{} results do not recover the best-fit values of the free parameters proposed by the original authors of the fitting functions. Next we discuss the results for each fitting function in detail.
We use the term \lq\lq{}universal\rq\rq{} in the sense that a given fitting function is insensitive to changes in the gravity theory underlying the spherical collapse of the haloes. Specifically, provided that we account for the changes in MG by using the correct collapse density in $\nu$, we expect a universal HMF to predict the same best-fit values for the free parameters in the fitting function. In this way, $\nu$ would account for changes in gravitational models, without requiring recalibration of the fitting function. This is entirely analogous to the definition of a universal HMF in GR, where the universality refers to the fitting function being insensitive to changes in redshift and cosmological parameters. One of the drivers behind the proliferation of fitting functions in \cref{tab:hmf_functions} is the quest for such au universal function in \lcdm{} (and earlier, in other CDM cosmologies). This would vindicate the current practice to derive and calibrate HMF functions in \lcdm{} $N$-body simulations alone---with the advantage that they could then be reliably applied to any scalar theory of MG with the same cosmological parameters.
Should we expect the LCDM results from our calculations to equal those from the original values of their respective papers? We do not expect to recover these results for a variety of reasons:
\begin{itemize}
\item The value of $\delta_{c}$ changes with $\Omega_{m0}$ and $H_{0}$
\item The value of $\sigma$ depends upon the power spectrum and value of $\sigma_{8}$
\item Differing halo finders and settings
\item The presence (or absence) of corrections to the $N$-body halo data
\end{itemize}
Let us examine each of these in turn.
\cref{tab:cosmologies} summarises the cosmological parameters used for each of the papers which provided "best-fit" parameters for particular fitting functions. The dependence of $\delta_{c}^{\Lambda}$ on $\Omega_{m0}$ is given in parametric form by \cite{1995MNRAS.274L..73E} as $\delta_{c} = 1.68/\Omega_{m0}^{0.28}$. Our value is $\delta_{c}^{\Lambda} \approx 1.675$. This contrasts with most of the papers (bar \cite{2011MNRAS.410.1911C}). Some \cite{Sheth:2001dp,Peacock2007,2010MNRAS.403.1353C} use the SCDM value $\delta_{c} \approx 1.686$ (following \PS{} \cite{1974ApJ...187..425P}) regardless of the actual cosmology of their simulations. Others \cite{2001MNRAS.321..372J,2008ApJ...688..709T,1203.3216,2013MNRAS.433.1230W,2006ApJ...646..881W} absorb the value of $\delta_{c}$ into their free parameters because they use $\sigma$ as the independent variable rather than $\nu$, so their choice---which we need to convert back to $\nu$---is ambiguous. In all cases we have assumed the exact SCDM collapse value $\delta_{c} = 3 \cdot (12 \pi)^{2/3} / 20$ when converting from $\sigma$ to $\nu$. Only \cite{2011MNRAS.410.1911C} vary $\delta_{c}$ according to the solution of the spherical collapse equation for a variety of cosmologies.
Thus we have a different numerator for $\nu$. Moreover, it is unlikely that our $P(k)$ is precisely equal to any of those used in the papers.
Even if it did match, the normalisation $\sigma_{8}$ differs. The variance $\sigma$ which forms the denominator of $\nu$ therefore also differs from previous publications. Since the independent variable $\nu$ (or more precisely, our mapping $\ln M \rightarrow \nu$ from counts to \fcd{}) differs in this paper (and indeed in all of the others), our free parameters must change to compensate. Even if we had exactly the same numbers of counts and the same bins, and our best-fit HMF had the same $n(M)$ values as a preceding paper, we would see a change in $f(\nu)$. This is a small contribution to the movement of the maximum-likelihood peaks from the published values.
The choice of halo finder further abstracts the problem. All of the published values except Peacock and Watson (both of which are FoF-only) arise from a compromise between the best-fit values for multiple halo finders. For example, Jenkins \cite{2001MNRAS.321..372J} uses both FoF and SO halo finders to find a fitting function which has a residual of within 20\% compared to their $N$-body HMFs simulated for a range of cosmologies (not just \lcdm{}). Even when restricting their data to FoF-only and SO-only (whereby they obtain different best-fit values), the FoF linking length is changed between cosmologies ($b=0.2$ for $\tau$CDM, $b=0.164$ for \lcdm{}), so we cannot disentangle the cosmological effects from the reduction of the $N$-body \lq\lq{}data\rq\rq{}. As we can see by comparing the results for \texttt{MatchMaker} and \texttt{RockStar} in \cref{fig:.plots/chains_new_mg/Jenkins_DGP1_lenm_30_nlive_100,fig:.plots/chains_new_mg/Peacock_DGP1_lenm_30_nlive_100,fig:.plots/chains_new_mg/SMT-Courtin_DGP1_lenm_30_nlive_100,fig:.plots/chains_new_mg/Warren-Crocce_DGP1_lenm_30_nlive_100}, the same distribution of mass within the $N$-body simulations produces a different HMF according to the halo finder used. \cite{2011MNRAS.415.2293K} discuss this in great detail for \lcdm{} and we have no reason to disagree with their findings.
Differing authors also treat the distribution $n(M)$ obtained from their halo finder(s) in a variety of ways. As we briefly covered in \cref{sec:algorithms}, the mass cutoffs are controlled by the simulation box size and the mass resolution. \cref{tab:hmf_functions} summarises the mass range used by each paper. This shifts the sampling along a subsection of the actual HMF, changing the influence of each parameter in the likelihood function. An extreme example of this is the Reed 2003 fit, whose parameter $c$ is completely unconstrained by masses $M \leq 10^{15} M_{\odot}$ \cite{2003MNRAS.346..565R}. Sometimes the individual halo masses are systematically \lq\lq{}corrected\rq\rq{}, \eg{} \cite{2017MNRAS.467.3454B} or (more relevantly here) the Warren correction: a halo of $N$ particles was updated to $N(1-N^{-0.6})$ to account for perceived flaws in the halo finders. These factors all affect the final \fcd{} of the data, against which is compared the \fcd{} of the model (converted to counts per bin) which contains the values of the free parameters at a chosen point in the parameter volume.
Given these factors, it is not surprising that our LCDM values do not always align with the published values. In \cref{fig:.plots/chains_new_mg/Jenkins_DGP1_lenm_30_nlive_100,fig:.plots/chains_new_mg/Peacock_DGP1_lenm_30_nlive_100,fig:.plots/chains_new_mg/SMT-Courtin_DGP1_lenm_30_nlive_100,fig:.plots/chains_new_mg/Warren-Crocce_DGP1_lenm_30_nlive_100} the grey lines show our \lcdm{} posteriors, while the black dashed lines show the values from one of the relevant papers respectively. (We do not show all of the published values for every fitting function because this would lead to up to five lines on some plots, which would be confusing.) Of particular interest is the variety of best-fit values for the SMT fitting function. Our changes in all HMFs are the same order of magnitude as the changes other authors have found when altering the cosmological parameters, halo finders and data reduction techniques used to derive the HMF. Since we are utilising a single cosmology and separating the effects of our two halo finders, it is not surprising that our values do not reproduce the existing ones.
Now we extend our discussion to our MG results. \cref{fig:.plots/chains_new_mg/Warren-Crocce_DGP1_lenm_30_nlive_100,fig:.plots/chains_new_mg/Peacock_DGP1_lenm_30_nlive_100,fig:.plots/chains_new_mg/SMT-Courtin_DGP1_lenm_30_nlive_100,fig:.plots/chains_new_mg/Jenkins_DGP1_lenm_30_nlive_100} show the 1,2-$\sigma$ credible regions for the free parameters in each of the fitting functions which passed our tests in \cref{app:calibration_of_multinest}. As in the previous subsection, the main diagonal shows the 1-$d$ posteriors marginalised over all other parameters, while the off-diagonal plots show correlations between pairs of parameters via the 2-$d$ credible regions at $1,2$-$\sigma$. The black dashed lines show the published values for each HMF, \ie{} those proposed by their original authors. (Where it is not visible, the value is outside the range of the plot provided by \texttt{GetDist}.) We examined the behaviour of both halo finders, showing the \texttt{MatchMaker} in top subplots and \texttt{RockStar} in the bottom subplots.
Some HMFs have precisely the same \lq\lq{}good behaviour\rq\rq{} (or lack thereof) in both cases. The Jenkins HMF has the same good behaviour in \cref{fig:.plots/chains_new_mg/Jenkins_DGP1_lenm_30_nlive_100} (full MG $\nu$) as we already saw in \cref{fig:.plots/chains_new_mg/Jenkins_DGP1_lenm_30_nlive_100} (\lcdm{} $\nu$). Similarly, the Peacock HMF is well-behaved in both \cref{fig:.plots/chains_new_mg/Peacock_DGP1_lenm_30_nlive_100} (full MG $\nu$) and \cref{fig:.plots/chains_new_lcdm/Peacock_DGP1_lenm_30_nlive_100} (\lcdm{} $\nu$). It is notable that the Peacock HMF---an empirically-derived fit to the Watson-FoF HMF accurate to 1\% in \lcdm{}---does behave well when generalised, whereas the original Watson fit does not. (In fact, we discarded the Tinker-Angulo-Watson fitting function in calibration in \cref{app:calibration_of_multinest}.) This suggests that the behaviour is not caused by over-simplifying the gravitational collapse, but by the underlying form of the fitting function itself. Despite the fact that we have used the full MG modifications to $\nu \left( S \given \denv, \Senv \right)$, the behaviour of these HMFs mirrors that of the previous subsection, in which we assumed $\nu = \delta_{c}^{\Lambda} / S$. In order to vindicate our more complex hypothesis as the correct gravitational model underlying the $N$-body data, we would need to calculate the evidence factors for our hypotheses.
In some cases, accounting for the mass dependence of the critical density does improve the behaviour of the HMF. While SMT-Courtin has the same good behaviour for $A$ (left column in triangle plot) and $a$ (centre column) in \cref{fig:.plots/chains_new_mg/SMT-Courtin_DGP1_lenm_30_nlive_100} (full MG $\nu$) and \cref{fig:.plots/chains_new_mg/SMT-Courtin_DGP1_lenm_30_nlive_100}, the behaviour of $p$ improves. Whereas in our \lcdm{}-$\nu$ results, all of the MAP values for $p$ were zero (bottom-right of triangle plot), the posteriors peak away from zero for SymA, SymB and F6 in our MG-$\nu$ results, regardless of halo finder. Warren-Crocce shows some improvement as well. Again, we see good behaviour for $A$ (left column in triangle plot) and $c$ (right column) in \cref{fig:.plots/chains_new_mg/SMT-Courtin_DGP1_lenm_30_nlive_100} (full MG $\nu$) and \cref{fig:.plots/chains_new_lcdm/SMT-Courtin_DGP1_lenm_30_nlive_100} (\lcdm{} $\nu$). The imrpovement in the remaining parameters $a$ and $b$ depends upon the MG theory. Instead of the posteriors peaking at $b \approx 0$ in \cref{fig:.plots/chains_new_lcdm/Warren-Crocce_DGP1_lenm_30_nlive_100}, the ones in \cref{fig:.plots/chains_new_mg/Warren-Crocce_DGP1_lenm_30_nlive_100} are well-defined for F5,6 models using the \texttt{MatchMaker} halo finder (top plots); moreover \texttt{RockStar} behaves (relatively) well for everything apart from SymB in the bottom plot. Thus we see that generalising $\nu$ from its \lcdm{} value (proportional to $\sigma^{-1}$) to MG does improve the overall behaviour of the HMFs, when the barrier density is not a constant.
We can use the HMFs to examine the universality of $\nu$: specifically, whether accounting for the excursion set behaviour of the MG models is sufficient to render our fitting functions independent of MG. While it is evident from \cref{fig:.plots/chains_new_mg/Jenkins_DGP1_lenm_30_nlive_100,fig:.plots/chains_new_mg/Peacock_DGP1_lenm_30_nlive_100,fig:.plots/chains_new_mg/SMT-Courtin_DGP1_lenm_30_nlive_100} that no single value works for every MG model, we do find clustering between families. The Jenkins posteriors (\cref{fig:.plots/chains_new_mg/Jenkins_DGP1_lenm_30_nlive_100}) show that the credible regions for the two symmetron models are quite distinct from the other five models, all of which have overlapping credible regions.
The Peacock function shows a higher degree of universality than Jenkins: particularly in \cref{chains_new_mg/Peacock_DGP1_lenm_30_nlive_500_MM_}, but slightly less in \cref{chains_new_mg/Peacock_DGP1_lenm_30_nlive_500_RS_}. In particular $c$ is practically universal, but there is a spread of overlapping values for the other two parameters.
SMT-Courtin (\cref{fig:.plots/chains_new_mg/SMT-Courtin_DGP1_lenm_30_nlive_100}) has very similar behaviour to the Jenkins HMF, with the two Symmetron models distinct from each other as well as the clustering--but here it is four of the other five models, with F6 closer to the Symmetron results.
Unlike the preceding fitting functions, Warren-Crocce (\cref{fig:.plots/chains_new_mg/Warren-Crocce_DGP1_lenm_30_nlive_100}) has posteriors which largely overlap, perhaps with the exception of F6, and more clustering in \texttt{MatchMaker} than \texttt{RockStar}. This fitting function has no credible regions which are isolated from one other. Thus, the fitting functions display a range of behaviours, but most show that the best-fit parameter values for multiple MG models do overlap.
Where the symmetron models are distinguishable from the others, the differences are driven by the \lq\lq{}normalisation\footnote{While $A$ originally played this role in the SMT-Courtin function \cite{Sheth:2001dp}, we no longer require the cumulative mass fraction to tend to unity.} factor:\rq\rq{} the symmetron models underpredict $n(M)$, requiring a systematic upwards shift by increasing the multiplicative factor $A$. These models have a drifting barrier $\delta_{c}(S)$ which includes mass dependence. However, this dependence via the collapse density ODE is not accounting for all of the actual behaviour of the haloes in non-linear collapse. Considering that we use a simple model of a collapsing spherical top-hat, we may be oversimplifying the effect of the symmetron fifth-force.
There is also the issue of the F5 and F6 models exhibiting greater spread (albeit still with overlapping credible regions) than the LCDM and DGP results. This is particularly visible in Warren-Crocce. Recall that here we fully account for the excursion-set barrier density using the Volterra solution. However, we use this to scale the unconditional HMF, so we approximate the integral over the environmental dependence of the HMF via the value at the peak of the environment distribution (which happens to be $\denv = 0$). Under these circumstances, it is remarkable that our approximation does produce such a universal result.
The two DGP models cluster strongly with \lcdm{} in all the HMFs. This is possibly because all of these models use a flat barrier, so there is no additional mass- or environment-dependence to be included in $\nu$, so no additional excursion set behaviour which needs to be approximated by the change in the independent variable.
While we do not find a strong degree of overall universality, we can see that the different screening mechanisms do behave similarly. This is independent of the choice of halo finder, so this clustering is not caused by systematic effects or scatter in the $N$-body data.
This illustrates the caution which must be employed when using fitting function originally calibrated in \lcdm{} in the context of MG. Although the generalisation of the fitting functions from \lcdm{} to MG does not produce a completely universal fitting function, there is a degree of universality in the clustering of the credible regions for different screening mechanisms. This is because the effects of the fifth-force are largely encapsulated by the modified Poisson equation which appears in the ODE for gravitational collapse. The resulting $\delta_{c}$ clearly does not contain all of the non-linear collapse information (otherwise we \textit{would} have a universal HMF) but it does incorporate enough into $\nu$ that the resulting fitting function depends only on the type of screening, rather than the values of the fifth-force parameters.
\section{Conclusions}
\label{sec:conclusions}
This section reiterates the salient points of this paper. We outline the method we have used, before describing avenues for generalisation and other possibilities for further work. We conclude by summarising the key results of this paper.
\subsection{Summary}
\label{sub:summary}
In this paper, we explored the use of the halo mass function in screened MG theories. We selected a range of theories which have different screening mechanisms (\cref{sub:dgp_gravity,sub:fr_gravity,sub:symmetron_gravity}) and derived their additional contribution to the Poisson equation. We summarised a variety of HMFs and described the nature of their universality in GR and how to transform this into the equivalent in MG (\cref{sec:the_halo_mass_function}). The $N$-body simulations from which we extracted halo catalogues to compare to our empirical fits are described in \cref{sub:n_body_simulations_haloes}. Similarly, the Bayesian methodology for estimating maximum-likelihood parameters and the relative likelihood of the different models is outlined in \cref{sub:bayesian_inference}. The key steps of our method are:
\begin{enumerate}
\item Conversion of the GR HMF from $\sigma$ to $\nu$ (if necessary).
\item Calculation of the effective fifth-force $F_{\textrm{eff}}$ to insert into the spherical collapse ODE.
\item Calculations of the collapse density $\delta_{c}(S,\denv,\Senv)$ to incorporate into $\nu$.
\item Use of an appropriate excursion-set technique to account for the barrier density $\delta_{c}$.
\item MCMC estimation of the best-fit free parameter values and their credible regions.
\item Output of the corresponding best-fit HMF.
\end{enumerate}
Our main results (\cref{sec:results_and_discussion}) are as follows.
We found a broad spectrum of possible methods---some newly-proposed in this paper---by means of which we can incorporate MG into fitting functions originally designed for \lcdm{} alone. Of the various techniques for extending the fitting functions to chameleon MG, we have found that some are more suited to certain applications (\eg{} the cosmic web approach) or halo finders (\eg{} the two density-marginalised methods) than others. This demonstrates the additional complexity which environment dependence produces in chameleon screening compared to symmetron- and Vainshtein-screened theories. We cannot neglect this and simply substitute the unconditional HMF if we wish to produce a useful empirical function to use in lieu of deriving one from $N$-body simulations.
We found that the effects of MG can be interpreted as a change in best-fit parameters in the \lcdm{} HMF for all of the fitting functions. Alternatively, the relation can be inverted to judge the universality of the HMF, \ie{} its independence on the underlying theory of gravity. Although we found no completely universal HMF, the parameter values did cluster according to the type of screening mechanism, with Jenkins, Peacock and SMT-Courtin being the least universal and Warren-Crocce the most. The former group required very different best-fit parameters for the two Symmetron models, whereas in the latter all of the models had overlapping credible regions. The results suggest that a single, best-fit HMF might be used for each type of screening, independent of the parameters in the MG model. This demonstrates that the additional complexity of the gravitational collapse in screened MG theories cannot always be accounted for using the techniques developed in GR. However, it is unnecessary to develop new fitting functions and calibrate them on a case-by-case basis.
We have demonstrated that it is possible to generalise some of the halo mass functions in common use in GR to incorporate MG theories with a variety of screening mechanisms. However, the calibration of these fitting functions has a number of caveats which are not encountered in the \lcdm{} framework for which they were initially developed. Nonetheless, it is remarkable that our method can incorporate much of the non-linear collapse behaviour of screened MG in a simple and efficient mechanism. This is in direct contrast to the difficulties encountered in performing $N$-body simulations in screened MG. Thus we have provided an excursion-set-motivated alternative in MG to the need to replicate the time-consuming development of accurate halo mass functions which took place (and is ongoing) in GR.
\subsection{Further work}
\label{sub:further_work}
The method presented in this paper for calculation of the MG halo mass function using the fitting functions derived from \lcdm{} has many avenues for generalisation. Most straightforward of these is the application to other fitting functions as they become available, provided that these functions can be expressed in terms of the \lq\lq{}universal parameter\rq\rq{} $\nu$ rather than the variance $\sigma$.
The universality of the halo mass function can be further extended to higher redshifts. The collapse ODE (derived in \cite{2012MNRAS.421.1431L}) has a new stopping condition that $y_{h}(z_{c}) = 0$, but the same bijection scheme can be applied to calculate the collapse density $\delta_{c}(z_{c})$. The variance $\sigma(z)$ is obtained from the present-day value via the growth factor $D(z)$. However, to a good approximation, these modifications cancel, leaving $\nu$ independent of $z$ \cite{Zentner:2006vw}. This generalisation is particularly relevant given the ongoing discussion on the $z$-independence of $f(\nu)$ in \lcdm{}. It would be particularly interesting to determine the influence of the fifth-force on the evolution of the HMF.
The calibration techniques are applicable to any MG theory which satisfies the following:
\begin{itemize}
\item Existence of a modified Poisson equation to approximate the modifications to gravity
\item Well-posedness of the corresponding spherical-collapse ODE
\item Background expansion similar to \lcdm{} (so that the \lcdm{} growth factor can be used and the halo environment treated as \lcdm{} in the collapse ODE)
\end{itemize}
Galileon MG is an example of a screened theory for which this technique may be used. However, it may also be applied to MG theories which do not involve screening, but have some other method of being observationally-viable. It would be interesting to investigate whether the results we have found are unique to screening models, or whether they extend to non-screened theories.
The cosmology-dependence of these results can also be explored. This would be a daunting task, requiring $N$-body simulations for a grid of cosmological parameters, especially given the additional complexity of incorporating a fifth-force into the simulations. Nonetheless, this would permit comparison with the investigation of the cosmological-dependence of the HMF in GR (\eg{} \cite{2011MNRAS.410.1911C}). Moreover, if using changes in the best-fit GR parameters for a given fitting function to suggest a deviation from \lcdm{}, it would highlight the potential degeneracy between a change in MG and a change in the GR cosmological parameters. This is important if we are to use the HMF as a probe of MG in future surveys.
The many avenues for generalisation illustrate that the same attention to detail can be applied to the HMF in both \lcdm{} and MG. Having illustrated a number of caveats---the choice of fitting function, likelihood and the dependence of the results on both halo finder and bin width---we nonetheless show that three common HMFs can be used and calibrated in both GR and MG. Without applying the same calibration techniques in both theories, we are not making a like-for-like comparison when analysing the behaviour of the HMF, especially when constructing theoretical HMFs to compare to observations.
\section*{Acknowledgements}
We would like to thank Hans Winther for useful feedback on the many drafts and Pedro Ferreira for useful comments and discussions.
|
{
"timestamp": "2018-11-16T02:10:42",
"yymm": "1804",
"arxiv_id": "1804.05387",
"language": "en",
"url": "https://arxiv.org/abs/1804.05387"
}
|
\section{Introduction}
Inspired by constructions from Arakelov geometry and Archimedean cohomology,
Consani and Marcolli
develop in \cite{consani-marcolli} spectral triples associated to certain Cuntz--Krieger algebras.
In this note, we expand the applicability of these spectral triples by generalizing the construction of \cite{consani-marcolli} to the setting of higher-rank graphs. We also establish the compatibility of these spectral triples with the representations and wavelets for higher-rank graphs which were developed in \cite{FGKP}. Indeed, both spectral triples and wavelets are algebraic structures which encode geometrical information, so it is natural to ask about the relationship between wavelets and spectral triples.
Our earlier paper \cite{FGJKP2} was the first to establish a connection between wavelets and spectral triples in the setting of higher-rank graphs $\Lambda$. In that paper, we linked the representations of $C^*(\Lambda)$ from \cite{FGKP}, and their associated wavelets, to the eigenspaces of the Laplace--Beltrami operators which arise from the spectral triples of Pearson and Bellissard \cite{pearson-bellissard}. The present article establishes that the wavelets from \cite{FGKP} can also be identified with the eigenspaces of the Dirac operator of a Consani--Marcolli type spectral triple for $C^*(\Lambda)$.
Higher-rank graphs (also called $k$-graphs) were introduced by Kumjian and Pask in \cite{kp} to provide a combinatorial model to the higher-dimensional Cuntz-Krieger algebras given by Robertson and Steger in \cite{robertson-steger}. The $C^*$-algebras $C^*(\Lambda)$ of $k$-graphs $\Lambda$ have been studied by many authors and provided concrete, computable examples of many classifiable $C^*$-algebras. The graphical character of $k$-graphs has also facilitated the analysis of structural properties of $C^*(\Lambda)$, such as simplicity and ideal structure \cite{rsy2, robertson-sims, davidson-yang-periodicity, kang-pask, ckss}, quasidiagonality \cite{clark-huef-sims} and KMS states \cite{aHLRS, aHLRS1, aHKR}.
However, the analysis of the noncommutative geometry of $C^*(\Lambda)$ is in its infancy.
Although Pask, Rennie, and Sims establish in \cite{pask-rennie-sims-manifold} that higher-rank graph $C^*$-algebras often provide tractable examples of noncommutative manifolds, the current literature contains only one class of (semifinite) spectral triples for $C^*(\Lambda)$, namely those studied in \cite{pask-rennie-sims}. In the Pearson--Bellissard spectral triples $(\mathcal A, \H, D)$ which were associated to higher-rank graphs in \cite{FGJKP2}, the algebra $\mathcal A = C_{Lip}(\Lambda^\infty)$ is commutative. Thus, the spectral triples for the noncommutative $C^*$-algebra $C^*(\Lambda)$, which we construct in Theorem \ref{thm-Consani-Marcolli-spectral-triples-k-graphs} below, constitute an important step forward in our understanding of the noncommutative geometry of $C^*(\Lambda)$, in particular because of the link we establish between these spectral triples and wavelet theory for $C^*(\Lambda)$.
Wavelets for higher-rank graphs $\Lambda$ were introduced by four of the authors of the current paper in \cite{FGKP}, building on work of Marcolli and Paolucci \cite{marcolli-paolucci} for Cuntz--Krieger algebras, which in turn was inspired by the wavelets for fractal spaces developed by Jonsson \cite{jonsson} and Strichartz \cite{strichartz}. In all of these settings, the wavelets give an orthogonal decomposition of $L^2(X, \mu)$ for a fractal space $X$, which arises from applying dilation and translation operators to a finite family of ``mother wavelets'' $f_i \in L^2(X, \mu)$. The dilation and translation operators are determined by the underlying geometry. In Jonsson and Strichartz' work, the self-similar structure of the fractal space $X$ dictates the dilation and translation operators, while in the higher-rank graph case, the dilation and translation operators arise from the graph structure. (See Section~\ref{sec:wavelets} for more details.)
To further our understanding of the noncommutative geometry of $C^*(\Lambda)$,
we construct in Theorem~\ref{thm-Consani-Marcolli-spectral-triples-k-graphs} a spectral triple $(\mathcal{A}_\Lambda, L^2(\Lambda^\infty, M), D)$, where $\mathcal A_\Lambda$ is a dense (noncommutative) subalgebra of $C^*(\Lambda)$. This spectral triple was inspired by the spectral triples for Cuntz--Krieger algebras constructed in \cite{consani-marcolli}, and offers a very different perspective on the noncommutative geometry of $C^*(\Lambda)$ than the spectral triples of \cite{pask-rennie-sims}. Theorem \ref{thm:CM-Dirac-wavelets} then establishes our link between spectral triples and wavelets for higher-rank graphs by showing that the eigenspaces of the Dirac operator $D$ of this spectral triple agree with the wavelet decomposition of \cite{FGKP}.
\subsection*{Acknowledgments} E.G.~was partially supported by the SFB 878 ``Groups, Geometry, and Actions'' of the Westf\"alische-Wilhelms-Universit\"at M\"unster. C.F.~and J.P.~were partially supported by two individual grants from the Simons Foundation (C.F. \#523991; J.P. \#316981).
S.K.~was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (\#2017R1D1A1B03034697).
\section{Background material}\label{sec:background}
We begin by detailing some foundational material needed for our results, and in particular reviewing the definition of a higher-rank graph $\Lambda$, the definition of its $C^*$-algebra $C^*(\Lambda)$, and associated wavelets.
\subsection{Higher-rank graphs and their $C^*$-algebras}
\label{sec:kgraph}
Throughout this paper, we will view
$\mathbb{N}: =\{0,1,2,\dots\}$ as a monoid under addition, or as a category. In this interpretation, the natural numbers are the morphisms in $\mathbb{N}$. Thus, for consistency with the standard notation $n \in \mathbb{N}$, we will write
\[ \lambda \in \Lambda \]
to indicate that $\lambda $ is a morphism in the category $\Lambda$.
\begin{defn}
\label{def:k-graph}
A {\em higher-rank graph} or {\em $k$-graph} by definition is a countable small category $\Lambda$ with a degree functor $d:\Lambda\to \mathbb{N}^k$ satisfying the {\em factorization property}: for any morphism $\lambda\in\Lambda$ and any $m, n \in \mathbb{N}^k$ such that $d(\lambda)=m+n \in \mathbb{N}^k$, there exist unique morphisms $\mu,\nu\in\Lambda$ such that $\lambda=\mu\nu$ and $d(\mu)=m$, $d(\nu)=n$.
We often think of $k$-graphs as a generalization of directed graphs, so we call objects $v \in\Lambda^0$ ``vertices'' and morphisms $\lambda\in\Lambda$ are called ``paths.'' We write $r,s:\Lambda\to \Lambda^0$ for the range and source maps and $v \Lambda w=\{\lambda\in \Lambda: r(\lambda)=v, s(\lambda)=w\}$. Similarly, for any $n\in \mathbb{N}^k$, we write $v\Lambda^n=\{\lambda\in\Lambda: r(\lambda)=v, d(\lambda)=n\}$.
\end{defn}
For $m,n\in\mathbb{N}^k$, we denote by $m\vee n$ the coordinatewise maximum of $m$ and $n$. Given $\lambda,\eta\in \Lambda$, we write
\begin{equation*
\Lambda^{\operatorname{min}}(\lambda,\eta):=\{(\alpha,\beta)\in\Lambda\times\Lambda\,:\, \lambda\alpha=\eta\beta,\; d(\lambda\alpha)=d(\lambda)\vee d(\eta)\}.
\end{equation*}
We say that a $k$-graph $\Lambda$ is \emph{finite} if $\Lambda^n$ is a finite set for all $n\in\mathbb{N}^k$ and say that $\Lambda$ \emph{has no sources} or \emph{is source-free} if $v\Lambda^n\ne \emptyset$ for all $v\in\Lambda^0$ and $n\in\mathbb{N}^k$. It is well known that this is equivalent to the condition that $v\Lambda^{e_i}\ne \emptyset$ for all $v\in \Lambda$ and all basis vectors $e_i$ of $\mathbb{N}^k$. Also we say that a $k$-graph is \emph{strongly connected} if, for all $v,w\in\Lambda^0$, $v\Lambda w\ne \emptyset$.
\begin{defn}\cite{kp}
If $\Lambda$ is a finite $k$-graph with no sources, write $C^*(\Lambda)$ for the universal $C^*$-algebra generated by partial isometries $\{s_\lambda\}_{\lambda \in \Lambda}$ satisfying the Cuntz--Krieger conditions:
\begin{itemize}
\item[(CK1)] $\{ s_v: v \in \Lambda^0\}$ is a family of mutually orthogonal projections;
\item[(CK2)] Whenever $s(\lambda) = r(\eta)$ we have $s_\lambda s_\eta = s_{\lambda \eta}$;
\item[(CK3)] For any $\lambda \in \Lambda, \ s_\lambda^* s_\lambda = s_{s(\lambda)}$;
\item[(CK4)] For all $v \in \Lambda^0$ and all $n \in \mathbb{N}^k$, $\sum_{\lambda \in v\Lambda^n} s_\lambda s_\lambda^* = s_v$.
\end{itemize}
\end{defn}
Condition (CK4) implies that for any $\lambda, \eta \in \Lambda$ we have
\[ s_\lambda^* s_\eta = \sum_{(\alpha, \beta) \in \Lambda^{min}(\lambda, \eta)} s_\alpha s_\beta^*,\]
where we interpret empty sums as zero. Consequently, $C^*(\Lambda) = \overline{\text{span}}\{ s_\lambda s_\eta^*: \lambda, \eta \in \Lambda\}$.
\begin{defn}
\label{def:dense-subalg}
Let $\mathcal{A}_\Lambda$ denote the dense $*$-subalgebra of $C^*(\Lambda)$ spanned by $\{ s_\lambda s_\eta^*\}_{\lambda, \eta \in \Lambda}$.
\end{defn}
An important example of a $k$-graph is the category $\Omega_k$, where
\[ \text{Obj}(\Omega_k) = \mathbb{N}^k, \qquad \text{Mor}(\Omega_k) = \{ (p, q) \in \mathbb{N}^k: p \leq q\}.\]
The range and source maps $r,s$ in $\Omega_k$ are given by $r(p,q)=p$, $s(p,q)=q$, and the degree map $d: \Omega_k \to \mathbb{N}^k$ is given by
\[ d(p, q) = q-p.\]
\begin{defn}
\label{def:infinite-path}
An {\em infinite path} in a $k$-graph $\Lambda$ is a degree preserving functor $x:\Omega_k\to \Lambda$. We write $\Lambda^\infty$ for the set of infinite paths in $\Lambda$.
Given $\lambda \in \Lambda$, we define the {\em cylinder set}
$ [\lambda] \subseteq \Lambda^\infty$ by
\[[\lambda] := \{ x \in \Lambda^\infty: x(0, d(\lambda)) = \lambda\}\]
to be the infinite paths with initial segment $\lambda$. It is well-known (cf.~\cite{kp}) that the collection of cylinder sets $\{[\lambda]\}_{\lambda \in \Lambda}$ forms a compact open basis for a locally compact Hausdorff topology on $\Lambda^\infty$. If a $k$-graph $\Lambda$ is finite, then $\Lambda^\infty$ is compact in this topology.
For each $m\in \mathbb{N}^k$, we have a shift map $\sigma^m$ on $\Lambda^\infty$ given by
\begin{equation}\label{eq:shift-map}
\sigma^m(x)(p,q)=x(p+m, q+m).
\end{equation}
for $x\in \Lambda^\infty$ and $(p,q)\in \Omega_k$.
In duality to the shift map $\sigma^m$, for each $\lambda \in \Lambda$ we also have a prefixing map $\sigma_\lambda: [s(\lambda)] \to [\lambda]$ given by
\begin{equation}\label{eq:prefix-map}
\sigma_\lambda(x) = \lambda x = \left[ (p, q) \mapsto \begin{cases} \lambda(p, q), & q \leq d(\lambda) \\
x(p-d(\lambda), q-d(\lambda)), & p \geq d(\lambda) \\
\lambda (p, d(\lambda))\, x(0, q-d(\lambda)), & p < d(\lambda) < q
\end{cases} \right]
\end{equation}
\end{defn}
According to \cite[Proposition 8.1]{aHLRS}, for any finite and strongly connected $k$-graph $\Lambda$, there is a unique self-similar Borel probability measure $M$ on $\Lambda^\infty$. To describe $M$, we require more definitions.
\begin{defn}
\label{def:vertex-matrix}
For a finite $k$-graph $\Lambda$ and $1 \leq i \leq k$, the {\em vertex matrix}
$A_i \in M_{\Lambda^0}(\mathbb{N})$ is
\[ A_i(v,w) = \# (v\Lambda^{e_i} w).\]
\end{defn}
Lemma 3.1 of \cite{aHLRS} establishes that if $\Lambda$ is finite and strongly connected, then there exists a unique vector $\kappa^\Lambda \in (0, \infty)^{\Lambda^0}$, called the {\em Perron--Frobenius eigenvector} of $\Lambda$, such that
\[ \sum_{v\in \Lambda^0} \kappa^\Lambda_v =1 \qquad \text{ and } \qquad A_i \kappa^\Lambda = \rho_i \kappa^\Lambda \quad \forall \, 1 \leq i \leq k.\]
The unique self-similar Borel probability measure $M$ of \cite{aHLRS} is given on cylinder sets by
\[
M([\lambda])=(\rho(\Lambda))^{-d(\lambda)}\kappa^{\Lambda}_{s(\lambda)}\quad\text{for}\;\; \lambda\in\Lambda.
\]
Here $\rho(\Lambda)=(\rho_1,\dots \rho_k)$, where $\rho_i$ denotes the spectral radius of the vertex matrix $A_i \in M_{\Lambda^0}(\mathbb{N})$,
and $(\rho(\Lambda))^n:=\rho_1^{n_1}\dots \rho_k^{n_k}$ for $n=(n_1,\dots n_k)\in \mathbb{R}^k$. We call the measure $M$ the \emph{Perron--Frobenius measure} on $\Lambda^\infty$.
\subsection{Wavelets on higher-rank graphs}
\label{sec:wavelets}
According to Proposition~3.4 and Theorem~3.5 of \cite{FGKP}, there is a separable representation $\pi$ of $C^*(\Lambda)$ on $L^2(\Lambda^\infty, M)$ when $\Lambda$ is a finite, strongly connected $k$-graph. Theorem \ref{thm-Consani-Marcolli-spectral-triples-k-graphs} below identifies a Dirac operator $D$ for which this representation gives a spectral triple $(\mathcal{A}_\Lambda, L^2(\Lambda^\infty, M), D)$.
{Before stating Theorem \ref{thm-Consani-Marcolli-spectral-triples-k-graphs}, we review the definition of the representation $\pi$ and the associated wavelet decomposition of $L^2(\Lambda^\infty, M)$.
For $p\in \mathbb{N}^k$ and $\lambda\in \Lambda$, let $\sigma^p$ and $\sigma_\lambda$ be the shift and prefixing maps on $\Lambda^\infty$ given in \eqref{eq:shift-map} and \eqref{eq:prefix-map}.
If we let $S_\lambda:=\pi(s_\lambda)$, the image of the standard generator $s_\lambda$ of $C^*(\Lambda)$ under the representation $\pi$, then \cite[Theorem 3.5]{FGKP} tells us that $S_\lambda$ is given on characteristic functions of cylinder sets by
\begin{equation}\label{eq:S_lambda}
\begin{split}
S_\lambda\chi_{[\eta]} (x) &=\chi_{[\lambda]}(x)\rho(\Lambda)^{d(\lambda)/2}\chi_{[\eta]}(\sigma^{d(\lambda)}(x))=\begin{cases} \rho(\Lambda)^{d(\lambda)/2}\quad \text{if $x=\lambda\eta y$ for some $y \in \Lambda^\infty$}\\
0 \quad\quad\quad \text{otherwise}\end{cases}\\
&= \rho(\Lambda)^{d(\lambda)/2} \chi_{[\lambda \eta]}(x).
\end{split}
\end{equation}
Moreover, the adjoint $S^*_\lambda$ of $S_\lambda$ is given on characteristic functions of cylinder sets by
\begin{equation}
\label{eq:S-lambda-star}
\begin{split}
S^*_\lambda \chi_{[\eta]}(x) &=\chi_{[s(\lambda)]}(x) \rho(\Lambda)^{-d(\lambda)/2}\chi_{[\eta]}(\sigma_\lambda(x))=\begin{cases} \rho(\Lambda)^{-d(\lambda)/2}\quad\text{if $\lambda x=\eta y$ for some $y\in \Lambda^\infty$}\\ 0 \quad\quad\quad\text{otherwise}\end{cases}\\
&= \rho(\Lambda)^{-d(\lambda)/2}\sum_{(\zeta, \xi) \in \Lambda^{min}(\lambda, \eta)} \chi_{[\zeta]}(x).
\end{split}\end{equation}
}
We can think of the operators $S_{\lambda}$ as combined ``scaling and translation'' operators, since they change both the size and the range of a cylinder set $[\eta]$, and are intimately tied to the geometry of the $k$-graph $\Lambda$.
This perspective enabled four of the authors of the current paper to use the representation $\pi$ to construct a wavelet decomposition of $L^2(\Lambda^\infty, M)$; we recall the details from \cite[Section 4]{FGKP}.
For each vertex $v$ in $\Lambda$, let
\[ D_v = v\Lambda^{(1,\ldots, 1)} .\]
One can show (cf.~\cite[Lemma 2.1(a)]{aHLRS}) that $D_v$ is always nonempty when $\Lambda$ is strongly connected.
Enumerate the elements of $D_v$ as $D_v = \{ \lambda_0, \ldots, \lambda_{\#(D_v) -1}\}.$
Observe that if $D_v = \{ \lambda\}$ is a 1-element set, then $[v] = [\lambda]$. If $\#(D_v) > 1$,
then for each $1 \leq i \leq \#(D_v) -1$, we define
\begin{equation}\label{eq:f_iv}
f^{i,v} = \frac{1}{M[\lambda_0]} \chi_{[\lambda_0]} - \frac{1}{M[\lambda_i]} \chi_{[\lambda_i]}.
\end{equation}
One easily checks that in $L^2(\Lambda^\infty, M)$, $\langle f^{i,v} , \chi_{[w]} \rangle = 0$ for all $i$ and all vertices $v, w$, and that \[ \{ f^{i, v}: v \in \Lambda^0, 1 \leq i\leq \#(D_v)-1\}\]
is an orthogonal set.
Therefore, the functions $\{f^{i,v}\}_{i, v}$ span the subspace $\mathcal W_{0, \Lambda}\subseteq L^2(\Lambda^\infty, M)$ from \cite[Theorem 4.2]{FGKP}, which we will henceforth call $\mathcal W_0$.
The following Theorem, which was proved in \cite{FGKP}, justifies our labeling of the orthogonal decomposition \eqref{eq:wavelet-decomp} as a wavelet decomposition: the subspaces $\mathcal W_n$ are given by applying ``scaling and translation'' operators $S_\lambda$ to the finite family of ``mother functions'' $\{ f^{i,v}\}_{i,v}$.
\begin{thm}\cite[Theorem 4.2]{FGKP}
\label{thm:wavelets}
Let $\Lambda$ be a finite, strongly connected $k$-graph and define $\mathscr V_0 := \text{span} \{\chi_{[v]}: v \in \Lambda^0\}$. Let $\mathscr V_0 := \text{span} \{ \chi_{[v]}: v \in \Lambda^0\}$, and
set
\[\mathcal W_n = \text{span}\{ S_\lambda f^{i, s(\lambda)}: d(\lambda) = (n, \ldots, n), 1 \leq i \leq \#(D_{s(\lambda)}) - 1\}\]
for each $n \in \mathbb{N}$. Then $\{ S_\lambda f^{i, s(\lambda)}: d(\lambda) = (n, \ldots, n), 1 \leq i \leq \#(D_{s(\lambda)}) - 1\}$ is a basis for $\mathcal W_n$ and
\begin{equation}
\label{eq:wavelet-decomp}
L^2(\Lambda^\infty, M) \cong \mathscr V_0 \oplus \bigoplus_{n=0}^\infty \mathcal W_n.\end{equation}
\end{thm}
\section{Spectral triples of Consani-Marcolli type for strongly connected finite higher-rank graphs}
\label{sec-Consani-Marcolli-spectral-triples-for-k-graphs}
In Section 6 of \cite{consani-marcolli}, Consani and Marcolli construct a spectral triple for the Cuntz-Krieger algebra $\mathcal{O}_A$ associated to a matrix $A \in M_n(\mathbb{N})$.
Recall from \cite{kprr} that if $E$ is the 1-graph with adjacency matrix $A$, then $\mathcal{O}_A \cong C^*(E)$.
In this section, we generalize the construction of Consani and Marcolli to build spectral triples for higher-rank graph $C^*$-algebras $C^*(\Lambda)$.
For these spectral triples (described in Theorem~\ref{thm-Consani-Marcolli-spectral-triples-k-graphs} below), it is shown in Theorem \ref{thm:CM-Dirac-wavelets} that the eigenspaces of the Dirac operator agree with the wavelet decomposition from \cite{FGKP}. We also discuss in Remark \ref{rmk:CM-rectangle-wavelets} at the end of the section how to modify the construction of the spectral triple to make the eigenspaces of the Dirac operator compatible with the $J$-shape wavelets of \cite{FGKP2}.
\begin{defn} Let $\Lambda$ be a finite, strongly connected $k$-graph. Define $\mathcal{R}_{-1} \subset L^2(\Lambda^\infty, M)$ to be the linear subspace of constant functions on $\Lambda^\infty$.
For $s\in\mathbb{N}$, define $\mathcal{R}_s \subset L^2(\Lambda^\infty, M)$ by
\[
\mathcal{R}_s=\text{span} \left\{ \chi_{[\eta]} : \ \eta \in \Lambda, \ \sup \{ d(\eta)_i: 1 \leq i \leq k\} \leq s \right\},
\]
where $d(\eta) = (d(\eta)_1, \ldots, d(\eta)_k) \in \mathbb{N}^k$.
Let $\Xi_s$ be the orthogonal projection in $L^2(\Lambda^\infty, M)$ onto the subspace $\mathcal{R}_s$.
For a pair $(s,r)\in \mathbb{N}\times ( \mathbb{N} \cup \{-1\}) $ with $s>r$, let
\begin{equation*}\label{eq:ortho-proj}
\widehat{\Xi}_{s,r} =\Xi_s - \Xi_{r}.
\end{equation*}
Since $\mathcal{R}_r \subset \mathcal{R}_s$, $\widehat{\Xi}_{s,r}$ is the orthogonal projection onto the subspace $\mathcal{R}_s \cap ({\mathcal{R}_{r} })^{\perp}$.
Given an increasing sequence $\alpha= \{ \alpha_q\}_{q\in \mathbb{N}}$ of positive real numbers with $\lim_{q \to \infty} \alpha_q = \infty $, we define an operator $D$ on $L^2(\Lambda^\infty, M)$ by
\begin{equation} \label{eq:Dirac}
D:=\sum_{q\in \mathbb{N}} \alpha_q\;\widehat{\Xi}_{q, q-1}.
\end{equation}
\end{defn}
Note first that the operator $D$ has eigenvalues $\alpha_q$ with eigenspaces $\mathcal{R}_{q}\cap \mathcal{R}_{(q-1)}^\perp$ by construction. Also note that when $\Lambda$ has one vertex, $\mathcal{R}_{-1}=\mathcal{R}_0$ and the orthogonal projection $\widehat{\Xi}_{0,-1}$ is the zero projection.
\begin{prop}\label{pr:dirac-self-adjoint}
The operator $D$ on $L^2(\Lambda^\infty, M)$ of Equation \eqref{eq:Dirac} is unbounded and self-adjoint.
\end{prop}
\begin{proof}
The fact that $D$ is unbounded follows from the hypothesis that $\lim_{q \to \infty} \alpha_q = \infty$.
Thus, to see that $D$ is self-adjoint we must first check that it is densely defined, and then show that $D$ and $D^*$ have the same domain. For the first assertion, recall from Lemma 4.1 of \cite{FGKP} that
\[ \{[\eta]: d(\eta) = (n, \ldots, n)\text{ for some }n \in \mathbb{N}\} \]
generates the topology on $\Lambda^\infty$, and hence $
\text{span} \{\chi_{[\eta]}:d(\eta)=(n,n,\dots,n),\; n\in \mathbb{N}\}
$
is dense in $L^2(\Lambda^\infty, M)$.
Given such a ``square'' cylinder set $[\eta]$ with $d(\eta) =(s, \ldots, s)$, since $\chi_{[\eta]} \in \mathcal{R}_s$, we can write $\chi_{[\eta]} = \sum_{r\leq s} \widehat{\Xi}_{r, r-1}(\chi_{[\eta]})$. Then,
\begin{equation*}
D (\chi_{[\eta]}) = \sum_{r\leq s} \alpha_r \widehat{\Xi}_{r,r-1} (\chi_{[\eta]}) ,
\end{equation*}
which is a finite linear combination of vectors with finite $L^2$-norm, and hence is in $L^2(\Lambda^\infty, M)$.
In other words, for any finite linear combination $\xi$ of characteristic functions of square cylinder sets, $D\xi$ is in $L^2(\Lambda^\infty, M)$. Thus $D$ is defined on (at least) the finite linear combinations of square cylinder sets, which form a dense subspace of $L^2(\Lambda^\infty, M)$.
Moreover, our definition of $D$ as a diagonal operator on $L^2(\Lambda^\infty, M)$ with real eigenvalues implies that $D = D^*$ formally; since the operators $D$ and $D^*$ are given by the same diagonal formula, their domains also agree, and hence we do indeed have $D = D^*$ as unbounded operators.
\end{proof}
\begin{prop}
Let $D$ be the operator on $L^2(\Lambda^\infty, M)$ given in \eqref{eq:Dirac}. For all complex numbers $\lambda \not\in \{ \alpha_n \}_{n \in \mathbb{N}}$, the resolvent $R_\lambda(D) := (D - \lambda)^{-1}$ is a compact operator on $L^2(\Lambda^\infty, M)$.
\label{pr:cpt-resolvent}
\end{prop}
\begin{proof}
By definition, $D$ is given by multiplication by $\alpha_q$ on $\mathcal{R}_{q}\cap \mathcal{R}_{(q-1)}^\perp$. Consequently, for all $q
\in \mathbb{N}$, $(D- \lambda)^{-1}$ is given by multiplication by $\frac{1}{\alpha_q - \lambda}$ on $\mathcal{R}_{q}\cap \mathcal{R}_{(q-1)}^\perp$.
Since $\lambda \not \in \{\alpha_n\}_{n\in \mathbb{N}}$ and $\lim_{n\to \infty} \alpha_n=\infty$, given $\epsilon > 0 ,$ we can choose $N$ so that for all $n \geq N$, $\frac{1}{|\alpha_n -\lambda |} < \epsilon$. Fix $s\in \mathbb{N}$; then for any $f \in \mathcal{R}_s \cap \mathcal{R}_{s-1}^\perp$ of norm 1,
\begin{align*}
\| \left( \sum_{q=1}^N \frac{1}{\alpha_q - \lambda}\widehat \Xi_{q, q-1} (f) \right)& - (D-\lambda)^{-1}(f) \|
= \| \sum_{q > N} \frac{1}{\alpha_q - \lambda} \widehat{\Xi}_{q, q-1}(f) \| \\
&= \begin{cases}
\left| \frac{1}{\alpha_s - \lambda} \right| \, \|f\| & \text{ if } s > N \\
0 & \text{ if } s \leq N
\end{cases} \\
& < \epsilon,
\end{align*}
since $\|f \| = 1$ by hypothesis. Since the subspaces $\{ \mathcal{R}_s \cap \mathcal{R}_{s-1}^\perp: s \in \mathbb{N}_0\}$ span $L^2(\Lambda^\infty, M)$,
it follows that $(D - \lambda)^{-1}$ is the norm limit of finite rank operators and hence is compact.
\end{proof}
\begin{thm} \label{thm-Consani-Marcolli-spectral-triples-k-graphs}
Let $\Lambda$ be a finite, strongly connected $k$-graph, and denote by $\pi$ the representation of $C^*(\Lambda)$ on $L^2(\Lambda^\infty, M)$ given in Equations \eqref{eq:S_lambda} and \eqref{eq:S-lambda-star}. Let $\mathcal{A}_\Lambda$ be the dense $\ast$-subalgebra of $C^*(\Lambda)$ given in Definition~\ref{def:dense-subalg} and let $D$ be the operator given in \eqref{eq:Dirac}.
If {there exists a constant $C\ge 0$ such that} the sequence $\alpha= \{\alpha_q\}_{q \in \mathbb{N}}$ satisfies
\[
| \alpha_{q+1} - \alpha_q | \leq C,\ \forall q \in \mathbb{N},
\]
then the commutator $[D,\pi(a)]$ is a bounded operator on $L^2(\Lambda^\infty, M)$ for any $a \in \mathcal{A}_\Lambda$.
Combined with the above results, this implies that the data $(\mathcal{A}_\Lambda, L^2(\Lambda^\infty, M), D)$ gives a spectral triple for $C^*(\Lambda)$.
\end{thm}
\begin{proof}
To prove that $(\mathcal{A}_\Lambda, L^2(\Lambda^\infty, M), D)$ is a spectral triple we need to show that $D$ is self-adjoint, $(D^2+1)^{-1}$ is compact and $[D,\pi(a)]$ is bounded for all $a \in \mathcal{A}_\Lambda$. The first statement is the content of Proposition \ref{pr:dirac-self-adjoint}, and the second follows from Proposition \ref{pr:cpt-resolvent}, thanks to the fact that $\pm i \not \in \{ \alpha_n\}_{n\in \mathbb{N}}$ and hence $(D \pm i )^{-1}$ is compact.
Thus, to complete the proof of the Theorem, we will now show that $[D, \pi(a)]$ is bounded for all finite linear combinations $a = \sum_{i \in F } c_i s_{\lambda_i} s_{\eta_i}^*\in \mathcal{A}_\Lambda$, {where $c_i\in \mathbb{C}$.}
Given $\lambda \in \Lambda$, write $\max_\lambda = \max_j \{ d(\lambda)_j\}$ and $\min_\lambda = \min_j \{d(\lambda)_j\}$. Then the formula \eqref{eq:S_lambda} implies immediately that, for any fixed $s\in \mathbb{N}$,
the operator $S_\lambda$ on $L^2(\Lambda^\infty, M)$ takes $\mathcal{R}_s$ to $\mathcal{R}_{s+\max_\lambda}$.
Moreover, Equation \eqref{eq:S-lambda-star} implies that the operator $S_\lambda^*$ on $L^2(\Lambda^\infty, M)$ takes $\mathcal{R}_s$ to $\mathcal{R}_{s-\min_\lambda}$ if $\min_\lambda \leq s$, and to $\mathcal{R}_0$ otherwise.
To see this, suppose $\chi_{[\eta]} \in \mathcal{R}_s$ and $d(\eta) = (n_1, \ldots, n_k)$.
Then $S_\lambda^* \chi_{[\eta]}$ is a linear combination of cylinder sets $\chi_{[\zeta]}$ with
\[d(\zeta)_i = \begin{cases} 0, & d(\lambda)_i \geq d(\eta)_i \\
d(\eta)_i - d(\lambda)_i, & d(\lambda)_i < d(\eta)_i
\end{cases}\]
Consequently, we see that (as desired)
\begin{align*}
\max\{ d(\zeta)_i \} & = \max\{0, n_i - d(\lambda)_i: 1 \leq i \leq k \} \leq s - \textstyle{\min_\lambda}. \end{align*}
If $s<\min_\lambda$, then $n_i - d(\lambda)_i \leq 0$ for all $i$, so $S^*_\lambda \chi_{[\eta]}\in \mathcal{R}_0$ for all $\chi_{[\eta]}\in \mathcal{R}_s$.
Similarly, if $f \in \mathcal{R}_s^\perp$, then $S_\lambda f \in \mathcal{R}_{s + \min_\lambda}^\perp$.
Namely, if $\langle f, h \rangle = 0$ for all $h \in \mathcal{R}_s$, then our description of $S_\lambda^*$ above yields
\[ \langle f, S_\lambda^* g\rangle = 0 \ \forall \ g \in \mathcal{R}_{s + \min_\lambda}.\]
An analogous argument shows that $S_\lambda^*$ takes $\mathcal{R}_s^\perp$ to $\mathcal{R}_{s-\max_\lambda}^\perp$ if $s \geq \max_\lambda$.
Now fix $q \in \mathbb{N}, \ f\in \mathcal{R}_q \cap \mathcal{R}_{q-1}^{\perp}$, and fix $\lambda, \mu \in \Lambda$ with $s(\lambda) = s(\mu)$.
We use the reasoning of the previous paragraphs to identify the subspaces $\mathcal{R}_s, \mathcal{R}_t^\perp$ which contain $S_\lambda S_\mu^* f$.
If $ \max_\mu \geq q$, then we cannot guarantee that $S_\mu^* f$ is orthogonal to any $\mathcal{R}_t$ with $t \geq 0$; in order to do so, we must have $\langle S_\mu^* f, \xi \rangle = \langle f, S_\mu \xi \rangle = 0$ for all $\xi \in \mathcal R_t$. In other words, we must have $S_\mu \xi \in \mathcal R_{q-1}$ for all $\xi \in \mathcal R_t$. However, $S_\mu $ takes $ R_t $ into $ \mathcal R_{t + \max_\mu} \supsetneqq \mathcal R_{q-1}$ if $\max_\mu \geq q$ and $t \geq 0$.
Moreover, if $q < \min_\mu$, then $S_\mu^* f\in \mathcal{R}_0$. Thus,
\[ q < \textstyle{\min_\mu} \Rightarrow S_\lambda S_\mu^* f \in \mathcal{R}_{\max_\lambda}; \quad \textstyle{\min_\mu} \leq q \leq \max_\mu \Rightarrow S_\lambda S_\mu^* f \in\mathcal{R}_{q + \max_\lambda - \min_\mu}; \]
\[ q > \textstyle{\max_\mu} \Rightarrow S_\lambda S_\mu^* f \in \mathcal{R}_{q + \max_\lambda - \min_\mu} \cap \mathcal{R}_{(q-1) + \min_\lambda - \max_\mu}^\perp.\]
For now, assume $q > \max_\mu$. Writing $g= S_\lambda S_\mu^* f$,
we have
\begin{align*}
g & = \Big( \Xi_{q+\max_\lambda - \min_\mu }-\Xi_{(q-1)+\min_\lambda - \max_\mu} \Big) g =\sum_{w=q+\min_\lambda - \max_\mu }^{q+\max_\lambda - \min_\mu } \Big( \Xi_{w}-\Xi_{w-1} \Big) g
\end{align*}
and consequently
$$ D(S_\lambda S_\mu^* f) =: D g = \sum_{w=q+\min_\lambda - \max_\mu }^{q+\max_\lambda - \min_\mu } D \Big( \Big( \Xi_{w}-\Xi_{w-1} \Big) g \Big) = \sum_{w=q+\min_\lambda-\max_\mu}^{q+\max_\lambda-\min_\mu} \, \alpha_w\, \Big( \Big( \Xi_{w}-\Xi_{w-1} \Big) g \Big). $$
It now follows that, if $f \in \mathcal R_q \cap \mathcal R_{q-1}^\perp$ for $q > \max_\mu$,
\[
[D, S_\lambda S_\mu^* ] f= DS_\lambda S_\mu^* f - S_\lambda S_\mu^* D f = \sum_{w=q+\min_\lambda - \max_\mu }^{q+\max_\lambda - \min_\mu } \, ( \alpha_w - \alpha_q) \Big( \Big( \Xi_{w}-\Xi_{w-1} \Big) S_\lambda S_\mu^* f \Big) .
\]
Consequently, since $|\alpha_w - \alpha_{w-1}| \leq C$ for all $w$,
\[
\begin{split}
\Vert [D, S_\lambda S_\mu^* ] f \Vert &\leq \sum_{w=q+\min_\lambda - \max_\mu }^{q+\max_\lambda - \min_\mu } \, |\alpha_w-\alpha_q|\, \Vert S_\lambda S_\mu^* f \Vert \\
&\leq \|S_\lambda S_\mu^* f\| \sum_{w= q+ \min_\lambda - \max_\mu}^{q + \max_\lambda - \min_\mu} C |w-q| = \|S_\lambda S_\mu^* f\| C \sum_{t = \min_\lambda - \max_\mu }^{\max_\lambda - \min_\mu } | t| .
\end{split}
\]
Since $S_\lambda S_\mu^*$ is a partial isometry and hence norm-preserving,
whenever $f \in \mathcal R_q \cap \mathcal R_{q-1}^\perp$ for $q > \max_\mu$, $\| [D, S_\lambda S_\mu^*] f\|$ is bounded above by a constant which depends only on $\lambda$ and $\mu$.
If we have $\min_\mu \leq q \leq \max_\mu$, since we no longer know that $S_\lambda S_\mu^* f \in \mathcal{R}_t^\perp$ for any $t$, in calculating $\| [D, S_\lambda S_\mu^* f\ \|$ we have to begin our summation over $w$ at zero, rather than at $q + \min_\lambda - \max_\mu$.
In this case, the final (in)equality above becomes
\[ \| [D, S_\lambda S_\mu^*] f \| \leq \sum_{t = 1}^{\max_\lambda - \min_\mu} C t \|S_\lambda S_\mu^* f\| + \sum_{t=1}^q C t \| S_\lambda S_\mu^* f \|.\]
In this case, $q \leq \max_\mu$, so we obtain the norm bound
\begin{align*} \| [D, S_\lambda S_\mu^*] f \| & \leq \| S_\lambda S_\mu^* f\| C \left( \frac{(\max_\lambda - \min_\mu)(\max_\lambda - \min_\mu +1)}{2} + \frac{\max_\mu (\max_\mu +1)}{2} \right) .
\end{align*}
In other words, $\|[D, S_\lambda S_\mu^*] f\|$ is again bounded by a constant which only depends on $\lambda$ and $\mu$. A similar argument shows that if $q < \min_\mu$, $\Vert [D,S_\lambda S^*_\mu] f\Vert$ is bounded by a constant which only depends on $\lambda$ and $\mu$. Since $\{\mathcal R_q \cap \mathcal R_{q-1} \}_{q\in \mathbb{N}}$ densely spans $L^2(\Lambda^\infty, M)$, it follows that $[D, S_\lambda S_\mu^* ]$ is a bounded operator for all $(\lambda, \mu) \in \Lambda \times \Lambda$ with $s(\lambda) = s(\mu)$.
By linearity, it follows that $[D, \pi(a)]$ is bounded for all finite linear combinations $a = \sum_{i \in F} c_i s_{\lambda_i} s_{\eta_i}^*$ of the generators $s_\lambda s_\eta^*$ of $C^*(\Lambda)$. Since every element of the dense $*$-subalgebra $\mathcal{A}_\Lambda$ of $C^*(\Lambda)$ is given by such a finite linear combination, it follows that $(\mathcal{A}_\Lambda, L^2(\Lambda^\infty, M), D)$ is a spectral triple, as claimed.
\end{proof}
\begin{thm}
\label{thm:CM-Dirac-wavelets}
Let $(\mathcal{A}_\Lambda, L^2(\Lambda^\infty, M), D)$ be the spectral triple described in Theorem~\ref{thm-Consani-Marcolli-spectral-triples-k-graphs}.
The eigenspaces of the Dirac operator $D$ given in \eqref{eq:Dirac} agree with the wavelet decomposition
\[L^2(\Lambda^\infty, M) = \mathscr{V}_0 \oplus \bigoplus_{q=0}^\infty \mathcal{W}_q\]
of Theorem \ref{thm:wavelets} above (also see \cite[Theorem 4.2]{FGKP}). In particular,
\[ \mathscr{V}_0 = \mathcal{R}_0 \supseteq \mathcal{R}_{-1}\;\; \text{ and }\;\;
\mathcal{W}_q = \mathcal{R}_{q+1} \cap \mathcal{R}_{q}^\perp,\ q\geq 0.\]
\end{thm}
\begin{proof}
By definition, $\mathcal R_{-1} \subseteq \mathcal{R}_0 = \mathscr{V}_0 = \text{span}\{ \chi_{[v]}: v \in \Lambda^0\}$.
For the second assertion,
recall that $\mathcal W_n = \text{span}\{ S_\lambda f: f \in \mathcal{W}_0, \ d(\lambda) = (q, q, \ldots, q)\}$. Since $\max_\lambda = \min_\lambda = q$ for all such $\lambda$, each such $S_\lambda$ takes $\mathcal{R}_s \cap \mathcal{R}_{s-1}^\perp$ to $\mathcal{R}_{s+q} \cap \mathcal{R}_{s+q-1}^\perp$. Thus, it suffices to see that $\mathcal W_0 \subseteq \mathcal R_1 \cap \mathcal R_0^\perp$, and that $\mathcal W_q$ and $\mathcal R_q \cap \mathcal R_{q-1}^\perp$ have the same dimension for all $q \in \mathbb{N}$.
For the first statement, recall that $\mathcal{W}_0$ was constructed precisely to be the span of a family $\{f^{i,v}\}$ of functions (see Equation \eqref{eq:f_iv}) which were orthogonal to $\mathscr{V}_0 = \mathcal{R}_0$. Moreover, every function $f^{i,v}$ is a linear combination of characteristic functions $\chi_\eta$ with $d(\eta) = (1, \ldots, 1)$, and therefore lies in $\mathcal R_1 \cap \mathcal R_0^\perp$.
From the fact that $\{ S_\lambda f^{i, s(\lambda)}: d(\lambda) = (q, q, \ldots, q), \ 1 \leq i \leq \#(D_{s(\lambda)}) -1\}$ is a basis for $\mathcal W_q$, and the factorization rule in $\Lambda$, it follows that $\mathcal{W}_q$ has dimension
\[
{\sum_{v \in \Lambda^0} \#( \Lambda^{(q,\ldots, q)}v) \cdot \left( \#( v \Lambda^{(1, \ldots, 1)})-1 \right) = \#(\Lambda^{(q+1, \ldots, q+1)}) - \#(\Lambda^{(q, \ldots, q)}). }
\]
Moreover, we know from \cite[Lemma 4.1]{FGKP} that ``square'' cylinder sets generate the topology on $\Lambda^\infty$; it follows that $\mathcal{R}_s$ is spanned by $\{ \chi_{[\lambda]}: d(\lambda) = (s, \ldots, s) \}$. Indeed, this set forms a basis for $\mathcal R_s$: if $d(\lambda) = d(\mu) = (s, \ldots, s)$, then the factorization rule implies that
\[ \langle \chi_{[\lambda]}, \chi_{[\mu]} \rangle = \int_{\Lambda^\infty} \chi_{[\lambda]} \chi_{[\mu]} \, dM = \delta_{\lambda, \mu} M([\lambda]).\]
Consequently, $\mathcal R_{q+1} \cap \mathcal R_q^\perp$ also has dimension $\#( \Lambda^{(q+1, \ldots, q+1)} ) - \#(\Lambda^{(q,\ldots, q)}).$ Hence, $\mathcal W_q = \mathcal R_{q+1} \cap \mathcal R_q^\perp$ for all $q \in \mathbb{N}$, as desired.
\end{proof}
\begin{rmk} \label{rmk:CM-rectangle-wavelets}
Fix $J \in \mathbb{N}^k$ with $J_i > 0 $ for all $i$. We described in Section 5 of \cite{FGKP2} how to construct wavelets with ``fundamental domain'' $J$ -- the original construction in Section 4 of \cite{FGKP} used $J=(1, \ldots, 1)$. By defining
\[ \widetilde{\mathcal{R}}_s = \text{span} \{ \chi_{[\eta]}: d(\eta) \leq sJ\}\]
we can construct a Dirac operator $\widetilde{D}$ on $L^2(\Lambda^\infty, M)$ which gives rise to a spectral triple $(\mathcal{A}_\Lambda, L^2(\Lambda^\infty, M), \widetilde D)$ whose eigenspaces agree with the wavelet decomposition given in Theorem 5.2 of \cite{FGKP2}. We omit the details here as they are completely analogous to the proofs of Theorems \ref{thm-Consani-Marcolli-spectral-triples-k-graphs} and \ref{thm:CM-Dirac-wavelets} above.
\end{rmk}
\bibliographystyle{amsplain}
|
{
"timestamp": "2018-04-17T02:06:09",
"yymm": "1804",
"arxiv_id": "1804.05209",
"language": "en",
"url": "https://arxiv.org/abs/1804.05209"
}
|
\section{Introduction}
Distributed vector representations of words, henceforth referred to as word embeddings, have been shown to exhibit strong performance on
a variety of NLP tasks~\cite{turian2010word, zou2013bilingual}. Methods for producing word embedding sets exploit the distributional
hypothesis to infer semantic similarity between words within large bodies of text, in the process they have been found to additionally capture
more complex linguistic regularities, such as analogical relationships~\cite{mikolov2013linguistic}. A variety of methods now exist for the
production of word embeddings~\cite{collobert2008unified, mnih2009scalable, huang2012improving, pennington2014glove, mikolov2013efficient}. Comparative work has illustrated a variation in performance between methods across evaluative
tasks~\cite{chen2013expressive, yin2015learning}.
Methods of ``meta-embedding'', as first proposed by~\citet{yin2015learning}, aim to conduct a complementary combination of information
from an ensemble of distinct word embedding sets, each trained using different methods, and resources, to yield an embedding set with
improved overall quality.
Several such methods have been proposed. 1\texttt{TO}N~\cite{yin2015learning}, takes an ensemble of $K$ pre-trained word embedding sets, and employs a linear neural network to learn a set of meta-embeddings along with $K$ global projection matrices, such that through projection, for every word in the meta-embedding set, we can recover its corresponding vector within each source word embedding set. 1\texttt{TO}N+~\cite{yin2015learning}, extends this method by predicting embeddings for words not present within the intersection of the source word embedding sets. An unsupervised locally linear meta-embedding approach has since been taken~\cite{bollegala2017think}, for each source embedding set, for each word; a representation as a linear combination of its nearest neighbours is learnt. The local reconstructions within each source embedding set are then projected to a common meta-embedding space.
The simplest approach considered to date, has been to concatenate the word embeddings across the source sets~\cite{yin2015learning}.
Despite its simplicity, concatenation has been used to provide a good baseline of performance for meta-embedding.
A method which has not yet been proposed is to conduct a direct averaging of embeddings. The validity of this approach may perhaps not seem obvious, owing to the fact that no correspondence exists between the dimensions of separately trained word embedding sets.
In this paper we first provide some analysis and justification that, despite this dimensional disparity, averaging can provide an approximation of the performance of concatenation without increasing the dimension of the embeddings. We give empirical results demonstrating the quality of average meta-embeddings. We make a point of comparison to concatenation since it is the most comparable in terms of simplicity, whilst also providing a good baseline of performance on evaluative tasks. Our aim is to highlight the validity of averaging across distinct word embedding sets, such that it may be considered as a tool in future meta-embedding endeavours.
\section{Analysis}
To evaluate semantic similarity between word embeddings we consider the Euclidean distance measure.
For $\ell_{2}$ normalised word embeddings, Euclidean distance is a monotonically decreasing function of the cosine similarity, which is a popular choice in NLP tasks that use word embeddings such as semantic similarity prediction and analogy detection~\cite{Levy:TACL:2015,Levy:CoNLL:2014}.
We defer the analysis of other types of distance measures to future work.
By evaluating the relationship between the Euclidean distances of pairs of words in the source embedding sets and their corresponding Euclidean distances in the meta-embedding space we can obtain a view as to how the meta-embedding procedure is combining semantic information. We begin by examining concatenation through this lens, before moving on to averaging.
\subsection{Concatenation}
We can express concatenation by first zero-padding our source embeddings, before combining them through addition.
Without loss of generality, we consider both concatenation and averaging over only two source word embedding sets for ease of exposition.
Let ${\cS_{1}}$ and ${\cS_{2}}$ be unique embedding sets of real-valued continuous embeddings. We make no assumption that ${\cS_{1}}$ and ${\cS_{2}}$ were trained using the same method or resources. Consider two semantically similar words $\vec{u}$ and $\vec{v}$ such that $\vec{u}, \vec{v} \in {\cS_{1}}$ $\cap$ ${\cS_{2}}$. Let $\vec{u}_{\cS_{1}}$ and $\vec{v}_{\cS_{1}}$, and $\vec{u}_{\cS_{2}}$ and $\vec{v}_{\cS_{2}}$ denote the specific word embeddings of $\vec{u}$ and $\vec{v}$ within the embeddings ${\cS_{1}}$, and ${\cS_{2}}$ respectively.
Let the dimensions of embeddings ${\cS_{1}}$, and ${\cS_{2}}$ be denoted $d_{\cS_{1}}$, and $d_{\cS_{2}}$ respectively. We zero-pad embeddings from ${\cS_{1}}$ by front-loading $d_{\cS_{2}}$ zero entries to each word embedding vector. In contrast, we zero-pad embeddings from ${\cS_{2}}$ by adding $d_{\cS_{1}}$ zero entries to the end of each embedding vector. The resulting embeddings from ${\cS_{1}}$ and ${\cS_{2}}$ now share a common dimension of $d_{\cS_{1}} + d_{\cS_{2}}$. Denote the resulting embeddings of any word $\vec{u} \in {\cS_{1}} \cap {\cS_{2}}$, as $\vec{u}_{{\cS_{1}}}^{zero}$ and $\vec{u}_{{\cS_{2}}}^{zero}$ respectively.
Now, combining our source embeddings through addition we obtain equivalency to concatenation.
\begin{align} \vec{u}_{{\cS_{1}}}^{zero} + \vec{u}_{{\cS_{2}}}^{zero} = \begin{bmatrix}
\vec{u}_{{\cS_{2}}_{(1)}} \\
\vec{u}_{{\cS_{2}}_{(2)}} \\
\vdots \\
\vec{u}_{{\cS_{2}}_{(d_{\cS_{2}})}} \\
\vec{u}_{{\cS_{1}}_{(1)}} \\
\vec{u}_{{\cS_{1}}_{(2)}} \\
\vdots \\
\vec{u}_{{\cS_{1}}_{(d_{\cS_{1}})}}
\end{bmatrix}
= \begin{bmatrix}
\vec{u}_{\cS_{2}} \\
\vec{u}_{\cS_{1}}
\end{bmatrix}
\end{align}
Note that the zero-padded vectors are orthogonal.
Let the Euclidean distance between these words in each embedding be denoted by $E_{\cS_{1}}$ and $E_{\cS_{2}}$. Note that for any vector $\vec{u} \in \mathbb{R}^n$ the addition of zero-valued dimensions does not affect the value of its $\ell_{2}$-norm. So we have
\begin{align}
& E_{\cS_{1}} = \norm{\vec{u}_{\cS_{1}} - \vec{v}_{\cS_{1}}}_{2} = \norm{\vec{u}_{\cS_{1}}^{zero} - \vec{v}_{\cS_{1}}^{zero}}_{2} \\
& E_{\cS_{2}} = \norm{\vec{u}_{\cS_{2}} - \vec{v}_{\cS_{2}}}_{2} = \norm{\vec{u}_{\cS_{2}}^{zero} - \vec{v}_{\cS_{2}}^{zero}}_{2}
\end{align}
Consider the Euclidean distance between $\vec{u}$ and $\vec{v}$ after concatenation.
\begin{align*}
& E_{CONC} \\
& = \norm{ \begin{bmatrix}\vec{u}_{\cS_{2}} \\ \vec{u}_{\cS_{1}}\end{bmatrix} - \begin{bmatrix}\vec{v}_{\cS_{2}} \\ \vec{v}_{\cS_{1}}\end{bmatrix} }_{2} \\
& = \norm{(\vec{u}_{\cS_{1}}^{zero} + \vec{u}_{\cS_{2}}^{zero}) - (\vec{v}_{\cS_{1}}^{zero} + \vec{v}_{\cS_{2}}^{zero})}_{2} \\
& = \norm{(\vec{u}_{\cS_{1}}^{zero} - \vec{v}_{\cS_{1}}^{zero}) - (\vec{v}_{\cS_{2}}^{zero} - \vec{u}_{\cS_{2}}^{zero})}_{2} \\
& = \sqrt{ (E_{\cS_{1}})^{2} + (E_{\cS_{2}})^{2} - 2E_{\cS_{1}}E_{\cS_{2}}cos(\theta) } \\
& = \sqrt{ (E_{\cS_{1}})^{2} + (E_{\cS_{2}})^{2} - 2E_{\cS_{1}}E_{\cS_{2}} (0) } \\
& = \sqrt{ (E_{\cS_{1}})^{2} + (E_{\cS_{2}})^{2} }
\end{align*}
For any two words belonging to the resultant embedding obtained by concatenation, the distance between these words in the resultant space is the root of the sum of squares of Euclidean distances between these words in ${\cS_{1}}$ and ${\cS_{2}}$.
\subsection{Average word embeddings}
\label{sect:avg-sem}
\par Here we now make the assumption that ${\cS_{1}}$ and ${\cS_{2}}$ have common dimension $d$.\footnote{Without loss of generality, source embeddings with different dimensionality can be appropriately padded to have the same dimensionality.}
\par Despite there being no obvious correspondence between dimensions of ${\cS_{1}}$ and ${\cS_{2}}$ we can show that the average embedding set retains semantic information through preservation of the relative distances between words.
Consider the positioning of words $\vec{u}$, and $\vec{v}$ after performing a word-wise average between the source embedding sets. The Euclidean distance between $\vec{u}$ and $\vec{v}$ in the resultant meta-embedding is given by
\begin{align*}
& E_{AVG} \\
& = \norm{\frac{(\vec{u}_{\cS_{1}} + \vec{u}_{\cS_{2}})}{2} - \frac{(\vec{v}_{\cS_{1}} + \vec{v}_{\cS_{2}})}{2}}_{2} \\
& = \frac{1}{2}\norm{(\vec{u}_{\cS_{1}} - \vec{v}_{\cS_{1}}) - (\vec{v}_{\cS_{2}} - \vec{u}_{\cS_{2}})}_{2} \\
& \propto \sqrt{ (E_{\cS_{1}})^{2} + (E_{\cS_{2}})^{2} - 2E_{\cS_{1}}E_{\cS_{2}}\cos(\theta) }
\end{align*}
Now in this case, unlike concatenation, we have not designed our source embedding sets such that they are orthogonal to each other, and so it seems we are left with a term dependant on the angle between $(\vec{u}_{\cS_{1}} - \vec{v}_{\cS_{1}})$ and $(\vec{v}_{\cS_{2}} - \vec{u}_{\cS_{2}})$.
However, \newcite{cai2013distributions} showed that, if $\cX$ is a set of random points $\in \mathbb{R}^{n}$ with cardinality $|\cX|$, then the limiting distribution of angles, as $|\cX| \rightarrow \infty$, between pairs of elements from $\cX$, is Gaussian with mean $\pi/2$. In addition, \newcite{cai2013distributions} showed that the variance of this distribution shrinks as the dimensionality increases.
Word embedding sets typically contain in the order of ten thousand or more points, and are typically of relatively high dimension. Moreover, assuming the difference vector between any two words in an embedding set is sufficiently random, we may approximate the limiting Gaussian distribution described by~\citet{cai2013distributions}. In such a case the expectation would then be that the vectors $(\vec{u}_{\cS_{1}} - \vec{v}_{\cS_{1}})$ and $(\vec{v}_{\cS_{2}} - \vec{u}_{\cS_{2}})$ are orthogonal, leading to the following result.
\begin{align}
\label{eq:avg}
\mathbb{E} [ E_{AVG} ] = \frac{1}{2} \sqrt{ (E_{\cS_{1}})^{2} + (E_{\cS_{2}})^{2} } \propto E_{CONC}
\end{align}
To summarise, if word embeddings can be shown to be approximately orthogonal, then averaging will approximate the same information as concatenation, without increasing the dimensionality of the embeddings.
\section{Experiments}
\par We first empirically test our theory that word embeddings are sufficiently random and high dimensional, such that they are approximately all orthogonal to each other. We then present an empirical evaluation of the performance of the meta-embeddings produced through averaging, and compare against concatenation.
\subsection{Datasets}
\label{sect:datasets}
We use the following pre-trained embedding sets that have been used in prior work on meta-embedding learning~~\cite{yin2015learning,bollegala2017think} for experimentation.
\begin{itemize}
\item {\bf GloVe}~\cite{pennington2014glove}. 1,917,494 word embeddings of dimension 300.
\item {\bf CBOW}~\cite{mikolov2013distributed}. Phrase embeddings discarded, leaving 929,922 word embeddings of dimension 300.
\item {\bf HLBL}~\cite{turian2010word}. 246,122 hierarchical log-bilinear~\cite{mnih2009scalable} word embeddings of dimension 100.
\end{itemize}
Note that the purpose of this experiment is not to compare against previously proposed meta-embedding learning methods, but to empirically verify averaging as a meta-embedding method and validate the assumptions behind the theoretical analysis.
By using three pre-trained word embeddings with different dimensionalities and empirical accuracies, we can evaluate the averaging-based meta-embeddings in a robust manner.
We pad HLBL embeddings to the rear with 200 zero-entries to bring their dimension up to 300.
For GloVe, we $\ell_{2}$ normalise each dimension of the embedding across the vocabulary, as recommended by the authors. Every individual word embedding from each embedding set is then $\ell_{2}$-normalised. The proposed averaging operation, as well as concatenation, operate only on the intersection of these embeddings. The intersectional vocabularies GloVe $\cap$ CBOW, GloVe $\cap$ HLBL, and CBOW $\cap$ HLBL contain 154,076; 90,254; and 140,479 word embeddings respectively.
\subsection{Empirical distribution analysis}
We conduct an empirical analysis of the distribution of the angle $\sphericalangle [(\vec{u}_{\cS_{1}} - \vec{v}_{\cS_{1}})$, $(\vec{v}_{\cS_{2}} - \vec{u}_{\cS_{2}})]$ for each pair of datasets. Table~\ref{tbl:angle-table} shows the mean and variance of these distributions, obtained from samples of 200,000 random pairs of words from each intersectional vocabulary. We find that the angles are approximately normally distributed around $\pi/2$.
\begin{table}[h!]
\small
\centering
\begin{tabular}{l | l l}
Embeddings & $\mu$ & $\sigma^2$ \\
\midrule
GloVe \& CBOW & 1.5609 & 0.0121 \\
GloVe \& HLBL & 1.5709 & 0.0129 \\
CBOW \& HLBL & 1.5740 & 0.0126 \\
\midrule
\end{tabular}
\caption{Observed distribution parameters.}
\label{tbl:angle-table}
\end{table}
Figure~\ref{fig:angledistributions} shows a normalised histogram of the results for GloVe $\cap$ CBOW, along with a normal distribution characterised by the sample mean and variance. GloVe $\cap$ HLBL, and CBOW $\cap$ HLBL plots are not shown due to space limitations, but are similarly normally distributed. This result shows that the pre-trained word embeddings approximately satisfy the predictions made by \newcite{cai2013distributions}, thereby empirically justifying the assumption made in the derivation of \eqref{eq:avg}.
\begin{figure}
\centering
\includegraphics[width=1.0\columnwidth]{{angledistributions.pdf}}
\caption{Distribution of angles between embeddings within GloVe $\cap$ CBOW. }
\label{fig:angledistributions}
\vspace{-5mm}
\end{figure}
\begin{table}[t]
\small
\centering
\begin{tabular}{ p{25mm} p{4mm} p{4mm} p{4mm} p{4mm} p{4mm} p{4mm} p{4mm}}
Embeddings & RG & MC & WS & RW & SL & GL \\
\midrule
\textbf{sources} & & & & & & \\
HLBL 100 & 35.3 & 49.3 & 35.7 & 19.1 & 22.1 & 15.0 \\
CBOW 300 & 76.0 & 82.2 & 69.8 & 53.4 & 44.2 & 67.1 \\
GloVe 300 & 82.9 & 87.0 & 75.4 & 48.7 & 45.3 & 68.7 \\
\midrule
\textbf{AVG} & & & & & & \\
CBOW+HLBL 300 & 69.2 & 81.0 & 60.1 & 48.7 & 37.3 & 49.4 \\
GloVe+CBOW 300 & 82.2 & 87.0 & 74.5 & 52.9 & \textbf{46.5} & 73.8 \\
GloVe+HLBL 300 & 73.7 & 74.1 & 64.2 & 44.6 & 38.8 & 49.5 \\
\midrule
\textbf{CONC} & & & & & & \\
CBOW+HLBL 400 & 68.7 & 80.2 & 62.9 & 49.1 & 39.6 & 53.2 \\
GloVe+CBOW 600 & \textbf{83.0} & \textbf{88.8} & \textbf{76.4} & \textbf{54.8} & 46.3 & \textbf{75.5} \\
GloVe+HLBL 400 & 73.7 & 80.1 & 65.5 & 46.4 & 40.0 & 53.8 \\
\end{tabular}
\caption{Results on word similarity, and analogical tasks. Best performances bolded per task. Dimensionality of the meta embedding is shown next to the source embedding names.}
\label{tbl:results-table}
\end{table}
\subsection{Evaluation Tasks}
\label{sext:evaluation}
\subsubsection{Semantic Similarity }
We measure the similarity between words by calculating the cosine similarity between their embeddings; we then calculate Spearman correlation against human similarity scores. The following datasets are used: \textbf{RG} \cite{rubenstein1965contextual}, \textbf{MC} \cite{miller1991contextual}, \textbf{WS} \cite{finkelstein2001placing}, \textbf{RW} \cite{luong2013better}, and \textbf{SL} \cite{Hill2015}.
\subsubsection{Word Analogy}
Using the Google dataset \textbf{GL} \cite{mikolov2013distributed} (19544 analogy questions), we solve questions of the form \textit{a is to b as c is to what?}, using the CosAdd method \cite{mikolov2013linguistic} shown in \eqref{eqn:cosadd}. Specifically, we determine a fourth word \textit{d} such that the similarity between $(b - a + c)$ and $d$ is maximised.
\begin{equation}
\label{eqn:cosadd}
\mathrm{CosAdd}(a:b,c:d) = \cos(b - a + c, d)
\end{equation}
\subsection{Discussion of results}
\label{sect:discussion}
Table~\ref{tbl:results-table} shows task performance for each source embedding set, and for both methods on every pair of datasets. In our experiments concatenation obtains better overall performance. However, averaging offers improvements over the source embedding sets for semantic similarity task \textbf{SL} and word analogy task \textbf{GL}, on the combination of CBOW and GloVe. HLBL has a negative effect on CBOW and GloVe, but the performance of averaging is close to that of concatenation.
An advantage of averaging when compared against concatenation, is that the dimensionality of the produced meta-embedding is not increased beyond the maximum dimension present within the source embeddings, resulting in a meta-embedding which is easier to process and store.
\section{Conclusion}
We have presented an argument for averaging as a valid meta-embedding technique, and found experimental performance to be close to, or in some cases better than that of concatenation, with the additional benefit of reduced dimensionality. We propose that when conducting meta-embedding, both concatenation and averaging should be considered as methods of combining embedding spaces, and their individual advantages considered.
|
{
"timestamp": "2018-04-17T02:08:01",
"yymm": "1804",
"arxiv_id": "1804.05262",
"language": "en",
"url": "https://arxiv.org/abs/1804.05262"
}
|
\section{Introduction}
Quantum Chromodynamics, the theory of the strong interaction, is a vast field with a plethora of diverse phenomena still unexplored. Despite being the object of study of the high energy and nuclear physics communities both theoretically and experimentally, the true nature of a proton, its constituents, their interactions and how they come together to conform it remain elusive. In the collinear framework at a given resolution scale $Q^{2}$, the proton can be described as composed of quarks and gluons carrying a fraction $x$ of the proton momentum. Once known at an initial scale $Q^{2}_{0}$, the partonic densities can be determined at any scale $Q^{2}$ using the DGLAP evolution equations \cite{Gribov:1972ri,Gribov:1972rt,Altarelli:1977zs,Dokshitzer:1977sg}. This picture is successfully supported by extensive experimental evidence; however it can not be valid for all the kinematic range: as one explores lower $x$ values the DGLAP equations predict an infinite rise of the gluon density which would break unitarity. It follows that some phenomena, e.g. gluon recombination, has to enter in order to tame this dangerous behavior. This dynamically generates the \emph{saturation scale} $Q_s^2$, which determines when the transition to the non-linear regime of QCD takes place. Moreover, if $Q_s^2$ is large, perturbative calculations become possible as the strong coupling constant is weak. A successful theoretical framework to describe QCD in this region is known as the Color Glass Condensate~\cite{Gelis:2010nm}.
There are many theoretical models that incorporate saturation to QCD calculations with different approaches and considerations, a popular one being the IPsat parametrization \cite{Kowalski:2003hm,Kowalski:2006hc,Rezaeian:2012ji,Luszczak:2013rxa,Luszczak:2016bxd}. The parameters that provide the necessary non-perturbative input to these models are determined through fits to available data, the bulk of them being high precision deep inelastic scattering (DIS) data measured at HERA in electron-proton ($e+p$) collisions \cite{Aaron:2009aa,Abramowicz:1900rp,Abramowicz:2015mha,H1:2018flt}. Despite the wide kinematic range covered by this collider there are no spectacular signals of deviation from the DGLAP predictions and some observed discrepancies might be due to reasons other than saturation. Recently a possible hint of a non-linear regime from HERA data at low $x$ was shown to be
feasibly explainable
by the inclusion of resumed logarithmic corrections \cite{Ball:2017otu}. Saturation model calculations have also been able to provide a natural description for the nearly flat center-of-mass energy dependence of the diffractive to total cross section ratio at HERA~\cite{GolecBiernat:1998js,GolecBiernat:1999qd} (see also Ref.~\cite{Kowalski:2008sa}).
In general, there is no clear consensus on whether or not the onset of saturation has been reached and it will be necessary to perform a thorough and detailed exploration of the kinematic space beyond our current knowledge in order to observe the non-linear regime of QCD. Future facilities such as the Electro-Ion Collider (EIC) \cite{Accardi:2012qut,Aschenauer:2017jsk}, the Large electron-Hadron Collider (LHeC) \cite{AbelleiraFernandez:2012cc} and the Future Circular Collider (FCC-eh)~\cite{Zimmermann:2014qxa} hold the key to this door.
In this study we present a new determination of the IPsat and its linearized version (``IPnonsat'') description of the HERA combined data \cite{Aaron:2009aa,Abramowicz:1900rp,Abramowicz:2015mha,H1:2018flt} in the framework of the dipole model. What is new here, compared to the previous literature~\cite{Kowalski:2003hm,Kowalski:2006hc,Rezaeian:2012ji}, is that we also fit the IPnonsat model parametrization to the precise combined HERA data which allows us to explore the expected magnitude of saturation effects in current and future collider experiments. In addition, by simultaneously fitting the total cross section and the charm contribution to it, it becomes possible to determine the quark masses in this framework. For consistency, a variable flavor number scheme is also applied.
This work is organized as follows. In Sec. \ref{dipole} we describe the inclusive photon-proton interaction in terms of the dipole model for both IPsat and IPnonsat formulations. The analysis of the combined inclusive and charm data from HERA is the subject of Sec. \ref{HERA}, while in Sec. \ref{amplitude} we present an analysis of the obtained dipole-proton scattering amplitude. The application of our determined parameters to the exclusive production of vector mesons is discussed in Sec. \ref{EVM}. In Sec. \ref{Future} we consider the potential of the EIC and the LHeC to provide a signal of saturation in both $e+p$ and $e+A$ collisions. Finally Sec. \ref{conclusions} summarizes our findings.
\section{Photon-proton scattering in the dipole picture}
\label{dipole}
The most precise study of the proton structure has been performed in
$e+p$ DIS experiments at HERA~\cite{Aaron:2009aa,Abramowicz:2015mha,Abramowicz:1900rp,H1:2018flt}. The process is described as the electron emitting a virtual photon with momentum $q$, which then probes the proton with a resolution scale $Q^{2}=-q^{2}$. In the dipole picture, applicable at high energy and not too large $Q^2$, the virtual photon-proton scattering process can be factorized in two parts: the $\gamma^{*}$ splitting into a $q\bar{q}$ pair, and the dipole-target scattering. The total $\gamma^* p$ cross section is subsequently obtained as the imaginary part of the forward elastic $\gamma^* p \to \gamma^* p$ scattering amplitude using the optical theorem. The photon splitting into a dipole with transverse separation ${\mathbf{r}}$ is described in terms of the photon wave function $\Psi^f_{L,T}(r,z,Q^2)$, where $f$ is the quark flavor, $L$ and $T$ refer to transverse and longitudinal polarizations, $|{\mathbf{r}}|=r$, and $z$ is the fraction of the longitudinal momentum of the photon carried by the quark. The total photon-proton cross section is then given by
\begin{equation}
\sigma^{\gamma^* p}_{L,T}(x,Q^2) = \int \mathrm{d}^2 {\mathbf{b}} \mathrm{d}^2 {\mathbf{r}} \int_0^1 \frac{\mathrm{d} z}{4\pi} |\Psi_{L,T}^f(r,z,Q^2)|^2 \frac {\, \mathrm{d} { \sigma_\textrm{dip} }}{\, \mathrm{d}^2 {\mathbf{b}}},
\end{equation}
where $\frac {\, \mathrm{d} { \sigma_\textrm{dip} }}{\, \mathrm{d}^2 {\mathbf{b}}}$ is the proton-dipole cross-section with ${\mathbf{b}}$ denoting the impact parameter, and one has to sum over all the quark flavors included in the analysis ($u,d,s,c$ and $b$ in this work). The photon wave functions for the transverse and longitudinal polarizations summed over spins and helicities read~\cite{Kovchegov:2012mbw}
\begin{multline}
\label{eq:photon_t}
|\Psi_T(r,z,Q^2)|^2 = \frac{2{N_\mathrm{c}}}{\pi} \alpha_{\mathrm{em}} e_q^2 \Big\{ [ z^2+(1-z)^2 \varepsilon^2 K_1^2(\varepsilon r) \\
+ m_f^2 K_0^2(\varepsilon r) ] \Big\}
\end{multline}
and
\begin{equation}
\label{eq:photon_l}
|\Psi_L(r,z,Q^2)|^2 = \frac{8 {N_\mathrm{c}}}{\pi} \alpha_{\mathrm{em}} e_q^2 Q^2 z^2(1-z)^2 K_0^2(\varepsilon r),
\end{equation}
with $\varepsilon^2 = z(1-z)Q^2 + m_q^2$. Here, $e_q$ is the fractional charge of the quark $q$ and $m_q$ is the quark mass.
The proton structure functions $F_2$ and $F_L$ can be written in terms of the total photon-proton cross section as
\begin{align}
F_2 &= \frac{Q^2}{4\pi^2 \alpha_{\mathrm{em}}} (\sigma^{\gamma^*p}_L + \sigma^{\gamma^*p}_T), \\
F_L &= \frac{Q^2}{4\pi^2 \alpha_{\mathrm{em}}} \sigma^{\gamma^*p}_L.
\end{align}
The most precise combined HERA data is released in the form of the reduced cross section
\begin{equation}
\sigma_r(x,y,Q^2) = F_2(x,Q^2) - \frac{y^2}{1+(1-y)^2} F_L(x,Q^2),
\end{equation}
where $y=Q^2/(xs)$ is the inelasticity of the $e+p$ scattering with center-of-mass energy $\sqrt{s}$.
In the IPsat model the saturation scale of the target depends on the impact parameter, and the cross section is written as
\begin{equation}
\label{eq:ipsat}
\frac {\, \mathrm{d} { \sigma_\textrm{dip} }}{\, \mathrm{d}^2 {\mathbf{b}}} = 2\left[1 - \exp \left( -r^2 F(x,r) T_p({\mathbf{b}}) \right) \right].
\end{equation}
The proton density profile $T({\mathbf{b}}) = e^{-{\mathbf{b}}^2/(2B_p)} / (2\pi B_p)$ is assumed to be Gaussian, and we use fixed $B_p=4\ \textrm{GeV}^{-2}$ based on HERA exclusive $J/\Psi$ production data. Thus, the effective transverse area of the proton is not a free parameter in the model, and the root mean square radius of the proton is $\sqrt{2 B_p}$. However, we note that this parametrization describes only part of the observed growth of the proton already at the HERA energies~\cite{Aktas:2005xu,Chekanov:2002xi} due to the Gribov diffusion. Including this effect would require us to either parametrize the proton width $B_p$ and try to fit it simultaneously to the HERA data, or solving impact parameter dependent small-$x$ evolution equations such as in Refs.~\cite{Berger:2011ew,Berger:2012wx,Schlichting:2014ipa}. We do not want to include exclusive data into our fit due to additional model uncertainties, and we leave it for a future work.
At the lowest order in perturbation theory the function $F$ is proportional to the DGLAP evolved gluon distribution function
\begin{equation}
F(x, r^2) = \frac{\pi^2}{2 {N_\mathrm{c}}} \alpha_{\mathrm{s}}(\mu^2) xg(x, \mu^2),
\end{equation}
with $x$ being the momentum fraction of the proton carried by the gluon, and the scale $\mu^2$ is a function of $r^2$. This parametrization gives the correct pQCD limit for the dipole cross section, ${ \sigma_\textrm{dip} } \sim r^2$. At large dipoles, the saturation effects are described by having an eikonalized gluon distribution function, which gives $\, \mathrm{d} { \sigma_\textrm{dip} } / \, \mathrm{d}^2 {\mathbf{b}} \to 2$ at large $r$, corresponding with the interaction probability being unity at large dipoles.
The scale at which the gluon density and the strong coupling constant are evaluated is chosen to be $\mu^2 = \mu_0^2 + C/r^2$. Here, unlike in previous fits~\cite{Kowalski:2006hc,Rezaeian:2012ji}, we fix $\mu_{0}^{2}=1.1\ \textrm{GeV}^2$ and let $C$ be a free parameter. This allows us to force $\mu^2$ to remain in the perturbative region. In our fit we only include data in the $Q^2$ bins that satisfy $Q^2>\mu_0^2$, which guarantees the applicability of the perturbative calculation. We also consider data in the kinematical region $x<0.01$ where the dipole picture can be considered to be most reliable.
The gluon density at the initial scale $\mu_0$ is parametrized as
\begin{equation}
\label{eq:xg_ic}
xg(x,\mu_0^2) = A_g x^{-\lambda_g} (1-x)^{6},
\end{equation}
where $A_g$ and $\lambda_g$ are free parameters to be determined by the fit. Unlike previous works, we use a variable flavor number scheme when evaluating the strong coupling constant $\alpha_{\mathrm{s}}$ and when solving the DGLAP evolution for the gluon distribution. Neglecting the change in the number of flavors and using a fixed number of quark flavors as the data moves in $Q^{2}$ is not the most adequate strategy from the theoretical point of view but, in practice, it is solely reflected in different values of the fitted parameters, without a sizable effect in the description of the data in the currently probed kinematic range.
For simplicity we refrain from including the quark singlet contribution to the DGLAP evolution which should also be present. However we have checked that the fit quality and the resulting dipole amplitude are not significantly affected by its inclusion, and that the fit drives the quark singlet contribution to zero at the initial scale. Furthermore we choose the high-$x$ behavior to be an integer exponent ($6$) instead of the standard $5.6$ in order to speed up the DGLAP evolution performed in Mellin space. We have checked that this exponent does not have a significant impact on the determination of the parameters. The strong coupling constant is required to satisfy $\alpha_{\mathrm{s}}(M_z = 91.1876\ \textrm{GeV}) = 0.1183$~\cite{Abramowicz:2015mha}. When evaluating the heavy quark contribution, the Bjorken $x$ is replaced by
\begin{equation}
\label{eq:xshift}
x_q= x\left(1 + \frac{4 m_q^2}{Q^2} \right)
\end{equation}
in order to take into account the kinematical shift, where $q$ refers to the quark flavor $c$ or $b$.
As we stay in the perturbative (large $Q^2$) region in this work, the shift \eqref{eq:xshift} would have negligible effect in case of light quarks.
The quark masses $m_f$ for the light and charm quark are kept as free parameters and constrained by the fit. In the fit we only include data that satisfy $x_c<0.01$ for the charm quark. The $b$ quark mass is set to $4.75\ \textrm{GeV}$, and the $b$ quark contribution to the structure function is included if the corresponding Bjorken-$x$ satisfies $x_b<0.1$.
Effectively in the IPsat model we fit the $x$ dependence of the cross section to the HERA data, and extrapolate it down to smaller values of Bjorken-$x$ when calculating predictions e.g. for the future DIS experiments. The other approach used in small-$x$ phenomenology is to evolve the dipole amplitude in $x$ using the perturbatively derived Balitsky-Kovchegov evolution equation~\cite{Kovchegov:1999yj,Balitsky:1995ub}, and incorporate some of the DGLAP effects in the initial condition of the evolution, which is fitted to the HERA data. Currently these fits are done at the leading logarithmic accuracy with running coupling corrections, and very good description of the HERA data is obtained if the $x$ evolution speed is also adjusted in the fit process by fitting the coordinate space scale at which the running coupling constant is evaluated~\cite{Albacete:2010sy,Lappi:2013zma} (note that our fit parameter $C$ has a similar effect by controlling the scale at which we evaluate $\alpha_{\mathrm{s}}(\mu^2)$ and $xg(x,\mu^2)$). Recently, there has been a lot of progress in developing the theory to NLO accuracy, see e.g. Refs~\cite{Balitsky:2008zza,Lappi:2015fma,Lappi:2016fmu,Lappi:2016oup,Boussarie:2016bkq,Beuf:2017bpd,Ducloue:2017ftk,Hanninen:2017ddy}.
In order to quantify the magnitude of the saturation effects, we also study the linearized version of the IPsat parametrization~\eqref{eq:ipsat}:
\begin{equation}
\label{eq:ipnonsat}
\frac {\, \mathrm{d} { \sigma_\textrm{dip} }}{\, \mathrm{d}^2 {\mathbf{b}}} = 2 r^2 F(x,r) T_p({\mathbf{b}}),
\end{equation}
to which we refer as the \emph{IPnonsat} model.
We emphasize that the rigorous way to look for the saturation effects is to compare saturation model calculations with the perturbative QCD results obtained by applying collinear factorization. In practice, however, comparing IPsat and IPnonsat results can be used as a first estimate for the strength of the saturation effects in the given process. In order to enable such a comparison, we fit the IPnonsat model parameters to the same HERA data.
\section{Description of the HERA reduced cross section data}
\label{HERA}
\begin{table*}[htbp]
\centering
\begin{tabular}{@{} cccccccccccc @{}}
\toprule
Type & HERA & $\chi^2/N$ & $N$ & $Q^2_\text{min}$ & $Q^2_\text{max}$ & $m_l$ & $m_c$ & $C$ & $A_g$ & $\lambda_g$ \\
\hline
IPsat & I & 1.0978 & $156+33$ & 1.5 & 50 & 0.03 & 1.3528 & 2.2894 & 2.1953 & 0.08289 \\
IPsat & I+II & 1.2781 & $410+33$ & 1.5 & 50 & 0.03 & 1.3210 & 1.8178 & 2.0670 & 0.09575\\
IPsat & I & 1.2634 & $229+42$ & 1.5 & 500 & 0.03 & 1.3296 & 2.6477 & 2.2097 & 0.07795 \\
IPsat & I+II & 1.3014 & $609+42$ & 1.5 & 500 & 0.03 & 1.3113 & 2.3700 & 2.1394 & 0.08388 \\
\hline
IPnonsat & I & 1.122 & $156+33$ & 1.5 & 50 & 0.1516 & 1.3504 & 4.2974 & 3.0391 & -0.006657 \\
IPnonsat & I+II & 1.3023 & $410+33$ & 1.5 & 50 & 0.1497 & 1.3180 & 3.5445 & 2.8460 & 0.008336 \\
IPnonsat & I & 1.2194 & $229+42$ & 1.5 & 500 & 0.1332 & 1.3187 & 5.6510 & 3.2820 & -0.03460 \\
IPnonsat & I+II & 1.2944 & $609 + 42$& 1.5 & 500 & 0.1359 & 1.3047 & 4.7328 & 3.0573 & -0.01656
\end{tabular}
\caption{All dimensionfull parameters in $\ \textrm{GeV}$. $X+Y$ points means $X$ points for $\sigma_r$ and $Y$ points for $\sigma_{r,\text{charm}}$. Bottom mass is $m_b=4.75\,\ \textrm{GeV}$, and $B_p=4\ \textrm{GeV}^{-2}$. In the IPsat fit the light quark mass is fixed to prevent numerical instability. The starting scale for the DGLAP evolution is $\mu_0^2=1.1\ \textrm{GeV}^2$. Fit results with HERA I data~\cite{Aaron:2009aa,Abramowicz:1900rp} and HERA I+II data~\cite{Abramowicz:2015mha,H1:2018flt} are shown separately. }
\label{tab:fits}
\end{table*}
The H1 and ZEUS experiments from HERA have published two combined datasets for the reduced cross section: the first one in Ref.~\cite{Aaron:2009aa} with the charm contribution in Ref.~\cite{Abramowicz:1900rp} where the HERA-I data are combined, and the latest final combined result for the inclusive reduced cross section including all HERA (HERA I+II) data in Ref.~\cite{Abramowicz:2015mha}. Recently the latest charm reduced cross section data from HERA I+II have been made available~\cite{H1:2018flt}. We will perform fits to both HERA I and HERA I+II datasets, but we will consider the fit to HERA I data to be our main result, as the charm cross section from HERA I+II is not yet published and the additional data in the newer dataset mostly affects the high-$Q^2$ region not included in the analysis. Moreover, the newer total reduced cross section dataset has more than twice as many data points in the region of interest of this work, which renders that data less sensitive to the charm quark if one does not introduce artificial weight factors (the HERA I+II charm reduced cross section data has as many points as in the HERA I results).
We include data in the region $1.5 < Q^2 < 50 \ \textrm{GeV}^2$. The lower limit, which we require to be larger than $\mu_0^2$, guarantees that there is a large scale justifying the perturbative calculation. As the validity of the dipole picture becomes questionable at high $Q^2$, we only include data up to $Q^2=50\ \textrm{GeV}^2$ in our main fit, though we also show results for fits done in the larger virtuality range with $Q^2_\text{max}=500\ \textrm{GeV}^2$.
The free parameters in this work are $A_{g}$ and $\lambda_g$ that describe the gluon distribution at $\mu^2=\mu_0^2$, and $C$ that controls the momentum space scale corresponding to the given dipole size $|{\mathbf{r}}|$. In addition, and as mentioned in the previous section, the light quark and charm quark masses are to be determined by the fit. However, as the bottom quark contribution is small, the fit can not reliably determine the $b$ quark mass and we set it to $m_b=4.75\ \textrm{GeV}$.
\begin{figure}[tb]
\begin{center}
\includegraphics[width=0.49\textwidth]{figs/chisqr_lightmass.pdf}
\end{center}
\caption{
Fit quality to HERA inclusive~\cite{Aaron:2009aa} and charm~\cite{Abramowicz:1900rp} reduced cross section data as a function of the light quark mass.
}\label{fig:light_mass}
\end{figure}
The dependence of the fit quality on the light quark mass $m_\text{light}$ is shown in Fig.~\ref{fig:light_mass} when we fit the HERA I data~\cite{Aaron:2009aa,Abramowicz:1900rp}. Throughout this work we show results obtained by using the fit to the HERA I data. Here, the charm mass and all the other parameters are allowed to vary freely with the fixed light quark mass. As one moves to lower values of $m_\text{light}$ in the IPsat fits, the $\chi^2$ reaches a plateau, making it hard to determine a best fit extraction of it's value, similarly as in Ref.~\cite{Rezaeian:2012ji}. Therefore, and in order to have finite quark mass to act as an infrared regulator, we fix $m_\text{light}=0.03 \ \textrm{GeV}$ for the IPsat case. The situation is different in the IPnonsat fit, where a relatively large light quark mass $\sim 0.14\ \textrm{GeV}$ is preferred. This can be interpreted as an effective confinement requirement. The effect of a nonzero quark mass is to suppress dipoles larger than $\sim 1/m_\text{light}$, that in IPnonsat model have an unphysically large (unitarity violating) cross section. The final fit quality in both IPsat and IPnonsat models is similar, suggesting that, when describing the inclusive DIS data, the effective confinement effect in the IPnonsat has a comparable effect as the gluon saturation in the IPsat parametrization.
Unlike previous works~\cite{Kowalski:2006hc,Rezaeian:2012ji} we find that the fit clearly prefers a charm quark mass $\sim 1.35\ \textrm{GeV}$, with the $\chi^2$ presenting a clear minimum. One main difference of our work with that of Ref.~\cite{Rezaeian:2012ji} is that the charm data are included in the fit, which allows us to constrain the charm mass simultaneously with the other parameters. The quality of the fit as a function of the charm mass is shown in Fig.~\ref{fig:charm_mass}, where all parameters are allowed to vary (except the light quark mass which is fixed to $0.03\ \textrm{GeV}$ in the IPsat model), while keeping $m_{c}$ constant.
\begin{figure}[tb]
\begin{center}
\includegraphics[width=0.49\textwidth]{figs/chisqr_charmmass.pdf}
\end{center}
\caption{
Fit quality to HERA inclusive~\cite{Aaron:2009aa} and charm~\cite{Abramowicz:1900rp} reduced cross section data as a function of the charm quark mass. In the IPsat fit the light quark mass is fixed to $0.03\ \textrm{GeV}$ (see text).
}\label{fig:charm_mass}
\end{figure}
\begin{figure*}[tb]
\begin{center}
\includegraphics[width=0.7\textwidth]{figs/sigmar.pdf}
\end{center}
\caption{
Inclusive reduced cross section from the fit including data up to $50\ \textrm{GeV}^2$, compared with the HERA data~\cite{Aaron:2009aa}. }
\label{fig:total_sigmar}
\end{figure*}
In Table \ref{tab:fits} we present the fitted parameters for two different values of $Q^{2}_\text{max}$. As can be seen from the $\chi^2$, the description of the precise HERA data is excellent, as already noted in previous works~\cite{Kowalski:2003hm,Kowalski:2006hc,Rezaeian:2012ji}. What is new here compared to the previous literature is the IPnonsat model parametrization, that we find to describe the combined HERA data equally well\footnote{In Ref.~\cite{Kowalski:2003hm} the IPnonsat model was fitted to older H1 and ZEUS data that have much larger uncertainties than the combined dataset used in this work}. This is demonstrated in Figs. \ref{fig:total_sigmar} and \ref{fig:charm_sigmar} where the reduced cross section and the charm contribution to it are shown and compared to the IPsat and IPnonsat model results, the curves being practically one on top of the other. We are also able to determine the quark masses from the fit. For the charm contribution, we note that there is some tension, the HERA data suggesting slightly slower $Q^2$ evolution that what is the outcome of our combined fit. This could be due to shortcomings of the model in describing the heavy quarks, as it happens in the collinear factorized framework where higher order corrections are needed for a proper description of the charm and bottom data~\cite{Ball:2017nwa}. It also might be influenced by the fact that the HERA collider was not particularly tuned to measuring heavy quarks, an issue that will be addressed in future colliders such as the EIC.
The fits done to HERA I and HERA I+II combined datasets result in comparable parameters (the largest difference being the scale parameter $C$, on which the results depend only logarithmically). Also, the newer dataset prefers a slightly smaller charm quark mass, but the differences are small. Thus the difference at the level of an observable will be negligible between the two different fits performed to different datasets. We will demonstrate this in Appendix~\ref{appendix:hera_i_ii}. We consider the top row of Table.~\ref{tab:fits} to be our main fit, as it relies on published datasets and only includes measurements in the kinematical domain where the applied model can be considered to be most reliable. The bottom quark reduced cross section included in the latest combination of HERA heavy quark data~\cite{H1:2018flt} is discussed in Appendix~\ref{appendix:bquark}.
\begin{figure*}[tb]
\begin{center}
\includegraphics[width=0.7 \textwidth]{figs/sigmar_cc.pdf}
\end{center}
\caption{
Charm reduced cross section calculated from the fit that includes data up to $50\ \textrm{GeV}^2$, compared with the HERA data~\cite{Abramowicz:1900rp}. }
\label{fig:charm_sigmar}
\end{figure*}
\subsection*{Contribution from large dipoles}
As discussed above, especially in the IPnonsat model, a nonzero effective light quark mass is needed in order to obtain a satisfactory description of the HERA data. This means that the reduced cross section data is sensitive to the contributions from (possibly non-perturbatively) large dipoles, whose formation should be suppressed by confinement scale effects. In the IPsat parametrization, the imposed unitarity requirement limits the scattering probability not to exceed unity at large dipoles, but large dipoles can still have a numerically significant contribution.
The fractional contribution from large dipoles to the total $F_2$ and $F_L$ is shown in Figs.~\ref{fig:f2_maxr} and \ref{fig:fl_maxr}, respectively. These calculations are done using the IPsat fit (first line of Tab.~\ref{tab:fits}), using the IPnonsat fit would result in very similar $r_\text{max}$ dependence. For $F_2$, even at relatively large $Q^2 \sim 500\ \textrm{GeV}^2$, 10\% of the total structure function originates from dipoles larger than $1 \ \textrm{fm}$. On the other hand, the HERA reduced cross section data have relative uncertainties at the percentage level, much smaller than the contribution we obtain from the non-perturbatively large dipoles.
The reason for this large contribution is that there is a large so called \emph{aligned jet} contribution: in the limits $z\to 0$ and $z\to 1$ the large dipole contribution to the transverse photon-proton cross section is only suppressed by the light quark mass as $\sim e^{-m_\text{light} r}$. This can be seen from the virtual photon wave function, Eq.~\eqref{eq:photon_t}. On the other hand, as can be seen from Fig.~\ref{fig:fl_maxr}, in the case of $F_L$ at moderate $Q^2$ the contribution from the region $r\gtrsim 1\ \textrm{fm}$ is negligible. This is due to the extra factor $z^2(1-z)^2$ in the longitudinal photon wave function, Eq.~\eqref{eq:photon_l}, which suppresses the endpoint contributions. Thus, we would prefer to fit the $F_L$ data instead of the reduced cross section measurements which is dominated by $F_2$. However, the HERA $F_L$ data~\cite{Aaron:2008ad,Chekanov:2009na} are not precise enough for a detailed comparison with the dipole model calculations.
Future DIS facilities EIC and LHeC have plans to measure proton structure functions (including $F_L$) at an unprecedented accuracy~\cite{AbelleiraFernandez:2012cc,Accardi:2012qut,Aschenauer:2017jsk}. Similarly, studying only the charm contribution to the total cross section limits the contribution from large dipoles as demonstrated in Fig.~\ref{fig:f2_maxr_charm}. In case of the $F_{2,\text{charm}}$, even at small $Q^2$ contribution from dipoles larger than $\sim 0.6\ \textrm{fm}$ is negligible (but very small dipoles are not sensitive to the saturation effects either). In general we find that $F_2$, $F_L$ and $F_{2,\text{charm}}$ are sensitive to different dipole sizes, and future DIS data covering all these structure functions will provide much more precise constraints.
In order to further study how much large dipoles affect the fit result, we perform the fits to HERA I inclusive and charm cross section data up to $Q_\text{max}^2=50\ \textrm{GeV}^2$ with different cutoffs for large dipoles $r_\text{max}$. The resulting fit quality is shown in Fig.~\ref{fig:chisqr_maxr}. We find that in order to obtain a good fit to the HERA data, we have to include dipoles up to $\sim 2 \dots 2.5 \ \textrm{fm}$ in the IPsat model. In the case of the IPnonsat parametrization, the fit can compensate the effect of the $r_\text{max}$ cutoff as the light quark mass is also a fit parameter, thus the fit quality is more stable with respect to the infrared cutoff. The IPnonsat fit drives the light quark mass to zero when $r_\text{max}$ becomes $\sim 1.6\ \textrm{fm}$, and it is not possible to fit the HERA data with a much smaller cutoff. The dependence of the light quark mass on $r_\text{max}$ is shown in Fig.~\ref{fig:rmax_lightmass} which further demonstrates that in the IPnonsat model the inclusion of large dipoles requires a larger light quark mass to suppress the contribution from this unphysical region. In the IPsat model, the fits prefer a zero light quark mass at all $r_\text{max}$.
\begin{figure}[tb]
\begin{center}
\includegraphics[width=0.49\textwidth]{figs/ipsat_f2_maxr.pdf}
\end{center}
\caption{
Contribution to $F_2$ at $x=0.01$ from the IPsat model from dipoles smaller than $r_\text{max}$ at different $Q^2$. Results for IPnonsat are similar. }
\label{fig:f2_maxr}
\end{figure}
\begin{figure}[tb]
\begin{center}
\includegraphics[width=0.49\textwidth]{figs/ipsat_fl_maxr.pdf}
\end{center}
\caption{
Contribution to $F_L$ at $x=0.01$ from dipoles smaller than $r_\text{max}$ at different $Q^2$. }
\label{fig:fl_maxr}
\end{figure}
\begin{figure}[tb]
\begin{center}
\includegraphics[width=0.49\textwidth]{figs/ipsat_charm_f2_maxr.pdf}
\end{center}
\caption{
Contribution to charm structure function $F_{2,c}$ at $x=0.01$ from dipoles smaller than $r_\text{max}$ at different $Q^2$. Note that the scales are different than in Figs.~\ref{fig:f2_maxr} and \ref{fig:fl_maxr}.}
\label{fig:f2_maxr_charm}
\end{figure}
\begin{figure}[tb]
\begin{center}
\includegraphics[width=0.49\textwidth]{figs/chisqr_maxr.pdf}
\end{center}
\caption{
Fit quality with different cutoffs for large dipoles. In the IPsat model the light quark mass is fixed to $m_\text{light}=0.03\ \textrm{GeV}$. }
\label{fig:chisqr_maxr}
\end{figure}
\begin{figure}[tb]
\begin{center}
\includegraphics[width=0.49\textwidth]{figs/rmax_lightmass.pdf}
\end{center}
\caption{
Light quark mass obtained as a fit result with the IPnonsat model as a function of the infrared cut for the large dipoles. }
\label{fig:rmax_lightmass}
\end{figure}
\section{Dipole scattering amplitude}
\label{amplitude}
\begin{figure}[tb]
\begin{center}
\includegraphics[width=0.49\textwidth]{figs/dipoleamplitude.pdf}
\end{center}
\caption{
The obtained dipole amplitudes at $x=10^{-6}$ (red), $x=10^{-4}$ (blue) and $x=10^{-2}$ (black). RSKV refers to the previous fit~\cite{Rezaeian:2012ji}.
}\label{fig:dipoleamplitude}
\end{figure}
The resulting dipole amplitude at $b=0$ is shown in Fig.~\ref{fig:dipoleamplitude}, where we compare it with the previous results from \cite{Rezaeian:2012ji} labeled as ``RSKV''. Even thought our study has incorporated some refinements (variable flavor number scheme in the DGLAP evolution, quark masses determined by the fit, inclusion of the charm reduced cross section data), the base model is essentially the same and, by fitting similar data sets one is expected to obtain compatible dipole amplitudes, despite some numerical differences in the fitted parameters. In the IPnonsat model the evolution at the initial scale in $x$ is very slow ($\lambda_g$ being close to $0$), thus there is basically no evolution at large $r$, in the region where the IPsat parametrization has already reached unity (and where the IPnonsat model gives unphysical results).
To demonstrate the evolution of the gluon distribution function we plot $xg(x,\mu^2=\mu_0^2+C/r^2)$ in Fig.~\ref{fig:xg} as a function of $r$ using both the IPsat and IPnonsat fitted parameters to initialize the DGLAP evolution. At large scales the two parametrizations have small differences, as the effect of the initial condition is washed out in the evolution. Close to the initial scale $\mu_0^2$ there is basically no evolution if the IPnonsat model parametrization is used ($\lambda_g \approx 0$, which forces the dipole scattering amplitude not to grow in the region where it is already violating unitarity), unlike in the case of the IPsat model.
At large scales and at sufficiently large $x \gtrsim 10^{-3}$ it is also visible that the gluon density starts to decrease as the scale $\mu^2 \sim C/r^2$ increases. This is expected, as the momentum conservation in the DGLAP evolution removes the larger-$x$ gluons as they are splitting into the smaller-$x$ ones. Similar results were already found in Ref.~\cite{Kowalski:2003hm}. This effect is strong close to $x\sim 10^{-2}$, where the decreasing gluon density is probed already by dipoles that have large enough sizes to contribute significantly on $F_2$.
The point at which the non-linear effects become relevant is characterized by the saturation scale $Q_s^2$. To determine it we use the definition
\begin{equation}
\label{eq:satscale}
N(r^2=2/Q_s^2,x,b) = 1-e^{-1/2}.
\end{equation}
The extracted saturation scale as a function of $x$ is shown in Fig.~\ref{fig:qs}. Here, $Q_s^2$ is extracted at the central impact parameter $b=0$, and at the average $\langle b \rangle$ defined such that
\begin{equation}
\int_0^{\langle b \rangle} \mathrm{d} b \, b \, T_p(b) = \frac{1}{2} \int_0^\infty \mathrm{d} b \, b \, T_p(b).
\end{equation}
This definition gives $\langle b \rangle \approx 0.46\ \textrm{fm}$. The difference between the IPsat and IPnonsat parametrizations remains small at all values of Bjorken-$x$, the IPnonsat model having in general slightly faster evolution. As expected based on previous analyses (e.g.~\cite{Rezaeian:2012ji,Lappi:2013zma}), the saturation scale of the proton is at the $\Lambda_{\mathrm{QCD}}$ range in the region $x\sim 10^{-2}$, and the region of $Q_s^2$ being perturbative is reached below $x \lesssim 10^{-4} \dots 10^{-5}$ (note that the absolute value of $Q_s^2$ depends on the definition \eqref{eq:satscale}).
\begin{figure}[tb]
\begin{center}
\includegraphics[width=0.49\textwidth]{figs/xg.pdf}
\end{center}
\caption{
Gluon density $xg$ as a function of the dipole size $r$ for $x=10^{-4}$ (red), $x=10^{-3}$ (blue) and $x=10^{-2}$ (black) from top to bottom.
}\label{fig:xg}
\end{figure}
\begin{figure}[tb]
\begin{center}
\includegraphics[width=0.49\textwidth]{figs/satscale.pdf}
\end{center}
\caption{
Saturation scale at the center (thick black lines) of the proton and at average impact parameter $\langle b \rangle \approx 0.46\ \textrm{fm}$ (thin blue lines).
}\label{fig:qs}
\end{figure}
\section{Exclusive vector meson production}
\label{EVM}
Additional information about the proton structure can be obtained by studying exclusive processes. They are particularly powerful in probing the gluonic structure, as at leading order in collinear factorization the vector meson production cross section is proportional to the \emph{squared} gluon distribution~\cite{Ryskin:1992ui}. In addition to being sensitive to the total gluonic density, in exclusive process the momentum transfer $\Delta = \sqrt{-t}$ is the Fourier conjugate to the impact parameter, which makes it possible to probe also the impact parameter dependence.
In the dipole picture, the scattering amplitude for exclusive vector meson production reads (see e.g. Ref.~\cite{Kowalski:2006hc})
\begin{multline}
{\mathcal{A}}^{\gamma^* p \to V p} = \int \mathrm{d}^2 {\mathbf{r}} \mathrm{d}^2 {\mathbf{b}} \frac{\mathrm{d} z}{4\pi} (\Psi^* \Psi_V)({\mathbf{r}}, Q^2,z) \\
\times e^{-i {\mathbf{b}} \cdot {\bf \Delta}} \frac {\, \mathrm{d} { \sigma_\textrm{dip} }}{\, \mathrm{d}^2 {\mathbf{b}}}.
\end{multline}
This expression has a straightforward interpretation. First, the incoming virtual photon splits into a quark-antiquark pair as described by the virtual photon wave function $\Psi$. The dipole then scatters elastically off the proton with cross section $\sigma_\text{dip}$, and ultimately forms the final state vector meson described by the wave function $\Psi_V$. The scattering amplitude in the momentum space is obtained by calculating the Fourier transform from the coordinate space, with the momentum transfer ${\bf \Delta}$ being the Fourier conjugate to the impact parameter ${\mathbf{b}}$.
Here we have neglected the off-forward correction to the vector meson wave function~\cite{Bartels:2003yj}.
The exclusive vector meson production cross section reads
\begin{equation}
\frac{\mathrm{d} \sigma^{\gamma^* p \to Vp}}{\mathrm{d} t} = \frac{1}{16\pi} \left| {\mathcal{A}}^{\gamma^* p \to V p} \right|^2.
\end{equation}
In addition, we include the corrections due to the real part of the scattering amplitude neglected when deriving the above result, and the so called ``skewedness correction" which takes into account the fact that in the two-gluon exchange the two gluons carry different amount of longitudinal momentum~\cite{Shuvaev:1999ce}. These corrections are included as in Ref.~\cite{Lappi:2010dd} and, to a good approximation, only affect the overall normalization of the diffractive cross section.
Unlike the virtual photon wave function used to calculate inclusive cross sections, the vector meson wave function can not be calculated perturbatively. We use here the Boosted Gaussian parametrization as in Ref.~\cite{Kowalski:2006hc}, where one assumes that the vector meson is a quark-antiquark state with spin and polarization structure the same as in the case of the photon. This assumption makes it possible to write the overlap between the vector meson $V$ and the virtual photon wave function in case of the transverse polarization as
\begin{multline}
(\Psi^*_V\Psi)_T = \hat e_f e \frac{{N_\mathrm{c}}}{\pi z(1-z)} \left\{ m_f^2 K_0(\varepsilon r) \phi_T(r,z) \right. \\
\left. - [z^2+(1-z)^2] \varepsilon K_1(\varepsilon r) \partial_r \phi_T(r,z) \right\},
\end{multline}
and for the longitudinal polarization
\begin{multline}
(\Psi^*_V\Psi)_L =\hat e_f e \frac{{N_\mathrm{c}}}{\pi} 2 Q z(1-z) K_0(\varepsilon r) \\
\times \left[ M_V \phi_L(r,z) + \frac{m_f^2 - \nabla^2}{M_V z(1-z)} \phi_L(r,z) \right].
\end{multline}
The scalar part of the vector meson is parametrized as
\begin{multline}
\phi_{T,L}(r,z) = N_{T,L} z(1-z) \exp \left( -\frac{m_f^2 R^2}{8z(1-z)} \right. \\
- \frac{2z(1-z)r^2}{R^2} \left. + \frac{m_f^2 R^2}{2} \right).
\end{multline}
The advantage of this parametrization is that the wave function has the proper short-distance behavior $\sim z(1-z)$ in the limit of massless quarks. The normalization factors $N_{T,L}$ and the width $R$ are fixed by requiring that the decay width to the electron channel (calculated using the longitudinal polarization as in Ref.~\cite{Kowalski:2006hc}) reproduces the experimental value, and that the wave function is properly normalized. As these parameters depend on the quark masses, we calculate them for $J/\Psi$, $\rho$ and $\phi$ for the same values obtained in the fits to inclusive data for the IPsat and IPnonsat parametrizations. The obtained values and the comparison with the results from \cite{Kowalski:2006hc} are shown in Table.~\ref{table:boostedgaussian}. We remark that if one were to calculate the decay width using the transverse polarization, the final numbers would be slightly different as noted in Ref.~\cite{Kowalski:2006hc}.
\begin{table*}[tb]
\centering
\begin{tabular}{ccccccc}
\toprule
Meson & $M_V$ [GeV] & Type & $m_f$ [GeV] & $R$ $[\mathrm{GeV}^{-1}]$ & $N_T$ & $N_L$ \\
\hline
$J/\Psi$ & 3.097 & IPsat & 1.3528 & 1.5070 & 0.5890 & 0.5860 \\
$J/\Psi$ & 3.097 & IPnonsat & 1.3504 & 1.5071 & 0.5899 & 0.5868 \\
$J/\Psi$ & 3.097 & KMW & 1.4 & 1.5166 & 0.578 & 0.575 \\
\hline
$\phi$ & 1.019 & IPsat & 0.03 & 3.3922 & 0.9950 & 0.8400 \\
$\phi$ & 1.019 & IPnonsat & 0.1516 & 3.3530 & 0.9072 & 0.8196 \\
$\phi$ & 1.019 & KMW & 0.14 & 3.347 & 0.919 & 0.825 \\
\hline
$\rho$ & 0.776 &IPsat & 0.03 & 3.6376 & 0.9942 & 0.8926 \\
$\rho$ & 0.776 & IPnonsat & 0.1516 & 3.5750 & 0.8978 & 0.8467 \\
$\rho$ & 0.776 & KMW & 0.14 & 3.592 & 0.911 & 0.853 \\
\end{tabular}
\caption{Parameters for the Boosted Gaussian wave function corresponding to the quark masses obtained for the IPsat and IPnonsat parametrizations. For comparison we include the results from \cite{Kowalski:2006hc} (labeled as KMW), also determined using the longitudinal polarization. }
\label{table:boostedgaussian}
\end{table*}
\begin{figure}[tb]
\begin{center}
\includegraphics[width=0.49\textwidth]{figs/jpsi_tspectra.pdf}
\end{center}
\caption{
Differential J/$\Psi$ photoproduction cross section as a function of momentum transfer $-t={\bf \Delta}^2$ at two different center-of-mass energies. The dashed line is obtained by using the wave function provided in Ref.~\cite{Kowalski:2006hc} where the charm quark mass is $m_c=1.4\ \textrm{GeV}$. The other IPsat and IPnonsat curves use the wave function parametrizations from Table~\ref{table:boostedgaussian}. The $W=75\ \textrm{GeV}$ data are from Ref.~\cite{Alexa:2013xxa} and the $W=100\ \textrm{GeV}$ data from Refs.~\cite{Aktas:2005xu,Chekanov:2002xi}. The high-energy results are scaled by $5$ for illustrational purposes.
}\label{fig:jpsi_spectra}
\end{figure}
\begin{figure}[tb]
\begin{center}
\includegraphics[width=0.49\textwidth]{figs/totxs_jpsi_w.pdf}
\end{center}
\caption{
Total exclusive $\gamma p \to J/\Psi p$ production cross section as a function of $W$.
}\label{fig:jpsi_totxs}
\end{figure}
Due to the small light quark mass especially in the IPsat model parametrization, the photoproduction of $\rho$ and $\phi$ mesons can not be reliably calculated from our model. In the case of J/$\Psi$, the charm quark mass provides the necessary large scale that cuts out large dipoles, and makes it possible to calculate exclusive J/$\Psi$ production down to $Q^2=0 \ \textrm{GeV}^2$. This is advantageous, as recently it has become possible to measure exclusive vector meson photoproduction in ultraperipheral collisions at RHIC and the LHC~\cite{Bertulani:2005ru}.
In the literature there are some inconsistencies using the vector meson wave function parametrization from Ref.~\cite{Kowalski:2006hc} together with dipole model fits with different choices for the charm quark mass. In order to quantify the effect of having a consistent quark mass in the dipole model fit and in the vector meson wave function, we show in Fig.~\ref{fig:jpsi_spectra} the J/$\Psi$ production cross section using our IPsat fit (where $m_c\approx1.35\ \textrm{GeV}$) and the widely used wave function from Ref.~\cite{Kowalski:2006hc} (where $m_c=1.4\ \textrm{GeV}$, referred as KMW). The larger quark mass in the KMW parametrization reduces the cross section by approximately 14\%. We note that the uncertainties related to the modeling of the vector meson wave function are larger than this, see e.g. Refs.~\cite{Kowalski:2006hc,Lappi:2013am}. The IPsat and IPnonsat results are practically on top of each other at small $|t|$.
The agreement with the HERA data is good, except that we can not reproduce the small change of the $t$ slope at $|t| \lesssim 0.1\ \textrm{GeV}^2$ visible in the $W=75\ \textrm{GeV}$ data\footnote{Which is described accurately in the IP-Glasma model calculation in Refs.~\cite{Mantysaari:2016ykx,Mantysaari:2016jaz}}.
At large $|t|$ the different form factors generate different spectra. The Fourier transform of the IPnonsat dipole amplitude is exactly Gaussian, and the spectra goes like $e^{-B_p |t|}$. In the IPsat parametrization, the proton density profile is actually $\sim \exp(-e^{-b^2/2B_p})$, thus its Fourier transform is a more complicated function which generates diffractive dips at large $-t$. At $W\sim 100\ \textrm{GeV}$, we get the location for the first diffractive minimum to be $|t|\sim 2.5\ \textrm{GeV}^2$ (see also Ref.~\cite{Armesto:2014sma} for discussion about the energy dependence of the dip location).
The total $J/\Psi$ production cross section, calculated using our IPsat and IPnonsat model fits, is shown in Fig.~\ref{fig:jpsi_totxs}. The results are compared with the HERA data from the H1~\cite{Aktas:2005xu,Alexa:2013xxa} and ZEUS~\cite{Chekanov:2002xi} collaborations, and with the recent measurement by the ALICE collaboration~\cite{TheALICE:2014dwa} on ultraperipheral proton-lead collision (which can be seen as a photon-proton scattering due to the $Z^2$ enhancement for the photon flux emitted from the nucleus). The models are found to be in agreement with the current data\footnote{Compare with Ref.~\cite{Armesto:2014sma} where the IPnonsat model is not separately fitted to the HERA data, instead the same parameters are used in both parametrizations}, but the future more precise LHC data at even higher $W$ (requiring larger center-of-mass energy for the ultraperipheral proton-nucleus scattering or more forward rapidities) will be in the region where the difference between the models is large.
\begin{figure}[tb]
\begin{center}
\includegraphics[width=0.49\textwidth]{figs/rho_tspectra.pdf}
\end{center}
\caption{
Differential $\rho$ electroproduction cross section as a function of momentum transfer $-t={\bf \Delta}^2$. The dashed line is obtained by using the $\rho$ wave function provided in Ref.~\cite{Kowalski:2006hc} where the light quark mass is $m_\text{light}=0.14\ \textrm{GeV}$. The other IPsat and IPnonsat curves use the wave function parametrizations from Tab.~\ref{table:boostedgaussian}. The experimental data are from the H1 collaboration~\cite{Aaron:2009xp}.
}\label{fig:rhospectra}
\end{figure}
Let us then study the production of light mesons. The differential $\rho$ electroproduction cross section in different $Q^2$ bins is shown in Fig.~\ref{fig:rhospectra} and compared with the H1 data~\cite{Aaron:2009xp}. In the lowest $Q^2$ bins the applicability of our framework is questionable as, again, we do not have a large scale suppressing nonperturbative contributions. The agreement with the H1 data is good especially at higher $Q^2$ using both IPsat and IPnonsat parametrizations. Note that the light quark mass is much larger in the IPnonsat model, which explains why the cross section at small $-t$ is actually smaller in the IPnonsat calculation. Again, by calculating the $\rho$ production with the IPsat model parametrization and the KMW wave function~\cite{Kowalski:2006hc} we find that the larger light quark mass suppresses the cross section at small $Q^2$ bins similarly to the case of J/$\Psi$ production, and that the different impact parameter profile causes diffractive dips at large $|t|$ when we use the IPsat parametrization.
\section{Future experiments}
\label{Future}
\subsection{Proton targets}
As we saw in Sec. \ref{HERA}, both the IPsat and IPnonsat parametrizations give equally good descriptions of the HERA data. Due to the different behavior of the gluon distribution $xg$ at small $x$, and especially as the gluon distribution is eikonalized in the IPsat parametrization, differences are expected to arise when extrapolating to smaller values of Bjorken-$x$. This we already found in the case of exclusive J/$\Psi$ production in Fig.~\ref{fig:jpsi_totxs}, where both parametrizations give comparable results in the range covered by the HERA data, but differ by $\sim 50\%$ in the kinematics covered by recent and near-future LHC experiments. On the contrary, for inclusive DIS the structure function $F_2$ predicted by both, the IPsat and IPnonsat, prametrizations overlap almost perfectly for a wide range of $x$ values, as shown in Fig.~\ref{fig:f2}. Even at very small $x\sim 10^{-6}$ the two models differ only at the level of a few percent.
To see if the future high-energy DIS experiments can measure the structure functions with an accuracy lower or comparable to the difference between the two models, we show in Fig.~\ref{fig:f2ratio} the ratio of $F_2$ obtained using the IPnonsat and IPsat parametrizations compared with the projected accuracy of the LHeC measurement~\cite{AbelleiraFernandez:2012cc}.
As the uncertainty estimates for the LHeC consist of projected absolute, not relative uncertainties, the relative uncertainties shown as a colorfull bands are obtained by comparing the projected uncertainty to the result obtained by applying the IPsat parametrization. As already seen in Fig.~\ref{fig:f2}, the differences are at a few percent level, and only slightly larger than the projected experimental accuracy at the LHeC. FCC-eh would probe $x$ values down to $x\sim 10^{-7}$ with comparable precision, thus at least in $F_2$ there would not be a striking difference between the IPsat and IPnonsat extrapolations. For $F_L$ the model differences are similar, but the experimental accuracy is much lower.
The data from future DIS machines on $F_2$, $F_{2,\text{charm}}$ and $F_L$ will thus make it possible to constrain dipole-proton scattering much more accurately, thanks to the fact that different observables are sensitive to different dipole sizes. However, in inclusive $e+p$ scattering the IPsat model predicts the non-linear effects to be small. It thus becomes necessary to study nuclear DIS, where one expects the saturation effects to be enhanced by a large factor $A^{1/3}$.
\begin{figure}[tb]
\begin{center}
\includegraphics[width=0.49\textwidth]{figs/f2.pdf}
\end{center}
\caption{Proton structure function $F_2$ computed using the IPsat and IPnonsat fits. For $F_L$, the difference between IPsat and IPnonsat parametrizations is similar. }
\label{fig:f2}
\end{figure}
\begin{figure}[tb]
\begin{center}
\includegraphics[width=0.49\textwidth]{figs/ratio_f2.pdf}
\end{center}
\caption{
Ratio of the structure function $F_2$ computed using the IPsat and IPnonsat fits. The bands show relative uncertainty projected for the LHeC assuming the IPsat parametrization from small (left) to large (right) $Q^2$ (see text).}
\label{fig:f2ratio}
\end{figure}
\subsection{Nuclear targets}
\label{sec:nuclei}
Using the Optical Glauber model one can extend the dipole-proton scattering amplitude to the dipole-nucleus one.
Calculating the dipole-nucleus scattering by averaging over the positions of the individual nucleons from the Woods Saxon distribution following Ref.~\cite{Kowalski:2003hm} one obtains
\begin{equation}
\label{eq:ipsatlumpynuke}
\frac {\, \mathrm{d} { \sigma^A_\textrm{dip} }}{\, \mathrm{d}^2 {\mathbf{b}}} = 2\left[1 - \left( 1 - \frac{1}{2} T_A({\mathbf{b}}) { \sigma_\textrm{dip} } \right)^A \right],
\end{equation}
where ${ \sigma_\textrm{dip} }$ is the total dipole-proton cross section integrated over impact parameter, see Eq.~\eqref{eq:ipsat}. For large nuclei, this gives
\begin{equation}
\label{eq:ipsatlumpynuke_largeA}
\frac {\, \mathrm{d} { \sigma^A_\textrm{dip} }}{\, \mathrm{d}^2 {\mathbf{b}}} = 2\left[1 - \exp \left( 1 - \frac{1}{2} AT_A({\mathbf{b}}) { \sigma_\textrm{dip} } \right) \right].
\end{equation}
Only if, in addition to having a large $A$, the dipole-proton cross section is small (which requires small $r$ as ${ \sigma_\textrm{dip} } \sim \ln r$) one obtains the \emph{smooth nucleus} result
\begin{equation}
\label{eq:ipsatnuke}
\frac {\, \mathrm{d} { \sigma^A_\textrm{dip} }}{\, \mathrm{d}^2 {\mathbf{b}}} = 2\left[1 - \exp \left( -r^2 F(x,r) A T_A({\mathbf{b}}) \right) \right].
\end{equation}
In practice, as large dipoles have numerically significant contribution to $F_2$, this approximation is not a realistic and results in too small nuclear suppression as discussed in Ref.~\cite{Kowalski:2003hm}.
Here $T_A$ is the Woods Saxon distribution integrated over the longitudinal coordinate, and the nuclear radius is $R_A=1.13A^{1/3} - 0.86 A^{-1/3}$ fm. The normalization is chosen such that $\int \mathrm{d}^2 {\mathbf{b}} T_A({\mathbf{b}})=1$. The corresponding dipole-nucleus amplitude in the IPnonsat model is the first term from the series expansion
\begin{equation}
\label{eq:ipnonsatnuke}
\frac {\, \mathrm{d} { \sigma^A_\textrm{dip} }}{\, \mathrm{d}^2 {\mathbf{b}}} = 2 r^2 F(x,r) A T_A({\mathbf{b}}) .
\end{equation}
\begin{figure}[tb]
\begin{center}
\includegraphics[width=0.49\textwidth]{figs/f2_suppression_q2_exp.pdf}
\end{center}
\caption{Nuclear suppression for the structure function $F_2$ compared with the NMC and E665 data~\cite{Amaudruz:1995tq,Adams:1995is,Arneodo:1995cs}. The lead results are at $x=0.006185$. The Calcium data points cover $x$ values $0.005 \dots 0.0085$, and our calculation is done at average $x=0.0068$. By construction this ratio is exactly 1 with the IPnonsat parametrization. }
\label{fig:suppression_f2_exp}
\end{figure}
The nuclear suppression factor for the structure function $F_2$ is shown in Fig.~\ref{fig:suppression_f2_exp}, where we calculate
\begin{equation}
R=\frac{F_{2,A}}{AF_{2,p}}.
\end{equation}
For comparison, the experimental data points for Calcium and Lead from \cite{Amaudruz:1995tq,Adams:1995is,Adams:1995is} are shown\footnote{Part of the fixed target data is obtained by comparing total cross sections for nuclear and deuteron targets, which is not exactly the same as our structure function ratio}.
\begin{figure}[tb]
\begin{center}
\includegraphics[width=0.49\textwidth]{figs/f2fl_suppression_x.pdf}
\end{center}
\caption{Bjorken-$x$ dependence of the nuclear suppression factor for $F_2$ (thick lines) and $F_L$ (thin lines). By construction this ratio is exactly 1 with the IPnonsat parametrization. }
\label{fig:suppression_f2fl_x}
\end{figure}
The Bjorken-$x$ dependence of the $F_2$ and $F_L$ nuclear suppression factors is studied in Fig.~\ref{fig:suppression_f2fl_x}. We find that in the case of $F_2$, the $x$ dependence is weak due to the fact that a significant part of $F_2$ originates from large saturated dipoles, whose scattering cross section is not affected by the relatively slow evolution of the saturation scale $Q_s^2 \sim xg(x,\mu^2)$. In the case of $F_L$, which is dominated by smaller dipoles, a significantly faster $x$ dependence is found.
We also find that when $x$ increases, at some point the nuclear suppression starts to \emph{increase}, which is counter intuitive.
A similar observation was already seen in Ref.~\cite{Kowalski:2007rw}. We note that if we were to do a BK evolution for the nucleus similarly as in Ref.~\cite{Lappi:2013zma}, we would get a suppression factor for $F_2$ that always increases with increasing $x$. We consider the fact that the $F_2$ suppression factor has a maximum as a function of $x$ to be an artefact of the shortcomings of the IPsat parametrization (e.g. decreasing $xg$ with increasing scale in Fig.~\ref{fig:xg} which effectively decreases the saturation scale probed by larger dipoles).
Let us next study nuclear suppression in exclusive vector meson production. Here, we analyze the $Q^2$ dependence of light $\rho$ and $\phi$ meson production that allows us to scan the transition from saturation to dilute region (see also Ref.~\cite{Mantysaari:2017slo}). In addition, we include $J/\Psi$, which is significantly smaller and heavier, and should experience less non-linear effects. As the total coherent cross section scales like $A^2$, and the width of the first diffractive peak is proportional to $1/R_A^2 \sim A^{-2/3}$, we study the
suppression factor
\begin{equation}
R=\frac{\sigma^{\gamma^* A \to V A}} {c A^{4/3} \sigma^{\gamma p \to Vp}},
\end{equation}
where $V$ refers to the vector meson species. Note that diffractive cross sections are enhanced more strongly by the large nucleus compared to the inclusive scattering which scales linearly in $A$. The numerical factor $c$ in the denominator can be obtained as a ratio of the form factors for the nucleus and the proton
\begin{equation}
c = \frac{A^2 \int \mathrm{d} t \left| \tilde T_A(\sqrt{t}) \right|^2 } {A^{4/3} \int \mathrm{d} t \left | \tilde T_p(\sqrt{t}) \right|^2 },
\end{equation}
as the form factors determine the $t$ spectra.
Here the form factors are $\tilde T_A(\sqrt{t}) = \int \mathrm{d}^2 {\mathbf{b}} e^{-i {\mathbf{b}} \cdot {\boldsymbol{\Delta}}} T_A({\mathbf{b}})$ and $\tilde T_p(\sqrt{t})=\int \mathrm{d}^2 {\mathbf{b}} e^{-i {\mathbf{b}} \cdot {\boldsymbol{\Delta}}} T_p({\mathbf{b}})$ with $t={\boldsymbol{\Delta}}^2$. For the Gold nucleus with $A=197$, this gives $c\approx 0.5011$. By construction, this definition gives $R=1$ in case of the linear IPnonsat parametrization.
The $Q^2$ dependence of the suppression factor is relatively weak as shown in Fig.~\ref{fig:suppression_rho_phi}, with $R$ reaching unity only at $Q^2 \sim 1000\ \textrm{GeV}^2$. Physically the reason here is that even if at large $Q^2$ the photon preferably splits into a small dipole which does not see nonlinear effects, the requirement that a light (and large) vector meson is formed at the final state gives a small weight for the small dipoles. Instead, the specified final state requires the dipole to be relatively large, and it becomes necessary to go to very large $Q^2$ to give enough weight on small dipoles so that the suppression factor becomes close to 1. In case of J/$\Psi$ the vector meson wave function always picks up relatively small dipoles, thus the maximum suppression is only around $0.7$. For a discussion about the nuclear suppression in incoherent scattering, the reader is referred to Ref.~\cite{Lappi:2010dd}.
In Fig.~\ref{fig:suppression_rho_phi} we also show the $W$ dependence of the suppression factor, which is found to be relatively modest. However, in the future Electron Ion Collider it will be useful to have maximally large $Q^2$ lever arm to study the evolution of the suppression factor from the saturated to the dilute region (see also Refs.~\cite{Mantysaari:2017slo,Aschenauer:2017jsk}). For example, in the case of $\rho$ production and at ${x_\mathbb{P}}=10^{-2}$ (which is around the maximum $x$ where our model can be considered to be valid), the maximum $Q^2$ that can be reached at an EIC with $\sqrt{s_{NN}}=90\ \textrm{GeV}$ is $Q^2_\text{max}\approx 100\ \textrm{GeV}^2$, which would make it possible to observe the evolution of the nuclear suppression from $R\sim 0.2$ to $R\sim 0.8$.
\begin{figure}[tb]
\begin{center}
\includegraphics[width=0.49\textwidth]{figs/vm_suppression.pdf}
\end{center}
\caption{Nuclear suppression for total coherent J/$\Psi$, $\rho$ and $\phi$ vector meson production as a function of $Q^2$. The upper black lines refer to the case where $W=100\ \textrm{GeV}$ and the lower blue lines are calculated at $W=1000\ \textrm{GeV}$. The results are shown in the range where ${x_\mathbb{P}} < 0.02$.
}
\label{fig:suppression_rho_phi}
\end{figure}
\section{Conclusions}
\label{conclusions}
The possibility of finding new and exciting QCD phenomena is just around the corner, with the next generation of DIS colliders to come soon. In view of this it is timely to exploit to the fullest the available data. In this path this work bring us one step forward, by determining not only the IPsat model with the modern data sets but also for the first time the linearized IPnonsat parametrization is determined from the same data in a fully consistent way.
The main differences to previous works are the inclusion of the charm data to the global fit, which allows us to constrain the quark masses, and the use of a variable flavor number scheme. Also, for the first time, the combined HERA I+II datasets are used in dipole model fits with a similar outcome than in case of the HERA I combined results. We find that both models, with and without saturation, result in almost identical cross sections at HERA kinematics, and that the differences in
$e+p$ scattering are expected to be small even in the LHeC or FCC-eh kinematics. The nonlinear effects, however, become significant if a nuclear target/beam is used and should be easily observed in the future Electron-Ion Collider.
Despite some differences in the setup with the previous literature, the resulting dipole amplitude and calculated cross sections are similar than in the previous work~\cite{Rezaeian:2012ji}. This is a consequence of performing the fits using comparable data sets, which extrapolates to similar dipole amplitudes. Therefore for the IPsat case the models found in the literature will provide reasonable numerical results in the kinematical range accessible in current and future colliders. We emphasize that having a linearized ``IPnonsat'' model independently constrained by the HERA data is necessary for estimating the size of the saturation effects in these experiments.
The similar cross sections obtained from both IPsat and IPnonsat parametrizations are understood in terms of the effective description of the confinement scale physics. The linearized dipole cross section violates unitarity at large dipoles, and the fit compensates that by imposing an effective confinement effect damping dipoles larger than $\sim 1/m_{\text{light}}$. In the IPsat model, the unitarity requirement limits the contribution from unphysically large dipoles and a large light quark mass is not required in order to obtain a good description of the HERA data.
The inclusive structure function $F_2$ (and thus the reduced cross section $\sigma_r$) is especially sensitive to the dipoles expected to be heavily influenced by confinement effects not completely included in this work. On the other hand, $F_L$ and $F_{2,\text{charm}}$ are not sensitive at all to dipoles larger than $\sim 1/\Lambda_{\mathrm{QCD}}$. Thus, the future more precise $F_L$ and $F_{2,\text{charm}}$ data, together with inclusive structure function measurements, will allow us to perform a much more precise test of the saturation picture.
Both IPsat and IPnonsat parametrizations give comparable predictions for structure functions at the energies available in future DIS experiments such as LHeC and FCC-eh. Slightly larger differences are seen when calculating predictions for exclusive vector meson production, but in order to really see the onset of non-linear nature of QCD, we find that it is crucial to perform DIS with nuclear targets. These effects should become clearly visible already at the EIC energies. Additionally, the potential for inclusive diffraction to separate between the linear and non-linear parametrizations could be studied in future work.
\section*{Acknowledgements}
We thank T. Lappi, T. Toll, T. Ullrich and R. Venugopalan for discussions and are grateful to Max Klein for providing LHeC and FCC-eh pseudodata for the present work. H. M. was supported under DOE Contract No. DE-SC0012704 and European Research Council, Grant ERC-2015-CoG-681707, and wishes to thank the Nuclear Theory Group at BNL for hospitality during the preparation of this manuscript. P. Z. acknowledges the support by the U.S. Department of Energy under contract number No. DE-SC0012704.
|
{
"timestamp": "2018-08-07T02:17:56",
"yymm": "1804",
"arxiv_id": "1804.05311",
"language": "en",
"url": "https://arxiv.org/abs/1804.05311"
}
|
\section{Introduction}
Exchange spring magnets provide an interesting approach to enable synthesis of rare-earth free permanent magnets with comparable magnetic properties to commerialized rare-earth based magnets. The price instability of rare-earth resources has made the applications of Nd and Sm based magnets economically critical. There is an urgent need to develop permanent magnets with reduced rare-earth contents or to explore rare-earth free permanent magnetic materials where Mn-based intermetallics are considered to be potential candidates \cite{Poudyal:13, Kramer:12,DLi:16}. In order to be qualified as replacements for rare-earth magnets, it is required that new candidates have high magnetic anisotropy, high energy product, and high temperature stability \cite{Gutfleisch:11}.
The low temperature phase (LTP) of MnBi is one of the materials with a particularly high intrinsic magnetic anisotropy in the order of $10^7$\,erg/cm$^3$ as well as a large coercivity (about 1.6\,T), which rather uniquely shows a positive temperature coefficient \cite{Sabet:17,Park:14,Suzuki:15}.
Moreover, the relatively high Curie-temperature of 630\,K also makes MnBi an interesting candidate for high temperature applications \cite{Guillaud:51a}.
However, in spite of such extraordinary magnetic properties, the main drawback of MnBi for permanent magnet application
is its comparably low saturation magnetization of 710\,emu/cm$^3$ (0.71\,MA/m) limiting the maximum achievable energy product \cite{Park:14}.
As suggested back in 1991 by Kneller and Hawig, one way to overcome this barrier and further improve the energy product is through the synthesis of exchange spring magnets with coupled hard/soft magnetic phases (see schematics in Fig. \ref{EX}) \cite{Kneller:91,Fullerton:98,Fullerton:99,Leineweber:97,Coey:93,Skomski:94,Lewis:94}.
Such composite magnets, $e.g.$ coupled bilayers of MnBi in combination with FeCo as the soft phase, will possess a much higher saturation magnetization and thus an increased overall energy product.
\begin{figure}[!th]
\centering
\includegraphics[width=0.49\textwidth]{ex1.jpg}
\caption{ Schematic of typical hysteresis loops for a hard magnetic phase (e.g. MnBi, blue solid line), a soft magnetic phase (e.g. FeCo, red solid line), and the exchange spring composite magnet (e.g. MnBi/FeCo, green solid line) resulting from hard/soft exchange coupling. The four insets show a one-dimensional configuration of magnetic moments at the hard/soft magnetic interface under varying external magnetic field ($H$) assuming an out-of-plane easy axis direction. Red open arrows represent the magnetic moments in the soft phase and blue filled arrows represent the magnetic moments in the hard phase. The length of the arrows represents the magnetization and the width of the arrows represents the coercivity.}
\label{EX}
\end{figure}
For exchange spring heterostructures with MnBi as hard magnetic phase, to our knowledge there have been only a few recent studies investigating the synthesis and magnetic properties of the resulting bilayers \cite{Li:14,Gao:16,Yan:16,Sabet:17a}.
Although theoretical calculations have proven the concept of exchange spring magnets to increase the overall magnetic properties, according to the few available studies the coupling between the MnBi and Fe$_{x}$Co$_{1-x}$ layers is incoherent for magnetic layers thicker than $\sim$ 4\,nm \cite{Li:14,Gao:16,Yan:16,Sabet:17a}. This is also evident from the small shoulder observed around zero field on the hysteresis loops measured in all above mentioned studies where the two layers do not behave as a single magnetic phase. It is important to understand the interfacial effects responsible for an incoherent interlayer exchange coupling in order to make further advances to effectiveness of exchange spring magnets.
Based on the model suggested by Kneller \cite{Kneller:91}, there is a critical thickness (volume) of soft magnetic phase which is limited by the domain wall width (or exchange length) of the hard magnetic phase \cite{Fullerton:99, Leineweber:97}.
For thicker soft magnetic layers, the coupling between hard and soft magnets begins to deteriorate and hence the layers will switch independently during the magnetic reversal process.
However, if the thickness of the soft magnetic layer is less than twice of the domain wall width of the hard magnetic phase, the bilayer is expected to behave as a single hard phase with increased magnetization in which both soft and hard phases switch coherently during magnetic reversal under opposing field ($H<0$).
Experimentally, even for sufficiently thin soft magnetic layers incomplete exchange coupling is reported indicating that other factors are involved.
Beside the thickness of the soft and hard magnetic layers, structural factors such as degree of crystallinity and growth orientation are expected to affect the strength of exchange coupling. For instance, the lattice mismatch at the interface can act either in favor or against magnetic coupling. The hard/soft interface roughness resulting from the growth quality of the layers can influence the coupling between the layers as well. In addition, the effect of composition of the Fe$_{x}$Co$_{1-x}$ soft magnetic layer has been considered as a controlling factor affecting the interlayer exchange coupling. Based on their calculations, Gao {\it et al.} have also argued that the formation of a Co-rich Fe$_{x}$Co$_{1-x}$ layer at the interface with MnBi is beneficial for exchange coupling where according to their experimental data the strongest coupling occurs in MnBi/Co bilayers with an optimum Co thickness of $\sim$ 3\,nm \cite{Gao:16}.
In this work we combined theoretical and experimental methods to study the exchange coupling behavior in the MnBi/FeCo bilayer system, focusing on the structural factors including the effect of degree of crystallinity, interface roughness and composition of the soft magnetic phase to identify which of these factors control the strength of exchange spring effect in this system.
\section{Experimental procedure}
Exchange coupled bilayers of MnBi/FeCo have been deposited onto quartz glass substrates in a DC magnetron sputtering system with a base pressure of $\sim$ 4.0$\times10^{-6}$\,Pa and were capped with 4\,nm thick aluminium layer to protect them against oxidation. First, a MnBi layer with a typical thickness of 40\,nm was deposited from a Mn-Bi alloy target with a composition of Mn$_{55}$Bi$_{45}$ (at.\%) at room temperature. The optimized growth parameters in our setup were 0.7\,Pa Ar gas pressure at 20\,W sputtering power with a substrate to source distance of 15\,cm leading to a growth rate of 0.04\,nm/s. The MnBi film was subsequently annealed for 1\,hr (dwelling time) {\em in situ} under vacuum ($\sim 1.0\times10^{-5}$\,Pa) at the annealing temperature of $T_{\text{ann}}= 365$\,\degree C. The temperature was ramped up and down with a rate of 20\,\degree C/min and 10\,\degree C/min, respectively. After cooling to 100\degree C, the soft magnetic layers using either a Fe-rich or a Co-rich, Fe$_{x}$Co$_{1-x}$ ($x=0.35$ or $0.65$) alloy target with various thicknesses of 1\,nm-3\,nm were deposited on top of the MnBi layer. The growth parameters for FeCo deposition were 2.5\,Pa Ar gas pressure, 80\,W sputtering power and 8\,cm substrate to source distance leading to a deposition rate of 0.008\,nm/s. Then the substrate temperature was increased to 120\degree C for deposition of an aluminium capping layer. The capping layer was deposited under 3\,Pa Ar gas pressure at 20\,W sputtering power with 15\,cm substrate to source distance at a rate of $\sim$0.008\,nm/s. The phase composition and degree of texture for the MnBi layer were determined by X-ray diffraction with Cu-$K_{\alpha}$ radiation using a Rigaku SmartLab thin film diffractometer. The film thickness was determined by a Bruker Dektak-XT stylus surface profiling system. The magnetic properties were measured by a SQUID magnetometer (MPMS, QuantumDesign).
For cross-sectional High Resolution Transmission Electron Microscopy (HR-TEM) investigations, TEM lamella was prepared by Focused Ion Beam (FIB) using a FEI Strata 400S equipped with an OmniProbe 200 micromanipulator for $in$-$situ$ lift-out. TEM sample preparation was initially performed at 30\,kV with an ion beam current of 16\,nA, followed by cleaning with a 6.5\,nA ion beam current. The final thinning step of the area of interest at the interface was performed at a low voltage-low current regime starting from 8\, to 2\,kV with ion beam current ranging from 56\,pA to 3\,pA. An aberration (image) corrected FEI Titan 80-300 operating at 300\,kV acceleration voltage and equipped with a US1000 slow-scan CCD camera (Gatan Inc.), a high-angle annular dark-field (HAADF) detector (Fischione), and an S-UTW EDX detector (EDAX Inc.) were used to evaluate the crystallinity, interface quality and composition.
\section{Theoretical procedure}
\subsection{Density functional theory}
Density functional theory (DFT) calculations were performed using the projected augmented wave method as implemented in the
\texttt{VASP} code~\cite{vasp}.
The exchange correlation functionals were parameterized using the generalized gradient approximation (GGA) as in Ref.~\cite{PBE}. The effect of the GGA+U approximation, which is important to understand the bulk anisotropy of MnBi \cite{pbe+u} was investigated and no significant influence on the interface properties was observed.
Two MnBi/FeCo models were considered in our calculations, namely MnBi(001)/FeCo(110) and MnBi(001)/FeCo(111), each in both crystalline and amorphous states.
The amorphous structures were generated using {\it ab initio}-based molecular dynamics (AIMD) calculations implemented in \texttt{VASP} after 10\,ps run for the FeCo(111) and 2\,ps run for the FeCo (110) surfaces, respectively, both at 500$^{\circ}$C.
All models were constructed as symmetric and non-stoichiometric slabs for their interface formation energies would be comparable.
Moreover, Fe$_x$Co$_{(1-x)}$ layers with two different stoichiometries, {\it i.e.} Fe$_3$Co$_5$ and Fe$_5$Co$_3$, were considered in order to study the influence of chemical composition on the interfacial properties.
The thickness of the MnBi layer in MnBi(001)/FeCo(110) and MnBi(001)/FeCo(111) interfaces was 10 \AA ~and 15 \AA, respectively.
During the calculations, the first two layers of MnBi were kept fixed while the rest of the MnBi and the whole FeCo layers were fully relaxed. At least 14 \AA~vacuum was considered when constructing the supercells using slab models to minimize the interaction between periodic images.
Interface formation energy $\gamma_{\rm int}$, which is a measure of the stability of the corresponding interface, was calculated using the following equation
\begin{equation} \label{eq1}
\gamma_{\rm int}=\frac{1}{2S}\bigg[E_{\rm int}-n_1E^{\rm MnBi}_{\rm bulk}-n_2E^{\rm FeCo}_{\rm bulk}+\sum_i\mu_i\bigg]
\end{equation}
where $S$ and $E_{\rm int}$ are the area and the total energy of the whole interface, $E^{\rm MnBi}_{\rm bulk}$ and $E^{\rm FeCo}_{\rm bulk}$ are the total energies per formula unit of MnBi and FeCo, and $n_1$ and $n_2$ are the number of bulk units of MnBi and FeCo in the models, respectively. $\mu_i$ is the chemical potential of any missing atoms summation of which maintains the stoichiometry. The chemical potentials were considered as the total energies per atom in metallic bulks.
The interface exchange coupling energy $J^\text{int}$ was obtained using the following relation
\begin{equation} \label{eq2}
J^\text{int}=(E^\text{APMA}-E^\text{PMA})/S
\end{equation}
where $E^\text{APMA}$ and $E^\text{PMA}$ are the
DFT total energies for antiparallel magnetization alignment (APMA) and parallel magnetization alignment (PMA), respectively.
Based on the optimized lattice parameters using DFT calculations,
the lattice mismatch between the MnBi(001) and FeCo(110) is 4.8\% with a 10.5$^{\circ}$ angular misfit while the lattice mismatch between MnBi(001) and FeCo(111) is 7.1\% with zero angular misfit.
Our MnBi(001)/FeCo(111) model consisted of a 2$\times$2 supercell of MnBi(001) and a 1$\times$1 supercell of FeCo(111) with 96 atoms in total.
In our MnBi(001)/FeCo(110) model, we used a 6$\times$6 supercell of MnBi(001) together with a 5$\times$5 supercell of FeCo(110) with 752 atoms in total to achieve the 4.8\% lattice misfit.
For the calculation of MnBi(001)/FeCo(111) interface, 3$\times$3$\times$1 $k$-point mesh was used and the calculation of MnBi(001)/FeCo(110) was performed with gamma-point. The energy cutoff for all the calculations was 360 eV. The convergence tests of the energy cutoff and $k$-point mesh with respect to the magnetic moments of the elements in their bulk states and energy per atom were conducted.
\subsection{Micromagnetic simulation}
Using the results from DFT calculations as input, micromagnetic simulations were performed within a simplified model to investigate the mechanism of exchange coupling in MnBi/FeCo magnets by using the 3D NIST OOMMF (Object Oriented Micromagnetic Framework) code \cite{oommf}. In our model, the thicknesses of the hard MnBi and soft FeCo layers (initially) were set as 40\,nm and 2\,nm, respectively. The lateral size is chosen as $8\times 8$ nm$^2$ and an in-plane periodic boundary condition was applied. The model was discretized by 0.4\,nm$\times$0.4\,nm$\times$0.1\,nm cuboid cells. Magnetic reversal curves were calculated by setting the initial magnetization along a positive $z$ axis and changing the external magnetic field along $z$ axis from 2.5\,T to $-2.5$\,T.
The exchange stiffness $A^\text{int}$ which characterizes the exchange coupling between MnBi and FeCo layers through the interface was used. According to the method of calculating exchange energy in OOMMF code \cite{oommf}, interface exchange stiffness $A^\text{int}$ was estimated by the following expression
\begin{equation}
A^{\rm int} \cong I^\text{vol}\Delta z^2/2
\end{equation}
in which $\Delta z=0.1$ nm is the cell size in the $z$ direction and $I^\text{vol}$ is the equivalent volumetric energy density calculated by $J^\text{int}$ divided by the average interface distance measured from the crystal structures after relaxation from our DFT calculations.
The resulting $\gamma^{\rm int}$, $J^{\rm int}$ and $I^{\rm int}$ values are shown in Table \ref{t1}.
The bulk parameters for exchange stiffness $A$ and uniaxial anisotropy constant $K$ were set as: $A^\text{FeCo}=10$ pJ/m, $A^\text{MnBi}=8$ pJ/m, $K^\text{FeCo}=0$ MJ/m$^3$, and $K^\text{MnBi}=1.86$ MJ/m$^3$~\cite{Rana:16,Sabet:17}.
The saturation magnetizations, $M_\text{s}^\text{FeCo}$ and $M_\text{s}^\text{MnBi}$, were also obtained from the DFT results.
\section{Results and discussion}
\begin{figure}[!th]
\centering
\includegraphics[width=0.62\textwidth]{XRD-MB-FC.pdf}
\caption{ The XRD patterns from exchange spring bilayers of Mn$_{55}$Bi$_{45}$/Fe$_{35}$Co$_{65}$ (at.\%) with different thickness of FeCo soft magnetic layer between 1\,nm-3\,nm. The LTP-MnBi thin films were annealed at $T_{\rm ann}$=365\,\degree C followed by deposition of FeCo layer at a substrate temperature of $T_{\rm sub}$=100\,\degree C. The spectra have vertical offset for clarity. The peaks originating from residual bismuth in the films or Al capping layer are labelled with (*) and (+), respectively.}
\label{XRD-MB-FC}
\end{figure}
The XRD patterns collected from different Mn$_{55}$Bi$_{45}$/Fe$_{35}$Co$_{65}$ (at.\%) exchange spring bilayers with various thicknesses of Co-rich soft magnetic FeCo layer are shown in Fig.~\ref{XRD-MB-FC}. The peak indexing shows hexagonal MnBi (002) and (004) peaks in agreement with space group of $P$63/$mmc$ along with some small traces of residual bismuth resulting from annealing of the MnBi films at $T_{\rm ann}=360$\,\degree C. Fig.~\ref{XRD-MB-FC} clearly demonstrates the formation of LTP MnBi with strong $c$-axis texture. As expected, because of the very low thicknesses, no peaks are observed for the FeCo layer. Comparing the intensities of the MnBi (002) and (004) peaks in bilayers samples to that of the single layer MnBi thin film, all the XRD patterns show similar peak intensities implying that the MnBi hard magnetic layer in all the bilayer samples had the same high crystalline quality.
\begin{figure}[!th]
\centering
\includegraphics[width=0.55\textwidth]{MB-FC-M-H.pdf}
\includegraphics[width=0.55\textwidth]{MB-CF-M-H.pdf}
\caption{Out-of-plane magnetization data for MnBi/FeCo bilayers with different FeCo thicknesses from 0\,nm to 3\,nm measured at 300\,K, (a) with a Fe-rich and (b) with a Co-rich soft magnetic FeCo layer. The dashed line in (b) shows the in-plane magnetization for a single MnBi layer. }
\label{MB-FC-M-H}
\end{figure}
Room-temperature out-of-plane hysteresis loops for MnBi/FeCo bilayer samples with various thicknesses and two compositions of the soft magnetic layer are shown in Fig.~\ref{MB-FC-M-H}-a and -b. For comparison, the out-of-plane hysteresis loop for a single layer MnBi thin film sample is also included in the same graph. As expected, by addition of 1\,nm, 2\,nm and 3\,nm FeCo layers for both Fe-rich and Co-rich compositions, the saturation magnetization of exchange spring bilayer increased.
According to the graphs in Fig.~\ref{MB-FC-M-H}, the addition of Fe-rich soft magnetic FeCo layers improved the saturation magnetization more than the addition of Co-rich FeCo layers, since the Fe$_{65}$Co$_{35}$ (at.\%) phase has a $\sim$ 20\% larger saturation magnetization than the Fe$_{35}$Co$_{65}$ (at.\%) phase \cite{Kuhrt:93}. The total magnetization in a bilayer is given by the volume average of magnetization in the hard and soft magnetic layer \cite{Fullerton:99, Skomski:94}. The deposition of a 1\,nm and 2\,nm thick soft magnetic layer on top of MnBi retains the coercivity of the LTP-MnBi layer (about 15\,kOe, even with a slight increase), while regardless of the composition of the soft magnetic layer the addition of 3\,nm FeCo decreases the coercivity down to 12\,kOe.
The exchange coupling effect between the hard and soft magnetic layers can be considered complete when the bilayer sample shows a magnetically single phase behavior.
The small shoulder which was observed during the demagnetization process around zero field in the measured out-of-plane hysteresis curves of the double layers indicates that the exchange coupling between the layers is incoherent. As a descriptor to quantify the change in the degree of exchange coupling, the slope ($\frac{\Delta M}{\Delta H}$) of the hysteresis loop around zero-field crossing has been evaluated.
This will be explained in the following with more details using the micromagnetic simulations. As this slope increases with growing a thicker soft magnetic layer, it implies that the bilayers behave more as two separate magnetic layers instead of one single magnetic phase.
Although such decrease in degree of coupling is predicted with increasing thickness of the soft magnetic layer, it is also expected that the critical soft layer thickness, above which the exchange coupling begins to deteriorate, is roughly twice the domain wall width of the hard magnetic layer ($2\times \delta_{h}\simeq 2\times\pi\sqrt{\frac{A_{h}}{K_{h}}})$ \cite{Kneller:91, Fullerton:99,Leineweber:97} in which $\delta_{h}$ is domain wall width, $A_h$ is exchange stiffness constant and $K_h$ is magnetocrystalline anisotropy for the hard magnetic phase.
For a MnBi-FeCo bilayer with $A_h$ and $K_h$ equal to $\sim 1.0\times10^{-6}$\,erg/cm \cite{Rana:16} and $\sim 1.86\times10^{7}$\,erg/cm$^{3}$ \cite{Sabet:17} respectively, the critical thickness is predicted to be as high as $\sim$15\,nm. It becomes obvious that one needs to take into account a more detailed interface description to explain the experimental observations.
The observed incoherent coupling in the MnBi/FeCo exchange spring system can be attributed to different structural factors including: (i) a non-epitaxial hard magnetic layer which most likely results in subsequent growth of a polycrystalline or disordered soft magnetic layers on top, (ii) high interface roughness which could also be a side effect of non-epitaxial growth of the layers, caused by lattice mismatch at the interface, and (iii) composition gradients in the FeCo layer in the vicinity of the interface. In addition, according to Fig. \ref{MB-FC-M-H}, there exists a finite in-plane component of total magnetization in the MnBi hard magnetic layer. The in-plane components of the magnetization when incompletely or not coupled lead to a kink at the coercive field of the soft magnetic layer($H$ close to zero).
To examine the interface between MnBi and FeCo, cross-sectional HR-TEM and STEM investigations have been performed on a MnBi/FeCo bilayer sample. The degree of crystallinity was evaluated for the layers by capturing HR-TEM images from cross-section of the layers. Moreover, the distribution of different elements in each layer was examined in STEM mode.
Fig.~\ref{MB-FC-TEM}-a shows a cross-sectional HR-TEM image of the layers along with Fast Fourier Transform (FFTs) collected from each layer in a bilayer sample with a Co-rich soft layer. The HR-TEM image and the sharp diffraction spots in FFTs collected from the MnBi layer confirm the high crystallinity with out-of-plane orientation. The Co-rich FeCo layer, in contrary, shows polycrystalline structure.
Three different surface areas have been analyzed in the FeCo layer with various crystallinity. Only the examined area in the middle FFT shows high crystallinity and the two other investigated regions are disordered.
The reflections in the middle FFT pattern (FFT-A2 in Fig.~\ref{MB-FC-TEM}-a) of the crystalline region in FeCo layer can be indexed as (110) lattice plane.
\begin{figure}[!ht]
\centering
\includegraphics[width=0.49\textwidth]{MB-FC-TEM-a.png}
\includegraphics[width=0.30\textwidth]{MB-FC-TEM-b.png}
\caption{(a) Cross-sectional High Resolution Transmission Electron Microscopy (HR-TEM) image from a MnBi/FeCo bilayer sample( $c$-axis textured MnBi hard magnetic layer with a thickness of $\sim$ 50\,nm and polycrystalline Co-rich FeCo soft magnetic layer with a thickness of $\sim$ 5\,nm), (b) STEM image from cross section of the layers along with EDX elemental map from Mn, Bi, Fe, Co, Al, Pt and O across the layers.}
\label{MB-FC-TEM}
\end{figure}
As it can be seen in the cross-section HR-TEM image of the MnBi/FeCo bilayer, a few atomic layers of FeCo layer grown on MnBi are highly disordered. This was expected since FeCo and the (001) textured MnBi layer have different crystal structures, i.e. hexagonal structure in MnBi and bcc structure in FeCo, which results in the growth of polycrystalline FeCo layer because of the induced lattice misfit. The imperfection of crystallinity and the existence of grain boundaries in the FeCo layer also leads to the formation of a rough interface.
To check the elemental distribution in the bilayer sample, an EDX mapping was performed on the enclosed area in Fig.~\ref{MB-FC-TEM}b. The result of the EDX mapping is consistent with the phases present in each layer. Close to the interface between the two layers the Bi concentration starts to decrease earlier than the Mn concentration.
According to the quantitative EDX analysis from this specific area on the cross-section of the bilayer sample, the MnBi layer shows a stoichiometry of Mn:Bi$\sim$ 1.4 which is slightly higher than the starting stoichiometry of 1.2 in the alloy sputtering target. This value corresponds to a final stoichiometry of Mn$_{58}$Bi$_{42}$ (at.\%) which is slightly richer in Mn. The measured stoichiometry for the FeCo layer shows a Co:Fe ratio of $\sim$ 1.84 which is consistent with the starting stoichiometry of 1.86 in the alloy sputtering target. This confirms a fairly precise stoichiometry transfer from MnBi and FeCo alloy sputtering targets during film deposition.
In order to shed light on the possible mechanism which affects the performance of the MnBi/FeCo exchange spring magnets, density functional theory (DFT) calculations and micromagnetic simulations were carried out, with a focus on the interface properties.
Fig.~\ref{str} shows the atomic structures of the most favorable configurations after the atomic relaxation
for MnBi(001)/crystalline FeCo(110) (after atomic relaxation MnBi(001)/disordered FeCo(110)) and MnBi(001)/amorphous FeCo(111) interfaces.
\begin{figure}[!th]
\centering
(a)\\
\includegraphics[width=0.45\textwidth]{str-110-3D-eps-converted-to.jpg}\\
\vspace*{0.5cm}
(b)\\
\includegraphics[width=0.45\textwidth]{str-111-3D-2-eps-converted-to.jpg}\\
\caption{Schematic of atomic structures of relaxed (a) MnBi(001)/crystalline FeCo(110) and (b) MnBi(001)/amorphous FeCo(111) interfaces demonstrated using VESTA~\cite{vesta}. Mn, Bi, Fe and Co are shown with small dark violet, large light violet, gold and blue colors, respectively. The 1$\times$1$\times$1 unit cell of each orientation is indicated with dashed line.}
\label{str}
\end{figure}
\begin{table*}[!tbh]
\centering
\caption{{Calculated values of interface formation energy $\gamma^\text{int}$, interface exchange coupling energy $J^\text{int}$ , exchange constant $A^\text{int}$ and lattice misfit (linear and angular) obtained from DFT calculations.
Disordered structure is reconstructed while amorphous one is completely irregular.}}
\begin{tabular}{ccccccc}
\hline\hline\\
Composition & Lattice misfit & Orientation & \makecell {Final phase \\ after relaxation } & \makecell {$\gamma^{\rm int}$ \\ (eV/\AA$^2$)} & \makecell{$J^\text{int}$ \\ (J/m$^2$)} & \makecell{$A^\text{int}$ \\ (pJ/m)} \\ \hline\\
Fe$_3$Co$_5$ & \multirow{2}{*}{7.1\%, 0$^{\circ}$} &(111) & crystalline & 0.137 & 0.112 & 4.6 \\
Fe$_3$Co$_5$ & & (111) & amorphous & 0.130 & 0.093 & 3.3 \\
\cline{1-7}\\
Fe$_3$Co$_5$ & \multirow{2}{*}{4.8\%, 10.5$^{\circ}$} &(110) & crystalline & 0.129 & 0.195 & 4.5 \\
Fe$_3$Co$_5$ & & (110) & disordered & 0.127 & 0.073 & 1.7 \\
\hline\hline\\
\end{tabular}
\label{t1}
\end{table*}
First, it is found that in the MnBi/FeCo bilayer system, the most favourable atomic configuration at the interface and 0 K temperature forms with Bi-termination MnBi and Co-termination FeCo which is obtained with a symmetric non-stoichiometric model. These findings are in agreement with the cohesive energies of these elements \cite{kittel}. However, our DFT calculations show that the interchange of one Mn atom from MnBi and one Fe atom from FeCo requires only 0.3 eV energy which is a rather low energy barrier and can show the possibility of Mn migration across the interface. The presence of Mn atoms close to the interface is also observed in the EDX elemental map (see Fig. \ref{MB-FC-TEM}-b).
Second, based on the results shown in Tab.~\ref{t1}, it is demonstrated that
the interface formation energies of the MnBi(001)/crystalline Fe$_3$Co$_5$(110) and MnBi(001)/disordered Fe$_3$Co$_5$(110) interfaces are almost the same and the formation energy of the former is lower than MnBi(001)/crystalline Fe$_3$Co$_5$(111) case. This can be related to the fact that MnBi(001)/ Fe$_3$Co$_5$(110) interface has lower lattice mismatch compared to MnBi(001)/ Fe$_3$Co$_5$(111) case.
It should also be noted that during the atomic relaxation of the MnBi(001)/ crystalline FeCo(110) interface, the FeCo layer undergoes an atomic reconstruction with a peculiar spiral fashion which has the minimum energy configuration. Moreover, the final ground state structure of the MnBi(001)/amorphous Fe$_3$Co$_5$(110) interface obtained from AIMD calculation is not stable and transforms into a more ordered (almost crystalline) state which is opposite to the case of the MnBi(001)/amorphous Fe$_3$Co$_5$(111) interface.
Third, the similar and low interface formation energies of the MnBi(001)/amorphous Fe$_3$Co$_5$(111), MnBi(001)/crystalline Fe$_3$Co$_5$(110) and MnBi(001)/disordered Fe$_3$Co$_5$(110) suggest the possible coexistence of the crystalline(110) and disordered structures at the interface region on FeCo side. Interestingly, both crytsalline FeCo(110) and disordered (randomly oriented) regions have been observed in our cross-sectional HR-TEM image (Fig. \ref{MB-FC-TEM}-a) at the FeCo side which is in agreement with the result of DFT calculations.
\begin{figure*}[!th]
\centering
\includegraphics[width=1.0\textwidth]{figYi.jpg}
\caption{Micromagnetic simulation results of a MnBi/Fe$_3$Co$_5$ model system with crystalline FeCo(110) and (111) interfaces. Magnetic reversal curves:(a) No interface roughness with the $A^\text{int}$ value listed in Table \ref{t1}; (b) Interface roughness with the same $A^\text{int}$ as in (a); (c) Interface roughness with $A^\text{int}$ reduced to 10$\%$ of that in (a). The external magnetic field $\mu_0H_\text{ex}$ is applied along $z$ direction. Inset of (a): Model geoemetry with in-plane periodic boundary condition. Inset of (b): Interfacial roughness of MnBi with a maximum dent height of 0.4 nm. a-i and a-ii, b-i and b-ii, and c-i and c-ii present the magnetic configurations ($yz$ surface at $x=0$) corresponding to the marked circles of reversal curves in (a), (b), and (c), respectively which belongs to the most favorable case, {\it i.e.} crystalline Fe$_3$Co$_5$(110).}
\label{figYi}
\end{figure*}
Considering the values of lattice misfit and interface formation energy in Tab. \ref{t1}, it is postulated that MnBi(001)/crystalline FeCo(111) interface is slightly less probable to form.
Moreover, crystalline Fe$_3$Co$_5$(110) phase has higher values of $J^\text{int}$ and $A^\text{int}$ compared to other configurations which favors a more coherent interfacial exchange coupling.
However, the coexistence of disordered phases with lower values of $J^\text{int}$ and $A^\text{int}$ has deteriorated the magnetic exchange coupling at the interface in our experimental measurements (see Fig. \ref{MB-FC-M-H}).
In order to examine the influence of interfacial properties on the exchange coupling behavior, micromagnetic simulations are performed. Since the microstructure of the experimental sample is very complicated and cannot be fully implemented into any simulations, here we concentrate on a rather simplified model based on single crystalline structures for evaluating the exchange behavior.
The employed model with an in-plane size 8\,nm$\times$8\,nm, 40\,nm thick MnBi, and 2\,nm thick Fe$_3$Co$_5$ is shown in the inset of Fig.~\ref{figYi}-a. In-plane periodic boundary conditions are applied. Apart from the interface exchange coupling energy the interface roughness as a critical factor, which can influence the interfacial exchange coupling behavior, is evaluated in our micromagnetic simulation analysis (see Fig.~\ref{figYi}).
Here we take the interfaces with disordered Fe$_3$Co$_5$(110) and crystalline Fe$_3$Co$_5$(111) orientations as the model systems for micormagnetic simulations to consider the phases which could be responsible for incoherent exchange coupling observed in magnetic measurements. The following cases are considered:\\
i) Perfect flat interface with the interface exchange stiffness $A^\text{int}_\text{(111)}=4.6$ pJ/m for Fe$_3$Co$_5$(111) orientation and $A^\text{int}_\text{(110)}=1.7$ pJ/m for Fe$_3$Co$_5$(110) orientation, as shown in Fig.~\ref{figYi}-a;\\
ii) Rough interface with a random distribution of dent height (maximum 0.4\,nm, inset of Fig. \ref{figYi}-b) in MnBi and the same values of $A^\text{int}$ in the case i), as shown in Fig.~\ref{figYi}-b;\\
iii) The same rough interface as in the case ii, but with reduced $A^\text{int}_\text{(111)}=0.46$ pJ/m and $A^\text{int}_\text{(110)}=0.17$ pJ/m, as shown in Fig.~\ref{figYi}-c.\\
The simulated magnetic reversal curves in Fig.~\ref{figYi} do not show the shoulder which was observed in the measured hysteresis loops of the experimental samples.
As mentioned above, this shoulder is due to the residual in-plane magnetization component of the hard magnetic phase which was not considered in the micromagnetic simulations but rather assuming a full out-of-plane magnetization vector. By increasing the thickness of the soft magnetic layer a reduced rectangularity was observed in the hysteresis loops. Using micromagnetic simulations we examined the magnetic configuration and its evolution around the interface at different external fields, as shown in the second and third rows of Fig.~\ref{figYi}. When the interface is assumed to be perfect and $A^\text{int}_\text{(110)}=1.7$ pJ/m from Tab.~\ref{t1} is used, the magnetization vectors near the interface in FeCo tend to rotate coherently with those in MnBi, as shown in Figs.~\ref{figYi}-a-i and a-ii. This indicates a rather strong interface exchange coupling. When a rough interface was assumed and $A^\text{int}_\text{(110)}$ remained the same, Figs.~\ref{figYi}-b-i and b-ii still suggest strong interface exchange coupling. However, the interface magnetization vectors are much easier to be reversed.
This can be verified by comparing the distribution of the $z$ component of magnetization ($\mu_0M_z$). For instance, at $\mu_0H_\text{ex}=0.5$ T, the model with rough interface showed a minimum $\mu_0M_z$ ($\mu_0M_z^\text{min}$) of 0.68\,T around the interface (Fig.~\ref{figYi}-b-i), but the model without roughness showed a little higher $\mu_0M_z^\text{min}$ (Fig.~\ref{figYi}-a-i).
The premature reversal in Fig.~\ref{figYi}-b-i and b-ii could be attributed to the local higher demagnetization field induced by the sharp corners or irregularities in the rough interface \cite{oommfyi1,oommfyi2}. Accordingly, the simulated coercivity in Fig.~\ref{figYi}-b was also slightly smaller than that of Fig.~\ref{figYi}-a.
When the interface roughness was assumed to reduce $A^\text{int}_\text{(110)}$ to 0.17\,pJ/m, the magnetic reversal curve were a simple straight line, as shown in Fig.~\ref{figYi}-c.
From the magnetic configurations in Fig.~\ref{figYi}-c-i and c-ii, it can also be found that the magnetization vectors around the interface cross each other and the magnetization in FeCo almost rotates freely, indicating a very poor interface exchange coupling. From Fig.~\ref{figYi} we realize that the interface exchange coupling strength evaluated from DFT calculations of smooth interfaces provides useful insight into the atomic or compositional design of the MnBi/FeCo system. The micromagnetic modeling reveals in addition that the interface roughness and irregular occurrence of defects are also important parameters since it can induce locally premature reversal and, as a consequence, deteriorates the interface exchange coupling.
\begin{figure}[!th]
\centering
\includegraphics[width=0.41\textwidth]{hys-no-rough-new.jpg}
\includegraphics[width=0.41\textwidth]{hys-rough-new-2.jpg}
\vspace*{0.2cm}
\includegraphics[width=0.41\textwidth]{origin-MB-FC-final.jpg}
\caption{Hysteresis plots obtained from micromagnetic simulations for MnBi(001)/FeCo(110) double layers (a) without and (b) with interface roughness. (c) Variation of the magnetization with respect to the applied field around zero field for the theoretical and experimental hysteresis plots as a function of FeCo thickness. For the case of interfaces without roughness two regions are evident in which at 1\,nm FeCo thickness incoherent coupling between the hard and soft magnetic layers appears. The rough interfaces, in both theoretical and experimental results, behave incoherently from the beginning. For comparison, the experimental data for the epitaxial case of MnGa(001)/FeCo(001) bilayer are also presented.
}
\label{origin}
\end{figure}
Fig.~\ref{origin} summarizes the results of thickness analysis based on experimental measurements and theoretical modeling of MnBi/FeCo interface. In Fig.~\ref{origin}-a and -b,
hysteresis plots are shown for the two interfaces, namely without and with interface roughness corresponding to the information provided in Fig.~\ref{figYi} for the case of MnBi(001)-disordered Fe$_3$Co$_5$(110).
For each case in Fig.~\ref{origin} the hysteresis is plotted for different thicknesses of the FeCo layer. In the case of a 0.5\,nm thick FeCo layer, it is evident that for the interface without roughness the hysteresis loop is more rectangular (the hysteresis slope at zero-field crossing is close to zero) showing a more coherent coupling compared to the same thickness of FeCo with a rough interface. In order to quantitatively distinguish the changes in exchange coupling considering the effect of interface roughness and increasing the soft layer thickness, the first derivative of the corresponding hysteresis loops has been calculated. The slope around zero-field crossing showing the variation
of magnetization with respect to the applied field ($\frac{\Delta M}{\Delta H}$) for each hysteresis plots as a function of different FeCo thicknesses are shown in Fig.~\ref{origin}-c.
It should be noted that for the cases of experimental data, the shoulder observed at zero field has been excluded from the derivative plots in order to keep the consistency of the graphs in comparison to the simulation data.
From Fig.~\ref{origin}-c, it can be seen that for structures without interface roughness (blue circle points) two regions are observable. The first region is present up to 1\,nm of FeCo thickness and the second one starts above 1\,nm. It can be seen that for the sample with less than 1\,nm FeCo thickness without roughness, the hard and soft layers are coherently exchanged coupled since the first derivatives are close to zero. However, considering a rough interface (red triangles with continuous line), exchange coupling is incoherent regardless of the soft layer thickness as the slope is continuously increasing.
Using the same method, the first derivatives of our experimental hysteresis loops corresponding to Fig.~\ref{MB-FC-M-H}-b are plotted as a function of soft layer thicknesses in Fig.~\ref{origin}-c (green triangle with dashed line). In addition, the first derivatives of our experimental magnetization data for epitaxial MnGa(001)/FeCo(001) bilayers have also been included in Fig.~\ref{origin}-c (orange square with dashed line). A complete theoretical and experimental study
on the MnGa/FeCo exchange spring system is currently under preparation. As can be seen from the plots in Fig.~\ref{origin}-c, our theoretical and experimental findings for the case of MnBi/FeCo bilayer are in agreement and show that the effect of interface roughness on the incoherency of exchange coupling is significant. In addition, it can be concluded that the effect of the lattice misfit between the hard and soft layers is decisive since even in the case of the interfaces without roughness (blue solid line) using a single crystalline model, the coherent coupling is only observed below 1 nm of FeCo thickness.
These are important findings which provide a better understanding of exchange coupling and go beyond existing knowledge on exchange spring systems.
Comparing the trend of derivative plots for MnBi/FeCo and MnGa/FeCo bilayer systems, it can be seen that as MnGa/FeCo bilayers show much decreased interface roughness due to epitaxial growth, their magnetic data is used here to confirm the micromagnetic modeling approach. The shoulder observed in the slope at zero crossing indicates the transition from coherent to incoherent exchange coupling. While in case of MnBi/FeCo system due to the thin film growth properties only incoherent bilayers were obtained, nevertheless the corresponding graph shows an increasing slope as a function of soft layer thickness which is in agreement with the modeling. In contrast, in the case of MnGa/FeCo epitaxial bilayer a coherent exchange coupling can be obtained up to 2\,nm. Our study shows that not only interface roughness is limiting the interfacial exchange coupling but also
a reduced lattice misfit at interface will greatly improve the coupling behavior.
As a result, finding a suitable single crystal substrate with a small lattice misfit to enable growth of an epitaxial MnBi layer could be one way to improve the exchange coupling behavior in this system. Not only it will result in a better quality of the exchange interface but the total magnetic properties can also be improved by obtaining a higher degree of crystallinity in both hard and soft magnetic layers. Unfortunately, in case of MnBi it is hard to find such single crystalline substrate which matches the crystalline structure and lattice constants of the LTP-MnBi hexagonal phase which makes it very difficult to study the effect of epitaxial growth of MnBi thin films on the exchange coupling in MnBi/FeCo system. Preliminary results on the MnGa/FeCo system show how the combined experimental and theoretical approach described here is of great importance to improve synthesis and performance of future exchange spring material systems.
\section{Summary and conclusion}
In summary, exchange spring $\text{MnBi/Fe}_{x}\text{Co}_{1-x}$ ($x=0.65$ and $0.35$) bilayers with different soft magnetic layer thicknesses were fabricated by DC magnetron sputtering from alloy targets. The magnetic measurements revealed that a Co-rich FeCo soft magnetic layer results in more coherent exchange properties with an optimum soft layer thickness of $\sim$ 1\,nm leading to $\sim$ 3\% increase of the saturation magnetization, however, a complete single-phase hysteresis cannot be obtained for higher FeCo thickness.
A combined theoretical and experimental approach showed that in the MnBi(001)/FeCo system a partially incoherent interface with crystalline and disordered phases is both expected and observed which considerably limits the exchange coupling effect.
As the most important result, micromagnetic simulations showed that the thickness of the soft magnetic layer and the interface roughness between the hard and soft magnetic layers control the effectiveness of exchange coupling. The incomplete exchange coupling observed in MnBi/FeCo bilayers can be correlated with the high interfacial roughness (reducing the exchange constant). Other controlling structural factors include large lattice misfit and coexistence of crystalline and disordered phases in soft magnetic layer.
Our study suggests that a strong single phase exchange coupling can be extended to higher FeCo thicknesses only through epitaxial growth of both hard and soft magnetic layers with atomically smooth interfaces. Preliminary experimental results show that the MnGa/FeCo system could be a more suited exchange coupling material combination with a critical soft layer thickness of about 2\,nm.
\section{Acknowledgement}
The authors thank the LOEWE project RESPONSE funded by the Ministry of Higher Education, Research and the Arts (HMWK) of the state of Hessen, Germany. We also acknowledge the computer time given by the high performance computer of Hessen ``Lichtenberg'', and K.~Albe from Technische Universit\"at Darmstadt.
|
{
"timestamp": "2018-04-24T02:06:49",
"yymm": "1804",
"arxiv_id": "1804.05383",
"language": "en",
"url": "https://arxiv.org/abs/1804.05383"
}
|
\section{Introduction}
Let $A=(a_{ij})$ be an $n\times n$ Cartan matrix. By Kac\cite{Kac_68} and Moody\cite{Moody_68}, it is well known that there is a Kac-Moody Lie algebra $g(A)$ associated to $A$. In \cite{Kac_Peterson_83}\cite{Kac_Peterson_84}\cite{Kac_85} Kac and Peterson constructed the corresponding Kac-Moody group $G(A)$. In this paper for convenience we consider the derived Lie algebra $g'(A)$ and the associated simply connected group $G'(A)$. But we still use the symbols $g(A)$ and $G(A)$.
Cartan matrices are divided into three types, i.e. finite type, affine type and indefinite type. A Cartan matrix $A$ is indecomposable if $A$ can't be written as a direct sum of two Cartan matrices $A_1$ and $A_2$. $A$ is symmetrizable if there exists an invertible diagonal matrix $D$ and a symmetric matrix $B$ such that $A=DB$, see Kac\cite{Kac_82} for details. A Cartan matrix $A$ is generic if $a_{ij}a_{ji}\geq 4$ for all $i,j$. $A$ is generic if and only if all its principal sub-matrices of rank $2$ are not of finite type. All these properties for Cartan matrices can be used for the associated Kac-Moody Lie algebras and Kac-Moody groups. For example a generic Kac-Moody group is indecomposable.
In \cite{Zhao_Jin_15}, the authors determined the rational homotopy type of the indefinite Kac-Moody group $G(A)$. Since $G(A)$ has a multiplication, it is a Hopf space. It is important to determine the rational Hopf homotopy type of $G(A)$. This is equivalent to determine the rational cohomology Hopf algebra $H^*(G(A))$ or the dual rational homology Hopf algebra $H_*(G(A))$. It is further equivalent to determine the rational cohomology algebra $H^*(BG(A))$ of the classifying space $BG(A)$.
On the rational homotopy group $\pi_*(G(A))$, the Samelson product $[\ , \ ]:\pi_p(G(A))\times \pi_q(G(A))\to \pi_{p+q}(G(A))$ is defined as
$$[\alpha,\beta](s\wedge t)=\alpha(s)\beta(t)\alpha(s)^{-1}\beta(t)^{-1},s\in S^p,t\in S^q.$$
$(\pi_*(G(A)),[\ , \ ])$ is a rational graded Lie algebra.
Let $\chi: \pi_*(G(A))\to H_*(G(A))$ be the Hurewicz morphism of graded Lie algebras. By Milnor and Moore\cite{Milnor_Moore_65}, the induced morphism $\widetilde \chi: U(\pi_*(G(A)))\to H_*(G(A))$ is an isomorphism of Hopf algebras, where $U(\pi_*(G(A)))$ is the universal enveloping algebra of $\pi_*(G(A))$. And $H_*(G(A))$ is primitively generated by $\pi_*(G(A))$.
So to determine the Hopf algebra structure on $H_*(G(A))$, it is enough to compute the graded Lie algebra $\pi_*(G(A))$. By combining rational homotopy theory(see \cite{Sullivan_77}) with the cohomology ring $H^*(G(A))$(see \cite{Zhao_Jin_13}), one knows that for a generic Cartan matrix $A$, $\pi_{odd}(G(A))\cong \mathbb{Q}$ or $\{0\}$ depending on whether $A$ is symmetrizable or not. And the decomposition $\pi_*(G(A))=\pi_{even}(G(A))\oplus \pi_{odd}(G(A))$ is a decomposition of Lie algebras. $H_{even}(G(A))$(i.e. the rational Chow ring of $G(A)$) is isomorphic to the universal enveloping algebra $U(\pi_{even}(G(A)))
$. By \cite{Kac_85} the Poincar\'{e} series of $H_{even}(G(A))$ is $$C_A(q)=P_{F(A)}(q)(1-q^2)^n(1-q^4)^{-\epsilon(A)}.$$
Here $P_{F(A)}(q)$ is the Poincar\'{e} series of the flag manifolds $F(A)$ associate to Kac-Moody group $G(A)$ and $\epsilon(A)=\dim \pi_{odd}(G(A))$. By \cite{Zhao_Jin_13}, $\D{P_{F(A)}(q)=\frac{1+q^2}{1-(n-1)q^2}}$. Hence we can compute the Poincar\'{e} series of $H_{even}(G(A))$.
For a non-symmetrizable Cartan matrix $A$, we write the Poincar\'{e} series $\D{C_A(q)=\frac{1}{\frac{1-(n-1)q^2}{(1-q^2)^{n-1} (1-q^4)}}}$ as
$$\D{\frac{1}{1-a_4 q^4-a_4 q^6-\cdots-a_{2i} q^{2i}-\cdots}},$$ and for a symmetrizable Cartan matrix $A$, we write $\D{C_A(q)=\frac{1}{\frac{1-(n-1)q^2}{(1-q^2)^{n-1}}}}$ as $$\D\frac{1}{1-b_4 q^4-b_6 q^6-\cdots-b_{2i} q^{2i}-\cdots},$$
where for $i\geq 2$, $a_{2i}=\sum\limits_{k=0}^{[\frac{i}{2}]}(i-1-2k){{n+i-2k-3} \choose{n-3}}, b_{2i}= (i-1){{n+i-3}\choose{n-3}}$ are natural numbers depending on $n$.
These two Poincar\'{e} series are the same as the Poincar\'{e} series of the tensor Hopf algebra with $a_{2i}$ and $b_{2i}$ generators of degree $2i$ for each $i\geq 2$. In \cite{Zhao_Jin_15} we gave the following conjecture.
\noindent{\bf Conjecture: }For a non-symmetrizable(or symmetrizable) generic Kac-Moody group $G(A)$, the graded Lie algebra $\pi_{even}(G(A))$ is a free Lie algebra with $a_{2i}$(or $b_{2i}$) generators of degree $2i$ for each $i\geq 2$.
Since the universal enveloping algebra of an even graded free Lie algebra is a tensor algebra, if the conjecture is true, then $H_{even}(G(A))$ is a tensor Hopf algebra with $a_{2i}$(or $b_{2i}$) generators of degree $2i$ for each $i\geq 2$.
In this paper we compute the rational cohomology ring $H^*(BG(A))$ at first. This determines the rational homotopy type of $BG(A)$. Then we compute the graded Lie algebra $\pi_{*}(BG(A))$ with Whitehead product. Since the graded Lie algebra $\pi_{*}(G(A))$ with Samelson product is determined by $\pi_*(BG(A))$ with Whitehead product, we determine the graded Lie algebra $\pi_*(G(A))$ and prove the conjecture at last.
The main results in this paper are the following two theorems.
\noindent{\bf Theorem 1:}
If $A=(a_{ij})_{n\times n}$ is a non-symmetrizable generic Cartan matrix, $n\geq 3$,
then the Poincar\'{e} series of $BG(A)$ is $\D{P_n(q)=q\left [\frac{(n-1)q^2-1}{(1-q^2)^{n-1} (1-q^4)} +1\right ]+1}$.
\noindent{\bf Theorem 2:}
If $A=(a_{ij})_{n\times n}$ is a symmetrizable generic Cartan matrix, $n\geq 2$, then the Poincar\'{e} series of $BG(A)$ is $\D{Q_n(q)=\frac{1}{1-q^4}(q\left [ \frac{(n-1)q^2-1}{(1-q^2)^{n-1}} +1\right ]+1)}$.
The contents of this paper are as follows: in section 2 we give some preparing lemmas, in section 3 we prove Theorem 1 and 2, in section 4 we give some results derived from these theorems, including the above conjecture.
\section{Some preparing lemmas}
In the following all the Cartan matrices are assumed to be generic. All the homology and cohomology are of rational coefficients.
Let $S$ be the set of integers $1,2,\cdots,n$, and $\Pi=\{\alpha_1,\alpha_2,\cdots,\alpha_n\}$ be the simple root system of $G(A)$. For each $I\subset S$, the matrix $A_I=(a_{ij})_{i,j\in I}$ is also a Cartan matrix. Corresponding to $I\subset S$, there is a parabolic subgroup $G_I(A)$ of $G(A)$ whose simple root system is $\Pi_I=\{\alpha_i|i\in I\}$. All the proper subsets of $S$ form a category $\mathrm{C}$ with object $I\subset S$ and morphism $I\subset J$. By constructing classifying spaces we have a functor $F: \mathrm{C}\to Top$ which sends $I$ to $BG_I(A)$ and $I\subset J$ to map $BG_I(A)\to BG_J(A)$.
Since we only consider the homotopy type of the Kac-Moody group we replace the group $G(A)$(or $G_I(A)$) by its unitary form and use the same symbol.
We need the following lemmas to prove the main theorems.
\begin{lemma}
For a Kac-Moody group $G(A)$ and $I\subset S$, the subgroup $G_I(A)$ is isomorphic to $G(A_I)\widetilde \times T^{n-|I|}$, the semi-direct product of $G(A_I)$ and $T^{n-|I|}$. As a result there is an isomorphism $H^*(BG_I(A))\cong H^*(BG(A_I))\times H^*(BT^{n-|I|})$.
\end{lemma}
By this lemma, the Poincar\'{e} series of $BG_I(A)$ is obtained from the Poincar\'{e} series of $BG(A_I)$ by multiplying a factor $\D{\frac{1}{(1-q^2)^{n-|I|}}}$.
By Kichiloo\cite{Kitchloo_98}\cite{Broto_Kitchloo_02}, for a Cartan matrix of infinite type, the homotopy colimit of the functor $F$ gives the homotopy type of $BG(A)$. For any $I\in \mathrm{C}$, let $\mathrm{C}_I$ be the full subcategory of $\mathrm{C}$ whose objects are proper subsets of $I$. If $|I|\geq 2$, then $G_I(A)$
is of infinite type. By using the result of Kichiloo to $BG_I(A)$, we get $H^*(BG_I(A))\simeq \textrm{colimit} F|_{\mathrm{C}_I}$. As a consequence we have
\begin{lemma}
Let $\mathrm{C}'$ be the full subcategory of $\mathrm{C}$ which contains only objects $\emptyset,\{1\},\{2\},\cdots,\{n\}$, then for a generic Kac-Moody group $G(A)$,
$BG(A)\simeq \mathrm{colimit}F|_{\mathrm{C}'}$
$\ \ \ \ \ \ \ \ \ \ \simeq BG_1(A)\cup_{BT} BG_2(A)\cup_{BT}\cdots \cup_{BT} BG_n(A)$
$\ \ \ \ \ \ \ \ \ \ \simeq BG_{\{1,2,\cdots,n-1\}}(A)\cup_{BT} BG_n(A)$.
\end{lemma}
The action of Weyl group $W(A)$(or $W_I(A)$) of $G(A)$(or $G_I(A)$) on the maximal torus $T$ induces the action of $W(A)$(or $W_I(A)$) on $H^*(BT)$.
\begin{lemma}
For a generic Kac-Moody group $G(A)$, the image of the homomorphism $Bi_I^*: H^*(BG_I(A))\to H^*(BT)$ induced by the inclusion $i_I:T\subset G_I(A)$ is $H^*(BT)^{W_I(A)}$, i.e. the $W_I(A)$ invariants. In particularly the image of the homomorphism $H^*(BG(A))\to H^*(BT)$ is $H^*(BT)^{W(A)}$.
\end{lemma}
This lemma is the generalization of a result of Borel\cite{Borel_53_1} for compact Lie groups. It can be proved in the inductive procedure of the proofs for the main theorems. see \cite{Zhao_Gao_Ruan_18} for details.
\begin{lemma}
If $A$ is a non-symmetrizable generic Cartan matrix, then there exist $i,j,k\in S,i<j<k$ such that $A_{\{i,j,k\}}$ is non-symmetrizable.
\end{lemma}
\noindent {\bf Proof: }Suppose this lemma is not true for Cartan matrix $A$. Then for any $i<j<k$, $A_{\{i,j,k\}}$ is symmetrizable. Hence $a_{ij}a_{jk}a_{ki}=a_{ik}a_{kj}a_{ji}$. We set $d_{i}=\D{\frac{a_{1i}}{a_{i1}}}$. For $1<i<j$, we have $a_{1i}a_{ij}a_{j1}=a_{1j}a_{ji}a_{i1}$, hence $\D{\frac{a_{ij}}{a_{ji}}=\frac{d_j}{d_i}}$. But this means that $A$ is symmetrizable, a contradiction.
\begin{lemma}
Let $A$ be a generic Cartan matrix. If $A$ is symmetrizable, then $H^*(BT)^{W(A)}\cong \mathbb{Q}[\psi]$, where $\psi$ is the Killing form; If $A$ is non-symmetrizable, then $H^*(BT)^{W(A)}\cong\mathbb{Q}$.
\end{lemma}
This result was proved in Zhao-Jin\cite{Zhao_Jin_12}. In fact it is valid for an arbitrary indefinite and indecomposable Cartan matrix.
\begin{lemma}
Let $X=X_1\cup_{X_0} X_2$ be the push-out of the diagram $X_1 \stackrel{j_1} \leftarrow X_0\stackrel{j_2}\rightarrow X_2$. The homomorphism $j: H^*(X_1)\oplus H^*(X_2)\to H^*(X_0)$ is given by $j(u,v)=j_1^*(u)-j_2^*(v)$. If $X_1,X_2$ are deformation retracts of some open subspaces of $X$, then there exists a short exact sequence $$0\to \Sigma \mathrm{coker}j\to H^*(X)\to \ker j\to 0.$$
\end{lemma}
\noindent{\bf Proof: }We have the Mayer-Vietoris exact sequence
$$\cdots\to H^{*-1}(X_{1})\oplus H^{*-1}(X_{2})\stackrel{j}\to H^{*-1}(X_0)\stackrel{\delta}\to H^*(X)\stackrel{i}\to H^*(X_{1})\oplus H^*(X_{2})\stackrel{j}\to H^*(X_0)\to \cdots$$
From this sequence we get the short exact sequence $0\to \mathrm{im}\delta \to H^*(X)\to \mathrm{im}i\to 0$. By the exactness of this sequence, we have $\mathrm{im}i\cong \ker j$ and $\mathrm{im} \delta\cong H^{*-1}(X_0)/\ker \delta\cong H^{*-1}(X_0)/ \mathrm{im}j\cong \mathrm{coker} j$. This proves the lemma.
\begin{lemma}
Let $A$ be a generic $2\times 2$ Cartan matrix, then $H^*(BG(A))\cong \mathbb{Q}[\psi]$, where $\psi$ corresponds to Killing form which has degree 4. The Poincar\'{e} series of $BG(A)$ is $\D{\frac{1}{1-q^4}}$.
\end{lemma}
\noindent {\bf Proof: }By Lemma 2.6, we have the short exact sequence
$0\to \Sigma\mathrm{coker}j\to H^*(BG(A))\to \ker j\to 0$ with $j: H^*(BG_{\{1\}}(A))\oplus H^*(BG_{\{2\}}(A))\to H^*(BT)$.
The Poincar\'{e} series of $\textrm{coker}j$ is $\D{\frac{1}{(1-q^2)^2}-\frac{2}{(1-q^2)(1-q^4)}+\frac{1}{1-q^4}=0}$. Since $\ker j$ is isomorphic to $H^*(BT)^{W(A)}=\mathbb{Q}[\psi]$, its Poincar\'{e} series is $\D{\frac{1}{1-q^4}}$. Hence $H^*(BG(A))\cong \ker j\cong \mathbb{Q}[\psi]$.
This lemma is the special case of $n=2$ for Theorem 2.
\section{The proofs of the main theorems}
In this section we denote $BG_I(A)$ by $X_I$ for simplicity. For $I=\{i_1,i_2,\cdots,i_k\}$, we always denote $X_I$ by $X_{i_1i_2\cdots i_k}$. So we have $X_{\emptyset}=BT$ and $X_{12\cdots k} \simeq X_{12\cdots k-1}\cup_{ BT} X_k $.
\noindent{\bf Proof of Theorem 1:}
For $n=3$, $BG(A)$ is homotopic equivalent to $X_{12}\cup_{BT} X_{3}$. By Lemma 2.6 we have the short exact sequence
$0\to \Sigma \mathrm{coker}j\to H^*(BG(A))\to \ker j\to 0$. The homomorphism $j: H^*(X_{12})\oplus H^*(X_{3})\to H^*(BT)$ is given by $j(u,v)=Bi_{12}^*(u)-Bi_3^*(v)$, where $Bi^*_{12}$ and $Bi^*_3$ are induced by the homomorphisms $i_{12}: T\to G_{12}(A)$ and $i_3: T\to G_3(A)$. By Lemma 2.3, we observe that $\ker j$ is the sub-ring of Weyl group invariants. Since $A$ is non-symmetrizable, by Lemma 2.5, $\ker j\cong \mathbb{Q}$.
We have $\mathrm{im}j=\mathrm{im}(Bi^*_{12})+\mathrm{im}(Bi^*_3)$ and $\mathrm{im}(Bi^*_{12})\cap\mathrm{im}(Bi^*_3)=\ker j\cong\mathbb{Q}$. By Lemma 2.1, the Poincar\'{e} series of $\mathrm{im}(Bi^*_{12})$ and $\mathrm{im}(Bi_3)^*$ are $\D{\frac{1}{(1-q^2)(1-q^4)}}$ and $\D{\frac{1}{(1-q^2)^2(1-q^4)}}$ respectively. Combining these results, the Poincar\'{e} series of $\mathrm{coker} j$ is
$\D{\frac{1}{(1-q^2)^3}-\frac{1}{(1-q^2)(1-q^4)}-\frac{1}{(1-q^2)^2(1-q^4)}+1=\frac{2q^2-1}{(1-q^2)^2(1-q^4)}+1}$.
Hence for $n=3$ the Poincar\'{e} series of $H^*(BG(A))$ is $\D{q[\frac{2q^2-1}{(1-q^2)^2(1-q^4)}+1]+1}$.
For $n\geq 4$, we prove this theorem by induction on $n$. We assume the theorem is true for $n-1$. Since $A$ is non-symmetrizable, by Lemma 2.4, without loss of generality we can assume $A_{\{123\}}$ is non-symmetrizable.
Then $A'=A_{12\cdots n-1}$ is also non-symmetrizable. By Lemma 2.1, $H^*(X_{1,2\cdots n-1})\cong H^*(BG(A'))\otimes H^*(BS^1)$. By the induction assumption, the Poincar\'{e} series of $BG(A')$ is $\D{P_{n-1}(q)=q[\frac{(n-2)q^2-1}{(1-q^2)^{n-2}(1-q^4)}+1])+1}$. This means that the reduced cohomology $\widetilde H^*(BG(A'))$ concentrates in odd dimensions. Since $X\simeq X_{12\cdots n-1}\cup_{BT} X_n$, we use Lemma 2.6 to compute $H^*(BG(A))$. By the decomposition $H^*(X_{12\cdots n-1})\cong \widetilde H^*(BG(A'))\otimes H^*(BS^1)\oplus \mathbb{Q}\otimes H^*(BS^1)$, we have
$\ker j\cong \widetilde H^*(BG(A'))\otimes H^*(BS^1) \oplus \mathbb{Q}$. $\mathrm{im}j= \mathrm{im} Bi^*_{1,2\cdots n-1}+\mathrm{im} Bi^*_n$. And the intersection of $\mathrm{im} Bi^*_{1,2\cdots n-1}$ and $\mathrm{im} Bi^*_n $ is the sub-ring of Weyl group invariants. It is isomorphic to $\mathbb{Q}$.
The Poincar\'{e} series of $\mathrm{coker}j$ is $\D{\frac{1}{(1-q^2)^n}-\frac{1}{(1-q^2)^{n-1}(1-q^4)}-\frac{1}{1-q^4}+1}$. Therefore the Poincar\'{e} series of $BG(A)$ is
$\D{\frac{q}{(1-q^2)^n}-\frac{q}{(1-q^2)^{n-1}(1-q^4)}-\frac{q}{1-q^4}+q +\frac{q}{1-q^2}(P_{n-1}-1) +1}$
which is equal to $\D{q\left [\frac{(n-1)q^2-1}{(1-q^2)^{n-1} (1-q^4)} +1\right ]+1}$.
\noindent{\bf Proof of Theorem 2: }The proof of this theorem is similar to that of the Theorem 1. The difference is that for the symmetrizable case, the invariants of Weyl group is generated by the Killing form which is in degree 4.
We use induction on $n$. If $n=2$, by Lemma 2.7, the theorem is true. We assume this theorem is true for $n-1$. For an $n\times n$ symmetrizable generic Cartan matrix $A$, $A'=A_{12\cdots n-1}$ is a symmetrizable generic Cartan matrix. By induction assumption, the Poincar\'{e} series of $BG(A')$ is $\D{Q_{n-1}=\frac{1}{1-q^4}(q[\frac{(n-2)q^2-1}{(1-q^2)^{n-1}}+1]+1)}$. Since $BG(A)$ is homotopy equivalent to $X_{12\cdots n-1}\cup_{BT} X_n$. By a similar Mayer-Vietoris sequence computation, we get that the Poincar\'{e} series of $BG(A)$ is
\noindent $\D{Q_n=\frac{q}{(1-q^2)^n}-\frac{q}{(1-q^2)^{n-1}(1-q^4)}-\frac{q}{(1-q^2)(1-q^4)}+\frac{q}{(1-q^4)} +\frac{1}{(1-q^2)(1-q^4)}(Q_{n-1}-1)} $
\noindent $\D{+\frac{1}{1-q^4}=\frac{1}{1-q^4}\left [q(\frac{(n-1)q^2-1}{(1-q^2)^{n-1}} +1)+1\right ]}$.
\begin{remark}
In the proof of Theorem 2, we need the $H^*(BG(A))$-module structure on the Mayer-Vietoris sequences. In fact all the cohomology groups appear in the sequence are free $\mathbb{Q}[\psi]$-modules.
\end{remark}
\section{Some results derived from the main theorems}
In this section we need some general results in algebraic topology. For details see Whitehead\cite{Whitehead_78}.
From the expressions of the Poincar\'{e} series of $BG(A)$ in Theorem 1, we can see that for the non-symmetrizable case, the even dimensional cohomology group is $H^0(BG(A))$. Hence the cup product on $H^*(BG(A))$ is trivial and we have
\begin{coro}
For a generic $n\times n$ non-symmetrizable Cartan matrix $A$, the rational homotopy type of $BG(A)$ is $\bigvee\limits_{i=2}^\infty \bigvee\limits_{j=1}^{\alpha_{i}} S^{2i+1}$ with $P_n(q)=1 + \alpha_2 q^5+\alpha_3 q^7 +\cdots +\alpha_{i} q^{2i+1}+\cdots $.
\end{coro}
\begin{lemma}
For all $i\geq 2$, $\alpha_i=a_{2i}$.
\end{lemma}
\noindent{\bf Proof: }By definition we have
$$\D{\alpha_2 q^4+\alpha_3 q^6+\cdots=\frac{P_n(q)-1}{q}=\frac{(n-1)q^2-1}{(1-q^2)^{n-1} (1-q^4)} +1=1-\frac{1}{C_A(q)}=a_{4}q^4+a_6 q^6+\cdots+}$$
This proves the lemma.
By Milnor-Hilton theorem(see\cite{Whitehead_78}) and the homotopy equivalence $G(A)\simeq \Omega BG(A)$, we get
\begin{coro}
The homotopy Lie algebra $\pi_*(G(A))$ with Samelson product is the free graded Lie algebra generated by $\Sigma^{-1}\widetilde{H}_*(BG(A))$.
The Hopf algebra $H_*(G(A))$ is isomorphic to the tensor algebra $T(\Sigma^{-1} \widetilde{H}_*(BG(A)))$.
\end{coro}
From the expressions of the Poincar\'{e} series in Theorem 2, we can see that for the symmetrizable case as $\mathbb{Q}[\psi]$-module the only even dimensional generator of $H^*(BG(A))$ is $1\in H^0(BG(A))$. The cup product on $H^*(BG(A))$ can be determined by the following lemma.
\begin{lemma}
Let $R=R_0\oplus R_1$ be a $Z_2$ graded ring and $R_1$ be a free $R_0$ module, if all the elements in $R_1$ are nilpotent, then $R_1 R_1=0$.
\end{lemma}
\noindent{\bf Proof: }Let $b_1,b_2\not=0$ be two elements in $R_1$ and $b_1b_2=a\in R_0$. We show $a=0$. Since $b_2$ is nilpotent, there exists integer $k>0$ such that $b_2^{k-1}\not=0$ but $b_2^k=0$. Then we have $a b_2^{k-1} =b_1 b_2^k=0$. Hence we get $a=0$ from the fact that $R_1$ is a free $R_0$ module.
Set $R_0=H^{even}(BG(A))$ and $R_1=H^{odd}(BG(A))$, by this lemma, we get
\begin{coro}
For a generic $n\times n$ symmetrizable Cartan matrix $A$, the rational homotopy type of $BG(A)$ is $BS^3\times\bigvee\limits_{i=2}^\infty \bigvee\limits_{j=1}^{\beta_{i}} S^{2i+1}$ with $(1-q^4)Q_n=1 + \beta_2 q^5+\beta_3 q^7 +\cdots +\beta_{i} q^{2i+1}+\cdots $.
\end{coro}
Similarly we have
\begin{lemma}
For all $i\geq 2$, $\beta_i=b_{2i}$.
\end{lemma}
\begin{coro}
The homotopy Lie algebra $\pi_*(G(A))$ with Samelson product is the direct sum of $\pi_*(S^3)$ and the free graded Lie algebra generated by $\Sigma^{-1}\bar{H}_*(BG(A))$, where $\bar H_*(BG(A))\cong \widetilde H_*(\bigvee\limits_{i=2}^\infty \bigvee\limits_{j=1}^{\beta_{i}} S^{2i+1}) $.
The Hopf algebra $H_*(G(A))$ is isomorphic to the $\mathbb{Q}$-algebra $H_*(S^3)\times T(\Sigma^{-1} \bar{H}_*(BG(A)))$.
\end{coro}
Now we have the following result.
\begin{prop}
If $A_1,A_2$ are two generic Cartan matrices of size $n_1$ and $n_2$, then $G(A_1)$ and $G(A_2)$ are rational homotopy equivalent Hopf spaces if and only if $n_1=n_2$ and $\epsilon(A_1)=\epsilon(A_2)$.
\end{prop}
|
{
"timestamp": "2018-04-17T02:13:13",
"yymm": "1804",
"arxiv_id": "1804.05491",
"language": "en",
"url": "https://arxiv.org/abs/1804.05491"
}
|
\section{Introduction}
Cosmic masers act as probes into heavily obscured astrophysical sources providing observers with information about the dynamics and physics of a region. Shortly after the discovery of masers in 1963 \citep{wwdl65}, temporal variability of the 1665 and 1667\, MHz groundstate OH maser lines in the star-forming complex \ngc\ was reported by \citet{wdw68}. Some velocity channels displayed variations by an order of magnitude between 1965 July and 1966 February. Evidently, their observations represented the first reported maser flare though they referred to it only as `a new phenomenon in microwave spectroscopy'. In subsequent infrared continuum observations, \ngc\/ was identified as one of the most active regions of massive star-formation in the Galaxy \citep[see][and references therein]{hg83}. More recent higher resolution, multi-wavelength studies have shown that it contains many young, deeply-embedded clusters and protoclusters \citep{Hunter14,Feigelson2009,PT08}, numerous massive dense cores and filaments \citep{Tige2017,Fukui17,Andre16,Russeil13}, and appears to be undergoing a `mini-starburst' event \citep{willis13}.
\begin{table*}
\centering
\caption{Spectral transitions observed, monitoring start month, and receiver packages used at HartRAO.}
\label{tab:transitions}
\begin{tabular}{cccccccccc}
\hline
Mol. & Receiver & \multicolumn{2}{c}{Maser} &Beam & Band &\multicolumn{2}{c}{Velocity} &Sensitivity & Monitoring\\
& & Transition & Freq. & Width & Width$^{1}$ &Range &Resolution &3-$\sigma$ rms & Start\\
& (cm) & & (MHz) &($\arcmin$) &(MHz) &(\kms) &(\kms) &(Jy) & Month\\
\hline
OH & 18 & $^2\Pi_{J=3/2} ~F = 1 \rightarrow 2$ & 1612.231 & 29.6 & 0.25 & 22.5 & 0.045 & 0.4 & 2015 Sept \\
& 18 & $^2\Pi_{J=3/2} ~F = 1 \rightarrow 1$ & 1665.402 & 29.6 & 0.25$^{2}$ & 45.0 & 0.045 & $0.8 - 1.2$& 2011 Oct \\
& 18 & $^2\Pi_{J=3/2} ~F = 2 \rightarrow 2$ & 1667.359 & 29.6 & 0.25 & 22.5 & 0.044 & $0.4 - 0.8$& 2015 Sept \\
& 18 & $^2\Pi_{J=3/2} ~F = 2 \rightarrow 1$ & 1720.530 & 29.6 & 0.25 & 22.5 & 0.043 & 0.4 & 2015 Sept \\
& 6 & $^2\Pi_{J=1/2} ~F = 0 \rightarrow 1$ & 4660.242 & 9.6 & 1.0 & 33.0 & 0.063 & $0.2 - 0.4$& 2015 Sept \\
& 6 & $^2\Pi_{J=1/2} ~F = 1 \rightarrow 1$ & 4750.656 & 9.6 & 1.0 & 33.0 & 0.062 & 0.2 & 2015 Sept \\
& 6 & $^2\Pi_{J=1/2} ~F = 1 \rightarrow 0$ & 4765.562 & 9.6 & 1.0 & 33.0 &0.061 & $0.2 - 0.3$ & 2015 Sept\\
& 5 & $^2\Pi_{J=5/2} ~F = 2 \rightarrow 3$ & 6016.746 & 7.5 & 1.0 & 24.5 & 0.049 & 0.2 & 2015 Sept \\
& 5 & $^2\Pi_{J=5/2} ~F = 2 \rightarrow 2$ & 6030.747$^{3}$ & 7.5 & 1.0 & 24.5 & 0.049 & 0.4& 2015 Sept \\
& 5 & $^2\Pi_{J=5/2} ~F = 3 \rightarrow 3$ & 6035.092$^{3}$ & 7.5 & 1.0 & 24.5 & 0.049 & $0.3 - 0.4$& 2015 Aug \\
& 5 & $^2\Pi_{J=5/2} ~F = 3 \rightarrow 2$ & 6049.084 & 7.5 & 1.0 & 24.5 & 0.048 &$0.2 - 0.3$& 2015 Sept \\
\form & 6 & $J = 1_{11} \rightarrow 1_{10}$ & 4829.660$^{4}$ & 9.6 & 1.0 & 33.0 & 0.060 & 0.4& 2016 Nov \\
\meth & 4.5 & $J = 5_{1} \rightarrow 6_{0} ~A^{+}$ & 6668.518 & 7.0 &0.64$^{5}$ & 14.5 & 0.112 & $1.0 - 1.5$ & 1999 Feb\\
& & & & & 1.0 & 22.5 & 0.044 & $0.8 - 1.2$ & 2003 Mar\\
& 2.5 & $J = 2_{0} \rightarrow 3_{-1} ~E$ & 12178.593$^{6}$ & 4.0 & 0.64$^{5}$ & 7.75 & 0.061 & $1.5 - 2.5$ & 2000 Jan\\
& & & & & 2.0 & 24.5 & 0.048 & $0.6 - 0.9$& 2003 Mar\\
& 1.3 & $J = 9_{2} \rightarrow 10_{1} ~A^{+}$ & 23121.024 & 2.2 & 8.0$^{2}$ & 107.9 & 0.101 & $1.1 - 2.3$& 2015 Aug\\
\water & 1.3 & $J = 6_{16} \rightarrow 5_{23}$ &22235.120 & 2.2 & 8.0$^{2}$ & 107.9 & 0.105 & $2.3 - 2.9$& 2011 Apr \\
\hline
\multicolumn{10}{l}{$^{1}$A dual polarisation, 1024 channel each, spectrometer was employed after 2003 March 27.}\\
\multicolumn{10}{l}{$^{2}$The method of position switching was employed.}\\
\multicolumn{10}{l}{$^{3}$Monitoring was discontinued after 2016 June 01.}\\
\multicolumn{10}{l}{$^{4}$Only a single observation was made on 2016 November 12.}\\
\multicolumn{10}{l}{$^{5}$A single polarisation, 256 channel, spectrometer at LCP was employed until 2003 March 27.}\\
\multicolumn{10}{l}{$^{6}$The receiver was offline from 2016 January 01 to August 09.}\\
\end{tabular}
\end{table*}
Among the various centres of star formation in \ngc, one of the youngest contains a bright ultracompact H\,{\sc ii} (\UCHII) region \ngcf\/ \citep[G351.42+0.64,][]{rcm82,gm87,enm96}, which is associated with the far-infrared/millimeter source \ngci\ \citep{DeBuizer02,Gezari82,MFSW79,Cheung78,Emerson73}. The millimeter emission from this region has been resolved by interferometers into multiple continuum sources MM1$-$4 \citep{hbm06, bhccfi16}, two of which contain hot molecular cores accompanied by strong dust emission -- MM1 and MM2 \citep{bwtz07,zetal12,metal17,Bogelund18}. The combination of these four objects (hereafter referred to as \ngci, see Fig. \ref{fig:spots}) produce maser radiation from hydroxyl (OH), water (\water) at 22.2\,GHz \citep{mcbswb69}, and methanol (\meth) at 12.2\,GHz \citep{bmmw87}, 6.7\,GHz \citep{m91} and various other higher frequencies \citep[see][]{valtts99,Haschick90,mb89,Haschick89,hb89}. Many of these masers have been found to vary, as noted in several published works (see section~\ref{discussion}). Such variations have motivated more regular monitoring of multiple maser lines at the Hartebeesthoek Radio Astronomy Observatory \mbox{(HartRAO)}.
In this paper, we present the results of multi-year monitoring of 16 maser transitions (11 hydroxyl, 3 methanol, 1 formaldehyde, and 1 water) associated with \ngci\ since 1999. During 2015, 10 of these transitions started flaring, some of which have since dropped below our detection limits while others persist through 2017. We determine the onset of the flare in different velocity channels and when peaks occur, and discuss the possible physical causes of the flares and their association with the recent (sub)millimeter continuum outburst within the massive protostellar system NGC~6334I-MM1 detected with the Atacama Large Millimeter/submillimeter Array (ALMA) in July 2015 \citep{hetal17}. The methanol maser data from \citet{ggv04} have been re-assessed, leading to the identification of an earlier flare in 1999, possibly arising from the same object. Finally, we make a prediction for the date of a future outburst and suggest further observations to promote the ongoing study of this very interesting massive star-forming region.
\section{Observations}
The observations reported here were made using the 26\,m telescope of Hartebeesthoek Radio Astronomy Observatory (HartRAO)\footnote{See http://www.hartrao.ac.za/spectra/ for further information.}. Information for each transition observed, and receiver used, is listed in Table~\ref{tab:transitions}. Typically, observations were made every 10 to 15\,d commencing during the start month listed in Table~\ref{tab:transitions}. However, the cadence of observations varied depending on the availability of the telescope, and the weather conditions. At times observations were done daily, but there are also observations separated by weeks. The coordinates that the telescope pointed to were R.A.~=~17$^{h}$~20$^{m}$~53$\fs$4 and Dec.~=~$-35\degr$~47$\arcmin$~01$\farcs5$ (J2000).
Prior to 2003 March 27, only left circularly polarised (LCP) feeds were installed on the telescope and spectra were obtained using a 256-channel spectrometer, as described in \citet{ggv04}. From 2003 March 27 onward, each receiver system consisted of left and right circularly polarised (RCP) feeds. Dual polarization spectra were obtained using a 1024-channel (per polarisation) spectrometer. The 2.5\,cm receiver operates at ambient temperature, the others are cryogenically cooled. Each polarisation was calibrated independently relative to Hydra~A and 3C123 (and Jupiter in the case of the 1.3\,cm receiver), assuming the flux scale of \citet{Ott94}. Typical sensitivities achieved per observation are presented in Table~\ref{tab:transitions}. Observations made with the 1.3\,cm receiver were corrected for atmospheric absorption, the other observations were not (the effect is less than 3 per cent in these transitions).
\begin{figure*}
\includegraphics[width=\textwidth]{m67_122_ct.pdf}
\caption{Dynamic spectra of a sub-set of the methanol maser observations for: (a) 6.7\,GHz, and (b) 12.2\,GHz. The solid (blue), dashed (black) and dot-dashed (red) vertical lines represent the estimated onset (2015 January 01/MJD 57023 = day 0), peak (2015 August 15/MJD 57249) and a recent epoch (2017 August 31/MJD 57997) of Kitty. The horizontal red line demarcates the last red shifted channel with 6.7\,GHz emission.}
\label{fig:m67_ct}
\end{figure*}
Because of their large velocity extents, both \water\ and 1665\,MHz OH emissions were observed in position switching mode. Frequency switching was employed for observations at all other frequencies. Some contamination from an absorption feature at $+$6.4\kms\ was introduced into the 1667\,MHz maser emission spectra at $-$16\kms\ resulting from using frequency switching. The 23.1\,GHz \meth\ observations were initially done in position switching mode but were changed to frequency switching mode later. Because of the poor sensitivity of the receiver system at 23.1\,GHz, several observations had to be averaged to obtain a sufficient signal-to-noise ratio. Pointing observations were performed for the 1665\,MHz OH transition on 2017 January 21; no pointing observations were observed for any other OH transition. Water and methanol observations included pointing observations. The 4829\,MHz formaldehyde transition was observed only once on 2016 November 12. Because there was no detection, no further observations were done. Monitoring of 6031 and 6035\,MHz OH masers was discontinued on 2016 June 01 because radio frequency interference became increasingly worse after 2016 February. No observations were taken between mid-2008 and early-2010 while the 26\,m antenna underwent repairs \citep{Gaylard10}. The 12.2\,GHz \meth\ receiver was offline for repairs between 2016 January 10 and August 09.
\section{Results}
We present the results of an unprecedented maser flaring event in \ngci\/ that started in 2015. Many of our times are presented using Modified Julian Date (MJD = Julian Date $-$ 240\,0000.5). Because \ngci\ is found in the Cat's Paw Nebula, we refer to these 10 flaring maser transitions (from three molecules) in 2015 collectively as `Kitty'. The transitions are 6.7, 12.2 and 23.1\,GHz of \meth, 1665, 1667, 1720, 4660, 6031, and 6035\,MHz of OH, and the 22.2\,GHz line of \water.
\subsection{Methanol Masers}
The clearest evidence of the flaring event comes from the methanol data. The archival regular monitoring data of the 6.7 and 12.2\,GHz methanol observations for \ngci\ were used to produce dynamic spectra, which are shown in Fig.~\ref{fig:m67_ct}. The flux density colour scales have been chosen to include the strongest components of the flare in each plot, and are consequently different for the 6.7 and 12.2\,GHz maps. Velocity channels with persistent maser emission are seen as horizontal trails across the image; hereafter, we refer to these as `contrails'. Contrails can vary in flux density ($F$) with time. Note that no interpolation is done between epochs of observation.
The contrails in the 6.7\,GHz plot indicate that there are a number of velocity channels in which there are masers that are present prior to 2015. Between $v = -10$ and $-11.5$\kms\/ there are some strong masers ($F(6.7) > 2\,000$\,Jy, where $F(6.7)$ is the flux density of the maser emission at 6.7\,GHz) which do vary but do not flare during this time period. Between $v = -4.6$ (horizontal red line) and $-10$\kms\/ after the start of 2015 (MJD 57023, vertical blue line) a number of new velocity channels suddenly exhibit new emission, while some of the existing channels increase dramatically in brightness, all of which peak around 2015 August 15 (MJD 57249). Eventually, the emission in most of these channels either disappears or returns to its quiescent level, but some channels undergo a re-brightening in 2016.
The 12.2\,GHz dynamic spectrum also has strong contrails between $v = -10$ and $-11.5$\kms, some weak contrails before 2015, and a group of velocity channels that brighten and reach a peak at a similar time to those at 6.7\,GHz. The 12.2\,GHz emission spans a larger velocity range than the 6.7\,GHz lines which is obvious by the presence of components above the red horizontal line. Note that although the scales are different, we checked the processed data and found a lack of 6.7\,GHz emission at these velocities.
The vertical lines in the dynamic spectra indicate times for which spectra are plotted in Fig.~\ref{fig:meth_sp}. The top (a) and middle (b) plots are for 6.7 and 12.2\,GHz respectively, while the bottom (c) plot shows spectra for two epochs from the 23.1\,GHz line of \meth. In the latter panel, the spectrum labelled 2015 August 15 is an average of spectra taken over three days around the given date. The 2016 August 13 spectrum is an average of 13 observations taken between 2016 April 15 and 2017 January 01. In all of these spectra, the profiles are complex and cannot be resolved (in velocity space) into individual features of maser emission. They are dominated by the components with $v < -10$\kms\ which display variability, but are similar to the spectra when 6.7\,GHz emission was first discovered in \ngci\ by \citet{m91} and in 12.2 and 23.1\,GHz by \citet{mb89}. Between velocities of $-10$ and $-4.6$\kms, the 6.7\,GHz emission shows considerable variation, as does the 12.2\,GHz line but that has emission with velocity components that extend further to $v \sim -3.25$\kms. There was also an increase in the 23.1\,GHz emission in this velocity range.
\begin{figure}
\includegraphics[clip,width=\columnwidth]{meth_sp.pdf}
\caption{Spectra of methanol masers associated with \ngci\ at or near onset of Kitty (dashed green spectra), at or near the peak of Kitty (solid black spectra), and recent observations (dotted red spectra). Methanol spectra are presented for three transitions: (a) 6.7\,GHz, (b) 12.2\,GHz, and (c) 23.1\,GHz. Note that the 2016 August 13 spectra for 23.1\,GHz methanol is an average of the 13 observations taken from 2016 April 15 to 2017 January 01. The vertical red line demarcates the last red shifted channel with 6.7\,GHz emission.}
\label{fig:meth_sp}
\end{figure}
To investigate the temporal behaviour of the flare, we examined individual velocity channels in the spectra and generated time series plots. For the 6.7\,GHz methanol masers we present the time series plots for selected velocity channels in Fig.~\ref{fig:m67_ts_Kitty}. We show a variety of channels displaying different evolutionary patterns. The profiles in (a) are roughly symmetrical but the start and peak times occur at different times. In (b) the profiles are asymmetric and both display significant emission after minimum. The $v = -7.26$\kms\/ channel in (c) was the component with the strongest flux density, while the other component reached its peak later and decays very rapidly. In (d) the shape of the flare has a triangular shape which is different to the other profiles, while in (e) which has velocity components with $v < -10$\kms\/ there is very little variation associated with the flare. This stable behaviour suggests that these velocity components are in regions outside the influence of the flare, or the flaring components are too weak to be detected. The asymmetric shapes of the light curves shown in panels (b) and (c) of Fig.~\ref{fig:m67_ts_Kitty} can be due to more than one spot of maser emission turning on at different times in a particular velocity channel.
\begin{figure}
\includegraphics[width=\columnwidth]{m67_ts_Kitty.pdf}
\caption{Time series of the flux density for selected velocity channels (in \kms) of 6.7\,GHz methanol spectra. The vertical lines are defined in Fig.~\ref{fig:m67_ct}.}
\label{fig:m67_ts_Kitty}
\end{figure}
The start time of the flare in each velocity channel was determined by looking for the first data point that lay immediately above the quiescent level, and thereafter monotonically increased, while peaks represent the date with the maximum measured flux density. The cadence of the observations, which varied depending on telescope availability, determines the uncertainty in the timing measurements. This method was applied to all transitions studied here. The results for each velocity channel for all the transitions that were observed, are presented in Table~\ref{tab:Kitty}.
\begin{table}
\caption{Flare information of individual maser velocity channels associated with Kitty. The time lags of flare onsets and peaks of channels are determined against the onset (2015 January 01/MJD 57023 = day 0) of the $-$5.24\kms\ 6.7\,GHz methanol maser velocity channel. Estimated error margins are shown in parenthesis.}
\label{tab:Kitty}
\begin{tabular}{ccccc}
\hline
Vel. & \multicolumn{2}{c}{Offsets} & \multicolumn{2}{c}{Characteristics} \\
& Onset & Peak & $F_{Peak}$ & $F_{peak}/F_{onset}$ \\
(\kms)&(days) &(days)& (Jy) & \\
\multicolumn{5}{c}{6.7 GHz Methanol Masers}\\
$-$4.72 & 38(13) & 236(11) & 48 & $>$17 \\
$-$5.24 & 0(1) & 226(9) & 907 & 95 \\
$-$5.33 & 0(1) & 236(11) & 968 & 124 \\
$-$5.73 & 88(1) & 263(19) & 558 & 46 \\
$-$5.82 & 79(14) & 263(19) & 609 & 86 \\
$-$6.74 & 0(1) & 236(11) & 493 & 3 \\
$-$7.00 & 0(1) & 248(14) & 1558 & 16 \\
$-$7.26 & 47(9) & 263(19) & 1793 & 21 \\
$-$7.53 & 56(9) & 263(19) & 1316 & 29 \\
$-$7.70 & 0(1) & 263(19) & 920 & 26 \\
$-$7.92 & 47(9) & 263(19) & 526 & 9 \\
$-$8.62 & 0(1) & 263(19) & 621 & 4 \\
$-$9.06 & 189(36) & 340(9) & 353 & 3 \\
$-$9.28 & 189(36) & 295(5) & 493 & 4 \\
$-$9.50 & 88(1) & 370(5) & 347 & 3 \\
\hline
\multicolumn{5}{c}{12.2 GHz Methanol Masers}\\
$-$3.54 & 142(7) & 255(4) & 30 & 13 \\
$-$3.83 & 155(14) & 262(14) & 21 & $>$21 \\
$-$4.55 & 110(6) & 282(16) & 64 & $>$60 \\
$-$5.12 & 80(12) & 255(4) & 107 & $>$107 \\
$-$5.65 & 89(9) & 255(4) & 30 & $>$16 \\
$-$6.18 & 110(6) & 282(16) & 63 & 23 \\
$-$7.00 & 80(12) & 255(4) & 387 & 66 \\
$-$7.91 & 89(9) & 255(4) & 297 & 145 \\
$-$8.64 & 80(12) & 262(14) & 106 & 11 \\
$-$8.97 & 80(12) & 255(4) & 377 & 39 \\
$-$9.69 & 80(12) & 262(14) & 102 & 3 \\
\hline
\multicolumn{5}{c}{22.2 GHz Water Masers}\\
$-$3.00 & 83(21) & 337(9) & 1420 & 34 \\
$-$5.00 & 22(10) & 229(11) & 2158 & 15 \\
$-$7.11 & 22(10) & 219(13) & 12109 & 20 \\
$-$8.37 & 55(9) & 219(13) & 5491 & 27 \\
$-$24.49 & 62(14) & 344(9) & 1133 & $>$304 \\
\hline
\multicolumn{5}{c}{1665 MHz Hydroxyl Masers}\\
$-$7.40R & 188(45) & 309(7) & 32 & 4 \\
$-$8.10L & 113(14) & 396(8) & 205 & 26 \\
\hline
\end{tabular} \end{table}
From Table~\ref{tab:Kitty} it can be seen that flaring began in several 6.7\,GHz methanol velocity channels on 2015 January 01 (MJD 57023); we refer to this date as `day 0'. Fortuitously, an observation with a null detection was recorded on the previous day so the uncertainty in the onset of the flare in our data is one day. This is the date we have used as the starting time for the flare and represented by the vertical line (blue in the online version) in Figs.~\ref{fig:m67_ct} and \ref{fig:m67_ts_Kitty}.
Based on time of onset, there are four groupings of components associated with Kitty, those that (1) began on 2015 January 01, (2) about 50\,d later, (3) about 85\,d later, and (4) about 190\,d later. For comparison, a fifth set of velocity channels not associated with Kitty are plotted. The first velocity channel of Kitty to reach a maximum was $-$5.24\kms, then there is a weak correlation between progressively blue shifted velocity channels and increasing date of peak flux density in the velocity range $-6.1$ to $-5.1$. We see a similar progression between $-7.8$ to $-6.8$\kms; no such progression is obvious in the velocity range $-10$ to $-8.0$\kms. For some of the 6.7\,GHz velocity channels the flare was roughly symmetrical; the temporal behaviour is 220$\pm$50\,d for the rise and 170$\pm$30\,d for the decay and lasted for a total of 385$\pm$50\,d. A simple measure of variability, e.g. $F_{peak}/F_{onset}$, is also listed for each component in Table~\ref{tab:Kitty}. This ratio varied from 3 to 124 for all the channels associated with Kitty. The strongest feature at $-$7.26\kms\ increased by a factor of 21. Several velocity channels continue to flare.
On average, at our sensitivity levels, the 12.2\,GHz velocity channels began flaring about 6 weeks after their 6.7\,GHz counterparts, but they all reached peak emission within a few weeks of each other. No observations were taken between 2016 January to August; this made it difficult to determine the termination dates of some channels or if they experienced secondary flares. The average emission rise times of the 12.2\,GHz channels are 160$\pm$25\,d and we estimate the decay of some are 95$\pm$20\,d. The ratio of flare peak emission to quiescent values in the 12.2\,GHz channels varied from 3 to 145 and the strongest component at $-$7.0\kms\ increased by a factor of 66, three times more than its 6.7\,GHz counterpart.
\subsection{Water Masers}
A dynamic spectrum made from the regular monitoring data of the 22.2\,GHz line from \water\ masers is shown in Fig.~\ref{fig:h2o_ct}. Because the flux scale in this plot reaches 15\,000\,Jy, it is difficult to distinguish features below $\sim$1\,000\,Jy. There is some emission at the level of $< 1\,200$\,Jy between $\sim -7$ and $-8$\kms\ in 2014, but this drops to $\sim 300$\,Jy prior to the start of 2015. The first indication of the flare occurred on 2015 January 23 (MJD 57046). The first peak, which reached $F$(22.2) = 12\,100\,Jy, occurred at approximately the same time as the first peak in the methanol 6.7\,GHz data (see Table~\ref{tab:Kitty}). The water masers rebrightened in mid-2016, reaching another peak of $F$(22.2) = 15\,200\,Jy. In 2017 March a short burst occurred at $v = -3$\kms\ which we have classified as a super-outburst (MacLeod et al. in preparation 2018). At the end of 2017 another brightening phase occurred, during which the peak flux density reached 15\,800\,Jy.
\begin{figure*}
\includegraphics[width=\textwidth]{h2o_ct.pdf}
\caption{Dynamic spectrum of 22.2\,GHz water masers in \ngci. The solid, dashed and dot-dashed vertical lines are defined in Fig.~\ref{fig:m67_ct}.}
\label{fig:h2o_ct}
\end{figure*}
\begin{figure}
\includegraphics[width=\columnwidth]{h2o_sp.pdf}
\caption{Spectra of water masers associated with \ngci\ at the onset (2015 January 23) of Kitty (blue), at or near the peak (2015 August 08) of Kitty (black), and recent (2017 August 26) observations (red). The entire flux density range is shown in (a) and components with flux density less than 625\,Jy in (b).}
\label{fig:h2o_sp}
\end{figure}
The emission from water masers in \ngci\ has a larger velocity range than the methanol masers, as can be seen in the spectra shown in Fig.~\ref{fig:h2o_sp}. Maser emission occurs from $v = -50$ to above 0\kms\ in this region with all of the velocity channels showing variability on some time scale. The spectra are complex, consisting of the emission of numerous sources in the beam that cannot be resolved into individual velocity components. The peak flux density in 2015 occurred in the $v = -7.11$\kms\ velocity channel, which is similar to the 6.7 and 12.2\,GHz methanol lines that peaked at $v=-7.26$ and $-7.00$\kms, respectively. The flux in this channel increased by a factor of 20 which is similar to the 6.7\,GHz \meth\ line. The water velocity channel $v = -24.49$\kms\ varied the most (of all transitions observed), it increased by a factor of $>$300 and is located in CM2 (Brogan et al. 2018, in preparation).
\subsection{Hydroxyl Masers}
The spectra of the 1665\,MHz groundstate OH masers are complex and contain many features that are variable and have been present since before 2013. Absorption features, at $v > -7$\kms, from foreground clouds are seen in each spectrum. The LCP spectrum had a velocity spread of $-7$ to $-11$\kms\ and the brightest flux density measured prior to 2015 was $\sim 500$\,Jy, whereas the RCP spectrum was spread over $v = -6$ to $-13$\kms and peaked at $\sim 100$\,Jy. To identify components associated with this flaring event, dynamic spectra were made (not shown) and we looked for any velocity channels with sudden increases in brightness around the time of the methanol flare. Only one channel in each polarisation was definitively identified. In LCP flaring activity started in the $v = -8.1$\kms\ channel 113 days after 2015 January 01, and in RCP the $v=-7.40$\kms\ channel shows some activity after 188 days. Spectra nearest to the commencement, peak and a recent time are shown in Fig.~\ref{fig:oh1665_ts}, together with time series plots of the identified channels.
\begin{figure*}
\includegraphics[clip, width=\textwidth]{oh1665_67_ts.pdf}
\caption{Spectra of the 1665 and 1667\,MHz OH masers at or near onset, at or near peak, and recent observations of Kitty. Also time series of selected velocity channels are plotted. (a) RCP spectra, (b) LCP spectra, (c) RCP and LCP time series are presented for 1665\,MHz and similarly for 1667\,MHz in (d), (e) and (f). The vertical lines in (c) and (f) are defined in Fig.~\ref{fig:m67_ct} while the velocity channels plotted are those identified by the dashed vertical lines in the RCP and LCP spectra plots. In each spectrum an absorption feature, the result of a foreground cloud, is present at $v > -7$ \kms.}
\label{fig:oh1665_ts}
\end{figure*}
The regular monitoring programme did not include 1667\,MHz groundstate OH observations and, hence, we only have data after 2015 September (MJD 57275), which is 251 days after the start of Kitty. As for the 1665\,MHz lines, the 1667\,MHz LCP and RCP spectra are complex, but both occur over velocities from $-15$ to $-7$\kms. There are some consistent features (based on spectra obtained on 2012 July 05/MJD 56114), and some variable components. RCP and LCP spectra are shown in Fig.~\ref{fig:oh1665_ts} (d) and (e) respectively, and time series (f) of the two velocity channels that varied around the time of the flare and attained the highest peak in each polarisation. The flux densities in the spectra, like those for 1665\,MHz, are stronger in LCP than RCP\@. Note that neighbouring velocity channels also varied, but their peaks were smaller than those plotted, although after the event in some cases they have become stronger than the chosen channels. The peaks in the 1667\,MHz spectra occur even later than those in the 1665\,MHz spectra.
\begin{figure*}
\includegraphics[width=0.9\textwidth]{other_ts.pdf}
\caption{Spectra of the various OH transitions' observations taken: near or before 2015 January 01 (if available), nearest to 2015 August 15, and recent are presented for 1720\,MHz in (a) RCP and (b) LCP. Time series plots of selected velocity channels for 1720\,MHz masers are shown in (c). Similar plots for 4660\,MHz are presented in (d), (e), and (f), for 6031\,MHz in (g), (h) and (i), and for 6035\,MHz in (j), (k), and (l). The vertical line in each spectrum corresponds to the velocity channel plotted in the time series in the rightmost column. The vertical lines in the time series plots are defined in Fig.~\ref{fig:m67_ct}.}
\label{fig:oh_other_ts}
\end{figure*}
We looked for the satellite groundstate OH lines at 1612 and 1720\,MHz. Nothing was detected at 1612\,MHz over multiple epochs. A 1720\,MHz observation on 2015 September 09 revealed a previously known Zeeman pair at $v = -10.2$\kms\ and a new feature at $v = -7.7$\kms\ in RCP and LCP with similar flux densities. In Fig.~\ref{fig:oh_other_ts} we show spectra for RCP (a) and LCP (b) and time series (c) of the peak velocity channel in each polarisation. The time series decays monotonically, but we only started observing at, or after, the methanol peak (vertical dashed line in (c)) had been reached. The signal disappeared below our detection limit after 247\,d. The line profile is simple and can be fitted using a single Gaussian. The velocity difference between the peak of the RCP and LCP profiles corresponds to a Zeeman splitting due to a magnetic field of $+1.3\pm 0.2$\,mG\/ using a Zeeman splitting coefficient published in \citet{fram03}.
A surprising discovery was the detection of a 4660\,MHz excited OH maser with a flux density of 4.4\,Jy at $v =-7.8$\kms, that died away after 71\,d. The LCP and RCP observations, shown in Fig.~\ref{fig:oh_other_ts} (e) and (f), indicate no circular polarisation for this source. The time series for the peak velocity channels are shown in (g). This is only the fifth 4660\,MHz maser ever detected. Searches did not detect any 4765\,MHz emission which is usually the strongest 4.7\,GHz line found, and had been found at a velocity of $v = -10.3$\kms\ by \citet{cmc95}. The rarity of 4660\,MHz masers suggests unusual conditions for their existence. Multiple epoch observations subsequently have not detected any of the 4.7\,GHz OH masers in this source.
Masers from the first rotationally excited level of OH at 6031 and 6035\,MHz have been found in this source since their original detection by \citet{grg70}. The strongest features lie around $v = -10.5$\kms, but weaker emission has been reported in the range from $v = -12$ to $-7$\kms\ \citep{cv95}. In the RCP and LCP 6031 and 6035\,MHz spectra shown in Figs.~\ref{fig:oh_other_ts} (g), (h), (j), and (k) respectively, these persistent features can be seen. Significant new peaks had developed around $v \sim -7.7$\kms\ when we observed them on 2015 September 08 (MJD 57274) and August 30 (MJD 57265). The time series of the peak velocity channels at $v = -7.8$, $-7.65$, $-7.7$ and $-7.6$\kms\ are shown in (i) and (l) respectively, and show that the flux density decayed over the following several months. Unfortunately, due to radio frequency interference (RFI), observations became untenable after 2016 June, which explains the truncated time series.
The peak flux density of the new feature in the 6031\,MHz spectra was higher initially than the persistent lines around $v = -10.5$\kms. The flux density of the 6035\,MHz lines were slightly stronger than the 6031\,MHz lines but were much weaker than the long-term lines. In both of these excited OH lines the LCP flux density was consistently weaker than the RCP, the same as for the 1720\,MHz lines, whereas for the OH mainlines (see Fig.~\ref{fig:oh1665_ts}) the LCP flux density was significantly stronger than RCP\@. The asymmetric shape of the 6035\,MHz profiles at $v = -7.7$\kms\ indicates that there is more than one component of maser emission.
\subsection{An earlier event}
We have re-analysed the 6.7\,GHz methanol data for \ngci\ presented in \citet{ggv04}. A dynamic spectrum of the data from 1999 February 28 (MJD 51238) to 2001 March 31 (MJD 52000) is shown in Fig.~\ref{fig:m67_ct_Mini}. Based on an analysis of the time series for individual velocity channels (see Fig.~\ref{fig:m67_ts_Mini} in Appendix~\ref{Appendix A}), the brightest part of the flare is in velocity channel $v=-5.99$\kms, but it starts in the $v=-8.46$\kms\ channel at the time indicated by the vertical line. This event only reached a maximum flux density of $\sim 400$\,Jy and lasted for 340\,d. Channels that flared covered a velocity range from $-3$ to $-8.6$\kms, which is similar to the range for Kitty. There is no evidence of flaring in the persistent masers with $v < -8.6$\kms. Unfortunately, there is no data at other maser frequencies for this event. We refer to this event as `Mini'.
\begin{figure}
\includegraphics[width=\columnwidth]{m67_ct_Mini.pdf}
\caption{Dynamic spectrum of 6.7\,GHz methanol masers in \ngci. The solid vertical line is the start time, 1999 September 25 (MJD 51447), of a small flaring event (Mini) in 1999.}
\label{fig:m67_ct_Mini}
\end{figure}
\subsection{Summary of results}
To summarise the results discussed above, in Fig.~\ref{fig:4trans_ts} we compare time series of the velocity channels for water, methanol and hydroxyl that contained the peak maser emission. The vertical solid line indicates the start of the event as defined by one 6.7\,GHz methanol velocity channel (see Fig.~\ref{fig:m67_ct}). Data from the long term monitoring programme shows that there was some quiescent maser flux density in each of the chosen channels prior to 2015, but then the flux density increases by more than an order of magnitude in each maser line. The increases do not all occur simultaneously. The 12.2\,GHz methanol line returned to its quiescent level after $\sim 260$\,d. The 6.7\,GHz line decayed significantly in some channels which passed through a dip before rebrightening, while the water and 1665\,MHz hydroxyl masers were active throughout the observation period reported here. The total extent of the velocity range in which all these masers resided in all transitions is $-3$ to $-10$\kms, with the exception of the high velocity water maser at $v = -24.49$\kms.
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{4trans_ts.pdf}
\caption{Time series of selected velocity channels from 6.7\,GHz ($-$7.26\kms) and 12.2\,GHz ($-$7.00\kms) methanol, 22.2\,GHz ($-$7.11\kms) water, and 1665\,MHz ($-$8.10\kms\ LCP) hydroxyl. For comparison purposes, 4660\,MHz ($-$7.80\kms\ LCP+RCP) and 6031\,MHz ($-$7.65\kms\ RCP) OH velocity channels are plotted. The vertical lines are defined in Fig.~\ref{fig:m67_ct}. Note that a logarithmic scale has been used for the flux density.}
\label{fig:4trans_ts}
\end{figure*}
The lines of OH that were not part of a long term monitoring programme (1720/4660/6035/6031\,MHz), were only observed after 2015 August 15 (MJD 57249). They all declined monotonically after detection; the time series for selected channels of the 4660 and 6031\,MHz transitions are shown in Fig.~\ref{fig:4trans_ts} for comparison purposes.
\section{Discussion}
\label{discussion}
As a well known source of maser emission in the Galaxy, \ngci\/ has been found to host numerous masers from methanol, water and hydroxyl. The spectral appearance of some of these masers have been fairly consistent since their discovery, varying to some extent over the intervening decades. In contrast, for many other regions in the literature, the general variability of masers in all species has been reported extensively and regular monitoring programmes \citep[e.g., in water by][]{Felli07} have led to the discovery of flaring events in certain transitions. What is new about this study is that we have long term monitoring data of 6.7 and 12.2\,GHz \meth, 22.2\,GHz \water\ and 1665\,MHz OH maser lines that show contemporaneous flaring activity after a long period of relative stability. The velocities of these flaring masers all occur in the range from $v = -3$ to $-10$\kms. We also found new OH masers at 1720, 4660, 6031 and 6035\,MHz and for \meth\ at 23.1\,GHz. The 1667\,MHz groundstate OH maser follows the behaviour of the 1665\,MHz mainline, but we do not have enough data to conclusively claim a starting date for its activity.
To orient the subsequent discussion, a comprehensive list of maser positions from various interferometric measurements made prior to the outburst are plotted in Fig.~\ref{fig:spots}, overlaid with 1.3\,mm dust continuum emission observed in 2008 by \citet{hetal17} with the Submillimeter Array (SMA), and 6\,cm continuum observed in 2011 with the National Science Foundation's Karl G. Jansky Very Large Array (VLA) by \citet{bhccfi16}. Using ALMA,
\citet{bhccfi16} found that of the four primary millimetre sources that make up \ngci, the brightest, MM1, is comprised of several compact components named MM1A, B, etc. These observations were taken coincidentally when the methanol masers initially peaked in 2015 August. \citet{hetal17} reported that the dust luminosity of MM1 increased by a factor of $\sim$70 between 2011 May and 2015 July/August with the increase centered on MM1B. We report the maser emission similarly increased by factors of 10s to 100s between 2015 January and August (see Table~\ref{tab:Kitty}) suggesting a direct relationship. In recent VLA observations, \citet{hunter18} confirmed our suspicion that these flaring masers arise from new spatial regions of maser activity. In particular, they report six new regions of 6.7\,GHz methanol maser activity with overlapping velocity ranges, four in MM1 and two in CM2, and new 6.0\,GHz excited OH maser activity associated with CM2.
For the purpose of our discussion below we take the distance to \ngci\ to be 1.3\,kpc based on maser parallax measurements of the adjacent core NGC~6334I(N) \citep{chibueze14,Reid14}.
\subsection{Temporal behaviour}
Six velocity channels in the 6.7\,GHz \meth\ spectra started flaring on 2015 January 01 (day 0 in Table~\ref{tab:Kitty} and indicated by the solid vertical line in Figs.~\ref{fig:m67_ct}, \ref{fig:m67_ts_Kitty}, \ref{fig:h2o_ct}, \ref{fig:oh1665_ts}, and \ref{fig:4trans_ts}). The other 6.7\,GHz channels all started flaring later than this. About half of the channels reached their peak on day 263, some of which started flaring after day 0. The earliest date on which the 12.2\,GHz methanol masers started flaring was on day 80, and more than half of the velocity channels peaked on day 255. Given the cadence of the observations, it appears that the 6.7 and 12.2\,GHz \meth\ masers peaked around the same time. Observations of the excited OH masers and the 1720\,MHz groundstate OH masers only started around day 250, but then decayed monotonically. These observations are consistent with these OH masers peaking around the same time as the methanol masers. The 22.2\,GHz \water\ masers started flaring on day 22 and have remained strong. Because of the rapid fluctuations in these masers, there is no clear date when they reached a peak. The OH 1665\,MHz LCP and RCP channels only started flaring 113 and 188\,d after day 0. The 1665 and 1667\,MHz groundstate OH masers have remained strong with no obvious peak in the light curve.
After peaking some of the 6.7\,GHz methanol decayed back to quiescent levels, while others went through a rebrightening. The 12.2\,GHz methanol velocity channels decayed rapidly back to their quiescent levels with no rebrightening. If the methanol masers are radiatively pumped by the object in MM1B then the 12.2\,GHz masers only start flaring when the source becomes bright or hot while the 6.7\,GHz masers are pumped over a broader range of conditions. The rebrightening of the 6.7\,GHz methanol masers could be due to another (weaker) flaring event in MM1B, which does not pump the 12.2\,GHz masers. The 6.7\,GHz flare in 1999 was possibly also due to a similar type of event which was weaker than this flare. The start of flaring in some velocity channels after day 0 can be explained by conditions in the cloud not supporting masers pointing towards us, and, likewise, the different times at which the masers peaked could be due to geometrical effects or the masers dying out before the source peaked.
The methanol spot maps presented in \citet{hunter18} were taken in 2016 November (MJD 57711) during the peak of the rebrightening of the methanol masers reported here. The dominant new maser in their observations, at $v = -7.25$\kms, is about 1000\,au in projected distance from MM1B. Other areas of MM1 are at projected distances of between 650 to 2800\,au from MM1B. If MM1B is the source of this flaring event, as proposed by \citet{hetal17}, then the projected light travel time ranges from 4 to 16\,d; much shorter than our estimated time lags. Similarly, the light travel time to MM2 (3$\farcs$5) and MM3 ($4\farcs$5) is 25 and 33\,d in the sky plane, but there is no evidence of associated flaring in the masers of the latter two regions, even after 1100\,d. It is possible that the geometric dissipation of the energy in the event was sufficient that it had no impact on the masers in MM2 and MM3. Somewhat less likely is that MM2 and MM3 have not yet witnessed the event, but this requires them to be at least 0.5\,pc in the background relative to MM1. Regardless, the new masers in MM1 and CM2 are highly variable in comparison to the decades long stable masers in MM2 and MM3, possibly the result of the energy injected into it by the event at MM1B causing instabilities in the masing conditions.
\begin{figure*}
\includegraphics[width=0.9\linewidth]{fig10.pdf}
\caption{Image of \ngci\ with all associated maser spots reported prior to the 2015 outburst. The SMA 1.3\,mm continuum image (epoch 2008.6) is plotted as solid contours with intensity values: $-$0.024, 0.024, 0.06, 0.12, 0.24, and 0.48\,Jy\,beam$^{-1}$ from \citet{hetal17}. The 5\,cm VLA continuum image (epoch 2011.5) is plotted as dotted contours with intensity values: 0.012, 0.6, 6.0, 12.0, 24.0\,mJy~beam$^{-1}$ from \citet{bhccfi16}. Maser spots presented are: 6.7\,GHz methanol masers from \citet{kevb13} and \citet{bhccfi16} (magenta x and + respectively), 6.7\,GHz methanol masers from \citet{Walsh98} (brown +), 12.2\,GHz methanol masers from \citet{becgvfqa12} (orange +), 19.9 and 23.1\,GHz methanol masers from \citet{kevb13} (red + and x respectively), 44\,GHz methanol masers from \citet{glhkha10} (yellow dot), 22.2\,GHz water masers from \citet{fc89} and \citet{bhccfi16} (cyan and blue + respectively), OH masers from \citet{fc89} and \citet{arm00} (black + and x respectively), and excited OH masers from \citet{gcm15} (green +) and \citet{ckr11} (black x). The \citet{fc89} water positions have been shifted by $-0\farcs7,-1\farcs0$ for the reason described in \citet{bhccfi16}. The 5\,cm source (CM2) and the four millimetre source (MM1$-$4) are labeled in black; MM3 is the \UCHII\ region \ngcf. \citet{elletal96} identified 3 regions of methanol maser activity, C, S and NW, labeled in red. The new 6.7\,GHz methanol masers that appeared toward and surrounding MM1 and CM2 after the outburst (not shown) are reported by \citet{hunter18}.
}
\label{fig:spots}
\end{figure*}
\subsection{6.0\,GHz excited OH masers}
Maps of the 6.0\,GHz excited OH masers in \ngci\ made by \citet{ckr11} and \citet{hunter18} show that the persistent masers around $v \sim -10.5$\kms\ are located in MM3. These masers have not shown any variations associated with the 2015 flare.
The new masers we detected at $v \sim -7.7$\kms\ in 2015 September were not present in single dish observations in 2012 or 2013. We do not have data for their onset, but observed them from at (or near) their peak, after which they decayed. There is a gap of $\sim$160\,d between the termination of our single dish observations and the VLA observations of \citet{hunter18}, who found new 6.0\,GHz masers at the same velocity ($-7.7$\kms) in CM2 that are stronger than when the single dish observations ceased. It is not clear whether the flares in our data are from CM2, which had rebrightened by the time of the VLA observations, or from some region closer to MM1 that have subsequently faded below the sensitivity limit of the VLA. If these masers are radiatively pumped, then the light curves we obtained and the CM2 masers in the VLA observations could be the same. \citet{hunter18} detected a new 6.7\,GHz methanol feature in CM2 at $v = -7.6$\kms, a similar velocity to the 6.0\,GHz OH masers, increasing our confidence of their relationship to Kitty. We note that a temporal correlation between 6.7\,GHz methanol and 6.0\,GHz OH masers in a flaring event has been reported in the past in a massive protostar \citep[IRAS~18566+0408,][]{AlMarzouk12}.
\subsection{4.7\,GHz OH masers}
A 4765\,MHz excited OH maser feature with a flux density of 0.3\,Jy at $-$10.3\kms\ was detected in 1991 by \citet{cmc95} but it was not detected in 2000 by \citet{de02} down to a level of 0.09\,Jy. Unfortunately, the position determined has very large error bars which cover more than the whole map shown in Fig.~\ref{fig:spots}. Neither group detected emission at 4660 or 4750\,MHz. We did not detect 4750 or 4765\,MHz emission.
The 4660\,MHz line, which is not subject to observable Zeeman shifts in weak magnetic fields, has a velocity $v = -7.8$\kms, which is similar to the 1720, 6031 and 6035\,MHz masers (which can be Zeeman shifted by a few 0.1\kms). Although we did not see these masers turn on, we assume that all these OH lines are formed in the same region and are part of the 2015 flaring event. The VLA observations of \citet{hunter18} included a high resolution spectral window on the 4660\,MHz line, but did not detect it.
What is unusual about this maser is that 4660\,MHz masers are rare (only four are reported in the literature) while $\sim30$ masers in the 4765\,MHz line have been found \citep{Qiao14}. In general, models predict that flux densities $F(4765) > F(4750) > F(4660)$. Clearly, this is not the case in Kitty. Also there has been some discussion in the literature suggesting correlations between 4765\,MHz excited OH masers and either the ground state 1720 or 1612\,MHz lines, depending on the density and temperature of the gas. Our observations show no evidence of 4765\,MHz emission relative to 1720\,MHz masers by a factor of $\sim 200$.
The 4660\,MHz emission died out rapidly, faster than the 6\,GHz excited OH and 1720\,MHz lines (see Figs.~\ref{fig:4trans_ts} and \ref{fig:oh_trans_ts}). This behavior possibly suggests that the conditions under which the 4660\,MHz masers form are much more sensitive to the conditions in the gas than the other lines.
\subsection{6.7 vs. 12.2\,GHz methanol masers}
Pumping models predict that $F(6.7) > F(12.2)$ but we find examples of velocity channels in which the converse is true. This situation occurs for velocities $v > -4.6$\kms. There are some narrow regions with very specific conditions where $F(12.2)$ can be greater than $F(6.7)$ \citep{csg05}, but in dynamic regions where the conditions are changing we would not expect these conditions to exist for any significant length of time. No interferometric observations were taken at 12.2\,GHz during the period when the 12.2\,GHz masers were detectable. \citet{hunter18} do report 6.7\,GHz masers in this velocity range but at an epoch after the 12.2\,GHz masers had faded back to their quiescent levels. The survey of 400 6.7\,GHz maser targets by \citet{becgvfqa12} found only about three percent of objects where the 12.2\,GHz methanol masers were stronger than their 6.7\,GHz counterparts.
\subsection{6.7\,GHz methanol vs 22.2\,GHz water masers}
The contemporaneous increase in the methanol and water maser flux densities provides a strong constraint on the physical explanation of their respective flares. Recently imaged with the VLA in late 2016, the flaring 6.7~GHz methanol emission is localized to the area toward and surrounding MM1 and CM2 where strong masers now appear for the first time \citep{hunter18}. The infrared photons required to pump these masers \citep{scg97a}, as well as the OH mainline masers \citep{Gray07}, naturally would have become more abundant when the surrounding dust was rapidly heated by the increase in radiation from the proposed \citep{hetal17} protostellar outburst source MM1B. In contrast, prior to the outburst, water masers were already known to be associated with both MM1 and CM2 (see Fig.~\ref{fig:spots}), a non-thermal radio continuum source located $\sim$2\,$''$ north, which had the strongest water masers but no associated compact dust emission \citep{bhccfi16}. The projected extent of the water masers has a mechanical crossing time of many decades, for even a high velocity protostellar jet. Therefore, it seems unlikely that the creation of new shocks containing new water masers close to the central protostar would produce a rate of increase in the water maser emission that so closely parallels that of the radiatively-pumped methanol masers (see Fig.~\ref{fig:4trans_ts}). Instead, we find it more likely that some property of the existing water masing gas was changed by the excess radiation produced during Kitty. One possibility is a small increase in the ionisation fraction, which can result in more intense water masers \citep{kn87}. Alternatively, the phenomenon of superradiance that has been invoked to model the methanol and water maser flares in G107.298+5.639 \citep{rh17} could be contributing to the observed behavior of the light curves.
\subsection{Possible associated earlier events}
Using single-dish observations beginning in 1965 July, \citet{wdw68} noted changes in the OH mainline intensity of \ngc\/ by almost an order of magnitude on time scales of a few days to months. Adjacent velocity channels showed great variability. In particular, their 1665\,MHz observations at $v =-8.1$ and $-7.7$\kms\ decreased significantly in amplitude between two sets of observations, after which these channels were relatively constant at low amplitude. There was no activity in the corresponding 1667\,MHz channels. The velocity range in which the 1965 flare occurred is $-$12.7 to $-$6.6\kms\ with the largest variation occurring in the $-$7.7\kms\ channel. These channels are the same as variable features reported for Kitty presented in this paper. It is possible that shortly after the discovery of OH masers in 1963 a flaring event occurred, peaking around 1965 July, on par with Kitty reported here, and ending 1965 October seen in \citet{wdw68}.
Another flaring event that occurred in 1999 was found in the 6.7\,GHz data from the HartRAO monitoring programme. See Appendix~\ref{Appendix A} for analysis. It was a weaker and shorter-lived flare, but the velocity range and factors of increase from quiescent flux densities ($\sim$20 to 200) were similar to Kitty. Unfortunately no other transitions were observed during this event. It is possible that this event and the earlier event in \citet{wdw68} are both related to Kitty, i.e. arising from the same physical location.
\subsection{Physical Cause of the Outburst}
It is not obvious from these observations what caused the outburst that powered Kitty. Based on (sub)millimeter observations, \citet{hetal17} speculate that the unprecedented increase in dust temperature resulted in an increase in the radiative pumping of the various maser emissions in the region. While the association between the dust emission flare and the maser flares might explain their correlated timing, it does not directly explain the cause. \citet{hetal17} suggest the occurrence of an accretion event analogous to an FU Ori outburst -- a phenomenon in low-mass protostars in which they experience rapid increases in accretion luminosity followed by long decays that occur over tens of years \citep{Hartmann96}. The first such event in a massive protostar was recently reported \citep[S255~NIRS3,][]{Caratti17} and it also coincided with an extended methanol maser flare \citep{Moscadelli17}. As noted by \citet{hetal17}, the timescale for such an accretion outburst to heat the dust in MM1 by the observed amount is $\sim$200\,d based on the equations of \citet{jhhb13}. Qualitatively, one should expect the different maser transitions and velocity components to turn on at different times during the course of this heating due to differences in the local geometry and opacity of the dense medium. While it remains uncertain how many days before the first maser flare (2015 January 01) the accretion outburst began, this
theoretical heating timescale combined with the observed 190 d spread in maser flare start dates, suggests that it was likely only up to a few weeks.
Indeed, the VLA observations of \citet{hunter18} confirmed that 6.7\,GHz methanol and 6.0\,GHz hydroxyl masers are spatially associated with MM1 and CM2. Features of the 1665 and 1667~MHz OH and water maser spectra associated with Kitty are also consistent with the variation originating from MM1 and CM2.
Further support for an FU Ori-like origin hypothesis for Kitty comes from a comparison with the 1720\,MHz OH maser outburst associated with the 5.5\,mag optical flaring event in the FU~Ori object V1057~Cyg that began in 1969 \citep{Herbig77}. The OH maser emission originally detected by \citet{Lo73} declined exponentially over a two year interval and became undetectable by 1975 \citep{Andersson79}. A subsequent OH maser outburst was observed in 1979 which then diminished by a factor of 2 over a two month period \citep{wetal81}. Several features of the methanol masers of Kitty exhibited a rapid flare followed by an equally rapid decline, an initial flare in which the rise and fall times are similar followed by a secondary flaring event, or a rise which remained high. If Mini and Kitty indeed arose from the same spatial location, then we have recorded two such events in 15\,yr, analogous to the repeated maser outbursts in V1057~Cyg. This conclusion is further supported by the fact that 1720\,MHz OH maser flares have been imaged toward massive protostars in the past \citep[e.g. W75N,][]{Fish11}.
As an alternative origin hypothesis, a supernova can produce sufficient energy to heat the surrounding dust that then pumps the masers. It can also produce ejecta that might result in the non-thermal radio source CM2 $\sim$2\,$''$ north of MM1; however, CM2 already existed in the VLA observations of 2011. Also, while shock-excited 1720\,MHz OH masers are often seen towards supernova remnants \citep[e.g.,][]{brogan2013,hoffman2005}, the other groundstate lines are usually found only in absorption \citep[see e.g.,][and references therein]{Green02}, unlike in Kitty. But perhaps the most obvious characteristic that rules out a deeply-embedded supernova is the lack of new strong centimeter continuum emission. Using previous VLA observations of the optically obscured supernova SN 2008iz in M82 as a model \citep{Brunthaler10}, the 5~GHz flux density should have reached $\sim$100~kJy after a year with a diameter of $\sim8''$, which it quite obviously did not.
Finally, \citet{hetal17} noted the possibility that a near encounter with another star or a merger might have triggered the accretion event leading up to Kitty. \citet{bz05} propose that stellar mergers in our Galaxy will occur at the same rate as the birth rate of massive stars and will produce high luminosity infrared flares. Such flares could pump hydroxyl and methanol masers and associated shocks could pump the water masers. \citet{Bally02} presents a cartoon of the possible light curve of a binary stellar merger; he predicts progressively stronger flares at each periastron passage over decades to centuries until the actual merger occurs. However, the flare profile is different than that of the time series of the flaring masers presented here; he suggests that each flare will experience a rapid rise followed by a slower decay perhaps lasting years. Mini and Kitty may represent flares resulting from successive periastron passages with the re-brightened masers being the result of other mechanisms not considered in \citet{bz05}.
\subsection{Prediction of a future flare}
If we assume the peaks of the OH flare reported by \citet{wdw68}, Mini, and Kitty occur at successive periastrons of an embedded binary star, then using equation 55 of \citet{s10} we can fit for the orbital parameters and predict the next flare event. For our best-fit solution, shown in Fig.~\ref{fig:orbit}, we estimate the coalescence time is $\sim$156\,yr, the initial orbital period is $\sim$92\,yr, and the coalescence date is in $\sim$2059. In this scenario, the next flare is predicted to occur in late 2026. If we further assume that the cloud sound speed is $\sim$1.6\kms, and the ambient cloud H$_{2}$ density is $\sim6\times10^{7}$\,\pcmm, then using equation 56 of \citet{s10} the total mass of the system is $\sim$4.9\,M$_{\odot}$. This value is roughly consistent with a (proto) B3 zero age main sequence (ZAMS) star being the dominant member \citep{Hanson97}, and a ZAMS star of this type could power the measured properties of the hypercompact HII region surrounding MM1B \citep{bhccfi16,brogan17}. However, the derived mass from the orbital model is strongly dependent on the assumed sound speed in the gas. In any case, \citet{s10} suggest that the energy of interactions at periastron will rival that of the combined stellar output only during the final few per cent of the merger process. Nevertheless, we expect the next flare may be more powerful than Kitty.
\begin{figure}
\includegraphics[width=\columnwidth]{orbit_decay_155_91.pdf}
\caption{Panel (a) shows the theoretical orbital decay of a binary system in \ngci\ using equations in \citet{s10} fit to the dates of the three known maser flares: Weaver (1965), Mini (1999), and Kitty (2015). The solid line is the orbital period in years and the dashed line is the major axis of the orbit in au. In this scenario, panel (b) shows the predicted date of the next flare to be at the next completed orbit in late 2026.}
\label{fig:orbit}
\end{figure}
\section{Summary and Future Work}
We report here a significant flaring event in 10 transitions in three molecular species associated with \ngci\ that began 2015 January 01. The 6.7\,GHz methanol and 22.2\,GHz water masers began flaring contemporaneously within $\pm$22\,d of each other, while the 12.2\,GHz methanol and 1665\,MHz hydroxyl masers flared 80 and 113\,d later respectively. The flaring for all transitions occurred in the velocity range $-$10 to $-2$\kms. The strongest flaring methanol and water features increased, by $\sim$20 times above quiescent levels; the strongest water maser feature reached $\sim$15,000\,Jy. The weak emission in some velocity channels increased by factors up to 145. This flare coincides in time with an unprecedented increase in the millimeter continuum and inferred dust temperature reported by \citet{hetal17}.
We report the detection of only the fifth 4660\,MHz excited OH maser. We also report new maser emission at 1667, 1720, 6031 and 6035\,MHz hydroxyl, and 23.1\,GHz methanol in \ngci.
We report an earlier flare in 1999; it was only observed at 6.7\,GHz and was at most 5 times weaker than Kitty. We also highlight a 1965 OH maser flare reported by \citet{wdw68}. We hypothesize that these three flares could be related, analogous to the repeated OH maser flares in the FU~Ori star V1057~Cyg. Such repeated flares could be due to the orbital decay of a binary protostar, with radiative outbursts growing in strength with each successive periastron passage. If so, we predict a future maser flare in 2026.
We note that future observations of this event may provide constraints on maser pumping models. Further interferometric studies will also be fruitful, and we predict that new cluster(s) of OH masers will be detected with sufficiently high resolution observations and that evidence of features associated with Kitty may be found in historical interferometric data. Finally, we will continue to monitor this interesting source and HartRAO is upgrading its spectrometer to allow for simultaneous observations at many transitions and for more sources with even shorter cadences.
\section*{Acknowledgments}
We thank Dr. Alet de Witt and Jonathan Quick for their efforts to schedule time around various other observing programmes at HartRAO. Correspondence with Professor John Bally was greatly appreciated. We thank an anonymous referee for their comments, they improved the quality of this paper.
The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under agreement by the Associated Universities, Inc. The Submillimeter Array is a joint project between the Smithsonian Astrophysical Observatory and the Academia Sinica Institute of Astronomy and Astrophysics, and is funded by the Smithsonian Institution and the Academia Sinica. Finally, we acknowledge Dr. Michael Gaylard posthumously for his leadership in maintaining these long-term monitoring programmes at HartRAO; without which this work would not have been possible.
\bibliographystyle{mnras}
|
{
"timestamp": "2018-04-17T02:09:39",
"yymm": "1804",
"arxiv_id": "1804.05308",
"language": "en",
"url": "https://arxiv.org/abs/1804.05308"
}
|
\section{Introduction}
The problem at the heart of computational chemistry is electronic structure calculation. This problem concerns calculating the properties of the stationary state describing many electrons interacting with some external potential and between each other via Coulomb repulsion.
The ability to efficiently solve these problems for the cases of many body systems can have huge effects in pharmaceutical development, materials engineering, and all areas of chemistry. Quantum computing proposes the possibility to efficiently solve this problem for molecules with many more electrons than what can currently be simulated by classical computers\cite{kais_book}.
The ability to calculate properties of large quantum systems using precise control of some other quantum system was first proposed by Feynman\cite{feynman1982simulating}. He pointed out that if you have enough control over the states of some quantum system, you can create an analogy to some other quantum system. Using the example of spin in a lattice imitating many properties of bosons in quantum field theory, he conjectured that if you have enough individual quantum systems you could simulate any arbitrary quantum mechanical system. Simulation of the electronic structure Hamiltonian works very similar to this. Using the Jordan-Wigner or Bravyi-Kitaev transformation \cite{bravyi2002fermionic, setia2018bravyi, steudtner2017lowering} you can map an electronic structure Hamiltonian to a spin-type Hamiltonian which preserves energy eigenvalues \cite{xia2017electronic}. Evolution under this spin-type Hamiltonian, $e^{-iHt}$, can then be approximately simulated on quantum computers.
Quantum simulation provides a new and efficient way to calculate eigenenergies of a given molecule. Classically the problem would have a computational cost which grows exponentially with the system size, $n$, the number of orbital basis functions \cite{troyer2005computational}.
However, based on the phase estimation algorithm \cite{lloyd1996universal,abrams1997simulation}, the molecular ground state energies can be calculated with gate depth $O(poly(n))$ \cite{aspuru2005simulated,whitfield2011simulation,wecker2014gate}.
The quantum circuit for the Hamiltonian is generally approximated through a Trotter-Suzuki decomposition.
It is shown that the Hamiltonian dynamics can also be simulated through a truncated Taylor series \cite{berry2015simulating}.
This method is generalized as quantum signal processing\cite{low2016hamiltonian}.
Babbush et al. \cite{babbush2017low} further shows that it is possible to reduce the gate depth of the circuit to $O(n)$ by using plane wave orbitals.
Recently, a direct circuit implementation of the Hamiltonian within the phase estimation (Direct-PEA) is presented by authors of paper \cite{daskin2017ancilla,daskin2017direct,daskin2018generalized}: the circuit designs are provided to the time evolution operator by using the truncated series such as $U = I-\frac{iH}{\kappa}$ and $U = tH+i(I-\frac{t^2H^2}{2})$, in which $\kappa$ and $t$ are parameters to restrict truncation error.
These unitary operators are much simpler to implement than those of a Trotter decomposition, and can be also used to calculate ground state energies of molecular Hamiltonians.
Another approach called variational quantum eigensolver (VQE) has been introduced by Aspuru-Guzik and coworkers\cite{peruzzo2014variational,mcclean2016theory}: This method combines classical and quantum algorithms together and significantly reduces the gate complexity at the cost of a large amount of measurements.
It has also been applied on real-world quantum computers to solve ground state energies of molecules such as: H$_2$, LiH and BeH$_2$ \cite{o2016scalable,kandala2017hardware}.
This paper explores all these above mentioned methods for calculating the ground state energies of the water molecule and presents a comparison study, in terms of both the accuracy and the gate complexity dependent on error.
The next section explains the method by which the electronic Hamiltonian for water is calculated and the method by which to reduce the number of qubits required to simulate the transformed spin-type Hamiltonian.
Then, Section III discusses five methods of electronic structure simulation on quantum computers: the phase estimation using first order Trotter-Suzuki decomposed propagator (Trotter PEA), two direct implementations of the spin-type Hamiltonian (Direct PEA), a direct measurement and a specific variational quantum eigensolver method(Pairwise VQE). Section IV shows results for these methods with comparison to the exact energy from direct diagonalization of the spin-type Hamiltonian. It also gives qubit requirement and gate complexity for different methods asymptotically. Spin-type Hamiltonian for H$_2$O at equilibrium bond length is derived in Appendix A. Details of both error and complexity analyses are given in Appendix B and Appendix C.
\section{Hamiltonian Derivation}
In this section we provide details for calculating the spin-type Hamiltonian describing electronic structures of the water molecule using STO-3G basis set that will be used in later methods. This derivation can be generalized to an arbitrary molecular Hamiltonian.
To obtain the Hamiltonian of the water molecule, we start by considering the 1s orbital of each hydrogen atom along with the 1s, 2s, 2p$_x$, 2p$_y$, 2p$_z$ orbitals for the oxygen atom. This leads to a total of 14 molecular orbitals considering spin. To make our simulations more efficient, the number of qubits is reduced by considering orbital energies and exploiting the symmetry of the system \cite{kandala2017hardware}.
It can be initially assumed that the two molecular orbitals of largest energies are unoccupied. Consequently, the calculation of the Hamiltonian of the water molecule then only requires the consideration of 12 spin-orbitals. After second quantization, the Hamiltonian can be expressed as \cite{lanyon2010towards}:
\begin{equation}
H= \sum_{i,j =1}^{12}h_{ij}a_i^\dagger a_j+\frac{1}{2}\sum_{i,j,k,l=1}^{12} h_{ijkl}a_i^\dagger a_j^\dagger a_ka_l.
\end{equation}
\noindent
Here $a_i^\dagger$ and $a_i$ are fermionic creation and annihilation operators, and $h_{i,j}$ and $h_{i,j,k,l}$ are one-body and two-body interaction coefficients. In this work the molecular orbitals are calculated from the Hartree-Fock method and represented by the STO-3G basis functions. The numerical integration obtaining the one and two electron integrals for molecular water is performed by the PyQuante package \cite{pyquante}. The expressions for these integrations are: \begin{equation}h_{ij} = \int{d\vec{r}_1\chi_i^*(\vec{r}_1)(-\frac{1}{2}\nabla_1^2-\sum\limits_{\sigma} \frac{Z_\sigma}{|\vec{r}_1-\vec{R}_\sigma|})\chi_j(\vec{r}_1)},\end{equation}
\begin{equation}h_{ijkl} = \int{d\vec{r}_1d\vec{r}_2\chi_{i}^*(\vec{r}_1)\chi_{j}^*(\vec{r}_2)\frac{1}{r_{12}}\chi_{k}(\vec{r}_2)\chi_{l}(\vec{r}_1)}.\end{equation}
\noindent
Here we have defined $\chi_i(\vec{r})$ as the $i^{th}$ spin-orbital, which is calculated from a spatial orbital obtained by the Hartree-Fock method and the electron spin states. $Z_\sigma$ is the $\sigma^{th}$ nuclear charge, $\vec{r_{i}}$ is the position of electron $i$, $r_{12}$ is the distance between the two points $r_1$ and $r_2$, and $\vec{R}_\sigma$ is the position of $\sigma^{th}$ nucleus.
We have ordered our spin-orbitals from 1 to 12 as follows: \{$1\uparrow,2\uparrow...,6\uparrow,1\downarrow,2\downarrow,...6\downarrow$\}, with first spin-up orbitals ordered from lowest to highest energy and continuing into spin-down orbitals ordered from lowest to highest energy. Now introduce an ad hoc set $F= \{1,2,7,8\}$ corresponding to the 4 lowest energy spin orbitals $\{1\uparrow,2\uparrow,1\downarrow,2\downarrow\}$. For the H$_2$O ground state, it can be assumed the spin orbitals in the set $F$ will be filled with electrons. The following one-body single electron interaction operators then become:
\begin{align}
&a_{1}^\dagger a_{1} = 1,
\text{\ \ }a_{2}^\dagger a_{2} = 1,\text{\ \ }a_{7}^\dagger a_{7} = 1,
\text{\ \ }a_{8}^\dagger a_{8} = 1,\nonumber\\
&a_{i}^\dagger a_{j} = 0,\text{\ if\ } i\neq j \text{\ , and\ } i \in F\text{\ or \ }j \in F.
\end{align}
\noindent
This assumption also allows us to simplify the two-electron interaction terms under certain conditions:
\begin{align}
a_i^\dagger a_j^\dagger a_ka_l =\left\{
\begin{aligned}
a_j^\dagger a_k & , & i=l,i\in F, \{j,k\} \notin F, \\
a_i^\dagger a_l & , & j=k,j\in F, \{i,l\} \notin F, \\
-a_j^\dagger a_l & , & i=k,i\in F, \{j,l\} \notin F, \\
-a_i^\dagger a_k & , & j=l,j\in F, \{i,k\} \notin F.
\end{aligned}
\right.
\end{align}
\noindent
Moreover, this ability to neglect creation or annihilation operator with subscript from $\{1,2,7,8\}$, along with the ability to neglect two-body operators containing an odd number of modes in F, allows us to relabel our orbital set 1 to 8, corresponding to spin-orbitals: $\{3\uparrow,4\uparrow,5\uparrow,6\uparrow,3\downarrow,4\downarrow,5\downarrow,6\downarrow\}$
Using the parity basis and taking advantage of particle and spin conversation, the required qubit number can be further reduced\cite{seeley2012bravyi}. In the parity basis:
\begin{align}
a_j^\dagger = X^\leftarrow_{j+1} \otimes \frac{1}{2}(X_j\otimes Z_{j-1}-iY_j),\\
a_j = X^\leftarrow_{j+1} \otimes \frac{1}{2}(X_j\otimes Z_{j-1}+iY_j),
\end{align}
where
\begin{align}
X^{\leftarrow}_{i} \equiv X_{n-1}\otimes X_{n-1} \otimes \cdots \otimes X_{i+1} \otimes X_i,\text{\ }n=8.
\end{align}
\noindent
This fermionic Hamiltonian can now be mapped to an 8-local Hamiltonian represented as a weighted sum of tensor products of Pauli matrix $\{I_i,X_i,Y_i,Z_i\}$, which almost preserves the ground state energy value. The new Hamiltonian in the electronic occupation number basis set can be mapped to the parity basis set as:
\begin{align}
\ket{f_1 f_2 ... f_8} \rightarrow \ket{q_1} \otimes \ket{q_2} \otimes ... \otimes \ket{q_8},
\end{align}
\noindent
where
\begin{align}
q_i = \sum_{k=1}^i f_k \text{\ mod 2} \in \{0,1 \}.
\end{align}
\noindent
Here $f_k$ represents the number of electrons occupying the $k^{th}$ spin-orbital, and $q_k$ represents the sum of electron numbers from $1^{st}$ to $k^{th}$ spin-orbital.
We can now assume that half of the left 6 electrons are spin-up and the other half are spin-down. If this is the case, $\ket{q_4} = \ket{1}$, and $\ket{q_8} = \ket{0}$, which means only $Z_4,I_4,Z_8,I_8$ will apply on these states\cite{bravyi2017tapering}. Since $Z_4\ket{q_4} = -\ket{q_4}, Z_8\ket{q_8} = \ket{q_8}$, all $Z_4,Z_8$ can be substituted by $-I_4$ and $I_8$, with this assumption we can now reduce this problem to a 6-local Hamiltonian.
\section{Methods of Simulation}
After the parity transformation and simplifications made above we now have a reduced 6-local Hamiltonian describing H$_2$O in the form: $H = \sum_{i=1}^{L}\alpha_ih_i$, where $\{\alpha_i\}$ is a set of coefficients, and $\{ h_i\}$ is a set of tensor products of the Pauli matrices $\{I_i,X_i,Y_i,Z_i\}$. Method A,B,C tries to evolve quantum system state by approximating the propagator $e^{-iHt}$, and then extract the ground state energy from the phase. Method D implements the Hamiltonian, H, directly into quantum circuit, and evaluate ground state energy by multiple measurements. Method E produces the ground state energies by iterations.
\subsection{Trotter Phase Estimate Algorithm (Trotter-PEA)}
For each term in a Hamiltonian, $H$, the propagator, $e^{-i\alpha_i h_it}$, can be easily constructed in a circuit. However, since most of the time the set of $h_i$ do not commute, the propagator cannot be implemented term by term: i.e., $e^{-iHt}\ne\prod_{i=1}^{L}e^{-i\alpha_ih_it}$. The first order Trotter-Suzuki decomposition \cite{trotter1959product,suzuki1976generalized,dhand2014stability} provides an easy way to decompose a propagator for the spin-type Hamiltonian given as a sum of non-commuting terms into a product of each non commuting term exponentiated for a small time $t$:
\begin{align}
U = \prod_{i=1}^{L}e^{-i\alpha_ih_i t} = e^{-iHt} + O(A^2t^2).
\end{align}
Here $A = \sum_{i=1}^L |\alpha_i|$, and we have an error of order $O(A^2t^2)$. Here we don't consider time slicing as the original Trotter-Suzuki decompositoin does, as $t$ can be adjusted to be as small as necessary for error control. This method requires only multi-qubit rotations, and therefore $U$ can be implemented easily on a state register.
After $U$ is obtained PEA can be applied to extract the phase. We can use extra ancilla qubits to achieve wanted accuracy by iterative measurements\cite{kitaev1995quantum,dobvsivcek2007arbitrary,aspuru2005simulated}. We call this PEA based on first order Trotter-Suzuki decomposition Trotter-PEA.
Higher order Trotter-Suzuki decompositions are also available, however they have more complicated formulations, especially for order higher than 2. Here we only discuss first order case for simplicity. For simulation, a forward iterative PEA \cite{daskin2018generalized}-which estimates the phase starting from the most significant bit-can be used to save more time. The circuit for the forward iterative PEA is shown in FIG. \ref{Fig1} which needs only 1 qubit for measurement.
\begin{figure}[H]
\centering
\includegraphics[]{f1.eps}
\caption{Forward iterative PEA circuit with initial state $\ket{0}\ket{\psi}_s$. Here $\ket{\psi}_s$ is the ground state of the Hamiltonian, $H$ is the Hadamard gate, $U$ is the approximate propagator and $R_z(-\frac{\pi}{2})$ is a Z rotation gate. \label{Fig1}}
\end{figure}
\noindent
Then the generated state before the measurement is:
\begin{align}
\frac{1+e^{i2\pi(0.\phi_{k+1}\phi_{k+2}...-0.01)}}{2}e^{i\frac{\pi}{4}}\ket{0}\ket{\psi}_s + \frac{1-e^{i2\pi(0.\phi_{k+1}\phi_{k+2}...-0.01)}}{2}e^{i\frac{\pi}{4}}\ket{1}\ket{\psi}_s,
\end{align}
\noindent
Note decimals above are in binary. It can be checked if the measurement qubit has a greater probability of output 1, $\phi_{k+1} = 1$, otherwise $\phi_{k+1} = 0$. Then the ground state energy can be calculated as $E = -2\pi \times 0.\phi_1\phi_2\phi_3...$.
\subsection{Direct Implementation of Hamiltonian in First Order (Direct-PEA ($1^{st}$ order))}
It was proposed\cite{daskin2017direct} that given a Hamiltonian $H$ and large $\kappa$ we can construct an approximated unitary operator $U$ such that:
\begin{align}
U=I-i\frac{H}{\kappa}, \text{\ } \kappa \gg \sum_{i=1}^L |\alpha_i| \geq ||H||.
\end{align}
\noindent
If $\ket{\psi}_s$ is an eigenvector of $H$ and $E$ is the corresponding eigenvalue, then:
\begin{align}
U\ket{\psi}_s = \left(I-i\frac{H}{\kappa}\right)\ket{\psi}_s \approx e^{-i\frac{H}{\kappa}} \ket{\psi}_s = e^{-i\frac{E}{\kappa}} \ket{\psi}_s.
\end{align}
\noindent
The eigenvalue of $\ket{\psi}_s$ would be encoded directly in the approximate phase. This is the motivation behind directly implementing the Hamiltonian in quantum simulation.\\
\noindent
To implement this non-unitary matrix $U$, we can enlarge the state space and construct a unitary operator $U_r$\cite{berry2015simulating}. Rewrite U as:
\begin{align}
U = I - \frac{i}{\kappa} \sum_{j=1}^L \alpha_j h_j = \sum_{j=0}^{L} \beta_j V_j,
\end{align}
in which $\beta_j \geq 0$ and $V_j$ is unitary. By introducing a $m$-qubit ancilla register, where $m = \lceil \log_2 L \rceil$, we can construct a multi-control gate, $V$, such that:
\begin{align}
V\ket{j}_a\ket{\psi}_s=\ket{j}_a V_j \ket{\psi}_s.
\label{gateV}
\end{align}
\noindent
Define $\beta_j = 0$ when $ L < j \leq 2^m$ and $B$ as a unitary operator that acts on ancilla qubits as:
\begin{align}
B\ket{0}_a=\frac{1}{\sqrt{s}}\sum_{j=0}^{2^m}\sqrt{\beta_j}\ket{j}_a,\ s=\sum_{j=0}^{2^m}\beta_j.
\label{gateB}
\end{align}
\noindent
Define $U_r$ and $\Pi$ such that:
\begin{align}
&U_r = (B^\dagger \otimes I^{\otimes n}) V (B \otimes I^{\otimes n}),\\
&\Pi = \ket{0}_a \bra{0}_a \otimes I^{\otimes n}.
\end{align}
\noindent
Apply $U_r$ on input state $\ket{0}_a \ket{\psi}_s$:
\begin{align}
U_r \ket{0}_a \ket{\psi}_s &= (B^\dagger \otimes I^{\otimes n}) V (B \otimes I^{\otimes n}) \ket{0}_a \ket{\psi}_s \nonumber\\
&= (B^{\dagger} \otimes I^{\otimes n}) V \frac{1}{\sqrt{s}}\sum_{j=0}^{2^m}\sqrt{\beta_j}\ket{j}_a \ket{\psi}_s \nonumber\\
&= (B^{\dagger} \otimes I^{\otimes n}) \frac{1}{\sqrt{s}}\sum_{j=0}^{2^m}\sqrt{\beta_j}\ket{j}_a V_j \ket{\psi}_s \nonumber\\
&= \Pi (B^{\dagger} \otimes I^{\otimes n}) \frac{1}{\sqrt{s}}\sum_{j=0}^{2^m}\sqrt{\beta_j}\ket{j}_a V_j \ket{\psi}_s+ (I^{\otimes m+n} - \Pi) (B^{\dagger} \otimes I^{\otimes n}) \frac{1}{\sqrt{s}}\sum_{j=0}^{2^m}\sqrt{\beta_j}\ket{j}_a V_j \ket{\psi}_s \nonumber\\
&= (B\ket{0}_a)^\dagger \frac{1}{\sqrt{s}}\sum_{j=0}^{2^m}\sqrt{\beta_j}\ket{j}_a V_j \ket{\psi}_s + \sum_{j=1}^{j=2^m}\ket{j}_a \ket{u_j}_s \nonumber\\
&= \frac{1}{s} \ket{0}_a U \ket{\psi}_s + \ket{\Phi^\perp_1},
\end{align}
\noindent
where $\ket{\Phi^\perp_1}$ is orthogonal to $\ket{0}_a \ket{\psi}_s$. Then the approximated unitary operator $U$ is implemented by unitary operator $U_r$, which can be seen in FIG. \ref{ur.png}.
\begin{figure}[H]
\centering
\includegraphics[]{f2.eps}
\caption{Gate $U_{r}$ in Direct PEA circuit, gates $V$ and $B$ are shown in Eq. (\ref{gateV}) and Eq. (\ref{gateB})}
\label{ur.png}
\end{figure}
\noindent
Since $\kappa \gg ||H|| \geq E$, energy of eigenstate $\ket{\psi}_s$ is successfully implemented in phase:
\begin{align}
U_r \ket{0}_a \ket{\psi}_s &= \frac{1-i\frac{E}{\kappa}}{s} \ket{0}_a \ket{\psi}_s + \ket{\Phi^\perp_1} \nonumber\\
&= \frac{\sqrt{1+\frac{E^2}{\kappa^2}}}{s} e^{-i\tan^{-1} \frac{E}{\kappa}} \ket{0}_a \ket{\psi}_s + \ket{\Phi^\perp_1} \nonumber\\
&= pe^{-i\tan^{-1}\frac{E}{\kappa}} \ket{0}_a \ket{\psi}_s + \sqrt{1-p^2} \ket{\Phi^\perp}.
\end{align}
Here $p$ is defined by $\frac{\sqrt{1+\frac{E^2}{\kappa^2}}}{s}$, and $\ket{\Phi^\perp}$ is normalized.\\
This $U_r$ gate would then be used for PEA or iterative PEA process. For an accurate output, $p$ is required to be as close to 1 as possible. Using oblivious amplitude amplification\cite{berry2017exponential}, we can amplify that probability without affecting phase. Define the operator $U_0 =2\ket{0}_a \bra{0}_a-I^{\otimes m}$ and rotational operator:
\begin{align}
Q =U_r (U_0 \otimes I^{\otimes n}) U_r^\dagger (U_0 \otimes I^{\otimes n}).
\end{align}
\noindent
Iterating this operator $N$ times, we can achieve $U_q = Q^NU_r$ which brings $p$ close to $1$ by performing rotations within the space $span\{\ket{0}_a \ket{\psi}_s, \ket{\Phi^\perp}\}$. The details are in Supplementary Materials. Take the same circuit and the same procedure in Trotter-PEA, except replacing $U$ by $U_q$, we are able to get ground state energy of water molecule.
\subsection{Direct Implementation of Hamiltonian in Second Order (Direct-PEA ($2^{nd}$ order))}
Propagator $e^{-iHt}$ can also be approximated up to second order\cite{daskin2018generalized}:
\begin{align}
&U = I-iHt-\frac{H^2t^2}{2} = e^{-iHt} + O((At)^3).
\end{align}
\noindent
When $At$ is very small, $U$ would be a good approximation. Since $U$ is nonunitary, we have to construct a unitary operator $U_{r2}$ to implement it into a quantum circuit. With $U_r$ in method B, $B_2$ defined with the property:
\begin{align}
B_2 \ket{00} = \frac{\sqrt{t}\ket{00} + \ket{01} + \frac{t}{\sqrt{2}}\ket{10}}{\sqrt{1+t+\frac{t^2}{2}}},
\label{gateB2}
\end{align}
and gate $P$ constructed as:
\begin{equation}
P =
\begin{bmatrix}
I^{\otimes n}& 0 & 0 & 0\\
0 & 0 & I^{\otimes m} & 0\\
0 & I^{\otimes m} & 0 & 0\\
0 & 0 & 0 & I^{\otimes n}
\label{gateP}
\end{bmatrix}.
\end{equation}
\noindent
We can construct $U_{r2}$ as in FIG \ref{ur2.png}:
\begin{figure}[H]
\centering
\includegraphics[]{f3.eps}
\caption{Gate $U_{r2}$ in Second order Direct PEA circuit, with $B_2$ and $P$ defined in Eq. (\ref{gateB2}) and Eq. (\ref{gateP})}
\label{ur2.png}
\end{figure}
\noindent
which satisfies:
\begin{align}
U_{r2}\ket{00}\ket{0}_a\ket{\psi}_s &=\frac{1-i\frac{Et}{A}+\frac{E^2t^2}{2A^2}}{1+t+\frac{t^2}{2}} \ket{00}\ket{0}_a\ket{\psi}_s + \sum_{j=1}^{2^{m+2}}\ket{j}\ket{v_j}_{s} \nonumber\\
&= \frac{\sqrt{1+\frac{E^4t^4}{4A^2}}}{1+t+\frac{t^2}{2}} e^{-i\tan^{-1}{\frac{\frac{Et}{A}}{1+\frac{E^2t^2}{2A}}}} \ket{00}\ket{0}_a\ket{\psi}_s + \ket{\Psi_1^\perp}.
\end{align}
\noindent
In the formula, $A = \sum_{i=1}^{2^m-1} \beta_i = \sum_{i=1}^L |\alpha_i| \geq |E|$, and $\ket{\Psi_1^{\perp}}$ is perpendicular to $\ket{00}\ket{0}_a \ket{\psi}_s$. Just as in last section, we can rotate the final state to make the proportion of $\ket{00}\ket{0}_a\ket{\psi}_s$ as close to 1 as possible. Then we can apply PEA or iterative PEA to get the phase, $-\tan^{-1}{\frac{\frac{Et}{A}}{1+\frac{E^2t^2}{2A}}}$, which leads to ground state energy corresponding to ground state $\ket{\psi}_s$.
\subsection{Direct Measurement of Hamiltonian}
Another way to calculate the ground state energy is by direct measurement after implementing a given Hamiltonian as a circuit. Since Direct-PEA ($1^{st}$ order) method has already introduced a way to implement non-unitary matrix $U$ into circuit, Hamiltonian implementation is straightforward. We can just replace $U$ in method B by $U' = H = \sum_{j=1}^{L} \alpha_j h_j$, and obtain $U_r'$ such that:
\begin{align}
U_r'\ket{0}_a\ket{\psi}_s &= \frac{1}{s'}\ket{0}_a U_r' \ket{\psi}_s + \ket{\Phi_1^{'\perp}} \nonumber\\
&= \frac{E}{A} \ket{0}_a \ket{\psi}_s + \ket{\Phi_1^{'\perp}}.
\end{align}
\noindent
By measuring ancilla qubits multiple times, we can get the energy of the ground state $\ket{\psi}_s$ by multiplying $A$ by the square root of probability of getting all 0s.
This method can also be used for non-hermitian Hamiltonians. If now the eigenvalue for $\ket{\psi}_s$ is a complex number $E = |E|e^{i\theta}$, by replacing $U$ by $U' = H$ in method $B$, we would have:
\begin{align}
U_r'\ket{0}_a\ket{\psi}_s = \frac{|E|e^{i\theta}}{A} \ket{0}_a \ket{\psi}_s + \ket{\Phi_1^{'\perp}},
\end{align}
and can obtain $|E|$ through measurements. Then by replacing $U$ by $U'' = \frac{|E|}{A}I + H$ in method $B$, we would have:
\begin{align}
U_r''\ket{0}_a\ket{\psi}_s = \frac{|E|}{A}(1+e^{i\theta}) \ket{0}_a \ket{\psi}_s + \ket{\Phi_1^{'\perp}},
\end{align}
and can measure the absolute value of $\frac{|E|}{A} (1+e^{i\theta})$, which is $2\frac{|E|}{A} \cos \theta$. This helps determine the phase of a complex eigenenergy.
\subsection{Variational Quantum Eigensolver}
Recently the variational quantum eigensolver method has been put forward by Aspuru-guzik and coworkers to calculate the ground state energies\cite{peruzzo2014variational,mcclean2016theory,o2016scalable,kandala2017hardware,gilyen2017optimizing}, which is a hybrid method of classical and quantum computation. According to this method, an adjustable quantum circuit is constructed at first to generate a state of the system. This state is then used to calculate the corresponding energy under the system's Hamiltonian. Then by a classical optimization algorithm, like Nelder-Mead method, parameters in circuit can be adjusted and the generated state will be updated. Finally, the minimal energy will be obtained. The detailed circuit for the quantum part of our algorithm is shown in FIG.\ref{vqe1.png}. To make the expression more clear, we represent parameters in vector form, as follows: $\boldsymbol{\theta} = ( \boldsymbol{\theta_1}, \boldsymbol{\theta_2}..., \boldsymbol{\theta_D} ) $, $\boldsymbol{\theta_i} = (\boldsymbol{\theta_{i,0}}, \boldsymbol{\theta_{i,1}}..., \boldsymbol{\theta_{i,11})} $, $\boldsymbol{\theta_{i,j}} = (\theta_{i,j,1}, \theta_{i,j,2}, \theta_{i,j,3}, )$, $\boldsymbol{\varphi} = ( \boldsymbol{\varphi_1}, \boldsymbol{\varphi_2}..., \boldsymbol{\varphi_n} )$, $\boldsymbol{\varphi_k} = (\varphi_{k,1}, \varphi_{k,2}, \varphi_{k,3})$.
\begin{figure}[H]
\centering
\small
\includegraphics[]{f4.eps}
\caption{Circuit for state preparation and corresponding energy evaluation. $G({\boldsymbol\theta}_i)$ is entangling gate, in this paper we are taking the gate like FIG. \ref{vqe_v5.png}. $U(\boldsymbol{\varphi}_k)$ is an arbitrary single-qubit rotation and is equal to $R_{z}(\varphi_{k,1})R_{x}(\varphi_{k,2})R_{z}(\theta_{k,3})$ with parameters $\varphi_{k,1}$,$\varphi_{k,2}$ and $\varphi_{k,3}$ that can be manipulated. By increasing the number of layers, $d$, of our circuit, we are able to produce more complex states.}
\label{vqe1.png}
\end{figure}
\begin{figure}[H]
\centering
\footnotesize
\includegraphics[]{f5.eps}
\caption{Example entangling circuit $G(\boldsymbol\theta_i)$ for 4-qubit system. There are 12 arbitrary single-qubit gates $U_j$, a simplified written way for $U(\boldsymbol{\theta}_{i,j})$, which is $R_{z}(\theta_{i,j,1})R_{x}(\theta_{i,j,2})R_{z}(\theta_{i,j,3})$ with parameters $\theta_{i,j,1}$,$\theta_{i,j,2}$ and $\theta_{i,j,3}$ that can be manipulated. Each 2 qubits are entangled sequentially. Entangling gate $G(\boldsymbol\theta_i)$ for $n$-qubit system is similar to this gate, but then it has $n(n-1)$ arbitrary single-qubit gates and $\boldsymbol{\theta}_i$ has $3n(n-1)$ parameters.}
\label{vqe_v5.png}
\end{figure}
\noindent
We are using $d$ layers of gate $G({\boldsymbol{\theta}}_i)$ in FIG. \ref{vqe1.png} to entangle all qubits together. Here we introduce a hardware-efficient $G({\boldsymbol{\theta}}_i)$, and we call this method Pairwise VQE. The example gate of $G({\boldsymbol{\theta}}_i)$ for 4 qubits is shown in FIG. \ref{vqe_v5.png}. The entangling gate for 6-qubit system H$_2$O is similar: every 2 qubits are modified by single-qubit gates and entangled by $CNOT$ gate. By selecting initial value of all $\boldsymbol\theta_i$ and $\boldsymbol\varphi_k$, system state can be prepared by $d$ layers $G({\boldsymbol{\theta}}_i)$ gates and arbitrary single gates $U({\boldsymbol{\varphi}}_j)$. Then average value of each term in Hamiltonian $H$, $\langle h_j \rangle$ , can be evaluated by measuring qubits many times after going through gates like $I$ or $R_{x_j}(\frac{\pi}{2})$ or $R_{y_j}(-\frac{\pi}{2})$. For example, if $h_j = I_0X_1Y_2Z_3$, then
\begin{align*}
\langle h_j \rangle &= \langle I_0X_1Y_2Z_3\rangle_{\psi} = \bra{\psi} I_0X_1Y_2Z_3 \ket{\psi}\\
&= (\bra{\psi} R_{y_1}(\frac{\pi}{2})R_{x_2}(-\frac{\pi}{2})) I_0 (R_{y_1}(-\frac{\pi}{2})X_1R_{y_1}(\frac{\pi}{2})) (R_{x_2}(\frac{\pi}{2})Y_2R_{x_2}(-\frac{\pi}{2})) Z_3 (R_{y_1}(-\frac{\pi}{2})R_{x_2}(\frac{\pi}{2})\ket{\psi})\\
&= \langle I_0 Z_1 Z_2 Z_3\rangle_{\psi'} \text{\quad, where} \ket{\psi'} = R_{y_1}(-\frac{\pi}{2})R_{x_2}(\frac{\pi}{2})\ket{\psi},
\end{align*}
So we can let the quantum state after $U(\boldsymbol{\varphi_j})$ go through gates $R_{y_1}(-\frac{\pi}{2})$ and $R_{x_2}(\frac{\pi}{2})$ and then measure the result state multiple times to get $\langle h_j \rangle$. The energy corresponding to the state can be obtained by $ \langle H \rangle (\boldsymbol{\theta},\boldsymbol{\varphi}) = \sum_{j=1}^{L} \alpha_j \langle h_j\rangle (\boldsymbol{\theta},\boldsymbol\varphi)$. Then $\boldsymbol{\theta}$ and $\boldsymbol{\varphi}$ can be updated by classical optimization method and $\langle H \rangle(\boldsymbol\theta, \boldsymbol\varphi)$ can reach the minimal step by step.
\section{Results and Method Comparison}
The Hamiltonian of the water molecule is calculated for O-H bond lengths ranging from 0.5 a.u. to 2.9 a.u., using the methods introduced in Section II. This Hamiltonian is used in all five of the methods discussed within this paper. For the methods A-D, the input state of system is the ground state of the H$_2$O molecule. For each of these methods, the resulting ground state energy curve can be calculated to arbitrary accuracy (for details of error analysis see Appendix B). The results from each method is compared with result from a direct diagonalization of the Hamiltonian, as shown below. From FIG. \ref{result1.png} it can be seen that all of these methods are effective in obtaining the ground state energy problem of the water molecule. We also use method E (Pairwise VQE) to obtain the ground state energy. These results can be seen in FIG. \ref{vqe_result.eps}. Energy convergence at 1.9 a.u. can be seen in FIG. \ref{vqe_v5_iteration.eps} and the ground state energy curve calculated by this method is in FIG. \ref{vqe_energy.eps}. In this simulation, $d$ is selected to be 1, and $G(\boldsymbol{\theta_i})$ is constructed as described above, and it can already give a very accurate result. This shows Pairwise VQE a very promising method for solving electronic structure problems. Furthermore, Pairwise VQE has only $O(n^2d)$ gate complexity and doesn't require initial input of the ground state, which makes it more practical for near-term applications on a quantum computer.
\begin{figure}[h]
{
\begin{minipage}[t]{0.45\linewidth}
\centering
\includegraphics[width=3in]{f6a.eps}
\caption*{(a)}
\label{fig:side:a}
\end{minipage}%
\begin{minipage}[t]{0.45\linewidth}
\centering
\includegraphics[width=3in]{f6b.eps}
\caption*{(b)}
\label{fig:side:b}
\end{minipage}
\begin{minipage}[t]{0.45\linewidth}
\centering
\includegraphics[width=3in]{f6c.eps}
\caption*{(c)}
\label{fig:side:c}
\end{minipage}
\begin{minipage}[t]{0.45\linewidth}
\centering
\includegraphics[width=3in]{f6d.eps}
\caption*{(d)}
\label{fig:side:d}
\end{minipage}
}
\captionsetup{justification=raggedright,singlelinecheck=false}
\caption{
Ground State Energy Curve for H$_2$O, as a function of the bond length O-H in a.u. for (a) the Trotter-PEA, (b) the Direct-PEA ($1^{st}$ order), (c) the Direct-PEA ($2^{nd}$ order) and (d) Direct Measurement method ($1.6\times 10^8$ measurements), compared with the exact diagonalization. Errors are shown in the window of each figure. One thing to mention is that we can not tell whether one method have better property over another directly from these figures, because they have different parameters, gates etc. For comparison, we have to turn to gate complexity analysis in TABLE \ref{complexity_table}.
}
\label{result1.png}
\end{figure}
\begin{figure}
\begin{subfigure}{0.4\linewidth}
\includegraphics[scale=0.45]{f7a.eps}
\captionsetup{justification=raggedright,singlelinecheck=false}
\caption{Convergence of ground state energy of H$_2$O for fixed O-H bond length = 1.9 a.u., as number of iterations increases. The lines for exact ground state energy and for the limit almost overlap.}
\label{vqe_v5_iteration.eps}
\end{subfigure}
\begin{subfigure}{0.4\linewidth}
\includegraphics[scale=0.45]{f7b.eps}
\captionsetup{justification=raggedright,singlelinecheck=false}
\caption{Ground state energy curve for H$_2$O, as a function of O-H bond length in a.u. for variational quantum eigensolver. Errors are shown in the window of the figure.}
\label{vqe_energy.eps}
\end{subfigure}
\captionsetup{justification=raggedright,singlelinecheck=false}
\caption{Result from Pairwise VQE using the entangling gates in FIG. \ref{vqe_v5.png}. We take $\ket{0}_s$ as initial input, $d=1$ layer and use Nelder-Mead algorithm for optimization. }
\label{vqe_result.eps}
\end{figure}
\newpage
\noindent
Qubit requirement, gate complexity and number of measurements of different methods are analyzed in Appendix C and shown in TABLE \ref{complexity_table}. When counting gate complexities, we decompose all gates into single qubit gates and CNOT gates. While Pairwise VQE needs only n qubits, the other methods require extra number of qubits. In terms of gate scaling, Pairwise VQE also needs the least gates, which enables it to better suit the applications on near and intermediate term quantum computers. Among the remaining four methods, Direct Measurement requires less number of gates than the others. PEA-type methods have an advantage that they can give an accurate result under only $O(1)$ measurements. However, they need more qubits compared with the previous two methods and demands many more gates if smaller error is required. Due to huge gate complexity, these PEA-type algorithms would be put into practice only when the decoherence problem has been better solved. Among these three PEA based methods, in terms of the gate complexity, Direct-PEA(2$^{nd}$ order) requires less number of gates than the traditional Trotter-PEA and Direct-PEA(1$^{st}$ order) which is proved in Appendix C. One more thing to mention is that here the second quantization form Hamiltonian is based on STO-3G, so there are $O(n^4)$ terms. If a more recent dual form of plane wave basis \cite{babbush2017low} is used, the number of terms can be reduced to $O(n^2)$, and the asymptotic scaling in TABLE \ref{complexity_table} would also be reduced. To be specific, for PEA-type methods, upper bounds of gate complexities would be proportional to $n^3$ rather than $n^5$, and Number of Measurements for Pairwise VQE would be proportional to $n^4$ rather than $n^8$. As can be seen, these reductions wouldn't influence the comparison made above.\\
\begin{table}
\begin{tabular}{ |p{3.6cm}||p{3.5cm}|p{3cm}|p{3.9cm}| }
\hline
Method & Qubits Requirement & Gate Complexity & Number of Measurements\\
\hline
Trotter-PEA & $O(n)$ &$O(\frac{n^5}{(\epsilon/A)^2})$ & $O(1)$\\
Direct-PEA(1$^{st}$ order)& $O(n)$ & $O(\frac{n^5}{(\epsilon/A)^{2.5}})$ &$O(1)$\\
Direct-PEA(2$^{nd}$ order)& $O(n)$ & $O(\frac{n^5}{(\epsilon/A)^{1.3}})$ &$O(1)$\\
Direct Measurement & $O(n)$ & $O(n^5)$ & $O(\frac{E^2}{\epsilon^2}) $\\
Pairwise VQE & $n$ & $O(n^2d)$ & $O(\frac{A^2n^8}{\epsilon^2}N_{iter})$\\
\hline
\end{tabular}
\captionsetup{justification=raggedright,singlelinecheck=false}
\caption{Complexity of different methods. $n$ is the number of qubits for molecular system, 6 for water in this paper. $ A = \sum_{i=1}^L |\alpha_i|$ can serve as the scale of energy. E is the exact value of ground energy. $\epsilon$ is the accuracy of energy we want to reach. $d$ is the number of layers we used in Pairwise VQE. $N_{iter}$ is the number of iterations for optimization in Pairwise VQE. See Appendix C for details.}
\label{complexity_table}
\end{table}
\section{Excited states and resonances}
All the aforementioned methods can also be applied for the excited state energy calculation. For PEA-type methods and Direct Measurement method, it can be simply done by replacing the input system state by an excited state. The complexity for the calculation is the same. The energy accuracies for excited states are also similar to that for the ground state. For VQE, a recent publication \cite{colless2018computation,sim2018quantum} presents a quantum subspace expansion algorithm (QSE) to calculate excited state energies. They approximate a ``subspace'' of low-energy excited states from linear combinations of states of the form $O_i\ket{\psi}_s$, where $\ket{\psi}_s$ is the ground state determined by VQE and $O_i$ are chosen physically motivated quantum operators. By diagonalizing the matrix with elements $\bra{\psi}_sO_i^\dagger H O_j \ket{\psi}_s$ calculated by VQE, one is able to find the energies of excited states.
FIG. \ref{result_excited.png} shows the simulation of the first six excited states' energy curves of the water molecule from our 6-qubit Hamiltonian, calculated by PEA-type methods and Direct Measurement method. It can be seen that the $5^{th}$ excited energy curve indicates a shape resonance phenomenon, which can be described by a non-Hermitian Hamiltonian with complex eigenvalues. The life time of the resonance state is associated with the imaginary part of the eigenvalues. In this way, to solve the resonance problem, we can seek to solve the eigenvalues of non-Hermitian Hamiltonians.
\begin{figure}[]
{
\includegraphics[]{f8.eps}
}
\captionsetup{justification=raggedright,singlelinecheck=false}
\caption{
Excited states' energy curves for H$_2$O, as a function of the bond length O-H in a.u.. Markers with different colors represent data points calculated from different methods. Only a few points for each method are drawn for illustration. Energy curves in different line styles are calculated from exact diagonalization of Hamiltonian matrix.}
\label{result_excited.png}
\end{figure}
Some work has been done on this track to solve the resonance problem by quantum computers. By designing a general quantum circuit for non-unitary matrices, Daskin et al.\cite{daskin2014universal} explored the resonance states of a model non-Hermitian Hamiltonian. To be specific, he introduced a systematic way to estimate the complex eigenvalues of a general matrix using the standard iterative phase estimation algorithm with a programmable circuit design. The bit values of the phase qubit determines the phase of eigenvalue, and the statistics of outcomes of the measurements on the phase qubit determines the absolute value of the eigenvalue. Other approaches for solving complex eigenvalues can also be applied for this resonance problem. For example, Wang et al. \cite{wang2010measurement} proposed a measurement-based quantum algorithms for finding eigenvalues of non-unitary matrices. Terashima et al.\cite{terashima2005nonunitary} introduced a universal nonunitary quantum circuit by using a specific type of one-qubit non-unitary gates, the controlled-NOT gate, and all one-qubit unitary gates, which is also useful for finding the eigenvalues of a non-hermitian Hamiltonian matrix.
Method D in section III can also be used for solving complex eigenvalues and the complexity is polynomial in system size. After applying complex-scaling method\cite{simon1973resonances} to water molecule's Hamiltonian and obtaining a non-Hermitian Hamiltonian, we can make enough quantum measurements to get an accurate resonance width $\Gamma$, which is actually the imaginary part of Hamiltonian's eigenvalue\cite{moiseyev1979autoionizing}. Another easier way to solve this resonance problem is, we can first choose proper $a$ and $J$ to fit the potential energy in a widely studied Hamiltonian \cite{moiseyev1978resonance,moiseyev1984resonances,serra2001crossover}:
\begin{align}
H(x) = \frac{p^2}{2} + (\frac{x^2}{2}-J)e^{-ax^2}
\end{align}
to our energy curve. Then by complex-scaling method, the internal coordinates of the Hamiltonian is dilated by a complex factor $\eta = \alpha e^{-i\theta}$ such that $H(x) \rightarrow H(x/\eta) \equiv H_\eta(x)$. We can solve the complex eigenvalue of $H_{\eta}(x)$ by the method D or using our previous method \cite{daskin2014universal}.
\section{Conclusion}
In this study we have compared several recently proposed quantum algorithms when used to compute the electronic state energies of the water molecule. These methods include first order Trotter-PEA method based on the first order Trotter decomposition, first and second order Direct-PEA methods based on direct implementation of the truncated propagator, Direct Measurement method based on direct implementation of the Hamiltonian and Pairwise PEA method, a VQE algorithm with a designed ansatz.
After deriving the Hamiltonian of the water molecular using the STO-3G basis set, we have explained in detail how each method works and derived their qubit requirements, gate complexities and measurement scalings. We have also calculated the ground state energy of the water molecular and shown the ground energy curves from all five methods. All methods are able to provide an accurate result. We have compared these methods and concluded that the second order Direct-PEA provides the most efficient circuit implementations in terms of gate complexity. With large scale quantum computation, the second order direct method seems to better suit large molecule systems. In addition, since Pairwise VQE requires the least qubit number, it is the most practical method for near-term applications on the current available quantum computers.
Moreover, we have applied our PEA-type methods and Direct Measurement method to solve excited state energy curves for water molecule. The fifth excited state energy curve implies shape resonance. We have introduced recent work on quantum algorithms for solving the molecular resonance problems and given two possible ways to solve the water molecule resonance properties, including our Direct Measurement method which is able to solve the problem efficiently.\\
\noindent
\textbf{Acknowledgements}:\\
This material is based upon work supported by the U.S. Department of Energy, Office of Basic Energy Sciences, under Award Number DE-SC0019215. Author Daniel Murphy acknowledges the support of NSF REU award PHY-1460899.
|
{
"timestamp": "2019-01-29T02:05:21",
"yymm": "1804",
"arxiv_id": "1804.05453",
"language": "en",
"url": "https://arxiv.org/abs/1804.05453"
}
|
\section{Introduction}
Collective communication~\cite{MPICollective} is frequently used by the programmers and designers of parallel programs, especially in high performance computing (HPC) applications related to scientific simulations and data analysis, including machine learning calculations. Usually, collective operations, e.g. implemented in MPI~\cite{MPI}, are based on algorithms optimized for the simultaneous entering of all participants into the operation, i.e. they do not take into consideration possible differences in process arrival times (PATs), thus, in real environment, where such imbalances are ubiquitous, they can have significant performance issues. It is worth to note that well performing algorithms for the balanced times, work poorly in the opposite case~\cite{Faraj2008}.
Fig.~\ref{fig:pap-ex} presents an example of a typical execution of a distributed program, where after the computation phase all processes exchange data with each other using some kind of collective communication operation, e.g. all-reduce. We can observe that even for the same computation volume, different processes arrive at the communication phase in different time, in the example processes $1$ is the slowest. Sometimes such differences can be observed in the communication, where the data exchange time is shorter for process 1 than for the other processes.
\begin{figure}[!h]
\includegraphics[width=12cm]{Fig1}
\caption{Example of a process arrival pattern with the process \#1 delayed, where $P$ is the number of cooperating processes, $a_i$, $f_i$ and $e_i$ are respectively arrival, finish and elapsed times of process $i$ for the performed collective communication operation}
\label{fig:pap-ex}
\end{figure}
As a contribution of this paper, we present two new algorithms for the all-reduce operation, optimized for imbalanced process arrival patterns (PAPs): sorted linear tree (SLT) and pre-reduced ring (PRR). We described their idea, pseudo-code, implementation details, benchmark for their evaluation as well as a real case example related to machine learning. Additionally we introduced a new way of on-line PAP detection, including PAT estimations and their distribution among cooperating processes.
The following section presents the related works in the subject, the next one describes the used computation and communication model, section~\ref{sec:algs} presents the proposed algorithms, section~\ref{sec:experiments} provides the evaluation of the algorithms using a benchmark, section~\ref{sec:ml} shows a real case example of the algorithms' utilization, and the last section presents the final remarks.
\section{Related works}
We grouped the related works into three areas: the all-reduce operation in general, i.e. the review of the currently used algorithms in different implementations of MPI~\cite{MPI}, then we describe the current state of the art in process arrival patterns (PAPs), and finally we present the works related to process arrival times (PATs) on-line monitoring and estimation.
\subsection{All-reduce operation}
All-reduce operation is one of the most common collective operations used in HPC software~\cite{Faraj2008}. We can define it as a reduction in a vector of numbers using a defined operation, e.g. sum, which needs to be cumulative, and then distribution of the result into all participating processes, in short: all-reduce = reduce + broadcast. There are plenty of all-reduce algorithms, Table~\ref{tab:ar-mpi} summarizes the ones used in two currently most popular, open source MPI implementations: OpenMPI~\cite{OpenMPI} and MPICH~\cite{MPICH}.
\begin{table}[h]
\caption{All-reduce algorithms implemented in OpenMPI~\cite{OpenMPI} and MPICH~\cite{MPICH}}
\label{tab:ar-mpi}
\begin{tabular}{|p{3.2cm}|p{3.8cm}|p{3.8cm}|}
\hline
Selection criteria & OpenMPI & MPICH\\\hline
Short messages & Recursive doubling~\cite{Thakur2005} & Recursive doubling~\cite{Thakur2005}\\\hline
Long messages & Ring~\cite{Thakur2005} or Segmented ring~\cite{OpenMPI} & Rabenseifner~\cite{Rabenseifner2004}\\\hline
Non-commutive reduce operation & Nonoverlapping~\cite{OpenMPI} & Recursive doubling~\cite{Thakur2005}\\\hline
Others & Basic linear~\cite{OpenMPI}, Segmented linear tree~\cite{OpenMPI} & {} \\\hline
\end{tabular}
\end{table}
The \emph{basic linear} algorithm performs linear reduce (flat tree: the root process gathers and reduces all data from the other processes) followed by the broadcast without any message segmentation~\cite{OpenMPI}. \emph{Segmented linear tree} creates pipeline between the participating processes, where the data are split into segments and sent from process $0$ to $1$ to $2$~\ldots to $P-1$. The \emph{nonoverlapping} algorithm uses default reduce followed by the broadcast, these operations are not overlapping: the broadcast is performed sequentially after the reduce, even if both use segmentation and some segments are ready after the reduce~\cite{OpenMPI}. \emph{Recursive doubling} (a.k.a. \emph{butterfly} and \emph{binary split}) is performed in rounds, in every round each process exchanges and reduces its data with another, corresponding process, whose rank changes round by round, after $log_2P$ rounds ($P$ is the total number of participating processes) all data are reduced and distributed to the processes~\cite{Thakur2005}. \emph{Ring} is also performed in rounds, the data are divided into $P$ segments and each process sends one segment per round (the segment index depending on the round number and the process rank) to the next process (the processes are usually ordered according to their ranks). At the beginning, there are performed $P-1$ rounds, where a segment delivery is followed by its data reduction, thus after these rounds, each process has exactly one segment fully reduced, and needs to forward it to other processes. This is performed during next $P-1$ rounds, where the data are gathered, i.e. the processes transfer the reduced segments, thus the final result is delivered to all processes~\cite{Thakur2005}. Fig.~\ref{fig:ring} presents an example of a ring execution. \emph{Segmented ring} is similar to the ring, but after the segmentation related to process number, an additional one is performed for pipeline effect between corresponding processes~\cite{OpenMPI}. In the \emph{Rabenseifner} algorithm the reduction is performed in two phases, first the scatter and reduce of data are executed (using recursive doubling on divided data) and then the data gathering takes place (using recursive halving on gathered data), where the reduced data come back to all processes~\cite{Rabenseifner2004}.
\begin{figure}[!h]\sidecaption
\includegraphics[width=8cm]{Fig2}
\caption{Example of a ring algorithm execution, showing data distribution between the processes (one square corresponds to one data segment), a solid arrow represents send and reduce a data segment, an empty arrow represents send and override a data segment, and $\tau$ is the time of transfer and reduction/overriding of a data segment}
\label{fig:ring}
\end{figure}
\subsection{Imbalanced process arrival times}
Not much work has been done for imbalanced process arrival patterns analysis. To the best of our knowledge, the following four papers cover the research performed in this area.
In~\cite{Faraj2008} Faraj et al. performed advanced analysis of \emph{process arrival patterns} (PAPs) observed during execution of typical MPI programs. They executed a set of tests proving that the differences between \emph{process arrival times} (PATs) in operations of the collective communication are significantly high and they influence the performance of the underlying computations. The authors defined a PAP to be imbalanced for a given collective operation with a specific message length when its imbalance factor (a ratio between the highest difference between the arrival times of the processes and time of the simple (point-to-point) message delivery between each other) is larger than 1. The authors provided examples of typical HPC benchmarks, e.g. NAS, LAMMPS or NBODY, where imbalance factor, during their execution in a typical cluster environment, equals 100 or even more. They observed that, such behavior usually cannot be controlled directly by the programmer, and the imbalances are going to occur in any typical HPC environment. The authors proposed a mini-benchmark for testing various collective operations and found out the conclusion that the algorithms which perform better with balanced PAPs tend to behave worse when dealing with imbalanced ones. Finally, they proposed solution: their self-tuning framework -- STAR-MPI~\cite{Faraj2006}, which includes a large set of various implementations of collective operation and can be used for different PAPs, with automatic detection of the most suitable algorithm. The framework efficiency was proved by an example of tuned all-to-all operations, where the performance of the set of MPI benchmarks was significantly increased.
As a continuation of the above work Patarasuk et al. proposed a new solution for broadcast operation used in MPI application concerning imbalanced PAPs of the cooperating processes~\cite{Patarasuk2008}. The authors proposed a new metric for the algorithm performance: competitive ratio -- $PERF(r)$, which describes the influence of the imbalanced PATs for the algorithm execution, regarding the behavior for its worst case PAT. They evaluated well known broadcast algorithms, using the above metric, and presented two new algorithms, which have constant (limited) value of the metric. The algorithms are meant for large messages, and use the subsets of cooperating processes to accelerate the overall process: the data are sent to the earliest processes first. One of the algorithms is dedicated for non-blocking (\emph{arrival\_nb}) and other for blocking (\emph{arrival\_b}) message passing systems. The authors proposed a benchmark for algorithms’ evaluation, which introduces random PAPs and measures their impact on the algorithm performance. The experiments were performed using two different 16-node compute clusters (one with Infiniband and other with Ethernet interconnecting network), and 5 broadcast algorithms, i.e. \emph{arrival\_b}, flat, linear, binomial trees and the one native to the machine. The results of the experiments showed the advantage of the \emph{arrival\_b} algorithm for large messages and imbalanced PAPs.
In~\cite{Marendic2012} Marendic et al. focused on an analysis of reduce algorithms working with imbalanced PATs. They assumed atomicity of reduced data (the data cannot be split into segments and reduced piece by piece), as well as the Hockney~\cite{Hockney1994} model of message passing (time of message transmission depends on the link bandwidth: $\beta$ and constant latency: $\alpha$, with an additional computation speed parameter: $\gamma$) and presented related works for typical reduction algorithms. They proposed a new static load balancing optimized reduction algorithm requiring a priori information about current PATs of all cooperating processes. The authors performed a theoretical analysis proving the algorithm is nearly optimal for the assumed model. They showed that the algorithm gives the minimal completion time under the assumption that the corresponding point-to-point operations start exactly at the same time for any two communicating processes. However if the model introduces a delay of the receive operation in comparison with the send one, which seems to be the case in real systems, the algorithm does not utilize this additional time in receiving process, although, in some cases, it could slightly improve the performance of the overall reduce operation. The other proposed algorithm, presented by the authors: a dynamic load balancing can operate under the limited knowledge about PATs, being able to atomically reconfigure the message passing tree structure while performing reduce operation using auxiliary short messages for signaling the PATs between the cooperating processes. The overhead is minimal in comparison with the gains of the PAP optimization. Finally a mini-benchmark was presented and some typical PAPs were examined, the results showed the advantage of the proposed dynamic load balancing algorithm versus other algorithms: binary tree and all-to-all reduce.
In~\cite{Marendic2016} Marendic et al. continued the work with optimization of the MPI reduction operations dealing with the imbalanced PAPs. The main contribution is a new algorithm, called Clairvoyant, scheduling the exchange of data segments (fixed parts of reduced data) between reducing processes, without assumption of data atomicity, and taking into account PATs, thus causing as many as possible segments to be reduced by the early arriving processes. The idea of the algorithm bases on the assumption that the PAP is known during process scheduling. The paper provided a theoretical background for the PAPs, with its own definition of the time imbalances, including a PAT vector, absolute imbalance, absorption time as well as their normalized versions, followed by the analysis of the proposed algorithm, and its comparison to other typically used reduction algorithms. Its pseudo-code was described and the implementation details were roughly provided with two examples of its execution for balanced and imbalanced PAPs. Afterwards the performed experiments were described, including details about used mini-benchmark and the results of practical comparison with other solutions (typical algorithms with no support for imbalanced PAPs) were provided. Finally, the results of the experiments showing advantage of the proposed algorithm were presented and discussed.
\subsection{Process arrival time estimation}
The PAP collective communication algorithms require some knowledge about the PATs for their execution. Table~\ref{tab:pat-detection} presents the summary of the approaches used in the works described above.
\begin{table}[h]
\caption{Approaches for PAP detection}
\label{tab:pat-detection}
\begin{tabular}{|p{4cm}|p{7cm}|}
\hline
Operation & PAP detector\\\hline
All-to-all~\cite{Faraj2008} & Uses STAR-MPI overall efficiency indicators\\\hline
Broadcast~\cite{Patarasuk2008} & Uses its own messaging for PAT signaling\\\hline
Local redirect~\cite{Marendic2012} & Uses its own messaging for PAT signaling\\\hline
Clairvoyant~\cite{Marendic2016} & None, suggested usage of the static analysis or SMA\\\hline
\end{tabular}
\end{table}
In~\cite{Faraj2008} (dedicated for an all-to-all collective operation), there is assumption about the call site (a place in the code where the MPI collective operation is called) paired with the message size that they have a similar PAPs for the whole program execution, or at least their behavior changes infrequently. The proposed STAR-MPI~\cite{Faraj2006} system periodically assesses the call site performance (exchanging measured times between processes) and adapts a proper algorithm, trying one after another. The authors claim that it requires 50--200 calls to provide desired optimization. This is a general approach and it can be used even for other performance issues, e.g. network structure adaptation.
In~\cite{Patarasuk2008} (dedicated for a broadcast operation), the algorithm uses additional, short messages sent to root process signaling process readiness for the operations. In case the some processes are ready, the root performs sub-group broadcast, thus the a priori PATs are not necessary for this approach. Similar idea is used in~\cite{Marendic2012} (dedicated for a reduce operation) where the additional messages are used not only to indicate readiness, but also to redirect delayed processes.
In~\cite{Marendic2016} (dedicated for a reduce operation), the algorithm itself does not include any solution for the PAT estimation, that is why it is called Clairvoyant, but the authors assume recurring PAPs and give the suggestion that there can be used simple moving averages (SMA) approximation. This solution requires the additional communication to exchange the PAT values, what is performed every $k$ iterations, thus introducing the additional communication time. The authors claim that the speedup introduced by the usage of the algorithm overcomes this cost, and provide some experimental results showing the total time reduction in the overall computations.
\section{Computation and communication model}\label{sec:model}
We assume usage of the message-passing paradigm, within a homogeneous environment, where each communicating process is placed on a separated compute node. The nodes are connected by a homogeneous communication network. Every process can handle one or more threads of control communicating and synchronizing with each other using shared memory mechanisms. However there is no shared memory accessible simultaneously by different processes.
As a \emph{process arrival pattern} (PAP) we understand the timing of different processes arrivals for a concrete collective operation, e.g. all-reduce in an MPI program. We can evaluate a given PAP by measuring the \emph{process arrival time} (PAT) for each process. Formally, a PAP is defined as the tuple $(a_0, a_1, \ldots a_{P-1})$, where $a_i$ is a measured PAT for process $i$, while $P$ is the number of processes participating in the collective operation. Similarly \emph{process exit pattern} (PEP) is defined as the tuple $(f_0, f_1, \ldots f_{P-1})$, where $f_i$ is the time when process $i$ finishes the operation~\cite{Faraj2008}. Fig.~\ref{fig:pap-ex} presents an example of arrival and exit patterns.
For each process participating in a particular operation Faraj et al. define the \emph{elapsed time} as $e_i=f_i-a_i$, and the \emph{average elapsed time} for the whole PAP: $\bar{e}=\frac{1}{P}\sum_{i=0}^{P-1}e_i$~\cite{Faraj2008}. This is a mean value of time spent for communication by each process, the rest of the time is used for computations. Thus minimizing the elapsed times of the participating processes decreases the total time of program execution and, in our case, is the goal of the optimization.
For an all-reduce operation, assuming $\delta$ to be the time of sending the reduced data between any two processes and only one arbitrary chosen process $k$ is delayed (others have the same arrival time, see an example in Fig.~\ref{fig:pap-ex}, where $k=1$), we can estimate the lower bound of $\bar{e}$ as $\bar{e}_{lo}=a_k-a_o+\delta$, where $a_o$ is the time of arrival of all processes except $k$. On the other hand, assuming $\Delta$ to be time of an all-reduce operation for perfectly balanced PAT ($a_i=a_j$, for all $i,j\in\langle 0, k-1\rangle$), we can estimate the upper bound of $\bar{e}$ as $\bar{e}_{up}=a_k-a_o+\Delta$. Thus using PAP optimized all-reduce algorithm can decrease the elapsed time by $\Delta-\delta$ or less. E.g. for a typical ring all-reduce algorithm, working on 16~processes, 4\,MB data size and 1\,Gbps Ethernet network we can measure $\Delta=$45.7\,ms and $\delta=$18.2\,ms, thus using PAP optimized algorithm can save at most 27.5\,ms of average elapsed time, no matter how slow the delayed process is.
Furthermore, we assume a typical iterative processing model with two phases: the computation phase where every process performs independent calculations, and the communication phase where the processes exchange the results, in our case, using all-reduce collective operation. These two phases are combined into an iteration, which is repeated sequentially during the processing. We assume that the whole program execution consists of $N$ iterations.
Normally, during a computation phase, the message communication between processes/nodes is suspended. Nevertheless, each process can contain many threads carrying on the parallel computations exploiting shared memory for synchronization and data exchange. Thus, during this phase the communication network connecting nodes is unused and can be utilized for exchange of additional messages, containing information about a progress of computations or other useful events (e.g. a failure detection).
Thus, we introduce an additional thread which is responsible for inter-node communication during computation phase. It monitors the progress of the phase on its node, estimates the remaining computation time and exchanges this knowledge with other processes on the cooperating nodes. Gathering this information, every thread can approximate a process arrival pattern (PAP) for itself and other processes (see Fig.~\ref{fig:processing-model}).
\begin{figure}[!h]
\includegraphics[width=12cm]{Fig3}
\caption{Iterative processing model (solid lines), extended by progress monitoring (dashed lines)}
\label{fig:processing-model}
\end{figure}
For monitoring purposes the computation threads need to pass the status of current iteration processing calling a special function: edge(). We assume that for all processes this call is made after a defined part of performed computations e.g. 50\%, while the exact value is passed as the function parameter. In our implementation the edge() function is executed in some kind of callback function, reporting a status of the iteration progress.
Beside the PAP monitoring and estimation functions the additional thread can be used for other purposes. In our case, the proposed all-reduce algorithms, described in section~\ref{sec:algs}, are working much better (performing faster message exchange) when connections between the communicating processes/nodes are already established, thus we introduced an additional warm-up procedure where a series of messages are transferred in the foreseen directions of communication. This operation is performed by the thread after the exchange of the progress data.
Thus the analysis of the above requirements implemented in the additional thread, shows that the thread should react accordingly to the following events (see Fig.~\ref{fig:add-thread}):
\begin{itemize}
\item the beginning of processing, when the thread is informed about the computation phase start and when it stores the timestamp for the further time estimation,
\item the edge, in the middle of processing, when the thread estimates the ongoing computation phase time for its process, exchanges this information with other processes and performs the warm up procedure for establishing connections to speed up the message transfer in the coming collective operation,
\item the finish of processing, when the thread is informed about the end of the computation phase and when it stores the timestamp for the further time estimations.
\end{itemize}
\begin{figure}[!h]\sidecaption
\includegraphics[width=8cm]{Fig4}
\caption{Events and actions performed in the additional thread}
\label{fig:add-thread}
\end{figure}
\section{New all-reduce algorithms optimized for PAPs}\label{sec:algs}
In this section we introduce two new algorithms for all-reduce operations, optimized for a PAP observed during the computation phase: (i) sorted linear tree (SLT), (ii) pre-reduced ring (PRR). Both of them are based on the well known, and widely used regular all-reduce algorithms: linear tree~\cite{OpenMPI} and ring~\cite{Thakur2005}, respectively, and have similar communication and time complexity.
\subsection{Sorted linear tree (SLT)}
The algorithm is an extension of the linear tree~\cite{OpenMPI}, which transfers the data segments sequentially through processes exploiting the pipeline parallel computation model. The proposed modification causes the processes to be sorted by their arrival times. While the faster processes start the communication earlier, the later ones have more time to finish the computations.
Fig.~\ref{fig:slt} presents pseudo-code of the algorithm. At the beginning new identifiers are assigned to the processes according to the arrival order, then the data are split into equal segments, in our case we assume $P$ segments, where $P$ is a number of the cooperating processes. Afterwards the reduce loop is started (lines: 1-6), where the data segments are transferred and reduced, and then the override loop is executed (lines: 7-12), where the segments are distributed back to all processes.
\begin{figure}[!h]
\fbox{\parbox{.97\textwidth}{\sffamily
input parameters:\\
$P$ -- number of processes or nodes, we assume one process per node\\
$a_i$ -- arrival time of process $i$\\
$d_x$ -- $x$-th of $P$ data segments to be reduced, e.g. $d_0$ - first data segment\\
$id$ -- a new process rank after sorting according to arrival times\\
variables:\\
$si$ -- segment index\\
// reduce loop\\
1.~~for $i:=0$ to $P-1$\\
2.~~~~$si := i\ mod\ P$\\
3.~~~~if $id \neq 0$ then // not the first process\\
4.~~~~~~$rd$ = receive\_segment() // blocking\\
5.~~~~~~reduce($d_{si}$, $rd$)\\
6.~~~~send\_segment($d_{si}$, $(id+1)\ mod\ P$) // non-blocking\\
// override loop\\
7.~~for $i:=0$ to $P-1$\\
8.~~~~$si := i\ mod\ P$\\
9.~~~~if $id \neq P-1$ then // not the last process\\
10.~~~~~$d_{si} :=$ receive\_segment() // blocking\\
11.~~~if $id < P-2$ then // not the last or penultimate process\\
12.~~~~~send\_segment($d_{si}$, $(id+1)\ mod\ P$) // non-blocking
}}
\caption{Pseudo-code of the sorted linear tree (SLT) algorithm}
\label{fig:slt}
\end{figure}
Let's assume $\tau$ is a time period required for transfer and reduction/overriding of one data segment between any two processes. Fig.~\ref{fig:sltex}{}b presents an example of SLT execution, where the process $0$ arrival time is delayed for $4\tau$ in comparison with all other processes. Using the regular linear tree reduction, the total time of the execution would be $14\tau$ (see Fig.~\ref{fig:sltex}{}a), while the knowledge of the PAP and procedure of sorting the processes by their arrival times make the delayed process to be the last one in the pipeline and cause the whole operation time to be decreased to $12\tau$.
\begin{figure}[!h]
\includegraphics[width=12cm]{Fig6a}
~\\
\includegraphics[width=12cm]{Fig6b}
\caption{Example of (a) linear tree and (b) sorted linear tree (SLT) algorithm executions, showing data distribution between the processes (one square corresponds to one data segment), where process \#0 is delayed by $4\tau$, a solid arrow represents send and reduce a data segment, an empty arrow represents send and override a data segment, and $\tau$ is the time of transfer and reduction/overriding of a data segment}
\label{fig:sltex}
\end{figure}
\subsection{Pre-reduced ring (PRR)}
This algorithm is an extended version of ring~\cite{Thakur2005}, where each data segment is reduced and then passed to other processes in synchronous manner. The idea of the algorithm is to perform a number of so-called, reducing pre-steps, between faster processes (with lower arrival times), and then the regular processing, like in the typical ring algorithm, is performed.
Fig.~\ref{fig:prr} presents pseudo-code of the algorithm. First of all, the processes are sorted by their arrival times and the new ids are assigned, i.e. initially the processes perform communication in direction from the earliest to the latest ones. Then data are split into $P$ equal segments and for each such a segment the number of pre-steps is calculated (lines: 1-6), its value depending directly on the estimated process arrival times (PATs): $a_i$ and the time of transfer and reduction/overriding of one segment: $\tau$. Knowing the above, the algorithm resolves where each segment starts and finishes its processing, thus two process ids are assigned for each segment: $sp_i$ and $rp_i$, respectively (lines: 7-13).
\begin{figure}[!h]
\fbox{\parbox{.97\textwidth}{\sffamily
input parameters:\\
$\tau$ -- time of transfer and reduction/overriding of one segment\\
$P$ -- number of processes or nodes, we assume one process per node\\
$a_i$ -- arrival time of process $i$\\
$d_x$ -- $x$-th of $P$ data segments to be reduced, e.g. $d_0$ - first data segment\\
$id$ -- a new process rank after sorting according to arrival times\\
variables:\\
$k_i$ -- number of pre-steps in process $i$\\
$si$ -- segment index\\
$sp_j$ -- id of the first process where the slice $j$ is sent\\
$rp_j$ -- process id where the slice j is reduced last time\\
// calculate a number of pre-steps for all processes\\
1.~~$k_{P-1} := 0$\\
2.~~for $i:=P-2$ to 0\\
3.~~~~if $a_{P-1}-a_{i+1} \ge (k_{i+1}+1)\times\tau$ then\\
4.~~~~~~$k_i:=k_{i+1}+1$\\
5.~~~~else\\
6.~~~~~~$k_i:=k_{i+1}$\\
// calculate ids of the processes where a slice id send first time\\
// and ids of the processes where a slice is reduced last time\\
7.~~$i := 0$\\
8.~~for $j:=0$ to $P-1$\\
9.~~~~if $i+k_i \ge j$ then\\
10.~~~~~$sp_j := i$\\
11.~~~else\\
12.~~~~~$i := i+1$\\
13.~~~$rp_j := (sp_j+P-1)\ mod\ P$\\
// calculate what segment start for\\
14. $si := id+k_{id}$\\
// reduce loop\\
15.~for $i:=0$ to $P-1$\\
16.~~~if $sp_{si} \neq id$ then\\
17.~~~~~$rd :=$ receive\_segment() // blocking\\
18.~~~~~reduce($d_{si}$, $rd$)\\
19.~~~if $(rpsi+P-1)\ mod\ P \neq is$ then\\
20.~~~~~send\_segment($d_{si}$, $(id+1)\ mod\ P$) // non-blocking\\
21.~~~$si := (si+P-1)\ mod\ P$\\
// override loop\\
22.~for $i:=0$ to $P-1$\\
23.~~~if $rp_{si} \neq id$ then\\
24.~~~~~$d_{si} :=$ receive\_segment() // blocking\\
25.~~~if $(rp_{si}+P-1)\ mod\ P \neq id$ then\\
26.~~~~~send\_segment($d_{si}$, $(id+1)\ mod\ P$) // non-blocking\\
27.~~~$si := (si+P-1)\ mod\ P$
}}
\caption{Pseudo-code of the pre-reduced ring (PRR) algorithm}
\label{fig:prr}
\end{figure}
Afterwards, initial value of the segment index: $s_i$ is calculated from which every process starts processing. In the case of the regular ring algorithm, its value depends on the process $id$ only, however for PRR it also regards the pre-steps performed by the involved processes (line: 14). Then the reduce loop is started, the pre-steps and the regular steps are performed in one block, where the variable $s_i$ controls which segments are sent, received and reduced (lines: 15-21). Similarly, the override loop is performed, being controlled by the same variable (lines: 22-27).
In the regular ring every process performs $2P-2$ receive and send operations, thus total number is $P\times(2P-2)$. In case of PRR, this total number is the same, however the faster processes (the ones which finished computation earlier) tend to perform more communication while executing the pre-steps. In the case, when the arrival of the last process is largely delayed to the next one (i.e. more than $P\times\tau$) it performs only $P-1$ send and receive operations, while every other process does $2P-1$ ones.
Fig.~\ref{fig:prrex}{}b presents an example of PRR execution, where the process $0$ arrival time is delayed for $2\tau$ ($2\times$ time of transfer and reduction/overriding of one data segment) in comparison with all other processes. Using the regular ring reduction, the total time of the execution would be $8\tau$ (see Fig.~\ref{fig:prrex}b), while the knowledge of the PAP, performing two additional reduction pre-steps and setting the delayed process to be the last one in the pipeline, causes the whole operation time to be reduced to $7\tau$.
\begin{figure}[!h]
\includegraphics[width=9.5cm]{Fig8a}
~\\
\includegraphics[width=9.5cm]{Fig8b}
\caption{Example of (a) ring and (b) pre-reduced ring (PRR) algorithm executions, showing data distribution between the processes (one square corresponds to one data segment), where process \#0 is delayed by $2\tau$, a solid arrow represents send and reduce a data segment, an empty arrow represents send and override a data segment, and $\tau$ is the time of transfer and reduction/overriding of a data segment}
\label{fig:prrex}
\end{figure}
\section{The experiments}\label{sec:experiments}
The above algorithms were implemented and tested in a real HPC environment, the following subsections describe a proposed benchmark (including its pseudo-code and implementation details), the experiment's setup and provide the discussion about the observed results.
\subsection{The benchmark}
Fig.~\ref{fig:bench} presents pseudo-code of the proposed benchmark evaluating the performance of the proposed algorithms in the real HPC environment using MPI~\cite{MPI} for communication purposes. For every execution there is a sequence of repeated iterations consisting of the following actions:
\begin{description}
\item[line 2:] data generation, where the data are randomly assigned,
\item[line 3:] calculation of the emulated computation time, ensuring that the progress monitoring communication (the additional 100\,ms) is completely covered by the computation phase,
\item[lines 4--8:] the delay mode is applied, in `one-late' mode only one process (id: 1) is delayed for $maxDelay$, while in `rand-late' mode all processes are delayed for random time up to $maxDelay$,
\item[lines 9--10:] two MPI\_Barrier() calls, making sure all the processes are synchronized,
\item[lines 11--13:] the emulation of computation phase by using sleep() (usleep() in the implementation) function, including call to edge() function (see section~\ref{sec:model}), providing the progress status to the underlying monitoring thread,
\item[lines 14--16:] call to the all-reduce algorithm implementation, including the commands for the time measurements, for the benchmark purposes we assumed sum as the reduce operator,
\item[line 17:] checking the correctness of the performed all-reduce operation, using regular MPI\_Allreduce() function,
\item[lines 18--19:] calculation of the elapsed time for the current and for all processes using MPI\_Allgather() function,
\item[line 20:] saving the average elapsed time into the $results$ vector.
\end{description}
\begin{figure}[!h]
\fbox{\parbox{.97\textwidth}{\sffamily
input parameters:\\
$size$ -- number of elements (floats) in reduced data\\
$N$ -- number of iterations\\
$mode$ -- mode of delay: one-late/rand-late\\
$maxDelay$ -- maximal delay of the process(es)\\
$algorithm$ -- tested algorithm, one of ring, Rabenseifner, PRR, SLT\\
$P$ -- number of processes\\
$id$ -- process id -- MPI rank: $0$\dots$P-1$\\
output:\\
$results$ -- vector of measured average elapsed times for the given algorithm\\
variables:\\
$halfTime$ -- 50\% of the emulated computation time\\
$startTime$ -- start time of measurement\\
$endTime$ -- end time of measurement\\
$myElapsedTime$ -- elapsed time measured in the current process\\
$allElapsedTimes$ -- vector of the measured times of all processes\\
$data$ -- vector of data to be reduced\\
1.~~for $i := 1$ to $N$\\
2.~~~~$data$ := generateRandomData($size$)\\
3.~~~~$halfTime = (maxDelay+100\,ms)/2$\\
4.~~~~if $mode =$ "one-late" then\\
5.~~~~~~if $id = 1$ then\\
6.~~~~~~~~$halfTime := halfTime+maxDelay$\\
7.~~~~else // $mode =$ "rand-late" \\
8.~~~~~~$halfTime := halfTime +$ random($0$\dots$maxDelay$)$/2$\\
9.~~~~MPI\_Barrier()\\
10.~~~MPI\_Barrier()\\
11.~~~sleep($halfTime$)\\
12.~~~edge($0.5$)\\
13.~~~sleep($halfTime$)\\
14.~~~$startTime :=$ MPI\_Wtime()\\
15.~~~makeAllReduce($algorithm$, $data$)\\
16.~~~$endTime :=$ MPI\_Wtime()\\
17.~~~checkCorrectness($data$)\\
18.~~~$myElapsedTime := endTime-startTime$\\
19.~~~MPI\_Allgather($allElapsedTimes$, $myElapsedTime$, \ldots)\\
20.~~~results[i] := average($allElapsedTimes$)
}}
\caption{Pseudo-code of the performance benchmark}
\label{fig:bench}
\end{figure}
For the comparison purposes we used two typical all-reduce algorithms: ring~\cite{Thakur2005} and Rabenseifner~\cite{Rabenseifner2004}, they are implemented in the two most popular open source MPI implementations: OpenMPI~\cite{OpenMPI} and MPICH~\cite{MPICH} respectively, and used for large input data size. To the best knowledge of the author there are no PAP optimized all-reduce algorithms described in the literature.
The benchmark was implemented in C language v.~C99, compiled using GCC v.~7.1.0, with the maximal code optimization (-O3). The program uses OpenMPI v.~2.1.1 for processes/nodes message exchange and POSIX Threads v.~2.12 for intra-node communication and synchronization, moreover for managing of dynamic data structures GLibc v.~2.0 library was used. In this implementation the reduce operation is based on a sum (equivalent of using MPI\_SUM for $op$ parameter in MPI\_Allreduce()).
\subsection{Environment and test setup}\label{sec:bench-setup}
The benchmark was executed in a real HPC environment using cluster supercomputer Tryton, placed in Centre of Informatics - Tricity Academic Supercomputer and NetworK (CI TASK) at Gdansk University of Technology in Poland~\cite{Krawczyk2015}. The cluster consists of homogeneous nodes, where each node contains 2 processors (Intel Xeon Processor E5 v3, 2.3GHz, Haswell), with 12~physical cores (24~logical ones, due to Hyperthreading technology) and 128\,GB RAM memory. In total the supercomputer consists of 40 racks with 1,600~servers (nodes), 3,200~processors, 38,400~compute cores, 48~accelerators and 202\,TB~RAM memory. It uses fast FDR 56\,Gbps InfniBand in fat tree topology and 1\,Gbps Ethernet for networking. Total computing power is 1.48\,PFLOPS. The cluster weighs over 20 metric tons.
The experiments were performed using a subset of the nodes, with HT switched off, grouped in one rack and connected to each other trough a 1\,Gbps Ethernet switch (HP J9728A 2920-48G). The benchmark input parameters were set up to the following values:
\begin{description}
\item[$algorithm$:] ring, Rabenseifner, pre-reduced ring (PRR) and sorted linear tree (SLT);
\item[$size$:] (of data vector) 128\,K, 512\,K, 1\,M, 2\,M, 4\,M, 8\,M of floats (4~bytes long);
\item[$mode$:] (of process delay) one-late (where only one process is delayed by $maxDelay$) and rand-late (where all processes are delayed randomly up to $maxDelay$);
\item[$maxDelay$:] (of processes arrival times) 0, 1, 5, 10, 50, 100, 500, 1000\,ms;
\item[$P$:] (number of processes/nodes) 4, 6, 8, 10, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48;
\item[$N$:] (number of iterations) 64--256, depending on $maxDelay$ (more for lower delay);
\end{description}
\subsection{Benchmark results}
Table~\ref{tab:alg-cmp} presents the results of the benchmark execution for 1\,M of floats of reduced data, 1\,Gbps Ethernet network, where only one process was delayed on 48~nodes in a cluster environment of Tryton~\cite{Krawczyk2015} HPC computer. The results are presented as absolute values of average elapsed time: $\bar{e}_{alg}$ and speedup: $s_{alg}$, in comparison with ring algorithm $s_{alg}=\frac{\bar{e}_{ring}}{\bar{e}_{alg}}$, where $alg$ is the evaluated algorithm.
In this setup the ring algorithm seems to be more efficient than the Rabenseifner, thus in further analysis we use the former for reference purposes. For more balanced PAPs, where the delay is below 5\,ms the proposed algorithms perform worse than the ring, it is especially visible for SLT (about 25\% slower), however PRR shows only slight difference (below 5\%). On the other hand, when the PAP is more imbalanced with the delay over 10\,ms, both algorithms perform much better (up to 15\% faster than the ring). For the really high delays, the results stabilize providing about 17\,ms of average elapsed time savings, causing the total speedup to be lower. The comparison of influence of changes in the (maximum) delay on the algorithms' performance is presented in Fig.~\ref{fig:bench-delay}.
\begin{table}[h]
\caption{Benchmark results for 1\,M of floats of reduced data, 1\,Gbps Ethernet network, only one process delayed and 48~processes/nodes. Maximum delay is measured in ms and each result consists of two values: the average elapsed times in ms and speedup in comparison with the ring algorithm ($\frac{\bar{e}_{ring}}{\bar{e}_{alg}}$). The bold values indicate better performance in comparison with the ring algorithm}
\label{tab:alg-cmp}
\begin{tabular}{|p{1.7cm}|p{0.8cm}|p{0.8cm}|p{0.8cm}|p{0.8cm}|p{0.8cm}|p{0.8cm}|p{0.8cm}|p{0.8cm}|}\hline
Max delay$\rightarrow$ & 0 & 1 & 5 & 10 & 50 & 100 & 500 & 1000\\
Algorithm$\downarrow$ & & & & & & & & \\\hline
Ring & 67.8 & 67.6 & 70.1 & 75.6 & 115.9 & 165.5 & 558.6 & 1047.0\\
\ & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00\\\hline
Rabenseifner & 76.3 & 75.9 & 82.9 & 85.7 & 125.0 & 181.6 & 572.0 & 1061.8\\
\ & 0.89 & 0.89 & 0.85 & 0.88 & 0.93 & 0.91 & 0.98 & 0.99\\\hline
SLT & 90.3 & 90.3 & 89.7 & 89.7 & \bf 101.0 & \bf 149.8 & \bf 542.1 & \bf 1031.9\\
\ & 0.75 & 0.75 & 0.78 & 0.84 & \bf 1.15 & \bf 1.10 & \bf 1.03 & \bf 1.01\\\hline
PRR & 70.9 & 70.8 & 71.1 & \bf 73.0 & \bf 101.0 & \bf 149.7 & \bf 541.7 & \bf 1031.5\\
\ & 0.96 & 0.95 & 0.99 & \bf 1.04 & \bf 1.15 & \bf 1.11 & \bf 1.03 & \bf 1.01\\\hline
\end{tabular}
\end{table}
\begin{figure}[!h]\sidecaption
\includegraphics[width=8cm]{Fig10}
\caption{Benchmark results showing influence of the increasing maximum delay on average elapsed times of the tested algorithms. The experiments were performed on 48~nodes for 1\,Gbps Ethernet network, one process delayed, and message size: 1024\,K of float numbers. The error bars are set to $\pm{}2\sigma$ (95\% of the measurements for the normal distribution)}
\label{fig:bench-delay}
\end{figure}
Table~\ref{tab:prr-eth} presents the benchmark results related to PRR in comparison with the ring algorithm, executed on 48~nodes with 1\,Gbps Ethernet interconnecting network. While the absolute average elapsed time savings of PRR algorithm are higher for longer messages (up to 82\,ms for 8\,M), the high speedup values occur for all message sizes providing relative savings up to 15\%. We can observe that the lower delays cause lager speedup losses (up to 22\%, but for absolute time: 1.5\,ms only), which is especially visible for the smallest message size: 128\,K. In general, for the lower delays, the PRR is comparable with the ring (see Fig.~\ref{fig:bench-delay} for 1\,M of float numbers reduced data size).
\begin{table}[h]
\caption{Comparison of ring and PRR algorithms for 1\,Gbps Ethernet and 48~processes/nodes. Maximum delay is measured in ms and size in Kfloats (4$\times$KB). Each entry consists of two values: a difference of the average elapsed times in ms ($\bar{e}_{PRR}-\bar{e}_{ring}$) and speedup: a quotient of elapsed times ($\frac{\bar{e}_{ring}}{\bar{e}_{PRR}}$). The bold values indicate better performance in comparison with the ring algorithm}
\label{tab:prr-eth}
\begin{tabular}{|p{1.7cm}|p{0.8cm}|p{0.8cm}|p{0.8cm}|p{0.8cm}|p{0.8cm}|p{0.8cm}|p{0.8cm}|p{0.8cm}|}\hline
Max delay$\rightarrow$ & 0 & 1 & 5 & 10 & 50 & 100 & 500 & 1000\\\hline
Size$\downarrow$ &\multicolumn{8}{c|}{Only one process delayed}\\\hline
\ & 1.0 & 1.5 & \bf -5.0 & \bf -4.1 & \bf -4.3 & \bf -3.2 & \bf -0.9 & \bf -0.8\\
128 & 0.85 & 0.78 & \bf 1.53 & \bf 1.26 & \bf 1.08 & \bf 1.03 & \bf 1.00 & \bf 1.00\\\hline
\ & \bf -1.0 & \bf -0.3 & \bf -10.3 & \bf -12.6 & \bf -16.4 & \bf -11.6 & \bf -1.5 & \bf -1.7\\
512 & \bf 1.04 & \bf 1.01 & \bf 1.37 & \bf 1.41 & \bf 1.26 & \bf 1.10 & \bf 1.00 & \bf 1.00\\\hline
\ & 2.9 & 3.7 & 1.2 & \bf -3.2 & \bf -14.1 & \bf -15.2 & \bf -15.8 & \bf -16.0\\
1024 & 0.96 & 0.95 & 0.98 & \bf 1.04 & \bf 1.14 & \bf 1.10 & \bf 1.03 & \bf 1.02\\\hline
\ & 7.8 & 6.4 & 3.4 & \bf -1.7 & \bf -8.5 & \bf -23.9 & \bf -27.1 & \bf -27.7\\
2048 & 0.93 & 0.94 & 0.97 & \bf 1.02 & \bf 1.06 & \bf 1.13 & \bf 1.05 & \bf 1.03\\\hline
\ & 7.6 & 6.0 & 5.2 & 2.8 & \bf -20.9 & \bf -29.6 & \bf -38.1 & \bf -46.8\\
4096 & 0.96 & 0.97 & 0.97 & 0.99 & \bf 1.10 & \bf 1.12 & \bf 1.06 & \bf 1.04\\\hline
\ & 5.8 & 17.0 & 13.7 & 11.7 & \bf -29.0 & \bf -52.1 & \bf -75.7 & \bf -82.1\\
8192 & 0.98 & 0.95 & 0.96 & 0.97 & \bf 1.08 & \bf 1.14 & \bf 1.10 & \bf 1.07\\\hline
Size$\downarrow$ &\multicolumn{8}{c|}{All processes delayed randomly}\\\hline
\ & 1.2 & 1.0 & \bf -1.1 & \bf -1.1 & \bf -2.7 & \bf -1.0 & \bf -0.9 & \bf -1.3\\
128 & 0.82 & 0.85 & \bf 1.11 & \bf 1.09 & \bf 1.09 & \bf 1.02 & \bf 1.00 & \bf 1.00\\\hline
\ & 2.3 & \bf -0.3 & \bf -9.5 & \bf -9.3 & \bf -16.3 & \bf -11.5 & \bf -2.3 & \bf -2.9\\
512 & 0.91 & \bf 1.01 & \bf 1.37 & \bf 1.35 & \bf 1.38 & \bf 1.18 & \bf 1.01 & \bf 1.01\\\hline
\ & 2.9 & 3.8 & 1.7 & 0.7 & \bf -13.0 & \bf -13.2 & \bf -14.1 & \bf -14.5\\
1024 & 0.96 & 0.95 & 0.98 & 0.99 & \bf 1.17 & \bf 1.13 & \bf 1.05 & \bf 1.03\\\hline
\ & 7.1 & 8.3 & 4.8 & 3.0 & \bf -8.8 & \bf -21.2 & \bf -23.9 & \bf -24.5\\
2048 & 0.94 & 0.92 & 0.96 & 0.97 & \bf 1.08 & \bf 1.16 & \bf 1.07 & \bf 1.04\\\hline
\ & 6.1 & 0.1 & 3.2 & 5.6 & \bf -21.8 & \bf -19.0 & \bf -42.6 & \bf -39.8\\
4096 & 0.97 & 1.00 & 0.98 & 0.97 & \bf 1.11 & \bf 1.09 & \bf 1.11 & \bf 1.06\\\hline
\ & 13.8 & 16.3 & 6.8 & 7.1 & 0.4 & \bf -34.4 & \bf -75.9 & \bf -75.1\\
8192 & 0.96 & 0.95 & 0.98 & 0.98 & 1.00 & \bf 1.10 & \bf 1.15 & \bf 1.10\\\hline
\end{tabular}
\end{table}
The mode of the introduced PAT delay influences slightly the measured values. For larger delays and message sizes, when only one process is delayed the PRR algorithm provides slightly smaller relative savings than in the case when the delays were introduced for all processes, with the uniform probabilistic distribution, it is especially visible for 500--1000\,ms delays and message sizes of 2--8\,M of floats. However, in general, the PRR works fine for both modes of PAT delay.
Fig.~\ref{fig:bench-size} presents the behavior of the tested algorithms in a context of the changing data size. While the PRR algorithm performs well for a wide range of size values, the SLT lags for the larger data, the threshold depends on the (maximum) delay, e.g. for 50\,ms the SLT is worse than the regular ring for 4+\,M of float numbers data size.
\begin{figure}[!h]\sidecaption
\includegraphics[width=8cm]{Fig11}
\caption{Benchmark results showing the algorithms' behavior related to the reduced data size. The experiments were performed on 48~nodes for 1\,Gbps Ethernet network, one process delayed, and the (maximum) delay equals 50\,ms. The error bars are set to $\pm{}2\sigma$ (95\% of the measurements for the normal distribution)}
\label{fig:bench-size}
\end{figure}
Fig.~\ref{fig:bench-proc} shows the tested algorithms performance related to the increase of the number of the nodes/processes exchanging reduced data. For 32 nodes, we observed interesting behavior of the Rabenseifner algorithm, which was designed for power of 2 node/process numbers. Both the PRR and SLT algorithms present stable speedup over the regular ring algorithm, proving good usability and high scalability for imbalanced PAPs.
\begin{figure}[!h]\sidecaption
\includegraphics[width=8cm]{Fig12}
\caption{Benchmark scalability results for 1\,GB Ethernet network, one process delayed for 100\,ms, 1\,M of float numbers data size. The error bars are set to $\pm{}2\sigma$ (95\% of the measurements for the normal distribution)}
\label{fig:bench-proc}
\end{figure}
\section{All-reduce PAP optimization for training of a deep neural network}\label{sec:ml}
In this section we present a practical application of the proposed method for a deep learning iterative procedure, implemented using tiny-dnn open-source library~\cite{tiny-dnn}. The example is focused on training of a convolutional neural network to classify graphical images (photos).
For the experiments we used a training dataset usually utilized for benchmarking purposes: CIFAR-10~\cite{cifar10}, it contains 60,000 32$\times$32 color images grouped in 10~classes. There are 50,000 training images, exactly 5,000 images per class. The other 10,000 test images are used to evaluate training results. In our example we assess the performance of the distributed processing, especially collective communication, thus we did not need to use the test set.
We test the network architecture 8~layers: 3~convolutional, 3~average pooling, and 2~fully connected ones, the whole model has 145,578~parameters of float size. Each process trains such a network and after each training iteration it is averaged over the other processes, a similar procedure was described in~\cite{Dean2012}, with the distinction in using a separated parameter server. The mini-batch size was set to 8~images per node, what gives 390~iterations in total.
The training program was implemented in C++ language (v. C++11) using tiny-dnn library~\cite{tiny-dnn}. It was chosen because it is very lightweight: header only, easy to install, dependency-free and has an open-source license: BSD 3-Clause. The above features made it easy to introduce the required modification: a callback function for each neural network layer, called during the distributed SGD (Stochastic Gradient Descent~\cite{Dean2012}) training. The function is used for progress monitoring, where the egde() function (see section~\ref{sec:model}) is called just before the last layer is processed, which takes about 44\% of computation time of the iteration. Additionally, the computation part of each iteration is performed in parallel, using POSIX threads~\cite{pthreads} executed on available (24) cores (provided by two processors with Hyper Threading switched off).
We tested 3~all-reduce algorithms: ring, sorted linear tree (SLT) and pre-reduced ring (PRR). The PAP framework, including estimation of computation time and warm-up was initiated only for the latter two. The benchmark was executed 128~times for each algorithm, and each execution consists of 390~training iterations with all-reduce function calls. The tests were executed in an HPC cluster environment, consisting of 16~nodes with 1\,Gbps Ethernet interconnecting network (the configuration is described in section~\ref{sec:bench-setup}).
The results of the benchmark are presented in Table~\ref{tab:cifar10}. The PRR has the lowest time of communication: 31.9\,ms (average elapsed time of all-reduce) as well as the total time of training: 78.6\,s, SLT is slightly worse having 33.1\,ms and 78.8\,s respectively. These algorithms were compared to a typical ring~\cite{Thakur2005} implementation, where the average elapsed time equals 35.7\,ms and the total training took 81.9\,s. In context of the average elapsed time, the PRR and SLT algorithms are faster for 12.1\% and 7.9\%, while for the training total time 4.2\% and 4.0\% respectively.
\begin{table}[h]
\caption{Times and speedup of Cifar10~\cite{cifar10} benchmark execution. The times are presented in ms, and the speedup is calculated in comparison with a ring algorithm implementation}
\label{tab:cifar10}
\begin{tabular}{|p{1.4cm}|p{2.1cm}|p{2.1cm}|p{2.1cm}|p{2.1cm}|}
\hline
Algorithm & All-reduce average elapsed time & All-reduce speedup & Training total time & Training speedup\\\hline
Ring & 35.7 & 1.000 & 81,900 & 1.000\\\hline
SLT & 33.1 & 1.079 & 78,768 & 1.040\\\hline
PRR & 31.9 & 1.121 & 78,604 & 1.042\\\hline
\end{tabular}
\end{table}
The above results seem to be just a slight improvement, however in the current, massive processing systems, e.g. neural networks, which are trained using thousands of compute nodes, consuming MegaWatts of energy, with budgets of millions of dollars, introducing 4\% computing time reduction, without additional resource demand can provide great cost savings.
\section{Final remarks}
The proposed algorithms provide optimizations for all-reduce operations executed in environment of imbalanced PAPs. The experimental results show improved performance and good scalability of the proposed solution over currently used algorithms. The real case example (machine learning using distributed SGD~\cite{Dean2012}) shows usability of the prototype implementation with sum as the reducing operation. The solution can be used in a wide spectrum of applications using iterative computation model, including many machine learning algorithms.
The future works cover the following topics:
\begin{itemize}
\item evaluation of the method for a wider range of interconnecting network speeds and larger number of nodes using a simulation tool e.g.~\cite{Czarnul2017,Proficz2016},
\item expansion of the method for other collective communication algorithms, e.g. all-gather,
\item a framework for automatic PAP detection and proper algorithm selection, e.g. providing a regular ring for balanced PAPs and PRR for imbalanced ones,
\item introduction of the presented PAT estimation method for other purposes e.g. asynchronous SDG training~\cite{Dean2012} or deadlock and race detection in distributed programs~\cite{Krawczyk2000},
\item deployment of the solution in a production environment.
\end{itemize}
We believe that the ubiquity of the imbalanced PAPs in HPC environments~\cite{Faraj2008} will cause a fast development of new solutions related to this subject.
\bibliographystyle{plain}
|
{
"timestamp": "2018-04-17T02:10:17",
"yymm": "1804",
"arxiv_id": "1804.05349",
"language": "en",
"url": "https://arxiv.org/abs/1804.05349"
}
|
\section{Introduction}
\label{sec:intro}
Extracting feature descriptors from local image patches is a common stage in many computer vision tasks involving alignment or matching.
To replace handcrafted feature engineering, recently much attention has been paid to learning local feature descriptors.
Despite exciting progress, certain levels of handcrafting are currently present in the design of learning objectives for local feature descriptors, making it difficult to have performance guarantees when the learned descriptors are integrated into larger pipelines.
Indeed, according to a recent study \cite{comparative}, traditional handcrafted features such as SIFT \cite{SIFT} can still outperform learned ones in complicated tasks such as 3D reconstruction.
In this paper, we aim to improve the learning of local feature descriptors by optimizing better objective functions.
Our thesis is that local feature descriptor learning is not a standalone problem, but rather a component in the optimization of larger pipelines. Therefore, the learning objectives should be designed in accordance with other pipeline components.
Upon inspection of common local feature matching pipelines, we find that feature matching can be exactly formulated as nearest neighbor retrieval.
Thus, we propose a novel listwise \emph{learning to rank} formulation for learning local feature descriptors,
based on the direct optimization of a ranking-based retrieval performance metric: Average Precision.
Our formulation uses deep neural networks, and works for both binary and real-valued descriptors.
Compared to recent approaches, our method optimizes a commonly adopted evaluation metric, and eliminates complex optimization heuristics.
Descriptors learned with our formulation achieve state-of-the-art results in benchmarks including UBC Phototour \cite{DAISY}, HPatches \cite{HPatches}, RomePatches \cite{RomePatches}, and the Oxford dataset \cite{Mikolajczyk2005}.
An important feature of our proposed formulation is that it is general-purpose, as it optimizes the performance of the task-independent nearest neighbor matching stage,
rather than a task-specific pipeline.
Nevertheless, to better tailor the learned descriptors for feature matching, we also augment our formulation with task-specific improvements.
First, we make use of the Spatial Transformer module \cite{STN} to effectively handle geometric noise and improve the robustness of matching, without requesting extra supervision.
Also, for the challenging HPatches dataset, we design a clustering-based technique to mine additional patch-level supervision, which improves the performance of learned descriptors in the image matching task.
In summary, we propose a general-purpose learning to rank formulation that optimizes local feature descriptors for nearest neighbor matching.
Our learned descriptors achieve state-of-the-art performance, and are further enhanced by task-specific improvements.
We believe that our contribution can serve as a stepping stone for the direct optimization of larger computer vision pipelines.
\begin{figure*}[ht]
\centering
\includegraphics[width=.83\linewidth]{pipeline.pdf}
\caption{An example local feature-based image matching pipeline, where the task is to estimate the fundamental matrix $\mathbf{F}$ between images $\mathbf{I}=(I_1,I_2)$, using robust estimation techniques such as RANSAC \cite{RANSAC}.
We model the {feature descriptor extractor} using {deep neural networks}, and directly optimize {a ranking-based objective (Average Precision)} for the subsequent stage of descriptor matching.
}
\label{fig:pipeline}
\end{figure*}
\section{Related Work}
\label{sec:related}
\noindent\textbf{Learning Local Features}
Parallel with the long history of handcrafted computer vision pipelines (the most prominent example being SIFT \cite{SIFT}), numerous researchers have attempted to replace handcrafted components with learned counterparts.
There exist many formulations for learning different components in local feature based pipelines.
For example, interest point detectors are learned in \cite{TILDE,covariant,QuadNets}, LIFT \cite{LIFT} learns three components separately in a feature matching pipeline, and DSAC \cite{DSAC} approximately learns a camera localization pipeline end-to-end.
For learning local feature descriptors, some early works use simple architectures \cite{DAISY,BinBoost} and convex optimization \cite{Simonyan14}.
Later approaches use deep neural networks: PhilippNet \cite{PhilippNet} learns by fitting pseudo-classes, DeepDesc \cite{DeepDesc} applies Siamese networks, MatchNet \cite{MatchNet} and DeepCompare \cite{DeepCompare} learn nonlinear distance metrics for matching, and \cite{RomePatches} uses Convolutional Kernel Networks.
A series of recent works have considered more advanced model architectures and triplet-based deep metric learning formulations, including UCN \cite{UCN}, TFeat \cite{TFeat}, GLoss \cite{Gloss}, L2Net \cite{L2Net}, HardNet \cite{HardNet}, and GOR \cite{GOR}.
Instead of optimizing triplet-based surrogate losses, we employ listwise learning to rank to directly optimize the performance of the matching stage.
Although end-to-end optimization of the pipeline is attractive, it is unfortunately highly difficult and task-dependent.
By focusing on the two task-independent stages (descriptor extraction and matching), our solution is general-purpose and can be potentially integrated into larger optimization pipelines.
\vspace{.2em}
\noindent\textbf{Evaluating Local Feature Descriptors}
Local features ideally should be evaluated in terms of final task performance, \eg Mikolajczyk and Schmid \cite{Mikolajczyk2005} use precision and recall derived from image matching, and Schonberger \etal \cite{comparative} use a benchmark based on 3D reconstruction.
However, in complex vision pipelines, final task performance can be affected by individual components.
For example, \cite{HPatches} observes that without controlling for components such as interest point detection in image-based benchmarks, different conclusions can be drawn when comparing the relative performance of feature descriptors.
Patch-based benchmarks provide unambiguous evaluation for local feature descriptors.
The \emph{patch verification} task is first proposed in \cite{DAISY}, formulated as binary classification on the relationship between patch pairs.
RomePatches \cite{RomePatches} and HPatches \cite{HPatches} both consider the \emph{patch retrieval} task, which simulates nearest neighbor matching, and is shown \cite{HPatches} to be more realistic and challenging compared to patch verification.
A ranking-based evaluation metric, Average Precision, is adopted in both benchmarks.
\vspace{.2em}
\noindent\textbf{Ranking Optimization in Metric Learning}
Metric learning \cite{metric_survey} is a general family of methods that learn distance functions from data.
While much previous effort focused on learning Mahalanobis distances, recently the metric learning community has focused on learning vector embeddings to be used with standard (\eg Euclidean) distance metrics.
In this light, the problem of learning local feature descriptors is an instance of metric learning.
Learning vector embeddings necessarily calls for task-dependent formulations.
For nearest neighbor retrieval, optimization of ranking performance has been studied in metric learning.
For example, learning to rank formulations for Mahalanobis distances are proposed in \cite{MLR,Lim2014}.
Triplet-based deep metric learning approaches \cite{Law2017ICML,histloss,LiftStruct,samplingmatters} can also be viewed as optimizing surrogate ranking losses.
In the ``learning to hash'' subcommunity that considers the special case of learning binary embeddings, He \etal \cite{TALR} directly optimize ranking-based retrieval performance measures with deep neural networks, based on an approximation to histogram binning originally proposed in \cite{histloss}, which is also adopted in learning binary descriptors by \cite{mihash}.
We make use of their optimization technique in the learning of binary and real-valued descriptors for our problem.
\section{Optimizing Descriptors for Matching}
\label{sec:method}
In this section, we motivate our approach by analyzing the descriptor matching stage, and point out that it corresponds to nearest neighbor retrieval.
Then we discuss a learning to rank formulation to optimize ranking-based retrieval performance.
\subsection{Nearest Neighbor Matching}
\label{sec:method:nn}
Consider Fig.~\ref{fig:pipeline}, which depicts a pipeline for estimating the fundamental matrix between matching images $I_1$ and $I_2$.
It consists of four stages: feature detection, descriptor extraction, descriptor matching, and robust estimation.
Suppose we detect and extract $M$ local features from each image.
The descriptor matching stage operates as follows:
it computes the pairwise distance matrix with $M^2$ entries,
and for each feature in $I_1$, looks for its nearest neighbor in $I_2$, and vice versa.
Feature pairs that are mutual nearest neighbors\footnote{For simplicity, the distance ratio check \cite{SIFT} is not considered.} become candidate matches in the robust estimation stage, such as RANSAC \cite{RANSAC}.
We point out that this matching process is exactly performing nearest neighbor retrieval: each feature in $I_1$ is used to query a database, which is the set of features in $I_2$.
For good performance, true matches should be returned as top retrievals, while false matches are ranked as low as possible.
Performance of the matching stage also directly reflects the quality of the learned descriptors, since it has no learnable parameters (only performs distance computation and sorting).
To assess nearest neighbor matching performance, we adopt Average Precision (AP), a commonly used evaluation metric.
AP evaluates the performance of retrieval systems under the \emph{binary relevance} assumption: retrievals are either ``relevant'' or ``irrelevant'' to the query.
This naturally fits the local feature matching setup, where given a reference feature, features in a target image are either its true match or false match.
Next, we learn binary and real-valued local feature descriptors to optimize AP.
\subsection{Optimizing Average Precision}
\label{sec:method:AP}
We first introduce mathematical notation.
Let $\mathcal{X}$ be the space of image patches, and $S\subset\mathcal{X}$ be a database.
For a query patch $q\in\mathcal{X}$, let $S_q^+$ be the set of its matching patches in $S$, and let $S_q^-$ be the set of non-matching patches.
Given a distance metric $D$, let $(x_1,x_2,\ldots,x_n)$ be a ranking of items in $S_q^+\cup S_q^-$ sorted by increasing distance to $q$, \ie $D(x_1,q)\leq D(x_2,q) \ldots \leq D(x_n,q)$.
Given the ranking, AP is the average of precision values ($Prec@K$) evaluated at different positions:
\vspace{-.5em}
\begin{align}
Prec@K&=\frac{1}{K}\sum_{i=1}^K \boldsymbol{1}[x_i\in S_q^+],\\
AP & = \frac{1}{|S_q^+|}\sum_{K=1}^n \boldsymbol{1}[x_K\in S_q^+]Prec@K,
\end{align}
where $\boldsymbol{1}[\cdot]$ is the binary indicator.
AP achieves its optimal value if and only if every patch from $S_q^+$ is ranked above all patches from $S_q^-$.
The optimization of AP can be cast as a metric learning problem, where the goal is to learn a distance metric $D$ that gives optimal AP when used for retrieval.
Ideally, if all the above steps can be formulated in differentiable forms, then AP can be optimized by exploiting chain rule.
However, this is not possible in general: the sorting operation, required in producing the ranking, is non-differentiable, and continuous changes in the input distances induce discontinuous ``jumps" in the value of AP.
Thus, appropriate smoothing is necessary to derive differentiable approximations of AP.
Our solution is based on a recent result in the metric learning community.
For the problem of learning binary image-level descriptors for image retrieval, He \etal \cite{TALR} observe that sorting on integer-valued Hamming distances can be implemented as histogram binning,
and employ a differentiable approximation to histogram binning \cite{histloss} to optimize ranking-based objectives with gradient descent.
We use this optimization framework to optimize AP for both binary and real-valued local feature descriptors.
In the latter case, the optimization is enabled by a novel quantization-based approximation that we develop.
\vspace{.3em}
\noindent\textbf{Binary Descriptors}
Binary descriptors offer compact storage and fast matching, which are useful in applications with speed or storage restrictions.
Although binary descriptors can be learned one bit at a time \cite{BinBoost}, here we take a gradient-based relaxation approach to learn fixed-length ``hash codes".
Formally, a deep neural network $F$ is used to model a mapping from patches to a low-dimensional Hamming space: ${F}\!:\!\mathcal{X}\rightarrow\{-1,1\}^b$.
For the Hamming distance $D$, which takes integer values in $\{0,1,\ldots,b\}$, AP can be computed in closed form using entries of a histogram $\mathbf{h}^+=(h_0^+,\ldots,h_b^+)$, where $h_k^+=\!\sum_{x\in{S_q^+}}\boldsymbol{1}[D(q,x)\!=\!k]$.
The closed-form AP can further be continuously relaxed, and differentiated with respect to $\mathbf{h}^+$ \cite{TALR}.
The next step in the chain rule is to differentiate entries of $\mathbf{h}^+$ with respect to the network $F$.
Usnitova and Lempitsky \cite{histloss} approximate the histogram binning operation as
\vspace{-1em}
\begin{align}
h_k^+ \approx \sum_{x\in{S_q^+}}\delta(D(q,x),k),
\label{eq:histbin}
\end{align}
\vspace{-.2em}
\noindent replacing the binary indicator with a differentiable function $\delta$ that peaks when $D(q,x)=k$.
This allows to derive approximate gradients as
\vspace{-.2em}
\begin{align}
\frac{\partial h_{k}^+}{\partial F(q)} & \approx \sum_{x\in {S_q^+}}
\frac{\partial \delta(D(q,x),k)}{\partial D(q,x)}
\frac{\partial D(q,x)}{\partial F(q)},
\label{eq:chainrule1}\\
\frac{\partial h_{k}^+}{\partial F(x)} & \approx \boldsymbol{1}[x\in{S_q^+}]
\frac{\partial \delta(D(q,x),k)}{\partial D(q,x)}
\frac{\partial D(q,x)}{\partial F(x)}.
\label{eq:chainrule2}
\end{align}
Note that the partial derivative of the Hamming distance is obtained via this differentiable formulation:
\begin{align}
D(x,x') = \frac{1}{2}\left(b-F(x)^\top F(x')\right). \label{eq:hamming_dist}
\end{align}
Finally, the thresholding operation used to produce binary bits is smoothed using the $\tanh$ function,
\vspace{-.2em}
\begin{align}
{F}(x) & = (\mathop{\mathrm{sgn}}(f_1(x)),\ldots,\mathop{\mathrm{sgn}}(f_b(x))) \\
& \approx (\tanh(f_1(x)),\ldots,\tanh(f_b(x))), \label{eq:sigphi}
\end{align}
\vspace{-.2em}
\noindent where $f_i$ are real-valued neural network activations.
With these relaxations, the network can be trained end-to-end.
\vspace{.3em}
\noindent\textbf{Real-Valued Descriptors}
To complete our formulation, we next consider real-valued descriptors, which are preferred in high-precision scenarios.
We model the the descriptor as a vector of real-valued network activations, and apply $L_2$ normalization: $\|F(x)\|=1,\forall x$.
In this case, the Euclidean distance $D$ is given as
\vspace{-.6em}
\begin{align}
D(x,x') =\sqrt{2-2F(x)^\top F(x')}.
\label{eq:Euclidean}
\end{align}
\vspace{-.3em}
The main challenge in optimizing AP for real-valued descriptors is again the non-differentiable sorting, but real-valued sorting has no simple alternative form.
However, histogram binning can be used as an approximation: we \emph{quantize} real-valued distances using histogram binning, obtain the histograms $\mathbf{h}^+$, {and then reduce the optimization problem to the previous one.}
With $L_2$-normalized vectors, the quantization is easy to implement as the Euclidean distance has closed range $[0,2]$: we simply uniformly divide $[0,2]$ into $b+1$ bins.
To derive the chain rules in this case, only the partial derivatives of the distance function needs modification in \eqref{eq:chainrule1} and \eqref{eq:chainrule2}.
The differentiation rules for the $L_2$ normalization operation are well known, and we give full derivations in the appendix.
Differently from the case of binary descriptors, the number of histogram bins $b$ is now a free parameter, which involves a trade-off. On the one hand, a large $b$ reduces quantization error, which in fact achieves zero if each histogram bin contains at most one item.
On the other hand, gradient computation for approximate histogram binning has linear complexity in $b$. Nevertheless, in our experiments, we consistently obtain good results using $b\leq 25$.
\subsection{Comparison with Other Ranking Approaches}
\label{sec:method:LTR}
We would like to contrast our approach with others in the learning to rank context.
Some recent methods, \eg \cite{L2Net,HardNet,TFeat,GOR}, learn feature descriptors by optimizing losses defined on triplets in the form of $(a,p^+,p^-)$, where $a$ is an anchor patch, $p^+$ is its matching patch, and $p^-$ is a non-matching patch.
The loss typically encourages the learned distance metric $D$ to satisfy $D(a,p^+)<D(a,p^-)-\rho$, where $\rho$ is a margin.
Triplet losses have a long history in metric learning \cite{triplet2004,triplet2010}, and are better suited for ranking tasks than pair-based losses used in Siamese networks (\eg \cite{DeepDesc}).
In learning to rank terminology \cite{learn_rank}, triplets define local \emph{pairwise} ranking losses,
while our approach is \emph{listwise} since the evaluation metric that we optimize (AP) is defined on a ranked list.
Despite their simplicity, triplet losses can be very challenging to optimize.
For $N$ training examples, the set of triplets is of size $O(N^3)$, but most of them get classified correctly early on during learning.
To maintain stable progress, carefully tuned heuristics such as hard negative mining \cite{HardNet}, anchor swap \cite{TFeat}, or distance-weighted sampling \cite{samplingmatters} are crucial.
We note that these optimization difficulties stem from a fundamental mismatch between triplet losses and listwise evaluation.
As shown in Fig.~\ref{fig:AP}, listwise metrics are \emph{position-sensitive}, while local losses are insensitive; an error made on a single triplet may have a big impact on the result if it occurs near the top of the list.
Therefore, heuristics are needed to focus on reducing high-rank errors.
In contrast, our method directly optimizes the listwise evaluation metric, Average Precision, and is free of such heuristics.
The listwise optimization also implicitly encodes hard negative mining: it requires matching patches to be ranked above all non-matching patches, which automatically enforces correct classification of the hardest triplet in the batch without explicitly finding it.
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{triplets.pdf}
\caption{Comparison between triplet-based and listwise ranking approaches.
Top: in triplet-based training, most triplets get correctly classified early (first row), and it is crucial to find and correct high-rank errors (red dashed box), with a heuristic known as hard negative mining.
Bottom: in listwise ranking which is \emph{position-sensitive}, the high-rank error would reduce AP from $1$ to $0.5$, thus automatically receiving a heavy penalty.
Our listwise optimization corrects such errors without using complex mining heuristics.
Best viewed in color.
}
\label{fig:AP}
\end{figure}
\section{Task-Specific Improvements}
\label{sec:addon}
In addition to the general-purpose learning to rank formulation, we develop two improvements that take the nature of local feature matching into account.
\subsection{Handling Geometric Noise}
\label{sec:addon:STN}
To improve the robustness of local features for matching, it is key to build invariance to {geometric} noise into the descriptor:
SIFT \cite{SIFT} estimates orientation and affine shape to normalize input patches, and LIFT \cite{LIFT} includes a learned orientation estimation module.
Likewise, we {can} also include a geometric alignment module in our descriptor networks.
Our choice is the Spatial Transformer \cite{STN}, which aligns input patches by predicting a 6-DOF affine transformation, without requiring extra supervision.
In our experiments, this module is able to correct geometric distortion, and consistently improve performance.
In contrast to the image-based UCN \cite{UCN}, which also includes Spatial Transformers,
our patch-based networks have limited input size, and the predicted affine transformation can often lead to out-of-boundary sampling, which corrupts sampled patches.
We address this challenge by using appropriate boundary padding.
Details are given in the appendix.
\subsection{Label Mining for Image Matching}
\label{sec:addon:kmeans}
While our formulation directly optimizes for the task of \emph{patch retrieval},
it is also possible to address higher-level tasks.
We demonstrate this with the \emph{image matching} task in the challenging HPatches dataset \cite{HPatches}, which contains patches extracted from matching image sequences.
{The image matching task in HPatches is formulated similarly as patch retrieval, which involves retrieving matching patches in a pool of ``distractors".
However, the distractors are defined differently.
In patch retrieval, distractors do not include patches in the same image sequence as the query, due to concern of repeating structures in images.
In image matching, images are matched against others in the {same} sequence, which means that all distractors are actually in-sequence.
Thus, image matching performance can be improved by including in-sequence distractors when optimizing patch retrieval.
We perform \emph{label mining} to augment the set of distractors when optimizing patch retrieval in HPatches. To avoid noisy labels in the presence of repeating structures, we use a simple heuristic: clustering.
For each image sequence, we cluster all patches based on visual appearance. Then, patches having high inter-cluster distance are marked as distractors for each other (with 3D verification).
Note that label mining is not related to the hard negative mining heuristic, since its goal is to add additional supervision.
Please see Sec.~\ref{sec:exp:HPatches} and appendix for more details.
\section{Experiments}
\label{sec:exp}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{patches.pdf}
\caption{Examples from three patch-based datasets (top to bottom): RomePatches \cite{RomePatches}, UBC Phototour \cite{DAISY}, HPatches \cite{HPatches}.
In all datasets, patches are grouped such that patches in the same group correspond to the same 3D point.
}
\label{fig:patches}
\end{figure}
We experiment with three patch-based datasets (examples are in Fig.~\ref{fig:patches}): UBC Phototour \cite{DAISY}, HPatches \cite{HPatches}, and RomePatches \cite{RomePatches}.
We use the CNN architecture recently proposed in L2Net \cite{L2Net}, which consists of seven convolution layers, and is regularized with Batch Normalization and Dropout.
We do not use the more complex ``Center Surround'' architecture.
The input to the network is 32x32 grayscale, and we resize input patches to this size.
When adding the Spatial Transformer module, we increase the input size to 42x42, and use 3 convolution layers to predict a 6-DOF affine transformation, which is then used to sample a 32x32 patch.
\begin{table*}[ht]
\centering
\begin{tabular}{c|c|cc|cc|cc|c}
\hline
\multirow{2}{*}{\bf Method} & {Train} & Notredame & Yosemite & Liberty & Yosemite & Liberty & Notredame & FPR95 \\
\cline{2-8}
& {Test} & \multicolumn{2}{c|}{Liberty} & \multicolumn{2}{c|}{Notredame} & \multicolumn{2}{c|}{Yosemite} & {Mean} \\
\hline
\multicolumn{9}{c}{\it Real-valued descriptors} \\
\hline
SIFT \cite{SIFT} & 128
& \multicolumn{2}{c|}{29.84} & \multicolumn{2}{c|}{22.53} & \multicolumn{2}{c|}{27.29} & 26.55 \\
MatchNet\cite{MatchNet} & 128
& 7.04 & 11.47 & 3.82 & 5.65 & 11.6 & 8.70 & 8.05 \\
TFeat-M* \cite{TFeat} & 128
& 7.39 & 10.31 & 3.06 & 3.80 & 8.06 & 7.24 & 6.64 \\
TL-AS-GOR \cite{GOR} & 128
& 4.80 & 6.45 & 1.95 & 2.38 & 5.40 & 5.15 & 4.36 \\
DC-2ch2st+ \cite{DeepCompare} & 512
& 4.85 & 7.20 & 1.90 & 2.11 & 5.00 & 8.39 & 4.19 \\
CS-SNet-GLoss+ \cite{Gloss} & 256
& 3.69 & 4.91 & 0.77 & 1.14 & 3.09 & 2.67 & 2.71 \\
L2Net+ \cite{L2Net} & 128
& 2.36 & 4.7 & 0.72 & 1.29 & 2.57 & 1.71 & 2.23 \\
HardNet+ \cite{HardNet} & 128
& 2.28 & 3.25 & 0.57 & 0.96 & 2.13 & 2.22 & 1.90 \\
HardNet-GOR+ \cite{HardNet,GOR} & 128
& 1.89 & 3.03 & 0.54 & 0.90 & 2.41 & 2.39 & 1.86 \\
CS-L2Net+ \cite{L2Net} & 256
& 1.71 & 3.87 & 0.56 & 1.09 & 2.07 & 1.30 & 1.76 \\
DOAP{}+ & 128
& 1.54 & 2.62 & 0.43 & 0.87 & 2.00 & \textbf{1.21} & 1.45 \\
DOAP{}-ST+ & 128
& \textbf{1.47} & \textbf{2.29} & \textbf{0.39} & \textbf{0.78} & \textbf{1.98} & 1.35 & \textbf{1.38} \\
\hline
\multicolumn{9}{c}{\it Binary descriptors} \\
\hline
BinBoost \cite{BinBoost} & 64
& 20.49 & 21.67 & 16.90 & 14.54 & 22.88 & 18.97 & 19.24 \\
L2Net+ \cite{L2Net} & 128
& 7.44 & 10.29 & 3.81 & 4.31 & 8.81 & 7.45 & 7.01 \\
CS-L2Net+ \cite{L2Net} & 256
& 4.01 & 6.65 & 1.90 & 2.51 & 5.61 & 4.04 & 4.12 \\
DOAP{}+ & 256
& 3.18 & 4.32 & 1.04 & \textbf{1.57} & {4.10} & 3.87 & 3.01 \\
DOAP{}-ST+ & 256
& \textbf{2.87} & \textbf{4.17} & \textbf{0.96} & {1.76} & \textbf{3.93} & \textbf{3.64} & \textbf{2.89} \\
\hline
\end{tabular}
\vspace{.5em}
\caption{Patch verification performance on UBC Phototour, where metric is false positive rate at 95\% recall (FPR95). The best results are in \textbf{bold}.
Second column shows dimensionality, and methods with suffix ``+'' are trained with data augmentation.
Both the binary and real-valued versions of DOAP{} and DOAP{}-ST achieve state-of-the-art results.
}
\label{table:ubc}
\end{table*}
We name our {descriptor} DOAP{} ({\bf D}escriptors {\bf O}ptimized for {\bf A}verage {\bf P}recision), and test its binary and real-valued versions.
Our networks are trained using SGD with momentum 0.9 and weight decay $10^{-4}$, and {the} learning rate is decayed linearly to zero within a fixed number of epochs.
The initial learning rate (always on the order of 0.1) and number of epochs are tuned during training.
Input normalization is as follows: patches are normalized by subtracting the mean pixel value in the patch and then dividing by the standard deviation.
\subsection{UBC Phototour}
\label{sec:exp:UBC}
We first conduct experiments on the UBC Phototour dataset \cite{DAISY}, a classical benchmark of descriptor performance.
Patches are extracted from Difference-of-Gaussian detections in three image sequences: \emph{Liberty}, \emph{Notre Dame}, and \emph{Yosemite}.
Following the standard setup, we use six training/test combinations formed by the three sequences,
and report patch verification performance in terms of false positive rate at 95\% recall (FPR95).
We train our models on UBC Phototour with data augmentation, in the form of random flipping and 90-degree rotations, which showed consistent performance improvement in previous work.
We compare to a range of existing descriptors, including both binary and real-valued, listed in Table~\ref{table:ubc}.
L2Net \cite{L2Net} and HardNet \cite{HardNet} are two leading methods, which optimize triplet-based losses with the same CNN architecture as ours.
We also include methods that use the ``Center Surround'' architecture: CS-SNet-Gloss \cite{Gloss} and CS-L2Net,
and we have applied the recent global regularization technique in \cite{GOR} to HardNet, resulting in a more competitive method which we call HardNet-GOR.
Compared to existing approaches, DOAP{} achieves state-of-the-art performance with both binary and real-valued descriptors, and results are further improved by DOAP-ST, which includes the Spatial Transformer module.
We attribute the performance of DOAP{} and DOAP-ST to the listwise AP optimization.
As mentioned in Sec.~\ref{sec:method:LTR}, listwise optimization automatically includes the ``hard negative mining" heuristic in local ranking approaches, since it implicitly enforces the correct classification of all induced pairs and triplets.
We then expect performance to improve when increasing training batch size, as larger batches lead to longer lists and increased likelihood of including hard negatives.
We validate this by training the 128-dimensional DOAP{} model on \emph{Liberty}, varying batch size between 256 and 4096, and monitoring the average of FPR95 on \emph{Notre Dame} and \emph{Yosemite}.
Indeed, Fig.~\ref{fig:batchsize} shows that performance improves with batch size and saturates after 2048.
Similar trends are also observed in HardNet \cite{HardNet}, with saturation occurring at batch size 512.
In contrast, the listwise optimization allows the performance of DOAP to saturate at a later stage.
\begin{figure}
\vspace{-.5em}
\centering
\includegraphics[width=.8\linewidth]{batchsize.pdf}
\caption{Influence of training batch size for the 128-d DOAP{} descriptor trained on \emph{Liberty}, with data augmentation. Vertical axis: average of FPR95 on \emph{Notre Dame} and \emph{Yosemite}.
}
\label{fig:batchsize}
\vspace{-.5em}
\end{figure}
\subsection{HPatches}
\label{sec:exp:HPatches}
\begin{figure*}[ht]
\centering
\includegraphics[height=1.7em]{marker_leg1.pdf}
\includegraphics[height=1.7em]{marker_leg3.pdf}
\includegraphics[height=1.7em]{marker_leg2.pdf}
\vspace{-1.2em}
\includegraphics[width=.33\linewidth]{hp-verification.pdf}
\includegraphics[width=.33\linewidth]{hp-retrieval.pdf}
\includegraphics[width=.33\linewidth]{hp-matching.pdf}
\vspace{-.5em}
\caption{Results on the HPatches dataset, evaluated on the test set of the ``a'' split. No ZCA normalization \cite{HPatches} is used.
Suffix indicates training set used (Lib: \emph{Liberty}, no suffix: HPatches). HardNet++ is trained on the union of \emph{Liberty} and HPatches.
DOAP{} outperforms competing methods in all tasks, and all of its variants excel in handling tough test cases.
}
\label{fig:HPatches}
\end{figure*}
HPatches \cite{HPatches} consists of a total of over 2.5 million patches extracted from 116 image sequences, each with 6 images with known homography.
Both viewpoint and illumination changes are included, and test cases have levels of difficulty \emph{easy}, \emph{hard}, and \emph{tough}, according to the amount of geometric noise.
Three evaluation tasks are considered (in increasing order of difficulty): patch verification, patch retrieval, and image matching.
In this experiment, we focus on comparing real-valued descriptors.
We first include four baselines reported in \cite{HPatches}: SIFT \cite{SIFT}, RootSIFT \cite{RootSIFT}, DeepDesc \cite{DeepDesc}, and TFeat \cite{TFeat}.
Next, as results for L2Net and HardNet trained on the \emph{Liberty} sequence of UBC Phototour are reported in \cite{HardNet}, for fair comparison, we also report results for our models trained on \emph{Liberty}.
Finally, we train and evaluate three versions of our descriptor on HPatches: DOAP{}, DOAP{}-ST with the Spatial Transformer, and DOAP{}-ST-LM, which additionally uses label mining.
We compare to the L2Net model trained on HPatches, and HardNet++, trained on the union of \emph{Liberty} and HPatches.
Note that CS-L2Net is excluded as it performs worse than L2Net in this more realistic dataset, which is consistent with the observations in \cite{L2Net,Gloss}.
When determining training/test sets, we use the ``a'' split:
the test set contains 40 image sequences (20 viewpoint and 20 illumination), and the training set contains the other 76 sequences.
Fig.~\ref{fig:HPatches} presents results
on HPatches.\footnote{
Results for L2Net and HardNet are obtained using their publicly released models and may slightly differ from those reported in \cite{HardNet}.
}
Our descriptors achieve state-of-the-art results for all three tasks, and all variants are better at handling \emph{tough} test cases than competing methods.
Specifically, DOAP{} and DOAP{}-ST obtain the best patch retrieval performance, which directly results from the optimization of patch retrieval mAP.
This optimization also gives state-of-the-art performance in patch verification.
For the most challenging task of image matching, as mentioned in \cite{HPatches}, patch retrieval performance is well correlated.
However, due to the difference in task definition that we mentioned in Sec.~\ref{sec:addon:kmeans}, all methods see lower performance when tested for image matching.
With the clustering-based label mining, DOAP-ST-LM significantly improves image matching mAP compared to the next best models: around {6}\% and {10}\% over DOAP-ST and L2Net, respectively.
Notably, it achieves \textbf{over 50\% mAP} even in the toughest test cases (\emph{tough} geometric noise, illumination change).
The inclusion of extra supervision also boosts patch retrieval performance, since in-sequence distractors provide harder negatives to learn from.
\begin{table}
\centering
\begin{tabular}{c|cc|cc}
\hline
\textbf{Method} & \textbf{Coverage} & \textbf{Dim.} & \textbf{Train} & \textbf{Test} \\
\hline
SIFT \cite{SIFT} & 51x51 & {128} & 91.6 & 87.9 \\
AlexNet-conv3 \cite{AlexNet} & 99x99 & 384 & 81.6 & 79.2 \\
PhilippNet \cite{PhilippNet} & 64x64 & 512 & 86.1 & 81.4 \\
CKN-grad \cite{RomePatches} & 51x51 & 1024 & 92.5 & 88.1 \\
DOAP{} & 51x51 & {128} & \textbf{95.9} & \textbf{88.4} \\
Binary DOAP{} & 51x51 & {256} & 95.2 & 86.8 \\
\hline
\end{tabular}
\vspace{.2em}
\caption{Patch retrieval mAP comparison on RomePatches.
SIFT is a strong baseline, previously only surpassed by the high-dimensional CKN-grad \cite{RomePatches}.
DOAP is the first descriptor to outperform SIFT with the same dimensionality.
}
\label{table:rome}
\end{table}
\begin{figure*}[t]
\centering
\includegraphics[width=\linewidth]{chart.png}
\caption{Image matching performance on the Oxford dataset \cite{Mikolajczyk2005}.
Suffixes indicate the training set used (Lib: \emph{Liberty}, HP: HPatches).
Here, all versions of DOAP include the Spatial Transformer.
}
\label{fig:matching}
\end{figure*}
\subsection{RomePatches}
\label{sec:exp:Rome}
We next consider the RomePatches dataset \cite{RomePatches}, which contains 20,000 image patches of size 51x51, split equally into training and test sets.
The task is patch retrieval.
This dataset is constructed by performing SIFT matching on images taken in Rome, and keeping matching patches that satisfy 3D constraints.
With such tailored construction, SIFT is unsurprisingly a strong baseline on RomePatches.
In fact, in terms of test set mAP, previous methods,
including pretrained AlexNet \cite{AlexNet} and PhilippNet \cite{PhilippNet}, could not surpass SIFT.
The only method to do so was the CKN-grad variant proposed in \cite{RomePatches}, using 1024-dimensional descriptors.
Due to the small size of RomePatches,
we found it necessary to increase weight decay in SGD to $5\times 10^{-4}$, and Dropout rate from $0.1$ to $0.5$ in the L2Net architecture.
Also, adding Spatial Transformers did not improve results, possibly because the patches are already well aligned (see examples in Fig.~\ref{fig:patches}); therefore we only report results for the binary and real-valued DOAP.
As seen in Table~\ref{table:rome}, the real-valued DOAP{} outperforms SIFT and other descriptors with \textbf{88.4\% mAP} on the test set, while the binary version also performs competitively.
The comparison between DOAP and SIFT is fair, since they have the same input coverage and output dimensionality.
Note that the closest competitor to DOAP, CKN-grad \cite{RomePatches}, is unsupervised and needs high dimensionality to perform well.
By exploiting supervised learning and directly optimizing the evaluation metric, we are able to get better training and test performance while using 8x fewer dimensions (128 \vs 1024).
\subsection{Image Matching in Oxford Dataset}
\label{sec:exp:retrieval}
Lastly, we use our learned descriptors to perform image matching in six image sequences from the classical Oxford dataset \cite{Mikolajczyk2005}, where the matching pipeline also includes interest point detection.
We use the implementation from \texttt{VL-Benchmarks} \cite{vlbenchmarks}; features are detected by the Harris-Affine detector, and then patches are extracted with a magnification factor of 3 relative to the detected feature frames.
The evaluation metric is mean Average Precision (mAP), computed as the area under the precision-recall curve derived from nearest neighbor matching.
We compare to SIFT, LIOP \cite{LIOP} (the best-performing handcrafted descriptor in \cite{L2Net}'s experiment), and 128-d real-valued versions of L2Net and HardNet with different training sets.
We use the 256-bit binary and 128-d versions of DOAP trained on \emph{Liberty}, and the 128-d version trained on HPatches.
From the results in Fig.~\ref{fig:matching}, we can see that SIFT is indeed difficult to beat, and good results on the UBC benchmark does not guarantee high-level task performance, especially in the case of HardNet.
{The real-valued DOAP consistently outperforms SIFT and other descriptors with significant margins, especially in the more challenging sequences such as \emph{graf} and \emph{boat}.
The binary DOAP trained on \emph{Liberty} also outperforms other real-valued descriptors on average, including L2Net trained on HPatches, and HardNet trained on the union of \emph{Liberty} and HPatches.
}
\subsection{Discussion}
\vspace{-.3em}
\noindent\textbf{Minibatch Sampling.}
We discuss the minibatch sampling strategy used in training our models.
First, note that in all datasets considered, patches are provided in groups: patches within a group correspond to the same 3D point and thus match each other (see Fig.~\ref{fig:patches}).
The group size, denoted $n$, is between 2 and 3 on average in UBC Phototour, and equals 10 in RomePatches.
For HPatches, $n=16$, as each patch has a reference version, and five variations from each difficulty level.
{Our sampling strategy differs from those in local ranking approaches, where} patch groups are often broken up to form pairs or triplets in a pre-processing step before training.
Instead, we directly sample \emph{groups} to construct training minibatches, so that patches belonging to the same group are always in the same batch.
This allows our listwise optimization to utilize supervision with maximum efficiency.
Let minibatch size be $M$, every training patch is associated with a listwise ranking constraint, that its $n-1$ matches need to be ranked at the top of a list of size $M-1$.
This constraint alone needs $(n-1)(M-n)$ triplets to fully capture.
Take UBC Phototour as an example, assuming $n=2.5$ on average, a single minibatch of size $1024$ induces about $1.6\times 10^6$ triplets, which is already $1/32$ of the total number of training triplets used in HardNet.
For HPatches ($n=16$), this number would be $1.5\times 10^7$.
However, triplets do not need to be explicitly generated in our listwise optimization.
\vspace{.1em}
\noindent\textbf{Time Complexity.}
For a minibatch of size $M$, the pairwise distances between all examples are computed, and then binned into $b$-bin histograms. The time complexity is $O(bM^2)$.
The quadratic dependency on $M$ is in fact optimal, due to distance computation.
There is also a trade-off involving the batch size $M$.
A larger batch size leads to longer lists and better performance, but slows training.
Nevertheless, even with $M=4096$, a single training epoch on \emph{Liberty} takes less than 4 minutes on an Nvidia Titan X Pascal GPU.
Similar to the case of UBC (Fig.~\ref{fig:batchsize}), performance saturation is also observed around $M=2048$ in HPatches and RomePatches.
\vspace{-.3em}
\section{Conclusion}
\vspace{-.2em}
In this work, we use deep neural networks to learn binary and real-valued local feature descriptors that optimize nearest neighbor matching performance.
This is achieved through a listwise learning to rank formulation that directly optimizes Average Precision. Our formulation is general-purpose, and is superior to recent local ranking approaches.
We further enhance our formulation with task-specific components:
handling geometric noise with the Spatial Transformer, and mining labels using clustering.
The learned descriptors achieve state-of-the-art performance in patch verification, patch retrieval, and image matching.
Future work will explore the optimization of larger portions in vision pipelines, for example, by incorporating differentiable versions of robust estimation.
\vspace{-.2em}
\section*{Acknowledgements}
\vspace{-.2em}
A major part of this work was done during KH's internship at Honda Research Institute.
This work is also partly conducted at Boston University, supported by a BU IGNITION award, NSF grant 1029430, and gifts from Nvidia.
{\small
\bibliographystyle{plain}
|
{
"timestamp": "2018-04-19T02:02:23",
"yymm": "1804",
"arxiv_id": "1804.05312",
"language": "en",
"url": "https://arxiv.org/abs/1804.05312"
}
|
\section{Introduction}
The $AdS/CFT$ duality provides a correspondence between a strongly coupled
conformal field theory (CFT) in $d$-dimensions and a weakly coupled gravity
theory in ($d+1$)-dimensional anti-de Sitter ($AdS$) spacetime \cite{1,2,3}.
Since it is a duality between two theories with different dimensions, it is
commonly called holography. The idea of holography has been employed in
condensed matter physics to study various phenomena such as superconductors
\cite{4,5,6,7,8}. For describing the properties of low temperature
superconductors, the BCS theory can work very well \cite{9,10}. However,
this theory fails to describe the mechanism of high temperature
superconductor. In latter regime, the holography was suggested to study the
properties of superconductors \cite{11,12}. Hortnol $et$ $al.$ have
represented the first model of holographic superconductors~\cite{11,12}.
After that, holographic superconductors have attracted a lot of attention
and investigated from different point of views~\cite{4,13,14,15,16}.
BTZ black holes play a significant role in many of recent developments in
string theory~\cite{46,47,48}. BTZ-like solutions are dual of ($1+1
)-dimensional holographic systems such as one-dimensional holographic
superconductors. Distinctive features of normal and superconducting phases
of one-dimensional systems have been studied studied in~\cite{49}. The
latter study was done in probe limit. Considering the effects of
backreaction, the properties of one-dimensional holographic superconductors
have been studied both~numerically \cite{50,50-1} and analytically~\cit
{51,52}.
It is interesting to investigate the effect of nonlinear electrodynamic
models on holographic systems including holographic superconductors \cit
{17,18,19,20,21,22,23,24,34,30,31,32,38,Doa,n3,n5,n1,n2,n4,SheyPM}.
Nonlinear models carry more information than the usual Maxwell case and also
are considered as a possible way for avoiding the singularity of the
point-like charged particle at the origin \cite{25,26,27,28,29}. The oldest
nonlinear electrodynamic model is Born-Infeld (BI) model. There are also two
BI-like nonlinear electrodynamics namely logarithmic~\cite{33,nn,34} and
exponential~\cite{35,36,n5} electrodynamics. It has been found that the
exponential electrodynamics has stronger effect on the condensation~than
other models \cite{37}.
In the present work, we will study the one-dimensional holographic
superconductors both analytically and numerically in the presence of
exponential electrodynamics. To bring rich physics in holographic model, we
consider the backreaction of the scalar and gauge fields on the metric
background \cite{39,40,41,42,43,44,45}. To perform the analytical study, we
employ the Sturm-Liouville eigenvalue problem. We will study the effects of
nonlinear exponential electrodynamics model as well as backreaction on
critical temperature. We shall also use the numerical shooting method to
investigate the features of our holographic superconductors and make
comparison between analytical and numerical results.
This paper is organized as follows: In next section, we introduce the action
and basic field equations governing ($1+1$)-dimensional holographic
superconductors in the presence of exponential electrodynamics. In section
\ref{analst}, we study the properties of holographic superconductors
applying the analytical method based on Sturm-Liuoville eigenvalue problem.
In section \ref{numst}, we study holographic superconductors numerically by
employing the shooting method. We also compare our numerical and analytical
results. Finally, in last section we will summarize our results.
\section{Holographic Set-up}
To study a ($1+1$)-dimensional holographic superconductor, we consider a (
2+1$)-dimensional bulk action of AdS gravity coupled to a charged scalar
field $\psi
\begin{eqnarray}
S &=&\frac{1}{2\kappa ^{2}}\int d^{3}x\sqrt{-g}\left(R+\frac{2}{l^{2}}\right)
\notag \\
&&+\int d^{3}x\sqrt{-g}\left[ L\left( F\right) -|\nabla \psi -iqA\psi
|^{2}-m^{2}|\psi |^{2}\right] , \notag \\
&& \label{Act}
\end{eqnarray
where $g$ is the determinant of metric, $R$ is Ricci scalar, $l$ is the AdS
radius, $A$ is electromagnetic potential and $F=F_{\mu \nu }F^{\mu \nu }$ in
which $F_{\mu \nu }=\nabla _{\lbrack \mu }A_{\mu ]}$. In action (\ref{Act}),
$\kappa ^{2}=8\pi G_{3}$ where $G_{3}$ is the ($2+1$)-dimensional Newtonian
constant and $m$ and $q$ represent the mass and charge of scalar field,
respectively. $L\left( F\right) $ stands for the Lagrangian of
electrodynamics model. ($1+1$)-dimensional holographic superconductors in
the presence of linear Maxwell electrodynamics presented by $L\left(
F\right) =-F/4$ have been studied in \cite{50, 53}. The linear model is an
idealization of reality. In principle, other powers of $F$ may play role.
There are different nonlinear electrodynamics models which exhibit the
electrodynamics interaction. In this paper, we suppose that electrodynamics
interaction is governed by exponential nonlinear electrodynamics model \cit
{35
\begin{equation}
L\left( F\right) =\frac{1}{4b}\left( {\mathrm{e}}^{-bF}-1\right),
\label{eleclag}
\end{equation
where $b$ determines the nonlinearity. For small values of $b$, Lagrangian
\ref{eleclag}) recovers the linear Maxwell Lagrangian. The parameter $\kappa
$ in (\ref{Act}) also determines the backreaction. When $\kappa $ goes to
zero, we are in the probe limit, meaning that the gravity part of action
\ref{Act}) is stronger than the matter field part. Physically, this implies
that the gauge and matter fields do not back react on the metric background.
In superconducting language, implies that the Cooper pairs have negligible
interaction with background system. In the presence of backreaction, the
dual black hole solution may be given by the ansat
\begin{equation}
ds^{2}=-f(r){\mathrm{e}}^{-\chi (r)}dt^{2}+\frac{dr^{2}}{f(r)}+\frac{r^{2}}
l^{2}}dx^{2}.
\end{equation
The Hawking temperature of above black hole solution is given b
\begin{equation}
T=\frac{f^{\prime }(r_{+}){\mathrm{e}}^{-\chi (r_{+})}}{4\pi }, \label{3}
\end{equation
where $r_{+}$ is event horizon which could be obtained as the greatest root
of $f(r)=0$. Varying the action (\ref{Act}) with respect to $\psi $, $A_{\nu
}$ and $g_{\mu \nu }$, the field equations read, respectively,
\begin{eqnarray}
0 &=&\left( \nabla _{\mu }-i{q}A_{\mu }\right) \left( \nabla ^{\mu }-i{q
A^{\mu }\right) \psi -m^{2}\psi \,, \label{01} \\
&& \notag \\
0 &=&\nabla ^{\mu }\left( 4L_{F}F_{\mu \nu }\right) \notag \\
&&-i{q}\left[ -\psi ^{\ast }(\nabla _{\nu }-i{q}A_{\nu })\psi +\psi (\nabla
_{\nu }+i{q}A_{\nu })\psi ^{\ast }\right] \,, \label{02} \\
&& \notag \\
0 &=&\frac{1}{2\kappa ^{2}}\left[ R_{\mu \nu }-g_{\mu \nu }\left( \frac{R}{2
+\frac{1}{l^{2}}\right) \right] +2F_{ac}F_{b}{}^{c}L_{F} \notag \\
&&-\frac{g_{\mu \nu }}{2}\left[ L\left( F\right) -m^{2}|\psi |^{2}-{|\nabla
\psi -i{q}A\psi |^{2}}\right] \notag \\
&&-\frac{1}{2}\left[ (\nabla _{\mu }\psi -i{q}A_{\mu }\psi )(\nabla _{\nu
}\psi ^{\ast }+i{q}A_{\nu }\psi ^{\ast })+\mu \leftrightarrow \nu \right] ,
\notag \\
&& \label{03}
\end{eqnarray
where $L_{F}=\partial L/\partial F$. Adopting the ansatz $A_{\mu }=\phi
(r)\delta _{\mu }^{0}$ and $\psi =\psi (r)$, field equations (\ref{01})-(\re
{03}) lead to
\begin{eqnarray}
0 &=&\psi ^{\prime \prime }+\psi ^{\prime }\left[ \frac{1}{r}+\frac
f^{\prime }}{f}-\frac{\chi ^{\prime }}{2}\right] +\psi \left[ \frac
q^{2}\phi ^{2}{\mathrm{e}}^{\chi }}{f^{2}}-\frac{m^{2}}{f}\right] ,
\label{f1} \\
&& \notag \\
0 &=&\phi ^{\prime \prime }+\phi ^{\prime }\left( \frac{1}{r}+\frac{\chi
^{\prime }}{2}\right) \notag \\
&&-{\frac{2{q}^{2}\psi ^{2}\phi }{{\mathrm{e}^{2b\phi ^{\prime 2}{\mathrm{e
^{\chi }}}}f}+2b\phi ^{\prime 2}{\mathrm{e}^{\chi }}\left( 2\phi ^{\prime
\prime }+\phi ^{\prime }\chi ^{\prime }\right) }, \label{f2} \\
&& \notag \\
0 &=&f^{\prime }+{\frac{\kappa ^{2}r}{2b}}\left[ 1+{\mathrm{e}^{2b\phi
^{\prime 2}{\mathrm{e}^{\chi }}}}\left( 4b\phi ^{\prime 2}{\mathrm{e}^{\chi
}-1\right) \right] \notag \\
&&+2\,\kappa ^{2}r\left[ {\frac{q^{2}\phi ^{2}\psi ^{2}{\mathrm{e}^{\chi }}}
f}}+\left( m\psi \right) ^{2}+\psi ^{\prime 2}f\right] -{\frac{2r}{{l}^{2}}},
\label{f3} \\
&& \notag \\
0 &=&\chi ^{\prime }+4\kappa ^{2}r\left[ \frac{q^{2}\phi ^{2}\psi ^{2}
\mathrm{e}}^{\chi }}{f^{2}}+\psi ^{\prime 2}\right] . \label{f4}
\end{eqnarray
where the prime denotes the derivative with respect to $r$. Obviously, the
above equations reduce to the corresponding equations in Ref.~\cite{50} when
$b\rightarrow 0$ while in the absence of the backreaction ($\kappa
\rightarrow 0$), Eqs. (\ref{12}) and (\ref{13}) reduce to ones in Ref.~\cit
{37}. By virtue of symmetries of field equations (\ref{12})-(\ref{15}
\begin{gather}
q\rightarrow q/a,\text{ \ \ \ \ }\phi \rightarrow a\phi ,\text{ \ \ \ \
\psi \rightarrow a\psi , \notag \\
\kappa \rightarrow \kappa /a,\text{ \ \ \ \ }b\rightarrow b/a^{2}, \\
\notag \\
l\rightarrow al,\text{ \ \ \ \ }r\rightarrow ar,\text{ \ \ \ \ }q\rightarrow
q/a, \notag \\
m\rightarrow m/a,\text{ \ \ \ \ }b\rightarrow a^{2}b,
\end{gather
one can set $q=l=1$. In the following sections, we will study the
superconding phase transition both analytically and numerically.
\section{Analytical Study\label{analst}}
The behaviors of model functions governed by field equations (\ref{12})-(\re
{15}) near the boundary $r\rightarrow \infty $ are \footnote
Near the boundary, $\chi $ could be a constant but by using the symmetry of
field equation $\mathrm{e}^{\chi }\rightarrow a^{2}\mathrm{e}^{\chi },$
\phi \rightarrow \phi /a$, one can set it to zero there.
\begin{gather}
\chi (r)\rightarrow 0,\qquad f(r)\sim r^{2}, \notag \\
\phi (r)\sim \rho +\mu \ln (r),\qquad \psi (r)\sim \frac{\psi _{-}}
r^{\Delta _{-}}}+\frac{\psi _{+}}{r^{\Delta _{+}}}, \label{boundval}
\end{gather
where $\mu $ and $\rho $ are chemical potential and charge density of dual
field theory and $\Delta _{\pm }=1+\sqrt{1\pm m^{2}}$. The superconducting
phase transition is characterized by growing the expectation value of order
parameter $\left\langle O\right\rangle $ as temperature decreases. In normal
phase, $\left\langle O\right\rangle $ vanishes. According to holographic
dictionary, the expectation value of order parameter $\left\langle
O\right\rangle $ is dual to ${\psi _{+}}$ or ${\psi _{-}}$ while the other
one can be considered as the source. Therefore, near the critical point
\left\langle O_{\pm }\right\rangle $ is small and one can define it a
\begin{equation}
\epsilon \equiv \left\langle O_{i}\right\rangle ,
\end{equation
where $i=+$ or $-$. Since $\epsilon $ is so small, we can expand the model
functions as~\cite{39,54,55,56}\footnote
It is expected that when the sign of $\epsilon $ changes, the sign of scalar
filed which leads to order parameter, changes too. So, the expansion powers
of $\psi $ is considered odd. For other functions, even powers is used
because they should not change when the sign of order parameter changes.}
\begin{gather}
f=f_{0}+\epsilon ^{2}f_{2}+\epsilon ^{4}f_{4}+\cdots , \label{f} \\
\chi =\epsilon ^{2}\chi _{2}+\epsilon ^{4}\chi _{4}+\cdots , \label{chi} \\
\psi =\epsilon \psi _{1}+\epsilon ^{3}\psi _{3}+\epsilon ^{5}\psi
_{5}+\cdots , \label{psi} \\
\phi =\phi _{0}+\epsilon ^{2}\phi _{2}+\epsilon ^{4}\phi _{4}+\cdots .
\label{phi}
\end{gather
Also we can expand the chemical potential as~\cite{55}
\begin{equation}
\mu =\mu _{0}+\epsilon ^{2}\delta \mu _{2}+...,
\end{equation
where $\delta \mu _{2}>0$. Thus, the order parameter as a function of
chemical potential can be obtained a
\begin{equation}
\epsilon \approx \frac{(\mu -\mu _{0})^{1/2}}{\delta \mu _{2}}.
\end{equation
When $\mu \rightarrow \mu _{0}$, phase transition occurs and the order
parameter is zero at the critical value $\mu _{c}=\mu _{0}$. Above equation
also indicates the critical exponent $\beta =1/2$ which is the same as the
universal result from the mean field theory. Hereafter, we define the
dimensionless coordinate $z=r_{+}/r$ instead of $r,$ since it is easier to
work with it. In terms of this new coordinate, $z=0$ and $z=1$ correspond to
the boundary and horizon respectively. The field equations (\ref{12})-(\re
{15}) can be rewritten in terms of $z$ as
\begin{eqnarray}
0 &=&\psi ^{\prime \prime }+\left[ \frac{1}{z}+\frac{f^{\prime }}{f}-\frac
\chi ^{\prime }}{2}\right] \psi ^{\prime }+\frac{r_{+}^{2}}{z^{4}}\left[
\frac{q^{2}\phi ^{2}{\mathrm{e}}^{\chi }}{f^{2}}-\frac{m^{2}}{f}\right] \psi
, \notag \\
&& \label{12} \\
0 &=&\phi ^{\prime \prime }+\left( \frac{\chi ^{\prime }}{2}+\frac{1}{z
\left[ \frac{1+4\Upsilon }{1+2\Upsilon }\right] \right) \phi ^{\prime }
\frac{2q^{2}r_{+}^{2}\psi ^{2}}{z^{4}f}\left( \frac{{\mathrm{e}}^{-\Upsilon
}{1+2\Upsilon }\right) \phi , \notag \\
&& \label{13} \\
0 &=&f^{\prime }-{\frac{\kappa ^{2}{r_{+}}^{2}}{2bz^{3}}}\left[ 1+{\mathrm{e
^{\Upsilon }}\left( {2\Upsilon }-1\right) \right] \notag \\
&&-\frac{2\kappa ^{2}{r_{+}^{2}}}{z^{3}}\left[ \frac{q^{2}\phi ^{2}{\psi ^{2
}{\mathrm{e}^{\chi }}}{f}+m^{2}\psi ^{2}\right] +{\frac{2{r_{+}^{2}}}{z^{3}}
, \label{14} \\
&& \notag \\
0 &=&\chi ^{\prime }-\frac{4\kappa ^{2}r_{+}^{2}}{z^{3}}\left[ \frac
q^{2}\phi ^{2}\psi ^{2}{\mathrm{e}}^{\chi }}{f^{2}}+\frac{z^{4}\psi ^{\prime
2}}{r_{+}^{2}}\right] , \label{15}
\end{eqnarray
where $\Upsilon =2bz^{4}{\mathrm{e}}^{\chi }\phi ^{\prime 2}/{r_{+}}^{2}$.
The field equation of $\phi $ (Eq. (\ref{13})) at zeroth order with respect
to $\epsilon $ reduces t
\begin{equation}
\phi ^{\prime \prime }(z)+\frac{\phi ^{\prime }(z)(r_{+}^{2}+8{b}{z^{4}}\phi
^{\prime 2}(z))}{z(r_{+}^{2}+4{b}{z^{4}}\phi ^{\prime 2}(z))}=0.
\end{equation
The solution of above equation read
\begin{equation}
\phi (z)=\int_{1}^{z}\frac{r_{+}\sqrt{L_{W}(\frac{4bz^{2}C_{0}^{2}}{{r_{+}
^{2}})}}{2{z^{2}}\sqrt{b}}dz, \label{24}
\end{equation
where $C_{0}$ is an integration constant and $L_{W}(x)=LambertW(x)$ is the
Lambert function which satisfies~\cite{57}
\begin{equation}
L_{W}(x){\mathrm{e}}^{L_{W}(x)}=x,
\end{equation
and can be expanded a
\begin{equation}
L_{W}(x)=x-x^{2}+\frac{3}{2}x^{3}-\frac{8}{3}x^{4}.
\end{equation
Expanding Eq. (\ref{24}) for small $b$ and keeping the terms up to first
order of $b$ we fin
\begin{equation}
\phi (z)=C_{1}+C_{0}\ln (z)+\frac{C_{0}^{3}}{r_{+}^{2}}(1-z^{2})b+O\left(
b^{2}\right) . \label{27}
\end{equation
Comparing the above equation with Eq. (\ref{boundval}), we find $C_{0}=-\mu
. Also, $C_{1}=0$, since at the horizon $\phi(r_{+})=0$. Inserting $C_{0}$
into Eq. (\ref{27}) we have
\begin{equation}
\phi _{0}(z)=\lambda r_{+}\left[ -\ln (z)+\lambda ^{2}(z^{2}-1)b\right]
\text{ \ \ \ \ }{b}\lambda ^{2}<1, \label{phi0}
\end{equation
where $\lambda =\mu /r_{+}$. Substituting $\phi (z)$ into the field equation
(\ref{14}), we find the metric function at zeroth order with respect to
\epsilon $, $f_{0}(z)=r_{+}^{2}g(z)$ wher
\begin{equation}
g(z)=\left( \frac{1}{z^{2}}-1+\kappa ^{2}\lambda ^{2}\ln (z)+\frac{1}{2
\kappa ^{2}\lambda ^{4}b(1-z^{2})\right).
\end{equation
Note that at the horizon $f_{0}(1)=0$. The asymptotic behavior of scalar
field $\psi $ near the boundary ($z=0$) is given by Eq. (\ref{boundval}). In
order to match this behavior near the boundary, we introduce a trial
function $F(z)$ as $\psi (z)=\left\langle O_{i}\right\rangle \left(
z/r_{+}\right) ^{\Delta _{i}}F(z)$ which satisfies the boundary condition
F(0)=1$ and $F^{\prime }(0)=0$. Inserting the functions obtained above and
the trial function $F(z)$ into Eq. (\ref{12}), one receive
\begin{align}
& F^{\prime \prime }(z)+F^{\prime }(z)\left[ \frac{2\Delta +1}{z}+\frac
g^{\prime }(z)}{g(z)}\right] \notag \\
& +\left[ -\frac{m^{2}}{g(z)z^{4}}+\frac{\Delta ^{2}}{z^{2}}+\frac{\Delta
g^{\prime }(z)}{zg(z)}\right] F(z)+ \notag \\
& +\frac{q^{2}\lambda ^{2}F(z)}{g(z)^{2}}\left[ \frac{{\ln (z)}^{2}}{z^{4}}
\frac{2\lambda ^{2}{b}{\ln (z)}}{z^{2}}+\frac{2\lambda ^{2}{b}{\ln (z)}}
z^{4}}\right] =0.
\end{align
It is a matter of calculations to show that the above equation satisfies the
following second order Sturm-Liouville equation~\cite{58}
\begin{equation}
\left[ T(z)F^{\prime }(z)\right] ^{\prime }-Q(z)F(z)+\lambda ^{2}P(z)F(z)=0,
\label{35}
\end{equation
where
\begin{eqnarray}
T(z) &=&{g(z)}z^{2\Delta +1}, \\
P(z) &=&\frac{q^{2}T(z)}{g^{2}(z)}\left[ \frac{{\ln (z)}^{2}}{z^{4}}-\frac
2\lambda ^{2}{b}{\ln (z)}}{z^{2}}+\frac{2\lambda ^{2}{b}{\ln (z)}}{z^{4}
\right] , \notag \\
&& \\
Q(z) &=&-T(z)\left[ -\frac{m^{2}}{g(z)z^{4}}+\frac{\Delta ^{2}}{z^{2}}+\frac
\Delta g^{\prime }(z)}{zg(z)}\right] .
\end{eqnarray
Considering the trial function as $F(z)=1-\alpha z^{2}$ and using the
Sturm-Liouville eigenvalues problem, the eigenvalues of Eq. (\ref{35}) can
be obtained by minimizin
\begin{equation}
\lambda ^{2}=\frac{\int_{0}^{1}{dz}T(z)\left[ F^{\prime 2}(z)+Q(z)F^{2}(z
\right] }{\int_{0}^{1}{dz}.T(z)P(z)F^{2}(z)}, \label{39}
\end{equation
with respect to $\alpha $~\cite{59}. Here, we use a perturbative expansion
b\lambda ^{2}$ up to the first order of $b
\begin{equation}
b\lambda ^{2}=b(\lambda ^{2}|_{b=0})+O(b^{2}).
\end{equation
For backreaction parameter, we use iteration method and tak
\begin{equation}
\kappa _{n}=n\Delta \kappa ,\qquad \ n=0,1,2,3,\cdots ,
\end{equation
where $\Delta \kappa =\kappa _{n+1}-\kappa _{n}$. Here we chose $\Delta
\kappa =0.005$. So, the effect of the nonlinear corrections on the
backreaction term can be obtained a
\begin{equation}
\kappa ^{2}\lambda ^{2}={\kappa _{n}}^{2}\lambda ^{2}={\kappa _{n}
^{2}(\lambda ^{2}|_{\kappa _{n-1}})+O[(\Delta \kappa )^{4}].
\end{equation
By taking $\kappa _{-1}=0$ and $\lambda ^{2}|_{\kappa _{-1}}=0$, the minimum
eigenvalue of Eq. (\ref{39}) can be calculated. To calculate the critical
temperature, we obtain the latter value by variation of Eq. (\ref{39}) with
respect to $\alpha $ where the other parameters such as $b,\kappa
,m,q,\cdots $ are fixed. Using the definition of $T$ (Eq. (\ref{3})), the
critical temperature is given by $T_{c}=f^{\prime }(r_{+})/4\pi $ wher
\footnote
Note that at zeroth order with respect to $\epsilon $, $\chi $ is zero. So,
{\mathrm{e}}^{\chi }$ in temperature formula disappears.}
\begin{equation}
f^{\prime }(r_{+})={{2{r_{+}}}}-{\frac{\kappa ^{2}{r_{+}}}{2b}}\left[ 1+
\mathrm{e}^{2b\phi _{0}^{\prime 2}(r_{+})}}\left( 4b\phi _{0}^{\prime
2}(r_{+})-1\right) \right] ,
\end{equation
and $r_{+}=\mu /\lambda $ so we have
\begin{equation}
T_{c}=\frac{1}{4\pi }\frac{\mu }{\lambda }\left[ 2-{\kappa _{n}}^{2}(\lambda
^{2}|_{\kappa _{n-1}})+3b{\kappa _{n}}^{2}(\lambda ^{4}|_{\kappa _{n-1},b=0}
\right] . \label{45}
\end{equation
As an example, for $b=0.01$ and $\kappa =0$ the equation (\ref{39}) reduce
to
\begin{equation}
\lambda ^{2}=\frac{0.667\alpha ^{2}-1.333\alpha +1}{0.0276-0.0165\alpha
+0.0035\alpha ^{2}},
\end{equation
which has a minimum $\lambda _{\min }=21.6337$, with respect to $\alpha $,
at $\alpha =0.8258$ and thus according to Eq. (\ref{45}) we achieve
T_{c}=0.03421\mu $. We employ the iteration method to obtain the critical
temperature for different values of $\kappa $ and $b$.
In table \ref{Table21}, we present our results. This table shows that by
increasing nonlinear parameter ($b$), the value of $T_{c}$ decreases. As it
can be understood from this table, for a fixed value of $b$, with increasing
the backreaction parameter $\kappa $, the value of the critical temperature
decreases. We also reproduce the results of the linear Maxwell case without
backreaction (i.e. $b\rightarrow 0$ and $\kappa \rightarrow 0$) presented in
\cite{51}.
\begin{table}[t]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
$b$ & $\kappa^{2}=0$ & $\kappa^{2}=0.005$ & $\kappa^{2}=0.01$ &
\kappa^{2}=0.015$ & $\kappa^{2}=0.02$ \\ \hline
0 & 0.0429 & 0.0393 & 0.0379 & 0.0369 & 0.0359 \\ \hline
0.01 & 0.0342 & 0.0321 & 0.0311 & 0.0301 & 0.0288 \\ \hline
0.02 & 0.0274 & 0.0231 & 0.0216 & 0.0208 & 0.0199 \\ \hline
\end{tabular
\end{center}
\caption{Analytical results of ${T_{c}}/{\protect\mu }$ for different values
of $\protect\kappa $ and $b$.}
\label{Table21}
\end{table}
\begin{table}[t]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
$b$ & $\kappa^{2}=0$ & $\kappa^{2}=0.05$ & $\kappa^{2}=0.1$ &
\kappa^{2}=0.15$ & $\kappa^{2}=0.2$ \\ \hline
0 & 0.046 & 0.0368 & 0.0295 & 0.0236 & 0.0189 \\ \hline
0.01 & 0.0415 & 0.033 & 0.0262 & 0.0209 & 0.0166 \\ \hline
0.04 & 0.0325 & 0.0252 & 0.0197 & 0.0154 & 0.0121 \\ \hline
0.09 & 0.0236 & 0.0176 & 0.0133 & 0.0107 & 0.0078 \\ \hline
\end{tabular
\end{center}
\caption{Numerical results of ${T_{c}}/{\protect\mu }$ for different values
of $\protect\kappa $ and $b$.}
\label{Table31}
\end{table}
\section{Numerical Study\label{numst}}
\begin{figure*}[t]
\centering
\subfigure[~$b=0$]{
\label{fig1a}\includegraphics[width=.46\textwidth]{fig1}\qquad}
\subfigure[~$b=0.04$]{
\label{fig1b}\includegraphics[width=.46\textwidth]{fig2}\qquad}
\caption{The behavior of order parameter versus temperature for $m^{2}=0$.}
\label{fig1}
\end{figure*}
\begin{figure*}[t]
\centering
\subfigure[~$\protect\kappa ^{2}=0$]{
\label{fig2a}\includegraphics[width=.46\textwidth]{fig3}\qquad}
\subfigure[~$\protect\kappa ^{2}=0.1$]{
\label{fig2b}\includegraphics[width=.46\textwidth]{fig4}\qquad}
\caption{The behavior of order parameter versus temperature for $m^{2}=0$.}
\label{fig2}
\end{figure*}
\begin{table*}[t]
\centerin
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}
\hline
& \multicolumn{2}{|c|}{$\kappa ^{2}=0$} & \multicolumn{2}{|c|}{$\kappa
^{2}=0.005$} & \multicolumn{2}{|c|}{$\kappa ^{2}=0.01$} &
\multicolumn{2}{|c|}{$\kappa ^{2}=0.015$} & \multicolumn{2}{|c|}{$\kappa
^{2}=0.02$} \\ \cline{2-11}\hline
$b$ & $T_{c}$(An) & $T_{c}$(Nu) & $T_{c}$(An) & $T_{c}$(Nu) & $T_{c}$(An) &
T_{c}$(Nu) & $T_{c}$(An) & $T_{c}$(Nu) & $T_{c}$(An) & $T_{c}$(Nu) \\ \hline
0 & 0.0429 & 0.046 & 0.0393 & 0.0449 & 0.0379 & 0.0439 & 0.0369 & 0.0430 &
0.0359 & 0.0409 \\ \hline
0.01 & 0.0342 & 0.0415 & 0.0321 & 0.0406 & 0.0311 & 0.0396 & 0.0301 & 0.0387
& 0.0288 & 0.0378 \\ \hline
0.02 & 0.0274 & 0.0380 & 0.0231 & 0.0371 & 0.0216 & 0.0362 & 0.0208 & 0.0353
& 0.0199 & 0.0345 \\ \hline
\end{tabular}
\caption{Analytical and numerical results of ${T_{c}}/{\protect\mu }$ for
different values of $\protect\kappa $ and $b$.}
\label{Table1}
\end{table*}
In this section, we employ the shooting method \cite{4} to numerically
investigate the superconducting phase transition. Besides setting $q$ and $l$
to unity, we also set $r_{+}=1$ in the numerical calculation which may be
justified by virtue of the field equation symmetry \cite{50
\begin{equation*}
r\rightarrow ar,\text{ \ \ \ \ }f\rightarrow a^{2}f,\text{ \ \ \ \ }\phi
\rightarrow a\phi ,
\end{equation*
First, we expand Eqs. (\ref{12})-(\ref{15}) near black hole horizon ($z=1$
\begin{eqnarray}
\psi &\approx &\psi (1)+\psi ^{\prime }(1)(1-z)+\frac{\psi ^{{\prime }
\prime }}}{2}(1-z)^{2}+\cdots , \\
\phi &\approx &\phi ^{\prime }(1)(1-z)+\frac{\phi ^{{\prime }{\prime }}}{2
(1-z)^{2}+\cdots , \\
f &\approx &f^{\prime }(1)(1-z)+\frac{f^{{\prime }{\prime }}}{2
(1-z)^{2}+\cdots , \\
\chi &\approx &\chi (1)+\chi ^{\prime }(1)(1-z)+\frac{\chi ^{{\prime }
\prime }}}{2}(1-z)^{2}+\cdots .
\end{eqnarray
In above equations, we have imposed $f(1)=\phi (1)=0$.\footnote{$\phi $
should vanish at horizon so that the norm of gauge potential is regular
there.} In our numerical process, we will find $\psi (1)$, $\phi ^{\prime
}(1)$ and $\chi (1)$ such that the desired values for boundary parameters in
Eq. (\ref{boundval}) are attained. At boundary, one can set either $\psi _{-}
$ or $\psi _{+}$ to zero as source and find the value of the other one as
the expectation value of order parameter $\left\langle O\right\rangle $. We
will focus on $m^{2}=0$ case for our numerical calculations. For this case,
the behavior of $\psi $ near boundary is (see Eq. (\ref{boundval})
\begin{equation}
\psi (z)\approx \psi _{-}+{\psi _{+}}{z^{2}}.
\end{equation
We consider $\psi _{+}$ as holographic dual to the order parameter
\left\langle O_{+}\right\rangle $ at the boundary field theory.
In table \ref{Table31}, our numerical results for critical temperature with
different values of backreaction parameter $\kappa $ and nonlinear parameter
$b$ are presented. In the Maxwell limit ($b\rightarrow 0)$, our numerical
results reproduce the ones of \cite{50}. It can be seen that in the absence
of nonlinearity effect, the critical temperature decreases as $\kappa $
increases~\cite{50}. In the presence of nonlinearity parameter i.e. for any
nonvanishing value of $b$, it can be found that as $\kappa $ enhances, the
critical temperature $T_{c}$ decreases. Similar behavior can be found for
different values of $b$ when $\kappa $ is fixed. As the nonlinear parameter
b$ becomes larger, the critical temperature decreases i.e. the condensation
process becomes harder. This behavior have been reported previously in \cit
{60} for ($2+1$)-dimensional holographic superconductors too. Figs. \re
{fig1} and \ref{fig2} confirm above results. As it can be seen from Fig. \re
{fig1}, the scalar hair forms harder as $\kappa $ increases i.e. the gap in
graph of $\left\langle O_{+}\right\rangle $ becomes larger. The latter means
that the condensation of the operator $\left\langle O_{+}\right\rangle $
starts at larger values for stronger values of backreaction parameter. It
shows that the scalar hair can be formed more difficult when the
backreaction is stronger. Fig. \ref{fig2} shows the same effect for
nonlinearity parameter $b$. We compare the analytical and numerical results
in table \ref{Table1}. Table \ref{Table1} shows that there is a good
agreement between analytical and numerical results for small values of
\kappa $ and $b$. For larger values of these parameters, analytical and
numerical results separate more from each other.
\section{Conclusion}
In this work, we have studied the properties of one-dimensional holographic
superconductor in the presence of nonlinear exponential electrodynamics. We
have also considered the backreaction effect of scalar and gauge fields on
the background metric. We have performed both analytical and numerical
methods for studying our superconductors. To investigate the problem
analytically, we have used the Sturm-Lioville while our numerical study was
based on shooting method. It was shown that the enhancement in both
nonlinearity of electrodynamics model as well as the backreaction causes the
superconducting phase more difficult to be appeared. This result is
reflected in two ways from our data. From one side, we observed that the
increasement in the effects of nonlinearity and backreaction makes the
critical temperature of superconductor lower. From another side, for larger
values of nonlinear and backreaction parameters, the gap in condensation
parameter is larger which in turn exhibits that the condensation is formed
harder. We have also observed that, for small values of backreaction
parameter $\kappa $ and nonlinear parameter $b$, the analytic results are in
a good agreement with numerical ones whereas for larger values they separate
more from each other.
Finally, we would like to stress that in this work, we have only studied the
basic properties of one-dimensional backreacting holographic $s$-wave
superconductors in the presence of exponential nonlinear electrodynamics. It
is also interesting to investigate other characteristics of these systems
such as the behaviour of real and imaginary parts of conductivity or optical
features. One may also consider $(1+1)$-dimensional $p$-wave and $d$-wave
holographic superconductors in the background of BTZ black holes and
disclose the effects of nonlinearity as well as backreaction on the the
phase transition and conductivity of these models. These issue are now under
investigations and the result will be appeared soon.
\begin{acknowledgments}
MKZ would like to thank Shahid Chamran University of Ahvaz, Iran for
supporting this work. AS thanks the research council of Shiraz University.
The work of MKZ has been supported financially by Research Institute for
Astronomy \& Astrophysics of Maragha (RIAAM) under research project No.
1/5237-55.
\end{acknowledgments}
|
{
"timestamp": "2018-04-17T02:12:11",
"yymm": "1804",
"arxiv_id": "1804.05442",
"language": "en",
"url": "https://arxiv.org/abs/1804.05442"
}
|
\section{Introduction}
This paper is motivated by a former result of Lusztig \cite{lus83:left}. Let $W$ be the Weyl group of type $A_{n-1}$ and $\tc{W}$ be the centralizer of the longest element in $W$. Then $\tc{W}$ is naturally the Weyl group of type $C_m$ where $m=\floor{n/2}$. If $L: \tc{W} \rightarrow \mathbb{N}$ is the restriction of the usual length function $l: W \rightarrow \mathbb{N}$,
then \cite{lus83:left} gives a description of left cells in $\tc{W}$ with the weight function $L$, and showed that each left cell carries an irreducible representation of $\tc{W}$.
One of our goals in this paper is to extend his method to an affine setting. Let ${W}_a$ be the affine Weyl group of type $\tilde{A}_{n-1}$ and let $\omega$ be the involution of ${W}_a$ which corresponds to the nontrivial automorphism of the (finite) Dynkin diagram of type $A$. Then, the main theorem in this paper states that the number of left cells fixed by $\omega$ in each two-sided cell of ${W}_a$ is given by a certain Green polynomial of type $A$, originally defined by Green \cite{gre55}, evaluated at -1.
This paper is considered as a companion of \cite{cfkly}. There exists a bijection defined by Shi \cite{shi86, shi91} between left cells of ${W}_a$ and row-standard Young tableaux, called the generalized Robinson-Schensted correspondence. Under this bijection, the involution $\omega$ corresponds to an affine analogue of the usual Sch{\"u}tzenberger involution. The combinatorics of this involution is extensively studied in \cite{cfkly}, and the main theorem in this paper also follows from a more general result therein. Here, instead we explain this involution $\omega$ in a representation-theoretic view, and also provide another proof of our main theorem in terms of representation theory.
This paper is organized as follows: in Section \ref{section:notation} we cover basic notations and definitions used in this paper; in Section \ref{section:involution} we define and describe the involution $\omega$, called the affine Sch{\"u}tzenberger involution; in Section \ref{section:mainthm} we state the main theorem of this paper and remark some related facts; in Section \ref{section:proof} we prove the main theorem modulo some combinatorial reduction; in Section \ref{sec:comb} we introduce a combinatorial model associated with our main objects and complete the proof of the main theorem.
\begin{ack} The author thanks Roman Bezrukavnikov for kindly explaining his theory to him; it would have been impossible to write this paper without his help. The author is indebted to Michael Chmutov, Gabriel Frieden, and Joel Brewster Lewis for discussions about combinatorial descriptions of this subject, including contents in Section 6. He also thanks Sam Hopkins and Thomas McConville for making such discussions possible. Finally, he is grateful to George Lusztig and Devra Johnson for their helpful comments.
\end{ack}
\section{Notations and definitions} \label{section:notation}
\subsection{Basic notations} For a set $X$, we define $|X|$ to be the cardinal of $X$. If there is a map $f\colon X \rightarrow X$, we denote by $X^f$ the set of elements in $X$ fixed by $f$. Similarly, if there exists a group $G$ acting on $X$, then we denote by $X^G$ the set of elements in $X$ fixed by the action of $G$.
For a variety $X$, we denote by $H^i(X)=H^i(X, {\overline{\mathbb{Q}_\ell}})$ its $i$-th $\ell$-adic cohomology group, and define $H^*(X) \colonequals \bigoplus_{i\in \mathbb{Z}} (-1)^iH^i(X)$ to be their alternating sum (as a virtual ${\overline{\mathbb{Q}_\ell}}$-vector space).
For an abelian category $\mathcal{C}$, we write $D^b(\mathcal{C})$ to be its bounded derived category, and $K(\mathcal{C})=K(\mathcal{C})_\mathbb{C}$ to be the complexified Grothendieck group of $\mathcal{C}$. Likewise, for a variety $X$, we write $K(X)=K(X)_\mathbb{C}$ to be the complexified $K$-theory of $X$, i.e. the complexified Grothendieck group of the category of coherent sheaves on $X$. If there is an action of a group $G$ on $X$, then we denote by $K^G(X)=K^G(X)_\mathbb{C}$ the complexified $G$-equivariant $K$-theory of $X$.
\subsection{General setup}
We fix $n \in \mathbb{Z}_{>0}$ throughout this paper.
Let $G\colonequals GL_{n}$ be the general linear group of rank $n$ defined over $\mathbb{C}$, $B\subset G$ be the Borel subgroup consisting of upper-triangular matrices, and $T\subset B$ be the maximal torus consisting of diagonal matrices. Also let $\mathfrak{g} \colonequals \Lie G$, $\mathfrak{b}\colonequals \Lie B$, and $\mathfrak{h} \colonequals \Lie T$ be corresponding Lie algebras. Define $\mathcal{B}\colonequals G/B$ to be the flag variety of $G$. For a nilpotent element $N \in \mathfrak{g}$, let $\mathcal{B}_N$ be the Springer fiber of $N$, defined by $\mathcal{B}_N \colonequals \{ gB \in \mathcal{B} \mid \Ad(g)^{-1}(N) \in \mathfrak{b}\}$.
\subsection{Weyl groups of $G$}
Let $W$ and $\widetilde{W}$ be the Weyl group and the extended affine Weyl group of $G$, respectively. They are defined as follows.
\begin{gather*} W\colonequals N(G, T)/T, \qquad \widetilde{W} \colonequals N(G_{\mathbb{C}((t))}, T_{\mathbb{C}((t))}) / T_{\mathbb{C}[[t]]}.
\end{gather*}
Here, $N(X, Y)$ denotes the normalizer of $Y$ in $X$, and $G_R$ (resp. $T_R$) is the base change of $G$ (resp. $T$) from $\mathbb{C}$ to $R$. Also, we define ${W}_a \subset \widetilde{W}$ to be the subgroup generated by elements in $N(G_{\mathbb{C}((t))}, T_{\mathbb{C}((t))})$ whose determinant is contained in $\mathbb{C}[[t]]^\times$, called the affine Weyl group of $G$. Then, $W$ is naturally a subgroup of ${W}_a$.
We choose $\{s_1, \ldots, s_{n-1}\} \subset W$ such that $s_i$ corresponds to swapping $i$-th and $(i+1)$-th entries of diagonal matrices. Also we let $s_0, \tau \in \widetilde{W}$ be the images of
$$
\begin{pmatrix}0 & 0 & 0&\cdots&0 & 0 & t
\\0 & 1 &0& \cdots &0& 0 & 0
\\0 & 0 &1& \cdots &0& 0 & 0
\\\cdots & \cdots & \cdots & \cdots & \cdots& \cdots & \cdots
\\0 & 0 &0& \cdots &1& 0 & 0
\\0 & 0 &0& \cdots &0& 1 & 0
\\t^{-1} & 0 & 0&\cdots & 0 & 0&0
\end{pmatrix} \textup{ and } \begin{pmatrix}0 & 0 & 0&\cdots&0 & 0 & t
\\1 & 0 &0& \cdots &0& 0 & 0
\\0 & 1 &0& \cdots &0& 0 & 0
\\\cdots & \cdots & \cdots & \cdots & \cdots& \cdots & \cdots
\\0 & 0 &0& \cdots &0& 0 & 0
\\0 & 0 &0& \cdots &1& 0 & 0
\\0 & 0 & 0&\cdots & 0 & 1&0
\end{pmatrix},$$
respectively. Then,
\begin{enumerate}[label=$\bullet$]
\item $(W,\{s_1, \ldots, s_{n-1}\})$ and $(W_a, \{s_0, s_1, \ldots, s_{n-1}\})$ are Coxeter groups,
\item $\tau s_i \tau^{-1} = s_{i+1}$ for $0\leq i \leq n-2$ and $\tau s_{n-1} \tau^{-1} = s_0$, and
\item $\widetilde{W} \simeq {W}_a \rtimes \br{\tau}, \br{\tau} \simeq \mathbb{Z}$.
\end{enumerate}
We define $S\colonequals \{s_0, s_1, \ldots, s_{n-1}\}$ to be the set of simple reflections of ${W}_a$.
\subsection{Involution $\omega$} Define $J \in G$ to be the matrix whose anti-diagonal entries are 1 and other entries are 0. We define an involutive automorphism $\omega$ on $G$ by
$$\omega: G \mapsto G : g \mapsto J(^tg^{-1})J^{-1}.$$
We abuse notation and write $\omega$ for an involution on any object which is naturally induced from the above automorphism. Clearly, it induces an involution on $\mathfrak{g}$ defined by $\omega(X) = -\Ad(J)(^tX)$. Also, it induces involutions on $W, {W}_a,$ and $\widetilde{W}$, respectively. Indeed, direct calculation shows that
$$\omega(s_i) = s_{n-i} \textup{ for } 1 \leq i \leq n-1,\qquad \omega(s_0)=s_0,\qquad \omega(\tau) = \tau^{-1}.$$
On the other hand, since $\omega$ fixes $B$, it defines an action on the flag variety of $G$. If $N \in \mathfrak{g}$ is a nilpotent element fixed by $\omega$, then $\omega$ acts on its Springer fiber $\mathcal{B}_N$ and thus acts on the cohomology and the $K$-theory of $\mathcal{B}_N$ as well.
\subsection{Kazhdan-Lusztig cells} \label{sec:cells}
We usually use the symbol ${\underline{c}}$ (resp. $\Gamma$, $\Gamma^{-1}$) to denote a two-sided (resp. left, right) cell of $\widetilde{W}$. (The notion of Kazhdan-Lusztig cells is first defined in \cite{kalu79} for Coxeter groups, and is generalized to extended affine Weyl groups in \cite{lus89:cell}.) Then each two-sided (resp. left, right) cell of ${W}_a$ is of the form ${\underline{c}} \cap {W}_a$ (resp. $\Gamma \cap {W}_a$, $\Gamma^{-1} \cap {W}_a$), and this gives a bijection between two-sided (resp. left, right) cells of ${W}_a$ and $\widetilde{W}$. Note that each two-sided (resp. left, right) cell of $\widetilde{W}$ is stable under multiplication (resp. left multiplication, right multiplication) by $\tau \in \widetilde{W}$.
In \cite{lus89:cell}, a canonical bijection between two-sided cells of $\widetilde{W}$ and nilpotent orbits in $\mathfrak{g}$ is constructed. (Here, we identify $\mathfrak{g}$ with its Langlands dual.) We write ${\underline{c}}_\lambda$ to be the two-sided cell that corresponds to the nilpotent orbit in $\mathfrak{g}$ of Jordan type $\lambda \vdash n$ under this bijection. Then $a({\underline{c}}_\lambda)$ is equal to the dimension of the Springer fiber corresponding to this nilpotent orbit, where $a$ is Lusztig's $a$-function defined in \cite{lus85:cell}.
\subsection{Partitions, tableaux, and tabloids} For a partition $\lambda$, we write $\lambda = (\lambda_1, \ldots, \lambda_r)$ for $\lambda_1 \geq \lambda_2 \geq \cdots\geq \lambda_r >0$ or $\lambda=(1^{m_1}2^{m_2} \cdots)$ to describe its parts. Let $l(\lambda)$ be the length of $\lambda$, and put $\lambda_i=0$ if $i > l(\lambda)$. Define $|\lambda| \colonequals \sum_{i\geq 1} \lambda_i = \sum_{i\geq 1}im_i$. If $|\lambda|=k$, we also write $\lambda \vdash k$. We say $\lambda$ is strict if $\lambda_1 > \lambda_2> \cdots > \lambda_r >0$, or equivalently each $m_i$ is either 0 or 1. For another partition $\mu$, we write $\lambda \cup \mu$ to be the partition of $|\lambda|+|\mu|$ whose parts are the union (as a multiset) of parts of $\lambda$ and $\mu$.
\ytableausetup{smalltableaux}
For a partition $\lambda$, define $SYT(\lambda)$ to be the set of standard Young tableaux of shape $\lambda$ with entries $1, 2, \ldots, |\lambda|$. Similarly, we define $RSYT(\lambda)$ to be the set of row-standard Young tableaux of shape $\lambda$ with entries $1, 2, \ldots, |\lambda|$, which is defined by dropping the condition from $SYT(\lambda)$ that each column is strictly increasing. If $\lambda$ is a finite sequence of positive integers which is not necessarily a partition, we still write $RSYT(\lambda)$ to denote the set of row-standard Young tabloids of shape $\lambda$. For a tabloid $T$, we write $T=(T^{(1)}, \ldots, T^{(r)})$ where $T^{(i)}$ is the $i$-th part of $T$. For example, if $T\in RSYT((3,4))$ is $\begin{ytableau}1&3&4\\2&5&6&7\end{ytableau}$ with respect to the English notation, then $T^{(1)}=(1,3,4)$ and $T^{(2)}=(2,5,6,7)$.
\section{The affine Sch{\"u}tzenberger involution} \label{section:involution}
The involution $\omega$ on $G$ induces an involution on $\widetilde{W}$, which also permutes Kazhdan-Lusztig cells of $\widetilde{W}$.
\begin{defn} \emph{The affine Sch{\"u}tzenberger involution} is the involution induced by $\omega$ on the set of left cells of $\widetilde{W}$, again denoted by $\omega$.
\end{defn}
If we restrict $\omega$ to the Coxeter group $(W, S-\{s_0\})$, then it corresponds to the nontrivial involutive automorphism on the Dynkin diagram of type $A$, which is the same as conjugation by the longest element of $W$. This involution clearly permutes the left cells of $W$, and it is equivalent to the usual Sch{\"u}tzenberger involution on standard Young tableaux under the Robinson-Schensted correspondence. This is why we call $\omega$ the affine Sch{\"u}tzenberger involution.
Note that the usual Sch{\"u}tzenberger involution preserves the shape of each standard Young tableau. It means that the corresponding involution on $W$ stabilizes each two-sided cell of $W$. The same is true for $\omega$, as the following lemma shows.
\begin{lem}\label{lem:2sstab} Suppose that ${\underline{c}}$ is a two-sided cell of $\widetilde{W}$. Then $\omega({\underline{c}})={\underline{c}}$.
\end{lem}
\begin{proof}
It is clear that $\omega({\underline{c}})$ is also a two-sided cell of $\widetilde{W}$. Recall that ${\underline{c}}$ is stable under multiplication by $\tau$, thus in particular under its conjugation. Thus by \cite[Theorem 4.8(d)]{lus89:cell}, ${\underline{c}}$ intersects nontrivially with $W$. As we observed already that $\omega$ stabilizes each two-sided cell of $W$, the result follows.
\end{proof}
Therefore, it is possible to restrict $\omega$ to each two-sided cell of $\widetilde{W}$. We are interested in the number of left cells in each two-sided cell that are fixed by $\omega$.
\section{Main theorem and some remarks} \label{section:mainthm}
\subsection{Statement of the main theorem}
For $k \in \mathbb{N}$, let $\rho_2(k)$ be the partition $(2, 2, \ldots, 2) \vdash k$ (resp. $(2, 2, \ldots, 2, 1)\vdash k$) if $k$ is even (resp. odd). The main result of this paper is as follows.
\begin{thm}[Main theorem] \label{thm:main} Suppose that a two-sided cell ${\underline{c}}_\lambda\subset \widetilde{W}$ corresponds to the partition $\lambda \vdash n$. Then the number of left cells in ${\underline{c}}_\lambda$ that are fixed by $\omega$ is given by $\fgr^\lambda_{\rho_2(n)}(-1)$, where $\fgr^\lambda_{\rho_2(n)}(t)$ is the Green polynomial (for type $A$) originally defined in \textup{\cite{gre55}}.
\end{thm}
It is equivalent to \cite[Theorem 4.2]{cfkly}, which is proved by a more general combinatorial result. In this paper we give another proof of this theorem using representation theory.
\subsection{Affine Weyl groups of type $C$}
Let $\tc{{W}_a}$ be the set of elements in ${W}_a$ fixed by $\omega$ and $L$ be the restriction of $l: {W}_a \rightarrow \mathbb{N}$ to $\tc{{W}_a}$. Then $\tc{{W}_a}$ is the affine Weyl group of type $C$ whose simple reflections are given by
\begin{align*}
&s_0, s_1s_{n-1}, \ldots, s_{\frac{n-3}{2}}s_{\frac{n+3}{2}}, s_{\frac{n-1}{2}}s_{\frac{n+1}{2}}s_{\frac{n-1}{2}} &&\textup{ when $n$ is odd, and}
\\&s_0, s_1s_{n-1}, \ldots, s_{\frac{n-2}{2}}s_{\frac{n+2}{2}}, s_{\frac{n}{2}} &&\textup{ when $n$ is even}.
\end{align*}
The pair $(\tc{{W}_a}, L)$ is said to be in the quasisplit case in the sense of \cite{lus14:hecke}. There exists a strong connection between cells in ${W}_a$ and $(\tc{{W}_a}, L)$ as the following lemma shows.
\begin{lem} Suppose that ${\underline{c}}$ (resp. $\Gamma$) is a two-sided (resp. left) cell of $\widetilde{W}$. Thus ${\underline{c}} \cap {W}_a$ (resp. $\Gamma \cap {W}_a$) is a two-sided (resp. left) cell of ${W}_a$.
\begin{enumerate}[label=\textup{(\alph*)}]
\item $\Gamma \cap \tc{{W}_a}$ is nonempty if and only if $\Gamma$ is stable under $\omega$.
\item If $\Gamma \cap \tc{{W}_a}$ is nonempty, then it is also a left cell of $(\tc{{W}_a},L)$.
\item ${\underline{c}}$ is always $\omega$-stable, and ${\underline{c}}\cap \tc{{W}_a}$ is a (nonempty) union of two-sided cells of $(\tc{{W}_a},L)$.
\end{enumerate}
\end{lem}
\begin{proof}
For (a), one direction is clear since $({W}_a)^\omega = \tc{{W}_a}$. For the other direction, first note that there exists $\mathcal{D}\subset {W}_a$ (the set of distinguished involutions), such that $\omega(\mathcal{D}) = \mathcal{D}$ and $\mathcal{D}\cap \Gamma$ consists of only one element for each $\Gamma$. Therefore, if $\omega(\Gamma) = \Gamma$ then the unique element $\mathcal{D}\cap \Gamma$ is fixed by $\omega$, and thus $\tc{{W}_a} \cap \Gamma$ is nonempty.
For (b), we rely on the results of \cite{lus14:hecke}. Since ${W}_a$ is tame (see \cite[1.11, 1.15]{lus14:hecke}), it is bounded in the sense of \cite[13.2]{lus14:hecke} (also see \cite[13.4]{lus14:hecke} and \cite[Theorem 7.2]{lus85:cell} for its proof). Therefore, the argument in \cite[Chapter 16]{lus14:hecke} is applicable and (b) follows from \cite[Lemma 16.21]{lus14:hecke}.
For (c), the first part is exactly Lemma \ref{lem:2sstab}.
To show that ${\underline{c}}\cap \tc{{W}_a}$ is nonempty, we consider the canonical left cell $\Gamma \subset {\underline{c}}$ defined in \cite{luxi88}. Then clearly $\Gamma$ is $\omega$-stable, thus ${\underline{c}} \cap \tc{{W}_a} \supset \Gamma \cap \tc{{W}_a}$ is nonempty by (a). Finally, ${\underline{c}} \cap \tc{{W}_a}$ is a union of two-sided cells of $(\tc{{W}_a}, L)$ by \cite[Lemma 16.20(b)]{lus14:hecke} (and its right analogue).
\end{proof}
\begin{rmk} In general, ${\underline{c}} \cap \tc{{W}_a}$ is not a single two-sided cell. For example, if $n=4$ and $m=2$, there are 6 two-sided cells in $(\tc{{W}_a}, L)$, but $\widetilde{W}$ has only 5 two-sided cells. Indeed, the second highest two-sided cell of $\widetilde{W}$ (which contains $S$) splits into two two-sided cells $\{s_0\}$, $\{s_2\}$ of $\tc{{W}_a}$. (ref. \cite[p.40]{gui08})
\end{rmk}
Now the following corollary is an immediate consequence.
\begin{cor} \label{cor:equiv} Let ${\underline{c}}$ be a two-sided cell of $\widetilde{W}$. Then the number of left cells of $(\tc{{W}_a},L)$ in ${\underline{c}} \cap \tc{{W}_a}$ is equal to the number of left cells of $\widetilde{W}$ in ${\underline{c}}$ fixed by the involution $\omega$.
\end{cor}
\subsection{Relation to Domino tableaux and Springer theory}
Let $\tc{G}$ be $SO_{n}$ (resp. $Sp_{n}$) over $\mathbb{C}$ if $n$ is odd (resp. even) and $\tc{\mathfrak{g}}$ be its Lie algebra. Regard $\tc{W} \colonequals W^\omega$ as the Weyl group of $\tc{G}$ in a natural way. For a nilpotent element $\tc{N} \in \tc{\mathfrak{g}}$, let $A_{\tc{N}}$ be the component group of the stabilizer of $\tc{N}$ in $\tc{G}$. Let $\tc{\mathcal{B}}$ be the flag variety of $\tc{G}$ and $\tc{\mathcal{B}}_{\tc{N}}$ be the Springer fiber of $\tc{N}$.
There exists a canonical bijection between two-sided cells in $\tc{W}$ (with equal parameters) and special nilpotent orbits in $\tc{\mathfrak{g}}$.
Pick a two-sided cell $\tc{{\underline{c}}} \subset \tc{W}$ and let $\tc{N}\in \tc{\mathfrak{g}}$ be the nilpotent element in the corresponding special nilpotent orbit. Also let $\lambda$ be the Jordan type of $\tc{N}$. Then it follows from the results of Barbasch-Vogan \cite{bv82} and Garfinkle \cite{gar90, gar92, gar93} that the number of left cells in $\tc{{\underline{c}}}$ is equal to that of standard domino tableaux of shape $\lambda$. It is also the same as the number of $A_{\tc{N}}$-orbits in the set of irreducible components of $\tc{\mathcal{B}}_{\tc{N}}$, see \cite{mcg99, mcg00}.
This statement has an ``unequal'' analogue as follows. If we restrict $L: \tc{{W}_a} \rightarrow \mathbb{N}$ to $\tc{W}$, then again $(\tc{W}, L|_{\tc{W}})$ is in the quasisplit case in the sense of \cite{lus14:hecke}. Thus similarly to the affine case above, the left cells of $(\tc{W}, L|_{\tc{W}})$ are precisely an intersection of $\tc{W}$ and some left cell of $W$ fixed by $\omega$ (see \cite{lus83:left}). For a partition $\lambda \vdash n$, let ${\underline{c}}_\lambda \subset W$ be the two-sided cell of $W$ parametrized by $\lambda$. Then the number of left cells of $(\tc{W}, L|_{\tc{W}})$ contained in ${\underline{c}}_\lambda \cap \tc{W}$ is equal to the number of standard domino tableaux of shape $\lambda$. In particular, if $\lambda$ is the Jordan type of a nilpotent element $\tc{N} \in \tc{\mathfrak{g}}$ (not necessarily special), then it is again the same as the number of $A_{\tc{N}}$-orbits in the set of irreducible components of $\tc{\mathcal{B}}_{\tc{N}}$.
The statement above also has an ``affine'' analogue. For simplicity, let us assume that $n$ is odd, thus $\tc{G} = SO_{n}$ is of type $B$. Then $\tc{{W}_a}$ is naturally the affine Weyl group of the Langlands dual of $\tc{G}$. There exists a canonical bijection between nilpotent orbits in $\tc{\mathfrak{g}}$ and two-sided cells of $\tc{{W}_a}$ (with equal parameters) defined in \cite{lus89:cell}. Pick a two-sided cell $\tc{{\underline{c}}} \subset \tc{{W}_a}$ and let $\tc{N} \in \tc{\mathfrak{g}}$ be a nilpotent element in the corresponding nilpotent orbit. Then, a weaker version of \cite[Conjecture 10.5]{lus89:cell}, proved in \cite{bez04, beos04, bfo09}, implies that the number of left cells in $\tc{{\underline{c}}}$ is given by the dimension of $(H^*(\tc{\mathcal{B}}_{\tc{N}}))^{A_{\tc{N}}}$.
The main theorem in this paper should be considered as an ``affine unequal" analogue of the first statement.
Again let $G$ be $SO_n$ or $Sp_n$ depending on the parity of $n$, and let $\tc{N} \in \tc{\mathfrak{g}}$ be a nilpotent element of Jordan type $\lambda \vdash n$. Let ${\underline{c}}_\lambda \subset \widetilde{W}$ be the two-sided cell of $\widetilde{W}$ parametrized by $\lambda$. Then the result of \cite{kim:total} together with Theorem \ref{thm:main} implies that the number of left cells of $(\tc{{W}_a}, L)$ contained in ${\underline{c}}_\lambda\cap \tc{{W}_a}$ is equal to the Euler characteristic of $\tc{\mathcal{B}}_{\tc{N}}$, i.e. the dimension of $H^*(\tc{\mathcal{B}}_{\tc{N}})$.
\section{Proof of the main theorem} \label{section:proof}
\subsection{Reduction to strict partitions}
First, we claim that in order to prove the main theorem it suffices only to consider the case when a two-sided cell ${\underline{c}} \subset \widetilde{W}$ corresponds to a strict partition. This follows from two propositions below.
\begin{prop} \label{prop:str} Suppose that $\lambda=(\lambda_1, \ldots, \lambda_r)\vdash n-2k$ is a partition for some $k\geq 1$. Let $\varphi(\lambda\cup(k,k))$ be the number of left cells in ${\underline{c}}_{\lambda\cup (k,k)}$ fixed by $\omega$. We define $\varphi(\lambda)$ similarly (by replacing $n$ with $n-2k$, etc.) Then, for $m = \floor{n/2}$ we have
$$\varphi(\lambda\cup(k,k))=\binom{m}{k} 2^k\varphi(\lambda).$$
\end{prop}
Its proof relies on combinatorics, which we postpone until Section \ref{sec:comb}. We refer readers to \cite{cfkly} for detailed combinatorial descriptions of the affine Sch{\"u}tzenberger involution.
\begin{prop} \label{prop:grind}Suppose that $\lambda=(\lambda_1, \ldots, \lambda_r) \vdash n-2k$ is a partition for some $k\geq 1$. Then,
$$\fgr^{\lambda\cup(k,k)}_{\rho_2(n)}(-1)=\binom{m}{k} 2^k\fgr^{\lambda}_{\rho_2(n-2k)}(-1)$$
where $m = \floor{n/2}$ and $\fgr^{\lambda\cup(k,k)}_{\rho_2(n)}(t)$, $\fgr^{\lambda}_{\rho_2(n-2k)}(t)$ are the corresponding Green polynomials.
\end{prop}
\begin{proof} Let $Q'_\lambda(t)$ be the modified Hall-Littlewood $Q'$-function. (See \cite{kir98} for its definition and properties.) Note that $Q'_\lambda(t) = t^{b(\lambda)} \sum_{\rho\vdash |\lambda|} z_\rho^{-1}\fgr^\lambda_\rho(t^{-1})p_\rho$ where $b(\lambda) = \sum_{i\geq 1} (i-1)\lambda_i$, $z_\rho = \prod_{i\geq 1} i^{m_i} m_i!$ for $\rho = (1^{m_1}2^{m_2}\cdots)$, and $p_\rho$ is a power symmetric function. Thus we have
$$\br{Q'_{\lambda \cup (k,k)}(-1), p_{\rho_2(n)}} = (-1)^{b(\lambda \cup (k,k))}\fgr^{\lambda \cup (k,k)}_{\rho_2(n)}(-1) =(-1)^{b(\lambda)+k}\fgr^{\lambda \cup (k,k)}_{\rho_2(n)}(-1)$$
where $\br{\ , \ }$ is the usual scalar product on the ring of symmetric functions.
On the other hand, by \cite[Theorem 2.1 and 2.2]{llt94} we have $Q'_{\lambda \cup (k,k)}(-1) = (-1)^kQ'_{\lambda}(-1)s_k[p_2]$, where $s_k$ is the Schur function corresponding to the partition $(k)$, $p_2$ is the power symmetric function corresponding to the partition $(2)$, and $s_k[p_2]$ is their plethysm. Therefore, $\br{Q'_{\lambda \cup (k,k)}(-1), p_{\rho_2(n)}}$ is also equal to (here we use orthogonality of power symmetric functions with respect to $\br{\ , \ }$)
\begin{align*}
(-1)^k\br{Q'_{\lambda}(-1)s_k[p_2],p_{\rho_2(n)}}&=(-1)^{b(\lambda)+k}\frac{\fgr^\lambda_{\rho_2(n-2k)}(-1)}{z_{\rho_2(n-2k)}}\br{p_{\rho_2(n-2k)}s_k[p_2],p_{\rho_2(n)}}
\\&=(-1)^{b(\lambda)+k}\frac{\fgr^\lambda_{\rho_2(n-2k)}(-1)}{z_{\rho_2(n-2k)}k!}\br{p_{\rho_2(n)},p_{\rho_2(n)}}
\\&=(-1)^{b(\lambda)+k}\frac{\fgr^\lambda_{\rho_2(n-2k)}(-1)}{z_{\rho_2(n-2k)}k!}z_{\rho_2(n)}
\\&=(-1)^{b(\lambda)+k}\frac{m!2^k}{(m-k)!k!}\fgr^\lambda_{\rho_2(n-2k)}(-1).
\end{align*}
Hence the result follows.
\end{proof}
Combining two propositions above, we see that if the main theorem is true for ${\underline{c}}_\lambda$, then it is also true for ${\underline{c}}_{\lambda \cup (k,k)}$. Thus by inductive argument, the main theorem is valid if and only if it is valid for strict partitions.
\begin{rmk} We believe that this part is not essential to the proof of the main theorem; it is likely that argument in Section \ref{sec:morph} can be applied to the general cases without assuming that the corresponding partition is strict. However, this assumption is still useful as it simplifies our proof.
\end{rmk}
\subsection{Asymptotic Hecke algebra and the canonical basis of $K(\mathcal{B}_N)$} \label{sec:morph}
From now on, we fix a two-sided cell ${\underline{c}}={\underline{c}}_\lambda \subset \widetilde{W}$ where $\lambda$ is strict. Let $N \in \mathfrak{g}$ be a nilpotent element of Jordan type $\lambda$. By \cite[p.398]{car93}, the reductive part of $Z_G(N)$ (the stabilizer of $N$ in $G$) is a torus isomorphic to $(\mathbb{C}^\times)^{l(\lambda)}$, which we denote by $F_{\underline{c}}$. The idea we pursue here is motivated by the conjecture of Lusztig relating the asymptotic Hecke algebra ${\mathcal{J}}_{\underline{c}}$ attached to ${\underline{c}}$ and the $F_{\underline{c}}$-equivariant $K$-theory of a certain finite set, see \cite[Conjecture 10.5]{lus89:cell}.
We recall some results of \cite{bez16}. Let $D_{I^0I^0}$ be the category defined in \cite[p.4]{bez16} and $\mathcal{P}_{I^0I^0}$ be its subcategory of perverse sheaves. Also, define $\tilde{\mathfrak{g}} \colonequals \{ (X, gB) \in \mathfrak{g} \times \mathcal{B} \mid \Ad(g)^{-1}X \in \mathfrak{b}\}$ equipped with the obvious projection $\tilde{\mathfrak{g}} \rightarrow \mathfrak{g}$. Then we have a natural equivalence of categories \cite[Theorem 1]{bez16}
$$D_{I^0I^0} \simeq D^b(\mathcal{P}_{I^0I^0}) \simeq D^b(Coh^G_{\mathcal{N}}(\tilde{\mathfrak{g}}\times_\mathfrak{g} \tilde{\mathfrak{g}})),$$
where $Coh^G_{\mathcal{N}}(\tilde{\mathfrak{g}}\times_\mathfrak{g} \tilde{\mathfrak{g}})$ is the category of $G$-equivariant coherent sheaves which are set-theoretically supported on the nilpotent cone $\mathcal{N}\subset \mathfrak{g}$. (Here we identify $G$ with its Langlands dual.) This isomorphism respects the convolution structure on both sides.
Following \cite[11.2]{bez16}, it induces a canonical isomorphism
$$D_{I^0I^0} \simeq D^b(\mathcal{A} \otimes_{\mathcal{O}(\mathfrak{g})}\mathcal{A}-\textup{mod}^G_\mathcal{N}),$$
where $\mathcal{A} \otimes_{\mathcal{O}(\mathfrak{g})}\mathcal{A}-\textup{mod}^G_\mathcal{N}$ is the category of finitely generated $G$-equivariant $(\mathcal{A} \otimes_{\mathcal{O}(\mathfrak{g})}\mathcal{A})$-modules that are set-theoretically supported on $\mathcal{N} \subset \mathfrak{g}$. Here, $\mathcal{O}(\mathfrak{g})$ is the coordinate ring of $\mathfrak{g}$ and $\mathcal{A}$ is a noncommutative Grothendieck resolution of $\mathfrak{g} \times_{\mathfrak{h}/W}\mathfrak{h}$ defined in \cite[1.5]{bemi13}. (See also \cite{bez06}.)
Recall the bijection between two-sided cells and nilpotent orbits in $\mathfrak{g}$ in \cite{lus89:cell}. This bijection is order-preserving \cite{bez09}, and each order induces a filtration on each of two categories above. More precisely, let $D_{I^0I^0,\leq {\underline{c}}}$ (resp. $D_{I^0I^0,<{\underline{c}}}$) be the thick subcategory of $D_{I^0I^0}$ generated by irreducible objects $IC_w \in \mathcal{P}_{I^0I^0}$ for $w \in {\underline{c}}' \leq {\underline{c}}$ (resp. $w \in {\underline{c}}' <{\underline{c}}$), where $IC_w$ is defined as in \cite[Theorem 55]{bez16}. Then the quotient $D_{I^0I^0,\leq {\underline{c}}}/D_{I^0I^0,<{\underline{c}}}$, denoted $D_{I^0I^0,{\underline{c}}}$, is well-defined. Likewise, for a nilpotent orbit $O \subset \mathfrak{g}$ let $D^b(\mathcal{A} \otimes_{\mathcal{O}(\mathfrak{g})}\mathcal{A}-\textup{mod}^G_{\leq O})$ (resp. $D^b(\mathcal{A} \otimes_{\mathcal{O}(\mathfrak{g})}\mathcal{A}-\textup{mod}^G_{< O})$) be the full subcategory of $D^b(\mathcal{A} \otimes_{\mathcal{O}(\mathfrak{g})}\mathcal{A}-\textup{mod}^G_{\mathcal{N}})$ consisting of complexes whose cohomology is set-theoretically supported on $\overline{O}$ (resp. $\overline{O} - O$). Then the quotient $D^b(\mathcal{A} \otimes_{\mathcal{O}(\mathfrak{g})}\mathcal{A}-\textup{mod}^G_{\leq O})/D^b(\mathcal{A} \otimes_{\mathcal{O}(\mathfrak{g})}\mathcal{A}-\textup{mod}^G_{< O})$, denoted $D^b(\mathcal{A} \otimes_{\mathcal{O}(\mathfrak{g})}\mathcal{A}-\textup{mod}^G_{O})$, is well-defined.
\cite[Theorem 55]{bez16} states that the isomorphism above respects the filtrations on both sides. In particular, if we set $O$ to be the orbit of $N\in\mathfrak{g}$, then we have a canonical isomorphism
$$D_{I^0I^0,{\underline{c}}} \simeq D^b(\mathcal{A} \otimes_{\mathcal{O}(\mathfrak{g})}\mathcal{A}-\textup{mod}^G_O).$$
Also it sends the perverse $t$-structure on the left to the usual $t$-structure on the right shifted by $a({\underline{c}})=\dim \mathcal{B}_N$. (See \cite{lus85:cell} for the definition of Lusztig's $a$-function.) Now consider a full subcategory $\mathbb{I}_{\underline{c}}$ of $D_{I^0I^0,{\underline{c}}}$ whose objects are (the images of) direct sums of irreducible perverse sheaves and their shifts in $D_{I^0I^0,\leq {\underline{c}}}$ (under the quotient map). Under the isomorphism above, it is transferred to the full subcategory of $D^b(\mathcal{A} \otimes_{\mathcal{O}(\mathfrak{g})}\mathcal{A}-\textup{mod}^G_O)$ whose objects are (the images of) direct sums of irreducible $(\mathcal{A} \otimes_{\mathcal{O}(\mathfrak{g})}\mathcal{A})$-modules set-theoretically supported on $\overline{O}$ and their shifts (under the quotient map), which we denote by $\mathbb{I}_O$. Since all the irreducible $(\mathcal{A} \otimes_{\mathcal{O}(\mathfrak{g})}\mathcal{A})$-modules set-theoretically supported on $\overline{O}$ are also scheme-theoretically supported on $\overline{O}$, we may also identify $\mathbb{I}_O$ with the full subcategory of $D^b(\mathcal{A}_N \otimes \mathcal{A}_N-\textup{mod}^{Z_G(N)})$ (the bounded derived category of finitely generated $Z_G(N)$-equivalent $(\mathcal{A}_N \otimes \mathcal{A}_N)$-modules) whose objects are direct sums of irreducible objects and their shifts. Here, $\mathcal{A}_N$ is the fiber of $\mathcal{A}$ at $N \in \mathfrak{g}$.
From this description above, we have canonical isomorphisms
$$K(\mathbb{I}_{\underline{c}}) \simeq K(D_{I^0I^0,{\underline{c}}})\simeq K(D^b(\mathcal{A} \otimes_{\mathcal{O}(\mathfrak{g})}\mathcal{A}-\textup{mod}^G_O)) \simeq K(\mathbb{I}_O) \simeq K(D^b(\mathcal{A}_N \otimes \mathcal{A}_N-\textup{mod}^{Z_G(N)})).$$
We impose a $\mathbb{C}$-algebra structure on each term so that they are canonically isomorphic as $\mathbb{C}$-algebras. First, Lusztig \cite{lus97:cell} defined the truncated convolution on $K(\mathbb{I}_{\underline{c}}) \simeq K(D_{I^0I^0,{\underline{c}}})$, which is the usual convolution followed by applying ${}^p\mathcal{H}^{a({\underline{c}})}$, i.e. taking $a({\underline{c}})$-th perverse cohomology sheaf. Then $K(\mathbb{I}_{\underline{c}})$ equipped with this algebra structure is canonically isomorphic to the asymptotic Hecke algebra ${\mathcal{J}}_{\underline{c}}$ attached to ${\underline{c}}$ (defined over $\mathbb{C}$). On the other hand, this also induces a truncated convolution on $K(D^b(\mathcal{A} \otimes_{\mathcal{O}(\mathfrak{g})}\mathcal{A}-\textup{mod}^G_O)) \simeq K(\mathbb{I}_O)$, which is defined by the usual convolution followed by applying $\mathcal{H}^0$, i.e. taking $0$-th cohomology sheaf. (This is clear from the comparison of $t$-structures on $D_{I^0I^0,{\underline{c}}}$ and $D^b(\mathcal{A} \otimes_{\mathcal{O}(\mathfrak{g})}\mathcal{A}-\textup{mod}^G_O).$) Therefore, we have a canonical isomorphism of $\mathbb{C}$-algebras
$${\mathcal{J}}_{\underline{c}} \simeq K(D^b(\mathcal{A}_N \otimes \mathcal{A}_N-\textup{mod}^{Z_G(N)})).$$
It is also isomorphic to $K^{F_{\underline{c}}}(\mathcal{A}_N \otimes \mathcal{A}_N-\textup{mod})$ since $F_{\underline{c}}$ is the reductive part of $Z_G(N)$.
There exists a natural morphism (of $\mathbb{C}$-vector spaces)
$$K^{F_{\underline{c}}}(\mathcal{A}_N \otimes \mathcal{A}_N-\textup{mod}) \rightarrow K(\mathcal{A}_N \otimes \mathcal{A}_N-\textup{mod})$$
which is induced from the forgetful functor. We claim that this morphism is surjective and its kernel is a two-sided ideal of $K^{F_{\underline{c}}}(\mathcal{A}_N \otimes \mathcal{A}_N-\textup{mod})$ (with respect to the truncated convolution), thus it induces a $\mathbb{C}$-algebra structure on $K(\mathcal{A}_N \otimes \mathcal{A}_N-\textup{mod})$. Indeed, according to \cite[5.2.3]{bemi13}, every irreducible $(\mathcal{A}_N \otimes \mathcal{A}_N)$-module can be lifted to an $F_{\underline{c}}$-equivariant one and such two lifts are isomorphic up to characters of $F_{\underline{c}}$. (Here we use the assumption that $\lambda$ is strict and thus $F_{\underline{c}}$ is a torus.) Also, every irreducible $F_{\underline{c}}$-equivariant $\mathcal{A}_N \otimes \mathcal{A}_N$-module arises in this way. From this, the claim easily follows.
On the other hand, inspired by the conjecture of Lusztig \cite{lus89:cell}, Xi \cite{xi02} proved that ${\mathcal{J}}_{\underline{c}}$ is (non-canonically) isomorphic to $Mat_{\mathcal{X}\times \mathcal{X}}({\mathcal{J}}_{\Gamma \cap \Gamma^{-1}})$, where $\Gamma$ is some fixed left cell in ${\underline{c}}$, ${\mathcal{J}}_{\Gamma \cap \Gamma^{-1}}$ is the asymptotic Hecke algebra attached to $\Gamma \cap \Gamma^{-1}$, and $\mathcal{X}=\frac{n!}{\lambda_1!\cdots\lambda_r!}$ is the number of left cells in ${\underline{c}}$ which is also equal to the Euler characteristic of $\mathcal{B}_N$. Furthermore, ${\mathcal{J}}_{\Gamma \cap \Gamma^{-1}}$ is isomorphic to $Rep(F_{\underline{c}})$, which in our case is the $\mathbb{C}$-algebra of Laurent polynomials in $l(\lambda)$ variables, say $\mathbb{C}[x_1^{\pm1},x_2^{\pm1},\ldots,x_{l(\lambda)}^{\pm1}]$.
Let us fix the labeling of the left cells in ${\underline{c}}$ by $\Gamma_1, \Gamma_2, \ldots, \Gamma_\mathcal{X}$ once and for all. According to \cite{xi02}, we may choose an isomorphism ${\mathcal{J}}_{\underline{c}}\simeq Mat_{\mathcal{X}\times \mathcal{X}}({\mathcal{J}}_{\Gamma \cap \Gamma^{-1}})$ such that $(i,j)$-entries in $Mat_{\mathcal{X}\times \mathcal{X}}({\mathcal{J}}_{\Gamma \cap \Gamma^{-1}})$ corresponds to $\Gamma^{-1}_i \cap \Gamma_j$. Now consider the surjection $Mat_{\mathcal{X}\times \mathcal{X}}({\mathcal{J}}_{\Gamma \cap \Gamma^{-1}}) \twoheadrightarrow Mat_{\mathcal{X}\times \mathcal{X}}(\mathbb{C})$ induced from the evaluation morphism
$${\mathcal{J}}_{\Gamma \cap \Gamma^{-1}}\simeq \mathbb{C}[x_1^{\pm1},x_2^{\pm1},\ldots,x_{l(\lambda)}^{\pm1}] \rightarrow \mathbb{C}: f(x_1, x_2, \ldots, x_{l(\lambda)}) \mapsto f(1,1, \ldots, 1).$$ Then it is not hard to show that the composition ${\mathcal{J}}_{\underline{c}} \twoheadrightarrow Mat_{\mathcal{X}\times \mathcal{X}}(\mathbb{C})$ does not depend on the choice of the isomorphism ${\mathcal{J}}_{\underline{c}}\simeq Mat_{\mathcal{X}\times \mathcal{X}}({\mathcal{J}}_{\Gamma \cap \Gamma^{-1}})$ whenever it respects the fixed labeling of left cells in ${\underline{c}}$. In other words, there exists a canonical isomorphism ${\mathcal{J}}_{\underline{c}} \twoheadrightarrow Mat_{\mathcal{X}\times \mathcal{X}}(\mathbb{C})$ once the order of left cells in ${\underline{c}}$ is fixed.
So far, we have canonical morphisms
$$
\begin{tikzcd}
{\mathcal{J}}_{\underline{c}} \ar[r, "\simeq"] \ar[d,twoheadrightarrow]&K^{F_{\underline{c}}}(\mathcal{A}_N \otimes \mathcal{A}_N-\textup{mod}) \ar[d,twoheadrightarrow]
\\Mat_{\mathcal{X} \times \mathcal{X}}(\mathbb{C})&K(\mathcal{A}_N \otimes \mathcal{A}_N-\textup{mod})
\end{tikzcd}
$$
On the other hand, there exists a section map $Mat_{\mathcal{X} \times \mathcal{X}}(\mathbb{C})\rightarrow {\mathcal{J}}_{\underline{c}}$ which comes from the $\mathbb{C}$-algebra structure $\mathbb{C} \rightarrow {\mathcal{J}}_{\Gamma\cap\Gamma^{-1}}$. Composed with ${\mathcal{J}}_{\underline{c}}\simeq K^{F_{\underline{c}}}(\mathcal{A}_N \otimes \mathcal{A}_N-\textup{mod}) \twoheadrightarrow K(\mathcal{A}_N \otimes \mathcal{A}_N-\textup{mod})$, it induces a morphism $Mat_{\mathcal{X} \times \mathcal{X}}(\mathbb{C}) \rightarrow K(\mathcal{A}_N \otimes \mathcal{A}_N-\textup{mod})$ which makes the above diagram commute. In particular, this morphism is canonical (even though the section map $Mat_{\mathcal{X} \times \mathcal{X}}(\mathbb{C})\rightarrow {\mathcal{J}}_{\underline{c}}$ needs not be canonical).
Also, we claim that this morphism is an isomorphism. Indeed, since it is a $\mathbb{C}$-algebra morphism and $Mat_{\mathcal{X} \times \mathcal{X}}(\mathbb{C})$ is simple, $Mat_{\mathcal{X} \times \mathcal{X}}(\mathbb{C}) \rightarrow K(\mathcal{A}_N \otimes \mathcal{A}_N-\textup{mod})$ is injective. (This map is not zero as it preserves the multiplicative unit.) Now we recall one of the main results in \cite{bemi13}.
\begin{lem} \label{lem:canbasis} There exists a canonical isomorphism $K(\mathcal{A}_N-\textup{mod}) \simeq K(\mathcal{B}_N)$. Under this isomorphism, the basis $\Irr(\mathcal{A}_N)$ of $K(\mathcal{A}_N-\textup{mod})$ corresponds to the canonical basis of $K(\mathcal{B}_N)$ defined in \textup{\cite{lus99:kthy2}}.
\end{lem}
In particular, we have $\dim_\mathbb{C} K(\mathcal{A}_N \otimes \mathcal{A}_N-\textup{mod}) = \mathcal{X}^2 = \dim_\mathbb{C} Mat_{\mathcal{X} \times \mathcal{X}}(\mathbb{C})$, from which the claim follows. Thus we have a canonical commutative diagram
$$
\begin{tikzcd}
{\mathcal{J}}_{\underline{c}} \ar[r, "\simeq"] \ar[d,twoheadrightarrow]&K^{F_{\underline{c}}}(\mathcal{A}_N \otimes \mathcal{A}_N-\textup{mod}) \ar[d,twoheadrightarrow]
\\Mat_{\mathcal{X} \times \mathcal{X}}(\mathbb{C}) \ar[r,"\simeq"]&K(\mathcal{A}_N \otimes \mathcal{A}_N-\textup{mod})
\end{tikzcd}
$$
Now using the above lemma again, we obtain a canonical isomorphism (of vector spaces)
$$Mat_{\mathcal{X} \times \mathcal{X}}(\mathbb{C}) \simeq K(\mathcal{B}_N) \otimes K(\mathcal{B}_N).$$
\subsection{Involution $\omega$}
We recall the involution $\omega$ on $G$. Clearly, it also induces an involution on ${\mathcal{J}}_{\underline{c}}$ and $Mat_{\mathcal{X} \times \mathcal{X}}(\mathbb{C})$. It is clear that there exists a basis $\{v_{(\Gamma'^{-1}, \Gamma'')} \mid \Gamma', \Gamma'' \textup{ are left cells in } {\underline{c}}\}$ of $Mat_{\mathcal{X} \times \mathcal{X}} (\mathbb{C})$ such that $\omega(v_{(\Gamma'^{-1}, \Gamma'')}) = v_{(\omega(\Gamma'^{-1}), \omega(\Gamma''))}$. Therefore, we have
$$\tr(\omega, Mat_{\mathcal{X} \times \mathcal{X}}(\mathbb{C})) = |\{\textup{left cells in } {\underline{c}}\}^\omega|^2.$$
On the other hand, if $N\in \mathfrak{g}$ is $\omega$-stable, then $\omega$ also induces an action on $K(\mathcal{B}_N)$, and by \cite[Lemma 3.2]{hosp77} we have (note that $\rho_2$ is the cycle type of the longest element in $W$)
$$\tr(\omega, K(\mathcal{B}_N)\otimes K(\mathcal{B}_N))=\tr(\omega, H^*(\mathcal{B}_N)\otimes H^*(\mathcal{B}_N)) = (\fgr^\lambda_{\rho_2(n)}(-1))^2.$$
However, this still makes sense even when $N$ is not $\omega$-stable. Indeed, for any $g \in G$ such that $\Ad(g)(N) = \omega(N)$, we have an isomorphism $\Ad(g)^*: H^*(\mathcal{B}_{\omega(N)}) \rightarrow H^*(\mathcal{B}_N)$ which does not depend on the choice of $g$ since $Z_G(N)$ is connected. Also we have a commutative diagram
$$
\begin{tikzcd}[column sep=5em]
H^*(\mathcal{B}) \ar[r,"\omega"] \ar[d,twoheadrightarrow]& H^*(\mathcal{B}) \ar[d,twoheadrightarrow] \ar[r,"="]& H^*(\mathcal{B}) \ar[d,twoheadrightarrow]
\\H^*(\mathcal{B}_N) \ar[r,"\omega"]& H^*(\mathcal{B}_{\omega(N)}) \ar[r,"\Ad(g)^*"]& H^*(\mathcal{B}_N)
\end{tikzcd}
$$
Thus by identifying $H^*(\mathcal{B}_N)$ with the quotient of $H^*(\mathcal{B})$, the result above is still valid.
Recall that the $\mathbb{C}$-vector space isomorphism $Mat_{\mathcal{X} \times \mathcal{X}}(\mathbb{C}) \simeq K(\mathcal{B}_N) \otimes K(\mathcal{B}_N)$
is canonical (once the order of left cells in ${\underline{c}}$ is fixed). As $\omega$ on $Mat_{\mathcal{X} \times \mathcal{X}}(\mathbb{C})$ and $K(\mathcal{B}_N) \otimes K(\mathcal{B}_N)$ are both induced from the same automorphism $\omega$ on $G$, it follows that this isomorphism is $\omega$-equivariant. \footnote{To be precise, we should check that $\omega$ acts the same way on the Langlands dual of $G$ as on $G$, but it is also true since $\omega$ is self-dual on the root datum of $G$.} In particular, we have
$$|\{\textup{left cells in } {\underline{c}}\}^\omega|^2=\tr(\omega, Mat_{\mathcal{X} \times \mathcal{X}}(\mathbb{C}))=\tr(\omega, K(\mathcal{B}_N)\otimes K(\mathcal{B}_N)) = ( \fgr^\lambda_{\rho_2(n)}(-1))^2.$$
But since
$$\fgr^\lambda_{\rho_2(n)}(-1)= \tr(\omega, H^*(\mathcal{B}_N)) = \tr(\omega, K(\mathcal{A}_N-\textup{mod})) = |\Irr(\mathcal{A}_N)^\omega|\geq 0,$$
we have $|\{\textup{left cells in } {\underline{c}}\}^\omega| = \fgr^{\lambda}_{\rho_2(n)}(-1)$. Thus the main theorem is proved.
\begin{rmk} The canonical basis of $K(\mathcal{B}_N)$ in \cite{lus99:kthy2} is a signed basis, i.e. there is ambiguity on the choice of signs. On the other hand, $\Irr(\mathcal{A}_N) \subset K(\mathcal{A}_N-\textup{mod})$ is an actual basis, and $\pm\Irr(\mathcal{A}_N)$ is mapped to Lusztig's canonical basis under the isomorphism $K(\mathcal{A}_N-\textup{mod}) \simeq K(\mathcal{B}_N)$. In our proof, it is crucial that $\omega$ stabilizes not only the signed basis but also $\Irr(\mathcal{A}_N)$ itself.
\end{rmk}
\section{Proof of Proposition \ref{prop:str}: some combinatorics} \label{sec:comb}
This section is devoted to the proof of Proposition \ref{prop:str}. The argument in this section is explained in \cite{cfkly} in more detail and the proposition also follows from the results therein. However, we still provided its proof here for the sake of completeness.
\subsection{The generalized Robinson-Schensted algorithm}
First, we investigate the connection between left cells in $\widetilde{W}$ and row-standard Young tableaux under the generalized Robinson-Schensted algorithm originally defined by Shi \cite{shi86, shi91}. Following \cite{lus83:padic}, we identify $\widetilde{W}$ with the subgroup of $\Aut(\mathbb{Z})$ defined by
$$\{w \in \Aut(\mathbb{Z}) \mid \forall i \in \mathbb{Z}, w(n+i) =w(i)+n\}.$$
We express each $w\in \Aut(\mathbb{Z})$ in terms of the sequence $[w(1), w(2), \ldots, w(n)]$, called the window notation. Then we have
\begin{gather*}
s_i = [1, 2, \ldots, i-1, i+1, i, i+2, \ldots, n],
\\s_0 = [0, 2, \ldots, n-1, n+1], \qquad \tau = [2, 3, \ldots, n, n+1].
\end{gather*}
It is easy to check that they satisfy the defining relations of $\widetilde{W}$.
For the description of the generalized Robinson-Schensted algorithm, it is natural to consider an infinite version of (standard) Young tableaux. To that end, we define the notion of an infinite periodic sequence as follows.
\begin{defn} \label{def:piseq}
For each $i \in \mathbb{Z}_{>0}$, let $\tilde{r}_i$ be a finite sequence of integers. Let $\tilde{r}_i(1)$ (resp. $\tilde{r}_i(0)$) be the smallest positive (resp. largest nonpositive) integer in $\tilde{r}_i$ (if exists) and label each element in $\tilde{r}_i$ by its position relative to $\tilde{r}_i(1)$ or $\tilde{r}_i(0)$. Then we call $\tilde{r}=(\tilde{r}_1, \tilde{r}_2, \ldots)$ \emph{an infinite periodic sequence modulo $n$} (\emph{IP sequence mod $n$} for short) if it satisfies the following properties;
\begin{enumerate}
\item each $\tilde{r}_i$ is strictly increasing, thus in particular $\tilde{r}_i(k)>0$ if and only if $k>0$,
\item $\lim_{i \rightarrow \infty} |\tilde{r}_i| = \infty$,
\item for any $k \in \mathbb{Z}$, the limit $\lim_{i \rightarrow \infty}(\tilde{r}_i(k))$ exists,
\item there exists $M \in \mathbb{N}$ such that for any $i>0$ and $\tilde{r}_i=(\tilde{r}_i(s), \tilde{r}_i(s+1), \ldots, \tilde{r}_i(t-1), \tilde{r}_i(t))$,
$$\tilde{r}_i(k) = \lim_{j\rightarrow \infty} \tilde{r}_j(k) \quad \textup{ for any } s+M\leq k\leq t-M, \quad \textup{ and}$$
\item there exists $1 \leq l \leq n$ and $1\leq a_1<a_2<\cdots<a_l \leq n$ such that
\begin{gather*}
\lim_{i \rightarrow \infty}\tilde{r}_{i}(kl+r) =kn+a_r \quad \textup{ for any } k \in \mathbb{Z} \textup{ and } 1\leq r \leq l.
\end{gather*}
\end{enumerate}
For any such sequence $\tilde{r}$, it is clear that $l, a_1, \ldots, a_l$ in (5) are uniquely determined if they exist. We define $\Psi_n$ to be the function which sends $\tilde{r}$ to the finite sequence $(a_1, \ldots, a_l)$.
\end{defn}
Likewise, we define an infinite periodic tabloid as follows.
\begin{defn} \label{def:pitab}Let $\tilde{T} = (\tilde{T}_1, \tilde{T}_2, \ldots)$ be an infinite series of Young tabloids such that the following properties hold.
\begin{enumerate}
\item There exists $M' \in \mathbb{N}$ such that $\tilde{T}_i$ has $\leq M'$ rows. We define $l(\tilde{T})$ to be the smallest $M'$ which satisfies this property, called the length of $\tilde{T}$.
\item For each $1\leq j \leq l(\tilde{T})$, $\tilde{T}^{(j)} \colonequals \{\tilde{T}_{i}^{(j)}\}_{i\geq 1}$ is an IP sequence mod $n$ where $\tilde{T}_{i}^{(j)}$ is the $j$-th row of $\tilde{T}_i$.
\end{enumerate}
Then we call $\tilde{T}$ \emph{an infinite periodic tabloid modulo $n$} (\emph{IP tabloid mod $n$} for short). For such $\tilde{T}$, we similarly define $\Psi_n(\tilde{T})$ to be the Young tabloid whose rows are $\Psi_n(\tilde{T}^{(1)}),\ldots, \Psi_n(\tilde{T}^{(l(\tilde{T}))})$. Note that an IP sequence mod $n$ is an IP tabloid mod $n$ of length 1. We define $IP_n$ to be the set of infinite periodic tabloids modulo $n$.
\end{defn}
For any element $w \in \widetilde{W}$, consider the following sequence that is infinite in both ways:
$$\ldots, w(-3), w(-2), w(-1), w(0), w(1), w(2), w(3), \ldots$$
We consider a sequence $\{(a_i, b_i)\}_{i\geq 1}$ such that $a_i \leq b_i$, $a_i$ is decreasing, $b_i$ is increasing, $\lim_{i\rightarrow \infty}a_i = -\infty$, and $\lim_{i\rightarrow \infty} b_i = \infty$. For each $i$, we consider the standard Young tableaux $\tilde{T}_i$ which is the result of the usual Robinson-Schensted algorithm with input $(w(a_i), w(a_i+1), \ldots, w(b_i-1), w(b_i))$. Then we obtain a series of standard Young tableaux $\tilde{T}=\{\tilde{T}_i\}_{i\geq 1}$. Now we apply the argument in \cite[Section 7]{cpy15} to obtain the following.
\begin{prop} \label{prop:genrs} $\tilde{T} \in IP_n$ and $l(\tilde{T})\leq n$. Also, $\Psi_n(\tilde{T})$ is the same as the result of the generalized Robinson-Schensted algorithm defined in \textup{\cite{shi86, shi91}} applied to $w^{-1}$. (In particular, $\Psi_n(\tilde{T})$ does not depend on the choice of the sequence $\{(a_i, b_i)\}_{i\geq 1}$). Moreover, if $w$ is an element of $W \subset \widetilde{W}$, then $\Psi_n(\tilde{T})$ is the same as the output of the usual Robinson-Schensted algorithm applied to $w^{-1}$.
\end{prop}
Here $w^{-1}$ appears instead of $w$ since we consider the left action of $\widetilde{W}$ on $\mathbb{Z}$ instead of the right one. We define $Q(w)$ to be such $\Psi_n(\tilde{T})$ in the theorem and set $P(w) \colonequals Q(w^{-1})$. Then by \cite{shi86,shi91}, $P(w)$ and $Q(w)$ have the same shape. For $w, w' \in \widetilde{W}$, $Q(w)=Q(w')$ (resp. $P(w)=P(w')$) if and only if they are contained in the same left (resp. right) cell. Likewise, $Q(w)$ and $Q(w')$ have the same shape if and only if they lie in the same two-sided cell, which is parametrized by the shape of $Q(w)$.
\subsection{Affine Sch{\"u}tzenberger involution and combinatorial $R$-matrix}
Recall the involution $\omega$ acting on $\widetilde{W}$. Under the identification of $\widetilde{W}$ with the subset of $\Aut(\mathbb{Z})$, it corresponds to the conjugation by the element $k \mapsto 1-k$ in $\Aut(\mathbb{Z})$. It permutes left cells in $\widetilde{W}$, thus defines an involution on the set of row-standard Young tableaux under the generalized Robinson-Schensted correspondence. Also, since $\omega$ stabilizes each two-sided cell, it restricts to the involution on $RSYT(\lambda)$ (the set of row-standard Young tableaux of shape $\lambda$) for each $\lambda \vdash n$.
We claim that this action can be described in terms of combinatorial $R$-matrices. (This is originally proved by Chmutov-Lewis-Pylyavskyy.) First we recall the definition of combinatorial $R$-matrices on the tensor product of single-row Kirillov-Reshetikhin crystals (KR crystals for short). We refer readers to \cite{shi05} for a nice exposition on this subject. For $a, b \in \mathbb{Z}_{>0}$, we regard $RSYT((a,b))$ as a subset of the vertices of the crystal graph $\mathbb{B}^b \otimes \mathbb{B}^a$, where $\mathbb{B}^s$ is the KR crystal of shape $(s)$ (of $U_q(\widehat{\mathfrak{sl}_k})$ for a suitable choice of $k$). Then there exists a unique isomorphism $\mathbb{B}^b \otimes \mathbb{B}^a \rightarrow \mathbb{B}^a \otimes \mathbb{B}^b$, which we call the combinatorial $R$-matrix, and it restricts to a bijection
$$\mathcal{R}: RSYT((a,b)) \rightarrow RSYT((b,a)).$$
\cite[Example 4.10]{shi05} describes this operation using jeu-de-taquin and sliding process. Here we briefly explain his description with an example.
\begin{example} \label{ex:rmat}
\ytableausetup{smalltableaux}
Let $a=4, b=3$ and $T = \begin{ytableau}
3 &4 & 6 & 7 \\
1 & 2 & 5 \\
\end{ytableau} \in RSYT((a,b))$. To apply $\mathcal{R}$, we first draw the skew-shaped standard Young tableau $\begin{ytableau}
\none & \none& \none&3 &4 & 6 & 7 \\
1 & 2 & 5 \\
\end{ytableau}$
obtained from sliding the first row to the right, and apply jeu-de-taquin process until each row has the correct number of boxes.
$$\begin{ytableau}
\none & \none& \none&3 &4 & 6 & 7 \\
1 & 2 & 5 & \bullet \\
\end{ytableau} \rightarrow \begin{ytableau}
\none& \none&3 &4 & 6 & 7 \\
1 & 2 & 5 & \bullet \\
\end{ytableau}\rightarrow \begin{ytableau}
\none&\none&\none&4 & 6 & 7 \\
1 & 2 & 3 & 5 \\
\end{ytableau}$$
As a result, we have $\mathcal{R}(T) = \begin{ytableau}
4 & 6 & 7 \\
1 & 2 & 3 & 5 \\
\end{ytableau}$. Note that both $\begin{ytableau}
\none & \none& \none&3 &4 & 6 & 7 \\
1 & 2 & 5 \\
\end{ytableau}$ and $\begin{ytableau}
\none & \none& \none&\none &4 &6 & 7 \\
1 & 2 & 3&5 \\
\end{ytableau}$ are jeu-de-taquin equivalent to the standard Young tableau $\begin{ytableau}
1& 2& 3&4 &6 & 7 \\
5 \\
\end{ytableau}$. In general, the combinatorial $R$-matrix does not change the associated jeu-de-taquin equivalent standard Young tableau.
Or, first we again consider $\begin{ytableau}
\none & \none& \none&3 &4 & 6 & 7 \\
1 & 2 & 5 \\
\end{ytableau}$ and slide each box on the second row to the rightest with preserving semi-standard property to get $\begin{ytableau}
\none & \none&3 &4 & 6 & 7 \\
1 & 2 & \none &5 \\
\end{ytableau}$. Then, push down the correct number of leftmost boxes (in this case we push down $4-3=1$ box) from the first row to obtain $\begin{ytableau}
\none & \none&\none &4 & 6 & 7 \\
1 & 2 & 3 &5 \\
\end{ytableau}$. Thus we also see that $\mathcal{R}(T) = \begin{ytableau}
4 & 6 & 7 \\
1 & 2 & 3 & 5 \\
\end{ytableau}$.
If $a<b$, then we first slide each box in the first row to the leftest with preserving semi-standard property and push up the correct number of rightmost boxes from the second row.
\end{example}
This combinatorial $R$-matrix is generalized to any finite tensor product of single-row KR crystals. In particular, for any sequence of positive integers $\lambda=(\lambda_1, \ldots, \lambda_r)$ (not necessarily a partition) we similarly define
$$\mathcal{R}_i : RSYT(\lambda) \rightarrow RSYT((\lambda_1, \ldots, \lambda_{i-1}, \lambda_{i+1}, \lambda_i, \lambda_{i+2}, \ldots, \lambda_r))$$
to be the corresponding combinatorial $R$-matrix. From the theory of crystals, we easily deduce the following properties of $\mathcal{R}$.
\begin{enumerate}
\item $\mathcal{R}_i^2 = Id$.
\item $\mathcal{R}_i\mathcal{R}_{i+1}\mathcal{R}_i = \mathcal{R}_{i+1}\mathcal{R}_i\mathcal{R}_{i+1}$.
\item If $\lambda_i = \lambda_{i+1}$, then $\mathcal{R}_i = Id$.
\item In general, if $\lambda_1, \ldots, \lambda_r, \mu_1, \ldots, \mu_r\in \mathbb{N}$ such that $\{\lambda_1, \ldots, \lambda_r \} = \{\mu_1, \ldots, \mu_r \}, $ then all the compositions of combinatorial $R$-matrices from $RSYT((\lambda_1, \ldots, \lambda_r))$ to $RSYT((\mu_1, \ldots, \mu_r))$ give the same map.
\end{enumerate}
As promised, we illustrate $\omega$ in terms of combinatorial $R$-matrices as follows.
\begin{prop} \label{prop:rmat}Let $\lambda \vdash n$ be a partition. For a given $T \in RSYT(\lambda)$, $T \mapsto \omega(T)$ is equivalent to the following process:
\begin{enumerate}
\item rotate $T$ by $180^\circ$ and push each row to the left so that it becomes a Young tabloid,
\item substitute each entry $i$ by $n+1-i$ to make it row-standard, and
\item apply combinatorial $R$-matrices accordingly to retain the original shape $\lambda$.
\end{enumerate}
\end{prop}
First, we restrict our attention to $W \subset \widetilde{W}$ and $SYT(\lambda) \subset RSYT(\lambda)$ (the set of standard Young tableaux of shape $\lambda$). Then for any $w \in W$, $\omega(w) = w_0ww_0$ where $w_0 \in W$ is the longest element in $W$. Under the Robinson-Schensted algorithm, this corresponds to the usual Sch{\"u}tzenberger involution. Also, it follows from \cite[Appendix A]{sta86} that this involution is the same as the one described in Proposition \ref{prop:rmat}. Therefore, this proposition is true for elements in $SYT(\lambda)$.
In general, let $\tilde{T}$ be an IP tabloid mod $n$ such that $\Psi_n(\tilde{T})$ is a row-standard Young tabloid. We claim that combinatorial $R$-matrices and the function $\Psi_n$ behave well together as follows.
\begin{lem} \label{lem:comm}For $1 \leq i \leq l(\tilde{T})-1$, let $\mathcal{R}_i(\tilde{T})$ be the series $(\mathcal{R}_i(\tilde{T}_1), \mathcal{R}_i(\tilde{T}_2), \ldots)$. Then $\mathcal{R}_i(\tilde{T})$ is again an IP tabloid mod $n$ and we have $\Psi_n(\mathcal{R}_i(\tilde{T}))= \mathcal{R}_i(\Psi_n(\tilde{T})).$
\end{lem}
\begin{proof}It suffices to assume that $l(\tilde{T})=2$ and $i=1$. In this case, it is an easy combinatorial exercise using the description of $\mathcal{R}$ in terms of sliding process in \cite[Example 4.10]{shi05}.
\end{proof}
\begin{example} Suppose $\tilde{T}$ is an IP tabloid mod 7 such that $\Psi_7(\tilde{T}) = \begin{ytableau}
3 &4 & 6 & 7 \\
1 & 2 &5 \\
\end{ytableau}$. Then (the limit of) each row of $\tilde{T}$ looks like
\ytableausetup{nosmalltableaux}
\ytableausetup{boxsize=1.7em}
\begin{gather*}
\begin{ytableau}
\cdots &-11 &-10 & -8 & -7&-4 &-3 & -1 & 0& 3 &4 & 6 & 7 &10 &11 & 13 & 14&17 &18 & 20 & 21 & \cdots
\end{ytableau}, \quad \textup{ and}
\\\begin{ytableau}
\cdots&-13 &-12 & -9 &-6 &-5 & -2 & 1& 2 &5 & 8 & 9 &12 &15 & 16 &19 &\cdots
\end{ytableau}.
\end{gather*}
Now we put each box in the second row to the rightest with keeping semi-standard property. Then it looks like
\begin{gather*}
\begin{ytableau}
\none&\none&\cdots &-11 &-10 & -8 & -7&-4 &-3 & -1 & 0& 3 &4 & 6 & 7 &10 &11 & 13 & 14&17 &18 & 20 & 21 & \cdots
\\
\cdots&-13 &-12 &\none& -9 &-6 &-5 &\none& -2 & 1& 2 &\none& 5 & 8 & 9 &\none &12 &15 & 16 &\none &19 &\cdots
\end{ytableau}
\end{gather*}
\ytableausetup{smalltableaux}
But this is a concatenation of $\begin{ytableau}
\none&\none& 3 &4 & 6 & 7
\\1& 2 &\none& 5
\end{ytableau}$ and its shifts by a multiple of $7$. Note that $\begin{ytableau}
\none&\none& 3 &4 & 6 & 7
\\1& 2 &\none& 5
\end{ytableau}$ appears in the usual combinatorial $R$-matrix calculation on $\Psi_7(\tilde{T}) = \begin{ytableau}
3 &4 & 6 & 7 \\
1 & 2 &5 \\
\end{ytableau}$ (cf. Example \ref{ex:rmat}). Also, if one pushes down all the possible boxes from the first row, then it corresponds to pushing down the box of entry 3 in $\begin{ytableau}
\none&\none& 3 &4 & 6 & 7
\\1& 2 &\none& 5
\end{ytableau}$. Therefore, in this case we get $\Psi_7(\mathcal{R}(\tilde{T}))=\mathcal{R}(\Psi_7(\tilde{T}))$.
Indeed, it is not hard to show that the image of $\Psi_n\circ \mathcal{R}$ only depends on $\Psi_n(\tilde{T})$; if $\tilde{T}, \tilde{T}'$ are two IP tabloids mod $n$ such that $\Psi_n(\tilde{T}) = \Psi_n(\tilde{T}')$, then indeed we have $\Psi_n(\mathcal{R}(\tilde{T}))=\Psi_n(\mathcal{R}(\tilde{T}'))$. In other words, one may simply ignore ``finite error" part in both ends of IP sequences mod $n$ because of the condition (4) in Definition \ref{def:piseq}.
\end{example}
\begin{proof}[Proof of Proposition \ref{prop:rmat}] Suppose $w \in \widetilde{W}$ is given and $\tilde{T}=(\tilde{T}_1, \tilde{T}_2, \ldots)$ is an IP tabloid mod $n$ constructed in Proposition \ref{prop:genrs}. Here we choose $(a_i, b_i)_{i\geq 1}$ such that $a_i+b_i=1$. (This assumption is not necessary but it simplifies the proof.) In other words, each $\tilde{T}_i$ is the output of the usual Robinson-Schensted algorithm applied to the sequence $w(a_i), w(a_i+1), \ldots, w(b_i-1), w(b_i)$. If we apply $\omega$, then it corresponds to replacing the sequence $w(a_i), w(a_i+1), \ldots, w(b_i-1), w(b_i)$ with $\omega(w)(a_i), \omega(w)(a_i+1), \ldots, \omega(w)(b_i-1), \omega(w)(b_i), $ i.e.
$$1-w(1-a_i),1-w(-a_i), \ldots, 1-w(2-b_i), 1-w(1-b_i)$$
which is equal to
$$1-w(b_i),1-w(b_i-1), \ldots, 1-w(1+a_i), 1-w(a_i)$$
since $a_i+b_i=1$. According to \cite[Appendix A]{sta86}, this is similar to the usual Sch{\"u}tzenberger involution. Indeed, the output of usual Robinson-Schensted algorithm applied to $1-w(b_i),1-w(b_i-1), \ldots, 1-w(1+a_i), 1-w(a_i)$ can also be obtained from $\tilde{T}_i$ by the following process.
\begin{enumerate}
\item Rotate $\tilde{T}_i$ by $180^\circ$ and push each row to the left so that it becomes a Young tabloid,
\item substitute each entry $i$ by $1-i$, and
\item find a standard Young tableau which is jeu-de-taquin equivalent to the result of (2) with the same shape as $\tilde{T}_i$.
\end{enumerate}
From the properties of combinatorial $R$-matrices, step (3) is also equivalent to the following.
\begin{enumerate}[label=(3')]
\item apply combinatorial $R$-matrices accordingly to retain the original shape of $\tilde{T}_i$.
\end{enumerate}
Now we apply $\Psi_n$ on the output of each $\tilde{T}_i$ under this process. Step (1) obviously commutes with $\Psi_n$, and so does step (2) modulo $n$. Also, step (3') commutes with $\Psi_n$ by Lemma \ref{lem:comm}. Therefore, $\omega(\Psi_n(\tilde{T}))$ is obtained from $T=\Psi_n(\tilde{T})$ by applying the process below:
\begin{enumerate}
\item Rotate $T$ by $180^\circ$ and push each row to the left so that it becomes a Young tabloid,
\item substitute each entry $i$ by $1-i$ modulo $n$, say $n+1-i$, and
\item apply combinatorial $R$-matrices accordingly to retain the original shape of $T$.
\end{enumerate}
But this is what we want to prove.
\end{proof}
The description of $\omega$ in terms of combinatorial $R$-matrices has an advantage that it can be generalized to any row-standard Young tabloid, say $RSYT(\lambda)$ where $\lambda$ is a finite sequence of positive integers which is not necessarily a partition, since the method described in Proposition \ref{prop:rmat} does not rely on the condition that $\lambda$ is a partition. If we write such a generalization again by $\omega$, then the following lemma is easily proved.
\begin{lem}\label{lem:comm2} Let $\lambda = (\lambda_1, \ldots, \lambda_r)$ be a finite sequence of positive integers. Then for $1\leq i \leq r-1$, the two maps
$$\omega \circ \mathcal{R}_i, \mathcal{R}_{i}\circ \omega\colon RSYT(\lambda) \rightarrow RSYT((\lambda_1, \ldots, \lambda_{i-1}, \lambda_{i+1}, \lambda_{i}, \lambda_{i+2}, \ldots, \lambda_r))$$
coincide. In other words, $\mathcal{R}_i$ and $\omega$ ``commute".
\end{lem}
\subsection{Proof of Proposition \ref{prop:str}} By Proposition \ref{prop:rmat} and Lemma \ref{lem:comm2}, we may illustrate $T \mapsto \omega(T)$ for $T\in RSYT(\lambda\cup(k,k))$ by the following process.
\begin{enumerate}
\item Apply $\mathcal{R}$ accordingly to obtain a row-standard Young tabloid of shape $(k, \lambda_1, \ldots, \lambda_r, k)$.
\item Apply (the generalized version of) $\omega$ which is an involution on $RSYT((k, \lambda_1, \ldots, \lambda_r, k))$.
\item Apply $\mathcal{R}$ accordingly to obtain a row-standard Young tableau of shape $\lambda \cup (k,k)$.
\end{enumerate}
Note that step (1) and (3) are inverse to each other. Therefore by Lemma \ref{lem:comm2}, we have
$$|RSYT(\lambda\cup(k,k))^\omega| = |RSYT((k, \lambda_1, \ldots, \lambda_r, k))^\omega|.$$
Now, we consider the surjection
$$\Phi\colon RSYT((k, \lambda_1, \ldots, \lambda_r, k)) \rightarrow RSYT(\lambda)$$
which sends $T= (T^{(1)}, T^{(2)}, \ldots, T^{(r)}, T^{(r+1)}, T^{(r+2)})$ to the renormalization of $(T^{(2)}, \ldots, T^{(r)}, T^{(r+1)})$, i.e. removes the first and the last row (of length $k$) of $T$ and renormalizes the result so that the entries are $1, 2, \ldots, |\lambda|$. For example,
\ytableausetup{smalltableaux}
$$\textup{if } k=2, \lambda=(4,3), \textup{ and } T=\begin{ytableau}
3&9\\
1 & 2 &7&11\\
5 &6&10\\
4 &8\\
\end{ytableau}, \textup{ then } \Phi(T)= \begin{ytableau}
1 & 2 &5&7\\
3 &4&6\\
\end{ytableau}.
$$
Now, from the description of $\omega$ (Proposition \ref{prop:rmat}), $T$ is $\omega$-stable if and only if
\begin{enumerate}
\item there exists $a_1,a_2, \ldots, a_k \in \{1, 2, \ldots, n\}$ such that $a_1<a_2<\cdots<a_k$ and $\{a_1, \ldots, a_k\} \cap \{n-a_1, \ldots, n-a_k\}=\emptyset$, and the first (resp. last) row of $T$ is $(a_1, a_2, \ldots, a_k)$ (resp. $(n-a_k, n-a_{k-1}, \ldots, n-a_1)$),
\item $\Phi(T)$ is $\omega$-stable.
\end{enumerate}
Thus, $\varphi(\lambda \cup(k,k))$ equals $\varphi(\lambda)$ multiplied by the number of choices of such $\{a_1, a_2, \ldots, a_k\}$, which is equal to $\binom{m}{k}2^k$ where $m = \floor{n/2}$. But this is what we want to prove.
\bibliographystyle{amsalphacopy}
|
{
"timestamp": "2018-10-09T02:10:38",
"yymm": "1804",
"arxiv_id": "1804.05636",
"language": "en",
"url": "https://arxiv.org/abs/1804.05636"
}
|
\section{Introduction and Motivation}
\label{sec:intro}
We study optimal recovery of the regression function $\fo$ in the framework of reproducing kernel Hilbert space (RKHS) learning. Here we are given random and noisy
observations of the form
\[ Y_j= \fo(X_j) + \epsilon_j \;, \qquad j=1,...,n \]
at i.i.d. data points $X_1,...,X_n$, drawn according to some unknown distribution $\nu$ on some input space $\X$, taken as a standard Borel space.
More precisely, we assume that the observed data $(X_i,Y_i)_{1 \leq i \leq n} \in (\X \times \Y)^n$ are sampled i.i.d. from an unknown probability measure $\rho$ on
$\X \times \Y$, with $\E[Y_i|X_i]=\fo(X_i)$\,, so that the distribution of $\eps_i$ may
depend on $X_j$\,, while satisfying $\E[\eps_j|X_j]=0$\,.
For simplicity, we take the output space $\Y$ as the set of real numbers, but this could be generalized
to any separable Hilbert space, see \cite{optimalratesRLS}.
In our setting, an estimator $\hat f$ for $\fo$ lies in an hypothesis space $\h \subset L^2(\X, \nu)$, which we choose to be a separable reproducing kernel Hilbert
space (RKHS), having a measurable positive semi-definite kernel $K: \X \times \X \longrightarrow \R$, satisfying $\sup_{x \in \X}K(x,x) \leq \kappa^2$.
More precisely, we confine ourselves to estimators $\hat f^\lam$ arising from the fairly large class of
{\it spectral regularization methods}, see e.e. \cite{rosasco}, \cite{per}, \cite{DicFosHsu15}, \cite{BlaMuc16}.
This class of methods contains the well known Tikhonov regularization, Landweber iteration or spectral cut-off.
We recall that while tuning the regularization parameter $\lam$ is essential for spectral regularization to work well, an {\it a priori} choice of the
regularization parameter is in general not feasible in statistical problems since the choice necessarily depends on unknown structural properties
(e.g. smoothness of the target function or behavior of the statistical dimension).
This imposes the need for data-driven {\it a-posteriori} choices of the regularization parameter, which hopefully are optimal in some well defined sense.
An attractive approach is (some version of) the balancing principle going back to Lepskii's seminal paper \cite{lepskii90} in the context of Gaussian white noise,
having been elaborated by Lepskii himself in a series of papers and by other authors, see e.g. \cite{lepskii92}, \cite{lepskii93}, \cite{golden03}, \cite{birge01}, \cite{mathe06}
and references therein.
Before we present our somewhat abstract approach, we shall motivate the general idea in a specific example. Denoting by
$$B: f \in \h \mapsto \int_{\X} f(x) K(x,\cdot) d\nux(x) \in \h $$
the kernel integral operator associated to $K$ and the sampling measure $\nux$, we recall from \cite{BlaMuc16}
that the optimal regularization parameter (as well as the rate of convergence) is determined by the source condition
assumption $||B^{-r} \fo||_{\h} \leq R$ for some constants $r,R>0$ as well as by
an assumed power decay of the effective dimension
\[ \NN(\lam) = \tr{B(B + \lam)^{-1}} \leq C_b\lam^{-1/b} \]
with intrinsic dimensionality $b>1$ and by the noise variance $\sigma^2>0$. Error estimates are usually established by
deriving a {\it bias-variance} decomposition, which looks in this special case as
\begin{equation}
\label{eq:error_old_decomp}
\norm{ B^s (\fo-\hat f^{\lam}) }_{\h} \; \lesssim \;
C_{s}(\eta)\lam^s\paren{ R\lam^r + \frac{\sigma}{\sqrt n} \lam^{-\frac{b+1}{2b}} } \; ,
\end{equation}
holding with probability at least $1-\eta$, for any $\eta \in (0,1)$, provided $n$ is big enough.
Here, the function
$ \lam \mapsto R\lam^r$ is the leading order of an upper bound for the {\it approximation error} and $\lam \mapsto \frac{\sigma}{\sqrt n} \lam^{-\frac{b+1}{2b}} $ is
the leading order of an
upper bound for the {\it sample error}.
We combine all parameters in a vector
$(\gamma, \theta)$ with $\gamma=(\sigma, R) \in \Gamma = \R_+ \times \R_+ $ and $\theta=(r, b) \in \Theta = \R_+ \times (1, \infty)$.
The optimal regularization parameter $\lam_{n, (\gamma, \theta)} $ is chosen by balancing the two leading error terms, more precisely
by choosing $\lam_{n, (\gamma, \theta)}$ as the unique solution of
\begin{equation}
\label{eq:example_lam_choice}
R\lam^r = \sigma \lam^{-\frac{b+1}{2b}} \;,
\end{equation}
leading to the resulting error estimate
\[ ||B^s(\fo-\hat f^{\lam_{n, (\gamma, \theta)}} )||_{\h} \; \lesssim \; 2C_{s}(\eta) \lam_{n, (\gamma, \theta)}^{s+r} \; , \]
with probability at least $1-\eta$.
The associated sequence of estimated solutions
$(f_{\z}^{\lam_{n,(\gamma,\theta)}})_{n \in \N}$, depending on the regularization parameter $(\lam_{n,(\gamma, \theta)})_{(n,\gamma)\in \N\times \Gamma}$
was called weak/ strong minimax optimal over the model family $(\M_{(\gamma, \theta)})_{(\gamma, \theta) \in \Gamma \times \Theta}$ with rate of convergence
given by $(a_{n,(\gamma,\theta)})_{(n, \gamma)\in \N \times \Gamma}$, {\bf pointwisely for any fixed $\theta \in \Theta$}.
\\
\\
However, if the parameter $r$ in the source condition or the intrinsic dimensionality $b>1$ are unknown,
an {\it a priori} choice of the theoretically best value $\lam_{n, (\gamma, \theta)}$ as in \eqref{eq:example_lam_choice} is impossible. Therefore, it is necessary to use
some {\it a posteriori} choice of $\lam$, independent of the parameter $\theta=(r, b) \in \Theta$. Our aim is to construct an estimator
$f^{\hat \lam_{n, \gamma}(\z)}_{\z}$\;, i.e. to find a sequence of regularization parameters $(\hat \lam_{n, \gamma}(\z))_n$,
without knowledge of $\theta \in \Theta$, but depending on the data $\z$, on $\gamma \in \Gamma$ and on the confidence level,
such that $f^{\hat \lam_n(\z,\eta, \gamma)}_{\z}$ is {\it (minimax) optimal adaptive} in the sense of Definition \ref{def:weak_adaptive}.
{\bf Contribution: }
More generally, we derive adaptivity in the case where the approximation error is upper bounded by some increasing unknown function $\A(\cdot)$ and where
\[ \cals(n,\lam) = \sigma \sqrt{\frac{\NN(\lam)}{n\lam}} \]
is an upper bound for the sample error.
Crucial for our approach is a two-sided estimate of the effective dimension in terms of its empirical approximation. This in particular allows to
control the spectral structure of the covariance operator through the given input data.
In summary, our approach achieves:
\begin{enumerate}
\item A fully data-driven estimator for the whole class of spectral regularization algorithms, which does not use data splitting as e.g. Cross Validation.
\item Adaptation to unknown smoothness {\bf and} unknown covariance structure.
\item One for all: Balancing in $L^2$ (which is easiest) automatically gives optimal balancing in the stronger ${\mathcal H}$- norm (an analogous result is
open for other approaches to data dependent choices of the regularization parameter).
\end{enumerate}
The paper is organized as follows: In Section \ref{sec:empirical_effective_dimension} we provide a two-sided estimate of the effective dimension by its empirical counterpart.
The main results are presented in Section \ref{sec:balancing}, followed by some specific examples in Section \ref{sec:application_adaption}.
A more detailed discussion is given in Section \ref{sec:adapt_discussion}.
The proofs are collected in the Appendix.
\section{Empirical Effective Dimension}
\label{sec:empirical_effective_dimension}
The main point of this subsection is a two-sided estimate on the effective dimension by its empirical approximation which is crucial for our entire approach.
We recall the definition of the {\it effective dimension} and introduce its empirical approximation,
the {\it empirical effective dimension}: For $\lambda \in (0,1]$ we set
\begin{equation}
\label{def:empirical_eff_dim}
\NN(\lam) = \tr{\;(\bar B+ \lam)^{-1} \bar B\;} \;, \qquad \NN_{\x}(\lam) = \tr{\;(\bar B_{\x}+ \lam)^{-1} \bar B_{\x}\;} \; ,
\end{equation}
where we introduce the shorthand notation $\bar B_{x} := \kappa^{-2} B_{\x}$ and similarly $\bar B := \kappa^{-2} B$\,.
Here $\NN(\lam)$ depends on the marginal $\nu$ (through $B$), but is considered as deterministic, while $\NN_{\x}(\lam)$ is considered as a random variable.
\\
\begin{prop}
\label{prop:rel_bound}
For any $\eta \in (0,1)$, with probability at least $1-\eta$
\begin{equation}
\label{eq:main}
|\; \NN(\lam) - \NN_{\x}(\lam) \;| \; \leq \; 2\log(4\etainv)\big(1+ \sqrt{\NN_{\x}(\lam)}\big)\left( \frac{2}{\lam n} + \sqrt{\frac{\NN(\lam)}{n\lam}} \right)\; ,
\end{equation}
for all $n \in \N^*$ and $\lambda \in (0,1]$.
\end{prop}
\begin{cor}
\label{cor:rel_bound}
For any $\eta \in (0,1)$, with probability at least $1-\eta$,
one has
\[
\sqrt{\max(\NN(\lambda),1)} \leq (1 + 4\delta) \sqrt{\max(\NN_{\x}(\lambda),1)} \,,
\]
as well as
\[
\sqrt{\max(\NN_{\x}(\lambda),1)} \leq (1 + 4 (\sqrt{\delta} \vee \delta^2) ) \sqrt{\max(\NN(\lambda),1)} \,,
\]
where $\delta:= 2\log(4\etainv)/\sqrt{n\lambda}$\,. In particular, if $\delta \leq 1$, with probability at least $1-\eta$ one has
\[ \frac{1}{5} \; \sqrt{\max(\NN(\lambda),1)} \leq \sqrt{\max(\NN_{\x}(\lambda),1)} \leq 5 \sqrt{\max(\NN(\lambda),1)} \;.
\]
\end{cor}
\section{Balancing Principle}
\label{sec:balancing}
In this section, we present the main ideas related to the {\it Balancing Principle} and make the informal presentation from the Introduction more precise. Firstly a definition:
\begin{defi}
\label{def:weak_adaptive}
Let $\Gamma, \Theta$ be sets and let, for $(\gamma,\theta) \in \Gamma \times \Theta$, $\M_{(\gamma, \theta)}$ be a class of data generating distributions on $\X \times \Y$.
For each $\lam \in (0,1]$ let $(\X \times \Y)^n \ni \z \longmapsto f_{\z}^{\lam} \in \h$
be an algorithm.
If there is a sequence $(a_{n,(\gamma,\theta)})_{n \in \N}$
$(\gamma,\theta) \in \Gamma \times \Theta$
and a parameter choice $(\hat \lam_{n, \gamma, \tau}(\z))_{(n, \gamma)\in \N\times \Gamma}$ (not depending on $\theta \in \Theta$)
such that
\begin{small}
\begin{equation}
\label{eq:weak_adaptive}
\; \lim_{\tau \to \infty}\;
\limsup_{n \to \infty}\;
\sup_{\rho \in \M_{(\gamma,\theta)}} \rho^{\otimes n}
\left( \norm{ \bar B^s(f^{\hat \lam_{n, \gamma, \tau}(\z)}_{\z}-\fo)}_{\h} \geq \tau a_{n,(\gamma,\theta)} \right) = 0
\end{equation}
and
\begin{equation}
\label{lowern}
\lim_{\tau \to 0} \liminf _{n \to \infty }\inf_{\hat f} \sup_{\rho \in \M_{(\gamma,\theta)}} \rho^{\otimes n}
\left( \norm{ \bar B^s( \hat f-\fo)}_{\h} \geq \tau a_{n,(\gamma,\theta)} \right) > 0,
\end{equation}
\end{small}
where the infimum is taken over all estimators $\hat f$,
then the sequence of estimators $(f_{\z}^{\hat \lam_{n, \gamma, \eta}(\z)})_{n \in \N}$
is called
{\it minimax optimal adaptive over $\Theta$}
and the model family $(\M_{(\gamma,\theta)})_{(\gamma,\theta) \in \Gamma \times \Theta}$,
with respect to the family of rates $(a_{n,(\gamma,\theta)})_{(n, \gamma)\in \N \times \Gamma}$,
for the interpolation norm of parameter $s \in [0,\frac{1}{2}]$.
\end{defi}
We remind the reader from \cite{BlaMuc16} that
upper estimates typically hold on a class $\M^<_{(\gamma,\theta)}$
and lower estimates hold on a possibly different class $\M^>_{(\gamma,\theta)}$, the model class $\M_{(\gamma,\theta)}$ in the above definition being the
intersection of both.
To find such an adaptive estimator, we apply a method which is known in the statistical literature as
{\it Balancing Principle}. Throughout this section we need
\begin{assumption}
\label{assumption1}
Let $\M$ be a class of models. We consider a discrete set of possible values for the regularization parameter
\[ \Lam_m\; = \; \{ \; \lam_j \; : \; 0< \lam_0 < \lam_1 < ... < \lam_m \;\} \; .\]
for some $m \in \N$.
Let $s \in [0,\frac{1}{2}]$ and $\eta \in (0,1]$.
We assume to have the following error decomposition uniformly over the grid $\Lambda_m$:
\begin{equation}
\label{error_bound}
\norm{(\bar B_\x+\lam)^{s}(\fo-f_{\z}^{\lam} ) }_{\h} \; \leq \; C_{s}(m,\eta)\; \lam^s \left(\; \tilde \A(\lam) + \tilde \cals(n,\lam) \;\right) \; ,
\end{equation}
where
\begin{equation}
\label{def:constant}
C_{s}(m,\eta)=C_s\log^2(8|\Lam_m|\etainv)\;,\qquad C_s>0\;,
\end{equation}
with probability at least $1-\eta$, for all data generating distributions from $\M$.
The bounds $\tilde \A(\lam)$ and $ \tilde \cals(n,\lam)$ are given by
\[ \tilde \cals(n, \lam) = \cals (n ,\lam) + d_1(n, \lam)\;, \quad \cals (n ,\lam) = \sigma\sqrt{\frac{\tilde \NN(\lam)}{n\lam}}\;,
\qquad d_1(n, \lam)= \frac{M}{n \lam} \;, \]
with $\tilde \NN(\lam) = \max( \NN(\lam) ,1)$ and
\[ \tilde \A(\lam)=\A(\lam) + d_2(n)\;, \quad d_2(n)= \frac{C}{\sqrt{n}}\;, \]
where $\A(\lam)$ is increasing, satisfying $\lim_{\lam \to 0}\A(\lam)=0$ and
for some constants $C < \infty$, $M<\infty$. We further define $d(n, \lam) := d_1(n,\lam) + d_2(n)$.
\end{assumption}
We remark that it is actually sufficient to assume \eqref{error_bound} for $s=0$ and $s=\frac{1}{2}$. Interpolation via
inequality $||B^s f||_{\h} \leq ||\sqrt{B}f||^{2s}_{\h} \; ||f||^{1- 2s}_{\h}$ implies validity of \eqref{error_bound}
for any $s \in [0,\frac{1}{2}]$.
Note that for any $s \in [0,\frac{1}{2}]$, the map $\lam \mapsto \lam^{s} \cals(n, \lam)$ as well as $\lam \mapsto \lam^{s} d_1(n, \lam)$ are strictly decreasing in $\lam$.
Also, if $n$ is sufficiently large and if $\lam$ is sufficiently small, $\tilde \A(\lam) \leq \tilde \cals(n, \lam)$.
We let $$\lamopt(n):= \sup \{\lam:\; \tilde \A(\lam) \leq \tilde \cals(n, \lam) \} \;. $$
In this definition we have replaced $\A(\lam)$, $\cals(n, \lam)$ by $\tilde \A(\lam)$ and $\tilde\cals(n, \lam)$, thus including the
remainder terms $d_1(n, \lam)$ and $d_2(n)$ into our definition of $\lamopt(n)$. It will emerge {\it a-posteriori}, that the definition of $\lamopt(n)$ is not
affected, since the remainder terms are subleading. But {\it a priori}, this is not known. A correct proof of the crucial oracle inequality in Lemma
\ref{cor:oracle} below is much easier with this definition of $\lamopt(n)$. It will then finally turn out that the remainder terms are really subleading.
The grid $\Lambda_m$ has to be designed such that the optimal value $ \lamopt(n)$ is contained in $[\lam_0,\lam_m]$.
The best estimator for $\lamopt(n)$ within $\Lam_m$ belongs to the set
\[ \J(\Lam_m) \; = \; \left\{\lam_j \in \Lam_m \; :\; \tilde \A(\lam_j)\leq \tilde \cals(n, \lam_j) \right\} \]
and is given by
\begin{equation}
\label{lamstardef}
\lamstar := \max \; \J(\Lam_m) \; .
\end{equation}
In particular, since we assume that $\J(\Lam_m) \neq \emptyset$ and $ \Lam_m \setminus \J(\Lam_m) \neq \emptyset$, there is some $l \in \N$
such that $\lam_l = \lamstar \leq \lamopt(n) \leq \lam_{l+1}$.
Note also that the choice of the grid $\Lambda_m$ has to depend on $n$.
Before we define the balancing principle estimate of $ \lamopt(n)$, we give some intuition of its possible choice: For any
$\lam \leq \lamopt(n)$, we have $\tilde \A(\lam)\leq \tilde \cals(n,\lam)$. Moreover, for any $\lam_1 \leq \lam_2$ we have
\[ \norm{(\bar B_\x + \lam_1)^s f}_{\h} \leq \norm{(\bar B_\x+\lam_2)^sf}_{\h} \;. \]
Finally, since $\lam \mapsto \lam^s \tilde \cals(n,\lam)$ is decreasing, Assumption \ref{assumption1} gives for any two $\lam, \lam' \in \J(\Lam_m)$ satisfying
$\lam' \leq \lam$,
with probability at least $1-\eta$
\begin{align}
\label{monest}
\norm{(\bar B_\x+\lam')^{s}(f_{\z}^{\lam'}-f_{\z}^{\lam} ) }_{\h} &\leq \norm{(\bar B_\x+\lam')^{s}(\fo-f_{\z}^{\lam'} ) }_{\h} +
\norm{(\bar B_\x+\lam')^{s}(\fo-f_{\z}^{\lam} ) }_{\h} \nonumber \\
&\leq \norm{(\bar B_\x+\lam')^{s}(\fo-f_{\z}^{\lam'} ) }_{\h} +
\norm{(\bar B_\x+\lam)^{s}(\fo-f_{\z}^{\lam} ) }_{\h} \nonumber \\
&\leq C_{s}(m,\eta)\; \lam'^s \left(\; \tilde \A(\lam') + \tilde \cals(n,\lam') \;\right) \; + \nonumber \\
& \qquad \qquad + \; \;C_{s}(m,\eta)\; \lam^s \left(\; \tilde \A(\lam) + \tilde \cals(n,\lam) \;\right) \nonumber \\
&\leq 4 C_{s}(m,\eta)\; \lam'^s \tilde \cals(n,\lam') \;.
\end{align}
An essential step is to find an empirical approximation of the sample error. In view of Corollary \ref{cor:rel_bound} we define
\[ \tilde \cals_{\x}(n,\lam)= \cals_{\x}(n,\lam) + d_1(n,\lam)\;, \quad \cals_{\x}(n,\lam)=\sigma\sqrt{\frac{\tilde \NN_{\x}(\lam)}{n\lam}}\;, \]
with $\tilde \NN_{\x}(\lam) = \max(\NN_{\x}(\lam), 1)$ and $\NN_{\x}(\lam)$ the empirical effective dimension given in \eqref{def:empirical_eff_dim}.
Corollary \ref{cor:rel_bound} implies uniformly in $\lam \in \Lam_m$
\begin{equation}
\label{eq:var_bound}
\frac{1}{5}\tilde \cals_{\x}(n,\lam) \leq \tilde \cals(n,\lam) \leq 5 \tilde \cals_{\x}(n,\lam) \;,
\end{equation}
with probability at least $1-\eta$,
provided
\begin{equation}
\label{eq:lam_0}
n\lam_0 \geq 2 \;,\qquad 2\log(4|\Lam_m|\etainv) \leq \sqrt{n\lambda_0} \;.
\end{equation}
Substituting \eqref{eq:var_bound} into the rhs of the estimate \eqref{monest} motivates our definition of the balancing principle estimate of $ \lamopt(n)$ as
follows:
\begin{defi}
\label{def:lepest}
Given $s \in [0,\frac{1}{2}]$, $\eta \in (0,1]$
and $\z \in \Z^n$,
we set
\begin{small}
\begin{align*}
\J^+_{\z}(\Lambda_m) &= \{ \; \lam \in \Lambda_m \; :\;
|| (\bar B_{\x}+ \lam')^s(f_{\z}^{\lam} - f_{\z}^{\lam'}) ||_{\h}
\; \leq \; 20C_{s}(m,\eta/2) \; \lam'^s \; \tilde \cals_{\x}(n, \lam') \; , \\
& \qquad \; \forall \lam' \in \Lambda_m, \; \lam' \leq \lam\; \}
\end{align*}
\end{small}
and define
\begin{equation}
\label{tildelams}
\hat \lam_s(\z) :=\max \; \J^+_{\z}(\Lam_m) \;.
\end{equation}
\end{defi}
Notice that $\J^+_{\z}(\Lam_m)$ as well as $\hat \lam_s(\z)$ depend on the confidence level $\eta \in (0,1]$.
For the analysis it will be important that the grid $\Lambda_m$ has a certain regularity. We summarize all requirements needed in
\begin{assumption}(on the grid)
\label{assumption2}
\begin{enumerate}
\item Assume that $\J(\Lam_m) \not = \emptyset$ and $\Lam_m \setminus \J(\Lam_m) \not = \emptyset$.
\item (Regularity of the grid)
There is some $q>1$ such that the elements in the grid obey $1< \lam_{j+1}/\lam_j \leq q$, $j=0,...,m$.
\item Choose $\lam_0 = \lam_0(n)$ as the unique solution of $n\lam=\NN(\lam)$. We require that $n$ is sufficiently large, such that
$\NN(\lam_0(n))\geq 1$ (so that the maximum in the definition of $\tilde \NN(\lam)$ can be dropped).
We further assume that $n\lam_0\geq 2$.
\end{enumerate}
\end{assumption}
Note that
$\lam_0(n) \to 0$ as $n \to \infty$. Then, since $\NN(\lam) \to \infty$
as $ \lam \to 0$, we get that this $\lam_0=\lam_0(n)$ satisfies $\lam_0 n= \NN(\lam_0) \to \infty$.
Furthermore, a short argument shows that the optimal value $\lamopt(n)$ indeed satisfies $\lam_0 \leq \lamopt(n)$, if $n$ is big enough.
Since
$\A(\lam) \to 0$ as $\lam \to 0$, we get $\tilde \A(\lam_0(n)) \to 0 $ as $n \to \infty.$ Since $\tilde \cals(n,\lam_0(n))=1+ \frac{M}{n\lam_0(n)}$ by definition, it follows
$\tilde \A(\lam_0(n)) \leq \tilde \cals(n,\lam_0(n))$ for $n$ big enough. From the definition of $\lamopt(n)$ as a supremum, we actually have $\lam_0(n) \leq \lamopt(n)$, for $n$ sufficiently large.
Under the regularity assumption, we find that
\begin{equation}
\label{omega}
\tilde S(n,\lam_{j}) < q \tilde S(n,\lam_{j+1}) \;, \quad j=0,...,m \;.
\end{equation}
Indeed, while the effective dimension $\lam \to \NN(\lam)$ is decreasing, the related function $\lam \to \lam \NN(\lam)$ is non-decreasing. Hence we find that
\[ q^{-1} \NN(\lam) = (q\lam)^{-1} \lam \NN(\lam) < (q\lam)^{-1} (q\lam) \NN(q\lam) = \NN(q\lam) \]
and since $q > 1$
\[ q^{-1} \tilde \NN(\lam) = \max \left(q^{-1} \NN(\lam), q^{-1} \right) < \max(\NN(q\lam) , 1) = \tilde \NN(q\lam) \;.\]
Therefore
\[ q^{-1} \cals(n, \lam_{j})= \sigma \sqrt{\frac{q^{-1}\tilde \NN(\lam_{j})}{nq\lam_{j}}} < \sigma\sqrt{\frac{\tilde \NN(\lam_{j+1})}{n\lam_{j+1}}} = \cals(n, \lam_{j+1}) \;. \]
One also easily verifies that
\[ d_1(n, \lam_j) = \frac{M}{n\lam_j} \leq \frac{qM}{n\lam_{j+1}} = qd_1(n, \lam_{j+1})\;, \]
implying \eqref{omega}.
\begin{rem}
\label{rem:geom_grid}
The typical case for Assumption \ref{assumption2} to hold is given when the parameters $\lam_j$ follow a geometric progression, i.e., for some $q>1$ we let
$\lam_j:=\lam_0q^j$, $j=1,...,m$ and with $\lam_m =1$. In this case we are able to upper bounding the total number of grid points $|\Lam_m|$ in terms of $\log(n)$.
In fact, since $\lam_m=1= \lam_0q^m$, simple calculations lead to
\[ |\Lambda_m| = m+1 = 1-\frac{\log(\lam_0)}{\log(q)} \;. \]
Recall that the starting point $\lam_0$ is required to obey $\NN(\lam_0)=n\lam_0 \geq 2$ if $n$ is sufficiently large,
implying $-\log(\lam_0) \leq -\log\paren{\frac{2}{n}} \leq \log(n)$.
Finally, we obtain for $n$ sufficiently large
\begin{equation}
\label{est:gridlog}
|\Lambda_m| \leq C_q\log(n) \;,
\end{equation}
with $C_q = \log(q)^{-1} + 1$.
\end{rem}
We shall need an additional assumption on the effective dimension:
\begin{assumption}
\label{def:eff_dim_low}
\begin{enumerate}
\item
For some $\gamma_1 \in (0,1]$ and for any $\lam $ sufficiently small
\begin{equation*}
\NN(\lam ) \geq C_1 \lam^{-\gamma_1} \;,
\end{equation*}
for some $C_1>0$.
\item
For some $\gamma_2 \in (0,1]$ and for any $\lam $ sufficiently small
\begin{equation*}
\NN(\lam ) \leq C_2 \lam^{-\gamma_2} \;,
\end{equation*}
for some $C_2>0$.
\end{enumerate}
\end{assumption}
Note that such an additional assumption restricts the class of admissible marginals and shrinks the class $\M$ in Assumption \ref{assumption1} to a subclass $\M'$.
Such a lower and upper bound will hold in all examples which we encounter in Section \ref{sec:application_adaption}.
We further remark that Assumption \ref{def:eff_dim_low} ensures a precise asymptotic behavior for $\lam_0=n^{-1}\NN(\lam_0)$ of the form
\begin{equation}
\label{est:lam0}
C_{\gamma_1} \paren{ \frac{1}{n}}^{ \frac{1}{1+\gamma_1} } \; \leq \; \lam_0(n) \; \leq \; C_{\gamma_2} \paren{ \frac{1}{n}}^{ \frac{1}{1+\gamma_2} } \;,
\end{equation}
for some $C_{\gamma_1} >0$, $C_{\gamma_2}>0 $.
\subsubsection{Main Results}
The first result is of preparatory character.
\begin{prop}
\label{maintheorem_lepskii}
Let Assumption $\ref{assumption1}$ be satisfied. Define $\lamstar$ as in \eqref{lamstardef}.
Assume $n\lam_0 \geq 2$. Then for any
$$\eta \geq \eta_n:=\min\paren{1\;, \; 4|\Lam_m|\exp\paren{-\frac{1}{2}\sqrt{\NN(\lam_0(n))}} } \;,$$
uniformly over $\M$, with probability at least $1-\eta$
\[ \norm{ (\bar B_{\x} + \lamstar )^s( f_{\z}^{\hat\lam_s(\z)} - \fo )}_{\h} \; \leq \; 102C_s(m,\eta/2)\lamstar^s \; \tilde \cals(n,\lamstar) \;. \]
\end{prop}
We shall need
\begin{lem}
\label{cor:oracle}
If Assumption $\ref{assumption2}$ holds, then
\begin{equation}
\label{eq:oracle_step}
\lamstar^s \; \tilde \cals(n,\lamstar) \leq q^{1-s} \min_{\lam \in [\lam_0, \lam_m]} \{\; \lam^s\;(\tilde \A(\lam) + \tilde \cals(n, \lam) )\; \} \; .
\end{equation}
\end{lem}
We immediately arrive at our first main result of this section:
\begin{theo}
\label{cor:balancing_final}
Let Assumption $\ref{assumption1}$ be satisfied and
suppose the grid obeys Assumption $\ref{assumption2}$.
Then for any
$$\eta \geq \eta_n:=\min\paren{1\;, \; 4|\Lam_m|\exp\paren{-\frac{1}{2}\sqrt{\NN(\lam_0(n))}} } \;,$$
uniformly over $\M$, with probability at least $1-\eta$
\[ \snorm{f_{\z}^{\hat\lam_s(\z)} - \fo}_{\h} \; \leq \; q^{1-s}\; D_{s}(m,\eta) \;
\min_{\lam \in [\lam_0, \lam_m]} \{\; \lam^s( \tilde \A(\lam) + \tilde \cals(n, \lam) ) \;\} \;, \]
with
\[ D_{s}(m,\eta)= C'_{s} \log^{2(s+1)}(16|\Lam_m|\etainv) \;, \]
for some $C'_{s}>0$.
\end{theo}
In particular, choosing a geometric grid and assuming a lower and upper bound on the effective dimension, we obtain
\begin{cor}
\label{cor:balancing_final2}
Let Assumption $\ref{assumption1}$, Assumption \ref{assumption2} and Assumption \ref{def:eff_dim_low} be satisfied.
Suppose the grid is given by a geometric sequence $\lam_j=\lam_0q^j$, with $q>1$, $j=1,...,m$ and with $\lam_m=1$.
Then for any
$$\eta \geq \eta_n:=4C_q\log(n)\exp\paren{-C_{\gamma_1, \gamma_2}n^{\frac{\gamma_1}{2(1+\gamma_2)}}} \;,$$
uniformly over $\M'$, with probability at least $1-\eta$
\[ \snorm{f_{\z}^{\hat\lam_s(\z)} - \fo}_{\h} \; \leq \; \tilde D_{s,q}(n,\eta) \;
\min_{\lam \in [\lam_0, 1]} \{\; \lam^s( \tilde \A(\lam) + \tilde \cals(n, \lam) ) \;\} \;, \]
with
\[ \tilde D_{s,q}(n,\eta)= C_{s,q} \log^{2(s+1)}(\log(n)) \log^{2(s+1)}(16\etainv) \;, \]
for some $C_{\gamma_1, \gamma_2}>0$ and some $C_{s,q}>0$, provided $n$ is sufficiently large.
\end{cor}
Note that $\eta_n \to 0$ as $n\to \infty$.
\subsubsection{One for All: $L^2$-Balancing is sufficient !}
This section is due to an idea suggested by P. Math\'e (which itself was inspired by the work \cite{blahoffreiss16}) which we have worked out in detail.
We define the $L^2(\nu)-$ balancing estimate $\hat \lam_{1/2}(\z)$ according to Definition \ref{def:lepest} by explicitely choosing
$s=\frac{1}{2}$ (in contrast to Theorem \ref{cor:balancing_final}, where we choose $\hat \lam_{s}(\z)$ depending on the norm parameter $s$). Our main result states that balancing in the $L^2(\nu)-$ norm suffices to automatically give balancing in all other (stronger !) intermediate norms
$|| \cdot||_s$, for any $s \in [0, \frac{1}{2}]$.
\begin{theo}
\label{theo:oneforall}
Let Assumption $\ref{assumption1}$ and Assumption \ref{assumption2} be satisfied and
suppose the grid obeys Assumption $\ref{assumption2}$.
Then for any
$$\eta \geq \eta_n:=\min\paren{1\; ,\; 4|\Lam_m|\exp\paren{-\frac{1}{2}\sqrt{\NN(\lam_0(n))}} }\;,$$
uniformly over $\M$, with probability at least $1-\eta$
\[ \norm{\bar B^s( f_{\z}^{\hat \lam_{1/2}(\z)} - \fo ) }_{\h} \; \leq \;q^{1-s} \hat D_{s}(m,\eta)\;
\min_{\lam \in [\lam_0, \lam_m]} \{\;\lam^s( \tilde \A(\lam) + \tilde \cals(n, \lam)) \; \} \;, \]
with
\begin{align*}
\hat D_{s}(m, \eta)= C'_{s}\log^{2(s+1)}(16|\Lam_m|\etainv) \;,
\end{align*}
for some $C'_{s}>0$.
\end{theo}
In particular, choosing a geometric grid and assuming a lower and upper bound on the effective dimension, we obtain:
\begin{cor}
\label{theo:oneforall2}
Let Assumption $\ref{assumption1}$, Assumption \ref{assumption2} and Assumption \ref{def:eff_dim_low} be satisfied.
Suppose the grid is given by a geometric sequence $\lam_j=\lam_0q^j$, with $q>1$, $j=1,...,m$ and with $\lam_m=1$.
Then,
for $n$ sufficiently large and for any
$$\eta \geq \eta_n:=4C_q\log(n)\exp\paren{-C_{\gamma_1, \gamma_2}n^{\frac{\gamma_1}{2(1+\gamma_2)}}} \;,$$
uniformly over $\M'$, with probability at least $1-\eta$
\[ \norm{\bar B^s( f_{\z}^{\hat \lam_{1/2}(\z)} - \fo ) }_{\h} \; \leq \; q^{1-s}\hat D_{s,q}(n,\eta)\;
\min_{\lam \in [\lam_0, 1]} \{\;\lam^s( \tilde \A(\lam) + \tilde \cals(n, \lam)) \; \} \;, \]
with
\begin{align*}
\hat D_{s,q}(n, \eta)= C_{s,q}\log^{2(s+1)}(\log(n))\log^{2(s+1)}(16\etainv) \;,
\end{align*}
for some $C_{\gamma_1, \gamma_2}>0$ and some $C_{s,q}>0$.
\end{cor}
Note that $\eta_n \to 0$ as $n\to \infty$.
\begin{rem}
Still, our choice for $\lam_0$ is only a theoretical value which remains unknown as it depends on the unknown marginal $\nu$ through the effective dimension $\NN(\lam)$.
Implementation requires a data driven choice. Heuristically, it seems resonable to proceed as follows.
Let $q>1$ and $\tilde \lam_j = q^{-j}$, $j=0,1,...$ (we are starting from the right and reverse the order). Define the stopping index
\[ \hat j_0:= \min\{ \;j \in \N: \; \cals_{\x}(n,\tilde \lam_j) \geq 5 \; \} \]
and let $\Lambda = \{ \tilde \lam_{\hat j_0} < ... < \tilde \lam_0=1 \}$.
Here, $\cals_{\x}(n,\tilde \lam_j) $ depends on the empirical effective dimension $\NN_{\x}(\lam)$, see \eqref{def:empirical_eff_dim}, which by Corollary \ref{cor:rel_bound}
is close to the
unknown effective dimension $\NN(\lam)$. Thus we think that the above choice of $\lam_0$ is reasonable for implementing the dependence of $\lam_0$ on the unknown marginal.
A complete mathematical analysis is in development.
\end{rem}
\section{Specific Examples}
\label{sec:application_adaption}
We proceed by illustrating some specific examples of our method as described in the previous section.
In view of our Theorem \ref{theo:oneforall} and Corollary \ref{theo:oneforall2}
it suffices to only consider balancing in $L^2(\nu)$. We always choose a geometric grid as in Remark \ref{rem:geom_grid},
satisfying $\lam_m=1$.
{\bf (1) The regular case}
We consider the setting of \cite{BlaMuc16}, where the eigenvalues of $\bar B$ decay polynomially (with parameter $b>1$),
the target function $\fo$ satisfies a H\"older-type source condition
\[ \fo \in \Omega_\nu(r, R):= \{\; f \in \h \;: \; f=\bar B_{\nu}^rh \;, \; ||h||_{\h} \leq R \;\} \]
and the noise satisfies a Bernstein-Assumption
\begin{equation}
\label{bernstein_first}
\E[\; \abs{Y - \fo(X)}^{m} \; | \; X \;] \leq \frac{1}{2}m! \; \sigma^2 M^{m-2} \quad \nux - {\rm a.s.} \;,
\end{equation}
for any integer $m \geq 2$ and for some $\sigma > 0$ and $M>0$. We combine all structural parameters in a vector $(\gamma, \theta)$, with
$\gamma = (M, \sigma, R) \in \Gamma = \R_+^3$ and $\theta=(r, b) \in \Theta = (0, \infty) \times (1, \infty)$.
We are interested in adaptivity over $\Theta$.
It has been shown in \cite{BlaMuc16}, that the corresponding minimax optimal rate is given by
\begin{equation*}
a_n=a_{n,\gamma, \theta} = R \lam_{n,\gamma, \theta}^{r+s} = R \paren{\frac{\sigma^2}{R^2n}}^{\frac{b(r+s)}{2br+b+1}} \;.
\end{equation*}
We shall now check validity of our Assumption \ref{assumption1}.
In the following, we assume that the data generating distribution belongs to the class $\M=\M_{(\gamma, \theta)}$, defined in \cite{BlaMuc16}.
Recall that we let $\lam_0(n)$ be determined as the unique solution of $\NN(\lam) = n\lam$. Then,
we have uniformly for all data generating distributions from the class $\M$, with probability at least $1-\eta$, for any $\lam \in \Lambda_m$,
\[ || (\bar B_{\x}+\lam)^s(f_{\z}^{\lam} - \fo) ||_{\h} \leq C_{s}\log^{2}(8|\Lambda_m|\etainv)\;\lam^s\paren{ \; \tilde \A(\lam) + \tilde \cals(n, \lam) \; } \;,\]
for $n$ sufficiently large, with
\[ \tilde \A(\lam)=R\lam^r + \frac{Rr}{\sqrt n}1_{(1, \infty)}(r)\;, \quad \tilde \cals(n, \lam)= \sigma\sqrt{\frac{\NN(\lam)}{n\lam}} + \frac{M}{n \lam } \;, \]
where $C_{s}$ does not depend on the parameters $(\gamma , \theta ) \in \Gamma \times \Theta $.
Remember that the optimal choice for the regularization parameter $\lam_n$ is obtained by solving
\[ \A(\lam)= \sigma \sqrt{\frac{\lam^{- 1/b}}{n \lam} } \]
and belongs to the interval $[\lam_0(n) , 1]$. This can be seen
by the following argument: If $n$ is sufficiently large
\[ 1= \sqrt{ \frac{\NN(\lam_0(n))}{n\lam_0(n)} } \geq \sqrt{C_{\beta, b}}R\lam_n^r = \sigma \sqrt{\frac{C_{\beta, b}\lam_n^{-\frac{1}{b}}}{n\lam_n}} \geq \sqrt{ \frac{\NN(\lam_n)}{n\lam_n} } \;,\]
which is equivalent to $\cals(n, \lam_0(n)) \geq \cals(n, \lam_n)$. Since $\lam \mapsto \cals(n, \lam)$ is strictly decreasing we conclude $\lam_n \geq \lam_0(n)$. Here we use the bound
$\NN(\lam) \leq C_{\beta, b} \lam^{-\frac{1}{b}}$.
Recall that we also have corresponding lower bound $\NN(\lam) \geq C_{\alpha, b} \lam^{-\frac{1}{b}}$, since $\nu \in {\priorgr}(b, \alpha)$,
granting Assumption \ref{def:eff_dim_low}.
We adaptively choose the regularization parameter $\hat \lam_{1/2}(\z)$ according to Definition \ref{def:lepest} by $L^2(\nu)-$ balancing
(i.e. by choosing $s=\frac{1}{2}$) and independently from the parameters $b>1$, $r>0$.
Corollary \ref{theo:oneforall2} gives for any $s \in [0, \frac{1}{2}]$, if $n$ is sufficiently large, with probability at least $1-\eta$ (uniformly over
$\M$)
\begin{equation}
\label{eq:appl1}
\norm{\bar B^s( f_{\z}^{\hat \lam_{1/2}(\z)} - \fo ) }_{\h}
\leq \; C'_{s,q} C_s(\eta)\;\paren{ \; a_{n} + \lam_n^s d(n,\lam_n) \;} \;,
\end{equation}
where
$$ C_s( \eta)= \log^{2(s+1)}(\log(n))\log^{2(s+1)}(16\etainv),$$
provided that $\eta \geq \eta_n= 4C_q\log(n)\exp\paren{-C n^{\frac{1}{2(b+1)}}}$, for some $C>0$, depending on $\alpha, \beta$ and $b$.
Recall that $\eta_n \to 0$ as $n \to \infty$.
In \eqref{eq:appl1} we have used that
\begin{align*}
\min_{\lam \in [\lam_0(n), 1]} \{\;\lam^s( \tilde \A(\lam) + \tilde \cals(n, \lam)) \; \} &\leq \lam^s_n( \tilde \A(\lam_n) + \tilde \cals(n, \lam_n)) \\
&= \lam^s_n( \A(\lam_n) + \cals(n, \lam_n) + d(n, \lam_n)) \;.
\end{align*}
Then $\lam_n^{s}\A(\lam_n) \leq a_n $ and $\lam_n^{s} \cals(n, \lam_n)\leq C_{b}a_n $ give
equation \eqref{eq:appl1}
It remains to show that for $n$ sufficiently large, the remainder $\lam_n^s d(n,\lam_n)$ is of lower order than the rate $a_n$.
One finds that
\[
\frac{M}{n\lam_n} = o\paren{C_{ b}\sqrt{\frac{1}{n}\lam_n^{-\frac{b+1}{b}}}} \,, \quad \frac{r}{\sqrt n} = o(\lambda_n^r)\;.
\]
Summarizing the above findings gives
\begin{cor}[from Corollary \ref{theo:oneforall2}]
Let $s \in [0, \frac{1}{2}]$. Choose the regularization parameter $\hat \lam_{1/2}(\z)= \hat \lam_{n, \gamma, \eta}(\z)$ according to Definition \ref{def:lepest}
by choosing $s=\frac{1}{2}$.
Then, if $n$ is sufficiently large, for any
$$\eta \geq \eta_n= 4C_q\log(n)\exp\paren{-C n^{\frac{1}{2(b+1)}}}\;,$$
$(r, b) \in \R_+ \times (1, \infty)$, $(M, \sigma, R) \in \R_+^3$
\begin{small}
\[ \sup_{\rho \in \M} \rho^{\otimes n}\paren{ \norm{\bar B^s( f_{\z}^{\hat \lam_{1/2}(\z)} - \fo ) }_{\h} \leq C'_{s,q}\log^{2(s+1)}(16\etainv)\; b_{n} } \geq 1-\eta \;, \]
\end{small}
with $ b_n=\log^{2(s+1)}(\log(n))\; a_{n}$.
\end{cor}
Now defining $\tau = C'_{s,q} \log^{2(s+1)}(16\etainv)$ gives
\[ \eta = 16 \exp\paren{-\paren{\frac{\tau}{C'_{s,q}}}^{1/2(s+1)}}\;, \]
implying \eqref{eq:weak_adaptive}.
Observing that the results in \cite{BlaMuc16} imply validity of the lower bound \eqref{lowern}, this means:
\begin{cor}
In the sense of Definition \ref{def:weak_adaptive} the sequence of estimators $(f_{\z}^{\hat \lam_{1/2}(\z)})_{n \in \N}=(f_{\z}^{\hat \lam_{n, \gamma, \eta}(\z)})_{n \in \N}$ is adaptive over
$\Theta$ (up to log-term)
and the model family $(\M_{(\gamma,\theta)})_{(\gamma,\theta) \in \Gamma \times \Theta}$
with respect to the family of rates $(a_{n,(\gamma,\theta)})_{(n, \gamma)\in \N \times \Gamma}$,
for all interpolation norms of parameter $s \in [0,\frac{1}{2}]$.
\end{cor}
{\bf (2) General Source Condition, polynomial decay of eigenvalues}
Our approach also applies to the case where the smoothness is measured in terms of a {\it general source condition}, generated
by some index function, that is,
\[ \fo \in \Omega_{\nu}(\A):= \{\; f \in \h: \; f = \A(\bar B_{\nu})h, \; ||h||_{\h}\leq 1 \;\} \;,\]
where $\A: (0,1] \longrightarrow \R_+$ is a continuous non-decreasing function, satisfying $\lim_{t \to 0}\A(t) = 0$.
We keep the noise condition \eqref{bernstein_first} and we choose the parameter $\gamma=(M,\sigma) \in \Gamma=\R_+^2,$
$\theta=(\A,b) \in \Theta={\cal F} \times (1,\infty)$, where
${\cal F}$ denotes either the class of {\it operator monotone} functions or the class of functions decomposing into an operator monotone part and an
{\it operator Lipschitz} part. For more details, we refer the interested reader to \cite{per}, \cite{mathe16}.
We introduce the class of data-generating distributions
\begin{align*}
\M^<_{(\gamma,\theta)} &= \{ \rho(dx,dy)=\rho(dy|x) \nu(dx); \rho(\cdot|\cdot) \in \K(\Omega_{\nu}(\A)), \nu \in \priorle(b, \beta) \} \;,\\
\M^>_{(\gamma,\theta)} &= \{ \rho(dx,dy)=\rho(dy|x) \nu(dx); \rho(\cdot|\cdot) \in \K(\Omega_{\nu}(\A)), \nu \in \priorgr(b, \alpha) \} \;,
\end{align*}
where $\priorle(b, \beta)$ and $\priorgr(b, \alpha)$ are exactly defined as in \cite{BlaMuc16}.
Then $\M=\M_{(\gamma,\theta)}$ is defined as the intersection.
From \cite{rastogi17} and \cite{mathe16} (in particular Proposition 4.3) one then gets that Assumption \ref{assumption1} is satisfied:
Uniformly for all data generating distributions from the class $\M$, with probability at least $1-\eta$,
\[ || (\bar B_{\x}+\lam)^s(f_{\z}^{\lam} - \fo) ||_{\h} \leq C_{s}\log^{2}(8|\Lambda_m|\etainv)\;\lam^s\paren{ \; \tilde \A(\lam) + \tilde \cals(n, \lam)\; } \;,\]
for $n$ sufficiently large, with
\[ \tilde \A(\lam)= \A(\lam)+ \frac{C}{\sqrt n} \;, \quad \tilde \cals(n, \lam)= \sigma\sqrt{\frac{\NN(\lam)}{n\lam}} + \frac{M}{n \lam } \]
and
\[ d(n,\lam_n) = \frac{C}{\sqrt n} + \frac{M}{n \lam } \;. \]
Assuming $\NN(\lam) \leq C_{\beta, b} \lam^{-1/b} \;$, which as above is implied by polynomial asymptotics of the eigenvalues of the covariance operator $\bar B$ specified by the exponent $b$,
the sequence of estimators $(f^{\lam_{n,\A,b}}_{z})_n$ (defined via some spectral regularization having prescribed qualification) using the parameter choice
\begin{equation}
\label{def:psi}
\lam_n:=\lam_{n,\A,b}:= \psi_{\A,b}^{-1}\paren{\frac{1}{\sqrt n}}\;, \quad \quad \psi_{\A, b}(t):= \A(t)t^{\frac{1}{2}\left(\frac{1}{b} +1\right)}\;,
\end{equation}
is then minimax optimal, in both $\h-$norm ($s=0$) and $L^2(\nu)-$norm ($s=1/2$) (see \cite{rastogi17}, \cite{mathe16}), with rate
\begin{equation}
\label{def:rate_adaptive2}
a_n:= a_{n,\A,b}:= \lam_{n, \A,b}^{s}\;\A\left( \lam_{n, \A, b}\right) \;.
\end{equation}
This holds pointwisely for any $(\A, b) \in \Theta={\cal F} \times (1,\infty)$.
The crucial observation is that
equation \eqref{def:rate_adaptive2} is precisely the result obtained by balancing the leading order terms for sample and approximation error.
Arguments similar to those in the previous example show that $\lam_n \in [\lam_0(n), 1]$. Recall that $\NN(\lam) \leq C_{\beta,b} \lam^{-\frac{1}{b}}$ and that $\A(\lam) \to 0$ as $\lam \to 0$. Thus, if $n$ is big enough
\[ 1= \sqrt{ \frac{\NN(\lam_0(n))}{n\lam_0(n)} } \geq \sqrt{C_{\beta,b}} \;\A(\lam_n) = \sqrt{C_{\beta,b}} \psi(\lam_n)\lam_n^{-\frac{1}{2}(\frac{1}{b} +1)} \geq \sqrt{ \frac{\NN(\lam_n)}{n\lam_n} } \;,\]
which is equivalent to $\cals(n, \lam_0(n)) \geq \cals(n, \lam_n)$. Since $\lam \mapsto \cals(n, \lam)$ is strictly decreasing, we conclude that $\lam_n \geq \lam_0(n)$.
Recall that we also have corresponding lower bound $\NN(\lam) \geq C_{\alpha, b} \lam^{-\frac{1}{b}}$, since $\nu \in {\priorgr}(b, \alpha)$,
granting Assumption \ref{def:eff_dim_low}.
We again adaptively choose the regularization parameter $\hat \lam_{1/2}(\z)$ according to Definition \ref{def:lepest} by $L^2(\nu)-$ balancing
(i.e. by choosing $s=\frac{1}{2}$) and independently from the parameters $b>1$, $r>0$.
Corollary \ref{theo:oneforall2} gives for any $s \in [0, \frac{1}{2}]$, if $n$ is sufficiently large, with probability at least $1-\eta$ (uniformly over
$\M$)
\begin{equation}
\label{eq:appl2}
\norm{\bar B^s( f_{\z}^{\hat \lam_{1/2}(\z)} - \fo ) }_{\h}
\leq \; C'_{s,q} C_s(\eta)\;\paren{ \; a_{n} + \lam_n^s d(n,\lam_n) \;} \;,
\end{equation}
where
$$ C_s( \eta)= \log^{2(s+1)}(\log(n))\log^{2(s+1)}(16\etainv)\;,$$
provided that
$$\eta \geq \eta_n= 4C_q\log(n)\exp\paren{-C n^{\frac{1}{2(b+1)}}} \;, $$
for some $C>0$, depending on $\alpha, \beta$ and $b$.
One readily verifies also in this case that the remainder term $ d(n,\lam_n)$ is indeed subleading:
$$ n^{-1/2} =\psi_{\A,b}(\lam_n)=\lam_n^{\frac{1}{2}(1 + \frac{1}{b})} \A(\lam_n) =
o\left(\A(\lam_n) \right),$$
and moreover
\[
\frac{M}{n\lam_n} = o\paren{C_{ b}\sqrt{\frac{1}{n}\lam_n^{-\frac{b+1}{b}}}} \;.
\]
From Theorem 3.12 in \cite{rastogi17} one then obtains the lower bound \eqref{lowern}.
Thus, we have proved:
\begin{cor}[from Corollary \ref{theo:oneforall2}]
Let $s \in [0, \frac{1}{2}]$.
Choose the regularization parameter $\hat \lam_{1/2}(\z)= \lam_{n, \gamma, \eta}(\z) $ according to Definition \ref{def:lepest} by $L^2(\nu)-$ balancing.
Then, if $n$ is sufficiently large, for any
$$\eta \geq \eta_n= 4C_q\log(n)\exp\paren{-C n^{\frac{1}{2(b+1)}}} \;, $$
$\A \in {\cal F}$, $b>1$ and $(M, \sigma, R) \in \R_+^3$ one has
\begin{small}
\[ \sup_{\rho \in \M_{(\gamma,\theta)}} \rho^{\otimes n}\paren{ \norm{\bar B^s( f_{\z}^{\hat \lam_{1/2}(\z)} - \fo ) }_{\h} \leq C'_{s,q}\log^{2(s+1)}(16\etainv)\; b_{n} } \geq 1-\eta \;, \]
\end{small}
with
\[ b_n=\log^{2(s+1)}(\log(n))\; a_{n} \;.\]
This means that in the sense of Definition \ref{def:weak_adaptive} the sequence of estimators $(f_{\z}^{\hat \lam_{1/2}(\z)})_{n \in \N}=(f_{\z}^{\hat \lam_{n, \gamma, \eta}(\z)})_{n \in \N}$
is adaptive over $\Theta$ (up to log-term)
and the model family $(\M_{(\gamma,\theta)})_{(\gamma,\theta) \in \Gamma \times \Theta}$
with respect to the family of rates $(a_{n,\gamma,\theta})_{(n, \gamma)\in \N \times \Gamma}$ from \eqref{def:rate_adaptive2},
for all interpolation norms of parameter $s \in [0,\frac{1}{2}]$.
\end{cor}
{\bf (3) Beyond the regular case}
Recall the class of models considered in \cite{BlaMuc16beyond}: Let $\gamma=(M, \sigma, R) \in \Gamma = \R^3_+,$
$\Theta=\{ (r, \nu^*, \nu_*) \in \R_+ \times (1,\infty)^2; \nu^* \leq \nu_{*} \}$ and set
\begin{equation}
\label{measureclass:beyondn}
\M^<_{(\gamma,\theta)} \; := \; \{ \; \rho(dx,dy)=\rho(dy|x)\nux(dx)\; : \;
\rho(\cdot|\cdot)\in {\cal K}(\Omega_{\nux}(r,R)), \; \nux \in \priorle(\nu^{*}) \;\} \;,
\end{equation}
\begin{equation}
\M^>_{(\gamma,\theta)} \; := \; \{ \; \rho(dx,dy)=\rho(dy|x)\nux(dx)\; : \;
\rho(\cdot|\cdot)\in {\cal K}(\Omega_{\nux}(r,R)), \; \nux \in \priorgr(\nu_{*}) \;\} \;,
\end{equation}
and denote by $\M=\M_{(\gamma,\theta)}$ the intersection.
We shall verify validity of our Assumption \ref{assumption1}.
In the following, we assume that the data generating distribution belongs to the class $\M$.
Then,
we have uniformly for all data generating distributions from the class $\M$, with probability at least $1-\eta$, for any $\lam \in \Lambda_m$,
\[ || (\bar B_{\x}+\lam)^s(f_{\z}^{\lam} - \fo) ||_{\h} \leq C_{s, \nu^*}\log^{2}(8|\Lambda_m|\etainv)\;\lam^s\paren{ \; \tilde \A(\lam) + \tilde \cals(n, \lam) \; } \;,\]
with
\[ \tilde \A(\lam)=R\lam^r+ \frac{Rr}{\sqrt n}1_{(1, \infty)}(r) \;, \quad \tilde \cals(n, \lam)= \sigma \sqrt{\frac{\lam_n^{2r}}{n\G(\lam)}} + \frac{M}{n\lam} \;.
\]
As usual, we shall investigate adaptivity on the parameter space $\Theta$.
We upper bound the effective dimension by applying results from \cite{BlaMuc16beyond}, using the counting function $\F(\lam)$ defined in equation $(2.1)$.
We obtain
$$\NN (\lam) \leq C_{\nu^{*}} \F(\lam)\;,$$
for any $\lam$ sufficiently small.
We now follow the discussion in Example (1) above, with $\A(\lam)$, $\cals(n,\lam)$, $d_1(n)$, $d_2(n,\lam)$ remaining unchanged.
We shall only use the new
upper bound on $\cals(n,\lam)$ defined by
$$ \cals_{+}(n, \lam) = \sigma \sqrt{ \frac{\F(\lam)}{n \lam}} = \sigma \sqrt{ \frac{\lam^{2r}}{n \G(\lam)}} \;. $$
This gives, equating $R\lam^r = \cals_+(n, \lam)$, for $n$ sufficiently large
$$\lam_n=\lam_{n,\theta} = \G^{-1}\left(\frac{\sigma^2}{R^2n} \right)\; . $$
Also in this case, $\lam_n$ can shown to fall in the interval $[\lam_0(n), 1]$. Indeed, if $n$ is sufficiently large
\[ 1= \sqrt{ \frac{\NN(\lam_0(n))}{n\lam_0(n)} } \geq \sqrt{C_{\nu^*}}R\lam_n^r = \sqrt{C_{\nu^*}} \sigma \sqrt{ \frac{\F(\lam_n)}{n\lam_n} } \geq \sigma \sqrt{ \frac{\NN(\lam_n)}{n\lam_n} } \;,\]
which is equivalent to $\cals(n, \lam_0(n)) \geq \cals(n, \lam_n)$. Since $\lam \mapsto \cals(n, \lam)$ is strictly decreasing, we have $\lam_0(n) \leq \lam_n$, provided $n$ is big enough.
More refined bounds for the effective dimension follow from \cite{BlaMuc16beyond}.
We have
\[ C_{\nu_*} \lam^{-\frac{1}{\nu_*}} \leq \NN(\lam) \leq C_{\nu^*} \lam^{-\frac{1}{\nu^*}} \]
and Assumption \ref{def:eff_dim_low} is satisfied.
We adaptively choose the regularization parameter $\hat \lam_{1/2}(\z)$ according to Definition \ref{def:lepest} by $L^2(\nu)-$ balancing,
i.e. by choosing $s=\frac{1}{2}$.
Corollary \ref{theo:oneforall2} gives for any $s \in [0, \frac{1}{2}]$, if $n$ is sufficiently large, with probability at least $1-\eta$ (uniformly over
$\M$)
\begin{equation}
\label{eq:appl3}
\norm{\bar B^s( f_{\z}^{\hat \lam_{1/2}(\z)} - \fo ) }_{\h}
\leq \; C'_{s,q} C_s(\eta)\;\paren{ \; a_{n} + \lam_n^s d(n,\lam_n) \;} \;,
\end{equation}
where
$$ C_s( \eta)= \log^{2(s+1)}(\log(n))\log^{2(s+1)}(16\etainv),$$
provided that
$$\eta \geq \eta_n=4C_q\log(n)\exp\paren{-C_{\nu_* , \nu^*} n^{ \frac{\nu^*}{2\nu_*(1+\nu^*)} }} \;. $$
In \eqref{eq:appl3} we have used that $a_n=\lam_n^{r+s}$ and
\begin{align*}
\min_{\lam \in [\lam_0(n), 1]} \{\;\lam^s( \tilde \A(\lam) + \tilde \cals(n, \lam)) \; \} &\leq \lam^s_n( \tilde \A(\lam_n) + \tilde \cals(n, \lam_n)) \\
&= \lam^s_n( \A(\lam_n) + \cals(n, \lam_n) + d(n, \lam_n)) \;.
\end{align*}
As above, one readily checks that that the subleading term $d(n, \lam_n)$ is really subleading:
\[ n^{-\frac{1}{2}} = o(\lam_n^r)\;,\quad \frac{M}{n\lam_n} = o\paren{ \sqrt{\frac{\lam_n^{2r}}{n\G(\lam_n)}} } \;.\]
Summarizing, we have proved
\begin{cor}[from Corollary \ref{theo:oneforall2}]
Let $s \in [0, \frac{1}{2}]$.
Choose the regularization parameter $\hat \lam_{1/2}(\z)= \lam_{n, \gamma, \eta}(\z) $ according to Definition \ref{def:lepest} by choosing $s=\frac{1}{2}$.
Then, if $n$ is sufficiently large, for any
$$\eta \geq \eta_n=4C_q\log(n)\exp\paren{-C_{\nu_* , \nu^*} n^{ \frac{\nu^*}{2\nu_*(1+\nu^*)} }} \;. $$
for any $r>0$, $1<\nu^*\leq \nu_*$, $(M, \sigma, R) \in \R_+^3$, one has
\begin{small}
\[ \sup_{\rho \in \M_{(\gamma,\theta)}} \rho^{\otimes n}\paren{ \norm{\bar B^s( f_{\z}^{\hat \lam_{1/2}(\z)} - \fo ) }_{\h} \leq C'_{s,q}\log^{2(s+1)}(16\etainv)\; b_{n} } \geq 1-\eta \;, \]
\end{small}
with
\[ b_n=\log^{2(s+1)}(\log(n))\; a_{n} \;.\]
Moreover, in the sense of Definition \ref{def:weak_adaptive} the sequence of estimators $(f_{\z}^{\hat \lam_{1/2}(\z)})_{n \in \N}=(f_{\z}^{\hat \lam_{n, \gamma, \eta}(\z)})_{n \in \N}$
is adaptive over $\Theta$ (up to log-term)
and the model family $(\M_{(\gamma,\theta)})_{(\gamma,\theta) \in \Gamma \times \Theta}$
with respect to the family of rates $(a_{n,\gamma,\theta})_{(n, \gamma)\in \N \times \Gamma}$,
for all interpolation norms of parameter $s \in [0,\frac{1}{2}]$.
\end{cor}
\section{Discussion}
\label{sec:adapt_discussion}
\begin{enumerate}
\item
We have shown that it suffices to prove adaptivity only in $L^2(\nu)-$norm, which is the weakest of all our interpolating norms indexed by $s \in [0,1/2]$.
Similar results of this type (an estimate in a weak norm suffices to establish the estimate in a stronger norm) have been obtained e.g.
in \cite{blahoffreiss16} and also in the recent paper of Lepskii, see \cite{Lepski16}, in a much more general context.
\item
We shall briefly discuss where and how the presentation of the balancing principle in our work improves the results in the existing literature
on the subject. The first paper on the balancing principle for kernel methods, \cite{vitoperros}, did not yet introduce {\em fast rates}, i.e.
rates depending on the intrinsic dimensionality $b$. Within this framework the results give - in the wording of the authors - {\em an optimal adaptive
choice of the regularization parameter for the class of spectral regularization methods}.
In the sense of our Definition \ref{def:weak_adaptive} the obtained estimators are optimal adaptive on the parameter space $\Theta=\R_+$
with respect to minimax optimal rates, which depend on $r$ but not on $b$
(or more general, not on the effective dimension $\NN(\lam)$). Technically, the authors of \cite{vitoperros} define their optimal adaptive estimator
as the minimum of 2 estimators, corresponding to 2 different norms, namely, setting
\begin{small}
\[ \J^+_{\z}(\Lam_m)\; = \; \left\{ \lam_i \in \Lam_m\; : \; \norm{ \bar B^s_{\x}(f_{\z}^{\lam_i} - f_{\z}^{\lam_j}) }_{\h}
\; \leq \; 4 C_s (\eta)\; \lam_j^s \; \cals(n, \lam_j) \;,
\; j = 0,..., i-1 \right\} \]
\end{small}
and defining $\tilde \lam_s(\z) :=\max \; \J^+_{\z}(\Lam_m)$, their final estimator is given by
\begin{equation}
\label{def:old_adaptive}
\hat \lam_s(\z) := \min\{ \tilde \lam_s(\z), \tilde \lam_0(\z) \} \; .
\end{equation}
We encourage the reader to directly compare this definition with our definition in \eqref{tildelams}.
Using the minimum of two estimators in this way can be traced back
to the use of an additive error estimate of the form
\begin{equation}
\label{eq:old_approx}
\left| \norm{ \bar B^sf}_{\h} - \norm{\bar B^s_{\x}f}_{\h}\right| \; \leq \; \sqrt 6 \log(4/\eta) \; n^{-\frac{s}{2}} \norm{f}_{\h} \; ,
\end{equation}
holding for any $f \in \h$, $s \in [0,1/2]$ and $\eta \in (0,1)$, with probability at least $1-\eta$. Here we have slightly generalized the original estimate in \cite{vitoperros} to all values of $s \in [0,1/2]$.
\\
\\
In the setting of \cite{vitoperros}, where only slow rates are considered, the variance $\cals(n,\lambda)$ is fully known. However, when considering fast rates (polynomial decay of eigenvalues), $\cals(n,\lambda)$ additionally depends on the unknown parameter $b>1$ and we have to replace the variance by its empirical approximation
$\cals_{\x}(n,\lam)$. This can effectively achieved by our Corollary \ref{cor:rel_bound}, where we provide a two sided bound
\[ \frac{1}{5} \; \cals_{\x}(n,\lam) \leq \cals(n,\lam) \leq 5 \;\cals_{\x}(n,\lam) \;. \]
Our bound (in a slightly weaker form) is also used in \cite{mathe16}\; for bounding the variance by its empirical approximation.
In the preprint \cite{mathe16}\; the authors independently present the balancing principle for fast rates. More precisely, in the case of H\"older-type source conditions, it covers the range $\Theta_{hs}$ of parameters $(r,b)$ of {\em high smoothness} where $b>1$
and $r \geq 1/2(1-1/b)$, which excludes the region of {\em low smoothness}. In addition, their results include more general types of source conditions.
This work started independently from our work on the balancing principle.
A crucial technical difference is that \cite{mathe16} is still based on using
\eqref{eq:old_approx} in an essential way.
However,
the discussion proceeds essentially along the traditional lines of \cite{vitoperros}, using the above mentioned additive error estimates. This makes the region of low smoothness, i.e. $r < 1/2(1-1/b)$, much less accessible
and leads to an estimator obtained by balancing
only on the restricted parameter space $\Theta_{hs}$ (with respect to minimax optimal rates of convergence, which, however, are known on the larger parameter space $\Theta= \R_+ \times (0,\infty)$). As before, the final estimator is taken to be a minimum of 2 estimators corresponding to different norms.
Our modified
definition of the estimator defined by balancing, avoiding the additive error estimate in equation \eqref{eq:old_approx},
allows in the case of H\"older type source conditions to obtain an optimal adaptive estimator (up to $\log \log (n)$ term) on the
parameter space $\Theta= \R_+ \times (1,\infty)$.
The final estimator is constructed somewhat more directly. It is not taken as a minimum of 2 separately constructed estimators.
Furthermore, our discussion in Example (2) shows how the more general results of \cite{mathe16} on source conditions different from H\"older -type
can naturally be recovered in our approach.
\item
Finally we want to emphasize that this notion of optimal adaptivity is {\em not} quite the original approach of Lepskii. The paper \cite{birge01}
contains an approach to the optimal adaptivity problem in the white noise framework which is closer to the original Lepskii approach and thus somewhat stronger than the weak approach described above, where the optimal adaptive estimator depends on the confidence level. It seems to be a wide open question how to adapt this original approach to the framework of kernel methods, i.e. constructing an estimator which is {\it optimal adaptive in Lepskii-sense }(independent of the confidence level $\eta$) and satisfies
\begin{equation}
\label{new}
\sup_{\theta \in \Theta}\; \sup_{\gamma \in \Gamma}\; \limsup_{n \to \infty} \;
a^{-1}_{n,(\gamma,\theta)} \; R_n(\tilde f^{\lam_{n, \gamma} (\z)}, \gamma ) \; < \; \infty \,,
\end{equation}
with $R_n$ being the risk
\[ R_n(\tilde f^{\lam_{n, (\gamma,\theta)} (\z)}, \gamma)= \sup_{\rho \in {\M_{(\gamma, \theta)}}}
\E_{\rho^{\otimes n}}\big[ \| \bar B^s( \fo - \tilde f^{\lam_{n, \gamma} (\z)} )\|_{\h}^p \big]^{\frac{1}{p}} \;, \quad p>0\;, s\in[0,1/2] \;, \]
and $a_{n, (\gamma, \theta)}$ being a minimax optimal rate.
Here
we always want to take $\Theta$ as the maximal parameter space on which one has minimax optimal rates. For slow rates, i.e. $\Theta=\{r > 0 \}$, the supremum over $\Theta$ in equation \eqref{new} exists. For fast rates, the boundary of the open set $\{ b > 1 \}$ poses problems at $b=1$, since one looses the trace class condition on the covariance operator $\bar B$ (in which case minimax optimality as in this thesis is not even proved). We remark that, trying to only use the effective dimension and parametrizing it by
$$ \NN(\lambda) = O(\lambda^{-\frac{1}{b}}), $$
(thus redefining somewhat the meaning of $b$) possibly changes the nature of the boundary at $b=1$ and might give existence of the sup. We leave this question for future research. Furthermore we remark that a rigorous proof of non-existence of the sup for our (spectral) meaning of $b$ requires a suitable lower bound exploding as $b \downarrow 1$, similar to the example in \cite{lepskii90}.
A similar type of difficulty (related to the non-existence of the sup) has already been systematically investigated in \cite{lepskii90} and \cite{lepskii93}. In such a case Lepskii has introduced the weaker notion of {\em the adaptive minimax order of exactness} and he also discusses additional log terms. Such estimators (which are not optimally adaptive) are called simply {\em adaptive}. This is related to the situation which we encounter in this section. It is known that e.g. for point estimators, additional $\log$ terms are indispensable. Our situation, however, is different and one could expect to prove optimal adaptivity in future research.
\end{enumerate}
|
{
"timestamp": "2018-04-17T02:12:04",
"yymm": "1804",
"arxiv_id": "1804.05433",
"language": "en",
"url": "https://arxiv.org/abs/1804.05433"
}
|
\section{Introduction}
As e-learning systems become more prevalent they are accessed by students of varied backgrounds, learning styles and needs. There is thus a growing need for them to accommodate individual difference between students and adapt to their changing pedagogical needs over time.
There are mainly two families of approaches for adapting educational content: offline approaches build models from data (e.g. ~\cite{segal2014edurank}) while online approaches balance exploration and exploitation (e.g ~\cite{clement2013multi}).
We provide a novel algorithm for sequencing content in e-learning systems that combines both of these approaches. It integrates offline learning from students' past interactions with online mechanisms for sequencing questions to students in order to maximize their learning gains.
This approach is based on the concept of \textit{zone of proximal development} \cite{vygotsky1987zone} where students are presented with challenges that are neither too easy nor too difficult, but are slightly beyond their current abilities.
Our algorithm, called MAPLE (Multi-Armed Bandits based Personalization for Learning Environments),
extends prior multi-armed bandits approaches in education by explicitly considering question difficulty when initializing the online behavior of the algorithm and when updating its behavior over time. MAPLE treats successful and failed question attempts differently and becomes \mydoubleq{conservative} (i.e. decreases exploration and increases weights of easier questions) when students fail under the (pedagogically guided) assumption that repeated errors are detrimental to students' learning~\cite{pekrun2016academic}.
We first evaluated MAPLE in a simulation environment comparing its performance to a variety of sequencing algorithms, including an approach that sequenced questions according to educational expert guidelines, an approach based only on personalized difficulty ranking, and a multi-armed bandit approach without personalized difficulty ranking initialization. MAPLE outperformed all other approaches for average and strong simulated students while showing the need for further tuning for weak simulated students.
We then implemented MAPLE in the wild in an existing e-learning system in a school with 7th grade students. MAPLE's performance was compared to two other sequencing algorithms already implemented in the e-learning system: an approach that sequenced questions according to educational expert guidelines and a state of the art Bayesian Knowledge Tracing based algorithm \cite{david2016sequencing}. We found that our proposed approach showed promising results compared to the existing educational expert approach and the BKT based approach. Moreover, students reported being more satisfied with the questions posed to them by MAPLE.
The contribution of this paper is two-fold. First, we present a novel algorithm for sequencing questions to students in e-learning that extends multi-armed bandits with personalized difficulty ranking information. Second, we show the potential of the algorithm in simulations and in field trials in the wild.
\section{Related Work}
Our work relates to past research on using historical data to sequence content to students, and to work on multi-armed bandits for online adaptation of educational content.
Several approaches within the educational artificial intelligence community have used computational methods for sequencing content to students.
Pardos and Heffernan~\cite{pardos2009determining} inferred order over question presented to students by predicting their skill levels using Bayesian Knowledge Tracing (BKT)~\cite{corbett1994knowledge}. They showed the efficacy of their approach on simulated data as well as on a test-set comprising random sequences of three questions. Ben David et al.~\cite{david2016sequencing} developed a BKT based sequencing algorithm. Their algorithm (which we refer to in this paper as YBKT) uses knowledge tracing to model students' skill acquisition over time and sequence questions to students based on their mastery level and predicted performance. It was shown to enhance student learning beyond sequencing designed by pedagogical experts.
Champaign et al.~\cite{champaign2010model} used a peer-based approach for content sequencing in an intelligent tutor system by computing similarities between students and choosing questions that provide best benefits for similar students, measured by similar average performance on past questions. Segal et al.~\cite{segal2014edurank} developed EduRank, a sequencing algorithm that combines collaborative filtering with social choice theory to produce personalized learning sequences for students. The algorithm constructs a “difficulty” ranking over questions for a target student by aggregating the ranking of similar students when sequencing educational content.
Multi-armed bandits provides a fundamental model for tackling the \mydoubleq{exploration-exploitation} trade off ~\cite{thompson1933likelihood,bubeck2012regret}. Xu et al. \cite{xu2016personalized} used bandits to identify which sequences of courses lead students to obtain maximal GPAs. Lan and Baraniuk~\cite{lan2016contextual} used sparse factor analysis with bandits to identify sequences of educational content that could maximize students performance on subsequent assessments. Lomas et al. \cite{lomas2016interface} showed how bandits can be used to search a large space of design decisions in creating educational games. Williams et al.~\cite{williams2016axis} used Thompson Sampling to identify highly rated explanations for how to solve Math problems, and chose priors that assumed that every explanation was equally rated.
Clement et al.~\cite{clement2013multi} used human experts' knowledge to initialize a multi-armed bandit algorithm called EXP4, that discovered which activities were at the right level to push students learning forward. In our work we do not rely on human experts, but rather use personalized difficulty rankings to guide the initial exploitation and update steps of our algorithm.
\section{Problem Formulation and Approach}
We consider an e-learning setting with a group of students $S$ and a set of practice questions $Q$. At each time step in the practice session, a computer agent $A$ needs to choose a question in $Q$ to present to the student. The agent sequencing problem requires choosing at each time step the next question to present to the student so as to maximize her learning gains over the length of the practice session. The goal is to present students with challenging problems while ensuring a high likelihood that they will be able to solve these problems.
Our approach to solving the problem, called MAPLE, maintains a belief distribution over the expected learning gains to the student for solving each of the questions in $Q$. This belief distribution is initialized with a personalized difficulty ranking over the questions in $Q$. The algorithm samples the next question to the student from this distribution and updates it at each time step given the student's performance on the question and its inferred difficulty to the student.
MAPLE implements an exploration policy
similar to the one that is used by the EXP4 algorithm~\cite{auer2002nonstochastic,clement2013multi}. This approach
maintains a belief distribution over questions that is proportional to how much learning gain each question is expected to provide. Weights are decreased or increased based on how difficult a question is and whether the student successfully solves the question or not. When a student successfully solves a question, weights are adjusted to make harder questions more likely to be presented, and explore a broader range of questions. When a student fails to solve a question, weights are adjusted to make easier questions more likely to be presented, and explore a narrower set of questions. The weights are initialized to reflect the inferred difficulty level for the student.
\begin{algorithm}[H]
\KwData {Set of students $S$.
\\Set of $N$ questions $Q$.
\\Passing grade threshold $\eta$, exploration rate $\gamma$.
\\Normalization factors $\alpha_1,\ldots,\alpha_4$.
\\For each student in $S$, a partial difficulty ranking $\succ_k$ over $T_k \subseteq Q$.
\\Target student $s_i \in S$.}
\KwResult {Next question $q_s \in Q$ to present to student $s_i$.}
{\bf{Step 1: Initialization:}}
\begin{enumerate}[topsep=0pt]
\item For each student in $S$:\\
- Compute difficulty ranking $\succ_l$ for all $q_l \in Q$\\
- $w_1 > \ldots > w_N=initialize\_weights(\succ_l)$\\
\end{enumerate}
{\bf{Step 2: Get Next Question:}}
\begin{enumerate}[topsep=0pt]
\setcounter{enumi}{1}
\item For each $j=1, \ldots , N$:\\
- Calculate new weight: $\pi_j = w_j(1-\gamma) + \epsilon_j\gamma,$ $\epsilon_j \sim U(-1,1)$ \\
- Normalize: $\pi_j = \frac{\pi_j}{\sum_{j=1}^{N}\pi_j}$\\
\item Choose question $q_s$ randomly, with respect to question weights $\pi_j$
\item return $q_s$
\end{enumerate}
{\bf{Step 3: Update Question Grade:}}
\begin{enumerate}[topsep=0pt]
\setcounter{enumi}{4}
\item Get student grade $g_s$ after solving question $q_s$
\item reward $R = g_s-\eta$
\item {\bf{if}} $g_s>\eta$ {\bf{then}}
\begin{itemize}
\item Increase weights for questions more difficult than $q_s$:
\\ $w_j = \alpha_1 e^R w_j,$ for $j = s+1,\ldots,N$
\item Increase exploration factor: $\gamma = \alpha_2 \gamma$
\end{itemize}
\item {\bf{else}}
\begin{itemize}
\item Decrease weights for questions more difficult then $q_s$:\\
$w_j = \alpha_3 e^R w_j,$ for $j = s+1,\ldots,N$
\item Decrease exploration factor: $\gamma = \alpha_4\gamma$
\end{itemize}
{\bf{end if}}
\item For each $j=1, \ldots , N$:\\
- Normalize: $w_j = \frac{w_j}{\sum_{j=1}^{N} w_j}$\\
\end{enumerate}
\caption{The Maple Approach}
\label{alg:MAPLE}
\end{algorithm}
Algorithm~\ref{alg:MAPLE} formalizes the MAPLE approach.
The input to the algorithm is a set of students $S$ each with known solutions over a set of questions in $Q$, a target student $s_i$ and a set of initialization parameters which include $\eta$, a passing grade threshold, $\gamma$, the exploration factor, and $\alpha_1,\ldots,\alpha_4$, normalization factors. The algorithm returns the next question $q_s$ to present to student $s_i$.
The algorithm includes 3 steps: (1) The initialization step is performed once at the beginning of execution to obtain a personalized difficulty ranking for each student. During initialization the algorithm computes a difficulty ranking over questions per student. We used the EduRank approach for this purpose and describe it in the next section. Next, the question weights $w_j$ are initialized with values using a softmax function, with higher weights corresponding to easier questions per the difficulty ranking. As students solve questions and succeed or fail, these weights are updated for each student.
(2) Next Question Selection: performed at each time step by the agent to choose the next question to present to the student.
For this step MAPLE uses the distribution weights computed for the student. The algorithm first adds an exploration component per weight $w_j$ (line 2) generating new weights $\pi_j$, and then chooses the next question with respect to the $\pi_j$ weights (lines 3,4).
(3) Update Question Grade: performed at each time step after the question was presented to the student and her solution grade was obtained.
In line 6, a reward value $R$ is computed to reflect the magnitude of student's success or failure in the question with respect the the threshold value $\eta$. In line 7, the case of a successful solution is treated ($grade>\eta$): in this case, the exploration factor is increased, and weights of harder questions for this student are also increased proportionally to the reward value $R$. Thus, the probability of the student to get harder questions on next attempts increases, as well as the algorithm willingness to \mydoubleq{take risks} (exploration factor).
On the final step, 9, weights are re-normalized on the unit scale.
To summarize the update step, MAPLE weights update reflects the change in the algorithm's estimation as to the suitability of each question for the student based on expert guidelines. Nonetheless, the stochastic nature of the algorithm enables exploration of additional sequencing alternatives. We now move to describe our simulation and field trial results.
\section{Simulations with Synthesized Data}
We performed a set of simulations to compare four different sequencing algorithms:
(1) The \emph{MAPLE} approach which used the EduRank~\cite{segal2014edurank} algorithm for difficulty ranking. EduRank considers the history of students' actions in the system (including their grades and retries) and uses collaborative filtering~\cite{sarwar2001item} and voting aggregation approaches~\cite{nurmi1983voting} to compute a personalized difficulty ranking over questions. MAPLE's parameters for simulations were set empirically except for $\eta$ (the passing grade threshold) which was determined by an educational expert.
(2) The \emph{Ascending} approach sequenced questions according to an absolute difficulty ranking that was determined by pedagogical experts. Questions were labeled into one of 5 groups (from easy to hard) according to difficulty level. The algorithm selects questions in the following temporal order: The first 10\% of questions presented to students are level 1 questions (easiest level), followed by 20\% questions from level 2, 30\% questions from level 3, 30\% questions from level 4 and 10\% questions from level 5 (hardest level). This is the main sequencing approach used to sequence questions to students in the e-learning system tested in a school in the next section.
(3) The \emph{EduRank} approach provided a personalized difficulty ranking over questions for each student. Questions were sequenced from EduRank's easiest estimated question to its hardest estimated question per student.
(4) The \emph{Naive Maple} approach
sequenced questions using the multi-armed bandit algorithm with random weights initialization (without the EduRank based difficulty ranking component).
The algorithms were evaluated by comparing their performance in simulation. We model questions in the simulation using a $\langle$skill,difficulty$\rangle$ pair; students are modeled as a vector of skill values for each question type.
We now describe the details of the student model during simulation.
The probability of a student to successfully solve a question is based on Item Response Theory~\cite{hambleton1991fundamentals} and is as follows:
\begin{equation}
p(success) = \frac{1}{1+e^{\theta\cdot(ql - sl)}}
\label{eq:1}
\end{equation}
where $\theta$ is a constant influencing the shape of the probability distribution. The student's probability of learning is based on the difference between her skill level $sl$ and the level of the question $ql$.
To reflect the fact that students may differ in their answers due to factors beyond their skill level (e.g. guessing and slipping, affect condition) we add a stochastic component to Equation \ref{eq:1}. Thus, our student model behaves according to:
\begin{equation}
p(success) = \beta\cdot\frac{1}{1+e^{\theta\cdot(ql - sl)}}+(1-\beta) \cdot \epsilon_u
\label{eq:2}
\end{equation}
Where $\beta$ is a constant controlling the impact of the stochastic component and $\epsilon_u$ is drawn from a uniform distribution in the range [0,1].
Lastly we describe how the student skill level is updated upon solving a question. Intuitively, the skill level improves significantly following a correct response to a hard question when the student had a low level in that skill before answering the question. Similarly, the skill estimation was probably most exaggerated when the student failed in solving an easy question while having a high skill level before answering the question.
This leads to the following skill update rules: If the student is successful in solving the question, i.e. her grade is greater than predefined value $\eta$, her skill level is updated according to:
\begin{equation}
sl = sl+\delta_1\cdot ql \cdot(1-sl)
\label{eq:3}
\end{equation}
Otherwise, her skill level is updated according to:
\begin{equation}
sl = sl-\delta_2\cdot (1-ql)\cdot sl
\label{eq:4}
\end{equation}
The parameters $\theta, \beta, \delta_1, \delta_2$ are determined empirically. The parameter $\eta$ is determined by an educational expert and was set to 0.7.
In the simulations each algorithm was run with a group of 1000 students, and was required to sequence 200 questions for each student. Each question belonged to one of 10 possible skills, uniformly distributed. We generated questions according to an absolute difficulty level, uniformly distributed between 1 (easiest), and 5 (hardest). Students' initial competency level in each skill were also uniformly distributed between 0 (no skill knowledge) and 1 (full knowledge of skill). All the sequencing algorithms had access to \mydoubleq{historical} data generated by the simulation engine in a pre-simulation step so they can build their internal models. This data contained 1000 students each solving 500 randomly selected questions.
We now describe key simulation results.
We start by looking at MAPLE's sequencing behaviour. Figure~\ref{fig:seqalgs} shows how MAPLE adapted the question difficulty as time progressed. The x-axis presents the time and the y-axis presents the number of questions of a given difficulty level that were presented to students in that time frame. The colors represent five difficulty levels ranging from easy (level 1) to hard (level 5). We can see that MAPLE started with offering easy questions to most students and then moved to propose more difficult questions while at the same time continuing to propose easy questions for a long time. This behaviour demonstrates the adaptive nature of MAPLE and was not observed in the other conditions tested. Specifically, EduRank used mostly easy questions and was not able to adapt harder questions to students while Naive Maple did not exhibit adaptive behavior for the questions in the practice set. (For longer practice sets the algorithm may show adaptive behavior).
\begin{figure}[H]
\includegraphics[width=1.2\textwidth,center]{colors_hist_maple_all.png}
\caption{MAPLE questions sequencing}
\label{fig:seqalgs}
\end{figure}
To better understand MAPLE's adaptation behavior we next look at 3 different student groups: weak students (with initial skill competency levels under $0.33$), average students (with initial skill competency levels in the range [0.33, 0.67]), and strong students (with initial skill competency levels above $0.67$).
Figure~\ref{fig:simseq} shows MAPLE's behavior for the 3 types of students and the 5 types of questions. We can see that for weak students, MAPLE kept proposing easy questions for a long time and refrained from proposing hard questions. For strong students, MAPLE begun proposing more difficulty questions earlier while reaching the hardest questions during the 200 questions session. And for average students, MAPLE took a middle way approach focusing on questions with average difficulty level while challenging some students with more difficult questions.
\begin{figure}[H]%
\centering
\subfigure[Weak Students]{{\includegraphics[width=10cm]{colors_hist_maple_weak.png} }}%
\qquad
\subfigure[Average Students]{{\includegraphics[width=10cm]{colors_hist_maple_avg.png} }}%
\qquad
\subfigure[Strong Students]{{\includegraphics[width=10cm]{colors_hist_maple_strng.png} }}%
\caption{MAPLE's question sequencing for different student types.}%
\label{fig:simseq}%
\end{figure}
Finally, we look at the skill level progression during simulation for the 3 student types in the 4 algorithms tested.
Figure~\ref{fig:skillprog} shows the skill level progression of the different student types at each simulation step, for each algorithm. We can see that for strong and average students MAPLE outperformed the other three algorithms throughout the practice session. Both EduRank which relies only on the difficulty information and Naive Maple which relies only on the multi-armed bandit approach presented lower results. For weak students we can see that the Ascending and Naive Maple approaches failed altogether, probably since they offered questions that are too hard for these students. Both MAPLE and EduRank presented initial good progress but then experienced a decline in the estimated skill level. This implies that MAPLE's adaptation scheme needs to be improved for this segment of students to offer less challenging questions.
\begin{figure}
\centering
\includegraphics[width=1.4\textwidth,center]{median_vs_time.png}
\caption{Skill level progression per algorithm and student type.}
\label{fig:skillprog}
\end{figure}
\section{Deployment and Evaluation in the Classroom}
We next moved to conduct a field study in the wild where students used different sequencing approaches as part of their curriculum work in class (and not in a laboratory setting).
The MAPLE algorithm was implemented in an e-learning system used for Math education. In this system, K-12 students practice solving Math questions in various skill areas matching their curriculum studies. The study compared MAPLE in one school with 7th grade students, to two other existing sequencing algorithms already available in the e-learning system. The experiment was conducted between May 9th 2017 and June 19th 2017 (end of school year).
The students were randomly divided into 3 cohorts: (1) MAPLE Sequencing: Students in this group received questioned sequenced by the MAPLE algorithm when practicing with the system. (2) YBKT Sequencing\footnote{We note that the YBKT code was not available to us for the simulations.}: students in this group received questioned sequenced by the Bayesian Knowledge Trace based algorithm proposed by Ben David et al. \cite{david2016sequencing}. This sequencing approach always chooses the next question from the available skill set in a deterministic manner. (3) Ascending Sequencing: students in this group received questions sequenced by the Ascending algorithm described earlier.
All students in the experiment were initially exposed to a pretest session. In this session they solved 10 pretest questions that were hand picked by a pedagogical expert. The hand picked questions matched the expected level of 7th graders at that stage in the academic year. Ninety two students solved the pretest questions and there was no statistically significant difference between the three groups in the average score on this preliminary test. We thus concluded that the students in each group exhibited similar knowledge baselines of the material at hand.
The students then engaged in multiple practice sessions in the e-learning system for the next 35 days, solving 10 assignment questions at each such practice session. For each cohort, assignment questions were sequenced by the cohort's respective algorithm (i.e. MAPLE, YBKT or Ascending). At the end of this period, students were asked to complete a post test session, solving the same questions (in the same order) as in the pretest session. Twenty eight students completed the post test session. We attribute the decrease in students' response from pretest to post test to the pending end of the academic year (there was no difference in the dropout rates across the 3 cohorts).
\begin{table}[H]
\centering
\scalebox{1.1}{
\begin{tabular}{|l|c|c|}
\hline
Cohort & Time per Question (sec) & Average Grade \\
\hline
Ascending & 6.49 & 43.76 \\
MAPLE & 10.69 & 71.28 \\
YBKT & 12.86 & 67.08 \\
\hline
\end{tabular}
}
\caption{Post test results per cohort: time per question and average grade.}
\label{tab:posttest}
\end{table}
Breaking up results by the sequencing algorithms, Table~\ref{tab:posttest} shows students' average grade and time spent on post-test questions, while Table ~\ref{tab:pre2post} shows the pre- to post- test change. As measured by post test grades, students assigned to the MAPLE condition achieved higher post test results than students assigned to the Ascending condition or to the YBKT condition.
This effect is also evident when observing the pre- to post- test change in grades. Students assigned to the MAPLE condition learned more than students assigned to the Ascending condition or to the YBKT condition. We note that further field trials with larger student groups are needed to evaluate statistical significance.
In addition to objective measures of learning, we examined students' satisfaction from interacting with the various sequencing approaches. Students rated their degree of agreement on a 3 point scale ((1) I do not agree (2) I partially agree (3) I strongly agree) with ten statements about issues like the system helpfulness, ease of use, adaptivity, match to class material, and future appeal.
Students using the MAPLE sequencing algorithm demonstrated higher satisfaction as represented by their answers to the subjective experience questions. The average score on the agreement with positive characteristics of the system was significantly different across the three conditions, with the highest score for the MAPLE algorithm at 2.39, compared to the Ascending algorithm (2.23) and the YBKT algorithm (2.20) (one way Anova, $p<0.05$).
\begin{table}[H]
\centering
\scalebox{1.1}{
\begin{tabular}{|l|c|c|}
\hline
Cohort & Time per Question Diff & Average Grade Diff \\
\hline
Ascending & -9.2 & -2.83 \\
MAPLE & -9.12 & 4.99 \\
YBKT & -8.65 & 1.26 \\
\hline
\end{tabular}
}
\caption{Change from pre-test to post-test: time per question and average grade.}
\label{tab:pre2post}
\end{table}
\section{Conclusion}
We have provided a new computational method called MAPLE for sequencing questions to students in on-line educational systems. MAPLE combines difficulty ranking based on past student experiences with a multi-armed bandit approach using these difficulty rankings in on-line settings.
We tested our approach in simulations and verified its adaptive nature and its learning gain results compared to other algorithms as measured by the simulated skills' improvement.
We then performed a live experiment in the wild running MAPLE in a classroom in parallel to two baseline algorithms. We found that our proposed approach presented promising results in students' performance and satisfaction.
We mention several limitations of our work and subsequent suggestions for future work. First, the simulation results showed that MAPLE's performance needs to be fine tuned for weaker students. We plan to further investigate this issue in a followup research. Second, our student simulation model is IRT based~\cite{hambleton1991fundamentals} with assumptions taken about the behaviour of skill progression. While these assumptions follow past work (see ~\cite{clement2013multi}), one can consider other skill progression assumptions as well as other models for student simulation (e.g. Performance Factor Analysis~\cite{pavlik2009performance}). We plan to extend the simulation code with other students' models.
Finally, we had a small number of students performing the post test in the field experiment and plan to run larger scale field trials in future work to verify significance.
\bibliographystyle{plain}
\section{Introduction}
As e-learning systems become more prevalent they are accessed by students of varied backgrounds, learning styles and needs. There is thus a growing need for them to accommodate individual difference between students and adapt to their changing pedagogical needs over time.
There are mainly two families of approaches for adapting educational content: offline approaches build models from data (e.g. ~\cite{segal2014edurank}) while online approaches balance exploration and exploitation (e.g ~\cite{clement2013multi}).
We provide a novel algorithm for sequencing content in e-learning systems that combines both of these approaches. It integrates offline learning from students' past interactions with online mechanisms for sequencing questions to students in order to maximize their learning gains.
This approach is based on the concept of \textit{zone of proximal development} \cite{vygotsky1987zone} where students are presented with challenges that are neither too easy nor too difficult, but are slightly beyond their current abilities.
Our algorithm, called MAPLE (Multi-Armed Bandits based Personalization for Learning Environments),
extends prior multi-armed bandits approaches in education by explicitly considering question difficulty when initializing the online behavior of the algorithm and when updating its behavior over time. MAPLE treats successful and failed question attempts differently and becomes \mydoubleq{conservative} (i.e. decreases exploration and increases weights of easier questions) when students fail under the (pedagogically guided) assumption that repeated errors are detrimental to students' learning~\cite{pekrun2016academic}.
We first evaluated MAPLE in a simulation environment comparing its performance to a variety of sequencing algorithms, including an approach that sequenced questions according to educational expert guidelines, an approach based only on personalized difficulty ranking, and a multi-armed bandit approach without personalized difficulty ranking initialization. MAPLE outperformed all other approaches for average and strong simulated students while showing the need for further tuning for weak simulated students.
We then implemented MAPLE in the wild in an existing e-learning system in a school with 7th grade students. MAPLE's performance was compared to two other sequencing algorithms already implemented in the e-learning system: an approach that sequenced questions according to educational expert guidelines and a state of the art Bayesian Knowledge Tracing based algorithm \cite{david2016sequencing}. We found that our proposed approach showed promising results compared to the existing educational expert approach and the BKT based approach. Moreover, students reported being more satisfied with the questions posed to them by MAPLE.
The contribution of this paper is two-fold. First, we present a novel algorithm for sequencing questions to students in e-learning that extends multi-armed bandits with personalized difficulty ranking information. Second, we show the potential of the algorithm in simulations and in field trials in the wild.
\section{Related Work}
Our work relates to past research on using historical data to sequence content to students, and to work on multi-armed bandits for online adaptation of educational content.
Several approaches within the educational artificial intelligence community have used computational methods for sequencing content to students.
Pardos and Heffernan~\cite{pardos2009determining} inferred order over question presented to students by predicting their skill levels using Bayesian Knowledge Tracing (BKT)~\cite{corbett1994knowledge}. They showed the efficacy of their approach on simulated data as well as on a test-set comprising random sequences of three questions. Ben David et al.~\cite{david2016sequencing} developed a BKT based sequencing algorithm. Their algorithm (which we refer to in this paper as YBKT) uses knowledge tracing to model students' skill acquisition over time and sequence questions to students based on their mastery level and predicted performance. It was shown to enhance student learning beyond sequencing designed by pedagogical experts.
Champaign et al.~\cite{champaign2010model} used a peer-based approach for content sequencing in an intelligent tutor system by computing similarities between students and choosing questions that provide best benefits for similar students, measured by similar average performance on past questions. Segal et al.~\cite{segal2014edurank} developed EduRank, a sequencing algorithm that combines collaborative filtering with social choice theory to produce personalized learning sequences for students. The algorithm constructs a “difficulty” ranking over questions for a target student by aggregating the ranking of similar students when sequencing educational content.
Multi-armed bandits provides a fundamental model for tackling the \mydoubleq{exploration-exploitation} trade off ~\cite{thompson1933likelihood,bubeck2012regret}. Xu et al. \cite{xu2016personalized} used bandits to identify which sequences of courses lead students to obtain maximal GPAs. Lan and Baraniuk~\cite{lan2016contextual} used sparse factor analysis with bandits to identify sequences of educational content that could maximize students performance on subsequent assessments. Lomas et al. \cite{lomas2016interface} showed how bandits can be used to search a large space of design decisions in creating educational games. Williams et al.~\cite{williams2016axis} used Thompson Sampling to identify highly rated explanations for how to solve Math problems, and chose priors that assumed that every explanation was equally rated.
Clement et al.~\cite{clement2013multi} used human experts' knowledge to initialize a multi-armed bandit algorithm called EXP4, that discovered which activities were at the right level to push students learning forward. In our work we do not rely on human experts, but rather use personalized difficulty rankings to guide the initial exploitation and update steps of our algorithm.
\section{Problem Formulation and Approach}
We consider an e-learning setting with a group of students $S$ and a set of practice questions $Q$. At each time step in the practice session, a computer agent $A$ needs to choose a question in $Q$ to present to the student. The agent sequencing problem requires choosing at each time step the next question to present to the student so as to maximize her learning gains over the length of the practice session. The goal is to present students with challenging problems while ensuring a high likelihood that they will be able to solve these problems.
Our approach to solving the problem, called MAPLE, maintains a belief distribution over the expected learning gains to the student for solving each of the questions in $Q$. This belief distribution is initialized with a personalized difficulty ranking over the questions in $Q$. The algorithm samples the next question to the student from this distribution and updates it at each time step given the student's performance on the question and its inferred difficulty to the student.
MAPLE implements an exploration policy
similar to the one that is used by the EXP4 algorithm~\cite{auer2002nonstochastic,clement2013multi}. This approach
maintains a belief distribution over questions that is proportional to how much learning gain each question is expected to provide. Weights are decreased or increased based on how difficult a question is and whether the student successfully solves the question or not. When a student successfully solves a question, weights are adjusted to make harder questions more likely to be presented, and explore a broader range of questions. When a student fails to solve a question, weights are adjusted to make easier questions more likely to be presented, and explore a narrower set of questions. The weights are initialized to reflect the inferred difficulty level for the student.
\begin{algorithm}[H]
\KwData {Set of students $S$.
\\Set of $N$ questions $Q$.
\\Passing grade threshold $\eta$, exploration rate $\gamma$.
\\Normalization factors $\alpha_1,\ldots,\alpha_4$.
\\For each student in $S$, a partial difficulty ranking $\succ_k$ over $T_k \subseteq Q$.
\\Target student $s_i \in S$.}
\KwResult {Next question $q_s \in Q$ to present to student $s_i$.}
{\bf{Step 1: Initialization:}}
\begin{enumerate}[topsep=0pt]
\item For each student in $S$:\\
- Compute difficulty ranking $\succ_l$ for all $q_l \in Q$\\
- $w_1 > \ldots > w_N=initialize\_weights(\succ_l)$\\
\end{enumerate}
{\bf{Step 2: Get Next Question:}}
\begin{enumerate}[topsep=0pt]
\setcounter{enumi}{1}
\item For each $j=1, \ldots , N$:\\
- Calculate new weight: $\pi_j = w_j(1-\gamma) + \epsilon_j\gamma,$ $\epsilon_j \sim U(-1,1)$ \\
- Normalize: $\pi_j = \frac{\pi_j}{\sum_{j=1}^{N}\pi_j}$\\
\item Choose question $q_s$ randomly, with respect to question weights $\pi_j$
\item return $q_s$
\end{enumerate}
{\bf{Step 3: Update Question Grade:}}
\begin{enumerate}[topsep=0pt]
\setcounter{enumi}{4}
\item Get student grade $g_s$ after solving question $q_s$
\item reward $R = g_s-\eta$
\item {\bf{if}} $g_s>\eta$ {\bf{then}}
\begin{itemize}
\item Increase weights for questions more difficult than $q_s$:
\\ $w_j = \alpha_1 e^R w_j,$ for $j = s+1,\ldots,N$
\item Increase exploration factor: $\gamma = \alpha_2 \gamma$
\end{itemize}
\item {\bf{else}}
\begin{itemize}
\item Decrease weights for questions more difficult then $q_s$:\\
$w_j = \alpha_3 e^R w_j,$ for $j = s+1,\ldots,N$
\item Decrease exploration factor: $\gamma = \alpha_4\gamma$
\end{itemize}
{\bf{end if}}
\item For each $j=1, \ldots , N$:\\
- Normalize: $w_j = \frac{w_j}{\sum_{j=1}^{N} w_j}$\\
\end{enumerate}
\caption{The Maple Approach}
\label{alg:MAPLE}
\end{algorithm}
Algorithm~\ref{alg:MAPLE} formalizes the MAPLE approach.
The input to the algorithm is a set of students $S$ each with known solutions over a set of questions in $Q$, a target student $s_i$ and a set of initialization parameters which include $\eta$, a passing grade threshold, $\gamma$, the exploration factor, and $\alpha_1,\ldots,\alpha_4$, normalization factors. The algorithm returns the next question $q_s$ to present to student $s_i$.
The algorithm includes 3 steps: (1) The initialization step is performed once at the beginning of execution to obtain a personalized difficulty ranking for each student. During initialization the algorithm computes a difficulty ranking over questions per student. We used the EduRank approach for this purpose and describe it in the next section. Next, the question weights $w_j$ are initialized with values using a softmax function, with higher weights corresponding to easier questions per the difficulty ranking. As students solve questions and succeed or fail, these weights are updated for each student.
(2) Next Question Selection: performed at each time step by the agent to choose the next question to present to the student.
For this step MAPLE uses the distribution weights computed for the student. The algorithm first adds an exploration component per weight $w_j$ (line 2) generating new weights $\pi_j$, and then chooses the next question with respect to the $\pi_j$ weights (lines 3,4).
(3) Update Question Grade: performed at each time step after the question was presented to the student and her solution grade was obtained.
In line 6, a reward value $R$ is computed to reflect the magnitude of student's success or failure in the question with respect the the threshold value $\eta$. In line 7, the case of a successful solution is treated ($grade>\eta$): in this case, the exploration factor is increased, and weights of harder questions for this student are also increased proportionally to the reward value $R$. Thus, the probability of the student to get harder questions on next attempts increases, as well as the algorithm willingness to \mydoubleq{take risks} (exploration factor).
On the final step, 9, weights are re-normalized on the unit scale.
To summarize the update step, MAPLE weights update reflects the change in the algorithm's estimation as to the suitability of each question for the student based on expert guidelines. Nonetheless, the stochastic nature of the algorithm enables exploration of additional sequencing alternatives. We now move to describe our simulation and field trial results.
\section{Simulations with Synthesized Data}
We performed a set of simulations to compare four different sequencing algorithms:
(1) The \emph{MAPLE} approach which used the EduRank~\cite{segal2014edurank} algorithm for difficulty ranking. EduRank considers the history of students' actions in the system (including their grades and retries) and uses collaborative filtering~\cite{sarwar2001item} and voting aggregation approaches~\cite{nurmi1983voting} to compute a personalized difficulty ranking over questions. MAPLE's parameters for simulations were set empirically except for $\eta$ (the passing grade threshold) which was determined by an educational expert.
(2) The \emph{Ascending} approach sequenced questions according to an absolute difficulty ranking that was determined by pedagogical experts. Questions were labeled into one of 5 groups (from easy to hard) according to difficulty level. The algorithm selects questions in the following temporal order: The first 10\% of questions presented to students are level 1 questions (easiest level), followed by 20\% questions from level 2, 30\% questions from level 3, 30\% questions from level 4 and 10\% questions from level 5 (hardest level). This is the main sequencing approach used to sequence questions to students in the e-learning system tested in a school in the next section.
(3) The \emph{EduRank} approach provided a personalized difficulty ranking over questions for each student. Questions were sequenced from EduRank's easiest estimated question to its hardest estimated question per student.
(4) The \emph{Naive Maple} approach
sequenced questions using the multi-armed bandit algorithm with random weights initialization (without the EduRank based difficulty ranking component).
The algorithms were evaluated by comparing their performance in simulation. We model questions in the simulation using a $\langle$skill,difficulty$\rangle$ pair; students are modeled as a vector of skill values for each question type.
We now describe the details of the student model during simulation.
The probability of a student to successfully solve a question is based on Item Response Theory~\cite{hambleton1991fundamentals} and is as follows:
\begin{equation}
p(success) = \frac{1}{1+e^{\theta\cdot(ql - sl)}}
\label{eq:1}
\end{equation}
where $\theta$ is a constant influencing the shape of the probability distribution. The student's probability of learning is based on the difference between her skill level $sl$ and the level of the question $ql$.
To reflect the fact that students may differ in their answers due to factors beyond their skill level (e.g. guessing and slipping, affect condition) we add a stochastic component to Equation \ref{eq:1}. Thus, our student model behaves according to:
\begin{equation}
p(success) = \beta\cdot\frac{1}{1+e^{\theta\cdot(ql - sl)}}+(1-\beta) \cdot \epsilon_u
\label{eq:2}
\end{equation}
Where $\beta$ is a constant controlling the impact of the stochastic component and $\epsilon_u$ is drawn from a uniform distribution in the range [0,1].
Lastly we describe how the student skill level is updated upon solving a question. Intuitively, the skill level improves significantly following a correct response to a hard question when the student had a low level in that skill before answering the question. Similarly, the skill estimation was probably most exaggerated when the student failed in solving an easy question while having a high skill level before answering the question.
This leads to the following skill update rules: If the student is successful in solving the question, i.e. her grade is greater than predefined value $\eta$, her skill level is updated according to:
\begin{equation}
sl = sl+\delta_1\cdot ql \cdot(1-sl)
\label{eq:3}
\end{equation}
Otherwise, her skill level is updated according to:
\begin{equation}
sl = sl-\delta_2\cdot (1-ql)\cdot sl
\label{eq:4}
\end{equation}
The parameters $\theta, \beta, \delta_1, \delta_2$ are determined empirically. The parameter $\eta$ is determined by an educational expert and was set to 0.7.
In the simulations each algorithm was run with a group of 1000 students, and was required to sequence 200 questions for each student. Each question belonged to one of 10 possible skills, uniformly distributed. We generated questions according to an absolute difficulty level, uniformly distributed between 1 (easiest), and 5 (hardest). Students' initial competency level in each skill were also uniformly distributed between 0 (no skill knowledge) and 1 (full knowledge of skill). All the sequencing algorithms had access to \mydoubleq{historical} data generated by the simulation engine in a pre-simulation step so they can build their internal models. This data contained 1000 students each solving 500 randomly selected questions.
We now describe key simulation results.
We start by looking at MAPLE's sequencing behaviour. Figure~\ref{fig:seqalgs} shows how MAPLE adapted the question difficulty as time progressed. The x-axis presents the time and the y-axis presents the number of questions of a given difficulty level that were presented to students in that time frame. The colors represent five difficulty levels ranging from easy (level 1) to hard (level 5). We can see that MAPLE started with offering easy questions to most students and then moved to propose more difficult questions while at the same time continuing to propose easy questions for a long time. This behaviour demonstrates the adaptive nature of MAPLE and was not observed in the other conditions tested. Specifically, EduRank used mostly easy questions and was not able to adapt harder questions to students while Naive Maple did not exhibit adaptive behavior for the questions in the practice set. (For longer practice sets the algorithm may show adaptive behavior).
\begin{figure}[H]
\includegraphics[width=1.2\textwidth,center]{colors_hist_maple_all.png}
\caption{MAPLE questions sequencing}
\label{fig:seqalgs}
\end{figure}
To better understand MAPLE's adaptation behavior we next look at 3 different student groups: weak students (with initial skill competency levels under $0.33$), average students (with initial skill competency levels in the range [0.33, 0.67]), and strong students (with initial skill competency levels above $0.67$).
Figure~\ref{fig:simseq} shows MAPLE's behavior for the 3 types of students and the 5 types of questions. We can see that for weak students, MAPLE kept proposing easy questions for a long time and refrained from proposing hard questions. For strong students, MAPLE begun proposing more difficulty questions earlier while reaching the hardest questions during the 200 questions session. And for average students, MAPLE took a middle way approach focusing on questions with average difficulty level while challenging some students with more difficult questions.
\begin{figure}[H]%
\centering
\subfigure[Weak Students]{{\includegraphics[width=10cm]{colors_hist_maple_weak.png} }}%
\qquad
\subfigure[Average Students]{{\includegraphics[width=10cm]{colors_hist_maple_avg.png} }}%
\qquad
\subfigure[Strong Students]{{\includegraphics[width=10cm]{colors_hist_maple_strng.png} }}%
\caption{MAPLE's question sequencing for different student types.}%
\label{fig:simseq}%
\end{figure}
Finally, we look at the skill level progression during simulation for the 3 student types in the 4 algorithms tested.
Figure~\ref{fig:skillprog} shows the skill level progression of the different student types at each simulation step, for each algorithm. We can see that for strong and average students MAPLE outperformed the other three algorithms throughout the practice session. Both EduRank which relies only on the difficulty information and Naive Maple which relies only on the multi-armed bandit approach presented lower results. For weak students we can see that the Ascending and Naive Maple approaches failed altogether, probably since they offered questions that are too hard for these students. Both MAPLE and EduRank presented initial good progress but then experienced a decline in the estimated skill level. This implies that MAPLE's adaptation scheme needs to be improved for this segment of students to offer less challenging questions.
\begin{figure}
\centering
\includegraphics[width=1.4\textwidth,center]{median_vs_time.png}
\caption{Skill level progression per algorithm and student type.}
\label{fig:skillprog}
\end{figure}
\section{Deployment and Evaluation in the Classroom}
We next moved to conduct a field study in the wild where students used different sequencing approaches as part of their curriculum work in class (and not in a laboratory setting).
The MAPLE algorithm was implemented in an e-learning system used for Math education. In this system, K-12 students practice solving Math questions in various skill areas matching their curriculum studies. The study compared MAPLE in one school with 7th grade students, to two other existing sequencing algorithms already available in the e-learning system. The experiment was conducted between May 9th 2017 and June 19th 2017 (end of school year).
The students were randomly divided into 3 cohorts: (1) MAPLE Sequencing: Students in this group received questioned sequenced by the MAPLE algorithm when practicing with the system. (2) YBKT Sequencing\footnote{We note that the YBKT code was not available to us for the simulations.}: students in this group received questioned sequenced by the Bayesian Knowledge Trace based algorithm proposed by Ben David et al. \cite{david2016sequencing}. This sequencing approach always chooses the next question from the available skill set in a deterministic manner. (3) Ascending Sequencing: students in this group received questions sequenced by the Ascending algorithm described earlier.
All students in the experiment were initially exposed to a pretest session. In this session they solved 10 pretest questions that were hand picked by a pedagogical expert. The hand picked questions matched the expected level of 7th graders at that stage in the academic year. Ninety two students solved the pretest questions and there was no statistically significant difference between the three groups in the average score on this preliminary test. We thus concluded that the students in each group exhibited similar knowledge baselines of the material at hand.
The students then engaged in multiple practice sessions in the e-learning system for the next 35 days, solving 10 assignment questions at each such practice session. For each cohort, assignment questions were sequenced by the cohort's respective algorithm (i.e. MAPLE, YBKT or Ascending). At the end of this period, students were asked to complete a post test session, solving the same questions (in the same order) as in the pretest session. Twenty eight students completed the post test session. We attribute the decrease in students' response from pretest to post test to the pending end of the academic year (there was no difference in the dropout rates across the 3 cohorts).
\begin{table}[H]
\centering
\scalebox{1.1}{
\begin{tabular}{|l|c|c|}
\hline
Cohort & Time per Question (sec) & Average Grade \\
\hline
Ascending & 6.49 & 43.76 \\
MAPLE & 10.69 & 71.28 \\
YBKT & 12.86 & 67.08 \\
\hline
\end{tabular}
}
\caption{Post test results per cohort: time per question and average grade.}
\label{tab:posttest}
\end{table}
Breaking up results by the sequencing algorithms, Table~\ref{tab:posttest} shows students' average grade and time spent on post-test questions, while Table ~\ref{tab:pre2post} shows the pre- to post- test change. As measured by post test grades, students assigned to the MAPLE condition achieved higher post test results than students assigned to the Ascending condition or to the YBKT condition.
This effect is also evident when observing the pre- to post- test change in grades. Students assigned to the MAPLE condition learned more than students assigned to the Ascending condition or to the YBKT condition. We note that further field trials with larger student groups are needed to evaluate statistical significance.
In addition to objective measures of learning, we examined students' satisfaction from interacting with the various sequencing approaches. Students rated their degree of agreement on a 3 point scale ((1) I do not agree (2) I partially agree (3) I strongly agree) with ten statements about issues like the system helpfulness, ease of use, adaptivity, match to class material, and future appeal.
Students using the MAPLE sequencing algorithm demonstrated higher satisfaction as represented by their answers to the subjective experience questions. The average score on the agreement with positive characteristics of the system was significantly different across the three conditions, with the highest score for the MAPLE algorithm at 2.39, compared to the Ascending algorithm (2.23) and the YBKT algorithm (2.20) (one way Anova, $p<0.05$).
\begin{table}[H]
\centering
\scalebox{1.1}{
\begin{tabular}{|l|c|c|}
\hline
Cohort & Time per Question Diff & Average Grade Diff \\
\hline
Ascending & -9.2 & -2.83 \\
MAPLE & -9.12 & 4.99 \\
YBKT & -8.65 & 1.26 \\
\hline
\end{tabular}
}
\caption{Change from pre-test to post-test: time per question and average grade.}
\label{tab:pre2post}
\end{table}
\section{Conclusion}
We have provided a new computational method called MAPLE for sequencing questions to students in on-line educational systems. MAPLE combines difficulty ranking based on past student experiences with a multi-armed bandit approach using these difficulty rankings in on-line settings.
We tested our approach in simulations and verified its adaptive nature and its learning gain results compared to other algorithms as measured by the simulated skills' improvement.
We then performed a live experiment in the wild running MAPLE in a classroom in parallel to two baseline algorithms. We found that our proposed approach presented promising results in students' performance and satisfaction.
We mention several limitations of our work and subsequent suggestions for future work. First, the simulation results showed that MAPLE's performance needs to be fine tuned for weaker students. We plan to further investigate this issue in a followup research. Second, our student simulation model is IRT based~\cite{hambleton1991fundamentals} with assumptions taken about the behaviour of skill progression. While these assumptions follow past work (see ~\cite{clement2013multi}), one can consider other skill progression assumptions as well as other models for student simulation (e.g. Performance Factor Analysis~\cite{pavlik2009performance}). We plan to extend the simulation code with other students' models.
Finally, we had a small number of students performing the post test in the field experiment and plan to run larger scale field trials in future work to verify significance.
\bibliographystyle{plain}
|
{
"timestamp": "2018-04-17T02:06:12",
"yymm": "1804",
"arxiv_id": "1804.05212",
"language": "en",
"url": "https://arxiv.org/abs/1804.05212"
}
|
\section{Introduction}
Realization spaces of matroids are well studied objects \cite{BLSWZ, weird_bernd_book, mnev} which encode not only whether or not the matroid is realizable, but also carry additional information about the structure of the matroid. A realization (or representation) of a rank $d+1$ matroid $M$ is a set of vectors in $\mathbbm{k}^{d+1}$ which captures its independence structure. Roughly speaking, a realization space is the set of all such choices of vectors.
Fundamental questions in the study of realization spaces of matroids include discovering whether or not a given matroid is realizable, determining over which field it is realizable, finding the structure of the set of realizations, and characterizing when realizations exist.
A celebrated theorem of Mn\"ev states that every semialgebraic set defined over the integers is stably equivalent to the realization space of some oriented matroid. That is, realization spaces of matroids can become arbitrarily complicated. In light of this, we aim to connect the combinatorics of the matroid to properties of its realization space.
We generalize a construction of \cite{slack_paper} in which they describe a model for the realization space of a polytope using the {\em slack matrix} of the polytope. This model gave a new framework for answering questions about the realizability of polytopes. We extend these results to the setting of matroids, creating the beginnings of a dictionary between the combinatorial properties of the matroid and the algebraic description of its realization space.
In Section~\ref{SEC:bg} we introduce the main objects of study, as well as preliminary results and notation. In Section~\ref{SEC:realsp} we discuss two models for the realization space of a matroid. One of our main theorems, Theorem~\ref{THM:mothervariety}, shows how the two realization space models can be described via a single overarching variety. In Section~\ref{SEC:nonreal} we show how the slack realization model can be used to determine whether a matroid has a realization over a certain field. We also reframe the tools of final polynomials \cite{weird_bernd_book} in terms of slack ideals, and show how they can be used to improve computational efficiency of realizability checking. In Section~\ref{SEC:toric} we introduce a toric ideal associated to a matroid and study its relationship to the projective uniqueness of the matroid. In Appendix \ref{AP:notation} we include a table of notation used throughout the paper.
The computations in this paper are done in \texttt{Macaulay2} \cite{M2} with the help of the \texttt{Matroids} package \cite{M2matroids}; the code we used can be found at \href{http://sites.math.washington.edu/~awiebe}{\texttt{http://sites.math.washington.edu/$\sim$awiebe}}.
\section{The slack matrix of a matroid}
\label{SEC:bg}
Much of this section is analogous to \cite[\S2]{slack_paper} to which we refer the reader for further details and excluded proofs.
We assume the reader has familiarity with the basic definitions from matroid theory, see \cite{Oxley} or \cite{GM}. Throughout the paper, we assume all matroids are simple (having no loops or parallel elements).
Let $\mathbbm{k}$ be a field. Let $M = (E,\mathcal{B})$ be a matroid of rank $d+1$ with ground set $E = \{\mathbf{v}_1,\ldots, \mathbf{v}_n\}$, where each $\mathbf{v}_i\in\mathbbm{k}^{d+1}$ and $\mathcal{B}$ is its set of bases. If $V$ is the matrix with columns $\mathbf{v}_1,\ldots, \mathbf{v}_n$, then the independent sets of $M$ are the linearly independent subsets of columns, and we write $M = M[V]$.
Let $\mathcal{H}(M)$ denote the set of hyperplanes of $M$, which are the closed subsets (flats) of rank $d$. In $M[V]$, each hyperplane $H\in\mathcal{H}(M)$ corresponds to a linear subspace of $\mathbbm{k}^{d+1}$, so is determined by some linear equation; that is,
$H = \{\mathbf{x}\in E: \alpha_H^\top\mathbf{x} = 0\}$.
For $\mathcal{H}(M) = \{H_1,\ldots, H_h\}$, let $W$ be the matrix whose columns are the hyperplane defining normals $\alpha_1, \ldots, \alpha_h$, or some multiple, $\lambda_j\alpha_j$ for $\lambda_j\in\mathbbm{k}^*$, thereof.
\begin{definition}
The {\em slack matrix} of the matroid $M = M[V]$ over $\mathbbm{k}$ is the $n\times h$ matrix $S_{M[V]} = V^\top W.$
\label{DEFN:slackmatrix}
\end{definition}
We wish to parametrize the set of realizations of a matroid by its slack matrices. So, we must determine the characteristics which define the set of all possible slack matrices of a given matroid. We begin by considering the rank of a slack matrix.
\begin{lemma} If $S$ is a matrix having the same support as the slack matrix of some rank $d+1$ matroid $M = M[V]$, then $\rk(S)\geq d+1$. (See \cite[Lemma~3.1]{slack_paper}.) \label{LEM:trianglesubmat} \end{lemma}
\begin{corollary} If $M=M[V]$ is a rank $d+1$ matroid then $\rk(S_M) = d+1$. \label{COR:rank} \end{corollary}
\begin{proof} The factored form of $S_M \in \mathbbm{k}^{n\times (d+1)}\times\mathbbm{k}^{(d+1)\times h}$ implies that $\rk(S_M)\leq d+1$. The result then follows from Lemma~\ref{LEM:trianglesubmat}.
\end{proof}
Now, let $M = (E,\mathcal{B})$ be an abstract matroid of rank $d+1$.
Unless otherwise stated, we take $E = [n]=\{1,\ldots, n\}$.
A {\em realization} of $M$ over $\mathbbm{k}$ is a collection of vectors $V = \{\mathbf{v}_1,\ldots, \mathbf{v}_n\} \subset \mathbbm{k}^\ell$ such that the independent subsets of $V$ are indexed by the independent sets of the matroid, so $M=M[V]$.
A matroid with a realization is called {\em realizable} (also {\em representable, linear} or {\em coordinatizable}).
\begin{lemma} The rows of a slack matrix $S_M$ form a realization of the matroid $M$. \label{LEM:rowrealiz} \end{lemma}
\begin{proof} It suffices to show that if we label the rows of $S_M$ with $[n]$, the subsets indexing linearly independent rows of $S_M$ are the independent sets of $M$.
Since $S_M = V^TW$, if a subset $J$ of $E$ is dependent, then there exists a vector $\beta\in\mathbbm{k}^n$ with support indexed by $J$ such that $V\beta = 0$. But now, $\beta^\top S_M = (V\beta)^\top W = 0$, so $J$ also indexes a dependent subset of the rows of $S_M$.
Conversely, suppose $J$ indexes a dependent subset of the rows of $S_M$. Then for some $\beta\in\mathbbm{k}^n$ with support indexed by $J$, we have
$0 = \beta^\top S_M = (V\beta)^\top W$. Since $W$ has full rank by Corollary~\ref{COR:rank}, it must be the case that $V\beta = 0$, so that $J$ also indexes a dependent set of $M$. \end{proof}
From now on we assume that realizations come with a fixed labelling of ground set elements and hyperplanes, so that two slack matrices of the same matroid cannot differ by permutations of rows and columns. This also allows us to identify hyperplanes of a realization by vectors or the indices of those vectors.
Now, we characterize the set of matrices which correspond to slack matrices of a matroid $M$.
\begin{theorem} Let $M$ be a rank $d+1$ matroid with $n$ elements and hyperplanes $\mathcal{H}(M) = \{H_1, \ldots, H_h\}$. A matrix $S \in \mathbbm{k}^{n \times h}$ is the slack matrix of some realization of~$M$ if and only if both of the following hold:
\begin{enumerate}[label=(\roman{enumi})]
\hspace{0.7 in}
\begin{minipage}{0.3 \textwidth}
\item{$\text{supp}(S) = \text{supp}(S_M)$}
\end{minipage}
\hspace{0.5 in}
\begin{minipage}{0.3 \textwidth}
\item{$\rk(S) = d+1$.}
\end{minipage}
\end{enumerate}
\label{THM:slackconditions}
\end{theorem}
\begin{proof}
Suppose $S$ is the slack matrix of some realization of $M$. Then (i) holds trivially, and (ii) holds by Corollary~\ref{COR:rank}.
Conversely, suppose $S$ is a matrix satisfying (i) and (ii). By (ii), $S$ has some rank factorization $S=AB$, where $A\in\mathbbm{k}^{n\times(d+1)}$ and $B\in\mathbbm{k}^{(d+1)\times h}$.
Let $\mathbf{a}_1,\ldots, \mathbf{a}_n \in \mathbbm{k}^{d+1}$ be the rows of $A$ and $\mathbf{b}_1,\ldots, \mathbf{b}_h \in \mathbbm{k}^{d+1}$ be the columns of $B$. Then we claim that the rows of $A$ give a realization of $M$; that is $M = M[A^\top]$.
To see this, we show that the hyperplanes of $M$ are also hyperplanes of $M[A^\top]$, and that $M[A^\top]$ can not contain more hyperplanes.
By (i), for each hyperplane $H_j$ of $M$, there is a column of $S$ with zeros in positions indexed by elements of $H_j$. Since $S= AB$, we have $\mathbf{b}_{j}^\top \mathbf{a}_i = 0$ if and only if $i\in H_j$. Thus
$$\{\mathbf{x}\in\mathbbm{k}^{d+1} : \mathbf{b}_j^\top \mathbf{x} = 0\}\cap \{\text{rows}(A)\} = \{\mathbf{a}_i\}_{i\in H_j}$$
so that $H_j$ is also a hyperplane of the matroid $M[A^\top]$.
Now suppose $M[A^\top]$ has an extra hyperplane $H \not \in \mathcal{H}(M)$. Let $\{i_1, \ldots, i_d\} \subset H$ be any $d$ distinct independent elements of $H$. Since $i_1,\ldots, i_d$ are also elements of matroid $M$, the flat $\overline{\{i_1, \ldots, i_d\}}$ is a hyperplane $H'$ of $M$, and thus also a hyperplane of $M[A^\top]$, but $H' \neq H$. However, this means that $\{i_1, \ldots, i_d\}$ are contained in two distinct hyperplanes of $M[A^\top]$, which is not possible, so we arrive at a contradiction.
\end{proof}
We now recall two equivalence relations on the set of realizations of a matroid $M$, and illustrate how these equivalences are reflected in slack matrices. For $A\in GL(\mathbbm{k}^{d+1})$, it is easy to check that $V$ and $AV$ define the same matroid. We call these realizations {\em linearly equivalent}.
If $P\in\mathbbm{k}^{n\times n}$ is a permutation matrix which sends $i\mapsto \sigma(i)$, then $V$ and $VP$ define the same matroid up to relabelling the ground set $E=[n]$ with $\sigma(1),\ldots, \sigma(n)$. Thus if $A\in GL(\mathbbm{k}^{d+1})$ and $B$ is a permutation matrix with any element of $\mathbbm{k}^*$ in the $1$'s positions, then $V, AVB$ define the same matroid. We call the realizations $V, AVB$ {\em projectively equivalent}. Call a matroid $M$ {\em projectively unique} (over $\mathbbm{k}$) if all realizations are projectively equivalent.
\begin{lemma} Two realizations of a matroid $M$ are projectively equivalent if and only if their slack matrices are the same up to row and column scaling.
\label{LEM:pe} \end{lemma}
\begin{proof} Suppose we have projectively equivalent representations $U,V$ of $M$. Then $U = AVB$, where $A\in GL(\mathbbm{k}^{d+1})$ and without loss of generality $B$ is an invertible $n\times n$ diagonal matrix (since we have assumed a fixed labelling of our matroid)
If $H=\{\mathbf{v}_{i_1},\ldots, \mathbf{v}_{i_k}\}$ be a hyperplane of $M[V]$,
then $H' = \{\mathbf{u}_{i_1},\ldots, \mathbf{u}_{i_k}\}$ is a hyperplane of $M[U]$. Furthermore, if $\alpha_H\in\mathbbm{k}^{d+1}$ is normal to $H$, then since {${\mathbf{u}_i = A\mathbf{v}_i\cdot b_i}$}, $A^{-\top}\alpha_H$ is normal to $H'$,
so that a slack matrix of $M[U]$ is
$$
S_{M[U]} = U^\top \begin{bmatrix} A^{-\top}\alpha_H \end{bmatrix}_{H\in\mathcal{H}}
= B^\top V^\top A^\top\begin{bmatrix} A^{-\top}\alpha_H \end{bmatrix}_{H\in\mathcal{H}}
= B^\top V^\top W
= B^\top S_{M[V]}.
$$
Since we can always scale columns of a slack matrix, this completes the proof.
Conversely, suppose we have realizations $U$ and $V$ of the matroid $M$ such that
${S_{M[U]} = D_n S_{M[V]} D_h}$ for invertible diagonal matrices $D_n\in\mathbbm{k}^{n\times n},D_h\in\mathbbm{k}^{h\times h}$.
By definition, $S_{M[U]} = U^\top W$ and $S_{M[V]} = V^\top W'$. Multiplying both sides of the above equation on the right by $W^\top(WW^\top)^{-1}$, we find
$$
U^\top = D_n V^\top W' D_h W^\top(WW^\top)^{-1}.
$$
We see that $W' D_h W^\top(WW^\top)^{-1}$ is a $(d+1) \times (d+1)$ invertible matrix, which makes $U$ and $V$ projectively equivalent, as desired. \end{proof}
By taking $B,D_n$ each to be the $n\times n$ identity matrix in the above proof, we recover the following lemma.
\begin{lemma}
\label{LEM:columnscale}
Two realizations of a matroid $M$ are linearly equivalent if and only if their slack matrices are the same up to column scaling.
\end{lemma}
We now define an analog of the slack matrix which can be constructed for any abstract matroid, even ones which are not realizable, as follows.
\begin{definition} \label{DEFN:symbslack}
Define the {\em symbolic slack matrix} of matroid $M$ to be the matrix $S_M(\mathbf{x})$ with rows indexed by elements $i\in E$, columns indexed by hyperplanes $H_j\in \mathcal{H}(M)$ and $(i,j)$-entry
$$S_M(\mathbf{x})_{ij} = \begin{cases} x_{ij} &\text{ if } i\notin H_j \\ 0 &\text{ if } i\in H_j. \end{cases}$$
The {\em slack ideal} of $M$ is the saturation of the ideal generated by the $(d+2)$-minors of $S_M(\mathbf{x})$, namely
\begin{align*}
I_M :&=\Big\langle (d+2)-\text{minors of }S_M(\mathbf{x})\Big\rangle { :} \left(\prod_{i=1}^n\prod_{j:i\not \in H_j} x_{ij}\right)^\infty
&\subset \mathbbm{k}[\mathbf{x}].
\end{align*}
Suppose there are $t$ variables in $S_M(\mathbf{x})$. The {\em slack variety} is the variety $\mathcal{V}(I_M)\subset \mathbbm{k}^t$. The saturation of $I_M$ by the product of all the variables guarantees that there are no components of $\mathcal{V}(I_M)$ that live entirely in coordinate hyperplanes. If $\mathbf{s}\in\mathbbm{k}^t$ is a zero of $I_M$, then we identify it with the matrix $S_M(\mathbf{s})$.
\end{definition}
\begin{example}
\label{EG:four_lines}
Consider the rank 3 matroid $M_4 = M[V]$ for $V$ whose columns are
$\mathbf{v}_1 = (-2,-2,1)^\top$,
$\mathbf{v}_2 = (-1,1,1)^\top$,
$\mathbf{v}_3 = (0,4,1)^\top$,
$\mathbf{v}_4 = (2,-2,1)^\top$,
$\mathbf{v}_5 = (1,1,1)^\top$,
$\mathbf{v}_6 = (0,0,1)^\top$.
Projecting onto the plane $z=1$, this can be visualized as the points of intersection of four lines in the plane, as in Figure \ref{FIG:four_lines}.
\begin{figure}
\begin{center}
\begin{minipage}{0.25 \textwidth}
\includegraphics[height = 2 in]{four_lines_ex}
\end{minipage}
\begin{minipage}{0.6 \textwidth}
\
\begin{blockarray}{cccccccc}
&H_1 & H_2& H_3 & H_4 & H_5 & H_6 & H_7 \\
&123 & 246 & 345 & 156 & 25 & 14 & 36\\
\begin{block}{c[ccccccc]}
1 & 0 & x_{12} & x_{13} & 0 & x_{15} &0 & x_{17} \\
2 & 0 & 0 & x_{23} & x_{24} &0 &x_{26} & x_{27} \\
3 & 0 & x_{32} & 0 & x_{34} &x_{35} & x_{36} &0 \\
4 & x_{41} & 0 & 0 & x_{44} &x_{45} & 0& x_{47} \\
5 & x_{51} & x_{52} & 0 & 0 & 0& x_{56} & x_{57} \\
6 & x_{61} & 0 & x_{63} & 0 &x_{65} & x_{66} & 0\\
\end{block}
\end{blockarray}
\]
\end{minipage}
\caption{The point-line configuration of Example \ref{EG:four_lines}, and its symbolic slack matrix.
\label{FIG:four_lines}
\end{center}
\end{figure}
A slack matrix for this realization is then
\fontsize{10.5}{12}\selectfont
\begin{align*}
S_{M_4} &=
\begin{blockarray}{ccc}
\begin{block}{@{\;\;}[*{3}{@{\;}c@{\;}}]}
-2 & -2 & 1 \\
-1 & 1 & 1 \\
0 & 4 & 1 \\
2 & -2 & 1 \\
1 & 1 & 1 \\
0 & 0 & 1\\
\end{block}
\end{blockarray}\;\;
\begin{blockarray}{*{7}{@{\,}c@{\,}}}
H_1 & H_2& H_3 & H_4 & H_5 & H_6 & H_7 \\
\scriptstyle{123} & \scriptstyle{246} & \scriptstyle{345} & \scriptstyle{156} & \scriptstyle{25} & \scriptstyle{14} & \scriptstyle{36}\\
\begin{block}{[*{7}{@{\,}c@{\,}}]}
-3 & 3 & 6& -3 & 0 & 0 & 4\\
1 & 3 & 2 & 3 & 2 & 4 & 0 \\
-4 & 0 & -8 & 0 & -2 & 8 & 0\\
\end{block}
\end{blockarray}
&=
\begin{blockarray}{c*{7}{@{\,}c@{\,}}}
&H_1 & H_2& H_3 & H_4 & H_5 & H_6 & H_7 \\
& \scriptstyle{123} & \scriptstyle{246} & \scriptstyle{345} & \scriptstyle{156} & \scriptstyle{25} & \scriptstyle{14} & \scriptstyle{36}\\
\begin{block}{c@{\;\;}[*{7}{@{\,}c@{\,}}]@{\;}}
1& 0 & -12 & -24 & 0 & -6 & 0 & -8 \\
2& 0 & 0 & -12 & 6 & 0 & 12 & -4 \\
3& 0 & 12 & 0 & 12 & 6 & 24 & 0 \\
4& -12 & 0 & 0 & -12 & -6 & 0 & 8 \\
5& -6 & 6 & 0 & 0 & 0 & 12 & 4 \\
6& -4 & 0 & -8 & 0 & -2 & 8 & 0 \\
\end{block}
\end{blockarray},
\end{align*}
}
where using $\{\mathbf{v}_{j_1}, \ldots, \mathbf{v}_{j_d}\}\subset H$ independent, we calculate each $\alpha_H$ as
\begin{equation}\label{EQ:pluckerdet}
\text{det}
\begin{bmatrix}
\widehat{e_1}& | & \cdots & |\\
\vdots & \mathbf{v}_{j_1} & \cdots & \mathbf{v}_{j_d}\\
\widehat{e_{d+1}} & | & \cdots & |\\
\end{bmatrix}.
\end{equation}
The symbolic slack matrix of $M_4$ is in Figure \ref{FIG:four_lines}.
We take the ideal of $4$-minors of this matrix, and saturate with respect to the product of all of the variables to get the slack ideal $I_{M_4}$. This has codimension~12, degree~293 and is generated by the 72~binomial generators in Table \ref{TAB:72things}. In Section \ref{SEC:toric} we will see these correspond to the 72 cycles in the bipartite non-incidence graph of this configuration (Figure \ref{FIG:fourlinesgraph}).
\begin{table}
\begin{center}
\footnotesize
\begin{tabular}{| l | l |}
\hline
$\!\!\deg 2\!\!$ & $x_{36}x_{65}+x_{35}x_{66},\ x_{26}x_{63}-x_{23}x_{66},\ x_{15}x_{63}-x_{13}x_{65},\ x_{56}x_{61}-x_{51}x_{66},\
x_{45}x_{61}-x_{41}x_{65},$\\
& $x_{27}x_{56}+x_{26}x_{57},\ x_{36}x_{52}-x_{32}x_{56},\ x_{17}x_{52}-x_{12}x_{57},\ x_{47}x_{51}-x_{41}x_{57},\ x_{17}x_{45}+x_{15}x_{47},$\\
& $x_{35}x_{44}-x_{34}x_{45},\ x_{27}x_{44}-x_{24}x_{47},\ x_{26}x_{34}-x_{24}x_{36},\ x_{15}x_{32}-x_{12}x_{35},\ x_{17}x_{23}-x_{13}x_{27}$\\
\hline
$\!\!\deg 3\!\!$ & $x_{47}x_{56}x_{65}-x_{45}x_{57}x_{66},\ x_{17}x_{56}x_{65}+x_{15}x_{57}x_{66},\ x_{12}x_{56}x_{65}+x_{15}x_{52}x_{66},\ x_{26}x_{47}x_{65}+x_{27}x_{45}x_{66},$\\
&$x_{26}x_{44}x_{65}+x_{24}x_{45}x_{66},\ x_{17}x_{26}x_{65}-x_{15}x_{27}x_{66},\ x_{17}x_{56}x_{63}+x_{13}x_{57}x_{66},\ x_{12}x_{56}x_{63}+x_{13}x_{52}x_{66},$\\
&$x_{27}x_{45}x_{63}+x_{23}x_{47}x_{65},\ x_{24}x_{45}x_{63}+x_{23}x_{44}x_{65},\ x_{12}x_{36}x_{63}+x_{13}x_{32}x_{66},\ x_{24}x_{35}x_{63}+x_{23}x_{34}x_{65},$\\
&$x_{23}x_{57}x_{61}+x_{27}x_{51}x_{63},\ x_{15}x_{57}x_{61}+x_{17}x_{51}x_{65},\ x_{13}x_{57}x_{61}+x_{17}x_{51}x_{63},\ x_{35}x_{52}x_{61}+x_{32}x_{51}x_{65},$\\
&$x_{15}x_{52}x_{61}+x_{12}x_{51}x_{65},\ x_{13}x_{52}x_{61}+x_{12}x_{51}x_{63},\ x_{26}x_{47}x_{61}+x_{27}x_{41}x_{66},\ x_{23}x_{47}x_{61}+x_{27}x_{41}x_{63},$\\
&$x_{13}x_{47}x_{61}+x_{17}x_{41}x_{63},\ x_{36}x_{44}x_{61}+x_{34}x_{41}x_{66},\ x_{26}x_{44}x_{61}+x_{24}x_{41}x_{66},\ x_{23}x_{44}x_{61}+x_{24}x_{41}x_{63},$\\
&$x_{35}x_{47}x_{56}+x_{36}x_{45}x_{57},\ x_{34}x_{47}x_{56}+x_{36}x_{44}x_{57},\ x_{17}x_{35}x_{56}-x_{15}x_{36}x_{57},\ x_{35}x_{47}x_{52}+x_{32}x_{45}x_{57},$\\
&$x_{34}x_{47}x_{52}+x_{32}x_{44}x_{57},\ x_{27}x_{34}x_{52}+x_{24}x_{32}x_{57}, \ x_{13}x_{26}x_{52}+x_{12}x_{23}x_{56},\ x_{36}x_{45}x_{51}+x_{35}x_{41}x_{56},$\\
&$x_{32}x_{45}x_{51}+x_{35}x_{41}x_{52},\ x_{12}x_{45}x_{51}+x_{15}x_{41}x_{52},\ x_{36}x_{44}x_{51}+x_{34}x_{41}x_{56},\ x_{32}x_{44}x_{51}+x_{34}x_{41}x_{52},$\\
&$x_{26}x_{44}x_{51}+x_{24}x_{41}x_{56},\ x_{27}x_{36}x_{45}-x_{26}x_{35}x_{47},\ x_{17}x_{32}x_{44}+x_{12}x_{34}x_{47},\ x_{15}x_{23}x_{44}+x_{13}x_{24}x_{45},$\\
&$x_{17}x_{26}x_{35}+x_{15}x_{27}x_{36},\ x_{13}x_{26}x_{35}+x_{15}x_{23}x_{36},\ x_{15}x_{27}x_{34}+x_{17}x_{24}x_{35},\ x_{15}x_{23}x_{34}+x_{13}x_{24}x_{35},$\\
&$x_{17}x_{26}x_{32}+x_{12}x_{27}x_{36},\ x_{13}x_{26}x_{32}+x_{12}x_{23}x_{36},\ x_{17}x_{24}x_{32}+x_{12}x_{27}x_{34},\ x_{13}x_{24}x_{32}+x_{12}x_{23}x_{34}$\\
\hline
$\!\!\deg 4\!\!$ & $x_{27}x_{35}x_{52}x_{63}-x_{23}x_{32}x_{57}x_{65},\ x_{17}x_{36}x_{44}x_{63}-x_{13}x_{34}x_{47}x_{66},\ x_{24}x_{35}x_{57}x_{61}-x_{27}x_{34}x_{51}x_{65},$\\
&$x_{23}x_{34}x_{52}x_{61}-x_{24}x_{32}x_{51}x_{63},\ x_{12}x_{36}x_{47}x_{61}-x_{17}x_{32}x_{41}x_{66},\ x_{13}x_{32}x_{44}x_{61}-x_{12}x_{34}x_{41}x_{63},$\\
&$x_{15}x_{26}x_{44}x_{52}-x_{12}x_{24}x_{45}x_{56},\ x_{13}x_{26}x_{45}x_{51}-x_{15}x_{23}x_{41}x_{56},\ x_{12}x_{23}x_{44}x_{51}-x_{13}x_{24}x_{41}x_{52}$\\
\hline
\end{tabular}
\end{center}
\caption{The 72 generators of $I_{M_4}$.}
\label{TAB:72things}
\end{table}
\end{example}
\begin{remark} In \cite{slack_paper}, given a set of $n$ vertices $V\subset\mathbbm{k}^d$ defining a $d$-polytope $P=\text{conv}(V)$, they include only the facet defining hyperplanes in the slack matrix. We can also form a matroid associated to this polytope by considering all the hyperplanes; that is, we define the matroid $M = M[V']$ where $V' \subset \mathbbm{k}^{(d+1)\times n} $ is the matrix obtained from $V$ by appending a 1 to each vector. Then the symbolic slack matrix of~$P$ defined in \cite{slack_paper} is the restriction of the symbolic slack matrix of matroid $M$ to the subset of columns corresponding to facet-defining hyperplanes.
Thus the slack ideal of the polytope is always contained in the slack ideal of the matroid, $I_P\subseteq I_M$. We illustrate with the following example
\end{remark}
\begin{example} We consider the triangular prism $P$ labelled as in Figure \ref{FIG:toblerone}. As a 3-polytope, its facets are given by the hyperplanes $1234,1256,3456,135,246$ and the symbolic slack matrix is in Figure \ref{FIG:toblerone}. Its slack ideal $I_P$ is generated by 3 binomials.
\begin{figure}
\begin{minipage}{2.15in}
\includegraphics[width=1.8in]{toblerone2}
\end{minipage}\begin{minipage}{3.5in}
\[ S_P(\mathbf{x})=
\begin{blockarray}{cccccc}
&H_1 & H_2 & H_3 & H_4 & H_5 \\
&1234 & 1256 & 3456 & 135 & 246 \\
\begin{block}{c[ccccc]}
1 &0 &0 &x_{13} &0 &x_{15} \\
2 &0 &0 &x_{23} &x_{24} &0 \\
3 & 0 &x_{32} & 0 & 0 &x_{35} \\
4 & 0 &x_{42} & 0 &x_{44} &0 \\
5 & x_{5,1} & 0 & 0 & 0 &x_{55} \\
6 & x_{6,1} & 0 & 0 &x_{64} & 0 \\
\end{block}
\end{blockarray}
\]
\end{minipage}
\caption{The triangular prism and its slack matrix as a polytope.}
\label{FIG:toblerone}
\end{figure}
Considering $P$ as a rank 4 matroid which has the 3 facets $1234$, $1256$, $3456$ of $P$ as its non-bases, we obtain following symbolic slack matrix.
\[
S_M(\mathbf{x})=
\begin{blockarray}{cccccc|cccccc}
&H_1 & H_2 & H_3 & H_4 & H_5 &H_6 &H_7 &H_8 &H_9 &H_{10} &H_{11}\\
&1234 & 1256 & 3456 & 135 & 246 &136 & 146 &145 &245 &235 &236\\
\begin{block}{c@{\;}[*{5}{@{\;}c@{\;}}|@{\hspace{-3.3pt}}*{6}{@{\;}c@{\;}}]}
1 &0 &0 &x_{13} &0 &x_{15} &0 &0 &0 &x_{19} &x_{1,10} &x_{1,11} \\
2 &0 &0 &x_{23} &x_{24} &0 &x_{26} &x_{27} &x_{28} &0 &0 &0 \\
3 & 0 &x_{32} & 0 & 0 &x_{35} &0 &x_{37} &x_{38} &x_{39} &0 &0 \\
4 & 0 &x_{42} & 0 &x_{44} &0 &x_{46} &0 &0 &0 &x_{4,10} &x_{4,11} \\
5 & x_{51} & 0 & 0 & 0 &x_{55} &x_{56} &x_{57} &0 &0 &0 &x_{5,11} \\
6 & x_{61} & 0 & 0 &x_{64} & 0 &0 &0 &x_{68} &x_{69} &x_{6,10} &0 \\
\end{block}
\end{blockarray}
\]
Not only is $I_P\subseteq I_M$ but in this case $I_P$ is the elimination ideal given by eliminating the variables in the columns indexed by the additional hyperplanes $H_6,\ldots,H_{11}$.
\end{example}
\section{Realization space models}
\label{SEC:realsp}
A realization space for a rank $d+1$ matroid $M$ with $n$ elements is, roughly speaking, a space whose points are in correspondence with (equivalence classes of) collections of vectors $V = \{\mathbf{v}_1,\ldots, \mathbf{v}_n\}\subset\mathbbm{k}^{d+1}$ whose matroid $M[V]$ is $M$. In this section we show that the slack variety defined in the \S2 provides such a realization space, and we relate it to another realization space called the Grassmannian of the matroid.
Theorem~\ref{THM:slackconditions} characterizes the slack matrices of realizations of a matroid. The next theorem shows that the slack variety captures exactly these matrices.
\begin{theorem} Let $M$ be a rank $d+1$ matroid. Then $V$ is a realization of $M$ if and only if $S_{M[V]} = S_M(\ss)$ where $\ss \in \mathcal{V}(I_M)\cap(\mathbbm{k}^*)^t$.
\label{THM:realizvariety}
\end{theorem}
\begin{proof} Let $V$ be a realization of $M$. Then $S_{M[V]} = S_M(\ss)$ for some $\ss \in (\mathbbm{k}^*)^t$ by Theorem~\ref{THM:slackconditions} (i). Furthermore,
$\rk(S_{M[V]}) = d+1$ by Corollary~\ref{COR:rank}, so that its $(d+2)$-minors vanish and thus $\ss\in\mathcal{V}(I_M)$, as desired.
Let $V \in\mathbbm{k}^{(d+1)\times n}$ be such that $S_{M[V]}=S_M(\ss)$ for some $\ss\in\mathcal{V}(I_M)\cap(\mathbbm{k}^*)^t$. Then $\supp(S_{M[V]}) = \supp(S_M)$ and $\rk(S_{M[V]})\leq d+1$. But now by Lemma~\ref{LEM:trianglesubmat}, $\rk(S_{M[V]})\geq d+1$, making $V$ a realization of $M$ by Theorem~\ref{THM:slackconditions}.
\end{proof}
Since we know that the set of realizations of a matroid is closed under row and column scalings, Theorem~\ref{THM:realizvariety} implies the following corollary. We denote the torus of row and column scalings, $(\mathbbm{k}^*)^n\times(\mathbbm{k}^*)^h$, by $T_{n,h}$.
\begin{corollary}The slack variety is closed under the action of the group $T_{n,h}$, where $(\mathbbm{k}^*)^n$ acts by row scaling (left multiplication by diagonal matrices) and $(\mathbbm{k}^*)^h$ acts by column scaling (right multiplication by diagonal matrices). \label{COR:scaleclosed}\end{corollary}
Theorem~\ref{THM:realizvariety} and Corollary~\ref{COR:scaleclosed} tell us that the slack variety is a realization space for matroid $M$ and the slack variety modulo the action of $T_{n,h}$ is a realization space for the projective equivalence classes of realizations of $M$.
\begin{definition} \label{DEFN:slackreal} The \emph{slack realization space} of a rank $d+1$ matroid $M$ on $n$ elements with $h$ hyperplanes is the image of the slack variety inside a product of projective spaces
$\mathcal{V}(I_M)\cap(\mathbbm{k}^*)^t \hookrightarrow (\mathbb{P}^{n-1})^h,
$
where $\ss$ is sent to the columns of $S_M(\ss)$.
\end{definition}
\begin{prop}
Let $M$ be a rank $d+1$ matroid on $n$ elements with $h$ hyperplanes. Then the points of its slack realization space are in one-to-one correspondence with linear equivalence classes of realizations of $M$.
\end{prop}
\begin{proof}
Under this embedding, two slack matrices which differ by column scaling are the same point in $(\mathbb{P}^{n-1})^h$. So, the result follows by Lemma~\ref{LEM:columnscale}.
\end{proof}
Next we describe a known model for the realization space of a matroid arising from a subvariety of a Grassmannian.
The \emph{Grassmannian} $Gr(d+1,n)$ is a variety whose points correspond to ${(d+1)}$-dimensional linear subspaces of a fixed $n$-dimensional vector space $\Lambda$.
It embeds into $\mathbb{P}^{{n \choose d+1}-1}$ as follows. Any $(d+1)$-dimensional linear subspace of $\Lambda$ can be described as the row space of a $(d+1)\times n$ matrix of rank $d+1$. However, two matrices $A$ and $B$ have the same row space when then there is a matrix $G \in GL(d+1,\Lambda)$ such that $A = GB$. Thus, to ensure that we have a one-to-one correspondence between subspaces and points in the Grassmannian, we record a $(d+1) \times n$ matrix by its vector of $(d+1)$-minors. We call this the \emph{Pl\"{u}cker vector}, and it has coordinates indexed by subsets $\sigma$ of $[n]$ of size $d+1$.
The \emph{Pl\"{u}cker ideal} $P_{d+1,n} \subseteq \mathbbm{k}[\mathbf{p}] = \mathbbm{k}[p_\sigma\ |\ \sigma \subset [n],\ |\sigma| = d+1]$ is the set of all polynomials which vanish on every vector of $(d+1)$-minors coming from some $(d+1)\times n$ matrix. It is generated by the homogeneous quadratic Pl\"ucker relations and cuts out the Grassmannian as a variety inside $\mathbb{P}^{{n \choose d+1} -1}$.
If we have a rank $d+1$ matroid $M=(E,\mathcal{B})$ with realization $V\in\mathbbm{k}^{(d+1)\times n}$, the Pl\"{u}cker coordinate $p_\sigma$ of $V$ is zero if and only if $\sigma$ is a dependent set of $M$.
Thus, realizations of $M$ correspond to the subvariety of $Gr(d+1,n)$ defined by setting the Pl\"ucker coordinates of non-bases to 0. This variety is also cut out by an ideal, namely,
the \emph{Grassmannian ideal} $P_M \subset \mathbbm{k}[\mathbf{p}_\mathcal{B}] $ of $M$,
$$
P_M := (P_{d+1,n} + \langle p_\sigma \ :\ \sigma \not \in \mathcal{B} \rangle) \cap \mathbbm{k}[\mathbf{p}_\mathcal{B}],
$$
where $\mathbbm{k}[\mathbf{p}_\mathcal{B}] := \mathbbm{k}[p_\sigma\ |\ \sigma \text{ is a basis of $M$}]$ is the ring with one variable for each basis $\sigma\in\mathcal{B}$.
This is the ideal obtained from $P_{d+1,n} \subset \mathbbm{k}[\mathbf{p}]$ by setting the variables indexed by non-bases of $M$ to 0.
\begin{definition}\label{DEFN:Grassmannian} The \emph{Grassmannian of $M$}, denoted $Gr(M)$, is $\mathcal{V}(P_M) \cap (\mathbbm{k}^*)^{|\mathcal{B}|}.$ The points in $Gr(M)$ correspond to $GL(\mathbbm{k}^{d+1})$ equivalence classes of $(d+1)\times n$ matrices which realize the matroid $M$.
\end{definition}
\subsection{Universal realization ideal}
Given a matroid $M$, we now define an ideal whose variety contains pairs $(\ss,\mathbf{q})$, where $\mathbf{q}$ is a Pl\"{u}cker vector and $\ss$ the nonzero entries of a slack matrix, and both come from the same realization of $M$.
If $V$ is a realization of a rank $d+1$ matroid $M = (E,\mathcal{B})$, then a slack matrix $S_{M[V]}$ can be filled with the Pl\"ucker coordinates of $V$, which can be seen from \eqref{EQ:pluckerdet}. Given a hyperplane $H_j\in\mathcal{H}(M)$, we record all possible substitutions of Pl\"ucker variables for slack variables using a matrix $M_{H_j}$ whose rows are indexed by $i\in E\backslash H_j$, and whose columns are indexed by subsets $J=\{j_1,\ldots, j_d\} \subset E$ with $\overline{J\,} = H$; that is,
$$M_{H_j} = \begin{bmatrix}
x_{i_1j} & \sgn(i_1,J_1)\cdot p_{i_1\cup J_1} & \cdots & \sgn(i_1,J_k)\cdot p_{i_1\cup J_k}\\
\vdots & \vdots & \cdots & \vdots \\
x_{i_mj} & \sgn(i_m,J_1)\cdot p_{i_m\cup J_1}& \cdots & \sgn(i_m,J_k)\cdot p_{i_m\cup J_k}
\end{bmatrix},
$$
where $\sgn(i,J)$ is the sign of the permutation putting $(i,j_1,\ldots,j_d)$ in increasing order.
\begin{example}
\label{EG:four_lines2}
Recall the matroid $M_4$ of Example \ref{EG:four_lines} pictured in Figure \ref{FIG:four_lines}. Consider the hyperplane $H_2 = 246$. It corresponds to slack variables $x_{i,2}$ for $i=1,3,5$ and its independent subsets are $24$, $26$, and $46$. So the matrix $M_{246}$ has the form
$$
M_{246}=
\left[\begin{array}{rrrr}
x_{1,2} & p_{124} & p_{126} & p_{146} \\
x_{3,2} & -p_{234} & -p_{236} & p_{346} \\
x_{5,2} & p_{245} & -p_{256} & -p_{456} \\
\end{array}\right].
$$
\end{example}
\begin{definition} \label{DEFN:mother}
Let $M=(E,\mathcal{B})$ be a matroid, $P_M$ be the Grassmannian ideal of $M$ and $I_2(M_{H_j})$ be the ideal of $2$-minors of the matrix $M_{H_j}$.
The \emph{universal realization ideal} of $M$ is
$$U_M := P_M + \sum_{H_j \in \mathcal{H}(M)} I_2(M_{H_j}) \subseteq \mathbbm{k}[\mathbf{x},\mathbf{p]}.$$
\end{definition}
Intuitively, insisting that the matrices $M_{H_j}$ have rank 1 corresponds to ensuring the columns of the slack matrix are simply scaled versions of the appropriate Pl\"ucker coordinates.
We now state the main result of this section.
\begin{theorem} Let $M = ([n],\mathcal{B})$ be a rank $d+1$ matroid with universal realization ideal $U_M\subseteq\mathbbm{k}[\mathbf{x},\mathbf{p}]$. Then $\mathcal{V}(U_M)\in \mathbbm{k}^t\times\mathbbm{k}^{|\mathcal{B}|}$ with
\begin{enumerate}[label=(\roman{enumi})]
\item the projection of $\mathcal{V}(U_M)$ onto the Pl\"ucker coordinates, $\pi_\mathbf{p}:\mathcal{V}(U_M)\to \mathbbm{k}^{|\mathcal{B}|},$ is the Grassmannian of the matroid,
$$\overline{\pi_\mathbf{p}(\mathcal{V}(U_M))} = \overline{Gr(M)};$$
\item the projection of $\mathcal{V}(U_M)\cap\left( (\mathbbm{k}^*)^t\times (\mathbbm{k}^*)^{|\mathcal{B}|}\right)$ onto the slack coordinates, $\pi_\mathbf{x}: \mathcal{V}(U_M)\to \mathbbm{k}^t,$ is the set of slack matrices of realizations of $M$,
$$\pi_\mathbf{x}\left(\mathcal{V}(U_M)\cap \left( (\mathbbm{k}^*)^t\times (\mathbbm{k}^*)^{|\mathcal{B}|}\right)\right) = \mathcal{V}(I_M)\cap(\mathbbm{k}^*)^t.$$
\end{enumerate}
\label{THM:mothervariety}\end{theorem}
The proof of this theorem requires several preliminary lemmas.
We first have the following result on Gr\"obner bases. (For notation and further details see \cite{CLOS}.)
\begin{lemma} Fix an elimination order on $\mathbbm{k}[\mathbf{x},\mathbf{p}]$. Given two $\mathbf{x}$-homogeneous polynomials $f,g$ and an $\mathbf{x}$-homogeneous set $\mathcal{G}\subset\mathbbm{k}[\mathbf{x},\mathbf{p}]$, if $h:=\overline{S(f,g)}^{\mathcal{G}}$ with $h\neq 0$, then $\deg_\mathbf{x}(h) \geq \max\{\deg_\mathbf{x}(f),\deg_\mathbf{x}(g)\}$.
\label{LEM:sausage} \end{lemma}
\begin{lemma} The Grassmannian ideal of a matroid can be obtained by eliminating the slack variables from its universal realization ideal. That is,
$P_M= U_M \cap \mathbbm{k}[\mathbf{p}].$
\label{LEM:pluckerproj}
\end{lemma}
\begin{proof}
We obtain one containment by the definition of $U_M$, since
$$
P_M = P_M \cap \mathbbm{k}[\mathbf{p}]
\subseteq \left(P_M+\sum_{H\in\mathcal{H}(M)} I_2(M_{H})\right) \cap \mathbbm{k}[\mathbf{p}]
= U_M\cap \mathbbm{k}[\mathbf{p}].
$$
It remains to show the reverse containment.
Fix an elimination order on $\mathbbm{k}[\mathbf{x},\mathbf{p}]$ eliminating the $\mathbf{x}$ variables. Let $\mathcal{G}$ be a Gr\"{o}bner basis for $U_M$ with respect to this ordering. Then, it suffices to show that
\begin{equation}
\label{EQN:buchberger}
\mathcal{G} \cap \mathbbm{k}[\mathbf{p}] \subset P_M.
\end{equation}
If we start with a generating set satisfying \eqref{EQN:buchberger}, then by
by Lemma~\ref{LEM:sausage}, any terms which are added to $\mathcal{G}$ after applying each step of Buchberger's algorithm with $\mathbf{x}$-degree 0 must be the reduction of an $S$-pair of elements which also have $\mathbf{x}$-degree 0, and are therefore also contained in $P_M$.
It remains to show that an initial generating set of $U_M$ satisfies \eqref{EQN:buchberger}. Taking the generating set of the definition, it is enough to show that any minor in $I_2(M_H)$ not containing a slack variable is already in $P_M$. It is not hard to check that any such minor already arises in $P_M$ as some 3-term Pl\"ucker relation having a term $p_\sigma p_\tau$ for some $\sigma\notin\mathcal{B}$ which gets set to zero.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{THM:mothervariety}]\hfill
\begin{enumerate}[label=(\roman{enumi})]
\item This follows from the definition of $Gr(M)$ and Lemma~\ref{LEM:pluckerproj}.
\item[(ii, $\subset$)]
Let $(\ss,\mathbf{q}) \in \mathcal{V}(U_M)\cap \left((\mathbbm{k}^*)^t\times (\mathbbm{k}^*)^{|\mathcal{B}|}\right)$. From $\mathbf{q}$ we can obtain a $(d+1)\times n$ matrix~$V$ with Pl\"ucker vector whose nonzero coordinates come from~$\mathbf{q}$. We claim that $V$ is a realization of $M$, so that $S_{M[V]}\in\mathcal{V}(I_M)\cap(\mathbbm{k}^*)^t$ by Theorem~\ref{THM:realizvariety}. Furthermore, we claim that $S_{M[V]}$ and $S_M(\ss)$ are the same up to column scaling, so that $\ss\in\mathcal{V}(I_M)\cap(\mathbbm{k}^*)^t$ by Corollary~\ref{COR:scaleclosed}.
A subset of $d+1$ columns of $V$ is independent if and only if the
the corresponding Pl\"ucker coordinate is nonzero. Since $\mathbf{q}\in(\mathbbm{k}^*)^{|\mathcal{B}|}$, the nonzero Pl\"ucker coordinates correspond to bases $B\in \mathcal{B}$ of $M$. Hence $V$ is a realization of $M$.
Fix a hyperplane $H$ and an independent subset $J=\{j_1,\ldots, j_d\}\subset H$. Then the first column of $M_H$ is $(s_{iH})_{i\notin H}^\top$, column $J$ is $(\sgn(i,J)\cdot q_{i\cup J})_{i\notin H}^\top$, and by definition of $U_M$, there exists $\lambda_H\in\mathbbm{k}^*$ such that
$s_{iH} = \lambda_H\sgn(i,J)\cdot q_{i\cup J}$ for all~$i$. Since $\mathbf{q}$ is the Pl\"ucker vector of $V$, this gives
$$s_{iH} = \lambda_H\sgn(i,J)\cdot q_{i\cup J} = \lambda_H\det(\mathbf{v}_i,\mathbf{v}_{j_1},\ldots,\mathbf{v}_{j_d}), \quad \forall i$$
so that using subset $J$ to calculate $\alpha_H$, and hence the entries of $S_{M[V]}$, as in Example~\ref{EG:four_lines}, we see that $S_{M[V]}D_h = S_{M}(\ss)$, for $D_h$ the diagonal matrix with entries $\lambda_H$.
\item[(ii, $\supset$)] Let $\ss\in\mathcal{V}(I_M)\cap(\mathbbm{k}^*)^t$. By Theorem~\ref{THM:realizvariety}, $S_M(\ss)$ is the slack matrix of some realization $V$ of $M$. Let $\mathbf{q}$ be the vector of Pl\"ucker coordinates of $V$. We claim $(\ss,\mathbf{q})\in\mathcal{V}(U_M)\cap {\left((\mathbbm{k}^*)^t\times (\mathbbm{k}^*)^{|\mathcal{B}|}\right)}$. To see this it suffices to show that the columns of $M_H$, with entries defined using $(\ss,\mathbf{q})$, are scalar multiples of each other for each $H\in\mathcal{H}(M)$; that is, the $I_2(M_H)$ ideals are satisfied, since the Pl\"ucker coordinates necessarily satisfy the Pl\"ucker ideal equations, and hence the equations of $P_M$.
The $(i,H)$ slack entry $s_{iH}$ is of the form $\det(\mathbf{v}_i,\mathbf{w}_1,\ldots, \mathbf{w}_d)$ for some choice of $\mathbf{w}_1,\ldots, \mathbf{w}_d$ which span the hyperplane $H$. Each subsequent column of $M_H$ has entries $\det(\mathbf{v}_i,\mathbf{v}_{j_1},\ldots,\mathbf{v}_{j_d})$, obtained from $\mathbf{q}$ as above, for $j_1<\cdots<j_d$ spanning $H$. Since each $\mathbf{v}_{j_k}$ lies on hyperplane the $H$, there is a sequence of elementary column operations that takes the matrix with columns $\mathbf{v}_i,\mathbf{w}_1,\ldots, \mathbf{w}_d$ to the one with columns $\mathbf{v}_i,\mathbf{v}_{j_1},\ldots,\mathbf{v}_{j_d}$ for each $i$. These column operations change the determinant by some scale factor $\lambda\in\mathbbm{k}^*$ for all $i$, so that
each column of $M_H$ is a scalar multiple of the first column of slack entries as required.
\end{enumerate} \vspace{-10pt}
\end{proof}
\begin{corollary} Let $M=(E,\mathcal{B})$ be a rank $d+1$ matroid. Then
$$\sqrt{I_M} = \sqrt{U_M:\left(\prod_{i\in E, H\in\mathcal{H}} x_{iH}\prod_{\sigma\in\mathcal{B}}p_\sigma\right)^\infty}\bigcap \mathbbm{k}[\mathbf{x}].$$
\label{COR:motherradical}\end{corollary}
By universality \cite{mnev}, we do not expect that $I_M$ is radical for every matroid. Thus, Corollary \ref{COR:motherradical} may be the strongest relationship between $I_M$ and $U_M$.
\begin{example}
\label{EG:four_lines3}
We now continue Example \ref{EG:four_lines}, for which we can verify Theorem~\ref{THM:mothervariety} at the level of ideals.
Let $P_{M_4}$ be the Grassmannian ideal of this matroid, which is generated by the Pl\"{u}cker relations with variables $p_{123},p_{246},p_{345},p_{156}$ set to 0. Let $I$ be the ideal which guarantees each of the 7 matrices $M_{123}$, $M_{246}$, $M_{345}$ $M_{156}$, $M_{25}$, $M_{14}$, $M_{36}$ have rank 1; that is, $I = \sum_{H\in\mathcal{H}(M_4)} I_2(M_H)$.
Then, the universal realization ideal is $U_M = I + P_{M_4}$.
In this case, we can compute that $P_M = U_M\cap\mathbbm{k}[\mathbf{p}]$ and $I_M = U_M:\left(\prod x_{iH}\prod p_\sigma\right)^\infty \cap\mathbbm{k}[\mathbf{x}]$.
\end{example}
\section{Non-realizability}
\label{SEC:nonreal}
In this section we illustrate how the slack ideal can be used to determine matroid realizability over a given field.
This is a well-studied problem \cite{BLSWZ,weird_bernd_book,mnev} for which a complete characterization is only known in a very limited number of cases. Observe that Theorem~\ref{THM:realizvariety} gives us the following criterion for realizability.
\begin{corollary} \label{COR:varietytest}
A matroid $M$ is realizable over $\mathbbm{k}$ if and only if $\mathcal{V}(I_M)\cap(\mathbbm{k}^*)^t \not = \emptyset$.
\end{corollary}
We now recast this into a test for realizability in terms of the slack ideal.
\begin{prop}
Let $M$ be an abstract matroid and $\mathbbm{k}$ be a field. If the slack ideal $I_M = \langle 1\rangle$ over $\mathbbm{k}$, then $M$ is not realizable over $\mathbbm{k}$. If $\mathbbm{k}$ is algebraically closed and $M$ is not realizable over $\mathbbm{k}$, then $I_M=\langle 1 \rangle$.
\label{PROP:nonreal}
\end{prop}
\begin{proof}
If $M$ is realizable over $\mathbbm{k}$, then there exists a slack matrix $S_M$ which is an element of the variety $\mathcal{V}(I_M)$ by Theorem \ref{THM:realizvariety}. Then we cannot have $I_M= \langle1\rangle$. On the other hand, if $I_M \not = \langle1\rangle$, then $\mathcal{V}(I_M)$ is not empty by the Nullstellensatz, and since $I_M$ is saturated with respect to the product of the variables, $\mathcal{V}(I_M)$ is not contained entirely in the coordinate hyperplanes.
Therefore, by Theorem~\ref{THM:realizvariety} there is a slack matrix $S_M$, and the rows of $S_M$ give a realization of $M$ by Lemma \ref{LEM:rowrealiz}.
\end{proof}
\begin{example}
\label{EX:fano}
Consider the Fano plane $M_F$. It is a rank~3 matroid on 7 elements $E = \{0,1,2,3,4,5,6\}$, depicted in Figure~\ref{FIG:fano} with its symbolic slack matrix. It is known that $M_F$ is only realizable in characteristic 2.
Using \texttt{Macaulay2}~\cite{M2}, we find $I_{M_F} = \langle 1 \rangle$ over~$\mathbb{Q}$. So, the Fano plane is not realizable over $\mathbb{Q}$ by Proposition \ref{PROP:nonreal}.
Over $\mathbb{F}_2$, the slack ideal is generated by $126$ binomials of degrees 2,3, and 4. So, setting all variables to 1 in the slack matrix gives the realization of $M_F$ in $\mathbb{F}_2^7$.
\begin{figure}
\begin{center}
\begin{minipage}{0.3 \textwidth}
\includegraphics[height = 2 in]{fano_pic}
\end{minipage}
\hspace{0.1 in }
\begin{minipage}{0.65 \textwidth}
\[
\begin{blockarray}{cccccccc}
&H_1 & H_2 & H_3 & H_4 & H_5 & H_6 & H_7 \\
&126 &014 &456 &025 &036 &234 &135\\
\begin{block}{c[ ccccccc]}
0&{x}_{01} &0 &{x}_{03} &0 &0 &x_{06} &{x}_{07}\\
1&0 &0 &x_{13} &{x}_{14} &{x}_{15} &x_{16} &0\\
2&0 &x_{22} &{x}_{23} &0 &{x}_{25} &0 &{x}_{27}\\
3&{x}_{31} &x_{32} &{x}_{33} &{x}_{34} &0 &0 &0\\
4&x_{41} &0 &0 &{x}_{44} &{x}_{45} &0 &{x}_{47}\\
5&x_{51} &x_{52} &0 &0 &x_{55} &x_{56} &0\\
6&0 &{x}_{62} &0 &x_{64} &0 &x_{66} &x_{67}\\
\end{block}
\end{blockarray}
\]
\end{minipage}
\caption{The Fano plane with its non-bases drawn as lines and a circle, and its symbolic slack matrix.}
\label{FIG:fano}
\end{center}
\end{figure}
\end{example}
\begin{example}
Consider the \emph{complex matroid} $M_8$ from \cite[p. 33]{weird_bernd_book}. It is a rank~3 matroid on 8 elements with non-bases $124$, $235$, $346$, $457$, $568$, $167$, $278$, and $138$. It is depicted with its symbolic slack matrix in Figure \ref{FIG:complex_matroid}
\begin{figure}
\begin{center}
\begin{minipage}{0.26\textwidth}
\includegraphics[height=1.6in]{complex_pic2}
\end{minipage}\;\;\begin{minipage}{0.69\textwidth}
{\fontsize{8}{10}\selectfont
\[
\begin{blockarray}{*{13}{@{\;\;}c@{\;\;}}}
&H_1 & H_2 & H_3 & H_4 & H_5 & H_6 & H_7 &H_8 &H_9 &H_{10} &H_{11} &H_{12} \\
&167 &235 &568 &346 &124 &278 &138 &457 &48 &37 &26 &15\\
\begin{block}{c@{\;\;}[*{12}{@{}c@{}}]}
1 &0 &x_{12} &x_{13} &x_{14} &0 &x_{16} &0 &x_{18} &x_{19} &x_{1,10} &x_{1,11} &0 \\
2 &x_{21} &0 &x_{23} &x_{24} &0 &0 &x_{27} &x_{28} &x_{29} &x_{2,10} &0 &x_{2,12} \\
3 &x_{31} &0 &x_{33} &0 &x_{35} &x_{36} &0 &x_{38} &x_{39} &0 &x_{3,11} &x_{3,12} \\
4 &x_{41} &x_{42} &x_{44} &0 &0 &x_{46} &x_{47} &0 &0 &x_{4,10} &x_{4,11} &x_{4,12} \\
5 &x_{51} &0 &0 &x_{54} &x_{55} &x_{56} &x_{57} &0 &x_{59} &x_{5,10} &x_{5,11} &0 \\
6 &0 &x_{62} &0 &0 &x_{65} &x_{66} &x_{67} &x_{68} &x_{69} &x_{6,10} &0 &x_{6,12} \\
7 &0 &x_{72} &x_{73} &x_{74} &x_{75} &0 &x_{77} &0 &x_{79} &0 &x_{7,11} &x_{7,12} \\
8 &x_{81} &x_{82} &0 &x_{84} &x_{85} &0 &0 &x_{88} &0 &x_{8,10} &x_{8,11} &x_{8,12} \\
\end{block}
\end{blockarray}.
\]}
\end{minipage}
\caption{The complex matroid $M_8$ and its symbolic slack matrix $S_{M_8}(\mathbf{x})$.}
\label{FIG:complex_matroid}
\end{center}
\end{figure}
To simplify the computation, we use Corollary~\ref{COR:scaleclosed} and note that we can select a representative of each projective equivalence class by fixing certain variables in the slack matrix to be 1 (see \S\ref{SSEC:scaledslack} for more details). Fixing the variables $x_{14}$, $x_{27}$, $x_{28}$, $x_{3,12}$, $x_{41}$, $x_{46}$, $x_{47}$, $x_{4,10}$, $x_{57}$, $x_{67}$, $x_{72}$, $x_{73}$, $x_{74}$, $x_{75}$, $x_{77}$, $x_{79}$, $x_{7,11}$, $x_{7,12}$, $x_{84}$ to 1 and computing the slack ideal $I_{M_8}$ in \texttt{Macaulay2} \cite{M2}, we find that it is not the unit ideal. However, it contains the polynomial $x_{8,12}^2+ x_{8,12}+1$. Since this polynomial has only the complex roots $\frac{-1\pm i\sqrt{3}}{2}$, we get by Corollary~\ref{COR:varietytest} that $M_8$ is not realizable over $\mathbb{R}$, but it is realizable over $\mathbb{C}$ by Proposition~\ref{PROP:nonreal}.
\end{example}
\subsection{Final Polynomials}
The method of final polynomials introduced in \cite[\S4.2]{weird_bernd_book} certifies when a matroid has no realization. We define an analogous polynomial for the slack ideal, and show how it can be used to improve computational efficiency of checking non-realizability.
\begin{definition} \label{DEFN:final} Let $M$ be any matroid of rank $d+1$. Let $\mathcal{S}$ be the multiplicatively closed set generated by taking finite products of the variables $\mathbf{x}$ in the symbolic slack matrix. A polynomial $f \in \mathbbm{k}[\mathbf{x}]$ is a \emph{slack final polynomial} if
$$
f \in I_{d+2}(S_M(\mathbf{x})) \cap \left(\mathcal{S} + I_M \right)
$$
where $I_{d+2}(S_M(\mathbf{x}))$ is the ideal of $(d+2)$-minors of the symbolic slack matrix of $M$.
\end{definition}
We now have the following result, which demonstrates that the existence of slack final polynomials gives a certificate for non-realizability.
\begin{prop} Let $M$ be a matroid of rank $d+1$. The following are equivalent.
\begin{enumerate}[label = (\roman{enumi})]
\item $1 \in I_M \subseteq \mathbbm{k}[\mathbf{x}]$,
\item There is a monomial $m \in \mathbbm{k}[\mathbf{x}]$ such that $m \in I_{d+2}(S_M(\mathbf{x}))$,
\item A slack final polynomial $f \in I_{d+2}(S_M(\mathbf{x})) \cap \left(\mathcal{S} + I_M \right)$ exists for $M$.
\end{enumerate}
\label{PROP:finaleq}
\end{prop}
\begin{remark} Over an algebraically closed field, these conditions are equivalent to the matroid being non-realizable by Proposition \ref{PROP:nonreal}. When $\mathbbm{k}$ is not algebraically closed, these conditions imply non-realizability, but if a matroid is not realizable there may not be a slack final polynomial.
\end{remark}
\begin{example} Recall that the complex matroid $M_8$ has a complex realization, as $1\notin I_{M_8} \subseteq \mathbb{Q}[\mathbf{x}]$. However, by the above proposition, this means that even though $M_8$ is not realizable over $\mathbb{Q}$, it does not have a slack final polynomial.
\end{example}
\begin{proof} \hfill
\begin{enumerate}
\item[(i) $\Rightarrow$ (iii)] Suppose $1 \in I_M$. Since $I_M$ is the saturation of $I_{d+2}(S_M(\mathbf{x}))$, this implies that there exists a monomial $m \in \mathbbm{k}[\mathbf{x}]$ such that $m \cdot 1 \in I_{d+2}(S_M(\mathbf{x}))$. Then, we observe the $m$ is already a slack final polynomial for $M$.
\item[(ii) $\Rightarrow$ (i)] If there is a monomial $m \in \mathbbm{k}[\mathbf{x}]$ such that $m \in I_{d+2}(S_M(\mathbf{x}))$, then after saturation we find $1 \in I_M$.
\item[(iii) $\Rightarrow$ (ii)] Suppose $f$ is a slack final polynomial for $M$. Since $f \in (\mathcal{S} + I_M)$, there exists a monomial $m$ and a $g \in I_M$ with $f = m + g$. Since $g \in I_M$, there exists a monomial $n$ such that $ng \in I_{d+2}(S_M(\mathbf{x}))$, so $nm = nf-ng \in I_{d+2}(S_M(\mathbf{x}))$ is a monomial in $I_{d+2}(S_M(\mathbf{x}))$.
\end{enumerate} \vspace{-20pt}
\end{proof}
\begin{remark} In practice saturation of the ideal $I_{d+2}(S_M(\mathbf{x}))$ can be quite slow, which often makes testing realizability via checking $1\in I_M$ infeasible. Thus the real power of Proposition~\ref{PROP:finaleq} is that one often finds relatively small monomials which are already contained in $I_{d+2}(S_M(\mathbf{x}))$. So, if one simply wants to certify non-realizability, a faster method is to compute $I_{d+2}(S_M(\mathbf{x}))$ and check, for example, if $\prod \mathbf{x} \in I_{d+2}(S_M(\mathbf{x}))$. In the following example we exhibit how this method can be useful for certifying non-realizability.
\end{remark}
\begin{example} Consider the Fano matroid $M_F$ of Example \ref{EX:fano}. If we compute $I_{4}(S_{M_F}(\mathbf{x}))$, then we can verify that the product of all of the variables is contained in this ideal. In fact, even the monomial
$
x_{07}x_{16}x_{25}x_{33}x_{41}x_{52}x_{64}
$
is contained in $I_{4}(S_{M_F}(\mathbf{x}))$. Verifying this containment (using the laptop of one of the authors) in \texttt{Macaulay2} took 0.000067 seconds, while testing $1\in I_M$ took 3.40765 seconds, indicating a speed up by a factor of 50,000.
\end{example}
\begin{example} Consider the V\'amos matroid pictured in Figure~\ref{FIG:vamos}. It is a rank~4 matroid $M_v$ on 8 elements whose non-bases are given by the sets $1234$, $1256$, $3456$, $3478$, and $5678$. It is one of the smallest matroids known to be non-realizable over every field. However, the V\'amos matroid has 41 hyperplanes, so that its slack matrix is an $8\times 41$ matrix containing 200 distinct variables. Even computing the full set of minors of this matrix is computationally impractical.
We note though, that it always suffices to show that Proposition~\ref{PROP:finaleq} (ii) holds for some subideal of the ideal of $(d+2)$-minors. In particular, we can look at the minors of a submatrix of $S_{M_v}(\mathbf{x})$. Consider the submatrix of the V\'amos symbolic slack matrix in Figure \ref{FIG:vamos}.
One can easily check with \texttt{Macaulay2} that the monomial given by the product of all the variables in this submatrix is already in the minor ideal of this submatrix (over $\mathbb{Q}$ and various finite fields), making $M_v$ non-realizable over these fields by Propositions~\ref{PROP:finaleq} and \ref{PROP:nonreal}.
\begin{figure}
\begin{center}
\begin{minipage}{0.3 \textwidth}
\includegraphics[height=2.5in]{Vamos.pdf}
\end{minipage}
\hspace{0.1 in}
\begin{minipage}{0.65 \textwidth}
\[
\begin{blockarray}{ccccccccc}
&H_1 & H_2 & H_3 & H_4 & H_5 & H_6 & H_7 &H_8 \\
&3456 &567 8 &1234 &3478 &1256 &467 &267 &127 \\
\begin{block}{c[ cccccccc]}
1& x_{1,1} & x_{1,2} & 0 & x_{1,4} & 0 & x_{1,6} & x_{1,7} & 0 \\
2& x_{2,1} & x_{2,2} & 0 & x_{2,4} & 0 & x_{2,6} & 0 & 0 \\
3& 0 & x_{3,2} & 0 & 0 & x_{3,5} & x_{3,6} & x_{3,7} & x_{3,8} \\
4& 0 & x_{4,2} & 0 & 0 & x_{4,5} & 0 & x_{4,7} & x_{4,8} \\
5& 0 & 0 & x_{5,3} & x_{5,4} & 0 & x_{5,6} & x_{5,7} & x_{5,8} \\
6& 0 & 0 & x_{6,3} & x_{6,4} & 0 & 0 & 0 & x_{6,8} \\
7& x_{7,1} & 0 & x_{7,3} & 0 & x_{7,5} & 0 & 0 & 0 \\
8& x_{8,1} & 0 & x_{8,3} & 0 & x_{8,5} & x_{8,6} & x_{8,7} & x_{8,8} \\
\end{block}
\end{blockarray}
\]
\end{minipage}
\caption{The V\'amos matroid $M_v$ with non-bases pictured as planes, and a submatrix its slack matrix.}
\label{FIG:vamos}
\end{center}
\end{figure}
\end{example}
\section{Projective uniqueness of matroids}
\label{SEC:toric}
The simplest slack realization spaces are those belonging to projectively unique matroids. In this case, we know that there is a single realization up to projective transformations; in other words, $\mathcal{V}(I_M)$ is the toric variety which is the closure of the orbit of some realization under the action of $T_{n,h}$. This implies $\sqrt{I_M} = \mathcal{I}(\mathcal{V}(I_M))$ is a toric ideal; however, universality suggests that $I_M$ need not be radical. A natural question which arises is whether projectively unique matroids correspond exactly to matroids with toric slack ideals. To study this question we introduce an intermediate toric ideal associated to a matroid.
\begin{definition}
\label{DEFN:graph}
Define the {\em non-incidence graph of matroid~$M$} as the bipartite graph~$G_M$ with one node for each element of the ground set of $M$, one node for each hyperplane, and an edge between element $i$ and hyperplane $H_j$ if and only if $i\notin H_j$. Notice that $G_M$ records the support of the slack matrix $S_M$, and so we can think of its edges as being labelled by the corresponding entry of $S_M(\mathbf{x})$. (See Figure~\ref{FIG:fourlinesgraph} for an example of the graph $G_{M_4}$ for the matroid $M_4$ of Example~\ref{EG:four_lines}, and Figure~\ref{FIG:nonfano_fig} for the non-incidence graph of the non-Fano matroid.)
\end{definition}
Let $\mathcal{A}_G$ be the set of vectors forming the columns of the vertex-edge incidence matrix of the graph $G$, and let $T_G$ be the toric ideal of the vector configuration $\mathcal{A}_G$. The toric ideal of a bipartite graph is a well-studied object \cite{OH99, villarreal}. If a matroid $M$ has a slack matrix $S_M$ which is a 0-1 matrix, then the toric ideal $T_{G_M}$ associated to the graph $G_M$ is the ideal of the orbit of $S_M$ under the action of the torus $T_{n,h}$. So, $T_{G_M}$ describes one projective equivalence class of slack matrices of $M$. We now define an analogous toric ideal for any projective equivalence class.
\begin{definition} \label{DEFN:cycleideal}
Let $M$ be an abstract matroid with realization $V$. Let $\ss \in (\mathbbm{k}^*)^t$ be such that $S_{M[V]} = S_M(\ss)$, where $t$ is the number of variables in $S_M(\mathbf{x})$, the symbolic slack matrix of $M$. We define the \emph{cycle ideal} $C_V$ of $M[V]$ to be the ideal
\begin{equation}
C_V = \left\langle \mathbf{x}^{c+} - \alpha_c \mathbf{x}^{c-} : \text{$c$ is a cycle in $G_M$ and } \alpha_c = \frac{\ss^{c+}}{\ss^{c-}} \right\rangle \subseteq \mathbbm{k}[\mathbf{x}]
\label{EQ:cycleideal}\end{equation}
where $c+$ and $c-$ are alternating edges from the cycle $c$.
\end{definition}
\begin{theorem} Let $M$ be a matroid of rank $d+1$ on $n$ elements with $h$~hyperplanes. Let $V$ be a realization of $M$ with slack matrix $S_{M[V]} = [s_{i,j}]_{i=1,j=1}^{n,h}$. Then the ideal $C_V$ is the (scaled) toric ideal which is the kernel of the $\mathbbm{k}$-algebra homomorphism
$
\phi: \mathbbm{k}[\mathbf{x}] \to \mathbbm{k}[\mathbf{r},\tt, \mathbf{r}^{-1},\tt^{-1}],$
which sends
$
x_{ij} \mapsto s_{i,j}r_it_j.
$
\label{THM:cycleistoric}
\end{theorem}
We note that the cycle ideal of a realization provides a way to distinguish projective equivalence classes of realizations of $M$, as well as detect projective uniqueness.
\begin{lemma}
\label{LEM:cycleproj}
Let $M$ be a matroid with realizations $U$ and $V$. Then $U,V$ are projectively equivalent if and only if $C_V = C_U$. \end{lemma}
\begin{proof}
First suppose that $U$ and $V$ are projectively equivalent.
Let $S_{M[V]}$ have entries $s_{i,j}$ for elements $i$ of the ground set of $M$ and hyperplanes $H_j$ of $M$. By Lemma~\ref{LEM:pe} we know that $S_{M[U]}$ is obtained from $S_{M[V]}$ by scaling the rows by $r_1, \ldots, r_n\in\mathbbm{k}^*$ and scaling the columns by $t_1, \ldots, t_h\in\mathbbm{k}^*$. Then the entries of $S_{M[U]}$ are $r_it_js_{i,j}$. Since $S_{M[U]}$ and $S_{M[V]}$ have the same support, they have the same cycles. One may then show via elementary calculation that the coefficients $\alpha_c$ are the same when calculated from $S_{M[U]}$ and from $S_{M[V]}$ for every cycle $c \in G_M$.
Conversely, suppose $C_V = C_U$. Then $\mathcal{V}(C_V)=\mathcal{V}(C_U)$, and in particular, $S_{M[V]} = S_M(\ss)$ and $S_{M[U]} = S_M(\mathbf{u})$ for $\ss,\mathbf{u}\in\mathcal{V}(C_V)\cap(\mathbbm{k}^*)^t$. We now argue that $S_{M[U]}$ can be row and column scaled to be equal to $S_{M[V]}$, which proves the results by Lemma~\ref{LEM:pe}.
Fix a spanning forest $T$ of $G_M$. By Lemma \ref{LEM:scaleforest} we may scale the entries in $S_M(\mathbf{u})$ corresponding to edges in $T$ to be equal to the corresponding entries of $S_{M[V]}$.
Any remaining entry $a$ of $S_{M}(\mathbf{u})$ will correspond to an edge $e$ in $G_M$ such that $T \cup \{e\}$ contains a unique cycle $c$, where $e \in c$. The equation $\mathbf{x}^{c+}-\alpha_c\mathbf{x}^{c-}\in C_V$ corresponding to this cycle must be satisfied by the scaling of $S_M(\mathbf{u})$, since $C_V=C_U$. Since all variables in the cycle except the one labelled by edge $e$ have been fixed, we find that there is only one possible value for $a$. Furthermore, this value equals the corresponding entry in $S_{M[V]}$, since $S_{M[V]}$ satisfies the equations of $C_V$ by definition. \end{proof}
\begin{corollary} Let $M$ be a matroid with realization $V$. Then, $\mathcal{V}(C_V) \cap (\mathbbm{k}^*)^t$ consists of exactly the slack matrices of realizations projectively equivalent to $V$.
\label{COR:varofc}
\end{corollary}
\begin{lemma} Let $\mathbbm{k}$ be algebraically closed and $M$ be a matroid with realization $V$. Then the slack ideal $I_M$ is contained in the cycle ideal $C_V$. \label{LEM:cyclecontain}
\end{lemma}
\begin{proof}
By Corollary \ref{COR:varofc} we know that $\mathcal{V}(C_V) \subset \mathcal{V}(I_M)$. Then, $\mathcal{I}(\mathcal{V}(C_V)) \supset \mathcal{I}(\mathcal{V}(I_M))$. Since $C_V$ is radical, and since ${I_M}\subset \sqrt{I_M}$, this gives $C_V\supset I_M$.
\end{proof}
\begin{prop} Let $M$ be a matroid with projectively unique realization~$V$. Then $\mathcal{V}(I_M) = \mathcal{V}(C_V)$. \label{PROP:PUcyclevar} \end{prop}
\begin{proof} By Corollary~\ref{COR:varofc} and Theorem~\ref{THM:realizvariety}, we get $\mathcal{V}(I_M)\cap(\mathbbm{k}^*)^t = \mathcal{V}(C_V)\cap(\mathbbm{k}^*)^t$. Then since both varieties are irreducible, the result follows.
\end{proof}
In fact, we can have
$I_M = C_V$ for a realization $V$ of $M$. In this case, call $I_M$~{\em cyclic}.
\begin{theorem}
If the slack ideal of a matroid is cyclic then $M$ is projectively unique and $I_M$ is radical. The converse also holds when $\mathbbm{k}$ is algebraically closed.
\label{THM:cyclic}
\end{theorem}
\begin{proof} Suppose that $I_M$ is cyclic. Then $M$ is projectively unique by Corollary~\ref{COR:varofc} and $I_M$ is radical, since it is prime by Theorem~\ref{THM:cycleistoric}.
Conversely, suppose that $M$ is projectively unique and $I_M$ is radical. By Proposition~\ref{PROP:PUcyclevar}, $\mathcal{I}(\mathcal{V}(I_M)) = \mathcal{I}(\mathcal{V}(C_V))$, so that $I_M = C_V$, as both ideals are radical.
\end{proof}
\begin{example} Recall the matroid $M_4$ from Example \ref{EG:four_lines}, whose non-incidence graph $G_{M_4}$ is displayed in Figure \ref{FIG:fourlinesgraph}. The matroid $M_4$ has a cyclic slack ideal. Hence $I_{M_4}$ is a radical slack ideal, and $M_4$ is projectively unique.
\begin{figure}
\begin{center}
\includegraphics[height=1.5in]{fourlines_graph}
\end{center}
\caption{The graph $G_{M_4}$ for matroid $M_4$ with the highlighted cycle corresponding to binomial $x_{3,6}x_{6,5}+x_{3,5}x_{6,6}$ of Table~\ref{TAB:72things}.}
\label{FIG:fourlinesgraph}
\end{figure}
\end{example}
\begin{example} Recall the Fano plane discussed in Example \ref{EX:fano}. Over $\mathbb{F}_2$, the 126 binomial generators of $I_{M_F}$ found in Example \ref{EX:fano} correspond to each of the cycles in the graph $G_{M_F}$. Over $\mathbb{F}_2$ this is the projectively unique representation of $M_F$, and this ideal is equal to the cycle ideal of the representation.
\end{example}
\subsection{Scaled Slack Matrices} \label{SSEC:scaledslack}
From Corollary~\ref{COR:scaleclosed} we know that quotienting by the action of $T_{n,h}$ on $\mathcal{V}(I_M)\cap(\mathbbm{k}^*)^t$ gives us a realization space for projective equivalence classes of representations of $M$. We now give an explicit way of computing the variety of these equivalence classes.
As in \cite[Lemma 5.5]{slack_paper}, we scale rows and columns of a slack matrix via the following lemma, to fix one representative of each projective equivalence class.
\begin{lemma} Given a realization of a matroid $M$, we may scale the rows and columns of its slack matrix $S_M$ so that it has ones in the entries indexed by the edges in a maximal spanning forest $F$ of the graph $G_M$; the resulting realization of $M$ is projectively equivalent to the original realization of $M$. \label{LEM:scaleforest}
\end{lemma}
\begin{definition} Given a matroid $M$ we can take a symbolic slack matrix
and set variables corresponding to edges in a maximal spanning forest $F$ to 1
as in Lemma~\ref{LEM:scaleforest} to obtain a \emph{scaled symbolic slack matrix}. Then, the \emph{scaled slack ideal} is obtained by taking the $(d+2)$-minors of this matrix and saturating with respect to the product of all the variables. \label{DEFN:scaled}
\end{definition}
Using the scaled symbolic slack matrix not only allows us to study the projective realization space of $M$, but also proves to be a useful tool for computations because this matrix will have considerably fewer variables.
\begin{example} \label{EX:non-Fano} Let $M_{NF}$ be the non-Fano matroid. It is a rank 3 matroid on 7 elements depicted in Figure~\ref{FIG:nonfano_fig} with its symbolic slack matrix. It differs from the Fano plane by the inclusion of $135$ as a basis.
\begin{figure}[h]
\begin{center}
\begin{minipage}{0.25 \textwidth}
\includegraphics[height = 1.6 in]{nonfano_pic}
\end{minipage}
\hspace{0.1 in}
\begin{minipage}{0.7 \textwidth}
\[
\begin{blockarray}{cccccccccc}
&H_1 & H_2 & H_3 & H_4 & H_5 & H_6 & H_7 &H_8 &H_9 \\
&126 &014 &456 &025 &036 &234 &35 &13 &15\\
\begin{block}{c[ccccccccc]}
0 &{x}_{01} &0 &{x}_{03} &0 &0 &x_{06} &{x}_{07} &x_{08} &x_{09}\\
1 &0 &0 &x_{13} &{x}_{14} &{x}_{15} &x_{16} &0 &0 &0\\
2 &0 &x_{22} &{x}_{23} &0 &{x}_{25} &0 &{x}_{27} &x_{28} &x_{29}\\
3 &{x}_{31} &x_{32} &{x}_{33} &{x}_{34}&0 &0 &0 &0 &x_{39}\\
4 &x_{41} &0 &0 &{x}_{44} &{x}_{45} &0 &{x}_{47} &x_{48} &x_{49}\\
5 &x_{51} &x_{52} &0 &0 &x_{55} &x_{56} &0 &x_{58} &0\\
6 &0 &{x}_{62} &0 &x_{64} &0 &x_{66} &x_{67} &x_{68} &x_{69}\\
\end{block}
\end{blockarray}
\]
\end{minipage}
\includegraphics[width = 0.7 \textwidth]{nonfanotree2}
\end{center}
\caption{The non-Fano matroid, with its non-bases depicted as lines, together with its symbolic slack matrix and the spanning tree $F$ selected of Example~\ref{EX:non-Fano}.}
\label{FIG:nonfano_fig}
\end{figure}
We now show that the non-Fano matroid is projectively unique, and write down a realization from the slack matrix. Let $F$ be the spanning tree of $G_{M_{NF}}$ depicted in Figure \ref{FIG:nonfano_fig}. We set the corresponding variables $x_{41},$ $x_{51},$ $x_{22},$ $x_{32},$ $x_{52},$ $x_{13},$ $x_{64},$ $x_{55},$ $x_{06},$ $x_{16},$ $x_{56},$ $x_{66},$ $x_{67},$ $x_{08},$ $x_{69}$ to 1 in the symbolic slack matrix in Figure \ref{FIG:nonfano_fig}.
Taking the ideal of $4$-minors and saturating, we find that the ideal consists of equations of the form $x_{i,j}-\alpha_{i,j}$, for $\alpha_{i,j}\in\mathbb{Q}$, so the configuration is projectively unique over $\mathbb{Q}$. The slack matrix corresponding to the single point in $\mathcal{V}(I_M)\cap(\mathbbm{k}^*)^{39}/T_{7,9}$ is
\scriptsize
$$\bgroup\begin{pmatrix}
1&0&1&0&0&1&1&1&1\\
0&0&1&1&-1&1&2&0&0\\
0&1&-1&0&1&0&-1&-1&1\\
-1&1&-1&1&0&0&0&-2&0\\
1&0&0&-1&1&0&-1&1&1\\
1&1&0&0&1&1&0&0&2\\
0&1&0&1&0&1&1&-1&1\\
\end{pmatrix}\egroup.$$
\end{example}
\begin{example} \label{EG:perles} Consider the \emph{Perles configuration} $M_\star$ of Figure~\ref{FIG:perles}.
It is a matroid on 9 elements with hyperplanes given by $0 6 7 8$, $3 4 7$, $1 5 6$, $1 2 8$, $0 45$, $3 5 8$, $0 1 3$, $2 4 6$, $2 5 7$, $48$, $1 7$, $3 6$, $1 4$, $2 3$, $0 2$. Its symbolic slack matrix is shown in Figure~\ref{FIG:perles} and
has the matrix $S(\mathbf{x})$ studied in \cite[\S4.3]{slack_paper} as a submatrix.
\begin{figure}[h]
\begin{center}
\includegraphics[height=1.7in]{perles}
\scriptsize
\[
\begin{blockarray}{cccccccccccccccc}
&H_1 & H_2 & H_3 & H_4 & H_5 & H_6 & H_7 &H_8 &H_9 &H_{10} &H_{11} &H_{12} &H_{13} &H_{14} &H_{15} \\
&0 6 7 8 &3 4 7 &1 5 6 &1 2 8 &0 45 &3 5 8 &0 1 3 &2 4 6 &2 5 7 &48 &1 7 &3 6 &1 4 &2 3 &0 2 \\
\begin{block}{c[ccccccccccccccc]}
0 &0 &x_{02} & x_{03} & x_{04} &0 & x_{06} &0 & x_{08} & x_{09} & x_{0,10} & x_{0,11} & x_{0,12} & x_{0,13} & x_{0,14}&0 \\
1 &x_{11} & x_{12} &0 &0 & x_{15} & x_{16} &0 & x_{18} & x_{19} & x_{1,10} &0 & x_{1,12} &0 & x_{1,14}& x_{1,15}\\
2 &x_{21} & x_{22} & x_{23} &0 & x_{25} & x_{26} & x_{27} &0 &0 & x_{2,10} & x_{2,11} & x_{2,12} & x_{2,13} &0 &0 \\
3 &x_{31} & 0 & x_{33} & x_{34} & x_{35} &0 &0 & x_{38} & x_{39} & x_{3,10} & x_{3,11} &0 & x_{3,13} &0 & x_{3,15}\\
4 &x_{41} &0 & x_{43} & x_{44} &0 & x_{46} & x_{47} &0 & x_{49} &0 & x_{4,11} & x_{4,12} &0 & x_{4,14}& x_{4,15}\\
5 &x_{51} & x_{52} &0 & x_{54} &0 &0 & x_{57} & x_{58} &0 & x_{5,10} & x_{5,11} & x_{5,12} & x_{5,13} & x_{5,14}& x_{5,15}\\
6 &0 & x_{62} &0 & x_{64} & x_{65} & x_{66} & x_{67} &0 & x_{69} & x_{6,10}& x_{6,11} &0 & x_{6,13} & x_{6,14}& x_{6,15}\\
7 &0 &0 & x_{73} & x_{74} & x_{75} & x_{76} & x_{77} & x_{78} &0 & x_{7,10} &0 & x_{7,12} & x_{7,13} & x_{7,14}& x_{7,15}\\
8 &0 & x_{82} & x_{83} &0 & x_{85} & 0 & x_{87} & x_{88} & x_{89} &0 & x_{8,11} & x_{8,12} & x_{8,13} & x_{8,14}& x_{8,15}\\
\end{block}
\end{blockarray}
\]
\caption{The Perles configuration matroid $M_\star$ with non-bases shown as lines, and its symbolic slack matrix. }
\label{FIG:perles}
\end{center}
\end{figure}
Since the ideal of $S(\mathbf{x})$ will be contained in the ideal of the whole matrix $S_{M_\star}(\mathbf{x})$, it follows from computation in \cite{slack_paper} that $M_\star$ is not realizable over $\mathbb{Q}$. However, it is realizable over $\mathbb{R}$, and computing its scaled slack ideal we find that the slack variety consists of the following matrices, where $\alpha$ is a root of the polynomial $\alpha^2-3\alpha+1$:
\tiny
\[
\begin{blockarray}{ccccccccccccccc}
\begin{block}{[ccccccccccccccc]}
0 &1 &\alpha-3 &3-\alpha &0 &3-\alpha &0 &-1 &1 &3-\alpha &1 &\alpha-2 &2-\alpha &2-\alpha &0\\
2\alpha-5&1 &0 &0 &3-\alpha &3-\alpha &0 &2-\alpha &3-\alpha &\alpha-2 &0 &\alpha-2 &0 &2-\alpha &5-2\alpha\\
1 &\alpha &1 &0 &\alpha &\alpha-1 &1 &0 &0 &\alpha &1-\alpha &1 &\alpha &0 &0\\
1 &0 &\alpha-1 &1-\alpha &\alpha-1 &0 &0 &\alpha-1 &-\alpha &1 &1-2\alpha&0 &\alpha &0 &-1\\
2-\alpha &0 &2-\alpha &\alpha-1 &0 &\alpha-2 &1 &0 &\alpha-1 &0 &\alpha &2-\alpha &0 &\alpha-1&\alpha-1\\
2-\alpha &1-\alpha &0 &1 &0 &0 &1 &\alpha-1 &0 &2-\alpha &1 &1-\alpha &1 &\alpha &\alpha-1\\
0 &1 &0 &1 &1 &1 &1 &0 &1 &1 &1 &0 &1 &1 &1\\
0 &0 &3-\alpha &\alpha-2 &1 &\alpha-2 &1 &1 &0 &\alpha-2 &0 &2-\alpha &\alpha-1 &\alpha-1 &1\\
0 &\alpha+1 &1 &0 &1 &0 &1 &\alpha &1-\alpha &0 &1-\alpha &1-\alpha &\alpha &\alpha&1\\
\end{block}
\end{blockarray}.
\]
\normalsize
Over any field where $5$ has a square root, the variety has degree 2 and dimension~0, so it consists of two points obtained by setting $\alpha = \frac{3 \pm \sqrt{5}}{2}$. Over $\mathbb{F}_5$, this polynomial has a double root $\alpha=4$. In particular, it is projectively unique here and the scaled slack variety is non-reduced.
\end{example}
\newpage
\subsection{Acknowledgements}
We would like to thank Bernd Sturmfels and Rekha Thomas for their guidance and helpful discussion. We also thank Dan Corey for pointing us to examples of matroids with interesting realization spaces.
The software package \texttt{Macaulay2} \cite{M2} was invaluable in calculating all of the examples from this paper. In addition, the package ``Matroids" written for \texttt{Macaulay2} by Justin Chen was indispensable.
We also acknowledge the Mathematical Sciences Research Institute, the Max-Planck-Institut f\"ur Mathematik in den Naturwissenschaften, and the University of Washington for facilitating our collaboration on this paper.
\newpage
|
{
"timestamp": "2018-04-17T02:08:03",
"yymm": "1804",
"arxiv_id": "1804.05264",
"language": "en",
"url": "https://arxiv.org/abs/1804.05264"
}
|
\section{Additional lemmata}
\paragraph{\textbf{Energy estimates in the interior}}
\begin{lemma}\label{lem:exponentialdecay}
Let $\Psi \in C_c^\infty(\mathcal H)$ and denote by $\psi$ its evolution in the interior. Then, the non-degenerate $N$ energy of $\Psi$ decays exponentially towards $i_+$ on every $\{r=r_0 \}$ hypersurface for $r_{red}< r_0 < r_+$. Here, $r_{red}$ only depends on the black hole parameters.
\begin{proof}
This argument is very similar to \cite[Proposition 4.2]{franzen2016boundedness}. We only prove it for the right component of $i^+$ and clearly only have to look at a neighborhood of $i^+$. First, recall the existence of the celebrated redshift vector field $N$ satisfying $K^N[\psi] \geq b J^N_\mu[\psi]n_v^\mu$ for $r_+ \geq r\geq r_{red}$, where $n_v$ is the normal to a $v=const.$ hypersurface.\footnote{The normal is fixed by making a choice of a volume form on the null hypersurface}
We set
\begin{align}
E(v_0) = \int_{v=v_0, r_{red}\leq r \leq r_+} J^N_\mu n_v^\mu \d\mathrm{vol},
\end{align}
and apply the energy identity with the redshift vector field $N$ in the region $\mathcal{R} = \{ r\in[r_{red}, r_+], v \in [v_0, v_1] \}$, where $v_0$ is large enough such that $v_0 > \sup\operatorname{supp}(\Psi)$.
This gives in view of the coarea formula that
\begin{align}
E(v_1) - E(v_0) + \tilde b \int_{v_0}^{v_1} E(v) \d v \leq 0 \label{eq:expbound}
\end{align}
for every $v_1 \geq v_0> \sup \text{supp}(\Psi)$. Inequality~\eqref{eq:expbound}, smoothness of $v\mapsto E(v)$ and a further application of the energy identity in the region $\{ v\geq v_0, r_+ \geq r \geq r_{red} \}$ finally shows
\begin{align}
\int_{v\geq v_0, r=r_{red}}J^N_\mu n_{r}^\mu \d{\text{vol}}\leq C \exp(-\tilde b v_0),
\end{align}where $C$ is a constant depending on $\Psi$. This concludes the proof.
\end{proof}
\end{lemma}
\begin{rmk}\label{rmk:approxwithcompactly}
By cutting off smoothly we can clearly approximate $\Psi$ on a $\{r=const.\}$ hypersurface with compactly supported functions for any fixed $r\in(r_{red}, r_+)$.
\end{rmk}
\begin{lemma}\label{lem:estimatetorconst}
Let $\psi$ be a smooth solution of the wave equation on $\mathcal{M}_\mathrm{RN}$ such that its restriction to the event horizon has compact support and let $r_0 \in (r_{red}, r_+)$. Then,
\begin{align}
\int_{\mathcal{H}} J^T_{\mu} n^\mu \d\mathrm{vol} \lesssim \int_{ \{r=r_0\} } J^N_\mu n^\mu \d \mathrm{vol}.
\end{align}
\begin{proof}
We shall use the vector field $S = r^{-2} \partial_{r_\ast}$. By potentially making $r_{red}$ larger, we can assure that the bulk term $K^S := \nabla^\mu J^{S}_\mu$ of the vector field $S$ has a fixed negative sign in $r_0\in(r_{red},r_+)$. This current is analogous to the current introduced in \cite[par.\ 4.1.3.2]{franzen2016boundedness}. Moreover, applying the energy identity in the region $\mathcal{R}=\{ r_0 \leq r \leq r_+ \}$ and noting that $J^N[\psi]_\mu n^\mu\vert_{r=r_0} \sim J^S[\psi]_\mu n^\mu\vert_{r=r_0}$ as well as $J^T[\psi]_\mu n^\mu \vert_\mathcal{H} \sim J^S[\psi]_\mu n^\mu\vert_\mathcal{H}$ yields
\begin{align}
\int_{ \{r=r_0\} } J^N[\psi]_\mu n^\mu \d\mathrm{vol} + \int_{\mathcal{R}} K^S \d\mathrm{vol} \gtrsim \int_{\mathcal{H}} J^T_\mu n^\mu \d\mathrm{vol}.
\end{align}
This concludes the proof.
\end{proof}
\end{lemma}
\section{Proof of \texorpdfstring{\cref{thm:cosmological}}{Theorem 6}: Breakdown of \texorpdfstring{$T$}{T} energy scattering for cosmological constants \texorpdfstring{$\Lambda\neq 0$}{Lambdaneq0}}
\label{sec:cosmo}
In the presence of a cosmological constant $\Lambda$, the situation regarding the $T$ energy scattering problem is changed radically. In this section we will consider the subextremal (anti-) de Sitter--Reissner--Nordström black hole interior $(\mathcal{M}_{\text{(a)dSRN}} , g_{Q,M,\Lambda})$ which is completely analogous to $(\mathcal{M}_{\text{RN}} , g_{Q,M}$). We will assume that $(M,Q,\Lambda) \in \mathcal{P}_{\mathrm{se}}$ as defined in \cref{subsec:nonex}.
Also, recall that in the presence of a cosmological constant it is natural to look at the Klein--Gordon equation
\begin{align}\label{eq:kg}
\Box_g \psi - \mu \psi = 0
\end{align} with mass $\mu = \frac{3}{2} \Lambda$ for the conformal invariant equation or more general $\mu = \nu \Lambda$ for fixed $\nu \in \mathbb R$.
This section is devoted to prove \cref{thm:cosmological} which relies on the fact that solutions of the corresponding radial o.d.e.\ in the vanishing frequency limit $\omega=0$ generically map bounded solutions at $r_\ast = -\infty$ to unbounded solutions at $r_\ast = +\infty$. More precisely, for $\Lambda \neq 0$ we obtain---after separation of variables for \eqref{eq:kg} and setting $\d r_\ast = h^{-1} \d r$---the o.d.e.\
\begin{align}
\label{eq:ODElambda}
-u^{\prime \prime } + V_{\ell,\Lambda} u = \omega^2 u
\end{align}
for $u(r_\ast) = r(r_\ast) R(r_\ast)$, where
\begin{align}\label{eq:potentiallam}
V_{\ell,\Lambda} = h \left(\frac{ h h^\prime}{r} + \frac{\ell(\ell+1)}{r^2} - \mu \right) = h \left(\frac{ \frac{\d h}{\d r}}{r} + \frac{\ell(\ell+1)}{r^2} - \mu \right)
\end{align}
and \begin{align}\label{eq:hlambda}h= \frac{\Delta}{r^2} = 1- \frac{2M}{r}- \frac 13 \Lambda r^2 + \frac{Q^2}{r^2}.\end{align}
Here, consider $r(r_\ast)$ as a function $r_\ast$ and recall that $~^\prime$ denotes the derivative with respect to $r_\ast$. The presence of the mass and the cosmological constant leads to a modification of the potential $V_{\ell,\Lambda}$.
Nevertheless, the potential $V_{\ell,\Lambda}$ still decays exponentially at $\pm \infty$ and we can define asymptotic states $u_1^{(\Lambda)},u_2^{(\Lambda)}$, and $ v_1^{(\Lambda)},v_2^{(\Lambda)}$ for $\omega\neq 0$ and $\tilde u_1^{(\Lambda)}$, $\tilde u_2^{(\Lambda)}$, and $\tilde v_1^{(\Lambda)},\tilde v_2^{(\Lambda)}$ for $\omega =0$ just as in the case where $\Lambda=\mu =0$ in \cref{defn:u1u2}.
In particular, $\tilde u_1^{(\Lambda)}$ and $\tilde v_1^{(\Lambda)}$ remain bounded as $r_\ast \to -\infty$ and $r_\ast \to + \infty$, respectively. In contrast to that, $\tilde u_2^{(\Lambda)}$ and $\tilde v_2^{(\Lambda)}$ grow linearly in their respective limits. The next proposition states that in the presence of a cosmological constant, solutions to \eqref{eq:kg} in the case $\omega=0$ which are bounded at $r_\ast =-\infty$ do not need to be bounded at $r_\ast = +\infty$.
\begin{prop}\label{prop:Bneq0} Fix $\nu \in \mathbb{R}$ (e.g.\ $\nu = \frac{3}{2}$ for the conformal invariant mass) and fix subextremal black hole parameters $(M,Q,\Lambda) \in \mathcal{P}_{\mathrm{se}}$. Assume moreover that $(M,Q,\Lambda) \notin D(\nu)$, where $D(\nu)\subset \mathcal{P}_{\mathrm{se}}$ is defined in the proof and has measure zero. Then, there exists an $\ell_0 = \ell_0(\nu) \in \mathbb{N}_0$ such that we have \begin{align}\label{eq:decompu1inlambda}\tilde u_1^{(\Lambda)} =A(\ell_0 ,\Lambda,M,Q) \tilde v_1^{(\Lambda)} + B(\ell_0,\Lambda,M,Q) \tilde v_2^{(\Lambda)},
\end{align}
with $B = B(\ell_0, \Lambda,M,Q) \neq 0$.
Moreover, $\mathcal{P}_{\mathrm{se}}^{\Lambda =0}\subset D(\nu)$ for all $\nu \in \mathbb R$ and there exists an open subset $U$ with $\mathcal{P}_{\mathrm{se}}^{\Lambda =0} \subset U\subset \mathcal{P}_{\mathrm{se}}$ and $\mathcal{P}_{\mathrm{se}} \cap U = \mathcal{P}_{\mathrm{se}}^{\Lambda =0}$.
\begin{proof} Let $\nu \in \mathbb R$ be fixed.
In the case $\Lambda = 0$ we can represent $\tilde u_1$ with Legendre polynomials and in particular we have that $B(\ell,\Lambda=0,M,Q ) =0$ for all $\ell$ and $0<|Q|<M$.
Note that we can write $B$ as
\begin{align}\label{eq:defnb}
B(\Lambda,\ell,M,Q) = \frac{\mathfrak W (\tilde v_2^{(\Lambda)}, \tilde u_1^{(\Lambda)})}{\mathfrak W(\tilde v_1^{(\Lambda)}, \tilde v_2^{(\Lambda)})} = \mathfrak W (\tilde v_2^{(\Lambda)}, \tilde u_1^{(\Lambda)})
\end{align}
for all $\Lambda$ such that $(M,Q,\Lambda) \in \mathcal{P}_{\mathrm{se}}$.
\paragraph{Step 1: \texorpdfstring{$\mathcal{P}_{\mathrm{se}}\subset \mathbb{R}^3$ is open and has two connected components where either $Q>0$ or $Q<0$}{Pse is connected}} For the sake of completeness we will give a proof of \emph{Step 1}, although this seems a quite well-known fact.
Note that $\mathcal{P}_{\mathrm{se}} = \mathcal{P}_{\mathrm{se}}^{\Lambda >0} \cup\mathcal{P}_{\mathrm{se}}^{\Lambda <0} \cup\mathcal{P}_{\mathrm{se}}^{\Lambda =0} $ is open which can be inferred from its definition.
For the second statement, first note that $\{Q=0\}\cap \mathcal{P}_{\mathrm{se}} =\emptyset$. We will now show that $ \{Q>0\}\cap \mathcal{P}_{\mathrm{se}}$ is connected. In \cref{prop:connectedness} in the appendix we show that $\mathcal{P}_{\mathrm{se}}^{\Lambda >0} \cap \{ Q>0\}$ and $\mathcal{P}_{\mathrm{se}}^{\Lambda <0} \cap \{ Q>0\}$ are path-connected.
To conclude, note that for every $(M_0,Q_0,\Lambda_0 = 0 )\in \mathcal{P}_{\mathrm{se}}^{\Lambda=0}$, there exist paths from $(M_0,Q_0,\Lambda_0)$ to both $(M_0,Q_0,\epsilon) \in \mathcal{P}_{\mathrm{se}}^{\Lambda >0}$ and $(M_0,Q_0,-\epsilon) \in \mathcal{P}_{\mathrm{se}}^{\Lambda <0}$ for some $\epsilon(M_0,Q_0)>0$. Together with the fact that $\mathcal{P}_{\mathrm{se}}^{\Lambda=0}\cap \{ Q >0\}$ is path-connected, this shows that $ \{Q>0\}\cap \mathcal{P}_{\mathrm{se}}$ is path-connected and similarly that $ \{Q<0\}\cap \mathcal{P}_{\mathrm{se}}$ is path-connected which proves the claim.
\paragraph{Step 2: \texorpdfstring{$\mathcal{P}_{\mathrm{se}}\ni (M,Q,\Lambda) \mapsto B(\ell,\Lambda,M,Q)$ is real analytic}{B is holomorphic}}
To show \emph{Step 2} we first express \eqref{eq:decompu1inlambda} in $r$ coordinates. Note that for $(M,Q,\Lambda)\in \mathcal{P}_{\mathrm{se}}$ equation \eqref{eq:decompu1inlambda} is equivalent to
\begin{align}\label{eq:p=ptilde}
\frac{r_+}{r_-} (-1)^\ell P_\ell^{(\Lambda)} (x(r)) = A(\ell,\Lambda) \tilde P^{(\Lambda)}_\ell (x(r)) + B(\ell,\Lambda) \tilde Q^{(\Lambda)}_\ell (x(r)),
\end{align}
where $r\in (r_-, r_+)$,
\begin{align}
x(r):= - \frac{2r}{r_+ - r_-} + \frac{r_+ + r_-}{r_+ - r_-},
\end{align}
\begin{align}
r(x) =- \frac{r_+ - r_-}{2} x+ \frac{r_+ + r_-}{2}
\end{align}
and $0<r_-<r_+$. Now, note that $\mathcal{P}_{\mathrm{se}} \ni (M,Q,\Lambda)\mapsto r_-$ and $\mathcal{P}_{\mathrm{se}}\ni (M,Q,\Lambda) \mapsto r_+$ are real analytic. Moreover, we can write $\Delta = (r - r_-) (r-r_+) p(r)$ for a second order polynomial $p(r)$, where $\mathcal{P}_{\mathrm{se}} \ni \Lambda \mapsto p(r) $ is also real analytic for fixed $r$. Now, $P_\ell^{(\Lambda)}$, $\tilde P_\ell^{(\Lambda)}$ and $ \tilde Q_\ell^{(\Lambda)}$ appearing in \eqref{eq:p=ptilde} are defined as the unique solutions of
\begin{align}\label{eq:legendreperturbed}
\frac{\d }{\d x} \left( (1-x^2) p(r(x)) \frac{\d R}{\d x}\right) + \ell (\ell+1) R - r(x)^2 \nu \Lambda R=0
\end{align}
satisfying
\begin{align}\label{eq:propertiesfirst}
&P_\ell^{(\Lambda)} = (-1)^\ell +O_\ell(1+x) \text{ as } x\to -1,\\
& \frac{\d P_\ell^{(\Lambda)}}{\d x} = O_\ell(1) \text{ as } x\to-1,\\
&\tilde P_\ell^{(\Lambda)} = 1 +O_\ell(1-x) \text{ as } x\to 1,\\
& \frac{\d \tilde P_\ell^{(\Lambda)}}{\d x} = O_\ell(1) \text{ as } x\to1,\\
&\tilde Q_\ell^{(\Lambda)} = - \frac 12 \log(1-x) + O_\ell(1) \text{ as } x\to 1,\\
& \frac{\d \tilde Q_\ell^{(\Lambda)}}{\d x} = \frac{1}{2(1-x)} + O_\ell( (1-x) \log(1-x)) \text{ as } x\to 1.\label{eq:propertieslast}
\end{align}
Note that \eqref{eq:legendreperturbed} depends real analytically on $(M,Q,\Lambda) \in \mathcal{P}_{\mathrm{se}}$ such that $ P_\ell^{(\Lambda)}(x)$, $\tilde P_\ell^{(\Lambda)}(x)$, $\tilde Q_\ell^{(\Lambda)}(x)$ are real analytic functions of $(M,Q,\Lambda) \in \mathcal{P}_{\mathrm{se}}$ for $x\in (-1,1)$. Hence, $\mathcal{P}_{\mathrm{se}} \ni (M,Q,\Lambda) \mapsto B(\ell, \Lambda,M,Q)$ is real analytic.
\paragraph{Step 3: \texorpdfstring{$B(\ell_0(\nu),\Lambda,M,Q)$}{Bdlambda} only vanishes on a set \texorpdfstring{$D(\nu) \subset \mathcal{P}_{\mathrm{se}}$}{D} of measure zero.} The claim follows from
\begin{align}\label{eq:dbdlam}\left.\frac{\partial B(\ell,\Lambda,M_0,Q_0)}{\partial \Lambda}\right|_{\Lambda = 0}\neq 0\end{align}
for some $0<|Q_0|<M_0$. Throughout \emph{Step 2} we fix $0<|Q_0|<M_0$ and avoid writing their explicit dependence.
First note that that for $\Lambda =0$ we obtain the Legendre functions of first and second kind, i.e.\ $P_\ell^{(0)} = \tilde P_\ell^{(0)} = P_\ell $ and $\tilde Q_\ell^{(0)} = Q_\ell$ and $B(0,\ell) =0$.
Now, define coefficients $\tilde A(\ell,\Lambda)$ and $\tilde B(\ell,\Lambda)$ to satisfy
\begin{align}
\label{eq:expansion11}
P_\ell^{(\Lambda)} = \tilde A(\ell,\Lambda) \tilde P_\ell^{(\Lambda)} + \tilde B (\ell,\Lambda)\tilde Q_\ell^{(\Lambda)},
\end{align}
and note that \eqref{eq:dbdlam} is equivalent (use that $B(\ell,0) = \tilde B(\ell,0) = 0 $) to
\begin{align}
\frac{\partial \tilde B(\ell,\Lambda)}{\partial \Lambda}\Big\vert_{\Lambda =0} \neq 0 .
\end{align}
By construction, $P_\ell^{(\Lambda)}$ solves \eqref{eq:legendreperturbed}. Multiplying
\begin{align}
\frac{\d }{\d x} \left( (1-x^2) p(r(x)) \frac{\d P_\ell^{(\Lambda)}}{\d x}\right) + \ell (\ell+1) P_\ell^{(\Lambda)} - r(x)^2 \nu \Lambda P_\ell^{(\Lambda)}=0
\end{align}
by $P_\ell^{(0)}$ and integrating from $x=-1$ to $x=1$ yields
\begin{align}
0= \int_{-1}^{1}P_\ell^{(0)} \left( \frac{\d }{\d x} \left( (1-x^2) p(r(x)) \frac{\d P_\ell^{(\Lambda)}}{\d x}\right) + \ell (\ell+1) P_\ell^{(\Lambda)} - r(x)^2\nu \Lambda P\ell^{(\Lambda)} \right)\d{x}.
\end{align}
Using the expansion \eqref{eq:expansion11} and the properties \eqref{eq:propertiesfirst} -- \eqref{eq:propertieslast} at the end points $x=-1$ and $x=1$ gives after an integration by parts
\begin{align}0= & \int_{-1}^{1}P_\ell^{(\Lambda)} \left( \frac{\d }{\d x} \left( (1-x^2) p(r(x)) \frac{\d P_\ell^{(0)}}{\d x}\right) + \ell (\ell+1) P_\ell^{(0)} - r(x)^2\nu \Lambda P_\ell^{(0)}\right)\d{x} + p(r(1)) \tilde B(\ell,\Lambda). \end{align}
Now, taking $\partial_\Lambda \big\vert_{\Lambda =0}$ and integrating by parts once again yields
\begin{align}
p(r(1)) \partial_{\Lambda} \big\vert_{\Lambda =0} \tilde B (\ell, \Lambda) & = \int_{-1}^1 \left[ \left| \frac{\d P_\ell^{(0)} }{\d x}\right|^2 (1-x^2) \partial_{\Lambda} \big\vert_{\Lambda =0} ( p(r(x)) )+ \left|P_\ell^{(0)}\right|^2\partial_{\Lambda}\big\vert_{\Lambda =0} ( \nu r(x)^2 \Lambda)\right] \d x \nonumber \\ & = \int_{-1}^1 \left[ \left| \frac{\d P_\ell^{(0)} }{\d x}\right|^2 (1-x^2) \partial_{\Lambda} \big\vert_{\Lambda =0}( p(r(x)) )+\nu \left|P_\ell^{(0)}\right|^2 r(x)^2\vert_{\Lambda =0} \right] \d x .
\end{align}
Recall that we are in the subextremal range which guarantees that $p(r(1))\neq 0$. We will now distinguish two cases, $\nu =0$ and $\nu \neq 0$.
\paragraph{Part I: $\nu =0$}In the case $\nu =0$ we have
\begin{align}
p(r(1)) \partial_\Lambda\vert_{\Lambda =0} \tilde B(\ell,\Lambda) = \partial_\Lambda\vert_{\Lambda =0} \int_{-1}^{1} \left|\frac{\d P_\ell}{\d x}\right|^2 (1-x^2)p(r(x)) \d x
\end{align}
In the case $\nu =0$ we will choose $\ell =1$ such that
\begin{align*} p(r(1)) \partial_\Lambda\vert_{\Lambda =0} \tilde B(1,\Lambda) =& \partial_\Lambda\vert_{\Lambda =0} \int_{-1}^{1} (1-x^2)p(r(x)) \d x \\ = &
\partial_\Lambda\vert_{\Lambda =0} \int_{-1}^{1} -\Delta(r(x)) \frac{4}{(r_+ - r_-)^2} \d x
\\ = &
\partial_\Lambda\vert_{\Lambda =0}\left( \frac{-8}{(r_+ - r_-)^3} \int_{r_-}^{r_+} \Delta(r) \d r\right)
\\= & -8\, \partial_\Lambda\vert_{\Lambda =0} \left(
\frac{\frac{r_+^3-r_-^3}{3} - M_0 (r_+^2 - r_-^2) +Q_0^2(r_+ - r_-) -\frac {1}{15} \Lambda (r_+^5 - r_-^5) }{(r_+ - r_-)^3}\right)\\ =&\,
\frac{8(r_+^5-r_-^5)}{15(r_+-r_-)^3} \Big\vert_{\Lambda =0} \\ & + 8
\frac{\frac{r_+^3-r_-^3}{3} - M_0 (r_+^2 - r_-^2) +Q_0^2(r_+ - r_-) }{(r_+ - r_-)^5} (r_+^4 + r_-^4)\Big\vert_{\Lambda =0}\\ &
- \frac{8}{3} \frac{ r_+^6 + r_-^6 - 2M_0 (r_+^5+r_-^5) + Q_0^2(r_+^4 + r_-^4)}{(r_+ - r_-)^4}\Big\vert_{\Lambda =0} \\ = &
\frac{-8}{15} \left( 3 r_+^3 + 3 r_-^2 + 4 r_+ r_- \right)\Big\vert_{\Lambda =0} \\ = & \frac{-8}{15} \left(6 M_0^2-Q_0^2\right) < - 24 M_0^2 .
\end{align*}
The last step is a long but direct computation using that $\Delta=r^2 - 2M_0r + Q_0^2 -\frac{\Lambda}{3} r^4$ and $r_{\pm}\vert_{\Lambda =0} = M_0\pm \sqrt{M_0^2-Q_0^2}$, i.e.\ $Q_0^2 = r_+ r_-\vert_{\Lambda =0}$ and $2M_0 = r_+\vert_{\Lambda =0} + r_-\vert_{\Lambda =0}$. Moreover, in view of the inverse function theorem we have
\begin{align}
\partial_\Lambda \vert_{\Lambda =0} r_+ = \frac{ r_+^4{}}{3(r_+-r_-) } \Big\vert_{\Lambda =0}
\end{align}
and
\begin{align}
\partial_\Lambda \vert_{\Lambda =0} r_- = - \frac{ r_-^4}{3 (r_+-r_-)}\Big\vert_{\Lambda =0}.
\end{align}
\paragraph{Part II: $\nu \neq 0$} In this case we choose $\ell=0$ such that $P_\ell^{(0)} = 1$ and $\frac{\d P_\ell^{(0)}}{\d x } = 0$. Hence,
\begin{align}\nonumber
p(r(1)) \partial_\Lambda\vert_{\Lambda =0} \tilde B(\ell,\Lambda) &= \partial_\Lambda\vert_{\Lambda =0} \int_{-1}^{1} r(x)^2 \nu \Lambda \d x = \nu \partial_{\Lambda}\vert_{\Lambda =0} \int_{-1}^{1} \left( -\frac{r_+ - r_-}{2}x + \frac{r_+ + r_-}{2}\right)^2 \Lambda \d x\\ & = \nu \left(\frac{1}{6} {(r_+ - r_-)^2} + \frac{1}{2} (r_+ + r_-)^2 \right)\Big\vert_{\Lambda =0} \neq 0.
\end{align}
This shows that $\mathcal{P}_{\mathrm{se}} \ni (M,Q,\Lambda) \mapsto B(\ell_0(\nu), M,Q,\Lambda)$ is a non-trivial real analytic function which zero set $D(\nu)$ has zero measure. The proof also shows that $\mathcal{P}_{\mathrm{se}}^{\Lambda =0}\subset D(\nu)$ and that there exists an open set $U\subset \mathcal{P}_{\mathrm{se}}$ with $\mathcal{P}_{\mathrm{se}}^{\Lambda =0} \subset U$ and $D(\nu) \cap U = \mathcal{P}_{\mathrm{se}}^{\Lambda =0}$.
\end{proof}
\end{prop}
\begin{prop}\label{cor:RTunbounded}Let $\nu \in \mathbb{R}$ be fixed.
Let $\omega\neq 0$, $(M,Q,\Lambda) \in \mathcal{P}_{\mathrm{se}}$, and $\ell \in \mathbb N_0$. Then, define completely analogously to \cref{defn:TandR} transmission and reflection coefficients $\mathfrak T(\omega,\ell,\Lambda)$ and $\mathfrak R(\omega,\ell,\Lambda)$ as the unique coefficients such that
\begin{align}
u_1^{(\Lambda)} = \mathfrak T (\omega,\ell,\Lambda) v_1^{(\Lambda)} + \mathfrak R(\omega,\ell,\Lambda) v_2^{(\Lambda)}
\end{align}
holds.
Now, assume further that $(M,Q,\Lambda) \in \mathcal{P}_{\mathrm{se}}\setminus D(\nu)$, where $D(\nu)$ is defined in \cref{prop:Bneq0}. Then, there exists an $ \ell_0 = \ell_0(\nu)$ such that
\begin{align}
\lim_{\omega \to 0} |\mathfrak R(\omega,\ell_0)| = \lim_{\omega \to 0} |\mathfrak T(\omega,\ell_0)| = +\infty.
\end{align}
This shows that $\mathfrak T$ and $\mathfrak R$ have a simple pole at $\omega=0$.
\begin{proof}Fix $\ell_0 = \ell_0(\nu)$ from \cref{prop:Bneq0} and $(M,Q,\Lambda) \in \mathcal{P}_{\mathrm{se}}$ such that $B(\ell_0,\Lambda,M,Q) \neq 0$. Now, note that the o.d.e.\ implies that $\frac{\d}{\d r_\ast}\text{Im}( \bar u u^\prime) = 0 $ which shows that $1 = |\mathfrak T|^2 - |\mathfrak R|^2$. In particular, either $|\mathfrak T|$ and $|\mathfrak R|$ are both bounded or both unbounded as $\omega \to 0$. Also note that as $\omega \to 0$, we have that $u_1^{(\Lambda)}\to \tilde u_1^{(\Lambda )}$ pointwise.
Now, assume for a contradiction that there exists a sequence $\omega_n \to 0$ such that $|\mathfrak T(\omega_n)|$ and $|\mathfrak R(\omega_n)|$ remain bounded. Thus, \begin{align}\nonumber\limsup_{\omega_n \to 0} \|u_1^{(\Lambda)} \|_{L^\infty(\mathbb{R})}\leq& \limsup_{\omega_n \to 0} \|u_1^{(\Lambda)} \|_{L^\infty((-\infty,0))}\\& + \limsup_{\omega_n \to 0} \|\mathfrak R v_1^{(\Lambda)} + \mathfrak T v_2^{(\Lambda)} \|_{L^\infty((0,\infty))} \leq C \end{align} for some constant $C>0$.
Now, using that $B(\ell_0,\Lambda,M,Q)\neq 0$ in \cref{prop:Bneq0}, we can choose a $r^\ast_{0} \in \mathbb{R}$ such that $|\tilde u_1^{(\Lambda)}(r^\ast_{0})| > C$ which contradicts the fact that $u_1^{(\Lambda)} \to \tilde u_1^{(\Lambda)}$ pointwise as $\omega_n \to 0$.
\end{proof}
\end{prop}
Finally, this allows us to prove \cref{thm:cosmological} which we restate in the following for the convenience of the reader.
\cosmological*
\begin{proof}Fix $\ell_0 = \ell_0(\nu)$ from \cref{cor:RTunbounded} such that the reflection and transmission coefficients blow up as $\omega \to 0$.
Define a sequence of compactly supported functions $\Psi_n $ on $\mathcal{H}_A$ by $\Psi_n(v,\theta,\varphi) = f_n(v) Y_{0\ell}(\theta,\varphi)$, such that $f_n \in C_c^\infty(\mathbb R)$, \begin{align}\int_{\mathbb R} \omega^2 |\hat f_n (\omega)|^2 \d \omega = 1 \text{ and } \int_{-\frac{1}{n}}^{\frac{1}{n}} \omega^2 |\hat f_n (\omega)|^2 \d \omega \geq \epsilon \int_{\mathbb R} \omega^2 |\hat f_n (\omega)|^2 \d \omega = \epsilon\end{align}
for some $\epsilon >0$.\footnote{ Such a function can be constructed by setting $f_n(v) := \frac{c}{\sqrt n} f(\frac{v}{n})$ for smooth $f\colon \mathbb R \to [0,1]$ with $\operatorname{supp}(f) \subset [-2,2]$, $f\restriction_{[-1, 1]} =1 $ and some normalization constant $c >0$. Indeed,
\begin{align}
\int_{-\frac 1n}^{\frac 1n} \omega^2 |\hat f_n(\omega)|^2 \d \omega = \int_{-\frac 1n}^{\frac 1n} \omega^2 |\sqrt n \hat f(n\omega)|^2\d \omega = \int_{-1}^{1} \omega^2 |\hat f (\omega)|^2 =: \epsilon >0
\end{align} in view of $\hat f (0) = \int_\mathbb R f(v) \d v >0$.} Imposing vanishing data on $\mathcal{H}_B$, this gives rise to a unique smooth solutions $\psi_n$ up to but excluding the Cauchy horizon.
Arguments completely analogous to those given in the proof of \cref{thm:thmbounds} show that
\begin{align}
\| \psi_n \restriction_{\mathcal{CH}}\|_{ \mathcal{E}^T_{\mathcal{CH}}}^2 = \frac{r_+^2}{r_-^2} \int_{\mathbb{R}} \omega^2 (|\mathfrak R(\omega,\ell)|^2 + |\mathfrak T(\omega,\ell)|^2) |\hat f_n(\omega)|^2\d \omega.
\end{align}
Thus, \begin{align}
\| \psi_n \restriction_{\mathcal{CH}}\|_{ \mathcal{E}^T_{\mathcal{CH}}}^2 \geq \frac{r_+^2}{r_-^2} \int_{-\frac 1n}^{\frac 1n} \omega^2 (|\mathfrak R(\omega,\ell)|^2 + |\mathfrak T(\omega,\ell)|^2) |\hat f_n(\omega)|^2\d \omega \geq \epsilon \,\frac{r_+^2}{r_-^2} \inf_{\omega \in [- \frac 1n, \frac 1n]}\left( |\mathfrak R|^2 + |\mathfrak T|^2 \right) .
\end{align}
Since $|\mathfrak R|,|\mathfrak T|\to\infty$ as $\omega \to 0$, also $\inf_{\omega \in [\frac {1}{2n}, \frac 1n]} |\mathfrak R| \to \infty$ and $\inf_{\omega \in [\frac {1}{2n}, \frac 1n]} |\mathfrak T| \to \infty$ as $n\to\infty$. Thus, as $n \to \infty$, we have
\begin{align}
\| \psi_n \restriction_{\mathcal{CH}}\|_{ \mathcal{E}^T_{\mathcal{CH}}}^2 \to \infty.
\end{align}
\end{proof}
\section{Proof of \texorpdfstring{\cref{thm:kleingordon}}{Theorem 7}: Breakdown of \texorpdfstring{$T$}{T} energy scattering for the Klein--Gordon equation}
\label{sec:kleingordonequation}
In this last section we will prove that for a generic set of Klein--Gordon masses, there does not exist a $T$ scattering theory on the interior of Reissner--Nordstr\"om for the Klein--Gordon equation. For the convenience of the reader, we have restated \cref{thm:kleingordon}.
\kleingordon*
\begin{proof}
The proof of this statement is easier than and similar to the proof of \cref{thm:cosmological} and the proofs of the propositions leading up to it. More precisely, similar to \cref{sec:cosmo} we define asymptotic states $\tilde u_1^{(\mu)}$, $\tilde v_1^{(\mu)}$ and $\tilde v_2^{(\mu)}$ and define $A(\ell,\mu)$ and $B(\ell,\mu)$ by
$\tilde u_1^{(\mu)} = A(\ell,\mu) \tilde v_1^{(\mu)} + B(\ell,\mu) \tilde v_2^{(\mu)}$. As in \cref{sec:cosmo}, $\mathbb R \ni \mu \mapsto B(\ell,\mu)$ is real analytic and from the o.d.e.\ $-u^{\prime\prime} + V_{\ell,\mu} u =0$ we obtain
\begin{align}
\left. \frac{\partial B(\ell,\mu)}{\partial \mu}\right|_{\mu = 0} = \int_{-\infty}^\infty\left. \frac{\partial V_{\ell,\mu}}{\partial \mu}\right|_{\mu =0} \tilde u_1^2 \d r_\ast,
\end{align}
where
\begin{align}
V_{\ell,\mu} = h \left(\frac{ h h^\prime}{r} + \frac{\ell(\ell+1)}{r^2} - \mu \right) = h \left(\frac{ \frac{\d h}{\d r}}{r} + \frac{\ell(\ell+1)}{r^2} - \mu \right)
\end{align}
and \begin{align}h= 1- \frac{2M}{r}+ \frac{Q^2}{r^2}\end{align}
as in \eqref{eq:h}. Now, note that
\begin{align}
\left. \frac{\partial V_{\ell,\mu}}{\partial \mu}\right|_{\mu =0} = - h >0
\end{align}
which is manifestly positive from which we can infer, by analyticity, that $B(\ell,\mu) \neq 0$ for all $\mu \in \mathbb R \setminus \tilde D$, where $\tilde D = \tilde D(M,Q)\subset \mathbb R$ is a discrete set. This proves the analogous statements to \cref{prop:Bneq0} and \cref{cor:RTunbounded}. The claim of \cref{thm:kleingordon} follows now as in the proof of \cref{thm:cosmological}.
\end{proof}
\section{Introduction}
\setcounter{page}{1}
One of the most stunning predictions of general relativity is the formation of \textbf{\textit{black holes}}, defined by the property that information cannot propagate from their interior region to outside far-away observers.
Fortunately, we can count ourselves among the latter; nevertheless, if a group of physicists were so courageous as to cross the \emph{event horizon} and enter a black hole, they could still very well perform experiments and compare the outcomes amongst themselves. Indeed, the problem of determining the fate of these black hole explorers (and their laboratories) has led to some of the most central conceptual puzzles in gravitational physics.
In view of the above, there has been
a lot of recent activity analyzing the Cauchy problem on black hole interiors, e.g.\ \cite{franzen,franzen2016boundedness,sbierski2014initial, luk2016instability,interiorschwarzschild}. However, for certain physical processes it is more natural to consider the \emph{\textbf{scattering problem}}
(see \cite{futterman1988scattering} for scattering on the exterior of black holes).
With this paper, we initiate the mathematical study of the finite energy scattering problem on black hole interiors.
Specifically, we will consider solutions of the wave equation on what can be viewed as the most elementary interior, that of Reissner--Nordstr\"om. The Reissner--Nordstr\"om metrics constitute a family of spacetimes, parametrized by mass $M$ and charge $Q$, which satisfy the Einstein--Maxwell system in spherical symmetry \cite{reissner1916eigengravitation,nordstrom1918energy} and admit an additional Killing vector field $T$. For vanishing charge $Q=0$, the family reduces to Schwarzschild. We shall moreover restrict in the following to the subextremal case where $0<|Q|<M$.
In addition to the bifurcate event horizon, these black hole interiors then admit an additional bifurcate inner horizon, the so-called \textit{Cauchy horizon}. Our past and future scattering states will be defined as suitable traces of the solution on the bifurcate event horizon and bifurcate Cauchy horizon, respectively, restricted to have finite $T$ energy flux on each component of the horizons.
In the rest of the introduction we will state our main results for the scattering problem on the interior of Reissner--Nordstr\"om ({Theorems~\ref{thm:forwardevolution} -- \ref{thm:c1instab}}),
relate them to existing literature in fixed frequency scattering, and draw links to various recent results in the interior and exterior of black holes.
Finally, we will see that the existence of a bounded scattering map for the wave equation on Reissner--Nordstr\"om turns out to be a very fragile property; we shall show that there does \underline{not} exist an analogous scattering theory in the presence of a cosmological constant ({\cref{thm:cosmological}}) or Klein--Gordon mass ({\cref{thm:kleingordon}}).
\paragraph{\textbf{The scattering problem on Reissner--Nordstr\"om interior}}
In this paper, we will establish a scattering theory for \underline{finite energy} solutions of the linear wave equation,
\begin{align}\label{eq:linearwave}
\Box_g \psi =0,
\end{align} on the interior of a Reissner--Nordstr\"om black hole, from the bifurcate event horizon $\mathcal H = \mathcal{H}_A \cup \mathcal{H}_B\cup \mathcal{B}_-$ to the bifurcate Cauchy horizon $\mathcal{CH} = \mathcal{CH}_A \cup \mathcal{CH}_B\cup \mathcal{B}_+$, as depicted in \cref{fig:penroseinterior}.
\begin{figure}[ht]\centering
\input{interior.pdf_tex}
\caption{Penrose diagram of the interior of the Reissner--Nordstr\"om black hole and visualization of the scattering map.}
\label{fig:penroseinterior}
\end{figure} The first main result of our paper is \textbf{\cref{thm:forwardevolution}} (see \cref{sec:existencescatteringmap}) in which we will show \textit{existence}, \textit{uniqueness} and \textit{asymptotic completeness} of finite energy scattering states. In this context, \textit{existence} and \textit{uniqueness} mean that for given finite energy data $\psi_0$ on the event horizon $\mathcal{H}$, there exist unique finite energy data on the Cauchy horizon $\mathcal{CH}$ arising from $\psi_0$ as the evolution of \eqref{eq:linearwave}. With \textit{asymptotic completeness} we denote the property that all finite energy data on the Cauchy horizon $\mathcal{CH}$ can indeed be achieved from finite energy data on the event horizon $\mathcal{H}$. This provides a way to construct solutions with desired asymptotic properties which is a necessary first step to properly understand quantum theories in the interior of a Reissner--Nordstr\"om black hole (cf.\ \cite{wald1994quantum,hafnerhawk,drouothawk}). The energy spaces on the event and Cauchy horizon are associated to the Killing field and generator of the time translation $T$. Indeed, $T$ is null on the horizons and, in particular, is the generator of the event and Cauchy horizon $\mathcal{H}$ and $\mathcal{CH}$. Because of the sign-indefiniteness of the energy flux of the vector field $T$ on the bifurcate event (resp.\ Cauchy) horizon (see already \eqref{eq:t-r=1}), we define our energy space by requiring the finiteness of the $T$ energy on both components separately of the event (resp.\ Cauchy) horizon. These define Hilbert spaces with respect to which the scattering map is proven to be bounded.
Finally, it is instructive to draw a comparison between the interior of Reissner--Nordstr\"om and the interior of Schwarzschild ($Q=0$). As opposed to Reissner--Nordstr\"om discussed above, the Schwarzschild interior terminates at a singular boundary at which solutions to \eqref{eq:linearwave} generically blow-up (see\ \cite{interiorschwarzschild}). In contrast, the non-singular and, moreover, Killing, Cauchy horizons (see\ \cref{fig:penroseinterior}) of Reissner--Nordstr\"om immediately yield natural Hilbert spaces of finite energy data to consider. In view of this, Reissner--Nordstr\"om with $Q\neq 0$ can be considered the most elementary interior on which to study the scattering problem. Furthermore, in view of the recent work \cite{dafermos2017interior}, we have that the causal structure of Reissner--Nordstr\"om is stable in a weak sense (see the discussion below about related works in the interior).
\paragraph{\textbf{Fixed frequency scattering}}
It is well known that the wave equation \eqref{eq:linearwave} on Reissner--Nordstr\"om spacetime allows separation of variables which reduces it to the radial o.d.e.
\begin{align}\label{eq:radialode1}
u^{\prime \prime} - V_\ell u + \omega^2 u =0,
\end{align} with potential $V_\ell$ (see already \eqref{eq:potential}), where $\omega\in \mathbb R $ is the time frequency and $\ell \in \mathbb{N}_0$ is the angular parameter. Indeed, most of the existing literature concerning scattering of waves in the interior of Reissner--Nordstr\"om mainly considers fixed frequency solutions, e.g.\ \cite{mcnamara1978behaviour,mcnamara1978instability,hartle1982crossing,gursel1979evolution,matzner1979instability,gursel1979final,MR686719}. For a purely \emph{incoming} (i.e.\ supported only on $\mathcal{H}_A$) fixed frequency solution with parameters $(\omega,\ell)$, we can associate transmission and reflection coefficients $\mathfrak T(\omega,\ell)$ and $\mathfrak R(\omega,\ell)$. The transmission coefficient $\mathfrak T (\omega,\ell)$ measures what proportion of the incoming wave is transmitted to $\mathcal{CH}_B$, whereas the reflection coefficient specifies the proportion reflected to $\mathcal{CH}_A$. An essential step to go from fixed frequency scattering to physical space scattering is to prove \textbf{\emph{uniform boundedness}} of $\mathfrak T(\omega,\ell)$ and $\mathfrak R(\omega,\ell)$. This is non-trivial in view of the discussion of the energy identity \eqref{eq:t-r=1} below.
In this paper, we indeed obtain this uniform bound in \textbf{\cref{thm:boundednesstrans}} (see \cref{subsec:scatteringcoefficients}).
In particular, the regime $\omega \to 0, \ell \to \infty$ is the most involved frequency range to prove uniform boundedness. As we shall see, the proof relies on an explicit computation at $\omega =0$ (see \cite{gursel1979evolution}) together with a careful analysis of special functions and perturbations thereof.
The uniform boundedness of the scattering coefficients is the main ingredient to prove the boundedness of the scattering map in \cref{thm:forwardevolution}. Moreover, it allows us to connect the separated picture to the physical space picture by means of a Fourier representation formula. This is stated as \textbf{\cref{thm:fouriertophysical}} (see \cref{subsec:connfourierphysical}).
A somewhat surprising, direct consequence of the Fourier representation of the scattered data on the Cauchy horizon is that purely incoming compactly supported data lead to a solution which vanishes at the future bifurcation sphere $\mathcal{B}_+$. This is moreover shown to be a necessary condition for the existence of a bounded scattering map (\textbf{\cref{cor:vanishingbifurcation}}).
\paragraph{\textbf{Comparison to scattering on the exterior of black holes}}On the exterior of black holes, the scattering problem has been studied more extensively; see the pioneering works \cite{dimock1985scattering,dimock1987classical,dimock1986classical2,bachelot1991gravitational,bachelot1994asymptotic}, the book \cite{futterman1988scattering} and related results on conformal scattering in \cite{conf1,conf2,conf3,conf4}. Note that for the exterior of a Schwarzschild or Reissner--Nordstr\"om black hole, the uniform boundedness of the scattering coefficients or equivalently, the boundedness of the scattering map, can be viewed a posteriori\footnote{Note that proving \eqref{eq:Tid1} requires first establishing some form of qualitative decay towards $i^+$ and $i^-$.} as a consequence of the global $T$ energy identity
\begin{align}\label{eq:Tid1}
\int_{\mathcal{H}^-} |T\psi|^2 + \int_{\mathcal{I}^-} |T\psi|^2 = \int_{\mathcal{H}^+} |T\psi|^2 + \int_{\mathcal{I}^+} |T\psi|^2.
\end{align} Considering only incoming radiation from $\mathcal{I}^-$, this statement translates into $|\mathfrak R|^2 + |\mathfrak T|^2 = 1$ for the reflection coefficient $ \mathfrak R$ and transmission coefficients $\mathfrak T$. In the interior, however, due to the different orientations of the $T$ vector field on the horizons (cf. \cref{fig:compare}), boundedness of the scattering map is not at all manifest.
\begin{figure}[ht]
\centering
\input{compare.pdf_tex}
\caption{Interior of Reissner--Nordstr\"om (left) and exterior of Schwarzschild or Reissner--Nordstr\"om (right).
\newline In both diagrams the arrows denote the direction of the $T$ Killing vector field. Note that we have the identifications $\mathcal{H}_A = \mathcal{H}^+ $ and $ \mathcal{B}_- = \mathcal{B}$.}
\label{fig:compare}
\end{figure}
In particular, the global $T$ energy identity on the interior of a Reissner--Nordstr\"om black hole reads
\begin{align}\label{eq:t-r=1}
\int_{\mathcal{H}_A} |T\psi|^2 - \int_{\mathcal{H}_B} |T\psi|^2 = \int_{\mathcal{CH}_B} |T\psi|^2 -\int_{\mathcal{CH}_A} |T\psi|^2
\end{align}
from which we cannot deduce boundedness of the scattering map even a posteriori. (Indeed, identity \eqref{eq:t-r=1} corresponds only to the ``pseudo-unitarity'' statement of \cref{thm:forwardevolution}.) Again, considering only ingoing radiation from $\mathcal{H}_A$, this translates to \begin{align}\label{eq:t-r=12}|\mathfrak T(\omega,\ell)|^2 - |\mathfrak R(\omega,\ell)|^2 = 1\end{align} for the reflection coefficient $\mathfrak R$ and the transmission coefficient $\mathfrak T$. Hence, while for fixed $|\omega| > 0$ and $\ell$, it is straightforward to show that $\mathfrak{T}$ and $\mathfrak{R}$ are finite, there is no a priori obvious obstruction from \eqref{eq:t-r=12} for these scattering coefficients to blow up in the limits $\omega \to 0,\pm\infty$ and $\ell \to \infty$.
Moreover, note that in the exterior, the Killing field $T$ is timelike, so the radial o.d.e.\ \eqref{eq:radialode1} should be considered as an equation for a fixed time frequency wave on a constant time slice. In the interior, however, the Killing field $T$ is spacelike so the radial o.d.e.\ \eqref{eq:radialode1} is rather an evolution equation for a constant spatial frequency.
The Schwarzschild family can be viewed as a special case ($a=0$) of the two parameter Kerr family, describing rotating black holes with mass parameter $M$ and rotation parameter $a$ \cite{MR0156674}.\footnote{Both Kerr and Reissner--Nordström can be viewed as special cases of the Kerr--Newman spacetime. For decay results on Kerr--Newman see \cite{civin2015stability}.} On the exterior of Kerr many other difficulties arise:~superradiance, intricate trapping, presence of ergoregion, etc.\ \cite{MR3488738}. Nevertheless, using the decay results in \cite{MR3488738}, a definitive physical space scattering theory for Kerr black holes has been established in \cite{dafermos2014scattering} (see also the earlier \cite{deSitterScatter}). The proof on the exterior of Kerr involved first establishing a scattering map from past null infinity $\mathcal{I}^-$ to a constant time slice $\Sigma$ and then concatenating that map with a second scattering map from $\Sigma$ to the future event horizon $\mathcal{H}^+$ and future null infinity $\mathcal{I}^+$. In the interior, however, we will directly show the existence of a ``global'' scattering map from the event horizon $\mathcal H$ to the Cauchy horizon $\mathcal{CH}$. Indeed, due to blue-shift instabilities (see \cite{dafermos2017time}), we do not expect that the analogous concatenation of scattering maps (event horizon $\mathcal{H}$ to spacelike hypersurface $\Sigma$ and then from $\Sigma$ to the Cauchy horizon $\mathcal{CH}$) as in the Kerr exterior, to be bounded in the interior of Reissner--Nordstr\"om.
\paragraph{\textbf{Injectivity of the reflection map and blue-shift instabilities}} We can also conclude from our work that there is always non-vanishing reflection to the Cauchy horizon $\mathcal{CH}_A$ arising from non-vanishing purely ingoing radiation at $\mathcal{H}_A$. This follows from the fact that in the separated picture and for fixed $\ell$, the reflection coefficient $\mathfrak R (\omega,\ell)$ can be analytically continued to the strip $|\operatorname{Im}(\omega)| < \kappa_+$ and hence, only vanishes on a discrete set of points on the real axis. This is shown in \textbf{\cref{thm:nonvanishingreflection}} (see \cref{subsec:reflection}).
We will also deduce from the Fourier representation of the scattered data on the Cauchy horizon $\mathcal{CH}$, and a suitable meromorphic continuation of the transmission coefficient, that there exist purely incoming compactly supported data on the event horizon $\mathcal{H}$ leading to solutions which fail to be $C^1$ on the Cauchy horizon $\mathcal{CH}$. This $C^1$-blow-up of linear waves puts on a completely rigorous footing the pioneering work of Chandrasekhar and Hartle \cite{hartle1982crossing}. We state this as \textbf{\cref{thm:c1instab}} (see \cref{sec:c1instab}).
For generic solutions arising from localized data on an asymptotically flat hypersurface, one expects a stronger instability, namely, non-degenerate energy blow-up at the Cauchy horizon $\mathcal{CH}$. Such non-degenerate energy blow-up was proven in \cite{lukohchblowup} for generic compactly supported data on an asymptotically flat Cauchy hypersurface. Later, for the more difficult Kerr interior, non-degenerate energy blow-up was proven in \cite{luk2016instability} assuming certain polynomial lower bounds on the tail of incoming data on the event horizon $\mathcal{H}$ and in~\cite{dafermos2017time} for solutions arising from generic initial data along past null infinity $\mathcal{I}^-$ with polynomial tails.
Finally, we mention the forthcoming work \cite{yakov} which studies the problem of non-degenerate energy blow-up from a scattering theory perspective and also uses the non-triviality of reflection to establish results related to mass inflation for the spherically symmetric Einstein--Maxwell--scalar field system (cf.\ \cite{luk2017strong,luk2017strong2}).
\paragraph{\textbf{Related results on the interior}}There has been a lot of recent progress studying the interior of black holes. In particular, new insights were gained concerning the stability of the Cauchy horizon and the strong cosmic censorship conjecture.
For the Cauchy problem for \eqref{eq:linearwave} on the interior of both a fixed Kerr and a Reissner--Nordstr\"om black hole, the works \cite{franzen2016boundedness,franzen,hintzinterior} establish uniform boundedness (in $L^\infty$) and continuity up to and including the Cauchy horizon for solutions arising from smooth and compactly supported data on an asymptotically flat spacelike hypersurface. Such data in particular give rise to solutions with polynomial decay along the event horizon.
In contrast, for the scattering problem considered in the present paper, we are required to work with spaces which are symmetric with respect to the event and Cauchy horizons. This naturally leads to the rougher class of finite $T$ energy data in the statement of \cref{thm:forwardevolution}. Note that for such data on the Cauchy horizon, continuity or boundedness (in $L^\infty$) does \underline{not} necessarily hold true.
Turning finally to the full nonlinear dynamics of the Einstein equations, it is shown in \cite{dafermos2017interior} that the Kerr Cauchy horizon is $C^0$-stable. Thus, the existence of a Cauchy horizon, a very natural setting parameterizing scattering data in the interior, is not a pure artifact of symmetry but rather a stable property at least in a weak sense. On the other hand, in \cite{luk2017strong,luk2017strong2,VandeMoortel2018} it is proven that for a suitable Einstein--matter system under spherical symmetry, the Cauchy horizon, while $C^0$-stable, is generically $C^2$-unstable.
Finally, we mention that for the Schwarzschild black hole ($Q=0$), which does not admit a Cauchy horizon, it is shown in \cite{interiorschwarzschild} that solutions to \eqref{eq:linearwave} generically blow up at the spacelike singularity $\{r=0\}$.
\paragraph{\textbf{Breakdown of $T$ energy scattering for $\Lambda \neq 0$ or $\mu\neq 0$}} If a cosmological constant $\Lambda \in \mathbb R$ is added to the Einstein--Maxwell system, we can consider the analogous (anti-) de Sitter--Reissner--Nordstr\"om family of solutions whose interiors have the same Penrose diagram as depicted in \cref{fig:penroseinterior}. For very slowly rotating Kerr--de Sitter and Reissner--Nordstr\"om--de Sitter spacetimes, boundedness, continuity, and regularity up to and including the Cauchy horizon has been shown for solutions to \eqref{eq:linearwave} arising from smooth and compactly supported data on a Cauchy hypersurface \cite{hintzvasyinterior}. However, remarkably, there is no analogous $T$ energy scattering theory for either the linear wave equation \eqref{eq:linearwave} or the Klein--Gordon equation with conformal mass. This is the statement of~\textbf{\cref{thm:cosmological}} (see \cref{subsec:nonex}). The reason for this failure is the unboundedness of the transmission coefficient $\mathfrak T$ and reflection coefficients $\mathfrak R$ in the vanishing frequency limit $\omega \to 0$. To be more precise, we will prove that there does not exist a $T$ energy scattering theory of the wave or Klein--Gordon equation in the interior of a (anti-) de Sitter--Reissner--Nordstr\"om black hole for generic subextremal black hole parameters $(M,Q,\Lambda)$. In particular, for fixed $0<|Q|<M$, there is an $\epsilon >0$ such that there does not exist a $T$ energy scattering theory for all $0\neq |\Lambda| < \epsilon$.
Similarly, we prove in \textbf{\cref{thm:kleingordon}} (see \cref{subsec:notkleingordon}) that there does not exist a $T$ energy scattering theory for the Klein--Gordon equation $\Box_g \psi - \mu \psi =0$ on the Reissner--Nordstr\"om interior for a generic set of masses $\mu$. This is in contrast to the exterior, where $T$ energy scattering theories were established for massive fields in \cite{bachelot1994asymptotic,MR2016993}.
It remains an open problem to formulate an appropriate scattering theory in the cosmological setting and for the Klein--Gordon equation as well as for the interior of Kerr.
\paragraph{\textbf{Outline}}
This paper is organized as follows. In \cref{prelims}, we shall set up the spacetime, introduce the relevant energy spaces, and define the scattering coefficients in the separated picture. In \cref{sec:mainthms} we state the main results of this paper, Theorems \ref{thm:forwardevolution} -- \ref{thm:kleingordon}.
Section~\ref{sec:radial} is devoted to the proof of \cref{thm:boundednesstrans}.
Then, the statement of \cref{thm:boundednesstrans} allows us to prove \cref{thm:forwardevolution} in \cref{sec:mainthm}.
Finally, in the last two sections are show our non-existence results: In \cref{sec:cosmo}, we prove \cref{thm:cosmological} and in \cref{sec:kleingordonequation}, we give the proof of \cref{thm:kleingordon}.
\paragraph{\textbf{Acknowledgement.}} The authors would like express their gratitude to Mihalis Dafermos for many valuable discussions and helpful remarks. The authors also thank Igor Rodnianski, Jonathan Luk, and Sung-Jin Oh for useful conversations. CK acknowledges support from the EPSRC and thanks Princeton University for hosting him as a VSRC.
YS acknowledges support from the NSF Postdoctoral Research Fellowship under award no.\ 1502569.
\section{Proof of \texorpdfstring{\cref{thm:forwardevolution}}{Theorem 1}: Existence and boundedness of the \texorpdfstring{$T$}{T} energy scattering map}
\label{sec:mainthm}
Having performed the analysis of the radial o.d.e.\ and having in particular proven uniform boundedness of the transmission coefficient $\mathfrak T$ and the reflection coefficients $\mathfrak R$, we shall prove \cref{thm:forwardevolution} in this section.
\subsection{Density of the domains \texorpdfstring{$\mathcal{D}^T_\mathcal{H}$}{DTH} and \texorpdfstring{$\mathcal{D}^T_\mathcal{CH}$}{CTCH}}
We start by proving that the domains $\mathcal{D}^T_\mathcal{H}$ and $\mathcal{D}^T_\mathcal{CH}$ are dense.
\begin{lemma}\label{lem:lemmadense}
The domains of the forward and backward evolution $\mathcal{D}_{\mathcal{H}}^T$ and $\mathcal{D}^T_{\mathcal{CH}}$ are dense in $\mathcal{E}^T_{\mathcal{H}}$ and $\mathcal{E}^T_{\mathcal{CH}}$, respectively.
\begin{proof}We will only prove that the domain of the forward evolution is dense since the other claim is analogous.
Recall that by definition $C_c^\infty(\mathcal{H})$ is dense in $\mathcal{E}^T_{\mathcal{H}}$. Now, let $\Psi \in C_c^\infty(\mathcal{H})$ be arbitrary and denote by $\psi$ its forward evolution. We will show that we can approximate $\Psi$ with functions of $\mathcal{D}^T_{\mathcal{H}}$ arbitrarily well. To do so, fix $r_{red}< r_0<r_+$. Then, using the red-shift effect (see \cref{lem:exponentialdecay} in the appendix) the $N$ energy of $\psi\restriction_{r=r_0}$ will have exponential decay towards $i_+$. Hence, it can be approximated with smooth functions $\phi_n$ of compact support on the hypersurface $r=r_0$ w.r.t.\ the norm induced by the non-degenerate $N$ energy (see \cref{rmk:approxwithcompactly} in the appendix). More precisely, on $\Sigma_{r_0} = \{r=r_0\}$ define a sequence $\phi_n \in C_c^\infty(\Sigma_{r_0}) $ by
\begin{align} \phi_n(t,\theta,\phi) = \psi\restriction_{r=r_0}(t,\theta,\phi)\chi(n^{-1} t ),
\end{align}
where $(\theta,\phi) \in \mathbb{S}^2$ and $\chi\colon \mathbb{R}\to [0,1]$ is smooth with $\operatorname{supp} \chi \subseteq [-2,2]$, $\chi\restriction_{[-1,1]} = 1$. Then, we obtain that $ \int_{\Sigma_{r_0}} J_\mu^N[\psi-\phi_n ] n_{\Sigma_{r_0}}^\mu \d\mathrm{vol} \to 0 $ as $n\to\infty$. By construction, the restriction to the event horizon of the backward evolution, $\Phi_n$ of each $\phi_n$ will lie in $\mathcal{D}_{\mathcal{H}}^T$. Finally, we can conclude the proof by applying \cref{lem:estimatetorconst} from the appendix, which yields
\begin{align}
\| \Psi - \Phi_n\|_{\mathcal{E}^T_\mathcal{H}}^2 = \int_{\mathcal{H}} J^T_\mu[\Psi-\Phi_n] T^\mu \d\mathrm{vol}\lesssim \int_{r=r_0} J^N_\mu [\psi- \phi_n] n_{\Sigma_{r_0}}^\mu \d\mathrm{vol} \to 0
\end{align}
as $n\to\infty$.
\end{proof}
\end{lemma}
\subsection{Boundedness of the scattering and backward map on \texorpdfstring{$\mathcal{D}^T_\mathcal{H}$}{DTH} and \texorpdfstring{$\mathcal{D}^T_\mathcal{CH}$}{DTCH}}
In the following proposition we shall lift the boundedness of the transmission and reflection coefficients (\cref{thm:boundednesstrans}) to the physical space picture on the dense domains $\mathcal{D}^T_\mathcal{H}$ and $\mathcal{D}^T_\mathcal{CH}$.
\begin{restatable}{prop}{thmbounds}
\label{thm:thmbounds}
Let $\psi$ be a smooth solution to \eqref{eq:linearwave} on $\mathcal{M}_{\mathrm{RN}} $ such that $\psi\restriction_{\mathcal{H}}\in \mathcal{D}_{\mathcal{H}}^T$ (or equivalently, $\psi\restriction_{\mathcal{CH}}\in \mathcal{D}_{\mathcal{CH}}^T$). Then,
\begin{align}
\|\psi\restriction_{\mathcal{CH}_A}\|_{\mathcal{E}^T_{\mathcal{CH}_A}}^2 + \|\psi\restriction_{\mathcal{CH}_B} \|_{\mathcal{E}^T_{\mathcal{CH}_B}}^2
\leq B \left( \| \psi\restriction_{\mathcal{H}_A}\|_{\mathcal{E}^T_{\mathcal{H}_A}}^2 + \|\psi\restriction_{\mathcal{H}_B}\|_{\mathcal{E}^T_{\mathcal{H}_B}}^2\right)
\end{align}
and
\begin{align}
\| \psi\restriction_{\mathcal{H}_A}\|_{\mathcal{E}^T_{\mathcal{H}_A}}^2 + \|\psi\restriction_{\mathcal{H}_B}\|_{\mathcal{E}^T_{\mathcal{H}_B}}^2
\leq \tilde B \left( \|\psi\restriction_{\mathcal{CH}_A}\|_{\mathcal{E}^T_{\mathcal{CH}_A}}^2 + \|\psi\restriction_{\mathcal{CH}_B} \|_{\mathcal{E}^T_{\mathcal{CH}_B}}^2 \right)
\end{align}
for constants $B$ and $\tilde B$ only depending on the black hole parameters.
\end{restatable}
\begin{proof}
Set $\phi:= T\psi$ and note that $\phi\restriction_{\mathcal H} \in \mathcal{D}^T_{\mathcal H}$ and $\phi$ also solves \eqref{eq:linearwave}. Since $\psi \in \mathcal{D}_{\mathcal H}^T \subset \mathcal{E}^T_{\mathcal{H}}$, we have that $\phi\restriction_{\mathcal{H}_A} = T \psi\restriction_{\mathcal{H}_A} \in L^2(\mathcal{H}_A)$ with respect to the unique volume form induced by the normal vector field $T$. Analogously, we also have $\phi\restriction_{\mathcal{H}_B} = T \psi\restriction_{\mathcal{H}_B} \in L^2(\mathcal{H}_B)$. Thus, we can define the Fourier transform on the event horizon with the charts \eqref{eq:ha} and \eqref{eq:hb} as \begin{align}
a_{\mathcal{H}_A}(\omega,\theta,\phi) := \frac{1}{\sqrt{2\pi} } \int_{\mathbb{R} } \phi\restriction_{\mathcal{H}_A} (v,\theta,\phi) e^{-i\omega v} \d v
\end{align}
and
\begin{align}
a_{\mathcal{H}_B}(\omega,\theta,\phi) := \frac{1}{\sqrt{2\pi} } \int_{\mathbb{R} } \phi\restriction_{\mathcal{H}_B} (u,\theta,\phi) e^{i\omega u} \d u.
\end{align}
We can further decompose the Fourier coefficients in spherical harmonics to obtain
\begin{align}
a^{\ell,m}_{\mathcal{H}_A}(\omega) = \langle Y_{\ell m} ,a_{\mathcal{H}_A} \rangle_{L^2(\mathbb S^2)}
\text{ and }
a^{\ell,m}_{\mathcal{H}_B}(\omega) = \langle Y_{\ell m} ,a_{\mathcal{H}_B} \rangle_{L^2(\mathbb S^2)}.
\end{align}
From Plancherel's theorem, we obtain
\begin{align}
& \| \psi\restriction_{\mathcal{H}_A} \|_{\mathcal{E}^T_{\mathcal{H}_A}}^2 = \sum_{|m|\leq \ell, \ell \geq 0}\int_{\mathbb{R}} |a_{\mathcal{H}_A}^{\ell,m} (\omega) |^2 \d \omega,\label{eq:psiT1} \\
& \| \psi\restriction_{\mathcal{H}_B} \|_{\mathcal{E}^T_{\mathcal{H}_b}}^2 = \sum_{|m|\leq \ell, \ell \geq 0}\int_{\mathbb{R}} |a_{\mathcal{H}_B}^{\ell,m} (\omega) |^2 \d \omega.\label{eq:psiT2}
\end{align}
Similarly, since $\phi\restriction_{\mathcal{CH}} \in \mathcal{D}^T_{\mathcal{CH}}$, we define
\begin{align}
b_{\mathcal{CH}_A}(\omega,\theta,\phi) := \frac{1}{\sqrt{2\pi} } \int_{\mathbb{R} } \phi\restriction_{\mathcal{CH}_A} (v,\theta,\phi) e^{-i\omega v} \d v
\end{align}
and
\begin{align}
b_{\mathcal{CH}_B}(\omega,\theta,\phi) := \frac{1}{\sqrt{2\pi} } \int_{\mathbb{R} } \phi\restriction_{\mathcal{CH}_B} (u,\theta,\phi) e^{i\omega u} \d u.
\end{align}
We can further decompose the Fourier coefficients in spherical harmonics to obtain
\begin{align}
b^{\ell,m}_{\mathcal{CH}_A}(\omega) = \langle Y_{\ell m} ,b_{\mathcal{CH}_A} \rangle_{L^2(\mathbb S^2)}
\text{ and }
b^{\ell,m}_{\mathcal{CH}_B}(\omega) = \langle Y_{\ell m} ,b_{\mathcal{CH}_B} \rangle_{L^2(\mathbb S^2)}.
\end{align}
Again, in view of Plancherel's theorem
\begin{align}\label{eq:psiT3}
&\|\psi\restriction_{\mathcal{CH}_A} \|^2_{\mathcal{E}^T_{\mathcal{CH}_A}} = \sum_{|m|\leq \ell, \ell \geq 0} \int_{\mathbb{R}} |b^{\ell,m}_{\mathcal{CH}_A} (\omega)|^2 \d\omega ,\\
&\|\psi\restriction_{\mathcal{CH}_B} \|^2_{\mathcal{E}^T_{\mathcal{CH}_B}} = \sum_{|m|\leq \ell, \ell \geq 0} \int_{\mathbb{R}} |b^{\ell,m}_{\mathcal{CH}_B} (\omega)|^2 \d\omega .\label{eq:psiT4}
\end{align}
and similarly for $\mathcal{CH}_B$.
We shall also decompose $\phi$ on a constant $r$ slice. Fix $r\in(r_-,r_+)$, then set
\begin{align}
\hat \phi_{m \ell }(\omega,r) = \frac{1}{\sqrt{2\pi }}\int_{\mathbb{R}} \int_{\mathbb{S}^2} Y_{m\ell }(\theta,\phi) \phi(t,r,\theta,\phi) e^{-i\omega t} \sin\theta \d\theta \d\phi \d t
\end{align}
such that
\begin{align}
\phi(t,r,\theta,\phi) = \frac{1}{\sqrt{2\pi}} \sum_{|m|\leq \ell, \ell \geq 0 } \int_{\mathbb{R}} \hat{\phi}_{m\ell} ( \omega, r) Y_{m\ell} (\theta,\phi) e^{i\omega t} \d \omega .
\end{align}
This is well-defined since $\phi(t,r,\theta,\phi)$ is compactly supported on each $r=const.$ slice.
Since $\phi$ is smooth, we also know that $\hat{\phi}_{m\ell}$ satisfies the radial o.d.e.\ \eqref{eq:radialode} and can be expanded as
\begin{align}
\hat{\phi}_{m\ell} (\omega,r(r_\ast)) = \alpha^{ \ell, m }_{\mathcal{H}_A} (\omega) \frac{r_+}{r} u_1(\omega, r_\ast) + \alpha^{\ell, m}_{\mathcal{H}_B} (\omega) \frac{r_+}{r} u_2(\omega, r_\ast), \label{eq:psihatrplus}
\end{align}
where \begin{align} &| u_1 - e^{i\omega r_\ast} | \lesssim_\ell e^{2 \kappa_+ r_\ast} \sim (r_+ - r),
\\ &| u_2 - e^{ - i\omega r_\ast} | \lesssim_\ell e^{2 \kappa_+ r_\ast} \sim (r_+ - r)
\end{align}
for $r_\ast \leq 0$. Note that this holds uniformly in $\omega$. We shall show in the following that indeed $\alpha_{\mathcal{H}_A}^{\ell,m} = a_{\mathcal{H}_A}^{\ell,m}$ and $\alpha_{\mathcal{H}_B}^{\ell,m} = a_{\mathcal{H}_B}^{\ell,m}$. To do so, note that for $r(r_\ast)$ with $r_\ast \leq 0$ we have for fixed $(m,\ell)$ that
\begin{align}\nonumber
\phi^{\ell,m}(t,r) = \langle \phi, Y_{m\ell} \rangle_{L^2(\mathbb S^2)} = \int_{\mathbb{R}} &\left( \alpha^{\ell,m}_{\mathcal{H}_A} (\omega) \frac{r_+}{r} u_1(\omega, r_\ast(r)) + \alpha^{\ell,m}_{\mathcal{H}_B} (\omega) \frac{r_+}{r} u_2(\omega, r_\ast(r)) \right) e^{i\omega t} \frac{ \d \omega }{\sqrt{2\pi}}.
\end{align}
We want to interchange the limit $r\to r_+$ with the integral. In order to use Lebesgue's dominated convergence theorem we will estimate $ \alpha_{\mathcal{H}_A}^{\ell,m}$ and $ \alpha_{\mathcal{H}_B}^{\ell,m}$. Note that
\begin{align}
| \alpha_{\mathcal{H}_A}^{\ell,m}| = \left| \frac{\mathfrak W(\frac{r}{r_+} \hat \phi_{m\ell}, u_2)}{\mathfrak W(u_1, u_2)} \right| = \left| \frac{\mathfrak W( \frac{r}{r_+} \hat {T\psi_{m\ell}}, u_2)}{\mathfrak W(u_1, u_2)} \right|\leq \frac{|\omega \mathfrak W( \frac{r}{r_+} \hat \psi_{m\ell} , u_2)|}{2|\omega|} \leq \left|\mathfrak W\left(\frac{r}{r_+} \hat \psi_{m\ell}, u_2\right)\right|,
\end{align}
which is independent of $r(r_\ast)$ and integrable since $\omega\mapsto \hat \psi_{m\ell}(\omega,r_\ast)$ is a Schwartz function.
Now, we shall fix $v= r_\ast+ t$ and let $r\to r_+$ such that $r_\ast \to -\infty$. Then, using Lebesgue's dominated convergence theorem, we obtain
\begin{align}\nonumber
\phi^{\ell,m} = \int_{\mathbb{R}} \left( \alpha^{\ell,m}_{\mathcal{H}_A} (\omega) e^{i\omega v} + \alpha^{\ell,m}_{\mathcal{H}_B} (\omega) e^{-2i \omega r_\ast}e^{i \omega v } \right) \frac{ \d \omega }{\sqrt{2\pi}} + O(r_+ - r)
\end{align}
as $r\to r_+$.
Finally, for $v$ fixed and letting $r\to r_+$ (or $r_\ast \to -\infty$), we obtain
\begin{align}
\phi^{\ell,m}\restriction_{\mathcal{H}_A} (v) = \int_{\mathbb{R}} \alpha^{\ell,m}_{\mathcal{H}_A} (\omega) e^{i\omega v} \frac{ \d \omega }{\sqrt{2\pi}}
\end{align}
in view of the Riemann--Lebesgue lemma.
Also, by definition of $a^{\ell, m}_{\mathcal{H}_A}$,
\begin{align}
\phi\restriction_{\mathcal{H}_A} (v,\theta,\phi)= \sum_{|m|\leq \ell, \ell \geq 0} \int_{\mathbb{R}} a^{\ell,m}_{\mathcal{H}_A}(\omega, \theta, \phi) e^{i\omega v } Y_{\ell m}(\theta, \phi) \frac{\d v}{\sqrt{2\pi}}.
\end{align}
In view of the Fourier inversion theorem and the fact that the spherical harmonics form a basis we conclude that \begin{align}\label{eq:aequalsalpha} \alpha_{\mathcal{H}_A}^{\ell,m} = a_{\mathcal{H}_A}^{\ell,m} \text{ and analogously, }\alpha_{\mathcal{H}_B}^{\ell,m} = a_{\mathcal{H}_B}^{\ell,m}.\end{align}
Similarly to \eqref{eq:psihatrplus}, we can expand $\hat \psi_{m\ell}$ in a fundamental pair of solutions corresponding to both Cauchy horizons $\mathcal{CH}_A$ and $\mathcal{CH}_B$. In particular, we can write
\begin{align}
\hat{\phi}_{m\ell} (\omega,r(r_\ast)) = \beta^{\ell,m}_{\mathcal{CH}_A} (\omega) \frac{r_+}{r} v_1(\omega, r_\ast) + \beta^{ \ell,m}_{\mathcal{CH}_A} (\omega) \frac{r_+}{r} v_2(\omega, r_\ast),
\end{align}
where
\begin{align} &| v_1 - e^{-i\omega r_\ast} | \lesssim_\ell e^{2 \kappa_- r_\ast} \sim (r- r_-) ,
\\ &| v_2 - e^{ i\omega r_\ast} | \lesssim_\ell e^{2 \kappa_- r_\ast} \sim (r- r_-).
\end{align}
for $r_\ast \geq 0$. Similarly to \eqref{eq:aequalsalpha}, we can prove
\begin{align}
\frac{r_+}{r_-}\beta^{\ell,m}_{\mathcal{CH}_A} (\omega)= b^{\ell,m}_{\mathcal{CH}_A} (\omega) \text{ and } \frac{r_+}{r_-}\beta^{\ell,m}_{\mathcal{CH}_B} (\omega) = b^{\ell,m}_{\mathcal{CH}_B} (\omega).
\end{align}
Moreover, from the uniform boundedness of the reflection and transmission coefficients (cf. \cref{thm:boundednesstrans}) we have the estimate
\begin{align}
| b^{\ell,m}_{\mathcal{CH}_A} (\omega)| + | b^{\ell,m}_{\mathcal{CH}_B} (\omega)| &= \frac{r_+}{r_-} | \beta^{\ell,m}_{\mathcal{CH}_A} (\omega)| + \frac{r_+}{r_-}| \beta^{\ell,m}_{\mathcal{CH}_B} (\omega) | = \frac{r_+}{r_-}\left(\left| \mathfrak R \alpha_{\mathcal{H}_A}^{\ell,m} +\bar{ \mathfrak T}\alpha_{\mathcal{H}_B}^{\ell,m}\right| + \left| \bar{\mathfrak{R}}\alpha_{\mathcal{H}_B}^{\ell,m} + \mathfrak T \alpha_{\mathcal{H}_A}^{\ell,m} \right| \right) \nonumber \\&\leq C ( | \alpha^{\ell,m}_{\mathcal{H}_A} (\omega)| + |\alpha^{\ell,m}_{\mathcal{H}_B} (\omega)| ) = C( | a^{\ell,m}_{\mathcal{H}_A} (\omega)| + |a^{\ell,m}_{\mathcal{H}_B} (\omega)|) \label{eq:estimatebeta}
\end{align}
for a constant $C$ which only depends on the black hole parameters. Here, we have used the fact that
\begin{align}\label{eq:scatteringcoef1}
\begin{pmatrix}\beta^{\ell,m}_{\mathcal{CH}_B} \\
\beta^{\ell,m}_{\mathcal{CH}_A}
\end{pmatrix} = \begin{pmatrix}
\mathfrak T & \bar{\mathfrak R}\\\mathfrak R & \bar{\mathfrak T}
\end{pmatrix}
\begin{pmatrix}
\alpha^{\ell,m}_{\mathcal{H}_A} \\ \alpha^{\ell,m}_{\mathcal{H}_B}.
\end{pmatrix}.
\end{align}
In view of $1=|\mathfrak T|^2 - |\mathfrak R|^2$, we also have
\begin{align}\label{eq:scatteringcoef2}
\begin{pmatrix}
\alpha^{\ell,m}_{\mathcal{H}_A} \\ \alpha^{\ell,m}_{\mathcal{H}_B}
\end{pmatrix} = \begin{pmatrix}
\bar{ \mathfrak T} & -\bar{\mathfrak R}\\ -\mathfrak R & {\mathfrak T}
\end{pmatrix}
\begin{pmatrix}\beta^{\ell,m}_{\mathcal{CH}_B} \\
\beta^{\ell,m}_{\mathcal{CH}_A}
\end{pmatrix}
\end{align}
from which we deduce
\begin{align}\label{eq:boundedbackwards}
| a^{\ell,m}_{\mathcal{H}_A} (\omega)| + |a^{\ell,m}_{\mathcal{H}_B} (\omega)| \lesssim | b^{\ell,m}_{\mathcal{CH}_A} (\omega)| + | b^{\ell,m}_{\mathcal{CH}_B} (\omega)|.
\end{align}
Estimate \eqref{eq:estimatebeta} and \eqref{eq:boundedbackwards} show the claim in view of \eqref{eq:psiT1}, \eqref{eq:psiT2}, \eqref{eq:psiT3}, and \eqref{eq:psiT4}. Finally, in view of the Fourier inversion theorem, note that the previous also justifies the Fourier representation of scattering map \eqref{eq:scatteringfourier}, and the Fourier representations \eqref{eq:fourierrep1} and \eqref{eq:fourierrep2}.
\end{proof}
\subsection{Completing the proof}
Having proven \cref{lem:lemmadense} and \cref{thm:thmbounds}, we can finally show \cref{thm:forwardevolution} in the following.
\begin{proof}[Proof of \cref{thm:forwardevolution}] Since $\mathcal{D}^T_{\mathcal{H}}\subset \mathcal{E}^T_\mathcal{H}$ is dense (\cref{lem:lemmadense}) and $S_0^T\colon \mathcal{D}^T_{\mathcal{H}}\subset \mathcal{E}^T_\mathcal{H} \to \mathcal{D}^T_{\mathcal{CH}}\subset \mathcal{E}^T_\mathcal{CH}$ is a bounded injective map (\cref{rmk:injectiv}, \cref{thm:thmbounds}), we can uniquely extend $S_0^T$ to the bounded injective scattering map
\begin{align}
S^T\colon \mathcal{E}^T_\mathcal{H} \to \mathcal{E}^T_\mathcal{CH}.
\end{align}
Analogously, in view of \cref{thm:time}, \cref{rmk:domain}, \cref{rmk:injectiv}, and \cref{thm:thmbounds}, we can uniquely extend the bounded injective map $B_0^T\colon \mathcal{D}^T_\mathcal{CH} \subset \mathcal{E}^T_\mathcal{CH} \to \mathcal{D}^T_\mathcal{CH}\subset \mathcal{E}^T_\mathcal{H}$ to the bounded injective backward map $B^T\colon \mathcal{E}^T_\mathcal{CH} \to \mathcal{E}^T_\mathcal{H}$ (\cref{lem:lemmadense}).
Since $B_0^T\circ S_0^T = \mathrm{Id}_{ \mathcal{D}^T_{\mathcal{H}} }$ and $S_0^T \circ B_0^T = \mathrm{Id}_{ \mathcal{D}^T_{\mathcal{CH}}}$ on dense sets, it also extends to $\mathcal{E}^T_\mathcal{H}$ and $\mathcal{E}^T_\mathcal{CH}$ from which \eqref{eq:inverses} follows.
Similarly, it suffices to check \eqref{eq:pseudounitary} for $\psi \in \mathcal{D}_{\mathcal{H}}^T$. Indeed, \eqref{eq:pseudounitary} holds true for $\psi \in \mathcal{D}_{\mathcal{H}}^T$ in view of the $T$ energy identity.
\end{proof}
\section{Preliminaries}
In this section we will define the background differentiable structure and metric for the Reissner--Nordström spacetime and introduce some convenient coordinate systems.
\label{prelims}
\subsection{Interior of the subextremal Reissner--Nordstr\"om black hole}
The global geometry of Reissner--Nordström was first described in \cite{MR0128960}. For completeness, we will explicitly construct in this section the coordinates for the region considered.
We start, in \cref{sec:interiorwoboundary}, by defining a Lorentzian manifold corresponding to the interior of the Reissner--Nordström black hole without the horizons. Then, in \cref{sec:interiorwboundary}, we will attach the boundaries which will correspond to the event horizon and Cauchy horizon.
\subsubsection{The interior without boundary}
\label{sec:interiorwoboundary}
We will now give an explicit description of the differential structure and metric. The Reissner--Nordström solutions \cite{reissner1916eigengravitation,nordstrom1918energy} are a two-parameter family of spherically symmetric spacetimes with mass parameter $M\in \mathbb{R}$ and electric charge parameter $Q\in \mathbb R$ solving the Einstein--Maxwell system
\begin{align}
\label{eq:einsteinmaxwell}
&{Ric}_{\mu\nu} - \frac{1}{2}g_{\mu\nu}{R} = 8 \pi T_{\mu\nu} := 8 \pi \left( \frac{1}{4\pi} \left( F_{\mu}^{~\lambda} F_{\lambda\nu} - \frac{1}{4} g_{\mu\nu} F_{\lambda \rho} F^{\lambda \rho} \right)\right),
\\& \nabla^\mu F_{\mu\nu} = 0 , \nabla_{[\mu} F_{\nu\lambda]} = 0.
\nonumber
\end{align}
In this paper, we consider the subextremal family of black holes with parameter range $0<|Q|<M$. Define the manifold $\mathcal M$ by
\begin{align}
\mathcal M = \mathbb{R} \times (r_-,r_+) \times \mathbb{S}^2,
\end{align}
where $r_\pm = M \pm \sqrt{M^2 - Q^2} >0$. The manifold is parametrized by the standard coordinates $t \in \mathbb{R}$, $r\in (r_-, r_+)$, and a choice of spherical coordinates $(\theta,\phi)$ on the sphere $\mathbb{S}^2$. We denote the global coordinate vector field $\partial_t$ by $T$:\begin{align}T := \frac{\partial}{\partial t}.\label{def:T}\end{align}
Using the above coordinates, we equip $\mathcal{M}$ with the Lorentzian metric
\begin{align}
g_{Q,M} = - \left(1-\frac{2M}{r} + \frac{Q^2}{r^2}\right) \d t \otimes \d t + \left(1-\frac{2M}{r} + \frac{Q^2}{r^2}\right)^{-1} \d r \otimes \d r + r^2 \slashed g_{\mathbb{S}^2},
\end{align}
where $\slashed g_{\mathbb{S}^2}$ is the round metric on the 2-sphere. We also define the quantities
\begin{align}
\Delta := r^2 - 2Mr + Q^2 = (r - r_+) (r-r_-)
\text{ and }h:= \frac{\Delta}{r^2}. \label{eq:h}
\end{align}
Furthermore, define $r_\ast$ by
\begin{align}
\d r_\ast := \frac{r^2}{\Delta} \d r,
\end{align}
where we choose $r_\ast (\frac{r_+ + r_-}{2}) = 0$ for definiteness. Thus,
\begin{align}
r_\ast (r) = r + \frac{1}{2\kappa_+} \log|r - r_+| + \frac{1}{2\kappa_-} \log| r-r_-| + C \label{defn:rstar}
\end{align}
for a constant $C$ only depending on the black hole parameters and
\begin{align}\label{eq:surface}
\kappa_\pm = \frac{r_\pm - r_\mp}{2r_\pm^2}.
\end{align}
We also introduce null coordinates
\begin{align}\label{eq:nullu}
v= r_\ast + t \text{ and } u = r_\ast - t
\end{align}
on $\mathcal{M}$. The Penrose diagram of $\mathcal M$ is depicted in \cref{fig:interior_wo_boundary}.
\begin{figure}[ht]
\centering
\input{interior_w_o_boundary.pdf_tex}
\caption{Penrose diagram of $\mathcal{M}$; formally we have denoted the boundary (not part of the manifold) by $\mathcal{H} = \mathcal{H}_A \cup \mathcal{H}_B$ and $\mathcal{CH}=\mathcal{CH}_A \cup \mathcal{CH}_B$. }
\label{fig:interior_wo_boundary}
\end{figure}
\subsubsection{Attaching the event and Cauchy horizon}
\label{sec:interiorwboundary}
Now, we will attach the Cauchy and event horizon to the manifold. The Cauchy horizon characterizes the future boundary up to which the spacetime is uniquely determined as a solution to the Einstein--Maxwell system arising from data on the event horizon. Although the metric is smoothly extendible beyond the Cauchy horizon, any such extension fails to be uniquely determined from initial data, leading to a severe failure of determinism.
Attaching the event and Cauchy horizon gives rise to a manifold with corners.
To do so, we first define the following two pairs of null coordinates.
Let $U_{\mathcal{H}}\colon \mathbb{R} \to (0,\infty)$ and $V_{\mathcal{H}}\colon \mathbb{R} \to (0,\infty)$ be smooth and strictly increasing functions such that
\begin{itemize}
\item $ U_{\mathcal{H}}(u) =u $ for $u\geq 1$, $V_\mathcal{H} (v) = v $ for $v\geq 1$,
\item $ U_\mathcal{H}(u) \to 0$ as $u \to -\infty$ , $V_\mathcal{H} (v) \to 0 $ as $v \to -\infty$,
\item there exists a $u_+ \leq 1$ such that $\frac{\d U_\mathcal{H} }{\d u} = \exp(\kappa_+u) $ for $u\leq u_+$,
\item there exists a $v_+ \leq 1$ such that $\frac{\d V_\mathcal{H} }{\d v} = \exp(\kappa_+v) $ for $v\leq v_+$.
\end{itemize}
This defines -- in mild abuse of notation -- the null coordinates $U_\mathcal{H} \colon \mathcal{M} \to (0,\infty)$ via $U_\mathcal{H} (u)$ and $V_\mathcal{H} \colon \mathcal{M} \to (0,\infty)$ via $V_\mathcal{H}(v)$, where $u,v$ are the null coordinates defined in \eqref{eq:nullu}.
Similarly, let $U_{\mathcal{CH}}\colon \mathbb{R}\to (-\infty,0)$ and $V_{\mathcal{CH}}\colon \mathbb R \to (-\infty,0)$ be smooth and strictly increasing functions such that
\begin{itemize}
\item $ U_{\mathcal{CH}}(u) =u $ for $u\leq -1$, $V_\mathcal{CH} (v) = v $ for $v\leq -1$,
\item $ U_\mathcal{CH}(u) \to 0$ as $u \to \infty$ , $V_\mathcal{CH} (v) \to 0 $ as $v \to \infty$,
\item there exists a $u_+ \geq -1$ such that $\frac{\d U_\mathcal{CH} }{\d u} = \exp(\kappa_-u) $ for $u\geq u_+$,
\item there exists a $v_+ \geq -1$ such that $\frac{\d V_\mathcal{CH} }{\d v} = \exp(\kappa_-v) $ for $v\geq v_+$.
\end{itemize}
As above, this defines null coordinates $U_\mathcal{CH} : \mathcal{M} \to (0,\infty)$ via $U_\mathcal{CH} (u)$ and $V_\mathcal{CH}\colon \mathcal{M} \to (0,\infty)$ via $V_\mathcal{CH}(v)$, where $u,v$ are the null coordinates defined in \eqref{eq:nullu}.
To define the event horizon, we consider the coordinate chart $(U_\mathcal{H}, V_\mathcal{H},\theta,\phi)$. We now define the event horizon without the bifurcation sphere as the union
\begin{align}
\mathcal{H}_0 := \mathcal{H}_A \cup \mathcal{H}_B,
\end{align}
where
\begin{align}\mathcal{H}_A :=\{ U_\mathcal{H} =0 \}\times (0,\infty)\times \mathbb{S}^2 \text{ and }\mathcal{H}_B :=(0,\infty)\times \{V_\mathcal{H} = 0 \}\times \mathbb{S}^2.
\end{align}
Analogously, we also define the Cauchy horizon without the bifurcation sphere in the coordinate chart $(U_\mathcal{CH}, V_\mathcal{CH}, \theta, \phi)$ as the union
\begin{align}
\mathcal{CH}_0 := \mathcal{CH}_A \cup \mathcal{CH}_B,
\end{align}
where
\begin{align}\mathcal{CH}_A :=(0,\infty)\times \{V_\mathcal{CH} = 0 \}\times \mathbb{S}^2 \text{ and }\mathcal{CH}_B :=\{ U_\mathcal{CH} =0 \}\times (0,\infty)\times \mathbb{S}^2.
\end{align}
Then, we define the interior of the Reissner--Nordstr\"om spacetime without the bifurcation sphere as the manifold with boundary \begin{align}
\tilde{\mathcal{M}}:=\mathcal{M} \cup \mathcal{H} \cup \mathcal{CH}.
\end{align}
The Lorentzian metric on $\mathcal{M}$ extends smoothly to $\tilde{\mathcal{M}}$. In particular, the boundary of $\tilde{\mathcal{M}}$ consists of four disconnected null hypersurfaces. In \cref{fig:penrosemrn} we have depicted its Penrose diagram.
\begin{figure}[ht]
\centering
\input{interior_w_boundary.pdf_tex}
\caption{Penrose diagram of $\tilde{\mathcal{M}}$.}
\label{fig:penrosemrn}
\end{figure}
In mild abuse of notation we shall also use the coordinate systems
\begin{align}
&(U_\mathcal{H}, v, \theta,\phi) \text{ on }\mathcal M\cup \mathcal{H}_A\label{eq:ha},\\
&(u, V_\mathcal{H}, \theta,\phi) \text{ on }\mathcal M\cup \mathcal{H}_B\label{eq:hb},\\
&(u, V_\mathcal{CH}, \theta,\phi) \text{ on }\mathcal M\cup \mathcal{CH}_A\label{eq:ca},\\
&(U_\mathcal{CH}, v, \theta,\phi) \text{ on }\mathcal M\cup \mathcal{CH}_B.\label{eq:cb}
\end{align} In particular, we can write \begin{align}
&\mathcal{H}_A = \{ U_\mathcal{H} = 0 \}\times\{v \in \mathbb{R}\}\times \mathbb{S}^2,\\
&\mathcal{H}_B = \{ u \in \mathbb{R} \}\times\{V_\mathcal{H} = 0 \}\times \mathbb{S}^2,\\
&\mathcal{CH}_A = \{ u \in \mathbb{R} \}\times\{V_\mathcal{CH} = 0 \}\times \mathbb{S}^2,\\
&\mathcal{CH}_B = \{ U_\mathcal{CH} = 0 \}\times\{v \in \mathbb{R}\}\times \mathbb{S}^2.
\end{align}
Note also that the vector field $T$, initially defined on $\mathcal{M}$ in \eqref{def:T}, extends to a smooth vector field on $\tilde{\mathcal{M}}$ with \begin{align}T\restriction_{\mathcal{H}_A} = \frac{\partial}{\partial v}\restriction_{\mathcal{H}_A},\end{align} where $\frac{\partial}{\partial v}$ is the coordinate derivative with respect to local chart defined in \eqref{eq:ha}.
Similarly, we have
\begin{align}
& T\restriction_{\mathcal{H}_B} = -\frac{\partial}{\partial u}\restriction_{\mathcal{H}_B} \text{ w.r.t. to the local chart \eqref{eq:hb}},\\
&T\restriction_{\mathcal{CH}_A} = -\frac{\partial}{\partial u}\restriction_{\mathcal{CH}_A} \text{ w.r.t. to the local chart \eqref{eq:ca}},\\
&T\restriction_{\mathcal{CH}_B} = \frac{\partial}{\partial v}\restriction_{\mathcal{CH}_B} \text{ w.r.t. to the local chart \eqref{eq:cb}}.
\end{align}
Note that $T$ is a Killing null generator of the Killing horizons $\mathcal{H}_A, \mathcal{H}_B, \mathcal{CH}_A$, and $\mathcal{CH}_B$. Recall also that $\nabla_T T \restriction_{\mathcal{CH}} = \kappa_- T\restriction_{\mathcal{CH}}$ and $\nabla_T T \restriction_{\mathcal{H}} = \kappa_+ T\restriction_{\mathcal{H}}$, where $\kappa_\pm$ is defined by~\eqref{eq:surface}.
At this point, we note that we can attach corners to $\mathcal{H}_0$ and $\mathcal{CH}_0$ to extend $\tilde{\mathcal{M}}$ smoothly to a Lorentzian manifold with corners. To be more precise, we attach the past bifurcation sphere $\mathcal{B}_-$ to $\mathcal{H}_0$ as the point $(U_\mathcal{H},V_\mathcal{H}) = (0,0)$. Then, define $\mathcal{H}:=\mathcal{H}_0 \cup \mathcal{B}_-$. Similarly, we can attach the future bifurcation sphere $\mathcal{B}_+$ to the Cauchy horizon which will be denoted by $\mathcal{CH}$. We call the resulting manifold ${\mathcal{M}_{\mathrm{RN}}}$.
Further details are not given since the precise construction is straightforward and the metric extends smoothly. Moreover, the $T$ vector field extends smoothly to $\mathcal{B}_+$ and $\mathcal{B}_-$ and vanishes there. Its Penrose diagram is depicted in \cref{fig:penrosembar}.
\begin{figure}[H]
\centering
\input{interior_w_boundary_wb.pdf_tex}
\caption{Penrose diagram of ${\mathcal{M}_\text{RN}}$ which includes the bifurcate spheres $\mathcal{B}_+$ and $\mathcal{B}_-$.}
\label{fig:penrosembar}
\end{figure}
Further details about the coordinate systems can be found in~\cite{o2014geometry}. From a dynamical point of view, we can also consider the spacetimes $(\mathcal M_\mathrm{RN} , g_{M,Q})$ as being contained in the Cauchy development of a spacelike hypersurface with two asymptotically flat ends solving the Einstein--Maxwell system in spherical symmetry.
\subsection{The characteristic initial value problem for the wave equation}
In the context of scattering theory we will be interested in solutions to the wave equation \eqref{eq:linearwave} arising from suitable characteristic initial data. Recall the following well-posedness result for \eqref{eq:linearwave} with characteristic initial data.
\begin{prop}\label{thm:welldefined} Let $\Psi \in C_c^\infty(\mathcal{H}) $ be smooth compactly supported data on the event horizon $\mathcal{H}$. Then, there exists a unique smooth solution $\psi$ to \eqref{eq:linearwave} on $\mathcal{M}_{\mathrm{RN}}\setminus \mathcal{CH}$ such that $\psi\restriction_{\mathcal{H}} = \Psi$.
\end{prop}
The previous proposition is well known, see \cite{characteristic,rendall1990reduction}.
Analogously, we have the following for the backward evolution.
\begin{prop}\label{thm:time}
Let $\Psi \in C_c^\infty(\mathcal{CH}) $ be smooth compactly supported data on the Cauchy horizon $\mathcal{CH}$. Then, there exists a unique smooth solution $\psi$ to \eqref{eq:linearwave} on $\mathcal{M}_{\mathrm{RN}}\setminus \mathcal{H}$ such that $\psi\restriction_{\mathcal{CH}} = \Psi$.
\end{prop}
\subsection{Hilbert spaces of finite \texorpdfstring{$T$}{T} energy on both horizon components}
Now, we are in the position to define the Hilbert spaces on the event $\mathcal{H}= \mathcal{H}_A \cup \mathcal{H}_B \cup \mathcal{B}_-$ and Cauchy horizon $\mathcal{CH}= \mathcal{CH}_A \cup \mathcal{CH}_B \cup \mathcal{B}_+$, respectively.
We will start with constructing the Hilbert space on the event horizon. Roughly speaking, it will be defined by requiring finiteness of the $T$ energy flux on $\mathcal{H}_A$ \underline{minus} the $T$ energy flux on $\mathcal{H}_B$.
More precisely, let $C^\infty_c(\mathcal{H})$ be the vector space of smooth compactly supported functions on $\mathcal{H}$.
Recall that the Killing vector field $T$ is also a null generator of $\mathcal{H}$ and vanishes at the past bifurcation sphere $\mathcal{B}_-$.
This allows us to define the norm $\| \cdot \|^2_{\mathcal{E}^T_{\mathcal H}}$ on the vector space $C^\infty_c(\mathcal{H})$ as
\begin{align}\label{eq:firstdefnflux}
\| \psi \|^2_{\mathcal{E}^T_{\mathcal H}}:= \int_{\mathcal{H}_A} J_\mu^T[\psi] n_{\mathcal{H}_A}^\mu \, \d\mathrm{vol}_{n_{\mathcal{H}_A}} -\int_{\mathcal{H}_B} J_\mu^T[\psi] n_{\mathcal{H}_B}^\mu \, \d\mathrm{vol}_{n_{\mathcal{H}_B}},
\end{align}
where $\psi \in C_c^\infty(\mathcal{H})$, $\mathbf T[\psi]$ is the energy momentum tensor
\begin{align}
\mathbf{T}[\psi]_{\mu\nu} := \mathrm{Re}(\partial_\mu\psi\overline{ \partial_\nu \psi}) - \frac{1}{2}g_{\mu\nu} \partial_\alpha\psi \overline{\partial^\alpha\psi},
\end{align} and $J^T[\psi] := \mathbf{T}[\psi](T,\cdot)$.
In \eqref{eq:firstdefnflux}, the fluxes are defined with respect to future directed normal vector fields $n_{\mathcal{H}_A}$ and $n_{\mathcal{H}_B}$ on $\mathcal{H}_A$ and $\mathcal{H}_B$, respectively.\footnote{A choice of such normal vectors fixes the volume form. Also note that this is the natural setup for energy estimates. } Moreover, recall from \cref{fig:compare} that $T$ is future (resp.\ past) directed on $\mathcal{H}_A$ (resp.\ $\mathcal{H}_B$). Thus, the terms arising in \eqref{eq:firstdefnflux} satisfy $\int_{\mathcal{H}_A} J_\mu^T[\psi] n_{\mathcal{H}_A}^\mu \, \d\mathrm{vol}\geq 0$ and $-\int_{\mathcal{H}_B} J_\mu^T[\psi] n_{\mathcal{H}_B}^\mu \, \d\mathrm{vol}\geq 0 $.
Again, in view of the fact that on the component $\mathcal{H}_B$ the normal vector field $T$ is past directed, we can choose $n_{\mathcal{H}_A}: = T\restriction_{\mathcal{H}_A}$ and $n_{\mathcal{H}_B}:= -T\restriction_{\mathcal{H}_B}$ as the future directed normal vector fields on $\mathcal{H}_A$ and $\mathcal{H}_B$, respectively, such that we can realize the norm \eqref{eq:firstdefnflux} as
(using the coordinate charts \eqref{eq:ha} and \eqref{eq:hb})
\begin{align}
\| \psi \|^2_{\mathcal{E}^T_{\mathcal H}} = \int_{\mathbb R \times \mathbb{S}^2} |\partial_v\psi\restriction_{\mathcal{H}_A}|^2 \d v \sin\theta\d\theta \d \varphi + \int_{\mathbb R \times \mathbb{S}^2} |\partial_u\psi\restriction_{\mathcal{H}_B}|^2 \d u \sin\theta\d\theta \d \varphi .
\end{align}
The norm \eqref{eq:firstdefnflux} defines an inner product, hence its completion is a Hilbert space.
\begin{definition}
We define the Hilbert space of finite $T$ energy $\mathcal{E}^T_{\mathcal{H}}$ on both components of the event horizon as the completion of smooth and compactly supported functions $C_c^\infty(\mathcal{H})$ on the event horizon $\mathcal{H}= \mathcal{H}_A \cup \mathcal{H}_B \cup \mathcal{B}_-$ with respect to the norm \eqref{eq:firstdefnflux}.
\end{definition}
Analogously, we can consider the vector space $C_c^\infty(\mathcal{CH})$ and define the norm $\| \cdot \|^2_{\mathcal{E}^T_{\mathcal{CH}}}$ as the $T$ energy flux on the component $\mathcal{CH}_B$ \underline{minus} the $T$ energy flux on the component $\mathcal{CH}_A$:
\begin{align}\label{eq:firstdefnfluxch}
\| \psi \|^2_{\mathcal{E}^T_{\mathcal{CH}}}:= \int_{\mathcal{CH}_B} J_\mu^T[\psi] n_{\mathcal{CH}_B}^\mu \, \d\mathrm{vol}_{n_{\mathcal{CH}_B}} -\int_{\mathcal{CH}_A} J_\mu^T[\psi] n_{\mathcal{CH}_A}^\mu \, \d\mathrm{vol}_{n_{\mathcal{CH}_A}}.
\end{align} Again, in view of the orientation of the $T$ vector field (cf.\ \cref{fig:compare}), this norm can be represented as
(using the coordinate charts \eqref{eq:ca} and \eqref{eq:cb})
\begin{align}
\| \psi \|^2_{\mathcal{E}^T_{\mathcal{CH}}} = \int_{\mathbb R \times \mathbb{S}^2} |\partial_v\psi\restriction_{\mathcal{CH}_B}|^2 \d v \sin\theta\d\theta \d \varphi + \int_{\mathbb R \times \mathbb{S}^2} |\partial_u\psi\restriction_{\mathcal{CH}_A}|^2 \d u \sin\theta\d\theta \d \varphi .
\end{align}
\begin{definition}
We define the Hilbert space of finite $T$ energy $ \mathcal{E}^T_{\mathcal{CH}}$ on both components of the Cauchy horizon as the completion of smooth and compactly supported functions $C_c^\infty(\mathcal{CH})$ the Cauchy horizon $\mathcal{CH}= \mathcal{CH}_A \cup \mathcal{CH}_B \cup \mathcal{B}_+$ with respect to the norm \eqref{eq:firstdefnfluxch}.
\end{definition}
\subsection{Separation of variables}
In this section we show how the radial o.d.e.\ \eqref{eq:radialode1} arises from decomposing a general solution in spherical harmonics and Fourier modes. For concreteness, let $\psi$ be a smooth solution to $\Box_g \psi = 0$ such that on each $\{r=const.\}$ slice, $\psi$ is compactly supported in the $t$ variable.\footnote{Note that we will prove later that such solutions arise from data which are dense in $\mathcal{E}_\mathcal{H}^T$.}
Then, we can define its Fourier transform in the $t$ variable and also decompose $\psi$ in spherical harmonics to end up with
\begin{align}
\hat \psi_{m\ell} ( r,\omega) := \int_{\mathbb{R}\times \mathbb{S}^2} e^{-i\omega t} Y_{m\ell}(\theta,\phi) \psi(t,r,\theta,\phi) \sin\theta \d\theta\d\phi \frac{\d t}{\sqrt{2\pi}}.
\end{align} Due to the compact support on constant $r$ slices, the wave equation $\Box_g \psi = 0$ implies that \begin{align}\hat \psi_{m\ell}(r, \omega) =: R_{m\ell}^{(\omega)}(r) =: R(r) \end{align} satisfies the radial o.d.e.
\begin{align}\label{eq:radialode}
\Delta \frac{\d}{\d r}\left(\Delta \frac{\d}{\d r} R\right) - \Delta \ell(\ell+1) R + r^4 \omega^2 R =0.
\end{align}
In \cref{sec:radial} we will analyze solutions to \eqref{eq:radialode} and denote a solution thereof with $R(r)$. Moreover, it is useful to introduce the function $u$ defined as
\begin{align}
u(r) := r R(r)
\end{align}
and consider $u = u(r(r_\ast))$ as a function of $r_\ast$, which is defined in \eqref{defn:rstar}.
Using the $r_\ast$ variable, the o.d.e.~\eqref{eq:radialode} finally reduces to
\begin{align}
u^{\prime \prime} + (\omega^2 - V_\ell ) u= 0 \label{ODE1}
\end{align}
on the real line with potential
\begin{align}\label{eq:potential}
V=V_\ell = \Delta\left( \frac{r(r_+ + r_-) - 2 r_+ r_-}{r^3} + \frac{\ell(\ell+1)}{r^4}\right).
\end{align}
In \cref{lem:asymptoticspotential} in the appendix it is proven that, as a function of $r_*$, the potential $V_\ell$ decays exponentially as $ r_\ast \to \pm \infty$. In particular, this indicates that we have asymptotic free waves (asymptotic states) near the event and Cauchy horizon of the form $e^{\pm i \omega r_\ast}$ as $|r_\ast | \to \infty$. In order to construct these solutions we use the following proposition for Volterra integral equations (see Lemma 2.4 of \cite{schlag2010decay}).
\begin{prop}\label{lem:volterra}
Let $x_0\in \mathbb{R}\cup \{+\infty\}$ and $g\in L^\infty(-\infty,x_0)$. Suppose the integral kernel $K$ satisfies
\begin{align}
\alpha := \int_{-\infty}^{x_0} \sup_{ \{x: y<x<x_0 \} } |K(x,y)| \d y < \infty.
\end{align}
Then, the Volterra integral equation
\begin{align}\label{avolterraeqn}
f(x) = g(x) + \int_{-\infty}^{x} K(x,y) f(y) \d y
\end{align}
has a unique solution $f$ satisfying
\begin{align}
\| f\|_{L^\infty(-\infty,x_0)}\leq e^{\alpha} \| g\|_{L^\infty(-\infty,x_0)}.
\end{align}
If in addition $K$ is smooth in both variables and
\begin{align}
\int_{-\infty}^{x_0} \sup_{\{x: y<x<x_0\}} |\partial_x^k K(x,y)| \d y < \infty
\end{align}
for all $k\in \mathbb{N}$, then the solution $f$ is smooth on $(-\infty, x_0)$ and the derivatives can be computed by formal differentiation of~\eqref{avolterraeqn}.
\end{prop}
\begin{rmk}\label{rmk:volterra}
Analogous results as in \cref{lem:volterra} also hold true for Volterra integral equations on intervals of the form $(x_0,x_1)$ or $(x_0,+\infty)$.
\end{rmk}
This allows us to define the following fundamental pairs of solutions to the o.d.e.~\eqref{ODE1}. In view of the exponential decay of the potential, it is straightforward to check that the assumptions of \cref{lem:volterra} are satisfied.
\begin{definition}\label{defn:u1u2}
Let $\omega \in \mathbb R$ and $\ell\in\mathbb N_0$ be fixed. Define asymptotic state solutions $u_1$ and $u_2$ of the radial o.d.e.\ \eqref{ODE1} as the unique solutions to the Volterra integral equations
\begin{align}\label{eq:constructionu1}
&u_1(\omega, r_\ast) = e^{i\omega r_\ast} + \int_{-\infty}^{r_\ast} \frac{\sin(\omega(r_\ast - y))}{\omega} V(y) u_1(\omega, y) \d y,
\\
&u_2(\omega, r_\ast) = e^{-i\omega r_\ast} + \int_{-\infty}^{r_\ast} \frac{\sin(\omega(r_\ast - y))}{\omega} V(y) u_2(\omega, y) \d y. \end{align}
Analogously, define $v_1$ and $v_2$ as the unique solutions to the Volterra integral equations
\begin{align}
&v_1(\omega, r_\ast) = e^{i\omega r_\ast} - \int_{r_\ast}^{\infty} \frac{\sin(\omega(r_\ast - y))}{\omega} V(y) v_1(\omega, y) \d y,
\\
&v_2(\omega, r_\ast) = e^{-i\omega r_\ast} - \int_{r_\ast}^{\infty} \frac{\sin(\omega(r_\ast - y))}{\omega} V(y) v_2(\omega, y) \d y. \end{align}
For $\omega =0$, we set $\frac{\sin(\omega(r_\ast - y))}{\omega}\restriction_{\omega =0} = r_\ast - y$ in the integral kernel in which case $u_1$ and $u_2$ coincide. We define
\begin{align} \tilde u_1(r_\ast) := u_1(0,r_\ast) = u_2(0,r_\ast) \end{align}
and similarly,
\begin{align}
\tilde v_1 (r_\ast) := v_1(0,r_\ast) = v_2(0,r_\ast).
\end{align}
Since $u_1(0,r_\ast) = u_2(0,r_\ast)$ for $\omega=0$, there exists another linearly independent fundamental solution $\tilde u_2$ solving the Volterra integral equation
\begin{align}
&\tilde u_2 (r_\ast) = r_\ast + \int_{-\infty}^{r_\ast} (r_\ast -y) V(y) \tilde u_2(y) \d y.
\end{align}
Similarly, we also have another fundamental solution, which is linearly independent from $\tilde v_1$, solving
\begin{align}
&\tilde v_2 (r_\ast) = r_\ast - \int_{r^\ast}^{\infty} (r_\ast -y) V(y) \tilde v_2(y) \d y.
\end{align}
Since $r_*$ is not uniformly bounded, we cannot apply \cref{lem:volterra} to construct $\tilde u_2$ and $\tilde v_2$. Nevertheless, after switching to coordinates which are regular at $\mathcal{H}$ or $\mathcal{CH}$, the existence of the desired solutions follows immediately from the usual local theory of regular singularities (see~\cite{olver2014asymptotics}).
\end{definition}
\begin{rmk}
\label{rmk:holo}
Due to the exponential decay of the potential $V_\ell$ (see \cref{lem:asymptoticspotential} in the appendix), it follows from standard theory that the solutions $u_1(\omega, r_\ast), u_2(\omega, r_\ast)$, $v_1(\omega, r_\ast)$ and $v_2(\omega, r_\ast)$ can be continued to holomorphic functions of $\omega$ in the strip $|\operatorname{Im}(\omega)| < \kappa_+$ for fixed $r_\ast \in \mathbb R$. Indeed, in \cite{hartle1982crossing} it is shown that $u_1(\omega,r_\ast)$ is analytic in $\mathbb C\setminus \{im\kappa_+\colon m\in \mathbb{N}\}$ with possible poles at $\{ i m \kappa_+ \colon m\in \mathbb N \}$ and similarly for $u_2, v_1$, and $v_2$. See also the proof of \cref{prop:rtboundedcomplex} in the appendix.
\end{rmk}
This allows us now to define the reflection and transmission coefficients $\mathfrak R$ and $\mathfrak T$.
\begin{definition} \label{defn:TandR} Let $\omega \neq 0$. Then we define the transmission coefficient $\mathfrak T(\omega, \ell)$ and reflection coefficient $\mathfrak R(\omega, \ell)$ as the unique coefficients such that
\begin{align}
u_1 = \mathfrak T v_1 + \mathfrak R v_2. \label{eq:defn1}
\end{align}
Using the fact that the Wronskian
\begin{align}\mathfrak W(f,g) := f g^\prime - f^\prime g\end{align} of two solutions $f$ and $g$ is independent of $r_\ast$, we can equivalently define the scattering coefficients as
\begin{align}
\mathfrak T : = \frac{ \mathfrak W(u_1,v_2)}{ \mathfrak W(v_1,v_2)} = \frac{\mathfrak W(u_1,v_2)}{-2i\omega}
\end{align}
and
\begin{align}
\mathfrak R : = \frac{\mathfrak W(u_1,v_1)}{\mathfrak W(v_2,v_1)} = \frac{ \mathfrak W(u_1,v_1)}{2i\omega}.
\end{align}
\end{definition}
The transmission and reflection coefficients satisfy a pseudo-unitarity property proven in the following.
\begin{prop}[Pseudo-unitarity in the separated picture]\label{prop:pseudouni}
The transmission and reflection coefficients satisfy\begin{align}\label{eq:pseudounitaryode}
1 = |\mathfrak T|^2 - |\mathfrak R|^2.
\end{align}
\begin{proof} First, note that any solution to the o.d.e.\ \eqref{ODE1} satisfies the identity \begin{align}
\operatorname{Im}(\bar u u^\prime ) = const.\
\end{align}
Applying this to the solution $u_1 = \mathfrak T v_1 + \mathfrak R v_2$ shows the claim.
\end{proof}
\end{prop}
In the following we shall see that the reflection and transmission coefficients are regular at $\omega =0$.
\begin{prop}\label{cor:rdoesntvanish}Let $\ell \in \mathbb N_0 $ be fixed. Then the scattering coefficients $\mathfrak R(\omega,\ell)$ and $\mathfrak T (\omega,\ell)$ are analytic functions of $\omega$ in the strip $
\{\omega \in \mathbb C \colon |\operatorname{Im}(\omega)| < \kappa_+
\}$ with values for $\omega =0$ given by
\begin{align}\label{eq:r0}
&\mathfrak R(0,\ell) = \frac{(-1)^\ell}{2} \left( \frac{r_-}{r_+} - \frac{r_+}{r_-}\right),\\
&\label{eq:t0} \mathfrak T(0,\ell) = \frac{(-1)^\ell}{2} \left(\frac{r_-}{r_+}+ \frac{r_+}{r_-} \right).
\end{align} In particular, the reflection coefficient $\mathfrak R(\omega,\ell)$ only vanishes on a discrete set of points $\omega$.
Moreover, the reflection and transmission coefficients $\mathfrak R(\omega,\ell)$ and $\mathfrak T(\omega,\ell)$ are analytic functions on $\mathbb C \setminus \mathbb P$ with possible poles at $\mathbb P = \{ i m \kappa_+\colon m \in \mathbb N\} \cup \{i k \kappa_- \colon k \in \mathbb Z \setminus \{ 0\} \}$.
\begin{proof}
From the analyticity of $u_1,u_2,v_1$, and $v_2$ in the strip $|\operatorname{Im}(\omega) |< \kappa_+$ (cf.\ \cref{rmk:holo}), we conclude that $\mathfrak T$ and $\mathfrak R$ are holomorphic in $\{\omega\neq 0 \in \mathbb C : |\operatorname{Im}(\omega) |< \kappa_+\}$ with a possible pole at $\omega =0$. In the following we shall show that $\{\omega =0\}$ is a removable singularity. Indeed, we will compute the explicit value of the reflection and transmission coefficient at $\omega=0$ and deduce that for fixed $\ell\in\mathbb{N}_0$, the transmission coefficient $\mathfrak T(\omega,\ell)$ and the reflection coefficient $\mathfrak R(\omega,\ell)$ are analytic functions on the strip $\{\omega \in \mathbb C \colon \operatorname{Im}(\omega) |< \kappa_+ \}$ (cf.\ unpublished work of McNamara cited in \cite{gursel1979evolution}). To do so, note that from \cref{prop:intermediate} in \cref{sec:boundedrefltransmi} we conclude the pointwise limits
\begin{align}
&u_1(\omega,r_\ast) \to \tilde u_1(r_\ast),\\
&v_1(\omega,r_\ast) \to \tilde v_1 (r_\ast) = (-1)^\ell \frac{r_+}{r_-}\tilde u_1(r_\ast),\\
&v_2(\omega,r_\ast) \to \tilde v_1 (r_\ast) = (-1)^\ell \frac{r_+}{r_-}\tilde u_1(r_\ast)
\end{align}
as $|\omega|\to 0$. Using the definition in \eqref{eq:defn1} of $\mathfrak T(\omega,\ell)$, $\mathfrak R(\omega,\ell)$, and the condition $1+|\mathfrak R|^2 = |\mathfrak T|^2$ (cf.~\cref{prop:pseudouni}), we deduce that the limits $\lim_{\omega\to 0} \mathfrak R(\omega, \ell)$ and $\lim_{\omega\to 0} \mathfrak T(\omega, \ell)$ exist and moreover can be computed to be \eqref{eq:r0} and \eqref{eq:t0}.
Note that \eqref{eq:r0} and \eqref{eq:t0} have been established in \cite{gursel1979final}.
Also note that in view of the analyticity properties of $u_1$, $v_1$, and $v_2$, the $\mathfrak R(\omega,\ell)$ and $\mathfrak T(\omega,\ell)$ are analytic functions on $\mathbb C \setminus \mathbb P$ with possible poles at $\mathbb P = \{ i m \kappa_+\colon m \in \mathbb N\} \cup \{i k \kappa_- \colon k \in \mathbb Z \setminus \{ 0\} \}$.
\end{proof}
\end{prop}
\subsection{Conventions} Let $X$ be a point set with a limit point $c$ (e.g.\ $X = \mathbb R, [a,b], \mathbb C$). Throughout this paper we will use the symbols $\lesssim$ and $\gtrsim$, where the implicit constants might depend on the black hole parameters $M$ and $Q$. In particular, for functions (or constants) $a(x),b(x) >0$ the notation $a\lesssim b$ means that there is a constant $C = C(M,Q) > 0 $ such that $a(x)\leq Cb(x) $ for all $x\in X$. We will also make use of the notation $\lesssim_\ell$ or $\gtrsim_\ell$ which means that the constant may additionally also depend on $\ell$. We also write $a\sim b$ if there are constants $C(M,Q),\tilde C(M,Q) >0$ such that $C a(x) \leq b(x) \leq \tilde Ca(x)$ for all $x\in X$.
We shall also make use of the standard Landau notation $O$ and $o$ \cite{NIST:DLMF,olver2014asymptotics}. To be more precise, as $x\to c$ in $X$
\begin{align}
f(x) = O(g(x)) &\text{ means } \left|\frac{f(x)}{g(x)}\right| \leq C(M,Q)\label{eq:O}
\end{align}
and
\begin{align}
f(x) = o(g(x)) &\text{ means } \frac{f(x)}{g(x)} \to 0\label{{eq:O1}}.
\end{align}
We will also use the notation $O_\ell$ if the constant $C$ in \eqref{eq:O} may additionally depend on $\ell$.
\section{Main theorems}
\label{sec:mainthms}
In this section we will formulate our main theorems.
\cref{thm:forwardevolution}, which we state in \cref{sec:existencescatteringmap}, establishes the existence of a scattering map $S^T$ of the form
\begin{align}
&S^T: \mathcal{E}^T_{\mathcal{H}}\to\mathcal{E}^T_{\mathcal{CH}},
\end{align}
which is a Hilbert space isomorphism, i.e.\ a bounded and invertible map with bounded inverse. \cref{thm:forwardevolution} will be proven in \cref{sec:mainthm}. In the separated picture, the boundedness of $S^T$ corresponds to the uniform boundedness of the transmission and reflection coefficients which is stated as \cref{thm:boundednesstrans} in \cref{subsec:scatteringcoefficients}. \cref{thm:boundednesstrans} will be proven in \cref{sec:radial} (and later used in the proof of \cref{thm:forwardevolution}).
\cref{subsec:connfourierphysical} is then devoted to \cref{thm:fouriertophysical}, which
connects our physical space scattering theory to the fixed frequency scattering theory. (We will infer \cref{thm:fouriertophysical} as a consequence of \cref{thm:forwardevolution}.) In \cref{subsec:reflection}, this connection allows us to prove that the reflection map is injective, which is the content of \cref{thm:nonvanishingreflection}. In \cref{thm:c1instab}, which is stated and proven in \cref{sec:c1instab}, we construct data which are incoming and compactly supported but nevertheless, lead to a solution which fails to be in $C^1$ on the Cauchy horizon.
We end this section with the statement of our two non-existence results. In \cref{subsec:nonex} we formulate \cref{thm:cosmological}, the non-existence of the $T$ energy scattering theory for the Klein--Gordon equation with conformal mass on the interior of (anti-) de~Sitter--Reissner--Nordstr\"om black holes. The proof of \cref{thm:cosmological} is given in \cref{sec:cosmo}. Finally, in \cref{thm:kleingordon}, stated in \cref{subsec:notkleingordon}, we show the non-existence of the $T$ energy scattering map for the Klein--Gordon equation on the interior of Reissner--Nordstr\"om. The proof of \cref{thm:kleingordon} is given in \cref{sec:kleingordonequation}.
\subsection{Existence and boundedness of the \texorpdfstring{$T$}{T} energy scattering map}
\label{sec:existencescatteringmap}
First, we define the forward (resp.\ backward) evolution on a dense domain.
\begin{definition}\label{defn:domainforward}
The domains of the forward and backward evolution are defined as \begin{align}\nonumber
\mathcal{D}^T_{\mathcal{H}}:= \{ \psi \in C_c^\infty(\mathcal H) &\subset \mathcal{E}_{\mathcal{H}}^T \text{ s.t. the Cauchy evolution of $\psi$ has}\\ &\text{ compact support on constant $r=const.$ hypersurfaces} \} \end{align}
and
\begin{align}\nonumber
\mathcal{D}^T_{\mathcal{CH}}:= \{ \psi \in C_c^\infty(\mathcal{CH})& \subset \mathcal{E}_{\mathcal{CH}}^T \text{ s.t. the backward evolution of $\psi$ has}\\ & \text{ compact support on constant $r=const.$ hypersurfaces} \}, \end{align}
respectively. Here, we consider $r_-< r < r_+$ and note that if $\psi$ is compactly supported on one $\{r=const.\}$ slice, then, as a direct consequence of the domain of dependence, its evolution will be compactly supported on all other $\{r=const.\}$ hypersurfaces for $r_-< r < r_+$.
We will prove in \cref{lem:lemmadense} in \cref{sec:mainthm} that $\mathcal{D}^T_{\mathcal{H}} \subset \mathcal{E}_{\mathcal{H}}^T$ and $ \mathcal{D}^T_{\mathcal{CH}} \subset \mathcal{E}_{\mathcal{CH}}^T $ are dense domains.
\end{definition}
These definitions of the domains are motivated by the following observation.
\begin{rmk}\label{rmk:domain}
Suppose we are given data in $\mathcal{D}_\mathcal{H}^T$ on the event horizon $\mathcal{H}$. Consider now the unique Cauchy development (cf.\ \cref{thm:welldefined}) and observe that its restriction to the Cauchy horizon $\mathcal{CH}$ will lie in $\mathcal{D}_{\mathcal{CH}}^T$. This holds true since we can first smoothly extend the metric beyond the Cauchy horizon $\mathcal{CH}$ and then use the compact support on a constant $r_\ast$ hypersurface to solve an equivalent Cauchy problem in an appropriate region which extends the Cauchy horizon $\mathcal{CH}$, includes the support of the solution, but does not include $i^+$. The smoothness of the solution up to and including the Cauchy horizon $\mathcal{CH}$ follows now from Cauchy stability.
\end{rmk}
In view of \cref{rmk:domain} we can define the forward and backward map on the domains $\mathcal{D}^T_\mathcal{H}$ and $\mathcal{D}^T_\mathcal{CH}$, respectively.
\begin{definition}
Define the forward map $S_0^T\colon \mathcal{D}^T_{\mathcal{H}} \subset \mathcal{E}_{\mathcal{H}}^T \to \mathcal{D}^T_{\mathcal{CH}} \subset \mathcal{E}_{\mathcal{CH}}^T $ as the unique forward evolution from data on the event horizon to data on the Cauchy horizon. More precisely, let $\psi$ be the solution to \eqref{eq:linearwave} arising from initial data $\Psi \in \mathcal{D}^T_{\mathcal{H}} \subset \mathcal{E}_{\mathcal{H}}^T$. Then, define $S_0^T(\Psi)$ as the restriction of $\psi$ to the Cauchy horizon, i.e.\ $S_0^T(\Psi) := \psi\restriction_{\mathcal{CH}} \in \mathcal{D}^T_{\mathcal{CH}}$.
Similarly, let $\phi$ be the unique backward evolution of \eqref{eq:linearwave} arising from $\Phi \in \mathcal{D}^T_{\mathcal{CH}}$. Then, define the backward map by $B_0^T (\Phi):= \phi\restriction_{\mathcal{H}} \in \mathcal{D}^T_\mathcal{H}$.
\end{definition}
\begin{rmk}\label{rmk:injectiv}
Note that by the uniqueness of the Cauchy evolution we have that $S_0^T$ and $B_0^T$ are inverses of each other, i.e.\ $B_0^T\circ S_0^T = \mathrm{Id}_{ \mathcal{D}^T_{\mathcal{H}} }, \; S_0^T \circ B_0^T = \mathrm{Id}_{ \mathcal{D}^T_{\mathcal{CH}}}. $
\end{rmk}
Now, we are in the position to state our main theorem.
\begin{theorem}\label{thm:forwardevolution}
The map $S_0^T\colon \mathcal{D}^T_\mathcal{H} \subset \mathcal{E}^T_\mathcal{H} \to \mathcal{D}^T_\mathcal{CH} \subset \mathcal{E}^T_\mathcal{CH} $ is bounded and uniquely extends to
\begin{align}
& S^T\colon \mathcal{E}^T_{\mathcal{H}} \to \mathcal{E}^T_{\mathcal{CH}},
\end{align}
called the ``scattering map''. The scattering map $S^T$ is a Hilbert space isomorphism, i.e.\ a \underline{bounded} and \underline{invertible} linear map with \underline{bounded inverse} $B^T\colon \mathcal{E}^T_{\mathcal{CH}} \to \mathcal{E}^T_{\mathcal{H}}$ satisfying
\begin{align}
& \label{eq:inverses}B^T\circ S^T = \mathrm{Id}_{ \mathcal{E}^T_{\mathcal{H}} }, \; S^T \circ B^T = \mathrm{Id}_{ \mathcal{E}^T_{\mathcal{CH}}}.
\end{align}
Here, $B^T\colon \mathcal{E}^T_{\mathcal{CH}} \to \mathcal{E}^T_{\mathcal{H}}$ is the ``backward map'', which is the unique bounded extension of $B_0^T$.
In addition, the scattering map $S^T$ is pseudo-unitary, meaning that for $\psi\in \mathcal{E}^T_{\mathcal H}$, we have
\begin{align}\label{eq:pseudounitary}
\int_{\mathcal{H}_A} |T\psi|^2 - \int_{\mathcal{H}_B} |T\psi|^2 = \int_{\mathcal{CH}_B} |TS^T\psi|^2 -\int_{\mathcal{CH}_A} |T S^T\psi|^2.
\end{align}
\end{theorem}
In more traditional language, \cref{thm:forwardevolution} yields existence, uniqueness, and asymptotic completeness of scattering states.
The proof of \cref{thm:forwardevolution} is given in \cref{sec:mainthm}. Let us note already that \cref{thm:forwardevolution} is a posteriori the physical space equivalent of the uniform boundedness of the scattering coefficients proven in \cref{thm:boundednesstrans} (see \cref{subsec:scatteringcoefficients}). This equivalence is made precise in \cref{thm:fouriertophysical} (see \cref{subsec:connfourierphysical}).
\begin{rmk}
Note that in general, neither initial data nor scattered data have to be bounded in $L^\infty$ or continuous. Indeed, we have that $\Phi_A(u,\theta,\varphi) = \log(u)\chi_{u\geq 1} \in \mathcal{E}^T_{\mathcal{CH}_A}$, where $\chi_{u\geq 1}$ is a smooth cutoff. Thus, there exist initial data $B^T(\Phi_A)\in \mathcal{E}^T_{\mathcal{H}}$ such that its image under the scattering map is not in $L^\infty$ and not continuous. We emphasize the contrast with the estimates from \cite{franzen2016boundedness} for which more regularity and decay along the event horizon $\mathcal{H}$ are necessary.
\end{rmk}
\subsection{Uniform boundedness of the transmission and reflection coefficients}
\label{subsec:scatteringcoefficients}
On the level of the o.d.e.~\eqref{ODE1} in the separated picture, the problem of boundedness of the scattering map reduces to proving that the transmission coefficient $\mathfrak T$ and the reflection coefficient $\mathfrak R$ are uniformly bounded over all parameter ranges of $\omega\in\mathbb R$ and $\ell\in\mathbb{N}_0$. This is stated as \cref{thm:boundednesstrans} below.
\begin{restatable}{theorem}{boundedness}
\label{thm:boundednesstrans}The reflection and transmission coefficients $\mathfrak R(\omega,\ell)$ and $\mathfrak T(\omega, \ell)$ are uniformly bounded, i.e.\ they satisfy
\begin{align}
\sup_{\omega \in \mathbb R , \ell \in \mathbb N_0 }( |\mathfrak R(\omega, \ell) | + |\mathfrak T (\omega, \ell)| ) \lesssim 1.
\end{align}
\end{restatable}
\cref{thm:boundednesstrans} is proved in \cref{sec:radial}. As discussed in the introduction, the proof relies on an explicit calculation for $\omega =0$ together with a careful analysis of the radial o.d.e.~\eqref{ODE1}, involving properties of special functions and perturbations thereof.
Let us note that, \emph{given} \cref{thm:forwardevolution}, we could infer \cref{thm:boundednesstrans} as a corollary (using the theory to be described in \cref{subsec:connfourierphysical}). We emphasize, however, that in the present paper we \emph{use} \cref{thm:boundednesstrans} to prove \cref{thm:forwardevolution} in \cref{sec:mainthm}.
\subsection{Connection between the separated and the physical space picture}
In this section, we will make the connection of the separated and physical space picture precise.
\label{subsec:connfourierphysical}
First, let us note that we have natural Hilbert space decompositions $\mathcal{E}^T_{\mathcal{H}} \cong \mathcal{E}^T_{\mathcal{H}_A} \oplus \mathcal{E}^T_{\mathcal{H}_B}$ and $ \mathcal{E}^T_{\mathcal{CH}}\cong \mathcal{E}^T_{\mathcal{CH}_B} \oplus \mathcal{E}^T_{\mathcal{CH}_A}$.
\begin{prop}\label{cor:decomposition}
The Hilbert spaces $\mathcal{E}^T_{\mathcal{H}}$ and $\mathcal{E}^T_{\mathcal{CH}}$ of finite $T$ energy on the event horizon $\mathcal H$ and on the Cauchy horizon $\mathcal{CH}$ admit the orthogonal decomposition
\begin{align}
\mathcal{E}^T_{\mathcal{H}} \cong \mathcal{E}^T_{\mathcal{H}_A} \oplus \mathcal{E}^T_{\mathcal{H}_B}\, \text{ and }\; \mathcal{E}^T_{\mathcal{CH}} \cong \mathcal{E}^T_{\mathcal{CH}_A} \oplus \mathcal{E}^T_{\mathcal{CH}_B}.
\end{align}
\begin{proof}
Clearly, the embedding $i\colon \mathcal{E}^T_{\mathcal{H}_A} \oplus \mathcal{E}^T_{\mathcal{H}_B} \hookrightarrow \mathcal{E}^T_{\mathcal{H}}$ is well-defined and isometric. It remains to show that $i$ is surjective. Let $\psi \in C_c^\infty(\mathcal{H})$. First, we show that we can approximate (in $T$-energy) $\psi\restriction_{\mathcal{H}_A}$ on $\mathcal{H}_A$ with functions $\psi_\epsilon \in C_c^\infty(\mathcal{H}_A)$ which are supported away from the past bifurcation sphere. On $\mathcal{H}_A $ choose non-degenerate coordinates $(V,\theta,\varphi) := (V_\mathcal{H}, \theta,\varphi)$ as in \cref{sec:interiorwboundary} and recall that the past bifurcation sphere is $\{V=0\}$. Then, for small $\epsilon >0$, set
\begin{align}\label{eq:logcut1}
\psi_\epsilon(V,\theta, \varphi) := \psi(U=0,V,\theta,\varphi) \chi(-\epsilon \log(V)),
\end{align}
where $\chi\colon \mathbb{R} \to [0,1]$ is smooth and such that $\operatorname{supp} (\chi) \subseteq (-\infty,2]$ and $\chi\restriction_{(-\infty,1]} = 1$.
Then, it is straightforward to check that $\psi_\epsilon \in C_c^\infty(\mathcal{H}_A)$ and
\begin{align}\label{eq:logcut2}
\int_{\mathcal{H}_A} J^T[\psi-\psi_\epsilon]_\mu n^\mu \d\mathrm{vol} \lesssim \int_{\mathbb{S}^2} \int_{0}^{\infty} V (\partial_V (\psi - \psi_\epsilon) )^2 \d{V} \sin\theta \d \theta \d\varphi \to 0
\end{align}
as $\epsilon \to 0$.
Analogously, we can do this for $\mathcal{H}_B$ from which the claim follows. \end{proof}
\end{prop}
We will use this identification to represent the scattering map also in the Fourier picture and show how these pictures connect. To do so we define the following.
\begin{definition}
For $(\Psi_A, \Psi_B) \in \mathcal{E}^T_{\mathcal{H}_A} \oplus \mathcal{E}^T_{\mathcal{H}_B}$ note that $\partial_v\Psi_A (v,\theta,\phi) \in L^2(\mathbb R \times \mathbb{S}^2; \mathbb C)$ and analogously for $\Psi_B$. Hence, in mild abuse of notation, we can define the Fourier and spherical harmonics coefficients $ \mathcal{F}_{\mathcal{H}_A}(\Psi_A)$ and $\mathcal{F}_{\mathcal{H}_B}(\Psi_B)$ as
\begin{align}\label{eq:defnF1}
i \omega \mathcal{F}_{\mathcal{H}_A}(\Psi_A) (\omega,m,\ell):= r_+ \int_{\mathbb R} \int_{\mathbb{S}^2} \partial_v\Psi_A(v,\theta,\varphi) e^{-i\omega v} Y_{\ell m}(\theta,\varphi) \sin\theta \d\theta \d\varphi \frac{\d v}{\sqrt{2\pi}}
\end{align}
and
\begin{align}
- i \omega \mathcal{F}_{\mathcal{H}_B}(\Psi_B)(\omega, m ,\ell) := r_+ \int_{\mathbb R} \int_{\mathbb{S}^2} \partial_u\Psi_B(u,\theta,\varphi) e^{i\omega u} Y_{\ell m}(\theta,\varphi)\sin\theta \d\theta \d\varphi \frac{\d u}{\sqrt{2\pi}}.
\end{align}
Similarly, for $(\Phi_A, \Phi_B) \in \mathcal{E}^T_{\mathcal{CH}_A} \oplus \mathcal{E}^T_{\mathcal{CH}_B}$ set
\begin{align}
-i \omega \mathcal{F}_{\mathcal{CH}_A}(\Phi_A) (\omega, m, \ell ):= r_-\int_{\mathbb R} \int_{\mathbb{S}^2} \partial_u\Phi_A(u,\theta,\varphi) e^{i\omega u} Y_{\ell m}(\theta,\varphi) \sin\theta \d\theta \d\varphi \frac{\d u}{\sqrt{2\pi}}
\end{align}
and
\begin{align}\label{eq:defnF2}
i \omega \mathcal{F}_{\mathcal{CH}_B}(\Phi_B) (\omega, m , \ell ):= r_- \int_{\mathbb R} \int_{\mathbb{S}^2} \partial_v\Phi_B(v,\theta,\varphi) e^{-i\omega v} Y_{\ell m}(\theta,\varphi)\sin\theta \d\theta \d\varphi \frac{\d v}{\sqrt{2\pi}}.
\end{align}
\end{definition}
Also, recall the Hilbert space decomposition $\mathcal{E}^T_{\mathcal{H}} \cong \mathcal{E}^T_{\mathcal{H}_A} \oplus \mathcal{E}^T_{\mathcal{H}_B}$ and $ \mathcal{E}^T_{\mathcal{CH}}\cong \mathcal{E}^T_{\mathcal{CH}_B} \oplus \mathcal{E}^T_{\mathcal{CH}_A}$. Thus, the scattering matrix can be also decomposed as
\begin{align}
S^T = \begin{pmatrix}
S^T_{BA}& S^T_{BB} \\
S^T_{AA} & S^T_{AB}
\end{pmatrix},
\end{align}
where \begin{align}S^T_{ij}\colon \mathcal{E}^T_{\mathcal{H}_j} \to \mathcal{E}^T_{\mathcal{CH}_i}\end{align} is a bounded linear map for $i,j \in \{ A,B\}$.\footnote{Note that $T$ does not denote the transpose but the fact that it is the scattering map associated with the $T$ vector field.}
\begin{definition}\label{def:hilbertspacesfourier}
Define the Hilbert spaces \begin{align*} &\hat{\mathcal{E}}^T_{\mathcal{H}_A} := \ell^2(Z;L^2(r_+^{-2} {\omega^2} \d \omega)),\, \; \hat{\mathcal{E}}^T_{\mathcal{H}_B} := \ell^2(Z;L^2(r_+^{-2} {\omega^2} \d \omega)),\\ &\hat{\mathcal{E}}^T_{\mathcal{CH}_A} := \ell^2(Z;L^2(r_-^{-2} {\omega^2} \d \omega)),\,\; \hat{\mathcal{E}}^T_{\mathcal{CH}_B} := \ell^2(Z;L^2(r_-^{-2} {\omega^2} \d \omega)),\end{align*}
where $Z = \{(m,\ell) \in \mathbb Z \times \mathbb N_0: |m|\leq \ell \}$.
\end{definition}
The Hilbert spaces defined in \cref{def:hilbertspacesfourier} are unitary isomorphic to their corresponding physical energy spaces. This is captured in
\begin{prop}
The linear maps defined in \eqref{eq:defnF1}--\eqref{eq:defnF2} \begin{align}&\mathcal{F}_{\mathcal{H}_A}\oplus \mathcal{F}_{\mathcal{H}_B}\colon \mathcal{E}^T_{\mathcal{H}_A} \oplus \mathcal{E}^T_{\mathcal{H}_B} \to \hat{\mathcal{E}}^T_{\mathcal{H}_A} \oplus \hat{\mathcal{E}}^T_{\mathcal{H}_B} \\ &\mathcal{F}_{\mathcal{CH}_B}\oplus \mathcal{F}_{\mathcal{CH}_A} \colon \mathcal{E}^T_{\mathcal{CH}_B} \oplus \mathcal{E}^T_{\mathcal{CH}_A} \to \hat{\mathcal{E}}^T_{\mathcal{CH}_B} \oplus \hat{\mathcal{E}}^T_{\mathcal{CH}_A} \end{align} are unitary.
\begin{proof}
This follows from the fact that the Fourier transform and the decomposition into spherical harmonics are unitary maps.
\end{proof}
\end{prop}
Now, we will define the scattering map in the separated picture and show that it is bounded.
\begin{prop}
The scattering map in the separated picture \begin{align}\hat{S^T} \colon \hat{\mathcal{E}}^T_{\mathcal{H}_A} \oplus \hat{\mathcal{E}}^T_{\mathcal{H}_B} \to \hat{\mathcal{E}}^T_{\mathcal{CH}_B} \oplus \hat{\mathcal{E}}^T_{\mathcal{CH}_A},\end{align}
defined as the multiplication operator
\begin{align}\label{eq:scatteringfourier}
\hat{S^T} = \begin{pmatrix}
\hat{S^T_{BA}}& \hat{S^T_{BB}} \\
\hat{S^T_{AA}} & \hat{S^T_{AB}}\end{pmatrix} := \begin{pmatrix}
\mathfrak T(\omega, \ell )& \bar{\mathfrak R} (\omega, \ell ) \\
\mathfrak R(\omega, \ell ) & \bar{\mathfrak T} (\omega, \ell )
\end{pmatrix},
\end{align}
is bounded. Moreover, the map $\hat S^T$ is invertible with bounded inverse given by
\begin{align}\label{eq:inverse}
{{}{\hat{S^{T}}}}^{-1} = \begin{pmatrix}
\bar{\mathfrak T}(\omega, \ell )& -\bar{\mathfrak R} (\omega, \ell ) \\
-\mathfrak R(\omega, \ell ) & {\mathfrak T} (\omega, \ell )
\end{pmatrix}.
\end{align}
\begin{proof}
Indeed, $\hat{S}^T$ is bounded in view of the uniform boundedness of the transmission and reflection coefficients $\mathfrak T$ and $\mathfrak R$ (cf. \cref{thm:boundednesstrans}). Also note that
$ |\mathfrak T|^2 = 1 + |\mathfrak R|^2 $
implies that \begin{align}\det \left( \hat S^T \right) = 1\end{align}
which shows \eqref{eq:inverse}. The boundedness of ${{}{\hat{S^{T}}}}^{-1}$ is again immediate since the scattering coefficients are uniformly bounded.
\end{proof}
\end{prop}
Using the previous definitions, we obtain the following connection for the scattering map between the physical space and the separated picture.
\begin{theorem}\label{thm:fouriertophysical} The following diagram commutes and each arrow is a Hilbert space isomorphism:
\[
\begin{tikzcd}
\mathcal{E}^T_{\mathcal{H}_A} \oplus \mathcal{E}^T_{\mathcal{H}_B} \arrow{r}{S^T} \arrow[swap]{d}{\mathcal{F}_{\mathcal{H}_A}\oplus \mathcal{F}_{\mathcal{H}_B}} & \mathcal{E}^T_{\mathcal{CH}_B} \oplus \mathcal{E}^T_{\mathcal{CH}_A} \arrow{d}{\mathcal{F}_{\mathcal{CH}_B}\oplus \mathcal{F}_{\mathcal{CH}_A}} \\%
\hat{\mathcal{E}}^T_{\mathcal{H}_A} \oplus \hat{\mathcal{E}}^T_{\mathcal{H}_B} \arrow{r}{\hat{S^T}} & \hat{ \mathcal{E}}^T_{\mathcal{CH}_B} \oplus \hat{ \mathcal{E}}^T_{\mathcal{CH}_A}.
\end{tikzcd}
\]
Moreover, the maps $S^T$ and $\hat S^T$ are pseudo-unitary satisfying \eqref{eq:pseudounitary} and \eqref{eq:pseudounitaryode}, respectively.
More concretely, for $(\Psi_A, \Psi_B) \in \mathcal{E}^T_{\mathcal{H}_A}\oplus \mathcal{E}^T_{\mathcal{H}_B}$, we can write
\begin{align}
\begin{pmatrix}
\Phi_B \\ \Phi_A
\end{pmatrix} = S^T \begin{pmatrix}
\Psi_A \\\Psi_B
\end{pmatrix}, \end{align}
where $ \partial_u \Phi_A \in L^2(\mathcal{CH}_A)$ and $\partial_v \Phi_B\in L^2(\mathcal{CH}_B) $ can be represented by
\begin{align}\nonumber
\partial_u \Phi_A (u,\theta,\varphi )= & \frac{1}{\sqrt {2\pi}r_-} \int_{\mathbb R} \sum_{|m|\leq\ell}-i\omega \mathfrak R(\omega,\ell) \,\mathcal{F}_{\mathcal{H}_A}(\Psi_A) (\omega,m,\ell) Y_{m\ell}(\theta,\varphi) e^{-i\omega u } \d\omega \\
&+\frac{1}{\sqrt {2\pi} r_-} \int_{\mathbb R} \sum_{|m|\leq \ell} -i\omega \bar{\mathfrak T}(\omega,\ell) \,\mathcal{F}_{\mathcal{H}_B}(\Psi_B) (\omega,m,\ell) Y_{m\ell}(\theta,\varphi) e^{-i\omega u } \d\omega \label{eq:fourierrep1}
\end{align}
and
\begin{align}\nonumber\partial_v \Phi_B (v,\theta,\varphi )=& \frac{1}{\sqrt {2\pi} r_-} \int_{\mathbb R} \sum_{|m|\leq \ell} i \omega \mathfrak T(\omega,\ell) \,\mathcal{F}_{\mathcal{H}_A}(\Psi_A) (\omega,m,\ell) Y_{m\ell}(\theta,\varphi) e^{i\omega v} \d\omega \\
&+\frac{1}{\sqrt {2\pi} r_-} \int_{\mathbb R} \sum_{|m|\leq \ell} i \omega \bar{\mathfrak R}(\omega,\ell) \,\mathcal{F}_{\mathcal{H}_B}(\Psi_B) (\omega,m,\ell) Y_{m\ell}(\theta,\varphi) e^{i\omega v } \d\omega\label{eq:fourierrep2}
\end{align}
as well as $ \Phi_A \in \mathcal{E}^T_{\mathcal{CH}_A} \cong \dot{H}^1(\mathbb R; L^2(\mathbb S^2)), \Phi_B \in \mathcal{E}^T_{\mathcal{CH}_B}\cong \dot{H}^1(\mathbb R; L^2(\mathbb S^2))$ can be represented by regular distributions as
\begin{align}\nonumber
\Phi_A (u,\theta,\varphi )= & \frac{1}{\sqrt {2\pi}r_-} \operatorname{p.v.}\int_{\mathbb R} \sum_{|m|\leq \ell} \mathfrak R(\omega,\ell) \,\mathcal{F}_{\mathcal{H}_A}(\Psi_A) (\omega,m,\ell) Y_{m\ell}(\theta,\varphi) e^{-i\omega u } \d\omega \\
&+\frac{1}{\sqrt {2\pi} r_-} \operatorname{p.v.}\int_{\mathbb R} \sum_{|m|\leq \ell} \bar{\mathfrak T}(\omega,\ell) \,\mathcal{F}_{\mathcal{H}_B}(\Psi_B) (\omega,m,\ell) Y_{m\ell}(\theta,\varphi) e^{-i\omega u } \d\omega
\end{align}
and
\begin{align}\nonumber \Phi_B (v,\theta,\varphi )=& \frac{1}{\sqrt {2\pi} r_-} \operatorname{p.v.}\int_{\mathbb R} \sum_{|m|\leq \ell} \mathfrak T(\omega,\ell) \,\mathcal{F}_{\mathcal{H}_A}(\Psi_A) (\omega,m,\ell) Y_{m\ell}(\theta,\varphi) e^{i\omega v} \d\omega \\
&+\frac{1}{\sqrt {2\pi} r_-} \operatorname{p.v.}\int_{\mathbb R} \sum_{|m|\leq \ell} \bar{\mathfrak R}(\omega,\ell) \,\mathcal{F}_{\mathcal{H}_B}(\Psi_B) (\omega,m,\ell) Y_{m\ell}(\theta,\varphi) e^{i\omega v } \d\omega.
\label{eq:scatteredcauchyb}
\end{align}
\begin{proof}
This is a direct consequence of \cref{thm:forwardevolution}, \cref{thm:boundednesstrans} and \eqref{eq:scatteringcoef1}, \eqref{eq:scatteringcoef2} in the proof of \cref{thm:thmbounds}.
\end{proof}
\end{theorem}
From the previous representation of the scattered solution we can draw a link between the boundedness of the scattering map and the fact that compactly supported incoming data will lead to solutions which vanish on the future bifurcation sphere $\mathcal{B}_+$. This is the content of the following
\begin{cor}\label{cor:vanishingbifurcation}
Let $\Psi= (\Psi_A,0)\in \mathcal{E}^T_{\mathcal{H}_A} \oplus \mathcal{E}^T_{\mathcal{H}_B}$ be purely incoming smooth data. Assume further that $\Psi_A$ is supported away from the past bifurcation sphere $\mathcal{B}_-$ and future timelike infinity $i^+$.
Then, the Cauchy evolution $\psi$ arising from $\Psi_A$ vanishes at the future bifurcation sphere $\mathcal{B}_+$.
On the other hand, if $\Psi$, as above, led to a solution which does not vanish at the future bifurcation sphere $\mathcal{B}_+$, then the scattering map $S^T: \mathcal{E}^T_{\mathcal{H}} \to \mathcal{E}^T_{\mathcal{CH}}$ could not be bounded.
\begin{proof}
The first claim is a direct consequence of \eqref{eq:scatteredcauchyb} in \cref{thm:fouriertophysical}.
For the second statement let $\Psi_A$ be compactly supported data on the event horizon and assume that its Cauchy evolution $\psi$ does not vanish at the future bifurcation sphere $\mathcal{B}_+$. Now take data $\tilde \Psi_A$ which is supported away from the past bifurcation sphere $\mathcal{B}_-$ and satisfies $T \tilde \Psi_A = \Psi_A$. Then, $\tilde \Psi_A \in \mathcal{E}^T$ but its Cauchy evolution $\tilde \psi $ satisfies $\tilde \psi\restriction_{\mathcal{CH}}\notin \mathcal{E}^T_{\mathcal{CH}}$ since
\begin{align}
\| \tilde \psi\restriction_{\mathcal{CH}_B} \|^2_{\mathcal{E}^T_{\mathcal{CH}_B}} = \int_{\mathbb R \times \mathbb S^2} |\psi\restriction_{\mathcal{CH}_B} (v,\theta,\varphi)|^2 \d v \sin\theta \d \theta \d \varphi = \infty,
\end{align}
as $\psi\restriction_{\mathcal{CH}_B} = T \tilde \psi \restriction_{\mathcal{CH}_B}$ does not vanish at the future bifurcation sphere $\mathcal{B}_+$. By cutting off smoothly, one can construct normalized (in $\mathcal{E}^T_{\mathcal{H}}$-norm) smooth compactly supported initial data on $\mathcal{E}^T_{\mathcal{H}}$ such that its Cauchy evolution has arbitrary large norm $\mathcal{E}^T_{\mathcal{CH}}$-norm at the Cauchy horizon.
\end{proof}
\end{cor}
\begin{rmk}
For convenience we have stated the second statement of \cref{cor:vanishingbifurcation} only for the interior of Reissner--Nordström. However, note that it holds true for more general black hole interiors (e.g.\ subextremal (anti-) de~Sitter--Reissner--Nordström) with Penrose diagram as depicted in \cref{fig:penrosembar}.
\end{rmk}
\subsection{Injectivity of the reflection map}
\label{subsec:reflection}
In this section, we define the reflection operator of purely incoming radiation (cf.\ \cref{fig:reflection}) and prove that it is has trivial kernel as an operator from $\mathcal{E}^T_{\mathcal{H}_A} \to \mathcal{E}^T_{\mathcal{CH}_A}$. \begin{figure}[ht]\centering
\input{reflection.pdf_tex}
\caption{Reflection $\mathscr{R}$ of purely incoming radiation.}
\label{fig:reflection}
\end{figure}
\begin{definition}[Reflection operator]\label{defn:reflectionoperator}
For purely incoming radiation $(\Psi_A,0)\in \mathcal{E}^T_{\mathcal{H}_A} \oplus \mathcal{E}^T_{\mathcal{H}_B}$, define the reflection operator
\begin{align}
\mathscr{R}: \mathcal{E}^T_{\mathcal{H}_A} \to \mathcal{E}^T_{\mathcal{CH}_A}
\end{align}
as
\begin{align}
\mathscr{R}(\Psi_A) = \Phi_A := \operatorname{pr}_{A} \left( S^T \begin{pmatrix}\Psi_A \\ 0
\end{pmatrix}\right),
\end{align}
where $\operatorname{pr}_{A}\colon \mathcal{E}^T_{\mathcal{CH}_B} \oplus \mathcal{E}^T_{\mathcal{CH}_A} \to \mathcal{E}^T_{\mathcal{CH}_A} $ is the orthogonal projection.
\end{definition}
\begin{theorem}\label{thm:nonvanishingreflection}
The reflection operator $\mathscr{R}$ defined in \cref{defn:reflectionoperator} has trivial kernel.
\begin{proof}
Assume $\mathscr{R}(\Psi_A) =0$ for some $\Psi_A \in \mathcal{E}^T_{\mathcal{H}_A}$. Then, in view of \cref{thm:fouriertophysical}, \begin{align}\mathfrak R(\omega,\ell) \mathcal{F}_{\mathcal{H}_A} (\Psi_A) (\omega,m,\ell) =0\end{align} for all $m,\ell$, and a.e.\ $\omega \in \mathbb {R} $. Moreover, since $\mathfrak R(\omega,\ell)$ only vanishes on a discrete set (cf.\ \cref{cor:rdoesntvanish}), we obtain that $\mathcal{F}_{\mathcal{H}_A} (\Psi_A) (\omega,m,\ell) =0$ for all $m,\ell$, and a.e.\ $\omega \in \mathbb {R} $. Again, in view of \cref{thm:fouriertophysical}, we conclude $\Psi_A=0$ as an element of $\mathcal{E}^T_{\mathcal{H}_A}$.
\end{proof}
\end{theorem}
\subsection{\texorpdfstring{$C^1$}{C1}-blow-up on the Cauchy horizon}\label{sec:c1instab}
In this section, we shall revisit and discuss the seminal work \cite{hartle1982crossing} of Chandrasekhar and Hartle. The Fourier representation of the scattered data on the Cauchy horizon in \cref{thm:fouriertophysical} serves as a good framework to provide a completely rigorous framework for the $C^1$-blow-up at the Cauchy horizon studied in~\cite{hartle1982crossing}.
\begin{theorem}[$C^1$-blow-up on the Cauchy horizon \cite{hartle1982crossing}]\label{thm:c1instab}
There exist smooth, compactly supported and purely incoming data $\Psi_A$ on the event horizon $\mathcal{H}_A$ for which the Cauchy evolution of \eqref{eq:linearwave} fails to be $C^1$ at the Cauchy horizon $\mathcal{CH}$. More precisely, the solution $\psi$ arising from $\Psi_A$ fails to be $C^1$ at every point on the Cauchy horizon $\mathcal{CH}_A\cup \mathcal{B}_+$. Moreover, the incoming radiation can be chosen to be only supported on any angular parameter $\ell_0$ which satisfies $\ell_0(\ell_0+1) \neq r_+^2 ( r_+ - 3 r_-)$.
\begin{proof}
Let $\ell_0$ be fixed and satisfy $\ell_0(\ell_0+1) \neq r_+^2 ( r_+ - 3 r_-)$. Define purely incoming smooth data $\Psi_A(v,\theta,\varphi) = f(v) Y_{\ell_0 0}(\theta,\varphi)$ on $\mathcal{H}_A$, where $f(v)$ is smooth and compactly supported. Moreover, assume that the entire function $\hat f$ satisfies $\hat f(i\kappa_+) \neq 0$. In view of the representation formula \eqref{eq:scatteredcauchyb} from \cref{thm:fouriertophysical}, the degenerate derivative of its Cauchy evolution $\Phi_B$ on the Cauchy horizon $\mathcal{CH}_B$ reads
\begin{align}\label{eq:contour}
\partial_v \Phi_B(v,\theta,\varphi) = \frac{r_+}{\sqrt{2\pi}r_-} \int_{\mathbb R} i \omega \mathfrak T(\omega,\ell_0) \hat f(\omega)e^{i\omega v} \d \omega Y_{\ell_0 0}(\theta,\varphi).
\end{align}
Since $\mathfrak T(\omega,\ell)$ has a simple pole at $\omega = i\kappa_+$ (cf.\ \cref{prop:rtboundedcomplex} in the appendix), we pick up the residue at $i\kappa_+$ when we deform the contour of integration in \eqref{eq:contour} from the real line to the line $\operatorname{Im}(\omega) = \kappa_+ + \delta$ for some $\kappa_+ >\delta >0$. Here we use that the compact support of $f(v)$ implies the bound $|\hat f(\omega)| \leq e^{|\operatorname{Im}(\omega)|\sup| \operatorname{supp}(f)|} \hat f(\operatorname{Re}(\omega))$ and that, in addition, by \cref{prop:rtboundedcomplex}, the transmission coefficient $\mathfrak T$ remains bounded as $|\operatorname{Re}(\omega)| \to \infty$. This ensures that the deformation of the integration contour is valid. Hence,
\begin{align}\nonumber
\partial_v \Phi_B(v,\theta,\varphi) =& \frac{i r_+}{\sqrt{2\pi}r_-} 2 \pi i (i\kappa_+) \hat f(i\kappa_+) e^{-\kappa_+ v } Y_{\ell_0 0} (\theta,\varphi) \operatorname{Res}(\mathfrak T(\omega,\ell_0), i\kappa_+) \\\nonumber& + i
\frac{r_+ e^{-(\kappa_+ +\delta )v}}{\sqrt{2\pi}r_-} \int_{\mathbb R}\Big[ (\omega_R + i(\kappa_+ + \delta)) \mathfrak T(\omega_R + i (\kappa_+ + \delta))
\\\nonumber& \hat f(\omega_R + i (\kappa_+ + \delta)) e^{i \omega_R v} Y_{\ell_0 0} (\theta,\varphi) \Big] \d \omega_R\\
&= C e^{-\kappa_+ v} Y_{\ell_0 0 }(\theta,\varphi) + o\left(e^{-(\kappa_+ + \delta) v}\right)
\end{align}
as $v\to\infty$,
where
\begin{align}
C = -i \kappa_+ \frac{r_+}{r_-}\sqrt{2\pi} \hat f(i\kappa_+) \operatorname{Res}(\mathfrak T(\omega,\ell_0), \omega = i\kappa_+)\neq 0
\end{align}
by construction.
Thus, $\Phi_B$ is not in $C^1$ at the future bifurcation sphere as the non-degenerate derivative diverges as $v \to \infty$:
\begin{align}
\frac{\partial}{\partial V_{\mathcal{CH}}} \Phi_B = e^{-\kappa_- v} \partial_v \Psi_B(v,\theta,\varphi) = C e^{-(\kappa_+ + \kappa_-) v} (1 + o(1)) ,
\end{align}
where we recall that $\kappa_- < - \kappa_+<0$. Finally, propagation of regularity gives that the solution is not in $C^1$ at each point on the Cauchy horizon $\mathcal{CH}_A$. More precisely, expressing \eqref{eq:linearwave} is $(u,v)$ coordinates gives
\begin{align}\label{eq:nullcords}
\partial_u \partial_v \psi = \frac{-\Delta}{2r^3} (\partial_v \psi + \partial_u \psi) + \frac{\Delta}{4r^4}\ell_0(\ell_0 +1) \psi,
\end{align}
where $\Delta$ is as in \eqref{eq:h} and where we have used that $\Delta_{\mathbb S^2} \psi = -\ell_0(\ell_0 +1) \psi$. Now, note that $|\psi|, |\partial_u\psi|$ and $|\partial_v\psi|$ are uniformly bounded in the interior by a higher order norm of $\Psi_A$. This follows from \cite{franzen2016boundedness}, commuting with $T$ and angular momentum operators as well as elliptic estimates. Finally, integrating \eqref{eq:nullcords} in $u$, using the estimate $|\Delta|\lesssim e^{\kappa_- (u+v)}$ for $r_\ast \geq 0$ (see \eqref{eq:decayindelta}) and using the non-degenerate coordinate $V_\mathcal{CH}$ gives the $C^1$ blow-up also everywhere on $\mathcal{CH}_A$.
\end{proof}
\end{theorem}
\subsection{Breakdown of \texorpdfstring{$T$}{T} energy scattering for cosmological constants \texorpdfstring{$\Lambda\neq 0$}{Lambdaneq0}}
\label{subsec:nonex}
Interestingly, the analogous result to \cref{thm:forwardevolution} on the interior of a subextremal (anti-) de Sitter--Reissner--Nordstr\"om black hole does not hold for almost all cosmological constants $\Lambda$. In the presence of a cosmological constant it is also natural to consider the Klein--Gordon equation with conformal mass $\mu = \frac{3}{2} \Lambda$. We will consider in fact a general
mass term of the form $\mu = \nu \Lambda $, where $\nu \in \mathbb{R}$. Note that $\nu = \frac{3}{2}$ corresponds to the conformal invariant Klein--Gordon equation. To be more precise, we prove that for generic subextremal black hole parameters $(M,Q,\Lambda)$, there exists a normalized (in $\mathcal{E}^T_{\mathcal{H}}$-norm) sequence of Schwartz initial data on the event horizon for which the $\mathcal{E}^T_{\mathcal{CH}}$-norm of the evolution restricted to the Cauchy horizon blows up.
We define a black hole parameter triple $(M,Q,\Lambda)$ to be \emph{subextremal} if \begin{align}\label{defn:subextremal}(M,Q,\Lambda) \in \mathcal{P}_{\mathrm{se}} := \mathcal{P}_{\mathrm{se}}^{\Lambda =0} \cup \mathcal{P}_{\mathrm{se}}^{\Lambda >0} \cup \mathcal{P}_{\mathrm{se}}^{\Lambda < 0},\end{align} where
\begin{align}
\nonumber \mathcal{P}_{\mathrm{se}}^{\Lambda =0} := &\{ (M,Q,\Lambda ) \in \mathbb{R}_+ \times \mathbb{R} \times \{ 0 \} \colon\\ & \Delta(r):= r^2 - 2Mr + Q^2 \text{ has two positive simple roots satisfying } 0 < r_- < r_+. \} ,
\\ \nonumber
\mathcal{P}_{\mathrm{se}}^{\Lambda >0} := &\{ (M,Q,\Lambda) \in \mathbb{R}_+ \times \mathbb{R} \times \mathbb{R}_+ \colon\\ & \Delta(r):= r^2 - 2Mr - \frac 1 3 \Lambda r^4 + Q^2 \text{ has three positive simple roots satisfying } 0 < r_- < r_+ < r_c\} \label{defn:subextremal>0},\\\nonumber
\mathcal{P}_{\mathrm{se}}^{\Lambda <0} := &\{ (M,Q,\Lambda) \in \mathbb{R}_+ \times \mathbb{R} \times \mathbb{R}_- \colon\\ & \Delta(r):= r^2 - 2Mr - \frac 13 \Lambda r^4 + Q^2 \text{ has two positive roots satisfying } 0 < r_- < r_+ \}. \label{defn:subextremal<0}
\end{align}
\begin{restatable}{theorem}{cosmological}
\label{thm:cosmological}Let $\nu\in\mathbb{R}$ be a fixed Klein--Gordon mass parameter. (In particular, we may choose $\nu = \frac{3}{2}$ to cover the conformal invariant case or $\nu =0$ for the wave equation \eqref{eq:linearwave}.) Consider the interior of a subextremal (anti-) de Sitter--Reissner--Nordstr\"om black hole with generic parameters $(M,Q,\Lambda) \in \mathcal{P}_{\mathrm{se}}\setminus D(\nu)$. (Here, $ D(\nu)\subset \mathcal{P}_{\mathrm{se}} $ is a set with measure zero defined in \cref{prop:Bneq0} (see \cref{sec:cosmo}). Moreover $D(\nu)$ satisfies $ \mathcal{P}_{\mathrm{se}}^{\Lambda =0} \subset D(\nu) $ and $U \cap D(\nu) = \mathcal{P}_{\mathrm{se}}^{\Lambda =0}$ for some open set $U\subset \mathcal{P}_{\mathrm{se}}$.)
Then, there exists a sequence $(\Psi_n)_{n\in \mathbb{N}}$ of purely ingoing and compactly supported data on $\mathcal{H}_A$ with
\begin{align}
\| \Psi_n \|_{ \mathcal{E}^T_{\mathcal{H}}} = 1 \text{ for all } n
\end{align}such that the solution $\psi_n$ to the Klein--Gordon equation with mass $\mu = \nu \Lambda$
\begin{align}
\Box_{g_{M,Q,\Lambda}}\psi - \mu \psi = 0
\end{align}
arising from $\Psi_n$ has unbounded $T$ energy at the Cauchy horizon
\begin{align}
\| \psi_n \restriction_{\mathcal{CH}} \|_{ \mathcal{E}^T_{\mathcal{CH}}} \to \infty \text{ as } n\to\infty.
\end{align}
\end{restatable}
\begin{proof}
See \cref{sec:cosmo}.
\end{proof}
\begin{rmk}
Note that from \cref{thm:cosmological} it also follows that for fixed $0<|Q|<M$, the $T$ energy scattering breaks down (in sense of \cref{thm:cosmological}) for all cosmological constants $0<|\Lambda|<\epsilon$, where $\epsilon = \epsilon(M,Q)>0$ is small enough.
\end{rmk}
\subsection{Breakdown of \texorpdfstring{$T$}{T} energy scattering for the Klein--Gordon equation}
\label{subsec:notkleingordon}
Finally, we will also prove that the $T$ energy scattering theory does not hold for the Klein--Gordon equation for a generic set of masses $\mu$, even in the case of vanishing cosmological constant $\Lambda =0$.
\begin{restatable}{theorem}{kleingordon}
\label{thm:kleingordon}Consider the interior of a subextremal Reissner--Nordstr\"om black hole.
There exists a discrete set $\tilde D(M,Q) \subset \mathbb R$ with $0\in \tilde D$ such that the following holds true.
For any $\mu \in \mathbb R \setminus \tilde D$ there exists a sequence $(\Psi_n)_{n\in \mathbb{N}}$ of purely ingoing and compactly supported data on $\mathcal{H}_A$ with
\begin{align}
\| \Psi_n \|_{ \mathcal{E}^T_{\mathcal{H}}} = 1 \text{ for all } n
\end{align}such that the solution $\psi_n$ to the Klein--Gordon equation with mass $\mu$
\begin{align}
\Box_{g_{M,Q,\Lambda}}\psi - \mu \psi = 0
\end{align}
arising from $\Psi_n$ has unbounded $T$ energy at the Cauchy horizon
\begin{align}
\| \psi_n \restriction_{\mathcal{CH}} \|_{ \mathcal{E}^T_{\mathcal{CH}}} \to \infty \text{ as } n\to\infty.
\end{align}
\end{restatable}
\begin{proof}
See \cref{sec:kleingordonequation}.
\end{proof}
The above \cref{thm:cosmological} and \cref{thm:kleingordon} show that the existence of a $T$ energy scattering theory for the wave equation \eqref{eq:linearwave} on the interior of Reissner--Nordstr\"om is in retrospect a surprising property. Implications of the non-existence of a $T$ energy scattering map and in particular, the unboundedness of the scattering map in the cosmological setting $\Lambda \neq 0$, are yet to be understood.
\section{Proof of \texorpdfstring{\cref{thm:boundednesstrans}}{Theorem 2}: Uniform boundedness of the transmission and reflection coefficients}
\label{sec:radial}
This section is doteevoted to the proof of \cref{thm:boundednesstrans}. We will analyze solutions to the o.d.e.\ (recall from \eqref{eq:radialode})
\begin{align*}
\Delta \frac{\d}{\d r}\left(\Delta \frac{\d}{\d r} R\right) - \Delta \ell(\ell +1) R + r^4 \omega^2 R =0.
\end{align*}
This o.d.e.\ can be written equivalently (recall from \eqref{ODE1}) as
\begin{align*}
u^{\prime \prime} +(\omega^2 - V_\ell) u =0,
\end{align*} in the $r_\ast$ variable, where $u=r R$.
For the convenience of the reader we recall the statement of \cref{thm:boundednesstrans}.
\boundedness*
The proof of \cref{thm:boundednesstrans} will involve different arguments for different regimes of parameters. Also, note that in view of \eqref{eq:r0} and \eqref{eq:t0} it is enough to assume $\omega \neq 0$.
The first regime for bounded frequencies ($|\omega|\leq \omega_0$, $\ell$ arbitrary) requires the most work. One of its main difficulties is to obtain estimates which are uniform in the limit $\ell \to \infty$. We shall use that the o.d.e.\ \eqref{ODE1} with $\omega =0$, which reads
\begin{align}
u^{\prime\prime} - V_\ell u = 0,
\end{align}
can be solved explicitly in terms of Legendre polynomials and Legendre functions of second kind. The specific algebraic structure of the Legendre o.d.e.\ leads to the feature that solutions which are bounded at $r_\ast=-\infty$ are also bounded at $r_\ast=+\infty$. For generic perturbations of the potential this property fails to hold. Nevertheless, for perturbations of the form as in \eqref{ODE1} for $\omega \neq 0$ and $|\omega|\leq |\omega_0|$, this behavior survives and most importantly, can be quantified. To prove this we will essentially divide the real line $\mathbb{R}\ni r_\ast$ into three regions.
The first region will be near the event horizon ($r_\ast = - \infty$), where we will consider the potential $V_\ell$ as a perturbation. The second region will be the intermediate region, where we will consider the term involving $\omega$ as a perturbation. Finally, in the third region near the Cauchy horizon ($r_\ast = +\infty$), we consider the potential $V_\ell$ as a perturbation again. This eventually allows us to prove the uniform boundedness of the reflection and transmission coefficients $\mathfrak R$ and $\mathfrak T$ in the bounded frequency regime $|\omega| < \omega_0$.
The second regime will be bounded angular momenta and $\omega$-frequencies bounded from below $(|\omega|\geq \omega_0, \ell\leq \ell_0)$. For this parameter range we will consider $V_\ell$ as a perturbation of the o.d.e.\ since $V_\ell$ might only grow with $\ell$, which is, however, bounded in that range. Again, this allows us to show uniform boundedness for the transmission and reflection coefficients $\mathfrak T$ and $\mathfrak R$.
The third regime will be angular momenta and frequencies both bounded from below ($|\omega| \geq \omega_0$, $\ell \geq \ell_0$). To prove boundedness of reflection and transmission coefficients $\mathfrak R$ and $\mathfrak T$, we will consider $\frac{1}{\ell}$ as a small parameter to perform a WKB-approximation.
\subsection{Low frequencies \texorpdfstring{($|\omega|\leq \omega_0$)}{wleqw0}}
\label{subsec:smallfreq}
We first analyze the o.d.e.\ for the special case of vanishing frequency. Then, we will summarize properties of special functions, which we will need to finally prove the boundedness of reflection and transmission coefficients in the low frequency regime. Let \begin{align}0<\omega_0 \leq \frac 12\end{align} be a fixed constant.
\subsubsection{Explicit solution for vanishing frequency (\texorpdfstring{$\omega = 0$}{w=0})}
For $\omega=0$ we can explicitly solve the o.d.e.\ with special functions.
In that case the o.d.e.\ reads
\begin{align}\label{eq:ODEw0}
\frac{\d}{\d r} \left( \Delta \frac{\d R}{\d r}\right) - \ell ( \ell +1) R =0.
\end{align} We define the coordinate $x(r)$ as
\begin{align}
\label{eq:x(r)}
x(r):= - \frac{2r}{r_+ - r_-} + \frac{r_+ + r_-}{r_+ - r_-}
\end{align}
or equivalently,
\begin{align}
r(x) =- \frac{r_+ - r_-}{2} x+ \frac{r_+ + r_-}{2}.
\end{align}
Then, we can write
\begin{align}
\Delta(x ) = \left(\frac{r_+ - r_-}{2}\right)^2(x+1)(x-1) = \left(\frac{r_+ - r_-}{2} \right)^2(x^2 -1).
\end{align}
Hence, \cref{eq:ODEw0} reduces to the Legendre o.d.e.\
\begin{align}\label{eq:legendreode}
\frac{\d}{\d x} \left((1-x^2) \frac{\d R}{\d x} \right) + \ell(\ell+1) R = 0.
\end{align}
We will denote by $P_\ell(x)$ and $Q_\ell(x)$ the two independent solutions, the Legendre polynomials and the Legendre functions of second kind, respectively \cite{olver2014asymptotics,NIST:DLMF}.
We will prove later in \cref{prop:intermediate} that $\tilde u_1$ and $\tilde u_2$ from \cref{defn:u1u2} satisfy
\begin{align}\label{eq:uP}
&\tilde u_1(r_\ast) = w_1(r_\ast) := (-1)^\ell \frac{r(r_\ast)}{r_+} P_\ell(x(r_\ast)),\\
&\tilde u_2(r_\ast) = w_2(r_\ast) := (-1)^\ell \frac{r(r_\ast)}{k_+ r_+} Q_\ell(x(r_\ast)).\label{eq:uQ}
\end{align}
These are a fundamental pair of solutions for the o.d.e.\ in the case $\omega =0$. We will perturb these explicit solutions for the regime of low frequencies ($|\omega|\leq \omega_0$). To do so, we will need properties about special functions which will be considered first.
In view of the fact that $\omega_0$ is fixed, constants appearing in $\lesssim$ and $\gtrsim$ may also depend on $\omega_0$.
Before we begin, we shall summarize the special functions we will use and list their relevant properties in the case $|\omega| \leq \omega_0$.
\subsubsection{Special functions}
Good references for the following discussion are \cite{abramowitz1964handbook,olver2014asymptotics,NIST:DLMF}.
First, we shall recall the definition of the Gamma and Digamma function.
\begin{definition}For $z\in \mathbb{C}$ with $\operatorname{Re}(z)>0$ we denote the Gamma function with $\Gamma(z)$ and will also make use of the Digamma function $ \digamma(z )$ defined as
\begin{align}
\digamma(z ) := \int_0^\infty \left( \frac{e^{-x}}{x} - \frac{e^{-zx}}{1-e^{-x}} \right) \d x.
\end{align}
Note that
\begin{align}
\digamma(z+1) - \digamma(z) = \frac 1 z
\end{align}
and \begin{align}
\digamma(n) = \sum_{k=1}^{n-1} \frac{1}{k} - \gamma = \log(n) + O(n^{-1}),
\end{align}
where $\gamma$ is the Euler--Mascheroni constant.
\end{definition}
As we mentioned above, we shall use the Legendre polynomials and the Legendre functions of second kind.
We will express them in terms of the hypergeometric function $\mathbf{F}(a,b;c;x)$ for $x\in (-1,1)$, $a,b,c \in \mathbb R$ as defined in \cite[Equation~(9.3)]{olver2014asymptotics}.
\begin{definition}[Legendre functions of first and second kind] We use the standard conventions which are used in \cite{olver2014asymptotics,NIST:DLMF}.
For $x\in(-1,1)$, we define the associated Legendre polynomials by
\begin{align}
P_\ell^m(x) =\left( \frac{1+x}{1-x}\right)^{\frac m2} \mathbf{F}\left(\ell + 1 , - \ell ; 1-m; \frac{1-x}{2}\right)
\end{align}
and the associated Legendre functions of second kind by
\begin{align}
Q^m_\ell(x) = - \frac{1}{2}\pi \sin\left(\frac 12 \pi (\ell +m) \right) w_1(\ell,x) + \frac 12 \pi \cos\left(\frac 12 (\ell +m) \pi \right) w_2(\ell,x).
\end{align}
Here,
\begin{align}
& w_1(\ell,x) = \frac{2^m \Gamma(\frac{\ell + m+ 1}{2} )}{\Gamma({1+\frac{\ell}{2}})} (1-x^2)^{-\frac m2} \mathbf{F}\left( -\frac{\ell + m}{2}, \frac{1+\ell-m}{2};\frac 12 ; x^2\right),\\
&w_2(\ell,x) = \frac{2^m\Gamma({1+\frac{\ell+m}{2}})}{\Gamma(\frac{\ell-m + 1}{2} )}x (1-x^2)^{-\frac m2} \mathbf{F}\left(\frac{1-\ell-m}{2}, 1 + \frac{\ell-m}{2}; \frac 32 ; x^2\right).
\end{align}
\end{definition}
We shall also use the convention $P_\ell = P_\ell^0$ and $Q_\ell^m = Q_\ell^0$. Also, recall the symmetry
\begin{align}
& P_\ell( x) = (-1)^\ell P_\ell(-x),\\
& Q_\ell(x) = (-1)^{\ell+1} Q_\ell(-x).
\end{align}In the asymptotic expansion in the parameter $\ell$ for the Legendre polynomials and functions we will make use of Bessel functions which we define in the following.
\begin{definition}[Bessel functions of first and second kind] Recall the Bessel functions of first kind
\begin{align}
&J_0(x) := \sum_{k=0}^\infty \frac{x^{2k} }{(-4)^k k!^2},\\
&J_1(x) := \frac{x}{2} \sum_{k=0}^\infty \frac{x^{2k} }{(-4)^k k! (k+1)!},
\end{align}
and the Bessel functions of second kind
\begin{align}
Y_0(x) := &\frac{2}{\pi} J_0(x) \left( \log\left( \frac{x}{2}\right) + \gamma \right) - \frac{2}{\pi} \sum_{k=1}^\infty H_k \frac{x^{2k} }{(-4)^k (k!)^2},
\\ \nonumber Y_1(x) :=& -\frac{1}{2\pi x} + \frac{2}{\pi} \log\left(\frac{x}{2}\right) J_1(x)\\&\;\;\; - \frac{x}{2\pi} \sum_{k=0}^\infty( \digamma(k+1) + \digamma(k+2)) \frac{x^{2k}}{(-4)^k k! (k+1)!},
\end{align}
where $H_k = \sum_{n=1}^k n^{-1}$ is the $k$-the harmonic number.
We have the asymptotic expansions
\begin{align}\label{eq:estimatea}
&J_0(x) = 1 + O(x^2),\\
&J_1(x) = \frac{x}{2} + O(x^3),\label{eq:estimateJ1} \\
&Y_0(x) = \frac{2}{\pi} \log\left(\frac{x}{2}\right) + O(1),\\
&Y_1(x) = -\frac{1}{2\pi x} + o(1) \text{ as } x\to 0.\label{eq:estimateY1}
\end{align}
Note that bounds deduced from \eqref{eq:estimatea} -- \eqref{eq:estimateY1} hold uniformly on any interval $(0,a]$ of finite length.
We shall also use the bounds
\begin{align}\label{eq:boundsonJY1}
&|J_0(x)|\leq 1 ,\,
|Y_0(x)| \lesssim 1+ |\log(x)|
\end{align}
for $0< x\leq 1$ and
\begin{align}\label{eq:boundsonJY2}
&|J_0(x)| \lesssim \frac{1}{\sqrt x}, \,
|Y_0(x)| \lesssim \frac{1}{\sqrt{x}} \;
\end{align}
for $x\geq 1$ \cite[p.\ 360, p.\ 364]{abramowitz1964handbook}.
\end{definition}
In the proof we will also use the following asymptotic formulae for $P_\ell$ and $Q_\ell$ for large $\ell$ in terms of Bessel functions.
\begin{lemma}\cite[\S 14.15(iii)]{NIST:DLMF} \label{lem:plandql}We have
\begin{align}
&P_\ell(\cos\theta ) = \left(\frac{\theta}{\sin\theta}\right)^{\frac 12} \left(J_0\left(\frac{\theta(2\ell + 1)}{2} \right) + e_{1,\ell} ( \theta) \right),\\
&Q_\ell(\cos\theta ) =-\frac{\pi}{2} \left(\frac{\theta}{\sin\theta}\right)^{\frac 12} \left(Y_0\left(\frac{\theta(2\ell + 1)}{2} \right) + e_{2,\ell} ( \theta) \right),\label{eq:estimateql}\\
& Q_\ell^1(\cos\theta) = - \frac{\pi}{2\ell}\left(\frac{\theta}{\sin\theta}\right)^{\frac 12} \left( Y_1\left(\frac{\theta(2\ell + 1)}{2}\right) + e_{3,\ell}(\theta) \right), \label{eq:estimateql1}
\end{align}
where the error terms can be estimated by
\begin{align}
& |e_{1,\ell}(\theta)| ,\, |e_{2,\ell}(\theta) | \lesssim \frac{1}{1+\ell}\left[ \left|J_0 \left( \frac{\theta ( 2\ell +1)}{2}\right)\right| + \left|Y_0\left( \frac{\theta ( 2\ell +1)}{2}\right)\right|\right] ,\label{eq:estimateerror}
\\
& |e_{3,\ell}(\theta) | \lesssim \frac{1}{1+\ell} \left[ \left| J_1 \left( \frac{\theta ( 2\ell +1)}{2}\right)\right| + \left|Y_1\left( \frac{\theta ( 2\ell +1)}{2}\right)\right| \right]\label{eq:errorestimatee3}
\end{align}
for $\theta\in ( 0, \pi - \delta)$ and for any fixed $\delta>0$. In particular, this holds uniformly as $\theta \to 0$.
\end{lemma}
We shall use the following asymptotic formulae for the Legendre functions at the singular endpoints.
\begin{lemma}\cite[\S 14.8]{NIST:DLMF} For $0<x<1$ we have
\begin{align}
& P_\ell(x) = 1 + f_1(x), \\
& Q_\ell(x) = \frac 12 (\log(2) - \log(1-x) ) - \gamma - \digamma(\ell +1) + f_1(x),\label{eq:largeellQ}
\end{align}
where $|f_1(x)|\lesssim_\ell (1-x)$. Moreover, analogous results hold true for $-1<x<0$ due to symmetry.
\end{lemma}
Now, we will estimate the derivatives of the Legendre polynomials and Legendre functions of second kind.
\begin{lemma} For $x\in (-1,1)$ we have
\begin{align}\label{eq:derivativelegendre1}
&\left|\frac{\d P_\ell }{\d x}\right| \leq \ell^2.\end{align}
For $x_{\alpha, \ell}:=1-\frac{\alpha}{1+\ell^2}$ with $0<\alpha<1$ and $\ell \in \mathbb N$ we have
\begin{align}
& ( 1- (\pm x_{\alpha,\ell})^2) \left| \frac{\d Q_\ell}{\d x}(\pm x_{\alpha, \ell})\right| \lesssim 1.\label{eq:derivativelegendre2}
\end{align}
\begin{proof}Inequality \eqref{eq:derivativelegendre1} is known as Markov's inequality and is proven in \cite[Theorem 5.1.8]{borwein2012polynomials}. We only have to prove \eqref{eq:derivativelegendre2} for $x=+x_{\alpha,\ell} $ due to symmetry.
From the recursion relation \cite[\S 14.10]{NIST:DLMF} we have
\begin{align}\nonumber
(\ell + 1)^{-1} & (1-x_{\alpha,\ell}^2) \frac{\d Q_\ell }{\d x} (x_{\alpha,\ell}) = x_{\alpha,\ell} Q_\ell(x_{\alpha,\ell}) - Q_{\ell + 1}(x_{\alpha,\ell}) \\ &= (x_{\alpha,\ell}-1)Q_\ell(x_{\alpha,\ell}) + (Q_\ell(x_{\alpha,\ell}) - Q_{\ell +1}(x_{\alpha,\ell})).\label{eq:derivativeQ}
\end{align}
We will consider both summands separately.
\textbf{Part 1}: Summand $(x_{\alpha,\ell}-1)Q_\ell(x_{\alpha,\ell})$
\noindent
First, consider $1-x_{\alpha,\ell}=\frac{\alpha}{1+\ell^2}$, where we implicitly define $\cos(\theta_{\alpha,\ell}) = x_{\alpha,\ell}$. Note that we have \begin{align}\nonumber
\theta_{\alpha,\ell}(x) = \sqrt{2(1-x_{\alpha,\ell})} + O((1-x_{\alpha,\ell})^\frac{3}{2}) &= \sqrt{\frac{2\alpha}{1+\ell^2}} + O\left(\left( \frac{\alpha}{1+\ell^2} \right)^{\frac 32 } \right)\\ & =
\sqrt{\frac{2\alpha}{1+\ell^2}} \left( 1+ O\left(\frac{\alpha}{1+\ell^2}\right)\right).\label{eq:theta}
\end{align}
In particular, we have $\theta_{\alpha,\ell} \ell \lesssim 1$.
This gives
\begin{align}
-Q_\ell(x_{\alpha,\ell}) =- Q_\ell(\cos\theta_{\alpha,\ell}) = \frac{\pi}{2} \left(\frac{\theta_{\alpha,\ell}}{\sin\theta_{\alpha,\ell}}\right)^{\frac 12 }\left( Y_0\left(\frac{\theta_{\alpha,\ell}(2\ell +1)}{2}\right) + e_{2,\ell}(\theta_{\alpha,\ell}) \right).
\end{align}
Again, we will look at the two terms independently. First, note that
\begin{align}\nonumber
\frac{\pi}{2} &\left( \frac{\theta_{\alpha,\ell}}{\sin \theta_{\alpha,\ell}}\right)^{\frac 12} \left( Y_0\left(\theta_{\alpha,\ell}\left(\ell + \frac 12 \right)\right) \right)\\
\nonumber & = \frac{\pi}{2} \left( \frac{\theta_{\alpha,\ell}}{\sin\theta_{\alpha,\ell}}\right)^\frac{1}{2} \left( \frac{2}{\pi} \log\left( \frac{ \theta_{\alpha,\ell}(2\ell + 1)}{4}\right) + O(1) \right)\\\nonumber& =
\left( 1+ O(\theta_{\alpha,\ell}^2)\right) \left( \log(\theta_{\alpha,\ell}) + \log\left(\ell + \frac 12 \right) + O(1) \right)\\\nonumber & =
\left( 1+ O\left(\frac{\alpha}{1+\ell^2}\right)\right) \left( \frac 12 \log\left(\frac{\alpha}{1+\ell^2}\right) + \log\left(\ell+ \frac 12 \right) + O(1) \right)\\ \nonumber & =
\left( 1+ O\left(\frac{\alpha}{1+\ell^2}\right)\right) \left( \frac{1}{2} \log(\alpha) + \frac{1}{2} \log\left(1 + \frac{\ell- \frac{3}{4}}{\ell^2 + 1 }\right) + O(1) \right)\\ & =
\frac{1}{2} \log(\alpha) + O(1).
\end{align}
In order to estimate $e_{2,\ell}(\theta_{\alpha,\ell})$ we shall recall inequality~\eqref{eq:estimateerror}. It works analogously to the previous estimate up to a good term of $\frac{1}{1+\ell}$. In particular, this shows
\begin{align}
|Q_\ell(x_{\alpha,\ell})| \lesssim | \log(\alpha) | + 1
\end{align}
and
\begin{align}
|(x_{\alpha,\ell}-1)Q_\ell(x_{\alpha,\ell})| \lesssim \frac{\alpha}{1+\ell^2} (|\log(\alpha) | + 1)\lesssim \frac{1}{1+\ell^2}.
\end{align}
\textbf{Part 2}: Summand $(Q_\ell(x_{\alpha,\ell}) - Q_{\ell +1}(x_{\alpha,\ell}))$
\noindent
Using the recursion relation for the difference of two Legendre function \cite[\S14.10]{NIST:DLMF}, we have
\begin{align}
(\ell +1) (Q_\ell(x_{\alpha,\ell}) - Q_{\ell +1}(x_{\alpha,\ell}) = - ( 1- x_{\alpha,\ell}^2)^\frac{1}{2} Q_\ell^1(x_{\alpha,\ell}) + (1-x_{\alpha,\ell}) Q_\ell(x_{\alpha,\ell}) .
\end{align}
We estimate the term $(1-x_{\alpha,\ell})Q_\ell(x_{\alpha,\ell})$ by what we have done above as
\begin{align}
| (1-x_{\alpha,\ell}) Q_\ell(x_{\alpha,\ell}) |\lesssim \frac{\alpha}{1+\ell^2}( |\log(\alpha) |+ 1) \lesssim 1.
\end{align}
For the term $-(1-x_{\alpha,\ell}^2)^{\frac 12} Q_\ell^1(x_{\alpha,\ell})$ we use \eqref{eq:estimateql1} to get
\begin{align}\nonumber
&\left|-(1-x_{\alpha,\ell}^2)^{\frac 12} Q_\ell^1(x_{\alpha,\ell})\right|\\ &\lesssim\sqrt{ \frac{\alpha}{\ell^2 + 1}} \frac{1}{1+\ell} \left(1+O\left(\frac{\alpha}{1+\ell^2}\right)\right) \left( Y_1\left( \left(\ell + \frac 12\right) \theta_{\alpha,\ell} \right) + e_{2,\ell} (\theta_{\alpha,\ell})\right).
\end{align}
As before, we shall start estimating the first term using \eqref{eq:estimateY1} and \eqref{eq:theta} to obtain
\begin{align}\nonumber
&\sqrt{ \frac{\alpha}{\ell^2 + 1}} \frac{1}{1+\ell} \left(1+O\left(\frac{\alpha}{1+\ell^2}\right)\right) Y_1\left( \left(\ell + \frac 12\right) \theta_{\alpha,\ell} \right) \\\nonumber &= \sqrt{ \frac{\alpha}{\ell^2 + 1}} \frac{1}{1+\ell} \left(1+O\left(\frac{\alpha}{1+\ell^2}\right)\right) \left( -\frac{1}{\pi(2 \ell +1 ) \theta_{\alpha,\ell} } + O(1) \right) \\&
\lesssim \sqrt{ \frac{\alpha}{\ell^2 + 1}} \frac{1}{1+\ell} \left( \frac{1}{\sqrt{\alpha} } + 1 \right)\lesssim 1.
\end{align}
We estimate the second term using \eqref{eq:errorestimatee3}, \eqref{eq:estimateJ1}, \eqref{eq:estimateY1}, and \eqref{eq:theta} to obtain
\begin{align}\nonumber
&\left|\sqrt{ \frac{\alpha}{\ell^2 + 1} } \frac{1}{1+\ell}\left(1+O\left(\frac{\alpha}{1+\ell^2}\right)\right) e_{2,\ell} (\theta_{\alpha,\ell} )\right|\\ & \lesssim \sqrt{ \frac{\alpha}{\ell^2 + 1} } \frac{1}{1+\ell^2} \left( \frac{1}{\sqrt \alpha} + 1 \right) \lesssim 1.
\end{align}
We have estimated that $|Q_\ell(x_{\alpha,\ell}) - Q_{\ell + 1}(x_{\alpha,\ell}) |\lesssim \frac{1}{1+\ell}$ which proves the claim in view of \eqref{eq:derivativeQ}.
\end{proof}
\end{lemma}
Finally, we prove asymptotics for the derivatives of the Legendre of functions of second kind near the singular points.
\begin{lemma}For $0<x<1$ and $x \to 1$ we have
\begin{align}
(1-x^2) \frac{\d Q_\ell}{\d x} = 1 + O_\ell ( (1-x) \log(1-x)).
\end{align}
By symmetry this also yields for $-1<x<0$ and $x\to -1$
\begin{align}
(1-x^2) \frac{\d Q_\ell}{\d x} = (-1)^\ell + O_\ell ( (1+x) \log(1+x)). \label{eq:dqdxlimit}
\end{align}
\begin{proof}
From the recursion relation \cite[\S 14.10]{NIST:DLMF} and \eqref{eq:largeellQ} we obtain
\begin{align}\nonumber
& (1-x^2) \frac{\d Q_\ell }{\d x} = (\ell +1)(x Q_\ell - Q_{\ell + 1} )
\\ \nonumber & = (\ell + 1) (x-1)Q_\ell + (\ell + 1)(Q_\ell - Q_{\ell +1})\\ \nonumber &=
(\ell + 1) ( Q_\ell - Q_{\ell + 1} ) + O_\ell ((1-x) \log(1-x))\\ \nonumber &
= (\ell + 1) ( \digamma(\ell + 2) - \digamma(\ell + 1) ) + O_\ell( (1-x) \log (1-x) ) \\ & = 1 + O_\ell( ( 1-x) \log(1-x)).
\end{align}
\end{proof}
\end{lemma}
Having reviewed the required facts about special functions, we shall now proceed to prove the uniform boundedness of the reflection and transmission coefficients.
\subsubsection{Boundedness of the reflection and transmission coefficients}
\label{sec:boundedrefltransmi}
As mentioned before, we will consider three different regions: a region near the event horizon, an intermediate region, and a region near the Cauchy horizon. In $r_\ast$ coordinates we separate these regions at \begin{align}R_1^\ast(\omega,\ell) := \frac{1}{2 \kappa_+} \log\left(\frac{\omega^2}{1+\ell^2}\right)\label{defn:R1}\end{align} and \begin{align}R_2^\ast(\omega,\ell):= \frac{1}{2\kappa_-}\log\left(\frac{\omega^2}{1+\ell^2 }\right)\label{defn:R2}\end{align}
for $0<|\omega| < \omega_0$ and $\ell \in \mathbb N_0$.
Note that $-\infty < R_1^\ast(\omega,\ell) < 0 < R_2^\ast(\omega,\ell) < \infty$.
\paragraph{\textbf{Region near the event horizon}}
\begin{prop}
\label{prop:firstregion}
Let $0<|\omega|<\omega_0$ and $\ell\in\mathbb N_0$. Then, we have
\begin{align}
& \| u_1^\prime \|_{L^\infty(-\infty, R_1^\ast)} \lesssim |\omega|,\\
& \|u_1\|_{L^\infty(-\infty,R_1^\ast)} \lesssim 1.
\end{align}
\end{prop}
\begin{proof}
Recall the defining Volterra integral equation for $u_1$ from \cref{defn:u1u2}
\begin{align}
u_1(r_\ast) = e^{i\omega r_\ast } + \int_{-\infty}^{r_\ast} \frac{\sin(\omega(r_\ast-y))}{\omega} V(y) u_1(y) \d{y}.
\end{align}
with integral kernel
\begin{align}
K(r_\ast , y) := \frac{\sin(\omega(r_\ast - y)) }{\omega} V(y).
\end{align}
From \cref{lem:asymptoticspotential} in the appendix, we obtain for $r_\ast \leq R_1^\ast$
\begin{align}
|V(r_\ast)| \lesssim e^{2k_+ r_\ast} (1 + \ell^2)
\end{align}
and in particular,
\begin{align}
|V(R_1^\ast) |\lesssim e^{2k_+ R_1^\ast} ( 1+ \ell^2)= \omega^2 .
\end{align}
This implies for $r_\ast \leq R_1^\ast$
\begin{align}
|K(r_\ast,y)| \leq \frac{1}{|\omega|}| V(y) | \lesssim
\frac{1}{|\omega|}( 1+ \ell^2 ) e^{2k_+ y}
\end{align}
and thus,
\begin{align}
\int_{-\infty}^{R_1^\ast} \sup_{y < r_\ast < R_1^\ast} |K(r_\ast,y)| \d{y} \lesssim \frac{\ell^2+1}{|\omega|}e^{2k_+ R_1^\ast} \lesssim 1.
\end{align}
The claim follows now from \cref{lem:volterra}.
\end{proof}
Now, we would like to consider $\omega$ as a small parameter and perturb the explicit solutions for the $\omega =0$ case in order to propagate the behavior of the solution through the intermediate region, where $V_\ell$ is large compared to $\omega$. In particular, $V_\ell$ can be arbitrarily large since $\ell$ is not bounded above in the considered parameter regime.
\paragraph{\textbf{Intermediate region}}
First, recall the following fundamental pair of solutions which is based on the Legendre functions of first and second kind
\begin{align}
& w_1(r_\ast) :=(-1)^\ell \frac{r(r_\ast)}{r_+} P_\ell(x(r_\ast)), \\
& w_2(r_\ast) := (-1)^\ell \frac{r(r_\ast)}{k_+ r_+} Q_\ell(x(r_\ast)),
\end{align}
where $P_\ell$ and $Q_\ell$ are the Legendre polynomials and Legendre functions of second kind, respectively. Our first claim is that we have constructed this fundamental pair $(w_1,w_2)$ to have unit Wronskian and moreover $\tilde u_1 = w_1$ and $\tilde u_2 = w_2$ holds true.
\begin{prop}\label{prop:intermediate}We have $w_1=\tilde u_1$ and $w_2= \tilde u_2$ and
the Wronskian of $u_1$ and $u_2$ satisfies\begin{align}
\mathfrak W(w_1,w_2 ) = \mathfrak W (\tilde u_1, \tilde u_2) = 1.
\end{align}
Similarly, we also have $\tilde v_1 = (-1)^\ell \frac{r_+}{r_-} w_1 = (-1)^\ell \frac{r_+}{r_-} \tilde u_1$.
\begin{proof}We first prove that $\mathfrak W(w_1,w_2) = 1$.
Since the Wronskian is independent of $r_\ast$, we will compute its value in the limit $r_\ast \to-\infty$. In this proposition $\ell$ is fixed and we shall allow implicit constants in $\lesssim$ to depend on $\ell$. Clearly,
\begin{align}
w_1(r_\ast ) \to 1 \text{ as } r_\ast \to-\infty.
\end{align}
Moreover, we have that for $r_\ast \leq 0$
\begin{align}
\left|\frac{\d}{\d r_\ast} w_1(r_\ast) \right| \lesssim e^{2k_+ r_\ast }| P_\ell(x(r_\ast)) | + \left|\frac{\d P_\ell( x) }{\d x}(r_\ast) \frac{\d x}{\d r_\ast } (r_\ast)\right| \lesssim e^{2k_+ r_\ast } ,
\end{align}
where we have used \eqref{eq:derivativelegendre1}. This, in particular, also shows that $w_1$ satisfies the same boundary conditions ($w_1\to 1$, $w_1^\prime \to 0$ as $r_\ast \to-\infty$) as $\tilde u_1$ defined in \cref{defn:u1u2} and thus, $w_1$ and $\tilde u_1$ have to coincide. Similarly, we can deduce $\tilde v_1 = (-1)^\ell \frac{r_+}{r_-} w_1$.
For $w_2$, we use \eqref{eq:largeellQ} to obtain
\begin{align}
|w_2(r_\ast) - r_\ast | \lesssim \left( - \frac{r(r_\ast) }{k_+ r_+} \left(\frac{1}{2} \log\left( \frac{2}{1+x(r_\ast)}\right) -\gamma - \digamma(\ell+1) \right) - r_\ast \right) + e^{2 k_+ r_\ast}.
\end{align}
For an intermediate step, we compute $\log( 1+ x(r_\ast))$ from \eqref{eq:x(r)} near $r_\ast = -\infty$. In particular, for the limit $r_\ast \to -\infty$, we can assume that $r_\ast \leq 0$ and thus, $r - r_- \gtrsim r_+ - r_- $. Hence,
\begin{align}\nonumber
\log( 1+ x(r_\ast) ) & = \log\left( 1 + \frac{(r_+ - r )+( r_- - r)}{r_+ - r_-}\right) \\ \nonumber & = \log\left( 1 + \frac{f(r_\ast) }{r_+ - r_-}e^{2k_+ r_\ast } + \frac{r_- - r}{r_+ - r_-}\right) \\ \nonumber &= \log\left( \frac{2 f(r_\ast) }{r_+ - r_-}e^{2k_+ r_\ast } \right) \\ &= 2 k_+ r_\ast + \log( 2 f(r_\ast) (r_+ - r_-)^{-1}),
\end{align}
where $f$ is defined in \eqref{eq:fofr}.
Thus, this directly implies
\begin{align}
|w_2(r_\ast) - r_\ast | \lesssim r_\ast e^{2 k_+ r_\ast } + 1 \lesssim 1.
\end{align}
Finally, we claim that $w_2^\prime \to 1$ as $r_\ast \to-\infty$. We shall use estimate \eqref{eq:dqdxlimit} near $x(r_\ast)=-1$ to obtain
\begin{align}\nonumber
|w_2^\prime&(r_\ast) - 1 |
\lesssim e^{2k_+ r_\ast}( |r_\ast | + 1) + \left| (-1)^\ell \frac{r(r_\ast) }{k_+ r_+} \frac{\d Q_\ell(x)}{\d x}\frac{\d x}{\d r_\ast} - 1 \right| \\ & \lesssim e^{2k_+ r \ast} + \left| \frac{ r(r_\ast) }{k_+ r_+} \left[1+ O\left((1+x(r_\ast))\log(1+x(r_\ast)) \right) \right]\frac{1}{1-x^2(r_\ast)}\frac{\d x}{\d r_\ast} - 1 \right| .
\end{align}
Now, in order to conclude that
\begin{align}
|w_2^\prime(r_\ast) - 1 | \to 0 \text{ as } r_\ast \to -\infty,
\end{align}
it suffices to check that
\begin{align}
\frac{1}{1-x^2(r_\ast)} \frac{\d x}{\d r_\ast} \to k_+ \text{ as } r_\ast \to-\infty.
\end{align}
But this holds true because
\begin{align}
\frac{1}{1-x^2(r_\ast)} \frac{\d x}{\d r_\ast} =
\frac{1}{1-x^2(r_\ast)} \frac{-2}{r_+ - r_- } \frac{\Delta}{r^2 } = \frac{r_+ -r_-}{2r^2} \to k_+ \text{ as } r_\ast \to -\infty.
\end{align}
Now, this implies that
\begin{align}
\mathfrak W(w_1,w_2) = \lim_{r_\ast \to -\infty} \left( w_1 w_2^\prime - w_1^\prime w_2\right)= 1,
\end{align}
and moreover, that $w_2= \tilde u_2$ as they satisfy the same boundary conditions at $r_\ast = - \infty$.
\end{proof}
\end{prop}
Having proved the Wronskian condition we are in the position to define the perturbations of $\tilde u_1$ and $ \tilde u_2$ to non-zero frequencies.
\begin{definition}\label{def:perurb}Define perturbations $\tilde u_{1,\omega}$ and $\tilde u_{2,\omega}$ of $\tilde u_1$ and $\tilde u_2$ (cf.\ \eqref{eq:uP} and \eqref{eq:uQ}) in the intermediate region by the unique solutions to the Volterra equations
\begin{align}\label{eq:u1}
\tilde u_{1,\omega} (r_\ast) =\tilde u_1(r_\ast) + \omega^2 \int_{R_1^\ast}^{r_\ast} \left(\tilde u_1(r_\ast) \tilde{u}_2(y) - \tilde u_1(y) \tilde {u}_2(r_\ast) \right) \tilde u_{1,\omega}(y) \d{y}
\end{align}
and
\begin{align}\label{eq:u2}
\tilde{u}_{2,\omega }(r_\ast) = \tilde{u_2}(r_\ast ) + \omega^2 \int_{R^\ast_1}^{r_\ast} \left(\tilde u_1(r_\ast) \tilde{u}_2(y) - \tilde u_1(y) \tilde{u}_2(r_\ast) \right) \tilde{u}_{2,\omega}(y) \d{y}.
\end{align}
\end{definition}
\begin{prop}\label{prop:boundedAB}
Let $0<|\omega|<\omega_0$ and $\ell\in\mathbb N_0$, then we have for $r_\ast \in [R_1^\ast,R_2^\ast]$
\begin{align}
u_1(\omega, r_\ast) = A(\omega,\ell)\tilde u_{1,\omega}(r_\ast) + B(\omega,\ell) \omega \tilde{u}_{2,\omega}(r_\ast),
\end{align}
where \begin{align}|A(\omega,\ell)| + |B(\omega,\ell)| \lesssim 1.\end{align}
\begin{proof}
First, note that by construction in \cref{def:perurb} we have
\begin{align}
& \tilde u_{1,\omega}(R_1^\ast ) = \tilde u_1(R_1^\ast),\\
& \tilde u_{1,\omega}^\prime (R_1^\ast ) =\tilde u_1^\prime(R_1^\ast),\\
& \tilde u_{2,\omega}(R_1^\ast) = \tilde{u}_2(R_1^\ast),\\
& \tilde u_{2,\omega}^\prime(R_1^\ast) = \tilde{u}_2^\prime(R_1^\ast).
\end{align}
Now, we want to estimate the previous terms.
By construction, we directly have that
\begin{align}
|\tilde u_1(R_1^\ast )| \leq 1.
\end{align}
Then, note that
\begin{align}\label{eq:1plusx} \frac{\omega^2}{\ell^2 + 1}\lesssim 1+x(R_1^\ast) \lesssim \frac{\omega^2}{\ell^2 + 1}.
\end{align}
Hence, from \eqref{eq:largeellQ}, we obtain
\begin{align}
| \tilde u_2(R_1^\ast) |\lesssim 1 + \left|-\frac 12 \log(1+x(R_1^\ast)) - \digamma(\ell+1) \right| \lesssim 1 + |\log(|\omega|)|\lesssim \log\left(\frac{1}{|\omega|}\right),
\end{align}
where we have used that for $\ell \geq 1$ we have $\digamma(\ell + 1) = \log(\ell) + \gamma + O(\ell^{-1})$.
For $\tilde u_2^\prime(R_1^\ast)$ we have the estimate
\begin{align}
|\tilde u_2^\prime(R_1^\ast)| & \lesssim | \Delta(R_1^\ast) Q_\ell(x(R_1^\ast))| +\left| \frac{\d Q_\ell}{\d x}(R_1^\ast) \frac{\d x}{\d r_\ast}(R_1^\ast)\right|
\lesssim 1 ,\label{eq:analogous}
\end{align}
where we have used \eqref{eq:derivativelegendre2} and \eqref{eq:1plusx} as well as the fact that \begin{align}\frac{\d x}{\d r_\ast} (1-x(r_\ast)^2)^{-1}\lesssim 1.\end{align}
Now, we can express $A$ via the Wronskian as \begin{align}
| A| =\left|\frac{ \mathfrak W(u_1,\tilde u_{2,\omega}) }{\mathfrak W(\tilde u_{1,\omega}, \tilde u_{2,\omega})} \right|.
\end{align}
By construction, we have $\mathfrak W(\tilde u_{1,\omega},\tilde u_{2,\omega})=\mathfrak W(\tilde u_1,\tilde u_2) = 1$. Hence, using \cref{prop:firstregion} we conclude
\begin{align}
|A| \leq | u_1(R_1^\ast ) \tilde u_{2,\omega}^\prime(R_1^\ast) | + | u_1^\prime(R_1^\ast) \tilde u_{2,\omega}(R_1^\ast )| \lesssim |\tilde u_{2}^\prime(R_1^\ast) | + |\omega\tilde u_{2}(R_1^\ast )|.
\end{align}
Thus, we conclude
\begin{align}
|A| \lesssim 1.
\end{align}
Note that from \eqref{eq:derivativelegendre1}, we have
\begin{align}\label{eq:u1primer1}
|\tilde u_1^\prime(R_1^\ast)| \lesssim \left|\left(1 +\frac{\d P_\ell}{\d x}\right) \frac{\d x}{\d r_\ast}\right| \lesssim (1+\ell^2) \frac{\omega^2}{1+\ell^2}\leq \omega^2.
\end{align}
Hence, we can also estimate $B$ by
\begin{align}\nonumber
|B| &= \frac{1}{|\omega|}| \mathfrak W(u_1,\tilde u_{1,\omega})|\lesssim \frac{1}{|\omega|} \left( | \tilde u_1^\prime(R_1^\ast) | +| \omega \tilde u_1(R_1^\ast)| \right)
\\&\lesssim 1 + \frac{1}{|\omega|} |\tilde u_1^\prime(R_1^\ast) |\lesssim 1,
\end{align}
where we used \cref{prop:firstregion} again.
\end{proof}
\end{prop}
For the intermediate region we will need the following result in order to get uniform bounds for the Volterra iteration.
\begin{lemma}\label{lem:u1u2}
Let $0<|\omega|<\omega_0$ and $\ell\in\mathbb N_0$, then \begin{align}
&\int_{R_1^\ast}^{R_2^\ast}|\tilde u_1(r_\ast)| \d{r_\ast} \lesssim \log^2\left(\frac{1}{|\omega|}\right)\label{eq:l1u1},\\
& \int_{R_1^\ast}^{R_2^\ast} |\tilde u_2(r_\ast)| \d{r_\ast} \lesssim \log^2\left(\frac{1}{|\omega|}\right)\label{eq:l1u2}.
\end{align}
\begin{proof}We first prove \eqref{eq:l1u1}.
We shall split the integral in two regions. The first region is from $r_\ast = R_1^\ast$ to $r_\ast =0$. In that region we define $\theta \in (0,\frac{\pi}{2}]$ such that $\cos(\theta) = - x(r_\ast)$. Using also \cref{lem:plandql} we obtain
\begin{align}\nonumber
|\tilde u_1(r_\ast)| &\lesssim |P_\ell (x(r_\ast)) | = | P_\ell( -x(r_\ast))| = | P_\ell (\cos\theta)| \\ & \lesssim \left| \left( \frac{{\theta}}{\sin\theta}\right)^{\frac 12} J_0( (\ell + \frac 12) \theta)\right| + | e_{1,\ell}(\theta)|.
\label{eq:estimateu1}
\end{align}
The last term shall be treated as an error term. Thus,
\begin{align}
\nonumber \int_{R_1^\ast}^{0}|\tilde u_1(r_\ast)| \d{r_\ast}&
\lesssim \int_{x(R_1^\ast)}^{0} |P_\ell(x)| \frac{1}{1+x} \d{x}\leq \int_{-1 + C \frac{\omega^2}{1+\ell^2}}^{0} |P_\ell(-x)| \frac{1}{1+x} \d{x}\\\nonumber& \lesssim \int_{\arccos(1-C\frac{\omega^2}{ 1+\ell^{2} })}^{\frac \pi 2}| P_\ell(\cos\theta)| \frac{1}{1-\cos\theta} \sin\theta\, \d{\theta}\\
& \leq \int_{C_1 \frac{|\omega|}{1+\ell}}^{\frac \pi 2} |P_\ell(\cos\theta)| \frac{\sin\theta}{1-\cos\theta} \d\theta.
\end{align}
Here, $C$ and $C_1$ are positive constants only depending on the black hole parameters.
We further estimate using equation~\eqref{eq:estimateu1}
\begin{align}\nonumber
&\int_{R_1^\ast}^{0}|\tilde u_1(r_\ast)| \d{r_\ast}\\
& \lesssim \int_{C_1 \frac{\omega}{1+\ell}}^{\frac \pi 2 }\left( \frac{\theta}{\sin\theta}\right)^{\frac 12}\left|J_0( (\ell + \frac 12) \theta)\right| \frac{\sin\theta}{1-\cos\theta}\d\theta + \text{Error} ,
\end{align}
where we will take care of the term
\begin{align}
\text{Error} = \int_{C_1 \frac{\omega}{1+\ell}}^{\frac \pi 2 } |e_{1,\ell}(\theta)|
\end{align}
later. First, we look at the term
\begin{align}\nonumber&\int_{C_1 \frac{\omega}{1+\ell}}^{\frac \pi 2 }\left( \frac{\theta}{\sin\theta}\right)^{\frac 12}\left|J_0\left( \left(\ell + \frac 12\right) \theta\right)\right| \frac{\sin\theta}{1-\cos\theta}\d\theta\\\nonumber
& \lesssim \int_{C_1 \frac{\omega}{1+\ell}}^{\frac \pi 2 } \frac{1}{\theta} \left|J_0\left( \left(\ell + \frac 12\right) \theta\right)\right| \d\theta\\\nonumber
&\lesssim \int_{C_1\omega }^{\frac \pi 2 (\ell +1) } \frac 1 \theta \left|J_0\left(\frac{\ell + \frac 12 }{\ell + 1} \theta\right)\right| \d\theta
\\ \nonumber & \lesssim \int_{C_1\omega}^{1} \frac{\left|J_0\left(\frac{\ell + \frac 12 }{\ell + 1} \theta\right)\right|}{\theta} \d\theta + \int_{1}^{\infty} \frac{\left|J_0\left(\frac{\ell + \frac 12 }{\ell + 1} \theta\right)\right|}{\theta} \d\theta \label{eq:J0}\\& \lesssim
\int_{C_1\omega}^1 \frac{1}{\theta} \d\theta + \int_1^\infty \frac{1}{\theta^\frac{3}{2}} \d\theta
\lesssim |\log(|\omega|)| ,
\end{align}
where we have used equation~\eqref{eq:boundsonJY1} and \eqref{eq:boundsonJY2}.
Now, we are left with the error term
\begin{align}\nonumber
\text{Error}& \leq \frac{1}{1+\ell} \int_{C_1 \frac{\omega}{\ell+1}}^{\frac \pi 2} \frac{\sin\theta}{1-\cos\theta} (|J_0((\ell+\frac 12 )\theta)| + |Y_0( ( \ell+\frac 12)\theta)| ) \d \theta \\ \nonumber & \lesssim \frac{1}{1+\ell} \int_{C_1 \frac{\omega}{\ell+1}}^{\frac \pi 2}\frac{\sin\theta}{1-\cos\theta} ( 1+ |\log(|\omega|)|)\d\theta
\lesssim \frac{|\log(|\omega|)| }{1+\ell }\int_{C_1 \frac{\omega}{\ell + 1}}^\frac{\pi}{2} \frac{1}{\theta} \d \theta
\\ &\lesssim \frac{\log^2(|\omega|) + \log(1+\ell)}{1+\ell}\lesssim \log^2\left(\frac{1}{|\omega|}\right).
\end{align}
Thus,
\begin{align}
\int_{R_1^\ast}^0 |\tilde u_1(r_\ast)| \d{r_\ast} \lesssim \log^2\left( \frac{1}{|\omega|}\right) .\end{align}
Completely analogously, we can compute
\begin{align}
\int_{0}^{R_2^\ast} |\tilde u_1(r_\ast)| \d{r_\ast} \lesssim \log^2\left( \frac{1}{|\omega|}\right) .\end{align}
The proof of equation \eqref{eq:l1u1} is completely similar up to a term which involves
\begin{align}
\int_{C_1 \omega}^1 \frac{\left| Y_0\left( \frac{\ell + \frac 12}{\ell + 1}\theta\right) \right|}{\theta} \d\theta \lesssim \log^2\left(\frac{1}{|\omega|}\right)
\end{align}
appearing in the estimate analogous to \eqref{eq:J0}.
\end{proof}
\end{lemma}
With the help of the previous lemma we can now bound our solution $u_1$ at $R_2^\ast$. This results in
\begin{prop}\label{prop:ur2} Let $0<|\omega|<\omega_0$ and $\ell\in\mathbb N_0$, then
\begin{align}
\| u_1 \|_{L^\infty(R_1^\ast, R_2^\ast)} \lesssim 1 \text{ and } |u_1^\prime|(R_2^\ast) \lesssim | \omega |.
\end{align}
\begin{proof}
Recall that we have from \cref{prop:boundedAB} for $r_\ast \in [R_1^\ast,R_2^\ast]$
\begin{align}
u_1(\omega, r_\ast ) = A(\omega,\ell) \tilde u_{1,\omega}(r_\ast) + \omega B(\omega,\ell) \tilde u_{2,\omega}(r_\ast)\label{eq:u1expansion}
\end{align}
for some uniformly bounded (in $|\omega | \leq \omega_0$ and $\ell$) constants $A,B$.
In particular, from \cref{lem:volterra} and \cref{rmk:volterra} we obtain the bound
\begin{align}
\|\tilde u_{1,\omega}\|_{L^\infty(R_1^\ast,R_2^\ast)} \leq e^{\alpha} \|\tilde u_1\|_{L^\infty(R_1^\ast, R_2^\ast)}
\end{align}
for
\begin{align}
\alpha =\omega^2 \int_{R_1^\ast}^{R_2^\ast} \sup_{ \{ r_\ast\vert y \leq r_\ast \leq R_{2}^\ast \}}|\tilde u_1(r_\ast) \tilde u_2(y) -\tilde u_1(y)\tilde u_2(r_\ast) | \d{y}.
\end{align}
First, we have the bound\begin{align}
\|\tilde u_1\|_{L^\infty(R_1^\ast, R_2^\ast)}\leq 1.\label{eq:u1tilde}
\end{align}
Secondly, for $r_\ast \in [R_1^\ast, R_2^\ast]$ we have \begin{align}1- x(r_\ast) \gtrsim \frac{\omega^2}{1+\ell^2}\end{align}
and
\begin{align}
1+x(r_\ast) \gtrsim \frac{\omega^2}{1+\ell^2}.
\end{align}
Consider the case $x(r_\ast) \geq 0$ first and implicitly define $\theta(r_\ast)$ by $\cos\theta(r_\ast) = x(r_\ast)$. Then, in view of \eqref{eq:estimateql} and $\theta(x(r_\ast)) = \sqrt{2-2x(r_\ast)} + O ((1-x(r_\ast)^{\frac 32} ))$, we estimate
\begin{align}
| \tilde u_2(r_\ast) |\lesssim |Q_\ell(\cos(\theta(r_\ast)))| \lesssim \left| Y_0\left( \frac{\theta(r_\ast) (2\ell + 1)}{2} \right) \right|\lesssim \left| Y_0 \left( C {|\omega|} \right) \right|
\end{align} for a $C=C(M,Q)>0$. Analogously, this also holds for $x(r_\ast) < 0$ such that \eqref{eq:boundsonJY1} and \eqref{eq:boundsonJY2} imply
\begin{align}
\|\tilde u_2\|_{L^\infty(R_1^\ast, R_2^\ast)}\lesssim \log\left(\frac{1}{|\omega|}\right). \label{eq:estu2}
\end{align}
Together with \cref{lem:u1u2} we obtain
\begin{align}
\alpha \lesssim 1.
\end{align}
Hence,
\begin{align}
\| \tilde u_{1,\omega}\|_{L^\infty(R_1^\ast, R_2^\ast)} \lesssim 1
\end{align}
and similarly,
\begin{align}
\|\tilde u_{2,\omega}\|_{L^\infty(R_1^\ast,R_2^\ast)} \lesssim \log\left(\frac{1}{|\omega|}\right). \label{eq:estimu2w}
\end{align}
This shows $\|u_1\|_{L^\infty(R_1^\ast,R_2^\ast)} \lesssim 1$ in view of \eqref{eq:u1expansion}.
Now, we are left with the derivative $u_1^\prime(R_2^\ast)$. To do so, we start by estimating $\tilde u_1^\prime(R_2^\ast)$ and $\tilde u_2^\prime(R_2^\ast)$. Using the analogous estimate as we did for $R_1^\ast$ in \eqref{eq:analogous} and \eqref{eq:u1primer1}, we obtain
\begin{align}\label{eq:u2r2}
| \tilde u^\prime_2 (R_2^\ast)| \lesssim 1 \text{ and } |\tilde u^\prime_{1} (R_2^\ast) | \lesssim \omega^2.
\end{align}
Note that
\begin{align}
\tilde u_{2,\omega}^\prime(R_2^\ast) = \tilde u_2^\prime(R_2^\ast) + \omega^2 \int_{R_1^\ast}^{R_2^\ast} \left( \tilde u_1^\prime(R_2^\ast) \tilde u_2(y) - \tilde u_1(y) \tilde u_2^\prime(R_2^\ast) \right) \tilde u_{2,\omega}(y) \d y
\end{align} and thus in view of \cref{lem:u1u2}, \eqref{eq:u2r2}, \eqref{eq:estimu2w}, \eqref{eq:estu2}, and \eqref{eq:u1tilde} we estimate
\begin{align}\nonumber
|\tilde u_{2,\omega}^\prime(R_2^\ast ) |&\leq|\tilde u_2^\prime(R_2^\ast)| + \omega^2 \log\left(\frac{1}{|\omega|}\right)\int_{R_1^\ast}^{R_2^\ast} |\tilde u_1^\prime(R_2^\ast)\tilde u_2(y)| + | \tilde u_1(y)\tilde u_2^\prime(R_2^\ast)| \d{y}\\ &\lesssim 1+ \omega^2 \, | \log(|\omega|)| \, ( \omega^2 \log^2(|\omega|) + \log^2(|\omega|)) \lesssim 1.
\end{align}
Similarly, we obtain
\begin{align}\nonumber
|\tilde u_{1,\omega}^\prime(R_2^\ast ) |&\leq|\tilde u_1^\prime(R_2^\ast)| + \omega^2 \int_{R_1^\ast}^{R_2^\ast} |\tilde u_1^\prime(R_2^\ast) \tilde u_2(y)| + | \tilde u_1(y) \tilde u_2^\prime(R_2^\ast)| \d{y}\\ &\lesssim \omega^2+ \omega^2 ( \omega^2 \log^2(|\omega|) + \log^2(|\omega|)) \lesssim | \omega|
\end{align}
which concludes the proof in the light of \eqref{eq:u1expansion}.
\end{proof}\end{prop}
\paragraph{\textbf{Region near the Cauchy horizon}}
Completely analogously to \cref{prop:firstregion}, we have
\begin{prop}
\label{prop:thirdregion}
Let $0<|\omega|<\omega_0$ and $\ell\in\mathbb N_0$. Then, we have
\begin{align}
& \| v_1^\prime \|_{L^\infty(R_2^\ast,\infty)} \lesssim |\omega|,\;\;
\|v_1\|_{L^\infty(R_2^\ast,\infty)} \lesssim 1
\end{align}
and
\begin{align}
& \| v_2^\prime \|_{L^\infty(R_2^\ast,\infty)} \lesssim |\omega|,\;\;
\|v_2\|_{L^\infty(R_2^\ast,\infty)} \lesssim 1.
\end{align}
\end{prop}
\paragraph{\textbf{Boundedness of the scattering coefficients}}
Finally, we conclude that the reflection and transmission coefficients are uniformly bounded for parameters $0<|\omega|<\omega_0$ and $\ell \in \mathbb{N}_0$.
\begin{prop}
\label{prop:easyregime}
We have
\begin{align}
\sup_{0<|\omega|<\omega_0, \ell \in \mathbb N_0} (|\mathfrak R(\omega, \ell)| + |\mathfrak T(\omega,\ell)| )\lesssim 1.
\end{align}
\begin{proof} Let $0<|\omega|<\omega_0$ and $\ell \in \mathbb N_0$ and recall \cref{defn:TandR}.
Then, \cref{prop:ur2} and \cref{prop:thirdregion} imply
\begin{align}
& |\mathfrak T| \lesssim \frac{|\mathfrak W(u_1,v_2)|}{|\omega|} \leq \frac{|u_1(R_2^\ast)v_2^\prime(R_2^\ast)| +|u_1^\prime(R_2^\ast)v_2(R_2^\ast)| }{|\omega|}\lesssim 1\end{align}
and
\begin{align}
& |\mathfrak R| \lesssim \frac{|\mathfrak W(u_1,v_1)|}{|\omega|} \leq \frac{|u_1(R_2^\ast)v_1^\prime(R_2^\ast)| +|u_1^\prime(R_2^\ast)v_1(R_2^\ast)| }{|\omega|}\lesssim 1.
\end{align}
\end{proof}
\end{prop}
\subsection{Frequencies bounded from below and bounded angular momenta (\texorpdfstring{$|\omega|\geq \omega_0, \ell \leq \ell_0$}{wgeqw0,lleql0})}
Now, we will consider parameters of the form $|\omega|\geq \omega_0$ and $\ell \leq \ell_0$, where $\omega_0$ is small and determined from \cref{subsec:smallfreq}. Also, the upper bound on the angular momentum $\ell_0$ will be determined from \cref{subsec:boundedbelow}. As before, constants appearing in $\lesssim$ and $\gtrsim$ may depend on $\omega_0$.
\begin{prop} \label{prop9}We have
\begin{align}
\sup_{\omega_0\leq|\omega|, \ell\leq \ell_0} (|\mathfrak R(\omega, \ell)| + |\mathfrak T(\omega,\ell)| )\lesssim 1.
\end{align}
\begin{proof}
Recall the definition of $u_1$ as the unique solution to
\begin{align}\label{eq:integralbound}
u_1(\omega, r_\ast ) = e^{i\omega r_\ast } + \int_{-\infty}^{r_\ast } \frac{\sin(\omega(r_\ast -y))}{\omega} V(y) u_1(\omega, y) \d y .
\end{align}
Note that in the regime $\ell \leq \ell_0$ we have a bound of the form
\begin{align}
| V(r_\ast)|\lesssim e^{-2 \min(k_+,|k_-|) |r_\ast|}
\end{align}
which implies the following bound on the integral kernel of the perturbation in \eqref{eq:integralbound}
\begin{align}
|K(r_\ast,y)| =\left| \frac{\sin(\omega(r_\ast - y))}{\omega} V(y)\right| \lesssim |V(y)|
\end{align}
in view of $|\omega| \geq \omega_0$. Thus,
\begin{align}
\int_{-\infty}^{\infty} \sup_{r_\ast \in \mathbb R} |K(r_\ast,y)| \d y\lesssim \int_{-\infty}^{\infty} |V(y)| \d y\lesssim 1.
\end{align}
Hence, from \cref{lem:volterra} we deduce
\begin{align}\label{eq:boundonu}
\|u_1\|_{L^\infty(\mathbb R)} \lesssim 1
\end{align}
and
\begin{align}\label{eq:boundsonuprime}
\|u_1^\prime \|_{L^\infty(\mathbb R)} \lesssim |\omega|.
\end{align}
Note that we have obtained similar, indeed even stronger bounds for $u_1$ as in \cref{prop:ur2}. An argument completely similar to \cref{prop:easyregime} allows us to conclude.
\end{proof}
\end{prop}
\subsection{Frequencies and angular momenta bounded from below (\texorpdfstring{$|\omega| \geq \omega_0$, $\ell \geq \ell_0$}{wgeqw0,lgeql0})}
\label{subsec:boundedbelow}
In this regime we assume $\omega\geq \omega_0$ and $\ell \geq \ell_0$, where we choose $\ell_0$ large enough such that $V_\ell < 0$ everywhere. Note that such an $\ell_0$ can be chosen only depending on the black hole parameters.
We write the o.d.e.\ as \begin{align}
u^{\prime \prime} = - (\omega^2 - V_\ell) u
\end{align}
and will represent the solution of the o.d.e.\ via a WKB approximation. For concreteness we will use the following theorem which is a slight modification of \cite[Theorem 4]{olver1961error}.
\begin{lemma}[Theorem 4 of \cite{olver1961error}]\label{thm:wkb}
Let $p\in C^2(\mathbb{R})$ be a positive function such that
\begin{align}
F(x) = \left|\int_{-\infty}^x p^{-\frac{1}{4}} \left|\frac{\d^2}{\d x^2} \left( p^{-\frac 14}\right)\right| \d y\right|
\end{align}
satisfies $\sup_{x\in\mathbb{R}}F(x)< \infty$.
Then, the differential equation
\begin{align}
\frac{\d^2 u (x) }{\d x^2} = - p(x) u(x)
\end{align}
has conjugate solutions $u$ and $\bar u$ such that
\begin{align}
&u(x) = p^{-\frac 14}\left( \exp\left(i \int^x_0 \sqrt p(y) \d y \right) +\epsilon \right), \\
&u^\prime (x) = i p^\frac{1}{4}\left[ \exp\left(i \int_0^x \sqrt{p}(y) \d y \right) -i \eta + \frac{i p^\prime}{4 p^\frac{3}{2}}\left( \exp\left(- i \int_0^x \sqrt p (y) \d y \right) + \epsilon \right) \right],
\end{align}
where
\begin{align}
|\eta(x)|, |\epsilon(x)|\leq \exp\left(F(x)\right) - 1.\label{eq:boundsonetaolver}
\end{align}
\end{lemma}
\begin{prop}\label{prop:reg3}Let $\omega_0 \leq |\omega|$ and $\ell \geq \ell_0$. Assume without loss of generality that $\omega >0$. Then,
\begin{align}
&u_1(\omega, r_\ast) = A \omega^{\frac 12} (\omega^2 - V(r_\ast))^{-\frac 14}\left( \exp\left(i \int_{0}^{r_\ast} (\omega^2 - V_\ell(y))^\frac{1}{2} \d{y}\right) + \epsilon(r_\ast)\right),\\
&u_1^\prime(\omega, r_\ast) = A {\omega}^\frac{1}{2} i (\omega^2 - V(r_\ast))^{\frac 14} \left[ \exp\left( i \int_0^{r_\ast} (\omega^2 - V_\ell(y))^{\frac{1}{2}}\d y\right) - i \eta(r_\ast) \right. \nonumber \\& \hspace{2cm} \left. - \frac{i V^\prime(r_\ast)}{4 (\omega^2 - V)^{\frac 32}(r_\ast)}\left( \exp\left(i \int_0^{r_\ast} (\omega^2 - V_\ell(y))^\frac{1}{2} \d y\right) + \epsilon(r_\ast) \right) \right] ,
\end{align}
where \begin{align}|A| =1 , \sup_{r_\ast \in \mathbb{R}}(|\epsilon|(r_\ast)+ |\eta|(r_\ast)) \lesssim 1\label{eq:boundsoneta}\end{align}
and
\begin{align}\label{eq:limits}
\lim_{r_\ast \to -\infty} \eta (r_\ast)= \lim_{r_\ast \to -\infty} \epsilon(r_\ast) = 0.
\end{align}
In particular, this proves
\begin{align}
& \limsup_{r_\ast \to\infty}|u(r_\ast)| \lesssim 1, \\
& \limsup_{r_\ast \to \infty} |u^\prime(r_\ast)|\lesssim |\omega|,
\end{align}
and uniform bounds on the reflection and transmission coefficients
\begin{align}
\sup_{\omega_0\leq|\omega|, \ell\geq \ell_0} (|\mathfrak R(\omega, \ell)| + |\mathfrak T(\omega,\ell)| )\lesssim 1.
\end{align}
\begin{proof}
We will apply \cref{thm:wkb}. First, we set \begin{align}p=(\omega^2 - V_\ell )\end{align} which is positive and smooth. Then, the o.d.e.\ reads
\begin{align}
u^{\prime \prime} = - p u.
\end{align} Now we have to show that $F$ is uniformly bounded on the real line.
Note that we have the following bounds on the potential and its derivatives
\begin{align}
|V_\ell(r_\ast)|, |V_\ell^\prime(r_\ast)|, |V_\ell^{\prime\prime}(r_\ast)|\lesssim \ell^2 e^{2\kappa_+ r_\ast} \text{ and } \ell^2 e^{2\kappa_+ r_\ast} \lesssim |V_\ell(r_\ast)|\text{ for } r_\ast \leq 0,\label{eq:boundsonV01}\\
| V_\ell(r_\ast)|,|V_\ell^\prime(r_\ast)|, |V_\ell^{\prime\prime}(r_\ast)|\lesssim \ell^2 e^{2\kappa_- r_\ast} \text{ and } \ell^2 e^{2\kappa_- r_\ast} \lesssim | V_\ell(r_\ast)| \text{ for } r_\ast \geq 0.\label{eq:boundsonV02}
\end{align}
Here, we might have to choose $\ell_0(M,Q)$ even larger ($r_+^2 (r_+ - 3r_-) + \ell(\ell+1)>0$, cf.\ \eqref{eq:cmV}) in order to assure the lower bounds on the potential.
Finally, we can estimate $F$ by
\begin{align}\nonumber
\sup_{r_\ast \in \mathbb{R}} F(r_\ast) &\leq \left| \int_{-\infty}^{\infty} p^{- \frac 14} \left| \frac{\d^2}{\d x^2} \left(p^{-\frac 14 } \right| \right)\d y\right| \\ \nonumber&= \int_{-\infty}^{\infty} p^{- \frac 14}\left( p^{- \frac 94} {p^\prime}^2 + p^{-\frac 54 } |p^{\prime \prime}| \right)\d y\\ \nonumber&
\lesssim \frac{1}{\ell}\int_{0}^{\infty}\left( \frac{e^{4\kappa_- y} }{(\ell^{-2} + e^{2\kappa_- y})^{\frac 52}} + \frac{e^{2\kappa_- y}}{(\ell^{-2} + e^{2\kappa_- y})^{\frac 32}} \right)\d y\\
& + \frac{1}{\ell} \int_{-\infty}^{0} \left( \frac{e^{4\kappa_+ y} }{(\ell^{-2} + e^{2\kappa_+ y})^{\frac 52}} + \frac{e^{2\kappa_+ y}}{(\ell^{-2} + e^{2\kappa_+ y})^{\frac 32}} \right) \d y,
\end{align}
where we have used the bounds from \eqref{eq:boundsonV01} and \eqref{eq:boundsonV02}.
We shall estimate both terms independently. After a change of variables $y \mapsto \frac{1}{2\kappa_-}\log(y)$, we can estimate the first term by
\begin{align} \nonumber &\frac{1}{\ell}\int_{0}^{\infty}\left( \frac{e^{4\kappa_- y} }{(\ell^{-2} + e^{2\kappa_- y})^{\frac 52}} + \frac{e^{2\kappa_- y}}{(\ell^{-2} + e^{2\kappa_- y})^{\frac 32}} \right)\d y\\ \nonumber & \lesssim \frac{1}{\ell}
\int_{0}^{1}\left( \frac{y}{(\ell^{-2} +y)^{\frac 52}} + \frac{1}{(\ell^{-2} + y)^{\frac 32}} \right)\d y\\ \nonumber &\lesssim \ell^2 \int_{0}^1 \frac{\ell^2 y}{(1+\ell^2 y)^{\frac 52} } + \frac{1}{(1+ \ell^2 y)^{\frac 32 }}\d y\\ & \lesssim \int_0^\infty \frac{y}{(1+y)^{\frac 52}} + \frac{1}{(1+y)^{\frac 32 }} \d y \lesssim 1.
\end{align}
Completely analogously, we get the bound for the second integral. In particular, this shows
\begin{align}
\sup_{\mathbb{R}} F \lesssim 1.
\end{align}
This implies the bounds on $\eta$ and $\epsilon$ in the statement of the theorem (cf.\ \eqref{eq:boundsoneta}) using \eqref{eq:boundsonetaolver}.
The limits in equation \eqref{eq:limits} follow from the fact that $F(r_\ast) \to 0$ as $r_\ast \to -\infty$ by construction.
The bound on the reflection and transmission coefficients follows now from
\begin{align}
|\mathfrak R| \lesssim \left|\frac{\mathfrak W(u_1,v_1)}{\omega}\right| \leq \frac{1}{|\omega|} \limsup_{r_\ast\to\infty} \left(| u_1^\prime v_1| +| u_1 v_1^\prime|\right) \lesssim 1
\end{align}
and analogously for $\mathfrak T$.
Finally, $A$ can be determined from the asymptotic behaviour $u \to e^{i\omega r_\ast}$ as $r_\ast \to - \infty$ and it is given by
\begin{align}\nonumber
A & =\lim_{r_\ast \to - \infty} \exp\left( i \omega r_\ast -i \int_0^{r_\ast} (\omega^2 - V(y))^{\frac 12 } \d y\right) \\ &=
\lim_{r_\ast \to - \infty} \exp\left( -i \int_0^{r_\ast} \left((\omega^2 - V(y))^{\frac 12 } - \omega \right) \d y\right)
\end{align}
which converges since $V$ tends to zero exponentially fast. In particular, this also shows that $|A| =1$.
\end{proof}
\end{prop}
Finally, \cref{thm:boundednesstrans} is a consequence of \cref{prop:easyregime}, \cref{prop9}, and \cref{prop:reg3}.
|
{
"timestamp": "2018-12-18T02:03:25",
"yymm": "1804",
"arxiv_id": "1804.05438",
"language": "en",
"url": "https://arxiv.org/abs/1804.05438"
}
|
\section{INTRODUCTION\label{intro}}
Modern unbiased transient surveys have revolutionized our understanding of various explosive phenomena in the Universe.
One of the remarkable results is the discovery of a special class of supernovae (SNe) characterized by their high luminosities ($10$--$100$ times higher than those of normal core-collapse SNe), which are now called superluminous supernovae (SLSNe; see \citealt{2012Sci...337..927G} for a review).
Although their volumetric rate is extremely small ($<0.1\%$ of normal core-collapse SNe) \citep[e.g.][]{2013MNRAS.431..912Q,2015MNRAS.448.1206M,2017MNRAS.464.3568P}, they could be detectable at high-z galaxies thanks to their extreme luminosities \citep{2012MNRAS.422.2675T,2013MNRAS.435.2483T,2014ApJ...796...87I}.
SLSNe are classified into a couple of subcategories based on the presence or absence of hydrogen features in their spectra.
SLSNe without any hydrogen feature are called hydrogen-poor or type-I SLSNe (hereafter SLSNe-I) and suggested to be explosions of massive stars without hydrogen and helium envelopes \citep[e.g.][]{2007ApJ...668L..99Q,2009ApJ...690.1358B,2010ApJ...724L..16P,2011Natur.474..487Q,2011ApJ...743..114C}.
Observations of individual SLSNe-I and their host galaxies suggest that SLSNe-I are likely produced by massive stars born in dwarf galaxies with low metallicities and high specific star formation rates \citep{2011ApJ...727...15N,2014ApJ...787..138L,2015MNRAS.449..917L,2015MNRAS.451L..65T,2016MNRAS.458...84A,2017MNRAS.470.3566C,2016ApJ...830...13P,2018MNRAS.473.1258S}.
Despite intensive observational and theoretical studies on SLSNe-I, the energy source of their bright emission is still poorly understood.
Currently, three different scenarios have been proposed, (1) core-collapse SNe interacting with massive hydrogen-poor circumstellar matter \citep{2011ApJ...729L...6C,2012ApJ...757..178G,2013MNRAS.428.1020M}, (2) pair-instability SNe \citep{1967PhRvL..18..379B,1967ApJ...148..803R,2002ApJ...567..532H,2007Natur.450..390W,2009Natur.462..624G}, and (3) central engine powered SNe (\citealt{2010ApJ...717..245K,2010ApJ...719L.204W}; see also \citealt{1971ApJ...164L..95O,1976SvA....19..554S,2007ApJ...666.1069M}).
The traditional and widely used way to asses the existing scenarios of the energy source is to examine whether light curves of SLSNe can successfully be explained in the framework of these scenarios.
Supernova light curves are well explained by diffusion of thermal photons in freely expanding spherical ejecta \citep{1980ApJ...237..541A,1982ApJ...253..785A,1996snih.book.....A}.
Therefore, the timescale of the luminosity evolution can be a key to constraining properties of exploding stars and the energy source.
One-zone light curve models with multiple energy supplies, i.e., radioactive decay, CSM interaction, and central engine, have been formulated \citep[e.g.][]{2012ApJ...746..121C} and applied for observed light curves of SLSNe and other extraordinary SNe \citep{2013ApJ...770..128I,2013ApJ...773...76C,2015MNRAS.452.3869N,2015ApJ...799..107W,2015ApJ...807..147W,2016ApJ...821...22W,2017ApJ...837..128W,2017ApJ...850...55N}.
Among the three scenarios, the pair-instability SN scenario requires extremely large nickel and ejecta masses, indicating slow light curve evolution.
This is in tension with some SLSNe showing rapid evolution \citep{2013Natur.502..346N}.
However, distinguishing these scenarios solely from IR-optical photometric observations of SLSNe is generally difficult because of a number of adjustable parameters in theoretical light curve models.
Spectroscopic observations have also been conducted and revealed that early spectra of SLSNe-I exhibit a blue continuum and absorption features by highly-ionized oxygen \citep{2007ApJ...668L..99Q,2010ApJ...724L..16P,2011Natur.474..487Q,2013ApJ...779...98H}.
Numerical investigations on the spectral formation in SLSNe-I have also been attempted \citep{2012MNRAS.426L..76D,2016MNRAS.458.3455M}.
Spectra of SLSNe-I at various epochs are available particularly for well-observed nearby events, such as SN 2007bi, PTF09cnd, 2010gx, PS1-11ap, PTF12dam, LSQ14an, LSQ14mo, and SN 2015bn \citep[e.g.][]{2009Natur.462..624G,2010ApJ...724L..16P,2011Natur.474..487Q,2013Natur.502..346N,2014MNRAS.437..656M,2015ApJ...807L..18N,2015MNRAS.452.1567C,2015ApJ...815L..10L,2017MNRAS.468.4642I,2017A&A...602A...9C}.
\cite{2017ApJ...845...85L} compared spectra of SLSNe-I and stripped-envelope CCSNe in a systematic way by using the Markov Chain Monte Carlo (MCMC)-based spectral fitting method developed by
\cite{2016ApJ...827...90L} and \cite{2016ApJ...832..108M}.
They found that the average photospheric velocity of SLSNe-I implied by FeII absorption lines ($\sim15000$ km s$^{-1}$ around 10 days after the peak) is higher than normal type Ic SNe ($\sim 7000$ km s$^{-1}$) and similar to type Ic SNe characterized by broad absorption features (SNe Ic-BL).
Recent observations of the SLSN-I 2015bn at $z=0.1136$ also showed that the nebular spectra at later epochs were remarkably similar to those of SNe Ic-BL \citep{2016ApJ...828L..18N,2017ApJ...835...13J}.
These findings may indicate a link between the two extraordinary classes of SNe, SLSNe-I and SNe Ic-BL.
However, theoretical understanding of characteristic spectral features associated with different scenarios has still been limited, which makes it difficult by optical spectra to distinguish different scenarios.
Furthermore, an SLSN-like bump is found in the afterglow light curve of the ultra-long gamma-ray burst (GRB) 111209A and named SN 2011kl \citep{2015Natur.523..189G,2016arXiv160606791K}.
These observations may further support the scenario that SLSNe-I and SNe Ic-BL (or associated GRBs) are powered by the same engine.
Actually, a fast rotating magnetized neutron star, which is the most popular power source for the central engine scenario of SLSNe-I, have also been considered as a potential central engine for GRBs \citep{1992Natur.357..472U,2000ApJ...537..810W,2004ApJ...611..380T,2008MNRAS.383L..25B,2009MNRAS.396.2038B,2011MNRAS.413.2031M}.
More recently, roles of a mili-second magnetized neutron star newly born in supernova ejecta are also paid a great attention in the context of fast radio bursts (FRBs).
The recently realized localization of the repeating FRB 121102 and the associated persistent radio source have stimulated intense discussion on its progenitor and emission mechanism \citep{2017Natur.541...58C,2017ApJ...834L...8M,2017ApJ...834L...7T}.
The similarity of the host galaxy of the FRB and those of SLSNe may indicate the possible FRB-SLSN or FRB-SLSN-GRB association \citep{2017ApJ...834L...7T,2017ApJ...841...14M,2017ApJ...843...84N}.
Another potential way to distinguishing energy sources of SLSNe-I is to identify emission signatures across a wide energy range from radio to gamma-rays.
Radio waves and high energy photons from young CCSNe are usually attributed to emission from non-thermal electrons produced by blast waves driven by supernova ejecta.
Such non-thermal emission is also naturally expected for SLSNe-I.
Multi-wavelength observations of SLSNe-I have been conducted particularly for nearby events, such as SN 2015bn \citep{2016ApJ...826...39N}, Gaia16apd \citep{2018ApJ...856...56C}, and SN 2017egm (also known as Gaia17biu) \citep{2018ApJ...853...57B,2017ApJ...845L...8N}.
Systematic searches for X-ray emission from SLSNe-I have been carried out and compiled by \cite{2013ApJ...771..136L} and \cite{2017arXiv170405865M}. Currently, possible detections of X-ray sources whose locations are consistent with SCP 06F6 \citep{2009ApJ...697L.129G} and PTF12dam \citep{2017arXiv170405865M} have been reported.
However, the origin of the X-ray emission is still debated.
Some SLSNe-I, e.g., Gaia 16apd, exhibit a significant UV excess, which should also be a key to revealing the energy source \citep{2017ApJ...840...57Y,2017ApJ...835L...8N,2017MNRAS.469.1246K,2017ApJ...845L...2T}.
Furthermore, a systematic search for gamma-ray emission associated with SLSNe has been conducted by \cite{2018A&A...611A..45R}, although they obtained only an upper limit for the gamma-ray luminosity by assuming a photon spectrum of $\nu^{-2}$.
If an SLSNe-I is powered by a relativistic wind from a fast-rotating magnetized neutron star in an analogy to Galactic pulsar wind nebulae, electron-positron pairs would be copiously produced in the downstream of the shock wave terminating the wind.
These high energy particles with non-thermal energy spectra can serve as an ionizing photon source for the supernova ejecta surrounding the neutron star.
Thus, the presence of a nascent neutron star at the centre of the expanding supernova ejecta could be probed by the ionization structure of the ejecta and/or radio, X-ray, and gamma-ray emission \citep{2013MNRAS.432.3228K,2014MNRAS.437..703M,2015ApJ...805...82M,2016MNRAS.461.1498M,2016ApJ...818...94K}.
\cite{2017ApJ...841...14M} considered radio emission associated with SLSNe in the context of the FRB-SLSNe connection and pointed out that the quiescent radio source found in the host galaxy of FRB 121102 could be an SLSN remnant having produced a magnetar.
They considered radio emission from the pulsar wind nebula and the forward shock driven by the supernova ejecta.
More recently, \cite{2018MNRAS.474..573O} present similar calculations based on the model developed by \cite{2015ApJ...805...82M} and \cite{2016ApJ...818...94K}.
Light curve modellings of central engine powered supernova ejecta in a wide range of wavelengths would greatly help us understand the emission mechanism of SLSNe-I and ultimately unveil their enigmatic origin.
Especially, radio and X-ray emission from SNe can probe the density and velocity structure of supernova ejecta and also is expected to play a role in distinguishing the existing scenarios of SLSNe-I.
Recent numerical studies on supernova ejecta with central energy injection have claimed that multi-dimensional effects are important in determining the density structure of the ejecta.
Multi-dimensional effects of the central energy injection into supernova ejecta, such as mixing and breakout of a hot bubble, have been considered by several authors \citep{2003ApJ...589..871A,2011MNRAS.411.2054L}.
However, it is only recently that such effects are investigated by using hydrodynamic simulations in the context of SLSNe \citep{2016ApJ...832...73C,2017MNRAS.466.2633S,2017ApJ...845..139B}.
The two-dimensional hydrodynamics simulation of the interaction between supernova ejecta and a relativistic wind from the central engine presented by \cite{2017MNRAS.466.2633S} revealed that the interaction leads to the creation of a hot bubble at the centre of the ejecta, which eventually blows out the whole ejecta and accelerates outermost layers to mildly relativistic speeds.
These features are very different from the conventional ejecta structure applied to normal CCSNe.
The mildly relativistic component of the supernova ejecta colliding with the CSM potentially produces bright non-thermal emission, which can serve as a signature of the hot bubble breakout.
In this paper, we present a theoretical model for photospheric, synchrotron, and inverse Compton emission from central engine powered supernova ejecta and the accompanying blast wave in the CSM.
The thermal and non-thermal emission are calculated by a method commonly used in supernova studies, while we adopt the ejecta profile indicated by the hydrodynamic simulation.
The dynamical evolution of the supernova ejecta expanding into the CSM is treated by the method developed by \cite{2017ApJ...834...32S}, who considered the hydrodynamic collision of trans-relativistic spherical ejecta with a steady stellar wind.
In Section \ref{sec:dynamics}, we describe our model for the dynamical evolution and photospheric emission of freely expanding supernova ejecta with a power-law density profile.
In Section \ref{sec:non_thermal}, details of the non-thermal emission model are presented.
In Section \ref{sec:results}, we compare our theoretical light curves with currently available observations of SLSNe-I and the two SNe Ic-BL 1998bw \citep{1998Natur.395..663K,1998Natur.395..670G} and 2009bb \citep{2010Natur.463..513S}, which are characterized by bright radio emission indicating the presence of a central engine activity.
Finally, we conclude this paper in Section \ref{sec:conclusion}
\section{Dynamical evolution and thermal emission of supernova ejecta}\label{sec:dynamics}
In this section, we describe our model for the dynamical evolution of supernova ejecta powered by a central engine.
\subsection{Energy injection from the central engine}
Our previous simulation \citep{2017MNRAS.466.2633S} is based on the central-engine scenario for SLSNe \citep{2010ApJ...717..245K} and has assumed energy injection at a constant rate around the centre.
On the other hand, the most popular model for central-engine powered supernovae adopts mili-second magnetar spin down as the primary power source.
The energy injection rate is usually assumed to be proportional to $(1+t/t_\mathrm{sd})^{-s}$, where $t_\mathrm{sd}$ is the spin down time of the magnetar and $s$ is an exponent (hereafter, $s=2$).
Therefore, while the energy injection rate is constant well before the characteristic spin down time $t_\mathrm{sd}$, it decays in a power-law fashion at $t\gg t_\mathrm{sd}$.
In order to incorporate the multi-dimensional picture revealed by the numerical simulation into calculations of thermal and non-thermal emission powered by the central engine, we assume that the structure of the ejecta has been fixed after the total energy of the ejecta reaches $E_\mathrm{ej}=10^{52}$ erg at $t=t_\mathrm{sd}$.
The radial density and velocity profiles of the ejecta are assumed to be those derived by \cite{2017MNRAS.466.2633S}, which are reviewed in the next subsection.
The normalization of the spin down energy deposition rate $L_\mathrm{sd}$ is determined so that the deposited energy reaches $E_\mathrm{ej}$ at $t=t_\mathrm{sd}$.
Thus, the spin down rate is expressed as follows,
\begin{equation}
L_\mathrm{sd}(t)=\frac{2E_\mathrm{ej}}{t_\mathrm{sd}}(1+t/t_\mathrm{sd})^{-2}
\label{eq:L_sd}
\end{equation}
With this normalization, the total deposited energy yields $2E_\mathrm{ej}$.
The energy deposited at $t>t_\mathrm{sd}$ serves as a power source for thermal emission from the ejecta.
Although a fraction of the energy may be used to accelerate the ejecta, it is smaller than the total energy of the ejecta and thus unlikely to significantly affect the subsequent dynamical evolution of the ejecta.
This treatment makes the dynamical model not fully self-consistent.
Nevertheless, important aspects of the dynamical evolution of the ejecta are certainly captured.
In this work, we are interested in thermal and non-thermal emission from the ejecta having experienced the hot bubble breakout.
We start the calculation of the emission at the initial time $t_\mathrm{i}=t_\mathrm{sd}$.
The spin down time, which is now equal to the initial time of the calculation, is a free parameter specifying the timescale of the central energy injection.
\subsection{Supernova ejecta with central energy injection}
We review the dynamical evolution of supernova ejecta powered by the central energy injection at a constant rate and how the subsequent energy redistribution throughout the ejecta shapes the density and velocity structure of the ejecta.
\subsubsection{Powering supernova ejecta}\label{sec:powering_sn_ejecta}
\cite{2017MNRAS.466.2633S} have considered the dynamical evolution of supernova ejecta powered by a relativistic wind injected from a central engine at a constant energy injection rate.
The ejecta are assumed to be expanding in a homologous way, i.e., the radial velocity $v$ of a layer is its radius divided by the elapsed time, $v=r/t$.
A widely used broken power-law model \citep{1989ApJ...341..867C}, where the density is proportional to the radial velocity, $\rho\propto v^{-\delta}$ for inner ejecta and $\rho\propto v^{-m}$ for outer ejecta, is employed for the density profile of the supernova ejecta.
The inner density gradient should be shallow, $\delta<3$, so that the mass of the ejecta should not diverge at the centre.
On the other hand, the outer density gradient is usually assumed to be steep, with a typical index of $m\sim 10$.
The numerical simulation adopted $\delta=1$ and $m=10$.
The important parameters characterizing the dynamical evolution of the ejecta are the original kinetic energy of the ejecta $E_\mathrm{sn}$ and the energy injection rate $\dot{E}_\mathrm{in}$.
These two parameters give the characteristic timescale $t_\mathrm{c}=E_\mathrm{sn}/\dot{E}_\mathrm{in}$, at which the injected energy reaches the original kinetic energy.
The dynamical evolution can be scaled by this critical timescale.
In other words, we can apply the following scenario for different energy injection rates by rescaling the time $t$.
The numerical simulation revealed that the dynamical evolution of supernova ejecta with an embedded relativistic wind can be divided into the following three stages (see the schematic representation in Figure \ref{fig:schematic}):
(1) The relativistic wind injected around the centre first creates a quasi-spherical, geometrically thin shell composed of the shocked wind and ejecta (quasi-spherical stage).
In this stage, the shocked gas forms a quasi-spherical hot bubble well confined by the ram pressure of the ejecta.
The dynamical evolution of the shell in this stage is described by a self-similar solution and the radius of the shell evolves as $t^{\alpha}$, where $\alpha=(6-\delta)/(5-\delta)$ \citep{1982ApJ...258..790C,1998ApJ...499..282J,2005ApJ...619..839C}.
(2) When the forward shock propagating in the ejecta reaches a layer above which the density gradient is steep, the ram pressure of the ejecta no longer confines the hot bubble.
As a result, the steep density gradient efficiently accelerates the forward shock and the whole ejecta are gradually overwhelmed by the shocked gas (hot bubble breakout).
This transition happens at $t_\mathrm{br}=f_\mathrm{br}t_\mathrm{c}$, when the total amount of the energy injected from the central engine exceeds a threshold value $f_\mathrm{br}E_\mathrm{sn}$.
The factor $f_\mathrm{br}$ depends on the structure of the ejecta.
We assume $f_\mathrm{br}=5$ (\citealt{2017MNRAS.466.2633S}; see also \citealt{2017ApJ...845..139B}).
(3) After the emergence of the forward shock from the outermost layer of the ejecta, the energy of the ejecta is gradually redistributed and the ejecta approach the homologous expansion stage.
The density structure in this stage is well represented by a power-law function of the radial velocity with an exponent $-6$ \citep{2017MNRAS.466.2633S}.
\begin{figure*}
\begin{center}
\includegraphics[scale=0.8,bb=0 0 565 225]{./central_engine.pdf}
\caption{Schematic views of the dynamical evolution of supernova ejecta with a relativistic wind.
The three stages, (1) quasi-spherical, (2) hot bubble breakout, and (3) homologous expansion stages, are depicted from left to right.
The reverse shock, the contact discontinuity, and the forward shock are denoted by RS, CD, and FS in the density profiles. }
\label{fig:schematic}
\end{center}
\end{figure*}
\subsubsection{Homologous expansion of supernova ejecta}\label{sec:ejecta}
We focus on the evolution of the supernova ejecta after the power-law density structure is realized ($t>t_\mathrm{sd}$).
The velocity distribution is again represented by
\begin{equation}
v(t,r)=\left\{\begin{array}{ccl}
r/t&\mathrm{for}&r\leq v_\mathrm{max}t,\\
0&\mathrm{for}&v_\mathrm{max}t<r,
\end{array}\right.
\label{eq:velocity_profile}
\end{equation}
with a maximum velocity $v_\mathrm{max}$.
We assume that the density profile of the freely expanding ejecta is described by a power-law function of the four-velocity with an exponent $-n$,
\begin{equation}
\rho(t,r)=\left\{
\begin{array}{cl}
\rho_0\left(\frac{t}{t_\mathrm{i}}\right)^{-3}\left(\frac{\Gamma v}{\Gamma_\mathrm{max}v_\mathrm{max}}\right)^{-n}
&\mathrm{for}\ \ \ v_\mathrm{min}t\leq r\leq v_\mathrm{max}t,\\
0&\mathrm{otherwise},
\end{array}
\right.
\label{eq:density_profile}
\end{equation}
where the Lorentz factor is given by
\begin{equation}
\Gamma=\frac{1}{\sqrt{1-(v/c)^2}}.
\end{equation}
In this study, we use a fixed value for the maximum four-velocity, $\Gamma_\mathrm{max}v_\mathrm{max}=c$, following our previous simulation \citep{2017MNRAS.466.2633S}.
The outermost layer travelling at the maximum velocity has been transparent to optical photons at the time of creation.
Thus, optical emission would not be affected by the adopted maximum velocity.
Furthermore, the layer will soon be swept by the reverse shock, making the subsequent dynamical evolution and non-thermal emission from the shocked gas insensitive to the assumed value \citep{2017ApJ...834...32S}.
Our previous study showed that the angle-averaged density structure of the supernova ejecta is well represented by a power-law profile with an exponent $n=6$ (see, Section \ref{sec:powering_sn_ejecta}).
We use $n=6$ as our fiducial value and examine how different values affect the non-thermal emission.
The normalization constant $\rho_0$ and the minimum velocity $v_\mathrm{min}$ are determined for a given set of the ejecta mass and energy, $M_\mathrm{ej}$ and $E_\mathrm{ej}$, as follows,
\begin{equation}
M_\mathrm{ej}=4\pi \rho_0(ct_\mathrm{i})^3\int_{v_\mathrm{min}}^{v_\mathrm{max}}
\Gamma\left(\frac{\Gamma v}{\Gamma_\mathrm{max}v_\mathrm{max}}\right)^{-n}v^2dv,
\end{equation}
and
\begin{equation}
E_\mathrm{ej}=4\pi \rho_0c^5t_\mathrm{i}^3\int_{v_\mathrm{min}}^{v_\mathrm{max}}
\Gamma(\Gamma-1)\left(\frac{\Gamma v}{\Gamma_\mathrm{max}v_\mathrm{max}}\right)^{-n}v^2dv.
\end{equation}
We assume $M_\mathrm{ej}=10M_\odot$ and $E_\mathrm{ej}=10^{52}$ erg in order to imitate the freely expanding ejecta realized in our previous numerical simulation \citep{2017MNRAS.466.2633S}.
\subsection{Photospheric emission}
The supernova ejecta powered by the central energy injection give rise to bright thermal emission \citep{2010ApJ...717..245K}.
We consider thermal photons diffusing out from the ejecta after $t=t_\mathrm{i}$.
The photospheric radius $R_\mathrm{ph}$ at time $t$ can be calculated in the following way.
The optical depth for a ray radially extending from a given radius $r$ to the outermost radius of the ejecta is calculated by
\begin{equation}
\tau(r,t)=\int_r^{v_\mathrm{max}t}\kappa\rho(t,r') dr',
\label{eq:tau}
\end{equation}
where $\kappa$ is the opacity for thermal photons and set to be $\kappa=0.1$ cm$^2$ g$^{-1}$.
Here we have ignored the motion of the ejecta while the ray is travelling.
In addition, the outermost layers of the ejecta would be swept by the reverse shock and thus the density structure is modified.
We also ignore the modification of the density structure for simplicity.
The photospheric radius at $t$ is determined so that the optical depth is equal to unity, $\tau(R_\mathrm{ph},t)=1$.
We particularly denote the photospheric radius at $t=t_\mathrm{i}$ by $R_\mathrm{i}$.
We calculate the photospheric emission from the ejecta being powered by the continuous energy injection at the centre.
We basically use the Arnett's solution for photon diffusion in freely expanding spherical ejecta \citep{1980ApJ...237..541A,1982ApJ...253..785A}.
The bolometric luminosity of the photospheric emission from the ejecta with energy input $L_\mathrm{in}(t)$ is given by
\begin{eqnarray}
L_\mathrm{ph}(t)&=&\frac{2}{t_\mathrm{d}}e^{-t(t+2t_\mathrm{h})/t_\mathrm{d}^2}
\int^t_{t_\mathrm{i}}e^{t'(t+2t_\mathrm{h})/t_\mathrm{d}^2}L_\mathrm{in}(t')
\left(\frac{t_\mathrm{h}}{t_\mathrm{d}}+\frac{t'}{t_\mathrm{d}}\right)dt'
\nonumber
\\&&
+\frac{E_\mathrm{th,0}}{t_0}e^{-t(t+2t_\mathrm{h})/t_\mathrm{d}^2},
\end{eqnarray}
where the timescales $t_0$, $t_\mathrm{h}$, and $t_\mathrm{d}$ are given by
\begin{equation}
t_\mathrm{0}=\frac{\kappa M_\mathrm{ej}}{\beta cR_\mathrm{i}},
\end{equation}
\begin{equation}
t_\mathrm{h}=\frac{R_\mathrm{i}}{v(R_\mathrm{i})},
\end{equation}
and
\begin{equation}
t_\mathrm{d}=\sqrt{2t_0t_\mathrm{h}},
\end{equation}
\citep{2012ApJ...746..121C,2013ApJ...770..128I}.
Here $E_\mathrm{th,0}$ is the initial thermal energy and $v(R_\mathrm{i})$ is the radial velocity at the photosphere $r=R_\mathrm{i}$, both given at $t=t_\mathrm{i}$.
The initial thermal energy can be obtained from the dynamical model.
The thermal energy of the ejecta in the quasi-spherical stage almost linearly increases with time \citep{2017MNRAS.466.2633S}.
The value at the end of the increase is given by
\begin{equation}
E_\mathrm{th}=\frac{2-\gamma}{1+3\alpha(\gamma-1)}E_\mathrm{ej},
\label{eq:E_th}
\end{equation}
where $\gamma=4/3$ is the adiabatic index.
The non-dimensional constant $\beta$ depending on the density structure is set to be a commonly used value $\beta=13.8$ \citep{1980ApJ...237..541A,1982ApJ...253..785A}.
We assume the energy injection at the rate given by Equation (\ref{eq:L_sd}), where the spin down time $t_\mathrm{sd}$ is one of our input parameters, the effects of which are to be examined in this paper.
The energy input into the ejecta is given by
\begin{equation}
L_\mathrm{in}(t)=L_\mathrm{sd}(t)(1-e^{-\tau_\gamma}),
\end{equation}
where the last factor takes into account the leakage of the injected energy from the ejecta as gamma-rays \citep{2015ApJ...799..107W}.
We calculate the optical depth by the following integration,
\begin{equation}
\tau_\gamma=\kappa_\gamma\int^{v_\mathrm{max}t}_{v_\mathrm{min}t}\rho(t,r)dr,
\end{equation}
which evolves as $\tau_\gamma\propto t^{-2}$.
The gamma-ray opacity $\kappa_\gamma$ should depend on the frequency of gamma-rays and the effective value may possibly be dependent the three-dimensional density distribution of the supernova ejecta.
Therefore, the value is highly uncertain.
Theoretical calculations by \cite{2013MNRAS.432.3228K}, who assumed simplified spherical ejecta, suggest that the gamma-ray opacity could be of the order of $\sim 0.1$ cm$^2$ g$^{-1}$ for photons with $h\nu\simeq 100$ keV because of Compton scattering and $\sim0.01$ cm$^2$ g$^{-1}$ for photons with $h\nu>10$ MeV because of pair production.
We should note that the effective opacity may be lower than these values when we take into account patchy density structure.
Recently, several authors have incorporated the gamma-ray leakage effect into their light curve fitting models and tried to constrain the gamma-ray opacity.
\cite{2017ApJ...842...26L} fitted light curves of 19 SLSNe-I by their light curve model and analyzed the results by an MCMC approach.
They reported that the best-fit value of the gamma-ray opacity ranges from $\kappa_\gamma\simeq 0.01$ cm$^{2}$ g$^{-1}$ to $\kappa_\gamma\simeq 0.82$ cm$^{2}$ g$^{-1}$.
\cite{2017ApJ...850...55N} systematically studied multi-colour light curves of 38 SLSNe-I.
For example, their analysis on SN 2015bn inferred $\kappa_\gamma\simeq 0.01$ gm$^{2}$ g$^{-1}$.
Other SLSNe with well-covered late-time evolutions also showed small values, indicating significant gamma-ray leakage especially at later epochs.
Keeping in mind that the value should be treated with caution, we adopt a constant value of $\kappa_\gamma=0.01$ cm$^2$ g$^{-1}$.
Finally, we determine the temperature of the photospheric emission.
Determining the colour temperature of the photospheric emission requires sophisticated treatments of radiative transfer, the ionization states of different layers of the ejecta, and numerous line opacities contributing to the thermal balance of the ejecta.
For simplicity, we assume that the spectrum of the emission is well represented by a Planck function.
Thus, we estimate the effective temperature $T_\mathrm{eff}$ of the photospheric emission from the photospheric radius and the bolometric luminosity given above,
\begin{equation}
L_\mathrm{ph}=4\pi R_\mathrm{ph}^2\sigma_\mathrm{SB}T_\mathrm{eff}^4,
\end{equation}
where $\sigma_\mathrm{SB}$ is the Stefan-Boltzmann constant.
\subsection{Ejecta-CSM interaction}
\cite{2017ApJ...834...32S} considered the hydrodynamical interaction between spherical supernova ejecta travelling at mildly relativistic speeds and a steady wind with a mass-loss rate $\dot{M}$ and a wind velocity $v_\mathrm{w}$,
\begin{equation}
\rho_\mathrm{csm}=\frac{\dot{M}}{4\pi v_\mathrm{w} r^2}\equiv Ar^{-2}
\end{equation}
where $A$ is a free parameter specifying the CSM density.
We introduce the non-dimensional parameter $A_\star=A/(5\times 10^{11}\ \mathrm{g\ cm}^{-1})$.
In this normalization, $A_\star=1$ corresponds to a mass-loss rate of $\dot{M}= 10^{-5}\ M_\odot$ yr$^{-1}$ for a wind velocity of $10^3$ km s$^{-1}$.
In the following, we assume CSM density parameters up to $A_\star=10$, with which the CSM is still transparent for electron scattering.
Thus we can safely assume that the ejecta-CSM interaction does not give rise to optically thick thermal radiation significantly contributing to the optical brightness of SNe.
We use this semi-analytic model to describe the evolution of the forward and reverse shocks developed as a result of the collision of the ejecta with the CSM.
For a given set of the parameters, $M_\mathrm{ej}$, $E_\mathrm{ej}$, and $n$, the density and radial velocity profiles of the ejecta are specified.
Under the assumption that the ejecta start interacting with the surrounding gas at $t=t_\mathrm{i}$, the semi-analytic model is used to calculate the shock radius, the rate of the energy dissipation via shock, and the swept mass, as a function of time for both the forward and reverse shocks.
The rate of the energy dissipation at the shock front and the mass swept by the shock are used to specify the number and the average energy of non-thermal electrons injected into the shocked region as we describe in the next section.
\begin{figure}
\begin{center}
\includegraphics[scale=0.5,bb=0 0 432 576]{./dynamics.pdf}
\caption{Temporal evolutions of the shock radius (top panel), shock velocity (middle panel), and the energy dissipation rate (bottom panel) calculated by the one-zone model.
In each panel, the solid and dashed lines correspond to the quantities at the forward and reverse shock fronts, respectively.
The parameters specifying the ejecta and CSM models are assumed to be $M_\mathrm{ej}=10\ M_\odot$, $E_\mathrm{ej}=10^{52}$ erg, $n=6$, and $A_\star=1.0$. }
\label{fig:dynamics}
\end{center}
\end{figure}
Figure \ref{fig:dynamics} shows an example of the semi-analytic calculation with $t_\mathrm{sd}=10^6$ s.
The temporal evolution of the shock radius, the shock velocity, the post-shock pressure, and the internal energy dissipation rate are plotted for the forward and reverse shocks.
After the beginning of the calculation at $t=t_\mathrm{i}$($=10^6$ s), the forward and reverse shock radii steadily increase with time.
The difference in the forward and reverse shock radii is much smaller than the shock radius, indicating that the shocked region can be described as a geometrically thin shell.
The shock velocities decrease to $\sim 0.2c$ by the end of the calculation at $t=10^8$ s.
The post-shock pressure at the forward and reverse shock fronts are similar because of the pressure balance across the contact discontinuity separating the shocked ejecta and CSM.
\section{Non-thermal emission model}\label{sec:non_thermal}
In this section, we describe our non-thermal emission model.
We treat the thin shocked region under a one-box approximation and calculate the temporal evolution of the isotropic momentum distribution, $dN/dp_\mathrm{e}$, of electrons uniformly distributed in the shocked region.
Non-thermal electrons with a power-law momentum distribution are injected through the forward and reverse shocks and then they lose their energies via radiative and adiabatic cooling.
We consider synchrotron and inverse Compton emission as radiative cooling processes.
From the temporal evolution of the electron distribution, we calculate the spectrum of the non-thermal emission from the shocked gas.
\subsection{Electron injection at the shock front}
The shock dissipation at the forward and reverse shock fronts creates non-thermal electrons.
Our treatment of the electron injection is similar to studies on non-thermal emission from CCSNe in the literature (e.g., \citealt{1998ApJ...499..810C,2006ApJ...651..381C}; see \citealt{2016arXiv161207459C} for a recent review).
We introduce two free parameters, $\epsilon_\mathrm{e}$ and $\epsilon_\mathrm{B}$, representing the efficiencies of the non-thermal electron acceleration and magnetic field generation at the shock front.
These values should ideally be self-consistently determined by microscopic plasma processes responsible for the energy equipartition in the shock downstream.
However, how exactly electrons are accelerated and magnetic fields are amplified are still debated even with state-of-the-art numerical computations based on first principle approaches \citep[e.g.,][]{2008ApJ...682L...5S}.
Thus, we fix their values to be constant.
The energy injected into the shocked region per unit time is obtained by the semi-analytic model described in Section \ref{sec:dynamics}.
We denote the energy injection rates at the forward and reverse shocks by $\dot{E}_\mathrm{fs}$ and $\dot{E}_\mathrm{rs}$.
For a given set of the post-shock density $\rho$ and the internal energy density $u_\mathrm{int}$, the internal energy density $u_\mathrm{ele}$ and the number density $n_\mathrm{ele}$ of the injected non-thermal electrons are
\begin{equation}
u_\mathrm{ele}=\epsilon_\mathrm{e}u_\mathrm{int},
\end{equation}
and
\begin{equation}
n_\mathrm{ele}=\frac{Z\rho}{Am_\mathrm{u}},
\end{equation}
where $A$ and $Z$ are the mass and atomic numbers of ions predominantly composing the ejecta and $m_\mathrm{u}$ is the atomic mass unit.
We assume $Z/A=0.5$ in the following.
The average energy of the injected non-thermal electrons is obtained as follows,
\begin{equation}
\bar{\gamma}m_\mathrm{e}c^2=\frac{u_\mathrm{ele}}{n_\mathrm{ele}}=\frac{\epsilon_\mathrm{e}Am_\mathrm{u}u_\mathrm{int}}{Z\rho},
\label{eq:electron_gamma}
\end{equation}
where $m_\mathrm{e}$ is the electron mass.
We assume that electrons are spontaneously accelerated by the shock passage and then obey the following simple power-law momentum distribution,
\begin{equation}
\left(\frac{d\dot{N}}{dp_\mathrm{e}}\right)_\mathrm{in}=
\left\{\begin{array}{cl}
0&\mathrm{for}\ \ p_\mathrm{min}\leq p_\mathrm{e}\leq p_\mathrm{in},\\
K(p_\mathrm{e}/p_\mathrm{in})^{-p}&\mathrm{for}\ \ p_\mathrm{in}\leq p_\mathrm{e}\leq p_\mathrm{max},\\
\end{array}\right.
\end{equation}
where $p_\mathrm{e}$ is the electron momentum and $p_\mathrm{min}$ and $p_\mathrm{max}$ are the minimum and maximum values.
The minimum and maximum momenta are set to be $p_\mathrm{max}=10^{-3}m_\mathrm{e}c$ and $p_\mathrm{max}=10^6m_\mathrm{e}c$.
We assume a power-law index of $p=3$, which is commonly employed to account for radio observations of stripped-envelope CCSNe \citep[e.g.,][]{2006ApJ...651..381C}.
Therefore, electrons at the minimum injection energy $c(m_\mathrm{e}^2c^2+p_\mathrm{in}^2)^{1/2}$ carry a considerable fraction of the internal energy of electrons.
The normalization constant $K$ and the injection momentum $p_\mathrm{in}$ characterizing the momentum distribution of the injected electrons are determined by equating the mass and energy injection rates and the following two integrals,
\begin{equation}
\int_{p_\mathrm{in}}^{p_\mathrm{max}}\left(\frac{d\dot{N}}{dp_\mathrm{e}}\right)_\mathrm{in}dp_\mathrm{e}=
\frac{\dot{E}}{\bar{\gamma}m_\mathrm{e}c^2},
\end{equation}
and
\begin{equation}
\int_{p_\mathrm{in}}^{p_\mathrm{max}}
c(m_\mathrm{e}^2c^2+p_\mathrm{e}^2)^{1/2}
\left(\frac{d\dot{N}}{dp_\mathrm{e}}\right)_\mathrm{in}dp_\mathrm{e}=
\dot{E},
\end{equation}
with $\dot{E}=\dot{E}_\mathrm{fs}$ or $\dot{E}_\mathrm{rs}$.
\subsection{Electron momentum distribution}
The electrons injected at the shock front experience synchrotron, inverse Compton, and adiabatic cooling.
The governing equation describing the temporal evolution of the electron momentum distribution is written as follows,
\begin{equation}
\frac{\partial }{\partial t}\left(\frac{dN}{dp_\mathrm{e}}\right)
=\frac{\partial }{\partial p_\mathrm{e}}
\left[
\left(\dot{p}_\mathrm{syn}+\dot{p}_\mathrm{ic}+\dot{p}_\mathrm{ad}\right)
\frac{dN}{dp_\mathrm{e}}
\right]
+\left(\frac{d\dot{N}}{dp_\mathrm{e}}\right)_\mathrm{in},
\label{eq:momentum_equation}
\end{equation}
where $\dot{p}_\mathrm{syn}$, $\dot{p}_\mathrm{ic}$, and $\dot{p}_\mathrm{ad}$ are the momentum loss rates by the three cooling processes.
The synchrotron and inverse Compton momentum loss rate are related to the corresponding energy loss rates, $\dot{E}_\mathrm{syn}$ and $\dot{E}_\mathrm{ic}$, as follows,
\begin{equation}
\dot{p}_{\{\mathrm{syn,ic}\}}=\frac{\sqrt{m_\mathrm{e}^2c^2+p_\mathrm{e}^2}}{p_\mathrm{e}c}
\dot{E}_{\{\mathrm{syn,ic}\}}.
\end{equation}
The synchrotron and inverse Compton energy loss rates are given by
\begin{equation}
\dot{E}_\mathrm{syn}=\frac{4}{3}\sigma_\mathrm{T}cu_\mathrm{B}\beta_\mathrm{e}^2\gamma_\mathrm{e}^2,
\label{eq:Esyn}
\end{equation}
and
\begin{equation}
\dot{E}_\mathrm{ic}=\frac{4}{3}\sigma_\mathrm{T}cu_\mathrm{rad}\beta_\mathrm{e}^2\gamma_\mathrm{e}^2,
\label{eq:Eic}
\end{equation}
\citep[e.g.][]{1979rpa..book.....R} where $\sigma_\mathrm{T}$ is the Thomson cross section.
The energy loss rates are proportional to the energy densities, $u_\mathrm{B}$ and $u_\mathrm{rad}$, of the magnetic field and the seed photons, which are described later.
The electron velocity $\beta_\mathrm{e}$ and the Lorentz factor $\gamma_\mathrm{e}$ are expressed in terms of the corresponding electron momentum $p_\mathrm{e}$ as follows,
\begin{equation}
\beta_\mathrm{e}=\frac{p_\mathrm{e}}{\sqrt{m_\mathrm{e}^2c^2+p_\mathrm{e}^2}},
\end{equation}
and
\begin{equation}
\gamma_\mathrm{e}=\sqrt{1+p_\mathrm{e}^2/(m_\mathrm{e}^2c^2)}.
\end{equation}
As we will see below, the photospheric emission serves as a dominant seed photon source for inverse Compton cooling.
Thus, the seed photon temperature is of the order of $1$ eV.
On the other hand, the injection momentum of electrons is typically $\sim 30m_\mathrm{e}c$ (see Section \ref{sec:electron_spectrum}).
Therefore, the energy of most seed photons in the rest frame of non-thermal electrons is much smaller than the electron rest energy $m_\mathrm{e}c^2$, allowing us to neglect several processes reducing the efficiency of inverse Compton cooling, such as the Klein-Nishina suppression and the electron recoil effect.
The electrons in the shell can also cool according to the expansion of the shell.
The adiabatic momentum loss rate is given by
\begin{equation}
\dot{p}_\mathrm{ad}=\frac{p_\mathrm{e}}{3}\frac{\dot{V}}{V},
\end{equation}
where $V$ and $\dot{V}$ are the volume of the shell and its expansion rate.
The temporal evolutions of these quantities are also obtained from the semi-analytic model.
The governing equation (\ref{eq:momentum_equation}) is numerically solved by a simple upwind scheme with first-order implicit time integration.
In the following calculations, the distributions of non-thermal electrons accelerated at the forward and reverse shock fronts are separately treated.
\subsection{Synchrotron spectrum}
The shock dissipation generates random magnetic field via some magnetohydrodynamics and/or plasma collective effects.
In a similar way to the energy density of electrons, we use a parameter $\epsilon_\mathrm{B}$ describing the fraction of the magnetic field energy density to the dissipated shock energy.
Thus, using the downstream internal energy densities $u_\mathrm{fs}$ and $u_\mathrm{rs}$ for the forward and reverse shocks, the corresponding magnetic energy densities are $u_\mathrm{B,fs}=\epsilon_\mathrm{B}u_\mathrm{fs}$ and $u_\mathrm{B,rs}=\epsilon_\mathrm{B}u_\mathrm{rs}$.
These magnetic energy densities are used to evaluate the synchrotron energy loss rate, Equation (\ref{eq:Esyn}).
The magnetic field strengths are given by
\begin{equation}
B_\mathrm{fs}=(8\pi u_\mathrm{B,fs})^{1/2}=(8\pi \epsilon_\mathrm{B}u_\mathrm{fs})^{1/2},
\end{equation}
and
\begin{equation}
B_\mathrm{rs}=(8\pi u_\mathrm{B,rs})^{1/2}=(8\pi \epsilon_\mathrm{B}u_\mathrm{rs})^{1/2}.
\end{equation}
For a given electron momentum distribution and a magnetic field strength, the synchrotron emissivity per unit frequency is calculated by the following formula,
\begin{equation}
j_{\nu,\mathrm{syn}}=\frac{1}{4\pi V}\int P_{\nu,\mathrm{syn}}(\gamma_\mathrm{e})\frac{dN}{dp_\mathrm{e}}\mathrm{d}p_\mathrm{e}.
\end{equation}
The synchrotron power per unit frequency $P_{\nu,\mathrm{syn}}(\gamma_\mathrm{e})$ as a function of electron Lorentz factor $\gamma_\mathrm{e}$ and frequency $\nu$ is described in Appendix \ref{sec:synchrotron}.
At low frequencies, synchrotron emission suffers from absorption by its inverse process.
The synchrotron self-absorption coefficient is given by
\begin{equation}
\alpha_{\nu,\mathrm{syn}}=\frac{c^2}{8\pi V\nu^2}
\int\frac{\partial}{\partial p_\mathrm{e}}
\left[
p_\mathrm{e}\gamma_\mathrm{e}P_{\nu,\mathrm{syn}}(\gamma_\mathrm{e})
\right]
\frac{1}{p_\mathrm{e}^2}\frac{dN_\mathrm{e}}{dp_\mathrm{e}}dp_\mathrm{e}.
\end{equation}
Using these quantities and assuming that the emitting region is geometrically thin, the synchrotron intensity is obtained as follows,
\begin{equation}
I_\mathrm{syn}(\nu)=\frac{j_{\nu,\mathrm{syn}}}{\alpha_{\nu,\mathrm{syn}}}(1-e^{-\tau_{\nu,\mathrm{syn}}}),
\end{equation}
\citep[e.g.][]{1979rpa..book.....R}, where $\tau_{\nu,\mathrm{syn}}$ is the corresponding optical depth.
\subsection{Inverse Compton spectrum}
We consider the photospheric emission from the ejecta as the dominant source of seed photons for inverse Compton emission.
The radiation energy density corresponding to the photospheric luminosity $L_\mathrm{ph}$ is
\begin{equation}
u_\mathrm{rad}=\frac{L_\mathrm{ph}}{4\pi cR_\mathrm{sh}^2},
\end{equation}
at the shell $r=R_\mathrm{sh}$, which is used to evaluate the inverse Compton energy loss rate, Equation (\ref{eq:Eic}).
We assume that the photospheric emission is well represented by a blackbody spectrum with the colour temperature identical with $T_\mathrm{eff}$.
Therefore, the photon spectrum is given by
\begin{equation}
\begin{aligned}
I_\mathrm{sn}(\nu)=&\frac{2u_\mathrm{rad}}{c^2a_\mathrm{r}T_\mathrm{eff}^4}
\frac{h\nu^3}{e^{h\nu/k_\mathrm{B}T_\mathrm{eff}}-1}
\\=&
\frac{L_\mathrm{sn}}{2\pi c^3a_\mathrm{r}T_\mathrm{eff}^4R_\mathrm{sh}^2}
\frac{h\nu^3}{e^{h\nu/k_\mathrm{B}T_\mathrm{eff}}-1},
\end{aligned}
\end{equation}
where $a_\mathrm{r}$ and $k_\mathrm{B}$ are the radiation constant and the Boltzmann constant.
We also consider synchrotron emission as the other source of seed photons.
We obtain the total intensity of seed photons by adding those of the photospheric emission and the synchrotron emission, $I_\mathrm{seed}(\nu)=I_\mathrm{sn}(\nu)+I_\mathrm{syn}(\nu)$.
The spectrum $I_\mathrm{ic}(\nu)$ of the inverse Compton emission is calculated by convolving the seed photon spectrum, the electron distribution, and the redistribution function of Compton scattering $\Delta G$ as follows,
\begin{equation}
I_\mathrm{ic}(\nu)=\int _{p_\mathrm{min}}^{p_\mathrm{max}}
\Delta G
I_\mathrm{seed}(\nu')\frac{dN}{dp_\mathrm{e}}d\nu' dp_\mathrm{e},
\end{equation}
where the function $\Delta G$ is described in Section \ref{sec:compton}.
\section{Results}\label{sec:results}
In this section, we show light curves and spectra calculated by the method described above.
In all the calculations below, the microphysics parameters are assumed to be $p=3$, $\epsilon_\mathrm{e}=0.1$, and $\epsilon_\mathrm{B}=0.02$.
We note that the electron spectral index $p=3$ is widely used for stripped-envelope CCSNe \citep[e.g.,][]{2006ApJ...651..381C}.
The parameter $\epsilon_\mathrm{e}$ of the order of $0.1$ is also used in radio light curve modellings of highly energetic SNe including relativistic SNe \citep[e.g.,][]{2010Natur.463..513S,2015MNRAS.448..417B,2015ApJ...805..164N}, while the values of $\epsilon_\mathrm{B}$ show a variety depending on the radio brightness \citep[e.g.,][for GRB afterglows]{2014ApJ...785...29S}.
We have determined the value of $\epsilon_\mathrm{B}$ so that the radio light curves of highly energetic SNe are reproduced by our fiducial model (see below).
The ejecta mass and energy are also fixed to be $M_\mathrm{ej}=10M_\odot$ and $E_\mathrm{ej}=10^{52}$ erg, while we examine how light curves at different frequencies depend on the other parameters, $t_\mathrm{sd}$, $n$, and $A_\star$.
Hereafter, the model with $t_\mathrm{sd}=10^{6}$ s, $n=6$, and $A_\star=1.0$ is called the fiducial model.
\subsection{Photospheric emission}\label{sec:photospheric_emission}
First, we show theoretical light curves for photospheric emission from the ejecta powered by the central energy injection.
In particular, we focus on the dependence of the light curves on the assumed spin down time.
\begin{figure}
\begin{center}
\includegraphics[scale=0.55,bb=0 0 432 576]{./Lsn.pdf}
\caption{Temporal evolutions of the photospheric luminosity (upper panel), the effective temperature (middle panel), and the photospheric radius (lower panel).
Models with different spin down times are plotted in each panel.
The solid, dashed, dash-dotted and dotted lines represent models with $t_\mathrm{sd}=10^{6}$, $10^{5}$, $10^{4}$, and $10^{3}$ s. }
\label{fig:Lsn}
\end{center}
\end{figure}
\begin{figure*}
\begin{center}
\includegraphics[scale=0.55,bb=0 0 850 566]{./espec_multi.pdf}
\caption{Electron momentum distributions multiplied by $p_\mathrm{e}^p$ ($p_\mathrm{e}^pdN/dp_\mathrm{e}$) at $t-t_\mathrm{i}=10$ (top left), $20$ (top right), $50$ (bottom left), and $100$ (bottom right) days.
In each panel, the solid and dashed curves correspond to the contributions from the forward and reverse shocks.
}
\label{fig:espec}
\end{center}
\end{figure*}
Figure \ref{fig:Lsn} shows the temporal evolutions of the photospheric luminosity, the effective temperature at the photosphere, and the photospheric radius.
The duration and the peak luminosity of the bolometric light curve become shorter and less luminous for shorter spin down times.
Therefore models with shorter $t_\mathrm{sd}$ result in small radiated energies.
In these models, the energy injection is terminated at early stages of the evolution of the ejecta.
The timescale of the energy injection is only a small fraction of the timescale at which the ejecta becomes transparent to thermal photons.
Therefore, the injected energy suffers from significant adiabatic loss before escaping into the interstellar space as radiation, leading to low radiative efficiencies.
In other words, the ratio of the spin down time $t_\mathrm{sd}$ to the diffusion time $t_\mathrm{d}$ plays a critical role in determining bolometric light curves of central engine-powered SNe \citep{2010ApJ...717..245K,2015MNRAS.454.3311M,2015MNRAS.452.3869N,2017ApJ...850...55N}.
The model with $t_\mathrm{sd}=10^{6}$ s can reproduce the timescale and the peak bolometric luminosity of SLSNe-I.
On the other hand, the model with $t_\mathrm{sd}=10^{3}$ s exhibit a shorter timescale and a lower peak luminosity than SLSNe-I.
Such models may be relevant to SNe associated with GRBs.
The over-luminous SN 2011kl associated with the ultra-long GRB 111209A exhibited a fast evolving light curve with a timescale of $10$--$20$ days and a peak luminosity of $\sim 3\times 10^{43}$ erg s$^{-1}$.
The timescale is similar to that of the model with $t_\mathrm{sd}=10^{3}$ s, while the peak luminosity is smaller by a factor of $\sim 6$.
The discrepancy could be resolved by adjusting the free parameters or considering jet-like energy injection rather than a quasi-spherical wind.
In our model, the photospheric radius at time $t$ is solely determined by the density structure of the ejecta.
Thus, the temporal evolutions of $R_\mathrm{ph}$ for different models are exactly same as shown in the bottom panel of Figure \ref{fig:Lsn}.
The entire ejecta become transparent at $t\simeq 190$ days, after which both the photospheric radius and the effective temperature cannot be well defined and thus we stop calculations.
The temporal evolution of the photospheric radius and the spectral evolution have been obtained for the nearby slowly evolving SLSN-I 2015bn \citep{2016ApJ...826...39N,2016ApJ...828L..18N,2017ApJ...835...13J}.
The measured photospheric radius of SLSNe-I 2015bn rose until $\sim 50$ days after the optical maximum and then started declining after reaching the maximum value of $R_\mathrm{ph}\simeq 10^{16}$ cm.
The decline of the bolometric light curve becomes steeper around $200$--$300$ days after the optical maximum, which is accompanied by a gradual transition from a continuum-dominated spectrum to a nebular one.
This epoch of the transition is roughly consistent with the time at which the ejecta used in our model become transparent at $t\simeq 190 $ days.
However, the temporal behavior of the photospheric radius should be calculated by a more sophisticated treatment of radiative transfer and the internal structure of the ejecta.
These values would be sensitive to the opacity for thermal photons and possibly to three-dimensional ejecta structure.
In reality, the ejecta structure would be patchy due to the ``shredding'' by gas flows at high Lorentz factors penetrating the entire ejecta \citep{2003ApJ...589..871A,2011MNRAS.411.2054L,2017MNRAS.466.2633S}.
Therefore, when taking the multi-dimensional effect into account, optical depths corresponding to radial rays with different directions could differ from each other.
This indicates that more thorough investigations including effects of three-dimensional ejecta structure and sophisticated radiative transfer in the ejecta are required.
\subsection{Electron momentum distribution}\label{sec:electron_spectrum}
Figure \ref{fig:espec} shows the electron momentum distributions at $t-t_\mathrm{i}=10$, $20$, $50$, and $100$ days.
The plotted distributions are multiplied by $p_\mathrm{e}^p$ ($p_\mathrm{e}^pdN/dp_\mathrm{e}$) so that the spectrum of the injected electrons appears to be flat.
In this model, electrons accelerated at the forward shock front are more abundant and have higher average energy than those at the reverse shock, which reflects the large energy dissipation rate at the forward shock (see Figure \ref{fig:dynamics}).
These electrons behind the forward shock thus predominantly contribute to the non-thermal emission.
The distributions are divided into two segments separated by a peak.
The peak momentum corresponds to the minimum injected momentum.
As Figure \ref{fig:dynamics} shows, the shock velocity is $v_\mathrm{sh}\simeq 0.3$--$0.4$c at several 10 days.
The kinetic energy density of the flow is roughly given by $\rho v_\mathrm{sh}^2$, and a considerable fraction of this energy is supposed to dissipate at the shock front, $u_\mathrm{int}\sim \rho v_\mathrm{sh}^2$.
Therefore, the average Lorentz factor of electrons is roughly estimated from Equation (\ref{eq:electron_gamma}),
\begin{equation}
\hat{\gamma}=\epsilon_\mathrm{e}\frac{Am_\mathrm{u}v_\mathrm{sh}^2}{Zm_\mathrm{e}c^2}\simeq 33
\left(\frac{\epsilon_\mathrm{e}}{0.1}\right)
\left(\frac{v_\mathrm{sh}/c}{0.3}\right)^2
\left(\frac{Z/A}{0.5}\right)^{-1},
\end{equation}
for the forward shock, which agrees with the peak momenta shown in Figure \ref{fig:espec}.
The lower peak momenta for the reverse shock are due to small shock velocities relative to the unshocked ejecta velocities.
At higher energies, the distributions are well represented by a power-law function with an index $-(p+1)$, $dN/dp_\mathrm{p}\propto p_\mathrm{e}^{-(p+1)}$.
This indicates that the fast cooling regime \citep{1998ApJ...497L..17S,2001ApJ...548..787S} is realized at earlier epochs.
This is because thermal photons abundantly produced by the photospheric emission can efficiently cool non-thermal electrons via inverse Compton scattering.
At later epochs, e.g., the distribution at $t-t_\mathrm{i}=100$ days, a relatively flat distribution at $p_\mathrm{e}/(m_\mathrm{e}c)=20$--$100$ indicates that injected electrons with lower energies remain uncooled because of the declining photospheric luminosity.
At lower energies than the peak, on the other hand, the electron momentum distribution shows a hard spectrum, which is composed of electrons having lost most of their energies.
\subsection{Radio light curve}\label{sec:radio_lc}
\begin{figure}
\begin{center}
\includegraphics[scale=0.50,bb=0 0 453 680]{./radio.pdf}
\caption{Radio light curves of the non-thermal emission model for $1.4$ (top), $4.8$ (middle), and $8.5$ (bottom) Hz.
The solid lines show the radio light curve calculated by the fiducial model with $t_\mathrm{sd}=10^{6}$ s, $n=6$ and $A_\star=1$.
The microscopic free parameters are set to be $p=3.0$, $\epsilon_\mathrm{e}=0.1$, and $\epsilon_\mathrm{B}=0.02$.
For comparison, light curves of radio-loud SNe Ic-BL, 1998bw (blue square) and 2009bb (red circle) at the corresponding frequency, are also plotted.
The star marks with arrows represent upper limits obtained by radio observations for two SLSNe-I.
The green star in the bottom panel represents the upper limit for SLSN-I 2015bn at $7.4$ GHz, the orange stars in the middle panel represent the upper limits for Gaia16apd at $6.6$ GHz, and the magenta stars in the top and bottom panels are those for SLSN-I 2017egm at $1.5$ and $10$ GHz. }
\label{fig:radio}
\end{center}
\end{figure}
Figure \ref{fig:radio} shows the radio light curves at different frequencies, $1.4$, $4.8$, and $8.5$ GHz, calculated by our fiducial model with $t_\mathrm{sd}=10^{6}$ s, $n=6$, and $A_\star=1$.
Because of the hot bubble breakout and the subsequent acceleration of the outermost layers of the ejecta, central engine powered SNe give rise to bright radio emission especially at early epochs.
The radio light curves of our fiducial model suggest that the radio luminosity exhibits a peak around $\sim 5$--$10$ days for $\nu\simeq 5$--$10$ GHz, while the peak at $\nu\simeq 1$ GHz appears around $50$--$100$ days.
These features are worth comparing with radio-loud SNe.
In Figure \ref{fig:radio}, we plot the radio light curves of SNe 1998bw and 2009bb for comparison.
SN 1998bw was a widely known SN Ic-BL associated with GRB 980425 \citep{1998Natur.395..663K,1998Natur.395..670G}.
SN 2009bb was the SN Ic-BL whose properties were remarkably similar to GRB-associated SNe, but lacking any signature of gamma-ray emission \citep{2010Natur.463..513S}.
As shown in Figure \ref{fig:radio}, their radio luminosities were similar to each other.
For SN 1998bw, the peak of the light curve was successfully observed at $1.4$, $4.8$ and $8.5$ GHz thanks to early observations triggered by the gamma-ray detection.
The peak was earlier at higher frequencies as is the case for radio emission from normal CCSNe interacting with their CSM \citep[e.g.][]{2016arXiv161207459C}.
For SN 2009bb, the peaks at higher frequencies, $\nu=4.8$ and $8.5$ Hz, were probably missed, while the peak at $1.4$ GHz was successfully observed.
The decline rates of the luminosities per unit frequency after the peak are similar for both events.
For SN 2009bb, the presence of an ultra-relativistic jet is unlikely because of the absence of emission indicating off-axis jet.
The radio emission is explained by trans-relativistic supernova ejecta (\citealt{2010Natur.463..513S}, see also \citealt{2015ApJ...805..164N}).
We also plot the upper limits obtained by radio observations of SN 2015bn \citep{2016ApJ...826...39N}, Gaia16apd \citep{2018ApJ...856...56C}, and 2017egm \citep{2018ApJ...853...57B} in Figure \ref{fig:radio}.
We should note that the frequency bands for SN 2015bn ($7.4$ GHz), Gaia16apd ($6.6$ GHz), and 2017egm ($1.5$ and $10$ GHz) are slightly different from the theoretical light curve and SNe 1998bw and 2009bb ($1.5$ and $8.5$ GHz).
However, the spectral energy distributions of the synchrotron emission (see Figure \ref{fig:sed}) suggest that the radio luminosities at the corresponding frequency bands are similar to the theoretical light curve shown in Figure \ref{fig:radio} within a factor of a few.
\begin{figure*}
\begin{center}
\includegraphics[scale=0.38,bb=0 0 1359 680]{./radio2.pdf}
\caption{Dependence of the radio light curves on the spin down time $t_\mathrm{sd}$ (left column), the power-law exponent $n$ of the density profile $n$ (middle column), and the CSM density $A_\star$ (right column).
In the left panels, we compare the models with $t_\mathrm{sd}=10^{6}$(solid), $10^{5}$(dashed), $10^{4}$(dotted), and $10^{3}$(dash-dotted) s.
The models with $n=5$ (dash-dotted) , $6$ (solid), $7$ (dashed), and $8$ (dotted) are shown in the middle panels.
The right panels represent the models with $A_\star=10.0$ (dash-dotted), $1.0$ (solid), $0.1$ (dashed), and $0.01$ (dotted).
The other free parameters are set to the same one as the fiducial model in Figure \ref{fig:radio}. }
\label{fig:radio2}
\end{center}
\end{figure*}
In Figure \ref{fig:radio2}, we show how the radio light curves depend on the free parameters, the spin down time $t_\mathrm{sd}$, the power-law exponent $n$ of the density profile, and the CSM density $A_\star$.
We first focus on the effect of the spin down time.
As is seen in the left column of Figure \ref{fig:radio2}, the models with longer $t_\mathrm{sd}$ exhibit bright radio emission in early epochs but are less luminous at later epochs than those with shorter $t_\mathrm{sd}$.
The power-law exponent more significantly affects the radio light curve than the spin down time, since it determines how much fraction of the kinetic energy is distributed in the outermost layers interacting with the CSM.
For shallower density slopes (smaller $n$), more energy is available in the outermost layer to produce non-thermal electrons, giving rise to brighter synchrotron emission.
This trend of brighter radio luminosities for shallower density slopes is seen in the middle column of Figure \ref{fig:radio2}.
The increase in the CSM density makes the emission brighter because a dense CSM can efficiently dissipate the kinetic energy of the ejecta.
The radio light curves of our fiducial model in Figure \ref{fig:radio} show good agreement with SNe 1998bw and 2009bb.
The radio luminosities at $8.5$, $4.8$, $1.4$ GHz show their peaks around $t\simeq 5$, $7$, and $50$ days.
After the peak, the model luminosity steadily declines at a rate similar to those observed for SNe 1998bw and 2009bb.
The radio non-detection of SN 2015bn is consistent with most of the models.
Since the upper limit of the radio luminosity is smaller than the peak luminosity of the fiducial model, earlier radio observations might have detected radio emission from the SLSN.
On the other hand, the fiducial model disagrees with the radio observations of SN 2017egm at $\sim30$ days after the detection.
Thanks to the proximity of the SLSN, the tightest constraint for radio emission of SLSNe-I is obtained.
Most of theoretical radio light curves exceed the upper limit for a range of the parameters.
In order to explain the radio emission from SN 2017egm, either a steep density gradient $(n\la 8)$ or a small CSM density $A_\star\la 0.01$ is required.
The upper limit for Gaia 16apd at 26 days after the detection also places a meaningful constraint on the radio luminosity.
In similar ways to SN 2017egm, steep density gradients and/or small CSM densities are required to reconcile the disagreement.
We will further discuss theoretical interpretations of the current radio constraints in Section \ref{sec:conclusion}.
\subsection{X-ray light curve}
\begin{figure*}
\begin{center}
\includegraphics[scale=0.50,bb=0 0 850 850]{./xray.pdf}
\caption{X-ray light curves of models with $t_\mathrm{sd}=10^{6}$ (top left), $10^{5}$ (top right), $10^{4}$ (bottom left), and $10^{3}$ (bottom right) s.
Theoretical $\nu L_\nu$ light curves are plotted for $0.3$, $1.0$, and $10$ keV and $0.3$--$10$ X-ray light curves of some SNe are compared.
Green stars represent the (possible) X-ray detection of SLSNe SCP 06F6 and PTF12dam.
We also plot SNe Ic-BL 1998bw (blue circles) and 2009bb (red square) for comparison.
SLSNe upper limits are plotted as gray triangles.
Plotted data adopted from \protect\cite{2017arXiv170405865M} for SLSNe, \protect\cite{2000ApJ...536..778P} and \protect\cite{2004ApJ...608..872K} for SN 1998bw, and \protect\cite{2010Natur.463..513S} for SN 2009bb.
}
\label{fig:xray}
\end{center}
\end{figure*}
Figure \ref{fig:xray} shows the X-ray light curves of the inverse Compton emission.
The curves in the panels of Figure \ref{fig:xray} present the temporal evolution of $\nu L_\nu$ at $h\nu=0.3$, $1.0$, and $10$ keV for models with $t_\mathrm{sd}=10^{6}$, $10^{5}$, $10^{4}$, and $10^{3}$ s.
The other parameters are fixed as $n=6$ and $A_\star=1.0$.
The luminosity of the inverse Compton emission from the ejecta can reach $\nu L_\nu\simeq 10^{41-42}$ erg s$^{-1}$, which is brighter than most of normal CCSNe.
The luminous X-ray emission is owing to the presence of the mildly relativistic ejecta and the optical photons abundantly provided by the photospheric emission.
We plot the currently available upper limits for SLSNe-I (including SN 2015bn) and $0.3$--$10$keV X-ray luminosities of SCP 06F6 and PTF12dam (we used the data compiled by \cite{2017arXiv170405865M}).
The luminous X-ray emission associated with SCP 06F6 have generated a lot of discussion on its origin \citep{2009ApJ...697L.129G,2013ApJ...771..136L,2014MNRAS.437..703M}.
However, the X-ray luminosity is well above most of the upper limits obtained for SLSNe-I so far (Figure \ref{fig:xray}), leading to the consensus that such luminous X-ray emission is not common among SLSNe-I.
\cite{2017arXiv170405865M} found an X-ray source at the location of PTF12dam.
However, as they mention in their paper, the X-ray source can also be explained by X-ray emission associated with the star-forming activity in the host galaxy with a relatively high star formation rate $\sim 5M_\odot$ yr$^{-1}$.
Therefore, further observations are required to see whether the X-ray emission is certainly associated with PTF12dam or not.
In Figure \ref{fig:xray}, we also plot the X-ray light curves of the SNe Ic-BL 1998bw \citep{2000ApJ...536..778P,2004ApJ...608..872K} and 2009bb \citep{2010Natur.463..513S} for comparison.
We first focus on our fiducial model in the top left panel of Figure \ref{fig:xray}.
The theoretical light curve is below most of the upper limits placed for the other SLSNe, suggesting that deeper observations are needed to further constrain the central engine scenario for SLSNe-I.
The X-ray luminosity of PTF12dam agrees with the theoretical value, but it may have to be treated as an upper limit because of the reason described above.
The theoretical light curves exhibit their peaks around 10-30 days.
Since the luminosity of inverse Compton emission is proportional to the product of the seed photon energy density and the energy of non-thermal electrons, the light curve is determined by the convolution of the bolometric light curve of the photospheric emission and the steadily declining energy dissipation rate at the shock front (see Figure \ref{fig:dynamics}).
Thus, the peak in the X-ray light curve slightly precedes the optical maximum.
The theoretical light curves of the other models exhibit similar X-ray luminosities but earlier peaks for shorter spin down times.
This is because the optical maximum shifts earlier for shorter $t_\mathrm{sd}$ as we have described in Section \ref{sec:photospheric_emission}.
For models with shorter $t_\mathrm{sd}$, the light curve exhibits a plateau rather than a peak and then the luminosity declines.
This feature is similar to the X-ray light curve of SN 1998bw, although the observed light curve shows a longer flat part.
Although only a single data point is available, SN 2009bb also show similar X-ray luminosity, which agrees with the declining theoretical light curve at $\sim$30 days.
One caveat on this comparison is that the corresponding theoretical bolometric luminosities of the photospheric emission (Figure \ref{fig:Lsn}) are brighter than those of SN 1998bw and SN 2009bb.
This discrepancy indicates that we should explore appropriate parameters satisfying both optical and X-ray observational constraints and/or an improved treatment of the photospheric emission with multi-colour radiation transfer and other sources of seed photons would be required.
We leave such improvements to future work.
As in the case of radio light curves, increasing the CSM density $A_\star$ makes the X-ray emission more luminous.
Since the theoretical X-ray light curve of the fiducial model with $A_\star=1.0$ in the upper left panel of Figure \ref{fig:xray} roughly matches the X-ray flux of PTF12dam, the CSM density much larger than this value would predict too bright X-ray emission.
This can place an upper limit on the CSM density by treating the X-ray flux as an upper limit.
The adopted value $A_\star=1.0$ corresponds to a steady wind at a mass-loss rate of $\dot{M}=10^{-5}\ M_\odot$ yr$^{-1}$ for a wind velocity $10^3$ km s$^{-1}$.
Therefore, mass-loss rates much larger than this value is unlikely.
\cite{2017arXiv170405865M} have already constrained the CSM density by using the X-ray upper limit and reached a similar conclusion, $\dot{M}<2\times 10^{-5}\ M_\odot$ yr$^{-1}$.
\subsection{Broad-band spectral energy distribution}
\begin{figure*}
\begin{center}
\includegraphics[scale=0.55,bb=0 0 850 566]{./sed_multi.pdf}
\caption{Spectral energy distribution at $t-t_\mathrm{i}=10$ (top left), $20$ (top right), $50$ (bottom left), and $100$ (bottom right) days.
In each panel, the red, blue, and green curves, whose peaks are at around $10^9$--$10^{10}$, $10^{15}$, and $10^{18}$ Hz, represent the synchrotron, photospheric, and inverse Compton components.
The solid and dashed curves correspond to the contributions from the reverse and forward shocks, while the gray thick curve is the sum of all the contributions.
}
\label{fig:sed}
\end{center}
\end{figure*}
Finally, we present spectral energy distributions at several epochs in Figure \ref{fig:sed}.
The spectral energy distributions at different epochs look similar to each other, and qualitatively similar to those of normal CCSNe interacting with their CSM.
Each distribution is composed of the synchrotron, photospheric, and inverse Compton components, whose peaks are located around $10^9$--$10^{10}$, $10^{15}$, and $10^{18}$ Hz.
The peak in the radio energy range divides the synchrotron component into optically thick (lower frequencies) and thin (higher frequencies) regimes.
The peak shifts toward lower frequencies with time because the non-thermal electrons in the shocked region gradually become transparent to radio waves with lower frequencies.
This temporal shift of the radio peak frequency creates the peak in the radio light curve described in Section \ref{sec:radio_lc}
The flux of the optically thick synchrotron emission follows a power-law function of $\nu$, $L_\nu\propto \nu^{5/2}$ \citep[e.g.][]{1979rpa..book.....R}.
On the other hand, the spectral slope of the optically thin synchrotron emission depends on the power-law exponent $p$ of the electron energy spectrum.
As we have described in Section \ref{sec:electron_spectrum}, the fast cooling regime can apply, $L_\nu\propto \nu^{-p/2}=\nu^{-1.5}$.
Since the inverse Compton spectrum is the convolution of the energy spectra of non-thermal electrons and seed photons, the peak frequency of the inverse Compton component is determined by the peak in the electron momentum distribution and the effective temperature of the photospheric emission.
As we have described in Section \ref{sec:electron_spectrum}, the injection energy of non-thermal electrons is $\hat{\gamma}\simeq 30$--$40$.
The inverse Compton scattering by electrons with this Lorentz factor increases the photon energy by a factor $\hat{\gamma}^2$,
\begin{equation}
\nu_\mathrm{ic}\simeq \frac{3\hat{\gamma}^2k_\mathrm{B}T_\mathrm{eff}}{h}=10^{18}\mathrm{Hz}
\left(\frac{\hat{\gamma}}{40}\right)^2
\left(\frac{T_\mathrm{eff}}{10^4\mathrm{K}}\right),
\end{equation}
which explains the X-ray peak in the spectral energy distribution.
The spectrum at frequencies higher than the X-ray peak frequency is well represented by a power-law distribution, whose exponent depends on the electron energy spectrum.
The spectral index is same as the optically thin synchrotron emission, $L_\nu\propto \nu^{-p/2}=\nu^{-1.5}$.
\section{Discussion and conclusions}\label{sec:conclusion}
In this study, we have calculated broad-band emission from supernova ejecta powered by a central engine based on the picture revealed by the recent two-dimensional special relativistic hydrodynamic simulation \citep{2017MNRAS.466.2633S}.
In the hydrodynamic simulation, the outermost layers of the ejecta are efficiently accelerated owing to a hot gas emerging from the central region of the ejecta and thus the maximum velocity of the ejecta can be mildly relativistic.
While the ejecta emit thermal photons by radiative diffusion, the outermost layers colliding with a CSM create the forward and reverse shocks propagating in the CSM and ejecta, respectively.
We model the photospheric emission from the ejecta by using the Arnett-type one-zone model for photon diffusion throughout the ejecta and at the same time we determine the photospheric radius and the effective temperature from the ejecta model.
Furthermore, we calculated non-thermal emission from the shocked gas by using the semi-analytic model for the propagation of the forward and reverse shocks \citep{2017ApJ...834...32S}.
We found that non-thermal electrons produced in the shocked region can give rise to bright radio and X-ray emission via synchrotron and inverse Compton processes.
When we adopt commonly assumed values for microphysics parameters, $\epsilon_\mathrm{e}$, $\epsilon_\mathrm{B}$, and $p$, and the CSM density corresponding to a steady mass-loss rate of $\dot{M}=10^{-5}\ M_\odot$ yr$^{-1}$ and a constant wind velocity of $10^3$ km s$^{-1}$, the theoretical radio light curves of central engine powered SNe well agree with those of radio-loud SNe Ic-BL, such as SNe 1998bw and 2009bb, consistent with the idea that they also harbour a central engine.
\subsection{Radio and X-ray emission from SLSNe-I as a probe of a central engine}
Our results suggest that SLSNe-I can also give rise to radio synchrotron emission with similar fluxes to radio-loud SNe Ic-BL.
This could also be explained naturally if the SLSNe-I and SNe Ic-BL would be certainly linked to each other as suggested by similarities in spectra of SNe Ic-BL and SLSNe-I \citep{2010ApJ...724L..16P,2017ApJ...835...13J,2016ApJ...828L..18N,2017ApJ...845...85L}.
As shown in Figure \ref{fig:radio2}, the radio brightness of the central-engine powered supernova ejecta highly depends on the density profile of the ejecta and the CSM density.
Our previous hydrodynamics simulation \citep{2017MNRAS.466.2633S} suggests that if an SN harbours a sufficiently energetic central engine to produce the hot bubble breakout, the impact of the blowout would create an ejecta component travelling at relativistic speeds.
This situation corresponds to models with shallow density gradient ($n=5$ and $6$), which produce bright radio emission.
Therefore, radio observations of SLSNe-I may probe whether the SLSN-I have experienced the hot bubble breakout.
The radio upper limit for SN 2015bn \citep{2016ApJ...826...39N} is well above the theoretical light curves of our fiducial model (Figure \ref{fig:radio}).
If radio emission of SN 2015bn was as bright as SN 1998bw and 2009bb, it could have been detected by observations conducted earlier (at several 10 days).
On the other hand, for the recently discovered SLSN-I 2017egm and Gaia16apd, tighter upper limits are available.
These upper limits rule out most of the models with shallow density gradients and dense CSM densities.
X-ray emission can also assess the presence of relativistic ejecta.
The predicted X-ray fluxes are much lower than most of the currently available upper limits for SLSNe-I.
The theoretical models cannot explain the X-ray luminosity of SCP 06F6, indicating a different origin for the unusually bright X-ray emission.
Our fiducial model agrees with the X-ray luminosity of PTF12dam, although it requires further observations to confirm the association of the X-ray source with PTF12dam.
Furthermore, models with short spin down times well explain the X-ray emission from the SNe Ic-BL, 1998bw and 2009bb.
Although X-ray observations of SLSNe-I are currently not so constraining as radio observations, future X-ray observations can also be used as a powerful tool for investigating the outermost ejecta structure of SLSNe-I and other extraordinary SNe.
We consider two possibilities to interpret the radio non-detections.
First, some SLSNe may not experience the hot bubble breakout because only a small amount of additional energy is injected from the central engine.
Although the kinetic energy of the supernova explosion preceding the central energy injection is not known, if we assume a typical kinetic energy of $10^{51}$ erg, the additional energy required to produce the hot bubble breakout ranges from a few $10^{51}$ to $10^{52}$ erg, depending on the density structure of the supernova ejecta \citep{2017MNRAS.466.2633S,2017ApJ...845..139B}.
The light curve fitting of multi-colour optical data of SN 2017egm \citep{2017ApJ...845L...8N} infers a relatively small kinetic energy, $1$--$2\times 10^{51}$ erg, for this nearby event.
The kinetic energy of Gaia16apd is estimated to be $3.69^{+1.38}_{-0.59}\times 10^{51}$ erg \citep{2017ApJ...850...55N}.
The estimated kinetic energy of the ejecta also depend on the density structure of the freely expanding ejecta assumed in the light curve fitting model.
Although there are uncertainties in the critical energy for the hot bubble breakout and the kinetic energy of the supernova ejecta, the injected energy may be smaller than the critical energy and thus relativistic ejecta may not be produced.
\cite{2017ApJ...850...55N} also performed light curve fitting to other SNSNe-I.
According to their results, the ejecta mass and the kinetic energy of SLSNe-I are distributed in the range of $2.2$--$12.9\ M_\odot$ and $(1.9$-$9.8)\times10^{51}$ erg, respectively.
Thus, the ejecta mass and energy assumed in our model are close to the upper end of the distributions.
On the other hand, the spin down time $t_\mathrm{sd}=10^6$ s employed in our fiducial model is typical among the SLSNe.
This may suggest the possibility that not all SLSNe-I experience the hot bubble breakout with bright radio emission.
The other possibility is that the density slope of the outermost layer of SLSNe ejecta is not so shallow as predicted by the hydrodynamic simulation by \cite{2017MNRAS.466.2633S}.
Although the hydrodynamic simulation suggests the presence of the mildly relativistic ejecta, it is still unclear whether the relativistic component is realized when correctly taking into account coupling between gas and radiation.
In the hydrodynamics simulation without radiative transfer, gas and radiation are assumed to be strongly coupled, which enables the efficient acceleration of the outermost layers by radiation pressure.
However, in reality, gas and radiation may be coupled only weakly in the outermost layers.
In other words, radiation in the outermost layer may simply escape into the surrounding space rather than accelerating gas in the layer, leading to smaller kinetic energy and maximum velocity of the ejecta in the homologous expansion stage.
This may be especially true for SLSNe-I, because they require spin down timescales comparable to the diffusion time of thermal photons in the ejecta.
\subsection{SN explosions with a central engine}
From the results of our broad-band light curve modelling combined with the dynamical evolution of SNe with central energy sources revealed by \cite{2017MNRAS.466.2633S}, we can speculate the following scenario for SNe with central energy sources.
The most important factor is the total amount of the injected energy.
For an injected energy exceeding a critical value depending on the original ejecta structure, the ejecta are significantly affected by the energy injection.
Even when the additional energy is deposited as thermal energy, a quasi-spherical relativistic wind would soon be created around the centre and start pushing the ejecta.
The hot bubble breakout and the associated energy redistribution throughout the ejecta potentially produce mildly relativistic ejecta with a shallow density gradient.
The presence of relativistic ejecta depends on the timescale of the energy injection compared with the diffusion timescale of the ejecta.
For central energy injection with much shorter duration than the diffusion timescale, the injected energy would predominantly be converted to the kinetic energy of the ejecta via adiabatic expansion rather than escaping as thermal photons.
This case likely produces supernova ejecta with a relatively large kinetic energy, which may be observed as SNe Ic-BL.
On the other hand, for energy injection timescales comparable to or longer than the diffusion timescale, the injected energy can easily escape into interstellar space as thermal photons, giving rise to bright thermal emission.
This may correspond to SLSNe-I.
In terms of their radio and X-ray properties, SLSNe-I produced in such a way can be divided into two classes.
One is the population harbouring sufficiently energetic central engine to produce the hot bubble breakout and thus they are radio-loud.
The other is the population whose central engine can give rise to bright optical emission but is not accompanied by relativistic ejecta.
\subsection{Other remarks}
Finally, we mention the following two remarks on the broad-band light curve modelling.
We should note that the expected radio and X-ray luminosities highly depend on the CSM density and the density profile of the supernova ejecta.
The circumstellar environments of SLSNe-I are poorly known.
They may explode in relatively clean environments, making non-thermal emission weak.
The radio and X-ray light curve modelling of SLSNe-I significantly suffer from these uncertainties.
Another potential caveat is that the non-thermal emission could also arise from the wind nebula of the nascent neutron star.
Recent theoretical modellings of non-thermal emission from the wind nebula embedded in spherical supernova ejecta suggest that the non-thermal emission start leaking the dense supernova ejecta after $\sim 100$ days \citep{2013MNRAS.432.3228K,2014MNRAS.437..703M,2015ApJ...805...82M,2016ApJ...818...94K,2018MNRAS.474..573O}.
Although how early X-ray and radio emission starts leaking depends on the multi-dimensional density structure of the supernova ejecta, non-thermal emission from the wind nebula would basically be preceded by that from the shock interaction at the ejecta-CSM interface.
Therefore, radio and X-ray detections at early epochs likely indicate non-thermal emission from the blast wave driven by the supernova ejecta.
|
{
"timestamp": "2018-04-17T02:11:31",
"yymm": "1804",
"arxiv_id": "1804.05397",
"language": "en",
"url": "https://arxiv.org/abs/1804.05397"
}
|
\section{Introduction}
\label{section:introduction}
In \cite{Tarnauceanu2018}, T{\u{a}}rn{\u{a}uceanu introduces a nilpotency criterion for finite groups by studying element orders. This criterion involves the following generalization of the Euler’s totient function \cite{Tarnauceanu2015},
\[
\varphi(G)=|\{g\in G|o(g)=\exp(G)\}|,
\]
where $G$ is a finite group. Obviously, if $G$ is nilpotent, then $\varphi(G)$ is necessarily different from zero. Moreover, subgroups and quotients of a nilpotent group are also nilpotent, so $\varphi(K)\neq 0$ for every section $K$ of the nilpotent group $G$. T\u arn\u auceanu proves that the converse holds, i.e., if $\varphi(K)\neq 0$ for all sections $K$ of $G$, then $G$ is nilpotent.
In this note, we give the following related criterion for $p$-nilpotency of finite groups, in which we consider instead $p'$-reduced sections.
\begin{thm}\label{thm:main}
Let $G$ be a finite group and let $p$ be a prime. Then the following are equivalent:
\begin{enumerate}
\item[(1)] $G$ is $p$-nilpotent.
\item[(2)] $\varphi(K)\neq 0$ for each $p'$-reduced section $K$ of $G$.
\item[(3)] $\varphi(K)\neq 0$ for each $p'$-reduced section $K$ of $N_G(Q)$, where $Q$ is any $p$-centric $p$-subgroup of $G$.
\item[(4)] $\varphi(K)\neq 0$ for each $p'$-reduced section $K$ of $N_G(Q)$, where $Q$ is any $p$-centric and $p$-radical $p$-subgroup of $G$.
\end{enumerate}
\end{thm}
Recall that $Q$ is $p$-centric if $Z(P)\in\operatorname{Syl}\nolimits_p(C_G(P))$ and that $Q$ is $p$-radical if $N_G(P)/P$ is $p$-reduced. Our proof is formulated in the modern language of fusion systems \cite{BLO2}, following the trend of the works \cite{BESW}, \cite{BGH}, \cite{CSV}, \cite{D}, \cite{DGPS}, \cite{DEV}, \cite{GRV}, \cite{KLN} and \cite{LZ}. Our proof of Theorem \ref{thm:main}, carried out in the following section, is independent of the classification of minimal non-nilpotent groups. Therefore, Theorem \ref{thm:main} provides an alternative proof of the main result in \cite{Tarnauceanu2018}.
\begin{cor}\label{cor:main}
Let $G$ be a finite group. Then the following are equivalent:
\begin{enumerate}
\item[(1)] $G$ is nilpotent.
\item[(2)] $\varphi(K)\neq 0$ for each section $K$ of $G$.
\end{enumerate}
\end{cor}
Regarding Theorem \ref{thm:main}$(2)$, one could wonder whether $\varphi(K)\neq 0$ for the particular $p'$-reduced section $K=G/O_{p'}(G)$ is enough for the $p$-nilpotency of $G$. As the next example shows, this is not the case.
\begin{exa}
We exhibit a finite group $G$ such that $O_{p'}(G)=1$, $\varphi(G)\neq 0$ and $G$ is not $p$-nilpotent.
We start considering a group $K$ such that $K$ is $p'$-reduced and $\varphi(K)=0$. Such group can be obtained as a semidirect product via Lemma \ref{lemma:1} below. Then $K$ is not $p$-nilpotent by Theorem \ref{thm:main}$(2)$. If $\exp(K)=p_1^{e_1}\cdots p_s^{e_s}$ with $p_i$'s different primes, there exist, for each $i$, an element $k_i\in K$ with $o(k_i)=p_i^{e_i}$. Define $G$ as the direct product of $s$ copies of $K$. Then clearly $\exp(G)=\exp(K)=o((k_1,\ldots,k_s))$ and hence $\varphi(G)\neq 0$. Moreover, $O_{p'}(G)=1$ and $G$ is not $p$-nilpotent as $K$ is neither.
\end{exa}
\begin{notation} We denote by $O_{p'}(G)$ the largest normal subgroup of $G$ of order prime to $p$, and by $O_p(G)$ the largest normal $p$-subgroup of $G$. We say that $G$ is $p'$-reduced if $O_{p'}(G)=1$ and that $G$ is $p$-reduced if $O_p(G)=1$. For terminology and results on fusion systems, we refer the reader to \cite{AKO} and \cite{Craven2011}.
\end{notation}
\section{Proof of the theorem}
\label{section:proof}
We need two preliminary results.
\begin{lem}\label{lemma:1}
Let $p$ and $q$ be different primes and let $V$ be a non-trivial simple $\BF_p[\BZ/q]$-module. Then $\varphi(V\rtimes \BZ/q)=0$.
\end{lem}
\begin{proof}
Note that $\exp(V\rtimes \BZ/q)=pq$. Assume that there exists $x\in V\rtimes \BZ/q$ satisfying $o(x)=pq$. Then $x^q\in V$ and $x^p$ centralizes $x^q$. As $x^p$ projects onto a generator of $\BZ/q$, $\BF_p\cong\langle x^q\rangle\leq V$ is a trivial $\BF_p[\BZ/q]$-submodule, which is not possible by hypothesis.
\end{proof}
\begin{lem}\label{lemma:2}
Consider a short exact sequence of groups,
\[
\xymatrix{
P\ar[r]& G \ar[r]^\pi & K,
}
\]
where $1\neq P$ is a $p$-group, $C_G(P)\leq P$ and $1\neq K$ is $p$-reduced. Then there exists a $p'$-reduced section $H$ of $G$ with $\varphi(H)=0$.
\end{lem}
\begin{proof}
As $K$ is non-trivial and $p$-reduced, there exists a prime $1<q\neq p$ such that $q$ divides $|K|$. Hence, we can choose $g\in G$ such that $o(\pi(g))=q$. Moreover, raising $g$ to an appropriate power of $p$, we can assume that $o(g)=q$.
The automorphism induced by $g$ on $P$, $c_g\in \mathrm{Aut}(P)$, cannot be trivial because $C_G(P)\leq P$ and $g\notin P$. Hence, $c_g$ must have order $q$. By Burnside's theorem on coprime actions \cite[Theorem 5.1.4]{Gorenstein}, $c_g$ induces a non-trivial order $q$ automorphism on the Frattini quotient $W=P/\Phi(P)\cong (\BF_p)^r$ ($r\geq 1$). Hence, $W$ is a non-trivial $\BF_p[\BZ/q]$-module.
By Maschke's theorem \cite[Theorem 3.3.1]{Gorenstein}, we can decompose $W$ as a direct sum of simple $\BF_p[\BZ/q]$-submodules. As $W$ is a non-trivial $\BF_p[\BZ/q]$-module, one of these simple submodules is non-trivial. Call it $V$. Then $H=V\rtimes \BZ/q$ is a $p'$-reduced section of $G$ and $\varphi(H)=0$ by Lemma \ref{lemma:1}.
\end{proof}
Now we can prove the main theorem.
\begin{proof}[Proof of Theorem \ref{thm:main}]
To prove $(1)\Rightarrow(2)$, let $G$ be a $p$-nilpotent group and recall that $p$-nilpotency is preserved under subgroups and quotients. Hence, any section $K$ of $G$ is $p$-nilpotent and may be written as $K=O_{p'}(K)S$, with $S\in \operatorname{Syl}\nolimits_p(K)$. If $K$ is $p'$-reduced, i.e., $O_{p'}(K)=1$, we must have $K=S$ and hence $\varphi(K)\neq 0$.
The remaining non-trivial implication is $(4)\Rightarrow (1)$. We use Frobenius's normal $p$-complement theorem \cite[Theorem 7.4.5]{Gorenstein} in its version for fusion systems \cite[Theorem 1.12]{Craven2011}. Thus, $G$ is $p$-nilpotent if and only if $\CF_S(G)=\CF_S(S)$, where $S\in\operatorname{Syl}\nolimits_p(G)$ and $\CF=\CF_S(G)$ is the fusion system of $G$ over $S$. In turn, by Alperin's fusion theorem for fusion systems \cite[Theorem A.10]{BLO2}, to show that $\CF=\CF_S(S)$, it is enough to show that $S$ is the only $\CF$-centric and $\CF$-radical subgroup of $S$, and that $\mathrm{Out}_\CF(S)=1$.
So let $Q\leq S$ be $\CF$-centric and $\CF$-radical. Then $C_G(P)=Z(P)\times C'_G(P)$ with $C'_G(P)=O_{p'}(C_G(P))$ \cite[Lemma A.4]{BLO1}, $O_p(\mathrm{Out}_G(Q))=1$, and $Q$ is $p$-centric and $p$-radical. Consider the short exact sequence,
\[
\xymatrix{
Q\ar[r]& N_G(Q)/C'_G(Q) \ar[r] & \mathrm{Out}_G(Q)=N_G(Q)/QC_G(Q).
}
\]
It satisfies the hypothesis of Lemma \ref{lemma:2} unless $\mathrm{Out}_G(Q)=1$. In the former case, we get a contradiction with hypothesis $(4)$. So $\mathrm{Out}_G(Q)=1$. As $\mathrm{Out}_S(Q)\leq \mathrm{Out}_G(Q)$, this cannot be the case for $Q<S$. So $Q=S$ and $\mathrm{Out}_\CF(S)=\mathrm{Out}_G(S)=1$.
\end{proof}
\begin{proof}[Proof of Corollary \ref{cor:main}]
By the comments in the introduction, it is left to prove that $(2)$ implies $(1)$. Recall that $G$ is nilpotent if and only it is $p$-nilpotent for every prime $p$. Finally, since every $p'$-reduced section of $G$ is obviously a section of $G$, then $(2)$ implies Theorem \ref{thm:main}$(2)$ and therefore $G$ is $p$-nilpotent for every prime $p$.
\end{proof}
|
{
"timestamp": "2018-04-17T02:14:11",
"yymm": "1804",
"arxiv_id": "1804.05530",
"language": "en",
"url": "https://arxiv.org/abs/1804.05530"
}
|
\section{Introduction}
\label{sec:Int}
Metal polycrystals are examples of heterogeneous materials in which heterogeneity results mainly from different orientations of local anisotropy axes in each grain. When the size of grains is sufficiently large the continuum mechanics framework is applicable and usually the boundaries between the grains are treated as perfect interfaces that do not require special treatment.
In order to estimate effective properties of a coarse-grained polycrystalline material a micromechanical methodology is used. Besides the classical Voigt and Reuss estimates, the self-consistent model due to Kroner \cite{Kroner58} and Hill \cite{Hill65} is most often employed. This methodology was successfully extended to the (visco)plastic crystals and large strain regime \cite{Molinari87,Lebensohn93}. Verification of such micromechanical mean-field estimates is usually performed by means of full-field simulations employing either the finite element method (FEM) \cite{Kamaya20092642,Fan20105008} or the fast Fourier transform (FFT) technique \cite{Prakash09,Liu10,Lebensohn12}.
Nanocrystalline metals are polycrystalline materials with a grain size smaller than 100 nm \cite{Gleiter00,Gao13}. In such materials two phases can be distinguished: grain cores and grain boundaries. The effect of the grain boundaries on the overall properties of a bulk material increases with decreasing grain size \cite{Sanders97,Gao13}. At the macroscopic level a nanograined material is still described in the framework of continuum theories. Since there is a need for quick and simple estimates of effective properties, which use only basic knowledge of the material behaviour: single grain properties, orientation distribution (texture) within the material volume and the averaged grain size, micromechanical mean-field estimates are also utilized in this case. Most of the mean-field schemes proposed for nanocrystalline materials describe them as composite media made of two or more phases. One phase is the grain core and the other phase (or phases) represents the behaviour of the grain boundary. In the early mixture-based model \cite{Carsley95} a nanophase metal was described as a mixture of a bulk intergranular region and a grain boundary built of amorphous metal. The overall strength of the material was obtained as a simple volume average of the strengths of the two phases. In \cite{Kim99} a refined mixture model was developed in which the intercrystalline phase was composed of three sub-phases: grain boundary, triple line junctions and quadratic nodes. Additionally pores were considered. All phases were modelled as isotropic and in order to obtain overall elastic moduli the implicit self-consistent relation was used. The concept of grain boundary subdivision into sub-phases was followed in \cite{Benson01,Qing06}, although using simple volume averages to obtain the overall Young's modulus and strength, while in the approach presented by \cite{Zhou07} the overall properties were assessed by using subsequently iso-strain and iso-stress assumptions for finding the average response of the grain core and the two grain boundary sub-phases. In \cite{Sharma03} a nano-grained material was described as a crystalline matrix with embedded ellipsoidal flat disks representing the grain boundary. To estimate the overall strength that accounts for grain boundary sliding the disks' stiffness moduli were anisotropic with one modulus vanishing, and the Mori-Tanaka (MT) scale-transition scheme was applied.
In \cite{Jiang04} the idea of the generalized self-consistent (GSC) model was used to find the strength and stiffness of a nanograined polycrystal: the coating of the spherical grain was assumed to be made of an isotropic material representing the grain boundaries. The grain cores were assumed to be elastically isotropic, while retaining their plastic anisotropy. This idea was followed by \cite{Capolungo07} and \cite{Ramtani09}. In the former model the implicit formulae of the GSC scheme \cite{Christensen79} were replaced by the explicit relations of the Mori-Tanaka model under the assumption that the grains are embedded in the matrix of the grain boundary phase. {Similarly, in \cite{Mercier07} a two-phase elastic-viscoplastic model combined with the Taylor-Lin homogenization theory was used to find the overall yield stress of nanocrystalline copper. The core-shell modelling framework was also used in the case of nanowires \cite{Chen06}.} The main difficulty of all these models is related to a valid description of the grain boundary phase(s) and quantification of its volume fraction.
Differently than in the case of coarse-grained materials, in the case of nano-grained polycrystals the use of full-field simulations based on continuum theory for the purpose of verification or calibration of the above-discussed models is questionable. Therefore atomistic simulations based on molecular dynamics are performed to assess the elastic moduli and strength of such materials \cite{Schiotz99,Chang03,Choi12,Gao13,Mortazavi2014,Fang16}. An alternative approach is to use refined constitutive models, for example those incorporating scale effects through gradient enhancements, within the framework of FEM calculations \cite{Kim20123942}.
In atomistic simulations of elastic properties, usually only the tensile Young's modulus is determined, see e.g. \cite{Schiotz99,Gao13}. It is observed that this modulus decreases with a decreasing grain size. This tendency is qualitatively and quantitatively reproduced by two- or multi-phase micromechanical models after proper adjustment of the grain boundary stiffness and the volume fraction, e.g. \cite{Jiang04,Ramtani09,Gao13}.
{The goal of the present paper is to estimate the effective elastic properties of nanocrystalline copper taking the anisotropy of a single crystallite fully into account. To this end, a series of static molecular simulations are performed to find all 21 components of the stiffness tensor and an anisotropic mean-field core-shell model of a nanograined polycrystal is formulated within the continuum mechanics framework. The results of both approaches are compared.}
The paper is constructed as follows. The next section describes the details of the performed atomistic simulations. Section \ref{sec:Cont}
presents the proposed formulation of the two-phase model of a nano-grain polycrystal. Additionally, the isotropisation procedure is outlined for the anisotropic stiffness acquired in the simulations. Section~\ref{sec:Res} discusses the results of the atomistic simulations and compares their outcomes with the respective mean-field estimates. The paper is closed with conclusions.
\section{Computational methods}
\label{sec:Cm}
All molecular simulations were performed using the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) \cite{Plimpton1995} and a scientific visualization and analysis software for atomistic simulation data OVITO was used to visualize and analyse simulation results \cite{Ovito2010}. Due to interest in the static behaviour of the material, the molecular statics (MS) approach at the temperature of 0K \cite{Tadmor2011,Maz2011,Maz2010} was used, which is more appropriate in such cases than molecular dynamics (MD).
Copper is metal arranged in a face-centered cubic (FCC) structure. The potentials based on the Embedded Atom Model (EAM) have been successful in the description of many metallic systems \cite{Tadmor2011}.
For analysed FCC copper, the well known EAM potential parametrized by \cite{Mishin2001} and taken from NIST Interatomic Potentials Repository \cite{Becker2013277} was utilized. This potential reproduces the copper monocrystal FCC lattice constant $a_{FCC}$=3.615\,\AA, the cohesive energy $E_{c}$=-3.54 eV, and the elastic constants in crystallographic axes coinciding with Cartesian coordinate system axes, $C_{1111}$=169.88\,GPa, $C_{1122}$=122.60\,GPa and $C_{2323}$=76.19\,GPa.
Polycrystal structures were generated using the Voronoi tessellation method by the Atomsk program which is designed for creating, converting and manipulating atomic systems \cite{Hirel2015212}. Data for the positions of seeds, orientations and size distribution of grains are attached in the \textit{Supplementary material} (polycrystals.txt and polycrystals.dat files).
Comprehensive study of the topological properties of the Voronoi tessellations of cubic systems can be found in \cite{Lucarini2009}. In physics of condensed matter, the Voronoi cell of the lattice point of a crystal is well known as the Wigner-Seitz cell, while the Voronoi cell of the reciprocal lattice point is called the Brillouin zone.
All the Voronoi tessellation cells are convex polyhedra. For the simple cubic (SC) crystal system these polyhedra are cubes (having 12 edges, 6 faces and 8 trivalent vertices) with a side length $V_{cell}^{-1/3}$ and total surface area, $A = 6\,V_{cell}^{-2/3}$, for the body-centered cubic (BCC) crystal system are truncated octahedra (having 36
edges, 14 faces and 24 trivalent vertices) of side length $\approx 0.445\,V_{cell}^{-1/3}$ and total, $A \approx 5.315\, V_{cell}^{-2/3}$ and for the face-centered cubic (FCC) crystal system these polygons are rhombic dodecahedra (having 24 edges, 12 faces, 6 trivalent vertices and 8 tetravalent vertices) of side length $\approx 0.687\,V_{cell}^{-1/3}$ and total, $A \approx 5.345\, V_{cell}^{-2/3}$. The standard isoperimetric quotient $IQ= 36\pi V_{cell}^{2}/A^3$,
non-dimensional measure of the surface-to-volume ratio of a solid ($IQ$ = 1 for a sphere), is 0.524, 0.753, and 0.74 for the SC, BCC and FCC system, respectively. Thus, $IQ$ decreases in value with increasing shape irregularity.
To examine the effect of the number and size of grains on mechanical properties of a polycrystalline material, eleven distinct copper samples were considered in this work: monocrystal, polycrystals of different sizes from 25$^3$ to 100$^3$ copper monocrystal unit cells with fixed number of 2x4$^3$ grains in BCC arrangement, a polycrystals with fixed size of 50$^3$ copper monocrystal unit cells with 2x2$^3$, 2x3$^3$, 2x4$^3$ and 2x5$^3$ grains in BCC arrangement, a polycrystal with fixed size of 50$^3$ copper monocrystal unit cells with 4x3$^3$ grains in FCC arrangement, a polycrystal of the same size and 5$^3$ grains in random arrangement and additionally polycrystal with a fixed size of 100$^3$ copper monocrystal unit cells with 2x2$^3$ grains in BCC arrangement, see Tab.\ref{tab:Samples} and Fig.\ref{fig:Samples}.
Structures created in such a geometrical way usually are not as dense as those observed in real materials.
In order to make the initial configurations more physical and to move atoms at grain boundaries to their energetically favourable positions the generated structures were prerelaxed.
Energy minimisation with a periodic boundary condition applied to all facets of the sample
was carried out with the non-linear conjugate gradient algorithm. Convergence was achieved when the
relative change in energy and forces, between two consecutive iterations were less than $10^{-8}$. After energy minimization process an external pressure equal to \textit{zero} was applied to the simulation box. This procedure allows the simulation box shape and volume to vary during the iterations of the minimizer so that the final configuration should be both an energy minimum for the potential energy of the atoms, as well as the system pressure tensor should be close to the prescribed external tensor \cite{Plimpton1995}.
The elastic constants, of all relaxed structures, $\bar{C}_{ijkl}$ were computed with the strain-stress approach with the maximum strain amplitude set to be 10$^{-4}$ \cite{Plimpton1995,Mazdziarz2015}.
There exist many computational analysis approaches to classify local atomic arrangements in computer-based large-scale atomistic simulations of crystalline solids that can be used to characterize when the atom is part of a perfect lattice, a local defect (e.g. a stacking fault or dislocation), or is positioned on a surface. The most common techniques include popular neighbor analysis (CNA), centrosymmetry parameter analysis (CPA), coordination analysis (CA), energy filtering, etc., see \cite{Stukowski2012}. We have found that in our case the cohesive energy/atom quantity and {CNA} are very useful techniques to analyse polycrystals.
\section{Continuum mechanics estimates of elastic moduli}
\label{sec:Cont}
Overall elastic properties of a coarse-grained polycrystalline material are usually found employing the micromechanics methodology set in a continuum mechanics framework.
According to the micromechanical theories polycrystals are one-phase heterogeneous materials in which heterogeneity roots from the different orientation of crystallographic axes
from grain-to-grain in the representative volume of polycrystalline continuum. Effective properties are assessed assuming that local elastic properties are known and then performing a micro-macro transition. The standard estimates for one-phase polycrystalline materials
{are collected in \ref{Ap:A}}.
None of these estimates are sensitive to the grain size.
\begin{figure}[h!]
\centering
\includegraphics[width=.7\textwidth]{core-shell-scheme-2-eps-converted-to.pdf}
\caption{Schematic of the core-shell model of the nanograin polycrystal.}
\label{fig:CoreShell}
\end{figure}
As discussed in Sec.\ref{sec:Int}, in order to account for the size effect, which is present in nano-crystalline aggregates, a family of models using the concept of a two- or multi-phase polycrystal were proposed. Two such models are described below that are called \emph{the MT and SC core-shell models}, respectively. The presented formulation follows the idea that has been proposed by \cite{Jiang04} and \cite{Capolungo07}. According to this approach an additional phase that represents a transient zone between the grains is introduced. This phase forms a coating of a specified thickness. An impact of this transient zone on the effective properties is more significant for smaller grains.
In opposition to the mentioned studies \cite{Jiang04,Capolungo07}, in order to estimate the overall stiffness tensor the elastic anisotropy of the grain core is thoroughly taken into account. The local constitutive relation between the stress tensor
$\boldsymbol{\sigma}$ and strain tensor
$\boldsymbol{\varepsilon}$ in the grain core is
\begin{equation}\label{locconst}
\boldsymbol{\sigma}=\mathbb{C}(\phi_c)\cdot\boldsymbol{\varepsilon},\quad
\boldsymbol{\varepsilon}=\mathbb{S}(\phi_c)\cdot\boldsymbol{\sigma},\quad\mathbb{S}=\mathbb{C}^{-1}\,,
\end{equation}
where $\mathbb{C}(\phi_c)$ and $\mathbb{S}(\phi_c)$ are the fourth order anisotropic elastic stiffness and compliance
tensors, respectively, and $\phi^c$ denotes orientation of local axes
$\{\mathbf{a}_k\}$ with respect to some macroscopic frame
$\{\mathbf{e}_k\}$ specified by e.g. three Euler angles. On the other hand the grain shell is isotropic with the stiffness tensor equal to the lower zeroth-order bound $\mathbb{C}_0$ (see \ref{Ap:A}). The latter specification is based on the results of atomistic simulations reported in Sec.\ref{ssec:ResAS} \comm{the next section}. Macroscopic relations for the averaged
fields $\mathbf{E}=\langle \boldsymbol{\varepsilon}\rangle $ and
$\boldsymbol{\Sigma}=\langle \boldsymbol{\sigma}\rangle $ in the polycrystal are then as follows
\begin{equation}
\boldsymbol{\Sigma}=\bar{\mathbb{C}}\cdot\mathbf{E},\quad
\mathbf{E}=\bar{\mathbb{S}}\cdot\boldsymbol{\Sigma},\quad
\bar{\mathbb{S}}=\bar{\mathbb{C}}^{-1}\,,
\end{equation}
where averaging is performed over the representative material volume so that $\langle . \rangle =\frac{1}{V}\int_V(.) dV $ and $\bar{\mathbb{C}}$ ($\bar{\mathbb{S}}$) is the effective stiffness (compliance) tensor of the polycrystal to be found by the mean-field model.
The effective stiffness $\bar{\mathbb{C}}$ is estimated by assuming that the coated grain is an inhomogeneity embedded in the infinite matrix of the stiffness $\mathbb{C}_m$ and using the methodology of the double-inclusion model \cite{HoriNematNasser93}. As a result it is obtained
\begin{equation}\label{eq:core-shell}
\bar{\mathbb{C}}_{\rm{CS}}=\left[f_0\mathbb{C}_0\mathbb{A}+(1-f_0)\left<\mathbb{C}(\phi^c)\mathbb{A}(\phi^c)\right>_{\mathcal{O}}\right]\left[f_0\mathbb{A}_0+(1-f_0)\left<\mathbb{A}(\phi^c)\right>_{\mathcal{O}}\right]^{-1}
\end{equation}
where
\begin{equation}
\mathbb{A}(\phi^c)=(\mathbb{C}(\phi^c)+\mathbb{C}_*(\mathbb{C}_{\rm{m}}))^{-1}(\mathbb{C}_{\rm{m}}+\mathbb{C}_*(\mathbb{C}_{\rm{m}}))\,,
\end{equation}
\begin{equation}
\mathbb{A}_0=(\mathbb{C}_0+\mathbb{C}_*(\mathbb{C}_{\rm{m}}))^{-1}(\mathbb{C}_{\rm{m}}+\mathbb{C}_*(\mathbb{C}_{\rm{m}}))\,,
\end{equation}
and $f_0$ is the volume fraction of the transient zone, $\mathbb{C}_*(\mathbb{C}_{\rm{m}})$ is the Hill tensor \cite{Hill65} depending on the stiffness $\mathbb{C}_{\rm{m}}$ of matrix
material and the shape of the coated grain (here assumed to be spherical) and $\langle . \rangle_{\mathcal{O}}$ denotes the averaging over grain crystallographic orientations. Two variants of the core-shell model are considered. For the first variant $\mathbb{C}_{\rm{m}}=\mathbb{C}_0$ is assumed (the infinite matrix has shell properties), while for the second variant $\mathbb{C}_{\rm{m}}=\bar{\mathbb{C}}_{\rm{CS}}$ (the infinite matrix has the effective properties to be found and formula (\ref{eq:core-shell}) is implicit). Due their correspondence to the Mori-Tanaka (MT) method and the self-consistent (SC) model, widely used for the two-phase composites, these two variants are referred as MT and SC core-shell models, respectively. Note that for a coarse-grained polycrystal ($f_0\rightarrow 0$) the effective properties $\bar{\mathbb{C}}_{\rm{CS/MT}}$ and $\bar{\mathbb{C}}_{\rm{CS/SC}}$ approach the Hashin-Shtrikman lower bound $\bar{\mathbb{C}}_{\rm{HS}}$ (Eq. \ref{HSbound}) and the self-consistent estimate $\bar{\mathbb{C}}_{\rm{SC}}$ (Eq. \ref{sc1}) for a one-phase polycrystal, respectively. The latter result is usually recommended in the literature \cite{Walpole81}. For very small grains ($f_0\rightarrow 1$) the estimates approach each other and coincide with $\mathbb{C}_0$. Consequently, for the same $f_0$ the stiffness moduli predicted by the MT variant of the core-shell model will be always lower then those predicted by the SC variant and the gap between estimates will increase with a grain size.
Two cases are analysed. The first one for which the finite set of $N_g$ grain orientations is known, and the second one for which an infinite set of orientations of random uniform distribution in the orientation space \cite{Bunge} is assumed. The latter case results in exactly isotropic overall properties of polycrystalline material. It should be highlighted that in both cases it is assumed that the volume fraction of grains with a specified orientation is the same and that the grains are equiaxed so that the shape of a grain core and its coating can be approximated as spherical.
For two analysed cases of orientation distribution, i.e. discrete and infinite, the averaging operation $\langle . \rangle_{\mathcal{O}}$ present in Eq. (\ref{eq:core-shell}) reduces to \cite{Bunge}:
\begin{equation}\label{Eq:orient}
\langle . \rangle_{\mathcal{O}}=\frac{1}{N_g}\sum_{i=1}^{N_g}(.)_i\quad\textrm{or}\quad
\langle . \rangle_{\mathcal{O}}=\frac{1}{8\pi^2}\int_0^{2\pi}\int_0^{\pi}\int_0^{2\pi}(\quad\cdot\quad)\sin\psi
\mathrm{d}\varphi_1\mathrm{d}\psi\mathrm{d}\varphi_2\,,
\end{equation}
respectively. In the second case, for the averaging over the fourth order tensor field $\mathbb{T}(\phi^c)$, it can be simplified to \cite{Forte96,Rychlewski01,Kowalczyk09}:
\begin{displaymath}
\langle \mathbb{T}(\phi_c) \rangle_{\mathcal{O}} = \frac{1}{3}T(\phi_c)_{iijj}\mathbb{I}^{\rm{P}}+\frac{1}{5}\left(T(\phi_c)_{ijij}-\frac{1}{3}T(\phi_c)_{iijj}\right)\mathbb{I}^{\rm{D}}\,,
\end{displaymath}
where $\mathbb{I}^{\rm{P}}$ and $\mathbb{I}^{\rm{D}}$ are fourth order orthogonal projectors to the hydrostatic and deviatoric subspaces of the second order tensor space. In any orthonormal basis they have components: $I^{\rm{P}}_{ijkl}=1/3\delta_{ij}\delta_{kl}$ and $I^{\rm{D}}_{ijkl}=1/2(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk})-I^{\rm{P}}_{ijkl}$. The scalars $T(\phi_c)_{iijj}$ and $T(\phi_c)_{ijij}$ are invariants of the fourth order tensor, such that their values are independent of the selected basis, therefore they can be calculated for the tensor representations in the local crystal frame $\{\mathbf{a}_k\}$. The overall stiffness tensor is then isotropic and specified by two parameters: the bulk modulus $\bar{K}$ and shear modulus $\bar{G}$; namely
\begin{displaymath}
\bar{\mathbb{C}}=3\bar{K}\mathbb{I}^{\rm{P}}+2\bar{G}\mathbb{I}^{\rm{D}}\,.
\end{displaymath}
Eq. (\ref{eq:core-shell}) simplifies further if the crystal has locally cubic lattice symmetry. In such a case, the local stiffness tensor can be expressed in the following spectral form \cite{Walpole81,Rychlewski95,Kowalczyk09}
\begin{equation}\label{Eq:SpekCubic}
\mathbb{C}(\phi^c)=3K\mathbb{I}^{\rm{P}}+2G_1(\mathbb{K}(\phi^c)-\mathbb{I}^{\rm{P}})+2
G_2(\mathbb{I}-\mathbb{K}(\phi^c))\,,
\end{equation}
where
\begin{equation}
\mathbb{K}=\sum_{k=1}^3\mathbf{a}_k\otimes\mathbf{a}_k\otimes\mathbf{a}_k\otimes\mathbf{a}_k\,,
\end{equation}
and $3K$, $2G_1$ and $2G_2$ are Kelvin moduli specified as
\begin{equation}\label{Eq:moduli}
3K=C_{1111}+2C_{1122}\,,\quad 2G_1=C_{1111}-C_{1122}\,,\quad 2G_2=2C_{2323}\,,
\end{equation}
and $C_{ijkl}$ are the components of the local stiffness tensor in the basis $\mathbf{a}_k$.
For the second case of orientation distribution in the representative volume, it can be easily verified that all discussed estimates for the polycrystal bulk modulus are equal: $\bar{K}=K$.
With regard to the shear modulus the respective expressions for classical estimates are collected in Table \ref{tab:LinearEstimates}. The zeroth-order lower and upper bounds for $\bar{G}$ are equal to $\min{(G_1,G_2)}$ and $\max{(G_1,G_2)}$, respectively, such that the $\bar{G}$-estimate by the MT core-shell model for random orientation distribution, is specified as
\begin{equation}
{{\bar{G}_{\rm{CS/MT}}=G_1\frac{4G_1(4 G_1 + 21G_2) + 3K(6 G_1 + 19 G_2) -
3(G_2 - G_1) (8 G_1 + 9 K) f_0}{
4G_1(19 G_1 + 6 G_2) + 3K(21 G_1 + 4 G_2) +
18 (G_2 - G_1) (2 G_1 + K) f_0}}}
\end{equation}
for $G_1<G_2$, or,
\begin{equation}
{{\bar{G}_{\rm{CS/MT}}=G_2 \frac{4G_2(19 G_1 + 6G_2) + 3K(16 G_1 + 9 G_2) -
2(G_1 - G_2) (8 G_2 + 9 K) f_0}{
4G_2(9 G_1 + 16 G_2) + 3K(6 G_1 + 19 G_2) +
12 (G_1 - G_2) (2 G_2 + K) f_0}}}
\end{equation}
for $G_1>G_2$. The $\bar{G}$-estimate by the SC core-shell model, again for random orientation distribution, is obtained as a solution of the following third order equation
\begin{equation}\label{G-CS-SC-1}
\begin{split}
8\bar{G}^3_{\rm{CS/SC}}+[9K+4G_1+12f_0(G_2-G_1)]\bar{G}^2_{\rm{CS/SC}}+\qquad\qquad\quad\quad\\
-[3G_2(4G_1+K) - 9 f_0 K(G_2-G_1)]\bar{G}_{\rm{CS/SC}}-6 G_1 G_2 K=0
\end{split}
\end{equation}
for $G_1<G_2$ or
\begin{equation}\label{G-CS-SC-2}
\begin{split}
8\bar{G}^3_{\rm{CS/SC}}+[9K+4G_1+8f_0(G_1-G_2)]\bar{G}^2_{\rm{CS/SC}}+\qquad\qquad\quad\quad\\
-[3G_2(4G_1+K) - 6 f_0 K(G_1-G_2)]\bar{G}_{\rm{CS/SC}}-6 G_1 G_2 K=0
\end{split}
\end{equation}
for $G_1>G_2$. By analysing the coefficients of the cubic polynomials (\ref{G-CS-SC-1}) and (\ref{G-CS-SC-2}) it can be easily proved that they have a single positive root, such that the solution is unique. It can be specified by a closed formula using known relations for roots of the cubic polynomial.
In the case of polycrystalline materials composed of a finite set of $N_g$ grains with randomly selected orientations, the overall stiffness tensor can be estimated by either of Eqs. (\ref{lowerL}-\ref{sc1}) or Eq. (\ref{eq:core-shell}) and is in general anisotropic. With an increasing number of orientations the overall properties approach isotropy. The discrepancy between the current stiffness obtained for a given set of orientations and its isotropic limit can be quantified using some anisotropy measure. Two definitions of the anisotropy factor used in this study result from the spectral decomposition of the overall stiffness.
The first anisotropy measure $\zeta_1$ is motivated by the Zener parameter proposed for cubic crystals \cite{zener1948elasticity}. This is defined as
\begin{equation}\label{Eq:zeta1}
\zeta_1=\frac{\rm{min}(\lambda_K)}{\rm{max}(\lambda_K)}\leq 1\,,\quad K=2,\ldots,6\,,
\end{equation}
where $\lambda_K$ are the eigenvalues (the Kelvin moduli) of $\bar{\mathbb{C}}$ obtained by means of its spectral decomposition \cite{Cowin95,Rychlewski95} and ordered according to the value of the invariant $P_{Kiijj}$ of the corresponding eigenprojector $\mathbb{P}_K$ \cite{Kowalczyk09}. Note that $\zeta_1$ is equal to 1 for isotropic $\bar{\mathbb{C}}$ and equal to the Zener parameter for cubic $\bar{\mathbb{C}}$ if for such $\bar{\mathbb{C}}$ we have $G_1>G_2$ (see Eq. \ref{Eq:SpekCubic}). The fulfilment of $\zeta_1=1$ is necessary to ensure elastic isotropy, however, it is not sufficient in the general-anisotropy case (it is sufficient for volumetrically isotropic materials \cite{Kowalczyk12}).
The second anisotropy measure $\zeta_2$ is defined by a norm of difference between the closest isotropic approximation of $\bar{\mathbb{C}}$ and the actual $\bar{\mathbb{C}}$. The difference is normalized by the norm of $\bar{\mathbb{C}}$. The closest isotropic approximation of $\bar{\mathbb{C}}$ is established employing the Log-Euclidean metric as proposed in \cite{Moakher06}. The anisotropy factor is then calculated as \cite{Kowalczyk11b}
\begin{equation}\label{Eq:zeta2}
\zeta_2=\frac{||\mathrm{Log}\bar{\mathbb{C}}-\mathrm{Log}\bar{\mathbb{C}}^{\mathcal{L}}_{\rm{iso}}||}{||\mathrm{Log}\bar{\mathbb{C}}||}\geq 0\,,
\end{equation}
where $||\mathbb{A}||=\sqrt{A_{ijkl}A_{ijkl}}$ and $\mathrm{Log}\mathbb{A}=\sum_{K}\mathrm{log}\lambda_L\mathbb{P}_K$ ($\lambda_K$ - eigenvalues of $\mathbb{A}$, $\mathbb{P}_K$ - eigenprojectors of $\mathbb{A}$, both resulting from its spectral decomposition \cite{Kowalczyk09}), and
\begin{equation}
\bar{\mathbb{C}}^{\mathcal{L}}_{\rm{iso}}=\exp\left(\frac{1}{3}Z_{iijj}\right)\mathbb{I}^{\rm{P}}+\exp\left(\frac{1}{5}\left(Z_{ijij}-\frac{1}{3}Z_{iijj}\right)\right)\mathbb{I}^{\rm{D}}\,\quad\textrm{for}\quad
\mathbb{Z}=\mathrm{Log}\bar{\mathbb{C}}\,.
\end{equation}
Naturally, $\zeta_2$ is equal to 0 for isotropic $\bar{\mathbb{C}}$. The fulfilment of $\zeta_2=0$ is sufficient to ensure elastic isotropy. In further analysis of results the following scalars:
\begin{equation}\label{Eq:IsoBulkShear}
\bar{K}^{\mathcal{L}}_{\rm{iso}}=\frac{1}{3}\exp\left(\frac{1}{3}Z_{iijj}\right)\,,\quad
\bar{G}^{\mathcal{L}}_{\rm{iso}}=\frac{1}{2}\exp\left(\frac{1}{5}\left(Z_{ijij}-\frac{1}{3}Z_{iijj}\right)\right)
\end{equation}
will be used as estimates of isotropic bulk and shear moduli corresponding to, in general, an anisotropic estimate of effective stiffness tensor obtained for a finite set of $N_g$ grain orientations.
It should be stressed that for a polycrystal of cubic grains (i.e. made of cubic symmetry material) all classical continuum-mechanics (\ref{lowerL}-\ref{sc1}) and core-shell model estimates (\ref{eq:core-shell}) -- if shell properties are defined as zeroth-order lower bound -- deliver exactly the same value of the isotropic bulk modulus: $\bar{K}^{\mathcal{L}}_{\rm{iso}}=K$, independently of the number of orientations \cite{Walpole85}.
\section{Results}
\label{sec:Res}
\subsection{Results of atomistic simulations}
\label{ssec:ResAS}
The computational samples analysed by atomistic approach are denoted as $$N_{\rm{UC}}-N_g-\rm{SYS}$$ where $N_{\rm{UC}}$ is a number of unit cells, $N_g$ - a number of grain orientations and $\rm{SYS}$ denotes the system of grain distribution, i.e.: BCC, FCC or random, see Table \ref{tab:Samples}.
The finite set of $N_g$ orientations has been considered, namely polycrystalline representative volumes composed of copper grains with 16, 54, 128 or 250 randomly selected orientations have been analysed. The orientation was defined in terms of three Euler angles. The random selection of a set of Euler angles was done in a standard way, i.e. assuming uniform distribution of orientations in the orientation space and taking into account the specific features of the non-Euclidean space of Euler angles \cite{Bunge}.
Elasticity tensors $\bar{\mathbb{C}}$\, derived from molecular simulations of analysed samples are listed in the Table \ref{tab:Cij}.
Visualizations of atomistic computational samples and cohesive energy $E_{c}$\,(eV/atom) are depicted in Fig. \ref{fig:Samples}, while cohesive energy density $E_{c}$\,(eV/atom) histograms in Fig. \ref{fig:Histograms}. It can be seen that as the average grain size decreases, the fraction of transient shell atoms in the sample and the average cohesive energy rises.
Fig. \ref{fig:Samples} indicates that polycrystals can be treated as crystalline cores of monocrystal pattern surrounded by amorphous sheaths. In solid-state systems the CNA pattern is a useful measure of the local crystal structure around an atom \cite{Plimpton1995}. There are five kinds of CNA patterns: FCC, HCP, BCC, Icosohedral and Other. Sometimes it may be difficult to choose the right \textit{cutoff} radius for the conventional CNA and therefore an adaptive version of the CNA has been developed that works without a fixed \textit{cutoff}. The adaptive common neighbor analysis (a-CNA) method determines the optimal \textit{cutoff} radius automatically for each individual particle \cite{Stukowski2012}. On the basis of this analysis for FCC copper we can define fraction of non-FCC structure atoms $f_{CNA}$ in the polycrystalline sample, see Table \ref{tab:Samples}.
In a core-shell model proposed in Sec.\ref{sec:Cont} the grain is composed of a core and of the shell that has different properties (see also \cite{Palosz2002}). In conjunction with atomistic simulations the shell thickness $\Delta$ can be preliminary assessed as equal to \textit{the cutoff radius} of the used potential \cite{Mishin2001}, i.e. 5.5\,\AA.{The value is consistent with the general premises for the non-bulk layers of atoms in a 3D FCC crystals \cite{Park2007}.} Using such definition and assuming spherical shape of coated grains the fraction of transient shell atoms $f_0$ in the sample is calculated by the formula
\begin{equation}\label{def:fsa}
f_0=1-\left(1-\frac{2\Delta}{d}\right)^3\,,\quad\Delta=5.5\,\text{\AA}
\end{equation}
where $d$ is an averaged grain diameter. The values of $d$ and $f_0$ for ten analysed samples are collected in Table \ref{tab:Samples}.
\begin{figure}[H]
\centering
\begin{tabular}{ccc}
\includegraphics[width=0.34\linewidth]{Monocrystal-eps-converted-to.pdf} &
\includegraphics[width=0.34\linewidth]{253-128-BCC-eps-converted-to.pdf} &
\includegraphics[width=0.34\linewidth]{503-250-BCC-eps-converted-to.pdf} \\
\includegraphics[width=0.34\linewidth]{753-128-BCC-eps-converted-to.pdf} &
\includegraphics[width=0.34\linewidth]{1003-128-BCC-eps-converted-to.pdf} &
\includegraphics[width=0.34\linewidth]{1003-16-BCC-eps-converted-to.pdf} \\
\includegraphics[width=0.34\linewidth]{503-16-BCC-eps-converted-to.pdf} &
\includegraphics[width=0.34\linewidth]{503-54-BCC-eps-converted-to.pdf} &
\includegraphics[width=0.34\linewidth]{503-250-BCC-eps-converted-to.pdf} \\
\includegraphics[width=0.34\linewidth]{503-108-FCC-eps-converted-to.pdf} &
\includegraphics[width=0.34\linewidth]{503-125-RANDOM-eps-converted-to.pdf}
\end{tabular}
\caption{Visualization of atomistic computational samples and cohesive energy $E_{c}$\,(eV/atom). }
\label{fig:Samples}
\end{figure}
\begin{figure}[H]
\centering
\begin{tabular}{ccc}
\includegraphics[width=0.34\linewidth]{HMonocrystal-eps-converted-to.pdf} &
\includegraphics[width=0.34\linewidth]{H253-128-BCC-eps-converted-to.pdf} &
\includegraphics[width=0.34\linewidth]{H503-128-BCC-eps-converted-to.pdf} \\
\includegraphics[width=0.34\linewidth]{H753-128-BCC-eps-converted-to.pdf} &
\includegraphics[width=0.34\linewidth]{H1003-128-BCC-eps-converted-to.pdf} &
\includegraphics[width=0.34\linewidth]{H1003-16-BCC-eps-converted-to.pdf} \\
\includegraphics[width=0.34\linewidth]{H503-16-BCC-eps-converted-to.pdf} &
\includegraphics[width=0.34\linewidth]{H503-54-BCC-eps-converted-to.pdf} &
\includegraphics[width=0.34\linewidth]{H503-250-BCC-eps-converted-to.pdf} \\
\includegraphics[width=0.34\linewidth]{H503-108-FCC-eps-converted-to.pdf} &
\includegraphics[width=0.34\linewidth]{H503-125-RANDOM-eps-converted-to.pdf} &
\includegraphics[width=0.34\linewidth]{EdivVsAgd-eps-converted-to.pdf}
\end{tabular}
\caption{Cohesive energy density $E_{c}$\,(eV/atom) histograms (vertical dashed line represents the average cohesive energy) and $\Delta E_{c}/atom = E_{c}/atom-E^{Monocrystal}_{c}/atom$\,(eV/atom) as a function of average grain diameter $d$ (\AA). }
\label{fig:Histograms}
\end{figure}
\begin{table}[H]
\caption{Volume (\AA$^3$), box lengths (\AA), number of atoms, average grain diameter $d$ (\AA), fraction of transient shell atoms $f_{0}$ (\ref{def:fsa}), {fraction of non-FCC structure atoms $f_{CNA}$}, average cohesive energy $E_{c}$\,(eV/atom) of analysed computational samples.}
\label{tab:Samples}
\centering
\renewcommand{\arraystretch}{1.5}
\tiny
\begin{tabular}{|c c c c c c c c|}
\hline Sample & V & L & No.of atoms & $d$ & $f_{0}$ & $f_{CNA}$ & E$_{c}$\\
\hline Monocrystal & 47.24 & 3.615 & 4 & & & & -3.54 \\
25$^3$-128-BCC & 750452.29 & 90.87 & 62302 & 22.4 & 0.87 & 0.71 &-3.473 \\
50$^3$-128-BCC & 5985342.9 & 181.56 & 499984 & 44.7 & 0.57 & 0.36 &-3.497 \\
75$^3$-128-BCC & 20129512 & 272.03 & 1687719 & 67 & 0.42 & 0.25 &-3.509 \\
100$^3$-128-BCC & 47605657 & 362.43 & 3999934 & 89.2 & 0.33 & 0.19 & -3.516 \\
100$^3$-16-BCC & 47432793 & 361.99 & 4000010 & 178.2 & 0.17 & 0.09 & -3.528 \\
50$^3$-16-BCC & 5950371.6 & 181.21 & 500020 & 89.2 & 0.33 & 0.18 & -3.516 \\
50$^3$-54-BCC & 5969399.2 & 181.40 & 500058 & 59.6 & 0.46 & 0.28 &-3.506 \\
50$^3$-250-BCC & 5997791.1 & 181.69 & 500008 & 35.8 & 0.67 & 0.46 &-3.489 \\
50$^3$-108-FCC & 5981645.7 & 181.53 & 500021 & 47.3 & 0.55 & 0.35 &-3.499 \\
50$^3$-125-Random & 5985474.7 & 181.57 & 499836 & 45.1 & 0.57 & 0.39 &-3.495 \\
\hline
\end{tabular}
\end{table}
\begin{table}[H]
\caption{{{Elasticity tensors $\bar{\mathbb{C}}$\,[GPa] of analysed samples (for notation used see Eq. (\ref{eqn:CuCij}) in appendix)}}.}
\label{tab:Cij}
\begin{threeparttable}[b]
\centering
\tiny
\begin{tabular}{ c c }
{Monocrystal} & {25$^3$ unit cells, 128 grains in BCC system, 62302 atoms} \\
& {(25$^3$-128-BCC)} \\
$\begin{bmatrix} {169.88} & {122.60} & {122.60} & 0 & 0 & 0 \\
{122.60} & {169.88} & {122.60} & 0 & 0 & 0 \\
{122.60} & {122.60} & {169.88} & 0 & 0 & 0 \\
0 & 0 & 0 & {76.19} & 0 & 0 \\
0 & 0 & 0 & 0 & {76.19} & 0 \\
0 & 0 & 0 & 0 & 0 & {76.19} \end{bmatrix}$ &
$\begin{bmatrix} {166.50} & {120.68} & {120.46} & {-0.56} & {0.53} & {0.65} \\
{120.68} & {167.56} & {120.64} & {-1.64} & {-0.15} & {0.76} \\
{120.46} & {120.64} & {166.60} & {3.42} & {-0.57} & {-0.32} \\
{-0.56} & {-1.64} & {3.42} & {22.62} & {0.06} & {-0.39} \\
{0.53} & {-0.15} & {-0.57} & {0.06} & {23.96} & {-0.56} \\
{0.65} & {0.76} & {-0.32} & {-0.39} & {-0.56} & {22.09} \end{bmatrix}$\\
\hline {50$^3$ unit cells, 128 grains in BCC system, 499984 atoms} & {75$^3$ unit cells, 128 grains in BCC system, 1687719 atoms} \\
{(50$^3$-128-BCC)} & {(75$^3$-128-BCC)} \\
$\begin{bmatrix} {174.61} & {116.92} & {119.13} & {-0.34} & {1.11} & {-1.54} \\
{116.92} & {177.10} & {117.18} & {-1.80} & {1.09} & {1.17} \\
{119.13} & {117.18} & {174.8778} & {0.18} & {-0.27} & {0.08} \\
{-0.34} & {-1.80} & {0.18} & {32.96} & {0.17} & {-1.63} \\
{1.11} & {1.09} & {-0.27} & {0.17} & {28.10} & {1.05} \\
{-1.54} & {1.17} & {0.08} & {-1.63} & {1.05} & {29.19} \end{bmatrix}$ &
$\begin{bmatrix} {180.71} & {114.88} & {115.79} & {-1.074} & {-1.39} & {-0.57} \\
{114.88} & {181.95} & {115.82} & {-0.94} & {-0.62} & {1.40} \\
{115.79} & {115.82} & {180.43} & {1.58} & {0.64} & {-0.85} \\
{-1.074} & {-0.94} & {1.58} & {34.986} & {-0.44} & {-0.07} \\
{-1.39} & {-0.62} & {0.64} & {-0.44} & {36.91} & {0.39} \\
{-0.57} & {1.40} & {-0.85} & {-0.07} & {0.39} & {35.74} \end{bmatrix}$\\
\hline {100$^3$ unit cells, 128 grains in BCC system, 3999934 atoms} & {100$^3$ unit cells, 16 grains in BCC system, 4000010 atoms} \\
{(100$^3$-128-BCC)} & {(100$^3$-16-BCC)} \\
$\begin{bmatrix} {184.74} & {113.60} & {115.07} & {-0.61} & {0.34} & {-0.87} \\
{113.60} & {184.55} & {114.18} & {-0.69} & {-0.92} & {1.83} \\
{115.07} & {114.18} & {183.48} & {1.54} & {0.22} & {-0.82} \\
{-0.61} & {-0.69} & {1.54} & {37.90} & {-0.43} & {-0.44} \\
{0.34} & {-0.92} & {0.22} & {-0.43} & {37.57} & {-0.57}\\
{-0.87} & {1.83} & {-0.82} & {-0.44} & {-0.57} & {37.93} \end{bmatrix}$ &
$\begin{bmatrix} {186.97} & {114.79} & {112.47} & {-0.88} & {-0.07} & {-6.21} \\
{114.79} & {186.11} & {113.14} & {-1.20} & {1.42} & {2.40} \\
{112.47} & {113.14} & {187.21} & {2.26} & {-1.08} & {3.56} \\
{-0.88} & {-1.20} & {2.26} & {42.93} & {2.71} & {1.77} \\
{-0.07} & {1.42} & {-1.08} & {2.71} & {42.50} & {-1.00} \\
{-6.21} & {2.40} & {3.56} & {1.77} & {-1.00} & {44.38} \end{bmatrix}$\\
\hline {50$^3$ unit cells, 16 grains in BCC system, 500020 atoms} & {50$^3$ unit cells, 54 grains in BCC system, 500058 atoms} \\
{(50$^3$-16-BCC)} & {(50$^3$-54-BCC)} \\
$\begin{bmatrix} {182.57} & {115.48} & {115.55} & {-0.65} & {0.03} & {-6.23} \\
{115.48} & {182.04} & {115.56} & {-1.44} & {0.09} & {3.82} \\
{115.55} & {115.56} & {180.12} & {1.32} & {-0.69} & {2.28} \\
{-0.65} & {-1.44} & {1.32} & {38.88} & {3.44} & {0.88} \\
{0.03} & {0.09} & {-0.69} & {3.44} & {38.67} & {-1.52} \\
{-6.23} & {3.82} & {2.28} & {0.88} & {-1.52} & {40.46} \end{bmatrix}$ &
$\begin{bmatrix} {177.33} & {116.96} & {114.38} & {-1.49} & {-0.49} & {-2.32} \\
{116.96} & {179.06} & {116.59} & {-1.01} & {-1.65} & {2.03} \\
{114.38} & {116.59} & {175.49} & {3.30} & {3.30} & {0.30} \\
{-1.49} & {-1.01} & {3.30} & {36.85} & {1.42} & {-0.36} \\
{-0.49} & {-1.65} & {3.30} & {1.42} & {36.20} & {-1.05}\\
{-2.32} & {2.03} & {0.30} & {-0.36} & {-1.05} & {35.09} \end{bmatrix}$\\
\hline {50$^3$ unit cells, 250 grains in BCC system, 500008 atoms} & {50$^3$ unit cells, 108 grains in FCC system, 500021 atoms} \\
{(50$^3$-250-BCC)} & {(50$^3$-108-FCC)} \\
$\begin{bmatrix} {172.29} & {118.51} & {120.44} & {-0.14} & {0.45} & {-1.32} \\
{118.51} & {173.39} & {119.08} & {-0.48} & {0.46} & {0.35} \\
{120.44} & {119.08} & {173.41} & {1.24} & {-0.55} & {0.75} \\
{-0.14} & {-0.48} & {1.24} & {28.53} & {-2.40} & {-1.77} \\
{0.45} & {0.46} & {-0.55} & {-2.40} & {27.57} & {-1.19} \\
{-1.32} & {0.35} & {0.75} & {-1.77} & {-1.19} & {29.23} \end{bmatrix}$ &
$\begin{bmatrix} {177.94} & {117.26} & {116.47} & {-1.255} & {0.43} & {1.01} \\
{117.26} & {174.99} & {119.01} & {-0.92} & {-1.33} & {1.377} \\
{116.47} & {119.01} & {175.09} & {1.67} & {1.60} & {-0.49} \\
{-1.255} & {-0.92} & {1.67} & {33.06} & {-0.14} & {0.46} \\
{0.43} & {-1.33} & {1.60} & {-0.14} & {29.84} & {-0.75} \\
{1.01} & {1.377} & {-0.49} & {0.46} & {-0.75} & {33.20} \end{bmatrix}$\\
\hline \multicolumn{2}{ c }{50$^3$ unit cells, 125 grains in random system, 500008 atoms} \\
\multicolumn{2}{ c }{(50$^3$-125-Random)} \\
\multicolumn{2}{ c }{$\begin{bmatrix} {176.96} & {115.67} & {118.06} & {3.00} & {2.65} & {-2.05} \\
{115.67} & {174.31} & {118.15} & {-0.71} & {-1.01} & {-0.10} \\
{118.06} & {118.15} & {175.02} & {3.97} & {1.02} & {0.04} \\
{3.00} & {-0.71} & {3.97} & {31.71} & {2.56} & {-2.30} \\
{2.65} & {-1.01} & {1.02} & {2.56} & {30.66} & {0.05} \\
{-2.05} & {-0.10} & {0.04} & {-2.30} & {0.05} & {29.92} \end{bmatrix}$ }\\
\end{tabular}
\end{threeparttable}
\end{table}
\subsection{Comparison of atomistic and continuum-mechanics estimates}
\label{ssec:CompACest}
Table~\ref{tab:BoundsCM} contains classical continuum-mechanics mean-field estimates (\ref{lowerL})-(\ref{sc1}) of the effective isotropic shear modulus (\ref{Eq:IsoBulkShear})$_2$ and anisotropy factors (\ref{Eq:zeta1}-\ref{Eq:zeta2}) calculated for the respective sets of $N_g$ orientations assumed in the atomistic simulations. As discussed in \ref{Ap:A}, such estimates neglect the grain size effect and treat the polycrystal as a one-phase material. Table~\ref{tab:BoundsCM} demonstrates that the mean-field estimates $\bar{G}^{\mathcal{L}}_{\rm{iso}}$ are close to the corresponding estimates obtained for the infinite set of orientations. When the number of orientations increases anisotropy factors approach limit values characterizing isotropy. Moreover, for a given $N_g$ the factors are close to each other for different estimates.
Therefore, in further analysis of atomistic predictions the estimates of the bulk and shear moduli obtained for an infinite set of orientations will be used as reference values.
For comparison purpose, the anisotropy degree of effective stiffness $\bar{\mathbb{C}}$, obtained for a given sample in atomistic simulations were also analysed (see Table \ref{tab:AtomEstimates}).
It was observed that anisotropy of effective properties acquired in atomistic simulations, when assessed using the anisotropy factors (\ref{Eq:zeta1}) and (\ref{Eq:zeta2}), is higher than the corresponding values obtained for the same number of orientations $N_g$ employing the continuum mechanics (see Table \ref{tab:BoundsCM}). Additionally, for the same number of orientations anisotropy of atomistic estimates decreases with increasing size of the grain specified by the number of atoms per grain. On the other hand, for the same number of unit cells $N_{UC}$ (i.e. the similar number of atoms per sample) anisotropy of atomistic estimates decreases with an increasing number of orientations.
\begin{table}[!htp]
\caption{The overall shear modulus $\bar{G}^{\mathcal{L}}_{\rm{iso}}$\,[GPa] and anisotropy factors $\zeta_1$ and $\zeta_2$\,[\%] obtained for the Voigt, Reuss and Hashin-Shtrikman (H-S) bounds of effective stiffness tensor for copper polycrystal composed of $N_g$ grains. The local elastic stiffness (\ref{Eq:SpekCubic}) is specified by: $K=138.36$\,GPa, $G_1=23.64$\,GPa, $G_2=76.19$\,GPa.
}
\label{tab:BoundsCM}\vspace{.05in}
\centering
\begin{tabular}{|c|cccccc|cccccc|}
\hline
$N_g$ & $\bar{G}^{\mathcal{L}}_{\rm{iso}}$ & $\zeta_1$ & $\zeta_2$ & $\bar{G}^{\mathcal{L}}_{\rm{iso}}$ & $\zeta_1$ & $\zeta_2$& $\bar{G}^{\mathcal{L}}_{\rm{iso}}$ & $\zeta_1$ & $\zeta_2$ & $\bar{G}^{\mathcal{L}}_{\rm{iso}}$ & $\zeta_1$ & $\zeta_2$\\ \hline
1& 47.71 & 0.31 & 10.76 & -- & -- & --& -- & -- & -- & -- & -- & --\\ \hline
&\multicolumn{3}{c}{Voigt}& \multicolumn{3}{c|}{Reuss}&\multicolumn{3}{c}{H-S (U)}& \multicolumn{3}{c|}{H-S (L)}\\
16& 54.98 & 0.77 & 1.54 & 40.54 & 0.73 & 1.98& 49.84 & 0.75 & 1.79 & 46.39 & 0.73 & 1.91\\
54& 55.06 & 0.83 & 1.15 & 40.45 & 0.80 & 1.47& 49.87 & 0.81 & 1.34 & 46.37 & 0.80 & 1.42\\
128& 55.15 & 0.93 & 0.49 & 40.35 & 0.91 & 0.63& 49.89 & 0.91 & 0.57 & 46.35 & 0.91 & 0.61\\
250& 55.16 & 0.94 & 0.39 & 40.34 & 0.93 & 0.50 & 49.89 & 0.93 & 0.45 & 46.35 & 0.93 & 0.49\\ \hline
$\infty$ & 55.17 & 1 & 0 & 40.33 & 1 & 0 & 49.90 & 1 & 0 & 46.35 & 1 & 0 \\ \hline
\end{tabular}\\[.051in]
\end{table}
\begin{table}[!htp]
\caption{The overall bulk and shear moduli $\bar{K}^{\mathcal{L}}_{\rm{iso}}$\,[GPa] and $\bar{G}^{\mathcal{L}}_{\rm{iso}}$\,[GPa] and anisotropy factors $\zeta_1$ and $\zeta_2$\,[\%] calculated for the effective stiffness tensors resulting from the atomistic simulations and from the MT and SC core-shell models for copper polycrystal ($\Delta$=5.5\,\AA).
}
\label{tab:AtomEstimates}\vspace{.05in}
\centering
\begin{tabular}{|c|cccc|ccc|ccc|}
\hline
Sample & $\bar{K}^{\mathcal{L}}_{\rm{iso}}$ & $\bar{G}^{\mathcal{L}}_{\rm{iso}}$ & $\zeta_1$ & $\zeta_2$ & $\bar{G}^{\mathcal{L}}_{\rm{iso}}$ & $\zeta_1$ & $\zeta_2$& $\bar{G}^{\mathcal{L}}_{\rm{iso}}$ & $\zeta_1$ & $\zeta_2$\\ \hline
&\multicolumn{4}{c|}{Atomistic}&\multicolumn{3}{c|}{MT Core-Shell}&\multicolumn{3}{c|}{SC Core-Shell}\\ \hline
$25^3$-128-BCC& 136.02 & 22.91 & 0.78 & 1.80 &25.80& 0.99 & 0.10&25.95&0.99&0.11\\
$50^3$-128-BCC& 137.00 & 29.50 & 0.78 & 1.71 &31.52& 0.96& 0.32&32.45&0.96&0.34\\
$75^3$-128-BCC& 137.34 & 34.56 & 0.84 & 1.19 &34.84& 0.95& 0.42&36.35&0.95&0.44\\
$100^3$-128-BCC& 137.61 & 36.62 & 0.86 & 1.10 &37.00& 0.94& 0.48&38.91&0.94&0.49\\ \hline
$50^3$-16-BCC& 137.54 & 36.43 & 0.69 & 2.86 & 37.01& 0.82& 1.51&38.90&0.81&1.55\\
$100^3$-16-BCC & 137.90 & 40.26 & 0.70 & 2.59 &41.25& 0.78& 1.85&43.82&0.78&1.83\\
$50^3$-54-BCC & 136.41 & 33.65 & 0.73 & 2.38 & 33.92& 0.89& 0.92&35.26&0.89&0.97\\
$50^3$-250-BCC & 137.24 & 27.69 & 0.79 & 1.84 & 29.49& 0.98& 0.20&30.09&0.97&0.21\\ \hline
$50^3$-108-FCC & 137.05 & 30.79 & 0.79 & 1.74 & 31.94& 0.95& 0.43&32.94&0.95&0.45\\
$50^3$-125-RAN & 136.63 & 29.91 & 0.73 & 2.16 & 31.52& 0.96& 0.32&32.45&0.96&0.34\\ \hline
\end{tabular}\\[.051in]
\end{table}
\comm{ \begin{table}[!htp]
\caption{The overall bulk and shear moduli $\bar{K}^{\mathcal{L}}_{\rm{iso}}$\,[GPa] and $\bar{G}^{\mathcal{L}}_{\rm{iso}}$\,[GPa] and anisotropy factors $\zeta_1$ and $\zeta_2$\,[\%] calculated for the effective stiffness tensors resulting from the atomistic simulations and from the optimized MT and SC core-shell models for copper polycrystal ($\Delta_{\rm{MT}}=6.387$\,\AA, $\Delta_{\rm{SC}}$=6.837\,\AA).
}
\label{tab:AtomEstimates0}\vspace{.05in}
\centering
\begin{tabular}{|c|cccc|ccc|ccc|}
\hline
Sample & $\bar{K}^{\mathcal{L}}_{\rm{iso}}$ & $\bar{G}^{\mathcal{L}}_{\rm{iso}}$ & $\zeta_1$ & $\zeta_2$ & $\bar{G}^{\mathcal{L}}_{\rm{iso}}$ & $\zeta_1$ & $\zeta_2$& $\bar{G}^{\mathcal{L}}_{\rm{iso}}$ & $\zeta_1$ & $\zeta_2$\\
&\multicolumn{4}{c|}{Atomistic}&\multicolumn{3}{c|}{MT Core-Shell}&\multicolumn{3}{c|}{SC Core-Shell}\\ \hline
$25^3$-128-BCC& 136.02 & 22.91 & 0.78 & 1.80 &24.93& 0.99&0.06&24.65&0.99&0.05\\
$50^3$-128-BCC& 137.00 & 29.50 & 0.78 & 1.71 &30.17&0.97 &0.27&30.19&0.97&0.27\\
$75^3$-128-BCC& 137.34 & 34.56 & 0.84 & 1.19 &33.69&0.95 &0.39&34.32&0.95&0.39\\
$100^3$-128-BCC& 137.61 & 36.62 & 0.86 & 1.10 &36.00&0.94 &0.45&37.11&0.94&0.46\\ \hline
$50^3$-16-BCC& 137.54 & 36.43 & 0.69 & 2.86 &36.00&0.83 &1.42&37.10&0.83&1.43\\
$100^3$-16-BCC & 137.90 & 40.26 & 0.70 & 2.59 &40.42&0.79 &1.79&42.45&0.79&1.76\\
$50^3$-54-BCC & 136.41 & 33.65 & 0.73 & 2.38 &32.69&0.83 &0.90&33.12&0.90&0.84\\
$50^3$-250-BCC & 137.24 & 27.69 & 0.79 & 1.84 &28.25&0.98 &0.16&28.05&0.98&0.15\\ \hline
$50^3$-108-FCC & 137.05 & 30.79 & 0.79 & 1.74 &30.67&0.96 &0.37&30.76&0.96&0.37\\
$50^3$-125-RAN & 136.63 & 29.91 & 0.73 & 2.16 &30.24&0.97 &0.27&30.27&0.97&0.27\\ \hline
\end{tabular}\\[.051in]
\end{table} }
\begin{figure}[H]
\centering
\includegraphics[width=.8\textwidth]{BulkA-eps-converted-to.pdf}
\caption{The isotropic bulk modulus $\bar{K}^{\mathcal{L}}_{\rm{iso}}$ calculated for the overall stiffness tensor acquired in atomistic simulations for different polycrystalline samples with respect to the $\bar{K}$ value predicted by the continuum mechanics methodology. Samples are ordered according to the increasing value of grain diameter $d$.}
\label{fig:BulkA}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[width=.93\textwidth]{ShearA-eps-converted-to.pdf}
\caption{The isotropic shear modulus $\bar{G}^{\mathcal{L}}_{\rm{iso}}$ calculated for the overall stiffness tensor acquired in atomistic simulations for different polycrystalline samples with respect to the selected $\bar{G}$ estimates obtained by continuum mechanics methodology for ideal random polycrystal (horizontal lines: R - Reuss, HS$_{\rm{L}}$ - lower Hashin-Shtrikman bound, SC - self-consistent) and the core-shell model (MT-CS and SC-CS with $\Delta$=5.5\,\AA) estimates for a finite set of orientations (non-filled bars). Samples are ordered according to the increasing value of grain diameter.}
\label{fig:ShearA}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[width=.8\textwidth]{Approx2-eps-converted-to.pdf}\\
\includegraphics[width=.8\textwidth]{Approx2A-eps-converted-to.pdf}
\caption{The isotropic shear modulus $\bar{G}^{\mathcal{L}}_{\rm{iso}}$ calculated for the overall stiffness tensor acquired in atomistic simulations as a function of the average grain diameter $d$ for ten polycrystalline samples and by the two variants of the core-shell model. Exponential approximation (\ref{Eq:approx}) of dependency is also shown. (a) $\Delta$=5.5\,{\AA} (b) $\Delta=\Delta_{MT}$=6.387\,{\AA} and $\Delta=\Delta_{SC}$=6.837\,{\AA} for MT and SC variants, respectively.}
\label{fig:Approx}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[width=.8\textwidth]{YoungModulus-eps-converted-to.pdf}
\caption{The isotropic Young's modulus $\bar{E}^{\mathcal{L}}_{\rm{iso}}$ calculated for the overall stiffness tensor acquired in atomistic simulations for ten polycrystalline samples (green stars) and by the MT and SC core-shell models with $\Delta$=5.5\,{\AA} (green line) as a function of the average grain diameter $d$. Additionally, results of other experimental and atomistic studies reported in \cite{Gao13} are shown. In particular, black dots represent the results of simulations of \cite{Gao13} (performed at 300\,K), while squares represent the experimental data (for detailed references see \cite{Gao13})}
\label{fig:ApproxE}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[width=.8\textwidth]{Poisson-eps-converted-to.pdf}
\caption{The Poisson's ratio $\bar{\nu}^{\mathcal{L}}_{\rm{iso}}$ calculated for the isotropized overall stiffness tensors acquired in atomistic simulations for ten polycrystalline samples (green stars) and by the core-shell model (green line) as a function of the average grain diameter $d$.}
\label{fig:ApproxNu}
\end{figure}
As concerns the estimated values of elastic moduli, presented results (see Fig. \ref{fig:BulkA}) indicate that the isotropic bulk modulus $\bar{K}^{\mathcal{L}}_{\rm{iso}}$, established in atomistic calculations for all samples is similar and close to the continuum-mechanics estimate $\bar{K}$=138.36\,GPa. On the contrary, the value of the isotropized shear modulus $\bar{G}^{\mathcal{L}}_{\rm{iso}}$ established in atomistic simulations strongly depends on the grain size. It increases with the average grain diameter used in the simulations, approaching the Reuss continuum-mechanics estimate for the largest grains (see Fig. \ref{fig:ShearA}). The observed dependence can be approximated by the following exponential function:
\begin{equation}\label{Eq:approx}
\bar{G}^{\mathcal{L}}_{\rm{iso}}(d)=G_0 + (G_{\infty} - G_0) (1 - \exp(-\beta d))
\end{equation}
where $G_0$ is a shear modulus for
$d\rightarrow 0$, $G_{\infty}$ - a shear modulus for a coarse-grained polycrystal and $\beta$ governs the rate of change of $\bar{G}^{\mathcal{L}}_{\rm{iso}}$ with the grain size. It is found that $G_0=11.27$\,GPa, $G_{\infty}=40.65$\,GPa and $\beta=0.023$ for the considered copper polycrystal (see Fig. \ref{fig:Approx}). Using the presented results obtained for the bulk and shear moduli the isotropic tensile Young's modulus $\bar{E}^{\mathcal{L}}_{\rm{iso}}$ and Poisson's ratio $\bar{\nu}^{\mathcal{L}}_{\rm{iso}}$ can be derived by well-known formulas:
\begin{equation}\label{Eq:E-nu}
\bar{E}^{\mathcal{L}}_{\rm{iso}}=\frac{9\bar{K}^{\mathcal{L}}_{\rm{iso}}\bar{G}^{\mathcal{L}}_{\rm{iso}}}{3\bar{K}^{\mathcal{L}}_{\rm{iso}}+\bar{G}^{\mathcal{L}}_{\rm{iso}}}\,,\quad\bar{\nu}^{\mathcal{L}}_{\rm{iso}}=\frac{3\bar{K}^{\mathcal{L}}_{\rm{iso}}-2\bar{G}^{\mathcal{L}}_{\rm{iso}}}{6\bar{K}^{\mathcal{L}}_{\rm{iso}}+2\bar{G}^{\mathcal{L}}_{\rm{iso}}}\,.
\end{equation}
The obtained atomistic estimate of Young's modulus shows similar dependence on the grain size as the shear modulus and is in line with the outcomes of other literature studies \cite{Gao13} (see Fig. \ref{fig:ApproxE}). The calculated Poisson's ratio $\bar{\nu}^{\mathcal{L}}_{\rm{iso}}$ decreases with an increasing grain size (Fig. \ref{fig:ApproxNu}). Note that similar predictions concerning Poisson's ratio were obtained by \cite{Kim20123942} with use of the modified continuum theory that includes scale effects and a non-conforming finite element method.
The tendencies observed in atomistic simulations are well captured, both qualitatively and quantitatively, by two variants of core-shell model formulated in Section \ref{sec:Cont}. Similarly to atomistic results, in the model the bulk modulus does not depend on the grain size and is equal to the local bulk modulus. The results concerning other elastic constants are compared in figures \ref{fig:Approx}-\ref{fig:ApproxNu}. The volume fraction of the transient zone $f_0$ was calculated using the \textit{cutoff} radius of atomistic potential (Eq. (\ref{def:fsa})). Although the specific values of $\bar{G}$ and $\bar{E}$ are slightly over-predicted for very small grains, the quality of obtained estimates is satisfactory in the whole range of grain sizes. It is worth to recall and underline that no fitting of model parameters was performed to achieve this remarkable agreement, therefore such selection of $\Delta$ can be recommended for the preliminary assessment of effective elastic properties of nanocrystalline copper. Even better agreement is obtained if the value $\Delta$ in Eq. (\ref{def:fsa}) is established to fit best the atomistic results by Eq. (\ref{eq:core-shell}). These optimized values are found to be 6.387\,{\AA} for MT and 6.837\,{\AA} for the SC variant, respectively (see Fig. \ref{fig:Approx}b).
\section{Conclusions}
\label{sec:Con}
The paper discusses the applicability of continuum-mechanics mean-field one-phase and two-phase models to estimating effective elastic properties of bulk nanocrystalline FCC copper. The methods used to verify such micromechanical mean-field estimates in the case of coarse-grained materials, such as full-field simulations on representative aggregates using the FEM or FFT framework, when they are combined with the classical linearly elastic constitutive law, are not applicable in the present context. Therefore, in this paper a set of atomistic simulations have been performed on the generated grain aggregates with randomly selected orientations. At variance with the available literature, all 21 components of the elasticity tensor $\bar{\mathbb{C}}$ are acquired by performing six independent numerical tests for ten samples of the polycrystalline material. The obtained results are then analysed with respect to the anisotropy degree of the overall stiffness tensor $\bar{\mathbb{C}}$, resulting from the limited number of grain orientations and their spatial distribution. The closest isotropic approximation of this tensor is found using the Log-Euclidean norm \cite{Moakher06}. The dependence of the obtained overall bulk and shear moduli on the average grain diameter is analysed. It is found that, while the shear modulus decreases with the grain size, the bulk modulus shows negligible dependence on the grain diameter and is close to the bulk modulus of a single crystal. When another pair of parameters: Young's modulus and Poisson's ratio are calculated, it is found that with a decreasing grain size the former decreases (in agreement with other numerical and experimental studies), while the latter increases.
Two variants of an anisotropic two-phase model of a nano-grained polycrystal, built in the spirit of \cite{Jiang04,Capolungo07} and called MT and SC core-shell models, have been formulated. In the model the thickness of the shell is specified by the \emph{cutoff radius} of a corresponding atomistic potential, while the grain shell has the stiffness tensor corresponding to the lower zero-order bound of $\bar{\mathbb{C}}$. Under such assumptions, in the case of grain cores with cubic elastic symmetry, the effective stiffness tensor of a bulk polycrystal is specified by an explicit formula. It has been shown that the obtained estimates are in satisfactory qualitative and quantitative agreement with the results of atomistic simulations performed for nano-crystalline copper. In particular, in accordance with the atomistic simulations, the predicted bulk modulus does not depend on the grain size, while the shear modulus decreases with it.
In future studies it should be checked whether the observed difference in the dependence of the bulk and shear moduli on the grain size is valid for other metals with cubic lattice symmetry, especially when they differ in their Zener anisotropy factor. Such studies have been already initiated. The proposed mean-field two-phase model can be extended to estimate a non-linear response of a nano-grained polycrystal (compare \cite{Jiang04,Capolungo07}) and specifically the yield strength. The extension will also require verification based on atomistic simulations.
|
{
"timestamp": "2018-04-17T02:13:29",
"yymm": "1804",
"arxiv_id": "1804.05503",
"language": "en",
"url": "https://arxiv.org/abs/1804.05503"
}
|
\section{Introduction}\label{sec:1}
The fullerene molecule was discovered in 1986 by R. Smalley, R. Curl, J.
Heath, S. O'Brien, and H. Kroto, and since then continues to attract great
deal of attention in the scientific community (e.g. the original work
\cite{Kr} has currently 15 thousand citations). Its importance led in 1996 to
awarding the Nobel prize in chemistry to Smalley, Curl and Kroto. Since
its discovery, applications of fullerene C60 have been extensively explored in biomedical research due to their unique structure and physicochemical properties.
\begin{figure}[ptb]
\begin{center}
\resizebox{11cm}{!}{\includegraphics{0.png} }
\end{center}
\caption{The fullerene molecule}%
\label{fig:fullerene}%
\end{figure}
The fullerene molecule is composed of $60$ carbon atoms at the vertices of a
truncated icosahedron (see Figure \ref{fig:fullerene}). Various mathematical
models for the fullerene molecule are built in the framework of classical
mechanics. Many force fields have been proposed for the fullerene in terms of
bond stretching, bond bending, torsion and van der Waals forces. Some force
fields were optimized to duplicate the normal modes obtained using IR or Raman
spectroscopy, but only few of these models reflect the nonlinear
characteristics of the fullerene. For instance, the force field implemented in
\cite{Wa} for carbon nanotubes and in \cite{Be} for the fullerene, assumes
bond deformations that exceed very small fluctuations about equilibrium
states, while the force fields proposed in \cite{We} and \cite{Wu} are
designed only to consider small fluctuations in the fullerene model. Although
the linear vibrational modes of fullerene can be measured using IR and Raman
spectroscopy, increasing attention has been given to the mathematical study of
other vibrational modes that cannot be measured experimentally (see
\cite{He,Ho,Ji,Wa,We,Wu,Sc} and the large bibliography therein).
\paragraph{Mathematical Models.}
In molecular dynamics, a fullerene molecule model consists of a Newtonian
system
\begin{equation}
\label{eq:1}m\ddot u=-\nabla V(u),
\end{equation}
where the vector $u(t)\in\widetilde{\mathscr V}:=\mathbb{R}^{180}$ represents
the positions of 60 carbon atoms in space, $m$ is the carbon mass (which by
rescaling the model can be assumed to be one) and $V (u)$ is the energy given
by a force field. This force field is symmetric with respect to the action of the
group $\Gamma:=I\times O(3)$, where $I$ denotes the full icosahedral group.
This important property allows application of various equivariant methods to
analyze its dynamical properties. In order to make the system \eqref{eq:1}
reference point-depended, we define the subspace $\mathscr V$ of
$\mathbb{R}^{180}$ by
\begin{equation}
\label{eq:space}\mathscr V:=\{x=(x_{1},x_{2},\dots, x_{60}): x_{k}%
\in\mathbb{R}^{3},\; \sum_{k=1}^{60} x_{k}=0\},
\end{equation}
from which we exclude the collision orbits, i.e. we consider the restriction
of the system \eqref{eq:1} to the set $\Omega_{o}:=\{x\in\mathscr V:
x_{j}\not = x_{k}, \; j\not =k\}$.
\paragraph{Analysis of Nonlinear Molecular Vibrations.}
The mathematical analysis of a molecular model includes two objectives:
identification of the \textit{normal frequencies} and the classification of different
families of \textit{periodic solutions with various spatio-temporal symmetries}
emerging from the equilibrium configuration of the
molecule (nonlinear normal modes). Let us emphasize that the classification of normal modes
is a central problem of the molecular spectroscopy.
The study of
periodic orbits in Hamiltonian systems can be traced back to Poincare and
Lyapunov, who proved the existence of nonlinear normal modes (periodic orbits)
near an elliptic equilibrium under non-resonant conditions. Later on,
Weinstein (cf. \cite{Weinstein}) extended this result to the case with
resonances. However, in general, the existence of nonlinear normal modes is
not guaranteed under the presence of resonances. Indeed, Moser presented in
\cite{Mos} an example of a Hamiltonian systems with resonances, where the
linear system is full of periodic solutions, but the nonlinear system has
none.
Let us point out that due to the icosahedral symmetries, the fullerene molecule
$C_{60}$ has resonances of multiplicities 3, 4 and 5, i.e. the existence of linear modes does not guarantee the
existence of nonlinear normal modes in fullerene $C_{60}$ due to resonances. Therefore, one needs
a good method that takes in consideration these symmetries. Standard methods
for such analysis may use reductions to the $H$-fixed point spaces, normal
form classification, center manifold theorem, averaging method and/or
Lyapunov--Schmidt reduction.
\paragraph{Variational Reformulation.}
The problem of finding periodic solutions for the fullerene can be
reformulated as a variational problem on the Sobolev space $H_{2\pi}%
^{1}(\mathbb{R};{\mathscr V})$ (of $2\pi$-periodic ${\mathscr V}$-valued
functions) with the functional
\[
J_{\lambda}(u):=\int_{0}^{2\pi}\left[ \frac{1}{2}|\dot{u}(t)|^{2}-\lambda
^{2}V(u(t))\right] dt,\quad u\in H_{2\pi}^{1}(\mathbb{R};\Omega_{o}),
\]
where $V$ is the force field, $\lambda^{-1}$ the frequency and $u$ the
renormalized $2\pi$-periodic solution. The existence of periodic solutions
(with fixed frequency $\lambda^{-1}$) is equivalent to the existence of
critical points of $J_{\lambda}$. It follows from the construction of the
force field that the functional $J_{\lambda}$ is invariant under the action of
the group
\[
G:=\Gamma\times O(2)=(I\times O(3))\times O(2),
\]
which acts as permutations of atoms, rotations in space, and translations and
reflection in time, respectively.
\begin{figure}[ptb]
\begin{center}
\resizebox{7cm}{!}{\includegraphics{1.png} }
\end{center}
\caption{Local bifurcation.}\label{fig:bif}
\end{figure}
\paragraph{Gradient Equivariant Degree Method.}
To provide an alternative to the equivariant singularity theory (cf.
\cite{Golubitsky}) and other geometric methods that have been used to analyze molecules (see \cite{Hoy3},
\cite{Mon1}, \cite{Mon2} and references), we proposed (see
\cite{GaTe16,BeKr,BeGa}) the method based on the equivariant gradient degree
(fundamental properties of the gradient equivariant degree are collected in
Appendix \ref{sec:equi-degree}) -- a generalization of the
Brouwer/Leray-Schauder degree that was developed in \cite{Geba} for the
gradient maps (see also \cite{DaKr} and \cite{RY2}). The gradient equivariant
degree is just one of many equivariant degrees that were introduced in the
last three decades for various types of differential equations (see
\cite{survey}, \cite{BaKr06}, \cite{IzVi03}, \cite{KW} and references therein).
To describe the main idea of this method, let us point out that the gradient
equivariant degree satisfies all the standard properties expected from a
degree theory (i.e. existence, additivity, homotopy and multiplicativity
properties). The $G$-equivariant gradient degree $\nabla_{G}\text{-Deg}(\nabla
J_{\lambda},\mathscr U)$ of $\nabla J_{\lambda}$ on $\mathscr U$ can be
expressed elegantly as an element of the Euler ring $U(G)$ (which is the free
$\mathbb{Z}$-module generated by the conjugacy classes $(H)$ of closed
subgroups $H\leq G$) in the form
\[
\nabla_{G}\text{-Deg}(\nabla J_{\lambda},\mathscr U)=n_{1}(H_{1})+n_{2}%
(H_{2})+\dots+n_{m}(H_{m}),\;\;\;n_{k}\in\mathbb{Z},
\]
where $\mathscr U$ is a neighborhood of the $G$-orbit of the equilibrium
$u_{o}$ (for some non-critical frequency $\lambda^{-1}$) and $(H_{j})$ are the
orbit types in $\mathscr U$. The changes of $\nabla_{G}\text{-deg}(\nabla
J_{\lambda},\mathscr U)$ when $\lambda^{-1}$ crosses a critical frequency
$\lambda_{o}^{-1}$ allow to establish the existence of various families of
orbits of periodic molecular vibrations and their symmetries emerging from an equilibrium. In fact, the equivariant topological invariant
\begin{equation}\label{eq:top-inv}
\omega_{G}(\lambda_{o}):=\nabla_{G}\text{-Deg\thinspace}(\nabla J_{\lambda
_{-}},\mathscr U)-\nabla_{G}\text{-Deg\thinspace}(\nabla J_{\lambda_{+}%
},\mathscr U)
\end{equation}
provides a full topological
characterization of the families of periodic solutions (together with their
symmetries) emerging from an equilibrium at $\lambda_{o}$ (cf. \cite{DGR}).
More precisely, for every non-zero coefficient $m_{j}$ in
\[
\omega_{G}(\lambda_{o})=m_{1}(K_{1})+m_{2}(K_{2})+\dots m_{r}(K_{r}),
\]
there exists a global family of periodic molecular vibrations with symmetries
at least $K_{j}$ (see Figure \ref{fig:bif} below). Moreover, if $(K_{j})$ is a maximal
orbit type then this family has exact symmetries $K_{j}$.
\paragraph{Global Bifurcation Result.}
The so-called classical Rabinowitz Theorem \cite{Ra} establishes occurrence of
a global bifurcation from purely local data for compact perturbations of the
identity. Its main idea is that if the maximal connected set $\mathcal{C}$
bifurcating from a trivial solution is compact (i.e. bounded), then the sum of
the local Leray-Schauder degrees at the set of bifurcation points of
$\mathcal{C}$ is zero. Since such maximal connected set $\mathcal{C}$ is
either unbounded or comes back to another bifurcation point, this result is
also referred to as the \emph{global Rabinowitz alternative} (we refer to
Nirenberg's book \cite{Ni} where a simplified proof of this statement is
presented in Theorem 3.4.1).
The classical Rabinowitz's global bifurcation argument can be easily adapted
in the equivariant setting for the gradient $G$-equivariant degree (cf.
\cite{GolRyb}). That is, for any $G$-orbit of a compact (bounded) branch
$\mathcal{C}$ in $\mathbb{R}_{+}\times H_{2\pi}^{1}(\mathbb{R};\Omega_{o})$
containing $(\lambda_{0},u_{o})$ we have
\begin{equation}
\sum_{k=0}^{m}\omega_{G}(\lambda_{k})=0,\label{eq:int-global}%
\end{equation}
(see Figure \ref{fig:glob-bif}), where $\lambda_{k}^{-1}$ are the normal modes
belonging to $\mathcal{C}$. In this context the \textbf{global property} means
that a family of periodic solutions, represented by continuous branch
$\mathcal{C}$ in $\mathbb{R}_{+}\times H_{2\pi}^{1}(\mathbb{R};\Omega_{o})$,
is not compact or comes back to another bifurcation point from the equilibrium.
The non-compactness of $\mathcal{C}$ implies that the norm or period of
solutions from $\mathcal{C}$ goes to the infinity, $\mathcal{C}$ ends in a
collision orbit or goes to a different equilibrium point.
\begin{figure}
\begin{center}
\resizebox{11cm}{!}{\includegraphics{2.png} }
\end{center}
\caption{Global bifurcation}\label{fig:glob-bif}
\end{figure}
By applying formula \eqref{eq:int-global} one can establish an effective
criterium allowing to determine the existence of the non-compact
branches of nonlinear normal modes with particular (e.g. maximal) orbit types.
To be more precise, it is sufficient to consider all the critical frequencies
$\lambda_{k}^{-1}$ corresponding to the first Fourier mode and simply show,
that for some of them, say $\lambda_{0}^{-1}$, the sum in
\eqref{eq:int-global} can never be zero. For the fullerene molecule such non-compact global
branches exist.
\paragraph{Novelty.}
In this paper, we apply equivariant gradient degree
to the classification of the global nonlinear modes in a model of the
fullerene molecule $C_{60}$. By taking advantage of various properties of the gradient
equivariant degree, the approximate values (obtained numerically) of the
normal frequencies can be used to determine
(under plausible isotypical non-resonance assumption) the exact values of the topological
invariants $\omega_{G}(\lambda_{o})$. In particular, the information contained
in the topological invariants can be applied to obtain the presence of such
global branches of periodic solutions with the maximal orbit types, as it is
presented in our main theorem (Theorem \ref{th:main}).
Let us point out that (to the best of our
knowledge) all the previous studies of the fullerene molecule have considered
only the existence of the linear modes (which have constant frequency), while the
occurrence of the non-linear normal modes have frequencies depending on
the amplitudes of the oscillation. \emph{Such an
analysis of nonlinear normal modes for the fullerene molecule was never
done before}. We complement our results with numerical computations using Newton's method
and pseudo-arclength procedure to continue some of these nonlinear
normal modes.
It is important to notice that the icosahedral symmetries appear also in adenoviruses with icosahedral capsid, or
other icosahedral molecules considered in \cite{We}. The methods presented
here are applicable to these cases as well.
\vskip.3cm
\paragraph{Contents.}
The rest of the paper is arranged as follows. In section
\ref{sec:fullerene} we present the model equations
appropriate for studying the dynamics of the fullerene molecule. In subsection
\ref{sec:coordinates}, we propose a new indexation for the fullerene atoms
which greatly simplifies the description of symmetries in the molecule. Then, in subsection \ref{sec:forcefield} we discuss the
choice of the force field for the fullerene molecule that seems to be the most
appropriate in order to model nonlinear vibrations and in subsection
\ref{sec:iso-symm} we describe the action of the group $I\times O(3)$ on the
space $\mathscr V$. Then, in subsection \ref{sec:minimizer}, we find the
minimizer of the potential $V$ among the configurations with icosahedral
symmetries by applying Palais criticality principle. In subsection
\ref{sec:iso-decomp}, we identify the $I$-isotypical decomposition of the
space $\mathscr V$ and use it to determine the spectrum of the operator
$\nabla^{2}V(u_{o})$ and the $I$-isotypical types of the corresponding
eigenspaces. In Section \ref{sec:eq-bif} we prove the bifurcation of periodic
solutions from the equilibrium configuration $u_{o}$ of the fullerene
molecule. In Section \ref{sec:symmetries} we describe the symmetries of the
periodic solutions. In addition to the theoretical results stated Theorem \ref{th:main},
several of these symmetric periodic solutions were obtained
by numerical continuation, for which the numerical data is shown graphically.
In Appendix \ref{sec:equi-degree}, we include
a short review of the gradient degree, including computational algorithms, and
the computations of the $I\times O(2)$-equivariant gradient degree.
\vskip.3cm
\section{Fullerene Model}
\label{sec:fullerene}
\subsection{Equations for Carbons}
\label{sec:coordinates}
In this section, we propose a new indexation for the fullerene atoms
which greatly simplifies the description of symmetries in the molecule: to
each atom we assigned two indices -- one (being a $5$-cycle in $S_{5}$)
indicating in which the side of the dodecahedron the atom is located and the
second indicating its position in that side (as it is illustrated on Figure
\ref{fig:fullerene})
Let $A_{5}$ be the alternating group of permutations of five elements
$\{1,2,3,4,5\}$. The five conjugacy classes in $A_{5}$ are listed in Table
\ref{tab:conjugacy}. The $C_{60}$ molecule is arranged in $12$ unconnected
pentagons of atoms. We implement the following notation for the indices of the
$60$ atoms (see Figure \ref{fig:fullerene}):
\begin{itemize}
\item $\tau\in\mathcal{C}_{4}$ is used to denote each of the $12$ pentagonal faces.
\item $k\in\{1,...,5\}=:\mathbb{Z}[1,5]$ is used to denote each of the $5$
vertices in the $12$ pentagonal faces.
\end{itemize}
\begin{table}[ptb]
\vskip.4cm
\par
\begin{center}%
\begin{tabular}
[c]{|c|c|c|c|c|}%
\noalign{\vskip2pt\hrule\vskip2pt} $\mathcal{C}_{1}$ & $\mathcal{C}_{2}$ &
$\mathcal{C}_{3}$ & $\mathcal{C}_{4}$ & $\mathcal{C}_{5}$\\
\noalign{\vskip2pt\hrule\vskip2pt} $(1)$ & $(12)(34)$, $(13)(24)$,$(14)(23)$ &
$(123)$, $(132)$ & $(12345)$ & $(12354)$\\
& $(12)(35)$, $(13)(25)$,$(15)(23)$ & $(124)$, $(142)$ & $(12453)$ &
$(12435)$\\
& $(12)(45)$, $(14)(25)$,$(15)(24)$ & $(125)$, $(152)$ & $(12534)$ &
$(12543)$\\
& $(13)(45)$, $(14)(35)$, $(15)(34)$ & $(134)$, $(143)$ & $(13254)$ &
$(13245)$\\
& $(23)(45)$, $(24)(35)$,$(25)(34)$ & $(135)$, $(153)$ & $(13542)$ &
$(13524)$\\
& & $(145)$, $(154)$ & $(13425)$ & $(13452)$\\
& & $(234)$, $(243)$ & $(14235)$ & $(14253)$\\
& & $(235)$, $(253)$ & $(14352)$ & $(14325)$\\
& & $(245)$, $(254)$ & $(14523)$ & $(14532)$\\
& & $(345)$, $(354)$ & $(15243)$ & $(15234)$\\
& & & $(15432)$ & $(15423)$\\
& & & $(15324)$ & $(15342)$\\\hline
\end{tabular}
\end{center}
\caption{Conjugacy classes of elements in $A_{5}$.}%
\label{tab:conjugacy}%
\end{table}
We define the set of indices as
\[
\Lambda=\mathcal{C}_{4}\times\mathbb{Z}[1,5]~\text{.}%
\]
With these notations each index $(\tau,k)\in\Lambda$ represents a face and a
vertex in the face of the truncated icosahedron as it is shown on Figure
\ref{fig:fullerene}. The vectors that represent the positions of the carbon
atoms are%
\[
u_{\tau,k}\in\mathbb{R}^{3},
\]
and the vector for the $60$ positions is
\[
u=(u_{\tau,k})_{(\tau,k)\in\Lambda}\in\left( \mathbb{R}^{3}\right)
^{60}\text{.}%
\]
The space $\left( \mathbb{R}^{3}\right) ^{60}$ is a representation of the
group $I\times O(3)$, where $I=A_{5}\times\mathbb{Z}_{2}$ stands for the full
icosahedral group. With this notation, the action of $I\times O(3)$ on $V$ has
a simple definition: the action of $\sigma\in A_{5}$ and $-1\in\mathbb{Z}_{2}$
in $u$ is defined in each component by%
\begin{equation}
\rho(\sigma)u_{\tau,k}=u_{\sigma^{-1}\tau\sigma,\sigma^{-1}(k)}~,\qquad
\rho(-1)u_{\tau,k}:=u_{\tau^{-1},k}~. \label{eq:act1}%
\end{equation}
And the action of the group $A\in O(3)$ is defined by
\[
Au=(Au_{\tau,k})_{(\tau,k)\in\Lambda}.
\]
\subsection{Force Field}
\label{sec:forcefield}
The system for the fullerene molecule is given by \eqref{eq:1}. To describe
the force field $\nabla V$ we recognize that:
\begin{itemize}
\item The $60$ edges in the $12$ pentagonal faces represent single bonds. For
these single bonds we define the function $S:\Lambda\rightarrow\Lambda,$%
\[
S(\tau,k)=(\tau,\tau(k))~, \quad\tau\in\mathcal{C}_{4},\; k\in\mathbb{Z}%
[1,5].
\]
\item The $30$ remaining edges in the hexagon, which connect the different
pentagonal faces, represent double bounds. For these double bonds we define
the function $D:\Lambda\rightarrow\Lambda$,%
\[
D(\tau,k)=(\sigma,k)\text{ with }\sigma=\left( k,\tau^{2}(k),\tau(k),\tau
^{4}(k),\tau^{3}(k)\right) .
\]
\end{itemize}
Using the above notation, the force field energy is elegantly expressed by
\[
V(u)=\sum_{\left( \tau,k\right) \in\Lambda}\left( U(\left\vert u_{\tau
,k}-u_{S(\tau,k)}\right\vert )+\frac{1}{2}U(\left\vert u_{\tau,k}%
-u_{D(\tau,k)}\right\vert )+U_{\left( \tau,k\right) }(u)\right) \text{,}%
\]
where the coefficient $\frac12$ before the second term is to eliminate the
double count bonds, and the term $U_{\left( \tau,k\right) }(u)$ includes the
bending and torsion forces.
Bond stretching is represent by potential%
\[
U(x)=E_{0}\left( \left( 1-e^{-\beta(x-r_{0})}\right) ^{2}-1\right) ,
\]
where $r_{0}$ is the equilibrium bond length, $E_{0}$ is the bond energy and
$\beta^{-1}$ is the width of the energy. The term $U_{\left( \tau,k\right)
}(u)$ includes bending and torsion forces given by%
\[
U_{\left( \tau,k\right) }(u)=E_{B}(\theta_{1})+E_{B}(\theta_{2}%
)+E_{B}(\theta_{3})+E_{T}(\phi_{1})+E_{T}(\phi_{2})+E_{T}(\phi_{3}),
\]
where the bending $E_{B}(\theta)$ around each atom in a molecule is governed
by the hybridization of orbitals and is given by%
\[
E_{B}(\theta)=\frac{1}{2}k_{0}(\cos\theta-\cos\theta_{0})^{2}=\frac{1}%
{2}k_{\theta}(\cos\theta+1/2)^{2},\quad\theta_{0}:=2\pi/3,
\]
(here $\theta_{o}$ is the equilibrium angle and $k_{0}$ is the bending force
constant), and the torsion energy $E_{T}(\phi)$ (which describes the energy
change associated with rotation around a bond with a four-atom sequence) is
given by%
\[
E_{T}(\phi)=\frac{1}{2}k_{\phi}\left( 1-\cos2\phi\right) =k_{\phi}\left(
1-\cos^{2}\phi\right) \text{.}%
\]
The torsion energy reaches a maximum value at angles $\phi=\pm\pi/2$.
Each carbon $\left( \tau,k\right) \in\Lambda$ has three angles,%
\begin{align*}
\cos\theta_{1} & =\frac{u_{\tau,k}-u_{S(\tau,k)}}{\left\vert u_{\tau
,k}-u_{S(\tau,k)}\right\vert }\bullet\frac{u_{\tau,k}-u_{S^{-1}(\tau,k)}%
}{\left\vert u_{\tau,k}-u_{S^{-1}(\tau,k)}\right\vert },\\
\cos\theta_{2} & =\frac{u_{\tau,k}-u_{D(\tau,k)}}{\left\vert u_{\tau
,k}-u_{D(\tau,k)}\right\vert }\bullet\frac{u_{\tau,k}-u_{S(\tau,k)}%
}{\left\vert u_{\tau,k}-u_{S(\tau,k)}\right\vert }~,\\
\cos\theta_{3} & =\frac{u_{\tau,k}-u_{D(\tau,k)}}{\left\vert u_{\tau
,k}-u_{D(\tau,k)}\right\vert }\bullet\frac{u_{\tau,k}-u_{S^{-1}(\tau,k)}%
}{\left\vert u_{\tau,k}-u_{S^{-1}(\tau,k)}\right\vert }.
\end{align*}
Clearly, the bond bending at each atom $\left( \tau,k\right) \in\Lambda$ is
$E_{B}(\theta_{1})+E_{B}(\theta_{2})+E_{B}(\theta_{3})$. Let
\[
n=\frac{u_{D(\tau,k)}-u_{S(\tau,k)}}{\left\vert u_{D(\tau,k)}-u_{S(\tau
,k)}\right\vert }\times\frac{u_{D(\tau,k)}-u_{S^{-1}(\tau,k)}}{\left\vert
u_{D(\tau,k)}-u_{S^{-1}(\tau,k)}\right\vert }%
\]
be the unit normal vector to the plane passing by $u_{D(\tau,k)}$,
$u_{S(\tau,k)}$ and $u_{S^{-1}(\tau,k)}$. Each carbon $\left( \tau,k\right)
\in\Lambda$ has three torsion energies given by%
\begin{align*}
\cos\phi_{1} & =n\bullet n_{1}~,\qquad n_{1}=\frac{u_{(\tau,k)}%
-u_{S(\tau,k)}}{\left\vert u_{(\tau,k)}-u_{S(\tau,k)}\right\vert }\times
\frac{u_{(\tau,k)}-u_{S^{-1}(\tau,k)}}{\left\vert u_{(\tau,k)}-u_{S^{-1}%
(\tau,k)}\right\vert }~,\\
\cos\phi_{2} & =n\bullet n_{2}~,\qquad n_{2}=\frac{u_{(\tau,k)}%
-u_{D(\tau,k)}}{\left\vert u_{(\tau,k)}-u_{D(\tau,k)}\right\vert }\times
\frac{u_{(\tau,k)}-u_{S^{-1}(\tau,k)}}{\left\vert u_{(\tau,k)}-u_{S^{-1}%
(\tau,k)}\right\vert }~,\\
\cos\phi_{3} & =n\bullet n_{3}~,\qquad n_{3}=\frac{u_{(\tau,k)}%
-u_{D(\tau,k)}}{\left\vert u_{(\tau,k)}-u_{D(\tau,k)}\right\vert }\times
\frac{u_{(\tau,k)}-u_{S(\tau,k)}}{\left\vert u_{(\tau,k)}-u_{S(\tau
,k)}\right\vert }~.
\end{align*}
Then, the bond bending at each atom $\left( \tau,k\right) \in\Lambda$ is
$E_{T}(\phi_{1})+E_{T}(\phi_{2})+E_{T}(\phi_{3})$.
For this work, we use the parameters given in \cite{Be} , which are
$E_{0}=6.1322~eV$, $\beta=1.8502~A^{-1}$, $r_{0}=01.4322$~$A$, $k_{\theta
}=10~eV$, $k_{\phi}=0.346~eV$. In this paper we use exactly these values.
\subsection{Icosahedral Symmetries}
\label{sec:iso-symm} In order to make the system \eqref{eq:1} reference
point-depended, we define the subspace
\begin{equation}
\mathscr V:=\{u\in\left( \mathbb{R}^{3}\right) ^{60}:\sum_{(\sigma
,k)\in\Lambda}u_{\sigma,k}=0\} \label{eq:V}%
\end{equation}
and
\[
\Omega_{o}=\left\{ u\in\mathscr V:u_{\sigma,k}\not =u_{\tau,j}\right\} .
\]
We have that $\mathscr V$ and $\Omega_{o}$ are flow-invariant for \eqref{eq:1}
By the properties of functions $S$ and $D$, the potential
\begin{equation}
V:\Omega_{o}\rightarrow\mathbb{R~} \label{eq:mol}%
\end{equation}
is well defined and $I$-invariant. Moreover, the potential $V$ is invariant by
rotations and reflections of the group $O(3)$ because bonding, bending and
torsion forces depend only on the norm of the distances among pairs of atoms.
Therefore, the potential $V$ is $I\times O(3)$-invariant,%
\[
V((\sigma,A)u)=V(u),\qquad(\sigma,A)\in I\times O(3)~.
\]
Let
\begin{equation}
\label{eq:infinitezimal}J_{1}:=\left[
\begin{array}
[c]{ccc}%
0 & 0 & 0\\
0 & 0 & -1\\
0 & 1 & 0
\end{array}
\right] ,\quad J_{2}:=\left[
\begin{array}
[c]{ccc}%
0 & 0 & -1\\
0 & 0 & 0\\
1 & 0 & 0
\end{array}
\right] ,\quad J_{3}:=\left[
\begin{array}
[c]{ccc}%
0 & -1 & 0\\
1 & 0 & 0\\
0 & 0 & 0
\end{array}
\right]
\end{equation}
be the three infinitesimal generators of rotations in $O(3)$, i.e., $e^{\phi
J_{1}}$, $e^{\theta J_{2}}$ and $e^{J_{3}\psi}$, where $\phi$, $\theta$ and
$\psi$ are the Euler angles. The angle between two adjacent pentagons in a
dodecahedron is $\theta_{0}=\arctan2$. Then, the rotation by $\pi$ that fixes
a pair of antipodal edges is%
\begin{equation}
A=e^{-\left( \theta_{0}/2\right) J_{2}}e^{\pi J_{3}}e^{\left( \theta
_{0}/2\right) J_{2}}=%
\begin{bmatrix}
-\frac{1}{\sqrt{5}} & 0 & \frac{2}{\sqrt{5}}\\
0 & -1 & 0\\
\frac{2}{\sqrt{5}} & 0 & \frac{1}{\sqrt{5}}%
\end{bmatrix}
\text{.}%
\end{equation}
The rotation by $2\pi/5$ of the upper pentagonal face of a dodecahedron is%
\begin{equation}
B=e^{\frac{2\pi}{5}J_{3}}=%
\begin{bmatrix}
\frac{-1+\sqrt{5}}{4} & -\sqrt{\frac{5+\sqrt{5}}{8}} & 0\\
\sqrt{\frac{5+\sqrt{5}}{8}} & \frac{-1+\sqrt{5}}{4} & 0\\
0 & 0 & 1
\end{bmatrix}
~\text{.}%
\end{equation}
The subgroup of $O(3)$ generated by the matrices $A$ and $B$ is isomorphic to
icosahedral group $A_{5}$. Indeed, the generators $A$ and $B$ satisfy the
relations%
\[
A^{2}=B^{5}=(AB)^{3}=\text{\textrm{Id\,}}\ .
\]
On the other hand, the group $A_{5}$ is generated by
\begin{equation}
a=(23)(45),\qquad b=\left( 12345\right) \text{,}%
\end{equation}
and we have similar relations%
\[
a^{2}=b^{5}=(ab)^{3}=(1).
\]
Therefore, the explicit homomorphism $\rho:A_{5}\rightarrow\mathrm{SO}(3)$
defined on generators by $\rho(a):= A$ and $\rho( b):= B$ is the required
isomorphism $A_{5}\simeq\rho(A_{5})\subset SO(3)$. We extend
\[
\rho:A_{5}\times\mathbb{Z}_{2}\rightarrow O(3)
\]
with $\rho(-1)=-\text{\textrm{Id\,}}\in O(3)$, and consequently we obtain an
explicit identification of the full icosahedral group $I$ with $\rho(I)\subset
O(3)$.
\subsection{Icosahedral Minimizer}
\label{sec:minimizer}
Let $\tilde{I}$ be the icosahedral subgroup of $I\times O(3)$ given by
\[
\tilde{I}=\left\{ (\sigma,\rho(\sigma))\in I\times O(3):\sigma\in I\right\}
~.
\]
The fixed point space
\[
\Omega_{0}^{\tilde{I}}=\mathscr V^{\tilde{I}}\cap\Omega_{o}=\{\left(
a_{\tau,k}\right) _{(\tau,k)\in\Lambda}\in\Omega_{0}:a_{\tau,k}=(\sigma
,\rho(\sigma))a_{\tau,k}\},
\]
consist of all the truncated icosahedral configurations. An equilibrium of the
fullerene molecule can be found as a minimizer of $V$ on these configurations.
More precisely, since $V$ is $I\times O(3)$-invariant, by Palais criticality
principle, the minimizer of the potential $V$ on the fixed-point space of
$\tilde{I}$ is a critical point of $V$.
To find the minimizer among configurations with symmetries $\tilde{I}$, we
parameterize the carbons positions by fixing the position of one of them. Let
\[
u_{b,1}=(x,0,z), \quad x,z\in\mathbb{R},
\]
then we have
\[
u_{b,1}=(\sigma,\rho(\sigma))u_{b,1}=\rho(\sigma)u_{\sigma^{-1}b\sigma
,\sigma^{-1}(1)}\text{~,}%
\]
and relations
\begin{equation}
u_{\sigma^{-1}b\sigma,\sigma^{-1}(1)}=\rho(\sigma)^{-1}u_{b,1}=\rho
(\sigma)^{-1}(x,0,z)^{T},\quad\sigma\in A_{5}, \label{rel}%
\end{equation}
allow us to determine the positions of all other coordinates of the vector
$u=\left( u_{\tau,k}\right) $.
Therefore, the representation of $u(x,z)$ given by (\ref{rel}) provides us a
parametrization of a connected component of $\Omega_{0}^{\tilde{I}}/O(3)$ with
the domain
\[
\mathcal{D}=\{(x,z)\in\mathbb{R}^{2}:0<z,\;x<Cz\}
\]
where $C>0$ is a number determined by the geometric restrictions. We define
$v:\mathcal{D}\rightarrow\mathbb{R}$ by
\[
v(x,z)=V(u(x,z)).
\]
Since $v(x,z)$ is exactly the restriction of $V$ to the fixed-point subspace
$\Omega_{0}^{\tilde{I}}/O(3)$, then by the symmetric criticality condition, a
critical point of $v:\mathcal{D}\rightarrow\mathbb{R}$ is also a critical
point of $V$.
We implemented a numerical minimizing procedure to find the minimizer
$(x_{o},z_{o})$ of $v(x,z)$. We denote the truncated icosahedral minimizer
corresponding to the fullerene $C_{60}$ as
\[
u_{o}=u\left( x_{o},z_{o}\right) \in\mathscr V.
\]
The lengths of single and double bonds for this minimizer are given by%
\begin{align*}
d_{S}=\left\vert u_{b,1}-u_{S(b,1)}\right\vert =1.438084,\\
d_{D}=\left\vert u_{b,1}-u_{D(b,1)}\right\vert =1.420845\text{,}%
\end{align*}
respectively. These results are in accordance with the distances measured in
the paper \cite{He}.
An advantage of the notation $u=\left( u_{\tau,k}\right) $ is that it is
easy to visualize the elements associated to the rotations $\rho(\sigma)$ in
the truncated icosahedron (Figure \ref{fig:fullerene}). In these
configurations we have $\rho(\sigma)u_{\tau,k}=\sigma^{-1}u_{\tau,k}%
=u_{\sigma\tau\sigma^{-1},\sigma(k)}$, then $\rho(\sigma)$ is identified with
the rotation that sends the face $\tau$ to $\sigma\tau\sigma^{-1}$ and the
carbon atom identified by $k$ to $\sigma(k)$; for example, under the $\pi
$-rotation given by $\rho(a)=A$, face $b=(12345)$ goes to the face
$aba^{-1}=(13254)$, and the element $k=1$ to $a(1)=1$. In this manner, we
conclude that the elements of the conjugacy classes $\mathcal{C}_{2}$,
$\mathcal{C}_{3}$, $\mathcal{C}_{4}$ and $\mathcal{C}_{5}$ correspond to the
$15$ rotations by $\pi$, the $20$ rotations by $2\pi/3$, the $12$ rotations by
$2\pi/5$ and the $12$ rotations by $\pi/5$, respectively.
\subsection{Isotypical Decomposition and Spectrum of $\nabla^{2}V(u_{o})$}
\label{sec:iso-decomp}
The space $\mathscr V$ is an orthogonal complement in $\widetilde
{\mathscr V}=(\mathbb{R}^{3})^{60}$ of the subspace $\{(v,v,\dots,v):
v\in\mathbb{R}^{3}\}$, thus it is $\tilde{I}$-invariant. Given that the system
\eqref{eq:1}, $u(t)\in\mathscr V$, is symmetric with respect the group action
of $I\times O(3)$, the orbit of equilibria $u_{o}$ is a $3$-dimensional
submanifold in $\mathscr V$. To describe the tangent space, we define
\[
\mathcal{J}_{j}u=(J_{j}u_{\sigma,k}),
\]
where $J_{j}$ are the three infinitesimal generators of the rotations defined
in \eqref{eq:infinitezimal}. Then, the slice $S_{o}$ to the orbit of $u_{o}$
is
\[
S_{o}:=\{x\in\mathscr V:x\bullet\mathcal{J}_{j}u_{0}=0,\quad j=1,2,3\}.
\]
Since $u_{o}$ has the isotropy group $\tilde{I}$, then $S_{o}$ is an
orthogonal $\tilde{I}$ representation.
In this section we identify the $\tilde{I}$-isotypical components of $S_{o}$.
In order to simplify the notation, hereafter we identify $\tilde{I}$ with the
group $A_{5}\times\mathbb{Z}_{2}$,%
\[
\tilde{I}=A_{5}\times\mathbb{Z}_{2}~.
\]
Put $\varphi_{\pm}=\frac{1}{2}(1\pm\sqrt{5})$, and consider the permutations
$a=(2,3)(4,5)$, $b=\left( 1,2,3,4,5\right) $ and $c=(1,2,3)$. The character
table of $A_{5}$ is:%
\[%
\begin{tabular}
[c]{|cc|ccccc|}\hline
Rep. & Character & $(1)$ & $a$ & $c$ & $b$ & $b^{2}$\\\hline
$\mathcal{V}_{1}$ & $\chi_{1}$ & $1$ & $1$ & $1$ & $1$ & $1$\\
$\mathcal{V}_{2}$ & $\chi_{2}$ & $4$ & $0$ & $1$ & $-1$ & $-1$\\
$\mathcal{V}_{3}$ & $\chi_{3}$ & $5$ & $1$ & $-1$ & $0$ & $0$\\
$\mathcal{V}_{4}$ & $\chi_{4}$ & $3$ & $-1$ & $0$ & $\varphi_{+}$ &
$\varphi_{-}$\\
$\mathcal{V}_{5}$ & $\chi_{5}$ & $3$ & $-1$ & $0$ & $\varphi_{-}$ &
$\varphi_{+}$\\\hline
\end{tabular}
\ \ \ \ \
\]
The character table of $\tilde{I}\simeq A_{5}\times\mathbb{Z}_{2}$ is obtained
from the table of $A_{5}$. We denote the irreducible representations of $I$
by\ $\mathcal{V}_{\pm n}$ for $n=1,...,5$, where the element $-1\in
\mathbb{Z}_{2}$ acts as $\pm\text{\textrm{Id\,}}$ in $\mathcal{V}_{\pm n}$ and
elements $\gamma\in A_{5}$ act as they act on $\mathcal{V}_{n}$. Notice that
all the representations $\mathcal{V}_{\pm n}$ are absolutely irreducible.
Therefore, the character table for $A_{5}\times\mathbb{Z}_{2}$ is as follows:
\begin{table}[ptb]
\begin{center}%
\begin{tabular}
[c]{|c|cccccccccc|}\hline
& (1) & $a$ & $c$ & $b$ & $b^{2}$ & $(-1)$ & $(a,-1) $ & $(c, -1) $ & $(b,-1)$
& $(b^{2},-1) $\\\hline
$\chi_{1}$ & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\
$\chi_{2}$ & 4 & 0 & 1 & -1 & -1 & 4 & 0 & 1 & -1 & -1\\
$\chi_{3}$ & 5 & 1 & -1 & 0 & 0 & 5 & 1 & -1 & 0 & 0\\
$\chi_{4}$ & 3 & -1 & 0 & $\varphi_{+} $ & $\varphi_{-}$ & 3 & -1 & 0 &
$\varphi_{+}$ & $\varphi_{-}$\\
$\chi_{5}$ & 3 & -1 & 0 & $\varphi_{-}$ & $\varphi_{+}$ & 3 & -1 & 0 &
$\varphi_{-}$ & $\varphi_{+}$\\
$\chi_{-1}$ & 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1\\
$\chi_{-2}$ & 4 & 0 & 1 & -1 & -1 & -4 & 0 & -1 & 1 & 1\\
$\chi_{-3}$ & 5 & 1 & -1 & 0 & 0 & -5 & -1 & 1 & 0 & 0\\
$\chi_{-4}$ & 3 & -1 & 0 & $\varphi_{+} $ & $\varphi_{-}$ & -3 & 1 & 0 &
$-\varphi_{+} $ & $-\varphi_{-}$\\
$\chi_{-5}$ & 3 & -1 & 0 & $\varphi_{-}$ & $\varphi_{+} $ & -3 & 1 & 0 &
$-\varphi_{-} $ & $-\varphi_{+}$\\\hline
\end{tabular}
\end{center}
\caption{Character table for $A_{5}\times\mathbb{Z}_{2}$}%
\label{tab:I}%
\end{table}\vskip.3cm By comparing the character $\chi_{_{\mathscr V}}$ withe
the characters in Table \ref{tab:I}, we obtain the following $I$-isotypical
decomposition of $\mathscr V$
\[
\mathscr V=\bigoplus_{n=1}^{5} \mathscr V_{n}\oplus\mathscr V_{-n},
\]
where $\mathscr V_{\pm n}$ is modeled on $\mathcal{V}_{\pm n}$.
\vskip.3cm
We numerically computed the spectrum $\{\mu_{j}: j=1,2,\dots, 47\}$ of the
Hessian $\nabla^{2}V(u_{o})$ at this minimizer $u_{o}$. Since $\nabla
^{2}V(u_{o}):\mathscr V\rightarrow\mathscr V$ is $\tilde{I}$-equivariant,
\[
\nabla^{2}V(u_{o})|_{\mathscr{V}_{n}\cap E(\mu_{j})}=\mu_{j}%
\,\text{\textrm{Id}}:\mathscr{V}_{n}\cap E(\mu_{j})\rightarrow\mathscr{V}_{n}%
\cap E(\mu_{j}),
\]
where $E(\mu_{j})$ stands for the eigenspace corresponding to $\mu_{j}$. We
found that each of the eigenspaces $E(\mu_{j})$ is an irreducible
subrepresentation of $\mathscr V$ , i.e. the isotypical multiplicity of
$\mu_{j}$ is simple. Including the zero eigenspace, we have $47$ different
components. Thus we have that $\sigma(\nabla^{2}V(u_{o})|_{{S_{o}}})=\{\mu
_{1},...,\mu_{46}\}$ with $\mu_{j}>0$, so
\begin{equation}
S_{o}=\bigoplus_{j=1}^{46}\mathcal{V}_{n_{j}} \quad\text{ and } \quad
\nabla^{2}V(u_{o})|_{\mathcal{V}_{n_{j}}}=\mu_{j}\,\text{\textrm{Id}.}
\label{eq:S4-iso-S}%
\end{equation}
In order to determine in which isotypical component $\mathscr V_{n_{j}}$ the
eigenspace $E(\mu_{j})$ is contained for a given eigenvalue $\mu_{j}$, we
apply the isotypical projections
\[
P_{{n}}v:=\frac{\dim(\mathcal{V}_{n})}{120}\sum_{g\in\tilde{I}}\chi
_{n}(g)~gv,\quad v\in\mathscr V,
\]
where $\mathcal{V}_{n}$, $n=\pm1,\dots,\pm5$, is the irreducible
representation identified by the character Table \ref{tab:I}. Then the
component $\mathscr V_{n_{j}}$ can be clearly identified by the projection
$P_{n_{j}}$.
\begin{table}[t]
\begin{center}
\scalebox{.7}{
\begin{tabular}
[c]{|c|cccc|}\hline
$j$ & \text{{\small Mult.}} & $\mu_{j}$ & $\lambda_j$&${\small n}_{j}$\\
\hline
{\small 1} & {\small 5} & {\small 176.536} &{\small $ 0.075263 $}& -{\small 3}\\
{\small 2} & {\small 5} & {\small 176.366} &{\small $ 0.075300 $}& {\small 3}\\
{\small 3} & {\small 4} & {\small 164.083} &{\small $ 0.078067 $}& {\small 2}\\
{\small 4} & {\small 4} & {\small 160.292} &{\small $ 0.078985 $}& -{\small 2}\\
{\small 5} & {\small 3} & {\small 159.290} &{\small $ 0.079233$}& -{\small 5}\\
{\small 6} & {\small 5} & {\small 148.597} &{\small $ 0.082034 $}& {\small 3}\\
{\small 7} & {\small 3} & {\small 141.071} &{\small $ 0.084194 $}& -{\small 4}\\
{\small 8} & {\small 3} & {\small 140.573} &{\small $ 0.084343 $}& {\small 5}\\
{\small 9} & {\small 1} & {\small 135.632} &{\small $ 0.085866 $}& {\small 1}\\
{\small 10} & {\small 4} & {\small 134.935} &{\small $ 0.086087$}& -{\small 2}\\
{\small 11} & {\small 4} & {\small 129.544} &{\small $ 0.087860 $}& {\small 2}\\
{\small 12} & {\small 5} & {\small 125.431} &{\small $ 0.089289 $}& -{\small 3}\\
{\small 13} & {\small 3} & {\small 107.719} &{\small $ 0.096350 $}& {\small 4}\\
{\small 14} & {\small 5} & {\small 98.5525} &{\small $ 0.100732 $}& {\small 3}\\
{\small 15} & {\small 3} & {\small 93.4648} &{\small $ 0.103437 $}& -{\small 5}\\
{\small 16} & {\small 5} & {\small 87.7541} &{\small $0.106750 $}& -{\small 3}\\
{\small 17} & {\small 3} & {\small 83.9718} &{\small $ 0.109127 $}& -{\small 4}\\
{\small 18} & {\small 4} & {\small 71.6288} &{\small $ 0.118156 $} &{\small 2}\\
{\small 19} & {\small 5} & {\small 67.1181} &{\small $ 0.122062 $} &{\small 3}\\
{\small 20} & {\small 1} & {\small 59.3865} &{\small $ 0.129765$}& -{\small 1}\\
{\small 21} & {\small 3} & {\small 50.4797} &{\small $0.140748 $}& -{\small 5}\\
{\small 22} & {\small 4} & {\small 47.5646} &{\small $ 0.144997$}& -{\small 2}\\
{\small 23} & {\small 3} & {\small 42.2947} &{\small $ 0.153765$}& {\small 4}\\
\hline
\end{tabular}
\qquad\begin{tabular}
[c]{|c|cccc|}\hline
$j$ & Mult. & ${\small \mu}_{j}$ & $\lambda_j$&$n_{j}$\\
\hline
{\small 24} & {\small 3} & {\small 41.3918} &{\small $ 0.155433$}& {\small 5}\\
{\small 25} & {\small 4} & {\small 33.9885} &{\small $ 0.171528$}& -{\small 2}\\
{\small 26} & {\small 5} & {\small 28.8031} &{\small $ 0.186329 $}& -{\small 3}\\
{\small 27} & {\small 5} & {\small 27.4795} &{\small $ 0.190764 $}& {\small 3}\\
{\small 28} & {\small 3} & {\small 27.3153} &{\small $ 0.191336 $}& {\small 5}\\
{\small 29} & {\small 4} & {\small 25.5388} &{\small $ 0.197879 $}& -{\small 2}\\
{\small 30} & {\small 4} & {\small 22.7212} &{\small $ 0.209790$}& {\small 2}\\
{\small 31} & {\small 5} & {\small 19.4536} &{\small $ 0.226725 $}& -{\small 3}\\
{\small 32} & {\small 5} & {\small 19.3377} & {\small $ 0.227404 $}&{\small 3}\\
{\small 33} & {\small 3} & {\small 19.2379} &{\small $ 0.227993 $}& -{\small 5}\\
{\small 34} & {\small 4} & {\small 16.5356} &{\small $ 0.245918 $} &{\small 2}\\
{\small 35} & {\small 3} & {\small 16.5255} & {\small $ 0.245993 $}&{\small 5}\\
{\small 36} & {\small 3} & {\small 15.1033} & {\small $ 0.257314 $}&-{\small 4}\\
{\small 37} & {\small 3} & {\small 10.3908} & {\small $0.310224 $}&{\small 4}\\
{\small 38} & {\small 1} & {\small 10.2520} &{\small $ 0.312317 $} &{\small 1}\\
{\small 39} & {\small 5} & {\small 10.1098} &{\small $ 0.314506 $} &-{\small 3}\\
{\small 40} & {\small 3} & {\small 9.03077} &{\small $ 0.332765 $} &-{\small 4}\\
{\small 41} & {\small 4} & {\small 9.02666} &{\small $ 0.332841 $}& {\small 2}\\
{\small 42} & {\small 5} & {\small 6.99929} & {\small $ 0.377984 $}&{\small 3}\\
{\small 43} & {\small 5} & {\small 6.95354} & {\small $ 0.379225$}&-{\small 3}\\
{\small 44} & {\small 3} & {\small 5.42311} &{\small $0.429414 $} &-{\small 5}\\
{\small 45} & {\small 4} & {\small 5.26429} & {\small $ 0.435843$}&-{\small 2}\\
{\small 46} & {\small 5} & {\small 3.04384} &{\small $ 0.573177$} &{\small 3}\\
\hline
\end{tabular}}
\end{center}
\caption{Eigenvalues $\mu_{j}$ of $\nabla^{2}V(u_{o})$ and critical numbers
$\lambda_{j}$ according to their isotypical type $\mathcal{V}_{n_{j}}$.}\label{tab:crit-values}
\end{table}
In the following Table \ref{tab:crit-values}, we show the number $n_{j}%
\in\{-5,...,-1,1,...,5\}$ that identifies the irreducible representation
corresponding to the eigenvalue $\mu_{j}$ for $j=1,..,46$. The numerical
computations strongly indicate that all the eigenvalues $\mu_{j}$,
$j=1,..,46$, are non-resonant.
\begin{remark}
\textrm{The models proposed in \cite{Wa} and \cite{Be}, consider the presence
of van der Waals forces among carbon atoms, which are modeled by the
potential
\[
W(x)=\varepsilon\left( \frac{\sigma^{12}}{x^{12}}-2\frac{\sigma^{6}}{x^{6}%
}\right) ~,
\]
where $\sigma=3.4681~A^{-1}$ is the minimum energy distance and $\epsilon
=0.0115~eV$ the depth of this minimum. Our numerical computations indicate
that the models with van der Waals forces do not produce acceptable bond
lengths between the atoms (as it is given in \cite{He}), neither the spectrums
fit the experimental data (cf. \cite{Ho}). Actually, the models \cite{Wa} and
\cite{Be} without van der Waals forces lead to results which correctly
approximate the measurements in \cite{He} and \cite{Ho} (for bond lengths
$d_{S}$ and $d_{D}$ and frequencies $\sqrt{\mu_{j}/m}$, which are within the
range $100$ to $1800$ $cm^{-1}$). }
\end{remark}
\section{Equivariant Bifurcation}
\label{sec:eq-bif}
In what follows, we are interested in finding non-trivial $T$-periodic
solutions to \eqref{eq:1}, bifurcating from the $G$-orbit of the equilibrium point
$u_{o}$. By normalizing the period, i.e. by making the substitution
$v(t):=u\left( \lambda t\right) $ in \eqref{eq:1}, we obtain the system
\begin{equation}%
\begin{cases}
\ddot{v}=-\lambda^{2}\nabla V(v),\\
v(0)=v(2\pi),\;\;\dot{v}(0)=\dot{v}(2\pi),
\end{cases}
\label{eq:mol1}%
\end{equation}
where $\lambda^{-1}=2\pi/T$ is the frequency.
\subsection{Equivariant Gradient Map}
Since $\mathscr V$ is an orthogonal $I\times O(3)$- representation, we can
consider the first Sobolev space of $2\pi$-periodic functions from
$\mathbb{R}$ to $\mathscr V$, i.e.,
\[
H_{2\pi}^{1}(\mathbb{R},\mathscr V):=\{u:\mathbb{R}\rightarrow\mathscr
V\;:\;u(0)=u(2\pi),\;u|_{[0,2\pi]}\in H^{1}([0,2\pi];\mathscr V)\},
\]
equipped with the inner product
\[
\langle u,v\rangle:=\int_{0}^{2\pi}(\dot{u}(t)\bullet\dot{v}(t)+u(t)\bullet
v(t))dt~.
\]
Let $O(2)=SO(2)\cup\kappa SO(2)$ denote the group of $2\times2$-orthogonal
matrices, where
\[
\kappa=\left[
\begin{array}
[c]{cc}%
1 & 0\\
0 & -1
\end{array}
\right] ,\qquad\left[
\begin{array}
[c]{cc}%
\cos\tau & -\sin\tau\\
\sin\tau & \cos\tau
\end{array}
\right] \in SO(2)~.
\]
It is convenient to identify a rotation with $e^{i\tau}\in S^{1}%
\subset\mathbb{C}$. Notice that $\kappa e^{i\tau}=e^{-i\tau}\kappa$, i.e.,
$\kappa$ as a linear transformation of $\mathbb{C}$ into itself, acts as
complex conjugation.
Clearly, the space $H_{2\pi}^{1}(\mathbb{R},\mathscr V)$ is an orthogonal
Hilbert representation of
\[
G:=I\times O(3)\times O(2).
\]
Indeed, we have for $u\in H_{2\pi}^{1}(\mathbb{R},\mathscr V)$ and
$(\sigma,A)\in I\times O(3)$ (see \eqref{eq:act1})
\begin{align}
\left( \sigma,A\right) u(t) & =(\sigma,A)u(t),\label{eq:ac}\\
e^{i\tau}u(t) & =u(t+\tau),\nonumber\\
\kappa u(t) & =u(-t).\nonumber
\end{align}
It is useful to identify a $2\pi$-periodic function $u:\mathbb{R}\rightarrow
V$ with a function $\widetilde{u}:S^{1}\rightarrow\mathscr V$ via the map
{$\mathfrak{e}(\tau)=e^{i\tau}:\mathbb{R}$}$\rightarrow S^{1}$. Using this
identification, we will write $H^{1}(S^{1},\mathscr V)$ instead of $H_{2\pi
}^{1}(\mathbb{R},\mathscr V)$. Let
\[
\Omega:=\{u\in H^{1}(S^{1},\mathscr V):u(t)\in\Omega_{o}\}.
\]
We define $J:\mathbb{R}\times\Omega\rightarrow\mathbb{R}$ by
\begin{equation}
J(\lambda,u):=\int_{0}^{2\pi}\left[ \frac{1}{2}|\dot{u}(t)|^{2}-\lambda
^{2}V(u(t))\right] dt. \label{eq:var-1}%
\end{equation}
Then, the system \eqref{eq:mol1} can be written as the following variational
equation
\begin{equation}
\nabla_{u}J(\lambda,u)=0,\quad(\lambda,u)\in\mathbb{R}\times\Omega.
\label{eq:bif1}%
\end{equation}
Consider $u_{o}\in\mathscr V$ -- the equilibrium point of \eqref{eq:mol} (i.e.
symmetric ground state) described in previous section. Then $u_{o}$ is a
critical point of $J$. We are interested in finding non-stationary $2\pi
$-periodic solutions bifurcating from $u_{o}$, i.e. non-constant solutions to
system \eqref{eq:bif1}. We consider the orbit $G(u_{o})$ of $u_{o}$ in
$H^{1}(S^{1},\mathscr V)$. We denote by $\mathcal{S}_{o}$ the slice to
$G(u_{o})$ in $H^{1}(S^{1},\mathscr V)$. We consider the $G_{u_{o}}$-invariant
restriction $J:\mathbb{R}\times\left( \mathcal{S}_{o}\cap\Omega\right)
\rightarrow\mathbb{R}$ of $J$ to the set $\mathcal{S}_{o}\cap\Omega$. This
restriction will allow us to apply the Slice Criticality Principle (see
Theorem \ref{thm:SCP}) in order to compute the gradient equivariant degree of
$\nabla J_{\lambda}$ on a small neighborhood $\mathscr U$ of $G(u_{o})$
needed for evaluation of the equivariant invariant $\omega_{G}(\lambda)$.
Consider the operator $L:H^{2}(S^{1};\mathscr V)\rightarrow L^{2}%
(S^{1};\mathscr V)$, given by
\[
Lu=-\ddot{u}+u
\]
for $u\in H^{2}(S^{1},\mathscr
V)$. Then the inverse operator $L^{-1}$ exists and is bounded. Let
$j:H^{2}(S^{1};\mathscr V)\rightarrow H^{1}(S^{1},\mathscr V)$ be the natural
embedding operator. Clearly, $j$ is a compact operator. Then, one can easily
verify that
\begin{equation}
\nabla_{u}J(\lambda,u)=u-j\circ L^{-1}(\lambda^{2}\nabla V(u)+u),
\label{eq:gradJ}%
\end{equation}
where $u\in H^{1}(S^{1},\mathscr V)$. Consequently, the bifurcation problem
\eqref{eq:bif1} is equivalent to $\nabla_{u}J(\lambda,u)=0$. Moreover, we
have
\begin{equation}
\nabla_{u}^{2}J(\lambda,u_{o})v=v-j\circ L^{-1}(\lambda^{2}\nabla^{2}%
V(u_{o})v+v)~, \label{eq:D2J}%
\end{equation}
where $v\in H^{1}(S^{1},\mathscr V)$.
Notice that
\[
\mathscr A(\lambda):=\nabla_{u}^{2}J(\lambda,u_{o})|_{\mathcal{S}_{o}%
}:\mathcal{S}_{o}\rightarrow\mathcal{S}_{o}.
\]
Thus, by implicit function theorem, $G(u_{o})$ is an isolated orbit of
critical points, whenever $\mathscr A(\lambda)$ is an isomorphism. Therefore,
if a point $(\lambda_{o},u_{o})$ is a bifurcation point for \eqref{eq:bif1},
then $\mathscr A(\lambda_{o})$ cannot be an isomorphism. In such case we
define
\[
\Lambda:=\{\lambda>0:\mathscr A(\lambda_{o})\text{ is not an isomorphism}\}~,
\]
and call this set the \textit{critical set} for the trivial solution $u_{o}$.
\subsection{Critical Numbers}
Consider the $S^{1}$-action on $H^{1}(S^{1},\mathscr V)$, where $S^{1}$ acts
on functions by shifting the argument (see \eqref{eq:ac}). Then, $(H^{1}%
(S^{1},\mathscr V))^{S^{1}}$ is the space of constant functions, which can be
identified with the space $\mathscr V$, i.e.,
\[
H^{1}(S^{1},\mathscr V)=\mathscr V\oplus\mathscr W,\quad\mathscr W:=\mathscr
V^{\perp}.
\]
Then, the slice $\mathcal{S}_{o}$ in $H^{1}(S^{1},\mathscr V)$ to the orbit
$G(u_{o})$ at $u_{o}$ is exactly
\[
\mathcal{S}_{o}=S_{o}\oplus\mathscr W.
\]
Consider the $S^{1}$-isotypical decomposition of $\mathscr W$, i.e.,
\[
\mathscr W=\overline{\bigoplus_{l=1}^{\infty}\mathscr W_{l}},\quad\mathscr
W_{l}:=\{\cos(l\cdot)\mathfrak{a}+\sin(l\cdot)\mathfrak{b}:\mathfrak{a}%
,\,\mathfrak{b}\in\mathscr V\}
\]
In a standard way, the space $\mathscr W_{l}$, $l=1,2,\dots$, can be naturally
identified with the complexification $\mathscr V^{\mathbb{C}}$ on which $S^{1}$ acts by
$l$-folding,
\[
\mathscr W_{l}=\{e^{il\cdot}z:z\in\mathscr V^{\mathbb{C}}\}.
\]
Since the operator $\mathscr A(\lambda)$ is $G_{u_{o}}$-equivariant with
\[
G_{u_{o}}=\tilde{I}\times O(2),
\]
it is also $S^{1}$-equivariant and thus $\mathscr A(\lambda)(\mathscr
W_{l})\subset\mathscr W_{l}$. Using the $\tilde{I}$-isotypical decomposition
of $\mathscr V^{\mathbb{C}}$, we have the $G_{u_{o}}$-invariant decomposition
\[
\mathscr W_{l}=\bigoplus_{j=1}^{46}\mathcal{V}_{{j},l},\quad \mathcal{V}_{{j},l}:=\{e^{il\cdot}z:z\in E(\mu_j)^\mathbb C \},
\]
where $\mathcal{V}_{n_{j},l}=\mathcal V_{n_j}^\mathbb C=\mathbb C\otimes_\mathbb R \mathcal V_{n_j}$ is the $I\times O(2)$-irreducible representation with $O(2)$ acting on $\mathbb C$ by $l$-folding and complex conjugation.
We have
\[
\mathscr A(\lambda)|_{\mathcal{V}_{{j},l}}=\left( 1-\frac{\lambda^{2}%
\mu_{j}+1}{l^{2}+1}\right) \text{\rm Id\,}.
\]
Thus $A(\lambda_{o})|_{\mathcal{V}_{{j},l}}=0$ if and only if
$\lambda_{o}^{2}=l^{2}/\mu_{j}$ for some $l=1,2,3,\dots$ and $j=0,1,2, \dots, 46$.
We will write
\[
\lambda_{j,l}=\frac{l}{\sqrt{\mu_{j}}}
\]
to denote the critical numbers in $\Lambda$. Then the critical set for the equilibrium $u_{o}$ of system \eqref{eq:mol}
is
\[
\Lambda= \{ \lambda_{j,l}:j=0,...,46,\quad l=1,2,3,\dots \} .
\]
Let us point out that in the case of isotypical resonances, the critical numbers may not be uniquely identified by indices $(j,l)$. The first and last critical numbers for $l=1$ are
$\lambda_{1,1}=.07526\,3$ and $\lambda_{46,1}=0.573\,18$, respectively. We
computed numerically (with precision $10^{-5}$) all the different values $\lambda_{j,l}$ from $\lambda_{1,1}$ to
$\lambda_{46,1}$. We obtain that among these approximations there is no-resonance
with harmonic critical number from $\lambda_{1,1}$ to $\lambda_{46,1}$, i.e.,%
\begin{equation}\label{eq:reson-no}
\lambda_{1,1}<\lambda_{2,1}<\lambda_{3,1}<\lambda_{4,1}%
<...<\lambda_{5,7}<\ \lambda_{26,3}<\lambda_{21,4}<\lambda_{27,3}%
<\lambda_{46,1}.
\end{equation}
Therefore, a plausible assumption under the numerical evidence is that all the eigenvalues $\lambda_{j,1}$ are isotypical non-resonant for $j=1,...,46$.
\subsection{Conjugacy Classes of Subgroups in $I\times O(2)$}\label{sec:conjugacy}
In order to simplify the notation, in what follows, instead of using the symbol $\tilde{I}$, we will write $I$. Under this notation the isotropy group $G_{u_o}$ is
\[
G_{u_{o}}=I\times O(2).
\]
The notation in this section is useful to obtain the classification of all
conjugacy classes $(\mathscr H)$ of closed subgroups in $I\times O(2)$.
The representatives of the conjugacy classes of the subgroups in $A_5\times \mathbb Z_2$ consisting of proper nontrivial subgroups of $A_5$ are:
\begin{align*}
\mathbb Z_2&=\{ ((1),1),\, ((12)(34),1)\},\\
\mathbb Z_3&=\{ ((1),1)\,, ((123),1),\, ((132),1)\},\\
V_4&=\{((1),1),\, ((12)(34),1),\, ((13)(24),1),\, ((23)(14),1)\},\\
\mathbb Z_5&=\{((1),1), \,((12345),1),\, ((13524),1),\, ((14253),1),\, ((15432),1)\},\\
D_3&=\{((1),1),\, ((123),1),\, ((132),1),\, ((12)(45),1),\, ((13)(45),1),\, ((23)(45),1)\},\\
A_4&=\{((1),1),\, ((12)(34),1),\, ((13)(24),1),\, ((14)(23),1),\,((123),1),\, ((132),1),\, ( (124),1),\\
&\hskip.5cm ((142),1,\, ((134),1),\, ((143),1),\, ( (234),1),\, ((243),1)\},\\
D_5&=\{((1),1),\, ((12345),1),\, ((13524),1),\, ((14253),1),\, ((15432),1),\, ((12)(35),1),\, ((13)(45),1),\\
&\hskip.5cm ((14)(23),1),
\,( (15)(24),1),\, ((25)(34)\}.
\end{align*}
The representatives of the additional conjugacy classes of the subgroups in $A_5\times \mathbb Z_2$ will be used to describe the symmetries of nonlinear vibrations. Besides of the product subgroups $H^{p}:=H\times\mathbb{Z}_{2}$, we have the also following twisted subgroups $H^\varphi$ of $A_5\times \mathbb Z_2$ (where $H$ is a subgroup of $A_5$):
{\small\begin{align*}
\mathbb Z_2^z&=\Big\{\big((1),1\big),\, \big((12)(34),-1\big)\Big\},\\
V_4^z&=\Big\{\big((1),1\big),\, \big((12)(34),-1\big),\, \big((13)(24),-1\big),\, \big((23)(14),1\big)\Big\},\\
D_3^z&=\Big\{\big((1),1\big),\, \big((123),1\big),\, \big((132),1\big),\,\big((12)(45),-1\big),\, \big((13)(45),-1\big),\\
&\hskip.5cm \, \big((23)(45),-1\big)\Big\},\\
D_5^z&=\Big\{\big((1),1\big),\, \big((12345),1\big),\, \big((13524),1\big),\, \big(14253),1\big),\, \big((15432),1\big),\\
&\hskip.5cm \, \big((12)(35),-1\big),\, \big((13)(45),-1\big),\, \big((14)(23),-1\big),\, \big((15)(24),-1\big),\\
&\hskip.5cm \, \big((25)(34),-1\big)\Big\}.
\end{align*}} With these definitions the subconjugacy lattice for $A_5\times \mathbb Z_2$ is shown in Figure \ref{fig:sub-A5}.
\begin{figure}
\vglue-4cm
\begin{center}
\includegraphics[height=15cm]{lattice}
\end{center}
\vskip-3.4cm
\caption{Lattice of conjugacy classes of subgroups in $A_5\times \mathbb Z_2$. The square boxes indicate that the related subgroup is normal in $A_5\times \mathbb Z_2$.}\label{fig:sub-A5}
\end{figure}
The result (see \cite{DaKr,Goursat}) provides a description
of subgroups of the product group $I\times O(2)$. Namely, for any subgroup $\mathscr
H$ of the product group $I\times O(2)$, there exist subgroups $H\leq I$ and $K\leq O(2)$, a group $L$ and two epimorphisms $\varphi
:H\rightarrow L$ and $\psi:K\rightarrow L$ such that
\[\mathscr H=\{(h,k)\in H\times K:\varphi(h)=\psi(k)\}.\]
In order to make the notation self-contained, we will
assume that $L=K/\ker(\psi)$, so $\psi:K\rightarrow L$ is evidently the
natural projection. On the other hand, the group $L$ can be naturally
identified with a finite subgroup of $O(2)$ being either $D_{n}$ or
$\mathbb{Z}_{n}$. Since we are interested in describing conjugacy
classes of $\mathscr H$, we can identify different epimorphisms $\varphi,\psi:H\rightarrow
L$ by indicating
\[
Z=\text{Ker\thinspace}(\varphi)\quad\text{ and }\quad L=K/\ker(\psi).
\]
Therefore, to identify $\mathscr H$ we will
write
\begin{equation}
\mathscr H=:H{\prescript{Z}{}\times_{L}^{m}}K~, \label{eq:amalg}%
\end{equation}
where $H$ and $Z$ are subgroups of $I$ and $m$ is a number used to identify groups in different conjugacy classes.
In the case that all the epimorphisms $\varphi$ with the kernel $Z$ are
conjugate, there is no need to use the number $m$ in \eqref{eq:amalg}, so we
will simply write $\mathscr H=H{\prescript{Z}{}\times_{L}K}$. In addition, in
the case that all epimorphisms $\varphi$ from $H$ to $L$ are conjugate, we can also
omit the symbol $Z$, i.e. we will write $\mathscr H=H\times_{L}K$.
\subsection{Bifurcation Theorem}
\begin{theorem}
\label{th:main} Assume that the critical numbers $\lambda_{j,1}\in\Lambda$,
$j=1,2,\dots,46$, for the system \eqref{eq:bif1} are isotypically non-resonant. Then, there
exist multiple global bifurcations of solutions from the critical number $\lambda_{j,1}$ corresponding to the
irreducible representation $V_{n_{j}}$ in Table 3:
\begin{itemize}
\item For $n_{j}=1$ there exists a $G$-orbit of a branch of periodic solutions
with the orbit type ${(\amal{{A_5^p}}{D_{1}}{}{}{})}$;
\item For $n_{j}=2$ there exist $G$-orbits of branches of periodic solutions
with the orbit types ${(\amal{{D_3^p}}{D_{2}}{\mathbb Z_{2}}{{\mathbb Z_3^p}}{})}$,
${(\amal{{V_4^p}}{D_{2}}{\mathbb Z_{2}}{{\mathbb Z_2^p}}{})}$,
${(\amal{{A_4^p}}{D_{1}}{}{}{})}$, ${(\amal{{D_3^p}}{D_{1}}{}{}{})}$,
${(\amal{{D_5^p}}{D_{5}}{D_{5}}{{\mathbb Z_1^p}}{{1}})}$,
${(\amal{{D_5^p}}{D_{5}}{D_{5}}{{\mathbb Z_1^p}}{{2}})}$,
${(\amal{{D_3^p}}{D_{3}}{D_{3}}{{\mathbb Z_1^p}}{})}$;
\item For $n_{j}=3$ there exist $G$-orbits of branches of periodic solutions
with the orbit types ${(\amal{{V_4^p}}{D_{2}}{\mathbb Z_{2}}{{\mathbb Z_2^p}}{})}$,
${(\amal{{D_5^p}}{D_{1}}{}{}{})}$, ${(\amal{{D_3^p}}{D_{1}}{}{}{})}$,
${(\amal{{D_5^p}}{D_{5}}{D_{5}}{{\mathbb Z_1^p}}{{1}})}$,
${(\amal{{D_5^p}}{D_{5}}{D_{5}}{{\mathbb Z_1^p}}{{2}})}$,
${(\amal{A_4^p}{\mathbb Z_{3}}{\mathbb Z_3}{V_4^p}{})}$;
\item For $n_{j}=4$ there exist $G$-orbits of branches of periodic solutions
with the orbit types ${(\amal{{D_5^p}}{D_{2}}{\mathbb Z_{2}}{{\mathbb Z_5^p}}{})}$,
${(\amal{{D_3^p}}{D_{2}}{\mathbb Z_{2}}{{\mathbb Z_3^p}}{})}$,
${(\amal{{V_4^p}}{D_{2}}{\mathbb Z_{2}}{{\mathbb Z_2^p}}{})}$,
${(\amal{{D_5^p}}{D_{5}}{D_{5}}{{\mathbb Z_1^p}}{{1}})}$,
${(\amal{{D_3^p}}{D_{3}}{D_{3}}{{\mathbb Z_1^p}}{})}$,
\item For $n_{j}=5$ there exist $G$-orbits of branches of periodic solutions
with the orbit types ${(\amal{{D_5^p}}{D_{2}}{\mathbb Z_{2}}{{\mathbb Z_5^p}}{})}$,
${(\amal{{D_3^p}}{D_{2}}{\mathbb Z_{2}}{{\mathbb Z_3^p}}{})}$,
${(\amal{{V_4^p}}{D_{2}}{\mathbb Z_{2}}{{\mathbb Z_2^p}}{})}$,
${(\amal{{D_5^p}}{D_{5}}{D_{5}}{{\mathbb Z_1^p}}{{2}})}$,
${(\amal{{D_3^p}}{D_{3}}{D_{3}}{{\mathbb Z_1^p}}{})}$;
\item For $n_{j}=-1$ here exists a $G$-orbit of a branch of periodic solutions
with the orbit type ${(\amal{{A_5^p}}{D_{2}}{\mathbb Z_{2}}{{A_5}}{})}$;
\item For $n_{j}=-2$ there exist $G$-orbits of branches of periodic solutions
with the orbit types ${(\amal{{A_4^p}}{D_{2}}{\mathbb Z_{2}}{{A_4}}{})}$,
${(\amal{{D_3^p}}{D_{2}}{\mathbb Z_{2}}{{D_3^z}}{})}$,
${(\amal{{D_3^p}}{D_{2}}{\mathbb Z_{2}}{{D_3}}{})}$,\break
${(\amal{{V_4^p}}{D_{2}}{\mathbb Z_{2}}{{V_4^z}}{})}$,
${(\amal{{D_5^p}}{D_{10}}{D_{10}}{{\mathbb Z_1}}{{1}})}$,
${(\amal{{D_5^p}}{D_{10}}{D_{10}}{{\mathbb Z_1}}{{2}})}$,
${(\amal{{D_3^p}}{D_{6}}{D_{6}}{{\mathbb Z_1}}{})}$,
\item For $n_{j}=-3$ there exist $G$-orbits of branches of periodic solutions
with the orbit types ${(\amal{{D_5^p}}{D_{2}}{\mathbb Z_{2}}{{D_5}}{})}$,
${(\amal{{D_3^p}}{D_{2}}{\mathbb Z_{2}}{{D_3}}{})}$,
${(\amal{{V_4^p}}{D_{2}}{\mathbb Z_{2}}{{V_4^z}}{})}$,\break
${(\amal{{D_5^p}}{D_{10}}{D_{10}}{{\mathbb Z_1}}{{1}})}$,
${(\amal{{D_5^p}}{D_{10}}{D_{10}}{{\mathbb Z_1}}{{2}})}$,
${(\amal{A_4^p}{\mathbb Z_{6}}{\mathbb Z_6}{V_4}{})}$;
\item For $n_{j}=-4$ there exist $G$-orbits of branches of periodic solutions
with the orbit types ${(\amal{{D_5^p}}{D_{2}}{\mathbb Z_{2}}{{D_5^z}}{})}$,
${(\amal{{D_3^p}}{D_{2}}{\mathbb Z_{2}}{{D_3^z}}{})}$,
${(\amal{{V_4^p}}{D_{2}}{\mathbb Z_{2}}{{V_4^z}}{})}$,\break
${(\amal{{D_5^p}}{D_{10}}{D_{10}}{{\mathbb Z_1}}{{1}})}$,
${(\amal{{D_3^p}}{D_{6}}{D_{6}}{{\mathbb Z_1}}{})}$;
\item For $n_{j}=-5$ there exist $G$-orbits of branches of periodic solutions
with the orbit types ${(\amal{{D_5^p}}{D_{2}}{\mathbb Z_{2}}{{D_5^z}}{})}$,
${(\amal{{D_3^p}}{D_{2}}{\mathbb Z_{2}}{{D_3^z}}{})}$,
${(\amal{{V_4^p}}{D_{2}}{\mathbb Z_{2}}{{V_4^z}}{})}$,\break
${(\amal{{D_5^p}}{D_{10}}{D_{10}}{{\mathbb Z_1}}{{2}})}$,
${(\amal{{D_3^p}}{D_{6}}{D_{6}}{{\mathbb Z_1}}{})}$.
\end{itemize}
\end{theorem}
\begin{proof}
The critical numbers for system \eqref{eq:bif1} are $\lambda_{j,l}=l/\sqrt
{\mu_{j}}$ for $l=1,2,3,\dots,$ and $j=1,2,\dots,46$, where $\mu_{j}$
(together with its isotypical types) are listed in Table \ref{tab:crit-values}. We assumed (under the numerical evidence) that the critical frequencies $\lambda_{j,1}^{-1}$ are
isotypically non-resonant. Based on the ideas explained in section \ref{sec:1}, then
$\lambda_{o}:=\lambda_{j_{o},1}$ is an isolated critical point in $\Lambda$.
That is, there are $\lambda_{-}<\lambda_{o}<\lambda_{+}$ such that
$[\lambda_{-},\lambda_{+}]\cap\Lambda=\{\lambda_{o}\}$. Moreover, there exists
an isolating $G$-neighborhood $\mathscr U$ of $G(u_{o})$ such that no other
critical orbits of $J_{\lambda_{\pm}}$ belong to $\overline{\mathscr U}$.
Thus, we can define the topological invariant $\omega_{G}(\lambda_{o})$ by
\eqref{eq:top-inv}. Then, by the properties of the gradient equivariant degree,
if
\[
\omega_{G}(\lambda_{o})=n_{1}(H_{1})+n_{2}(H_{2})+\dots+n_{m}(H_{m})
\]
is non-zero, i.e. $n_{j}\not =0$ for a $j=1,2,\dots,m$, then there exists a
bifurcating branch of nontrivial solutions to \eqref{eq:bif1} from the orbit
$\{\lambda_{o}\}\times G(u_{o})$ with symmetries at least $(H_{j})$.
Next, by Theorem \ref{thm:SCP},
\[
\nabla_{G}\text{-deg}(\nabla J_{\lambda_{\pm}},\mathscr U)=\Theta\left(
\nabla_{G_{u_{o}}}\text{-deg}(\nabla{ J}_{\lambda_{\pm}},\mathscr
U\cap\mathscr S_{o})\right) ,
\]
where $G=I\times O(3)\times O(2)$, $G_{u_{o}}=\tilde{I}\times O(2)$ and
$\Theta:U(G_{u_{o}})\rightarrow U(G)$ is the homomorfism given by
$\Theta(H)=(H)$. For convenience, in what follows we will ignore the symbol
$\Theta$. Moreover, by standard linearization technique, we have
\[
\nabla_{G_{u_{o}}}\text{-deg}(\nabla{J}_{\lambda_{\pm}},\mathscr U\cap
\mathscr
S_{o})=\nabla_{G_{u_{o}}}\text{-deg}(\mathscr A_{\lambda_{\pm}},\mathscr
U\cap\mathscr S_{o}).
\]
By \eqref{eq:lin-GdegGrad}, since all the eigenvalues $\mu_{j}$ are
isotopically simple, we have
\begin{align*}
\nabla_{G_{u_{o}}}\text{-deg}(\mathscr A_{\lambda_{-}},\mathscr U\cap
\mathscr S_{o}) & =\prod_{\left\{ \left( j,l\right) \in\mathbb{N}%
^{2}:\lambda_{j,l}<\lambda_{o}\right\} }\nabla\text{-deg}_{\mathcal{V}%
_{n_{j},l}},\\
\nabla_{G_{u_{o}}}\text{-deg}(\mathscr A_{\lambda_{+}},\mathscr U\cap
\mathscr S_{o}) & =\nabla\text{-deg}_{\mathcal{V}_{n_{j_{o}},l}}%
\prod_{\left\{ \left( j,l\right) \in\mathbb{N}^{2}:\lambda_{j,l}%
<\lambda_{o}\right\} }\nabla\text{-deg}_{\mathcal{V}_{n_{j},l}},
\end{align*}
where $\nabla\text{-deg}_{\mathcal{V}_{n_{j},l}}$ are gradient $I\times
O(2)$-equivariant basic degrees listed in Appendix \ref{sec:basic}. Therefore,
we obtain
\[
\omega_{G}(\lambda_{o}):=\Big((I\times O(2))-\nabla\text{-deg}_{\mathcal{V}%
_{n_{j_{o}},1}}\Big)\prod_{\left\{ \left( j,l\right) \in\mathbb{N}%
^{2}:\lambda_{j,l}<\lambda_{o}\right\} }\nabla\text{-deg}_{\mathcal{V}%
_{n_{j},l}},
\]
where $(I\times O(2))$ is the unit element in $U(I\times O(2))$. For instance,
the first equivariant invariants are given by
\begin{align*}
\omega_{G}(\lambda_{1,1}) & =(I\times O(2))-\nabla\text{-deg}_{\mathcal{V}%
_{3,1}}\\
\omega_{G}(\lambda_{2,1}) & =\nabla\text{-deg}_{\mathcal{V}_{3,1}}%
\ast\Big((I\times O(2))-\nabla\text{-deg}_{\mathcal{V}_{-3,1}}\Big)\\
\omega_{G}(\lambda_{3,1}) & =\nabla\text{-deg}_{\mathcal{V}_{3,1}}\ast
\nabla\text{-deg}_{\mathcal{V}_{-3,1}}\ast\Big((I\times O(2))-\nabla
\text{-deg}_{\mathcal{V}_{2,1}}\Big).
\end{align*}
We will prove that a maximal orbit type $(H)$ that appears in the gradient
$I\times O(2)$-basic degree $\nabla\text{-deg}_{\mathcal{V}_{n_{j_{o}},1}}$
with non-zero coefficient $n_{H}$,
\[
\nabla\text{-deg}_{\mathcal{V}_{n_{j_{o}},1}}=(I\times O(2))+n_{H}(H)+\dots,
\]
also appears in $\omega_{G}(\lambda_{o})$ with non-zero coefficients.
Hereafter, dots indicate all the remaining terms corresponding to orbit types
strictly smaller than $(H)$. Notice that all such maximal orbit types (which
are indicated in subsection \ref{sec:basic} by red color) belong to $\Phi
_{0}(I\times O(2))$ (i.e. dim\thinspace$W(H)=0$) except for
$(\amal{A_4^p}{\mathbb Z_{3}}{\mathbb Z_3}{V_4^p}{})$ (in $\nabla\text{-deg}%
_{\mathcal{V}_{3,1}}$) and $(\amal{A_4^p}{\mathbb Z_{6}}{\mathbb Z_6}{V_4}{}))$ (in
$\nabla\text{-deg}_{\mathcal{V}_{-3,1}}$).
Now, assume that $(H)$ is a maximal orbit type such that dim\thinspace$W(H)=0$
and
\[
\nabla\text{-deg}_{\mathcal{V}_{n,1}}=(I\times O(2))+n_{H}(H)+\dots,
\]
with $n_{H}\not =0$. By maximality of $(H)$ in $\mathcal{V}_{n,1}$, formula
\eqref{eq:rec-Brouwer} gives
\[
n_{H}=\frac{(-1)^{k}-1}{|W(H)|},\quad k:=\text{dim\thinspace}\mathcal{V}%
_{n,1}^{H}.
\]
Then $k$ must be odd and consequently $n_{H}=-1$ when $|W(H)|=2$ or $n_{H}=-2$
when $|W(H)|=1$. Suppose now that $\nabla\text{-deg}_{\mathcal{V}_{\bar{n},1}%
}$ is another (not necessarily different) basic degree containing a non-zero
coefficient for $(H)$, i.e.
\[
\nabla\text{-deg}_{\mathcal{V}_{\bar{n},1}}=(I\times O(2))+n_{H}(H)+\dots.
\]
Then, we have,
\[
\nabla\text{-deg}_{\mathcal{V}_{n,1}}\ast\nabla\text{-deg}_{\mathcal{V}%
_{\bar{n},1}}=(I\times O(2))+2n_{H}(H)+n_{H}^{2}(H)^{2}+\dots
\]
However, by \eqref{eq:rec-coef}, we have $(H)\ast(H)=m_{H}(H)+\dots$, where
\[
m_{H}:=\frac{|W(H)|\cdot|W(H)|}{|W(H)|}=|W(H)|.
\]
Thus
\[
\nabla\text{-deg}_{\mathcal{V}_{n,1}}\ast\nabla\text{-deg}_{\mathcal{V}%
_{\bar{n},1}}=(I\times O(2))+\left( 2n_{H}+n_{H}^{2}m_{H}\right) (H)+\dots.
\]
One can easily notice that
\[
2n_{H}+n_{H}^{2}m_{H}=0,
\]
for both cases $n_{H}=-1$ and $n_{H}=-2$. Therefore, the coefficient of the
product $\nabla\text{-deg}_{\mathcal{V}_{n,1}}\ast\nabla\text{-deg}%
_{\mathcal{V}_{\bar{n},1}}$ is zero for the group $(H)$. Consequently, since
ether $\nabla_{G_{u_{o}}}\text{-deg}(\mathscr A_{\lambda_{-}},\mathscr
U\cap\mathscr S_{o})$ or $\nabla_{G_{u_{o}}}\text{-deg}(\mathscr
A_{\lambda_{+}},\mathscr U\cap\mathscr S_{o})$ (but not both) contains an even
number of factors $\nabla\text{-deg}_{\mathcal{V}_{n_{i},1}}$ with non-zero
coefficient $n_{H}$ of $(H)$, it follows that their difference also contains
non-zero $\pm n_{H}$ coefficient of $(H)$. Actually, the computation with GAP shows
that $n_{H}=-1$, then in these cases we have%
\[
\omega_{G}(\lambda_{o})=\pm(H)+\dots.\text{.}%
\]
Now, assume that $H=\amal{A_4^p}{\mathbb Z_{3}}{\mathbb Z_3}{V_4^p}{}$ and consider
$\nabla\text{-deg}_{\mathcal{V}_{3,1}}=(G)+n_{H}(H)+\dots$. Then we have
\[
\nabla\text{-deg}_{\mathcal{V}_{3,1}}\ast\nabla\text{-deg}_{\mathcal{V}_{3,1}%
}=(G)+2n_{H}(H)+n_{H}^{2}(H)^{2}+\dots.
\]
By functoriality property of the gradient equivariant degree we have that the
inclusion $\psi:I\times S^{1}\rightarrow I\times O(2)$ induces the Euler
homomorphism $\Psi:U(I\times O(2))\rightarrow U(I\times S^{1})$ such that
$\Psi(\nabla\text{-deg}_{\mathcal{V}_{3,1}})$ is also a gradient equivariant
basic degree (see \cite{DaKr}). This can be easily computed (cf. \cite{RR}) as
follows,
\begin{align*}
\Psi(\nabla\text{-deg}_{\mathcal{V}_{3,1}}) & =(I\times S^{1})-(D_{5}%
^{p})-(D_{3}^{p})-(A_{4}^{t_{1}}\times\mathbb Z_{2})-(A_{4}^{t_{2}}\times\mathbb Z_{2})\\
& -(V_{4}^{-}\times\mathbb Z_{2})-(\mathbb Z_{5}^{t_{1}}\times\mathbb Z_{2})-(\mathbb Z_{5}^{t_{2}%
}\times\mathbb Z_{2})+2(\mathbb Z_{2}^{p}).
\end{align*}
Thus $n_{H}=-1$. Notice that by \eqref{eq:Euler-hom}, $\Psi
(\amal{A_4^p}{\mathbb Z_{3}}{\mathbb Z_3}{V_4^p}{})=(A_{4}^{t_{1}}\times\mathbb Z_{2}%
)+(A_{4}^{t_{2}}\times\mathbb Z_{2})$. As it was shown in \cite{RR}, $(A_{4}^{t_{i}%
}\times\mathbb Z_{2})\ast(A_{4}^{t_{j}}\times\mathbb Z_{2})=0$. Thus we have
\[
0=\Psi((H)\ast(H))=\Psi(m_{H}(H)+\dots)=m_{H}\Big(((A_{4}^{t_{1}}\times
\mathbb Z_{2})+(A_{4}^{t_{2}}\times\mathbb Z_{2})\Big),
\]
which implies $m_{H}=0$. Therefore, $(H)\ast(H)=0$ and for $k\in\mathbb{N}$,
\[
\left( \nabla\text{-deg}_{\mathcal{V}_{3,1}}\right) ^{k}=(I\times
O(2))-k(H)+\dots.
\]
Clearly,
$\omega_{G}(\lambda_{o})$ has a non-zero coefficient,
\[
\omega_{G}(\lambda_{o})=(H)+\dots.
\]
For $(H)=(\amal{A_4^p}{\mathbb Z_{6}}{\mathbb Z_6}{V_4}{})$ the proof is similar. This
concludes the proof of our main theorem.
\end{proof}
\begin{remark}\rm
Since all the invariants are $\omega_{G}(\lambda_{o})=(H)+...$ for
$H=(\amal{A_4^p}{\mathbb Z_{3}}{\mathbb Z_3}{V_4^p}{})$ and
$H=(\amal{A_4^p}{\mathbb Z_{6}}{\mathbb Z_6}{V_4}{})$, then the sum of $\omega_{G}$'s can
never be zero, i.e. all the connected components $\mathcal{C}$ with symmetries
$(\amal{A_4^p}{\mathbb Z_{3}}{\mathbb Z_3}{V_4^p}{})$ and
$(\amal{A_4^p}{\mathbb Z_{6}}{\mathbb Z_6}{V_4}{})$ are non-compact. Similarly, notice that
there is an odd number of irreducible subrepresentations $\mathcal V_{-n}$ in the isotypical component $\mathscr V_{-n}$, for $n=1$, $3$, $4$, $5$,
and the topological invariant is $\omega_{G}(\lambda_{o})=\pm(H)+...$ (for
a maximal group $(H)$). This excludes a possibility that all the branches with the orbit type $(H)$, bifurcating from all the critical points $\lambda_{j,1}$ corresponding to $\mathcal V_{-n}$, are compact. Thus, for any maximal orbit type $(H)$ in $\mathcal V_{-n}$ for $n=1,3,4,5$ there exists a non-compact branch $\mathcal{C}$ with orbit type $(H)$.
\end{remark}
\begin{remark}\rm
All the gradient basic degrees $\nabla\text{-deg}_{\mathcal{V}_{\pm
n,1}}$, which were computed using G.A.P. programming, are included in Appendix
\ref{sec:equi-degree}. These degrees can be used to compute the exact value of topological
invariants $\omega_{G}(\lambda_{o})$ even in the case that $\lambda_{o}$ is isotypically
resonant, so a bifurcation result can be established in the resonant case as well. For example, such a resonant case was studied in \cite{BeGa} (to classify the nonlinear modes in a tetrahedral molecule).
\end{remark}
\section{Description of Symmetries and Numerical results}
\label{sec:symmetries}
For any maximal orbit type $(H)$ in $\mathcal V_{n,1}$ the element $-1\in\mathbb{Z}_{2}<I$ belongs to $ H$, and in $\mathcal V_{-n,1}$, the element $(-1,-1)\in\mathbb{Z}_{2}\times \mathbb Z_2<I\times S^1$ belongs to $ H$. A solution $u(t)$ in the fixed point space for a group containing $-1\in\mathbb{Z}_{2}$, satisfies
\[
u_{\tau,k}(t)=-u_{\tau^{-1},k}(t),
\]
while for a group containing $(-1,-1)\in\mathbb{Z}_{2}\times \mathbb Z_2$ satisfies
\[
u_{\tau,k}(t)=-u_{\tau^{-1},k}(t+\pi).
\]
Since the isotropy groups in $\mathcal V_{n,1}$ and $\mathcal V_{-n,1}$ differ only in this element, we only need to describe the symmetries of the maximal groups for
the representations $\mathcal{V}_{n,1}$.
The existence of the symmetry $\kappa\in O(2)$ in the maximal groups implies
that the solutions are brake orbits,%
\[
u_{\tau,k}(t)=u_{\tau,k}(-t)\text{,}%
\]
i.e., the velocity $\dot{u}$ of all the molecules are zero at times
$t=0,\pi$, i.e., $\dot{u}(0)=\dot{u}(\pi)=0$. We classify the maximal groups in two
classes: the groups that have the element $\kappa\in O(2)$ and the groups that
have the element $\kappa$ coupled with a rotation of $I$. That is, if there is an
element $\gamma\in\mathcal{C}_{2}$ such that $(\gamma,\kappa)$ is in the
second class of groups, then their solutions have the symmetry
\[
u_{\tau,k}(t)=\rho(\gamma)u_{\gamma\tau\gamma^{-1},\gamma^{-1}(k)}(-t).
\]
The maximal orbit type that does not have a symmetry $(\gamma,\kappa)$ is
$(\amal{A_4^p}{\mathbb Z_{6}}{\mathbb Z_6}{V_4}{})$ which is the only maximal group (in $\mathscr V_{3,1}$)
with Weyl group of dimension one.
\vskip.3cm
\subsection{Standing Waves (Brake Orbits)}
In this category we consider the groups that have the element $\kappa\in
O(2)$, which generate the group $D_{1}<O(2)$.
For the groups%
\[
{({A_{5}^{p}}\prescript{}{}\times_{{}}^{{}}D_{1}),({A_{4}^{p}}%
\prescript{}{}\times_{{}}^{{}}D_{1}),({D_{5}^{p}}\prescript{}{}\times_{{}}%
^{{}}D_{1}),({D_{3}^{p}}\prescript{}{}\times_{{}}^{{}}D_{1})}%
\]
the solutions have the following symmetries at all times: icosahedral symmetries for
${{A_{5}}}$, tetrahedral symmetries for ${{A_{4}}}$, pentagonal
symmetries for ${{D_{5}}}$ and triangular symmetries for
${{D_{3}}}$.
For the group%
\[
{({D_{3}^{p}}\prescript{{\mathbb Z_3^p}}{}\times_{\mathbb{Z}_{2}}^{{}}D_{2})}%
\]
the solutions are symmetric by the $2\pi/3$-rotations of $\mathbb{Z}_{3}%
<D_{3}<I$, while the reflection of $D_{3}<I$ is coupled with the $\pi$-time
shift of $-1\in\mathbb{Z}_{2}<S^{1}$. Therefore, the solutions in three faces have the exact dynamics, but these faces are not symmetric by reflection
such as in the symmetries of ${({D_{3}^{p}}\prescript{}{}\times_{{}%
}^{{}}D_{1})}$.
For the group%
\[
{({V_{4}^{p}}\prescript{{\mathbb Z_2^p}}{}\times_{\mathbb{Z}_{2}}^{{}}D_{2}%
)}\text{,}%
\]
the solutions are symmetric by the $\pi$-rotations of $V_{4}<I$, while the
other $\pi$-rotation of $\mathbb{Z}_{2}<V_{4}$ is coupled with the $\pi$-time
shift of $-1\in D_{2}<S^{1}$.
These seven symmetries give solutions which are standing waves in the sense
that each symmetric face has the exact dynamic repeated for all times.
\subsection{Discrete Rotating Waves}
In the groups%
\[
{({D_{5}^{p}}\prescript{{\mathbb Z_1^p}}{}\times_{D_{5}}^{1}D_{5})},{({D_{5}^{p}%
}\prescript{{\mathbb Z_1^p}}{}\times_{D_{5}}^{2}D_{5})}~,
\]
the spatial dihedral group $D_{5}<I$ is coupled with the temporal group
$D_{5}<O(2)$. Therefore, in these solutions we have $5$ faces with the same
dynamics, but there is a $2\pi/5$-time shift in time between consecutive
faces. In addition, $\kappa$ is coupled with a $\pi$-rotation, i.e., there is
an axis of symmetry in each face. In this sense, the solutions have the
appearance of a discrete rotating wave with a $2\pi/5$ delay along consecutive faces. There are two groups
because there are two different conjugacy classes, $\mathcal{C}_{4}$ and
$\mathcal{C}_{5}$, of $A_{5}$.
Similarly, in the solutions of the group%
\[
{({D_{3}^{p}}\prescript{{\mathbb Z_1^p}}{}\times_{D_{3}}^{{}}D_{3})~,}%
\]
we have $3$ faces with the same dynamics, but with a $2\pi/3$-time shift, i.e.,
the solutions have the appearance of a discrete rotating wave in $3$ faces
with a $2\pi/3$-time shift and each face has an axis of symmetry.
For the solutions of the group
\[
(\amal{A_4^p}{\mathbb Z_{6}}{\mathbb Z_6}{V_4}{})
\]
we have $3$ faces with the same dynamics with a $2\pi/3$-time shift.
Moreover, in these solutions the inversion is coupled with a $\pi$-time shift
in time. Therefore, there are a total of $6$ faces ($3$ faces and their
inversions) that have the same dynamics but with $2\pi/6$-time shift. In these
solutions the faces do not have an axis of symmetry, instead there are two
symmetries by a $\pi$-rotation.
\subsection{Numerical results}
In this section, we present the implementation of the numerical continuation of some families of periodic solutions. In order to compute numerically the families of periodic solutions, we use the
Hamiltonian formulation,%
\begin{equation}
\dot{x}=J\nabla H(x),\qquad x=(q,p),\label{ODE}%
\end{equation}
where $H(q,p)=\left\vert p\right\vert ^{2}/2-V(q)$ is the Hamiltonian and $J$
is the symplectic matrix%
\[
J=\left(
\begin{array}
[c]{cc}%
0 & -I\\
I & 0
\end{array}
\right) \text{.}%
\]
Since the Hamiltonian is invariant by the action of the group $\mathbb{R}%
^{3}$ that acts by translation, $O(3)$ by rotations and $\varphi\in S^{1}$ by
time shift, then the Hamiltonian satisfies the orthogonal relations%
\[
\left\langle H(x),A_{j}(x)\right\rangle =0,
\]
for $j=1,...,7$, where $A_{j}$ are the generators of the groups,%
\begin{align*}
A_{j}(q,p) & =\partial_{\tau}|_{\tau=0}(q+\tau\mathcal{E}_{j},p)=(\mathcal{E}%
_{j},0),\qquad\mathcal{E}_{j}=(e_{j},...,e_{j})\\
A_{j+3}(q,p) & =\partial_{\theta}|_{\theta=0}(e^{\theta\mathcal{J}%
}q,e^{\theta\mathcal{J}}p)=(\mathcal{J}_{j}q,\mathcal{J}_{j}p),\qquad
\mathcal{J}_{j}=diag(J_{j},...,J_{j})
\end{align*}
for $j=1,2,3$, and%
\[
A_{7}(q,p)=\partial_{\varphi}|_{\varphi=0}(q,p)(t+\varphi)=J\nabla H\text{.}%
\]
\begin{remark}
Actually, the conserved quantities $G_{j}$ are related to the generator fields
$A_{j}$ by
\[
A_{j}=J\nabla G_{j}.
\]
Using the Poisson bracket, the orthogonality relations are equivalent to%
\[
\{H,G_{j}\}=\left\langle \nabla H,J\nabla G_{j}\right\rangle =\left\langle
\nabla H,A_{j}\right\rangle =0\text{.}%
\]
The explicit conserved quantities are $G_{j}=-p\cdot\mathcal{E}_{j}$,
$G_{j+3}=p^{T}\mathcal{J}_{j}q$, for $j=1,2,3$, and $G_{7}=H$.
\end{remark}
To numerically continue a solution it is necessary to augment the
differential equation with Lagrange multipliers $\lambda_{j}\in\mathbb{R}$ for
$j=1,..,7$,%
\begin{equation}
\dot{x}=J\nabla H(x)+\sum_{j=1}^{7}\lambda_{j}JA_{j}(x)\text{.}\label{ODE2}%
\end{equation}
The solutions of equation \eqref{ODE2} are solutions of the original equations
of motion when the values of the seven parameters are zero. If $A_{j}(x)$ are
linearly independent, a solution $x$ to the equation (\ref{ODE2}) is a
solution to the equation (\ref{ODE}) because
\[
0=\left\langle \dot{x},JA_{i}(x)\right\rangle =\sum_{j=1}^{7}\lambda
_{j}\left\langle A_{j},A_{i}\right\rangle
\]
implies that $\lambda_{j}=0$ for $j=1,...,7$.
The period $T=2\pi\lambda$ can be obtained as parameter in equation
\eqref{ODE2} by rescaling time,%
\[
\dot{x}=TJ\nabla H+\sum_{j=1}^{7}\lambda_{j}JA_{j}\text{.}%
\]
Let $\varphi_{t}(x)$ be the flow of this equation. We can define the time one
map (for the rescaled time)\ as
\[
\varphi_{1}(x;\lambda_{1},...,\lambda_{7},T):V\times\mathbb{R}^{7}%
\times\mathbb{R}\rightarrow V\text{,}%
\]
where the period $T$ is a parameter. Therefore, a fixed point of $\varphi
_{1}(x)$ corresponds to a $T$-periodic solutions of the Hamiltonian system.
To numerically continue the fixed points of $\varphi_{1}(x)$ it is necessary
to implement Poincar\'{e} sections. For this we define the augmented map%
\begin{align*}
F(q,p,\lambda_{1},...,\lambda_{7};T) & :V\times\mathbb{R}^{7}\times
\mathbb{R}\rightarrow V\times\mathbb{R}^{7}\\
F & =\left( x-\varphi_{1}(x),A_{j}(\tilde{x})\cdot\left( x-\tilde
{x}\right) \right) .
\end{align*}
Then a solution of $F=0$ is a $T$-periodic solution of the Hamiltonian system.
The restrictions $A_{j}(\tilde{x}%
)\cdot\left( x-\tilde{x}\right) =0$ for $j=1,...,7$ represent the
Poincar\'{e} sections, where $\tilde{x}$ is a previously computed solutions on
the family of solutions of $F=0$. This map is a local submersion except for bifurcation points, see \cite{MuAl}.
\begin{figure}[H]
\begin{center}
\resizebox{11.0cm}{!}{\includegraphics{e02m1.png} }
\resizebox{11.0cm}{!}{\includegraphics{e03m1.png} }
\resizebox{11.0cm}{!}{\includegraphics{e03m1.png} }
\end{center}
\caption{ Top: Solutions from eigenvalue $j=2$ with symmetries $({D_{5}^{p}}\prescript{}{}\times_{{}}
^{{}}D_{1})$.
Middle: Solution from eigenvalue $j=3$ with symmetries $
{({D_{3}^{p}}\prescript{{\mathbb Z_3^p}}{}\times_{\mathbb{Z}_{2}}^{{}}D_{2})}$. Bottom: Solution from eigenvalue $j=3$ with symmetries
$(\amal{{A_4^p}}{D_{2}}{\mathbb Z_{2}}{{A_4}}{})$ }%
\label{fig-1}%
\end{figure}
\begin{figure}[H]
\begin{center}
\resizebox{11cm}{!}{\includegraphics{e04m1.png} }
\resizebox{11cm}{!}{\includegraphics{e05m1.png} }
\resizebox{11cm}{!}{\includegraphics{e13m1.png} }
\end{center}
\caption{ Top: Solutions from eigenvalue $j=4$ with symmetries $(\amal{{V_4^p}}{D_{2}}{\mathbb Z_{2}}{{V_4^z}}{})$. Middle: Solutions from eigenvalue $j=5$ with symmetries $(\amal{{D_3^p}}{D_{6}}{D_{6}}{{\mathbb Z_1}}{})$.
Bottom: Solutions from eigenvalue $j=9$ with symmetries ${({D_{3}^{p}}\prescript{{\mathbb Z_1^p}}{}\times_{D_{3}}^{{}}D_{3})}$. }%
\label{fig-2}%
\end{figure}
The map $\varphi_{1}(x)$ is computed numerically using a Runge-Kutta
integrator.\ A first solution of $F=0$ is obtained by applying a Newton method
in the approximating solution obtained by solving the linearized Hamiltonian system. The family of periodic
solutions is computed numerically using a pseudo-arclength procedure to
continue the solutions of $F=0$.
We present the results of our numerical computations in Figures \ref{fig-1} and \ref{fig-2}. The position of the atoms in space are in the right columns. The atoms with the same color have oscillations related by a rotation or inversion in $O(3)$. In addition, atoms with the same color but different texture describe oscillations that are related by the inversion coupled with a phase shift in time. In the left columns of Figures \ref{fig-1} and \ref{fig-2} we illustrate the norm of the atoms oscillating in time.
\vskip.3cm
|
{
"timestamp": "2018-10-09T02:13:07",
"yymm": "1804",
"arxiv_id": "1804.05455",
"language": "en",
"url": "https://arxiv.org/abs/1804.05455"
}
|
\section{Introduction}
\label{intro}
General relativity
is the most successful theory of gravity, which can explain various gravitational
phenomena including gravitational waves recently observed for the first time \cite{GW1,GW2}.
There is no
experimental
result which clearly contradicts
general relativity so far.
Nevertheless,
many people are fascinated by the fundamental question of how accurately general relativity describes our universe,
and attracted by seeking for an alternative gravitational theory
as a clue of quantum gravity or to elucidate the dark side of our universe.
Scalar-tensor theories of gravity are simple examples of the modified gravity,
which were originally proposed by Brans and Dicke in 1961~\cite{Brans1961}.
It contains an additional scalar field other than the Einstein-Hilbert term and the standard matter term.
One of the motivations to consider scalar-tensor theories is
to explain the accelerated expansion of the universe by adding the
scalar degree of freedom.
The scalar field usually couples to the standard model particles and
affects the motion of them
through the so-called fifth force.
Experimental tests of gravity in the solar system can give strong constraints on the fifth force
and thus parameters of scalar-tensor theories \cite{Will2014}.
In order to accord with the fifth force constraints,
scalar-tensor theories must have a mechanism that screens the fifth force mediated by the scalar field on small scales.
We may classify the scalar fields by mechanisms of the screening~\cite{Joyce2015}.
One example is the chameleon field introduced by
Khoury and Weltman~%
\cite{Khoury2004a,Khoury2004}.
The chameleon field has a large
value of the effective mass in a sufficiently high density region such as on the Earth
or in the solar system,
so that the fifth force mediated by the chameleon field becomes an unobservable short-range force.
In contrast, the chameleon field has a
smaller mass and long Compton wavelength in cosmological low density regions so that it could accelerate
the expansion of our universe.
This chameleon mechanism can be also applied to other types of modified gravity theories
such as $f(R)$ gravity (see, e.g. Ref. \cite{Felice2010}).
A lot of experimental tests have been
proposed and performed in order to seek such a field, e.g. astrophysical tests
such as those using distance indicators~\cite{Jain2013} or Galaxy rotation curve~\cite{Vikram2014},
and laboratory tests such as those using torsion pendulum~\cite{Upadhye2012}, atom interferometer~\cite{Hamilton2015}, and so on.
Calculations of the fifth force have been mainly done with
a spherically symmetric smooth density profile surrounded by a cosmological low density region
as the environment.
For instance, for a compact object,
we can estimate the scalar charge and show the fifth force can be much
weaker than the Newtonian gravitational force (see, e.g. Ref. \cite{Khoury2004} and Refs. \cite{Kobayashi2008, Upadhye2009, Tsujikawa:2009yf, Babichev:2009td, Babichev:2009fi} for relativistic stars).
Recently,
the chameleon mechanism in more general situations has been started to be investigated in numerical ways.
The screening effect on the structure formation is investigated by generalized N-body simulations \cite{Oyaizu2008f, Oyaizu2008s, Oyaizu2008t, Zhao2011, Zhao2012}
and
strong constraints on $f(R)$ parameters
are obtained from the modified gravity effects on galaxy clusters \cite{Terukina2014,Wilcox2016}.
Also, the screening for non-spherical sources is investigated in Ref. \cite{Burrage2018}.
In this paper, we focus on an aspect that has been overlooked in the above analyses.
Usually, the screening effect for a system is investigated by using the smoothly averaged density profile
over the system.
However, actual objects in the universe
do not necessarily have a smooth density profile but inhomogeneous in general.
If the Compton wavelength of the field is shorter than scales of the inhomogeneities,
the smoothing may not be justified and
effects of the inhomogeneities should be taken into account.
For example, in our galaxy, the upper bound on the Compton wavelength of the chameleon field can be obtained as $\lambda_{\phi} < 10^{7-12} {\rm m}$ \cite{Davis2014} by rescaling the terrestrial experimental upper bound,
which is smaller than the average interstellar distance.
Also in globular clusters,
the Compton wavelength $\sim 10^4 {\rm m}$, is much less than distances between the stars in the cluster.
This indicates that the chameleon field may vary
rapidly and be kicked by the inhomogeneity.
Then, a significant fifth force may be mediated inside an inhomogeneous object like a galaxy.
In order to
understand the essence of effects of inhomogeneities, as a first step, in this paper,
we keep the system as simple as possible with extremely large density contrasts.
Concretely, we assume a static spherically symmetric system composed of a
set of infinitely thin shells at regular intervals of radius,
where the inhomogeneity is controlled by the number of the shells.
The shell interval corresponds to the scale of the inhomogeneity in this system.
Thus, if we choose the parameters such that the Compton wavelength is shorter than the
shell interval, the scalar field is perceptible to the inhomogeneity and a
significantly large fifth force may appear inside the system.
Moreover, the fluctuations of the field inside the system may also affect the scalar charge of the overall system.
We calculate the field profile and the fifth force strength,
and investigate those dependence on the parameters of the system.
This paper is organized as follows.
In the section \ref{chamfield}, we introduce the chameleon field and the fifth force.
A brief review of the
uniform density case is given in the section \ref{constar} for comparison with our case.
Then, we introduce our model, the spherical shell system in the section \ref{SSS}.
The resultant fifth-force profiles are
shown in the section \ref{SinSSS}.
In the section \ref{thick},
we investigate how the fifth-force profiles change
as the shells become thicker.
Section \ref{conclusion} is devoted to a summary and conclusion.
In this paper, we use natural units in which both
the speed of light $c$ and the reduced Planck constant $\hbar$ are
one.
\section{Chameleon Field}
\label{chamfield}
A prototype of the chameleon field is given by a scalar field with a conformal coupling and a runaway-type potential
\cite{Khoury2004},
\b
\nabla_{\mu}\nabla^{\mu}\phi-\frac{\beta}{M_{\rm pl}}\rho - V'(\phi) =0 \,,
\e
where $\beta$ represents a dimensionless conformal coupling and the potential $V(\phi)$ is typically assumed to be
the inverse power-law potential: $V(\phi) = M^{4+n}/\phi^{n}$.
The prime means the derivative with respect to $\phi$.
Here, $M_{\rm pl}$ is the Planck mass and $M$ is bounded above as $M \lesssim 10^{-3} {\rm eV}$
to evade laboratory constraints on the fifth force \cite{Khoury2004},
where $\beta$ is assumed as ${\cal O} (1)$.
The second and the third terms can be combined into derivative of
the following effective potential:
\b \label{def:potential}
V_{\rm eff}(\phi) \equiv \frac{\beta}{M_{\rm pl}}\rho\ \phi+\frac{M^{4+n}}{\phi^n}.
\e
This effective potential has the minimum at
\b \label{def:minimum}
\phi_{\rm min}(\rho) \equiv M \({\frac{nM^3 M_{\rm pl}}{\beta\rho}}^{\frac{1}{n+1}},
\e
and the mass around this minimum is evaluated as
\begin{eqnarray}
m_{\rm eff}^2(\rho) &\equiv& V''_{\rm eff}(\phi_{\rm min}) \nonumber \\
&=& \frac{(n+1)\beta \rho}{M M_{\rm pl}} \left( \frac{\beta\rho}{nM^3 M_{\rm pl}} \right)^{\frac{1}{n+1}} \,. \label{def:mass}
\end{eqnarray}
The effective mass of the chameleon field increases with $\rho$.
In the static and spherically symmetric case,
the equation of motion (EoM) becomes
\b
\frac{\mathrm{d}^2\phi}{\mathrm{d} r^2}+\frac{2}{r}\frac{\mathrm{d}\phi}{\mathrm{d} r}-\rho\frac{\beta}{M_{\rm pl}}+n\frac{M^{4+n}}{\phi^{n+1}}=0.
\e
For later convenience,
we rewrite the above equation by
the following
dimensionless variable:
\b
\hat{\phi}\equiv\frac{\phi}{\phi_{\rm c}},
\e
where $\phi_{\rm c}$ is the field value at the potential minimum (\ref{def:minimum}) for the central density $\rho_c$,
that is,
$\phi_{\rm c} \equiv \phi_{\rm min}(\rho_{\rm c})$.
In addition, we introduce a length scale $L$ and use the normalized radius $x$ defined by $x \equiv r/L$.
Then, we obtain
\begin{eqnarray}
\frac{\mathrm{d}^2\hat{\phi}}{\mathrm{d} x^2}+\frac{2}{x}\frac{\mathrm{d}\hat{\phi}}{\mathrm{d} x}-\hat{\rho}\tilde{m}_{\rm c}^2 L^2 +\frac{\tilde{m}_{\rm c}^2 L^2}{\hat{\phi}^{n+1}}&=&0,
\label{chameq}
\end{eqnarray}
where $\tilde{m}_{\rm c}^2\equiv \beta\rho_{\rm c}/M_{\rm pl}\phi_{\rm c}$ and $\hat{\rho}\equiv\rho/\rho_{\rm c}$.
Note that
$m_{\rm eff}^2(\rho_{\rm c}) = (n+1)\tilde{m}_{\rm c}^2$.
As will be reviewed in the next section,
the large effective mass (\ref{def:mass}) can enforce the chameleon field to be approximately fixed at the minimum $\phi_{\rm c}$ in the interior of the star
and the matter inside the star does not contribute to the scalar charge except for the thin outer shell region
whose width is comparable to the Compton wavelength.
Therefore, the fifth force
\b
F_{\phi}=\frac{\beta}{M_{\rm pl}}\frac{\mathrm{d}\phi}{\mathrm{d} r}
=\frac{\beta\phi_{\rm c}}{M_{\rm pl}}\frac{\mathrm{d}\hat{\phi}}{\mathrm{d} r} \,,
\label{fforce}
\e
is screened by the $\rho$-dependent mass.
However, this argument is based on the smoothed density.
When density contrasts are high, the potential minimum (\ref{def:minimum}) and the effective mass (\ref{def:mass}) will vary rapidly.
For such a system, it will not be appropriate to solve the EoM with the smoothed density
and the inhomogeneity should be taken into account.
Unlike the smooth density case, the field value
may vary also in the interior of the system, which causes the appearance of a significant fifth force.
\section{Field Profile of Constant Density Stars}
\label{constar}
We review how the chameleon field is sourced by a star with the constant density $\rho_{\rm c}$
surrounded by the cosmological density $\rho_{\infty}$.
We take the radius of the star as the unit of the length scale $L$.
If the effective mass \eqref{def:mass} for the constant density $\rho_{\rm c}$ is sufficiently large,
the field value stays near the potential minimum $\phi_{\rm c}$ around the center of the star.
Then, we assume that there is a radius from which the field value starts to change and denote
this radius as $x_{\rm roll}$.
We can divide the whole region into
the following three pieces.
\begin{enumerate}
\item$x<x_{\rm roll}$\\
In this region, the value of the chameleon field does not change much,
and its value $\hat{\phi}$ and the first derivative $\mathrm{d}\hat{\phi}/{\mathrm{d} x}$ can be approximated by one and zero, respectively.
\item$x_{\rm roll}<x<1\ (r_{\rm roll}<r<L)$\\
The chameleon field rolls down the effective potential toward a larger value.
Then, the first term in the effective potential is dominant, so that
the EoM becomes
\b
\frac{\mathrm{d}^2\hat{\phi}}{\mathrm{d} x^2}+\frac{2}{x}\frac{\mathrm{d}\hat{\phi}}{\mathrm{d} x}=\tilde{m}_{\rm c}^2L^2.
\label{bound}
\e
The solution of the equation~(\ref{bound}) with the boundary conditions $\hat{\phi}=1$ and $\mathrm{d}\hat{\phi}/\mathrm{d} x=0$ at $x=x_{\rm roll}$ is given by
\b
\hat{\phi}=1+\frac{\tilde{m}_{\rm c}^2L^2}{6}\({\frac{2x_{\rm roll}^3}{x}+x^2-3x_{\rm roll}^2}.
\label{phic}
\e
\item$x>1\ (r>L)$\\
The chameleon field
quickly
falls into the value sufficiently
close to the minimum for the cosmological background $\hat{\phi}_{\infty}\equiv(\rho_{\rm c}/\rho_{\infty})^{\frac{1}{n+1}}$ outside the star.
Then, an approximate solution is obtained by linearizing the EoM (\ref{chameq})
and we obtain
\b
\hat{\phi}=\hat{\phi}_{\infty}+A\frac{{\rm e}^{-m_{\infty}Lx}}{x},
\label{phio}
\e
where $m^2_{\infty}\equiv m_{\rm eff}^2(\rho_{\infty})$.
\end{enumerate}
Matching $\hat{\phi}$ and $\mathrm{d}\hat{\phi}/\mathrm{d} x$ at $x=1$ by using the equations (\ref{phic}) and (\ref{phio}), we obtain
\b
A=-\frac{\tilde{m}_{\rm c}^2L^2}{3}\frac{1-x_{\rm roll}^3}{m_{\infty}L+1}{\rm e}^{m_{\infty}L} \,,
\label{A}
\e
and
\b
\hat{\phi}_{\infty}-1=\frac{\tilde{m}_{\rm c}^2L^2}{6}\({2\frac{1-x_{\rm roll}^3}{m_{\infty}L+1}+1+2x_{\rm roll}^3-3x_{\rm roll}^2}.
\label{phiinf}
\e
If the density of the object is sufficiently large, it is expected that the chameleon field stays near the minimum $\hat{\phi} \simeq 1$ in almost whole region inside the star,
and then $x_{\rm roll} \simeq 1$.
This limit is so-called
the thin shell regime because the only thin shell part of the star ($x_{\rm roll}<x<1$) contributes to the exterior field profile.
Then, the equations~(\ref{A}) and (\ref{phiinf}) can be approximated as follows:
\b
A\simeq-\tilde{m}_{\rm c}^2L^2\frac{1-x_{\rm roll}}{m_{\infty}L+1}{\rm e}^{m_{\infty}L},
\label{eq:A}
\e
\b
1-x_{\rm roll}\simeq\frac{1}{\tilde{m}_{\rm c}^2L^2}(m_{\infty}L+1)(\hat{\phi}_{\infty}-1).
\label{thinpara}
\e
We obtain the approximate form of $\hat{\phi}$ by substituting
equations \eqref{eq:A} and \eqref{thinpara} into the equation (\ref{phio})
as follows:
\b
\hat{\phi}=\hat{\phi}_{\infty}-(\hat{\phi}_{\infty}-1)\frac{{\rm e}^{-m_{\infty}L(x-1)}}{x}.
\label{ffcon}
\e
We can check that the equation (\ref{thinpara}) is consistent
with the assumption $x_{\rm roll} \simeq 1$ if the following condition is satisfied:
\b
\frac{\hat{\phi}_{\infty}-1}{\tilde{m}_{\rm c}^2L^2}(m_{\infty}L+1)\ll1 \,,
\label{shelcon}
\e
which is satisfied when the Compton wavelength $\lambda_\phi \equiv 1/m_{\rm c}~(m_{\rm c} \equiv \sqrt{n+1}\tilde{m}_{\rm c})$ is much shorter than the radius of the star $L$.
From the equation (\ref{fforce}), the fifth force for the constant density star is calculated as,
\b
F_{\phi}=F^{\rm con}_{\phi}:=\frac{\beta\phi_{\rm c}}{M_{pl}L}(\hat{\phi}_{\infty}-1)(m_{\infty}L x+1)\frac{{\rm e}^{-m_{\infty}L(x-1)}}{x^2}.
\e
The chameleon field has a sufficiently small effective mass (\ref{def:mass}) in the cosmological background unless $M$ is too small.
Then, the Compton wavelength of the chameleon field in the cosmological background is much longer than the radius of the star $L$, e.g. $\lambda_\phi \sim {\rm 1 Mpc}$ for $M \sim 10^{-3} {\rm eV}$.
Taking the limit $m_\infty L\rightarrow 0$,
we obtain the following expression:
\begin{equation}
\lim_{m_\infty L\rightarrow 0}F^{\rm con}_{\phi}=\frac{\beta\phi_{\rm c}}{M_{\rm pl}L}(\hat{\phi}_{\infty}-1)\frac{1}{x^2}.
\label{fiff_uni}
\end{equation}
Since the Newtonian gravitational force made by the constant density star is
given by
\b
F_{\rm Newton}=\frac{1}{8\pi M_{\rm pl}^2}\frac{4\pi L^3\rho_{\rm c}}{3L^2x^2}\n
=\frac{\rho_{\rm c}L}{6M_{\rm pl}^2}\frac{1}{x^2},
\e
we can evaluate the ratio $\mathcal R$ between the fifth force and the Newtonian gravitational force, which corresponds to the scalar charge in units of the stellar mass, as
\begin{eqnarray}
\mathcal R:=\left|\frac{F_{\phi}}{F_{\rm Newton}}\right|&=&6\frac{\beta M_{\rm pl}\phi_{\rm c}}{\rho_{\rm c}L^2}(\hat{\phi}_{\infty}-1)(m_{\infty}L x+1){\rm e}^{-m_{\infty}L(x-1)}\n\\
&\simeq&6\beta^2\frac{\phi_{\rm c}}{\tilde{m}_{\rm c}^2L^2}(\hat\phi_{\infty}-1),
\label{thin}
\end{eqnarray}
where we have taken the limit $m_\infty L\rightarrow 0$ in the second line.
We can find that, if the thin-shell assumption is valid, that is,
the equation \eqref{shelcon} is satisfied,
the value of $\left|F_{\phi}/F_{\rm Newton}\right|$ is suppressed.
On the other hand, in the case $x_{\rm roll}\simeq 0$, which is called the thick-shell limit,
the field value
and the fifth force are given by, respectively,
\b
\hat{\phi}=\hat{\phi}_{\infty}-\frac{\tilde{m}_{\rm c}^2L^2}{3(m_{\infty}L+1)}\frac{{\rm e}^{-m_{\infty}L(x-1)}}{x},
\e
\b
\mathcal R=2\beta^2\frac{m_{\infty}Lx+1}{m_{\infty}L+1}{\rm e}^{-m_{\infty}L(x-1)} = {\cal O}(\beta^2),
\e
where we have assumed $m_{\infty}L\simeq0$.
\section{Spherical Shell System}
\label{SSS}
\begin{figure}[t]
\begin{center}
\includegraphics[clip,width=8cm]{SSSdia.eps}
\caption{A schematic figure of the spherical shell system.}
\label{sss}
\end{center}
\end{figure}
In the previous section,
we have considered a spherical object with a constant density, where
the large effective mass can make the chameleon field stay at the minimum of the effective potential $V_{\rm eff}$ inside the object.
In this section, we consider a simple but non-trivial example of an inhomogeneous system:
$N$ pieces of concentric spherical shells
separated by vacuum regions with regular intervals $\Delta x$ and equal surface density $\sigma$ (see Fig.~\ref{sss}).
The shells are assumed to be infinitely thin, that is, the radial density profile of each shell is approximated by a delta function.
Moreover, in order to avoid running away of the chameleon field to infinity,
we assume that the shell system is surrounded by the cosmological density $\rho=\rho_{\infty}$ as usual.
Under these idealizations, we investigate how the inhomogeneities can have an impact on the field profile.
In this system,
neither the potential minimum nor the effective mass is defined at any radius and the previous intuitive argument cannot be applied.
In reality, we would
need to introduce a small density between the shells.
Nevertheless, if the field value does not reach the minimum of the effective potential in the intervals,
it is irrelevant whether the density is finite or zero as we assumed.
In addition, the infinitely thin shells are an idealization.
We will discuss how our argument here is affected when the shells are thicker in the section~\ref{thick}.
We assumed that
the shells regularly foliate the spherical region with a fixed interval of the radius,
and the surface density of each shell is identical to each other.
Then, given the radius of the outermost shell $L$, the interval $\Delta x$ can be written as $L/N$.
Denoting the total mass of the system by $M_{\rm tot}$,
the surface density $\sigma$ is given by,
\begin{eqnarray}
\sigma&=&\frac{M_{\rm tot}}{4\pi (L/N)^2(1^2+2^2+\cdots+N^2)}\n\\
&=&\frac{3M_{\rm tot}N}{2\pi L^2(N+1)(2N+1)} \,.
\end{eqnarray}
The smoothed density $\rho_{\rm c}$ can be written as
$\rho_{\rm c}=3M_{\rm tot}/4\pi L^3$ and is related to the surface density as
\b
\sigma = \rho_{\rm c}L\frac{N}{(N+1)(N+1/2)} \,.
\e
Hereafter, we use the radius of the outermost shell $L$ as the length scale $L$ in the section {\ref{chamfield}}.
Under these setup, the field equation (\ref{chameq}) becomes
\b
\frac{\mathrm{d}^2\hat{\phi}}{\mathrm{d} x^2}+\frac{2}{x}\frac{\mathrm{d}\hat{\phi}}{\mathrm{d} x}+\frac{\tilde{m}_{\rm c}^2L^2}{\hat{\phi}^{n+1}}=0,
\label{vacuum}
\e
in the vacuum regions,
and
\b
\frac{\mathrm{d}^2\hat{\phi}}{\mathrm{d} x^2}+\frac{2}{x}\frac{\mathrm{d}\hat{\phi}}{
\mathrm{d} x}+\frac{\tilde{m}_{\rm c}^2L^2}{\hat{\phi}^{n+1}}-\hat{\rho}_{\infty}\tilde{m}_{\rm c}^2L^2=0 \,,
\label{outside}
\e
in the outer cosmological region $r > L$.
The junction condition at each shell is given by \cite{Deruelle2008}
\b
[\phi]^+_-=0 \,,
\label{bfp}
\e
\b
\[\frac{\mathrm{d}\phi}{\mathrm{d} r}\]^+_-=\beta\frac{\sigma}{M_{\rm pl}}.
\e
The symbol $[~]^+_-$ on the left hand side of the equations is defined by
\b
[f(x)]^+_-\equiv \lim_{x\rightarrow x_{\rm shell}+0}f(x)-\lim_{x\rightarrow x_{\rm shell}-0}f(x).
\e
It will be more suggestive to rewrite the surface density in the second junction condition in terms of the smoothed density $\rho_{\rm c}$ or the effective mass $\tilde{m}_{\rm c} ~ (\equiv m_{\rm c}/\sqrt{n+1})$:
\begin{eqnarray}
\[\frac{\mathrm{d}\hat{\phi}}{\mathrm{d} x}\]^+_-&=&\beta\frac{\rho_{\rm c}L^2}{M_{\rm pl}\phi_{\rm c}}\frac{N}{(N+1)(N+1/2)}\n\\
&=&\tilde{m}_{\rm c}^2L^2\frac{N}{(N+1)(N+1/2)} \,. \label{bfdp}
\end{eqnarray}
The Newtonian gravitational force in the $i$-th region
is given by
\begin{eqnarray}
F_{\rm Newton}&=&\frac{1}{8\pi M^2_{\rm pl}}M_{\rm tot}\frac{i(i+1)(2i+1)}{N(N+1)(2N+1)}\frac{1}{r^2}\n\\
&=&\frac{\rho_{\rm c}L}{6M^2_{\rm pl}}\frac{i(i+1)(2i+1)}{N(N+1)(2N+1)}\frac{1}{x^2}\hspace{2cm}
{\rm for} \quad i/N<x<(i+1)/N,
\label{newton}
\end{eqnarray}
where
$i$
runs over $0$ to $N$.
As is well known, the Newtonian gravitational force $F_{\rm Newton}$ depends only on the enclosed mass at a given radius irrespective of its internal structures.
Then, from the equation (\ref{fforce}), we obtain
\begin{eqnarray}
\mathcal R
&=&6\beta\frac{M_{\rm pl}\phi_{\rm c}}{\rho_{\rm c}L^2}\frac{N(N+1)(2N+1)}{i(i+1)(2i+1)}\frac{\mathrm{d} \hat{\phi}}{\mathrm{d} x}x^2\n\\
&=&\frac{6\beta^2}{\tilde{m}_{\rm c}^2L^2}\frac{N(N+1)(2N+1)}{i(i+1)(2i+1)}\frac{\mathrm{d} \hat{\phi}}{\mathrm{d} x}x^2.
\end{eqnarray}
To see the impact of the inhomogeneities, in the next section, we will evaluate the value of $\mathrm{d}\phi/\mathrm{d} x$
for various values of the parameters $\tilde{m}_{\rm c}L$ and $N$,
which represent the ratio of the length scales in the system, $L/\lambda_\phi$ and $L/\Delta x$, respectively.
For the smoothed density,
we found $x^2 \mathrm{d} \hat{\phi}/\mathrm{d} x \sim \hat{\phi_\infty} - 1$,
and thus the small factor $1/\tilde{m}_{\rm c}^2L^2 \simeq (\lambda_\phi/L)^2$ ensures the screening.
On the other hand, in our case,
when the interval $\Delta x$, is large enough,
the chameleon field is expected to vary rapidly and the ratio ${\cal R}$ might become large.
\section{Screening in the Spherical Shell System}
\label{SinSSS}
We solve the field equations (\ref{vacuum})
and (\ref{outside}) with the junction conditions (\ref{bfp}) and (\ref{bfdp})
taking into account the thin-shell condition (\ref{shelcon}) for the smoothed density.
Here, as an example,
we consider
the averaged density of a galaxy for $\rho_{\rm c}$ as
$\rho_{\rm c}\simeq10^{7}\rho_{\infty}$, which corresponds to
\b
\hat{\phi}_{\infty}=10^{\frac{7}{n+1}}.
\e
Then, in the case of the smoothed density, the thin-shell condition is given by $ \tilde{m}_{\rm c}^2 L^2 > 10^{7/(n+1)}$.
\subsection{Numerical analysis}
First, we consider marginal cases $ \tilde{m}_{\rm c}^2 L^2 \gtrsim 10^{7/(n+1)}$ with a numerical method.
For simplicity, we choose the power $n$ of the potential as $n=2$ in the analysis.
Then, the thin-shell condition is given by $\tilde{m}_{\rm c}^2 L^2 \gtrsim 10^{7/3} \sim 200$.
We calculate the field profile by numerically solving the field equations using the shooting method with the junction conditions at each shell as well as the boundary conditions $\mathrm{d}\hat{\phi}/\mathrm{d} x|_{x=0} = 0$ and $\lim_{x\rightarrow\infty}\hat
{\phi}(x)=\hat{\phi}_{\infty}$.
We show the field profile and the value of $\mathcal R$ as functions of $x$ in Fig.~\ref{v102} for $\tilde{m}_{\rm c}^2L^2=10^2, 10^3,$ and $10^4$.
\begin{figure}[t]
\centering
\subfigure[profile of $\phi(x)$]{
\includegraphics[width=7.5cm]{n10mcc.eps}
}
\subfigure[$\mathcal R =F_{\phi}/F_{\rm Newton}$]{
\includegraphics[width=7.5cm]{n10fmcc.eps}
}
\caption{These figures show the profile of the chameleon field $\phi$ and the strength of the fifth force divided by the Newtonian gravitational force $\mathcal R$ for $N=10$ and $\tilde{m}_{\rm c}^2L^2=10^2$, $10^3$, $10^4$ respectively.}
\label{v102}
\centering
\end{figure}
In Fig.~\ref{v102}, it is clearly shown that,
for $\tilde{m}_{\rm c}^2L^2=10^2$, the fifth force is comparable to
the Newtonian gravitational force everywhere.
In contrast, for $\tilde{m}_{\rm c}^2L^2=10^4$, the fifth force is
suppressed compared with the Newtonian gravitational force outside the shell system.
We also check the dependence on the number of shells $N$.
In Fig.~\ref{v4nvary}, $\mathcal R$ is depicted as a function of $x$ outside the system for $N = 1, 5,$ and $10$
with $\tilde{m}_{\rm c}^2L^2=10^4$.
\begin{figure}[t]
\centering
\subfigure{
\includegraphics[width=7cm]{nvary1.eps}
}
\subfigure{
\includegraphics[width=7cm]{nvary2.eps}
}
\caption{This figure shows the variation of the fifth force for different numbers of shells $N = 1, 5,$ and $10$ with $\tilde{m}_{\rm c}^2L^2=10^4$.
We show an enlarged figure for the outside region in the right panel to
show the dependence on the number of the shells.
Each line almost coincides with the one for the smoothed density case in the outside of the outermost shell, $x>1$.
}
\label{v4nvary}
\centering
\end{figure}
The behavior of $\mathcal R$ is similar to the smoothed-density case
$\rho = \rho_{\rm c}$ irrespective of the number of shells
as shown in Fig.~\ref{v4nvary}.
Therefore,
the criterion of the thin-shell condition for the screening is applicable to the spherical shell system in the outside region.
It is worthy
of note that there is a small but finite deviation even in the outside region.
This finite deviation becomes larger for
the marginal case $\tilde{m}_{\rm c}^2L^2=10^3$
as shown in Fig.~~\ref{nvary103}.
\begin{figure}[t]
\begin{center}
\includegraphics[width=8cm]{nvary103.eps}
\end{center}
\caption{We plot the numerical result of the fifth-force strength divided by the Newtonian
gravitational force
for $\tilde{m}_{\rm c}^2L^2=10^3$ with $N=1$ and $5$.
The constant density case is also depicted for comparison. }
\label{nvary103}
\end{figure}
We will discuss it more quantitatively for a large value of $\tilde{m}_{\rm c}^2L^2$ in the subsection \ref{ss:ffout}.
On the other hand, in the inner region,
the fifth force is not screened well even when the thin-shell condition for the averaged density is satisfied.
In the subsection \ref{ss:ffin}, we will see that it is true for a larger value of $\tilde{m}_{\rm c}^2L^2$.
\subsection{Analytic approximate solution }
The thin-shell condition is well satisfied
a realistic situation for a galaxy as
$\tilde{m}_{\rm c}^2 L^2 \gtrsim 10^{28-22/(n+1)}$ with $L \sim 10 {\rm kpc}$ \cite{Davis2014}.
Numerical analyses for such a huge value of $\tilde{m}_{\rm c}^2L^2$
are very difficult \cite{Kobayashi2008, Upadhye2009}.
Instead of solving the EoM numerically, here, following Ref. \cite{Davis2014},
we use an approximation which is valid for a sufficiently large value of $\tilde{m}_{\rm c}^2L^2$.
We suppose that, for a large value of $\tilde{m}_{\rm c}^2L^2$,
the potential term
is much larger than the friction term $(2/x)\mathrm{d}\hat{\phi}/\mathrm{d} x$
between the shells.
Then, the EoM can be approximated as follows:
\b
\frac{\mathrm{d}^2\hat{\phi}}{\mathrm{d} x^2}+\frac{\tilde{m}_{\rm c}^2L^2}{\hat{\phi}^{n+1}}\simeq0.
\e
The solution for the above equation is given by
\b
\frac{\mathrm{d}\hat{\phi}}{\mathrm{d} x}\simeq\pm\sqrt{C+\frac{2\tilde{m}_{\rm c}^2L^2}{n\hat{\phi}^n}},
\label{dphi}
\e
where $C$ is an integration constant.
As is shown in Fig.~\ref{v4com}, the same shape is repeated between the shells.
The first derivative $\mathrm{d}\hat{\phi}/\mathrm{d} x$ vanishes at the middle point
and the profile of $\hat{\phi}$ has a symmetric shape with respect to this middle point.
Assuming a similar repeating structure in the solution for a large value of $\tilde{m}_{\rm c}^2L^2$,
we can estimate the first derivative at the shell positions
as
\b
\left.\frac{\mathrm{d}\hat{\phi}}{\mathrm{d} x}\right|_{x=i\cdot\Delta x+0}=\frac{1}{2}\[\frac{\mathrm{d}\hat{\phi}}{\mathrm{d} x}\]^+_- \,.
\label{fderiv:shell}
\e
Substituting the junction condition at a shell (\ref{bfdp}) into (\ref{fderiv:shell}),
we can determine the constant $C$ in terms of the field value at the shell position $\phi_{\rm s}$ as follows:
\b
\tilde{m}_{\rm c}^2L^2\frac{N}{(N+1)(2N+1)}\simeq\sqrt{C+\frac{2\tilde{m}_{\rm c}^2L^2}{n\hat{\phi_{\rm s}}^n}}.
\label{jcshell}
\e
The constant $C$ is written in a simpler form by using the field value at the middle point, $\phi_0$, as $C=-2\tilde{m}_{\rm c}^2L^2/(n\hat{\phi}_0^n)$.
Then, in order for the above approximation to be valid, we need to impose the following condition:
\b
\frac{1}{x}\frac{\mathrm{d}\hat{\phi}}{\mathrm{d} x}/\({\frac{\tilde{m}_{\rm c}^2L^2}{\hat{\phi}^{n+1}}}
\simeq\sqrt{\frac{2}{n}}\frac{\hat{\phi}^{n/2+1}}{\tilde{m}_{\rm c}Lx} \sqrt{1-\({\hat{\phi}/\hat{\phi}_0}^n}
\ll1.
\label{appvaconv}
\e
Our numerical results in Fig. \ref{v102} show that the field value $\hat{\phi}$ varies at most by $\Delta \hat{\phi}/\hat{\phi} = {\cal O}(1)$ in the inner regions.
Therefore, our approximation is valid for a sufficiently large value of $\tilde{m}_{\rm c}^2 L^2$.
The approximation \eqref{appvaconv} cannot be applied to the region near the center $\tilde{m}_{\rm c}Lx \ll 1$ and then neither the solution (\ref{dphi}).
In this region, we use the following asymptotic expansion of $\hat{\phi}$
inside the innermost shell:
\b
\hat{\phi}=c_0-\frac{1}{6}c_0^{-n-1}\tilde{m}_{\rm c}^2L^2x^2-\frac{1}{120}(n+1)c_0^{-2n-3}\tilde{m}_{\rm c}^4L^4x^4+\cdots,
\label{center}
\e
where $c_{0}$ is the field value at the origin.
This expansion is valid for sufficiently small $\tilde{m}_{\rm c}Lx$.
We can construct the field profile by jointing the
approximate solutions (\ref{center}), (\ref{dphi}) and (\ref{phio})
at each shell with the junction condition (\ref{bfp}).
\begin{figure}[t]
\begin{center}
\includegraphics[width=7cm]{anvsns.eps}
\end{center}
\caption{We plot the numerical result and analytical approximate one for $\tilde{m}_{\rm c}^2L^2=10^4$, $N=10$ on the same figure. The blue and green lines correspond
to the numerical result and
the analytical approximation, respectively.
}
\label{v4com}
\end{figure}
In Fig.~\ref{v4com}, we show that the analytic approximation
agrees well with
the numerical result for $\tilde{m}_{\rm c}^2L^2=10^4$ and $N=10$.
The deviation between the analytic approximation and the numerical result is less than several percents.
\subsection{Fifth Force outside the System}\label{ss:ffout}
Let us evaluate the value of $\mathcal R$
in the outside of the system
by using the analytic approximation given
in the previous subsection.
For simplicity, we concentrate on a specific form of the potential with $n=2$, then
Eq.~(\ref{dphi}) can be easily solved as
\b
\hat{\phi}=\sqrt{C_1-\frac{\tilde{m}_{\rm c}^2L^2}{C_1}(x+C_2)^2},
\e
where $C_1$ and $C_2$ are integration constants.
The integration constants can be rewritten by
using the field value at a shell $\hat{\phi}_s$ as
\b
\hat{\phi}(x)=\sqrt{\frac{\hat{\phi}_{\rm s}^2+\sqrt{\hat{\phi}_{\rm s}^4+\tilde{m}_{\rm c}^2L^2/N^2}}{2}-\frac{2\tilde{m}_{\rm c}^2L^2(x-x_0)^2}{\hat{\phi}_{\rm s}^2+\sqrt{\hat{\phi}_{\rm s}^4+\tilde{m}_{\rm c}^2L^2/N^2}}} \,,
\label{phiap}
\e
where $x_0$ is the value of $x$ at the middle of the interval and we have assumed $\mathrm{d} \hat{\phi}/\mathrm{d} x|_{x=x_0}=0$ ,which is suggested from the numerical calculation.
According to the junction condition (\ref{jcshell}), we can determine the field value at the shell position $\hat{\phi}_{\rm s}$ by the following equation:
\b
\hat{\phi}_{\rm s}\left(\hat{\phi}_{\rm s}^2+\sqrt{\hat{\phi}_{\rm s}^4+\tilde{m}_{\rm c}^2L^2/N^2}\right)=\frac{(N+1)(2N+1)}{N^2}.
\label{eqofphis}
\e
From the above equation, we obtain the following behavior depending on the value of the parameter $m_{\rm c}L/N=\Delta x/\lambda_{\phi}$:
\b
\hat{\phi}_{\rm s}\sim
\left\{
\begin{array}{cc}
1&(\tilde{m}_{\rm c}^2L^2/N^2\ll1)\\
2N/(\tilde{m}_{\rm c}L)&(\tilde{m}_{\rm c}^2L^2/N^2\gg1)
\end{array}
\right.
\label{phislimit}
\e
with estimating the right-hand side of Eq. \eqref{eqofphis} to be ${\cal O}(1)$.
Therefore, it is assured that $\hat{\phi}_{\rm s}$ is less than ${\cal O}(1)$.
If the approximation
is valid even at the outermost shell,
the field value at the outermost shell is also given by
$\hat{\phi}_{\rm s}$.
Then, we can estimate the fifth force outside the object from the equation (\ref{phio}) as
\b
\lim_{m_\infty L\rightarrow0}F_{\phi}=\frac{\beta \phi_{\rm c}}{M_{\rm pl}L}(\hat{\phi}_{\infty}-\hat{\phi}_{\rm s})\frac{1}{x^2} \,,
\label{ffap}
\e
in the limit $m_{\infty}L\rightarrow 0$.
The effect of the inhomogeneity
on the fifth force outside the object
can be calculated by taking the difference between
Eqs.~(\ref{ffap}) and
(\ref{fiff_uni}) as follows:
\b
\lim_{m_{\infty}L\rightarrow0}\left(F_{\phi}-F_{\phi}^{\rm con}\right)
=\frac{\beta \phi_{\rm c}}{M_{\rm pl}L}(1-\hat{\phi}_{\rm s})\frac{1}{x^2}.
\label{outf}
\e
We see that, from Eq.~(\ref{phislimit}), the value of $\phi_{\rm s}$ approaches to unity
and thus $F_\phi \to F_\phi^{\rm con}$ for $\tilde{m}_{\rm c}^2L^2/N^2 = (\Delta x/\sqrt{3}\lambda_{\phi})^2 \rightarrow 0$.
It is also noteworthy that
the difference between $F_\phi$ and $F_\phi^{\rm con}$ is suppressed by the
factor $1/(\tilde{m}_{\rm c}^2L^2)$ compared to the Newtonian gravitational force as follows:
\b
\lim_{m_{\infty}\rightarrow0}\left|\frac{F_{\phi}-F_{\phi}^{\rm con}}{F_{\rm Newton}}\right|=\frac{6\beta^2}{\tilde{m}_{\rm c}^2L^2}(1-\hat{\phi}_{\rm s}).
\label{diffef}
\e
The above expression is valid only for a large value of $\tilde{m}_{\rm c}^2L^2$ but suggests that
the difference between $F_\phi$ and $F_\phi^{\rm con}$ may be non-negligible for marginal cases such as $\tilde{m}^2_cL^2=10^3$.
\subsection{Fifth Force inside the System}\label{ss:ffin}
As we have already mentioned in Sec.~\ref{SinSSS}.B, the field profile is approximately symmetric at each shell,
so that the derivative of the field has the same absolute value but
the opposite sign at each side.
Then, the value of $\mathcal R$ at each shell can be straightforwardly evaluated by the
junction condition \eqref{fderiv:shell} and the form of Newtonian gravitational force \eqref{newton} as follows:
\b
\mathcal R=\frac{6\beta^2i}{(i+1)(2i+1)}.
\label{inf}
\e
The maximum value $\mathcal R_{\rm max} = \beta^2$ is realized at the innermost shell for $i=1$ irrespective of a value of $\tilde{m}_{\rm c}^2 L^2$.
It is to be noted that the value (\ref{inf}) is obtained without specifying the potential form.
We can confirm the validity of the approximation \eqref{inf} by comparing it with the numerical result~(see Fig.~\ref{ffanvsnu}).
\begin{figure}[t]
\begin{center}
\includegraphics[width=7cm]{ffanvsnu.eps}
\end{center}
\caption{
The value of $\mathcal R$ is depicted as a function of $x$.
The spiky blue lines show the result of numerical integration for $N=10$ and $\tilde{m}_{\rm c}^2L^2=10^4$.
The red points show the analytic approximation at the position of each shell given by Eq. \eqref{inf} with substituting $i=x/N$.
}
\label{ffanvsnu}
\end{figure}
This result is very suggestive in the following sense:
even if the Compton wavelength is sufficiently smaller than the size of the object,
so that the fifth force is screened outside the object,
the value of the fifth force can be comparable to the Newtonian gravitational force
in the shell system.
\section{Effect of the finite width and the origin of the fifth force enhancement in our model}
\label{thick}
In this section, we discuss how the fifth force appearing
in the previous section depends on the width of the shells.
\subsection{Fifth Force outside the System}
First, we examine the fifth force outside the object with changing the thickness $\delta$ of the outermost shell.
For simplicity, we divide the total mass of the system into the outermost thick shell and
the other inner thin shell at the radius $L/2$(see Fig.~\ref{sss2}).
\begin{figure}[htbp]
\begin{center}
\includegraphics[clip,width=8cm]{SSSdia2.eps}
\caption{A schematic figure of the thick outer shell and thin inner shell system.}
\label{sss2}
\end{center}
\end{figure}
The value of $\mathcal{R}=F_{\phi}/F_{\rm Newton}$ at the outer surface of the thick shell is depicted as a function of $\delta$ for each value of the outer shell mass $M_{\rm o}$ in the right panel of Fig.~\ref{outsideR} for $\tilde{m}_{\rm c}^2L^2=10^3$.
The field profile is also shown on the left panel for the same parameters with $\delta=0.25$ .
We note that, in this setup, $1-x_{roll}$ in Sec. III is estimated to be about $0.2$, which gives a typical length for the scalar field to settle down to the minimum.
Let us define $\delta_{\rm c}$ as the value of $\delta$ for which the density of the thick shell is equal to that of
the uniform density case.
The vertical line shows the value of $\delta_{\rm c}$ for each value of $M_{\rm o}$.
\begin{figure}[t]
\centering
\subfigure{
\includegraphics[width=6.4cm]{profile.eps}
}
\subfigure{
\includegraphics[width=8.6cm]{outside.eps}
}
\caption{
The field profile and the value of $\mathcal{R}=F_{\phi}/F_{\rm Newton}$ for $M_{\rm o}=0.8M, 0.6M$ and $0.4M$ with $\tilde{m}_{\rm c}^2L^2=10^3$ is depicted.
The field profile is shown with $\delta=0.25$ in the left panel and $\mathcal{R}$ is calculated at the same point, $x=2$, as a function of $\delta$ in the right panel.
The value of $\mathcal R$ for the uniform density case is shown by the dashed line. The vertical lines on the right panel correspond to the position of $\delta_c$ respectively.
}
\label{outsideR}
\centering
\end{figure}
As is shown in Fig.\ref{outsideR}, we obtain a smaller fifth force value for a thicker shell.
For $M_{\rm o}=0.6M$ and $0.8M$ cases, the lines intersect with the dashed line of the uniform density
when the density of the thick shell is equal to that of the uniform density case.
As can be easily seen from the field profile shown in the left panel of Fig.~\ref{outsideR}, in these cases, the scalar field is settle down to the minimum of the effective potential for the constant density and gives the same field profile in the outer region.
While, for $M_{\rm o}=0.4M$ case, we find $\delta_{\rm c}<1-x_{\rm roll}$, and
the value of $\mathcal R$ dose not exceed that for the uniform density case.
In summary,
the enhancement of the fifth force outside the system is due to the higher density near the surface.
Therefore, we may conclude that
the fifth force outside the system can be significantly different from the uniform density case
only if the density distribution within the damping depth
is significantly different.
\subsection{Fifth Force inside the System}
Next, we consider the effect of the thickness of the shell on the inside part.
For simplicity, we assume each shell has the identical
width and density. The shells regularly foliate the spherical region with a fixed interval.
It is also assumed that the size and total mass of the system is given by $L$ and $M_{\rm tot}$ respectively as before.
Then, the smoothed density $\rho_c$ is also fixed.
Denoting the width of the shells by $aL$, which should be less than $L/N$, the density of each shell is given by
\begin{align}
\rho&=\rho_c\frac{V_{\rm s}}{V_{\rm shell}}\n\\
&=\rho_c\frac{2N}{(N+1)(2N+1)a-3N(N+1)a^2+2N^2a^3} \sim \frac{\rho_c}{Na} \,,
\label{shellrho}
\end{align}
where $V_{\rm s}$ is the volume of the whole system and $V_{\rm shell}$ is the total volume of all shells.
The last equality is satisfied for large $N$ and small $a$.
We calculate the field profile with the same boundary conditions as those in the previous calculations.
The result has not changed qualitatively from the previous ones as shown in Fig.~\ref{thick_numeric}.
The large fifth force appears at each shell although its amplitude becomes smaller.
From Fig.~\ref{avary}, we can see the profile approaches to that for the infinitely thin shells as the width $aL$ decreases.
\begin{figure}[t]
\centering
\subfigure[profile of $\phi(x)$]{
\includegraphics[width=7.5cm]{shellwidth1.eps}
}
\subfigure[$\mathcal R =F_{\phi}/F_{\rm Newton}$]{
\includegraphics[width=7.5cm]{shellwidth2.eps}
}
\caption{These figures show the profile of the chameleon field $\phi$ and the strength of the fifth force divided by the Newtonian gravitational force $\mathcal R$ for $N=3$, $\tilde{m}_{\rm c}^2L^2=10^3$ and $a=1/12$.
The blue and red regions represent the shell regions. }
\label{thick_numeric}
\centering
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics[width=8cm]{avary.eps}
\end{center}
\caption{We plot the ratio $\mathcal{R}=F_{\phi}/F_{\rm Newton}$ for $a=0, 1/12, 1/30$
with $\tilde{m}_{\rm c}^2L^2=10^3$, $N=3$.
The profiles are plotted only in the vacuum regions to make the difference easier to see. }
\label{avary}
\end{figure}
Let us consider the origin of the large fifth force for thin shell cases.
From Eq. (\ref{def:mass}), we can find that the effective mass in the shell is enhanced by the factor $(Na)^{-\frac{1}{2}\frac{n+2}{n+1}}$ compared with that for the uniform-density object.
On the other hand,
the length scale of a shell $L_{\rm shell}$ along the radial direction is given by its width $aL$.
Then, we can roughly estimate the screening parameter for each shell as
\b
(\tilde m_{\rm eff} L_{\rm shell})^2 \sim N^{-\frac{n+2}{n+1}} a^{\frac{n}{n+1}} (\tilde m_{\rm c} L)^2 \,.
\label{thin_pm_shell}
\e
Therefore, when the number of the shells $N$, or the mass of each shell, is fixed,
the shell becomes totally unscreened for the
limit $a \to 0$.
This is the origin of the large fifth force in the inner region.
This argument indicates that, even when the chameleon screening mechanism is working for a total system,
the components of the system can be unscreened and then a large fifth force can appear in its inside, depending on the shape of the components.
\section{Conclusion}
\label{conclusion}
We investigated the chameleon screening mechanism for inhomogeneous density profiles.
For some specific density profile with high density contrasts,
it is expected that the chameleon field cannot trace the minimum of the
varying potential and the smoothing of the density may not be justified.
To explicitly show it, we considered one of the simplest examples, the spherical shell system composed of
a set of concentric shells, where there is no potential minimum at
any radius and the chameleon field cannot be stable
by a large mass as usually assumed for a successful screening of the fifth force.
The results show
that the fifth force can be screened outside the system if the so-called thin-shell
condition is satisfied for the smoothed average density as in the case of a constant density profile.
The screening mechanism successfully works for a cluster of unscreened objects
if the cluster satisfies the thin-shell condition on average.
However, we find the inhomogeneity near the surface can contribute to the fifth force value for the marginal screening case.
The field profile inside the system can be significantly different
from the smoothed density case for the shell system.
We derived an analytic approximate expression for the fifth force inside the system
with the help of insights from the numerical results.
In our simple toy model, irrespective of the other model parameters,
the maximum value of the ratio between the fifth force and
the Newtonian gravitational force is given by $\beta^2$ with $\beta$ being the dimensionless coupling constant for
the conformal coupling between the standard matter and the chameleon field.
Since the value of $\beta$ is usually assumed to be in the order of 1,
our result suggests the possibility that the fifth force can be significantly large inside an object with a highly inhomogeneous density profile.
Due to the fact that this result is irrelevant to the property of the effective Compton wavelength,
the same concern may exist in other fifth force models which
have a circumstance dependent screening mechanism, such as the
symmetron \cite{Hint2010}
and the environmentally dependent dilaton \cite{Brax2011}.
One should not feel complacent about the wellbehavedness of the fifth-force field
with an averaged density distribution.
A significant fifth force strength can be induced inside an inhomogeneous object
depending on the shape of inhomogeneity.
Since it does not follow the inverse square law, unlike the case of Newtonian gravitational force, the configuration of outer shell affects the fifth force inside it.
As shown in
Fig.~\ref{v102}, the fifth force works in the direction of collecting matters to each shell.
Then, it may cause new instability other than the one caused by the usual gravitational attraction
and should be investigated more carefully as a factor that may affect the structure formation.
\acknowledgements
We would like to thank Prof. Shin'ichi Nojiri for his useful comments.
We also would like to thank the anonymous referee for helpful suggestions.
This work was supported by JSPS
KAKENHI Grant Numbers JP16K17688, JP16H01097 (CY) and JP17K14286 (RS).
\bibliographystyle{apsrev4-1}
|
{
"timestamp": "2018-10-23T02:15:58",
"yymm": "1804",
"arxiv_id": "1804.05485",
"language": "en",
"url": "https://arxiv.org/abs/1804.05485"
}
|
\subsection{Experiment Configuration}
The experiments involved comparing different methods of learning user embeddings, all with a weighted bag-of-words modeling assumption:
\begin{itemize}
\item Weighted word2vec (W2V) using default\footnote{The default configuration uses
a window of $\pm7$ words. We also tried using a window of 50 words, which roughly matches the context used in other methods,
but community detection performance was significantly worse.}
skip-gram training \cite{mikolov2013distributed};
\item Latent Dirichlet allocation (LDA) \cite{Blei+03}, using default settings from the Scikit Learn library \cite{scikit-learn};
\item Person re-identification with random initialization (RE-ID); and
\item Person re-identification with W2V initialization (RE-ID, W2V init).
\end{itemize}
Both count-weighted W2V and LDA have been used as unsupervised representations in Twitter classification tasks, as noted in Section~\ref{sec:related}.
Default configurations are used because there is insufficient data to have a separate validation set.
For all methods, the same vocabulary, final dimension (128), unit vector normalization strategy, and logistic regression model training were used. The embeddings are trained on the 36k user general data, randomly sampling pairs of users $p_1$ and $p_2$ and then sampling 50 tweets at a time without replacement to create $u_{p_{1}^{1}}$, $u_{p_{1}^{2}}$, and $u_{p_{2}^{1}}$.
The logistic regression models are trained on the 1K user general training pool, using the 50 most recent tweets for each user.
Because there are so few labeled examples for most communities, training and evaluation is done using a leave-one-out strategy with the positive samples but including all of the 1K negative samples. For each of the $N$ classifiers (corresponding to $N$ labeled samples), the test set is the left-out positive example and the 43K general user test pool.
Also because of training limitations, there is no tuning of the regularization weight; the default weight of 1.0 is used. Tuning may be useful given a collection of training and testing communities.
Performance is averaged over the $N$ classifiers (corresponding to the $N$ labeled samples). Two evaluation criteria are used: a retrieval metric (inverse mean reciprocal rank or 1/MRR) \cite{voorhees1999trec} and a detection metric (area under the curve or AUC).
\subsection{Results}
Table \ref{table:stats} shows retrieval results averaged across all communities. The RE-ID model outperforms the W2V and LDA baselines for both criteria, with substantial gains in 1/MRR (lower is better).
Further, the version of RE-ID initialized with word2vec did better than the one that was initialized randomly even though the randomly initialized version was trained for twice as long.
\begin{table}[ht]
\centering
\begin{tabular}{lrr}
\textbf{Strategy} & \textbf{AUC} & \textbf{1/MRR} \\ \hline
W2V & 93.9 & 846 \\
LDA & 95.0 & 501 \\
RE-ID (rand. init) & 98.0 & 24 \\
RE-ID (W2V init) & \textbf{98.5} & \textbf{12} \\
\end{tabular}
\caption{Performance of different model variants.}
\label{table:stats}
\end{table}
A breakdown of the best model performance by community is given in Table \ref{table:breakdown}.
Sample size does not seem to be a good indicator of performance: the two smallest communities (Cartoonists, Fresno City Council) had the worst and one of the best results, respectively. Anecdotally, we observed that the sample of cartoonists were more likely to Tweet about topics outside their main interest (e.g., politics or sports). We hypothesize that the diversity of interests of the members of a community affects the difficulty of the retrieval task, but our test set is too small to confirm this hypothesis.
\begin{table}[ht]
\centering
\begin{tabular}{lrr}
\textbf{Community} & \textbf{Size} & \textbf{1/MRR} \\ \hline
Cartoonists & 8 & 58.1 \\
Chess Stars & 14 & 5.4 \\
Conan Show Writers & 12 & 4.7 \\
Fashion Commentators & 11 & 8.3 \\
Fresno City Council & 6 & 3.0 \\
Hedge Fund Managers & 11 & 25.7 \\
H.S Drama Departments & 18 & 2.3 \\
Mathematicians & 11 & 32.6 \\
NLP Researchers & 50 & 4.9 \\
Pittsburgh Food Trucks & 15 & 3.3 \\
Police Dogs & 16 & 2.7 \\
Professional Economists & 11 & 3.6 \\
SCOTUS Reporters\tablefootnote{People who write news articles about the Supreme Court of the United States.} & 16 & 1.9 \\
The Stranger Reporters\tablefootnote{The Stranger is a small weekly newspaper.} & 11 & 8.3 \\
Ultimate Frisbee Players & 130 & 6.7 \\
Ultramarathon Runners & 28 & 14.6
\end{tabular}
\caption{W2V+RE-ID results by community}
\label{table:breakdown}
\end{table}
These results may underestimate performance, because there is a chance that some users in the general population test data may actually belong to one or more of our test communities, i.e. there could be mislabeled data. To assess the potential impact, we manually checked the top ten false positives for each community for mislabeled users. We did discover some mislabeled examples for the economist, hedge fund manager, and ultramarathon runner communities. For the most part, the top ranked users from the general population tended to be people from related communities. For example, the top false ultimate frisbee users contained people who wrote about their participation in tournaments for other sports such as soccer.
\subsection{Analysis}
The finding that the W2V-initialized RE-ID model is significantly better than W2V raises the question: how do the embeddings learned by the re-identification task differ from the ones learned by the word2vec objective? To investigate this, we looked at the 1,000 words in the \mbox{RE-ID} model with embeddings that were farthest (in Euclidean distance) from its word2vec initialization. These top words disproportionately contain Twitter user handles, so some social network structure is captured. Using agglomerative clustering, we found groups of words that centered around frequent words used in particular regions (foreign words, dialects) or cultures (sociolects), associated with hobbies or interests (specific sports, music genres, gaming), or polarizing topics (political parties, controversial issues). At least one of the top tokens was the username of an account later identified as being sponsored by the Russian government to spread propaganda during the United States presidential election, e.g., ``ten\_gop'' in Table~\ref{table:changed_words} of the Appendix.
We also looked at which communities are closest in the embedding space. We
represent a community with the average of the member embeddings and use a normalized cosine distance for similarity.
The two nearest neighbors are Mathematicians and NLP researchers, which are also close to the next two nearest neighbors, Hedge Fund Managers and Professional Economists.
To interpret what the model as a whole captured, we found the top scoring tweets for each held-out user (creating an embedding for a single tweet) according to the logistic regression model.
Representative examples include ``recurrent neural\_network grammars simplified and analyzed'' for NLP Researchers, and ``we're looking\_forward to seeing you opening\_night may 24th love the cast of high\_school musical'' for High School Drama clubs. Examples for additional communities are included in the appendix.
The results provide insight into the community member identification decision.
\section{Introduction}
\input{intro-v2}
\section{Model}
\input{model-v2}
\section{Data}
\input{data-v2}
\section{Experiments}
\input{experiments-v2}
\section{Related Work}
\label{sec:related}
\input{related}
\section{Conclusion}
In summary, this paper defines a task of community member retrieval based on their tweets, introduces a person re-identification task to allow community definition with a small number of examples, and shows that that the method gives very good results compared to word2vec and LDA baselines. Analyses show that the user embeddings learned efficiently represent user interests. The text embeddings are largely complementary to the social network features used in other studies, so performance gains can be expected from feature combination.
While our experiments use a bag-of-words representation, as in most related work, the re-identification training objective proposed here can easily be used with other methods for deriving document embeddings, e.g.\ \cite{Le2014ICML,Kim2014}.
\section*{Acknowledgements}
The authors thank the anonymous reviewers for their feedback and helpful suggestions.
|
{
"timestamp": "2018-04-17T02:13:25",
"yymm": "1804",
"arxiv_id": "1804.05499",
"language": "en",
"url": "https://arxiv.org/abs/1804.05499"
}
|
\section{\label{sec:intro}Introduction}
Partially magnetized plasmas immersed in crossed $E\times B$ fields are used in various devices such as Hall thrusters for electric propulsion. Such plasmas are subject to a number of instabilities that affect device operation -- and in particular -- the level of anomalous transport that is typically found to be orders of magnitude larger than the classical (collisional) transport. The nature of the anomalous transport (mobility) is still poorly understood and has been attributed to several candidate instabilities that may interact with each other to bring about the observed levels of anomalous transport. The electron cyclotron drift instability (ECDI) driven by the electron $E\times B$ drift, and independent of any plasma gradients and collisions, has been recently actively discussed as a possible candidate \cite{Mikellides2016,LafleurPoP2016b,BoeufJAP2017}.
In earlier works \cite{ForslundPRL1970,GaryJPP1970,LampePRL1971,StenzelPRL1973}, electron cyclotron instabilities have been studied in relation to turbulent plasma heating by the electric current perpendicular due to the relative electron-ion drift. Electron cyclotron instability driven by ion beams was also identified as a possible source of anomalous resistivity explaining the width of collisionless shock waves, in particular in space conditions; for more recent work and references see Refs. \onlinecite{MatsukiyoJGR2009,MuschiettiJGR2013,MuschiettiAdvSpR2006}.
In the context of the anomalous transport in Hall thrusters, the cyclotron instability driven by the electron $E\times B$ drift was studied in 1D simulations \cite{DucrocqPoP2006,BoeufFP2014,BoeufJAP2017,LafleurPoP2016a}, 2D axial-azimuthal simulations \cite{AdamPoP2004,BoeufIEPC2017,CochePoP2014}, and 2D radial-azimuthal simulations \cite{CroesPSST2017,HaraIEPC2017,HeronPoP2013}.
Many of these works focused on the possibility that ECDI simply becomes the ion sound instability analogous to the case of unmagnetized plasma\cite{CavalierPoP2013,LafleurPoP2016a}. We have shown in our previous nonlinear simulations that the transition to ion sound (which for the 1D case is only possible due to nonlinear diffusion in the short wavelength regime) does not occur\cite{JanhunenPOP2018} and the instability is driven by the dominant $m=1$ cyclotron resonance. We also found in our previous 1D simulations that when larger azimuthal length is used, anomalous transport cascades to low-$k$ modes. Curiously in Ref.~\onlinecite{LafleurPoP2016a} the authors see the emergence of a large-scale structure in their simulation when they use a larger simulation box, but reject it as an artifact.
In this paper, using highly resolved particle-in-cell simulations, we study instabilities and transport in the 2D (azimuthal-radial) geometry. Periodic boundary conditions are used in the azimuthal direction, along the $E\times B$ drift. The magnetic field is in the radial direction bound by the dielectric wall boundaries. Curvature effects of the channel are not included in this work.
In 2D geometry a new class of unstable mode appears for finite values of the the wave number $k_z$ along the magnetic field, namely the Modified Two-Stream Instability (MTSI) \cite{McBridePF1972}. For larger values of $k_z$, the unstable mode looks similar to the unmagnetized ion sound \cite{GaryJPP1970a,GaryJPP1970,CavalierPoP2013}. In this paper, we study the linear and nonlinear evolution of the interacting ECDI and MTSI modes, their saturation and associated turbulent transport for typical conditions of a Hall thruster. Our simulations demonstrate that like in the 1D case, the instability is driven by nonlinear cyclotron resonance modes that dominate the anomalous transport, and that the cascade to long wavelengths observed in 1D simulations is further enhanced by the linear long wavelength instabilities that occur when finite $k_z$ is allowed. Moderate values of the anomalous electron current (of the order of $\Omega =\Omega_{ce}/\nu_{eff}\simeq 200$ are obtained in nonlinear stage similar to the 1D case\cite{JanhunenPOP2018}. An important new result is strong parallel electron heating due to the modes with a finite $k_z$.
The paper is structured as follows: we discuss the 2D linear regime and the unstable normal modes therein, and show that they appear as expected in fully non-linear simulations in the very early part of the simulation. The early non-linear saturation processes are discussed, such as mode competition between the ECDI harmonics and apparent coupling to the modified two-stream instability (MTSI). In the development of the strong turbulence regime, we show how the MTSI compresses the ECDI wave packet and produces large fluxes to the sheath with accompanied rapid heating of the parallel temperature. We discuss the spectral cascade of the anomalous current, the features of the anomalous current as a function of the radial (parallel to magnetic field) direction, and the time evolution of the overall anomalous current. Finally, the sheath losses and decay of the plasma column in the absence of sources are discussed, and a summary follows. We discuss technical details such as numerical parameters and analysis methods in the Appendix.
\section{\label{sec:instability}Linear features of the Electron Cyclotron Drift-Instability}
The electron drift cyclotron instability (ECDI) occurs in partially magnetized $E\times B$ plasma due to the significant $E\times B$ flow of electrons with respect of ions. It is convenient to discuss the characteristics of the ECDI with reference to the linear dispersion relation. We consider the electrostatic waves with $\mathbf{v}_{0}=\mathbf{E\times B}/B^2$ streaming of electrons across a uniform magnetic field $\mathbf{B}$, with unmagnetized ions, in homogeneous unbounded plasma. The two-dimensional linear dispersion equation has the form \cite{GaryJPP1970a}
\begin{equation}
\epsilon \left( \omega ,\mathbf{k}\right) =1+\epsilon _{i}\left( \omega ,\mathbf{k}\right) +\epsilon _{e}\left( \omega ,\mathbf{k}\right) =0,
\label{disp}
\end{equation}
where $\epsilon_{e}$ and $\epsilon_{i}$ are the electron and ion susceptibilities
\begin{gather}
\epsilon _{i}=-\frac{1}{2k^{2}\lambda _{D_{i}}^{2}}\mathrm{Z}^{^{\prime }}\hspace{-0.25ex}\left(\frac{\omega}{\sqrt{2}kv_{i}}\right),\label{ions}\\
\begin{aligned}
\epsilon_{e}=\frac{1}{k^{2}\lambda _{De}^{2}}\left[ 1+\frac{\omega -\mathbf{k}\cdot\mathbf{v}_{0}}{\sqrt{2}k_{z}v_{e}}\sum\limits_{m=-\infty}^{\infty}e^{-b}\mathrm{I}_{m}\left( b\right)\right.\\\left.\mathrm{Z}\left( \frac{\omega -\mathbf{k}\cdot \mathbf{v}_{0}+m\Omega_{ce}}{\sqrt{2}k_{z}v_{e}}\right) \right],
\end{aligned}\label{el}
\end{gather}%
where $b=k_{y}^{2}\rho_{e}^{2}$, $\rho_{e}^{2}=v_{e}^{2}/\Omega _{ce}^{2},$
$v_{e,i}^{2}=T_{e,i}/m_{e,i}$, $\lambda_{De,i}^{2}=\epsilon_{0} T_{e,i} / n_{0} q_{e,i}^{2}$, $Z(\xi )$ is the plasma dispersion function, $I_{m}(x)$ is the modified Bessel function of the 1st kind, $\mathbf{B} = B_{0} \widehat{\mathbf{z}}$ is the magnetic field in the $z-$ direction, $\mathbf{E}=E_{0}\widehat{ \mathbf{x}}$ is the external electric field in the x-direction, so that $\mathbf{v}=\mathbf{E\times B}/B^{2}=v_{0}\widehat{\mathbf{y}}$ is in the $y$ direction (or, against if $v_0=-E_0/B_0<0$). Then, $k_z$ and $k_y$ are the components of the wave vector $\mathbf{k}$ along the magnetic field and in the $\mathbf{E\times B/}$ directions, $k\equiv\left\vert \mathbf{k}\right\vert =\sqrt{k_{z}^{2}+k_{y}^{2}}$. Given the physical parameters, it remains to assign $k_z$ and $k_y$ values; solving the dispersion relation provides us with the frequency $\omega$, which generally gets a complex value.
For the geometry of the Hall thruster with a radial magnetic field, we define a few auxiliary quantities and relations that aid in the following discussions: $k_0=\Omega_{ce}/v_{0}$, $k_z=2\pi n_z/L_r$, $k_y=2\pi n_y/l_{\theta}$, where $L_r$ is the extent of the system in the radial (along the magnetic field) direction, $l_\theta$ is the extent of the system in azimuthal direction, so the quantum numbers $n_z$ and $n_y$ characterize the radial and azimuthal wave vectors that satisfy the boundary conditions. The local Cartesian coordinates $y,z$ correspond to the $\theta, r$ coordinates of the coaxial Hall thruster. It is important to emphasize however, that while the $y$-direction is periodic in our simulations, periodicity is not imposed in the radial direction. The eigenmode structure in radial direction is formed self-consistently by the mode parallel dynamics and by the sheath effects at $z=0$ and $z=L_r$. The ensuing mode structure will be discussed below. In this work we assume the gap to be a straight box, for simplicity.
In the limit of cold ions where $\omega>k v_{i}$, the ion response becomes
\begin{equation}
\epsilon_{i}=-\frac{\omega_{pi}^{2}}{\omega^{2}},
\label{ionc}
\end{equation}
where $\omega_{pi}^{2}=e^{2}n_{0}/\varepsilon_{0}m_{i}$ is the ion plasma frequency.
In one-dimensional case, $k_{z}\rightarrow 0$, $k=k_{y}$, the
dispersion equation (\ref{disp}) takes the form
\begin{equation}
\begin{aligned}
\epsilon_{e}&=\frac{1}{k^{2}\lambda _{De}^{2}}\left[ 1-\exp \left(
-k^{2}\rho _{e}^{2}\right) I_{0}\left( k^{2}\rho _{e}^{2}\right) \right.\\
&-2\left.\left( \omega -kv_{0}\right) ^{2}\sum\limits_{m=1}^{\infty }\frac{%
\exp \left( -k^{2}\rho _{e}^{2}\right) I_{m}\left( k^{2}\rho _{e}^{2}\right)
}{\left( \omega -kv_{0}\right) ^{2}-m^{2}\Omega _{ce}^{2}}\right].
\end{aligned}\label{eq:dispersion-magn}
\end{equation}
The form of Eq.~(\ref{eq:dispersion-magn}) emphasizes the role and the interaction of different cyclotron harmonics. Note that there is no resonance for the $m=0$ harmonic, while all higher harmonics with $m=1,2,..$ are resonant at $\left( \omega-kv_{0}\right) ^{2}=m^{2}\Omega_{ce}^{2}$. In the cold electrons limit $T_{e}\rightarrow 0$, only $m=0$ and $m=1$ harmonics contribute and the dispersion relation reduces to the Buneman magnetized plasma instability driven by transverse current \cite{Buneman1962}
\begin{equation}
1-\frac{\omega _{pi}^{2}}{\omega ^{2}}-\frac{\omega_{pe}^{2}}{(\omega
-kv_{0})^{2}-\Omega _{ce}^{2}}=0. \label{bi}
\end{equation}
The instability is a result of reactive coupling of the electron (Doppler shifted) upper hybrid mode $\left( \omega -kv_{0}\right) ^{2}=\omega_{pe}^2+ \Omega_{ce}^{2}$ with the short wavelength ion oscillations $\omega ^{2}=\omega_{pi}^{2}$. In the long wavelength low frequency limit, $\left( \omega,kv_{0}\right) < \Omega_{ce}$, equation (\ref{bi}) describes the lower-hybrid modes, $\omega _{LH}^{2}=\Omega _{ci}\Omega _{ce}. $ The contribution of higher $m>1$ harmonics (which are absent for $T_{e}=0$) grows with electron temperature and has the maximum at shorter wavelengths $k^{2}\rho _{e}^{2}\simeq 1$ due to the $\exp \left( -k^{2}\rho _{e}^{2}\right) I_{m}\left( k^{2}\rho _{e}^{2}\right)$ factors. The temperature effects also add dispersion to the lower hybrid modes $\omega ^{2}=\Omega _{LH}^{2}\left( 1+k^{2}\rho _{e}^{2}\right)$, which eventually becomes the high frequency ion sound for $k^{2}\rho _{e}^{2}\gg 1$, $\omega ^{2}=\Omega _{LH}^{2}k^{2}\rho _{e}^{2}\simeq k^{2}c_{s}^{2}$. In this limit, the contribution of the cyclotron harmonics decreases with $k \rho_{e}$: $\exp \left( -k^{2}\rho _{e}^{2}\right) I_{m}\left( k^{2}\rho_{e}^{2}\right) \rightarrow 1/\left( k\rho _{e}\right)$, so that the real part of the electron susceptibility becomes
\begin{equation}
\epsilon _{e}\simeq 1/\left( k^{2}\lambda _{De}^{2}\right), \label{ee}
\end{equation}
and equation (\ref{disp}) produces the ion sound mode
\begin{equation}
\omega ^{2}=k^{2}c_{s}^{2}/\left( 1+k^{2}\lambda _{De}^{2}\right).
\label{is}
\end{equation}
The imaginary part in $\epsilon_{e}$ (neglected so far) originate in the series of cyclotron resonances. In the limit $k^{2}\rho_{e}^{2}\gg 1$, the infinite series of cyclotron resonances can be summed resulting to the imaginary contribution equivalent to the pole contribution $1/\left(\omega -kv_{0}\right)$ as for the case of unmagnetized electrons. Thus, even for strictly perpendicular propagation, in the $k^{2}\rho _{e}^{2}\gg 1$ limit, one has the ion sound instability as in the unmagnetized plasma case. This case is directly related to the resolution of the Landau-Bernstein paradox: the sequence of Bernstein modes which are undamped in magnetized plasma result in collisionless Landau damping when $B\rightarrow 0$.
Another kind of instability occurs near the resonances $\left( \omega -kv_{0}\right) ^{2}\simeq m^{2}\Omega_{ce}^{2}$. This is a strong (fluid) reactive instability due to coupling of the ion and electron modes \citep{LashmorePhysA1970}, facilitated by the Doppler shift. For cold electrons, only the $m=1$ exists, resulting to the Buneman instability described by Eq.~(\ref{bi}). For finite $T_{e}$, all higher modes with $m>1$ are present. In our previous work, it was shown that in 1D case a set of modes with higher $m$ are excited, but eventually, via the linear (due to electron heating) and nonlinear effects, a dominant $m=1$ strongly coherent cnoidal wave appears.\cite{JanhunenPOP2018} The cyclotron resonance nature of the mode, defined by the condition $\omega <k_{y}v_{0}\simeq \Omega_{ce}$, extends far into the nonlinear stage. Similar result were also obtained in other simulations relevant to space plasma conditions \cite{MatsukiyoJGR2009}.
In 2D where plasma motion along the magnetic field is present, new regimes become possible due to finite values of the $k_{z}$. There have been a number of studies of the full linear dispersion equation with (\ref{ions}) and (\ref{el}), for example see Refs.~\onlinecite{ArefevTechPhys1970,GaryJPP1970,GaryJPP1970a,CavalierPoP2013}. One of the results of these studies is that for sufficiently large values of the parallel wave vector $k_{z}$, the solution of the dispersion relation (\ref{disp}) produces a mode which is close to the ion sound instability in unmagnetized plasma driven by the electron beam with $v_0$ velocity.
\begin{figure}[htp]
\includegraphics[width=\columnwidth,clip]{gamma-cyc.pdf}
\caption{Growth rate of instabilities for $k_z \protect\lambda_{De}$ of 0.01, 0.02, 0.04 and 0.08, respectively, found from the full dispersion equation (1)-(3). The first root from left is the MTSI (m=0), and subsequent roots are from $m=1,2,3,...$ ECDI resonances. Parameters for this figure are: $v_0=-369\,c_s$, $v_{Te}=489\,c_s$, $\Omega_{ce}=96.5\,\omega_{pi}$, $k_0\lambda_{De}=0.262$.}
\label{fig:gamma-kz}
\end{figure}
The general behavior of the growth rate of the instability is shown for several values of the $k_z$ parameter in Fig.~\ref{fig:gamma-kz} as a function of the azimuthal wave vector $k_y$.
We solve the dispersion relation numerically in Python\cite{Python} using the technique described in Ref.~\onlinecite{CavalierPoP2013}, where the solution is obtained through fixed point iteration using the relative error as a stopping condition. We use the convergence condition that $|1-\omega_{i+1}/\omega_{i}|<10^{-6}$ for $i\geq 15$.
The SciPy\cite{SciPy} Faddeeva function is used to get good numerical accuracy of the plasma dispersion function for a wide range of arguments. We show the first four roots obtained in the $(k_z,k_y)$ phase plane in figure~\ref{fig:full-dispersion} for a $10\,\text{eV}$ Xenon plasma with $n_e=10^{17}\,\text{m}^{-3}$, $B_0=0.02\,\text{T}$, and $E_0=20\,\text{kV/m}$, which are typical Hall thruster parameters. The normalization scheme for the dispersion relation solver code is the same as defined in Ref.~\onlinecite{CavalierPoP2013}, where frequencies are normalized to $\omega_{pi}$, velocities to $c_s$ and lengths to $\lambda_{De}$. The physical parameters given above are also used as the initial state of nonlinear simulations presented in this paper, unless otherwise noted. The Python code is also used for solving the dispersion relation in a simple case of $T_e\rightarrow 0$, as shown in Fig.~\ref{fig:sim-gamma} where the limit of the full dispersion relation is presented for $k_z\lambda_{De}=0.005$.
\begin{figure}[htp]
\includegraphics[width=\columnwidth,clip]{full.pdf}
\caption{Phase space plot of growth rate $\gamma(k_z,k_y)/\omega_{pi}$ for the full dispersion relation. Notice the modified two-stream instability (MTSI) in the sub-cyclotron low-$k$ region, indicating possible instability for simulations which are able to accommodate the mode in the azimuthal direction. Parameters for this figure are: $v_0=-369\,c_s$, $v_{Te}=489\,c_s$, $\Omega_{ce}=96.5\,\omega_{pi}$, $k_0\lambda_{De}=0.262$.}\label{fig:full-dispersion}
\end{figure}
\begin{figure}[htp]
\includegraphics[width=\columnwidth,clip]{simple-dr.pdf}
\caption{Comparison of the growth rate from a 2D simulation and a 1D simulation with 40000 particles/cell, using $T_e=0.001$, with the solution of the linear dispersion in the $T_e\rightarrow 0$ limit. Parameters for this figure are: $v_0=-36.9\cdot10^3\,c_s$, $v_{Te}=489\,c_s$, $\Omega_{ce}=96.5\,\omega_{pi}$, $k_z\lambda_{De}=0.00434$.}\label{fig:sim-gamma}
\end{figure}
For small values of the $k_z$ one can see in Fig.~\ref{fig:gamma-kz} the fluid type reactive instability peaks near the resonant values of $k_y=m\cdot k_0$. For larger values of $k_z$, the fluid resonance is broadened by the thermal effects and for $k_z v_{Te} \geq \omega$, the resonance becomes kinetic, resulting in the smooth curve corresponding to the kinetic resonance instability of the ion sound in unmagnetized plasma \cite{GaryJPP1970a}.
The regimes for different values of $k_{z}$ can be seen from the dispersion equation (\ref{disp}). In the limit $k_{z}\rightarrow 0$, $\xi _{m}\gg 1$, and using $\mathrm{Z}\left( \xi _{m}\right)\rightarrow -1/\xi _{m}$, the terms involving plasma dispersion functions result in cyclotron resonances:
\begin{gather}
\frac{\omega -\mathbf{k}\cdot \mathbf{v}_{0}}{\sqrt{2}k_{z}v_{e}}\mathrm{Z}%
\left( \xi _{m}\right) \rightarrow -\frac{\omega -\mathbf{k}\cdot \mathbf{v}_{0}}{\omega -\mathbf{k}\cdot \mathbf{v}_{0}+m\Omega _{ce}},\\
\xi _{m}=\frac{\omega -\mathbf{k}\cdot \mathbf{v}_{0}+m\Omega _{ce}}{\sqrt{2} k_{z}v_{e}}.
\end{gather}
This is the regime of the fluid (reactive) instability due to coupling of the ion sound and ion modes \cite{LashmorePhysA1970} which occurs for cold plasma (and in the limit of $k_z\rightarrow 0$).
The thermal broadening of the resonance and the transition to the kinetic ion sound instability can be clearly seen in the example when only $m=0$ is retained in the sum (\ref{el}):
\begin{equation}
1-\frac{\omega _{pi}^{2}}{\omega ^{2}}+\frac{1}{k^{2}\lambda _{De}^{2}}\left[
1+\frac{\omega -\mathbf{k}\cdot \mathbf{v}_{0}}{\sqrt{2}k_{z}v_{e}}e^{-b}%
\mathrm{I}_{0}\left( b\right) \mathrm{Z}\left( \frac{\omega -\mathbf{k}\cdot
\mathbf{v}_{0}}{\sqrt{2}k_{z}v_{e}}\right) \right] =0.\label{rede}
\end{equation}
The first three terms in this expression describe the ion sound mode (\ref{is}). Considering the last term as a small perturbation in Eq.~(\ref{rede}) one gets the so called modified ion sound instability \cite{ArefevTechPhys1970,ArefevNF1969}. Similar resonance broadening occurs for higher $m$ resonances, so that the resonant fluid type instability ($k_{z}\rightarrow 0$) is replaced by the kinetically driven mode for a finite $k_{z}$. This behavior is illustrated in Figs.~\ref{fig:gamma-kz} and \ref{fig:gamma-resonances}, where the mode frequencies and growth rates are calculated retaining only individual terms with different $m$ of the Bessel function series (lower panel) and partial sum up to the $m^\text{th}$ order (upper panel). Note that the real part of the mode frequency for individual $m$ and partial sum $m$ modes always remain close to the ion sound mode frequency from Eq.~\ref{is}, see Fig.~\ref{fig:gamma-resonances}. For sufficiently large $k_{z}$, the summation of several components in the Bessel series illustrates the transition to unmagnetized ion-sound instability as shown in Fig.~\ref{fig:gamma-resonances}, where the solutions of the partial sum slowly converge to the unmagnetized curve as terms are added.
\begin{figure}[htp]
\includegraphics[width=0.485\columnwidth,clip]{omega-ios125.pdf}
\includegraphics[width=0.485\columnwidth,clip]{gamma-ios125.pdf}
\includegraphics[width=0.485\columnwidth,clip]{omega-sing125.pdf}
\includegraphics[width=0.485\columnwidth,clip]{gamma-sing125.pdf}
\caption{Growth rate obtained from partial sums of the full dispersion relation (\ref{disp}); top row: terms up to $I_{\pm{k}}$ included solving for $\omega$ and $\gamma$; bottom row: only terms $I_{\pm{k}}$ of the sum included while solving for $\omega$ and $\gamma$. The solution from unmagnetized dispersion relation is included (labeled as IS); partial sum including terms up to $m=50$ is virtually indistinguishable from the unmagnetized solution. Parameters for this figure are: $v_0=-369\,c_s$, $v_{Te}=489\,c_s$, $\Omega_{ce}=96.5\,\omega_{pi}$.}\label{fig:gamma-resonances}
\end{figure}
\section{Modified Two-stream Instability and Modified Two-Stream Buneman Instability}
The discussion in the previous section remains silent on one important feature of the instability with finite $k_z$: namely on the regime of the Modified-Two-Stream Instability (MTSI)\cite{McBridePF1972,LashmoreNF1973,Stepanov1965} where the effect of the electron parallel motion is typically considered in the context of the following dispersion equation\cite{LashmoreNF1973}
\begin{equation}
1-\frac{\omega _{pi}^{2}}{\omega ^{2}}-\frac{\omega _{pe}^{2}k_{z}^{2}}{(\omega -k_{y}v_{0})^{2}k^{2}}+\frac{\omega _{pe}^{2}k_{y}^{2}}{\Omega
_{ce}^{2}k^{2}}=0. \label{mtsi}
\end{equation}
It is instructive to analyze the nature of this equation starting from the electron response in the form of Eq.~(\ref{ee}) which together with Eq.~\ref{ionc} results in the ion sound mode. The electron susceptibility in the form of Eq.~(\ref{ee}) is equivalent to the Boltzmann response for the perturbed electron density %
\begin{equation}
\widetilde{n}_{e}=\frac{e\widetilde{\phi }}{T_{e}}n_{0} \label{b}.
\end{equation}
This expression follows from the parallel electron balance in neglect of the electron inertia
\begin{equation}
0=en\nabla _{\Vert }\widetilde{\phi }-T_{e}\nabla _{\Vert }\widetilde{n}.
\end{equation}
The electron inertia however can be neglected in the parallel momentum balance only when the condition $\omega <k_{z}v_{e}$ is satisfied. In presence of the strong transverse electron flow $v_{0}$, apparent mode frequency has to be modified due to the Doppler shift: $\omega \rightarrow \omega -k_{y}v_{0}$. When the condition $\omega -k_{y}v_{0} <k_{z}v_{e}$ is violated, the electron inertia terms have to be included. In this case, the electron continuity and momentum balance equations
\begin{gather}
-i\left( \omega -k_{y}v_{0}\right) \tilde{n}+ik_{z}n_{0}v_{\Vert}=0,\\
-im_{e}\left( \omega -k_{y}v_{0}\right) v_{\Vert}=iek_{z}\tilde{\phi},
\label{mb}
\end{gather}
result in the electron density response in the form
\begin{equation}
n_{e}=-\frac{ek_{z}^{2}\widetilde{\phi }}{m_{e}(\omega -k_{y}v_{0})^{2}}.
\label{nmsi}
\end{equation}
Note that when $\left( \omega -k_{y}v_{0}\right) \simeq k_{z}v_{e}$, the
assumption of the isothermal electrons becomes invalid, and the only
consistent approximation in the fluid theory is to assume $T_{e}=0$, so the
pressure term in (\ref{mb}) was omitted.
In this equation, the third term corresponds to the electron density
perturbation from Eq.~(\ref{nmsi}) and the last term is due to the electron inertial perpendicular current. It is worth noting that the equation (\ref{mtsi}) is not fully consistent.
The second term in this equation is obtained under ordering $\left( \omega -k_{y}v_{0}\right) \simeq k_{z}v_{e}$, while the last term is obtained
with the low
frequency approximation $\left(\omega -k_{y}v_{0}\right) \ll \Omega_{ce}$.
The latter may not be satisfied for some applications such as Hall thrusters. A more accurate dispersion equation is obtained from Eqs.~(\ref{disp}) and (\ref{el}) by taking a rigorous limit $T_{e}\rightarrow 0$ which yields the relation
\begin{equation}
1-\frac{\omega _{pi}^{2}}{\omega ^{2}}-\frac{\omega _{pe}^{2}k_{z}^{2}}{%
(\omega -k_{y}v_{0})^{2}k^{2}}-\frac{\omega _{pe}^{2}k_{y}^{2}}{((\omega
-k_{y}v_{0})^{2}-\Omega _{ce}^{2})k^{2}}=0, \label{mbtsi}
\end{equation}%
which includes both the modified two-stream instability and the upper hybrid Buneman instability. In the following we will call this case the Modified Buneman Two-Stream Instability (MBTSI).
The MBTSI regime is of particular importance as a finite value of $k_z$ result in the long wavelength instability at small $k_y/k_0 \ll 1$, well below the cyclotron resonances with $k_y\simeq m\cdot k_0$. This long wavelength mode, the leftmost peak in Fig.~\ref{fig:gamma-kz}, also shown in Fig.~\ref{fig:mtsi-full}, has a small growth rate, but as discussed in Section~\ref{sec:nonlinear}, turns out to be important in the nonlinear saturation regime enhancing the tendency toward long wavelength condensation.
\begin{figure}[htp]
\includegraphics[width=\columnwidth,clip]{mtsi-sin.pdf}
\caption{Close-up of the modified two-stream instability (MTSI) growth rate $\gamma/\omega_{pi}$ given as a function of $\sin(\alpha)=k_z/k$ and $k / k_0$. Parameters for this figure are: $v_0=-369\,c_s$, $v_{Te}=489\,c_s$, $\Omega_{ce}=96.5\,\omega_{pi}$, $k_0\lambda_{De}=0.262$.}\label{fig:mtsi-full}
\end{figure}
In our 2D simulations, which are bounded in the $z$-direction, it is observed generally that up to non-linear regime the instability grows uniformly everywhere, with $k_z \simeq 0$, and only after non-linear regime is reached does the wave like structure develop in the parallel direction. This can be explained by the fact that the linear growth rates increase monotonically towards $k_z\rightarrow 0$, while the the sheath allows fluctuations to extend up to the boundary by "insulating" the perturbations from the wall.
\begin{figure}[htp]
\includegraphics[width=\columnwidth,clip]{omegamma-cyc.pdf}
\caption{Growth rate and frequency for ECDI instabilities for $k_z\lambda_{De}=0.00434$ at $T_e=10\,\text{eV}$. Growth rates as obtained from the 2D simulation (Fig.~\ref{fig:roots-sim}) are indicated by the symbols. Note the existence of the MTSI below the first cyclotron resonance. Refer to Table~\ref{table:rates} for the values. Parameters for this figure are: $v_0=-369\,c_s$, $v_{Te}=489\,c_s$, $\Omega_{ce}=96.5\,\omega_{pi}$, $k_0\lambda_{De}=0.262$.}\label{fig:omega-gamma}
\end{figure}
Generally in the literature the ECDI instability was considered\cite{CavalierPoP2013} for typical values of the parallel wave length of $k_z \lambda_{De}\simeq 0.01-0.09$ corresponding to a parallel wave length of $\lambda=7.4-0.8\,$cm, for typical Hall-effect thruster parameters. It could be misleading, however, to estimate $k_z$ and relevant dynamic regimes of the ECDI based on full wave lengths in a Hall-effect thruster gap. It has been noted earlier\cite{ChenJNE1965,ChenPF1965,ChenPF1979,BarrettPRL1972} that in a bounded plasma the sheath effects allow the wavelength along the magnetic field to be much longer than would otherwise follow from the the geometrical constraint $k_z=2\pi/L_r$, where $L_r$ is the plasma width between the boundaries. In our 2D simulations we observe that the parallel structure along the magnetic field is consistent with a half-wavelength fit between the boundaries. This corresponds to $k_z\lambda_{De}=0.00434$. We used this value for calculations of the linear growth rates in Figs.~\ref{fig:sim-gamma} and \ref{fig:omega-gamma}.
\section{Growth and saturation of the cyclotron harmonics of ECDI and MTSI modes}
PIC2D is a 2D3V PIC code developed by Dmytro Sydorenko at University of
Alberta based on his earlier 1D3V PIC code, EDIPIC.\cite{SydorenkoTh2006}
The initial state of the PIC2D simulation is a $v_0$-shifted Maxwellian distribution for the electrons, and a stationary Maxwellian for the ions. The initial state is quasi-neutral and homogeneous, equipped with the typical Hall-effect thruster parameters presented in the previous section (and Appendix). The magnetic field lines are terminated at a dielectric boundary, so a sheath potential develops soon after time starts running. The simulation box is $L_r=53.8\,\text{mm}$ by $l_\theta=13.45\,\text{mm}$, and the numerical parameters are detailed in Table~\ref{table:numerical_params} in Appendix~\ref{app:determine}. Even though to code allows to do so, no collisions were used in our simulations.
Chronologically, the evolution of the simulations goes as follows (see figures~\ref{fig:stages} and \ref{fig:roots-sim}). First, in the linear-like stage the fastest growing modes -- the ECDI with $m=3-5$ -- grow and introduce some heating to the electrons primarily in the perpendicular direction. Due to heating and nonlinearities the lowest $m=1$ resonance becomes the ECDI mode with the largest energy content, even though linearly it is the slowest growing mode. The resonance condition $k_y=m \Omega_{ce}/v_0$ gives, with our choice of $l_\theta=13.45\,\text{mm}$ the quantum numbers $n_y=m\cdot 7.534$, but due to kinetic effects the maximum growth rates are found at higher values of $n_y$. It is observed consistently with the linear dispersion relation that the maximum growth rates for the $m$ cyclotron resonances correspond to $n_y (m=1,2,3..)=\{10,17,24,31,...\}$ as ECDI modes. For higher resonances the up-shift in $k_y$ is lower, diminishing the gap between maximum growth rates of $m$-resonances to ${\Delta}n_y=7$. The mode amplitudes are shown in Fig.~\ref{fig:stages}, and growth rates are given in Table~\ref{table:rates}. As can be also seen from Fig.~\ref{fig:omega-gamma}, these locations are very close to the maximum growth rates obtained from the linear dispersion relation. After the $m=1$ and $m=2$ modes saturate, the MTSI mode starts growing, suggesting non-linear feedback between the modes. During the growth of the MTSI, mode competition between $m=1$ and $m=2$ ECDI modes is observed. The MTSI mode grows, heating the electrons predominantly in the parallel direction due to the parallel electric field. Heating in parallel direction results in enhanced losses to the sheath and saturation of the MTSI. The ECDI modes stay at their saturated level that was established earlier, but after the saturation of the MTSI the $m=1\,\&\,2$ modes grow by 5-10 \%, while $m=3\,\&\,4$ modes lose energy correspondingly. At this stage, we observe the saturation of the anomalous axial current, as shown in Fig.~\ref{fig:current-total}.
Linear growth rates may be determined directly from simulation data when the runs are performed with good enough resolution. The procedure for finding the linear growth rates from the spectrogram of a simulation is outlined in Appedix~\ref{app:determine}. The linear and early non-linear stage of the evolution of individual mode resonances is shown in figure~\ref{fig:roots-sim}, where we also provide the values for normalized wave number from the linear fits that can be made to the modes. The values obtained from simulations are given in table~\ref{table:rates}, and also plotted for the comparison with the linear dispersion relation solutions in figure~\ref{fig:omega-gamma}. As can be seen, the growth rates are of the right magnitude, although depressed due to the short time of growth available for fitting. The MTSI mode (first peak) is well represented though, showing the importance of good statistics.
The MTSI peak is well represented because it grows to a higher amplitude, giving a larger range for least-squares fitting. This explains why the ECDI growth rates are generally slightly underestimated by the 2D simulations; the mode energy for the ECDI modes has only four periods of growth (before nonlinear stage), whereas the MTSI has more than ten periods for fitting (see Fig.~\ref{fig:roots-sim}). Part of the ECDI growth curve is affected by early nonlinear saturation processes, decreasing the apparent growth rate.
To emphasize this point, we ran a case for $T_{e}\approx 0$ with the 2D code, and a case with 1D version of the code using very good statistics (Fig.~\ref{fig:sim-gamma}) that gives a remarkably good value for both the growth rate and frequency of the Modified Buneman Two-Stream instability (equation~\ref{mbtsi}).
\begin{figure}[htp]
\begin{tikzpicture}
\node (img1) {\includegraphics[width=0.95\columnwidth,clip]{nois-00214.pdf}};
\node[below of=img1,node distance=1.75cm] (img2) {\includegraphics[width=0.95\columnwidth,clip]{nois-00247.pdf}};
\node[below of=img2,node distance=1.75cm] (img3) {\includegraphics[width=0.95\columnwidth,clip]{nois-00321.pdf}};
\node[below of=img3,node distance=1.75cm] (img4) {\includegraphics[width=0.95\columnwidth,clip]{nois-00428.pdf}};
\node[below of=img4,node distance=1.75cm] (img5) {\includegraphics[width=0.95\columnwidth,clip]{nois-00535.pdf}};
\node[below of=img5,node distance=1.75cm] (img6) {\includegraphics[width=0.95\columnwidth,clip]{nois-00642.pdf}};
\node[below of=img6,node distance=1.75cm] (img7) {\includegraphics[width=0.95\columnwidth,clip]{nois-00856.pdf}};
\node[below of=img7,node distance=1.4cm]{$z~/~\text{mm}$};
\node[left of=img4,rotate=90,yshift=3.5cm]{$y~/~\text{mm}$};
\end{tikzpicture}
\caption{Time slices of the ion density fluctuation $n_i-\langle n_i\rangle_y(z,t)$, showing growth and saturation of the predominant $m=1$ cyclotron mode followed by the development of structure parallel to the magnetic field.}\label{fig:stages}
\end{figure}
\begin{figure}[htp]
\includegraphics[width=\columnwidth,clip]{linearanalys.pdf}
\caption{Logarithmic amplitude of the azimuthal electric field energy $|E_y|^2$, showing stages of linear growth and early nonlinear saturation of the cyclotron and MTSI modes. Linear least-squares fitting has been used to obtain values of the growth rates for each mode (labeled by their $k$ values wrt/$k_0$), given in Table~\ref{table:rates}.}\label{fig:roots-sim}
\end{figure}
\begin{table}[htp]
\begin{tabular}{|c|c|c|c|}\hline
Symbol & $n_y$& $k/k_0$&$\gamma / \omega_{pi}$\\\hline
$\blacktriangle$ &1 &0.1325&0.4957\\
\ding{58} &10 &1.3248&0.7133\\
$\blacktriangledown$ &17 &2.2521&0.7629\\
$\bullet$ &24 &3.1794&1.2568\\
$\blacksquare$ &31 &4.1068&1.1628\\\hline
\end{tabular}
\caption{The wave length and growth rate of unstable modes observed in the linear stage simulations as shown in Figs.~\ref{fig:roots-sim} and \ref{fig:omega-gamma}.}\label{table:rates}
\end{table}
A curious feature of the ECDI/MTSI energy in Fig.~\ref{fig:roots-sim} is how the growth of the MTSI commences only after the ECDI modes have saturated, and how the energy of the ECDI modes remains relatively constant while the MTSI is growing. This emphasizes that the modes are not in fact independent, but are nonlinearly coupled instead. It is observed that in the sheath-bounded radial direction the mode assumes a half-wave pattern with $n_z=1/2$, whereas in the azimuthal (periodic) direction the mode is a full wave with $n_y=1$. If we use this observation to prescribe $k_z=\pi/L_r$, we get Fig.~\ref{fig:omega-gamma} from the dispersion relation, and our MTSI peak aligns well with the peak of the MTSI in $k_y=2\pi/l_{\theta}$, and the fit to ECDI modes with $n_y=\{10,17,24,31,...\}$ is also satisfactory.
There are 3D simulations that indicate the existence of long-wavelength modes very similar to those found in our simulations\cite{Taccogna}, suggesting that the presence of the MTSI is a robust feature even in full Hall-effect thruster geometry.
\section{Nonlinear spectra of ion and electron density: short wavelength features}\label{sec:nonlinear}
We have earlier noted that the cyclotron resonances drive strongly coherent cnoidal type waves which are limited by saturation through ion dynamics\cite{JanhunenPOP2018}, while the waves retain their cyclotron resonance characteristics far into the simulation even with significant electron heating and nonlinear interactions between modes.
In 2D the dispersion relation admits a long wavelength instability that is absent in 1D, namely the modified two-stream instability of Section~\ref{sec:instability}. The latter mode is observed to modify the nonlinear dynamics compared to the 1D case, resulting in significantly faster evolution of the long wavelength components, and a lower energy content (or, fluctuation level) is retained in the ECDI modes.
After the early nonlinear stage, the convective nonlinearity compresses/expands the ECDI in the hills/troughs of the now-dominant MTSI mode (figure \ref{fig:condensation-slices}), causing jet-like injection to the sheath from the compressed maxima (particularly evident in the 4th figure of Fig.~\ref{fig:stages}) with accompanying faster decay of the plasma profile. Transient large fluctuations in the electron density predominantly in the parallel direction are observed to originate from the sheath at this stage, also seen as parallel ripples in the ion density, which likely act as a relaxation mechanism. At this stage (around $1\,\mu\text{s}$ into the simulation) the linear mode characteristics become a poor descriptor of the system: electron density fluctuations do not significantly increase, but ion density fluctuations grow (see figures~\ref{fig:delta-n}-\ref{fig:azim-sec}), and group velocity of the wave packet increases significantly. Feedback between the ECDI- and MTSI-scale modes is apparent, with expansion and compression of the wave crests as the wave packet progresses.
An important feature of the nonlinear dynamics observed in our 2D simulation is the the difference in the behavior of the ion and electron density. After heating, electron fluctuations become fairly uninteresting (and low-amplitude), but the ion density exhibits a wealth of non-linear phenomena. This difference is especially apparent in the short wavelength $k_y\lambda _{De}<1$ part of the spectrum. The ECDI still remains the dominant mode of energy injection into the short wavelength ion-sound fluctuations, whose frequency approach $\omega_{pi}$ in this limit, but strongly modified by signatures of nonlinear wave breaking \cite{Davidson_Nonlinear_methods} due to ion dynamics. In the strong turbulence state, the ion density fluctuations in the azimuthal direction become cnoidal-like, and are lead by a wave with a very sharp peak of positive amplitude, after which a train of crests of decreasing amplitude follow. As apparent from figure~\ref{fig:azim-sec}, where a radial section of the plasma at $5\,\text{mm}$ is shown, the crests do not propagate as much as exchange energy through elastic-like collisions (akin to soliton collisions), so the amplitude maximum travels at a higher speed than the individual crests do, as may be the case for envelope solitons\cite{Remoissenet2013Waves}. Similar features were observed in 1D simulations as well, although realized after a significantly longer time of simulation.
\begin{figure}[htp]
\includegraphics[width=0.8\columnwidth,viewport=45 89 719 507,clip]{deltan_comp.pdf}
\caption{Ion and electron density fluctuation levels over the whole simulation volume; standard deviation and maximum.}\label{fig:delta-n}
\end{figure}
\begin{figure}[htp]
\includegraphics[width=\columnwidth,clip]{ni_samples.pdf}
z\caption{Stages of the nonlinear development of the ion density fluctuations at four time slices. Large scale mode is formed, the wave crest compressing the EC waves, merging of the peaks with shift to lower $k$ and reshaping of the wave packet to a more triangular form. Plots are from $r=5\,\text{mm}$.}\label{fig:condensation-slices}
\end{figure}
\begin{figure}[htp]
\includegraphics[width=0.95\columnwidth,clip]{n_i_th.pdf}
\includegraphics[width=0.95\columnwidth,clip]{n_e_th.pdf}
\caption{Top: Azimuthal ion density fluctuations as a function of time. Individual wave crests propagate at different phase velocities with respect to the wave packet, exchanging energy with one another. Bottom: Azimuthal electron density fluctuations as a function of time. After saturation, electron density assumes a smooth profile. Volume-averaged root-mean-square fluctuation level is roughly half of the ion fluctuation level. The figures are scaled the same way, $n_0=10^{17}\,\text{m}^{-3}$ to fix units although $\delta\hspace{-0.25ex}n=n-\langle n\rangle_y(z,t)$, and both are measured at $z=5\,\text{mm}$.}\label{fig:azim-sec}
\end{figure}
\section{Parallel electron heating due to MTSI}
A new feature with respect to our earlier 1D simulations is the rapid parallel heating in the 2D simulations, observed to occur in the same pattern as the MTSI mode. It is well-known that the MTSI is an effective heating mechanism for electrons along the magnetic field \cite{McBridePF1972} and the heating is the likely saturation mechanism for the MTSI, as larger parallel temperature will induce large losses of high energy electrons into the sheath.
\begin{figure}[htp]
\includegraphics[width=\columnwidth,clip]{rms_etemp.pdf}
\caption{Anisotropic heating of the $T_{e\perp}$ and $T_{e\parallel}$ components. Parallel heating is due to the MTSI mode. The curves show the volume averaged parallel and perpendicular temperatures.}\label{fig:heating}
\end{figure}
\begin{figure}[htp]
\includegraphics[width=\columnwidth,clip]{Tpara.pdf}
\includegraphics[width=\columnwidth,clip]{Tperp.pdf}
\caption{Spatial structure of the perpendicular $T_{e\perp}$ and parallel $T_{e\parallel}$ electron temperatures.}\label{fig:heating-structure}
\end{figure}
After saturation of parallel temperature $T_{e\parallel}$ , electron heating in the perpendicular direction catches up to parallel heating (which has saturated through sheath losses), after which the state of strong turbulence is reached (see Fig.~\ref{fig:heating}). Even in this stage, the ECDI mode structure is prominent in density spectra, and is clearly affected into a ``wave street'' (alternating maxima and minima) characteristic to the MTSI mode, although the large scale structure is less apparent. These features are unchanged in simulations with half time-step, ruling out the possibility of numerical heating. The heating profile is shown in Fig.~\ref{fig:heating-structure}, clearly indicating the MTSI mode as the cause of parallel heating (and to a lesser degree, perpendicular). The growth of the MTSI mode terminates when the $T_{e\parallel}$ growth terminates in Fig.~\ref{fig:roots-sim}.
\section{Spectral cascade down to long wave lengths}
In the previous work \cite{JanhunenPOP2018}, linearly stable long wavelength components attributable to nonlinear processes were observed to arise in the 1D system. Energy cascade toward long wavelengths is also observed in 2D simulations, with the difference that in 2D the saturation and nonlinear stage is reached much more quickly due to the presence of the linearly unstable long wavelength MTSI mode. A salient feature observed in figure~\ref{fig:roots-sim} is that the MTSI becomes active only once the cyclotron modes have saturated, and the cyclotron modes respond to the saturation of the MTSI by resuming growth. This enhances the modulational nonlinear coupling through a faster linear response. The ion density $k_y$ spectrum at $r=1.35\,\text{cm}$, shown in figure~\ref{fig:ion-spectrum}, clearly demonstrates the progression towards lower-$k$ modes through inverse cascade, and emergence of the turbulent spectrum soon after the MTSI mode saturates.
\begin{figure}[htp]
\includegraphics[width=\columnwidth,viewport=63 105 700 486,clip]{ion-spectrum.pdf}
\includegraphics[width=\columnwidth,viewport=45 80 720 496,clip]{n_e-spectrum.pdf}
\caption{Evolution of the azimuthal ion density (top) and electron density (bottom) $k$-spectra over time at $L/4$ of the simulation, plotted as $\log_{10}{\tilde{n}_{i,e}(k_y)}$. Discrete peaks occur at the $\omega-k v_0-=m\Omega_{ce}$ resonances, and the lowest peak (after $0.5\mu\text{s}$) is the MTSI mode around $k_y=2\pi/l_\theta$, or $n_y=1$.}\label{fig:ion-spectrum}
\end{figure}
Cascade to low-$k$ is even more drastic in electron density, where the $m>1$ resonances become all but absent. This process coincides with the generation of low-$k$ components in the anomalous current, and subsequent growth of net current. For the anomalous current, a similar spectrum may be obtained (figure~\ref{fig:current-spectrum}) illustrating the initial cascade towards low-$k$ during the saturation of the ECDI modes.
\begin{figure}[htp]
\includegraphics[width=\columnwidth,clip]{current_spectrum.pdf}
\caption{Evolution of the anomalous current spectrum $\log_{10}{|\tilde{J}_z(k_y)|}$ over time at $z=L_r/4$ of the simulation. It is notable that the $n_y=1$ mode dominates soon after nonlinear saturation of the modes.}\label{fig:current-spectrum}
\end{figure}
The present simulations encompass only a small fraction of the total Hall-effect thruster gap in the azimuthal direction, due to their computational expense, and even the longest wavelength modes remain short compared to the total gap circumference.
\section{Anomalous current}
The axial anomalous current can be calculated from PIC2D as the relative charge flux between ions and electrons. The electric field exists only in the $(y,z)$-plane, but the axial anomalous current arises because of the cyclotron rotation in the $(x,y)$-plane, In particular, azimuthal electric field fluctuations (in $y$-direction) results in the electron displacement in $ x$-direction. The anomalous current $J_x$ obtained as the total charge flux in the $x$-direction is calculated directly from particles motion in the code.
The inverse cascade is particularly apparent in the spectrum of the anomalous current (Fig.~\ref{fig:current-spectrum}). Another feature of the spectrum is the dominance of the $n_y=1$ mode that corresponds to the MTSI scale length. The mode also has the same radial envelope as the current profile (Fig.~\ref{fig:current-mode-01}).
The spatial structure of the anomalous current in 2D has some special features which are not possible in 1D simulations. The net axial current is associated with the $k_y=0$ component, which is plotted in figure~\ref{fig:current-total} as a function of time. Like in the 1D simulations, we observe that the anomalous current experiences an overshoot in the saturation stage, and settles down to a much lower level. The anomalous current spectrum is dominated by low-$k$ modes, particularly by the lowest mode available to the system in azimuthal direction, but also has a strong radial variation that is illustrated in figure~\ref{fig:current-mode-01}. Even though the MTSI creates a large transient in the total anomalous current, after the strong turbulent regime is established the net anomalous current falls to levels that are similar to those observed in 1D simulations. The long-wavelength features in parallel direction do not contribute to the total volume-averaged current, but could perhaps be observed in localized measurements as large alternating axial jets in the anomalous current.
\begin{figure}[htp]
\includegraphics[width=\columnwidth,clip]{Jz_comp.pdf}
\caption{Evolution of the total anomalous current density $J_z$ averaged over the simulation region. The anomalous current is normalized to the inverse of the Hall parameter\cite{LafleurPoP2016a}. The value obtained from 1D presented in Ref.\onlinecite{JanhunenPOP2018}, 1/164.}\label{fig:current-total}
\end{figure}
\begin{figure}[htp]
\includegraphics[width=\columnwidth,clip]{currents.pdf}
\caption{Total anomalous current components $n_y=\{0,1\}$ for the nominal case. The total current is positive definite (blue curve), but the dominant $n_y=1$ component has a sheath-bounded half-wave radial profile corresponding to $n_z=1/2$.}\label{fig:current-mode-01}
\end{figure}
\section{Sheath losses and decay of the plasma column}\label{sec:boundary}
The plasma in these simulations is bounded by dielectric boundaries in the $z$-direction. Charge accumulation on the dielectric surface was allowed and the displacement current in the dielectric with $\epsilon=4.5$ was taken into account using the model of Ref.~\onlinecite{SmolyakovPRL2013}.
A sheath (potential) develops very quickly in the simulation before any modes have had time to grow. After having formed, the sheath does not in fact expand much into the plasma (evident in Fig.~\ref{fig:density} ), but once the MTSI saturates we observe a faster decay rate of the plasma, likely due to the rapid electron heating during wave breaking. At $0.5\,\mu\text{s}$ the transition to a different regime is apparent particularly in figure~\ref{fig:density-avg}, where the rate of sheath losses increases drastically and the pre-sheath begins to rapidly expand into the plasma. This is the reason why our simulation box is larger than typical Hall-effect thruster gaps in the magnetic field direction --- without sources the expansion of the pre-sheath would happen too soon, and physics present in a steady-state thruster gap would be dynamically obscured.
Because the simulations presented in this paper do not have sources, it is an important question whether the decay due to sheath losses is significant enough to alter interpretation of the simulation results, and if so, in what manner. Decay of the density profile is shown in figures~\ref{fig:density} and \ref{fig:density-avg}, where the electron density profile in the magnetic field direction and the total volume averaged electron density are shown as a function of time. Decay of the plasma column and its effect on the modes is evident from Fig.~\ref{fig:stages}, where the modes terminate in $z$-direction where the density drop-off of the pre-sheath begins, except at later stages for a more gradual profile. Even at $2\,\mu\text{s}$ it appears that the nonlinear mode structure has remained largely intact (exhibiting characteristics of the cyclotron mode) although the plasma column has already decayed significantly (40\%). The profile is largely unaffected by pre-sheath expansion up to $1\,\mu\text{s}$ as seen from figure~\ref{fig:density}, and about 10\% of the electron density is lost so far.
Hence, we contend that the plasma column decay appears to have fairly little effect on the modes. Larger effects could be expected from the increase in electron temperature, which tends to modify the linear spectrum towards lower cyclotron resonances, as well as with secondary electrons (not included) that may even reverse the sheath. It is therefore likely that the robust features observed in these simulations would remain in the presence of sources and sinks too if a steady state can be achieved. There are some inherent difficulties in using sources (axial feeding and ionization) to achieve a steady-state even in 1D simulations, because of feedback between the modes and the sources (heating for ionization, transport for axial feeding).
\begin{figure}[htp]
\includegraphics[width=\columnwidth,clip]{ne_time.pdf}
\caption{Evolution of the azimuthal mean of the electron density as a function of time.}\label{fig:density}
\end{figure}
\begin{figure}[htp]
\includegraphics[width=\columnwidth,clip]{ne_mean.pdf}
\caption{Evolution of the volume-averaged electron density over time.}\label{fig:density-avg}
\end{figure}
Particularly in contrast to earlier work by H\'eron et al.\onlinecite{HeronPoP2013}, it is somewhat disappointing that the parallel heating mechanism has not been investigated to a greater degree in the literature. This makes it difficult to assign relative importance to the inclusion of secondary emission.
\section{Summary and Conclusion}
In earlier numerical studies of the electron cyclotron instability in 1D geometry Refs.~\onlinecite{LampePRL1971,LampePF1972a} it was found that the linear (exponential growth) stage of the fast beam cyclotron instability is saturated due to nonlinear turbulent broadening, which smears out the cyclotron resonances and the instability transitions into much slower ion-sound instability much like in the ordinary unmagnetized plasma.
The authors of Refs.~\onlinecite{ForslundPRL1970,ForslundPF1972a} have performed analogous numerical studies for similar conditions and maintained that many properties of the observed instabilities are unlike those of the ion-sound mode of unmagnetized plasma. The apparent controversies from these simulations with regards to the importance and role of nonlinear electron diffusion, electron and ion trapping, as well as the role of the finite ion temperature have been discussed and contrasted at some length in two complementary papers, Refs.~\onlinecite{LampePF1972b} and \onlinecite{ForslundPF1972b}.
The electron cyclotron drift instability driven by the electron current has been suggested as a possible candidate for enhanced electron transport in Hall truster in more recent Refs.~\onlinecite{AdamPoP2004,DucrocqPoP2006}. The authors of Refs.~\onlinecite{BoeufFP2014,BoeufJAP2017,BoeufIEPC2017,LafleurPoP2016a,LafleurPoP2016b,CroesPSST2017} have performed a number of numerical simulations of ECDI and mostly concluded that the instability is analogous to the ion sound instability in absence of the magnetic field. Our recent 1D simulations\cite{JanhunenPOP2018}, performed with higher resolution and with longer azimuthal simulation box have not confirmed this conclusion. In our simulations, we found that the criteria for the nonlinear resonance broadening and the destruction of cyclotron resonances is not satisfied and the instability proceeds as a coherent mode driven at the main cyclotron resonance (of the reactive fluid type) $k_y v_0=\Omega_{ce}$ well into the nonlinear stage. Strong inverse energy cascade towards the longer wavelength was identified and it was shown that the anomalous current is dominated by the long wavelength modes.
In 1D case, the transition to the ion sound regime
may only occur due to the nonlinear resonance broadening and/or collisions. The role of numerical collisions was also discussed in Ref.~\onlinecite{ForslundPF1972b}. Numerical noise as well as particle re-injection (to limit the energy growth and mimic the axial direction) may also work as collisions. As it was discussed in the Section~\ref{sec:instability} above, in 2D geometry, when the direction along the magnetic field is resolved, the resonance thermal broadening due to a finite $k_z c_s$ may also facilitate the transition to the ion sound regime. However, arbitrary fluctuations are allowed at a sheath boundary, facilitating access to $k_z \approx 0$ regime where linear growth is largest.
In this paper, we have discussed the normal modes that are obtained from different limits of the full electrostatic dispersion relation, and compared them with the simulation results. Using the value of the effective $k_z$ obtained from simulations we have found that the linear dispersion relation predicts the discreet cyclotron resonance driven modes and the long wavelength MTSI modes which were also confirmed in the simulations. It appears therefore, that for our parameters full ion sound regime appears not to be realized, but instead a more effective regime of discrete cyclotron resonance instabilities and the long wavelength MTSI occur. The magnetic field remains to be a defining feature of these regimes (contrary to the unmagnetized ion sound regime) as also was concluded in Ref.~\onlinecite{ForslundPF1972b}.
An important feature of the strong turbulence regime observed in our simulations is the difference in the ion and electron density fluctuations. While the electron density perturbations are rather benign and lower amplitude, the ion density perturbations have much larger amplitude and contain much larger short wavelength (non-quasineutral) content with $k_y \lambda_{De}\geq 1$. Such short wavelength modes do exhibit some interesting ion sound -like characteristics, such as the nonlinear $\omega_{pi}$ harmonics, nonlinear wave breaking, tendency for wave crests to become shock-like and the elastic-like collision of wave crests without interference. It is important to note that the intense fluctuations in the short wavelength part of the spectrum are less effective in supporting the anomalous electron current as well as for the electron heating. We note that rich short wavelength features are observed in ion density but not in electron density, which is much more smooth and coherent. This difference in the electron and ion response can be important for interpretation of the fluctuation diagnostics data\cite{TsikataPRL2015}. The coherent nature of the electron density wave and coherent electron-wave interaction is crucial for the electron heating mechanism \cite{ForslundPF1972b} that also precludes the application of the quasilinear theory\cite{MikhailenkopoP2003}.
An interesting feature in our simulation is the strong $n_y=1$ component of the axial anomalous current, resulting in the alternating jets caused by the MTSI activity. The Modified Two-Stream Instability (MTSI) that was naturally absent from the 1D simulations is shown to amplify the inverse cascade tendency due to the long wavelength nature of the MTSI even in the linear regime. Development of the Modified Two-Stream Instability results in rapid parallel heating due to the finite electric field along the magnetic field. Intense parallel heating results in increased losses of high energy electrons into the sheath that serves as an additional saturation mechanism. Similar heating was also observed in Ref.~\cite{HeronPoP2013}. Their simulations also suggest that effects of secondary emission could significantly increase anomalous transport, but unfortunately restrict wave vector space for heating studies. They too observe modulations in the sheath, however.
In our simulations, the anomalous electron transport sets at the level similar to that in our 1D simulations\cite{JanhunenPOP2018}, perhaps due to the absence of secondary emission.
\section{Supplementary Material}
We provide movies of three quantities discussed in the paper: $\delta{n}_i$, $\delta{n}_e$ and $T_e$ over the simulation box $L_r\times l_\theta$. They illustrate the dynamics observed and reported in this paper.
\begin{acknowledgements}
This work was supported in part by NSERC Canada and the Air Force Office of Scientific Research under awards number FA9550-18-1-0132 and FA9550-15-1-0226. Computational resources from ComputeCanada/WestGrid were used in this work.
\end{acknowledgements}
|
{
"timestamp": "2018-07-24T02:01:25",
"yymm": "1804",
"arxiv_id": "1804.05450",
"language": "en",
"url": "https://arxiv.org/abs/1804.05450"
}
|
\section{Introduction}
\label{sec:intro}
The exploding multimedia content over the Internet, has created a new world of spoken content processing, for example the retrieval\cite{lee2015spoken, chelba2008retrieval, larson2012spoken, mandal2014recent, lee2014improved}, browsing\cite{lee2005spoken}, summarization\cite{lee2015spoken, lee2005spoken, shiang2013supervised, lee2013unsupervised}, and comprehension\cite{tseng2016towards, fang2016hierarchical,lee2014spoken, shen2015structuring} of spoken content.
On the other hand, we may realize there still exists a huge part of multimedia content not yet taken care of, i.e., the singing content or those with audio including songs. Songs are human voice carrying plenty of semantic information just as speech. It will be highly desired if the huge quantities of singing content can be similarly retrieved, browsed, summarized or comprehended by machine based on the lyrics just as speech. For example, it is highly desired if song retrieval can be achieved based on the lyrics in addition.
Singing voice can be considered as a special type of speech with highly flexible and artistically designed prosody: the rhythm as artistically designed duration, pause and energy patterns, the melody as artistically designed pitch contours with much wider range, the lyrics as artistically authored sentences to be uttered by the singer. So transcribing lyrics from song audio is an extended version of automatic speech recognition (ASR) taking into account these differences.
On the other hand, singing voice and speech differ widely in both acoustic and linguistic characteristics. Singing signals are often accompanied with some extra music and harmony, which are noisy for recognition. The highly flexible pitch contours with much wider range\cite{sasou2005auto, kawailyric}, the significantly changing phone durations in songs, including the prolonged vowels\cite{sasou2006singing, kawai2016speech} over smoothly varying pitch contours, create much more problems not existing in speech. The falsetto in singing voice may be an extra type of human voice not present in normal speech. Regarding linguistic characteristics\cite{hosoya2005lyrics, mesaros2010recognition}, word repetition and meaningless words (e.g.oh) frequently appear in the artistically authored lyrics in singing voice.
Applying ASR technologies to singing voice has been studied for long. However, not too much work has been reported, probably because the recognition accuracy remained to be relatively low compared to the experiences for speech. But such low accuracy is actually natural considering the various difficulties caused by the significant differences between singing voice and speech. An extra major problem is probably the lack of singing voice database, which pushed the researchers to collect their own closed datasets\cite{sasou2005auto, kawai2016speech, mesaros2010recognition}, which made it difficult to compare results from different works.
Having the language model learned from a data set of lyrics is definitely helpful\cite{kawai2016speech, mesaros2010recognition}. Hosoya et al.\cite{hosoya2005lyrics} achieved this with finite state automaton. Sasou et al.\cite{sasou2005auto} actually prepared a language model for each song. In order to cope with the acoustic characteristics of singing voice, Sasou et al.\cite{sasou2005auto, sasou2006singing} proposed AR-HMM to take care of the high-pitched sounds and prolonged vowels, while recently Kawai et al.\cite{kawai2016speech} handled the prolonged vowels by extending the vowel parts in the lexicon, both achieving good improvement. Adaptation from models trained with speech was attractive, and various approaches were compared by Mesaros el al.\cite{mesaros2009adaptation}.
In this paper, we wish our work can be compatible to more available singing content, therefore in the initial effort we collected about five hours of music-removed version of English songs directly from commercial singing content on YouTube. The descriptive term {\it "music-removed"} implies the background music have been removed somehow. Because many very impressive works were based on Japanese songs\cite{sasou2005auto, kawailyric, sasou2006singing, kawai2016speech, hosoya2005lyrics}, the comparison is difficult. We analyzed various approaches with HMM, deep learning with data augmentation, and acoustic adaptation on fragment, song, singer, and genre levels, primarily based on fMLLR\cite{gales1998maximum}. We also trained the language model with a corpus of lyrics, and modify the pronunciation lexicon and increase the transition probability of HMM for prolonged vowels. Initial results are reported.
\section{DATABASE}
\label{sec:data}
\subsection{Acoustic Corpus}
To make our work easier and compatible to more available singing content, we collected 130 music-removed (or vocal-only) English songs from www.youtube.com so as to consider only the vocal line.The music-removing processes are conducted by the video owners, containing the original vocal recordings by the singers and vocal elements for remix purpose. \footnote{Samples of our collected data: https://youtu.be/QA6x9MLgsc8}
After initial test by speech recognition system trained with LibriSpeech\cite{panayotov2015librispeech}, we dropped 20 songs, with WERs exceeding 95\%. The remaining 110 pieces of music-removed version of commercial English popular songs were produced by 15 male singers, 28 female singers and 19 groups. The term {\it group} means by more than one person. No any further preprocessing was performed on the data, so the data preserves many characteristics of the vocal extracted from commercial polyphonic music, such as harmony, scat, and silent parts. Some pieces also contain overlapping verses and residual background music, and some frequency components may be truncated. Below this database is called {\bf vocal data} here.
These songs were manually segmented into fragments with duration ranging from 10 to 35 sec primarily at the end of the verses. Then we randomly divided the vocal data by the singer and split it into training and testing sets. We got a total of 640 fragments in the training set and 97 fragments in the testing set. The singers in the two sets do not overlap. The details of the vocal data are listed in Table.\ref{tab: AcousticDatabase}.
Because music genre may affect the singing style and the audio, for example, hiphop has some rap parts, and rock has some shouting vocal, we obtained five frequently observed genre labels of the vocal data from wikipedia\cite{wiki} : pop, electronic, rock, hiphop, and R\&B/soul. The details are also listed in Table.\ref{tab: AcousticDatabase}. Note that a song may belong to multiple genres.
To train initial models for speech for adaptation to singing voice, we used 100 hrs of English clean speech data of LibriSpeech.
\definecolor{Gray}{gray}{0.85}
\begin{table}[]
\centering
\begin{tabular}{|c|c|c|c|c|}
\hline
& \cellcolor{Gray}\# songs
& \multicolumn{1}{c||}{\cellcolor{Gray}\# singers}
& pop & electronic\\
\hline
Training set
& \cellcolor{Gray}95
& \multicolumn{1}{c||}{\cellcolor{Gray}49}
& 202.2 & 85.8 \\
\hline
Testing set
&\cellcolor{Gray}15
& \multicolumn{1}{c||}{\cellcolor{Gray}13}
&20.3 & 22.0\\
\hline
\hline
& rock & hiphop &\multicolumn{1}{c||}{R\&B/soul} & \cellcolor{Gray}total \\ \hline
Training set & 51.1 & 30.0
& \multicolumn{1}{c||}{87.5}
& \cellcolor{Gray}271\\
\hline
Testing set & 17.7 & 8.4
& \multicolumn{1}{c||}{9.1}
& \cellcolor{Gray}42.8 \\
\hline
\end{tabular}
\caption{Information of training and testing sets in vocal data. The lengths are all measured in minutes.}
\label{tab: AcousticDatabase}
\end{table}
\subsection{Linguistic Corpus}
In addition to the data set from LibriSpeech (803M words, 40M sentences), we collected 574k pieces of lyrics text (totally 129.8M words) from {\it lyrics.wikia.com}, a lyric website, and the lyrics were normalized by removing punctuation marks and unnecessary words (like ’[CHORUS]’). Also, those lyrics for songs within our vocal data were removed from the data set.
\label{subsec:approach-flow}
\begin{figure}[htb]
\begin{minipage}[b]{1.0\linewidth}
\centering
\centerline{\includegraphics[angle=-90, width=8.5cm]{new1-approach-flow2.png}}
\end{minipage}
\caption{The overall structure for training the acoustic models.}
\label{fig:approach flow}
\end{figure}
\section{Recognition Approaches and System Structure}
\label{sec:approach}
Fig.\ref{fig:approach flow} shows the overall structure based on Kaldi\cite{povey2011kaldi} for training the acoustic models used in this work. The right-most block is the vocal data, and the series of blocks on the left are the feature extraction processes over the vocal data. Features \uppercase\expandafter{\romannumeral 1}, \uppercase\expandafter{\romannumeral 2}, \uppercase\expandafter{\romannumeral 3}, \uppercase\expandafter{\romannumeral 4} represent four different versions of features used here. For example, Feature \uppercase\expandafter{\romannumeral 4} was derived from splicing Feature \uppercase\expandafter{\romannumeral 3} with 4 left-context and 4 right-context frames, and Feature \uppercase\expandafter{\romannumeral 3} was obtained by performing fMLLR transformation over Feature \uppercase\expandafter{\romannumeral 2}, while Feature \uppercase\expandafter{\romannumeral 1} has been mean and variance normalized, etc.
The series of second right boxes are forced alignment processes performed over the various versions of features of the vocal data. The results are denoted as Alignment a, b, c, d, e. For example, Alignment a is the forced alignment results obtained by aligning Feature \uppercase\expandafter{\romannumeral 1} of the vocal data with the LibriSpeech SAT triphone model (denoted as Model A at the top middle).
The series of blocks in the middle of Fig.\ref{fig:approach flow} are the different versions of trained acoustic models. For example, model B is a monophone model trained with Feature \uppercase\expandafter{\romannumeral 1} of the vocal data based on alignment a. Model C is very similar, except based on alignment b which is obtained with Model B, etc. Another four sets of Models E, F, G, H are below. For example Model E includes models E-1, 2, 3, 4, Models F,G and H include F-1,2 , G-1,2,3, and H-1,2,3.
We take Model E-4 with fragment-level adaptation within model E as the example. Here every fragment of song (10-35 sec long) was used to train a distinct fragment-level fMLLR matrix, with which Feature \uppercase\expandafter{\romannumeral 3} was obtained. Using all these fragment-level fMLLR features, a single Model E-4 was trained with Alignment d. Similarly for Models E-1, 2, 3 on genre, singer and song levels. The fragment-level Model E-4 turned out to be the best in model E in the experiments.
\subsection{DNN, BLSTM and TDNN-LSTM}
\label{subsec:deep learning models}
The deep learning models (Models F,G,H) are based on alignment e, produced by the best GMM-HMM model. Models F-1,2 are respectively for regular DNN and multi-target, LibriSpeech phonemes and vocal data phonemes taken as two targets. The latter tried to adapt the speech model to the vocal model, with the first several layers shared, while the final layers separated.
Data augmentation with speed perturbation\cite{ko2015audio} was implemented in Models G, H to increase the quantity of training data and deal with the problem of changing singing rates. For 3-fold, two copies of extra training data were obtained by modifying the audio speed by 0.9 and 1.1. For 5-fold, the speed factors were empirically obtained as 0.9, 0.95, 1.05, 1.1. 1-fold means the original training data.
Models G-1,2,3 used projected LSTM (LSTMP)\cite{sak2014long} with 40 dimension MFCCs and 50 dimension i-vectors with output delay of 50ms. BLSTMs were used at 1-fold, 3-fold and 5-fold.
Models H-1,2,3 used TDNN-LSTM\cite{peddinti2017low}, also at 1-fold, 3-fold and 5-fold, with the same features as Model G.
\subsection{Special Approaches for Prolonged Vowels}
\label{subsec:lex}
Consider the many errors caused by the frequently appearing prolonged vowels in song audio, we considered two approaches below.
\subsubsection{Extended Lexicon}
\label{subsubsec:exten_lex}
The previously proposed approach \cite{kawai2016speech} was adopted here as shown by the example in Fig.\ref{fig:extened-vowels}(a). For the word ``apple'', each vowel within the word ( but not the consonants) can be either repeated or not, so for a word with $n$ vowels, $2^{n}$ pronunciations become possible. In the experiments below, we only did it for words with $n\leq3$.
\subsubsection{Increased Self-looped Transition Probabilities}
\label{subsubsec:self_loop}
This is also shown in Fig.\ref{fig:extened-vowels}. Assume an vowel HMM have $m+1$ states (including an end state). Let the original self-looped probability of state $i$ is denoted $1-p_{i}$ and the probability of transition to the next state is $p_{i},\ i=1,2,...,m$. We increased the self-looped transition probabilities by replacing $p_{i}$ by $rp_{i}$. This was also done for vowel HMMs only but not for consonants.
\begin{figure}[]
\begin{minipage}[b]{1.0\linewidth}
\centering
\centerline{\includegraphics[width=8.5cm]{prolonged_vowel_approach.png}}
\end{minipage}
\caption{Approaches for prolonged vowels: (a) extended lexicon (vowels can be repeated or not), (b) increased self-loop transition probabilities (transition probabilities to the next state reduced by $r$).}
\label{fig:extened-vowels}
\end{figure}
\section{EXPERIMENTS}
\label{sec:exp}
\subsection{Data Analysis}
\begin{figure}[htb]
\begin{minipage}[b]{1.0\linewidth}
\centering
\centerline{\includegraphics[width=7.5cm]{pitch_distribution_step_with_speech_Hz_log_scale2}}
\end{minipage}
\caption{Histogram of pitch distribution.}
\label{fig:pitch dist.}
\end{figure}
\subsubsection{Language Model (LM) statistics}
We analyzed the perplexity and out-of-vocabulary(OOV) rate of the two language models (trained with LibriSpeech and Lyrics respectively) tested on the transcriptions of the testing set of vocal data. Both models are 3-gram, pruned with SRILM with the same threshold. LM trained with lyrics was found to have a significantly lower perplexity(123.92 vs 502.06) and a much lower OOV rate (0.55\% vs 1.56\%).
\begin{table}[htb]
\centering
\begin{tabular}{|c|c|l|c|c|}
\hline
\multicolumn{2}{|c|}{} & Acoustic Models & WER(\%) & PER(\%)\\
\hline
\multicolumn{2}{|c|}{\multirow{2}{*}{ \rotatebox{90}{
\begin{tabular}[c]{@{}c@{}}
Libri\\Speech \\ LM
\end{tabular} } }} &
\begin{tabular}[c]{@{}l@{}}
(1) Model A:\\LibriSpeech(SAT)
\end{tabular} & 96.21 & 87.17\\
\cline{3-5}
\multicolumn{2}{|c|}{} & \begin{tabular}[c]{@{}l@{}}
(2) Model E-4:\\fragment-level
\end{tabular} & 88.26 & 77.18\\
\hline
\multicolumn{2}{|c|}{} & \begin{tabular}[c]{@{}l@{}}
(3) Model E-4:\\fragment-level
\end{tabular} & 80.40 & 68.80\\
\cline{2-5}
\multirow{9}{*}{ \rotatebox{90}{
\begin{tabular}[c]{@{}c@{}}
Lyrics Language Model
\end{tabular} } } &
\multirow{9}{*}{ \rotatebox{90}{
\begin{tabular}[c]{@{}c@{}}
Extended Lexicon
\end{tabular} } } &
\begin{tabular}[c]{@{}l@{}}
(4) Model B:\\Monophone
\end{tabular} & 86.57 & 76.10\\
\cline{3-5}
& & \begin{tabular}[c]{@{}l@{}}
(5) Model C:\\Triphone
\end{tabular} & 81.58 & 71.11\\
\cline{3-5}
& & \begin{tabular}[c]{@{}l@{}}
(6) Model D:\\Triphone
\end{tabular} & 82.02 & 72.10\\
\cline{3-5}
& & \begin{tabular}[c]{@{}l@{}}
(7) Model E-4:\\fragment-level
\end{tabular} & 77.08 & 66.04\\
\cline{3-5}
& & \begin{tabular}[c]{@{}l@{}}
(8) Model E-4:\\fragment-level\\
+Increased Trans. Prob.
\end{tabular} & 76.62 & 65.79\\
\cline{3-5}
& & \begin{tabular}[c]{@{}l@{}}
(9) Model F-1\\DNN (regular)
\end{tabular} & 75.56 & 65.64\\
\cline{3-5}
& & \begin{tabular}[c]{@{}l@{}}
(10) Model F-2\\DNN (multi-target)
\end{tabular} & 75.84 & 65.56\\
\cline{3-5}
& & \begin{tabular}[c]{@{}l@{}}
(11) Model G-1\\BLSTM (1-fold)
\end{tabular} & 79.94 & 70.27\\
\cline{3-5}
& & \begin{tabular}[c]{@{}l@{}}
(12) Model G-2\\BLSTM (3-fold)
\end{tabular} & 74.32 & 63.86 \\
\cline{3-5}
& & \begin{tabular}[c]{@{}l@{}}
(13) Model G-3\\BLSTM (5-fold)
\end{tabular} & 75.35 & 65.50\\
\cline{3-5}
& & \begin{tabular}[c]{@{}l@{}}
(14) Model H-1\\TDNN-LSTM (1-fold)\\
\end{tabular} & 79.01 & 69.20\\
\cline{3-5}
& & \begin{tabular}[c]{@{}l@{}}
(15) Model H-2\\TDNN-LSTM (3-fold)\\
\end{tabular} & {\bf 73.90} & 64.33\\
\cline{3-5}
& & \begin{tabular}[c]{@{}l@{}}
(16) Model H-3\\TDNN-LSTM (5-fold)\\
\end{tabular} & 74.53 & {\bf 63.70 }\\
\hline
\end{tabular}
\caption{Word error rate (WER) and phone error rate (PER) over the test set of vocal data.}
\label{tab: all results}
\end{table}
\begin{figure*}[tb]
\centering
\centerline{\includegraphics[width=\textwidth]{error_analysis2.png}
\caption{Sample recognition errors produced by Model E-4 : fragment-level in row(7) of Table.\ref{tab: all results}.}
\label{fig:error analysis}
\end{figure*}
\subsubsection{Pitch Distribution}
Fig.\ref{fig:pitch dist.} depicts the histogram for pitch distribution for speech and different genders of vocal. We can see the pitch values of vocal are significantly higher with a much wider range, and female singers produce slightly higher pitch values than male singers and groups.
\subsection{Recognition Results}
The primary recognition results are listed in Table.\ref{tab: all results}. Word error rate (WER) is taken as the major performance measure, while phone error rate (PER) is also listed as references. Rows (1)(2) on the top are for the language model trained with LibriSpeech data, while rows (3)-(16) for the language model trained with lyrics corpus. In addition, in rows (4)-(16) the lexicon was extended with possible repetition of vowels as explained in subsection \ref{subsubsec:exten_lex}. Rows (1)-(8) are for GMM-HMM only, while rows (9)-(16) with DNNs, BLSTMs and TDNN-LSTMs.
Row(1) is for Model A in Fig.\ref{fig:approach flow} taken as the baseline, which was trained on LibriSpeech data with SAT, together with the language model also trained with LibriSpeech. The extremely high WER (96.21\%) indicated the wide mismatch between speech and song audio, and the high difficulties in transcribing song audio. This is taken as the baseline of this work. After going through the series of Alignments a, b, c, d and training the series of Models B, C, D, we finally obtained the best GMM-HMM model, Model E-4 in Model E with fMLLR on the fragment level, as explained in section \ref{sec:approach} and shown in Fig.\ref{fig:approach flow}. As shown in row(2) of Table.\ref{tab: all results}, with the same LibriSpeech LM, Model E-4 reduced WER to 88.26\%, and brought an absolute improvement of 7.95\% (rows (2) vs. (1)), which shows the achievements by the series of GMM-HMM alone. When we replaced the LibriSpeech language model with Lyrics language model but with the same Model E-4, we obtained an WER of 80.40\% or an absolute improvement of 7.86\% (rows (3) vs. (2)). This shows the achievement by the Lyrics language model alone.
We then substituted the normal lexicon with the extended one (with vowels repeated or not as described in subsection \ref{subsubsec:exten_lex}), while using exactly the same model E-4, the WER of 77.08\% in row (7) indicated the extended lexicon alone brought an absolute improvement of 3.32\% (rows (7) vs. (3)). Furthermore, the increased self-looped transition probability ($r=0.9$) in subsection \ref{subsubsec:self_loop} for vowel HMMs also brought an 0.46\% improvement when applied on top of the extended lexicon (rows (8) vs. (7)). The results show that prolonged vowels did cause problems in recognition, and the proposed approaches did help.
Rows (4)(5)(6) for Models B, C, D show the incremental improvements when training the acoustic models with a series of improved alignments a, b, c, which led to the Model E-4 in row (7). Some preliminary tests with p-norm DNN with varying parameters were then performed. The best results for the moment were obtained with 4 hidden layers, 600 and 150 hidden units for p-norm nonlinearity\cite{zhang2014improving}. The result in rows (9) shows absolute improvements of 1.52\% (row (9) for Model F-1 vs. row (7)) for regular DNN. Rows(10) is for Models F-1 DNN (multi-target).
Rows (11)(12)(13) show the results of BLSTMs with different factors of data augmentation described in \ref{subsec:deep learning models}. Models G-1,2,3 used three layers with 400 hidden states and 100 units for recurrent and projection layer, however, since the amount of training data were different, the number of training epoches were 15, 7 and 5 respectively. Data augmentation brought much improvement of 5.62\% (rows (12) v.s.(11)), while 3-fold BLSTM outperformed 5-fold by 1.03\%. Trend for Model H (rows (14)(15)(16)) is the same as Model G, 3-fold turned out to be the best. Row (15) of Model TDNN-LSTM achieved the lowest WER(\%) of 73.90\%, with architecture $T^{130}T^{130}L^{130}T^{520}T^{520}L^{130}T^{520}T^{520}L^{130}$, while $T^{n}$ and $L^{m}$ denotes that the size of TDNN layer was $n$ and the size of hidden units of forward LSTM was $m$. The WER achieved here are relatively high, indicating the difficulties and the need for further research.
\subsection{Different Levels of fMLLR Adaptation}
In Fig.\ref{fig:approach flow} Model E includes different models obtained with fMLLR over different levels, Models E-1,2,3,4. But in Table.\ref{tab: all results} only Model E-4 is listed. Complete results for Models E-1,2,3,4 are listed in Table.\ref{tab: fMLLR GMM-HMM}, all for Lyrics Language Model with extended lexicon. Row (4) here is for Model E-4, or fMLLR over fragment level, exactly row (7) of Table.\ref{tab: all results}. Rows (1)(2)(3) are the same as row (5) here, except over levels of genre, singer and song. We see fragment level is the best, probably because fragment(10-35 sec long) is the smallest unit and the acoustic characteristic of signals within a fragment is almost uniform (same genre, same singer and the same song).
\begin{table}[]
\centering
\begin{tabular}{|c|l|c|c|}
\hline
& Acoustic Model & WER(\%) & PER(\%) \\
\hline
\multirow{5}{*}{ \rotatebox{90}{
\begin{tabular}[c]{@{}l@{}}
Lyrics \\Language Model\\
Extended Lexicon
\end{tabular} } }
& \begin{tabular}[c]{@{}c@{}}
(1) Model E-1,\\
genre-level
\end{tabular} & 84.24 & 68.92 \\
\cline{2-4}
& \begin{tabular}[c]{@{}c@{}}
(2) Model E-2, \\
singer-level
\end{tabular} & 78.53 & 68.48\\
\cline{2-4}
& \begin{tabular}[c]{@{}c@{}}
(3) Model E-3, \\
song-level
\end{tabular} & 78.80 & 68.24\\
\cline{2-4}
& \begin{tabular}[c]{@{}c@{}}
(4) Model E-4, \\
fragment-level
\end{tabular} & \textbf{77.08} & \textbf{66.04}\\
\hline
\end{tabular}
\caption{Model E : GMM-HMM with fMLLR over different levels.}
\label{tab: fMLLR GMM-HMM}
\end{table}
\subsection{Error Analysis}
From the data, we found errors frequently occurred under some specific circumstances, such as high-pitched voice, widely varying phone duration, overlapping verses (multiple people sing simultaneously), and residual background music.
Figure \ref{fig:error analysis} shows a sample recognition results obtained with Model E-4 as in row(7) of Table.\ref{tab: all results}, showing the error caused by high-pitched voice and overlapping verses. At first, the model successfully decoded the words, \textit{"what doesn't kill you makes"}, but afterward the pitch went high and a lower pitch harmony was added, the recognition results then went totally wrong.
\section{Conclusion}
\label{sec:conclusion}
In this paper we report some initial results of transcribing lyrics from commercial song audio using different sets of acoustic models, adaptation approaches, language models and lexicons. Techniques for special characteristics of song audio were considered. The achieved WER was relatively high compared to experiences in speech recognition. However, considering the much more difficult problems in song audio and the wide difference between speech and singing voice, the results here may serve as good references for future work to be continued.
\bibliographystyle{IEEEbib}
|
{
"timestamp": "2018-04-17T02:09:33",
"yymm": "1804",
"arxiv_id": "1804.05306",
"language": "en",
"url": "https://arxiv.org/abs/1804.05306"
}
|
\section{Introduction}
The rapid advancement of Internet of Things (IoT) and social networking applications results in an exponential growth of the data generated at the network edge. It has been predicted that the data generation rate will exceed the capacity of today's Internet in the near future \cite{ChiangFogOverview}.
Due to network bandwidth and data privacy concerns, it is impractical and often unnecessary to send all the data to a remote cloud. As a result, research organizations estimate that over $90\%$ of the data will be stored and processed locally \cite{KellyABI}.
Local data storing and processing with global coordination is made possible by the emerging technology of mobile edge computing (MEC) \cite{MaoEdgeComptSurvey, MachEdgeComputSurvey}, where edge nodes, such as sensors, home gateways, micro servers, and small cells, are equipped with storage and computation capability. Multiple edge nodes work together with the remote cloud to perform large-scale distributed tasks that involve both local processing and remote coordination/execution.
To analyze large amounts of data and obtain useful information for the detection, classification, and prediction of future events, machine learning techniques are often applied. The definition of machine learning is very broad, ranging from simple data summarization with linear regression to multi-class classification with support vector machines (SVMs) and deep neural networks \cite{shalev2014understanding, Goodfellow-et-al-2016}.
The latter have shown very promising performance in recent years, for complex tasks such as image classification. One key enabler of machine learning is the ability to learn (train) models using a very large amount of data.
With the increasing amount of data being generated by new applications and with more applications becoming data-driven, one can foresee that machine learning tasks will become a dominant workload in distributed MEC systems in the future. However, it is challenging to perform distributed machine learning on resource-constrained MEC systems.
In this paper, we address the problem of how to efficiently utilize the limited computation and communication resources at the edge for the optimal learning performance.
We consider a typical edge computing architecture where edge nodes are interconnected with the remote cloud via network elements, such as gateways and routers, as illustrated in Fig.~\ref{fig:architecture}. The raw data is collected and stored at multiple edge nodes, and a machine learning model is trained from the distributed data \emph{without} sending the raw data from the nodes to a central place. This variant of distributed machine learning (model training) from a federation of edge nodes is known as \emph{federated learning}~\cite{GoogleFederatedLearningBlog,mcmahan2016communication,WirelessNetworkIntelligence}.
\begin{figure}
\centering
\includegraphics[width=0.47\textwidth]{Architecture.pdf}
\caption{System architecture.}
\label{fig:architecture}
\end{figure}
We focus on gradient-descent based federated learning algorithms, which have general applicability to a wide range of machine learning models.
The learning process includes \emph{local update} steps where each edge node performs gradient descent to adjust the (local) model parameter to minimize the loss function defined on its own dataset. It also includes \emph{global aggregation} steps where model parameters obtained at different edge nodes are sent to an aggregator, which is a logical component that can run on the remote cloud, a network element, or an edge node. The aggregator aggregates these parameters (e.g., by taking a weighted average) and sends an updated parameter back to the edge nodes for the next round of iteration.
The frequency of global aggregation is configurable; one can aggregate at an interval of one or multiple local updates.
Each local update consumes computation resource of the edge node, and each global aggregation consumes communication resource of the network.
The amount of consumed resources may vary over time, and there is a complex relationship among the frequency of global aggregation, the model training accuracy, and resource consumption.
We propose an algorithm to determine the frequency of global aggregation so that the available resource is most efficiently used. This is important because the training of machine learning models is usually resource-intensive, and a non-optimal operation of the learning task may waste a significant amount of resources.
Our main contributions in this paper are as follows:
\begin{enumerate}
\item We analyze the convergence bound of gradient-descent based federated learning from a theoretical perspective, and obtain a novel convergence bound that incorporates non-independent-and-identically-distributed (non-i.i.d.) data distributions among nodes and an arbitrary number of local updates between two global aggregations.
\item Using the above theoretical convergence bound, we propose a control algorithm that learns the data distribution, system dynamics, and model characteristics, based on which it dynamically adapts the frequency of global aggregation in real time to minimize the learning loss under a fixed resource budget.
\item We evaluate the performance of the proposed control algorithm via extensive experiments using real datasets both on a hardware prototype and in a simulated environment, which confirm that our proposed approach provides near-optimal performance for different data distributions, various machine learning models, and system configurations with different numbers of edge nodes.
\end{enumerate}
\section{Related Work}
Existing work on MEC focuses on generic applications, where solutions have been proposed for application offloading~\cite{XiaoOffloading2017, TongOffloading2016}, workload scheduling \cite{TongHierarchical2016, TanInfocom2017}, and service migration triggered by user mobility \cite{WangICDCS2017, WangMicroCloudPredictedCost}. However, they do not address the relationship among communication, computation, and training accuracy for machine learning applications, which is important for optimizing the performance of machine learning tasks.
The concept of federated learning was first proposed in \cite{mcmahan2016communication}, which showed its effectiveness through experiments on various datasets. Based on the comparison of synchronous and asynchronous methods of distributed gradient descent in \cite{Chen2016}, it is proposed in \cite{mcmahan2016communication} that federated learning should use the synchronous approach because it is more efficient than asynchronous approaches. The approach in \cite{mcmahan2016communication} uses a fixed global aggregation frequency. It does not provide theoretical convergence guarantee and the experiments were not conducted in a network setting.
Several extensions have been made to the original federated learning proposal recently. For example, a mechanism for secure global aggregation is proposed in~\cite{SecurityAggregationFederatedLearning}. Methods for compressing the information exchanged within one global aggregation step is proposed in~\cite{Jakub2016,hardy2017distributed}. Adjustments to the standard gradient descent procedure for better performance in the federated setting is studied in~\cite{FederatedOptimization}. Participant (client) selection for federated learning is studied in~\cite{nishio2018client}. An approach that shares a small amount of data with other nodes for better learning performance with non-i.i.d. data distribution is proposed in~\cite{zhao2018federated}.
These studies do not consider the adaptation of global aggregation frequency, and thus they are orthogonal to our work in this paper. To the best of our knowledge, the adaptation of global aggregation frequency for federated learning with resource constraints has not been studied in the literature.
An area related to federated learning is distributed machine learning in datacenters through the use of worker machines and parameter servers~\cite{li2014scaling}. The main difference between the datacenter environment and edge computing environment is that in datacenters, shared storage is usually used. The worker machines do not keep persistent data storage on their own, and they fetch the data from the shared storage at the beginning of the learning process. As a result, the data samples obtained by different workers are usually independent and identically distributed (i.i.d.).
In federated learning, the data is collected at the edge directly and stored persistently at edge nodes, thus the data distribution at different edge nodes is usually non-i.i.d.
Concurrently with our work in this paper, optimization of synchronization frequency with running time considerations is studied in~\cite{WangSysML2019} for the datacenter setting. It does not consider characteristics of non-i.i.d. data distributions which is essential in federated learning.
Distributed machine learning across multiple datacenters in different geographical locations is studied in \cite{Gaia2017}, where a threshold-based approach to reduce the communication among different datacenters is proposed. Although the work in \cite{Gaia2017} is related to the adaptation of synchronization frequency with resource considerations, it focuses on peer-to-peer connected datacenters, which is different from the federated learning architecture that is not peer-to-peer. It also allows asynchronism among datacenter nodes, which is not the case in federated learning.
In addition, the approach in \cite{Gaia2017} is designed empirically and does not consider a concrete theoretical objective, nor does it consider computation resource constraint which is important in MEC systems in addition to constrained communication resource.
From a theoretical perspective, bounds on the convergence of distributed gradient descent are obtained in~\cite{agarwal2011distributed,lian2015asynchronous,pmlr-v70-zheng17b}, which only allow one step of local update before global aggregation. Partial global aggregation is allowed in the decentralized gradient descent approach in \cite{lian2017can,lian2017asynchronous}, where after each local update step, parameter aggregation is performed over a non-empty subset of nodes, which does not apply in our federated learning setting where there is no aggregation at all after some of the local update steps.
Multiple local updates before aggregation is possible in the bound derived in~\cite{Gaia2017}, but the number of local updates varies based on the thresholding procedure and cannot be specified as a given constant. Concurrently with our work, bounds with a fixed number of local updates between global aggregation steps are derived in~\cite{CooperativeSGD,YuAAAI2019}. However, the bound in~\cite{CooperativeSGD} only works with i.i.d. data distribution; the bound in~\cite{YuAAAI2019} is independent from how different the datasets are, which is inefficient because it does not capture the fact that training on i.i.d. data is likely to converge faster than training on non-i.i.d. data.
Related studies on distributed optimization that are applicable for machine learning applications also include \cite{zhang2012communication, arjevani2015communication, ma2017distributed}, where a separate solver is used to solve a local problem. The main focus of \cite{zhang2012communication, arjevani2015communication, ma2017distributed} is the trade-off between communication and optimality, where the complexity of solving the local problem (such as the number of local updates needed) is not studied.
In addition, many of the existing studies either explicitly or implicitly assume i.i.d. data distribution at different nodes, which is inappropriate in federated learning.
To our knowledge, the convergence bound of distributed gradient descent in the federated learning setting, {which captures both the characteristics of different (possibly non-i.i.d. distributed) datasets and a given number of local update steps between two global aggregation steps, has not been studied in the literature.
In contrast to the above research, our work in this paper formally addresses the problem of dynamically determining the global aggregation frequency to \emph{optimize the learning with a given resource budget} for federated learning in MEC systems.
This is a non-trivial problem due to the complex dependency between each learning step and its previous learning steps, which is hard to capture analytically.
It is also challenging due to non-i.i.d. data distributions at different nodes, where the data distribution is unknown beforehand and the datasets may have different degrees of similarities with each other, and the real-time dynamics of the system.
We propose an algorithm that is derived from theoretical analysis and adapts to real-time system dynamics.
We start with summarizing the basics of federated learning in the next section. In Section IV, we describe our problem formulation. The convergence analysis and control algorithm are presented in Sections V and VI, respectively. Experimentation results are shown in Section VII and the conclusion is presented in Section VIII.
\section{Preliminaries and Definitions \label{sec:distributedMachineLearning}}
\subsection{Loss Function}
Machine learning models include a set of parameters which are learned based on training data. A training data sample~$j$ usually consists of two parts. One is a vector $\mathbf{x}_j$ that is regarded as the input of the machine learning model (such as the pixels of an image); the other is a scalar $y_j$ that is the desired output of the model (such as the label of the image). To facilitate the learning, each model has a loss function defined on its parameter vector $\mathbf{w}$ for each data sample $j$. The loss function captures the error of the model on the training data, and the model learning process is to minimize the loss function on a collection of training data samples.
For each data sample $j$, we define the loss function as $f(\mathbf{w}, \mathbf{x}_j, y_j)$, which we write as $f_j(\mathbf{w})$ in short\footnote{Note that some unsupervised models (such as K-means) only learn on $\mathbf{x}_j$ and do not require the existence of $y_j$ in the training data. In such cases, the loss function value only depends on $\mathbf{x}_j$.}.
\begin{table} \protect\caption{Loss functions for popular machine learning models} \label{tab:learningModels}
\vspace{-0.15in}
\renewcommand{\arraystretch}{1.4}
\center{\footnotesize
\begin{tabularx}{\linewidth}
{>{\setlength\hsize{0.5\hsize}}X >{\setlength\hsize{1.5\hsize}}X}
\hline Model & Loss function $f(\mathbf{w}, \mathbf{x}_j, y_j)$ ($\triangleq f_j (\mathbf{w})$)\\
\hline
Squared-SVM & $\frac{\lambda}{2} \Vert \mathbf{w} \Vert^2 + \frac{1}{2} \max \left\{ 0; 1 - y_j \mathbf{w}^\mathrm{T} \mathbf{x}_j \right\}^2 $ ($\lambda$ is const.) \\
Linear regression & $\frac{1}{2} \Vert y_j - \mathbf{w}^\mathrm{T} \mathbf{x}_j \Vert^2 $ \\
K-means & $ \frac{1}{2} \min_l \Vert \mathbf{x}_j - \mathbf{w}_{(l)} \Vert^2 $ where $\mathbf{w} \triangleq [\mathbf{w}_{(1)}^\mathrm{T}, \mathbf{w}_{(2)}^\mathrm{T}, ...]^\mathrm{T}$ \\
Convolutional neural network & Cross-entropy on cascaded linear and non-linear transforms, see \cite{Goodfellow-et-al-2016} \\
\hline
\end{tabularx}
}
\end{table}
Examples of loss functions of popular machine learning models are summarized\footnote{While our focus is on non-probabilistic learning models, similar loss functions can be defined for probabilistic models where the goal is to minimize the negative of the log-likelihood function, for instance.} in Table~\ref{tab:learningModels} \cite{shalev2014understanding,Goodfellow-et-al-2016,bottou2010large}.
For convenience, we assume that all vectors are column vectors in this paper and use $\mathbf{x}^\mathrm{T}$ to denote the transpose of $\mathbf{x}$. We use ``$\triangleq$'' to denote ``is defined to be equal to'' and use $\Vert \cdot \Vert$ to denote the $\mathcal{L}^2$ norm.
Assume that we have $N$ edge nodes with local datasets $\mathcal{D}_{1},\mathcal{D}_{2},...,\mathcal{D}_{i},...,\mathcal{D}_{N}$. For each dataset $\mathcal{D}_{i}$ at node~$i$, the loss function on the collection of data samples at this node is
\begin{equation}
F_{i}(\mathbf{w}) \triangleq \frac{1}{\left|\mathcal{D}_{i}\right|}\sum_{j\in\mathcal{D}_{i}}f_{j}(\mathbf{w}).
\label{eq:localLossFuncAllSamples}
\end{equation}
We define $D_{i}\triangleq\left|\mathcal{D}_{i}\right|$, where $| \cdot |$ denotes the size of the set, and $D\triangleq\sum_{i=1}^{N}D_{i}$.
Assuming $\mathcal{D}_{i} \cap \mathcal{D}_{i'} = \emptyset$ for $i \neq i'$, we define the global loss function on all the distributed datasets as
\begin{equation}
F(\mathbf{w}) \triangleq \frac{\sum_{j\in\cup_i\mathcal{D}_{i}}f_{j}(\mathbf{w})}{\left|\cup_i\mathcal{D}_{i}\right|} = \frac{\sum_{i=1}^{N}D_{i}F_{i}(\mathbf{w})}{D}.
\label{eq:globalLossFuncAllSamples}
\end{equation}
Note that $F(\mathbf{w})$ \emph{cannot} be directly computed without sharing information among multiple nodes.
\subsection{The Learning Problem}
The learning problem is to minimize $F(\mathbf{w})$, i.e., to find
\begin{equation}
\mathbf{w}^{*} \triangleq \arg\min F(\mathbf{w}).
\label{eq:learningProblem}
\end{equation}
Due to the inherent complexity of most machine learning models, it is usually impossible to find a closed-form solution to (\ref{eq:learningProblem}). Thus, (\ref{eq:learningProblem}) is often solved using gradient-descent techniques.
\subsection{Distributed Gradient Descent}
\label{subsec:distGradDescent}
We present a canonical distributed gradient-descent algorithm to solve (\ref{eq:learningProblem}), which is widely used in state-of-the-art federated learning systems (e.g., \cite{mcmahan2016communication}).
Each node~$i$ has its local model parameter $\mathbf{w}_{i}{(t)}$, where $t=0,1,2,...$ denotes the iteration index. At $t=0$, the local parameters
for all nodes $i$ are initialized to the same value. For $t>0$, new values of $\mathbf{w}_{i}{(t)}$ are computed according to a gradient-descent update rule on the local loss function, based on the parameter value in the previous iteration $t-1$. This gradient-descent step on the local loss function (defined on the local dataset) at each node is referred to as the \emph{local update}.
After one or multiple local updates, a \emph{global aggregation} is performed through the aggregator to update the local parameter at each node to the weighted average of all nodes' parameters.
We define that each \emph{iteration} includes a local update step which is possibly followed by a global aggregation step.
After global aggregation, the local parameter $\mathbf{w}_{i}{(t)}$ at each node~$i$ usually changes. For convenience, we use $\widetilde{\mathbf{w}}_{i}{(t)}$ to denote the parameter at node~$i$ after possible global aggregation.
If no aggregation is performed at iteration $t$, we have $\widetilde{\mathbf{w}}_{i}{(t)}=\mathbf{w}_{i}{(t)}$.
If aggregation is performed at iteration $t$, then generally
$\widetilde{\mathbf{w}}_{i}{(t)} \neq \mathbf{w}_{i}{(t)}$ and we set $\widetilde{\mathbf{w}}_{i}{(t)} = \mathbf{w}{(t)}$, where $\mathbf{w}{(t)}$ is a weighted average of $\mathbf{w}_{i}{(t)}$ defined in (\ref{eq:globalAverage}) below. An example of these definitions is shown in Fig.~\ref{fig:wVariables}.
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{wVariables.pdf}
\caption{Illustration of the values of $\mathbf{w}_i(t)$ and $\widetilde{\mathbf{w}}_i(t)$ at node $i$.}
\label{fig:wVariables}
\end{figure}
The local update in each iteration is performed on the parameter after possible global aggregation in the previous iteration. For each node~$i$, the update rule is as follows:
\begin{align}
\mathbf{w}_{i}{(t)}=\widetilde{\mathbf{w}}_{i}{(t-1)}-\eta\nabla F_{i}\left(\widetilde{\mathbf{w}}_{i}{(t-1)}\right)\label{eq:localUpdate}
\end{align}
where $\eta > 0$ is the step size.
For any iteration $t$ (which may or may not include a global aggregation step), we define
\begin{equation}
\mathbf{w}{(t)}=\frac{\sum_{i=1}^{N}D_{i}\mathbf{w}_{i}{(t)}}{D}\label{eq:globalAverage}.
\end{equation}
This global model parameter $\mathbf{w}{(t)}$ is \emph{only observable to nodes in the system if global aggregation is performed at iteration $t$}, but we define it for all $t$ to facilitate the analysis later.
We define that the system performs $\tau$ steps of local updates at each node between every two global aggregations. We define $T$ as the total number of local iterations at each node.
For ease of presentation, we assume that $T$ is an integer multiple of $\tau$ in the theoretical analysis, which will be relaxed when we discuss practical aspects in Section~\ref{subsec:controlAlgImpl}.
The logic of distributed gradient descent is presented in Algorithm~\ref{alg:distGradDescent}, which ignores aspects related to the communication between the aggregator and edge nodes. Such aspects will be discussed later in Section~\ref{subsec:controlAlgImpl}.
\begin{algorithm}[t]
\caption{Distributed gradient descent (logical view)}
\label{alg:distGradDescent}
{\footnotesize
\KwIn{$\tau$, $T$}
\KwOut{Final model parameter $\mathbf{w}^\mathrm{f}$}
Initialize $\mathbf{w}^\mathrm{f}$, $\mathbf{w}_{i}{(0)}$ and $\widetilde{\mathbf{w}}_{i}{(0)}$ to the same value for all $i$;
\For{$t = 1,2,...,T$}
{
For each node $i$ \emph{in parallel}, compute local update using (\ref{eq:localUpdate}); \label{alg:distGradDescent:LocalUpdate}
\If{$t$ is an integer multiple of $\tau$}
{
Set $\widetilde{\mathbf{w}}_{i}{(t)} \leftarrow \mathbf{w}{(t)}$ for all $i$, where $\mathbf{w}{(t)}$ is defined in (\ref{eq:globalAverage}); //Global aggregation \label{alg:distGradDescent:TildeSetGlobalAgg}
Update $\mathbf{w}^\mathrm{f} \leftarrow \arg\min_{\mathbf{w}\in\{\mathbf{w}^\mathrm{f},\mathbf{w}{(t)}\}}F(\mathbf{w})$; \label{alg:distGradDescent:minStep}
}
\Else
{ Set $\widetilde{\mathbf{w}}_{i}{(t)} \leftarrow \mathbf{w}_{i}{(t)}$ for all $i$; //No global aggregation \label{alg:distGradDescent:TildeSet}
}
}
}
\end{algorithm}
The final model parameter $\mathbf{w}^\mathrm{f}$ obtained from Algorithm~\ref{alg:distGradDescent} is the one that has produced the minimum global loss after each global aggregation throughout the entire execution of the algorithm. We use $\mathbf{w}^\mathrm{f}$ instead of $\mathbf{w}(T)$, to align with the theoretical convergence bound that will be presented in Section~\ref{sec:mainResults}. In practice, we have seen that $\mathbf{w}^\mathrm{f}$ and $\mathbf{w}(T)$ are usually the same, but using $\mathbf{w}^\mathrm{f}$ provides theoretical rigor in terms of convergence guarantee so we use $\mathbf{w}^\mathrm{f}$ in this paper. Note that $F(\mathbf{w})$ in Line~\ref{alg:distGradDescent:minStep} of Algorithm~\ref{alg:distGradDescent} is computed in a distributed manner according to (\ref{eq:globalLossFuncAllSamples}); the details will be presented later.
The rationale behind Algorithm~\ref{alg:distGradDescent} is that when $\tau=1$, i.e., when we perform global aggregation after every local update step, the distributed gradient descent (ignoring communication aspects) is equivalent to the centralized gradient descent, where the latter assumes that all data samples are available at a centralized location and the global loss function and its gradient can be observed directly.
This is due to the linearity of the gradient operator.
\if\citetechreport1
Due to space limitation, see our online technical report \cite[Appendix~\ref{append:optimalityOfDistributedGradDescent}]{JournalTechReport} as well as \cite{FUSION2018} for detailed discussions about this.
\else
See Appendix~\ref{append:optimalityOfDistributedGradDescent} as well as \cite{FUSION2018} for detailed discussions about this.
\fi
The main notations in this paper are summarized in Table~\ref{tab:MainNotations}.
\begin{table}[t]
\caption {Summary of main notations} \label{tab:MainNotations}
\vspace{-0.15in}
{\footnotesize
\begin{center}
\begin{tabularx}{\linewidth}
{>{\setlength\hsize{0.2\hsize}}X >{\setlength\hsize{1.8\hsize}}X}
\hline
$F(\mathbf{w})$ & Global loss function \\
$F_i(\mathbf{w})$ & Local loss function for node $i$ \\
$t$ & Iteration index \\
$\mathbf{w}_i(t)$ & Local model parameter at node $i$ in iteration $t$ \\
$\mathbf{w}(t)$ & Global model parameter in iteration $t$ \\
$\mathbf{w}^\mathrm{f}$ & Final model parameter obtained at the end of learning process\\
$\mathbf{w}^*$ & True optimal model parameter that minimizes $F(\mathbf{w})$\\
$\eta$ & Gradient descent step size\\
$\tau$ & Number of local update steps between two global aggregations \\
$T$ & Total number of local update steps at each node \\
$K$ & Total number of global aggregation steps, equal to $T/\tau$ \\
$M$ ($m$)\! & Total number of resource types (the $m$-th type of resource) \\
$R_m$ & Total budget of the $m$-th type of resource \\
$c_m$ & Consumption of type-$m$ resource in one lo\underline{c}al update step\\
$b_m$ & Consumption of type-$m$ resource in one glo\underline{b}al aggregation step\\
$\rho$ & Lipschitz parameter of $F_i(\mathbf{w})$ ($\forall i$) and $F(\mathbf{w})$ \\
$\beta$ & Smoothness parameter of $F_i(\mathbf{w})$ ($\forall i$) and $F(\mathbf{w})$ \\
$\delta$ & Gradient divergence \\
$h(\tau)$ & Function defined in (\ref{eq:hFuncDef}), gap between the model parameters obtained from distributed and centralized gradient descents\\
$\varphi$ & Constant defined in Lemma~\ref{lemma:convergenceUpperBound}, control parameter \\
$G(\tau)$ & Function defined in (\ref{eq:GTauDef}), control objective \\
$\tau^*$ & Optimal $\tau$ obtained by minimizing $G(\tau)$ \\
\hline
\end{tabularx}
\end{center}
}
\end{table}
\section{Problem Formulation}
\label{sec:ProblemFormulation}
When there is a large amount of data (which is usually needed for training an accurate model) distributed at a large number of nodes, the federated learning process can consume a significant amount of resources. The notion of ``resources'' here is generic and can include time, energy, monetary cost etc. \emph{related to both computation and communication}.
One often has to limit the amount of resources used for learning each model, in order not to backlog the system and to keep the operational cost low.
This is particularly important in edge computing environments where the computation and communication resources are not as abundant as in datacenters.
Therefore, a natural question is how to make efficient use of a given amount of resources to minimize the loss function of model training.
For the distributed gradient-descent based learning approach presented above, the question narrows down to determining the optimal values of $T$ and $\tau$, so that the global loss function is minimized subject to a given resource constraint for this learning task.
We use $K$ to denote the total number of global aggregations within $T$ iterations. Because we assumed earlier that $T$ is an integer multiple of $\tau$, we have $K=\frac{T}{\tau}$.
We define
\begin{equation}
\label{eq:wminDef}
\mathbf{w}^\mathrm{f} \triangleq \argmin_{\mathbf{w} \in \{\mathbf{w}(k\tau): k=0, 1, 2, ..., K\}} F(\mathbf{w}).
\end{equation}
It is easy to verify that this definition is equivalent to $\mathbf{w}^\mathrm{f}$ found from Algorithm~\ref{alg:distGradDescent}.
To compute $F(\mathbf{w})$ in (\ref{eq:wminDef}), each node $i$ first computes $F_i(\mathbf{w})$ and sends the result to the aggregator, then the aggregator computes $F(\mathbf{w})$ according to (\ref{eq:globalLossFuncAllSamples}). Since each node only knows the value of $\mathbf{w}(k\tau)$ after the $k$-th global aggregation, $F_i(\mathbf{w}(k\tau))$ at node $i$ will be sent back to the aggregator at the $(k+1)$-th global aggregation, and the aggregator computes $F(\mathbf{w}(k\tau))$ afterwards.
To compute the last loss value $F(\mathbf{w}(K\tau)) = F(\mathbf{w}(T))$, an additional round of local and global update is performed at the end.
We assume that at each node, local update consumes the same amount of resource no matter whether only the local loss is computed (in the last round) or both the local loss and gradient are computed (in all the other rounds), because the loss and gradient computations can usually be based on the same intermediate result. For example, the back propagation approach for computing gradients in neural networks requires a forward propagation procedure that essentially obtains the loss as an intermediate step \cite{Goodfellow-et-al-2016}.
We consider $M$ different types of resources. For example, one type of resource can be time, another type can be energy, a third type can be communication bandwidth, etc. For each $m\in\{1,2,...,M\}$, we define that each lo\underline{c}al update step at \emph{all} nodes consumes $c_m$ units of type-$m$ resource, and each glo\underline{b}al aggregation step consumes $b_m$ units of type-$m$ resource, where $c_m\geq 0$ and $b_m\geq 0$ are both \emph{finite} real numbers.
For given $T$ and $\tau$, the total amount of consumed type-$m$ resource is $(T+1)c_m + \left(K + 1\right)b_m$, where the additional ``$+1$'' is for computing $F(\mathbf{w}(K\tau))$, as discussed above.
Let $R_m$ denote the total budget of type-$m$ resource.
We seek the solution to the following problem:
\begin{align}
\min_{\tau, K \in \{1,2,3,...\}} & \,\, F(\mathbf{w}^\mathrm{f}) \label{eq:optimizationProblem} \\
\textrm{s.t.} & \,\, (T\!+\! 1)c_m \!+\! \left(K\! +\! 1\right)b_m \leq R_m \, , \,\, \forall m\!\in\! \{\!1,...,M\!\} \nonumber \\
& \,\,T=K\tau . \nonumber
\end{align}
To solve (\ref{eq:optimizationProblem}), we need to find out how $\tau$ and $K$ (and thus $T$) affect the loss function $F(\mathbf{w}^\mathrm{f})$ computed on the final model parameter $\mathbf{w}^\mathrm{f}$.
It is generally impossible to find an exact analytical expression to relate $\tau$ and $K$ with $F(\mathbf{w}^\mathrm{f})$, because it depends on the convergence property of gradient descent (for which only upper/lower bounds are known \cite{convex}) and the impact of the global aggregation frequency on the convergence. Further, the resource consumptions $c_m$ and $b_m$ can be time-varying in practice which makes the problem even more challenging than (\ref{eq:optimizationProblem}) alone.
We analyze the convergence bound of distributed gradient descent (Algorithm~\ref{alg:distGradDescent}) in Section~\ref{sec:convergenceAnalysis}, then use this bound to approximately solve (\ref{eq:optimizationProblem}) and propose a control algorithm for adaptively choosing the best values of $\tau$ and $T$ to achieve near-optimal resource utilization in Section~\ref{sec:schedulingAlgorithm}.
\section{Convergence Analysis}
\label{sec:convergenceAnalysis}
We analyze the convergence of Algorithm~\ref{alg:distGradDescent} in this section and find an upper bound of $F(\mathbf{w}^\mathrm{f})-F(\mathbf{w}^*)$. To facilitate the analysis, we first introduce some notations.
\subsection{Definitions}
We can divide the $T$ iterations into $K$ different intervals, as shown in Fig.~\ref{fig:interval}, with only the first and last iterations in each interval containing global aggregation. We use the shorthand notations $[k]$ to denote the iteration interval\footnote{With slight abuse of notation, we use $[(k-1)\tau,k\tau]$ to denote the integers contained in the interval for simplicity. We use the same convention in other parts of the paper as long as there is no ambiguity.} $[(k-1)\tau,k\tau]$, for $k=1,2,...,K$.
\begin{figure}
\centering
\includegraphics[width=0.47\textwidth]{IntervalDescription.pdf}
\caption{Illustration of definitions in different intervals.}
\label{fig:interval}
\end{figure}
For each interval $[k]$, we use $\mathbf{v}_{[k]}(t)$ to denote an auxiliary parameter vector that follows a \emph{centralized} gradient descent according to
\begin{align}
\mathbf{v}_{[k]}{(t)}=\mathbf{v}_{[k]}{(t-1)}-\eta\nabla F(\mathbf{v}_{[k]}{(t-1)})\label{eq:updateV}
\end{align}
where $\mathbf{v}_{[k]}(t)$ is only defined for $t \in [(k-1)\tau,k\tau]$ for a given $k$.
This update rule is based on the global loss function $F(\mathbf{w})$ which is only observable when all data samples are available at a central place (thus we call it centralized gradient descent), whereas the iteration in (\ref{eq:localUpdate}) is on the local loss function $F_i(\mathbf{w})$.
We define that $\mathbf{v}_{[k]}(t)$ is ``synchronized'' with $\mathbf{w}(t)$ at the beginning of each interval $[k]$, i.e., $\mathbf{v}_{[k]}((k-1)\tau) \triangleq \mathbf{w}((k-1)\tau)$, where $\mathbf{w}(t)$ is the average of local parameters defined in (\ref{eq:globalAverage}).
Note that we also have $\widetilde{\mathbf{w}}_{i}{((k-1)\tau)}=\mathbf{w}{((k-1)\tau)}$ for all~$i$ because the global aggregation (or initialization when $k=1$) is performed in iteration $(k-1)\tau$.
The above definitions enable us to find the convergence bound of Algorithm~\ref{alg:distGradDescent} by taking a two-step approach. The first step is to find the gap between $\mathbf{w}(k\tau)$ and $\mathbf{v}_{[k]}(k\tau)$ for each $k$, which is the difference between the distributed and centralized gradient descents after $\tau$ steps of local updates without global aggregation. The second step is to combine this gap with the convergence bound of $\mathbf{v}_{[k]}(t)$ within each interval $[k]$ to obtain the convergence bound of $\mathbf{w}(t)$.
For the purpose of the analysis, we make the following assumption to the loss function.
\begin{assumption}
\label{assumption:Convex}
We assume the following for all $i$:
\begin{enumerate}
\item $F_{i}(\mathbf{w})$ is convex
\item $F_{i}(\mathbf{w})$ is $\rho$-Lipschitz, i.e., $\Vert F_{i}(\mathbf{w}) - F_{i}(\mathbf{w}') \Vert \leq \rho \Vert \mathbf{w} - \mathbf{w'} \Vert$ for any $\mathbf{w}, \mathbf{w}'$
\item $F_{i}(\mathbf{w})$ is $\beta$-smooth, i.e., $\left\Vert \nabla F_{i}(\mathbf{w}) - \nabla F_{i}(\mathbf{w}') \right\Vert \leq \beta \left\Vert \mathbf{w} - \mathbf{w}' \right\Vert$ for any $\mathbf{w}, \mathbf{w'}$
\end{enumerate}
\end{assumption}
Assumption~\ref{assumption:Convex} is satisfied for squared-SVM and linear regression (see Table~\ref{tab:learningModels}). The experimentation results that will be presented in Section~\ref{sec:experimentation} show that our control algorithm also works well for models (such as neural network) whose loss functions do not satisfy Assumption~\ref{assumption:Convex}.
\begin{lemma}
$F(\mathbf{w})$ is convex, $\rho$-Lipschitz, and $\beta$-smooth.
\end{lemma}
\begin{proof}
Straightforwardly from Assumption~\ref{assumption:Convex}, the definition of $F(\mathbf{w})$, and triangle inequality.
\end{proof}
We also define the following metric to capture the \emph{divergence} between the gradient of a local loss function and the gradient of the global loss function. This divergence is \emph{related to how the data is distributed at different nodes}.
\begin{definition}
\label{def:gradientDivergence}
(Gradient Divergence)
For any $i$ and $\mathbf{w}$, we define $\delta_i$ as an upper bound of $\left\Vert \nabla F_{i}(\mathbf{w})-\nabla F(\mathbf{w})\right\Vert$, i.e.,
\begin{equation}
\left\Vert \nabla F_{i}(\mathbf{w})-\nabla F(\mathbf{w})\right\Vert \leq \delta_i .
\label{eq:deltaiDef}
\end{equation}
We also define $\delta \triangleq \frac{\sum_{i}D_{i}\delta_{i}}{D}$.
\end{definition}
\subsection{Main Results}
\label{sec:mainResults}
The below theorem gives an upper bound on the difference between $\mathbf{w}(t)$ and $\mathbf{v}_{[k]}(t)$ when $t$ is within the interval $[k]$.
\begin{theorem} \label{theorem:wBound}
For any interval $[k]$ and $t\in [k]$, we have
\begin{equation}
\left\Vert \mathbf{w}{(t)}-\mathbf{v}_{[k]}{(t)}\right\Vert \leq h(t-(k-1)\tau)
\label{eq:wBound}
\end{equation}
where
\begin{equation}
h(x) \triangleq \frac{\delta}{\beta}\left((\eta\beta+1)^{x}-1\right)-\eta\delta x
\label{eq:hFuncDef}
\end{equation}
for any $x=0, 1, 2, ...$.
Furthermore, as $F(\cdot )$ is $\rho$-Lipschitz, we have
$F(\mathbf{w}{(t)})-F(\mathbf{v}_{[k]}(t))\leq \rho h(t - (k-1)\tau)$.
\end{theorem}
\begin{proof}
We first obtain an upper bound of $\left\Vert \widetilde{\mathbf{w}}_{i}{(t)}-\mathbf{v}_{[k]}{(t)}\right\Vert$ for each node $i$, based on which the final result is obtained.
\if\citetechreport1
See our online technical report \cite[Appendix~\ref{append:proofWBound}]{JournalTechReport} for details.
\else
For details, see Appendix~\ref{append:proofWBound}.
\fi
\end{proof}
Note that we always have $\eta>0$ and $\beta>0$ because otherwise the gradient descent procedure or the loss function becomes trivial. Therefore, we have $(\eta\beta + 1)^x \geq \eta\beta x + 1$ for $x = 0, 1, 2, ...$ due to Bernoulli's inequality. Substituting this into (\ref{eq:hFuncDef}) confirms that we always have $h(x) \geq 0$.
It is easy to see that $h(0) = h(1) = 0$.
Therefore, when $t = (k-1)\tau$, i.e., at the beginning of the interval $[k]$, the upper bound in (\ref{eq:wBound}) is zero. This is consistent with the definition of $\mathbf{v}_{[k]}((k-1)\tau) = \mathbf{w}((k-1)\tau)$ for any $k$. When $t = (k-1)\tau + 1$ (i.e., the second iteration in interval $[k]$), the upper bound in (\ref{eq:wBound}) is also zero. This agrees with the discussion at the end of Section~\ref{subsec:distGradDescent}, showing that there is no gap between distributed and centralized gradient descents when only one local update is performed after the global aggregation.
If $\tau = 1$, then $t - (k-1)\tau$ is either $0$ or $1$ for any interval $[k]$ and $t \in [k]$. Hence, the upper bound in (\ref{eq:wBound}) becomes exact for $\tau = 1$.
For $\tau > 1$, the value of $x = t - (k-1)$ can be larger. When $x$ is large, the exponential term with $(\eta\beta+1)^{x}$ in (\ref{eq:hFuncDef}) becomes dominant, and the gap between $\mathbf{w}(t)$ and $\mathbf{v}_{[k]}(t)$ can increase exponentially with $t$ for $t \in [k]$.
We also note that $h(x)$ is proportional to the gradient divergence $\delta$ (see (\ref{eq:hFuncDef})), which is intuitive because the more the local gradient is different from the global gradient (for the same parameter $\mathbf{w}$), the larger the gap will be. The gap is caused by the difference in the local gradients at different nodes starting at the second local update after each global aggregation. In an extreme case when all nodes have exactly the same data samples (and thus the same local loss functions), the gradients will be always the same and $\delta = 0$, in which case $\mathbf{w}(t)$ and $\mathbf{v}_{[k]}(t)$ are always equal.
Theorem~\ref{theorem:wBound} gives an upper bound of the difference between distributed and centralized gradient descents for each iteration interval $[k]$, assuming that $\mathbf{v}_{[k]}(t)$ in the centralized gradient descent is synchronized with $\mathbf{w}(t)$ at the beginning of each $[k]$.
Based on this result, we first obtain the following lemma.
\begin{lemma}
\label{lemma:convergenceUpperBound}
When all the following conditions are satisfied:
\begin{enumerate}
\item $\eta \leq \frac{1}{\beta}$
\item $\eta\varphi -\frac{\rho h(\tau)}{\tau\varepsilon^2} > 0$
\item $ F\left(\mathbf{v}_{[k]}(k\tau)\right)-F(\mathbf{w}^*) \geq \varepsilon$ for all $k$
\item $ F\left(\mathbf{w}(T)\right)-F(\mathbf{w}^*) \geq \varepsilon$
\end{enumerate}
for some $\varepsilon > 0$, where we define $\varphi \triangleq \omega \left(1-\frac{\beta \eta}{2}\right)$ and $\omega \triangleq \min_k \frac{1}{{\left\Vert \mathbf{v}_{[k]}((k-1)\tau)-\mathbf{w}^*\right\Vert^2 }}$,
then the convergence upper bound of Algorithm~\ref{alg:distGradDescent} after $T$ iterations is given by
\begin{equation}
F(\mathbf{w}(T))-F(\mathbf{w}^*) \leq \frac{1}{T\left(\eta\varphi -\frac{\rho h(\tau)}{\tau\varepsilon^2}\right)}.
\label{eq:convergenceUpperBound}
\end{equation}
\end{lemma}
\begin{proof}
We first analyze the convergence of $F\left(\mathbf{v}_{[k]}(t)\right)$ within each interval $[k]$. Then, we combine this result with the gap between $F(\mathbf{w}{(t)})$ and $F(\mathbf{v}_{[k]}(t))$ from Theorem~\ref{theorem:wBound} to obtain the final result.
\if\citetechreport1
See our online technical report \cite[Appendix~\ref{append:proofConvergenceUpperBound}]{JournalTechReport} for details.
\else
For details, see Appendix~\ref{append:proofConvergenceUpperBound}.
\fi
\end{proof}
We then have the following theorem.
\begin{theorem}
\label{theorem:convergenceUpperBoundFinal}
When $\eta \leq \frac{1}{\beta}$, we have
\begin{equation}
F(\mathbf{w}^\mathrm{f}) -F(\mathbf{w}^*)\leq \frac{1}{2\eta\varphi T} + \sqrt{\frac{1}{4\eta^2\varphi^2 T^2} + \frac{\rho h(\tau)}{\eta\varphi\tau}} + \rho h(\tau). \label{eq:convergenceUpperBoundFinal}
\end{equation}
\end{theorem}
\begin{proof}
Condition 1 in Lemma~\ref{lemma:convergenceUpperBound} is always satisfied due to the condition $\eta \leq \frac{1}{\beta}$ in this theorem.
When $\rho h(\tau) = 0$, we can choose $\varepsilon$ to be arbitrarily small (but greater than zero) so that conditions 2--4 in Lemma~\ref{lemma:convergenceUpperBound} are satisfied.
We see that the right-hand sides of (\ref{eq:convergenceUpperBound}) and (\ref{eq:convergenceUpperBoundFinal}) are equal in this case (when $\rho h(\tau) = 0$), and the result in (\ref{eq:convergenceUpperBoundFinal}) follows directly from Lemma~\ref{lemma:convergenceUpperBound} because $F(\mathbf{w}^\mathrm{f}) -F(\mathbf{w}^*)\leq F(\mathbf{w}(T))-F(\mathbf{w}^*)$ according to the definition of $\mathbf{w}^\mathrm{f}$ in (\ref{eq:wminDef}).
We consider $\rho h(\tau) > 0$ in the following.
Consider the right-hand side of (\ref{eq:convergenceUpperBound}) and let
\begin{equation}
\varepsilon_0 = \frac{1}{T\left(\eta\varphi -\frac{\rho h(\tau)}{\tau\varepsilon_0^2}\right)}.
\label{eq:convergenceUpperBoundFinalProof1}
\end{equation}
Solving for $\varepsilon_0$, we obtain
\begin{equation}
\varepsilon_0 = \frac{1}{2\eta\varphi T} + \sqrt{\frac{1}{4\eta^2\varphi^2 T^2} + \frac{\rho h(\tau)}{\eta\varphi\tau}}
\label{eq:convergenceUpperBoundFinalProof2}
\end{equation}
where the negative solution is ignored because $\varepsilon > 0$ in Lemma~\ref{lemma:convergenceUpperBound}.
Because $\varepsilon_0 > 0$ according to (\ref{eq:convergenceUpperBoundFinalProof2}), the denominator of (\ref{eq:convergenceUpperBoundFinalProof1}) is greater than zero, thus condition 2 in Lemma~\ref{lemma:convergenceUpperBound} is satisfied for any $\varepsilon \geq \varepsilon_0$, where we note that $\eta\varphi -\frac{\rho h(\tau)}{\tau\varepsilon^2}$ increases with $\varepsilon$ when $\rho h(\tau) > 0$.
Suppose that there exists $\varepsilon > \varepsilon_0$ satisfying conditions 3 and 4 in Lemma~\ref{lemma:convergenceUpperBound}, so that all the conditions in Lemma~\ref{lemma:convergenceUpperBound} are satisfied. Applying Lemma~\ref{lemma:convergenceUpperBound} and considering (\ref{eq:convergenceUpperBoundFinalProof1}), we have
\begin{equation*}
F(\mathbf{w}(T))-F(\mathbf{w}^*) \leq \frac{1}{T\!\left(\eta\varphi \!-\!\frac{\rho h(\tau)}{\tau\varepsilon^2}\right)} < \frac{1}{T\!\left(\eta\varphi \! -\!\frac{\rho h(\tau)}{\tau\varepsilon_0^2}\right)} = \varepsilon_0
\end{equation*}
which contradicts with condition 4 in Lemma~\ref{lemma:convergenceUpperBound}.
Therefore, there \emph{does not} exist $\varepsilon > \varepsilon_0$ that satisfy both conditions 3 and 4 in Lemma~\ref{lemma:convergenceUpperBound}. This means that either
1) $\exists k$ such that $ F\left(\mathbf{v}_{[k]}(k\tau)\right)-F(\mathbf{w}^*) \leq \varepsilon_0$ or 2) $ F\left(\mathbf{w}(T)\right)-F(\mathbf{w}^*) \leq \varepsilon_0$.
It follows that
\begin{equation}
\min\left\{\min_{k=1,2,...,K} F\left(\mathbf{v}_{[k]}(k\tau)\right); F\left(\mathbf{w}(T)\right)\right\} -F(\mathbf{w}^*) \leq \varepsilon_0.
\label{eq:convergenceUpperBoundFinalProof3}
\end{equation}
From Theorem~\ref{theorem:wBound}, $F(\mathbf{w}{(k\tau)}) \leq F(\mathbf{v}_{[k]}(k\tau)) + \rho h(\tau)$ for any $k$. Combining with (\ref{eq:convergenceUpperBoundFinalProof3}), we get
\begin{equation*}
\min_{k=1,2,...,K} F\left(\mathbf{w}{(k\tau)}\right) -F(\mathbf{w}^*) \leq \varepsilon_0 + \rho h(\tau)
\end{equation*}
where we recall that $T=K\tau$.
Using (\ref{eq:wminDef}) and (\ref{eq:convergenceUpperBoundFinalProof2}), we obtain the result in (\ref{eq:convergenceUpperBoundFinal}).
\end{proof}
We note that the bound in (\ref{eq:convergenceUpperBoundFinal}) has no restriction on how the data is distributed at different nodes. The impact of different data distribution is captured by the gradient divergence $\delta$, which is included in $h(\tau)$.
It is easy to see from (\ref{eq:hFuncDef}) that $h(\tau)$ is non-negative, non-decreasing in $\tau$, and proportional to $\delta$. Thus, as one would intuitively expect, for a given total number of local update steps $T$, the optimality gap (i.e., $F(\mathbf{w}^\mathrm{f}) -F(\mathbf{w}^*)$) becomes larger when $\tau$ and $\delta$ are larger. For given $\tau$ and $\delta$, the optimality gap becomes smaller when $T$ is larger.
When $\tau=1$, we have $h(\tau)=0$, and the optimality gap converges to zero as $T \rightarrow \infty$. When $\tau > 1$, we have $h(\tau) > 0$, and we can see from (\ref{eq:convergenceUpperBoundFinal}) that in this case, convergence is only guaranteed to a non-zero optimality gap as $T\rightarrow \infty$. This means that when we have unlimited budget for all types of resources (i.e., $R_m \rightarrow \infty, \forall m$), it is always optimal to set $\tau=1$ and perform global aggregation after every step of local update. However, when the resource budget $R_m$ is limited for some $m$, the training will be terminated after a finite number of iterations, thus the value of $T$ is finite. In this case, it may be better to perform global aggregation less frequently so that more resources can be used for local update, as we will see later in this paper.
\section{Control Algorithm}
\label{sec:schedulingAlgorithm}
We propose an algorithm that approximately solves (\ref{eq:optimizationProblem}) in this section. We first assume that the resource consumptions $c_m$ and $b_m$ ($\forall m$) are known, and we solve for the values of $\tau$ and $T$. Then, we consider practical scenarios where $c_m$, $b_m$, and some other parameters are unknown and may vary over time, and we propose a control algorithm that estimates the parameters and dynamically adjusts the value of $\tau$ in real time.
\subsection{Approximate Solution to (\ref{eq:optimizationProblem})}
\label{subsec:approxSolutionControlAlg}
We assume that $\eta$ is chosen small enough such that $\eta \leq \frac{1}{\beta}$, and use the upper bound in (\ref{eq:convergenceUpperBoundFinal}) as an approximation of $F(\mathbf{w}^\mathrm{f})-F(\mathbf{w}^*)$. Because for a given global loss function $F(\mathbf{w})$, its minimum value $F(\mathbf{w}^*)$ is a constant, the minimization of $F(\mathbf{w}^\mathrm{f})$ in (\ref{eq:optimizationProblem}) is equivalent to minimizing $F(\mathbf{w}^\mathrm{f})-F(\mathbf{w}^*)$.
With this approximation and rearranging the inequality constraints in (\ref{eq:optimizationProblem}), we can rewrite (\ref{eq:optimizationProblem}) as
\begin{align}
\min_{\tau, K \in \{1,2,3,...\}} & \,\, \frac{1}{2\eta\varphi T} + \sqrt{\frac{1}{4\eta^2\varphi^2 T^2} + \frac{\rho h(\tau)}{\eta\varphi\tau}} + \rho h(\tau) \label{eq:optimizationProblemApprox} \\
\textrm{s.t.} & \,\, K \leq \frac{R'_m }{c_m\tau + b_m}, \quad \forall m \in \{1,...,M\} \nonumber \\
& \,\, T = K\tau \nonumber
\end{align}
where $R'_m \triangleq R_m-b_m-c_m$.
It is easy to see that the objective function in (\ref{eq:optimizationProblemApprox}) decreases with $T$, thus it also decreases with $K$ because $T = K\tau$. Therefore, for any $\tau$, the optimal value of $K$ is $\left\lfloor \min_m \frac{R'_m }{c_m\tau + b_m} \right\rfloor$, i.e., the largest value of $K$ that does not violate any inequality constraint in (\ref{eq:optimizationProblemApprox}), where $\lfloor\cdot\rfloor$ denotes the floor function for rounding down to integer. To simplify the analysis, we approximate by ignoring the rounding operation and substituting $T = K\tau \approx \min_m \frac{R'_m \tau}{c_m\tau + b_m} = 1 \big/ \max_m \frac{c_m\tau + b_m}{R'_m \tau} $ into the objective function in (\ref{eq:optimizationProblemApprox}), yielding
\begin{equation}
G(\tau) \triangleq \frac{\max_m \!\frac{c_m\tau + b_m}{R'_m \tau}}{2\eta\varphi } + \sqrt{\!\frac{\left(\!\max_m \!\frac{c_m\tau + b_m}{R'_m \tau}\!\right)^{\!\!2}\!}{4\eta^2\varphi^2 }\! + \!\frac{\rho h(\!\tau\!)}{\eta\varphi\tau}} + \rho h(\!\tau\!)
\label{eq:GTauDef}
\end{equation}
and we can define the (approximately) optimal $\tau$ as
\begin{equation}
\tau^* = \argmin_{\tau \in \{1,2,3,...\}} G(\tau)
\label{eq:optimizationGTau}
\end{equation}
from which we can directly obtain the (approximately) optimal $K$ as $K^* =\left\lfloor \min_m\frac{R'_m}{c_m \tau^* + b_m} \right\rfloor$, and the (approximately) optimal $T$ as $T^* = K^* \tau^* =\left\lfloor \min_m \frac{R'_m}{c_m\tau^* + b_m} \right\rfloor \tau^*$.
\begin{proposition}
When $\eta \leq \frac{1}{\beta}$, $\rho > 0$, $\beta > 0$, $\delta > 0$, we have $\lim_{R_{\textnormal{min}} \rightarrow \infty} \tau^* = 1$, where $R_{\textnormal{min}} \triangleq \min_m R_m$.
\label{prop:TauOptOneInfR}
\end{proposition}
\begin{proof}
Because $R_{\textnormal{min}} \rightarrow \infty \iff R_{m} \rightarrow \infty, \forall m \iff R'_{m} \rightarrow \infty, \forall m$, we have $\lim_{R_{\textnormal{min}} \rightarrow \infty} \max_m \frac{c_m\tau + b_m}{R'_m \tau} = \max_m \lim_{R'_{m} \rightarrow \infty} \frac{c_m\tau + b_m}{R'_m \tau} = 0$. Thus,
$\lim_{R_{\textnormal{min}} \rightarrow \infty} G(\tau) = \sqrt{\frac{\rho h(\tau)}{\eta\varphi\tau}} + \rho h(\tau)$.
Let $B \triangleq \eta\beta + 1$. With a slight abuse of notation, we consider continuous values of $\tau \geq 1$. We have
\begin{align*}
d \left(\frac{ h(\tau)}{\tau}\right) \bigg/ d\tau & = \frac{\delta}{\beta\tau^2}\left( B^\tau \log B^\tau - (B^\tau - 1) \right) \\
& \geq \frac{\delta}{\beta\tau^2}\left( B^\tau \left( 1 - \frac{1}{B^\tau} \right) - B^\tau - 1 \right) \geq 0
\end{align*}
where the first inequality is from a lower bound of logarithmic function \cite{topsok2006some}. We also have
\begin{align*}
\frac{d h(\tau)}{d\tau} & = \frac{\delta}{\beta} (B^\tau \log B - \eta\beta) \geq \frac{\delta}{\beta} \left(\frac{2\eta\beta B^\tau}{2+\eta\beta} - \eta\beta \right) \\
& = \frac{\delta (2\eta\beta B^\tau - 2\eta\beta - \eta^2\beta^2 )}{\beta (2+\eta\beta )} \\
& \geq \frac{\delta (2\eta\beta B - 2\eta\beta - \eta^2\beta^2 )}{\beta (2+\eta\beta )} = \frac{\delta \eta^2 \beta^2 }{\beta (2+\eta\beta )} > 0
\end{align*}
where the first inequality is from a lower bound of $\log B$ \cite{topsok2006some}, the second inequality is because $B > 1$ and $\tau \geq 1$.
Thus, for any $\tau \geq 1$, $h(\tau)$ increases with $\tau$, and $\frac{h(\tau)}{\tau}$ is non-decreasing with $\tau$. We also note that $\sqrt{x}$ increases with $x$ for any $x\geq 0$, and $h(1)=0$. It follows that $\lim_{R_{\textnormal{min}} \rightarrow \infty} G(\tau)$ increases with $\tau$ for any $\tau \geq 1$. Hence, $\lim_{R_{\textnormal{min}} \rightarrow \infty} \tau^* = 1$.
\end{proof}
Combining Proposition~\ref{prop:TauOptOneInfR} with Theorem~\ref{theorem:convergenceUpperBoundFinal}, we know that using $\tau^*$ found from (\ref{eq:optimizationGTau}) guarantees convergence with zero optimality gap as $R_\textnormal{min} \rightarrow \infty$ (and thus $R'_m \rightarrow \infty, \forall m$ and $T^* \rightarrow \infty$), because $\lim_{R_\textnormal{min} \rightarrow \infty} \tau^* = 1$ and $h(1)=0$.
For general values of $R_m$ (and $R'_m$), we have the following result.
\begin{proposition}
When $\eta \! \leq \!\frac{1}{\beta}$, $\rho \!>\! 0$, $\beta \!>\! 0$, $\delta \!>\! 0$, there exists a \emph{finite} value $\tau_0$, which only depends on $\eta$, $\beta$, $\rho$, $\delta$, $\varphi$, $c_m$, $b_m$, $R'_m$ ($\forall m$), such that $\tau^* \leq \tau_0$. The quantity $\tau_0$ is defined as
\begin{align*}
\tau_0 \triangleq \max \Bigg\{ & \max_m \frac{b_m R'_\nu - b_\nu R'_m}{c_\nu R'_m -c_m R'_\nu} \, ; \,\, \frac{\varphi(2 \!+\! \eta\beta)}{2 \rho\delta}\! \left( \frac{2c_\nu b_\nu}{C_2} \!+\! \frac{2b_\nu^2}{C_2} \right) \!; \\
& \frac{1}{\rho\delta\eta \log B} \left(\!\frac{b_\nu}{C_1} \! + \!\rho\eta\delta\!\right) \! -\! \frac{1}{\eta\beta} ; \,\,\,\, \frac{1}{\eta\beta} + \frac{1}{2} \Bigg\}
\end{align*}
where index $\nu \triangleq \arg\max_{m\in V} \frac{b_m}{R'_m}$ (set $V \triangleq \arg\max_m \frac{c_m}{R'_m}$), $B \triangleq \eta\beta + 1$, $C_1 \triangleq 2\eta\varphi R'_\nu$, $C_2 \triangleq 4\eta^2\varphi^2 R'^2_\nu$. Here, for convenience, we allow $\arg\max$ to interchangeably return a set and an arbitrary value in that set, we also define $\frac{0}{0} \triangleq 0$.
We also note that $0 < \eta\beta \leq 1$, thus $\tau_0 \geq \frac{1}{\eta\beta} + \frac{1}{2} > 1$.
\label{prop:TauOptBounded}
\end{proposition}
\begin{proof}
We can show that $\max_m \frac{b_m R'_\nu - b_\nu R'_m}{c_\nu R'_m -c_m R'_\nu}$ is finite according to the definition of $\nu$ and $\frac{0}{0}$, then it is easy to see that $\tau_0$ is finite. We then show $\arg\max_m \frac{c_m\tau + b_m}{R'_m \tau} = \nu$ for any $\tau > \tau_0$, in which case the maximization over $m$ in (\ref{eq:GTauDef}) becomes fixing $m=\nu$. Then, the proof separately considers the terms inside and outside the square root in (\ref{eq:GTauDef}). It shows that the first order derivatives of both parts are always larger than zero when $\tau > \tau_0$. Because the square root is an increasing function, $G(\tau)$ increases with $\tau$ for $\tau > \tau_0$, and thus $\tau^* \leq \tau_0$.
\if\citetechreport1
See our online technical report \cite[Appendix~\ref{append:proofGTauOptBounded}]{JournalTechReport} for details.
\else
See Appendix~\ref{append:proofGTauOptBounded} for details.
\fi
\end{proof}
There is no closed-form solution for $\tau^*$ because $G(\tau)$ includes both polynomial and exponential terms of $\tau$, where the exponential term is embedded in $h(\tau)$. Because $\tau^*$ can only be a positive integer, according to Proposition~\ref{prop:TauOptBounded}, we can compute $G(\tau)$ within a finite range of $\tau$ to find $\tau^*$ that minimizes $G(\tau)$.
\subsection{Adaptive Federated Learning}
\label{subsec:controlAlgImpl}
In this subsection, we present the complete control algorithm for adaptive federated learning, which recomputes $\tau^*$ in every global aggregation step based on the most recent system state.
We use the theoretical results above to guide the design of the algorithm.
As mentioned earlier, the local updates run on edge nodes and the global aggregation is performed through the assistance of an aggregator, where the aggregator is a logical component that may also run on one of the edge nodes. The complete procedures at the aggregator and each edge node are presented in Algorithms~\ref{alg:protocolAggregator} and \ref{alg:protocolEdgeNode}, respectively, where Lines~\ref{alg:protocolEdgeNode:startLocalUpdt}--\ref{alg:protocolEdgeNode:endLocalUpdt} of Algorithm~\ref{alg:protocolEdgeNode} are for local updates and the rest is considered as part of global aggregation, initialization, or final operation. We assume that the aggregator initiates the learning process, and the initial model parameter $\mathbf{w}(0)$ is sent by the aggregator to all edge nodes.
We note that instead of transmitting the entire model parameter vector in every global aggregation step, one can also transmit compressed or quantized model parameters to further save the communication bandwidth, where the compression or quantization can be performed using techniques described in~\cite{Jakub2016,hardy2017distributed}, for instance.
\subsubsection{Estimation of Parameters in $G(\tau)$}
\label{subsec:ControlAlgParamEst}
The expression of $G(\tau)$, which includes $h(\tau)$, has parameters which need to be estimated in practice.
Among these parameters, $c_m$ and $b_m$ ($\forall m$) are related to resource consumption, $\rho$, $\beta$, and $\delta$ are related to the loss function characteristics. These parameters are estimated in real time during the learning process.
The values of $c_m$ and $b_m$ ($\forall m$) are estimated based on measurements of resource consumptions at the edge nodes and the aggregator (Line~\ref{alg:protocolAggregator:resourceConsumptionEst} of Algorithm~\ref{alg:protocolAggregator}). The estimation depends on the type of resource under consideration. For example, when the type-$m$ resource is energy, the sum energy consumption (per local update) at all nodes is considered as $c_m$; when the type-$m$ resource is time, the maximum computation time (per local update) at all nodes is considered as $c_m$.
The aggregator also monitors the total resource consumption of each resource type $m$ based on the estimates, and compares the total resource consumption against the resource budget $R_m$ (Line~\ref{alg:protocolAggregator:checkRemainingResource} of Algorithm~\ref{alg:protocolAggregator}). If the consumed resource is at the budget limit for some $m$, it stops the learning and returns the final result.
The values of $\rho$, $\beta$, and $\delta$ are estimated based on the local and global losses and gradients computed at $\mathbf{w}(t)$ and $\mathbf{w}_i(t)$, see Line~\ref{alg:protocolAggregator:receiveBetaGrad} and Lines~\ref{alg:protocolAggregator:firstEstimation}--\ref{alg:protocolAggregator:estDelta} of Algorithm~\ref{alg:protocolAggregator} and Lines~\ref{alg:protocolEdgeNode:RhoEstimate}, \ref{alg:protocolEdgeNode:BetaEstimate}, and \ref{alg:protocolEdgeNode:sendBetaAndGrad} of Algorithm~\ref{alg:protocolEdgeNode}. To perform the estimation, each edge node needs to have access to both its local model parameter $\mathbf{w}_i(t)$ and the global model parameter $\mathbf{w}(t)$ for the same iteration $t$ (see Lines~\ref{alg:protocolEdgeNode:RhoEstimate} and \ref{alg:protocolEdgeNode:BetaEstimate} of Algorithm~\ref{alg:protocolEdgeNode}), which is only possible when global aggregation is performed in iteration~$t$.
Because $\mathbf{w}(t)$ is only observable by each node after global aggregation,
estimated values of $\rho$, $\beta$, and $\delta$ are only available for recomputing $\tau^*$ starting from the second global aggregation step after initialization, which uses estimates obtained in the previous global aggregation step\footnote{See the condition in Line~\ref{alg:protocolAggregator:t0Cond} of Algorithm~\ref{alg:protocolAggregator} and Lines~\ref{alg:protocolEdgeNode:tCond} and \ref{alg:protocolEdgeNode:t0Cond} of Algorithm~\ref{alg:protocolEdgeNode}. Also note that the parameters $\hat{\rho}_i$, $\hat{\beta}_i$, $F_{i}(\mathbf{w}(t_0))$, $\nabla F_{i}(\mathbf{w}(t_0))$ sent in Line~\ref{alg:protocolEdgeNode:sendBetaAndGrad} of Algorithm~\ref{alg:protocolEdgeNode} are obtained at the previous global aggregation step ($t_0$, $\hat{\rho}_i$, and $\hat{\beta}_i$ are obtained in Lines~\ref{alg:protocolEdgeNode:t0Def}--\ref{alg:protocolEdgeNode:BetaEstimate} of Algorithm~\ref{alg:protocolEdgeNode}).}.
\begin{algorithm}[t]
\caption{Procedure at the aggregator}
\label{alg:protocolAggregator}
{\footnotesize
\KwIn{Resource budget $R$, control parameter $\varphi$, search range parameter $\gamma$, maximum $\tau$ value $\tau_\textrm{max}$}
\KwOut{$\mathbf{w}^\mathrm{f}$}
Initialize $\tau^* \leftarrow 1$, $t \leftarrow 0$, $s \leftarrow 0$; \quad //$s$ is a resource counter
Initialize $\mathbf{w}(0)$ as a constant or a random vector;
Initialize $\mathbf{w}^\mathrm{f} \leftarrow \mathbf{w}(0)$;
\RepeatNoEnd
{
Send $\mathbf{w}(t)$ and $\tau^*$ to all edge nodes, also send \texttt{STOP} if it is set; \label{alg:protocolAggregator:sendWTau}
$t_0 \leftarrow t$; \quad //Save iteration index of last transmission of $\mathbf{w}(t)$
$t \leftarrow t+\tau^*$; \quad //Next global aggregation is after $\tau$ iterations
Receive $\mathbf{w}_i(t)$, $\hat{c}_i$ from each node $i$;
Compute $\mathbf{w}(t)$ according to (\ref{eq:globalAverage}); \label{alg:protocolAggregator:computeWGlobal}
\If{$t_0 > 0$ \label{alg:protocolAggregator:t0Cond} }
{
Receive $\hat{\rho}_i$, $\hat{\beta}_i$, $F_{i}(\mathbf{w}(t_0))$, $\nabla F_{i}(\mathbf{w}(t_0))$ from each node $i$; \label{alg:protocolAggregator:receiveBetaGrad}
Compute $F(\mathbf{w}(t_0))$ according to (\ref{eq:globalLossFuncAllSamples})
\If{$F(\mathbf{w}(t_0)) < F(\mathbf{w}^\mathrm{f})$ \label{alg:protocolAggregator:minStepStart}}
{
$\mathbf{w}^\mathrm{f} \leftarrow \mathbf{w}(t_0)$; \label{alg:protocolAggregator:minStepEnd}
}
\If{\texttt{STOP} flag is set}
{
\textbf{break}; \quad //Break out of the loop here if \texttt{STOP} is set
}
Estimate $\hat{\rho} \leftarrow \frac{\sum_{i=1}^{N}D_{i}\hat{\rho}_{i}}{D}$; \label{alg:protocolAggregator:firstEstimation}
Estimate $\hat{\beta} \leftarrow \frac{\sum_{i=1}^{N}D_{i}\hat{\beta}_{i}}{D}$;
Compute $\nabla F(\mathbf{w}(t_0)) \leftarrow \frac{\sum_{i=1}^{N}D_{i}\nabla F_{i}(\mathbf{w}(t_0))}{D}$,
estimate $\hat{\delta}_i \leftarrow \Vert \nabla F_{i}(\mathbf{w}(t_0)) - \nabla F(\mathbf{w}(t_0)) \Vert$ for each $i$,
from which we estimate $\hat{\delta} \leftarrow \frac{\sum_{i=1}^{N}D_{i}\hat{\delta}_{i}}{D}$; \label{alg:protocolAggregator:estDelta}
Compute new value of $\tau^*$ according to (\ref{eq:optimizationGTau}) via linear search on integer values of $\tau$ within $[1, \tau_\textrm{m}]$, where we set $\tau_\textrm{m} \leftarrow \min\{\gamma\tau^*; \tau_\textrm{max}\}$; \label{alg:protocolAggregator:TauCompute}
}
\For{$m = 1,2,...,M$}
{
Estimate resource consumptions $\hat{c}_m$, $\hat{b}_m$, using $\hat{c}_{m,i}$ received from all nodes $i$ and local measurements at the aggregator; \label{alg:protocolAggregator:resourceConsumptionEst}
$s_m \leftarrow s_m + \hat{c}_m\tau + \hat{b}_m$;
}
\If{$\exists m$ such that $s_m + \hat{c}_m(\tau + 1) + 2\hat{b}_m \geq R_m$ \label{alg:protocolAggregator:checkRemainingResource} }
{
Decrease $\tau^*$ to the maximum possible value such that the estimated resource consumption for remaining iterations is within budget $R_m$ for all $m$, set \texttt{STOP} flag; \label{alg:protocolAggregator:decreaseTauToRemainWithinBudget}
}
}
Send $\mathbf{w}(t)$ to all edge nodes; \label{alg:protocolAggregator:FinalStart}
Receive $F_{i}(\mathbf{w}(t))$ from each node $i$;
Compute $F(\mathbf{w}(t))$ according to (\ref{eq:globalLossFuncAllSamples})
\If{$F(\mathbf{w}(t)) < F(\mathbf{w}^\mathrm{f})$ \label{alg:protocolAggregator:FinalLossCompare}}
{
$\mathbf{w}^\mathrm{f} \leftarrow \mathbf{w}(t)$; \label{alg:protocolAggregator:FinalEnd}
}
}
\end{algorithm}
\begin{algorithm}[t]
\caption{Procedure at each edge node $i$}
\label{alg:protocolEdgeNode}
{\footnotesize
Initialize $t \leftarrow 0$;
\Repeat{\texttt{STOP} flag is received}{
Receive $\mathbf{w}(t)$ and new $\tau^*$ from aggregator, set $\widetilde{\mathbf{w}}_i(t) \leftarrow \mathbf{w}(t)$; \label{alg:protocolEdgeNode:receiveWTauSetTilde}
$t_0 \leftarrow t$; \quad //Save iteration index of last transmission of $\mathbf{w}(t)$ \label{alg:protocolEdgeNode:t0Def}
\If{$t>0$ \label{alg:protocolEdgeNode:tCond} }
{
Estimate $\hat{\rho}_i \leftarrow \left\Vert F_{i}(\mathbf{w}_i(t)) - F_{i}(\mathbf{w}(t)) \right\Vert / \left\Vert \mathbf{w}_i(t) - \mathbf{w}(t) \right\Vert$; \label{alg:protocolEdgeNode:RhoEstimate}
Estimate $\hat{\beta}_i \leftarrow \left\Vert \nabla F_{i}(\mathbf{w}_i(t)) - \nabla F_{i}(\mathbf{w}(t)) \right\Vert / \left\Vert \mathbf{w}_i(t) - \mathbf{w}(t) \right\Vert$; \label{alg:protocolEdgeNode:BetaEstimate}
}
\For{$\mu = 1,2,..., \tau^*$ \label{alg:protocolEdgeNode:startLocalUpdt} }
{
$t \leftarrow t+1$; \quad //Start of next iteration
Perform local update and obtain $\mathbf{w}_i(t)$ according to (\ref{eq:localUpdate}); \label{alg:protocolEdgeNode:LocalUpdateGradDescent}
\If{$\mu < \tau^*$}
{
Set $\widetilde{\mathbf{w}}_i(t) \leftarrow \mathbf{w}_i(t)$; \label{alg:protocolEdgeNode:endLocalUpdt} \label{alg:protocolEdgeNode:LocalUpdateTildeSet}
}
}
\For{$m = 1,2,...,M$}
{
Estimate type-$m$ resource consumption $\hat{c}_{m,i}$ for one local update at node $i$;
}
Send $\mathbf{w}_i(t)$, $\hat{c}_{m,i}$ ($\forall m$) to aggregator;
\If{$t_0 > 0$ \label{alg:protocolEdgeNode:t0Cond} }
{
Send $\hat{\rho}_i$, $\hat{\beta}_i$, $F_{i}(\mathbf{w}(t_0))$, $\nabla F_{i}(\mathbf{w}(t_0))$ to aggregator; \label{alg:protocolEdgeNode:sendBetaAndGrad}
}
}
Receive $\mathbf{w}(t)$ from aggregator; \label{alg:protocolEdgeNode:FinalStart}
Send $F_{i}(\mathbf{w}(t))$ to aggregator; \label{alg:protocolEdgeNode:FinalEnd}
}
\end{algorithm}
\emph{Remark:} In the extreme case where $\mathbf{w}_i (t) = \mathbf{w}(t)$ in Lines~\ref{alg:protocolEdgeNode:RhoEstimate} and \ref{alg:protocolEdgeNode:BetaEstimate} of Algorithm~\ref{alg:protocolEdgeNode}, we estimate $\hat{\rho}_i$ and $\hat{\beta}_i$ as zero. When $\delta = \beta = 0$ and $\frac{\delta}{\beta}$ in $h(\tau)$ is undefined, we define that $h(\tau) = 0$ for all $\tau \geq 1$. This is because for $t>0$, $\mathbf{w}_i (t) = \mathbf{w}(t)$ only occurs when different nodes have extremely similar (often equal) datasets, in which case a large value of $\tau$ does not make the convergence worse than a small value of $\tau$, thus it makes sense to define $h(\tau) = 0$ in this case.
The parameter $\eta$ is the gradient-descent step size which is pre-specified and known. The remaining parameter $\varphi$ includes $\omega$ which is non-straightforward to estimate because the algorithm does not know $\mathbf{w}^*$, thus we regard $\varphi$ as a control parameter that is manually chosen and remains fixed for the same machine learning model\footnote{Although $\varphi$ is related to $\beta$ and we estimate $\beta$ separately, we found that it is good to keep $\varphi$ a constant value that does not vary with the estimated value of $\beta$ in practice, because there can be occasions where the estimated $\beta$ is large causing $\varphi < 0$, which causes abnormal behavior when computing $\tau^*$ from $G(\tau)$.}.
Experimentation results presented in the next section show that a fixed value of $\varphi$ works well across different data distributions, various numbers of nodes, and various resource consumptions/budgets.
If we multiply both sides of (\ref{eq:GTauDef}) by $\varphi$, we can see that a larger value of $\varphi$ gives a higher weight to the terms with $h(\tau)$, yielding a smaller value of $\tau^*$ (because $h(\tau)$ increases with $\tau$), and vice versa. Therefore, in practice, it is not hard to tune the value of $\varphi$ on a small and simple setup, which can then be applied to general cases. See also the results on the sensitivity of $\varphi$ in Section~\ref{subsubsec:sensitivityOfPhiResult}.
\subsubsection{Recomputing $\tau^*$}
The value of $\tau^*$ is recomputed by the aggregator during each global aggregation step, based on the most updated parameter estimations.
When searching for $\tau^*$, we use the following search range instead of the range in Proposition~\ref{prop:TauOptBounded} due to practical considerations of estimation error.
As shown in Line~\ref{alg:protocolAggregator:TauCompute} of Algorithm~\ref{alg:protocolAggregator}, we search for new values of $\tau^*$ up to $\gamma$ times the current value of $\tau^*$, and find $\tau^*$ that minimizes $G(\tau)$, where $\gamma > 0$ is a fixed parameter. The presence of $\gamma$ limits the search space and also avoids $\tau^*$ from growing too quickly as initial parameter estimates may be inaccurate. We also impose a maximum value of $\tau$, denoted by $\tau_\textrm{max}$, because if $\tau^*$ is too large, it is more likely for the system to operate beyond the resource budget due to inaccuracies in the estimation of local resource consumption, see Line~\ref{alg:protocolAggregator:checkRemainingResource} of Algorithm~\ref{alg:protocolAggregator}.
The new value of $\tau^*$ is sent to each node together with $\mathbf{w}(t)$ (Line~\ref{alg:protocolAggregator:sendWTau} of Algorithm~\ref{alg:protocolAggregator}).
\subsubsection{Distributed Gradient Descent}
The local update steps of distributed gradient descent at the edge node include Lines~\ref{alg:protocolEdgeNode:startLocalUpdt}--\ref{alg:protocolEdgeNode:endLocalUpdt} of Algorithm~\ref{alg:protocolEdgeNode}, where Line~\ref{alg:protocolEdgeNode:LocalUpdateGradDescent} of Algorithm~\ref{alg:protocolEdgeNode} corresponds to Line~\ref{alg:distGradDescent:LocalUpdate} of Algorithm~\ref{alg:distGradDescent} and Line~\ref{alg:protocolEdgeNode:LocalUpdateTildeSet} of Algorithm~\ref{alg:protocolEdgeNode} corresponds to Line~\ref{alg:distGradDescent:TildeSet} of Algorithm~\ref{alg:distGradDescent}. When global aggregation is performed, Line~\ref{alg:protocolAggregator:computeWGlobal} of Algorithm~\ref{alg:protocolAggregator} computes the global model parameter $\mathbf{w}(t)$ at the aggregator, which is sent to the edge nodes in Line~\ref{alg:protocolAggregator:sendWTau} of Algorithm~\ref{alg:protocolAggregator}, and each edge node receives $\mathbf{w}(t)$ in Line~\ref{alg:protocolEdgeNode:receiveWTauSetTilde} of Algorithm~\ref{alg:protocolEdgeNode} and sets $\widetilde{\mathbf{w}}_i(t) \leftarrow \mathbf{w}(t)$ to use $\mathbf{w}(t)$ as the initial model parameter for the next round of local update; this corresponds to Line~\ref{alg:distGradDescent:TildeSetGlobalAgg} of Algorithm~\ref{alg:distGradDescent}.
The final model parameter $\mathbf{w}^\mathrm{f}$ that minimizes $F(\mathbf{w})$ is obtained at the aggregator in Lines~\ref{alg:protocolAggregator:minStepStart}--\ref{alg:protocolAggregator:minStepEnd} of Algorithm~\ref{alg:protocolAggregator}, corresponding to Line~\ref{alg:distGradDescent:minStep} of Algorithm~\ref{alg:distGradDescent}.
As discussed in Section~\ref{sec:ProblemFormulation}, the computation of $\mathbf{w}^\mathrm{f}$ lags for one round of global aggregation, because for any iteration $t_0$ that includes a global aggregation step, $F(\mathbf{w}(t_0))$ can only be computed after each edge node has received $\mathbf{w}(t_0)$ and sent the local loss $F_i(\mathbf{w}(t_0))$ to the aggregator in the next round of global aggregation. To take into account the final value of $\mathbf{w}(t)$ in the computation of $\mathbf{w}^\mathrm{f}$, Lines~\ref{alg:protocolAggregator:FinalStart}--\ref{alg:protocolAggregator:FinalEnd} of Algorithm~\ref{alg:protocolAggregator} and Lines~\ref{alg:protocolEdgeNode:FinalStart}--\ref{alg:protocolEdgeNode:FinalEnd} of Algorithm~\ref{alg:protocolEdgeNode} perform an additional round of computation of the loss and $\mathbf{w}^\mathrm{f}$, as also discussed in Section~\ref{sec:ProblemFormulation}.
Overall, when global aggregation is executed for $K$ times in total, the computational complexity of Algorithm~\ref{alg:protocolAggregator} is $O(K(NM+\tau_\textrm{max}))$, because each global aggregation step includes the computation of global parameters from the local parameters collected from $N$ different nodes for $M$ resource types and the linear search step in Line~\ref{alg:protocolAggregator:TauCompute} of Algorithm~\ref{alg:protocolAggregator} which has at most $\tau_\textrm{max}$ steps. When $T$ steps of local updates are performed in total, Algorithm~\ref{alg:protocolEdgeNode} has a computational complexity of $O(T+KM)$, where the additional term $KM$ corresponds to the additional local processing (at each node) in global aggregation steps.
\subsection{Extension to Stochastic Gradient Descent}
When the amount of training data is large, it is usually computationally prohibitive to compute the gradient of the loss function defined on the entire (local) dataset. In such cases, stochastic gradient descent (SGD) is often used \cite{shalev2014understanding,Goodfellow-et-al-2016,bottou2010large}, which uses the gradient computed on the loss function defined on a randomly sampled subset (referred to as a mini-batch) of data to approximate the real gradient. Although the theoretical analysis in this paper is based on deterministic gradient descent (DGD), the proposed approach can be directly extended to SGD. As discussed in \cite{FUSION2018}, SGD can be seen as an approximation to DGD.
When using SGD with our proposed algorithm, all losses and their gradients are computed on mini-batches. Each local iteration step corresponds to a step of gradient descent where the gradient is computed on a mini-batch of local training data. The mini-batch changes for every step of local iteration, i.e., for each new local iteration, a new mini-batch of a given size is randomly selected from the local training data. However, to reduce errors introduced by random data sampling when estimating the parameters $\rho$, $\beta$, and $\delta$, the first iteration after global aggregation uses the same mini-batch as the last iteration before global aggregation. When $\tau=1$, the mini-batch changes if the same mini-batch has already been used in two iterations, to ensure that different mini-batches are used for training over time.
To avoid approximation errors caused by mini-batch sampling when determining $\mathbf{w}^\mathrm{f}$, when using SGD, the aggregator informs the edge nodes whether the current $\mathbf{w}(t_0)$ is selected as $\mathbf{w}^\mathrm{f}$ using an additional flag sent together with the message in Line~\ref{alg:protocolAggregator:sendWTau} of Algorithm~\ref{alg:protocolAggregator}. The edge nodes save their own copies of $\mathbf{w}^\mathrm{f}$. When an edge node computes $F_i(\mathbf{w}(t_0))$ that is sent in Line~\ref{alg:protocolEdgeNode:sendBetaAndGrad} of Algorithm~\ref{alg:protocolEdgeNode}, it also recomputes $F_i(\mathbf{w}^\mathrm{f})$ using the same mini-batch as for computing $F_i(\mathbf{w}(t_0))$. It then sends both $F_i(\mathbf{w}^\mathrm{f})$ and $F_i(\mathbf{w}(t_0))$ to the aggregator in Line~\ref{alg:protocolEdgeNode:sendBetaAndGrad} of Algorithm~\ref{alg:protocolEdgeNode}. The aggregator recomputes $F(\mathbf{w}^\mathrm{f})$ based on the most recently received $F_i(\mathbf{w}^\mathrm{f})$. In this way, the values of $F(\mathbf{w}^\mathrm{f})$ and $F(\mathbf{w}(t_0))$ used for the comparison in Lines~\ref{alg:protocolAggregator:minStepStart} and \ref{alg:protocolAggregator:FinalLossCompare} of Algorithm~\ref{alg:protocolAggregator} are computed on the same mini-batch at each edge node.
\section{Experimentation Results}
\label{sec:experimentation}
\subsection{Setup}
To evaluate the performance of our proposed adaptive federated learning algorithm, we conducted experiments both on networked prototype system with $5$ nodes and in a simulated environment with the number of nodes varying from $5$ to $500$.
The prototype system consists of three Raspberry Pi (version 3) devices and two laptop computers, which are all interconnected via Wi-Fi in an office building.
This represents an edge computing environment where the computational capabilities of edge nodes are heterogeneous. All these $5$ nodes have local datasets on which model training is conducted. The aggregator is located on one of the laptop computers, and hence co-located with one of the local datasets.
\subsubsection{Resource Definition}
\label{subsec:experimentation:resource}
For ease of presentation and interpretation of results, we let $M=1$ and consider time as the single resource type in our experiments.
For the prototype system, we train each model for a fixed amount of time budget. The values of $c$ and $b$ (we omit the subscript $m=1$ for simplicity) correspond to the actual time used for each local update and global aggregation, respectively.
The simulation environment performs model training with simulated resource consumptions, which are randomly generated according to Gaussian distribution with mean and standard deviation values
\if\citetechreport1
(see \cite[Appendix~\ref{append:ExperimentSimParam}]{JournalTechReport} for these values)
\else
(see Appendix~\ref{append:ExperimentSimParam} for these values)
\fi
obtained from measurements of the squared-SVM model on the prototype. See Section~\ref{subsec:experimentation:modelsDatasets} below for definitions of models and datasets.
\subsubsection{Baselines}
\label{subsec:experimentation:baseline}
We compare with the following baseline approaches:
\begin{enumerate}
\item[(a)] Centralized gradient descent \cite{shalev2014understanding, Goodfellow-et-al-2016}, where the entire training dataset is stored on a single edge node and the model is trained directly on that node using a standard (centralized) gradient descent procedure;
\item[(b)] Canonical federated learning approach presented in~\cite{mcmahan2016communication}, which is equivalent to using a fixed (non-adaptive) value of $\tau$ in our setting;
\item[(c)] Synchronous distributed gradient descent \cite{Chen2016}, which is equivalent to fixing $\tau = 1$ in our setting.
\end{enumerate}
For a fair comparison, we implement the estimation of resource consumptions for all baselines and the training stops when we have reached the resource (time) budget.
When conducting experiments on the prototype system, the centralized gradient descent is performed on a Raspberry Pi device. To avoid resource consumption related to loss computation, centralized gradient descent uses the last model parameter $\mathbf{w}(T)$ (instead of $\mathbf{w}^\mathrm{f}$) as the result, because convergence of $\mathbf{w}(T)$ can be proven in the centralized case \cite{convex}.
We do not explicitly distinguish the baselines (b) and (c) above because they both correspond to an approach with non-adaptive $\tau$ of a certain value. When $\tau$ is non-adaptive, we use the same protocol as in Algorithms~\ref{alg:protocolAggregator} and \ref{alg:protocolEdgeNode}, but remove any parts related to parameter estimation and recomputation of $\tau$.
\subsubsection{DGD and SGD}
We consider both DGD and SGD in the experiments to evaluate the general applicability of the proposed algorithm. For SGD, the mini-batch sampling uses the same initial random seed at all nodes, which means that when the datasets at all nodes are identical, the mini-batches at all nodes are also identical in the same iteration (while they are generally different across different iterations). This setup is for a better consideration of the differences between equal and non-equal data distributions (see Section~\ref{subsec:DataDistributionCases} below).
\begin{figure*}
\centering
\begin{subfigure}{0.2\textwidth}
\centering
\includegraphics[width=0.8\linewidth]{Legend_MultiRun1.pdf}
\end{subfigure}%
~
\begin{subfigure}{0.2\textwidth}
\centering
\includegraphics[width=0.8\linewidth]{Legend_MultiRun2.pdf}
\end{subfigure}%
~
\begin{subfigure}{0.2\textwidth}
\centering
\includegraphics[width=0.75\linewidth]{Legend_MultiRun3.pdf}
\end{subfigure}%
~
\begin{subfigure}{0.2\textwidth}
\centering
\includegraphics[width=0.75\linewidth]{Legend_MultiRun4.pdf}
\end{subfigure}%
\vspace{0.05in}
\begin{subfigure}[b]{0.17\textwidth}
\centering
\includegraphics[width=1\linewidth]{MNIST_DGD_MultiRun_Loss.pdf}
\end{subfigure}%
~\hspace{-0.1in}
\begin{subfigure}[b]{0.17\textwidth}
\centering
\includegraphics[width=1\linewidth]{MNIST_DGD_MultiRun_Acc.pdf}
\end{subfigure}%
~\hspace{-0.1in}
\begin{subfigure}[b]{0.17\textwidth}
\centering
\includegraphics[width=1\linewidth]{MNIST_SGD_MultiRun_Loss.pdf}
\end{subfigure}%
~\hspace{-0.1in}
\begin{subfigure}[b]{0.17\textwidth}
\centering
\includegraphics[width=1\linewidth]{MNIST_SGD_MultiRun_Acc.pdf}
\end{subfigure}%
~\hspace{-0.1in}
\begin{subfigure}[b]{0.17\textwidth}
\includegraphics[width=1\linewidth]{LinReg_SGD_MultiRun_Loss.pdf}
\end{subfigure}%
~\hspace{-0.1in}
\begin{subfigure}[b]{0.17\textwidth}
\includegraphics[width=1\linewidth]{KMeans_DGD_MultiRun_Loss.pdf}
\end{subfigure}%
\vspace{0.05in}
\begin{subfigure}[b]{0.17\textwidth}
\centering
\includegraphics[width=1\linewidth]{MNIST_Orig_CNN_MultiRun_Loss.pdf}
\end{subfigure}%
~\hspace{-0.1in}
\begin{subfigure}[b]{0.17\textwidth}
\centering
\includegraphics[width=1\linewidth]{MNIST_Orig_CNN_MultiRun_Acc.pdf}
\end{subfigure}%
~\hspace{-0.1in}
\begin{subfigure}[b]{0.17\textwidth}
\centering
\includegraphics[width=1\linewidth]{MNIST_Fashion_CNN_MultiRun_Loss.pdf}
\end{subfigure}%
~\hspace{-0.1in}
\begin{subfigure}[b]{0.17\textwidth}
\centering
\includegraphics[width=1\linewidth]{MNIST_Fashion_CNN_MultiRun_Acc.pdf}
\end{subfigure}%
~\hspace{-0.1in}
\begin{subfigure}[b]{0.17\textwidth}
\centering
\includegraphics[width=1\linewidth]{CIFAR_CNN_MultiRun_Loss.pdf}
\end{subfigure}%
~\hspace{-0.1in}
\begin{subfigure}[b]{0.17\textwidth}
\centering
\includegraphics[width=1\linewidth]{CIFAR_CNN_MultiRun_Acc.pdf}
\end{subfigure}%
\caption{Loss function values and classification accuracy with different $\tau$. Only SVM and CNN classifiers have accuracy values. The curves show the results from the baseline with different fixed values of $\tau$. Our proposed solution (represented by a single marker for each case) gives an average $\tau$ and loss/accuracy that is close to the optimum in all cases.}
\label{fig:LossAndAccuracyExperimentResults}
\vspace{-0.2in}
\end{figure*}
\subsubsection{Models and Datasets}
\label{subsec:experimentation:modelsDatasets}
We evaluate the training of four different models on five different datasets, which represent a large variety of both small and large models and datasets, as one can expect all these variants to exist in edge computing scenarios.
The models include squared-SVM, linear regression, K-means, and deep convolutional neural networks (CNN)\footnote{The CNN has $9$ layers with the following structure: $5 \times 5 \times 32$ Convolutional $\rightarrow$ $2 \times 2$ MaxPool $\rightarrow$ Local Response Normalization $\rightarrow$ $5 \times 5 \times 32$ Convolutional $\rightarrow$ Local Response Normalization $\rightarrow$ $2 \times 2$ MaxPool $\rightarrow$ $z \times 256$ Fully connected $\rightarrow$ $256 \times 10$ Fully connected $\rightarrow$ Softmax, where $z$ depends on the input image size and $z=1568$ for MNIST-O and MNIST-F and $z=2048$ for CIFAR-10. This configuration is similar to what is suggested in the TensorFlow tutorial \cite{TensorFlowTutorial}.}. See Table~\ref{tab:learningModels} for a summary of the loss functions of these models, and see \cite{shalev2014understanding, Goodfellow-et-al-2016,bottou2010large} for more details. Among them, the loss functions for squared-SVM (which we refer to as SVM in short in the following) and linear regression satisfy Assumption~\ref{assumption:Convex}, whereas the loss functions for K-means and CNN are non-convex and thus do not satisfy Assumption~\ref{assumption:Convex}.
SVM is trained on the original MNIST dataset (referred to as \emph{MNIST-O}) \cite{lecun1998gradient}, which contains gray-scale images of $70,000$ handwritten digits ($60,000$ for training and $10,000$ for testing). The SVM outputs a binary label that corresponds to whether the digit is even or odd.
We consider both DGD and SGD variants of SVM. The DGD variant only uses $1,000$ training and $1,000$ testing data samples out of the entire dataset in each simulation round, because DGD cannot process a large amount of data. The SGD variant uses the entire MNIST dataset.
Linear regression is performed with SGD on the energy dataset \cite{CANDANEDO201781}, which contains $19,735$ records of measurements from multiple sensors and the energy consumptions of appliances and lights. The model learns to predict the appliance energy consumption from sensor measurements.
K-means is performed with DGD on the user knowledge modeling dataset \cite{Kahraman}, which has $403$ samples each with $5$ attributes summarizing the user interaction with a web environment. The samples can be grouped into $4$ clusters representing different knowledge levels, but we assume that we do not have prior knowledge of this grouping.
CNN is trained using SGD on three different datasets, including MNIST-O as described above, the fashion MNIST dataset (referred to as \emph{MNIST-F}) which has the same format as MNIST-O but includes images of fashion items instead of digits \cite{FashionMNIST}, and the CIFAR-10 dataset which includes $60,000$ color images ($50,000$ for training and $10,000$ for testing) of $10$ different types of objects \cite{CIFAR10}.
A separate CNN model is trained on each dataset, to perform multi-class classification among the $10$ different labels in the dataset.
\subsubsection{Data Distribution at Different Nodes (Cases 1--\,4)}
\label{subsec:DataDistributionCases}
For the distributed settings, we consider four different ways of distributing the data into different nodes.
In \emph{Case~1}, each data sample is randomly assigned to a node, thus each node has uniform (but not full) information.
In \emph{Case~2}, all the data samples in each node have the same label\footnote{When there are more labels than nodes, each node may have data with more than one label, but the number of labels at each node is no more than the total number of labels divided by the total number of nodes rounded to the next integer.}. This represents the case where each node has non-uniform information, because the entire dataset has samples with multiple different labels.
In \emph{Case~3}, each node has the entire dataset (thus full information).
In \emph{Case~4}, data samples with the first half of the labels are distributed to the first half of the nodes as in Case 1; the other samples are distributed to the second half of the nodes as in Case 2. This represents a combined uniform and non-uniform case.
For datasets that do not have ground truth labels, such the energy dataset used with linear regression, the data to node assignment is based on labels generated from an unsupervised clustering approach.
\subsubsection{Training and Control Parameters}
\label{subsec:experiments:trainingCtrlParams}
In all our experiments, we set the search range parameter $\gamma=10$, the maximum $\tau$ value $\tau_\textrm{max} = 100$. Unless otherwise specified, we set the control parameter $\varphi=0.025$ for SVM, linear regression, and K-means, and $\varphi=5\times 10^{-5}$ for CNN. The gradient descent step size is $\eta=0.01$. The resource (time) budget is set as $R = 15$ seconds unless otherwise specified. Except for the instantaneous results in Section~\ref{subsec:results:instantaneous}, the average results of $15$ independent experiment/simulation runs are shown.
\subsection{Results}
\subsubsection{Loss and Accuracy Values}
\label{sec:LossAndAccuracyFromExperiment}
In our first set of experiments, the SVM, linear regression, and K-means models were trained on the prototype system.
Due to the resource limitation of Raspberry Pi devices, the CNN model was trained in a simulated environment of $5$~nodes, with resource consumptions generated in the way described in Section~\ref{subsec:experimentation:resource}.
We compare the loss function values of our proposed algorithm (with adaptive $\tau$) to baseline approaches, and also compare the classification accuracies for the SVM and CNN classifiers.
The results are shown in Fig.~\ref{fig:LossAndAccuracyExperimentResults}. We note that \emph{the proposed approach only has one data point (represented by a single marker in the figure) in each case}, because the value of $\tau$ is adaptive in this case and the marker location shows the average $\tau^*$ with the corresponding loss or accuracy. The centralized case also only has one data point but we show a flat line across different values of $\tau$ for the ease of comparison. We see that the proposed approach performs close to the optimal point for all cases and all models\footnote{Note that the loss and accuracy values shown in Fig.~\ref{fig:LossAndAccuracyExperimentResults} can be improved if we allow a longer training time. For example, the accuracy of CNN on MNIST data can become close to $1.0$ if we allow a long enough time for training. The goal of our experiments here is to show that our proposed approach can operate close to the optimal point with a \emph{fixed and limited} amount of training time (resource budget) as defined in Section~\ref{subsec:experiments:trainingCtrlParams}. }. We also see that the (empirically) optimal value of $\tau$ is different for different cases and models, so a fixed value of $\tau$ does not work well for all cases. In some cases, the distributed approach can perform better than the centralized approach, because for a given amount of time budget, federated learning is able to make use of the computation resource at multiple nodes.
For DGD approaches, Case~3 does not perform as well as Case~1, because the amount of data at each node in Case~3 is larger than that in Case~1, and DGD processes the entire amount of data thus Case~3 requires more resource for each local update.
Due to the high complexity of evaluating CNN models and the fact that linear regression and K-means models do not provide accuracy values, we focus on the SVM model in the following and provide further insights on the system.
\subsubsection{Varying Number of Nodes}
\begin{figure}
\centering
\begin{subfigure}{0.45\textwidth}
\centering
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=1\linewidth]{MNIST_SGD_Varying_Nodes_Tau.pdf}
\end{subfigure}%
~\,\,\,\,
\begin{subfigure}{0.18\textwidth}
\includegraphics[width=1\linewidth]{Varying_Tau_Legend.pdf}
\vspace{0.2in}
\end{subfigure}%
\caption{$\tau^*$ in proposed algorithm}
\end{subfigure}%
\vspace{0.1in}
\begin{subfigure}{0.45\textwidth}
\centering
\begin{subfigure}{0.24\textwidth}
\centering
\hspace{1\linewidth}
\end{subfigure}%
~
\begin{subfigure}{0.24\textwidth}
\centering
\includegraphics[width=0.9\linewidth]{Varying_EachCase_Legend1.pdf}
\end{subfigure}%
~
\begin{subfigure}{0.24\textwidth}
\centering
\includegraphics[width=0.9\linewidth]{Varying_EachCase_Legend2.pdf}
\end{subfigure}%
~
\begin{subfigure}{0.24\textwidth}
\centering
\hspace{1\linewidth}
\end{subfigure}%
\begin{subfigure}[b]{0.035\textwidth}
\centering
\includegraphics[width=1\linewidth]{LossLabel.pdf}
\end{subfigure}%
~
\begin{subfigure}[b]{0.23\textwidth}
\centering
\includegraphics[width=1\linewidth]{MNIST_SGD_Varying_Nodes_Loss1.pdf}
\end{subfigure}%
~
\begin{subfigure}[b]{0.23\textwidth}
\centering
\includegraphics[width=1\linewidth]{MNIST_SGD_Varying_Nodes_Loss2.pdf}
\end{subfigure}%
~
\begin{subfigure}[b]{0.23\textwidth}
\centering
\includegraphics[width=1\linewidth]{MNIST_SGD_Varying_Nodes_Loss3.pdf}
\end{subfigure}%
~
\begin{subfigure}[b]{0.23\textwidth}
\centering
\includegraphics[width=1\linewidth]{MNIST_SGD_Varying_Nodes_Loss4.pdf}
\end{subfigure}%
\begin{subfigure}[b]{0.035\textwidth}
\centering
\includegraphics[width=1\textwidth]{AccuracyLabel.pdf}
\end{subfigure}%
~
\begin{subfigure}[b]{0.23\textwidth}
\centering
\includegraphics[width=1\textwidth]{MNIST_SGD_Varying_Nodes_Acc1.pdf}
\end{subfigure}%
~
\begin{subfigure}[b]{0.23\textwidth}
\centering
\includegraphics[width=1\linewidth]{MNIST_SGD_Varying_Nodes_Acc2.pdf}
\end{subfigure}%
~
\begin{subfigure}[b]{0.23\textwidth}
\centering
\includegraphics[width=1\linewidth]{MNIST_SGD_Varying_Nodes_Acc3.pdf}
\end{subfigure}%
~
\begin{subfigure}[b]{0.23\textwidth}
\centering
\includegraphics[width=1\linewidth]{MNIST_SGD_Varying_Nodes_Acc4.pdf}
\end{subfigure}%
\caption{Loss function values and classification accuracy}
\end{subfigure}%
\caption{SVM (SGD) with different numbers of nodes.}
\label{fig:MultiNodeExperimentResults}
\end{figure}
Results of SVM (SGD) for the number of nodes varying from $5$ to $500$ are shown in Fig.~\ref{fig:MultiNodeExperimentResults}, which are obtained in the simulated environment. Our proposed approach performs better than or similar to the fixed $\tau=10$ baseline in all cases, where we choose fixed $\tau=10$ as the baseline in this and the following evaluations because it is empirically a good value for non-adaptive $\tau$ in different cases according to the results in Fig.~\ref{fig:LossAndAccuracyExperimentResults}.
\subsubsection{Varying Global Aggregation Time}
\begin{figure}
\centering
\begin{subfigure}{0.45\textwidth}
\centering
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=1\linewidth]{MNIST_SGD_Varying_GlobalAggTime_Tau.pdf}
\end{subfigure}%
~\,\,\,\,
\begin{subfigure}{0.18\textwidth}
\includegraphics[width=1\linewidth]{Varying_Tau_Legend.pdf}
\vspace{0.2in}
\end{subfigure}%
\caption{$\tau^*$ in proposed algorithm}
\end{subfigure}%
\vspace{0.1in}
\begin{subfigure}{0.45\textwidth}
\centering
\begin{subfigure}{0.24\textwidth}
\centering
\hspace{1\linewidth}
\end{subfigure}%
~
\begin{subfigure}{0.24\textwidth}
\centering
\includegraphics[width=0.9\linewidth]{Varying_EachCase_Legend1.pdf}
\end{subfigure}%
~
\begin{subfigure}{0.24\textwidth}
\centering
\includegraphics[width=0.9\linewidth]{Varying_EachCase_Legend2.pdf}
\end{subfigure}%
~
\begin{subfigure}{0.24\textwidth}
\centering
\hspace{1\linewidth}
\end{subfigure}%
\begin{subfigure}[b]{0.035\textwidth}
\centering
\includegraphics[width=1\linewidth]{LossLabel.pdf}
\end{subfigure}%
~
\begin{subfigure}[b]{0.23\textwidth}
\centering
\includegraphics[width=1\linewidth]{MNIST_SGD_Varying_GlobalAggTime_Loss1.pdf}
\end{subfigure}%
~
\begin{subfigure}[b]{0.23\textwidth}
\centering
\includegraphics[width=1\linewidth]{MNIST_SGD_Varying_GlobalAggTime_Loss2.pdf}
\end{subfigure}%
~
\begin{subfigure}[b]{0.23\textwidth}
\centering
\includegraphics[width=1\linewidth]{MNIST_SGD_Varying_GlobalAggTime_Loss3.pdf}
\end{subfigure}%
~
\begin{subfigure}[b]{0.23\textwidth}
\centering
\includegraphics[width=1\linewidth]{MNIST_SGD_Varying_GlobalAggTime_Loss4.pdf}
\end{subfigure}%
\begin{subfigure}[b]{0.035\textwidth}
\centering
\includegraphics[width=1\textwidth]{AccuracyLabel.pdf}
\end{subfigure}%
~
\begin{subfigure}[b]{0.23\textwidth}
\centering
\includegraphics[width=1\textwidth]{MNIST_SGD_Varying_GlobalAggTime_Acc1.pdf}
\end{subfigure}%
~
\begin{subfigure}[b]{0.23\textwidth}
\centering
\includegraphics[width=1\linewidth]{MNIST_SGD_Varying_GlobalAggTime_Acc2.pdf}
\end{subfigure}%
~
\begin{subfigure}[b]{0.23\textwidth}
\centering
\includegraphics[width=1\linewidth]{MNIST_SGD_Varying_GlobalAggTime_Acc3.pdf}
\end{subfigure}%
~
\begin{subfigure}[b]{0.23\textwidth}
\centering
\includegraphics[width=1\linewidth]{MNIST_SGD_Varying_GlobalAggTime_Acc4.pdf}
\end{subfigure}%
\caption{Loss function values and classification accuracy}
\end{subfigure}%
\caption{SVM (SGD) with different global aggregation times.}
\label{fig:VaryingGlobalAggSGDExperimentResults}
\end{figure}
To study the impact of different resource consumption (time) for global aggregation, we modify the simulation environment so that the global aggregation time is scaled by an \emph{adjustment factor}. The actual time of global aggregation is equal to the original global aggregation time multiplied by the adjustment factor, thus a small adjustment factor corresponds to a small global aggregation time. The results for SVM (SGD) are shown in Fig.~\ref{fig:VaryingGlobalAggSGDExperimentResults}.
\if\citetechreport1
Additional results for SVM (DGD) are included in \cite[Appendix~\ref{append:VaryingGlobalAggDGDExperimentResults}]{JournalTechReport}.
\else
Additional results for SVM (DGD) are included in Appendix~\ref{append:VaryingGlobalAggDGDExperimentResults}.
\fi
We can see that as one would intuitively expect, a larger global aggregation time generally results in a larger $\tau^*$ for the proposed algorithm, because when it takes more time to perform global aggregation, the system should perform global aggregation less frequently, to make the best use of available time (resource). The fact that $\tau^*$ slightly decreases when the adjustment factor is large is because in this case, the global aggregation time is so large that only a few rounds of global aggregation can be performed before reaching the resource budget, and the value of $\tau^*$ will be decreased in the last round to remain within the resource budget (see Line~\ref{alg:protocolAggregator:decreaseTauToRemainWithinBudget} of Algorithm~\ref{alg:protocolAggregator}).
Comparing to the fixed $\tau=10$ baseline, the proposed algorithm performs better in (almost) all cases.
\subsubsection{Varying Total Time Budget}
\begin{figure}
\centering
\begin{subfigure}{0.45\textwidth}
\centering
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=1\linewidth]{MNIST_SGD_Varying_TotalTime_Tau.pdf}
\end{subfigure}%
~\,\,\,\,
\begin{subfigure}{0.18\textwidth}
\includegraphics[width=1\linewidth]{Varying_Tau_Legend.pdf}
\vspace{0.2in}
\end{subfigure}%
\caption{$\tau^*$ in proposed algorithm}
\end{subfigure}%
\vspace{0.1in}
\begin{subfigure}{0.45\textwidth}
\centering
\begin{subfigure}{0.24\textwidth}
\centering
\hspace{1\linewidth}
\end{subfigure}%
~
\begin{subfigure}{0.24\textwidth}
\centering
\includegraphics[width=0.9\linewidth]{Varying_EachCase_Legend1.pdf}
\end{subfigure}%
~
\begin{subfigure}{0.24\textwidth}
\centering
\includegraphics[width=0.9\linewidth]{Varying_EachCase_Legend2.pdf}
\end{subfigure}%
~
\begin{subfigure}{0.24\textwidth}
\centering
\hspace{1\linewidth}
\end{subfigure}%
\begin{subfigure}[b]{0.035\textwidth}
\centering
\includegraphics[width=1\linewidth]{LossLabel.pdf}
\end{subfigure}%
~
\begin{subfigure}[b]{0.23\textwidth}
\centering
\includegraphics[width=1\linewidth]{MNIST_SGD_Varying_TotalTime_Loss1.pdf}
\end{subfigure}%
~
\begin{subfigure}[b]{0.23\textwidth}
\centering
\includegraphics[width=1\linewidth]{MNIST_SGD_Varying_TotalTime_Loss2.pdf}
\end{subfigure}%
~
\begin{subfigure}[b]{0.23\textwidth}
\centering
\includegraphics[width=1\linewidth]{MNIST_SGD_Varying_TotalTime_Loss3.pdf}
\end{subfigure}%
~
\begin{subfigure}[b]{0.23\textwidth}
\centering
\includegraphics[width=1\linewidth]{MNIST_SGD_Varying_TotalTime_Loss4.pdf}
\end{subfigure}%
\begin{subfigure}[b]{0.035\textwidth}
\centering
\includegraphics[width=1\textwidth]{AccuracyLabel.pdf}
\end{subfigure}%
~
\begin{subfigure}[b]{0.23\textwidth}
\centering
\includegraphics[width=1\textwidth]{MNIST_SGD_Varying_TotalTime_Acc1.pdf}
\end{subfigure}%
~
\begin{subfigure}[b]{0.23\textwidth}
\centering
\includegraphics[width=1\linewidth]{MNIST_SGD_Varying_TotalTime_Acc2.pdf}
\end{subfigure}%
~
\begin{subfigure}[b]{0.23\textwidth}
\centering
\includegraphics[width=1\linewidth]{MNIST_SGD_Varying_TotalTime_Acc3.pdf}
\end{subfigure}%
~
\begin{subfigure}[b]{0.23\textwidth}
\centering
\includegraphics[width=1\linewidth]{MNIST_SGD_Varying_TotalTime_Acc4.pdf}
\end{subfigure}%
\caption{Loss function values and classification accuracy}
\end{subfigure}%
\caption{SVM (SGD) with different total time budgets.}
\label{fig:VaryingTotalTimeSGDExperimentResults}
\end{figure}
We evaluate the impact of the total time (resource) budget on the prototype system. Results for SVM (SGD) are shown in Fig.~\ref{fig:VaryingTotalTimeSGDExperimentResults}.
\if\citetechreport1
Further results for SVM (DGD) are included in \cite[Appendix~\ref{append:VaryingTotalTimeDGDExperimentResults}]{JournalTechReport}.
\else
Further results for SVM (DGD) are included in Appendix~\ref{append:VaryingTotalTimeDGDExperimentResults}.
\fi
We see that except for Case 3 where all nodes have the same dataset, the value of $\tau^*$ of the proposed algorithm decreases with the total time budget. This aligns with the discussion in Section~\ref{subsec:approxSolutionControlAlg} that $\tau^*$ becomes close to one when the resource budget is large enough. We also see that the proposed algorithm performs better than or similar to the fixed $\tau=10$ baseline in all cases.
\subsubsection{Instantaneous Behavior}
\label{subsec:results:instantaneous}
We further study the instantaneous behavior of our system for a single run of $30$ seconds (for each case) on the prototype system. Results for SVM (DGD) is shown in Fig.~\ref{fig:InstantSVMDGDExperimentResults}.
\if\citetechreport1
Further results for SVM (SGD) are available in \cite[Appendix~\ref{append:InstantSVMSGDExperimentResults}]{JournalTechReport}.
\else
Further results for SVM (SGD) are available in Appendix~\ref{append:InstantSVMSGDExperimentResults}.
\fi
We see that the value of $\tau^*$ remains stable after an initial adaptation period, showing that the control algorithm is stable. The value of $\tau^*$ decreases at the end due to adjustment caused by the system reaching the resource budget (see Line~\ref{alg:protocolAggregator:decreaseTauToRemainWithinBudget} of Algorithm~\ref{alg:protocolAggregator}).
As expected, the gradient deviation $\delta$ is larger for Cases~2 and 4 where the data samples at different nodes are non-uniform. The same is observed for $\rho$ and $\beta$, indicating that the model parameter $\mathbf{w}$ is in a less smooth region for Cases~2 and 4. In Case 3, the data at different nodes are equal so we always have $\mathbf{w}_i (t) = \mathbf{w}(t)$ regardless of whether global aggregation is performed in iteration $t$. Thus, the estimated $\rho$ and $\beta$ values are zero by definition, as explained in the remark in Section~\ref{subsec:ControlAlgParamEst}.
Case 3 of SVM (DGD) has a much larger value of $c$ because it processes more data than in other cases and thus takes more time, as explained before. The value of $b$ exhibits fluctuations because of the randomness of the wireless channel.
\begin{figure}
\centering
\begin{subfigure}[b]{0.11\textwidth}
\centering
\includegraphics[width=0.9\linewidth]{SingleRun_Legend1.pdf}
\end{subfigure}%
\begin{subfigure}[b]{0.11\textwidth}
\centering
\includegraphics[width=0.9\linewidth]{SingleRun_Legend2.pdf}
\end{subfigure
\begin{subfigure}[b]{0.11\textwidth}
\centering
\includegraphics[width=0.9\linewidth]{SingleRun_Legend3.pdf}
\end{subfigure}%
\begin{subfigure}[b]{0.11\textwidth}
\centering
\includegraphics[width=0.9\linewidth]{SingleRun_Legend4.pdf}
\end{subfigure}%
\begin{subfigure}[b]{0.12\textwidth}
\centering
\includegraphics[width=1\linewidth]{SingleRun_DGD_Loss.pdf}
\end{subfigure}%
~\hspace{-0.1in}
\begin{subfigure}[b]{0.12\textwidth}
\centering
\includegraphics[width=1\linewidth]{SingleRun_DGD_Acc.pdf}
\end{subfigure}%
~\hspace{-0.1in}
\begin{subfigure}[b]{0.12\textwidth}
\centering
\includegraphics[width=1\linewidth]{SingleRun_DGD_c.pdf}
\end{subfigure}%
~\hspace{-0.1in}
\begin{subfigure}[b]{0.12\textwidth}
\centering
\includegraphics[width=1\linewidth]{SingleRun_DGD_b.pdf}
\end{subfigure}%
\vspace{-0.1in}
\begin{subfigure}[b]{0.12\textwidth}
\centering
\includegraphics[width=1\linewidth]{SingleRun_DGD_rho.pdf}
\end{subfigure}%
~\hspace{-0.1in}
\begin{subfigure}[b]{0.12\textwidth}
\centering
\includegraphics[width=1\linewidth]{SingleRun_DGD_beta.pdf}
\end{subfigure}%
~\hspace{-0.1in}
\begin{subfigure}[b]{0.12\textwidth}
\centering
\includegraphics[width=1\linewidth]{SingleRun_DGD_delta.pdf}
\end{subfigure}%
~\hspace{-0.1in}
\begin{subfigure}[b]{0.12\textwidth}
\centering
\includegraphics[width=1\linewidth]{SingleRun_DGD_tau.pdf}
\end{subfigure}%
\caption{Instantaneous results of SVM (DGD) with the proposed algorithm.}
\label{fig:InstantSVMDGDExperimentResults}
\end{figure}
\subsubsection{Sensitivity of $\varphi$}
\label{subsubsec:sensitivityOfPhiResult}
\begin{figure}[t]
\centering
\begin{subfigure}[b]{0.18\textwidth}
\centering
\includegraphics[width=1\linewidth]{MNIST_DGD_Varying_ControlParam_Tau.pdf}
\end{subfigure}%
~
\begin{subfigure}[b]{0.18\textwidth}
\centering
\includegraphics[width=1\linewidth]{MNIST_SGD_Varying_ControlParam_Tau.pdf}
\end{subfigure}%
~
\begin{subfigure}[b]{0.08\textwidth}
\centering
\includegraphics[width=1\linewidth]{Varying_ControlParam_Legend.pdf}
\vspace{0.4in}
\end{subfigure
\caption{Impact of $\varphi$ on the average value of $\tau^*$ in the proposed algorithm.}
\label{fig:sensitivityControlParameterExperimentationResult}
\end{figure}
The sensitivity of the control parameter $\varphi$ evaluated on the prototype system is shown in Fig.~\ref{fig:sensitivityControlParameterExperimentationResult}. We see that the relationship among $\tau^*$ in different cases is mostly maintained with different values of $\varphi$. The value of $\tau^*$ decreases approximately linearly with $\log\varphi$, which is consistent with the fact that there is an exponential term w.r.t. $\tau$ in $h(\tau)$ (and thus $G(\tau)$).
For Case~3, $\tau^*$ remains the same with different $\varphi$, because $h(\tau)=0$ in this case by definition (see the remark in Section~\ref{subsec:ControlAlgParamEst}) and the value of $\varphi$ does not affect $\tau^*$, as $G(\tau) \propto \frac{1}{\varphi}$ independently of $\tau$ in this case according to (\ref{eq:GTauDef}).
We also see that small changes of $\varphi$ does not change $\tau^*$ much, indicating that one can take big steps when tuning $\varphi$ in practice and the tuning is not difficult.
\subsubsection{Comparison to Asynchronous Distributed Gradient Descent}
Asynchronous gradient descent~\cite{Chen2016} is an alternative to the typically used synchronous gradient descent in federated learning. With asynchronous gradient descent, the edge nodes operate in an asynchronous manner. Each edge node pulls the most up-to-date model parameter from the aggregator, computes the gradient on its local dataset, then sends the gradient back to the aggregator. The aggregator performs gradient descent according to the step size $\eta$ weighted by the dataset sizes of each node, similar to the combination of (\ref{eq:localUpdate}) and (\ref{eq:globalAverage}). The process repeats until the training finishes.
Asynchronous gradient descent is able to fully utilize the available computational resource at each node by running more gradient descent steps at more powerful (faster) nodes. However, the asynchronism may hurt the overall performance.
It was shown in~\cite{Chen2016} that synchronous gradient descent has benefits over asynchronous gradient descent in a datacenter setting. Here, we study their differences in the edge computing setting with heterogeneous resources (laptops and Raspberry Pis in our experiment) and different data distributions (Cases 1--4). The results for DGD and SGD with SVM are shown in Figs.~\ref{fig:asyncDGD} and \ref{fig:asyncSGD}, respectively. We see that the performance of asynchronous gradient descent is much worse than synchronous gradient descent for non-uniform data distribution in Cases 2 and 4, with slower convergence, sudden changes (indicating instability of the training process), and convergence to higher loss and lower accuracy values. This is because the model tends overfit the datasets on the faster nodes, as many more steps of gradient descent are performed on these nodes compared to the slower nodes. With uniform data distribution (Cases 1 and 3), asynchronous gradient descent performs similar as or slightly better than synchronous gradient descent, because when the datasets at different nodes are similar (Case~1) or equal (Case 3), there is not much harm caused by overfitting the data on the faster nodes.
Considering the overall performance in all Cases 1--4, we can conclude that it is still better to perform federated learning with synchronous gradient descent as we do throughout this paper. However, how to make more efficient use of heterogeneous resources is something worth investigating in the future.
\begin{figure}[t]
\centering
\begin{subfigure}[b]{0.11\textwidth}
\centering
\includegraphics[width=1\textwidth]{Async_Legend1.pdf}
\end{subfigure}%
~
\begin{subfigure}[b]{0.11\textwidth}
\centering
\includegraphics[width=1\textwidth]{Async_Legend2.pdf}
\end{subfigure}%
~\,\,\,\,
\begin{subfigure}[b]{0.11\textwidth}
\centering
\includegraphics[width=1\textwidth]{Async_Legend3.pdf}
\end{subfigure}%
\begin{subfigure}[b]{0.015\textwidth}
\centering
\includegraphics[width=1\textwidth]{LossLabel.pdf}
\end{subfigure}%
~
\begin{subfigure}[b]{0.11\textwidth}
\centering
\includegraphics[width=1\textwidth]{Async_DGD_1.pdf}
\end{subfigure}%
~
\begin{subfigure}[b]{0.11\textwidth}
\centering
\includegraphics[width=1\linewidth]{Async_DGD_2.pdf}
\end{subfigure}%
~
\begin{subfigure}[b]{0.11\textwidth}
\centering
\includegraphics[width=1\linewidth]{Async_DGD_3.pdf}
\end{subfigure}%
~
\begin{subfigure}[b]{0.11\textwidth}
\centering
\includegraphics[width=1\linewidth]{Async_DGD_4.pdf}
\end{subfigure}%
\begin{subfigure}[b]{0.015\textwidth}
\centering
\includegraphics[width=1\textwidth]{AccuracyLabel.pdf}
\end{subfigure}%
~
\begin{subfigure}[b]{0.11\textwidth}
\centering
\includegraphics[width=1\textwidth]{Async_DGD_5.pdf}
\end{subfigure}%
~
\begin{subfigure}[b]{0.11\textwidth}
\centering
\includegraphics[width=1\linewidth]{Async_DGD_6.pdf}
\end{subfigure}%
~
\begin{subfigure}[b]{0.11\textwidth}
\centering
\includegraphics[width=1\linewidth]{Async_DGD_7.pdf}
\end{subfigure}%
~
\begin{subfigure}[b]{0.11\textwidth}
\centering
\includegraphics[width=1\linewidth]{Async_DGD_8.pdf}
\end{subfigure}%
\caption{Synchronous vs. asynchronous distributed DGD with SVM.}
\label{fig:asyncDGD}
\end{figure}
\begin{figure}[t]
\centering
\begin{subfigure}[b]{0.11\textwidth}
\centering
\includegraphics[width=1\textwidth]{Async_Legend1.pdf}
\end{subfigure}%
~
\begin{subfigure}[b]{0.11\textwidth}
\centering
\includegraphics[width=1\textwidth]{Async_Legend2.pdf}
\end{subfigure}%
~\,\,\,\,
\begin{subfigure}[b]{0.11\textwidth}
\centering
\includegraphics[width=1\textwidth]{Async_Legend3.pdf}
\end{subfigure}%
\begin{subfigure}[b]{0.015\textwidth}
\centering
\includegraphics[width=1\textwidth]{LossLabel.pdf}
\end{subfigure}%
~
\begin{subfigure}[b]{0.11\textwidth}
\centering
\includegraphics[width=1\textwidth]{Async_SGD_1.pdf}
\end{subfigure}%
~
\begin{subfigure}[b]{0.11\textwidth}
\centering
\includegraphics[width=1\linewidth]{Async_SGD_2.pdf}
\end{subfigure}%
~
\begin{subfigure}[b]{0.11\textwidth}
\centering
\includegraphics[width=1\linewidth]{Async_SGD_3.pdf}
\end{subfigure}%
~
\begin{subfigure}[b]{0.11\textwidth}
\centering
\includegraphics[width=1\linewidth]{Async_SGD_4.pdf}
\end{subfigure}%
\begin{subfigure}[b]{0.015\textwidth}
\centering
\includegraphics[width=1\textwidth]{AccuracyLabel.pdf}
\end{subfigure}%
~
\begin{subfigure}[b]{0.11\textwidth}
\centering
\includegraphics[width=1\textwidth]{Async_SGD_5.pdf}
\end{subfigure}%
~
\begin{subfigure}[b]{0.11\textwidth}
\centering
\includegraphics[width=1\linewidth]{Async_SGD_6.pdf}
\end{subfigure}%
~
\begin{subfigure}[b]{0.11\textwidth}
\centering
\includegraphics[width=1\linewidth]{Async_SGD_7.pdf}
\end{subfigure}%
~
\begin{subfigure}[b]{0.11\textwidth}
\centering
\includegraphics[width=1\linewidth]{Async_SGD_8.pdf}
\end{subfigure}%
\caption{Synchronous vs. asynchronous distributed SGD with SVM.}
\label{fig:asyncSGD}
\end{figure}
\section{Conclusion}
In this paper, we have focused on gradient-descent based federated
learning that include local update and global aggregation steps.
Each step of local update and global aggregation consumes
resources. We have analyzed the convergence bound for federated learning with non-i.i.d. data distributions. Using this theoretical bound, a control algorithm has been proposed to achieve the desirable trade-off between local update and global aggregation in order to
minimize the loss function under a resource budget constraint.
Extensive experimentation results confirm the effectiveness of our proposed
algorithm.
Future work can investigate how to make the most efficient use of heterogeneous resources for distributed learning, as well as the theoretical convergence analysis of some form of non-convex loss functions representing deep neural networks.
\bibliographystyle{IEEEtran}
|
{
"timestamp": "2019-02-19T02:13:44",
"yymm": "1804",
"arxiv_id": "1804.05271",
"language": "en",
"url": "https://arxiv.org/abs/1804.05271"
}
|
\section{INTRODUCTION}
Faced with large amounts of data, efficient parameter estimation remains one of the key bottlenecks in the application of probabilistic models. Once cast as an optimization problem, for example through the maximum likelihood principle, difficulties may arise from the size of the model, the number of observations, or the potential non-convexity of the objective functions, and often all three~\citep{daphne_koller_book,kevin_murphys_book}.
In this paper we focus primarily on situations where the number of observations is large; in this context, stochastic gradient descent (SGD) methods which look at one sample at a time are usually favored for their cheap iteration cost. However, finding the correct step-size (sometimes referred to as the learning rate) remains a practical and theoretical challenge, for probabilistic modeling but also in most other situations beyond maximum likelihood~\citep{bottou2016optimization}.
In order to preserve convergence, the step size $\gamma_n$ at the $n$-th iteration typically has to decay with the number of gradient steps (here equal to the number of data points which are processed), typically as $C / n^\alpha$ for $\alpha \in [1/2,1]$ \citep[see, e.g.,][]{gradsto,bottou2016optimization}. However, these often leads to slow convergence and the choice of $\alpha$ and $C$ is difficult in practice. More recently, constant step-sizes have been advocated for their fast convergence towards a neighborhood of the optimal solution~\citep{Bac_Mou_2013}, while it is a standard practice in many areas~\citep{deep_learning_book_bengio}. However, it is not convergent in general and thus small step-sizes are still needed to converge to a decent estimator.
Constant step-sizes can however be made to converge in one situation. When the functions to optimize are quadratic, like for least-squares regression, using a constant step-size combined with an averaging of all estimators along the algorithm can be shown to converge to the global solution with the optimal convergence rates~\citep{Bac_Mou_2013,Die_Bac_2015}.
The goal of this paper is to explore the possibility of such global convergence with a constant step-size in the context of probabilistic modeling with exponential families, e.g., for logistic regression or Poisson regression~\citep{mccullagh1984generalized}. This would lead to the possibility of using probabilistic models (thus with a principled quantification of uncertainty) with rapidly converging algorithms. Our main novel idea is to replace the averaging of the \emph{natural} parameters of the exponential family by the averaging of the \emph{moment} parameters, which can also be formulated as averaging \emph{predictions} instead of \emph{estimators}. For example, in the context of predicting binary outcomes in $\{0,1\}$ through a Bernoulli distribution, the moment parameter is the probability $p \in [0,1]$ that the variable is equal to one, while the natural parameter is the ``log odds ratio'' $\log \frac{p}{1-p}$, which is unconstrained. This lack of constraint is often seen as a benefit for optimization; it turns out that for stochastic gradient methods, the moment parameter is better suited to averaging. Note that for least-squares, which corresponds to modeling with the Gaussian distribution with fixed variance, moment and natural parameters are equal, so it does not make a difference.
More precisely, our main contributions are:
\begin{itemize}
\item For generalized linear models, we propose in \mysec{cond} averaging moment parameters instead of natural parameters for constant-step-size stochastic gradient descent.
\item For finite-dimensional models, we show in \mysec{finited} that this can sometimes (and surprisingly) lead to better predictions than the best linear model.
\item For infinite-dimensional models, we show in \mysec{kernels} that it always converges to optimal predictions, while averaging natural parameters never does.
\item We illustrate our findings in \mysec{exp} with simulations on synthetic data and classical benchmarks with many observations.
\end{itemize}
\section{CONSTANT STEP SIZE STOCHASTIC GRADIENT DESCENT}
\label{sec:sgd}
\begin{figure}
\includegraphics[width=0.5\textwidth]{Graph00.pdf}
\vspace*{-1cm}
\caption{Convergence of iterates $\theta_{n}$ and averaged iterates $\bar{\theta}_{n}$ to the mean $\bar{\theta}_\gamma$ under the stationary distribution~$\pi_\gamma$.}
\label{fig:mc}
\end{figure}
In this section, we present the main intuitions behind stochastic gradient descent (SGD) with constant step-size. For more details, see~\citet{dieuleveut2017bridging}.
We consider a real-valued function $F$ defined on the Euclidean space $\mathbb{R}^d$ (this can be generalized to any Hilbert space, as done in \mysec{kernels} when considering Gaussian processes and positive-definite kernels), and a sequence of random functions $(f_n)_{n \geqslant 1}$ which are independent and identically distributed and such that ${\mathbb E} f_n (\theta) = F(\theta)$ for all $\theta \in \mathbb{R}^d$. Typically, $F$ will the expected negative log-likelihood on unseen data, while $f_n$ will be the negative log-likelihood for a single observation. Since we require \emph{independent} random functions, we assume that we make single pass over the data, and thus the number of iterations is equal to the number of observations.
Starting from an initial $\theta_0 \in \mathbb{R}^d$, then SGD will perform the following recursion, from $n=1$ to the total number of observations:
\begin{equation}
\label{eq:sgd} \theta_n = \theta_{n-1} - \gamma_n \nabla f_n(\theta_{n-1}).
\end{equation}
Since the functions $f_n$ are independent, the iterates $(\theta_n)_n$ form a Markov chain. When the step-size $\gamma_n$ is constant (equal to $\gamma$) and the functions $f_n$ are identically distributed, the Markov chain is \emph{homogeneous}. Thus, under additional assumptions \citep[see, e.g.,][]{dieuleveut2017bridging,Mey_Twe_1993}, it converges in distribution to a stationary distribution, which we refer to as $\pi_\gamma$. These additional assumptions include that $\gamma$ is not too large (otherwise the algorithm diverges) and in the traditional analysis of step-sizes for gradient descent techniques, we analyze the situation of small $\gamma$'s (and thus perform asymptotic expansions around $\gamma=0$).
The distribution $\pi_\gamma$ is in general not equal to a Dirac mass, and thus, constant-step-size SGD is \emph{not} convergent.
However, averaging along the path of the Markov chain has some interesting properties. Indeed, several versions of the ``ergodic theorem'' \citep[see, e.g.,][]{Mey_Twe_1993} show that for functions $g$ from $\mathbb{R}^d$ to any vector space, then the empirical average $\frac{1}{n} \sum_{i=1}^n g(\theta_i)$ converges in probability to the expectation $\int g(\theta) d\pi_\gamma(\theta)$ of $g$ under the stationary distribution $\pi_\gamma$. This convergence can also be quantified by a central limit theorem with an error whichs tends to a normal distribution with variance proportional equal to a constant times $1/n$.
Thus, if denote $\bar{\theta}_n = \frac{1}{n+1} \sum_{i=0}^n \theta_i$, applying the previous result to the identity function $g $, we immediately obtain that $\bar{\theta}_n$ converges to $\bar{\theta}_\gamma = \int \theta d\pi_\gamma(\theta)$, with a squared error converging in $O(1/n)$. The key question is the relationship between $\bar{\theta}_\gamma $ and the global optimizer~$\theta_\ast$ of $F$, as this characterizes the performance of the algorithm with an infinite number of observations.
By taking expectations in \eq{sgd}, and taking a limit with~$n$ tending to infinity we obtain that
\begin{equation}
\label{eq:stat} \int \nabla F (\theta) d\pi_\gamma(\theta) = 0,
\end{equation}
that is, under the stationary distribution $\pi_\gamma$, the average gradient is zero. When the gradient is a linear function (like for a quadratic objective $F$), this leads to $ \nabla F ( \int\theta d\pi_\gamma(\theta) ) = \nabla F(\bar{\theta}_\gamma)= 0$, and thus $\bar{\theta}_\gamma$ is a stationary point of $F$ (and hence the global minimizer if $F$ is convex). However this is not true in general.
As shown by \citet{dieuleveut2017bridging}, the deviation $\bar{\theta}_\gamma - \theta_\ast$ is of order $\gamma$, which is an improvement on the non-averaged recursion, which is at average distance $O(\gamma^{1/2})$ (see an illustration in \myfig{mc}); thus, small or decaying step-sizes are needed. In this paper, we explore alternatives which are not averaging the estimators $\theta_1,\dots,\theta_n$, and rely instead on the specific structure of our cost functions, namely negative log-likelihoods.
\section{WARM-UP: EXPONENTIAL FAMILIES}
\label{sec:uncond}
In order to highlight the benefits of averaging moment parameters, we first consider unconditional exponential families. We thus consider the standard exponential family
$q(x|\theta) = h(x) \exp ( \theta^\top T(x) - A(\theta) )$, where $h(x)$ is the base measure, $T(x) \in \mathbb{R}^d$ is the sufficient statistics and $A$ the log-partition function. The function $A$ is always convex \citep[see, e.g.,][]{daphne_koller_book,kevin_murphys_book}. Note that we do not assume that the data distribution $p(x)$ comes from this exponential family. The expected (with respect to the input distribution $p(x)$) negative log-likelihood is equal to
\begin{eqnarray*}
F(\theta) & = & - {\mathbb E}_{p(x)} \log q(x|\theta) \\
& = &
A(\theta) - \theta^\top {\mathbb E}_{p(x)} T(x) - {\mathbb E}_{p(x)}\log h(x).
\end{eqnarray*}
It is known to be minimized by $\theta_\ast $ such that $\nabla A(\theta_\ast) = {\mathbb E}_{p(x)} T(x) $. Given i.i.d.~data $(x_n)_{n \geqslant 1}$ sampled from $p(x)$, then
the SGD recursion from \eq{sgd} becomes:
$$
\theta_n = \theta_{n-1} - \gamma \big[ \nabla A(\theta_{n-1}) - T(x_n) \big],
$$
while the stationarity equation in \eq{stat} becomes
$$ \int \big[ \nabla A(\theta) - {\mathbb E} _{p(x)}T(x)] d\pi_\gamma(\theta) = 0,$$
which leads to
$$ \int \nabla A(\theta) d\pi_\gamma(\theta) = {\mathbb E}_{p(x)}T(x) = \nabla A(\theta_\ast).$$
Thus, averaging $\nabla A(\theta_n)$ will converge to $\nabla A(\theta_\ast)$, while averaging $\theta_n$ will \emph{not} converge to $
\theta_\ast$. This simple observation is the basis of our work.
Note that in this context of unconditional models, a simpler estimator exists, that is, we can simply compute the empirical average $\frac{1}{n} \sum_{i=1}^n T(x_i)$ that will converge to $\nabla A(\theta_\ast)$. Nevertheless, this shows that averaging moment parameters $\nabla A(\theta)$ rather than natural parameters $\theta$ can bring convergence benefits. We now turn to conditional models, for which no closed-form solutions exist.
\section{CONDITIONAL EXPONENTIAL FAMILIES}
\label{sec:cond}
Now we consider the conditional exponential family $q(y|x,\theta)=h(y)\exp\big(y\cdot \eta_\theta(x) - a(\eta_\theta(x))\big)$. For simplicity we consider only one-dimensional families where $y \in \mathbb{R}$---but our framework would also extend to more complex models such as conditional random fields~\citep{lafferty2001conditional}. We will also assume that $h(y) = 1$ for all $y$ to avoid carrying constant terms in log-likelihoods. We consider functions of the form $\eta_\theta(x)=\theta^\top\Phi(x)$, which are linear in a feature vector $\Phi(x)$, where $\Phi: \mathcal{X} \to \mathbb{R}^d$ can be defined on an arbitrary input set $\mathcal{X}$. Calculating the negative log-likelihood, one obtains:
$$
f_n(\theta) = - \log q(y_n | x_n \theta) = -y_n\Phi(x_n)^\top\theta + a\big(\Phi(x_n)\theta\big) ,
$$
and, for any distribution $p(x,y)$, for which $p(y|x)$ may not be a member of the conditional exponential family,
\begin{eqnarray*}
F(\theta) & = & {\mathbb E}_{p(x_n,y_n)} f_n(\theta) \\
& = & {\mathbb E}_{p(x_n,y_n)}\Big[ -y_n\Phi(x_n)^\top\theta + a\big(\Phi(x_n)\theta\big) \Big].
\end{eqnarray*}
The goal of estimation in such generalized linear models is to find an unknown parameter $\theta$ given $n$ observations $(x_i,y_i)_{i=1,\dots,n}$:
\begin{equation}
\theta_{\ast} = \arg\min\limits_{\theta\in\mathbb{R}^d}F(\theta).
\label{thetastar}
\end{equation}
\subsection{FROM ESTIMATORS TO PREDICTION FUNCTIONS}
Another point of view is to consider that an estimator $\theta \in \mathbb{R}^d$ in fact defines a function
$\eta: \mathcal{X} \to \mathbb{R}$, with value a natural parameter for the exponential family $q(y) = \exp( \eta y - a(\eta))$.
This particular choice of function $\eta_\theta$ is linear in $\Phi(x)$, and we have, by decomposing the joint probability $p(x_n,y_n)$ in two (and dropping the dependence on $n$ since we have assumed i.i.d.~data):
\begin{eqnarray*}
F(\theta) \!\!\! & = & \!\!\! {\mathbb E}_{p(x)}\Big( {\mathbb E}_{p(y | x) } \big[
- y \Phi(x)^\top \theta + a ( \Phi(x)^\top \theta ) \big]
\Big) \\
\!\!\! & = &\!\!\! {\mathbb E}_{p(x)} \Big( - {\mathbb E}_{p(y | x) } y \Phi(x)^\top \theta + a ( \Phi(x)^\top \theta )
\Big) \\
\!\!\! & = &\!\!\! \mathcal{F} ( \eta_\theta),
\end{eqnarray*}
with $\mathcal{F}(\eta) = {\mathbb E}_{p(x)} \big( - {\mathbb E}_{p(y| x) } y \cdot \eta(x) + a ( \eta(x) )
\big) $ is the performance measure defined for a \emph{function} $\eta: {\mathcal X}\to \mathbb{R}$. By definition
$F(\theta) = \mathcal{F} ( \eta_\theta) = \mathcal{F}( \theta^\top \Phi(\cdot) )$.
However, the global minimizer of $\mathcal{F} ( \eta)$ over all functions $\eta: {\mathcal X} \to \mathbb{R}$ may not be attained at a linear function in~$\Phi(x)$ (this can only be the case if the linear model is well-specified or if the feature vector $\Phi(x)$ is flexible enough).
Indeed, the global minimizer of $\mathcal{F}$ is the function $\eta_{\ast\ast}: x \mapsto (a')^{-1}( {\mathbb E}_{p(y | x) } y )$ (starting from $\mathcal{F}(\eta) = \int\big[a(\eta(x)) - \mathbb{E}_{p(x|y)}y\cdot \eta(x) \big] p(x)dx$ and writing down the Euler - Lagrange equation: $\frac{\partial \mathcal{F}}{\partial \eta} - \frac{d}{dx}\frac{\partial F}{\partial \eta'} = 0 \Leftrightarrow \big[a'(\eta) - \mathbb{E}_{p(x|y)}y\big] p(x) =0 $ and finally $\eta \mapsto (a')^{-1}(\mathbb{E}_{p(x|y)}y)$) and is typically not a linear function in $\Phi(x)$ (note here that $p(y|x)$ is the conditional data-generating distribution).
\begin{figure}
\includegraphics[width=0.5\textwidth]{NicePic2.pdf}
\vspace*{-.415cm}
\caption{Graphical representation of reparametrization: firstly we expand the class of functions, replacing parameter $\theta$ with function $\eta(\cdot) = \Phi^\top(\cdot)\theta$ and then we do one more reparametrization: $\mu(\cdot) = a'(\eta(\cdot))$. Best linear prediction $\mu_{\ast}$ is constructed using $\theta_\ast$ and the global minimizer of $\mathcal{G}$ is $\mu_{\ast\ast}$. Model is well-specified if and only if $\mu_\ast = \mu_{\ast\ast}$. }
\label{Graph010}
\end{figure}
The function $\eta$ corresponds to the \emph{natural} parameter of the exponential family, and it is often more intuitive to consider the \emph{moment} parameter, that is defining functions $\mu: \mathcal{X} \to \mathbb{R}$ that now correspond to moments of outputs~$y$; we will refer to them as \emph{prediction functions}.
Going from natural to moment parameter is known to be done through the gradient of the log-partition function, and we thus consider for $\eta$ a function from $\mathcal{X}$ to $\mathbb{R}$, $\mu(\cdot) = a'(\eta(\cdot))$, and this leads to the performance measure
$$\mathcal{G}(\mu) = \mathcal{F}( (a')^{-1}(\mu(\cdot))).$$
Note now, that the global minimum of $\mathcal{G}$ is reached at $$\mu_{\ast\ast}(x) = {\mathbb E}_{p(y|x)} y.$$
We introduce also the prediction function $\mu_{\ast}(x)$ corresponding to the best $\eta$ which is linear in $\Phi(x)$, that is:
$$\mu_\ast(x) = a'\big(\theta_\ast^\top \Phi(x)\big).$$
We say that the model is well-specified when $\mu_\ast = \mu_{\ast \ast}$, and for these models,
$\inf_{\theta} F(\theta) = \inf_{\mu} \mathcal{G}(\mu)$. However, in general, we only have $\inf_{\theta} F(\theta) > \inf_{\mu} \mathcal{G}(\mu)$ and (very often) the inequality is strict (see examples in our simulations).
To make the further developments more concrete, we now present two classical examples: logistic regression and Poisson regression.
\paragraph{Logistic regression.} The special case of conditional family is logistic regression, where $y \in \{0,1\}$,
$a(t) = \log ( 1 + e^{-t})$ and $a'(t) = \sigma(t) = \frac{1}{1+e^{-t}}$ is the sigmoid function and the probability mass function is given by $p(y|\eta) = \exp(\eta y -\log(1+e^{\eta}))$.
\paragraph{Poisson regression.} One more special case is Poisson regression with $y \in \mathbb{N}$, $a(t) = \exp(t)$ and the response variable $y$ has a Poisson distribution. The probability mass function is given by $p(y|\eta) = \exp(\eta y - e^\eta - \log(y!))$. Poisson regression may be appropriate when the dependent variable is a count, for example in genomics,
network packet analysis, crime rate analysis, fluorescence microscopy, etc.~\citep{hilbe2011negative}.
\subsection{AVERAGING PREDICTIONS}
Recall from \mysec{sgd} that $\pi_\gamma$ is the stationary distribution of $\theta$.
Taking expectation of both parts of Eq. (\ref{eq:sgd}), we get, by using the fact that $\pi_\gamma$ is the limiting distribution
of $\theta_n$ and $\theta_{n-1}$:
\begin{eqnarray*}
& & {\mathbb E}_{\pi_\gamma(\theta_n)}\theta_n \\
& = & {\mathbb E}_{\pi_\gamma(\theta_{n-1})}\theta_{n-1} -\gamma {\mathbb E}_{\pi_\gamma(\theta_{n-1})}{\mathbb E}_{p(x_n,y_n)}f_n'(\theta_{n-1}),
\end{eqnarray*}
which leads to ${\mathbb E}_{\pi_\gamma(\theta)} {\mathbb E}_{p(x_n,y_n)} \nabla f_n(\theta)=0$, that is, now removing the dependence on $n$ (data $(x,y)$ are i.i.d.):
$${\mathbb E}_{\pi_\gamma(\theta)}{\mathbb E}_{p(x,y)}\Big[-y\Phi(x) + a'\big(\Phi(x)^\top\theta\big)\Phi(x) \Big] = 0 ,$$
which finally leads to
\begin{equation}
{\mathbb E}_{p(x)}\Big[{\mathbb E}_{\pi_\gamma(\theta)}a'\big(\Phi(x)^\top\theta\big) - \mu_{**}(x) \Big]\Phi(x) = 0.
\label{limitgrad}
\end{equation}
This is the core equation our method relies on. It does not imply that $b(x) = {\mathbb E}_{\pi_\gamma(\theta)}a'\big(\Phi(x)^\top\theta\big) - \mu_{**}(x)$ is uniformly equal to zero (which we want), but only that
${\mathbb E}_{p(x)} \Phi(x) b(x) = 0$, i.e., $b(x)$ is uncorrelated with $\Phi(x)$.
If the feature vector $\Phi(x)$ is ``large enough'' then this
is equivalent to $b=0$.\footnote{Let $\Phi(x) = (\phi_1(x),\dots, \phi_n(x))^\top$ be an orthogonal basis and $b(x) = \sum_{i=1}^n c_i\phi_i(x) + \varepsilon(x)$, where $\varepsilon(x)$ is small if the basis is big enough. Then $\mathbb{E}_{p(x)}\Phi(x)b(x) = 0 \Leftrightarrow \mathbb{E}\phi_i(x)\big[\sum_{i=1}^n c_i\phi_i(x) + \varepsilon(x)\big] = 0$ for every $i$, and due to the orthogonality of the basis and the smallness of $\varepsilon(x)$: $c_i\cdot\mathbb{E}_{p(x)}\phi^2(x) \approx 0$ and hence $c_i \approx 0$ and thus $b(x)\approx 0$.}
For example, when $\Phi(x)$ is composed of an orthonormal basis of the space of integrable functions (like for kernels in \mysec{kernels}), then this is exactly true.
Thus, in this situation,
\begin{equation}
\label{eq:consistency}\mu_{\ast \ast}(x) = {\mathbb E}_{\pi_\gamma(\theta)}a'\big(\Phi(x)^\top\theta\big),
\end{equation}
and averaging predictions $a'\big(\Phi(x)^\top\theta_n\big)$, along the path $(\theta_n)$ of the Markov chain should exactly converge to the optimal prediction.
This exact convergence is weaker (requires high-dimensional fatures) than for the unconditional family in \mysec{uncond} but it can still bring surprising benefits even when $\Phi$ is not large enough, as we present in \mysec{finited} and \mysec{kernels}.
\subsection{TWO TYPES OF AVERAGING}
\label{sec:avetwotype}
Now we can introduce two possible ways to estimate the prediction function $\mu(x)$.
\paragraph{Averaging estimators.}
The first one is the usual way: we first estimate parameter $\theta$, using Ruppert-Polyak averaging \citep{polyak1992acceleration}: $\bar{\theta}_n = \frac{1}{n+1}\sum_{i=0}^n\theta_i$ and then we denote
$$\bar{\mu}_n(x) = a'( \Phi(x)^\top \bar{\theta}_n )
= a' \Big(
\Phi(x)^\top\textstyle \frac{1}{n+1} \sum_{i=0}^n\theta_i
\Big) $$ the corresponding prediction. As discussed in \mysec{sgd} it converges to $\bar{\mu}_\gamma: x \mapsto
a'( \Phi(x)^\top \bar{\theta}_\gamma )$, which is \emph{not} equal to in general to $a'( \Phi(x)^\top \theta_\ast )$, where $\theta_\ast$ is the optimal parameter in $\mathbb{R}^d$. Since, as presented at the end of \mysec{sgd}, $\bar{\theta}_\gamma - \theta_\ast$ is of order $O(\gamma)$, $F(\bar{\theta}_\gamma) - F(\theta_\ast)$ is of order $O(\gamma^2)$ (because
$\nabla F (\theta_\ast) = 0$), and thus an error of $O(\gamma^2)$ is added to the usual convergence rates in~$O(1/n)$.
Note that we are limited here to prediction functions which corresponds to \emph{linear functions} in $\Phi(x)$ in the natural parameterization, and thus $ F(\theta_\ast) \geqslant \mathcal{G}(\mu_{\ast \ast})$, and the inequality is often strict.
\paragraph{Averaging predictions.}
We propose a new estimator
$$
\bar{\bar{\mu}}_n(x) = \frac{1}{n+1} \sum_{i=0}^n a'(\theta_i^\top \Phi(x) ).
$$
In general $ \mathcal{G} (\bar{\bar{\mu}}_n) -\mathcal{G}(\mu_{\ast\ast})$ does not converge to zero either (unless the feature vector $\Phi$ is large enough and \eq{consistency} is satisfied). Thus, on top of the usual convergence in $O(1/n)$ with respect to the number of iterations, we have an extra term that depends only on $\gamma$, which we will study in \mysec{finited} and \mysec{kernels}.
We denote by $\bar{\bar{\mu}}_\gamma(x)$ the limit when $n \to \infty$, that is, using properties of converging Markov chains,
$\bar{\bar{\mu}}_\gamma(x) = {\mathbb E}_{\pi_\gamma(\theta)}a'\big(\Phi(x)^\top\theta\big)$.
Rewriting Eq. (\ref{limitgrad}) using our new notations, we get:
$$
{\mathbb E} \big[
( \mu_{\ast\ast}(x) - \bar{\bar{\mu}}_\gamma(x))
\Phi(x_n) \big] = 0.
$$
When $\Phi : \mathbb{R} \to \mathbb{R}^d$ is high-dimensional, this leads to $\mu_{\ast\ast} = \bar{\bar{\mu}}_\gamma$ and in contrast to $\bar{\mu}_\gamma$, averaging predictions potentially converge to the optimal prediction.
\paragraph{Computational complexity.}
Usual averaging of estimators~\citep{polyak1992acceleration} to compute $\bar{\mu}_n(x) = a'( \Phi(x)^\top \bar{\theta}_n ) $ is simple to implement as we can simply update the average~$\bar{\theta}_n$ with essentially no extra cost on top the complexity $O(nd)$ of the SGD recursion. Given the number $n$ of training data points and the number $m$ of testing data points, the overall complexity is $O( d(n+m))$.
Averaging prediction functions is more challenging as we have to store all iterates $\theta_i$, $i\!=\!1,\dots,n$, and for each testing point $x$, compute $
\bar{\bar{\mu}}_n(x) = \frac{1}{n+1} \sum_{i=0}^n a'(\theta_i^\top \Phi(x) ).
$
Thus the overall complexity is $O( dn+mnd )$, which could be too costly with many test points (i.e., $m$ large).
There are several ways to alleviate this extra cost: (a) using sketching techniques~\citep{woodruff2014sketching}, (b) using summary statistics like done in applications of MCMC~\citep{gilks1995markov}, or (c) leveraging the fact that all iterates $\theta_i$ will end up being close to $\bar \theta_\gamma$ and use a Taylor expansion of $a'\big(\theta^\top \Phi(x)\big) $ around~$\bar \theta_\gamma$. This expansion is equal to:
$$ a'\big(\Phi(x)^\top \overline{\theta}_\gamma\big) + (\theta-\overline{\theta}_\gamma)^\top \Phi(x) \cdot a''\big(\Phi(x)^\top \overline{\theta}_\gamma\big)+$$
\vspace*{-1cm}
$$+\frac 12\big( (\theta-\overline{\theta}_\gamma )^\top \Phi(x) \big)^{2}\cdot a'''\big(\Phi(x)^\top \overline{\theta}_\gamma\big) + O\big(\Vert \theta-\overline{\theta}_\gamma\Vert^3\big). $$
Taking expectation in both sides above leads to:
$$\bar{\bar{\mu}}_\gamma(x) \approx \bar{\mu}_\gamma(x) + \frac{1}{2}\Phi(x)^\top \mbox{cov\,} (\theta)\cdot \Phi(x)\cdot a'''\big(\overline{\theta}_\gamma^\top\Phi(x)\big),$$
where $ \mbox{cov\,} (\theta)$ is the covariance matrix of $\theta$ under $\pi_\gamma$. This
provides a simple correction to $\bar\mu_\gamma$, and leads to an approximation of $\bar{\bar{\mu}}_n(x)$ as
$$\bar{\mu}_n(x) + \frac 12\ \Phi(x)^\top \mbox{cov}_n(\theta)\ \Phi(x) \cdot a'''\big(\overline{\theta}_n^\top\Phi(x)\big), $$
where $ \mbox{cov}_n(\theta)$ is the empirical covariance matrix of the iterates $(\theta_i)$.
The computational complexity now becomes $O(nd^2 + m d^2)$, which is an improvement when the number of testing points $m$ is large. In all of our experiments, we used this approximation.
\section{FINITE-DIMENSIONAL MODELS}
\label{sec:finited}
In this section we study the behavior of $ \bar{\bar{A}}(\gamma) = \mathcal{G} (\bar{\bar{\mu}}_\gamma) -\mathcal{G}(\mu_{\ast})$ for finite-dimensional models, for which it is usually not equal to zero. We know that our estimators $\bar{\bar{\mu}}_n$ will converge to
$\bar{\bar{\mu}}_\gamma$, and our goal is to compare it to $ \bar{A}(\gamma) = \mathcal{G} ({\bar{\mu}}_\gamma)
-\mathcal{G}(\mu_{\ast}) = F(\bar{\theta}_\gamma) - F(\theta_\ast)$ which is what averaging estimators tends to. We also consider for completeness the non-averaged performance $ {A}(\gamma) = {\mathbb E}_{\pi_\gamma(\theta)} \big[F( {\theta}) - F(\theta_\ast)
\big]$.
Note that we must have ${A}(\gamma) $ and $ \bar{A}(\gamma) $ non-negative, because we compare the negative log-likelihood performances to the one of of the best linear prediction (in the natural parameter), while $ \bar{\bar{A}}(\gamma)$ could potentially be negative (it will in certain situations), because the the corresponding natural parameters may not be linear in $\Phi(x)$.
We consider the same standard assumptions as~\citet{dieuleveut2017bridging}, namely smoothness of the negative log-likelihoods $f_n(\theta)$ and strong convexity of the expected negative log-likelihood $F(\theta)$. We first recall the results from~\citet{dieuleveut2017bridging}. See detailed explicit formulas in the supplementary material.
\subsection{EARLIER WORK}
\paragraph{Without averaging.}
We have that $A(\gamma) = \gamma B + O(\gamma^{3/2})$, that is $\gamma$ is \emph{linear} in $\gamma$, with $B$ non-negative.
\paragraph{Averaging estimators.} We have that $\bar{A}(\gamma) = \gamma^2 \bar{B} + O(\gamma^{5/2})$, that is $\bar{A}$ is \emph{quadratic} in $\gamma$, with $\bar B$ non-negative. Averaging does provably bring some benefits because the order in $\gamma$ is higher (we assume $\gamma$ small).
\subsection{AVERAGING PREDICTIONS}
We are now ready to analyze the behavior of our new framework of averaging predictions. The following result is shown in the supplementary material.
\begin{proposition}
Under the assumptions on the negative loglikelihoods $f_n$ of each observation from~\citet{dieuleveut2017bridging}:
\begin{itemize}
\item In the case of well-specified data, that is, there exists $\theta_\ast$ such that for all $(x,y)$, $p(y|x) = q(y|x,\theta_\ast)$, then $\bar{\bar{A}} \sim \gamma^2\bar{\bar{B}}^{\rm well}$, where $\bar{\bar{B}}^{\rm well}$ is a positive constant.
\item In the general case of potentially mis-specified data, $\bar{\bar{A}} = \gamma\bar{\bar{B}}^{\rm mis} + O(\gamma^2)$, where $\bar{\bar{B}}^{\rm mis}$ is constant which may be positive or negative.
\end{itemize}
\end{proposition}
Note, that in contrast to averaging parameters, the constant $\bar{\bar{B}}^{\rm mis}$ can be negative. It means, that we obtain the estimator better than the optimal linear estimator, which is the limit of capacity for averaging parameters. In our simulations, we show examples for which $\bar{\bar{B}}^{\rm mis}$ is positive, and examples for which it is negative. Thus, in general, for low-dimensional models, averaging predictions can be worse or better than averaging parameters. However, as we show in the next section, for infinite dimensional models, we always get convergence.
\section{INFINITE-DIMENSIONAL MODELS}
\label{sec:kernels}
Recall, that we have the following objective function to minimize:
\begin{equation}
F(\theta) = {\mathbb E}_{x,y}\Big[ -y \cdot \eta_\theta(x) + a\big(\eta_\theta(x)\big) \Big],
\end{equation}
where till this moment we consider unknown functions $\eta_\theta(x)$ which were linear in $\Phi(x)$ with $\Phi(x) \in \mathbb{R}^d$, leading to a complexity in $O(dn)$.
We now consider infinite-dimensional features, by considering that $\Phi(x) \in \mathcal{H}$, where $\mathcal{H}$ is a Hilbert space. Note that this corresponds to modeling the function $\eta_\theta$ as a Gaussian process~\citep{rasmussen2004gaussian}.
This is computationally feasible through the usual ``kernel trick'' \citep{scholkopf2001learning,shawe2004kernel}, where we assume that the kernel function $k(x,y) = \langle \Phi(x), \Phi(y) \rangle$ is easy to compute.
Indeed, following~\citet{bordes2005fast} and \citet{Die_Bac_2015}, starting from $\theta_0$, each iterate of constant-step-size SGD is of the form
$\theta_n = \sum_{t=1}^n\alpha_t\Phi(x_t)$, and the gradient descent recursion
$\theta_n = \theta_{n-1} - \gamma [ a' (\eta_{\theta_{n-1}}(x_n) )-y_n ] \Phi(x_n)$
leads to the following recursion on $\alpha_t$'s:\begin{eqnarray*}
\alpha_n & = &\textstyle -\gamma \big[a'\Big(\sum_{t=1}^{n-1}\alpha_t \langle \Phi(x_t), \Phi(x_n) \rangle\big) - y_n \big] \\
& = & \textstyle -\gamma \big[a'\big(\sum_{t=1}^{n-1}\alpha_t k(x_t,x_n) \big) - y_n \big].
\end{eqnarray*}
This leads to $\eta_{\theta_n}(x)=\langle\Phi(x),\theta_n \rangle$ and $\mu_{\theta_n}(x) = a'\big(\eta_{\theta_n}(x)\big)$
with
$$\eta_{\theta_n}(x) = \sum\limits_{t=1}^n\alpha_t \langle \Phi(x),\Phi(x_t) \rangle =\sum\limits_{t=1}^n\alpha_t k(x,x_t) , $$
and finally we can express $\bar{\bar{\mu}}_n(x)$ in kernel form as:
$$\bar{\bar{\mu}}_n(x) = \frac{1}{n+1}\sum\limits_{t=0}^na'\Big[\sum\limits_{l=1}^t\alpha_l\cdot k(x,x_l)\Big]. $$
There is also a straightforward estimator for averaging parameters, i.e., $\bar{\mu}_n(x) = a'\Big(\frac {1}{n+1}\sum\limits_{t=0}^n\sum\limits_{l=1}^t\alpha_l k(x,x_l)\Big).$
If we assume that the kernel function is \emph{universal}, that is, $\mathcal{H}$ is dense in the space of squared integrable functions, then it is known that if ${\mathbb E}_x b(x) \Phi(x) = 0$, then $b=0$~\citep{sriperumbudur2008injective}. This implies that we must have $\bar{\bar{\mu}}_\gamma =0$ and thus averaging predictions does always converge to the global optimum (note that in this setting, we must have a well-specified model because we are in a non-parametric setting).
\paragraph{Column sampling.}
Because of the usual high running-time complexity of kernel method in $O(n^2)$, we consider a ``column-sampling approximation''~\citep{williams2001using}. We thus choose a small subset $I = (x_1,\dots, x_m)$ of samples and construct a new finite $m$-dimensional feature map $\bar{\Phi}(x) = K(I,I)^{-1/2}K(I,x) \in {\mathbb R}^{m }$, where $K(I,I)$ is the $m \times m$ kernel matrix of the $m$ points and
$K(I,x) $ the vector composed of kernel evaluations $k(x_i,x)$. This allows a running-time complexity in $O(m^2n)$. In theory and practice, $m$ can be chosen small~\citep{bach2013sharp,rudi2017falkon}.
\paragraph{Regularized learning with kernels.}
Although we can use an unregularized recursion with good convergence properties~\citep{Die_Bac_2015}, adding a regularisation by the squared Hilbertian norm is easier to analyze and more stable with limited amounts of data. We thus consider the recursion (in Hilbert space), with $\lambda$ small:
\begin{eqnarray*}
\theta_n \!\!\!& = & \!\!\!\theta_{n-1} - \gamma \big[ f_n'(\theta_{n-1}) + \lambda \theta_{n-1} \big]
\\
& = & \!\!\! \theta_{n-1} + \gamma ( y_n - a'( \langle\Phi(x_n), \theta \rangle ) ) \Phi(x_n) - \gamma \lambda\theta_{n-1} .
\end{eqnarray*}
This recursion can also be computed efficiently as above using the kernel trick and column sampling approximations.
In terms of convergence, the best we can hope for is to converge to the minimizer $\theta_{\ast ,\lambda}$ of the regularized expected negative log-likelihood $F(\theta) + \frac{\lambda}{2} \| \theta\|^2$ (which we assume to exist). When $\lambda$ tends to zero, then
$\theta_{\ast ,\lambda}$ converges to $\theta_\ast$.
Averaging \emph{parameters} will tend to a limit $\bar{\theta}_{\gamma,\lambda}$ which is $O(\gamma)$-close to $\theta_{\ast ,\lambda}$, thus leading to predictions which deviate from the optimal predictions for two reasons: because of regularization and because of the constant step-size. Since $\lambda$ should decrease as we get more data, the first effect will vanish, while the second will not.
When averaging \emph{predictions}, the two effects will vanish as $\lambda$ tends to zero. Indeed, by taking limits of the gradient equation, and denoting by $\bar{\bar{\mu}}_{\gamma,\lambda}$ the limit of $\bar{\bar{{\mu}}}_n$, we have
\begin{equation}
{\mathbb E} \big[
( \mu_{\ast\ast}(x) - \bar{\bar{\mu}}_{\gamma,\lambda}(x))
\Phi(x) \big] = \lambda \bar{\theta}_{\gamma, \lambda}.
\label{RegKer}
\end{equation}
Given that $\bar{\theta}_{\gamma, \lambda}$ is $O(\gamma)$-away from $\theta_\ast$, if we assume that $\theta_\ast$ corresponds to a sufficiently regular\footnote{While our reasoning is informal here, it can be made more precise by considering so-called ``source conditions'' commonly used in the analysis of kernel methods~\citep{caponnetto2007optimal}, but this is out of the scope of this paper.} element of the Hilbert space $\mathcal{H}$, then the $L_2$-norm of the deviation satisfies $ \| \mu_{\ast\ast}(x) - \bar{\bar{\mu}}_{\gamma,\lambda} \| = O(\lambda)$ and thus as the regularization parameter $\lambda$ tends to zero, our predictions tend to the optimal one.
\section{EXPERIMENTS}
\label{sec:exp}
In this section, we compare the two types of averaging (estimators and predictions) on a variety of problems, both on synthetic data and on standard benchmarks. When averaging predictions, we always consider the Taylor expansion approximation presented at the end of \mysec{avetwotype}.
\subsection{SYNTHETIC DATA}
\paragraph{Finite-dimensional models.}
we consider the following logistic regression model:
$$q(y|x,\theta) = \exp\big(y\cdot \eta_\theta(x) - a(\eta_\theta(x))\big), $$
where we consider a linear model $\eta_\theta(x) = \theta^\top x$ in $x$ (i.e., $\Phi(x) = x$), the link function $a(t) = \log(1+e^t)$ and $a'(t) = \sigma(t)$ is the sigmoid function.
Let $x$ be distributed as a standard normal random variable in dimension $d=2$, $y\in \{0,1\}$ and ${\mathbb P}(y=1|x) = \mu_{\ast\ast}(x) = \sigma\big(\eta_{\ast\ast}(x)\big)$, where we consider two different settings:
\begin{itemize}
\item Model 1: $\eta_{\ast\ast}(x) = \sin x_1 + \sin x_2$,
\item Model 2: $\eta_{\ast\ast}(x) = x_1^3+x_2^3 $.
\end{itemize}
The global minimum ${\mathcal F}_{\ast\ast}$ of the corresponding optimization problem can be found as
$${\mathcal F}_{\ast\ast} = {\mathbb E}_{p(x)}\big[-\mu_{\ast\ast}(x)\cdot \eta_{\ast\ast}(x) + a(\eta_{\ast\ast}(x))\big]. $$
We also introduce the performance measure ${\mathcal F}(\eta)$
\begin{equation}
{\mathcal F}(\eta) = {\mathbb E}_{p(x)}\big[-\mu_{\ast\ast}(x)\cdot \eta(x) + a(\eta(x)) \big],
\label{Fofeta}
\end{equation}
which can be evaluated directly in the case of synthetic data. Note that in our situation, the model is misspecified because $\eta_{\ast\ast}(x)$ is not linear in $\Phi(x) = x$, and thus, $\inf_\theta F(\theta) > {\mathcal F}_{\ast\ast} $, and thus our performance measures ${\mathcal F}(\mu_n) - {\mathcal F}_{\ast\ast} $ for various estimators $\mu_n$ will not converge to zero.
The results of averaging 10 replications are shown in Fig.~\ref{Graph01} and Fig.~\ref{Graph02}.
We first observed that constant step-size SGD without averaging leads to a bad performance.
Moreover, we can see, that in the first case (Fig.~\ref{Graph01}) averaging predictions beats averaging parameters, and moreover beats the best linear model: if we use the best linear error ${\mathcal F}_{*}$ instead of ${\mathcal F}_{**}$, at some moment ${\mathcal F}(\eta_n)-{\mathcal F}_{\ast}$ becomes negative. However in the second case (Fig.~\ref{Graph02}), averaging predictions is not superior to averaging parameters. Moreover, by looking at the final differences between performances with different values of $\gamma$, we can see the dependency of the final performance in $\gamma$ for averaging predictions, instead of $\gamma^2$ for averaging parameters (as suggested by our theoretical results in \mysec{finited}). In particular in Fig.~\ref{Graph01}, we can observe the surprising behavior of a larger step-size leading to a better performance (note that we cannot increase too much otherwise the algorithm would diverge).
\begin{figure}
\includegraphics[width=0.5\textwidth]{Graph01.pdf}
\vspace*{-.415cm}
\caption{Synthetic data for linear model $\eta_\theta(x) = \theta^\top x $ and $\eta_{\ast\ast}(x) = \sin x_1 + \sin x_2$. Excess prediction performance vs. number of iterations (both in log-scale).}
\label{Graph01}
\end{figure}
\begin{figure}
\includegraphics[width=0.5\textwidth]{Graph02.pdf}
\vspace*{-.415cm}
\caption{Synthetic data for linear model $\eta_\theta(x) = \theta^\top x $ and $\eta_{\ast\ast}(x) = x_1^3+x_2^3$. Excess prediction performance vs. number of iterations (both in log-scale). }
\label{Graph02}
\end{figure}
\begin{figure}
\includegraphics[width=0.47\textwidth]{Graph03.pdf}
\vspace*{-.415cm}
\caption{Synthetic data for Laplacian kernel for $\eta_{\ast\ast}(x) = \frac{5}{5+x^\top x}$. Excess prediction performance vs. number of iterations (both in log-scale). }
\label{Graph03}
\end{figure}
\paragraph{Infinite-dimensional models} Here we consider the kernel setup described in \mysec{kernels}. We consider Laplacian kernels $k(s,t) = \exp\big(\frac{\Vert s-t\Vert_1}{\sigma}\big)$ with $\sigma = 50$, dimension $d=5$ and generating log odds ratio $\eta_{\ast\ast}(x) = \frac{5}{5+x^\top x}$. We also use a squared norm regularization with several values of $\lambda$ and column sampling with $m=100$ points. We use the exact value of ${\mathcal F}_{\ast \ast}$ which we can compute directly for synthetic data. The results are shown in Fig.~\ref{Graph03}, where averaging predictions leads to a better performance than averaging estimators.
\subsection{REAL DATA}
Note, that in the case of real data, one does not have access to unknown $\mu_{\ast\ast}(x)$ and computing the performance measure in Eq.~(\ref{Fofeta}) is inapplicable. Instead of it we use its sampled version on held out data:
$$\hat{{\mathcal F}}(\eta) = -\sum\limits_{i:y_i=1}\log\big(\mu(x_i)\big) - \sum\limits_{j:y_j=0}\log\big(1-\mu(x_i)\big). $$
We use two datasets, with $d$ not too large, and $n$ large, from \citep{Lichman2013}: the ``MiniBooNE particle identification'' dataset ($d=50$, $n=130\ 064$), the ``Covertype'' dataset ($d=54$, $n=581\ 012$).
We use two different approaches for each of them: a linear model $\eta_\theta(x)=\theta^\top x$ and a kernel approach with Laplacian kernel $k(s,t) = \exp\big(\frac{\Vert s-t\Vert_1}{\sigma}\big)$, where $\sigma = {d}$. The results are shown in Figures \ref{Graph04} to \ref{Graph07}.
Note, that for linear models we use $\hat{{\mathcal F}}_*$--the estimator of the best performance among linear models (learned on the test set, and hence not reachable from learning on the training data), and for kernels we use $\hat{{\mathcal F}}_{\ast\ast}$ (same definition as $\hat{{\mathcal F}}_*$ but with the kernelized model), that is why graphs are not comparable (but, as shown below, the value of $\hat{{\mathcal F}}_{\ast\ast}$ is lower than the value of $\hat{{\mathcal F}}_*$ because using kernels correspond to a larger feature space).
For the ``MiniBooNE particle identification'' dataset $\hat{{\mathcal F}}_{\ast} = 0.35$ and $\hat{{\mathcal F}}_{\ast\ast} = 0.21; $ for the``Covertype'' dataset $\hat{{\mathcal F}}_{\ast} = 0.46$ and $\hat{{\mathcal F}}_{\ast\ast} = 0.39$. We can see from the four plots that, especially in the kernel setting, averaging predictions also shows better performance than averaging parameters.
\begin{figure}
\includegraphics[width=0.5\textwidth]{Graph04.pdf}
\vspace*{-.415cm}
\caption{MiniBooNE dataset, dimension $d=50$, linear model. Excess prediction performance vs. number of iterations (both in log-scale).}
\label{Graph04}
\end{figure}
\begin{figure}
\includegraphics[width=0.5\textwidth]{Graph05.pdf}
\vspace*{-.415cm}
\caption{MiniBooNE dataset, dimension $d=50$, kernel approach, column sampling $m=200$. Excess prediction performance vs. number of iterations (both in log-scale).}
\label{Graph05}
\end{figure}
\section{CONCLUSION}
In this paper, we have explored how averaging procedures in stochastic gradient descent, which are crucial for fast convergence, could be improved by looking at the specifics of probabilistic modeling. Namely, averaging in the moment parameterization can have better properties than averaging in the natural parameterization.
\begin{figure}
\includegraphics[width=0.5\textwidth]{Graph06.pdf}
\vspace*{-.415cm}
\caption{CoverType dataset, dimension $d=54$, linear model. Excess prediction performance vs. number of iterations (both in log-scale).}
\label{Graph06}
\end{figure}
While we have provided some theoretical arguments (asymptotic expansion in the finite-dimensional case, convergence to optimal predictions in the infinite-dimensional case), a detailed theoretical analysis with explicit convergence rates would provide a better understanding of the benefits of averaging predictions.
\begin{figure}
\includegraphics[width=0.5\textwidth]{Graph07.pdf}
\caption{CoverType dataset, dimension $d=54$, kernel approach, column sampling $m=200$. Excess prediction performance vs. number of iterations (both in log-scale).}
\label{Graph07}
\end{figure}
\subsection*{Acknowledgements} The research leading to these results has received funding from the European Union's H2020 Framework Programme (H2020-MSCA-ITN-2014) under grant agreement n$^o$ 642685 MacSeNet, and from
the European Research Council (grant SEQUOIA 724063).
|
{
"timestamp": "2018-11-22T02:10:26",
"yymm": "1804",
"arxiv_id": "1804.05567",
"language": "en",
"url": "https://arxiv.org/abs/1804.05567"
}
|
\section{Introduction}
Let $G=(V(G),E(G))$ be an undirected graph. An \textit{independent set} in $G$ is a subset of pairwise non-adjacent vertices. The \emph{independence number} of $G$, denoted by $\alpha(G)$, is the largest possible size of an independent set in $G$. For two graphs $G$ and $H$, their \emph{strong product} $G\boxtimes H$ is a graph such that
\begin{enumerate}
\item the vertex set of $G\boxtimes H$ is the Cartesian product $V(G) \times V(H)$; and
\item any two distinct vertices $(u,u')$ and $(v,v')$ are adjacent in $G\boxtimes H$ if
$u\sim v$ and $u'=v'$, or $u=v$ and $u'\sim v'$, or $u\sim v$ and $u'\sim v'$.
\end{enumerate}
The graph $G^n$ is defined inductively by $G^{n}= G^{n-1} \boxtimes G$.
The \emph{Shannon capacity} of a graph $G$ is defined by
\begin{align}\label{eq:shannon_cap_def}
\Theta(G)
:=\sup_{n}\sqrt[\leftroot{-3}\uproot{3}n]{\alpha(G^n)}
=\lim_{n\to\infty}\sqrt[\leftroot{-3}\uproot{3}n]{\alpha(G^n)}
\end{align}
where the limit exists by the supermultiplicativity of $\alpha(G^n)$ and Fekete's Lemma.
This graph quantity was introduced by Shannon~\cite{Sh1956} as the zero-error capacity in the context of channel coding.
In this setup, a transmitter would like to communicate a message to a receiver through the channel, and the receiver must decode the message without error.
This problem can be equivalently cast in terms of the
\textit{confusion graph} $G$ associated with the channel. The vertices of the confusion graph are the input symbols, and two vertices are adjacent if the corresponding inputs can result in the same output.
It is easy to check that $G^n$ is the confusion graph for $n$ uses of the channel, and that $\alpha(G^n)$ is the maximum number of messages that can be transmitted without error over $n$ uses of the channel.
Despite the apparent simplicity of the problem, a general characterization of $\Theta(G)$ remains elusive. Several lower and upper bounds were obtained by Shannon~\cite{Sh1956}, Lov\'asz~\cite{Lo1979} and Haemers~\cite{Ha1979}. These bounds are briefly reviewed in Section~\ref{sec:known bounds}.
In Section~\ref{sec:new bound} we present a new bound on the Shannon capacity via a variation on the linear program pertaining to the fractional independence number of the graph.
Next, we show that the new bound can simultaneously outperform both the Lov\'asz theta number and the Haemers minimum rank bound.
In Section~\ref{sec:IndexCoding}, we leverage our bound to prove a new upper bound for the broadcast rate of Index Coding.
It should be noted that a fractional version of the Haemers minimum rank bound, denoted $\eta_f^{\FF}$, was introduced independently
by Blasiak~\cite{Blasiak2013} and Shanmugam et al.~\cite{ShanmugamAsterisDimakis2015}, and investigated in more detail by Bukh and Cox~\cite{BukhCox2018} very recently.
This bound is at least as good as $\eta_{\FF}^*$, one of our new bounds. Nevertheless $\eta_f^{\FF}$ is very difficult to compute,
and our $\eta_{\FF}^*$ bound is more tractable and provides a feasible way to approach $\eta_f^{\FF}$ (see Remark~\ref{FractionalHaemersBound} below for more details).
\section{Upper Bounds on the Shannon Capacity}\label{sec:known bounds}
In this section, we give a brief overview of three well-known upper bounds on the Shannon capacity. Throughout this section let $G$ be a graph with vertex set $V(G)=\{1,2,\dots,m\}$.
\subsection{Fractional Independence Number}
The fractional independence number is the linear programming relaxation of the $0$-$1$ integer linear programming that computes the independence number.
More precisely, the \emph{fractional independence number} $\alpha_f(G)$ is defined as the optimal value of the following linear program:
\begin{align}
\label{Primal}
\begin{split}
\textup{maximize } &\sum_{x} w(x) \\
\textup{subject to } & \sum_{x\in C}w(x) \le 1 \textup{ for every clique $C$ in $G$},\\
& w(x)\ge 0.
\end{split}
\end{align}
(A clique $C$ in $G$ is a subset of the vertices, $C\subseteq V(G)$, such that every two distinct vertices are adjacent in $G$.)
From the duality theorem of linear programming, $\alpha_f(G)$ can also be computed as follows:
\begin{align}
\label{Dual}
\begin{split}
\textup{minimize } &\sum_{C} q(C) \\
\textup{subject to } &\sum_{C\owns x}q(C) \ge 1 \textup{ for each vertex $x$ in $G$},\\
&q(C)\ge 0.
\end{split}
\end{align}
(The optimal value of~\eqref{Dual} is also called the \emph{fractional clique-cover number} of $G$, and denoted as $\overline{\chi}_f(G)$.)
The following bound was first given by Shannon~\cite{Sh1956}, and was also discussed in detail by Rosenfeld~\cite{Ro1967}.
\begin{theorem}\cite[Theorem 7]{Sh1956}
\label{ShannonBound}
$\Theta(G)\le\alpha_f(G)$.
\end{theorem}
\subsection{Lov\'asz Theta Number}
In his seminal paper~\cite{Lo1979}, Lov\'asz solved the long-standing problem of the Shannon capacity of the pentagon graph, by introducing an important new graph invariant, called the Lov\'asz theta number.
An \emph{orthonormal representation of $G$} is a system of unit vectors $\bfv_1,\dots,\bfv_m$ in some Euclidean space $\RR^d\ (d\ge 1)$ such that if $i$ and $j$ are nonadjacent then $\bfv_i$ and $\bfv_j$ are orthogonal (all vectors will be column vectors). The \emph{Lov\'asz theta number} of $G$ is defined as
\begin{align*}
\vartheta(G):=\min_{\bfc,\bfv_i}\max_{1\le i\le m}\frac{1}{(\bfc^T\bfv_i)^2}
\end{align*}
where the minimum is taken over all unit vectors $\bfc$ and all orthonormal representations $\{\bfv_1,\dots,\bfv_m\}$ of $G$.
The following bound is the main result of~\cite{Lo1979}.
\begin{theorem}\cite[Theorem 1]{Lo1979}
\label{LovaszThetaNumber}
$\Theta(G)\le\vartheta(G).$
\end{theorem}
In the sequel we will also need the following results from~\cite{Lo1979}. The theta number of odd cycles was calculated by Lov\'asz~\cite{Lo1979}.
\begin{proposition}\cite[Corollary 5]{Lo1979}
\label{LovaszThetaNumberOfOddCycles}
For odd $n$,
\begin{align*}
\vartheta(C_n) = \frac{n\cos(\pi/n)}{1+\cos(\pi/n)}.
\end{align*}
\end{proposition}
In particular, for the pentagon graph, Lov\'asz proved that $\Theta(C_5)\le\vartheta(C_{5})=\sqrt{5}$, which meets the lower bound given by Shannon~\cite{Sh1956}.
Also, there exists the following duality between $G$ and its complementary graph $\overline{G}$.
\begin{proposition}\cite[Theorem 5]{Lo1979}
\label{LovaszThetaDuality}
Let $\bfd$ range over all unit vectors and let $\{\bfu_1,\dots,\bfu_m\}$ range over all orthonormal representations of $\overline{G}$. Then
\begin{align}\label{LovaszDual}
\vartheta(G)=\max_{\bfd,\bfu_i}\sum_{i=1}^{m}(\bfd^T\bfu_i)^2.
\end{align}
\end{proposition}
For two graphs $G=(V(G),E(G))$ and $H=(V(H),E(H))$, their \emph{disjoint union}, denoted as $G+H$, is the graph whose vertex set is the disjoint union of $V(G)$ and $V(H)$ and whose edge set is the disjoint union of $E(G)$ and $E(H)$.
The Lov\'asz theta number is multiplicative with respect to the strong product,
and additive with respect to the disjoint union.
\begin{proposition}\label{prop:LovaszProperty}
\ \\[-5mm]
\begin{enumerate}
\item
\cite[Theorem 7]{Lo1979}
$\vartheta(G\boxtimes H)=\vartheta(G)\cdot\vartheta(H)$.
\item
\cite[Section 18]{Knuth1994}
$\vartheta(G+H)=\vartheta(G)+\vartheta(H)$.
\end{enumerate}
\end{proposition}
\subsection{Haemers Minimum Rank Bound}\label{Minrank}
Haemers~\cite{Ha1979,Ha1981} proved a very useful upper bound based on the matrix rank as follows.
An $m\times m$ matrix $B$ over some field is said to \emph{fit} $G$ if $B_{ii}\ne0$ for $1\le i\le m$, and $B_{ij}=0$ whenever vertices $i$ and $j$ are nonadjacent for $1\le i,j\le m$ and $i\ne j$. Let $B^{\otimes n}$ denote the Kronecker product of $n$ copies of $B$. It is easy to verify that if $B$ fits $G$, then $B^{\otimes n}$ fits $G^n$.
\begin{theorem}\cite{Ha1981}
\label{HaemersBound}
If a matrix $B$ fits a graph $G$, then $\Theta(G)\le\textup{rank}(B)$.
\end{theorem}
For a graph $G$, Haemers~\cite{Ha1981} introduced the following graph invariant
\begin{align*}
\eta(G):=\min\{\textup{rank}(B): B\textup{ fits }G\},
\end{align*}
where the minimization is taken over all fields.
By Theorem~\ref{HaemersBound} it follows that $\Theta(G)\le\eta(G)$. Moreover, for a fixed field $\FF$, define
\begin{align*}
\eta_{\FF}(G):=\min\{\textup{rank}(B): B\textup{ over $\FF$ fits }G\}.
\end{align*}
It is easy to verify that $\eta_{\FF}$ is submultiplicative with respect to the strong product and additive with respect to the disjoint union, i.e., for any two graphs $G$ and $H$,
\begin{align*}
\eta_{\FF}(G\boxtimes H)&\le\eta_{\FF}(G)\cdot\eta_{\FF}(H),\\
\eta_{\FF}(G + H)&=\eta_{\FF}(G)+\eta_{\FF}(H).
\end{align*}
The following example is provided by Haemers~\cite{Ha1979} to answer some problems raised in~\cite{Lo1979}.
\begin{example}{\rm\cite{Ha1979}\label{ex:Schlaefli}
Let $G$ be the complement of the Schl\"afli graph, which is the unique strongly regular
graph\footnote{A strongly regular graph with parameters $(v,k,\lambda,\mu)$ is a regular graph with $v$ vertices and degree $k$
such that every two adjacent vertices have $\lambda$ common neighbours, and
every two non-adjacent vertices have $\mu$ common neighbours.}
with parameters $(27,16,10,8)$. Let $A$ be the adjacency matrix of $G$, and let $I$ be the identity matrix of order $27$.
Then the matrix $A-I$ fits the graph $G$, and its rank over $\RR$ is equal to $7$. Hence $\eta(G)\le\eta_{\RR}(G)\le7<9=\vartheta(G)$. This improves the bound given by the Lov\'asz theta number. Moreover, Tims~\cite[Example 3.8]{Tims2013} proved that $\eta_{\FF}(G)\ge7$ over any field $\FF$, and therefore $\eta(G)=7$. Similarly, the rank of the matrix $A-I$ over the field $\FF_{11}$ is also equal to $7$, hence $\eta_{\FF_{11}}(G)=7$ (this fact will be used in Example~\ref{IndexCodingSchlaefliModify} and Remark~\ref{BukhCox}).
}
\end{example}
\section{A Linear Programming Variation}\label{sec:new bound}
In this section we will prove our main result, providing a new upper bound on the Shannon capacity by a variation of the linear programming bound given in~\eqref{Primal}-\eqref{Dual}.
For a subset $S\subset V(G)$, the \emph{induced subgraph} $G_S$ is the graph whose vertex set is $S$ and whose edge set consists of all of the edges in $E$ that have both endpoints in $S$.
Let $f$ be a real-valued function defined on graphs, and
let $f^*(G)$ be the optimal value of the following linear program:
\begin{align}\label{PrimalEx}
\begin{split}
\textup{maximize } &\sum_{x} w(x) \\
\textup{subject to } &\sum_{x\in S}w(x)\le f(G_S)\textup{ for each subset $S$ of } V(G),\\
& w(x)\ge 0.
\end{split}
\end{align}
By duality $f^*(G)$ can also be computed as follows:
\begin{align}\label{DualEx}
\begin{split}
\textup{minimize } &\sum_{S} q(S)f(G_S) \\
\textup{subject to } &\sum_{S\owns x}q(S) \ge 1 \textup{ for each vertex $x$ in $G$},\\
&q(S)\ge 0.
\end{split}
\end{align}
\begin{remark}\label{remark:1}
{\rm
The non-negative real-valued function $q:2^{V(G)}\rightarrow \RR$ in~\eqref{DualEx}, satisfying that $\sum_{S\owns x}q(S) \ge 1$ for each vertex $x$ in $G$, is called a \emph{fractional cover} of $G$ by
K\"orner, Pilotto and Simonyi~\cite{KornerPilottoSimonyi2005}. The fractional cover is used to generalize the local chromatic number to provide an upper bound for the Sperner capacity of directed graphs\footnote{The Sperner capacity of directed graphs is a natural generalization of the Shannon capacity of undirected graphs. See~\cite{KornerPilottoSimonyi2005} for definitions of the local chromatic number and the Sperner capacity.}, cf.~\cite[Theorem
6]{KornerPilottoSimonyi2005}. Note that for undirected graphs, the bounds in~\cite{KornerPilottoSimonyi2005} are always no stronger than the fractional independence number, and hence are not useful upper bounds for the Shannon capacity.
}
\end{remark}
By taking $q(S)=1$ for $S=V(G)$ and $q(S)=0$ otherwise, it is readily verified that $f^*(G)\leq f(G)$. In the next two lemmas, we show that if the function $f$ satisfies certain properties, then these properties are also inherited by $f^*$. We say that $f$ is an upper bound on the independence number if $\alpha(G)\leq f(G)$ for any graph $G$.
\begin{lemma}\label{UpperBounded}
If $f$ is an upper bound on the independence number, then so is $f^*$.
\end{lemma}
\begin{proof}
This result can be proved directly using the primal linear program~\eqref{PrimalEx}. However we would like to present a different proof using the dual linear program~\eqref{DualEx} and a counting argument as follows.
Let $\Gamma$ be any independent set in $G$ and $q$ a fractional cover of $G$. Since $f(G_S)\ge\alpha(G_S)$, we have
\begin{align*}
\sum_{S}q(S)f(G_S)
&\ge \sum_{S}q(S)\alpha(G_S) \\
&\ge \sum_{S}q(S)\alpha(G_{S\cap\Gamma}) \\
&= \sum_{x\in\Gamma}\sum_{S\owns x}q(S)\\
&\ge |\Gamma|.
\end{align*}
This proves the result.
\end{proof}
We say that $f$ is \emph{submultiplicative} (with respect to the strong product) if for any two graphs $G$ and $H$, $f(G\boxtimes H)\le f(G)f(H)$.
\begin{lemma}
\label{Submultiplicative}
If $f$ is submultiplicative, then so is $f^*$.
\end{lemma}
\begin{proof}
Let $q_1$ and $q_2$ be optimal solutions of the linear program~\eqref{DualEx} for $G$ and $H$ respectively. Now we assign weights $q(W)$ to each subset $W$ of $V(G\boxtimes H)$ as follows: if $W=S\times T$ for some $S\subset V(G)$ and $T\subset V(H)$, then we set $q(W)=q(S\times T)=q_1(S)q_2(T)$; otherwise we set $q(W)=0$. Then for each vertex $(x,y)$ in $V(G\boxtimes H)$, we have
\begin{align*}
\sum_{W\owns (x,y)} q(W) &= \sum_{S\owns x, T\owns y}q(S\times T)\\
&=\sum_{S\owns x}q_1(S)\sum_{T\owns y}q_2(T)\\
&\ge1.
\end{align*}
So $q$ is a feasible solution for~\eqref{DualEx}, and
\begin{align*}
f^*(G\boxtimes H) \le&\sum_{W}q(W)f((G\boxtimes H)_{W})\\
=&\sum_{S,T}q(S\times T)f((G\boxtimes H)_{S\times T})\\
=&\sum_{S,T}q_{1}(S)q_{2}(T)f(G_S\boxtimes H_T)\\
\le& \sum_{S}q_1(S)f(G_S)\sum_{T}q_2(T)f(H_{T})\\
=&f^*(G) \cdot f^*(H)
\end{align*}
in which the second inequality follows from the submultiplicativity of $f$.
This proves the result.
\end{proof}
Now we can prove the following upper bound on the Shannon capacity.
\begin{theorem}\label{NewBound}
Let $f$ be a submultiplicative upper bound on the independence number. Then, $$\Theta(G)\le f^*(G).$$
\end{theorem}
\begin{proof}
By Lemma~\ref{UpperBounded} and Lemma~\ref{Submultiplicative}, we get $\alpha(G^n)\le f^*(G^n)\le f^*(G)^n$.
\end{proof}
Any function $f$ that is a submultiplicative upper bound on the independence number forms by itself an upper bound on the Shannon capacity, i.e., $\Theta(G)\leq f(G)$. Combining this with Theorem~\ref{NewBound} and the fact that $f^*(G)\leq f(G)$ we get $\Theta(G) \leq f^*(G) \leq f(G).$
Simply put, this chain of inequalities shows that $f^*$
is a bound that is at least as good as the bound $f$ that we started with in the first place. An immediate question is, can we get the strict inequality $f^*(G)<f(G)$? In other words, can we improve the bound $f$ on the Shannon capacity by solving the corresponding linear programming problem? In the sequel, we give an affirmative answer to this question by providing several explicit examples where a strict inequality holds.
Furthermore, we answer the following two natural questions: 1) which functions $f$ should we use in Theorem~\ref{NewBound}? and 2) do we always get a tighter upper bound for any function $f$?
Before we proceed to answer those questions, we show some simple properties of $f^*$, which are used later.
We say that $f$ is \emph{superadditive} with respect to the disjoint union if $f(G+H)\ge f(G)+f(H)$ for any two graphs $G$ and $H$.
\begin{proposition}\label{prop:properties}
\ \\[-5mm]
\begin{enumerate}
\item If $f(C)=1$ for each clique $C$ in $G$, then $f^*(G)\le\alpha_f(G)$. In particular, $\eta_{\FF}^*(G)\le\alpha_f(G)$.
\item $f^*(G+H)\le f^*(G)+f^*(H)$.
\item If $f$ is superadditive, then $f^*(G+H)=f^*(G)+f^*(H)$. In particular, $\eta_{\FF}^*(G+H)=\eta_{\FF}^*(G)+\eta_{\FF}^*(H).$
\end{enumerate}
\end{proposition}
\begin{proof}
1) Follows directly from~\eqref{PrimalEx}.
2) Follows directly from~\eqref{DualEx}.
3) Let $w_1$ and $w_2$ be optimal solutions of the primal linear program~\eqref{PrimalEx} for $G$ and $H$ respectively. We define an assignment $w$ for $G+H$ as follows: $w(x)=w_1(x)$ if $x\in V(G)$ and $w(y)=w_2(y)$ if $y\in V(H)$. By the superadditivity of $f$, we can verify that $w$ is a feasible solution of~\eqref{PrimalEx} for $G+H$, and thus $f^*(G+H)\ge f^*(G)+f^*(H)$. Combining it with 2) proves that $f^*(G+H)=f^*(G)+f^*(H)$.
The second equality follows from the fact that $\eta_{\FF}$ is additive with respect to the disjoint union.
\end{proof}
\subsection{A New Bound $\eta_{\FF}^*$}
Now we take $f=\eta_{\FF}$ which is a submultiplicative upper bound on the Shannon capacity, and show that there exist graphs such that our new bound $\eta_{\FF}^*$ can outperform both $\eta$ and Lov\'asz theta number.
The following three examples show several instances of it.
Example~\ref{ex:C_n} shows a family of graphs where our bound outperforms $\eta$ but not Lov\'asz theta number.
\begin{example}\label{ex:C_n}
{\rm
For odd $n\ge5$, it is not hard to verify that $\eta(C_n)=\eta_{\RR}(C_n)=(n+1)/2$. By 1) of Proposition~\ref{prop:properties} we have $\eta_{\RR}^*(C_n)\le \alpha_f(C_n)=n/2<\eta(C_n)$.
If we let $w(x)=1/2$ for every vertex $x$ of $C_n$, then we can readily verify that $\{w(x):x\in V(C_n)\}$ is
a feasible solution of~\eqref{PrimalEx}. It follows that $\eta_{\RR}^*(C_n)\ge n/2$, and thus $\eta_{\RR}^*(C_n)=n/2$.
}
\end{example}
Example~\ref{ex:G+C_n} provides a family of graphs where our bound outperforms simultaneously both $\eta$ and Lov\'asz theta number, however it might seem a bit artificial since it is a disjoint union of two graphs.
\begin{example}\label{ex:G+C_n}
{\rm
Let $G$ be the complement of the Schl\"afli graph. Then for odd $n\ge5$, by~Proposition~\ref{LovaszThetaNumberOfOddCycles}, Proposition~\ref{prop:LovaszProperty}, and Examples~\ref{ex:Schlaefli}-\ref{ex:C_n},
\begin{align*}
\vartheta(G+C_n) &= \vartheta(G)+\vartheta(C_n)=9+\frac{n\cos(\pi/n)}{1+\cos(\pi/n)},\\
\eta(G+C_n) &= \eta(G)+\eta(C_n)=7+\frac{n+1}{2}.
\end{align*}
On the other hand, by 3) of Proposition~\ref{prop:properties},
\begin{align*}
\eta_{\RR}^*(G+C_n)=\eta_{\RR}^*(G)+\eta_{\RR}^*(C_n)\le 7+\frac{n}{2}.
\end{align*}
Hence $\Theta(G+C_n)\leq \eta_{\RR}^*(G+C_n) \le 7+\frac{n}{2} < 7+\frac{n+1}{2} = \min\{\eta(G+C_n),\vartheta(G+C_n)\}$.
}
\end{example}
\begin{figure}[t!]
\centering
\includegraphics[scale=0.5]{ModifiedSchlaefliGraph.eps}
\caption{The graph $G$ in Example~\ref{SchlaefliModify}}
\end{figure}
In Example~\ref{SchlaefliModify} we construct a connected graph for which our bound also outperforms both $\eta$ and Lov\'asz theta number.
\begin{example}\label{SchlaefliModify}
{\rm
Let $G$ be the graph as plotted in Fig. 1. Note that $G_T$, the induced subgraph of $G$ on the vertices $T=\{1,2,\dots,27\}$,
is the complement of the Schl\"afli graph, and the vertex $28$ of $G$ is connected to vertices $1,\ldots,5,11,12,23$ and $27$.
Using Sagemath~\cite{sagemath} one can verify that $\vartheta(G)=9$. From Proposition~\ref{thm:MinrankOfModifiedSchlaefliGraph} in Appendix we see that $\eta(G)=8$.
Take $f=\eta_{\RR}$, and consider the following linear program:
\begin{align}\label{lp:ModifiedSchlaefliGraph}
\begin{split}
\textup{maximize } &\sum_{x} w(x) \\
\textup{subject to } &\sum_{x\in C}w(x)\le \eta_{\RR}(C)= 1\textup{ for every clique $C$ in $G$},\\
&\sum_{x\in T}w(x)\le \eta_{\RR}(G_{T})= 7,\\
& w(x)\ge 0.
\end{split}
\end{align}
Using Sagemath~\cite{sagemath} one can compute that the optimal value of~\eqref{lp:ModifiedSchlaefliGraph} is equal to $71/9$.
Comparing~\eqref{lp:ModifiedSchlaefliGraph} with~\eqref{PrimalEx}, we have $\eta_{\RR}^*(G)\le 71/9<8=\min\{\eta(G),\vartheta(G)\}$.
}
\end{example}
The following result shows that we cannot always get a tighter bound through this linear programming variation.
\begin{proposition}\label{prop:minrkEx}
Fix a field $\FF$. Let $G=(V,E)$ be a graph such that $\eta_{\FF}(G)<\vartheta(G)$ and for any subset $S\subsetneq V$ we have $\eta_{\FF}(G_S)\ge \vartheta(G_S)$. Then $\eta_{\FF}^*(G)=\eta_{\FF}(G)$.
\end{proposition}
\begin{proof}
By definition $\eta_{\FF}^*(G)\le\eta_{\FF}(G)$. Now we show that $\eta_{\FF}^*(G) \ge \eta_{\FF}(G)$. Suppose that $\{q(S):S\subset V\}$ is an optimal solution for~\eqref{DualEx}.
It is easy to see that $q(V)\le 1$, otherwise $\{q(S):S\subset V\}$ is not an optimal solution for~\eqref{DualEx}. We have
\begin{align*}
\eta_{\FF}^*(G) &= \sum_{S}q(S)\eta_{\FF}(G_S)\\
&= q(V)\eta_{\FF}(G)+\sum_{S\subsetneq V}q(S)\eta_{\FF}(G_S)\\
&\ge q(V)\eta_{\FF}(G)+\sum_{S\subsetneq V}q(S)\vartheta(G_S).
\end{align*}
By Proposition~\ref{ThetaEx} below we have
$$\vartheta(G)=\vartheta^*(G)\le q(V)\vartheta(G)+\sum_{S\subsetneq V}q(S)\vartheta(G_S).$$
Hence
\begin{align*}
\eta_{\FF}^*(G)
\ge q(V)\eta_{\FF}(G)+(1-q(V))\vartheta(G)
\ge\eta_{\FF}(G).
\end{align*}
This concludes our proof.
\end{proof}
The following example shows that there exist graphs satisfying the conditions of Proposition~\ref{prop:minrkEx}.
\begin{example}{\rm
Fix a field $\FF$. Let $G$ be a graph such that $\eta_{\FF}(G)<\vartheta(G)$.
If $\eta_{\FF}(G_S)\ge \vartheta(G_S)$ for any subset $S\subsetneq V(G)$,
then $\eta_{\FF}^*(G)=\eta_{\FF}(G)$ by Proposition~\ref{prop:minrkEx}.
Otherwise, let $S$ be a subset of $V(G)$ with the smallest size among those subsets
such that $\eta_{\FF}(G_S)<\vartheta(G_S)$. Obviously, the induced subgraph $G_S$ satisfies
the conditions of Proposition~\ref{prop:minrkEx}, hence $\eta_{\FF}^*(G_S)=\eta_{\FF}(G_S)$.
(Note that there are many graphs for which $\eta_{\FF}(G)<\vartheta(G)$, e.g. the complement of the Schl\"afli graph for $\FF=\RR$.)
}
\end{example}
\subsection{Bounds for Disjoint Union of Graphs}
For the Shannon capacity of the disjoint union of two graphs, we have the following simple observation.
\begin{corollary}
$\Theta(G+H) \le \min\{\eta(G)+\alpha_f(H),\alpha_f(G)+\eta(H)\}.$
\end{corollary}
\begin{proof}
Suppose $\eta(G)=\eta_{\FF}(G)$ for some field $\FF$. By Theorem~\ref{NewBound} and Proposition~\ref{prop:properties},
we have $\Theta(G+H)\le\eta_{\FF}^*(G+H)=\eta_{\FF}^*(G)+\eta_{\FF}^*(H)\le\eta(G)+\alpha_f(H)$. Similarly, we can prove that $\Theta(G+H)\le\alpha_f(G)+\eta(H).$ This concludes the proof.
\end{proof}
Next, we shall combine the Lov\'asz theta number and $\eta_{\FF}$ through a weighted geometric mean
to get another upper bound on the Shannon capacity of the disjoint union.
Fix a field $\FF$ and suppose $0\le a\le 1$. Then we can easily verify that
$$\vartheta^a\eta_{\FF}^{*1-a}(G):=\vartheta(G)^a \eta^*_{\FF}(G)^{1-a}$$
is also a submultiplicative upper bound on the independence number.
\begin{corollary}\label{cor:GeometricAverage}
For a fixed field $\FF$ and a number $a\in [0,1]$,
$$\Theta(G+H)\le \vartheta^a\eta_{\FF}^{*1-a}(G)+\vartheta^a\eta_{\FF}^{*1-a}(H).$$
\end{corollary}
\begin{proof}
As $\vartheta^a\eta_{\FF}^{*1-a}$ is a submultiplicative upper bound on the independence number,
by Theorem~\ref{NewBound} we have
\begin{align*}
\Theta(G+H) &\le (\vartheta^a\eta_{\FF}^{*1-a})^*(G+H) \\
&\le (\vartheta^a\eta_{\FF}^{*1-a})^*(G)+(\vartheta^a\eta_{\FF}^{*1-a})^*(H) \\
&\le \vartheta^a\eta_{\FF}^{*1-a}(G)+\vartheta^a\eta_{\FF}^{*1-a}(H).
\end{align*}
Here the second inequality follows from (2) of Proposition~\ref{prop:properties}.
\end{proof}
\begin{example}\label{G+7C_5}
{\rm
Let $G$ be the complement of the Schl\"afli graph. Consider the graph $H=G+7 C_5$.
It is not hard to verify that $\vartheta(H)=9+7\sqrt{5}$ and $\eta_{\RR}(H)=28$.
By Corollary~\ref{cor:GeometricAverage},
\begin{align*}
\Theta(H)
&\le \vartheta^a\eta_{\RR}^{*1-a}(G)+7\cdot \vartheta^a\eta_{\RR}^{*1-a}(C_5)\\
&= \vartheta(G)^a \cdot \eta^*_{\RR}(G)^{1-a}+7\cdot \vartheta(C_5)^{a}\cdot\eta^*_{\RR}(C_5)^{1-a}\\
&\le 9^{a}7^{1-a}+7(\sqrt{5})^{a}\left(\frac{5}{2}\right)^{1-a}.
\end{align*}
For $a=0.287291$ the term $9^{a}7^{1-a}+7(\sqrt{5})^{a}\left(\frac{5}{2}\right)^{1-a}=24.4721$ achieving its minimum value on $[0,1]$.
Note that this value is strictly better than $\vartheta(H)$ ($a=1$) and $\eta_{\RR}(H)$.
}
\end{example}
Lastly, if we take $f$ to be the Lov\'asz theta number, our new bound cannot improve it.
\begin{proposition}\label{ThetaEx}
$\vartheta^*(G)=\vartheta(G)$.
\end{proposition}
\begin{proof}
From the primal linear program~\eqref{PrimalEx} we immediately get $\vartheta^*(G)\le\vartheta(G)$. On the other hand, let $\bfd$ and $\{\bfu_1,\dots,\bfu_m\}$ be an optimal solution for~\eqref{LovaszDual}. For each vertex $i,1\le i\le m$, we set $w(i)=(\bfd^T\bfu_i)^2$. Then for each subset $S$ of $V(G)$, we have
$$\sum_{i\in S}w(i)=\sum_{i\in S}(\bfd^T\bfu_i)^2\le\vartheta(G_S)$$
by Proposition~\ref{LovaszThetaDuality}. Hence $\{w(i):1\le i\le m\}$ is a feasible solution for~\eqref{PrimalEx},
and $\vartheta(G)=\sum_{1\le i\le m}w(i)\le \vartheta^*(G)$.
\end{proof}
\section{A New Upper Bound for Index Coding}\label{sec:IndexCoding}
In this section we show that our technique also allows us to derive a new bound for the Index Coding problem to be defined next.
In the Index Coding problem, a sender holds a set of messages to be broadcast to a group of receivers.
Each receiver is interested in one of the messages, and has some prior side information
comprising some subset of the other messages. This variant of source coding problem was first proposed in~\cite{BirkKol2006} by Birk and Kol,
and later investigated in~\cite{Bar-Yossefetal2011} by Bar-Yossef et al.
The Index Coding problem can be formalized as follows: the sender holds $m$ messages $x_1,x_2,\ldots,x_m\in\Sigma$
where $\Sigma$ is the set of possible messages, and wishes to send them to $m$ receivers $R_1,R_2,\ldots,R_m$. Receiver $R_j$ wants to receive the message $x_{j}$, and knows some subset $N(j)$ of the other messages. The goal is to construct an efficient encoding scheme $\cE:\Sigma^m\rightarrow\Omega$, where $\Omega$ is a finite alphabet to be transmitted by the sender, such that for any $(x_1,x_2,\ldots,x_m)\in\Sigma^m$, every receiver $R_j$ is able to decode the message $x_{j}$ from the value $\cE(x_1,x_2,\ldots,x_m)$
together with his own side information $N(j)$. We associate a directed graph $G$ with the side-information subset $N(j)$, whose vertex set is $[m]=\{1,2,\ldots,m\}$, and whose edge set consists of all ordered pairs $(i,j)$ such that $x_j\in N(i)$. Here and in what follows, we further assume that the side-information graph $G$ is undirected, that is, if $x_j\in N(i)$ then $x_i\in N(j)$. For messages that are $t$ bits long, i.e. $|\Sigma|=2^t$, we use $\beta_t(G)$ to denote the corresponding minimum possible encoding length $\lceil\log_2{|\Omega|}\rceil$. The \emph{broadcast rate} of the side-information graph $G$ is defined as
\begin{align*}
\beta(G)
:=\inf_{t}\frac{\beta_t(G)}{t}
=\lim_{t\rightarrow\infty}\frac{\beta_t(G)}{t},
\end{align*}
where the limit exists by subadditivity of $\beta_t(G)$ and Fekete's Lemma. That is to say that
$\beta(G)$ is the average asymptotic number of broadcast bits needed per bit of input. This quantity has received significant interest, and in this section we prove a new upper bound for it. In~\cite{BirkKol2006,Bar-Yossefetal2011,LubetzkyStav2009}, it was proved that
\begin{align}\label{IndexCodingUpperBounds}
\alpha(G)\le \beta(G)\le \eta_{\FF}(G)\le \overline{\chi}(G)
\end{align}
(here $\FF$ is an arbitrary finite field and $\overline{\chi}(G)$ is the clique-cover number of $G$). On the other hand, Blasiak et al.~\cite{BlasiakKleinbergLubetzky2013} proved that $\beta(G)\le \alpha_f(G)$.
For more background and details on the Index Coding problem,
see~\cite{BirkKol2006,Bar-Yossefetal2011,Alonetal2008,BlasiakKleinbergLubetzky2013} and references therein.
Similarly as in Section~\ref{sec:new bound}, let $f$ be a real-valued function defined on graphs, and let $f^*(G)$ be the optimal value of~\eqref{PrimalEx}.
Now we show that if $f$ is an upper bound on the broadcast rate, that is, $\beta(G)\le f(G)$ for any graph $G$, then $f^*$ is also an upper bound on the broadcast rate. The proof is a simple extension of~\cite[Claim 2.8]{BlasiakKleinbergLubetzky2013}.
\begin{theorem}\label{IndexCodingNewBound}
If $f$ is an upper bound on the broadcast rate, then so is $f^*$.
\end{theorem}
\begin{proof}
Let $q^*$ be an optimal solution of the linear program~\eqref{DualEx}.
Without loss of generality, we can assume that each $q^*(S), S\subset V(G)$
is a nonnegative rational number, otherwise we can choose a rational number arbitrarily close to $q^*(S)$.
By~\eqref{DualEx} we get $f^*(G)=\sum_{S}q^*(S)f(G_S)$ and $\sum_{S\owns x}q^*(S) \ge 1$ for every vertex $x$ in $G$. Let $t$ be a positive integer such that all the numbers $t\cdot q^*(S)$ are integers, and let $y_{S}=t\cdot q^*(S)$ for each $S\subset V(G)$. Then
\begin{align*}
t\cdot f^*(G)=\sum_{S}y_{S}f(G_S) \text{ and } \sum_{S\owns x}y_S \ge t \text{ for every vertex } x \text{ in } G.
\end{align*}
Namely, we cover the graph $G$ using a collection of $y_S$ copies of $S$ for each $S\subset V(G)$.
Set $p=\sum_{S}y_{S}$. Then, altogether we have a sequence of $p$ subsets $S_1,S_2,\dots,S_p$, in which each $S\subset V(G)$ appears $y_S$ times, such that every vertex in $G$ appears in at least $t$ of these subsets. By assumption, for each induced side-information graph $G_{S_i} (1\le i\le p)$, the average asymptotic number of broadcast bits needed per bit of input is upper bounded by $f(G_{S_i})$.
Concatenating these $p$ individual index codes for the graph $G_{S_i}$ (if for some vertex $x,\sum_{S\owns x}y_S>t$, then we may ignore extra bits), we can see that the average asymptotic number of broadcast bits needed per bit of input for graph $G$ is upper bounded by $\sum_{i}f(G_{S_i})/t=\sum_{S}y_{S}f(G_S)/t =f^*(G)$.
This concludes the proof.
\end{proof}
Let us now consider the function $\displaystyle f(G)=\inf_{\FF} \eta_{\FF}(G)$ for any graph $G$, where the infimum ranges over all finite fields $\FF$.
By~\eqref{IndexCodingUpperBounds} we see that $f$ is an upper bound on the broadcast rate, hence so is $f^*$ by Theorem~\ref{IndexCodingNewBound}.
Note that for different graphs, the value of $f$ may be obtained as the minimum rank over different fields.
Therefore, the achievable scheme given by an optimal solution of the corresponding linear program~\eqref{DualEx} might yield a scheme that uses several different fields simultaneously.
More simply, we can take $f(G)=\eta_{\FF}(G)$ for some fixed finite field $\FF$. As $\eta_{\FF}$ is an upper bound for the broadcast rate by~\eqref{IndexCodingUpperBounds}, we can get the following result directly from Theorem~\ref{IndexCodingNewBound}.
\begin{corollary}
For any graph $G$ and any finite field $\FF$, $\beta(G)\le\eta_{\FF}^*(G)$.
\end{corollary}
By 1) of Proposition~\ref{prop:properties}, $\eta_{\FF}^*(G)\le\alpha_f(G)$. Hence the bound $\eta_{\FF}^*$ is at least as good as $\eta_{\FF}$ and $\alpha_f$. The following example shows that sometimes $\eta_{\FF}^*$ can simultaneously outperform both $\eta_{\FF}$ and $\alpha_f$.
\begin{example}\label{IndexCodingSchlaefliModify}
{\rm
Let $G$ and $T$ be defined as in Example~\ref{SchlaefliModify}. From Example~\ref{ex:Schlaefli} we have $\eta_{\FF_{11}}(G_{T})=7$. Similarly as in Example~\ref{SchlaefliModify} we get $\eta_{\FF_{11}}^*(G)\le 71/9$, which is better than $\eta_{\FF}$ (as $\eta(G)=8$) and $\alpha_f$ (as $\alpha_f(G)\ge\vartheta(G)=9$).
}
\end{example}
\begin{remark}\label{FractionalHaemersBound}
{\rm
Blasiak~\cite{Blasiak2013} and Shanmugam et al.~\cite{ShanmugamAsterisDimakis2015} independently\footnote{Here we adopt the notion in Blasiak~\cite{Blasiak2013}, which is slightly different from that in Shanmugam et al.~\cite{ShanmugamAsterisDimakis2015}.} obtained expressions for the infimum of the broadcast rate of vector linear broadcasting schemes~\footnote{A vector linear broadcasting scheme over a finite field $\FF$ is a scheme in which the message alphabet $\Sigma$ is a finite dimensional vector space over $\FF$ and the encoding and decoding functions are linear.} over all finite fields as follows.
Let $G$ be a graph with vertex set $V(G)=\{1,2,\dots,m\}$, and let $B$ be an $m\times m$ matrix whose entries are $k\times k$ matrices over some field $\FF$. We say that $B$ \emph{fractionally represents} the side-information graph $G$ over $\FF^k$ if $B_{ii}$ is the identity matrix of size $k$, and $B_{ij}$ is the zero matrix of size $k$ whenever $i$ and $j$ are nonadjacent.
The \emph{fractional minrank} of $G$ is defined by
\begin{align}\label{FractionalMinrank}
\eta_{f}^{\FF}(G):= \inf_{k}\frac{\min\{\textup{rank}(B): B \text{ fractionally represents } G \text{ over } \FF^k\}}{k}
\end{align}
and
$$\eta_{f}(G):=\inf_{\FF}\eta_f^{\FF}(G).$$
It is shown in~\cite{Blasiak2013,ShanmugamAsterisDimakis2015} that $\eta_{f}^{\FF}(G)$ is the infimum of the broadcast rate of all vector linear broadcasting schemes over $\FF$.
On the other hand, we can obtain a vector linear broadcasting scheme over $\FF$ of rate $\eta_{\FF}^*(G)$, by using a vector linear broadcasting scheme of rate $\eta_{\FF}(G_{S_i})$ for each induced subgraph $G_{S_i}$ in the proof of Theorem~\ref{IndexCodingNewBound}.
Hence $\beta(G)\le\eta_{f}^{\FF}(G)\le\eta_{\FF}^*(G)$.
Note that it is very difficult to compute $\eta_{f}^{\FF}(G)$ via~\eqref{FractionalMinrank}.
But our graph invariant $\eta_{\FF}^*(G)$ provides a way to approach $\eta_{f}^{\FF}(G)$,
since we can always get an upper bound for $\eta_{\FF}^*(G)$, and thus for $\eta_{f}^{\FF}(G)$,
by solving the linear programming problem~\eqref{DualEx} or its subproblems obtained by removing some constraints from~\eqref{DualEx}.
Blasiak~\cite{Blasiak2013} and Shanmugam et al.~\cite{ShanmugamAsterisDimakis2015} also proved that $\Theta(G)\le\eta_{f}^{\FF}(G)$.
See~\cite{BukhCox2018} for more properties of $\eta_{f}^{\FF}$.
}
\end{remark}
\begin{remark}\label{BukhCox}
{\rm
In~\cite{BukhCox2018} Bukh and Cox asked the following question:
Are there graphs for which $\vartheta(G) < \eta(G)$, yet $\eta_f(G) < \vartheta(G)$?
Example~\ref{G+7C_5} gives an affirmative answer to this question.
Recall that we let $G$ be the complement of the Schl\"afli graph and consider the graph $G+7 C_5$.
Then $\vartheta(G+7C_5)=9+7\sqrt{5}<\eta(G+7C_5)=7+7\cdot 3=28$.
On the other hand, $\eta_f(G+7C_5)\leq \eta_{\FF_{11}}^*(G+7C_5) \leq 7 + 7\cdot 2.5 = 24.5$ which is strictly less than $\vartheta(G+7C_5)$.
}
\end{remark}
\begin{remark}{\rm
Shanmugam, Dimakis and Langberg~\cite{Shanmugametal2014} presented an upper bound for the broadcast rate of general side-information graphs using the local chromatic number. Later, this bound is further extended by Arbabjolfaei and Kim~\cite[Theorems 3--4]{ArbabjolfaeiKim2014} and Agarwal and Mazumdar~\cite[Theorems 3--5]{AgarwalMazumdar2016} via linear programming. Similarly as in Remark~\ref{remark:1}, for undirected graphs, it is not hard to check that those bounds are always no stronger than the fractional independence number.
}
\end{remark}
\section*{Appendix}
\begin{lemma}\cite[Theorem 3.6]{Tims2013}\label{MinrankTechnique}
Let $G$ be a graph, let $I=\{v_1,\dots,v_k\}$ be a maximum independent set in $G$, and let $u$ be a vertex not in $I$. Set $J=N(u)\cap I$, which is nonempty since $I$ is maximum. If there exists another vertex $w\in V(G)\backslash (I\cup\{u\})$ that is adjacent to $u$ but not adjacent to any vertex of $J$, then delete the edge $(u,w)$, and let $H$ be the resulting spanning subgraph of $G$. Then
$$\eta(G)=\alpha(G) \text{ if and only if } \eta(H)=\alpha(G).$$
\end{lemma}
\begin{proof}
(This proof was given by Tims in~\cite{Tims2013}).
It is easy to see that $\alpha(G)\le\eta(G)\le\eta(H)$. Hence if $\eta(H)=\alpha(G)$, then $\eta(G)=\alpha(G)$.
Now we assume that $\eta(G)=\alpha(G)=k$, and let $B$ be a matrix that fits $G$ with $\textup{rank}(B)=k$.
Without loss of generality, we can assume that $J=\{v_1,v_2,\ldots,v_l\}$ where $1\le l\le k$,
and all digonal entries of the matrix $B$ are equal to $1$.
Then we can write the matrix $B$ as follows:
$$
\kbordermatrix{
\mbox{}&v_1&v_2&\ldots&v_l&v_{l+1}&\ldots&v_k&u&w\\
v_1 &1 &0 &\ldots&0 &0 &\ldots&0 &*&0\\
v_2 &0 &1 &\ldots&0 &0 &\ldots&0 &*&0\\
\vdots &\vdots&\vdots&\ddots&\vdots&\vdots&\ddots&\vdots&\vdots\\
v_l &0 &0 &\ldots&1 &0 &\ldots&0 &*&0\\
v_{l+1}&0 &0 &\ldots&0 &1 &\ldots&0 &0&*\\
\vdots &\vdots&\vdots&\ddots&\vdots&\vdots&\ddots&\vdots&\vdots\\
v_{k} &0 &0 &\ldots&0 &0 &\ldots&1 &0&*\\
u &* &* &\ldots&* &0 &\ldots&0 &1&*\\
w &0 &0 &\ldots&0 &* &\ldots&* &*&1\\
}.
$$
(Here the entry indicated by $*$ can be any value, and we only present part of the matrix $B$.)
Since $\textup{rank}(B)=k$ and the first $k$ rows of $B$ are independent, all the rows of $B$
can be written as linear combinations of the first $k$ rows. In particular, we can easily verify that
the row indicated by $u$ must be a linear combination of the first $l$ rows, and hence $B_{uw}=0$.
Similarly, we have $B_{wu}=0$. Therefore matrix $B$ also fits $H$, and it follows that $\eta(H)=k=\alpha(G)$.
\end{proof}
\begin{proposition}\label{thm:MinrankOfModifiedSchlaefliGraph}
Let $G$ be the graph as plotted in Figure 1. Note that the induced subgraph of $G$ on the vertices $T=\{1,2,\dots,27\}$ is the complement of the Schl\"afli graph. Then $\eta(G)=8$.
\end{proposition}
\begin{proof}
It can be checked that the set $I=\{8, 9, 13, 15, 19, 25, 28\}$ is a maximum independent set of $G$. Hence $\eta(G)\ge|I|=7$. On the other hand, from the minimum rank of the complement of the Schl\"afli graph (see~Example~\ref{ex:Schlaefli}), we have $\eta(G)\le 7+1 =8$.
Now we use Lemma~\ref{MinrankTechnique} to show that $\eta(G)=8$.
First, let $u=6$. Then the neighbors of $u$ are $N(6)=\{5, 13, 14, 17, 18, 21, 22, 25, 26,27\}$, and $J=N(6)\cap I=\{13, 25\}$. Set $W =\{5, 17, 18, 21, 22,27\}$. We can check that every vertex $w$ in $W$ satisfies the conditions in Lemma~\ref{MinrankTechnique}. Secondly, let $u=17$. Then $N(17)= \{1, 4, 7, 9, 12, 19, 20, 22, 24\}$, and $J=N(17)\cap I=\{9, 19\}$. Set $W=\{1, 4, 12, 22, 24\}$. We can also check that every vertex $w$ in $W$ satisfies the conditions in Lemma~\ref{MinrankTechnique}.
Lastly, we delete the edges $(6,5), (6,17), (6,18), (6,21), (6,22), (6,27), (17,1), (17,4), (17,12), (17,22), (17,24)$ from $G$, and let $H$ be the resulting spanning subgraph of $G$. It can be checked that the set $\{6, 12, 15, 16, 17, 18, 24,27\}$ is a maximum independent set of $H$. Therefore, $\eta(H)\ge8>\alpha(G)$, and hence $\eta(G)>\alpha(G)=7$ by Lemma~\ref{MinrankTechnique}. This concludes the proof.
\end{proof}
\bibliographystyle{siamplain}
|
{
"timestamp": "2018-04-17T02:14:10",
"yymm": "1804",
"arxiv_id": "1804.05529",
"language": "en",
"url": "https://arxiv.org/abs/1804.05529"
}
|
\section{Introduction}
To date, about 200 quasi-stellar objects (QSOs) at z$\gtrsim 6$ are known, offering the opportunity to study the early growth and co-evolution of super massive black holes (SMBH) and their host galaxies.
These rare systems are the sites where large $10^{11.5} -10^{12}~M_{\odot}$ stellar masses are predicted to aggregate, and thus they are the likely cradles of local giant galaxies.
Optical and near-infrared (NIR) observations of these high-z QSOs lead to measured black hole masses often exceeding $10^9\rm~M_{\odot}$ (Jiang et al. 2007, De Rosa et al. 2014, Venemans et al. 2015, Ba\~nados, 2016, 2018).
Sub-millimeter observations revealed compact rotating disks, active star formation in their host galaxies, and large masses of highly excited molecular gas. Optical/NIR observations have led to estimations of SMBH-stellar mass ratios that are larger than the local universe
value by factors of between a few and ten (Walter et al. 2004, Maiolino et al. 2005, 2009, Carilli et al. 2007, Wang et al. 2010, 2013, 2016, Gallerani et al. 2014, Willott et al. 2015, Venemans et al. 2016, 2017ab, Decarli 2017, 2018).
Both simulations and semi-analytical models of galaxy formation showed that SMBHs can grow either from pop III star seeds at super-Eddington rates, or from massive seeds, and assemble BH masses of several times $10^9\rm ~ M_{\odot}$ observed at $z\gtrsim 6$ (Volonteri et al. 2016, Valiante et al. 2016, Pezzulli et al. 2016, 2017).
The [CII]~158 $\mu$m emission line is the preferred tracer used to study the host galaxies of high-z QSOs because its brightness makes it a powerful probe to survey large samples by snapshot observations (Decarli et al. 2018).
Conversely, the dense molecular ISM, where we expect most star formation to take place, remains relatively poorly explored at these high redshifts (Wang et al. 2010, 2013, Gallerani et al. 2014, Venemans et al. 2017a).
Finally, the host galaxies of high-z QSOs are mostly spatially unresolved in current [CII] and CO observations, making the estimate of dynamical masses very uncertain.
Today it is possible to resolve the molecular gas emission in the highest-redshift QSO host galaxies with the Atacama Large Millimeter/submillimeter Array (ALMA).
This work focuses on the QSO J231038.88+185519.7 (hereafter J2310+1855), the QSO with the brightest 250 GHz continuum at z$\sim$6 (8.29 mJy, Wang et al. 2013, hereafter W13).
Using the Northern Extended Millimeter Array (NOEMA) W13 detected the CO(6-5) emission line from the host galaxy of the QSO and measured an emission redshift of $z=6.0025$.
W13 also detected the [CII] emission line, and marginally resolved the rotation of a disk, from which they derived a dynamical mass of $M_{dyn}= (9.6\pm0.6) \times 10^{10}~\rm M_{\odot}$, and an inclination of 46 deg, estimated from major/minor axis ratio.
Jiang et al. (2016) detected weak C~IV and Mg~II emission lines in the GEMINI/GNIRS spectrum, from which they derived a redshift of $5.962\pm0.007$, which implies that the UV lines are blueshifted by $\sim 1200$ km/s with respect to CO and [CII] lines, similarly to what was found for other $z>6$ QSOs (Willott et al. 2015, Venemans et al. 2016, Wang et al. 2017), and common also at lower redshift
(Tytler \& Fan 1992).
They derived a BH mass of $(3.9\pm0.5)-(4.2\pm1.0)~\times 10^{9}\rm M_{\odot}$, based on Mg~II and C~IV, respectively.
In this work, we report on our ALMA observations of CO(6-5), the sub-mm continuum (Project 2015.1.00584.S), the [CII] emission line (Project 2015.1.00997.S) of the J2310+1855 host galaxy, and on archival X-SHOOTER/VLT (Programme 098.B-0537(A), P.I. Farina) observations of the rest-frame ultraviolet (UV)-optical spectrum of the QSO.
Compared to W13, we have used a synthetic
beam that is ten times smaller in angular size, and we have achieved ten times
better sensitivity in the CO(6-5) line, and two and half times better sensitivity
in the [CII] line, enabling us to resolve the molecular gas emission of the QSO host galaxy.
The observations are described in Section 2.
In section 3 we present results on CO and [CII] kinematics and their ratio, gas mass, dynamical mass, black hole mass, and detection of other line-emitting sources.
In Section 4 we discuss the results, and a summary is presented in Section 5.
A $\Lambda$-CDM cosmology with $H_0=70$ km s$^{-1}$ Mpc$^{-1}$, $\Omega_M=0.3$, and $\Omega_\Lambda=0.7$ is adopted throughout the paper. The angular scale is 5.835 kpc/\arcsec~ for the adopted cosmology.
\section{Observations}
\subsection{ALMA observations}
We observed J2310+1855 with ALMA band-3 receivers tuned to cover the frequency ranges 84.56-87.94, and 96.56-99.69 GHz.
Spectral window 1 was tuned at the expected redshifted frequency of CO(6-5), 98.75 GHz.
We performed calibration in the CASA environment (McMullin et al. 2007).
Mapping and data analysis were performed both in the CASA and in the GILDAS (Guilloteau et al. 2000) environments (the latter after converting CASA into GILDAS visibility tables).
Within GILDAS, we created two data cubes, one per baseband. We also created a continuum map using the four spectral windows and excluding channels within $\pm 1000$ km/s from the redshifted CO(6-5) frequency.
The continuum map and the data cubes were cleaned by adopting a natural weight scheme, with detection threshold equal to 0.5 times the noise (per channel), and without applying any mask for the detection. We used both the {\it hogbom} and the {\it mx} cleaning algorithms within GILDAS, and found consistent results.
In the following, we use the results from the {\it hogbom} cleaning, because it minimizes side-lobe residuals. Based on this setup we obtain a synthesized beam of $0.6 \times 0.4$ arcsec$^2$ at a PA$=-6$ deg in the cubes, which contains the CO(6-5) emission line.
The noise levels are 5.4 $\mu$Jy/beam in the continuum in the aggregated bandwidth, and 0.13 mJy/beam in the 23.7 km/s-wide channels (i.e., the maximum spectral resolution of the data).
By adopting a Briggs algorithm, the synthetic beam is $0.51 \times 0.28$ arcsec, at a PA$=-11$ deg,
and the noise level is 0.15 mJy/beam in the 23.7 km/s-wide channel.
[CII] observations were obtained with the ALMA 12-m array in project 2015.1.00997.S.
The data were calibrated and imaged in CASA v4.7 by applying a natural weighting with a detection threshold equal to 0.5 times the noise (per channel). The continuum-subtracted cube was obtained by combining two adjacent spectral windows after subtracting the continuum emission by fitting a UV-plane model with the {\it uvcontsub} task. The $1\sigma$ root mean square (r.m.s) sensitivity is 0.20 mJy/beam per 100 km/s channel. The synthesized beam is $0.9\times0.6$ arcsec$^2$ at a PA=49 deg.
In this paper we aim at comparing the kinematics and spatial distribution of CO- and [CII]-emitting gas.
A thorough description of [CII] luminosity, mass, and other host galaxy properties inferred from [CII] are the subject of a separate paper (Carniani et al. 2018 in prep.).
In order to compare CO and [CII] emissions, we imaged the CO(6-5) data with the same restoring beam as the [CII] data, and we registered the two cubes at the same reference system, that is, z$=6.0025$ in the LSRK reference frame, with the same spectral binning (23.7 km/s spectral resolution).
\subsection{X-SHOOTER spectrum}
We used archival data from X-SHOOTER at the VLT (Vernet et al. 2011) to investigate the optical/NIR spectrum of the QSO.
There are two frames available in the archive with an exposure time of 1200 s each, observed with a slit
of $0.9$ arcsec and a binning $2\times2$ in the VIS arm (550-1020 nm) and a slit of
$0.6$ arcsec in the NIR arm (1020-2480 nm).
The chosen slits correspond to nominal resolving powers of $R\simeq 8800$ and $8100$, respectively.
The spectra were reduced with the ESO pipeline (Modigliani et
al. 2010) with a manual localisation of the object, and adopting the sky subtraction
method BSPLINE1 in the VIS, and MEDIAN in the NIR. The one-dimensional (1D) flux
calibrated spectra produced by the pipeline were then corrected for
telluric absorption using the ESO tool {\tt Molecfit} (Smette et al. 2015, Kausch et al. 2015).
The final spectra, obtained combining the two frames,
were rebinned to a step of 0.4 and 0.6 \AA, respectively.
In order to fit the emission lines (see section 3.4), the NIR spectrum was then binned to a step of 7 \AA.
\section{Results}
\subsection{The 3.3 mm continuum}
Regarding the 91.5 GHz ($\sim 3.3$ mm) continuum,
we performed a fit of the continuum visibilities in both CASA and GILDAS. Both methods give consistent results and show that the continuum source is best fitted by a 2D Gaussian function with parameters reported in Table 1 (errors do not include the systematic error on the flux scale, which is of the order $\approx 5\%$ at this frequency).
The 91.5 GHz continuum flux density is $416\pm33~ \rm \mu Jy$, consistent with the measurement by W13 at a similar frequency.
The continuum has a beam--deconvolved FWHM size of $(0.24\pm0.04)\times(0.12\pm0.07)$ arcsec$^2$, at a PA=$138\pm24$ deg, which corresponds to a physical size of $1.4\pm0.2$ kpc (Table 1).
This measurement is in agreement with the continuum size measured by W13 at 263 GHz frequency (the 3 mm continuum is unresolved in the PdBI observations presented in W13).
\begin{table}
\caption{91.5 GHz continuum, CO(6-5) and [CII] best fit parameters.}
\label{table:1}
\centering
\begin{tabular}{l l r}
\hline
\hline
\smallskip
RA$_{\rm cont,91.5GHz}$ & 23:10:38.90 $\pm$ 0.02 & [J2000] \\
\smallskip
DEC$_{\rm cont,91.5GHz}$ & 18:55:19.82 $\pm$ 0.02 & [J2000] \\
\smallskip
Size$_{\rm cont,91.5GHz}$ & $(0.24\pm0.04)\times(0.12\pm0.07)$ & [arcsec$^2$] \\
\smallskip
PA$_{\rm cont,91.5GHz}$ & $138\pm24$ & [deg] \\
\smallskip
$S_{91.5GHz}$ & $416\pm33$ & $ \rm [\mu Jy]$ \\
\hline
\smallskip
RA$_{\rm CO(6-5)}$ & 23:10:38.900 $\pm$ 0.008 & [J2000] \\
\smallskip
DEC$_{\rm CO(6-5)}$ & 18:55:19.83 $\pm$ 0.01 &[J2000]\\
\smallskip
Size$_{\rm CO(6-5)}$ & $(0.33\pm0.06) \times (0.20\pm0.04)$ & [arcsec$^2$]\\
\smallskip
PA$_{\rm CO(6-5)}$ & $140\pm17$ & [deg]\\
\smallskip
$Sdv_{\rm CO(6-5)}$ & $1.26\pm0.06$ & [Jy km/s] \\
\hline
\smallskip
RA$_{\rm [CII]}$ & 23:10:38.929 $\pm0.09$ & [J2000] \\
\smallskip
DEC$_{\rm [CII]}$ & 18:55:18.1 $\pm0.1$ &[J2000]\\
\smallskip
Size$_{\rm [CII]}$ & $(0.86\pm0.10)\times(0.78\pm0.15)$ & [arcsec$^2$]\\
\smallskip
PA$_{\rm [CII]}$ & $147\pm68$ & [deg]\\
\smallskip
$Sdv_{\rm [CII]}$ & $8.75\pm0.51$ & [Jy km/s] \\
\hline
\hline
\end{tabular}
\end{table}
\subsection{CO(6-5) emission line}
We created a continuum-subtracted data cube by using the task {\it uvcontsub} in CASA.
To this aim, we combined spectral windows 1 and 2 (so as to have a broader bandwidth around the emission line for continuum estimate). The continuum was then estimated by fitting the line-free channels (at velocity $>|1000| ~\rm km/s$ from the line peak) with both a constant and a first-order polynomial. By eye, no slope is seen in the continuum, and we verified that a zero-order fit gives the best result.
The continuum-subtracted visibility set was then used to map the CO(6-5) emission and to create a visibility set averaged over the line channels, in the velocity range from $-500$ to 500 km/s from the systemic velocity.
We used the velocity-integrated visibility set to measure the line parameters in the {\it uv} plane. We fitted the visibilities with a point source and a circular and an elliptical Gaussian function. The best fit is given by the elliptical Gaussian function which delivers a source size (FWHM) of $(0.33\pm0.06) \times (0.20\pm 0.04)$ arcsec, at a $\rm PA=140\pm 17$ deg (Table \ref{table:1}).
This size is smaller than that derived for [CII] by W13, suggesting that the CO emitting gas is more centrally concentrated with respect to [CII].
The integrated flux density of the line is $1.26\pm0.06$ Jy km/s (consistent within 2$\sigma$ with W13).
The [RA, DEC] of the CO peak emission are consistent with the continuum position (Table \ref{table:1}).
We then extracted the spectrum in Fig. \ref{Fig-sp} using a mask that encompasses the $\geq2\sigma$ region in the velocity-integrated CO(6-5) map.
The CO(6-5) line peaks at a frequency corresponding to $z_{CO}=6.0028\pm0.0003$, consistent with previous CO and [CII]-based redshifts (W13).
A Gaussian fit with a single component gives a peak intensity of the line of $I=2.94\pm0.06$ mJy,
and FWHM$=361\pm9$ km/s. This gives an integrated flux density of $S_{CO}dv = 1.13\pm 0.06$ Jy km/s -- consistent with the estimate from the {\it uv fit} within $2\sigma$.
Figure \ref{moments-co} shows the continuum map, and the maps of the first three moments of CO(6-5).
The velocity map indicates a velocity gradient of $\approx 300$ km/s from north-east to south-west, at a $\rm PA\approx 60$ deg north of east, consistent with that estimated for [CII] (see Fig. 4, and W13).
The velocity dispersion map shows $\sigma \sim 300$ km/s at the center, which is likely affected by disk beam smearing effects. The intrinsic velocity dispersion may be significantly lower. According to Tacconi et al. (2013) and Davies et al. (2011) a less biased estimate of the velocity dispersion is given by the average value measured in the outer parts of the galaxy, assuming a flat profile. We then estimate $\sigma \approx 150-200$ km/s, similar to the value estimated for the [CII] (Fig. 4).
Under the assumption that the velocity gradient is due to a rotating disk, and based on the axis ratio (Table \ref{table:1}), we derive an inclination of the disk of $i\rm= cos^{-1}~ (minor/major) = 53$ deg (the lower limit inclination is $i_{min}\approx 25$ deg, by accounting for the statistical errors in minor/major axis).
The rotation velocity of the disk would then be $v_{rot} = 1.3 \times \Delta v / 2~sin(i) \sim 245$ km/s (e.g., Tacconi et al. 2013).
This is of the same order of the velocity dispersion given above, and in fact the ratio of the rotational velocity to the velocity dispersion, $v_{rot}/\sigma$, is in the range 1-2.
The large uncertainty on $v_{rot}/\sigma$ is driven by the uncertainty on the inclination and on the intrinsic unbiased $\sigma$.
Figure \ref{pv-co} shows position-velocity (PV) diagrams obtained along $\rm PA=30$ $\deg$ (i.e., along the velocity gradient) and orthogonal to it, through the peak CO position, with a slit of 0.5 arcsec in width.
Non-rotational motions are visible in Fig. \ref{pv-co}, at velocities $+200$ km/s and -300 km/s at an offset of 0.7-0.5 arcsec on the south-west side of the QSO.
The emission from this relatively high-velocity gas is seen as an elongated structure located south-west of the nucleus in the CO(6-5) moment maps (Fig. 2).
\subsection{[CII] and the CO(6-5)/[CII] ratio}
Figure \ref{moments-cii} shows the moment 0, 1, 2 maps of the [CII] emission line.
The [CII] velocity gradient is found approximately along the same direction as the CO one.
Figure 5 shows the PV diagram of the [CII] emission along the same directions as done for CO(6-5), with a larger slit width motivated by the larger beam of [CII] data.
This shows that [CII] gas kinematics are similar to those of the molecular gas traced by CO(6-5).
We computed the ratio of CO(6-5)/[CII] by combining the respective velocity-integrated data cubes. Figure \ref{ratio} shows the velocity-integrated [CII] map (range $\pm500$ km/s), the map of CO(6-5) obtained by degrading the CO data to the same resolution as [CII] data, and their ratio.
The CO(6-5)/[CII] ratio ranges from $\approx 0.1$ to 0.25, with a velocity-averaged mean ratio of 0.19.
We find that the CO(6-5)/[CII] ratio shows a local maximum close to the QSO position (indicated by the small black cross), and decreases at larger distances from the QSO (Fig. \ref{ratio}, right panel).
This suggests that the CO(6-5)-emitting gas may be more concentrated than the [CII]-emitting gas.
An excess in the CO(6-5)/[CII] ratio is seen at a spatial offset of $(-0.3,-0.5)$ arcsec
from the QSO (Fig. 6), at the position where a marginal disturbance is seen in both the velocity and velocity dispersion maps (Fig. 2).
We extracted the CO(6-5) and [CII] spectra from the respective data cubes, over a region centered at the QSO position (RA, DEC)=(23:10:38.90, 18:55:19.82), and with a radius of 0.5 arcsec for both lines, and compute their ratio (Fig. \ref{ratio-sp}).
We find that the line profiles are similar, although our data hint
at a broader velocity distribution of the [CII]-emitting gas compared to the CO(6-5)-emitting gas, within $\pm250$ km/s from the CO line peak.
Figure 8 shows PV diagrams of the CO(6-5)/[CII] ratio, obtained after combining the data cubes registered as detailed in Section 2.1. For consistency with the previous sections, we plot PV diagrams along the same PA used in the individual CO and [CII] diagrams.
The PV diagrams are consistent with the global trend of the CO(6-5)/[CII] ratio found in the spectra, and show an average value of $\approx 0.2$.
The PV diagrams suggest that the largest ratios are measured at a spatial offset of $(-0.3,-0.5)$ arcsec
from the QSO, that is, at the position where we also find a local maximum of the $\sigma$ (Fig. 2).
Confirmation of whether the small discrepancies in the line profiles are genuine would require additional observations.
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{j2310-co65-spectrum-2panel-22feb.pdf}
\caption{[Left panel]: Spectrum of spectral windows 1+2, including the CO(6-5) emission line and the continuum emission, extracted from the region included within the $\geq 2\sigma $ in the velocity-integrated map. [Right panel]: Zoom onto the continuum-subtracted CO(6-5) spectrum. The red line shows a fit with a Gaussian function. Spectra are plotted at the maximum spectral resolution of the data (23.7 km/s).}
\label{Fig-sp}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{moments-4panel-13apr.pdf}
\caption{From left to right: The dust continuum map of J2310+1855 (levels from 2 to 45$\sigma$ by 5$\sigma$, $\sigma=5.36 ~\mu $Jy/beam), the moment 0, 1, 2 maps of the CO(6-5) emission line. Color-scale units are mJy and km/s, respectively. The cross indicates the phase center. The synthesized beam is indicated in the lower-left part of the diagrams.}
\label{moments-co}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=8cm]{rot30.pdf}
\includegraphics[width=8cm]{rot120.pdf}
\caption{PV diagrams of the continuum-subtracted CO(6-5) emission line along the line of nodes (PA= 30 deg), and orthogonal to it (PA=120 deg). Contours are 2 to 20$\sigma$, by $\sigma$, $\sigma= 0.11$ mJy/beam. Slices are extracted from a slit of 0.5 arcsec in width.
Zero offset position is (RA, DEC)=(23:10:38.90 ,18:55:19.82) in these and the following PV diagrams.}
\label{pv-co}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{moments-cii-v3.pdf}
\caption{Moments of the [CII] emission line. (a) mean flux (levels are $-3,~ 3,~5,~ 10,~ 15,~ 20,~ 25,~ 30,~ 35\sigma,~\sigma=0.14 \rm ~ mJy/beam$); (b) velocity map; (c) velocity dispersion map. The synthesized beam is indicated in the left panel. The cross indicates the phase center.}
\label{moments-cii}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=8cm]{rot-cii30.pdf}
\includegraphics[width=8cm]{rot-cii120.pdf}
\caption{PV diagrams of the continuum-subtracted [CII] emission along the line of nodes (PA= 30 deg, left panel), and orthogonal to it (PA=120 deg, right panel). Contours are 2 to 20$\sigma$, by $\sigma$, $\sigma= 0.14$ mJy/beam. Slices are extracted from a slit of 1.1 arcsec in width. Zero offset position is (RA, DEC)=(23:10:38.90 ,18:55:19.82).}
\label{pv-cii}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{ratio-co-cii-same-beam-v2.pdf}
\caption{[Left panel]: The [-500,500] km/s velocity-integrated [CII] map. The beam is $0.9\times 0.6$ arcsec$^2$, at $\rm PA=49~ deg$. Levels are $-3,~ 3,~5,~ 10,~ 15,~ 20,~ 25\sigma,~\sigma=0.14 \rm ~ mJy/beam$. [Middle panel]: The velocity-integrated CO(6-5) map. The original angular resolution of CO data was degraded to match the same beam of [CII] data. Levels are $-3,~ 3,~5,~ 10,~ 15,~ 20,~ 25\sigma,~\sigma=27 \rm ~ \mu Jy/beam$. [Right panel]: The velocity-integrated CO(6-5)/[CII] ratio map. The small cross indicates (RA, DEC)=(23:10:38.90, 18:55:19.82). The large cross indicates the phase center.}
\label{ratio}
\end{figure*}
\begin{figure}
\centering
\includegraphics[width=7cm]{ratio-spectra-co-cii-22june.pdf}
\caption{[Upper panel]: The CO(6-5) (yellow filled histogram) and scaled [CII] (blue histogram) emission lines, extracted from 1 arcsec aperture centered at (RA, DEC)=(23:10:38.90, 18:55:19.82).
Zero velocity has been set to the rest frame corresponding to $z=6.0025$ for both emission lines.
[Lower panel]: The ratio of the CO(6-5) and [CII] spectra in the velocity range $\rm [-250,+250]$ km/s.
The dashed line indicates the velocity-averaged mean ratio. The binning is identical in the upper and lower panels.}
\label{ratio-sp}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=4.4cm]{casarot-ratio30.pdf}
\includegraphics[width=4.4cm]{casarot-ratio120.pdf}
\caption{The PV diagrams of the CO(6-5)/[CII] ratio along a $\rm PA = 30$ deg (left panel), and $\rm PA=120$ deg (right panel). Slit width is 0.9 arcsec. The original angular resolution of CO data was degraded to match that of the [CII] data.}
\label{pv-ratio}
\end{figure}
\subsection{BH mass, gas mass, and dynamical mass}
We measured the monochromatic luminosity from the rest frame 1350 \AA~ continuum in the X-SHOOTER spectrum, $\lambda~L_\lambda(1350 \AA)= (9.8 \pm 1.9) \times 10^{45}$ erg s$^{-1}$.
Mg~II is not detected or very weak in the X-SHOOTER spectrum, probably due to the reduced sensitivity of the instrument at 1960 nm; we therefore estimate the BH mass based on C~IV $\lambda$1549\AA.
We fit the CIV line with a single Gaussian component, from which we derive a FWHM=$11185\pm46$ km/s, and a $\chi^2=385.7$ for 181 degrees of freedom (d.o.f.). The C~IV line peak is located at 10727\AA.
We note that, because it is probably affected by strong outflows, the line profile of C~IV is strongly asymmetric in high-luminosity QSOs, and it has been shown to over-predict BH masses, when adopting the Vestergaard \& Peterson (2000) correlation.
The calibration by Coatman et al. (2016) provides a correction for these effects. By adopting the latter and
a systemic redshift of $z=6.0025$, we derive a $\rm M_{BH} = (1.8\pm 0.5) \times 10^{9}~ M_{\odot}$.
This is a factor of approximately two smaller than the estimate by Jiang et al. (2016) based on Gemini data and on the Vestergaard \& Peterson (2000) correlation. The discrepancy is probably due to the different calibrations used.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{civ_1g.pdf}
\caption{The C~IV emission line from X-SHOOTER/VLT spectrum (red points). The blue dashed line shows the best fit with a Gaussian function.}
\label{civ}
\end{figure}
Concerning the host galaxy mass budget, we can infer the molecular gas mass from the CO luminosity.
From the integrated intensity of CO(6-5), and using the relation $\rm L^\prime CO= 3.25\times 10^7 S_{CO}~\Delta v ~\nu^{-2}_{obs}~ D^2_L(1+z)^{-3}$ (Solomon \& Vanden Bout 2005),
we derive a luminosity of $\rm L^\prime CO(6-5)=(4.3\pm0.1) \times 10^{10}~ K~ km/s ~pc^{-2}$.
Stefan et al. (2015) and Wang et al. (2016) measured $\rm L^\prime CO (6-5) \sim 0.9~\times L^\prime CO (2-1)$ for the QSO J010013.02+280225.8 at $z=6.3$ and J114816.64+525150.3 at $z=6.4$, respectively. Since there is no measurement of lower J CO transition for J2310+1855, we adopt the same ratio.
We assume thermalized emission (i.e., $\rm L^\prime~CO(2-1)=L^\prime~CO(1-0)$), and a conversion factor of $\alpha_{CO}=0.8~M_{\odot}~ \rm K^{-1}~ (km/s)^{-1} ~pc^{2}$ to derive the molecular gas mass.
This conversion factor is currently thought to best represent QSO host galaxies (e.g., Carilli \& Walter 2013).
Under these assumptions, we find a molecular gas mass of $\rm M(H_2)=(3.2 \pm0.2) \times 10^{10}~(\alpha_{CO}/0.8)~~\rm M_{\odot}$.
We note that using a Milky Way conversion factor ($4.3~M_{\odot}~ \rm K^{-1}~ (km/s)^{-1} ~pc^{2}$) would imply $\rm M(H_2)=(1.8 \pm0.2) \times 10^{11}~\rm M_{\odot}$.
Under the assumption that the observed CO velocity gradient is due to an inclined rotating disk, we derive the dynamical mass by applying the relation
$\rm M_{dyn} \times sin^2(i) = 1.16~ 10^5 \times (0.75\times FWHM_{CO})^2 \times D$ (W13, Feruglio et al. 2014), where $\rm FWHM_{CO}=361\pm9$ km/s, and D is the source size in kiloparsecs (diameter).
We adopt the definition of source size used by W13, equal to 1.5 times the FWHM size, $D=2.9$ kpc (i.e., full width at 20\% of the peak intensity for a Gaussian profile; Table 1).
We find $\rm M_{dyn} \times sin^2(i) = (2.4\pm0.5) \times 10^{10}~ M_{\odot}$.
By using the virial relation $M_{dyn} =R~ v_{rot}^2 / G$ we find a similar value, $\rm M_{dyn} = 2.35 \times 10^{10}~ M_{\odot}$.
By applying the inclination derived from our data, we derive an inclination-corrected dynamical mass of $\rm M_{dyn} = 4.1^{+9.5}_{-0.5} \times 10^{10}~ M_{\odot}$. The upper limit $\rm M_{dyn}$ is derived from the lower limit inclination estimated from the minor/major axis ratio, $i\approx25$ deg.
\begin{table}
\caption{Derived properties of J2310+1855.}
\label{table:1}
\centering
\begin{tabular}{l l c }
\hline
\hline
$z_{CO}$ & & $6.0028\pm0.0003$ \\
L'CO(6-5) & $\rm [K~km/s~ pc^{-2}]$ & $(4.3\pm0.1)\times 10^{10}$ \\
M(H$_2$) & $[\rm M_\odot]$ & $(3.2\pm0.2)\times 10^{10}$ \\
i & [deg] & $53$ \\
Mol. disk diameter & [kpc] & $2.9\pm0.5$ \\
$\rm M_{dyn} sin^2(i)$ & $\rm [M_\odot]$ & $(2.4\pm0.5)\times10^{10}$\\
$\rm M_{dyn}$$^{(a)} $ & $\rm [M_\odot]$ & $(4.1^{+9.5}_{-0.5})\times10^{10}$ \\
$\rm M_{BH}$$^{(b)}$ & $\rm [M_\odot]$ & $(1.8\pm 0.5) \times10^9$ \\
$\rm M_{BH}/M_{dyn}$ & & $0.04^{+0.01}_{-0.035}$ \\
$\rm L_{bol}^{(c)}$ & $\rm [L_{\odot}$] & $9.3\times10^{13}$ \\
$\dot M_{acc}$ & $\rm [M_\odot/yr]$ & $63\pm11$ \\
$L_{FIR}$ & $\rm [L_{\odot}]$ & $(1.70\pm0.18)\times10^{13}$ \\
SFR $^{(d)}$ & $\rm [M_\odot/yr]$ & $1250\pm900$ \\
$\rm M(H_2)/M^*$ & & $\approx4.4$ \\
$\rm M(BH)/M^*$ & & $\approx0.25$ \\
\hline
\end{tabular}
\tablefoot{ The upper limit $\rm M_{dyn}$ is (a) derived using the lower limit inclination $i_{min}=25$ deg,
$ (b)$ derived from C~IV FWHM and the 1350\AA~ continuum luminosity,
$(c)$ derived from rest-frame 1450\AA~ magnitude and assuming a bolometric correction $L_{bol}=4.2~ L_{1450}$ (Runnoe et al. 2012a,b) and
$(d)$ derived from $\rm L_{FIR}$ (from W13) and the Kennicutt \& Evans (2012) relation.}
\end{table}
\subsection{Search for other line or continuum emitters.}
We scanned all spectral windows of the band 3 and band 6 data cubes in space and frequency, searching for continuum or line emitters. To this purpose, we degraded the original spectral resolution to 47.4 km/s.
Candidate detections are considered if signal is detected in at least three contiguous spectral channels with a global (i.e., velocity integrated) statistical significance of $\geq4.5\sigma$ in the velocity-integrated map.
In the following we describe the candidate emitters selected from this analysis.
One line emitter, detected in the band-3 data with a statistical significance of 5$\sigma$ and with prior information of its redshift, is presented in D'Odorico et al. (2018).
Other two line emitters are detected in band-3. Figure \ref{galaxies} shows the velocity-integrated line maps of these two candidate sources, which are detected with a significance of $4.5\sigma$ in the band-3 data.
The line peaks are detected at velocity $\sim200$ and $\sim1100$ km/s from the CO(6-5) of the QSO host galaxy.
Their positions are [RA, DEC]=[23:10:38.769,18:55:18.49], [RA, DEC]=[23:10:39.068,18:55:24.16], respectively.
Because the first candidate has a frequency similar to the emission line in the QSO, the possibility that this is due to an artifact (e.g., cleaning residual) cannot be ruled out.
By fitting the lines with a Gaussian function we find $\rm FWHM = 300\pm100$ and $220\pm60$ km/s,
and integrated flux densities of $0.036\pm0.008$ Jy km/s and $0.032\pm 0.007$ Jy km/s, respectively.
The underlying 3 mm continuum is undetected in both these cases.
At their position we do not detect any continuum or line in band 6 either.
A secure identification of these candidate lines requires the detection of at least another emission line.
Here we attempt to assess whether these candidate emission lines are genuine by comparing our findings with the results of the ASPECS Survey of the Hubble Ultra-Deep field (Walter et al. (2016), Decarli et al. 2016).
In their work, they provide molecular emission line number counts, and the CO luminosity function based on their ALMA observation covering the band 3 with a total bandwidht of 30.75 GHz, and with a sensitivity comparable to ours (0.17 mJy/beam per 23.7 km/s around 95 GHz, with the same field of view).
According to their number counts of molecular emission lines, and to their emission line luminosity function,
and by rescaling for our narrower bandwidth (3.38 GHz versus the $\sim30.75$ GHz covered by their survey),
0.2-0.8 CO-emitting sources with flux density in the range 0.03-0.04 Jy km/s are expected in the area covered by one ALMA antenna beam at 3 mm.
We therefore find a factor of $\sim 3-15$ excess in the number of line-emitting sources in our data compared to the expectation based on ASPECS. If we considered as genuine only the source detected with the highest
statistical significance (D'Odorico et al. 2018), we would still be at the upper boundary of the expected line counts.
We note, however, that the $log N -- log S$ of molecular lines is poorly known, and that the estimates based on the ASPECS survey have very large uncertainty. Their estimates are also based on a single field (one ALMA pointing at 3 mm), and therefore do not take into account the cosmic variance.
In this context, the UDF can be considered a random field, whereas our observation targets the field around a rare luminous QSO, and therefore likely probes a biased overdensity region (Decarli et al. 2016, Bischetti et al. 2018).
\begin{figure}
\centering
\includegraphics[width=4cm]{source1-17jul.pdf}
\includegraphics[width=3.95cm]{source5-17jul.pdf}
\caption{The velocity-integrated maps (upper panels) and the spectra (lower panels) of the candidate line emitters detected in band 3. In the upper-left panel the CO emission from the QSO host galaxy has been fitted and subtracted (the cross indicates the phase center).
The spectra were extracted using masks enclosing the $\geq2\sigma$ level on the velocity-integrated maps.
Contour levels are $-4,-3,-2,-1,1,2,3,4\times \sigma$, $\sigma=0.035, 0.059~ \rm mJy~ beam^{-1}$ in the upper--left and upper--right panels, respectively.
}
\label{galaxies}
\end{figure}
\begin{figure*}
\centering
\includegraphics[width=6cm]{mdyn-mbh-v8.pdf}
\includegraphics[width=6cm]{sfr-dotm-v5.pdf}
\includegraphics[width=6cm]{growth-v8.pdf}
\caption{[Left panel]: $\rm M_{BH}$ vs. $\rm M_{dyn}$ for J2310+1855 (red star) and a compilation of AGNs.
Violet circles: IBISCO hard X-ray selected sample at z$<0.05$ (Feruglio et al. in prep). Cyan circle: Brusa et al. (2017).
Green symbols: red QSOs at $z \sim2.5$, for these we plot the ranges given in Banerji et al. (2012, 2017). Blue squares: QSO at $z>4.7$ from Willott et al. (2013, 2015, 2017), Kimball et al. (2015), Wang et al. (2016), De Rosa et al. (2014), Venemans et al. (2016, 2017a), Trakhtenbrot et al. (2017), Decarli et al. (2018). Black symbol: $z=7.54$ QSO from Venemans et al. (2017b), Ba\~nados et al. (2018). For the QSO for which there is no estimate of the inclination available, we have assumed $i=50\deg$ (see e.g., Decarli et al. 2018). The solid line is the local $\rm M_{BH}-M_{bulge}$ relationship (Kormendy \& Ho 2013).
[Middle panel]: The BH accretion rate, $\rm \dot M_{acc}$, vs. the SFR. Same symbols as in the left panel, except purple squares = hyper-luminous QSO at $z=3-4$ from the WISSH sample (Bischetti et al. 2017, Duras et al. 2017, Vietri et al. 2018). The line is the locus of $\rm \eta=\dot M_{outflow}/SFR=1$, assuming the scaling relation of $\dot M_{outflow}$ with $L_{bol}$ derived for molecular outflows (Fiore et al. 2017).
[Right panel]: Black hole growth timescale, $\rm M_{BH}/\dot M_{acc}$ vs. the galaxy growth time, $\rm M_{dyn}/SFR$. The solid line is the 1:1 relation.
}
\label{mbh-mdyn}
\end{figure*}
\section{Discussion}
We mapped and resolved the CO(6-5) line emission in the host galaxy of the QSO J2310+1855.
We measured a molecular gas mass of $\rm M(H_2)=(3.2 \pm0.2) \times 10^{10}~(\alpha_{CO}/0.8)~\rm M_{\odot}$, and a size of the CO-emitting region of 2.9 kpc.
Under the assumption that the observed velocity gradient is due to a rotating disk, we derived
an inclination of $i=53$ deg from the minor/major axis ratio, and a dynamical mass of $\rm M_{dyn} = 4.1 \times 10^{10}~ M_{\odot}$ within the inner 2.9 kpc region, which is a factor of approximately two smaller than the W13 value.
For the lower limit inclination implied to our data, $i=25$ deg, the dynamical mass would be $\rm M_{dyn} = 1.3 \times 10^{11}~ M_{\odot}$.
We note that the estimate of the dynamical mass depends linearly on the disk size and on $sin^2(i)$, which in turn depend on the signal-to-noise ratio (S/N) of the data and on the angular resolution. Therefore data with a high S/N and good angular resolution are essential elements to derive accurate dynamical masses.
The discrepancy between our estimate and the W13 may be due to the smaller source size that we measured based on CO data ($(0.33\pm0.06) \times (0.20\pm0.04)$ arcsec vs. the $(0.56\pm0.03) \times (0.39\pm0.04)$ arcsec by W13), and to the slightly larger inclination.
We find that most of the mass within a region of 2.9 kpc in diameter is in the form of dense molecular gas.
By assuming that the stellar mass is $\rm M^*\sim M_{dyn}-M(H_2)-M(BH)=7.2\times 10^9~M_\odot$, we derive a molecular gas fraction of $\mu=M(H_2)/M^*\approx4.4$.
For comparison, the molecular gas fraction in z=3-4 main sequence galaxies is at most $\mu=2.5$ (Tacconi et al. 2018, Genzel et al. 2015).
The BH to stellar mass ratio is $\rm M(BH)/M_*\approx 0.25$, which is very large. We note that this value refers to the inner 2.9 kpc of the host galaxy only.
The ratio $v_{rot}/\sigma=1-2$ indicates that the molecular gas is turbulent, similarly to what is found in z=2 star-forming galaxies by Tacconi et al. (2013) (and references therein), and in z=3 star-forming galaxies by Gnerucci et al. (2011) and Williams et al. (2014).
The turbulence may be due to a thick, dynamically hot disk, and/or to outflows/inflows. The current observations do not allow us to discriminate between these two possibilities.
We derived the Toomre parameter $Q= a~ \sigma /f_{gas}~ v_{rot}$, where $a=1.4$ for flat rotation curves ($a=1$ for keplerian disks), and $f_{gas}$ is the gas fraction (Genzel et al. 2014).
We find $Q\sim0.2-0.5$, meaning that cloud fragmentation likely occurs in the disk.
We note, however, that small values of Q do not necessarily imply local gravitational instability (Romeo \& Agertz 2014, Romeo et al. 2010).
We caution that the estimated Q is an average value, does not exclude local variations through the disk, and is strongly dependent on angular resolution.
We studied the CO/[CII] ratio by comparing the CO and [CII] line profiles and their maps.
We find that the CO and [CII] lines show similar profiles.
Our analysis hints at a broader velocity distribution of the [CII]-emitting gas than the CO(6-5)-emitting gas, within $\pm250$ km/s from the CO line peak (Fig. \ref{ratio-sp}, \ref{pv-ratio}), suggesting that there may be differences in the kinematics and spatial distributions of dense clumps of molecular material traced by CO(6-5) ($n_{crit}\sim 10^6\rm cm^{-3}$), and the low-ionization gas traced by [CII] ($n\sim 10^3~\rm cm^{-3}$).
Additional observations are required to assess this finding.
We found a mean $\rm CO(6-5)/[CII]$ of $\sim 0.19$, while a lower value, of $\sim 0.05$, is expected from emission models of high-z galaxies that do not host a QSO and whose ionization field is due only to the stellar component (Pallottini et al. 2017, Vallini et al. 2018).
The larger [CII] extension may imply that some of the [CII] emission comes from the diffuse and less dense ISM.
The higher CO(6-5) concentration might also result from the presence of the QSO intense UV field, boosting high J CO lines.
We note, however, that the bulk of the [CII] emission from these bright QSOs at $z\sim6$ comes from the outer layers of photodissociation regions (PDR), and that a high CO(6-5)/[CII] ratio can also be explained within classical PDR modeling, assuming $\rm n_H>3~ 10^5$ cm$^{-3}$, and a radiation field intensity of $< 1000~ G_0$, or $\rm n_H>16~ \rm cm^{-3}$ and higher radiation fields ($>1000~ G_0$).
In the following we use $\rm M_{dyn}$ estimate derived from CO(6-5), which traces the densest regions of molecular clouds, where most of the SF should occur, and our new estimate of the BH mass, $\rm M_{BH} = (1.8\pm 0.5) \times 10^{9}~ M_{\odot}$.
Figure \ref{mbh-mdyn}, left panel, shows the BH mass versus the dynamical mass for J2310+1855 and a compilation of AGN and QSO for which both the BH mass and the dynamical mass has been measured, the latter through [CII] or CO emission lines (see caption of Fig. \ref{mbh-mdyn} for details about the literature samples).
In this plot we compare individual measurements of $\rm M_{dyn}$ with the local $\rm M_{BH}-M_{bulge}$ relationship (Kormendy \& Ho 2013).
For J2310+1855 we find a mass ratio $\rm M_{BH}/M_{dyn}=0.04^{+0.01}_{-0.035}$.
If the molecular disk is not seen at a very small inclination, the dynamical mass derived here places J2310+1855 significantly above the local $\rm M_{BH}-M_{bulge}$ correlation, similarly to other z=5-7 QSO.
From the QSO bolometric luminosity we derive the BH accretion rate, $\rm \dot M_{acc}=L_{bol}/\epsilon c^2 $, assuming $\epsilon=0.1$. The middle panel of
Figure \ref{mbh-mdyn} shows $\dot M_{acc}$ versus the SFR of the host galaxy.
Here we also plot hyper-luminous QSOs at $z=3-4$ from the WISSH sample (Bischetti et al. 2017, Duras et al. 2017, Vietri et al. 2018, note that these do not have a measured $\rm M_{dyn}$ and therefore they are not shown the left panel).
For QSOs with $\rm L_{bol}=10^{47}$ erg/s, Duras et al. (2017) found that about 50\% of the FIR luminosity is due to reprocessed radiation from the AGN, therefore we corrected SFRs by this amount in the relevant $L_{bol}$ range.
We plot the line corresponding to the mass loading factor of molecular winds equal to unity, $\rm \eta=\dot M_{outflow}/SFR =1$.
This was derived from the scaling relation of $\rm \dot M_{outflow}$ with $L_{bol}$, $\rm \log \dot M_{outflow} = 0.76 \log L_{bol} -32$ (Fiore et al. 2017).
Because $\rm L_{bol}= \epsilon \dot M_{acc} c^2$, and assuming $\rm \epsilon=0.1$, according to the scaling relation of Fiore et al. (2017),
$\rm \log \dot M_{acc} = 1.3~ log(SFR) - 3.65$, for $\eta=1$.
We find that, if the scaling relations hold at these extreme regimes and at high redshift, all the high-redshift QSOs, including J2310+1855, are in a regime where the QSO can drive massive outflows with $\rm \eta>1$. This suggest that appropriately sensitive observations have the potential to reveal molecular outflows in these QSOs (see e.g., Brusa et al. 2017).
$\rm \dot M_{acc}$ versus the SFR are the derivative quantities of BH mass and dynamical mass. Therefore, in principle, they are useful to understand how BH and galaxy grow to reach their location on the left-hand diagram of Fig. 11, if the sources are caught in the phase when most of both the stellar and the BH mass is being assembled.
This is realistic at $z\sim4-6$, but likely not the case at low redshift, therefore we omit the $z\sim 0 $ sample in this plot.
To verify this, we compare the growth timescale of the BH at the current accretion rate ($\rm M_{BH}/\dot M_{acc}$) with the growth timescale of the host galaxy at the current SFR ( $\rm M_{dyn}/SFR$).
The right panel of Figure \ref{mbh-mdyn} shows that the BH growth time is similar to the galaxy growth timescale if we consider the dynamical mass as a proxy for the stellar mass.
Concerning other continuum or line emitters, we detected two candidate emission lines within $\sim 1000$ km/s of the QSO redshift in the band-3 data, located at projected distances of 12 and 29 kpc from the QSO position.
A secure identification of these candidate sources requires the detection of at least one other emission line.
If the candidate lines were genuine, and if we identified them with CO(6-5), and under the same assumptions used for deriving the molecular gas mass of the QSO, these would correspond to molecular gas reservoirs of $\rm M(H_2)\approx (0.8- 1) \times 10^{9}~(\alpha_{CO}/0.8)$ M$_\odot$.
If we apply the calibration of Greve et al. (2014) between the total IR luminosity and $\rm L^\prime CO(6-5)$, and the Kennicutt (1998) relation corrected for a Kroupa (2011) IMF, these would convert into SFRs $\approx 50$ $\rm M_{\odot}/yr$.
This is a very rough estimate, but is consistent with the nondetection of these sources in 200 $\mu$m continuum in band 6. In fact, by assuming a typical SED of a star-forming galaxy with dust temperature in the range $\rm T_{dust}\sim 30-50~ K$, we would expect a $3\sigma$ detection of the continuum for a $\rm SFR=200 ~M_{\odot}/yr$ for a galaxy located at $ z\sim6$.
If confirmed, these line emitters, together with the galaxy presented in D'Odorico et al. (2018), would trace an overdensity of galaxies located close to the QSO, in a configuration similar to that found for other high -z QSOs (e.g., Decarli et al. 2017), and at $z\sim3$ (Fogasy et al. 2017, Bischetti et al. 2018).
We argue that the physical separations of these galaxies from the QSO are typical scales of the circum-galactic medium (CGM), therefore these galaxies may eventually be able to merge with the QSO host, thus contributing to the growth of a giant elliptical galaxy, in agreement with most models of hierarchical galaxy formation. Without confirmation of the two candidate detections presented here however, we are disinclined to make any firm conclusions on the close environment of this QSO.
\section{Conclusions}
Using ALMA we mapped the CO(6-5) and [CII] emission lines and the sub-millimeter continuum of the z$\sim6$ QSO SDSS J231038.88+185519.7. The angular resolution and sensitivity of our data allowed us to resolve the dense molecular gas emission in the host galaxy of the QSO.
Our findings are summarized below.
We measure a molecular gas mass of the QSO host galaxy of $\rm M(H_2)=(3.2 \pm0.2) \times 10^{10}\rm M_{\odot}$,
a size of the molecular disk of $2.9\pm0.5$ kpc, and an inclination of $i\approx53$ deg. We derived a dynamical mass of $\rm M_{dyn} = 4.1 \times 10^{10}~ M_{\odot}$, and a molecular gas fraction $\mu=M(H_2)/M_*\approx 4.4$, which is larger than that found for main sequence galaxies at z=3-4.
We derive a ratio $v_{rot}/\sigma=1-2$, suggesting high gas turbulence and/or outflows/inflows, and a Toomre parameter $Q\sim 0.2-0.5$, indicating cloud fragmentation.
We provide a new estimate of the BH mass based on C~IV emission line detected in the X-SHOOTER/VLT spectrum, $\rm M_{BH} = (1.8\pm 0.5) \times 10^{9}~ M_{\odot}$.
The dynamical mass derived here places J2310+1855 above the local $\rm M_{BH}-M_{bulge}$ correlation, similarly to other high-z QSOs.
We find that for J2310+1855 and for most QSOs, the current BH growth rate is similar to that of its host galaxy.
We argue that all the high-redshift QSOs, including J2310+1855, are in a regime where the QSO can drive massive outflows with loading factors $\eta>1$, if the AGN wind scaling relations hold at these extreme regimes and at high redshift.
We compare the CO(6-5) to the [CII] emission, finding that they have similar line profiles.
The observed slight discrepancies in the line profiles of CO(6-5) and [CII] near the line peaks, and the hints of a broader velocity distribution of the [CII]-emitting gas compared to the CO(6-5), require additional observations to be confirmed.
We detect two candidate emission lines within $\sim 1000$ km/s of the QSO CO line.
With the data used here, it is difficult to determine whether the two detections presented in this work are genuine. These candidate lines require confirmation by the detection of at least one additional emission line.
If they were genuine,
given their close projected distances from the QSO, these galaxies, together with the CO(6-5) emitter detected in the proximity of the QSO (D'Odorico et al. 2018), would trace an overdensity around the QSO, and may eventually be able to merge with the QSO host galaxy, thus contributing to the hierarchical growth of a giant elliptical galaxy, in agreement with most models of galaxy formation.
\begin{acknowledgements}
We thank the referee for his thorough review, and for his highly appreciated comments and suggestions.
This paper makes use of the following ALMA data: ADS/JAO.ALMA\#2015.1.00584.S and 2015.1.00997.S. ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada), MOST and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ.
Based on observations made with ESO Telescopes at the La Silla Paranal Observatory under programme ID 098.B-0537(A).
CF acknowledges support from the European Union Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 664931.
FF acknowledges financial support from INAF under the contract PRIN-INAF-2016 {\it FORECAST}.
R.M. and S.C. acknowledge support by the Science and Technology Facilities Council (STFC).
R.M. acknowledges ERC Advanced Grant 695671 {\it QUENCH}.
EP acknowledges financial support from INAF under the contract PRIN-INAF-2012.
LZ acknowledges financial support under ASI/INAF contract I/037/12/0.
\end{acknowledgements}
|
{
"timestamp": "2018-08-21T02:14:30",
"yymm": "1804",
"arxiv_id": "1804.05566",
"language": "en",
"url": "https://arxiv.org/abs/1804.05566"
}
|
\section{Introduction} \label{intro}
The notion of metallic structure (and, in particular, Golden structure) on a Riemannian manifold was initially studied in (\cite{CrHr},\cite{Hr2},\cite{Hr3},\cite{Hr4}).
In (\cite{Hr2}), the authors of the present paper studied the properties of the slant and semi-slant submanifolds in metallic or Golden Riemannian manifolds and obtained some integrability conditions for the distributions involved in the semi-slant submanifolds of Riemannian manifolds endowed with metallic or Golden Riemannian structures.
The geometry of slant submanifolds in complex manifolds, studied by B.Y. Chen in (\cite{Chen3},\cite{Chen4}) in the early 1990’s, was extended to semi-slant submanifold, pseudo-slant submanifold and bi-slant submanifold, respectively, in different types of differentiable manifolds. Semi-slant submanifolds in almost Hermitian manifolds were introduced by N. Papagiuc (\cite{Papaghiuc}). Semi-slant submanifolds in Sasakian manifolds were studies by J.L. Cabrerizo \textit{et al.} in (\cite{Cabrerizo1},\cite{Cabrerizo2}). A. Cariazzo \textit{et al.} (\cite{Carriazo}) studied bi-slant immersion in almost Hermitian manifolds and pseudo-slant submanifold in almost Hermitian manifolds. Slant and semi-slant submanifolds in almost product Riemannian manifolds were studied in (\cite{Atceken1},\cite{Li&Liu},\cite{Sahin}). The pseudo-slant submanifolds (also called hemi-slant submanifolds) in Kenmotsu or nearly Kenmotsu manifolds (\cite{Atceken3},\cite{Atceken4}), in LCS-manifolds (\cite{Atceken5}) or in locally decomposable Riemannian manifolds (\cite{Atceken6}) were studied by M. At\c{c}eken \textit{et al.} Properties of hemi-slant submanifolds in locally product Riemannian manifolds were studied by H.M. Ta\c{s}tan and F. Ozdem in \cite{Tastan}.
The purpose of the present paper is to investigate the properties of hemi-slant submanifolds in metallic (or Golden) Riemannian manifolds. We find some integrability conditions for the distributions which are involved in such types of submanifolds in metallic and Golden Riemannian manifolds and we give some examples of hemi-slant submanifolds in metallic (or Golden) Riemannian manifolds.
Using a polynomial structure on a manifold (\cite{Goldberg1},\cite{Goldberg2}) and the metallic numbers (\cite{Spinadel}), we defined the metallic structure $J$ (\cite{Hr4}).
The name of this structure is provided by the metallic number $\sigma _{p,q}=\frac{p+\sqrt{p^{2}+4q}}{2}$ (i.e. the positive solution of the equation $x^{2}-px-q=0$) for positive integer values of $p$ and $q$.
If $\overline{M}$ is an $m$-dimensional manifold endowed with a tensor field $J$ of type $(1,1)$ such that:
\begin{equation}\label{e1}
J^{2}= pJ+qI,
\end{equation}
for $p$, $q\in\mathbb{N}^*$, where $I$ is the identity operator on the Lie algebra $\Gamma(T\overline{M})$, then the structure $J$ is a \textit{metallic structure}. In this situation, the pair $(\overline{M},J)$ is called \textit{metallic manifold}.
In particular, if $p=q=1$ one obtains the \textit{Golden structure} (\cite{CrHr}) determined by a $(1,1)$-tensor field $J$ which verifies $J^{2}= J + I$. In this case, $(\overline{M},J)$ is called {\it Golden manifold} (\cite{CrHr}).
If ($\overline{M}, \overline{g})$ is a Riemannian manifold endowed with a metallic (or a Golden) structure $J$, such that the Riemannian metric $\overline{g}$ is $J$-compatible, i.e.:
\begin{equation} \label{e2}
\overline{g}(JX, Y)= \overline{g}(X, JY),
\end{equation}
for any $X, Y \in \Gamma(T\overline{M})$, then $(\overline{g},J)$ is called a {\it metallic} (or a {\it Golden) Riemannian structure} and $(\overline{M},\overline{g},J)$ is a {\it metallic (or a Golden) Riemannian manifold} (\cite{Hr4}).
Moreover, we have:
\begin{equation} \label{e3}
\overline{g}(JX, JY)=\overline{g}(J^{2}X, Y) =p \overline{g}(JX,Y)+q \overline{g}(X,Y),
\end{equation}
for any $X, Y \in \Gamma(T\overline{M})$ (\cite{Hr4}).
\normalfont
Any almost product structure $F$ on $\overline{M}$ induces two metallic structures on $\overline{M}$:
\begin{equation}\label{e4}
J= \frac{p}{2}I \pm \frac{2\sigma _{p, q}-p}{2}F,
\end{equation}
where $I$ is the identity operator on the Lie algebra $\Gamma(T\overline{M})$ (\cite{Hr4}).
\section{On the metallic Riemannian manifolds and its submanifolds}
Let $M$ be an $m'$-dimensional submanifold, isometrically immersed in the $m$-dimensional metallic (or Golden) Riemannian manifold ($\overline{M}, \overline{g},J)$ with $m, m' \in \mathbb{N}^{*}$ and $m > m'$.
Let $T_{x}M$ be the tangent space of $M$ in a point $x \in M$ and $T_{x}^{\bot }M$ the normal space of $M$ in $x$. The tangent space $T_x\overline{M}$ can be decomposed into the direct sum:
$$T_x\overline{M}=T_x M\oplus T_x^{\perp}M,$$ for any $x\in M$. Let $i_{*}$ be the differential of the immersion $i: M \rightarrow\overline{M}$.
The induced Riemannian metric $g$ on $M$ is given by $g(X, Y)=\overline{g}(i_{*}X, i_{*}Y)$, for any $X, Y \in \Gamma(TM)$. For the simplification of the notations, in the rest of the paper we shall note by $X$ the vector field $i_{*}X$, for any $X \in \Gamma(TM)$.
If we denote by $TX$ and $NX$, respectively, the tangential and normal parts of $JX$, for any $X \in \Gamma(TM)$, then we get:
\begin{equation}\label{e5}
JX = TX + NX,
\end{equation}
$T:\Gamma(TM)\rightarrow \Gamma(TM)$, $ TX:=(J X)^T$ and $N:\Gamma(TM)\rightarrow \Gamma(T^{\perp}M)$, $NX:=(J X)^{\perp}$.
For any $V \in \Gamma(T^{\perp}M)$, the tangential and normal parts of $JV$ satisfy:
\begin{equation}\label{e6}
JV = tV + nV,
\end{equation}
$t:\Gamma(T^{\perp}M)\rightarrow \Gamma(TM)$, $tV:=(J V)^{T}$ and $ n:\Gamma(T^{\perp}M)\rightarrow \Gamma(T^{\perp}M)$, $nV:=(J V)^{\perp}$.
We remark that the maps $T$ and $n$ are $\overline{g}$-symmetric (\cite{Blaga_Hr}):
\begin{equation}\label{e7}
(i)\: \overline{g}(TX,Y)=\overline{g}(X,TY), \quad (ii)\: \overline{g}(nU,V)=\overline{g}(U,nV),
\end{equation}
for any $X, Y\in \Gamma(TM)$ and $U, V \in \Gamma(T^{\perp}M)$. Moreover, we get
\begin{equation}\label{e8}
\overline{g}(NX,U)=\overline{g}(X,tU),
\end{equation}
for any $X\in \Gamma(TM)$ and $U\in \Gamma(T^{\perp}M)$.
By using (\ref{e5}), (\ref{e6}) and (\ref{e1}), we obtain:
\begin{remark}
If $M$ is a submanifold in a metallic Riemannian manifold $(\overline{M}, \overline{g}, J)$, then:
\begin{equation} \label{e99}
(i) \: T^{2}X = pTX+qX-tNX, \quad (ii) \: pNX= NTX+nNX
\end{equation}
and
\begin{equation} \label{e100}
(i) \: n^{2}V = pnV+qV-NtV, \quad (ii) \: ptV= TtV+tnV,
\end{equation}
for any $X \in \Gamma(TM)$ and $V \in \Gamma(T^{\bot}M)$.
For $p=q=1$ and $M$ is a submanifold in a Golden Riemannian manifold $(\overline{M}, \overline{g}, J)$ then, for any $X \in \Gamma(TM)$ we get
$ T^{2}X = TX+X-tNX$, $NX= NTX+ nNX$ and for any $V \in \Gamma(T^{\bot}M)$ we get $n^{2}V = nV+V-NtV$, $tV= TtV+tnV$.
\end{remark}
\begin{remark}(\cite{Hr6})
Let $(\overline{M}, \overline{g})$ be a Riemannian manifold endowed with an almost product structure $F$ and let $J$ be one of the two metallic structures induced by $F$ on $\overline{M}$. If $M$ is a submanifold in the almost product Riemannian manifold $(\overline{M}, \overline{g}, F)$, then:
\begin{equation} \label{e9}
(i) \: TX = \frac{p}{2}X \pm \frac{2\sigma-p}{2}fX, \quad (ii) \: NX= \pm \frac{2\sigma-p}{2}\omega X
\end{equation}
and
\begin{equation} \label{e10}
(i) \: tV = \pm \frac{2\sigma-p}{2}BV, \quad (ii) \: nV= \frac{p}{2}V \pm \frac{2\sigma-p}{2}CV,
\end{equation}
for any $X \in \Gamma(TM)$ and $V \in \Gamma(T^{\bot}M)$, where
$FX = fX + \omega X$, $FV = BV + CV$, with $fX:=(F X)^{T}$, $\omega X:=(FX)^{\perp}$, $BV:=(F V)^T$ and $CV:=(F V)^{\perp}$.
\end{remark}
In the next considerations we denote by $\overline{\nabla}$ and $\nabla $ the Levi-Civita connections on $(\overline{M},\overline{g})$ and its submanifold $(M,g)$, respectively.
The Gauss and Weingarten formulas are given by:
\begin{equation}\label{e11}
(i) \: \overline{\nabla}_{X}Y=\nabla_{X}Y+h(X,Y), \quad (ii) \: \overline{\nabla}_{X}V=-A_{V}X+\nabla_{X}^{\bot}V,
\end{equation}
for any $X, Y \in \Gamma(TM)$ and $V \in \Gamma(T^{\bot}M)$, where $h$ is the second fundamental form and $A_{V}$ is the shape operator. The second fundamental form $h$ and the shape operator $A_{V}$ are related by:
\begin{equation}\label{e12}
\overline{g}(h(X, Y),V)=\overline{g}(A_{V}X, Y).
\end{equation}
\begin{definition}\label{d1}(\cite{Hr5})
If $(\overline{M},\overline{g}, J)$ is a metallic (or Golden) Riemannian manifold and $J$ is parallel with respect to the Levi-Civita connection $\overline{\nabla}$ on $\overline{M}$ (i.e. $\overline{\nabla}J=0$), we say that $(\overline{M},\overline{g}, J)$ is a {\it locally metallic (or locally Golden) Riemannian manifold}.
\end{definition}
The covariant derivatives of the tangential and normal parts of $JX$ (and $JV$), $T$ and $N$ ($t$ and $n$, respectively) are given by (\cite{Hr5},\cite{Atceken3}):
\begin{equation}\label{e13}
(i) \: (\nabla_{X}T)Y=\nabla_{X}TY - T(\nabla_{X}Y), \quad (ii) \:(\overline{\nabla}_{X}N)Y=\nabla_{X}^{\bot}NY - N(\nabla_{X}Y),
\end{equation}
and
\begin{equation}\label{e14}
(i) \: (\nabla_{X}t)V=\nabla_{X}tV - t(\nabla_{X}^{\bot}V), \quad (ii) \:(\overline{\nabla}_{X}n)V=\nabla_{X}^{\bot}nV - n(\nabla_{X}^{\bot}V),
\end{equation}
for any $X$, $Y \in \Gamma(TM)$ and $V \in \Gamma(T^{\bot}M) $. From $\overline{g}(JX,Y)=\overline{g}(X,JY)$, it follows:
\begin{equation} \label{e15}
\overline{g}((\overline{\nabla}_XJ)Y,Z)=\overline{g}(Y,(\overline{\nabla}_XJ)Z),
\end{equation}
for any $X$, $Y$, $Z\in \Gamma(T\overline{M})$. Moreover, if $M$ is an isometrically immersed submanifold in the metallic Riemannian manifold $(\overline{M},\overline{g},J)$, then (\cite{Blaga2}):
\begin{equation}\label{e16}
\overline{g}((\nabla_X T)Y,Z)=\overline{g}(Y,(\nabla_X T)Z),
\end{equation}
for any $X$, $Y$, $Z\in \Gamma(TM)$.
\begin{prop}
If $M$ is a submanifold in a locally metallic (or Golden) Riemannian manifold $(\overline{M},\overline{g},J)$, then the covariant derivatives of $T$ and $N$ verify:
\begin{equation}\label{e17}
(i) (\nabla_{X}T)Y=A_{NY}X+th(X,Y), \quad (ii) \: (\overline{\nabla}_{X}N)Y=nh(X,Y)-h(X,TY),
\end{equation}
and
\begin{equation}\label{e18}
(i) (\nabla_{X}t)V=A_{nV}X - TA_{V}X, \quad (ii) \: (\overline{\nabla}_{X}n)V=-h(X,tV)-NA_{V}X,
\end{equation}
for any $X$, $Y \in \Gamma(TM)$ and $V \in \Gamma(T^{\bot}M)$.
\end{prop}
\begin{remark}
If $M$ is a totally geodesic submanifold in a locally metallic (or locally Golden) Riemannian manifold $(\overline{M},\overline{g},J)$, then:
$(\nabla_{X}T)Y=(\overline{\nabla}_{X}N)Y=(\nabla_{X}t)V=(\overline{\nabla}_{X}n)V=0,$
for any $X,Y \in \Gamma(TM)$ and $V \in \Gamma(T^{\perp}M)$.
\end{remark}
\begin{remark}
If $M$ is a submanifold in a locally metallic (or locally Golden) Riemannian manifold $(\overline{M},\overline{g},J)$, then we obtain:
\begin{equation}\label{e19}
\overline{g}((\overline{\nabla}_{X}N)Y,V )= \overline{g}((\nabla_{X}t)V,Y),
\end{equation}
for any $X$, $Y \in \Gamma(TM)$ and $V \in \Gamma(T^{\bot}M)$.
\end{remark}
\begin{proof}
From (\ref{e17})(ii) and (\ref{e7})(ii) we get
$$ \overline{g}((\overline{\nabla}_{X}N)Y,V )= \overline{g}(h(X,Y),nV )-\overline{g}(h(X,TY),V )= \overline{g}(A_{nV}X - TA_{V}X,Y) $$
and using (\ref{e18})(i) we obtain (\ref{e19}).
\end{proof}
\begin{prop}
Let $M$ be a submanifold in a locally metallic (or locally Golden) Riemannian manifold ($\overline{M},\overline{g},J$). Then $(\overline{\nabla}_{X}N)Y=0$ and $(\nabla_{X}t)V=0$, for any $X$, $Y \in \Gamma(TM)$, $V \in \Gamma(T^{\bot}M)$ if and only if the shape operator $A$ verifies:
\begin{equation}\label{e33}
A_{nV}X=TA_{V}X=A_{V}TX.
\end{equation}
\end{prop}
\begin{proof}
From (\ref{e7})(ii) we get $\overline{g}(nh(X,Y),V)=\overline{g}(h(X,Y),nV)$, for any $X, Y \in \Gamma(TM)$, $V \in \Gamma(T^{\bot}M)$. Thus, we obtain:
$$\overline{g}((\overline{\nabla}_{X}N)Y,V)=\overline{g}(h(X,Y),nV)-\overline{g}(h(X,TY),V)=\overline{g}(A_{nV}X,Y)-\overline{g}(A_{V}X,TY),$$
for any $X, Y \in \Gamma(TM)$, $V \in \Gamma(T^{\bot}M)$. From (\ref{e17})(ii) and (\ref{e12}) we have
\begin{equation}\label{e34}
\overline{g}((\overline{\nabla}_{X}N)Y,V)=\overline{g}(A_{nV}X-TA_{V}X,Y)=\overline{g}(A_{nV}Y-A_{V}TY,X),
\end{equation}
for any $X, Y \in \Gamma(TM)$, $V \in \Gamma(T^{\bot}M)$. Thus, from (\ref{e34}) and (\ref{e19}) we obtain the conclusion.
\end{proof}
\begin{prop}(\cite{Hr6})
If $M$ is a submanifold in a locally metallic (or locally Golden) Riemannian manifold $(\overline{M},\overline{g},J)$, then:
\begin{equation}\label{e20}
T([X,Y])=\nabla_{X}TY-\nabla_{Y}TX-A_{NY}X+A_{NX}Y
\end{equation}
and
\begin{equation}\label{e21}
N([X,Y])=h(X,TY)-h(TX,Y)+\nabla_{X}^{\bot}NY-\nabla_{Y}^{\bot}NX,
\end{equation}
for any $X, Y \in \Gamma(TM)$, where $\nabla$ is the Levi-Civita connection on $\Gamma(TM)$.
\end{prop}
\section{Hemi-slant submanifolds in metallic Riemannian manifolds}
In this section we recall the definition of a slant distribution and of a bi-slant submanifold in a metallic (or Golden) Riemannian manifold. Then, we define the hemi-slant submanifold and find some properties regarding the distributions involved in this type of submanifold, using a similar definition as for Riemannian product manifold (\cite{Tastan}).
\begin{definition}(\cite{Hr6})\label{d2}
Let $M$ be an immersed submanifold in a metallic (or Golden) Riemannian manifold $(\overline{M},\overline{g},J)$. A differentiable distribution $D$ on $M$ is called a {\it slant distribution} if the angle $\theta_{D}$ between $JX_{x}$ and the vector subspace $D_{x}$ is constant, for any $x \in M$ and any nonzero vector field $X_{x} \in \Gamma(D_{x})$. The constant angle $\theta_{D}$ is called the {\it slant angle} of the distribution $D$.
\end{definition}
\begin{prop}(\cite{Hr6})\label{pD}
Let $D$ be a differentiable distribution on a submanifold $M$ of a metallic (or Golden) Riemannian manifold $(\overline{M},\overline{g},J)$. The distribution $D$ is a slant distribution if and only if there exists a constant $\lambda \in [0, 1]$ such that:
\begin{equation}\label{e22}
(P_{D}T)^{2}X= \lambda(pP_{D}TX+qX),
\end{equation}
for any $ X \in \Gamma(D)$, where $P_{D}$ is the orthogonal projection on $D$. Moreover, if $\theta_{D}$ is the slant angle of $D$, then it satisfies $\lambda = \cos^{2} \theta_{D}$.
\end{prop}
\begin{definition}(\cite{Hr6})\label{d3}
Let $M$ be an immersed submanifold in a metallic (or Golden) Riemannian manifold $(\overline{M},\overline{g},J)$. We say that $M$ is a {\it bi-slant submanifold} of $\overline{M}$ if there exist two orthogonal differentiable distribution $D_{1}$ and $D_{2}$ on $M$ such that $TM = D_{1}\oplus D_{2}$, and $D_{1}$, $D_{2}$ are slant distributions with the slant angles $\theta_{1}$ and $\theta_{2}$, respectively. Moreover, $M$ is a {\it proper bi-slant submanifold} of $\overline{M}$ if $dim(D_{1})\cdot dim(D_{2}) \neq 0 $.
\end{definition}
A particular case of {\it bi-slant submanifold} in a metallic Riemannian manifold $(\overline{M},\overline{g},J)$ is the hemi-slant submanifold, defined in a similar manner as hemi-slant submanifold of the locally product Riemannian manifold (\cite{Tastan}):
\begin{definition}\label{d6}
An immersed submanifold $M$ in a metallic (or Golden) Riemannian manifold $(\overline{M},\overline{g},J)$ is a \textit{hemi-slant submanifold} if there exist two orthogonal distributions
$D^{\theta}$ and $D^{\perp}$ on $M$ such that:
(1) $TM$ admits the orthogonal direct decomposition $TM = D^{\theta}\oplus D^{\perp}$;
(2) The distribution $D^{\theta}$ is slant with angle $\theta \in [0,\frac{\pi}{2}]$;
(3) The distribution $D^{\perp}$ is anti-invariant distribution (i.e. $J(D^{\perp}) \subseteq \Gamma(T^{\perp}M$)).
Moreover, if $dim(D^{\theta})\cdot dim(D^{\perp}) \neq 0 $ and $\theta \in (0, \frac{\pi}{2})$, then $M$ is a proper hemi-slant submanifold.
\end{definition}
\begin{remark}
If $M$ is a hemi-slant submanifold in a metallic Riemannian manifold $(\overline{M},\overline{g},J)$, with $TM = D^{\theta}\oplus D^{\perp}$, for particular cases we get:
- if $\theta=0$ and $dim(D^{\perp})=0$, then $M$ is an invariant submanifold;
- if $dim(D^{\theta})=0$ or $\theta = \frac{\pi}{2}$, then $M$ is an anti-invariant submanifold;
- if $dim(D^{\perp})=0$ and $\theta \neq 0$, then $M$ is a slant submanifold;
- if $dim(D^{\theta}) \cdot dim(D^{\perp}) \neq 0$ and $\theta=0$, then $M$ is a semi-invariant submanifold.
\end{remark}
In a similar manner as in (\cite{Hr6}, Proposition 10), we obtain:
\begin{remark}
If $M$ is a hemi-slant submanifold in a metallic Riemannian manifold $(\overline{M},\overline{g},J)$, with $TM = D^{\theta}\oplus D^{\perp}$, then we get that $M$ is an anti-invariant submanifold if $\theta=\frac{\pi}{2}$ and $g(JX,Y)=0$, for any $X \in \Gamma(D^{\theta})$ and $X \in \Gamma(D^{\perp})$.
\end{remark}
Let $M$ be a hemi-slant submanifold in a metallic Riemannian manifold $(\overline{M},\overline{g},J)$, with $TM = D^{\theta}\oplus D^{\perp}$ and let $P_{1}$ and $P_{2}$ be the orthogonal projections on $D^{\theta}$ and $D^{\perp}$, respectively. Thus, for any $X \in \Gamma(TM)$, we can consider the decomposition of $X=P_{1}X + P_{2}X$, where $P_{1}X \in \Gamma(D^{\theta})$ and $P_{2}X \in \Gamma(D^{\perp})$. From $J(D^{\perp}) \subseteq \Gamma(T^{\perp}M)$ we obtain:
\begin{prop}
If $M$ is a hemi-slant submanifold in a metallic (or Golden) Riemannian manifold $(\overline{M},\overline{g},J)$, then:
\begin{equation}\label{e23}
JX= TP_{1}X+ NP_{1} X + NP_{2}X = TP_{1}X+ N X
\end{equation}
and
\begin{equation}\label{e24}
(i) JP_{2}X= NP_{2}X, \: (ii) TP_{2}X=0, \: (iii) TP_{1} X \in \Gamma(D^{\theta}),
\end{equation}
for any $X \in \Gamma(TM)$.
\end{prop}
\begin{remark}
If $M$ is a hemi-slant submanifold in a metallic (or Golden) Riemannian manifold $(\overline{M},\overline{g},J)$, then:
\begin{equation}\label{e101}
T^{\perp}M= N(D^{\theta}) \oplus N(D^{\perp}) \oplus \mu,
\end{equation}
where $\mu$ is an invariant subbundle of $T^{\perp}M$.
\end{remark}
\begin{proof}
For any $X \in \Gamma(D^{\theta})$ and $Z \in \Gamma(D^{\perp})$ we get
$$\overline{g}(NX,NZ)=\overline{g}(JX,JZ)=p \overline{g}(X,TZ)+q \overline{g}(X,Z)=0.$$
Thus, the distributions $N(D^{\theta})$ and $N(D^{\perp})$ are mutually perpendicular in $T^{\perp}M$. If we denote by $\mu$ the orthogonal complementary subbundle of $J(TM)$ in $T^{\perp}M$, then we obtain (\ref{e101}).
\end{proof}
\begin{remark}
If $M$ is a hemi-slant submanifold in a metallic (or Golden) Riemannian manifold $(\overline{M},\overline{g},J)$, then:
$$\overline{g}(JP_{1}X,TP_{1}X)=\cos \theta(X) \| TP_{1}X\| \cdot \|JP_{1}X\|$$
and the cosine of the slant angle $\theta(X)=:\theta$ of the distribution $D^{\theta}$ is constant, for any nonzero $X \in \Gamma(TM)$. Thus, we get:
\begin{equation}\label{e25}
\cos \theta =\frac{\overline{g}(JP_{1}X, TP_{1}X)}{\|TP_{1}X\| \cdot \|JP_{1}X\|}=\frac{\|TP_{1}X \|}{\|JP_{1}X\|},
\end{equation}
for any nonzero $X \in \Gamma(TM)$.
\end{remark}
\begin{prop}\label{p11}
If $M$ is a hemi-slant submanifold in a metallic Riemannian manifold $(\overline{M},\overline{g},J)$, then:
\begin{equation}\label{e26}
\overline{g}(TP_{1}X,TP_{1}Y)=\cos^2 \theta[p \overline{g}(TP_{1}X,P_{1}Y)+q \overline{g}(P_{1}X,P_{1}Y)]
\end{equation}
and
\begin{equation}\label{e27}
\overline{g}(NX,NY)=\sin^2 \theta[p \overline{g}(TP_{1}X,P_{1}Y)+q \overline{g}(P_{1}X,P_{1}Y)],
\end{equation}
for any $X$, $Y\in \Gamma(TM)$.
\end{prop}
\begin{proof}
Taking $X+Y$ in (\ref{e25}) we have:
$$
\overline{g}(TP_{1}X,TP_{1}Y)=\cos^{2}\theta \overline{g}(JP_{1}X,JP_{1}Y)= \cos^{2}\theta[p\overline{g}(JP_{1}X,P_{1}Y)+q\overline{g}(P_{1}X,P_{1}Y)],$$
for any $X$, $Y\in \Gamma(TM)$ and using (\ref{e24})(iii) we get (\ref{e26}).
From (\ref{e23}) we get $\overline{g}(TP_{1}X,TP_{1}Y)=\overline{g}(JP_{1}X,JP_{1}Y)-\overline{g}(NX,NY),$ for any $X$, $Y\in \Gamma(TM)$ and it implies (\ref{e27}).
\end{proof}
\begin{remark}
A hemi-slant submanifold $M$ in a Golden Riemannian manifold $(\overline{M},\overline{g},J)$ with the slant angle $\theta$ of the distribution $D^{\theta}$ verifies (\ref{e26}) and (\ref{e27}) with $p=q=1$.
\end{remark}
\begin{prop}
Let $M$ be a hemi-slant submanifold in a metallic Riemannian manifold $(\overline{M}, \overline{g},J)$ with the slant angle $\theta$ of the distribution $D^{\theta}$. Then:
\begin{equation}\label{e28}
(TP_{1})^2=\cos^2 \theta(p TP_{1}+qI),
\end{equation}
where $I$ is the identity on $\Gamma(D^{\theta})$ and
\begin{equation}\label{e29}
\nabla ((TP_{1})^2)=p \cos^2 \theta \nabla (TP_{1}).
\end{equation}
\end{prop}
\begin{remark}
Let $M$ be a hemi-slant submanifold in a metallic (or Golden) Riemannian manifold $(\overline{M},\overline{g},J)$, with $TM = D^{\theta}\oplus D^{\perp}$. Then $T(D^{\theta})=D^{\theta}$ and $T(D^{\perp})={0}$.
\end{remark}
\begin{proof}
By using (\ref{e7})(i), we get $\overline{g}(TX,Z)=\overline{g}(X,TZ)=0,$ for any $X \in \Gamma(D^{\theta})$, $Z \in \Gamma(D^{\perp})$. Thus, $T(D^{\theta}) \perp D^{\perp}$. Since $T(D^{\theta}) \subset \Gamma(TM)$ we obtain that $T(D^{\theta}) \subseteq D^{\theta}$.
Moreover, from (\ref{e28}) we obtain $X=\frac{1}{q} T(TX-p \cos^2 \theta X)$, for any $X \in \Gamma(D^{\theta})$ (i.e. $P_{1}X=X$), where $(\overline{M},\overline{g},J)$ is a metallic Riemannian manifold.
If $(\overline{M},\overline{g},J)$ is a Golden Riemannian manifold, then $X=T(TX- \cos^2 \theta X)$, for any $X \in \Gamma(D^{\theta})$.
Thus, $D^{\theta} \subseteq T(D^{\theta})$. Since $T(D^{\theta}) \subseteq D^{\theta}$, we get $T(D^{\theta})=D^{\theta}$.
By using (\ref{e24})(ii) we obtain that $D^{\perp}$ is anti-invariant with respect to $J$ and $T(D^{\perp})={0}$.
\end{proof}
\begin{prop}
Let $M$ be an immersed submanifold in a metallic Riemannian manifold $(\overline{M},\overline{g},J)$. Then $M$ is a hemi-slant submanifold in $\overline{M}$ if and only if there exists a constant $\lambda \in [0, 1]$ such that:
$$D=\{ X \in \Gamma(TM) | T^{2}X= \lambda(pTX+qX)\}$$ is a distribution and $TY=0$, for any $Y$ orthogonal to $D$, $Y\in \Gamma(TM)$, where $p$, $q\in\mathbb{N}^*$.
\end{prop}
\begin{proof}
If $M$ is a hemi-slant submanifold in a metallic Riemannian manifold $(\overline{M},\overline{g},J)$, with $D^{\theta}:=D$ and $TM = D^{\theta}\oplus D^{\perp}$ then, from (\ref{e28}) and $\theta(X) \neq 0$ we have $\lambda=\cos^2 \theta \in [0,1]$.
Conversely, if there exists a real number $\lambda\in[0,1]$ such that $T^2X=\lambda(pTX+qX)$, for any $X \in \Gamma(D)$, it follows that $\cos^2 \theta(X)=\lambda$ which implies that $\theta(X)=\arccos(\sqrt{\lambda})$ does not depend on $X$. If we consider the orthogonal direct sum $TM=D \oplus D^{\bot}$, since $T(D)\subseteq D$ and $TY=0$, for any $Y$ orthogonal to $D$, $Y\in \Gamma(TM)$, we obtain that $M$ is a hemi-slant submanifold in $\overline{M}$ with $D^{\theta}:=D$.
\end{proof}
\begin{remark}
An immersed submanifold $M$ in a Golden Riemannian manifold $(\overline{M},\overline{g},J)$ is a hemi-slant submanifold in $\overline{M}$ if and only if there exists a constant $\lambda \in [0, 1]$ such that $$D=\{ X \in \Gamma(TM) | T^{2}X= \lambda(TX+X)\}$$ is a distribution and $TY=0$, for any $Y \in \Gamma(TM)$ orthogonal to $D$.
\end{remark}
\textbf{Examples 1:}
Let $\mathbb{R}^{4}$ be the Euclidean space endowed with the usual Euclidean metric $<\cdot,\cdot>$.
Let $f: M \rightarrow \mathbb{R}^{4}$ be the immersion given by:
$$f(u,v)=(u \cos t, u \sin t, v,\frac{\sigma}{\sqrt{q}}v),$$
where $M :=\{(u,v) \mid u>0, t \in (0, \frac{\pi}{2})\}$ and $\sigma:=\sigma_{p,q}=\frac{p+\sqrt{p^{2}+4q}}{2}$ is the metallic number ($p, q \in N^{*}$).
We can find a local orthonormal frame on $TM$ given by:
$$Z_{1}= \cos t \frac{\partial}{\partial x_{1}} + \sin t \frac{\partial}{\partial x_{2}}, \quad
Z_{2}=\frac{\partial}{\partial x_{3}} + \frac{\sigma}{\sqrt{q}}\frac{\partial}{\partial x_{4}}.$$
We define the metallic structure $J : \mathbb{R}^{4} \rightarrow \mathbb{R}^{4} $ by:
$$
J(X_{1},X_{2},X_{3},X_{4})=(\sigma X_{1},\overline{\sigma} X_{2},\sigma X_{3},\overline{\sigma} X_{4}),
$$
and we can easily verify that $J^{2}X=p J + q I$ and $<JX, Y> = <X, JY>$, for any $X:=(X_{1},X_{2},X_{3},X_{4})$, $Y:=(Y_{1},Y_{2},Y_{3},Y_{4}) \in \mathbb{R}^{4}$.
Thus, we obtain:
$$JZ_{1}= \sigma \cos t \frac{\partial}{\partial x_{1}} + \overline{\sigma} \sin t \frac{\partial}{\partial x_{2}}, \quad
JZ_{2}= \sigma \frac{\partial}{\partial x_{3}} + \frac{\sigma\overline{\sigma}}{\sqrt{q}}\frac{\partial}{\partial x_{4}}.$$
We remark that $<JZ_{2}, Z_{1}> = <JZ_{2}, Z_{2}>=0$, thus $JZ_{2} \perp span \{Z_{1},Z_{2}\}$.
We find that $\|Z_{1}\|^{2}=1$, $\|JZ_{1}\|^{2}= \sigma^{2} \cos^{2}t +\overline{\sigma}^{2}\sin^{2}t=p (\sigma \cos^{2}t +\overline{\sigma}\sin^{2}t)+q$
and $<JZ_{1}, Z_{1}>=\sigma \cos^{2}t +\overline{\sigma}\sin^{2}t$. Thus, we get
$$ \cos \theta = \frac{<JZ_{1},Z_{1}>}{\|Z_{1}\|\cdot \|JZ_{1}\|}=\frac{\sigma \cos^{2}t +\overline{\sigma}\sin^{2}t}{\sqrt{\sigma^{2} \cos^{2}t +\overline{\sigma}^{2}\sin^{2}t}}=
\frac{\sigma \cos^{2}t +\overline{\sigma}\sin^{2}t}{\sqrt{p (\sigma \cos^{2}t +\overline{\sigma}\sin^{2}t)+q}}.$$
In particular, for $t=\frac{\pi}{4}$ we get $\cos \theta = \frac{\sigma +\overline{\sigma}}{\sqrt{\sigma^{2} +\overline{\sigma}^{2}}}$.
We define the distributions $D^{\theta}=span\{Z_{1}\}$ and $D^{\perp}=span\{Z_{2}\}$. We have $J(D^{\perp})\subset \Gamma(T^{\perp}M)$ and
$D^{\theta}$ is a slant distribution, with the slant angle $\theta = \arccos \frac{\sigma \cos^{2}t +\overline{\sigma}\sin^{2}t}{\sqrt{p (\sigma \cos^{2}t +\overline{\sigma}\sin^{2}t)+q}}.$
The Riemannian metric tensor of $D^{\theta} \oplus D^{\perp}$ is given by
$
g=du^{2} + \frac{p\sigma+2q}{q}d v^{2}.
$
Thus, $M$ is a hemi-slant submanifold in the metallic Riemannian manifold $(\mathbb{R}^{4}, <\cdot,\cdot>, J)$, with $TM=D^{\theta} \oplus D^{\perp}$.
In particular, for $p=q=1$ and $\phi:=\sigma_{1,1}=\frac{1+\sqrt{5}}{2}$ is the Golden number ($\overline{\phi}:=1-\phi$), the immersion $f: M \rightarrow \mathbb{R}^{4}$ is given by $f(u,v)=(u \cos t, u \sin t, v,\phi v)$ and the Golden structure $J : \mathbb{R}^{4} \rightarrow \mathbb{R}^{4} $ can be defined by
$$ J(X_{1},X_{2},X_{3},X_{4})=(\phi X_{1},\overline{\phi} X_{2},\phi X_{3},\overline{\phi} X_{4}).$$
The distribution $D^{\perp}=span\{Z_{2}\}$ verifies $J(D^{\perp})\subset \Gamma(T^{\perp}M)$ and $D^{\theta}=span\{Z_{1}\}$ is a slant distribution, with the slant angle $\theta = \arccos \frac{\phi \cos^{2}t +\overline{\phi}\sin^{2}t}{ \sqrt{(\phi \cos^{2}t +\overline{\phi}\sin^{2}t)+1}}$
and the Riemannian metric tensor of $D^{\theta} \oplus D^{\perp}$ is given by $g=du^{2} + (\phi+2)d v^{2}.$
Thus, $TM=D^{\theta} \oplus D^{\perp}$ and $M$ is a hemi-slant submanifold in the Golden Riemannian manifold $(\mathbb{R}^{4}, <\cdot,\cdot>, J)$.
If we consider the metallic structure $\overline{J} : \mathbb{R}^{4} \rightarrow \mathbb{R}^{4} $ given by
$$ \overline{J}(X_{1},X_{2},X_{3},X_{4})=(\sigma X_{1},\sigma X_{2},\sigma X_{3},\overline{\sigma} X_{4}),$$
then we obtain:
$\overline{J}Z_{1}= \sigma Z_{1}$ and $\overline{J}Z_{2}= \sigma \frac{\partial}{\partial x_{3}} + \frac{\sigma\overline{\sigma}}{\sqrt{q}}\frac{\partial}{\partial x_{4}}.$
In this case we obtain the distributions $D^{\perp}=span\{Z_{2}\}$ and
$D^{\theta}=span\{Z_{1}\}$ with the slant angle $\theta = \arccos 1 = 0.$ Thus, $TM=D^{\theta} \oplus D^{\perp}$ and $M$ is a semi-invariant submanifold in the metallic Riemannian manifold $(\mathbb{R}^{4}, <\cdot,\cdot>, \overline{J})$. Similarly, for $p=q=1$ we obtain that $M$ is a semi-invariant submanifold in the Golden Riemannian manifold $(\mathbb{R}^{4}, <\cdot,\cdot>, \overline{J})$.
\textbf{Examples 2:}
Let $\mathbb{R}^{7}$ be the Euclidean space endowed with the usual Euclidean metric $<\cdot,\cdot>$.
Let $f: M \rightarrow \mathbb{R}^{7}$ be the immersion given by:
$$f(u,v,w)=(\frac{1}{\sqrt{3}} u \cos t,\frac{1}{\sqrt{3}} u \sin t, v,\frac{\sigma}{\sqrt{q}}v, \frac{\sqrt{q}}{\sigma}w, w, \frac{\sqrt{2}}{\sqrt{3}}u),$$
where $M :=\{(u,v,w) \mid u>0, t \in (0, \frac{\pi}{2})\}$ and $\sigma:=\sigma_{p,q}$ is the metallic number ($p, q \in N^{*}$).
We can find a local orthonormal frame on $TM$ given by:
$$Z_{1}= \frac{1}{\sqrt{3}}\cos t \frac{\partial}{\partial x_{1}} + \frac{1}{\sqrt{3}}\sin t \frac{\partial}{\partial x_{2}}+\frac{\sqrt{2}}{\sqrt{3}}\frac{\partial}{\partial x_{7}},
\quad Z_{2}=\frac{\partial}{\partial x_{3}} + \frac{\sigma}{\sqrt{q}}\frac{\partial}{\partial x_{4}},
\quad
Z_{3}=\frac{\sqrt{q}}{\sigma} \frac{\partial}{\partial x_{5}} + \frac{\partial}{\partial x_{6}}.$$
We define the metallic structure $J : \mathbb{R}^{7} \rightarrow \mathbb{R}^{7} $ by:
$$
J(X_{1},X_{2},X_{3},X_{4},X_{5},X_{6},X_{7})=(\sigma X_{1},\overline{\sigma} X_{2},\sigma X_{3},\overline{\sigma} X_{4},\sigma X_{5},\overline{\sigma} X_{6},\sigma X_{7})
$$
and we can easily verify that $J^{2}X=p J + q I$ and $<JX, Y> = <X, JY>$, for any $X:=(X_{1},X_{2},X_{3},X_{4},X_{5},X_{6},X_{7})$, $Y:=(Y_{1},Y_{2},Y_{3},Y_{4},Y_{5},Y_{6},Y_{7}) \in \mathbb{R}^{7}$.
Thus, we obtain:
$$JZ_{1}= \frac{1}{\sqrt{3}}\sigma \cos t \frac{\partial}{\partial x_{1}} + \frac{1}{\sqrt{3}}\overline{\sigma} \sin t \frac{\partial}{\partial x_{2}}+\frac{\sqrt{2}}{\sqrt{3}}\sigma\frac{\partial}{\partial x_{7}},$$
$$ JZ_{2}= \sigma \frac{\partial}{\partial x_{3}} - \sqrt{q}\frac{\partial}{\partial x_{4}}, \quad
JZ_{3}= \sqrt{q} \frac{\partial}{\partial x_{5}} + \overline{\sigma}\frac{\partial}{\partial x_{6}}.$$
We find that $JZ_{2} \perp span \{Z_{1},Z_{2},Z_{3}\}$, $JZ_{3} \perp span \{Z_{1},Z_{2},Z_{3}\}$.
Moreover, we have $\|Z_{1}\|^{2}=1$, $\|Z_{2}\|^{2}=\frac{p\sigma+2q}{q}$ and $\|Z_{3}\|^{2}=\frac{p\sigma+2q}{p\sigma+q}$.
Thus, we get
$$ \cos \theta = \frac{<JZ_{1},Z_{1}>}{\|Z_{1}\|\cdot \|JZ_{1}\|}=\frac{\sigma (\cos^{2}t+2) +\overline{\sigma}\sin^{2}t}{\sqrt{3[\sigma^{2} (\cos^{2}t+2) +\overline{\sigma}^{2}\sin^{2}t]}}.$$
In particular, for $t=\frac{\pi}{4}$ we get $\cos \theta = \frac{5\sigma +\overline{\sigma}}{\sqrt{3(5\sigma^{2} +\overline{\sigma}^{2})}}$.
We define the distributions $D^{\theta}=span\{Z_{1}\}$ and $D^{\perp}=span\{Z_{2},Z_{3}\}$. We have $J(D^{\perp})\subset \Gamma(T^{\perp}M)$ and
$D^{\theta}$ is a slant distribution, with the slant angle
$\theta = \arccos \frac{\sigma (\cos^{2}t+2) +\overline{\sigma}\sin^{2}t}{\sqrt{3[\sigma^{2} (\cos^{2}t+2) +\overline{\sigma}^{2}\sin^{2}]}}.$
The Riemannian metric tensor of $D^{\theta} \oplus D^{\perp}$ is given by
$
g=du^{2} + \frac{p\sigma+2q}{q}d v^{2}+\frac{p\sigma+2q}{p\sigma+q}d w^{2}.
$
Thus, $TM=D^{\theta} \oplus D^{\perp}$ and $M$ is a hemi-slant submanifold in the metallic Riemannian manifold $(\mathbb{R}^{7}, <\cdot,\cdot>, J)$.
In particular, for $p=q=1$ and $\phi:=\sigma_{1,1}$ is the Golden number ($\overline{\phi}:=1-\phi$), the immersion $f: M \rightarrow \mathbb{R}^{7}$ is given by $$f(u,v,w)=(\frac{1}{\sqrt{3}} u \cos t,\frac{1}{\sqrt{3}} u \sin t, v, \phi v, \overline{\phi}w, w, \frac{\sqrt{2}}{\sqrt{3}}u),$$ and the Golden structure $J : \mathbb{R}^{7} \rightarrow \mathbb{R}^{7}$ can be defined by
by $$ J(X_{1},X_{2},X_{3},X_{4},X_{5},X_{6},X_{7})=(\phi X_{1},\overline{\phi} X_{2},\phi X_{3},\overline{\phi} X_{4},\phi X_{5},\overline{\phi} X_{6},\phi X_{7}).$$
The distributions $D^{\perp}=span\{Z_{2},Z_{3}\}$ verifies $J(D^{\perp})\subset \Gamma(T^{\perp}M)$ and slant distribution is $D^{\theta}=span\{Z_{1}\}$, with the slant angle
$\theta = \arccos \frac{\phi (\cos^{2}t+2) +\overline{\phi}\sin^{2}t}{\sqrt{3[\phi^{2} (\cos^{2}t+2) +\overline{\phi}^{2}\sin^{2}]}}.$
The Riemannian metric tensor of $D^{\theta} \oplus D^{\perp}$ is given by
$g=du^{2} + (\phi+2)d v^{2}+\frac{\phi+2}{\phi+1}d w^{2}.$
Thus, $TM=D^{\theta} \oplus D^{\perp}$ and $M$ is a hemi-slant submanifold in the metallic Riemannian manifold $(\mathbb{R}^{7}, <\cdot,\cdot>, J)$.
If we consider the metallic structure $\overline{J} :\mathbb{R}^{7} \rightarrow \mathbb{R}^{7}$ defined by
by
$$\overline{J}(X_{1},X_{2},X_{3},X_{4},X_{5},X_{6},X_{7})=(\sigma X_{1},\sigma X_{2},\sigma X_{3},\overline{\sigma} X_{4},\sigma X_{5},\overline{\sigma} X_{6},\sigma X_{7}),$$
then we obtain:
$\overline{J}Z_{1}= \sigma Z_{1}$, $\overline{J}Z_{2}= \sigma \frac{\partial}{\partial x_{3}} - \sqrt{q}\frac{\partial}{\partial x_{4}}$ and
$\overline{J}Z_{3}= \sqrt{q} \frac{\partial}{\partial x_{5}} + \overline{\sigma}\frac{\partial}{\partial x_{6}}.$
In this case we obtain the distributions $D^{\perp}=span\{Z_{2},Z_{3}\}$ and $D^{\theta}=span\{Z_{1}\}$ with the slant angle $\theta = \arccos 1 = 0.$ Thus, $TM=D^{\theta} \oplus D^{\perp}$ and $M$ is a semi-invariant submanifold in the metallic Riemannian manifold $(\mathbb{R}^{7}, <\cdot,\cdot>, \overline{J})$. Similarly, for $p=q=1$ we obtain that $M$ is a semi-invariant submanifold in the Golden Riemannian manifold $(\mathbb{R}^{7}, <\cdot,\cdot>, \overline{J})$.
\section{On the integrability of the distributions of a hemi-slant submanifold}
In this section we investigate the conditions for the integrability of the distributions of a hemi-slant submanifold in a metallic (or Golden) Riemannian manifold.
\begin{prop}
If $M$ is a hemi-slant submanifold in a locally metallic (or locally Golden) Riemannian manifold $(\overline{M},\overline{g},J)$, then
\begin{equation}\label{e30}
\nabla_{X}TY-\nabla_{Y}TX-A_{NY}X+A_{NX}Y \in \Gamma(D^{\theta}),
\end{equation}
for any $X,Y \in \Gamma(D^{\theta})$.
\end{prop}
\begin{proof}
By using (\ref{e7})(i), we obtain:
$\overline{g}(T([X,Y]),Z)=\overline{g}([X,Y],TZ)=0,$ for any $X,Y \in \Gamma(D^{\theta})$ and $Z \in \Gamma(D^{\perp})$ (i.e. $TZ=0$). Thus, $T([X,Y]) \in \Gamma(D^{\theta})$ and using (\ref{e20}) we obtain (\ref{e30}).
\end{proof}
\begin{prop}
If $M$ is a hemi-slant submanifold in a locally metallic (or locally Golden) Riemannian manifold $(\overline{M},\overline{g},J)$, then the distribution $D^{\theta}$ is integrable.
\end{prop}
\begin{proof}
By using (\ref{e3}), we have
$\overline{g}(\overline{\nabla}_{X}Y,Z) = \frac{1}{q} [\overline{g}(J\overline{\nabla}_{X}Y,JZ) - p \overline{g}(\overline{\nabla}_{X}Y,JZ)],$
for any $X,Y \in \Gamma(D^{\theta})$, $Z \in \Gamma(D^{\perp})$.
From $\overline{\nabla}J=0$ we get $J \overline{\nabla}_{X}Y = \overline{\nabla}_{X}JY$ and using $JZ=NZ$, for any $Z \in \Gamma(D^{\perp})$, we obtain
$q \overline{g}(\overline{\nabla}_{X}Y,Z) = \overline{g}(\overline{\nabla}_{X}JY,NZ) - p \overline{g}(\overline{\nabla}_{X}Y,NZ).$
Thus, from (\ref{e11}) and (\ref{e12}) we get
$q\overline{g}(\overline{\nabla}_{X}Y,Z) =\overline{g}(h(X,TY),NZ)+\overline{g}(\nabla_{X}^{\perp}NY,NZ) - p \overline{g}(h(X,Y),NZ).$
From (\ref{e13})(ii) and (\ref{e17})(ii) we obtain
$
\nabla_{X}^{\perp}NY =n h(X,Y) - h(X,TY) + N \nabla_{X}Y,
$
for any $X,Y \in \Gamma(D^{\theta})$.
Thus, we get
$$q\overline{g}(\overline{\nabla}_{X}Y,Z) =\overline{g}(nh(X,Y),NZ)+\overline{g}(N\nabla_{X}Y,NZ) - p \overline{g}(h(X,Y),NZ),$$
which implies
$$q \overline{g}([X,Y],Z)= \overline{g}(N \nabla_{X}Y,NZ) - \overline{g}(N \nabla_{Y}X,NZ)=\overline{g}(N[X,Y],NZ),$$
for any $X,Y \in \Gamma(D^{\theta})$ and $Z \in \Gamma(D^{\perp})$.
Thus, from (\ref{e27}) and (\ref{e7})(i) we have
$$q \overline{g}([X,Y],Z)= \sin^{2} \theta [p \overline{g}(P1[X,Y],TP_{1}Z)+ q \overline{g}(P1[X,Y],P_{1}Z)].$$
By using $P_{1}Z=0$ for any $Z \in \Gamma(D^{\perp})$ (where $P_{1}Z$ is the projection of $Z$ on $\Gamma(D^{\theta})$), we obtain $\overline{g}([X,Y],Z)=0$, for any $X,Y \in \Gamma(D^{\theta})$, $Z \in \Gamma(D^{\perp})$ which implies that $[X,Y] \in \Gamma(D^{\theta})$.
\end{proof}
\begin{prop}
Let $M$ be a hemi-slant submanifold in a locally metallic (or locally Golden) Riemannian manifold $(\overline{M},\overline{g},J)$. Then the distribution $D^{\perp}$ is integrable if and only if
\begin{equation}\label{e31}
A_{NZ}W=0,
\end{equation}
for any $Z,W\in \Gamma(D^{\perp})$.
\end{prop}
\begin{proof}
If $M$ is a hemi-slant submanifold in a locally metallic (or locally Golden) Riemannian manifold $(\overline{M},\overline{g},J)$ then, for any $Z,W \in \Gamma(D^{\perp})$ we have $TZ=TW=0$ which implies $\nabla_{Z}TW=\nabla_{W}ZX=0$. By using (\ref{e24})(ii) and (\ref{e20}) we get $T([Z,W])=0$ if and only if $A_{NZ}W=A_{NW}Z$ holds, for any $Z,W \in \Gamma(D^{\perp})$.
From (\ref{e17})(i), for any $X\in \Gamma(TM)$ and $Z,W \in \Gamma(D^{\perp})$, we get
$$ \overline{g}(A_{NZ}X,W)+\overline{g}(th(X,Z),W)=\overline{g}((\nabla_{X}T)Z,W)=-\overline{g}(\nabla_{X}Z,TW)=0,$$
which implies $ \overline{g}(A_{NZ}X,W) = -\overline{g}(th(X,Z),W)$.
From
$$\overline{g}(A_{NZ}X,W)=\overline{g}(A_{NZ}W,X)=\overline{g}(A_{NW}Z,X)=\overline{g}(h(X,Z),NW)=\overline{g}(th(X,Z),W),$$ we obtain $\overline{g}(A_{NZ}W,X)=0$ for any $X\in \Gamma(TM)$ and $Z,W \in \Gamma(D^{\perp})$. Thus, $A_{NZ}W=0$, for any $Z,W \in \Gamma(D^{\perp})$.
Conversely, if $A_{NZ}W=0$, for any $Z,W \in \Gamma(D^{\perp})$ then, from $\overline{g}(th(X,Z),W)=\overline{g}(h(X,Z),NW)=\overline{g}(A_{NW}Z,X)=0$ and (\ref{e17})(i), we get
$$0=\overline{g}((\nabla_{Z}T)W,X)=\overline{g}(T\nabla_{Z}W,X)=\overline{g}(\nabla_{Z}W,TX),$$ for any $Z,W \in \Gamma(D^{\perp})$, $X \in \Gamma(D^{\theta})$. From $T(D^{\theta})=D^{\theta}$, we obtain $\nabla_{Z}W \in \Gamma(D^{\perp})$ which implies $[Z,W] \in \Gamma(D^{\perp})$.
\end{proof}
\begin{prop}
Let $M$ be a hemi-slant submanifold in a locally metallic (or locally Golden) Riemannian manifold $(\overline{M},\overline{g},J)$. Then, the anti-invariant distribution $D^{\perp}$ is integrable if and only if
\begin{equation}\label{e32}
(\nabla_{Z}T)W=(\nabla_{W}T)Z
\end{equation}
for any $Z,W \in \Gamma(D^{\perp})$.
\end{prop}
\begin{proof}
By using (\ref{e17}) we get $(\nabla_{Z}T)W - (\nabla_{W}T)Z = A_{NW}Z - A_{NZ}W$, for any $Z,W \in \Gamma(D^{\perp})$ and using (\ref{e31}) we obtain the conclusion.
\end{proof}
\begin{remark}
Let $M$ be a hemi-slant submanifold in a locally metallic (or locally Golden) Riemannian manifold ($\overline{M},\overline{g},J)$. If $(\nabla_{Z}T)W =0$, for any $Z,W \in \Gamma(D^{\perp})$, then $D^{\perp}$ is integrable.
\end{remark}
\begin{prop}
Let $M$ be a hemi-slant submanifold in a locally metallic (or locally Golden) Riemannian manifold ($\overline{M},\overline{g},J)$. If $(\overline{\nabla}_{X}N)Y =0$, for any $X,Y \in \Gamma(D^{\theta})$ then, either $M$ is a $D^{\theta}$ geodesic submanifold (i.e $h(X,Y)=0$) or $h(X,Y)$ is an eigenvector of $n$, with eigenvalues
\begin{equation}\label{e35}
\lambda_{1}=\frac{p \cos^{2}\theta + \cos \theta \sqrt{p^{2}\cos^{2}\theta + 4q}}{2}, \quad \lambda_{2}=\frac{p \cos^{2}\theta - \cos \theta \sqrt{p^{2}\cos^{2}\theta + 4q}}{2}.
\end{equation}
\end{prop}
\begin{proof}
By using $(\overline{\nabla}_{X}N)Y =0$ for any $X,Y \in \Gamma(D^{\theta})$ and (\ref{e17})(ii) we obtain $nh(X,Y)=h(X,TY)$. From (\ref{e28}) we get, for any $X,Y \in \Gamma(D^{\theta})$:
$$ n^{2}h(X,Y)=h(X,T^{2}Y)=p\cos^{2}\theta nh(X,Y) + q\cos^{2}\theta h(X,Y).$$ Thus, we obtain either $M$ is a $D^{\theta}$ geodesic submanifold or $h(X,Y)$ is an eigenvector of $n$ with eigenvalue $\lambda$, which verifies the equation $\lambda^{2}-p\cos^{2}\theta \lambda - q\cos^{2}\theta =0$ and (\ref{e35}) holds.
\end{proof}
\section{Mixed totally geodesic hemi-slant submanifolds}
In the next propositions, we consider hemi-slant submanifolds in a locally metallic (or locally Golden) Riemannian manifold and we find some conditions for these submanifolds to be $D^{\theta} - D^{\perp}$ mixed totally geodesic (i.e. $h(X,Y)=0$, for any $X \in \Gamma(D^{\theta})$ and $Y \in \Gamma(D^{\perp})$).
\begin{prop}
If $M$ is a hemi-slant submanifold in a locally metallic (or locally Golden) Riemannian manifold $(\overline{M},\overline{g},J)$, then $M$ is a $D^{\theta}-D^{\perp}$ mixed totally geodesic submanifold if and only if $A_{V}X \in \Gamma(D^{\theta})$ and $A_{V}Y \in \Gamma(D^{\perp})$, for any $X \in \Gamma(D^{\theta})$, $Y \in \Gamma(D^{\perp})$ and $V \in \Gamma(T^{\bot}M)$.
\end{prop}
\begin{proof}
From $\overline{g}(A_{V}X,Y)=\overline{g}(A_{V}Y,X)=\overline{g}(h(X,Y),V)$, for any $X \in \Gamma(D^{\theta}), Y \in \Gamma(D^{\perp})$ and $V \in \Gamma(T^{\bot}M)$ we obtain that $M$ is a $D^{\theta}-D^{\perp}$ mixed totally geodesic submanifold in the locally metallic (or locally Golden) Riemannian manifold if and only if $A_{V}X \in \Gamma(D^{\theta})$ and $A_{V}Y \in \Gamma(D^{\perp})$, for any $X \in \Gamma(D^{\theta})$, $Y \in \Gamma(D^{\perp})$ and $V \in \Gamma(T^{\bot}M)$.
\end{proof}
\begin{prop}
Let $M$ be a proper hemi-slant submanifold in a locally metallic (or locally Golden) Riemannian manifold ($\overline{M},\overline{g},J)$. If $(\overline{\nabla}_{X}N)Z =0$, for any $X\in \Gamma(TM)$ and $Z \in \Gamma(D^{\perp})$, then $M$ is a $D^{\theta}-D^{\perp}$ mixed totally geodesic submanifold in $\overline{M}$ .
\end{prop}
\begin{proof}
If $X \in \Gamma(D^{\theta})$ and $Z \in \Gamma(D^{\perp})$ then, from $(\overline{\nabla}_{X}N)Z =0$, (\ref{e17})(ii) and $TZ=0$ we get $h(Z,TX)=nh(X,Z)=h(X,TZ)=0$.
From (\ref{e28}), we have
$$ 0=n^{2}h(Z,X)=h(Z,T^{2}X)=p\cos^{2}\theta nh(Z,TX) + q\cos^{2}\theta h(Z,X)$$ and we obtain $q\cos^{2}\theta h(Z,X)=0$. By using $\theta \neq \frac{\pi}{2}$ and $q\neq 0$,
we get $h(X,Z)=0$, for any $X \in \Gamma(D^{\theta})$ and $Z \in \Gamma(D^{\perp})$.
\end{proof}
|
{
"timestamp": "2018-05-03T02:10:04",
"yymm": "1804",
"arxiv_id": "1804.05229",
"language": "en",
"url": "https://arxiv.org/abs/1804.05229"
}
|
\section{Introduction}
Let $M$ be a $3$-connected matroid, and let $N$ be a $3$-connected minor of $M$.
We say that a pair $\{x_1,x_2\} \subseteq E(M)$ is \emph{$N$-detachable} if
either
\begin{enumerate}[label=\rm(\alph*)]
\item $M/x_1/x_2$ is $3$-connected and has an $N$-minor, or
\item $M \backslash x_1 \backslash x_2$ is $3$-connected and has an $N$-minor.
\end{enumerate}
This is the first in a series of three papers where we describe the structures that
arise when it is not possible to find an $N$-detachable pair. As a consequence, we show
that if $M$ has at least ten more elements than $N$, then
either $M$ has an $N$-detachable pair after possibly performing a single
$\Delta$-$Y$ or $Y$-$\Delta$ exchange, or $M$ is essentially $N$ with a single
spike attached. More precisely, we have the following theorem.
Formal definitions of $\Delta$-$Y$ exchange and ``spike-like $3$-separator'' are given
in Section~\ref{presec}.
\begin{theorem}
Let $M$ be a $3$-connected matroid,
and let $N$ be a $3$-connected minor of $M$ such that $|E(N)| \ge 4$ and $|E(M)|-|E(N)| \ge 10$.
Then either
\begin{enumerate}
\item $M$ has an $N$-detachable pair,
\item there is a matroid $M'$ obtained by performing a single $\Delta$-$Y$ or $Y$-$\Delta$ exchange on $M$ such that $M'$ has
an $N$-detachable pair, or
\item there is a spike-like $3$-separator~$P$ of $M$ such that at most one element of $E(M)-E(N)$ is not in $P$.
\end{enumerate}
\label{maintheorem}
\end{theorem}
In fact, we prove a stronger result that requires only that $|E(M)| - |E(N)| \ge 5$, but a handful of additional highly structured outcomes involving particular
$3$-separators of bounded size
arise. Describing these requires some preparation and we defer the full statement of the
stronger theorem until the third paper.
These papers had their genesis in the Ph.D.\ thesis of Alan Williams~\cite{Williams2015} where
the problem of finding a detachable pair without worrying about keeping a minor was solved.
In essence, the strategy here follows the strategy of \cite{Williams2015}, but with the additional
responsibility of always taking care to keep the minor.
\subsection*{Background and motivation}
The proof of \cref{maintheorem} is long; much longer than we originally anticipated.
Without a solid motivation, the case for going to the trouble of proving it is weak indeed.
In fact, the motivation is clear. It comes from a desire to find exact excluded-minor
characterisations of certain minor-closed classes of representable matroids.
What follows is a discussion of that motivation.
To some extent, progress in matroid theory
can be measured by success in finding excluded-minor characterisations of classes of
matroids. Results to date include Tutte's
excluded-minor characterisation of binary and regular matroids \cite{tutte1958homotopy}; Bixby's
and, independently, Seymour's excluded-minor characterisation of ternary matroids \cite{bixby1979reid,seymour1979matroid};
Geelen, Gerards and Kapoor's excluded-minor characterisation of GF$(4)$-representable
matroids \cite{Geelen2000excluded}; and Hall, Mayhew and van Zwam's excluded-minor characterisation of the
near-regular matroids, that is, the matroids representable over all fields
with at least three elements \cite{hall2011excluded}.
Recently Geelen, Gerards and Whittle announced a proof
of Rota's Conjecture \cite{geelen2014solving}. However, their techniques
are extremal and give no insight into how one might find the exact list of
excluded minors for such classes. Extending the range of known exact excluded-minor theorems
for basic classes of matroids remains a problem of genuine interest and, indeed, a significant
challenge that tests the state of the art of techniques in matroid theory.
At this stage we need to note that regular matroids, and many other naturally arising
classes of representable matroids such as near-regular,
dyadic and $\sqrt[6]{1}$-matroids \cite{whittle1997matroids},
can be described as classes of matroids representable
over an algebraic structure called a {\em partial field}. Of course, a field is an example
of a partial field, and classes of
matroids representable over partial fields enjoy many of the properties that hold for matroids
representable over fields \cite{pendavingh2010lifts,pendavingh2010confinement,semple1996partial}.
The immediate problem that looms large is that of finding the excluded minors for
the class of GF$(5)$-representable matroids. While this problem is beyond the range of
current techniques, a road map for an attack is outlined in \cite{pendavingh2010confinement}. In essence,
this road map reduces the problem to a finite sequence of problems of the following type.
We have the class of $\mathbb P$-representable matroids for some
fixed partial field $\mathbb P$. We have a $3$-connected matroid $N$ with the property that
every $\mathbb P$-representation of $N$ extends {\em uniquely} to a $\mathbb P$-representation
of any $3$-connected $\mathbb P$-representable matroid having $N$ as a minor. Such a matroid
$N$ is
called a {\em strong stabilizer} for the class of $\mathbb P$-representable matroids.
With these ingredients,
the goal is to bound the size of an excluded minor for the class of $\mathbb P$-representable
matroids having
the strong stabilizer $N$ as a minor.
This situation is a more general version of the one that arises in the proof of the excluded-minor characterisation
of GF$(4)$-representable matroids \cite{Geelen2000excluded}. There, the partial field is GF$(4)$
and the fixed minor $N$ is $U_{2,4}$.
For all of the classes described above we may attempt to generalise the
strategy developed by Geelen, Gerards and Kapoor. We have an excluded minor $M$, with
strong stabilizer $N$. We wish to bound the size of $M$ relative to $N$.
Assume, for a contradiction, that $M$ is large relative to $N$.
It is proved in \cite{Geelen2000excluded,stabilizers} that in this case, up
to duality, one can find a pair of elements $x,y\in E(M)$ such that $M\backslash x$, $M\backslash y$
and $M\backslash x\backslash y$ have $N$-minors and are $3$-connected up to series pairs. Finding such a
pair is the first step in the proofs given in \cite{Geelen2000excluded,hall2011excluded}. But there is the rub.
The possible presence of series pairs leads to a major complication in the subsequent analysis.
The current proofs for the excluded-minor characterisations of both GF$(4)$-representable and
near-regular matroids could be significantly shortened if we could replace
``$3$-connected up to series pairs'' by ``$3$-connected'' in the initial step. That is
precisely what Theorem~\ref{maintheorem} enables us to do.
If we are to succeed in finding the excluded minors for the classes of matroids
that would lead to an exact solution to Rota's Conjecture for GF$(5)$, eliminating
unnecessary technicalities in the analyses becomes more than just a convenience; it
becomes absolutely essential. Eliminating unnecessary technicalities is what this paper achieves. It gives a feasible first
step on the way to an explicit characterisation of the excluded minors for these classes.
Note that outcomes (ii) and (iii) of Theorem~\ref{maintheorem} do not limit its applicability for
finding excluded-minor characterisations of matroids representable over partial fields.
It is known that excluded minors for a partial field are closed under the $\Delta$-$Y$
exchange \cite{oxley2000generalized}. Moreover, it is not difficult to show that excluded minors have bounded-size spike-like $3$-separators.
Theorem~\ref{maintheorem} has already been applied to make further progress on excluded-minor problems.
For a fixed matroid $N$, a matroid $M$ is $N$-{\em fragile} if, for all $e\in E(M)$,
at most one of $M\backslash e$ and $M/e$ has an $N$-minor. It is shown in \cite{bcosw2019} that if
$M$ is a sufficiently large excluded minor for a partial field~$\mathbb P$ with a strong stabilizer $N$ as
a minor, then $M$ is $\Delta$-$Y$-equivalent to a matroid from which an $N$-fragile matroid can be obtained by deleting two elements. The proof of this result makes essential use of Theorem~\ref{maintheorem}.
In essence, this reduces the problem of bounding the size of
an excluded minor to understanding the class of $\mathbb P$-representable
$N$-fragile matroids. In general this appears to be a difficult problem, but progress
has been made for two genuinely interesting classes.
The Hydra-5 partial field captures the first layer of the hierarchy of
GF$(5)$-representable partial fields mentioned above. The 2-regular partial field
has the property 2-regular-representable matroids are representable over all fields
of size at least four. It turns out that $U_{2,5}$ is a strong stabilizer for
both these partial fields. Moreover, the $U_{2,5}$-fragile matroids that are either
2-regular or Hydra-5 representable are known \cite{clark2015structure}. Using this, it is
possible to obtain
an explicit bound for the size of an excluded minor for either of these partial fields \cite{ben}.
The current bound is too large to enable an exhaustive search for the excluded minors.
It is hoped that, in the not too distant future, we can refine this bound and
obtain an explicit list of the excluded minors. There would be some satisfaction in
achieving this. Finding the excluded minors would be an important first step on the
way to getting the excluded minors for GF$(5)$. Having said that, it seems likely that,
in the end, combinatorial explosion will make the full solution impossible. Nonetheless it
is interesting to know just where the boundary of infeasibility lies.
On the other hand, obtaining the excluded minors for the 2-regular matroids would be a
significant step towards understanding the matroids representable over all fields of size
at least 4. We know that this class contains the class of 2-regular matroids.
The excluded
minors for 2-regular that belong to the class would be interesting indeed and it is likely
that they could be exploited to obtain an explicit description of the class of matroids
representable over all fields of size at least four. Indeed it would also be a significant
step towards understanding the classes that arise when one considers matroids representable
over sets of fields that contain GF$(4)$. This would generalise analogous results for
GF$(2)$ and GF$(3)$ \cite{tutte1958homotopy,whittle1997matroids}.
\subsection*{The structure of these papers}
We now outline the approach taken to prove \cref{maintheorem} in this series of papers.
As is traditional, we begin by recalling Seymour's Splitter Theorem.
\begin{theorem}[Seymour's Splitter Theorem~\cite{pds}]
Let $M$ be a $3$-connected matroid that is not a wheel or a whirl,
and let $N$ be a $3$-connected proper minor of $M$.
Then there exists an element $e \in E(M)$ such that $M/e$ or $M\backslash e$ is $3$-connected and has an $N$-minor.
\end{theorem}
By Seymour's Splitter Theorem, we may assume, up to duality, that there is an element $d \in E(M)$ such that $M \backslash d$ is $3$-connected and has an $N$-minor.
Let $d' \in E(M \backslash d)$ such that $M \backslash d \backslash d'$ has an $N$-minor.
If $M \backslash d \backslash d'$ is $3$-connected, then $\{d,d'\}$ is an $N$-detachable pair.
On the other hand, if $M \backslash d \backslash d'$ is not $3$-connected, then $M \backslash d \backslash d'$ has a $2$-separation $(Y,Z)$ where
the $N$-minor is primarily contained in one side of the $2$-separation, $Z$ say.
The main result of this first paper of the series shows that if $M$ were to have no $N$-detachable pairs, and $|Y| \ge 4$, then either $Y$ contains a $3$-separating set $X$ with a number of strong structural properties, or $Y \cup d$ contains one of the handful of particular $3$-separator s that can appear in a matroid with no $N$-detachable pairs (we describe these particular $3$-separator s in \cref{sec-problematic}).
On the journey towards the proof of this result, we prove a number of lemmas about the existence of $N$-detachable pairs when $M$ or $M^*$ contains one of a few special structures: namely, triangles (\cref{sectris}), a $U_{3,5}$-restriction (\cref{secplanes}), or a single-element extension of a flan (\cref{secflans}).
The proof of the main result is in \cref{secunveil}.
In the second paper, we further analyse this structured set $X$, and show that if we cannot find an $N$-detachable pair, then $X \cup d$ is contained in one of the handful of particular $3$-separator s that can appear in a matroid with no $N$-detachable pairs.
In the third paper, the main hurdle that remains is handling the case where for any pair $\{d,d'\}$ such that $M \backslash d \backslash d'$ has an $N$-minor, the $2$-separation $(Y,Z)$ in $M \backslash d \backslash d'$ has $|Y|<4$.
\section{Preliminaries}
\label{presec}
The notation and terminology in the paper follow Oxley~\cite{oxbook}.
We write $x \in \cl^{(*)}(Y)$ to denote that either $x \in \cl(Y)$ or $x \in \cl^*(Y)$.
The phrase ``by orthogonality'' refers to the fact that a circuit and a cocircuit cannot intersect in exactly one element.
For a set~$X$ and element~$e$, we write $X \cup e$ instead of $X \cup \{e\}$, and $X-e$ instead of $X-\{e\}$.
We say that $X$ meets $Y$ if $X \cap Y \neq \emptyset$.
We denote $\{1,2,\dotsc,n\}$ by $\seq{n}$.
\subsection*{Connectivity}
Let $M$ be a matroid with ground set $E$. The \emph{connectivity function} of $M$, denoted by $\lambda_M$, is defined as follows, for all subsets $X$ of \nopagebreak $E$:
\begin{align*}
\lambda_M(X) = r(X) + r(E - X) - r(M).
\end{align*}
A subset $X$ or a partition $(X, E-X)$ of $E$ is \emph{$k$-separating} if $\lambda_M(X) \leq k-1$.
A $k$-separating partition $(X,E-X)$ is a \emph{$k$-separation} if $|X| \ge k$ and $|E-X|\ge k$.
A $k$-separating set $X$, a $k$-separating partition $(X,E-X)$ or a $k$-separation $(X,E-X)$ is \emph{exact} if $\lambda_M(X) = k-1$.
The matroid $M$ is \emph{$n$-connected} if, for all $k < n$, it has no $k$-separations.
When a matroid is $2$-connected, we simply say it is \emph{connected}.
The connectivity functions of a matroid and its dual are equal; that is,
$\lambda_M(X) = \lambda_{M^*}(X)$. In fact, it is easily shown that
\begin{align*}
\lambda_M(X) = r(X) + r^*(X) - |X|.
\end{align*}
\subsection*{Spike-like $3$-separators}
Let $M$ be a matroid with ground set $E$.
We say that a $4$-element set $Q \subseteq E$ is a \emph{quad} if it is both a circuit and a cocircuit of $M$.
\begin{definition}
\label{def-spike-like}
Let $P \subseteq E$ be an exactly $3$-separating set of $M$.
If there exists a partition $\{L_1,\dotsc,L_t\}$ of $P$ with $t\geq 3$ such that
\begin{enumerate}[label=\rm(\alph*)]
\item $|L_i|=2$ for each $i\in\{1,\dotsc,t\}$, and
\item $L_i\cup L_j$ is a quad for all distinct $i,j\in\{1,\dotsc,t\}$,
\end{enumerate}
then $P$ is a \emph{spike-like $3$-separator} of $M$.
\end{definition}
To illustrate the necessity for outcome (iii) of \cref{maintheorem} we describe the construction of a matroid that satisfies neither (i) nor (ii) of the theorem.
Let $F_7$ be a copy of the Fano matroid with a triangle $\{x,y,z\}$.
Let $F_7'$ be the matroid obtained from $F_7$ by adding elements $y'$ and $z'$ in parallel with $y$ and $z$ respectively, and relabelling the element~$x$ as $t$.
Now let $S$ be a spike with tip $t$, where $r(S) \ge 4$, and let $T=\{t,y',z'\}$ be a leg of $S$. Let $M = P_T(F_7',S) \backslash T$, the generalised parallel connection of $S$ and $F_7'$ along $T$ with the elements $T$ removed. Then $M$ has no $F_7$-detachable pairs.
Alternatively,
let $F_7''$ be the matroid obtained from $F_7$ by adding elements $y'$ and $z'$ in parallel with $y$ and $z$ respectively, and freely adding the element~$t$ on the line spanned by $\{x,y,z\}$.
Then, similarly, $P_T(F_7'',S) \backslash T$ has no $F_7$-detachable pairs.
A geometric illustration of a spike-like $3$-separator\ is given in \cref{spikelikefig1}.
We will see three more particular $3$-separator s, and how they can give rise to matroids without any $N$-detachable pairs, in \cref{sec-problematic}.
\begin{figure}
\centering
\begin{tikzpicture}[rotate=90,scale=0.8,line width=1pt]
\tikzset{VertexStyle/.append style = {minimum height=5,minimum width=5}}
\clip (-2.5,-6) rectangle (3.0,2);
\node at (-1,-1.4) {\large$E-P$};
\draw (0,0) .. controls (-3,2) and (-3.5,-2) .. (0,-4);
\draw (0,0) -- (2.5,0.5);
\draw (0,0) -- (2.25,-0.75);
\draw (0,0) -- (2,-2);
\draw (0,0) -- (1,-3);
\SetVertexNoLabel
\Vertex[x=1.25,y=0.25,LabelOut=true,L=$q_3$,Lpos=180]{c1}
\Vertex[x=2.25,y=-0.75,LabelOut=true,L=$q_2$,Lpos=90]{c2}
\Vertex[x=2.5,y=0.5,LabelOut=true,L=$q_1$,Lpos=180]{c3}
\Vertex[x=1.5,y=-0.5,LabelOut=true,L=$q_4$,Lpos=135]{c4}
\Vertex[x=1,y=-1,LabelOut=true,L=$q_1$,Lpos=180]{c5}
\Vertex[x=2,y=-2,LabelOut=true,L=$q_1$,Lpos=180]{c6}
\Vertex[x=1,y=-3,LabelOut=true,L=$q_4$,Lpos=135]{c7}
\Vertex[x=0.67,y=-2,LabelOut=true,L=$q_4$,Lpos=135]{c8}
\draw (0,0) -- (0,-4);
\SetVertexNoLabel
\tikzset{VertexStyle/.append style = {shape=rectangle,fill=white}}
\Vertex[x=0,y=0]{a1}
\end{tikzpicture}
\caption{An example of a spike-like $3$-separator\ in a matroid with rank $r(E-P)+3$}
\label{spikelikefig1}
\end{figure}
\subsection*{More connectivity}
The next lemma is a consequence of the easily verified fact that the connectivity function is submodular. We write ``by uncrossing'' to refer to an application of this lemma.
\begin{lemma}
\label{onetrick}
Let $M$ be a $3$-connected matroid, and let $X$ and $Y$ be $3$-separating subsets of $E(M)$.
\begin{enumerate
\item If $|X \cap Y| \ge 2$, then $X \cup Y$ is $3$-separating.
\item If $|E(M) - (X \cup Y)| \ge 2$, then $X \cap Y$ is $3$-separating.
\end{enumerate}
\end{lemma}
The following lemmas are well known.
\begin{lemma}
\label{swapSepSides}
Let $e$ be an element of a matroid $M$, and let $X$ and $Y$ be disjoint sets whose union is $E(M) - \{e\}$. Then $e \in \cl(X)$ if and only if $e \notin \cl^{*}(Y)$.
\end{lemma}
\begin{lemma}
\label{extendSep}
Let $X$ be an exactly $3$-separating set in a $3$-connected matroid $M$,
and suppose that $e \in E(M) - X$. Then $X \cup e$ is $3$-separating if and only if $e \in \cl^{(*)}(X)$.
\end{lemma}
\begin{lemma}
\label{exactSeps}
Let $(X, Y)$ be an exactly $3$-separating partition of a $3$-connected matroid $M$. Suppose $|X| \ge 3$ and $x \in X$. Then
$x \in \cl^{(*)}(X-x)$.
\end{lemma}
\begin{lemma}
\label{gutses}
Let $(X, Y)$ be an exactly $3$-separating partition of a $3$-connected matroid $M$, with $|X| \ge 3$ and $x \in X$. Then
$(X-x, Y \cup x)$ is exactly $3$-separating if and only if $x$ is in
one of $\cl(X-x) \cap \cl(Y)$ and $\cl^*(X-x) \cap \cl^*(Y)$.
\end{lemma}
If $(X, Y)$ and $(X-x, x \cup Y)$ are exactly $3$-separating partitions in a $3$-connected matroid, then we say $x$ is a \emph{guts element} if $x \in \cl(X-x) \cap \cl(Y)$, and $x$ is a \emph{coguts element} if $x \in \cl^*(X-x) \cap \cl^*(Y)$.
We also say $x$ is \emph{in the guts of $(X,Y)$} or $x$ is \emph{in the coguts of $(X,Y)$}, respectively.
\begin{lemma}
\label{gutsstayguts}
Let $(X,Y)$ be a $3$-separation in a $3$-connected matroid.
Then $\cl(X) \cap \cl^*(X) \cap Y = \emptyset$.
\end{lemma}
A $k$-separation $(X, E-X)$ of a matroid $M$ with ground set $E$ is \emph{vertical} if $r(X) \ge k$ and $r(E-X) \ge k$.
We also say a partition $(X, \{z\}, Y)$ of $E$ is a \emph{vertical $3$-separation} when $(X \cup \{z\}, Y)$ and $(X, Y \cup \{z\})$ are both vertical $3$-separations and $z \in \cl(X) \cap \cl(Y)$.
Note that, given a vertical $3$-separation $(X,Y)$ and some $z \in Y$, if $z \in \cl(X)$, then $(X,\{z\},Y)$ is a vertical $3$-separation, by \cref{extendSep,exactSeps}.
A vertical $k$-separation in $M^*$ is known as a cyclic $k$-separation in $M$.
More specifically, a $k$-separation $(X, E-X)$ of $M$ is \emph{cyclic} if $r^*(X) \ge k$ and $r^*(E-X) \ge k$; or, equivalently, if $X$ and $E-X$ contain circuits.
We also say that a partition $(X, \{z\}, Y)$ of $E$ is a \emph{cyclic $3$-separation} if $(X, \{z\}, Y)$ is a vertical $3$-separation in $M^*$.
We say that a partition $(X_1,X_2,\dotsc,X_m)$ of $E(M)$ is a \emph{path of $3$-separations} if $(X_1 \cup \dotsm \cup X_i, X_{i+1} \cup \dotsm \cup X_m)$ is a $3$-separation for each $i \in \seq{m-1}$.
Observe that a vertical, or cyclic, $3$-separation $(X, \{z\},Y)$ is an instance of a path of $3$-separations.
A proof of the following is in \cite{stabilizers}. We use this lemma, and its dual, freely without reference.
\begin{lemma}
\label{openVertSep}
Let $M$ be a $3$-connected matroid and let $z \in E(M)$.
The following are equivalent:
\begin{enumerate
\item $M$ has a vertical $3$-separation $(X, \{z\}, Y)$.\label{vsi}
\item $\si(M/z)$ is not $3$-connected.\label{vsii}
\end{enumerate}
\end{lemma}
A \emph{segment} in a matroid $M$ is a subset $S$ of $E(M)$ such that $M|S \cong U_{2,k}$ for some $k \ge 3$, while a \emph{cosegment} of $M$ is a segment of $M^*$.
\begin{lemma}
\label{rank2Remove}
Let $M$ be a $3$-connected matroid and let $S$ be a
segment
with at least four elements. If $s \in S$, then $M \backslash s$ is $3$-connected.
\end{lemma}
The next two lemmas will be referred to by name.
\begin{lemma}[Bixby's Lemma~\cite{bixby}]
\label{bixbyL}
Let $e$ be an element of a $3$-connected matroid $M$.
Then either $M/e$ is $3$-connected up to parallel pairs, or $M \backslash e$ is $3$-connected up to series pairs.
\end{lemma}
\begin{lemma}[Tutte's Triangle Lemma~\cite{tutte1966}]
\label{ttL}
Let $\{a,b,c\}$ be a triangle in a $3$-connected matroid $M$. If neither $M \backslash a$ nor $M \backslash b$ is $3$-connected, then $M$ has a triad which contains $a$ and exactly one element from $\{b,c\}$.
\end{lemma}
A proof of the following is in \cite{stabilizers}.
\begin{lemma
\label{r3cocircsi}
Let $C^*$ be a rank-$3$ cocircuit of a $3$-connected matroid $M$.
If $x \in C^*$ has the property that $\cl_M(C^*)-x$ contains a triangle of $M/x$, then $\si(M/x)$ is $3$-connected.
\end{lemma}
Proofs of the following two lemmas appear in \cite{bs2014}.
\begin{lemma
\label{r3cocirc}
Let $M$ be a $3$-connected matroid with $r(M) \ge 4$.
Suppose that $C^*$ is a rank-$3$ cocircuit of $M$.
If there exists some $x \in C^*$ such that $x \in \cl(C^*-x)$, then $\co(M \backslash x)$ is $3$-connected.
\end{lemma}
\begin{lemma
\label{presingle}
Let $(X,Y)$ be a $3$-separation of a $3$-connected matroid $M$. If $X\cap \cl(Y)\neq \emptyset$ and $X\cap \cl^*(Y)\neq \emptyset$, then $|X\cap \cl(Y)|=1$ and $|X\cap \cl^*(Y)|=1$.
\end{lemma}
Suppose $M$ is a $3$-connected matroid, there is an element $d \in E(M)$ such that $M \backslash d$ is $3$-connected, and $X \subseteq E(M\backslash d)$ is exactly $3$-separating in $M \backslash d$.
We say that $d$ \emph{blocks} $X$ if $X$ is not $3$-separating in $M$, and $d$ \emph{fully blocks} $X$ if neither $X$ nor $X \cup d$ is $3$-separating in $M$.
If $d$ blocks $X$, then $d \notin \cl(E(M\backslash d)-X)$, so $d \in \cl^*(X)$ by \cref{swapSepSides}.
It is easily shown that $d$ fully blocks $X$ if and only if $d \notin \cl(X) \cup \cl(E(M \backslash d)-X)$.
\subsection*{Full closure}
A set $X$ in a matroid $M$ is {\em fully closed} if it is closed and coclosed; that is, $\cl(X)=X=\cl^*(X)$.
The {\em full closure} of a set $X$, denoted $\fcl(X)$, is the intersection of all fully closed sets that contain $X$.
It is easily seen that the full closure is a well-defined closure operator, and that one way of obtaining the full closure of a set $X$ is to take the closure of $X$, then the coclosure of the result, and repeat until neither the closure nor coclosure introduces new elements.
We frequently use the following straightforward lemma.
\begin{lemma}
\label{fcllemma}
Let $(X,Y)$ be a $2$-separation in a connected matroid $M$ where $M$ contains no series or parallel pairs.
Then $(\fcl(X),Y-\fcl(X))$ is also a $2$-separation of $M$.
\end{lemma}
\subsection*{Fans}
Let $M$ be a $3$-connected matroid.
A subset $F$ of $E(M)$ having at least three elements is a \emph{fan} if there is an ordering $(f_1, f_2, \dotsc, f_k)$ of the elements of $F$ such that
\begin{enumerate}[label=\rm(\alph*)]
\item $\{f_1,f_2,f_3\}$ is either a triangle or a triad, and\label{fani}
\item for all $i \in \seq{k-3}$, if $\{f_i, f_{i+1}, f_{i+2}\}$ is a triangle, then $\{f_{i+1}, f_{i+2}, f_{i+3}\}$ is a triad, while if $\{f_i, f_{i+1}, f_{i+2}\}$ is a triad, then $\{f_{i+1}, f_{i+2}, f_{i+3}\}$ is a triangle.\label{fanii}
\end{enumerate}
An ordering of $F$ satisfying \ref{fani} and \ref{fanii} is a \emph{fan ordering} of $F$.
If $F$ has a fan ordering $(f_1, f_2, \dotsc, f_k)$ where $k \geq 4$, then $f_1$ and $f_k$ are the \emph{ends} of $F$, and $f_2, f_3, \dotsc, f_{k-1}$ are the \emph{internal elements} of $F$.
A fan ordering is unique, up to reversal, when $k \ge 5$.
Let $F$ be a fan with ordering $(f_1, f_2, \dotsc, f_k)$ where $k \geq 4$,
and let $i \in \seq{k}$ if $k \geq 5$, or $i \in \{1,4\}$ if $k=4$.
An element $f_i$ is a \emph{spoke element} of $F$ if $\{f_1, f_2, f_3\}$ is a triangle and $i$ is odd, or if $\{f_1, f_2, f_3\}$ is a triad and $i$ is even; otherwise $f_i$ is a \emph{rim element}.
The next lemma follows easily from Bixby's Lemma.
\begin{lemma}
\label{fanEnds}
Let $M$ be a $3$-connected matroid that is not a wheel or a whirl.
Suppose $M$ has a fan~$F$ of at least four elements, and let $f$ be an end of $F$.
\begin{enumerate
\item If $f$ is a spoke element, then $\co(M\backslash f)$ is $3$-connected and $\si(M/f)$ is not $3$-connected.
\item If $f$ is a rim element, then $\si(M/f)$ is $3$-connected and $\co(M \backslash f)$ is not $3$-connected.
\end{enumerate}
\end{lemma}
A fan $F$ is \emph{maximal} if it is not properly contained in any other fan. Oxley and Wu~\cite[Lemma~1.5]{ow2000} proved the following
result concerning the ends of a maximal fan.
\begin{lemma}
\label{fanEndsStrong}
Let $M$ be a $3$-connected matroid
that is not a wheel or a whirl.
Suppose $M$ has a maximal fan~$F$ of at least four elements, and let $f$ be an end of $F$.
\begin{enumerate
\item If $f$ is a spoke element, then $M\backslash f$ is $3$-connected.
\item If $f$ is a rim element, then $M/f$ is $3$-connected.
\end{enumerate}
\end{lemma}
\subsection*{Retaining an $N$-minor}
Let $M$ and $N$ be matroids.
Throughout, when we say that $M$ \emph{has an $N$-minor}, we mean that $M$ has an isomorphic copy of $N$ as a minor.
Let $X \subseteq E(M)$. To simplify exposition, we say $M$ \emph{has an $N$-minor with} $|X \cap E(N)| \le 1$, for example, to mean that $M$ has an isomorphic copy $N'$ of $N$ as a minor such that $|X \cap E(N')| \le 1$.
For a matroid $M$ with a minor $N$, we say an element $e \in E(M)$ is \emph{$N$-contractible} if $M/e$ has an $N$-minor, and $e$ is \emph{$N$-deletable} if $M \backslash e$ has an $N$-minor.
We also say a set $X \subseteq E(M)$ is \emph{$N$-contractible} if $M/X$ has an $N$-minor, and $X$ is \emph{$N$-deletable} if $M\backslash X$ has an $N$-minor.
An element $e \in E(M)$ is \emph{doubly $N$-labelled} if both $M/e$ and $M \backslash e$ have $N$-minors.
The next lemma has a straightforward proof.
\begin{lemma}
\label{m2.7}
Let $(X,Y)$ be a $2$-separation of a connected matroid $M$ and let $N$ be a $3$-connected minor of $M$. Then $\{X, Y\}$ has a member $U$ such that $|U \cap E(N)| \leq 1$. Moreover, if $u \in U$, then
\begin{enumerate
\item $M/u$ has an $N$-minor if $M/u$ is connected, and
\item $M \backslash u$ has an $N$-minor if $M \backslash u$ is connected.
\end{enumerate}
\end{lemma}
The dual of the following is proved in \cite{bcosw2019,bs2014}.
\begin{lemma}
\label{doublylabel}
Let $N$ be a $3$-connected minor of a $3$-connected matroid $M$. Let $(X, \{z\}, Y)$ be a cyclic $3$-separation of $M$ such that $M\backslash z$ has an $N$-minor with $|X \cap E(N)| \le 1$. Let $X' = X-\cl^*(Y)$
and $Y' = \cl^*(Y) - z$.
Then
\begin{enumerate
\item each element of $X'$ is $N$-deletable; and
\item at most one element of $\cl^*(X)-z$ is not $N$-contractible, and if such an element~$x$ exists, then $x \in X' \cap \cl(Y')$ and $z \in \cl^*(X' - x)$.
\end{enumerate}
\end{lemma}
Suppose $C$ and $D$ are disjoint subsets of $E(M)$ such that $M/C \backslash D \cong N$.
We call the ordered pair $(C,D)$ an \emph{$N$-labelling of $M$}, and say that each $c \in C$ is \emph{$N$-labelled for contraction}, and each $d \in D$ is \emph{$N$-labelled for deletion}.
We also say a set $C' \subseteq C$ is \emph{$N$-labelled for contraction}, and $D' \subseteq D$ is \emph{$N$-labelled for deletion}.
An element $e \in C \cup D$ or a set $X \subseteq C \cup D$
is \emph{$N$-labelled for removal}.
Let $(C,D)$ be an $N$-labelling of $M$, and let $c \in C$, $d \in D$, and $e \in E(M)-(C \cup D)$.
Then, we say that the ordered pair
$((C-c) \cup d,(D-d)\cup c)$ is
obtained from $(C,D)$ by \emph{switching the $N$-labels on $c$ and $d$}.
Similarly,
$((C-c) \cup e,D)$ (or $(C,(D-d) \cup e)$, respectively) is
obtained from $(C,D)$ by \emph{switching the $N$-labels on $c$} (respectively, $d$) \emph{and $e$}.
The following straightforward lemma,
which gives a sufficient condition for retaining a valid $N$-labelling after an $N$-label switch,
will be used freely.
\begin{lemma}
\label{freeswitch}
Let $M$ be a $3$-connected matroid, let $N$ be a $3$-connected minor of $M$ with $|E(N)| \ge 4$, and let $(C,D)$ be an $N$-labelling of $M$.
Suppose $\{d,e\}$ is a parallel pair in $M/c$, for some $c \in C$.
Let $(C',D')$ be obtained from $(C,D)$ by switching the $N$-labels on $d$ and $e$; then $(C',D')$ is an $N$-labelling.
\end{lemma}
\subsection*{Delta-wye exchange}
Let $M$ be a matroid with a triangle $\Delta=\{a,b,c\}$.
Consider a copy of $M(K_4)$ having $\Delta$ as a triangle with $\{a',b',c'\}$ as the complementary triad labelled such that $\{a,b',c'\}$, $\{a',b,c'\}$ and $\{a',b',c\}$ are triangles.
Let $P_{\Delta}(M,M(K_4))$ denote the generalised parallel connection of $M$ with this copy of $M(K_4)$ along the triangle $\Delta$.
Let $M'$ be the matroid $P_{\Delta}(M,M(K_4))\backslash\Delta$ where the elements $a'$, $b'$ and $c'$ are relabelled as $a$, $b$ and $c$ respectively.
This matroid $M'$
is said to be obtained from $M$ by a \emph{$\Delta$-$Y$~exchange} on the triangle~$\Delta$.
Dually, a matroid $M''$ is obtained from $M$ by a \emph{$Y$-$\Delta$~exchange} on the triad $\{a,b,c\}$
if $(M'')^*$ is obtained from $M^*$ by a $\Delta$-$Y$~exchange\ on $\{a,b,c\}$.
\section{Triangles and triads}
\label{sectris}
Let $M$ be a $3$-connected matroid and let $N$ be a $3$-connected minor of $M$.
If, for a triangle~$T$ and for all distinct $a,b \in T$, none of $M/a/b$, $M/a\backslash b$, $M\backslash a/b$, and $M\backslash a\backslash b$ have an $N$-minor, then $T$ is an \emph{$N$-grounded\ triangle}.
Similarly, a triad $T^*$ of $M$ is an \emph{$N$-grounded\ triad} if, for all distinct $a,b \in T^*$, none of $M/a/b$, $M/a\backslash b$, $M\backslash a/b$, and $M\backslash a\backslash b$ have an $N$-minor.
In this section, we show that if $M$ has a triangle or triad that is not $N$-grounded, then either $M$ or $M'$, which can be obtained from $M$ by performing a $\Delta$-$Y$ or $Y$-$\Delta$ exchange, has an $N$-detachable pair.
When $|E(N)| \ge 4$, no element of an $N$-grounded\ triangle is $N$-contractible. As we use this straightforward fact frequently, we state it as a lemma below.
\begin{lemma}
\label{basicunfortunate}
Let $M$ be a $3$-connected matroid with a $3$-connected minor $N$ where
$|E(N)| \ge 4$.
If $T$ is an $N$-grounded\ triangle of $M$ with $x \in T$, then $M/x$ does not have an $N$-minor.
\end{lemma}
\begin{proof}
Let $T = \{x,y,z\}$. Since $\{y,z\}$ is a parallel pair in $M/x$, and $|E(N)| \ge 4$, if $M/x$ has an $N$-minor, then $M /x \backslash y$ has an $N$-minor. Thus $T$ is not $N$-grounded; a contradiction.
\end{proof}
We now prove the main result of this section.
Subject to this \lcnamecref{unfortunatetri}, we can then focus on the case where every triangle or triad of $M$ is $N$-grounded.
\begin{theorem}
\label{unfortunatetri}
Let $M$ be a $3$-connected matroid,
and let $N$ be a $3$-connected minor of $M$ with $|E(N)| \ge 4$, where $|E(M)|-|E(N)| \ge 5$.
Then either
\begin{enumerate}
\item $M$ has an $N$-detachable pair, or\label{ut1}
\item there is a matroid $M'$ obtained by performing a single $\Delta$-$Y$ or $Y$-$\Delta$ exchange on $M$ such that $M'$ has
an $N$-detachable pair, or\label{ut2}
\item each triangle or triad of $M$ is $N$-grounded.\label{ut3}
\end{enumerate}
\end{theorem}
\begin{proof}
Suppose $M$ has a triangle or triad $T$ that is not $N$-grounded.
First, suppose that $M$ is a wheel or a whirl.
By taking the dual, if necessary, we may assume that $T$ is a triangle.
Let $T = \{x,y,z\}$ where $y$ is a rim element and $x$ and $z$ are spoke elements with respect to a fan ordering of $E(M)$.
Since $T$ is not $N$-grounded, it follows that either $M \backslash x$ or $M \backslash z$ has an $N$-minor.
If $M$ is a wheel (respectively, a whirl), then $M/y \backslash z$ is a wheel (respectively, a whirl) of rank $r(M)-1$. In particular, $M/y \backslash z$ is $3$-connected since $|E(M)| > 6$.
Let $M'$ be the matroid obtained from $M$ by performing a $\Delta$-$Y$~exchange\ on $T$. Then $M/y \backslash z \cong M'/z/x$.
As $M\backslash x$ or $M \backslash z$ has an $N$-minor, $N$ is a minor of a wheel or a whirl of rank $r(M)-1$, so $M'/z/x$ has an $N$-minor, and $\{x,z\}$ is an $N$-detachable pair of $M'$, satisfying \cref{ut2}.
\smallskip
Now, suppose $T$ is contained in a maximal fan~$F$ of size at least five.
We start by proving the following claim:
\begin{sublemma}
\label{dcpair}
Suppose there are distinct elements $c \in E(M)$ and $d \in F$ such that $M/c\backslash d$ is $3$-connected and has an $N$-minor.
Then \cref{ut2} holds.
\end{sublemma}
\begin{slproof}
Since $M\backslash d$ is $3$-connected, \cref{fanEnds} implies that if $d$ is an end of $F$, it is a spoke element.
Now $d$ is either an internal element or a spoke of $F$, so it is contained in a triangle $T_1$. Let $M'$ be the matroid obtained from $M$ by performing a $\Delta$-$Y$~exchange\ on $T_1$. Then $M \backslash d$ is isomorphic to $M' / d$. Hence $M' / d / c$ is $3$-connected and has an $N$-minor, as required.
\end{slproof}
By \cref{dcpair} and its dual, we can now look for a pair of elements, at least one of which is in $F$, whose removal in any way preserves $3$-connectivity and an $N$-minor.
\Cref{fanEndsStrong} provides one candidate element for removal; to find the second, we require that the resulting matroid, after the element is removed, is not a wheel or a whirl.
\begin{sublemma}
\label{nowheel}
The triangle or triad $T$ is contained in a maximal fan~$F'$ with ordering $(x_1,x_2,\dotsc,x_\ell)$, for $\ell \ge 5$, such that, up to duality, $\{x_1,x_2,x_3\}$ is a triangle, and $M \backslash x_1$ is $3$-connected and not a wheel or a whirl.
\end{sublemma}
\begin{slproof}
We have that $T$ is contained in a maximal fan~$F$ of size at least five.
We may assume, by reversing the ordering if necessary, that $T \subseteq F-x_\ell$, and, by duality, that $x_1$ is a spoke element of $F$, so $\{x_1,x_2,x_3\}$ is a triangle.
Then, by \cref{fanEndsStrong}, $M \backslash x_1$ is $3$-connected.
Towards a contradiction, suppose $M \backslash x_1$ is a wheel or a whirl.
Then $x_2$ is in a triangle of $M\backslash x_1$ that meets $x_3$ or $x_4$, by orthogonality with the triad $\{x_2,x_3,x_4\}$ of $M \backslash x_1$.
If $\{x_2,x_3\}$ is contained in a triangle of $M \backslash x_1$, then $\{x_1,x_2,x_3\}$ is contained in a $4$-element segment of $M$ that intersects the triad $\{x_2,x_3,x_4\}$ in two elements, which contradicts orthogonality.
So $M \backslash x_1$ has a triangle $\{x_2,x_4,q\}$, say, where $q \in E(M \backslash x_1) - x_3$.
Suppose $|F| > 6$. Then $\{x_4,x_5,x_6\}$ is a triad,
and, by \cite[Lemma~3.4]{ow2000}, the only triangle of $M$ containing $x_4$ is $\{x_3,x_4,x_5\}$.
Since $\{x_2,x_4,q\}$ is also a triangle of $M$, this is a contradiction.
So $|F|=5$.
Now $(x_1,x_3,x_2,x_4,q)$ is a fan ordering of $M$, and this fan contains $T$.
It follows from orthogonality that $\{x_4,q\}$ is not contained in a triad, so this fan ordering extends to a maximal fan~$F'$ where $q$ is an end.
As $M \backslash q$ is $3$-connected by \cref{fanEndsStrong}, if $M \backslash q$ is not a wheel or a whirl, then \cref{nowheel} holds for the fan~$F'$.
So we may assume that $M \backslash q$ is a wheel or a whirl.
Now $(x_1,x_2,x_3,x_4,x_5)$ is a fan ordering in $M \backslash q$ that extends to a fan ordering $(x_1,x_2,\dotsc,x_\ell)$ of $E(M \backslash q)$.
Observe that $\ell \ge 8$ and $\ell$ is even.
In $M \backslash x_1$, there is a fan with ordering $(q,x_2,x_4,x_3,x_5)$ that extends to a fan ordering of $E(M \backslash x_1)$.
So there is a triad containing $\{x_3,x_5\}$, and it meets $\{x_6,x_7\}$ by orthogonality, but if it contains $x_6$, then $\{x_3,x_4,x_5,x_6\}$ is a cosegment that intersects the triangle $\{x_5,x_6,x_7\}$ in two elements; a contradiction.
So $\{x_3,x_5,x_7\}$ is a triad.
If $\ell > 8$, then this triad intersects the triangle $\{x_7,x_8,x_9\}$ in a single element; a contradiction.
So $|E(M)|=9$, and hence $r(M)=4$.
It now follows that $q$ is in a triangle $\{q,x_6,x_8\}$.
By circuit elimination, $\{x_2,x_4,x_6,x_8\}$ contains a circuit.
As this set does not contain a triangle, $\{x_2,x_4,x_6,x_8\}$ is a circuit, so $M \backslash q$ is a wheel.
Since $\{x_2,x_4,q\}$ and $\{x_6,x_8,q\}$ are circuits of $M$, it follows that $M$ is binary.
So $M$ has no $U_{2,4}$-minor, in which case $|E(N)| \ge 5$, and $|E(M)| \ge 10$; a contradiction.
\end{slproof}
Let $F_1$ be the fan $F'$ of \cref{nowheel} with ordering $(x_1,\dotsc,x_\ell)$.
Now $M \backslash x_1$ is $3$-connected, and is neither a wheel nor a whirl.
\begin{sublemma}
\label{justdone}
There is an $N$-labelling such that $x_1$ is $N$-labelled for deletion, and either $x_2$ or $x_3$ is $N$-labelled for contraction.
\end{sublemma}
\begin{slproof}
First, observe that if either $x_2$ or $x_3$ is $N$-labelled for contraction, then, since $\{x_1,x_2,x_3\}$ is a triangle and $|E(N)| \ge 4$, it follows that
$x_1$ is $N$-labelled for deletion up to an $N$-label switch with $x_3$ or $x_2$ respectively, using \cref{freeswitch}.
So it suffices to show that either $x_2$ or $x_3$ is $N$-labelled for contraction.
Since $F_1$ contains the triangle or triad $T$ that is not $N$-grounded, there is an internal element $x_j$ of $F_1$ that is $N$-labelled for removal.
Suppose $x_2$ is $N$-labelled for deletion.
Then $\{x_3,x_4\}$ is a series pair in $M \backslash x_2$.
It follows that, after possibly performing an $N$-label switch on $x_3$ and $x_4$, the element $x_3$ is $N$-labelled for contraction.
Similarly, if $x_j$ is $N$-labelled for deletion for some $j \ge 3$, then, as $\{x_{j-1},x_j\}$ is contained in a triad, $x_{j-1}$ is $N$-labelled for contraction, up to a possible $N$-label switch.
Likewise, if $x_j$ is $N$-labelled for contraction, for some $j > 3$, then, there is a triangle containing $\{x_{j-1},x_j\}$; after a possible $N$-label switch, $x_{j-1}$ is $N$-labelled for deletion.
By repeating this process, we obtain an $N$-labelling where either $x_2$ or $x_3$ is $N$-labelled for contraction, as required.
This proves the claim.
\end{slproof}
Consider the matroid $M \backslash x_1$. By \cref{justdone}, this matroid has an $N$-labelling where either $x_2$ or $x_3$ is $N$-labelled for contraction.
The set $F_1 - x_1$ is a $4$-element fan
that is contained in a maximal fan $F_2$, with ordering $(y_1,y_2,\dotsc,y_t)$, for some $t \ge 4$.
If $x_2$ is $N$-labelled for contraction and $x_2$ is an end of $F_2$, then, as $x_2$ is a rim, the matroid $M \backslash x_1 / x_2$ is $3$-connected by \cref{fanEndsStrong}, and \cref{ut2} holds by \cref{dcpair}.
So we may assume that either $x_3$ is $N$-labelled for contraction, or $x_2$ is not an end of $F_2$. In either case, $F_2$ has an internal element that is $N$-labelled for contraction.
By \cref{fanEndsStrong}, either $y_1$ is a spoke and $M \backslash x_1 \backslash y_1$ is $3$-connected, or $y_1$ is a rim and $M \backslash x_1/y_1$ is $3$-connected.
Using a similar argument as in \cref{justdone}, we can iteratively switch $N$-labels so that $y_1$ is $N$-labelled for deletion if it is a spoke, or $N$-labelled for contraction if it is a rim.
It follows that \cref{ut2} holds.
\medskip
Now suppose $T$ is contained in a maximal $4$-element fan $F$.
Let $(f_1,f_2,f_3,f_4)$ be a fan ordering of $F$ where $\{f_1,f_2,f_3\}$ is a triangle.
Since $F$ contains $T$, which is not $N$-grounded, at least one of $f_2$ and $f_3$, is $N$-labelled for removal. Up to duality and switching labels on $f_2$ and $f_3$, we may assume that $f_2$ is $N$-labelled for deletion.
Since $\{f_3,f_4\}$ is a series pair in $M \backslash f_2$, we may also assume, up to an $N$-label switch, that $f_4$ is $N$-labelled for contraction.
Now $M / f_4$ is $3$-connected, by \cref{fanEndsStrong}, and has an $N$-minor.
Let $M'$ be the matroid obtained by $Y$-$\Delta$~exchange\ on the triad $\{f_2,f_3,f_4\}$. Then $M / f_4$ is isomorphic to $M' \backslash f_4$.
Now $\{f_1,f_2,f_3\}$ is a triangle of $M/f_4$ that does not meet a triad, so $M/f_4$ is not a wheel or a whirl.
Hence, by the Splitter Theorem, there is an element $e \in E(M/f_4)$ such that either $M/f_4/e$ or $M/f_4 \backslash e$ is $3$-connected and has an $N$-minor.
In the latter case, $M/f_4 \backslash e$ is isomorphic to $M' \backslash f_4 \backslash e$, so \cref{ut2} holds.
\medskip
Finally, we may assume that $T$ is a triangle that is not contained in a $4$-element fan.
Let $T= \{a,b,c\}$.
We claim that, up to relabelling, $M \backslash a$ and $M \backslash b$ have $N$-minors.
Indeed, if $c$ is $N$-labelled for contraction, then, since $\{a,b\}$ is a parallel pair in $M/c$, both $M \backslash a$ and $M \backslash b$ have $N$-minors.
On the other hand, if $T$ has no elements that are $N$-labelled for contraction, then, as $T$ is not $N$-grounded, it has at most one element that is not $N$-labelled for removal, and, by labelling this element $c$, we have that $M\backslash a$ and $M\backslash b$ have $N$-minors.
Since there is no triad meeting $T$, Tutte's Triangle Lemma implies that at least one of $M \backslash a$ and $M \backslash b$ is $3$-connected.
Without loss of generality, let $M \backslash a$ be $3$-connected.
Now $M \backslash a$ has a proper $N$-minor, so if $M \backslash a$ is not a wheel or a whirl, then, by the Splitter Theorem, there is some element $x \in E(M \backslash a)$ such that $M \backslash a \backslash x$ or $M \backslash a / x$ is $3$-connected and has an $N$-minor. In the first case, $M$ has an $N$-detachable pair as required, so assume the latter.
Let $M'$ be the matroid obtained by a $\Delta$-$Y$ exchange on $T$. Then $M \backslash a$ is isomorphic to $M' / a$.
In particular, $M' / a$ has an $N$-minor.
Thus $\{a,x\}$ is an $N$-detachable pair in $M'$, satisfying \cref{ut2}.
It remains to consider the case where $M\backslash a$ is a wheel or a whirl.
Since $M$ has no $4$-element fans, for every triad $T^*$ of $M \backslash a$,
we have that $T^* \cup a$ is a cocircuit of $M$.
By orthogonality, $T-a$
has non-empty intersection with each such $T^*$.
If a wheel or whirl has rank more than four, then no two elements meet every triad.
So $r(M \backslash a) \le 4$, and thus $|E(M \backslash a)| \leq 8$.
Thus, in the only remaining case $|E(M)| = 9$ and $|E(N)| = 4$, so $N \cong U_{2,4}$.
Since $M \backslash a$ has an $N$-minor, $M \backslash a$ is the rank-$4$ whirl.
Let $d$ be a spoke of $M \backslash a$. Then it is easily verified that $M \backslash d$ is $3$-connected and has an $N$-minor.
Moreover, $M \backslash d$ is not a wheel or a whirl, and $d$ is in distinct triangles $T_1$ and $T_2$ of $M$.
By the Splitter Theorem, there is some element $x \in E(M \backslash d)$ such that $M \backslash d \backslash x$ or $M \backslash d / x$ is $3$-connected and has an $N$-minor. In the first case, $M$ has an $N$-detachable pair as required.
In the latter case, observe that $x$ is not contained in either $T_1$ or $T_2$.
Say $x \notin T_1$.
Letting $M'$ be the matroid obtained by a $\Delta$-$Y$ exchange on $T_1$, we observe that $M \backslash d / x \cong M' /d /x$ is $3$-connected and has an $N$-minor, so \cref{ut2} holds.
\end{proof}
\section{$5$-element planes}
\label{secplanes}
In this section, we show that when $M$ has a $U_{3,5}$ restriction, and there are certain elements whose removal preserves an $N$-minor, then $M$ has an $N$-detachable pair.
For $P \subseteq E(M)$, we say that $P$ is a \emph{$5$-element plane} if $M|P \cong U_{3,5}$. We also say $P$ is a \emph{$5$-element coplane} if $M^*|P \cong U_{3,5}$.
The proofs of the first two lemmas are routine.
\begin{lemma}
\label{6pointplane}
Let $M$ be a $3$-connected matroid with $P \subseteq E(M)$ such that $M|P \cong U_{3,5}$.
Then $M\backslash p$ is $3$-connected for each $p \in \cl(P)-P$.
\end{lemma}
\begin{lemma}
\label{basicplanelemma}
Let $M$ be a $3$-connected matroid with a set $P$ such that $M|P \cong U_{3,5}$, and $|E(M)| \ge 6$.
If $P$ contains a triad~$T^*$, then $M \backslash p$ is $3$-connected for each $p \in P-T^*$.
\end{lemma}
\begin{lemma}
\label{planelemma}
Let $M$ be a $3$-connected matroid with $P \subseteq E(M)$ such that $M|P \cong U_{3,5}$.
Suppose that $\cl(P)$ contains no triangles and $P$ contains no triads.
If $M \backslash p$ is not $3$-connected for some $p \in P$, then
there is a labelling $\{p_1,p_2,p_3,p_4\}$ of $P-p$ such that $M \backslash p_i \backslash p_j$ is $3$-connected for each $i \in \{1,2\}$ and $j \in \{3,4\}$.
\end{lemma}
\begin{proof}
Let $P = \{p,p_1,p_2,p_3,p_4\}$, and suppose $M \backslash p$ is not $3$-connected.
If $p$ is in a triad, then this triad is contained in $P$, by orthogonality; a contradiction. So $M$ has a cyclic $3$-separation $(A,\{p\},B)$, where $(A,B)$ is a $2$-separation of $M \backslash p$.
Without loss of generality, we may assume that $\{p_1,p_2\} \subseteq A$.
If $p_3 \in A$ or $p_4 \in A$, then $p \in \cl(A)$, so $(A \cup p, B)$ is $2$-separating in $M$; a contradiction. So $\{p_3,p_4\} \subseteq B$.
Let $A' = A - \{p_1,p_2\}$ and $B' = B - \{p_3,p_4\}$.
Since $A$ and $B$ contain circuits and $\cl(P)$ contains no triangles, $|A'|,|B'|\ge 2$.
Now,
$(A', \{p_1,p_2\}, \{p\}, \{p_3,p_4\}, B')$ is a path of $3$-separations of $M$ where $p_1$ and $p_2$, and $p_3$ and $p_4$, are guts elements.
Again using that $\cl(P)$ contains no triangles, it follows that $r(A'),r(B') \ge 3$.
Furthermore, each $p_i$ is not in a triad, by orthogonality.
Thus, by Bixby's Lemma, $M \backslash p_i$ is $3$-connected for $i \in \{1,2,3,4\}$; and, moreover,
$M \backslash p_i \backslash p_j$ is $3$-connected up to series pairs for $i \in \{1,2\}$ and $j \in \{3,4\}$.
Suppose that $\{p_i,p_j\}$ is in a $4$-element cocircuit $C^*$ of $M$. Then $E(M)-C^*$ is closed, so $C^*$ meets $A'$ and $B'$, and contains an element of $P - \{p_i,p_j\}$. But this implies $|C^*| \ge 5$; a contradiction.
This proves \cref{planelemma}.
\end{proof}
\begin{lemma}
\label{6pointplane2}
Let $M$ be a $3$-connected matroid with $P \subseteq E(M)$ such that $M|P \cong U_{3,6}$, and $X \subseteq P$ such that $|X|=4$.
Suppose that $\cl(P)$ contains no triangles.
Then there are distinct elements $x_1,x_2 \in X$ such that $M\backslash x_1\backslash x_2$ is $3$-connected.
\end{lemma}
\begin{proof}
Pick any distinct $x_1,x_2 \in X$.
By \cref{6pointplane}, $M \backslash x_1$ is $3$-connected, and $M|(P-x_1) \cong U_{3,5}$.
If $M \backslash x_1 \backslash x_2$ is $3$-connected, then the \lcnamecref{6pointplane2} holds, so we may assume otherwise.
Observe that $P$ contains no triads, by orthogonality.
Now, by \cref{planelemma}, $M \backslash x_1 \backslash p \backslash p'$ is $3$-connected for $p,p' \in P - \{x_1,x_2\}$, where we can choose $p$ and $p'$ such that $p \in X$.
In particular, $M\backslash x_1\backslash p$ is $3$-connected for $\{x_1,p\} \subseteq X$, as required.
\end{proof}
The following lemma is useful for finding candidates for contraction in a $4$-element cocircuit, particularly in the case where the cocircuit is independent.
\begin{lemma}
\label{r4cocirc}
Let $M$ be a $3$-connected matroid and let $C^*$ be a $4$-element cocircuit of $M$. If there are distinct elements $c',c'' \in C^*$ such that neither $c'$ nor $c''$ is in a triangle, then
$M/c$ is $3$-connected for some $c \in C^*$.
\end{lemma}
\begin{proof}
Let $C^*=\{c_1,c_2,c_3,c_4\}$ and suppose that $c_1$ is one of two elements that is not contained in a triangle.
If $M/c_1$ is not $3$-connected, then $M$ has a vertical $3$-separation $(X,\{c_1\},Y)$.
We may assume that $c_2\in X$ and $c_3,c_4\in Y$.
Suppose that $c_2$ is not in a triangle.
If $X$ is a triad, then by the dual of \cref{r3cocirc}, $M/c_2$ is $3$-connected as required.
If $X$ is not a triad, then either $X$ is a cosegment with at least four elements, or $X$ contains a circuit.
In the first case, $M/c_2$ is $3$-connected by the dual of \cref{rank2Remove}.
In the second case, as $c_2 \in \cl^*(Y\cup c_1)$, the circuit contained in $X$ does not contain $c_2$.
Now $(X-c_2,\{c_2\},Y\cup c_1)$ is a cyclic $3$-separation of $M$, so $M/c_2$ is once again $3$-connected, by Bixby's Lemma.
So we may assume that $c_2$ belongs to some triangle $T$.
As $C^*$ is a cocircuit, $T\cap(C^*-c_2)\neq\emptyset$ by orthogonality, so we may assume that $c_3 \in T$ and $c_4$ is the other element of $C^*$ that is not contained in a triangle.
As $c_2\not\in\cl(Y)$, we have $|T\cap X|=2$, so $(Y- c_3,\{c_1\},X\cup c_3)$ is a vertical $3$-separation of $M$.
Note that $(Y-c_3) \cap C^* = \{c_4\}$.
Again, if $Y- c_3$ is not a triad or a cosegment, then $(Y-\{c_3,c_4\},\{c_4\},X\cup\{c_1,c_3\})$ is a cyclic $3$-separation of $M$, and $M/c_4$ is $3$-connected by Bixby's Lemma. On the other hand, if $Y- c_3$ is a triad, then $M/c_4$ is $3$-connected by the dual of \cref{r3cocirc}; while if $Y-c_3$ is a cosegment with at least four elements, then $M/c_4$ is $3$-connected by the dual of \cref{rank2Remove}.
\end{proof}
The next two results show the existence of $N$-detachable pairs when $M$ has a subset $P$ such that $M|P \cong U_{3,5}$. The first handles the case where $P$ is $3$-separating, whereas the second handles the case where $P$ is not $3$-separating.
\begin{proposition}
\label{basicplaneupgrade}
Let $M$ be a $3$-connected matroid with $|E(M)| \ge 9$ and $r(M) \ge 5$, and let $N$ be a $3$-connected minor of $M$ where $|E(N)| \ge 4$ and every triangle or triad of $M$ is $N$-grounded.
Suppose there exists some exactly $3$-separating set $P \subseteq E(M)$ such that $M|P \cong U_{3,5}$, and there are distinct elements $d^*,p \in P$ such that
\begin{enumerate}[label=\rm(\alph*)]
\item either $P$ or $P-p$ is a cocircuit, and
\item $M / d^* / p'$ has an $N$-minor for all $p' \in P-\{d^*,p\}$.
\end{enumerate}
Then $M$ has an $N$-detachable pair.
\end{proposition}
\begin{proof}
First, observe that for any $p' \in P-\{d^*,p\}$, the set
$P-\{d^*,p'\}$ is contained in a parallel class in $M/d^*/p'$.
Since $|E(N)| \ge 4$,
the matroid $M \backslash q_1 \backslash q_2$ has an $N$-minor for any distinct $q_1,q_2 \in P-\{d^*,p'\}$.
By an appropriate choice of $p'$, it follows that $M \backslash q_1 \backslash q_2$ has an $N$-minor for all distinct $q_1,q_2 \in P-d^*$.
Let $P= \{p_1,p_2,p_3,p_4,p_5\}$, where
$p_4=d^*$ and $p_5=p$.
For each $i \in \seq{4}$, $M/p_i$ has an $N$-minor, so $P$ does not contain an $N$-grounded\ triangle.
Similarly, $M \backslash p_i$ has an $N$-minor for each $i \in \{1,2,3,5\}$, so $P$ does not contain an $N$-grounded\ triad.
Suppose $\cl(P)$ contains an $N$-grounded\ triangle~$T$.
Then $T \subseteq \cl(P)-\{p_1,p_2,p_3,p_4\}$.
Since $M/p_1/p_4$ has an $N$-minor and
$\cl(P)-\{p_1,p_4\}$ is contained in a parallel class in $M/p_1/p_4$,
there is an $N$-labelling $(C,D)$ such that $T \subseteq C$; a contradiction.
So $\cl(P)$ does not contain any triangles.
By \cref{planelemma}, we may assume that $M \backslash p_i$ is $3$-connected for each
$i \in \seq{5}$.
Since $P$ does not contain any triads,
either $P$ is a cocircuit, or $P$ contains a $4$-element cocircuit.
Towards a contradiction, we now assume that $M$ does not have an $N$-detachable pair.
\begin{sublemma}
\label{claim2}
For each $i \in \seq{3}$,
there exists a cocircuit $\{p_i,p_i',p_5,z_i\}$ of $M$, where
$p_i' \in P-\{p_i,p_5\}$ and $z_i \in E(M)-P$, and $M/z_i$ is $3$-connected.
\end{sublemma}
\begin{slproof}
We claim that $\co(M \backslash p_5 \backslash p_i)$ is $3$-connected for each $i \in \seq{4}$.
First, suppose that $P-p_5$ is a cocircuit.
Then, for $i \in \seq{4}$,
$(P-\{p_i,p_5\},\{p_5\}, E(M)-P)$ is a vertical $3$-separation of $M \backslash p_i$.
Thus, by Bixby's Lemma, $\co(M \backslash p_5\backslash p_i)$ is $3$-connected.
Now suppose that $P$ is a cocircuit.
We will show that $\co(M \backslash p_5 \backslash p_i)$ is $3$-connected for $i=4$, but the argument is the same for $i \in \seq{3}$.
Let $(X,Y)$ be a $2$-separation of $M \backslash p_5 \backslash p_4$.
We may assume that $\{p_1,p_2\} \subseteq X$.
Now $\{p_1,p_2,p_3\}$ is a triad of $M \backslash p_5 \backslash p_4$, so either
$(X \cup p_3, Y-p_3)$ is a $2$-separation, or $Y$ is a series pair.
But
$p_5 \in \cl(X \cup p_3)$, so, in the former case, $(X \cup \{p_3,p_5\}, Y-p_3)$ is a $2$-separation of $M \backslash p_4$; a contradiction.
Thus $Y$ is a series pair, and it follows that
$\co(M \backslash p_5 \backslash p_4)$ is $3$-connected.
Let $i \in \seq{3}$, and
recall that $M \backslash p_i \backslash p_5$ has an $N$-minor.
Since $\{p_i,p_5\}$ is not an $N$-detachable pair, it follows that $p_5$ is in a triad~$T^*$ of $M \backslash p_i$.
By orthogonality, $T^*$ contains an element $p_i' \in P-\{p_i,p_5\}$, so let $T^* = \{p_5,p_i',z_i\}$.
If $P$ is a cocircuit, then $T^* \nsubseteq P$, so $z_i \in E(M)-P$,
whereas if $P-p_5$ is a cocircuit, then $p_5 \in \cl(E(M)-P)$, so, by orthogonality, $z_i \in E(M)-P$.
Since $T^*$ is not a triad of $M$, $\{p_i,p_i',p_5,z_i\}$ is a cocircuit.
Suppose $M/z_i$ is not $3$-connected.
If $z_i$ is in a triangle, then, by orthogonality with the cocircuit $\{p_i,p_i',p_5,z_i\}$, this triangle meets $P$; a contradiction.
So $\si(M/z_i)$ is also not $3$-connected. Let $(A,\{z_i\},B)$ be a vertical $3$-separation of $M$.
Without loss of generality, $|A \cap P| \ge 3$, so $(A \cup P, \{z_i\}, B-P)$ is also a vertical $3$-separation, by uncrossing.
But then $z_i \in \cl^*(A \cup P) \cap \cl(B-P)$; a contradiction.
So $M/z_i$ is $3$-connected.
\end{slproof}
\begin{sublemma}
\label{claim3*}
Suppose, up to relabelling $\{p_1,p_2,p_3\}$,
that $M$ has a cocircuit $\{p_1,p_2,p_5,z\}$,
for some $z \in E(M)-P$.
Then $M$ has an $N$-detachable pair.
\end{sublemma}
\begin{slproof}
Let $(C,D)$ be an $N$-labelling such that $\{p_3,p_4\} \subseteq C$; such an $N$-labelling exists since $M / p_3 / p_4$ has an $N$-minor.
Since $\{p_1,p_2,p_5\}$ is contained in a parallel class in $M/p_3/p_4$, we may assume, up to switching the $N$-labels on $p_5$ and $p_1$ or $p_2$, that $p_1$ and $p_2$ are $N$-labelled for deletion.
Moreover, as $\{z,p_5\}$ is a series pair in $M \backslash p_1 \backslash p_2$, we may also assume, by a possible $N$-label switch on $p_5$ and $z$, that $z$ is $N$-labelled for contraction.
In particular, $\{z,p_3\}$ is an $N$-contractible pair.
Since $P$ is exactly $3$-separating and $z \in \cl^*(P)$,
\cref{gutsstayguts} implies that $z \notin \cl(P)$.
So $P$ or $P-p_5$ is a rank-$3$ cocircuit in $M/z$.
By \cref{r3cocircsi},
$\si(M / z /p_3)$ is $3$-connected.
Now either $M / z /p_3$ is $3$-connected, or $\{z,p_3\}$ is contained in a $4$-element circuit.
In the former case, $M$ has an $N$-detachable pair.
So we may assume that $\{z,p_3\}$ is contained in a $4$-element circuit $C_z$.
By orthogonality, $C_z$ meets $\{p_1,p_2,p_5\}$; moreover, since $z \notin \cl(P)$, we have $|C_z \cap P| = 2$.
So $C_z = \{z,p_3,p'',f\}$ where $p'' \in \{p_1,p_2,p_5\}$ and $f \in E(M) - (P \cup z)$.
Note that $p_3$ and $z$ are $N$-labelled for contraction. Thus, after possibly switching the $N$-labels on $p''$ and $f$, the element $f$ is $N$-labelled for deletion.
Let $p''' \in \{p_1,p_2\}-p''$,
and note that $p'''$ is also $N$-labelled for deletion.
As $(P \cup z, \{f\}, E(M) - (P \cup \{z,f\}))$ is a vertical $3$-separation
and
$f$ is not in an $N$-grounded\ triad,
the matroid $M \backslash f$ is $3$-connected and has an $N$-minor.
Note that $f \notin \cl^*(P)$, so $P$ does not contain any triads in $M \backslash f$.
Thus, by \cref{planelemma}, $M \backslash f \backslash p'''$ is $3$-connected, so $\{f,p'''\}$ is an $N$-detachable pair.
\end{slproof}
Now, by \cref{claim2,claim3*}, we may assume that
$\{p_1,p_4,p_5,z_1\}$, $\{p_2,p_4,p_5,z_2\}$, and $\{p_3,p_4,p_5,z_3\}$ are cocircuits of $M$.
Suppose $z_i = z_j$ for some distinct $i,j \in \seq{3}$.
Then, by the cocircuit elimination axiom, $\{p_i,p_j,p_4,p_5\}$ contains a cocircuit; in fact, since
$P$ does not contain any $N$-grounded\ triads, this set is a $4$-element cocircuit.
Since $P$ is not a cocircuit, $P-p_5$ is also a cocircuit, by hypothesis.
But now $P-p_5$ is $3$-separating and
$p_5 \in \cl(\{p_1,p_2,p_3,p_4\}) \cap \cl^*(\{p_1,p_2,p_3,p_4\})$; a contradiction.
So $z_i \neq z_j$ for all distinct $i,j \in \seq{3}$.
For $j \in \{2,3\}$, the partition
$(P,\{z_1\},\{z_j\},E(M)-(Z \cup \{z_1,z_j\}))$ is a path of $3$-separations where $z_1$ and $z_j$ are coguts elements.
In particular, $z_j \in \cl^*(E(M)-(Z \cup \{z_1,z_j\}))$, so $z_j \notin \cl(P \cup z_1)$.
We now fix an $N$-labelling such that $p_1$ and $p_5$ are $N$-labelled for deletion and $p_2$ is $N$-labelled for contraction (such an $N$-labelling exists since $M/p_2/p_4$ has an $N$-minor and $\{p_1,p_3,p_5\}$ is contained in a parallel class in this matroid).
We may also assume that $z_1$ is $N$-labelled for contraction, since
$\{z_1,p_4\}$ is a series pair in $M \backslash p_2 \backslash p_4$.
Recall that $M/z_1$ is $3$-connected.
By \cref{r3cocircsi}, $\si(M / z_1 /p_2)$ is $3$-connected.
Thus, either $\{z_1,p_2\}$ is an $N$-detachable pair, or
$\{z_1,p_2\}$ is contained in a $4$-element circuit~$C_1$.
By orthogonality, $C_1$ meets $\{p_1,p_4,p_5\}$ and $\{p_4,p_5,z_2\}$.
Since $z_1 \notin \cl(P)$, we have $|C_1 \cap P|=2$.
If $p_4 \in C_1$ or $p_5 \in C_1$, then $C_1=\{z_1,p_2,p_\ell,z_3\}$ for $\ell \in \{4,5\}$, so $z_3 \in \cl(P \cup z_1)$; a contradiction.
On the other hand, if $\{p_4,p_5\} \cap C_1 = \emptyset$, then
$\{p_1,z_2\} \subseteq C_1$, so $C_1=\{p_1,p_2,z_1,z_2\}$ and $z_2 \in \cl(P \cup z_1)$; a contradiction.
This completes the proof.
\end{proof}
\begin{proposition}
\label{planeupgrade}
Let $M$ be a $3$-connected matroid with a $3$-connected matroid~$N$ as a minor, where $|E(N)| \ge 4$ and every triangle or triad of $M$ is $N$-grounded.
Suppose there exists $P \subseteq E(M)$ such that $M|P \cong U_{3,5}$ and $P$ is not $3$-separating,
and there are elements $d^*,p \in P$ such that
\begin{enumerate}[label=\rm(\alph*)]
\item $M / d^*$ is $3$-connected,\label{planeupgradec1
\item $M / d^* / p'$ has an $N$-minor for all $p' \in P-\{d^*,p\}$,
and\label{planeupgradec2}
\item
for any $p' \in P-d^*$ and distinct elements $u,v \in \cl^*(P-d^*)-P$, either $M \backslash p' \backslash u$ or $M \backslash p' \backslash v$ has an $N$-minor.\label{planeupgradec4}
\end{enumerate}
Then $M$ has an $N$-detachable pair.
\end{proposition}
\begin{proof}
Pick $p \in P$ such that $M /d^*/p'$ has an $N$-minor for each $p' \in P-\{d^*,p\}$.
Since $P-\{d^*,p'\}$ is contained in a parallel class in $M/d^*/p'$ and $|E(N)| \ge 4$,
the matroid $M \backslash q_1 \backslash q_2$ has an $N$-minor for any distinct $q_1,q_2 \in P-\{d^*,p'\}$.
As $p'$ is chosen arbitrarily among $P-\{d^*,p\}$, it follows that $M \backslash q_1 \backslash q_2$ has an $N$-minor for all distinct $q_1,q_2 \in P-d^*$.
As $M \backslash p'$ has an $N$-minor for each $p' \in P-d^*$, $P$ does not contain an $N$-grounded\ triad.
Suppose $\cl(P)$ contains an $N$-grounded\ triangle~$T$. Then $T$ does not meet $P-\{d^*,p\}$, since $M / p'$ has an $N$-minor for each $p' \in P-\{d^*,p\}$.
There exist distinct $p', p'' \in P-\{d^*,p\}$ such that $M/p'/p''$ has an $N$-minor, and $T$ is contained in a parallel class in this matroid. But this contradicts the fact that $T$ is $N$-grounded, so $\cl(P)$ does not contain any triangles.
\begin{sublemma}
\label{p0}
If there are distinct elements $q,q',q'' \in P-d^*$ such that neither $\{q,q'\}$ nor $\{q,q''\}$ is contained in a $4$-element cocircuit of $M$, then $M$ has an $N$-detachable pair.
\end{sublemma}
\begin{slproof}
Recall that $\cl(P)$ does not contain an $N$-grounded\ triangle and $P$ does not contain an $N$-grounded\ triad.
Thus, by \cref{planelemma}, either $M$ has an $N$-detachable pair, or $M \backslash q$ is $3$-connected.
By the dual of \cref{r4cocirc},
if there are distinct elements $q'$ and $q''$ in $P-\{d^*,q\}$ that are not contained in a triad of $M \backslash q$, then either
$\{q,q'\}$ or $\{q,q''\}$ is an $N$-detachable pair.
\end{slproof}
\begin{sublemma}
\label{p1}
There is a labelling $\{p_1,p_2,p_3,p_4\}$ of $P-d^*$ such that one of the following holds:
\begin{enumerate}
\item $\{p_1,p_2,p_3,u\}$ and $\{p_2,p_3,p_4,v\}$ are cocircuits of $M$, with $u,v \in E(M)-P$, or\label{p1a}
\item $\{p_1,p_2,p_3,u\}$, $\{d^*,p_2,p_4,u_2\}$ and $\{d^*,p_3,p_4,u_3\}$ are cocircuits of $M$, with $u, u_2, u_3 \in E(M)-P$, or\label{p1b}
\item each of $\{d^*,p_1,p_3\}$, $\{d^*,p_1,p_4\}$, $\{d^*,p_2,p_3\}$, $\{d^*,p_2,p_4\}$, and $\{d^*,p_3,p_4\}$ is contained in a $4$-element cocircuit of $M$.\label{p1c}
\end{enumerate}
\end{sublemma}
\begin{slproof}
By orthogonality, a $4$-element cocircuit that intersects $P$ must contain at least three elements of $P$; in fact, since $P$ is not $3$-separating, such a cocircuit contains exactly three elements of $P$.
If there are no cocircuits containing a $3$-element subset of $\{p_1,p_2,p_3,p_4\}$, then by repeated applications of \cref{p0}, it follows that \cref{p1c} holds.
On the other hand, if there are two cocircuits of $M$ containing distinct $3$-element subsets of $\{p_1,p_2,p_3,p_4\}$, then \cref{p1a} holds.
So assume that $\{p_1,p_2,p_3,u\}$ is a cocircuit of $M$ for $u \in E(M)-P$, and every other $4$-element cocircuit meeting $P$ contains $d^*$.
If neither $\{p_2,p_4\}$ nor $\{p_3,p_4\}$ is contained in a $4$-element cocircuit, then $M$ has an $N$-detachable pair by \cref{p0}; so we may assume that $\{p_3,p_4,v\}$ is a $4$-element cocircuit for some $v \in E(M)-P$.
But by repeating this argument with $\{p_1,p_4\}$ and $\{p_2,p_4\}$, we deduce that
$\{p_2,p_4,v'\}$ is a cocircuit for some $v' \in E(M)-P$.
Since \cref{p1b} holds in this case, this completes the proof.
\end{slproof}
Let $u$ and $v$ be elements in $E(M)-P$ contained in distinct $4$-element cocircuits that intersect $P$ in three elements.
If $u = v$, then $P$ contains a cocircuit by the cocircuit elimination axiom, contradicting the fact that $P$ is not $3$-separating.
So $u \neq v$.
\begin{sublemma}
\label{p3}
Let $u,v \in E(M)-P$ be distinct elements in $4$-element cocircuits $C^*_u$ and $C^*_v$, respectively, where $C^*_u \subseteq P \cup u$ and $C^*_v \subseteq P \cup v$.
Then $\si(M/u/v)$ is $3$-connected.
\end{sublemma}
\begin{slproof}
Suppose $(X,Y)$ is a $2$-separation of $M/u/v$ where neither $X$ nor $Y$ is contained in a parallel class.
We may assume that $|X \cap P| \ge 3$ and that $X$ is closed.
Thus $P \subseteq X$.
But $\{u,v\} \subseteq \cl^*(P) \subseteq \cl^*(X)$, so
$(X \cup \{u,v\}, Y)$ is a $2$-separation of $M$; a contradiction.
\end{slproof}
\begin{sublemma}
\label{p5}
Let $C^*_u$ be a $4$-element cocircuit with $u \in C^*_u \subseteq P \cup u$, for $u \in E(M)-P$,
and let $p' \in P-C^*_u$. Then
$\si(M / p' / u)$ is $3$-connected.
\end{sublemma}
\begin{slproof}
Suppose $\si(M/p'/u)$ is not $3$-connected, and let $(X,Y)$ be a $2$-separation in $M/p'/u$ where neither $X$ nor $Y$ is a parallel pair.
We may assume that $|X \cap (P-p')| \ge 2$ and that $X$ is closed.
Since $r_{M/p'}(P-p')=2$, we have $P-p' \subseteq X$.
Since $p' \notin C^*_u$, we have $u \in \cl^*_{M/p'}(P-p')$, and $(X\cup u,Y)$ is a $2$-separation of $M/p'$; a contradiction.
\end{slproof}
\begin{sublemma}
\label{p6}
If \cref{p1}\cref{p1a} holds, then $M$ has an $N$-detachable pair.
\end{sublemma}
\begin{slproof}
Let $u$ and $v$ be elements in $E(M)-P$ such that $\{p_1,p_2,p_3,u\}$ and $\{p_2,p_3,p_4,v\}$ are cocircuits of $M$.
Recall that $M \backslash p_2\backslash p_3$ has an $N$-minor.
Let $(C,D)$ be an $N$-labelling such that $\{p_2,p_3\} \subseteq D$.
Since $\{p_1,u\}$ and $\{p_4,v\}$ are series pairs in $M\backslash p_2 \backslash p_3$, we may assume that $\{u,v\} \subseteq C$.
If $M/u/v$ is $3$-connected, then $\{u,v\}$ is an $N$-detachable pair.
By \cref{p3}, $\si(M/u/v)$ is $3$-connected.
Since $u$ and $v$ are $N$-labelled for contraction, each is not in an $N$-grounded\ triangle.
So we may assume there is a $4$-element circuit~$C_{uv}$ of $M$ containing $\{u,v\}$.
By orthogonality, $C_{uv}$ contains at least one element in $P$.
Let $C_{uv} = \{u,v,p',z\}$ for some $p' \in P$
and $z \in E(M)-\{u,v,p'\}$.
We claim that $z \notin P$.
Let $Z = E(M) - (P \cup \{u,v\})$.
Since $\lambda(P)=3$ and $u,v \in \cl^*(P)$, we have $r(Z) = r(M)-2$.
Suppose $z \in P$.
Then $r(P \cup \{u,v\}) \le 4$, so $(Z, P\cup \{u,v\})$ is a $3$-separation.
Next we show that $(Z, \{d^*\}, (P -d^*) \cup \{u,v\})$ is a vertical $3$-separation. Clearly $d^* \in \cl(P-d^*)$.
If $d^*$ is in a cocircuit containing $v$ and elements of $P-d^*$, then cocircuit elimination with $\{v,p_2,p_3,p_4\}$ implies that $P$ contains a cocircuit; a contradiction.
So $d^* \notin \cl^*((P-d^*) \cup v)$; thus $d^* \in \cl(Z \cup u)$.
But if $d^* \notin \cl(Z)$, then $u \in \cl(Z \cup d^*)$ by the Mac~Lane-Steinitz exchange property, contradicting that $u \in \cl^*(\{p_1,p_2,p_3\})$.
So $d^* \in \cl(Z)$, and
$(Z, \{d^*\}, (P -d^*) \cup \{u,v\})$ is a vertical $3$-separation implying that $M/d^*$ is not $3$-connected, contradicting \ref{planeupgradec1}.
Now $C_{uv} \cap P = \{p'\}$, so $p' \in \{p_2,p_3\}$, by orthogonality.
Since $\{p',z\}$ is a parallel pair in $M/u/v$, by switching the $N$-labels on $p'$ and $z$, we have that $z$ is $N$-labelled for deletion.
In $M/u$, $\{v,p',z\}$ is a triangle, and, since $M \backslash z$ has an $N$-minor, $z$ is not in an $N$-grounded\ triad.
Thus Tutte's Triangle Lemma implies that $M/u \backslash z$ or $M/u \backslash v$ is $3$-connected.
Since $\{u,z\}$ and $\{u,v\}$ are not contained in triads, either $M \backslash z$ or $M \backslash v$ is $3$-connected.
Moreover, the same argument applies with the roles of $u$ and $v$ swapped, implying that either $M \backslash z$ or $M \backslash v$ is $3$-connected.
Thus, if $M \backslash z$ is not $3$-connected, then both $M \backslash u$ and $M \backslash v$ are $3$-connected.
Then,
since $(M \backslash u)|P \cong (M \backslash v)|P \cong U_{3,5}$, it follows from \cref{basicplanelemma} that $M \backslash u \backslash p_1$ and $M \backslash v \backslash p_1$ are $3$-connected.
Thus either $\{u, p_1\}$ or $\{v, p_1\}$ is an $N$-detachable pair, by \cref{planeupgradec4}.
Now we may assume that $M \backslash z$ is $3$-connected.
As $(M \backslash z)|P \cong U_{3,5}$,
if $P$ does not contain a triad of $M \backslash z$, then, by \cref{planelemma}, $M$ has an $N$-detachable pair.
So suppose that $z$ is in a $4$-element cocircuit $C^*_z$ with elements in $P$.
Let $Q=P \cup \{u,v,z\}$.
Observe that $Q$ is $3$-separating, as $r(Q) \le 5$ due to the circuit $\{u,v,p',z\}$, and $r(E(M)-Q) = r(M)-3$, as $r(E(M)-P) = r(M)$ and $\{u,v,z\} \subseteq \cl^*(P)$.
If $d^* \notin C^*_z$, then $$\lambda(Q-d^*) = r(Q-d^*) + r^*(Q-d^*) - |Q-d^*| \le 5+4-7 = 2.$$
It follows, by \cref{gutses}, that $d^*$ is a guts element in the path of $3$-separations $(Q-d^*, \{d^*\}, E(M)-Q)$.
But then $M/d^*$ is not $3$-separating; a contradiction.
So $d^* \in C^*_z$.
Now $T^* = C^*_z-z$ is a triad in $M \backslash z$ with $d^* \in T^*$.
Let $p'' \in P-(T^* \cup p')$.
By \cref{basicplanelemma}, $M \backslash z \backslash p''$ is $3$-connected.
Since $p'' \in P-\{d^*,p'\}$,
$M \backslash z \backslash p''$ has an $N$-minor, so $\{z, p''\}$ is an $N$-detachable pair.
\end{slproof}
\begin{sublemma}
\label{p7}
If \cref{p1}\cref{p1b} holds, then $M$ has an $N$-detachable pair.
\end{sublemma}
\begin{slproof}
If $M \backslash p_1 \backslash p_4$ is $3$-connected, then $M$ has an $N$-detachable pair, so assume otherwise.
Suppose $\{p_1,p_4\}$ is not contained in a $4$-element cocircuit.
Then $\co(M \backslash p_1 \backslash p_4)$ is also not $3$-connected.
Now $M \backslash p_1 \backslash p_4$ has a
$2$-separation $(X,Y)$ where
$|X \cap \{p_2,p_3,d^*\}| \ge 2$ and $X$ is fully closed.
But it follows that $\{p_2,p_3,d^*\} \subseteq X$, and hence $(X \cup \{p_1,p_4\}, Y)$ is a $2$-separation of $M$; a contradiction.
So $\{p_1,p_4\}$ is contained in a $4$-element cocircuit.
If this cocircuit also contains either $p_2$ or $p_3$, then, up to relabelling $\{p_1,p_2,p_3,p_4\}$, we are in case \cref{p1}\cref{p1a}. By \cref{p6}, we may assume otherwise.
So $\{d^*,p_i,p_4,u_i\}$ is a cocircuit for all $i \in \seq{3}$.
Let $i \in \seq{3}$ and $j \in \seq{3}-i$ such that $p_j \neq p$.
Then $M / d^*/p_j$ has an $N$-minor, and, as $P-\{d^*,p_j\}$ is contained in a parallel class in this matroid, $M /p_j \backslash p_i \backslash p_4$ also has an $N$-minor.
Since $\{d^*,u_i\}$ is a series pair in $M /p_j \backslash p_i \backslash p_4$, it follows that $\{p_j,u_i\}$ is $N$-contractible in $M$.
Now, by \cref{p5}, either $\{p_j,u_i\}$ is an $N$-detachable pair, or $\{p_j,u_i\}$ is contained in a $4$-element circuit $C_{i,j}$.
By orthogonality, $C_{i,j}$ meets $\{d^*,p_i,p_4\}$.
If $C_{i,j} \subseteq P \cup u_i$, then $u_i \in \cl(P)$.
Then $M|(P \cup u_i) \cong U_{3,6}$, and $M$ has an $N$-detachable pair by \cref{6pointplane,basicplanelemma}.
So let $C_{i,j} = \{p_j,u_i,q_{i,j},v_{i,j}\}$ where $q_{i,j} \in \{d^*,p_i,p_4\}$ and $v_{i,j} \in E(M)-(P \cup u_i)$ (for ease of notation, $q_{i,j} = q_{j,i}$ and $v_{i,j} = v_{j,i}$).
If $v_{i,j} = u$, then by letting $j' \in \seq{3}-\{i,j\}$,
orthogonality between $C_{i,j}$ and the cocircuit $\{d^*,p_{j'},p_4,u_{j'}\}$ implies that $q_{i,j} = p_i$, whereas orthogonality between $C_{i,j}$ and the cocircuit $\{d^*,p_j,p_4,u_j\}$ implies that $q_{i,j} \neq p_i$.
So $v_{i,j} \neq u$.
Observe that $p_j$ is a member of the cocircuit $\{p_1,p_2,p_3,u\}$, and recall that $u \neq u_i$.
Then, by orthogonality, $q_{i,j} = p_i$, so
$\{p_i,p_j,u_i,v_{i,j}\}$ is a circuit.
If $v_{i,j} \neq u_j$, then
$\{p_j,p_4,d^*,u_j\}$ is a cocircuit that intersects this circuit in one element;
a contradiction.
Without loss of generality we may now assume that $\{p_1,p_2,u_1,u_2\}$ is a circuit.
It follows that $(P \cup \{u_1, u_2\}, E(M)-(P \cup \{u_1,u_2\}))$ is $3$-separating in $M$.
Since $M / d^* \backslash p_1 \backslash p_2$ has an $N$-minor, and $\{p_3,u\}$ is a series pair in this matroid, $M/d^*/u$ has an $N$-minor up to an $N$-label switch.
Suppose $M/d^*/u$ is not $3$-connected.
If $\{d^*,u\}$ is contained in a $4$-element cocircuit, then, by orthogonality, this cocircuit is contained in $P \cup u$. It follows, by cocircuit elimination with $\{p_1,p_2,p_3,u\}$, that $P$ contains a cocircuit; a contradiction.
So $M/d^*/u$ has a $2$-separation $(U,V)$ for which we may assume $|P \cap U| \ge 2$ and $U$ is fully closed.
It follows that $P-d^* \subseteq U$, and hence $(U \cup u,V)$ is a $2$-separation of $M/d^*$. But $M/d^*$ is $3$-connected, so this is contradictory.
Hence $\{d^*,u\}$ is an $N$-detachable pair.
\end{slproof}
\begin{sublemma}
\label{p8}
If \cref{p1}\cref{p1c} holds, then $M$ has an $N$-detachable pair.
\end{sublemma}
\begin{slproof}
Consider $M \backslash p_1 \backslash p_2$.
If this matroid is $3$-connected, then $M$ has an $N$-detachable pair, so assume otherwise.
If $\co(M \backslash p_1 \backslash p_2)$ is not $3$-connected, then there is a $2$-separation $(X,Y)$ of $M \backslash p_1 \backslash p_2$ for which $|X \cap \{p_3,p_4,d^*\}| \ge 2$ and $X$ is fully closed.
It follows that
$(X \cup \{p_1,p_2\},Y)$ is a $2$-separation of $M$; a contradiction.
So $\{p_1,p_2\}$ is contained in a $4$-element cocircuit of $M$.
If this cocircuit also contains $p_3$ or $p_4$, then, up to relabelling, we are in case \cref{p1}\cref{p1b}. Hence, by \cref{p7}, we may assume $\{d^*,p_1,p_2\}$ is contained in a $4$-element cocircuit of $M$.
Now, for all distinct $i,j \in \seq{4}$, we have that $\{d^*,p_i,p_j,u_{i,j}\}$ is a cocircuit for some $u_{i,j} \in E(M)-P$,
where $u_{i,j} \neq u_{i',j'}$ if $i \neq i'$ or $j \neq j'$.
(For ease of notation, we let $u_{i,j} = u_{j,i}$.)
We may assume that $p = p_4$.
Then, for all $i \in \seq{3}$
we have that $M/d^*/p_i$ has an $N$-minor.
For all distinct $j,j' \in \seq{4}-i$,
since $P-\{d^*,p_i\}$ is contained in a parallel class in $M/d^*/p_i$,
the matroid $M/p_i\backslash p_j \backslash p_{j'}$ has an $N$-minor,
and it follows that $M/p_i/u_{j,j'}$ has an $N$-minor.
By \cref{p5}, if $\{p_i,u_{j,j'}\}$ is not contained in a $4$-element circuit,
then $M$ has an $N$-detachable pair.
So we may assume that $\{p_i,u_{j,j'}\}$ is contained in a $4$-element circuit,
for all $i \in \seq{3}$ and distinct $j,j' \in \seq{4}-i$.
Let $\{i,j,j',\ell\} = \seq{4}$ with $i \in \seq{3}$.
Consider the $4$-element circuit containing $\{p_i,u_{j,j'}\}$.
By orthogonality, this circuit meets $\{p_s,u_{i,s},d^*\}$ for each $s \in \seq{4}-i$.
Hence,
as the
$p_s$'s and $u_{i,s}$'s are distinct for $s \in \seq{4}-i$,
the circuit contains $d^*$.
That is, $\{d^*,p_i,u_{j,j'},v_{i,\ell}\}$ is a circuit for some $v_{i,\ell}$.
Since $M/d^*/p_i$ has an $N$-minor,
it follows that $\{p_t,u_{j,j'}\}$ is $N$-deletable for $t \in \seq{4}-i$.
If $v_{i,\ell} \in P$, then $u_{j,j'} \in \cl(P)$, so $M|(P \cup u_{j,j'}) \cong U_{3,6}$, and it follows from \cref{6pointplane} that $M$ has an $N$-detachable pair.
So we may assume that $v_{i,\ell} \in E(M)-P$.
Now, if $M \backslash u_{j,j'}$ is $3$-connected, then, as $(M \backslash u_{j,j'})|P \cong U_{3,5}$, and $\{d^*,p_j,p_{j'}\}$ is a triad in $M \backslash u_{j,j'}$, it follows, by \cref{basicplanelemma}, that
$\{p_\ell, u_{j,j'}\}$ is an $N$-detachable pair.
So we may assume that $M \backslash u_{j,j'}$ is not $3$-connected.
Again, we let $\{i,j,j',\ell\} = \seq{4}$ with $i \in \seq{3}$.
Consider $M/d^*$. Recall that this matroid is $3$-connected, and
observe that $\{p_i,u_{j,j'},v_{i,\ell}\}$ is a triangle, where $v_{i,\ell} \in E(M)-P$.
Suppose that $\{p_i,u_{j,j'},v_{i,\ell}\}$ is part of a $4$-element fan. Then there is a triad of $M$ that contains two elements of $\{p_i,u_{j,j'},v_{i,\ell}\}$.
But as $p_i$ and $u_{j,j'}$ are $N$-deletable, neither is contained in an $N$-grounded\ triad, so this is contradictory.
Now, by Tutte's Triangle Lemma, either $M/d^* \backslash u_{j,j'}$ or $M/d^* \backslash v_{i,\ell}$ is $3$-connected.
If $M/d^* \backslash u_{j,j'}$ is $3$-connected, then, as $d^*$ is not in a triangle since it is $N$-contractible, $M \backslash u_{j,j'}$ is $3$-connected; a contradiction.
So $M / d^* \backslash v_{i,\ell}$ is $3$-connected, and hence $M \backslash v_{i,\ell}$ is $3$-connected.
Observe that, for $i \in \seq{3}$ and $s \in \seq{4}-i$, the matroid $M \backslash p_s \backslash v_{i,\ell}$ has an $N$-minor, since $M/d^*/p_i\backslash p_s$ has an $N$-minor and $\{u_{j,j'},v_{i,\ell}\}$ is a parallel pair in this matroid.
As $(M \backslash v_{i,\ell})|P \cong U_{3,5}$,
if $P$ does not contain a triad of $M \backslash v_{i,\ell}$, then, by \cref{planelemma}, $M$ has an $N$-detachable pair.
So suppose $v_{i,\ell}$ is in a $4$-element cocircuit $C^*$ of $M$, where $C^* \subseteq P \cup v_{i,\ell}$.
Then, by orthogonality, $C^*$ contains $d^*$ or $p_i$.
Thus, there exists some $s \in \seq{4}-i$ such that $p_s \notin C^*$.
By \cref{basicplanelemma}, $M \backslash v_{i,\ell} \backslash p_s$ is $3$-connected,
so $\{v_{i,\ell},p_s\}$ is an $N$-detachable pair.
This completes the proof of \cref{p8}.
\end{slproof}
The \lcnamecref{planeupgrade} now follows from \cref{p1} and \cref{p6,p7,p8}.
\end{proof}
\section{Particular $3$-separator s}
\label{sec-problematic}
Throughout this series of papers, we will build up a collection of particular $3$-separators. Any such particular $3$-separator~$P$ will have the property that it can appear in a $3$-connected matroid~$M$, with a $3$-connected minor $N$, and with no $N$-detachable pairs, where
$E(M) - E(N) \subseteq P$.
Recall that the first example we have seen is a spike-like $3$-separator\ (see \cref{def-spike-like}).
In this section, we define three more particular $3$-separator s,
and, for each, describe the construction of a matroid containing the particular $3$-separator, and with no $N$-detachable pairs.
These particular $3$-separator s are illustrated in \cref{sfpspid,psfig2}.
Note that this is not a complete list of all such $3$-separators that can give rise to a matroid without an $N$-detachable pair.
Here, we first just consider those particular $3$-separator s that are either a single-element extension, or the dual of a single-element coextension, of a structure known as a flan, which we consider in the next section.
Let $M$ be a $3$-connected matroid with ground set~$E$.
\begin{definition}
\label{def-pspider}
Let $P \subseteq E$ be a $6$-element exactly $3$-separating set such that $P = Q \cup \{p_1,p_2\}$ and $Q$ is a quad. If there exists a labelling $\{q_1,q_2,q_3,q_4\}$ of $Q$ such that
\begin{enumerate}[label=\rm(\alph*)]
\item $\{p_1,p_2,q_1,q_2\}$ and $\{p_1,p_2,q_3,q_4\}$ are the circuits of $M$ contained in $P$, and
\item $\{p_1,p_2,q_1,q_3\}$ and $\{p_1,p_2,q_2,q_4\}$ are the cocircuits of $M$ contained in $P$,
\end{enumerate}
then $P$ is an \emph{elongated-quad $3$-separator} of $M$ with {\em associated partition} $(Q,\{p_1,p_2\})$.
\end{definition}
\begin{definition}
\label{def-twisted}
Let $P \subseteq E$ be a $6$-element exactly 3-separating set of $M$. If there exists a labelling $\{s_1,s_2,t_1,t_2,u_1,u_2\}$ of $P$ such that
\begin{enumerate}[label=\rm(\alph*)]
\item $\{s_1,s_2,t_2,u_1\}$, $\{s_1,t_1,t_2,u_2\}$, and $\{s_2,t_1,u_1,u_2\}$ are the circuits of $M$ contained in $P$; and
\item $\{s_1,s_2,t_1,t_2\}$, $\{s_1,s_2,u_1,u_2\}$, and $\{t_1,t_2,u_1,u_2\}$ are the cocircuits of $M$ contained in $P$;
\end{enumerate}
then $P$ is a \emph{skew-whiff $3$-separator} of $M$.
\end{definition}
\begin{figure}[hbtp]
\begin{subfigure}{0.45\textwidth}
\centering
\begin{tikzpicture}[rotate=90,scale=0.875,line width=1pt]
\tikzset{VertexStyle/.append style = {minimum height=5,minimum width=5}}
\clip (-2.5,2) rectangle (3.0,-6);
\node at (-1,-1.4) {$E-P$};
\draw (0,0) .. controls (-3,2) and (-3.5,-2) .. (0,-4);
\draw (0,0) -- (2,-2) -- (0,-4);
\draw (0,0) -- (2.5,0.5) -- (2,-2);
\draw (0,0) -- (2.25,-0.75);
\draw (2,-2) -- (1.25,0.25);
\Vertex[x=1.25,y=0.25,LabelOut=true,L=$q_3$,Lpos=180]{c1}
\Vertex[x=2.25,y=-0.75,LabelOut=true,L=$q_2$,Lpos=90]{c2}
\Vertex[x=2.5,y=0.5,LabelOut=true,L=$q_1$,Lpos=180]{c3}
\Vertex[x=1.5,y=-0.5,LabelOut=true,L=$q_4$,Lpos=135]{c4}
\Vertex[x=1.33,y=-2.67,LabelOut=true,L=$p_1$,Lpos=45]{c5}
\Vertex[x=0.67,y=-3.33,LabelOut=true,L=$p_2$,Lpos=45]{c6}
\draw (0,0) -- (0,-4);
\SetVertexNoLabel
\tikzset{VertexStyle/.append style = {shape=rectangle,fill=white}}
\end{tikzpicture}
\caption{An elongated-quad $3$-separator.}
\label{sfpspida}
\end{subfigure}
\begin{subfigure}{0.45\textwidth}
\centering
\begin{tikzpicture}[rotate=90,scale=0.7,line width=1pt]
\tikzset{VertexStyle/.append style = {minimum height=5,minimum width=5}}
\clip (-2.5,-6) rectangle (4.4,2);
\node at (-1,-1.4) {$E-P$};
\draw (0,0) .. controls (-3,2) and (-3.5,-2) .. (0,-4);
\draw (0,0) -- (4.0,0.9);
\draw (0,-2) -- (2.5,-2.2);
\draw (0,-4) -- (3.8,-4.9);
\Vertex[x=3.0,y=0.67,LabelOut=true,Lpos=180,L=$s_1$]{a2}
\Vertex[x=2.0,y=0.45,LabelOut=true,Lpos=180,L=$s_2$]{a3}
\Vertex[x=2.5,y=-2.2,LabelOut=true,Lpos=90,L=$t_2$]{b1}
\Vertex[x=0.64,y=-2.056,LabelOut=true,Lpos=-45,L=$t_1$]{b2}
\Vertex[x=3.8,y=-4.9,LabelOut=true,L=$u_1$]{c1}
\Vertex[x=2.8,y=-4.67,LabelOut=true,L=$u_2$]{c2}
\draw[dashed] (3.8,-4.9) .. controls (2.0,-2) .. (4.0,0.9);
\draw[dashed] (2.8,-4.67) .. controls (1.0,-2) .. (3.0,0.67);
\draw[dashed] (1.8,-4.45) .. controls (0.25,-2) .. (2.0,0.45);
\draw (0,0) -- (0,-4);
\SetVertexNoLabel
\tikzset{VertexStyle/.append style = {shape=rectangle,fill=white}}
\Vertex[x=4.0,y=0.9]{a1}
\Vertex[x=1.5,y=-2.12]{b3}
\Vertex[x=1.8,y=-4.45]{c3}
\end{tikzpicture}
\caption{A skew-whiff $3$-separator.}
\end{subfigure}
\caption{Two particular $3$-separator s. Each is a single-element extension of a $5$-element flan, and is in a matroid with rank $r(E-P)+2$.
\label{sfpspid}
\end{figure}
\begin{definition}
Let $P \subseteq E$ be an exactly $3$-separating set with $P=\{p_1,p_2,q_1,q_2,s_1,s_2\}$.
Suppose that
\begin{enumerate}[label=\rm(\alph*)]
\item $\{p_1,p_2,s_1,s_2\}$, $\{q_1,q_2,s_1,s_2\}$, and $\{p_1,p_2,q_1,q_2\}$ are the circuits of $M$ contained in $P$; and
\item $\{p_1,q_1,s_1,s_2\}$, $\{p_2,q_2,s_1,s_2\}$,
$\{p_1,p_2,q_1,q_2,s_1\}$ and $\{p_1,p_2,q_1,q_2,s_2\}$
are the cocircuits of $M$ contained in $P$.
\end{enumerate}
Then $P$ is a \emph{twisted cube-like $3$-separator} of $M$.
\end{definition}
\begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth}
\centering
\begin{tikzpicture}[rotate=90,xscale=1.21,yscale=0.605,line width=1pt]
\tikzset{VertexStyle/.append style = {minimum height=5,minimum width=5}}
\clip (-1.5,2) rectangle (2.7,-6);
\node at (-0.6,-1.4) {$E-P$};
\draw (0,0) .. controls (-1.6,2) and (-2,-2) .. (0,-4);
\draw (1.2,1) -- (0.8,-1);
\draw (0,-4) -- (1.2,-3);
\draw (1.2,1) -- (2.2,-1) -- (1.2,-3);
\draw (2.2,-1) -- (2.0,-3);
\draw (1.2,1) -- (1.2,-3);
\draw[white,line width=5pt] (0.8,-1) -- (2.0,-3) -- (0.8,-5);
\draw[white,line width=5pt] (0.8,-1) -- (0.8,-5);
\draw[white,line width=5pt] (0.8,-1) -- (0,0);
\draw (0.8,-1) -- (2.0,-3) -- (0.8,-5);
\draw (0.8,-1) -- (0.8,-5);
\draw (1.2,-3) -- (0.8,-5) -- (0,-4);
\draw (0.8,-1) -- (0,0);
\draw (0,0) -- (0,-4);
\draw (0,0) -- (1.2,1);
\Vertex[x=1.2,y=1,LabelOut=true,L=$q_2$,Lpos=180]{c1}
\Vertex[x=1.2,y=-3,LabelOut=true,L=$p_2$,Lpos=225]{c4}
\Vertex[x=0.8,y=-1,LabelOut=true,L=$q_1$,Lpos=-45]{c5}
\Vertex[x=0.8,y=-5,LabelOut=true,L=$p_1$,Lpos=0]{c6}
\Vertex[x=2.0,y=-3,LabelOut=true,L=$s_1$,Lpos=45]{c5}
\Vertex[x=2.2,y=-1,LabelOut=true,L=$s_2$,Lpos=45]{c6}
\end{tikzpicture}
\caption{A twisted cube-like $3$-separator\ of $M$.}
\end{subfigure}
\begin{subfigure}{0.50\textwidth}
\centering
\begin{tikzpicture}[rotate=90,scale=0.65,line width=1pt]
\tikzset{VertexStyle/.append style = {minimum height=5,minimum width=5}}
\clip (-2.78,-6) rectangle (5.0,2);
\node at (-1,-1.4) {$E-P$};
\draw (0,0) .. controls (-3,2) and (-3.5,-2) .. (0,-4);
\draw (0,0) -- (4.0,0.9);
\draw (0,-2) -- (2.5,-2.2);
\draw (0,-4) -- (3.8,-4.9);
\Vertex[x=2.0,y=0.45,LabelOut=true,Lpos=180,L=$p_2$]{a3}
\Vertex[x=2.5,y=-2.2,LabelOut=true,Lpos=90,L=$q_1$]{b1}
\Vertex[x=0.61,y=-2.05,LabelOut=true,Lpos=35,L=$q_2$]{b2}
\Vertex[x=3.00,y=-4.7,LabelOut=true,L=$s_1$]{c1}
\Vertex[x=2.15,y=-4.5,LabelOut=true,L=$s_2$]{c2}
\Vertex[x=4.0,y=0.9,LabelOut=true,Lpos=180,L=$p_1$]{a1}
\draw[dashed] (3.8,-4.9) .. controls (2.0,-2) .. (4.0,0.9);
\draw[dashed] (1.3,-4.3) .. controls (0.25,-2) .. (2.0,0.45);
\draw (0,0) -- (0,-4);
\SetVertexNoLabel
\tikzset{VertexStyle/.append style = {shape=rectangle,fill=white}}
\Vertex[x=3.8,y=-4.9]{c1}
\Vertex[x=1.3,y=-4.3]{c3}
\end{tikzpicture}
\caption{A twisted cube-like $3$-separator\ of $M^*$.}
\end{subfigure}
\caption{Geometric representations of a twisted cube-like $3$-separator\
of $M$ and $M^*$.}
\label{psfig2}
\end{figure}
For a $3$-connected matroid $M$, we say that a pair $\{x_1,x_2\} \subseteq E(M)$ is \emph{detachable} if either $M/x_1/x_2$ or $M \backslash x_1 \backslash x_2$ is $3$-connected.
In the former case, we will say that $\{x_1,x_2\}$ is a \emph{contraction pair}; in the latter case, $\{x_1,x_2\}$ is a \emph{deletion pair}.
Each particular $3$-separator~$P$ that we have seen in this section can be used to construct a $3$-connected matroid~$M$ with a $3$-connected matroid~$N$ as a minor, such that $M$ has no $N$-detachable pairs, and $E(M)-E(N) \subseteq P$.
For the elongated-quad $3$-separator\ and skew-whiff $3$-separator, this follows from the fact that for such a $3$-separator $P$, there is no detachable pair contained in $P$.
On the other hand, the twisted cube-like $3$-separator\ can contain a detachable pair, but appear in a matroid with no $N$-detachable pairs.
We first consider the elongated-quad $3$-separator.
Let $M$ be a $3$-connected matroid with a $3$-separation $(P,S)$ such that $P$ is an elongated-quad $3$-separator, the matroid $N = M \backslash P$ is $3$-connected, $\cl(P)$ does not contain any triangles, and $\cl^*(P)$ does not contain any triads.
Provided $N$ is sufficiently structured to ensure that $M/s$ and $M \backslash s$ have no $N$-minor for any $s \in S$, the matroid $M$ has no $N$-detachable pairs, even after first performing a $\Delta$-$Y$ or $Y$-$\Delta$ exchange.
Note that in this example $|E(M)|-|E(N)|=6$.
It is also possible that $|E(M)|-|E(N)| = 5$.
In this case, up to duality, the $3$-connected $N$-minor can be obtained by extending $M$ by an element $e$ in the guts of $(P,S)$, then restricting to $S \cup e$.
Different cases arise depending on where in the guts needs to be ``filled in'' in order to obtain $N$.
If $P$ is labelled as in \cref{sfpspida},
then $e$ is in either
\begin{itemize}
\item $\cl(\{q_1,q_3\}) \cap \cl(\{q_2,q_4\}) \cap \cl(S)$,
\item $\cl(\{p_1,p_2\}) \cap \cl(S)$, or
\item $\cl(S) - (\cl(\{q_1,q_3\}) \cup \cl(\{q_2,q_4\}) \cup \cl(\{p_1,p_2\}))$.
\end{itemize}
Here we have just focussed on cases where $|E(M)| - |E(N)| \ge 5$, though it is also possible that $|E(M)| - |E(N)| \in \{3,4\}$.
Similarly, a skew-whiff $3$-separator\ can appear in a $3$-connected matroid $M$ with a $3$-connected minor $N$ such that $E(M)-E(N) \subseteq P$ and $M$ has no $N$-detachable pairs, where $|E(M)|-|E(N)| \in \{3,4,5,6\}$.
\begin{figure}[htb]
\centering
\begin{tikzpicture}[rotate=90,xscale=1.331,yscale=0.6655,line width=1pt]
\tikzset{VertexStyle/.append style = {minimum height=5,minimum width=5}}
\clip (-1.5,2) rectangle (2.7,-6);
\draw (0,0) -- (-1.3,-2) -- (0,-4);
\draw (-0.65,-1) -- (0,-4);
\draw (0,0) -- (-0.65,-3);
\draw (0,-2) -- (-1.3,-2);
\draw (1.2,1) -- (0.8,-1);
\draw (0,-4) -- (1.2,-3);
\draw (1.2,1) -- (2.2,-1) -- (1.2,-3);
\draw (2.2,-1) -- (2.0,-3);
\draw (1.2,1) -- (1.2,-3);
\draw[dotted] (1.2,1) .. controls (-0.4,-0.3333) and (-0.4,-4.3333) .. (1.2,-3);
\draw[white,line width=5pt] (0.8,-1) -- (2.0,-3) -- (0.8,-5);
\draw[white,line width=5pt] (0.8,-1) -- (0.8,-5);
\draw[white,line width=5pt] (0.8,-1) -- (0,0);
\draw[white,line width=5pt] (0.8,-1) .. controls (-0.2666,0.2222) and (-0.2666,-3.7777) .. (0.8,-5);
\draw (0.8,-1) -- (2.0,-3) -- (0.8,-5);
\draw (0.8,-1) -- (0.8,-5);
\draw (1.2,-3) -- (0.8,-5) -- (0,-4);
\draw (0.8,-1) -- (0,0);
\draw (0,0) -- (0,-4);
\draw (0,0) -- (1.2,1);
\draw[dashed] (0.8,-1) .. controls (-0.2666,0.2222) and (-0.2666,-3.7777) .. (0.8,-5);
\Vertex[x=1.2,y=1,LabelOut=true,L=$q_2$,Lpos=180]{c1}
\Vertex[x=1.2,y=-3,LabelOut=true,L=$p_2$,Lpos=225]{c4}
\Vertex[x=0.8,y=-1,LabelOut=true,L=$q_1$,Lpos=-45]{c5}
\Vertex[x=0.8,y=-5,LabelOut=true,L=$p_1$,Lpos=0]{c6}
\Vertex[x=2.0,y=-3,LabelOut=true,L=$s_1$,Lpos=45]{c2}
\Vertex[x=2.2,y=-1,LabelOut=true,L=$s_2$,Lpos=45]{c3}
\Vertex[x=0,y=0,LabelOut=true,L=$t_1$,Lpos=180]{c7}
\Vertex[x=0,y=-4,LabelOut=true,L=$t_2$,Lpos=0]{c8}
\SetVertexNoLabel
\Vertex[x=-1.3,y=-2,LabelOut=true,L=$s_2$,Lpos=45]{c9}
\Vertex[x=-.65,y=-1,LabelOut=true,L=$s_2$,Lpos=45]{c10}
\Vertex[x=-.65,y=-3,LabelOut=true,L=$s_2$,Lpos=45]{c11}
\Vertex[x=-.45,y=-2,LabelOut=true,L=$s_2$,Lpos=45]{c12}
\SetVertexNoLabel
\tikzset{VertexStyle/.append style = {fill=white}}
\tikzset{VertexStyle/.append style = {shape=rectangle}}
\Vertex[x=0,y=-2]{v2}
\end{tikzpicture}
\caption{A matroid $M$ with a twisted cube-like $3$-separator\ and no $F_7^-$-detachable pairs.}
\label{tvamoseg}
\end{figure}
Finally, we consider the twisted cube-like $3$-separator.
We describe a matroid $M$ with a twisted cube-like $3$-separator\ and with no $F_7^-$-detachable pairs; this matroid is illustrated in \cref{tvamoseg}.
Let $U_8$ be the paving matroid on ground set $\{p_1,p_2,q_1,q_2,s_1,s_2,t_1,t_2\}$ whose non-spanning circuits are $\{t_1,t_2,p_1,q_1\}$, $\{t_1,t_2,p_2,q_2\}$, $\{p_1,p_2,q_1,q_2\}$, $\{p_1,p_2,s_1,s_2\}$, and $\{q_1,q_2,s_1,s_2\}$.
Let $U_8^+$ be the
single-element extension of $U_8$ by the element~$z$ such that $z$ is in the span of the lines $\{t_1,t_2\}$, $\{q_1,p_1\}$, and $\{q_2,p_2\}$ and $z$ is not a loop.
Label the triangle $T=\{t_1,t_2,z\}$.
Let $F_7^-$ be a copy of the non-Fano matroid with $E(F_7^-) \cap E(U_8^+) = T$ such that $T$ is a triangle of $F_7^-$.
Now let $M=P_{T}(U_8^+,F_7^-) \backslash z$, and observe that $M$ is $3$-connected and has an $F_7^-$-minor.
In particular, $F_7^- \cong M/p_1 \backslash \{s_1,s_2,p_2,q_2\}$, for example, so $|E(M)|-|E(F_7^-)| = 5$.
Let $X = \{p_1,p_2,q_1,q_2,s_1,s_2\}$; the set $X$ is a twisted cube-like $3$-separator\ of $M$.
The matroid $M$ has no $F_7^-$-detachable pairs.
To see this, first observe that neither $M\backslash y$ nor $M/y$ has an $F_7^-$-minor for any $y \in E(M)-X$.
Due to the $4$-element circuits and cocircuits contained in $X$, the only detachable pairs of $M$ contained in $X$ are the deletion pairs
$\{p,q\} \in \{\{p_1,q_2\}, \{p_2,q_1\} \}$.
But for any such $\{p,q\}$, the matroid $M \backslash p \backslash q$ has no $F_7^-$-minor.
Note that although here we have used $N=F_7^-$ as the minor, other choices of $N$ would work provided $N$ is sufficiently structured and has a triangle $T=\{t_1,t_2,z\}$.
\section{Flans}
\label{secflans}
Let $F$ be a set of elements in a $3$-connected matroid $M$, with $t=|F| \ge 4$.
Suppose $F$ has an ordering $(f_1,f_2,\dotsc,f_t)$ such that
\begin{enumerate}[label=\rm(\alph*)]
\item if $i \in \seq{t-2}$ is odd, then $\{f_i,f_{i+1},f_{i+2}\}$ is a triad; and
\item if $i \in \{4,5,\dotsc,t\}$ is even, then $f_i\in\cl(\{f_1,f_2,\ldots,f_{i-1}\})$.
\end{enumerate}
Then $F$ is a {\em flan} of $M$,
and $(f_1,f_2,\dotsc,f_t)$ is a \emph{flan ordering} (or just an \emph{ordering}) of $F$.
A flan $F$ is {\em maximal} if
it is not properly contained in another flan.
Note that $\{f_1,f_2,\dotsc,f_i\}$ is $3$-separating for any $i \in \seq{t}$.
In this section we consider the case where there is an $N$-deletable element $d \in E(M)$ such that $M \backslash d$ is $3$-connected, but $M \backslash d$ has a flan~$F$ with at least five elements.
In this case, we show that either $M$ has an $N$-detachable pair, or
$F \cup d$ is one of the $3$-separators defined in \cref{sec-problematic}.
We focus on the case where the flan has at least five elements, because
if $M \backslash d$ has a maximal $4$-element flan with ordering $(f_1,f_2,f_3,f_4)$, then $M \backslash d \backslash f_4$ is $3$-connected.
Note that a flan generalises the notion of a fan.
Note also that the definition of a flan used here is more restrictive than that often appearing in the literature (see \cite{Hall2005,Hall2007}, for example).
A geometric illustration of an example of a flan is given in \cref{figflan}.
\begin{figure}[htbp]
\centering
\begin{tikzpicture}[scale=0.8]
\draw[line width=1pt] (0,0) -- (2,3.464) -- (0,4);
\draw[line width=1pt] (0,0) .. controls (4,2) and (6.5,5.464) .. (2,3.464);
\SetVertexNoLabel
\tikzset{VertexStyle/.append style = {shape=rectangle,draw,minimum height=5,minimum width=5}}
\tikzset{VertexStyle/.append style = {fill=white}}
\Vertex[x=0,y=0,L=$c$]{c}
\tikzset{VertexStyle/.append style = {shape=circle,fill=black}}
\node at (3.05,3.25) {\small$E(M)-F$};
\SetVertexLabel
\Vertex[x=-2,y=3.464,LabelOut=true,Lpos=90,L=$f_4$]{y1}
\Vertex[x=1,y=1.732,LabelOut=true,Lpos=180,L=$f_8$]{y1p}
\Vertex[x=0,y=4,LabelOut=true,Lpos=90,L=$f_6$]{g2}
\Vertex[x=-3.464,y=2,LabelOut=true,Lpos=180,L=$f_1$]{x2}
\Vertex[x=-2.598,y=1.5,LabelOut=true,Lpos=-90,L=$f_2$]{b0}
\Vertex[x=-.75,y=2.8,LabelOut=true,Lpos=90,L=$f_5$]{yg}
\Vertex[x=-2.05,y=2.05,LabelOut=true,Lpos=90,L=$f_3$]{xy}
\Vertex[x=.75,y=2.8,LabelOut=true,Lpos=90,L=$f_7$]{ygp}
\Edge(x2)(y1)
\Edge(g2)(y1)
\Edge(c)(g2)
\Edge(c)(b0)
\Edge(x2)(b0)
\Edge(c)(y1)
\Edge(c)(y1p)
\end{tikzpicture}
\caption{An example of a flan $F$ with ordering $(f_1,f_2,\dotsc,f_8)$. Note that $r(M) = r(E(M)-F) + 3$.}
\label{figflan}
\end{figure}
We start with a lemma that demonstrates that certain elements in a flan are candidates for contraction.
\begin{lemma}
\label{flanend}
Let $F$ be a maximal flan in a $3$-connected matroid~$M$, with $|F| \ge 5$ and $F \neq E(M)$.
Let $i \in \seq{3}$,
let $j$ be an odd integer such that $5 \le j \le |F|$, and
suppose $F$ has an ordering $(f_1,f_2,f_3,\dotsc,f_{|F|})$
such that $f_i$ and $f_j$ are not contained in triangles.
Then
\begin{enumerate}
\item $M / f_i$, $M / f_j$ and $\si(M/f_i/f_j)$ are $3$-connected;
\item if $j \ge 7$, then $M / f_i / f_j$ is $3$-connected; and
\item if $|F|=5$, then $M / f_i / f_j$ is $3$-connected.
\end{enumerate}
\end{lemma}
\begin{proof}
Let $F' = \{f_1,f_2,\dotsc,f_j\}$.
Suppose that $(F'-f_j, \{f_j\}, E(M)-F')$ is a cyclic $3$-separation of $M$.
Then $\si(M/f_j)$ is $3$-connected by Bixby's Lemma.
Since
$f_j$ is not contained in a triangle,
$M/f_j$ is $3$-connected.
On the other hand, if $(F'-f_j, \{f_j\}, E(M)-F')$ is not a cyclic $3$-separation of $M$,
then $(E(M)-F')\cup f_j$ is independent.
Then $j = |F|$ and $F' = F$, otherwise $f_{j+1}$ is in a circuit contained in $E(M)-F'$.
If $|E(M)-F| < 3$, then, as $M$ is $3$-connected, $F$ spans $M$, contradicting the maximality of $F$.
It follows that $(E(M)-F) \cup f_j$ is a cosegment consisting of at least four elements, so $M/f_j$ is $3$-connected by the dual of \cref{rank2Remove}.
Now, by the dual of \cref{r3cocirc}, the matroids $\si(M /f_i)$ and $\si(M /f_i/f_j)$ are $3$-connected.
But $f_i$ is not in a triangle,
so $M / f_i$ is $3$-connected.
This proves (i).
Now suppose $\{f_i,f_j\}$ is contained in a $4$-element circuit $C$.
By orthogonality, $C$ must contain an element $f_i' \in \{f_1,f_2,f_3\}-f_i$ and an element $f_j' \in \{f_{j-2},f_{j-1}\}$. If the elements $\{f_i,f_i',f_j,f_j'\}$ are distinct, then $f_j \in \cl(F'-f_j)$; a contradiction.
It follows that if $j > 5$, then
$\{f_i,f_j\}$ is not contained in a $4$-element circuit, and thus $M/f_i/f_j$ is $3$-connected.
This proves (ii).
Now we may assume that $j=5$,
and $C \cap \{f_1,f_2,f_3,f_4,f_5\} = \{f_\ell,f_3,f_5\}$ for some $\ell \in \{1,2\}$.
Let $C - \{f_\ell,f_3,f_5\} = \{x\}$.
Then $\{f_1,f_2,f_3,f_4,f_5,x\}$ is a flan. Thus, if $|F|=5$,
then
$\{f_i,f_j\}$ is not contained in a $4$-element circuit, and thus $M/f_i/f_j$ is $3$-connected. This proves (iii).
\end{proof}
The next \lcnamecref{flannew} deals with the case where the flan
has at least six elements.
\begin{proposition}
\label{flannew}
Let $M$ be a $3$-connected matroid, and let $N$ be a $3$-connected minor of $M$, where $|E(N)| \ge 4$ and every triangle or triad of $M$ is $N$-grounded.
Suppose that $M \backslash d$ is $3$-connected
and has a maximal flan~$F$ with $|F| \ge 6$ and ordering $(f_1,f_2,f_3,\dotsc,f_{|F|})$,
where $M \backslash d \backslash f_5$ has an $N$-minor
with $|\{f_1,\dotsc,f_4\} \cap E(N)| \le 1$.
Then $M$ has an $N$-detachable pair.
\end{proposition}
\begin{proof}
Let $t = |F|$.
Observe that $(\{f_1,f_2,f_3,f_4\}, E(M \backslash d) - \{f_1,\dotsc,f_5\})$ is a $2$-separation of $M \backslash d \backslash f_5$.
\begin{sublemma}
\label{flannewnsl}
$M \backslash d/ f_i / f_j$ has an $N$-minor for $i \in \{1,2\}$ and
$j \in \{5,7\} \cap \seq{t}$.
\end{sublemma}
\begin{slproof}
First consider $j=5$.
Since $\{f_3,f_4\}$ is a series pair in $M \backslash d \backslash f_5$, we have that $M \backslash d \backslash f_5 / f_4$ is connected.
Thus
\cref{m2.7} implies that
$M \backslash d\backslash f_5 /f_4$, and hence $M\backslash d/f_4$
has an $N$-minor.
By further applications of \cref{m2.7}, we obtain that $M \backslash d / f_4 \backslash f_3$ has an $N$-minor, and that $M \backslash d / f_4 \backslash f_3 / f_i$ has an $N$-minor for $i \in \{1,2\}$.
Since $M \backslash d \backslash f_3 /f_i$ has an $N$-minor and $\{f_4,f_5\}$ is a series pair in this matroid, $M \backslash d \backslash f_3/f_i /f_5$, and in particular $M \backslash d/f_i / f_5$, has an $N$-minor, as required.
Now suppose $t \ge 7$, and consider $j=7$.
As $M \backslash d \backslash f_5$ has an $N$-minor and
$M \backslash d \backslash f_5 / f_i$ is connected, $M \backslash d \backslash f_5 / f_i$ has an $N$-minor by \cref{m2.7}. But $\{f_6,f_7\}$ is a series pair in this matroid, so we deduce that $M \backslash d/f_i /f_7$ has an $N$-minor.
\end{slproof}
\begin{sublemma}
\label{flansl7}
If $t \ge 7$, then $M$ has an $N$-detachable pair.
\end{sublemma}
\begin{slproof}
Let $i \in \{1,2\}$.
By \cref{flanend}(ii), $M \backslash d/f_i/f_7$ is $3$-connected, and, by \cref{flannewnsl}, $M \backslash d/f_i /f_7$ has an $N$-minor.
So either $\{f_i,f_7\}$ is an $N$-detachable pair, or $d$ is in a parallel pair in $M/f_i/f_7$.
Since neither $f_i$ nor $f_7$ is in an $N$-grounded\ triangle, $M$ has a $4$-element circuit $\{d,f_i,f_7,t_i\}$
where, by orthogonality, $t_i \in \{f_3,f_4,f_5\}$.
By circuit elimination, $\{f_1,f_2,t_1,t_2,f_7\}$ contains a circuit. But $f_7 \notin \cl(\{f_1,f_2,f_3,f_4,f_5\})$,
so this circuit is $\{f_1,f_2,t_1,t_2\}$,
and it follows that $\{t_1,t_2\} = \{f_3,f_4\}$.
We now work towards showing that either $\{f_5,f_7\}$ is an $N$-detachable pair, or $\{f_5,f_7\}$ is contained in a $4$-element circuit of $M\backslash d$.
We have that $\{d,f_4\} \subseteq \cl_{M/f_7}(\{f_1,f_2,f_3\})$,
but $f_5 \notin \cl_{M/f_7}(\{f_1,f_2,f_3\}) = \cl_{M/f_7}(\{f_1,f_2,f_3,f_4,d\})$.
Consider a triangle containing $f_5$ in $M/f_7$. It can contain at most one element in $\{f_1,f_2,f_3,f_4,d\}$. Thus, it cannot contain $d$, as $d \notin \cl_{M/f_7}(E(M/f_7) - \{f_1,f_2,f_3\})$ since $d$ blocks the triad $\{f_1,f_2,f_3\}$.
We claim that $M\backslash d/f_5/f_7$ has an $N$-minor.
Since $M\backslash d \backslash f_5$ has an $N$-minor and $M \backslash d \backslash f_5 / f_2 / f_3$ is connected, \cref{m2.7} implies that $M \backslash d \backslash f_5 / f_2 / f_3$ has an $N$-minor.
Fix an $N$-labelling $(C,D)$ with $\{f_2,f_3\} \subseteq C$ and $\{d,f_5\} \subseteq D$.
Since $\{f_6,f_7\}$ is a series pair in
$M\backslash d \backslash f_5$, we may assume that $f_7$ is $N$-labelled for contraction,
up to an $N$-label switch on $f_6$ and $f_7$.
Since $\{f_1,f_4\}$ is a parallel pair in $M / f_2 / f_3$, we may also assume, up to an $N$-label switch on $f_1$ and $f_4$, that $f_4$ is $N$-labelled for deletion.
In particular, observe that $M \backslash d \backslash f_4 / f_7$ has an $N$-minor.
Finally, $\{f_3,f_5\}$ is a series pair in $M \backslash d \backslash f_4$, so, after switching the $N$-labels on $f_3$ and $f_5$, the element $f_5$ is $N$-labelled for contraction, and $f_3$ is $N$-labelled for deletion.
To summarise, $\{f_2,f_5,f_7\} \subseteq C$ and $\{d,f_3,f_4\} \subseteq D$.
So $M\backslash d/f_5/f_7$ has an $N$-minor, as claimed.
Since $f_7$ is $N$-contractible, $f_7$ is not in a triangle of $M \backslash d$.
Thus, by \cref{flanend}, $M \backslash d /f_7$ is $3$-connected, and $(f_1,f_2,\dotsc,f_6)$ is a flan ordering in this matroid.
Hence, either $f_5$ is in a triangle of $M \backslash d / f_7$, or, by another application of \cref{flanend}, $M \backslash d /f_5/f_7$ is $3$-connected.
In the latter case, as $d$ is not in a triangle with $f_5$ in $M/f_7$,
the matroid $M/f_5/f_7$ is $3$-connected, so
$\{f_5,f_7\}$ is an $N$-detachable pair.
So we may assume that $\{f_5,f_7\}$ is contained in a $4$-element circuit of $M \backslash d$.
By orthogonality, this circuit meets $\{f_3,f_4\}$.
But if this circuit is contained in $\{f_1,f_2,\dotsc,f_7\}$, then $f_7 \in \cl(\{f_1,f_2,\dotsc,f_6\})$; a contradiction.
It follows, by orthogonality, that
$\{f_4,f_5,f_7,f_8\}$ is a circuit, where
$f_8 \in E(M\backslash d)-\{f_1,f_2,\dotsc,f_7\}$.
Note that $\{f_1,f_2,\dotsc,f_8\}$ is a flan.
Recall the $N$-labelling $(C,D)$ of $M$ with $d \in D$ and $\{f_5,f_7\} \subseteq C$.
Since
$\{f_4,f_8\}$ is a parallel pair in $M/f_5/f_7$,
the element $f_8$ is $N$-labelled for deletion
after switching the $N$-labels on $f_4$ and $f_8$.
In particular, $M \backslash d \backslash f_8$ has an $N$-minor.
Let $Z=E(M\backslash d)-\{f_1,\dotsc,f_8\}$,
and observe that
$(\{f_1,f_2,\dotsc,f_7\}, \{f_8\}, Z)$ is a
path of $3$-separations.
Suppose $|Z| = 1$. Then $E(M \backslash d)$ is a $9$-element flan,
$f_2$ and $f_7$ are $N$-labelled for contraction with respect to the $N$-labelling $(C,D)$,
and it is easily verified that $M/f_2/f_7$ is $3$-connected. So $\{f_2,f_7\}$ is an $N$-detachable pair.
We now may assume that $|Z| \ge 2$.
We claim that $\co(M \backslash d \backslash f_8)$ is $3$-connected.
If $r(Z) \ge 3$, then $(\{f_1,f_2,\dotsc,f_7\}, \{f_8\}, Z)$ is a vertical $3$-separation,
and the claim follows from Bixby's Lemma.
On the other hand, if $r(Z) \le 2$ and $|Z| \ge 3$, then $(M \backslash d)|(Z \cup f_8) \cong U_{2,4}$, and $M \backslash d \backslash f_8$ is $3$-connected by \cref{rank2Remove}.
Finally, if $|Z| = 2$, then
$Z \cup \{f_7,f_8\}$ is a rank-$3$ cocircuit, and $\co(M\backslash d \backslash f_8)$ is $3$-connected by \cref{r3cocirc}, thus proving the claim.
So either $\{d,f_8\}$ is an $N$-detachable pair, or $f_8$ is in a triad of $M \backslash d$.
We may now assume that $f_8$ is in a triad~$T^*$ of $M \backslash d$.
Since $f_8 \notin \cl^*(\{f_1,f_2,\dotsc,f_7\})$, the triad $T^*$ contains an element $q \in E(M)-\{f_1,f_2,\dotsc,f_8\}$.
By orthogonality, $T^*$ meets $\{f_4,f_5,f_7\}$.
But if $T^* = \{f_4,f_8,q\}$, then $T^*$ intersects the circuit $\{f_1,f_2,f_3,f_4\}$ in one element; a contradiction.
Similarly, if $T^* = \{f_5,f_8,q\}$,
then $\{d,f_5,f_8,q\}$ is a cocircuit of $M$ that intersects the circuit $\{d,f_1,f_7,t_1\}$ (where $t_1 \in \{f_3,f_4\}$) in one element; a contradiction.
We deduce that $T^* = \{f_7,f_8,q\}$.
After relabelling $q$ as $f_9$, we observe that $(f_1,f_2,\dotsc,f_9)$ is a flan ordering.
Next we claim that $M/f_i/f_9$ has an $N$-minor for $i \in \{1,2\}$.
Again we recall the $N$-labelling $(C,D)$ from earlier, which has $\{f_2,f_7\} \subseteq C$ and $\{d,f_8\} \subseteq D$.
Since
$\{f_7,f_9\}$ is a series pair in $M \backslash d \backslash f_8$,
the element $f_9$ is $N$-labelled for contraction after switching the $N$-labels on $f_7$ and $f_9$.
So $M/f_2/f_9$ has an $N$-minor.
Using a similar argument, but starting with an $N$-labelling $(C',D')$ that has $\{f_1,f_3\} \subseteq C'$ and $\{d,f_5\} \subseteq D'$, one can show that $M/f_1/f_9$ has an $N$-minor.
Let $i \in \{1,2\}$.
Since each of $f_i$ and $f_9$ is not contained in an $N$-grounded\ triangle,
$M \backslash d / f_i / f_9$ is $3$-connected,
by \cref{flanend}(ii).
Now either $\{f_i,f_9\}$ is an $N$-detachable pair, or $d$ is in a parallel pair in $M/f_i/f_9$.
Hence, we may assume that
$M$ has a $4$-element circuit containing $\{d,f_i,f_9\}$.
By orthogonality,
this circuit meets $\{f_3,f_4,f_5\}$ and $\{f_5,f_6,f_7\}$, so
$\{d,f_i,f_5,f_9\}$ is a circuit.
As $i \in \{1,2\}$ was chosen arbitrarily, we may now assume that
$\{d,f_1,f_5,f_9\}$ and $\{d,f_2,f_5,f_9\}$ are circuits.
By circuit elimination, $\{f_1,f_2,f_5,f_9\}$ contains a circuit.
But this set does not contain a triangle, and $f_9 \notin \cl(\{f_1,f_2,f_5\})$, so this is contradictory.
\end{slproof}
\begin{sublemma}
\label{flansl6}
If $t=6$, then $M$ has an $N$-detachable pair.
\end{sublemma}
\begin{slproof}
For each $i \in \{1,2\}$,
the matroid $\si(M \backslash d / f_i / f_5)$ is $3$-connected, by \cref{flanend}(i), and $M \backslash d /f_i /f_5$ has an $N$-minor, by \cref{flannewnsl}.
First, suppose that $\{d,f_i,f_5\}$ is not contained in a $4$-element circuit, for some $i \in \{1,2\}$.
It follows that if $M \backslash d/f_i/f_5$ is $3$-connected, then $M /f_i/f_5$ is $3$-connected, and $\{f_i,f_5\}$ is an $N$-detachable pair.
So
assume that $\{f_i,f_5\}$ is contained in a $4$-element circuit in $M \backslash d$.
By orthogonality, this circuit must also contain an element of $\{f_1,f_2,f_3\}-f_i$ and an element of $\{f_3,f_4\}$. Thus, if it does not contain $f_3$, then $f_5 \in \cl(\{f_1,f_2,f_3,f_4\})$; a contradiction.
It follows that this circuit
is $\{f_i,f_3,f_5,q\}$ for some $q \in E(M\backslash d)-\{f_1,f_2,f_3,f_4,f_5\}$.
Since $M \backslash d / f_i / f_5$ has an $N$-minor, and $\{f_3,q\}$ is a parallel pair in this matroid, $\{d,q\}$ is $N$-deletable.
Let $F' = \{f_1,f_2,f_3,f_4,f_5\}$.
Now $F'$ and $F' \cup q$ are $3$-separating in $M \backslash d$, by \cref{extendSep}.
As $|F' \cap E(N)| \le 1$ and $|E(N)| \ge 4$, we have $|E(M\backslash d)-F'| \ge 3$.
We claim that $\co(M \backslash d \backslash q)$ is $3$-connected.
If $r(E(M\backslash d)-F') \le 2$, then $(E(M\backslash d)-F') \cup f_5$ is a rank-$3$ cocircuit, and it follows, by \cref{r3cocirc}, that $\co(M \backslash d \backslash q)$ is $3$-connected.
On the other hand, if $r(E(M\backslash d)-F') \ge 3$,
then
$(F', \{q\},E(M\backslash d)-(F' \cup q))$ is a vertical $3$-separation of $M \backslash d$, so $\si(M \backslash d/q)$ is not $3$-connected, and hence $\co(M \backslash d\backslash q)$ is $3$-connected by Bixby's Lemma.
So $\{d,q\}$ is an $N$-detachable pair unless $q$ is in a triad $T^*$ of $M \backslash d$.
Note also that, by the foregoing, $q \in \cl(E(M \backslash d)-(F' \cup q))$.
As $q \notin \cl^*_{M \backslash d}(F')$, the triad $T^*$ contains at most one element of $F'$.
By orthogonality, $T^*$ meets $\{f_i,f_3,f_5\}$.
Now if $f_5 \notin T^*$, then, as $\{f_1,f_2,f_3,f_4\}$ is a circuit, $T^*$ intersects $\{f_1,f_2,f_3,f_4\}$ in two elements; a contradiction.
It follows that $T^* = \{f_5,q,q'\}$, for $q' \in E(M\backslash d)-F'$.
Now, as $f_6 \in \cl(F')$ but $f_6 \notin \cl(F'-f_5)$, there is a circuit containing $\{f_5,f_6\}$ that is contained in $F$.
Again by orthogonality, we deduce that $f_6 \in T^*$, so let $q=f_6$.
Now $F \cup q'$ is a flan, contradicting that $F$ is a maximal $6$-element flan.
\medskip
It remains to consider the case where $M$ has circuits $\{d,f_1,f_5,g_1\}$ and $\{d,f_2,f_5,g_2\}$ for some $g_1 \in E(M)-\{d,f_1,f_5\}$ and $g_2 \in E(M)-\{d,f_2,f_5\}$.
First, suppose that $g_1 = g_2 = f_3$.
Then $\{d,f_1,f_3,f_5\}$ and $\{d,f_2,f_3,f_5\}$ are circuits, so $\{f_1,f_2,f_3,f_5\}$ contains a circuit by circuit elimination. But $\{f_1,f_2,f_3,f_5\}$ does not contain a triangle, and $f_5 \notin \cl(\{f_1,f_2,f_3\})$, since $f_5 \in \cl^*_{M \backslash d}(\{f_1,f_2,f_3,f_4\})$, so this is contradictory.
Let $i \in \{1,2\}$ such that $g_i \neq f_3$.
We will show that either $\{f_3,g_i\}$ is an $N$-detachable pair, or
$g_i \in E(M \backslash d)-F$ and
there is a $4$-element cocircuit~$C^*_i$ with $\{f_3,g_i\} \subseteq C^*_i \subseteq F \cup g_i$.
Using a similar argument as in the proof of \cref{flannewnsl}, it follows from \cref{m2.7} that $M \backslash f_3 / f_i / f_5$ has an $N$-minor for $i \in \{1,2\}$.
Since $\{d,g_i\}$ is a parallel pair in $M/f_i/f_5$,
the pair $\{f_3,g_i\}$ is $N$-labelled for deletion,
up to swapping the $N$-labels on $d$ and $g_i$.
Suppose that $\co(M \backslash f_3 \backslash g_i)$ is not $3$-connected.
Then $M \backslash f_3 \backslash g_i$ has a $2$-separation $(X,Y)$ for which $(\fcl(X),Y-\fcl(X))$ and $(X-\fcl(Y),\fcl(Y))$ are also $2$-separations.
Thus, we may assume that
$\{f_1,f_2,d\} \subseteq X$ and $X$ is fully closed.
Pick $j$ such that $\{i,j\} = \{1,2\}$.
If $f_5 \in X$, then
$g_j \in X$ due to the circuit $\{f_j,f_5,d,g_j\}$, and
$f_4 \in X$ due to the cocircuit $\{f_4,f_5,d\}$ of $M \backslash f_3 \backslash g_i$.
Similarly, if $f_4 \in X$, then $\{f_5,g_j\} \subseteq X$.
But then
$\{f_3,g_i\} \subseteq \cl(X)$, so $(X \cup \{f_3,g_i\},Y)$ is a $2$-separation of $M$; a contradiction.
So $\{f_4,f_5\} \subseteq Y$.
Now, if $g_j \in X$, then $f_5 \in \cl(X)$, so $X$ is not fully closed; a contradiction.
So $g_j \in Y$.
Consider $\fcl(Y)$. As $d \in \fcl(Y)$, we have $f_j \in \fcl(Y)$, so $f_i \in \fcl(Y)$, and $(X-\fcl(Y), \fcl(Y) \cup \{f_3,g_i\})$ is a $2$-separation of $M$; a contradiction.
So $\co(M \backslash f_3 \backslash g_i)$ is $3$-connected.
So we may assume that $M$ has a $4$-element cocircuit $C^*_i$ containing $\{g_i,f_3\}$,
otherwise $M$ has an $N$-detachable pair.
By orthogonality, $C^*_i$ meets $\{f_1,f_2,f_4\}$ and $\{d,f_i,f_5\}$.
Suppose $d \in C^*_i$. If $f_4 \in C^*_i$, then $\{f_3,f_4,f_5,g_i\}$ is a cosegment of $M \backslash d$. If $g_i \in \{f_1,f_2\}$, then $r^*_{M \backslash d}(F-f_6) = 2$, so $\lambda_{M \backslash d}(F-f_6) = 4+2-5=1$; a contradiction. But on the other hand if $g_i \notin \{f_1,f_2\}$, then
this contradicts orthogonality with the circuit $\{f_1,f_2,f_3,f_4\}$.
So $\{f_1,f_2\}$ meets $C^*_i$, and thus $\{f_1,f_2,f_3,g_i\}$ is a cosegment of $M \backslash d$.
As before, $g_i \notin \{f_4,f_5\}$, otherwise $\lambda_{M \backslash d}(F)=1$; and $g_i \neq f_6$, since $f_6 \notin \cl^*_{M \backslash d}(F-f_6)$.
Observe that
there is a circuit contained in
$\{f_1,f_3,f_4,f_5,f_6\}$ with at least four elements.
But this contradicts orthogonality. So $d \notin C^*_i$.
Suppose $|C^*_i \cap F| \le 2$.
Then $C^*_i \cap F = \{f_i,f_3\}$.
Again, pick $j$ such that
$\{i,j\} = \{1,2\}$, and
observe that there is a circuit contained in $\{f_j,f_3,f_4,f_5,f_6\}$ that contains $f_3$. But this contradicts orthogonality, so $C^*_i \subseteq F \cup g_i$.
Now suppose $C^*_i \subseteq F$.
Since $F-f_6$ is exactly $3$-separating in $M \backslash d$ and $f_6 \in \cl(F-f_6)$, it follows that $f_6 \notin \cl^*_{M \backslash d}(F-f_6)$.
So $C^*_i \subseteq F-f_6$.
But, as $f_3 \in C^*_i$, and $r^*_{M \backslash d}(\{f_1,f_2,f_3\}) = r^*_{M \backslash d}(\{f_3,f_4,f_5\}) = 2$, the set $C^*_i$ is a $4$-element cosegment in $M \backslash d$. Thus $r^*_{M \backslash d}(F-f_6)=2$, and $\lambda_{M \backslash d}(F-f_6)=1$; a contradiction.
We deduce that $g_i \notin F$.
Finally, we recall that
$\{d,f_1,f_5,g_1\}$ and $\{d,f_2,f_5,g_2\}$ are circuits, so, by circuit elimination, there are circuits contained in the sets $\{d,f_1,f_2,g_1,g_2\}$ and $\{f_1,f_2,f_5,g_1,g_2\}$.
By the foregoing, if $f_3 \notin \{g_1,g_2\}$, then $\{g_1,g_2\} \cap F = \emptyset$, and $\{g_1,g_2\} \subseteq \cl^*_{M \backslash d}(F)$.
Since $\{d,f_3,f_4,f_5\}$ is a cocircuit, we deduce, by orthogonality, that $\{f_1,f_2,g_1,g_2\}$ is a circuit. Thus $g_1 \neq g_2$. Since $\{g_1,g_2\} \subseteq \cl^*_{M \backslash d}(F)$, we have
\begin{align*}
\lambda_{M \backslash d}(F \cup \{g_1,g_2\}) &= r_{M \backslash d}(F \cup \{g_1,g_2\}) + r^*_{M \backslash d}(F) - 8 \\
&\le (r(F) + 1) +4 - 8 = 1;
\end{align*}
a contradiction.
On the other hand, if $g_2 = f_3$, say, then $g_1 \neq f_3$, and there is a cocircuit $C_1^* \subseteq F \cup g_1$ of $M \backslash d$ with $g_1 \in E(M \backslash d)-F$.
Since $\{f_1,f_2,f_5,g_1,g_2\}=\{f_1,f_2,f_3,f_5,g_1\}$ contains a circuit and $f_5 \notin \cl(\{f_1,f_2,f_3\})$, this circuit must contain $g_1$. But then $g_1 \in \cl_{M \backslash d}(F) \cap \cl^*_{M \backslash d}(F)$; a contradiction.
\end{slproof}
The \lcnamecref{flannew} now follows from \cref{flansl6,flansl7}.
\end{proof}
Next we address the case where $M \backslash d$ has a maximal $5$-element flan~$F$. We can break this into two cases depending on whether or not $d$ fully blocks $F$. The next \lcnamecref{5flan} primarily deals with the case where $d$ fully blocks $F$. However, we use a more general hypothesis than this, as the same argument also applies in one situation that arises when $d$ does not fully block $F$.
More specifically, suppose $F$ has the ordering $(f_1,f_2,f_3,f_4,f_5)$.
The following \lcnamecref{5flan} applies when
any $4$-element circuit containing $\{f_i,f_5,d\}$ for $i \in \{1,2\}$ is not contained in $F \cup d$.
In particular, observe that this is the case when $d$ fully blocks $F$.
\begin{proposition}
\label{5flan}
Let $M$ be a $3$-connected matroid with a $3$-connected matroid~$N$ as a minor, where $|E(N)| \ge 4$ and every triangle or triad of $M$ is $N$-grounded.
Suppose that $M \backslash d$ is $3$-connected and has a maximal $5$-element flan~$F$ with ordering $(f_1,f_2,f_3,f_4,f_5)$, where
any $4$-element circuit containing $\{f_i,f_5,d\}$ for $i \in \{1,2\}$ is not contained in $F \cup d$,
and $M \backslash d \backslash f_3$ has an $N$-minor.
Then $M$ has an $N$-detachable pair.
\end{proposition}
\begin{proof}
We start by showing that either there is an $N$-detachable pair in $F$, or there are certain $4$-element circuits in $M$ containing $d$ and intersecting $F$.
\begin{sublemma}
\label{5flansb1}
For each $i \in \{1,2\}$, the matroids $M\backslash d/f_i/f_5$ and $M\backslash f_3/f_i/f_5$ have $N$-minors.
\end{sublemma}
\begin{slproof}
The matroid $M \backslash d \backslash f_3$ has an $N$-minor, and $\{f_1,f_2\}$ and $\{f_4,f_5\}$ are series pairs in this matroid, so $M \backslash d \backslash f_3 /f_i/f_5$ has an $N$-minor for $i \in \{1,2\}$.
\end{slproof}
Now it follows from \cref{5flansb1} that none of $f_1$, $f_2$, and $f_5$ is contained in an $N$-grounded\ triangle.
Hence, we can apply \cref{flanend} to deduce that $M \backslash d/f_i/f_5$ is $3$-connected for each $i \in \{1,2\}$.
Then $\{f_i,f_5\}$ is an $N$-detachable pair for $i \in \{1,2\}$ unless $d$ is in a parallel pair in $M/f_i/f_5$.
Since neither $f_i$ nor $f_5$ is contained in an $N$-grounded\ triangle, we have that $\{d,f_i,f_5,g_i\}$ is a circuit in $M$
for some $g_i \in E(M) - \{d,f_i,f_5\}$.
By hypothesis, $g_i \notin F$ for $i \in \{1,2\}$.
Moreover, if $g_1 = g_2$, then there is a circuit contained in $\{d,f_1,f_2,f_5\}$; a contradiction.
So $g_1$ and $g_2$ are distinct elements of $E(M\backslash d)-F$.
\begin{sublemma}
\label{predelcands}
$\{f_1,f_2,g_1,g_2\}$ is a circuit of $M$.
\end{sublemma}
\begin{slproof}
As $r(\{d,f_5,f_1,f_2,g_1,g_2\}) \le 4$ and $d \notin \cl(E(M)-\{f_3,f_4,f_5\})$, we have $r(\{f_1,f_2,g_1,g_2\}) \le 3$.
Since neither $f_1$ nor $f_2$ is in an $N$-grounded\ triangle, we deduce that $\{f_1,f_2,g_1,g_2\}$ is a circuit.
\end{slproof}
We now work towards showing that, for each $i \in \{1,2\}$, either $\{f_3,g_i\}$ is an $N$-detachable pair, or $\{f_3,g_i\}$ is contained in a $4$-element cocircuit.
To this end, we start by showing that $M \backslash f_3 \backslash g_i$ has an $N$-minor for $i \in \{1,2\}$.
Let $i \in \{1,2\}$.
By \cref{5flansb1}, $M \backslash f_3 / f_i / f_5$ has an $N$-minor.
Since $\{d,g_i\}$ is a parallel pair in this matroid and $|E(N)| \ge 4$, the matroid $M \backslash f_3 \backslash g_i$ has an $N$-minor.
\begin{sublemma}
\label{delcands2}
Let $i \in \{1,2\}$.
If $\{f_3,g_i\}$ is not contained in a $4$-element cocircuit of $M$, then $\{f_3,g_i\}$ is an $N$-detachable pair.
\end{sublemma}
\begin{slproof}
We give the proof for $i=2$; the argument is almost identical when $i=1$.
Observe that neither $f_3$ nor $g_2$ is in a triad of $M$, since $M \backslash f_3$ and $M \backslash g_2$ have $N$-minors.
Let $(P,Q)$ be a $2$-separation of $M \backslash f_3 \backslash g_2$.
Since $\{f_3,g_2\}$ is not contained in a $4$-element cocircuit of $M$,
we have that $(\fcl(P),Q-\fcl(P))$ and $(P-\fcl(Q),\fcl(Q))$ are $2$-separations in $M \backslash f_3 \backslash g_2$.
So we may assume that $P$ is fully closed.
As $\{f_1,f_2,d\}$ is a triad in $M \backslash f_3 \backslash g_2$,
without loss of generality $\{f_1,f_2,d\} \subseteq P$.
If $\{f_5,g_1\}$ meets $P$, then $\{f_5,g_1\} \subseteq P$, due to the circuit $\{f_1,f_5,d,g_1\}$, and $f_4 \in P$ as well, due to the triad $\{f_4,f_5,d\}$.
But then $(P \cup \{f_3,g_2\},Q)$ is a $2$-separation of $M$; a contradiction.
So $\{f_5,g_1\} \subseteq Q$, and, similarly, $f_4 \in Q$.
Now consider $\fcl(Q)$. Due to the triad $\{f_4,f_5,d\}$, we have $d \in \fcl(Q)$, and thus $\{f_1,f_2\} \subseteq \fcl(Q)$ due to the circuits $\{f_1,f_5,d,g_1\}$ and $\{f_2,f_5,d,g_2\}$.
Thus $(P-\fcl(Q),\fcl(Q) \cup \{f_3,g_2\})$ is a $2$-separation of $M$; a contradiction.
So $M \backslash f_3 \backslash g_2$ is $3$-connected.
Since $\{f_3,g_2\}$ is $N$-deletable, this completes the proof of \cref{delcands2}.
\end{slproof}
It remains to consider the case where $\{f_3,g_i\}$ is contained in a $4$-element cocircuit for each $i \in \{1,2\}$.
We next prove two claims regarding the elements that appear in such a cocircuit.
\begin{sublemma}
\label{delcands}
Let $i \in \{1,2\}$.
If $\{f_3,g_i\}$ is in a $4$-element cocircuit of $M$, then
either this cocircuit contains $f_i$, or $g_i \in \cl^*(F \cup d)$.
\end{sublemma}
\begin{slproof}
Let $C^*$ be a $4$-element cocircuit of $M$ containing $\{f_3,g_i\}$.
Pick $i'$ such that $\{i,i'\} = \{1,2\}$.
By orthogonality, $C^*$ meets $\{f_1,f_2,f_4\}$ and $\{f_i,f_5,d\}$.
Thus, either $f_i \in C^*$, or $C^*$ meets $\{f_{i'},f_4\}$ and $\{f_5,d\}$ in which case $g_i \in \cl^*(F \cup d)$.
\end{slproof}
\begin{sublemma}
\label{5flansc}
Suppose $\{f_i,f_3,g_i,h_i\}$ is a cocircuit of $M$, for some $i \in \{1,2\}$ and $h_i \in E(M) - (F \cup \{d,g_i\})$.
Then, either $M$ has an $N$-detachable pair, or $h_i \in \cl(F \cup \{d,g_i\})-\{g_1,g_2\}$.
\end{sublemma}
\begin{slproof}
First, we will show that if $\{f_5,h_i\}$ is not contained in a $4$-element circuit, then $\{f_5, h_i\}$ is an $N$-detachable pair.
Pick $i'$ such that $\{i,i'\} = \{1,2\}$.
Observe that $\{f_i,f_3,g_1,g_2\}$ is not a cocircuit, by orthogonality with the circuit $\{f_{i'},f_5,d,g_{i'}\}$. So $h_i \neq g_{i'}$.
Let $(P,Q)$ be a $2$-separation in $M / f_5 / h_i$, where neither $P$ nor $Q$ is contained in a parallel class. So $(\fcl(P), Q- \fcl(P))$ is also a $2$-separation.
Without loss of generality,
$\{f_i,d,g_i\} \subseteq P$. If $P$ meets $\{f_{i'},f_3\}$, then
$\{f_{i'},f_3\} \subseteq P$ due to the cocircuit $\{f_1,f_2,f_3,d\}$, and $f_4 \in P$ due to the circuit $\{f_1,f_2,f_3,f_4\}$.
But then $(P \cup \{f_5,h_i\},Q)$ is a $2$-separation of $M$; a contradiction.
So $\{f_{i'},f_3\} \subseteq Q$.
Since $\{f_1,f_2,g_1,g_2\}$ is a circuit, by \cref{predelcands}, we have $g_{i'} \in Q$, otherwise $f_{i'} \in \fcl(P)=P$.
Now consider the $2$-separation $(P',Q') = (P-\fcl(Q),\fcl(Q))$.
We have $d \in Q'$, due to the triangle $\{f_{i'},d,g_{i'}\}$, and it follows that
$f_i \in Q'$, due to the cocircuit $\{f_1,f_2,f_3,d\}$;
$f_4 \in Q'$, due to the circuit $\{f_1,f_2,f_3,f_4\}$; and
$g_i \in Q'$, due to the triangle $\{f_i,d,g_i\}$.
So $(P', Q' \cup \{f_5,h_i\})$ is a $2$-separation of $M$; a contradiction.
So $M / f_5 / h_i$ is $3$-connected up to parallel pairs.
We claim that $M /f_5 / h_i$ has an $N$-minor.
Since $M /f_i/f_5 \backslash f_3$ has an $N$-minor, there is an $N$-labelling $(C,D)$ with $\{f_i,f_5\} \subseteq C$ and $f_3 \in D$.
As $\{g_i,d\}$ is a parallel pair in $M/f_i/f_5$, we may assume, up to switching the $N$-labels on $g_i$ and $d$, that $g_i \in D$.
Now, in $M\backslash f_3 \backslash g_i$, we have that $\{f_i,h_i\}$ is a series pair, so, after switching the $N$-labels on $f_i$ and $h_i$, we obtain an $N$-labelling $(C',D')$ with $\{h_i,f_5\} \subseteq C'$. So $M / h_i / f_5$ has an $N$-minor, as claimed.
We may now assume that $\{f_5,h_i\}$ is contained in a $4$-element circuit,
otherwise $M$ has an $N$-detachable pair.
By orthogonality, this circuit meets $\{f_3,f_4,d\}$ and $\{f_i,f_3,g_i\}$.
If it does not contain $f_3$, then $h_i \in \cl(\{f_i,f_4,f_5,d,g_i\})$, as required.
So suppose it contains $f_3$. Then, again by orthogonality, it also meets $\{f_1,f_2,d\}$, in which case
$h_i \in \cl(F \cup d)$.
\end{slproof}
Observe that $\{g_1,g_2\} \nsubseteq \cl^*(F \cup d)$.
Indeed, if $\{g_1,g_2\} \subseteq \cl^*(F \cup d)$, then
\begin{align*}
\lambda(F \cup \{d,g_1,g_2\})
&= r(F \cup d) + r^*(F \cup d) - 8 \\
&\le 5 + 4 - 8 = 1;
\end{align*}
a contradiction.
So there exists some $\ell \in \{1,2\}$ such that $g_\ell \notin \cl^*(F \cup d)$.
Then, by \cref{delcands}, $M$ has a cocircuit $\{f_\ell,f_3,g_\ell, h_\ell\}$ for some $h_\ell \in E(M) - \{f_\ell,f_3,g_\ell\}$.
In fact, $h_\ell \notin F \cup d$, since $g_\ell \notin \cl^*(F \cup d)$.
Thus, by \cref{5flansc}, we may assume that $h_\ell \in \cl(F \cup \{d,g_\ell\})$ and $h_\ell \notin \{g_1,g_2\}$.
\begin{sublemma}
\label{5flanscomb}
For each $i \in \{1,2\}$, we have $g_i \notin \cl^*((F \cup \{d,g_1,g_2\})-g_i)$. Moreover,
there are distinct elements $h_1,h_2 \in E(M) -(F \cup \{d,g_1,g_2\})$ such that $\{f_1,f_3,g_1,h_1\}$ and $\{f_2,f_3,g_2,h_2\}$ are cocircuits of $M$.
\end{sublemma}
\begin{slproof}
Consider $\lambda(F \cup \{d,g_1,g_2,h_\ell\})$.
Observe that $\{g_1,g_2,h_\ell\} \subseteq \cl(F \cup d)$ and
$h_\ell \in \cl^*(F \cup g_\ell)$.
Thus,
\begin{align*}
\lambda(F \cup \{d,g_1,g_2,h_\ell\}) &= r(F \cup d) + r^*(F \cup \{d,g_1,g_2\}) - 9 \\
&\le r^*(F \cup \{d,g_1,g_2\}) -4.
\end{align*}
Now if either $g_1 \in \cl^*(F \cup \{d,g_2\})$ or $g_2 \in \cl^*(F \cup \{d,g_1\})$, then $$\lambda(F \cup \{d,g_1,g_2,h_\ell\}) \le (r^*(F \cup d)+1) -4 = 1;$$ a contradiction.
By \cref{delcands}, $\{f_1,f_3,g_1\}$ and $\{f_2,f_3,g_2\}$ are each contained in a $4$-element cocircuit of $M$. Let these cocircuits be $\{f_1,f_3,g_1,h_1\}$ and $\{f_2,f_3,g_2,h_2\}$ respectively.
Observe that, for each $i \in \{1,2\}$, we have $h_i \in E(M)- (F \cup \{d,g_1,g_2\})$, since $g_i \notin \cl^*((F \cup \{d,g_1,g_2\})-g_i)$.
Suppose that $h_1 = h_2$.
Then $\{f_1,f_2,f_3,g_1,g_2\}$ contains a cocircuit, by cocircuit elimination.
Since $g_1 \notin \cl^*(F \cup g_2)$ and $g_2 \notin \cl^*(F \cup g_1)$, it follows that $\{f_1,f_2,f_3\}$ is a cocircuit of $M$; a contradiction.
\end{slproof}
Now, by \cref{5flanscomb}, there are distinct elements $h_1,h_2 \in E(M) - (F \cup \{d,g_1,g_2\})$ such that
$\{h_1,h_2\} \subseteq \cl(F \cup \{d,g_1,g_2\}) = \cl(F \cup d)$.
Note that
$\{h_1,h_2\} \subseteq \cl^*(F \cup \{g_1,g_2\})$.
Thus,
\begin{align*}
\lambda(F \cup \{d,g_1,g_2,h_1,h_2\}) &= r(F \cup d) + r^*(F \cup \{d,g_1,g_2\}) - 10 \\
&\le 5 + (r^*(F \cup d)+2) -10 \\
&= 1;
\end{align*}
a contradiction.
This completes the proof.
\end{proof}
Next we handle the case where $M \backslash d$ has a maximal $5$-element flan and $d$ does not fully block $F$. Since $d$ blocks the triads of $M \backslash d$ contained in $F$, we have that $d \in \cl_M(F)$.
\begin{proposition}
\label{flan5structure}
Let $M$ be a $3$-connected matroid with a $3$-connected matroid~$N$ as a minor, where $|E(N)| \ge 4$, and every triangle or triad of $M$ is $N$-grounded.
Let $d$ be an element of $M$ such that $M\backslash d$ is $3$-connected
and has a maximal $5$-element flan~$F$ with ordering $(f_1,f_2,f_3,f_4,f_5)$, where $d \in\cl_M(F)$.
Suppose that either
\begin{enumerate}[label=\rm(\alph*)]
\item $M \backslash d \backslash f_5$ has an $N$-minor with $|\{f_1,\dotsc,f_4\} \cap E(N)| \le 1$, or
\item $M /f_i /f_{i'}$ has an $N$-minor for all distinct $i,i' \in \seq{3}$.
\end{enumerate}
Then one of the following holds:
\begin{enumerate}
\item $M$ has an $N$-detachable pair,
\item $F\cup d$ is a skew-whiff $3$-separator\ of $M$,
\item $F\cup d$ is an elongated-quad $3$-separator\ of $M$, or
\item $F\cup d$ is a twisted cube-like $3$-separator\ of $M^*$.
\end{enumerate}
\end{proposition}
\begin{proof}
First, we observe that each element in $F-f_5$ is $N$-deletable in $M \backslash d$.
Indeed, if (a) holds, then since $(F-f_5, \{f_5\}, E(M \backslash d)-F)$ is a cyclic $3$-separation of $M \backslash d$, and $F-f_5$ is a circuit, this follows from \cref{doublylabel}(i).
On the other hand, if (b) holds, then since $M / f_i / f_i'$ has an $N$-minor for all distinct $i,i' \in \seq{3}$, and $\{f_1,f_2,f_3,f_4\}$ is a circuit, it follows that each element in $F-f_5$ is $N$-deletable up to an $N$-label switch.
Now each triad of $M \backslash d$ contained in $F$ is not an $N$-grounded\ triad,
so $\{f_1,f_2,f_3,d\}$ and $\{f_3,f_4,f_5,d\}$ are cocircuits of $M$.
Moreover, as $M \backslash d \backslash f_3$ has an $N$-minor, and $\{f_1,f_2\}$ and $\{f_4,f_5\}$ are parallel pairs in this matroid,
$M \backslash d/f_i / f_5$ has an $N$-minor for $i \in \{1,2\}$.
By \cref{flanend}(iii), $M \backslash d / f_i / f_5$ is $3$-connected, for $i \in \{1,2\}$. Thus,
assuming (i) does not hold,
we deduce the existence of $4$-element circuits $\{f_1,f_5,d,g_1\}$ and $\{f_2,f_5,d,g_2\}$.
We claim that $\{g_1,g_2\} \subseteq F$ or $\{g_1,g_2\} \subseteq \cl(F \cup d)-(F \cup d)$.
Suppose $g_1 \notin F$.
Since $F$ is a maximal flan, $g_1 \notin \cl(F)$.
By circuit elimination, $\{f_1,f_2,f_5,g_1,g_2\}$ contains a circuit.
If this circuit contains $g_1$, then $g_1 \in \cl(F \cup g_2)$, so $g_2 \notin F$, and $\{g_1,g_2\} \subseteq \cl(F \cup d)-F$ as required.
So suppose $\{f_1,f_2,f_5,g_2\}$ is a circuit. Then $g_2 \in F$, since $F$ is a maximal flan, so
$g_2 \in \{f_3,f_4\}$. It follows that $F \subseteq \cl(\{f_1,f_2,g_2\})$; a contradiction.
If $g_1,g_2 \notin F$, then we can apply \cref{5flan}, so (i) holds.
So we may assume that $\{g_1, g_2\} \subseteq F$.
Observe that
since $\{f_1,f_2,f_3,f_4\}$ and $\{f_1,f_5,d,g_1\}$
are circuits,
every element of $F \cup d$ is in a circuit contained in $F \cup d$.
\begin{sublemma}
\label{anothercovering}
If $C_1$ and $C_2$ are distinct circuits of $M$ contained in $F \cup d$,
then $F \cup d = C_1\cup C_2$. Similarly, if
$C_1^*$ and $C_2^*$ are distinct cocircuits of $M$ contained in $F \cup d$,
then $F \cup d = C_1^*\cup C_2^*$.
\end{sublemma}
\begin{slproof}
The set $F \cup d$ is exactly $3$-separating in $M$, and $r(F \cup d)=4$, so
$r(E(M)-(F \cup d))=r(M)-2$.
Suppose that $C_1^*\subseteq F \cup d$ and $C_2^*\subseteq F \cup d$ are distinct cocircuits of $M$.
Then $E(M)-(C_1^*\cup C_2^*)$ is a flat of rank $r(M)-2$. Thus, if $x\in(F\cup d)-(C_1^*\cup C_2^*)$, then $x\in\cl(E(M)-(F \cup d))$.
But this contradicts the fact that every element of $F \cup d$ is contained in some cocircuit that is itself contained in $F \cup d$.
The proof of the dual result follows in the same manner due to the fact that $r^*_M(F \cup d)=4$ and every element of $F \cup d$ is contained in a circuit that is itself contained in $F \cup d$.
\end{slproof}
Now we assume that (i) does not hold, and show that either (ii), (iii), or (iv) holds.
As $\{f_1,f_5,d,g_1\}$ and $\{f_2,f_5,d,g_2\}$ are circuits of $M$ contained in $F \cup d$, either $g_1 = f_2$ and $g_2=f_1$ so that these circuits coincide, or, by \cref{anothercovering}, $\{g_1,g_2\} = \{f_3,f_4\}$.
We will show that in the former case (iii) or (iv) holds, whereas in the latter case (ii) holds.
\begin{sublemma}
\label{flan6sub1}
$M/f_1 \backslash f_2 \backslash f_5$ and $M/f_2 \backslash f_1 \backslash f_5$ have $N$-minors.
\end{sublemma}
\begin{slproof}
First suppose that (a) holds.
Since $M \backslash d \backslash f_5$ has an $N$-minor and $M \backslash d \backslash f_5 / f_1 /f_3$ is connected, \cref{m2.7} implies that
$M \backslash f_5 / f_1 /f_3$ has an $N$-minor. Since $\{f_2,f_4\}$ is a parallel pair in this matroid, $M /f_1 \backslash f_2 \backslash f_5$ has an $N$-minor, up to an $N$-label switch.
Similarly, $M \backslash f_5 / f_2/f_3$ has an $N$-minor, and, up to an $N$-label switch,
$M /f_2 \backslash f_1 \backslash f_5$ has an $N$-minor.
Now suppose (b) holds.
Recall that either $\{f_i,f_4,f_5,d\}$ and $\{f_{i'},f_3,f_5,d\}$ are circuits for some $\{i,i'\} = \{1,2\}$, or $\{f_1,f_2,f_5,d\}$ is a circuit. Assume the former.
Since $M/f_{i'}/f_3$ has an $N$-minor, and $\{f_i,f_4\}$ and $\{f_5,d\}$ are parallel pairs in this matroid, $M/f_{i'} \backslash f_i \backslash f_5$ has an $N$-minor.
Moreover, $M \backslash d \backslash f_3$ has an $N$-minor, where $\{f_i,f_{i'}\}$ and $\{f_4,f_5\}$ are series pairs in this matroid, so $M \backslash f_3 / f_i /f_4$ has an $N$-minor. But $\{f_5,d\}$ is a parallel pair in this matroid, so $M \backslash f_5 / f_i /f_4$ has an $N$-minor. Now $\{f_{i'},f_3\}$ is a parallel pair in this matroid, so $M / f_i \backslash f_{i'} \backslash f_5$ has an $N$-minor as required.
Now we assume that $\{f_1,f_2,f_5,d\}$ is a circuit.
Since, for any $\{i,i'\} = \{1,2\}$,
the matroid $M/f_i/f_{i'}$ has an $N$-minor,
and $\{f_3,f_4\}$ and $\{f_5,d\}$ are parallel pairs in this matroid, $M/f_{i'} \backslash f_4 \backslash f_5$ has an $N$-minor.
Since $\{f_3,d\}$ is a series pair in this matroid, $M/f_{i'}/f_3 \backslash f_5$ has an $N$-minor. Now $\{f_i,f_4\}$ is a parallel pair in this matroid, so $M/f_{i'} \backslash f_i \backslash f_5$ has an $N$-minor as required.
\end{slproof}
\begin{sublemma}
\label{flan6sub2}
Either $M/f_1\backslash f_2\backslash f_5$ or $M/f_2\backslash f_1\backslash f_5$ is $3$-connected.
\end{sublemma}
\begin{slproof}
Let $\{i,j\} = \{1,2\}$.
We start by showing that either $M/f_i\backslash f_j\backslash f_5$ is $3$-connected, or there is a $4$-element cocircuit $\{f_j, f_j', f_5, h_j\}$ where $f_j' \in \{f_3,f_4\}$ and $h_j \in E(M)-(F \cup d)$.
Consider the $3$-connected matroid $M/f_i$.
Observe that $r_{M/f_i}((F - f_i) \cup d) = 3$.
Since $r_M(F \cup d) = 4$, it follows that $\{f_i,f_3,f_4,d\}$ is independent in $M$.
So $\{f_3,f_4,f_5,d\}$ is a rank-$3$ cocircuit in $M/f_i$, with $f_5 \in \cl_{M/f_i}(\{f_3,f_4,d\})$.
Thus, by \cref{r3cocirc}, $\co(M/f_i\backslash f_5)$, and indeed $M/f_i\backslash f_5$, is $3$-connected.
Now $(\{f_3,f_4,d\},\{f_j\},E(M)-(F \cup d))$ is a vertical $3$-separation in $M/f_i \backslash f_5$.
By Bixby's Lemma,
$\co(M/f_i \backslash f_j \backslash f_5)$ is $3$-connected.
So
$M/f_i\backslash f_j\backslash f_5$ is $3$-connected unless $f_j$ is in a triad of $M/f_i\backslash f_5$
that meets both $\{f_3,f_4,d\}$ and $E(M)-(F \cup d)$.
By orthogonality, this triad does not contain $d$.
So $\{f_j,f_j',f_5,h_j\}$ is a cocircuit of $M$ where $f_j' \in \{f_3,f_4\}$ and $h_j \in E(M)-(F \cup d)$, as claimed.
Suppose neither $M/f_2\backslash f_1\backslash f_5$ nor $M/f_1\backslash f_2\backslash f_5$ is $3$-connected. Then $\{f_1,f_1',f_5,h_1\}$ and $\{f_2,f_2',f_5,h_2\}$ are cocircuits, where $f_1',f_2' \in \{f_3,f_4\}$.
Recall that $M /f_i \backslash f_j \backslash f_5$ has an $N$-minor
when $\{i,j\} = \{1,2\}$.
Since $\{f_j',h_j\}$ is a series pair in this matroid, it follows that $M/f_i/h_j$ has an $N$-minor.
Next, we claim that either $M\backslash d/f_1/h_2$ or $M\backslash d/f_2/h_1$ is $3$-connected.
Suppose not,
so $M\backslash d/f_i/h_j$ is not $3$-connected for $\{i,j\} = \{1,2\}$.
Observe that $(F-f_i,\{h_j\},E(M)-(F \cup \{d,h_j\}))$ is a cyclic $3$-separation of $M\backslash d/f_i$,
so $\si(M\backslash d/f_i/h_j)$ is $3$-connected, by Bixby's Lemma.
Thus $M\backslash d/f_i/h_j$ contains a parallel pair, implying that $\{f_i,h_j\}$ is contained in a $4$-element circuit in $M \backslash d$ that, by orthogonality, intersects $\{f_1,f_2,f_3\}$ in two elements.
But if $f_3$ is in this circuit, then it also meets $\{f_4,f_5\}$, by orthogonality, in which case $h_j \in \cl(F)$; a contradiction. We deduce that $\{f_1,f_2,h_j,q_j\}$ is a circuit for some $q_j \in E(M)-(F \cup d)$.
Now $\{f_1,f_2,h_1,q_1\}$ and $\{f_1,f_2,h_2,q_2\}$ are both circuits,
so $r(\{f_1,h_1,h_2,q_1,q_2\}) \le 4$.
Since $f_1 \in \cl^*(\{f_2,f_3,d\})$, the set $\{h_1,h_2,q_1,q_2\}$ contains a circuit.
But such a circuit intersects one of the cocircuits $\{f_1,f_5,f_1',h_1\}$ or $\{f_2,f_5,f_2',h_2\}$ in a single element, contradicting orthogonality.
Up to labels, we may now assume that $M\backslash d/f_1/h_2$ is $3$-connected.
So either $\{f_1,h_2\}$ is an $N$-detachable pair, contradictory to the assumption that (i) does not hold, or there is a $4$-element circuit of $M$ containing $\{d,f_1,h_2\}$. By orthogonality, such a circuit meets $\{f_3,f_4,f_5\}$. So $\{d,f_1,f_\ell,h_2\}$ is a circuit, for $\ell \in \{3,4,5\}$.
But then $h_2 \in \cl(F \cup d) \cap \cl^*(F \cup d)$ where $F \cup d$ is exactly $3$-separating; a contradiction.
Thus $M/f_1\backslash f_2\backslash f_5$ or $M/f_2\backslash f_1\backslash f_5$ is $3$-connected as required.
\end{slproof}
Now \cref{flan6sub1,flan6sub2},
together with the assumption that $M$ has no $N$-detachable pairs, implies that
$M$ has a $4$-element cocircuit $\{f_1,f_2,f_5,z\}$.
\begin{sublemma}
\label{flan6sub3}
If $z \notin F$, then $\{f_3,z\}$ is an $N$-detachable pair.
\end{sublemma}
\begin{slproof}
First we show that $M/f_3/z$ has an $N$-minor.
Suppose (a) holds.
Since $M \backslash d \backslash f_5$ has an $N$-minor and $M \backslash d \backslash f_5 / f_2 /f_3$ is connected, \cref{m2.7} implies that $M \backslash f_5 / f_2 /f_3$ has an $N$-minor.
Since $\{f_1,f_4\}$ is a parallel pair in this matroid, $M /f_3 \backslash f_1 \backslash f_5$ has an $N$-minor, up to an $N$-label switch.
Now suppose (b) holds.
Since $M/f_1/f_2$ has an $N$-minor,
and $\{f_3,f_4\}$ and $\{f_5,d\}$ are parallel pairs in this matroid, $M/f_2 \backslash f_4 \backslash f_5$ has an $N$-minor.
Since $\{f_3,d\}$ is a series pair in this matroid, $M/f_2/f_3 \backslash f_5$ has an $N$-minor. Now $\{f_1,f_4\}$ is a parallel pair in this matroid, so $M/f_3 \backslash f_1 \backslash f_5$ has an $N$-minor.
So in either case $M /f_3 \backslash f_1 \backslash f_5$ has an $N$-minor.
Now $\{f_1,f_2,f_5,z\}$ is a cocircuit of $M$, so $\{f_2,z\}$ is a series pair in $M /f_3 \backslash f_1 \backslash f_5$. It follows that $M /f_3 /z$ has an $N$-minor as required.
Next we show that $\si(M/f_3/z)$ is $3$-connected.
Evidently, $z \in \cl^*(F \cup d)$,
where $F \cup d$ is exactly $3$-separating, so $z \notin \cl(F \cup d)$, by \cref{gutsstayguts},
implying $z \in \cl^*(E(M)-(F \cup \{d,z\}))$, by \cref{swapSepSides}.
Note that $M/f_3$ is $3$-connected by \cref{flanend}(i). Now
$((F-f_3) \cup d, \{z\}, E(M)-(F \cup \{d,z\}))$ is a cyclic $3$-separation in $M/f_3$.
It follows that $\co(M/f_3 \backslash z)$ is not $3$-connected, so
$\si(M/f_3 /z)$ is $3$-connected by Bixby's Lemma, as required.
Now, if $M/f_3/z$ is not $3$-connected, then $M$ has a $4$-element circuit containing $\{f_3,z\}$. By orthogonality, such a circuit~$C$ intersects the cocircuits $\{f_1,f_2,f_5,z\}$, $\{f_1,f_2,f_3,d\}$, and $\{f_3,f_4,f_5,d\}$ in at least two elements. So $C \subseteq F \cup \{d,z\}$. But then $z \in \cl(F \cup d)$; a contradiction.
We deduce that $M/f_3/z$ is $3$-connected.
\end{slproof}
By \cref{flan6sub3}, we may now assume that $z \in \{f_3,f_4\}$.
Since $\{f_1,f_2,f_3,d\}$ is a cocircuit of $M$, it follows from \ref{anothercovering} that $z = f_4$ so that $\{f_1,f_2,f_4,f_5\}$ is a cocircuit.
Now we examine the potential configurations of the $4$-element circuits $\{f_1,f_5,d,g_1\}$ and $\{f_2,f_5,d,g_2\}$, each of which is contained in $F \cup d$. If $g_1=f_3$, then $g_2=f_4$ due to \ref{anothercovering}.
In this situation, it is easily checked that $F \cup d$ is a skew-whiff $3$-separator\ of $M$, so that (ii) holds, as illustrated in \cref{tw2}. Similarly, if $g_1=f_4$, we obtain a skew-whiff $3$-separator\ as shown in \cref{tw1}.
\begin{figure}
\begin{subfigure}{0.45\textwidth}
\begin{tikzpicture}[rotate=90,scale=0.7,line width=1pt]
\tikzset{VertexStyle/.append style = {minimum height=5,minimum width=5}}
\clip (-2.5,-6) rectangle (4.4,2);
\draw (0,0) .. controls (-3,2) and (-3.5,-2) .. (0,-4);
\draw (0,0) -- (4.0,0.9);
\draw (0,-2) -- (2.5,-2.2);
\draw (0,-4) -- (3.8,-4.9);
\Vertex[x=3.0,y=0.67,LabelOut=true,Lpos=180,L=$f_1$]{a2}
\Vertex[x=2.0,y=0.45,LabelOut=true,Lpos=180,L=$f_2$]{a3}
\Vertex[x=2.5,y=-2.2,LabelOut=true,Lpos=90,L=$f_3$]{b1}
\Vertex[x=0.64,y=-2.056,LabelOut=true,Lpos=-45,L=$d$]{b2}
\Vertex[x=3.8,y=-4.9,LabelOut=true,L=$f_4$]{c1}
\Vertex[x=2.8,y=-4.67,LabelOut=true,L=$f_5$]{c2}
\draw[dashed] (3.8,-4.9) .. controls (2.0,-2) .. (4.0,0.9);
\draw[dashed] (2.8,-4.67) .. controls (1.0,-2) .. (3.0,0.67);
\draw[dashed] (1.8,-4.45) .. controls (0.25,-2) .. (2.0,0.45);
\draw (0,0) -- (0,-4);
\SetVertexNoLabel
\tikzset{VertexStyle/.append style = {shape=rectangle,fill=white}}
\Vertex[x=4.0,y=0.9]{a1}
\Vertex[x=1.5,y=-2.12]{b3}
\Vertex[x=1.8,y=-4.45]{c3}
\end{tikzpicture}
\caption{When $g_1 = f_3$.}
\label{tw2}
\end{subfigure}
\begin{subfigure}{0.45\textwidth}
\begin{tikzpicture}[rotate=90,scale=0.7,line width=1pt]
\tikzset{VertexStyle/.append style = {minimum height=5,minimum width=5}}
\clip (-2.5,-6) rectangle (4.4,2);
\draw (0,0) .. controls (-3,2) and (-3.5,-2) .. (0,-4);
\draw (0,0) -- (4.0,0.9);
\draw (0,-2) -- (2.5,-2.2);
\draw (0,-4) -- (3.8,-4.9);
\Vertex[x=3.0,y=0.67,LabelOut=true,Lpos=180,L=$f_1$]{a2}
\Vertex[x=2.0,y=0.45,LabelOut=true,Lpos=180,L=$f_2$]{a3}
\Vertex[x=2.5,y=-2.2,LabelOut=true,Lpos=90,L=$f_4$]{b1}
\Vertex[x=0.64,y=-2.056,LabelOut=true,Lpos=-45,L=$f_5$]{b2}
\Vertex[x=3.8,y=-4.9,LabelOut=true,L=$f_3$]{c1}
\Vertex[x=2.8,y=-4.67,LabelOut=true,L=$d$]{c2}
\draw[dashed] (3.8,-4.9) .. controls (2.0,-2) .. (4.0,0.9);
\draw[dashed] (2.8,-4.67) .. controls (1.0,-2) .. (3.0,0.67);
\draw[dashed] (1.8,-4.45) .. controls (0.25,-2) .. (2.0,0.45);
\draw (0,0) -- (0,-4);
\SetVertexNoLabel
\tikzset{VertexStyle/.append style = {shape=rectangle,fill=white}}
\Vertex[x=4.0,y=0.9]{a1}
\Vertex[x=1.5,y=-2.12]{b3}
\Vertex[x=1.8,y=-4.45]{c3}
\end{tikzpicture}
\caption{When $g_1 = f_4$.}
\label{tw1}
\end{subfigure}
\caption{The two possible labellings of the skew-whiff $3$-separator\ when \cref{flan5structure}(ii) holds.}
\label{tws}
\end{figure}
The final possibilities arise when $g_1=f_2$. In this case, \ref{anothercovering} forces $g_2=f_1$.
First, suppose that $\{f_3,f_4,f_5,d\}$ is a circuit.
Then $F \cup d$ is an elongated-quad $3$-separator\ with associated partition $(\{f_3,f_4,f_5,d\},\{f_1,f_2\})$, as illustrated in \cref{pss}; so (iii) holds.
We may now assume $\{f_3,f_4,f_5,d\}$ is independent.
Then, since $r(F \cup d) = 4$, the element $f_1$ (respectively, $f_2$) is in a circuit contained in $\{f_1,f_3,f_4,f_5,d\}$ (respectively, $\{f_2,f_3,f_4,f_5,d\}$).
Since $\{f_1,f_2,f_5,d\}$ is also a circuit contained in $F \cup d$, \ref{anothercovering} implies that these circuits contain $\{f_3,f_4\}$.
Similarly, due to the circuit $\{f_1,f_2,f_3,f_4\}$, \ref{anothercovering} implies that these circuits contain $\{f_5,d\}$.
So $\{f_1,f_3,f_4,f_5,d\}$ and $\{f_2,f_3,f_4,f_5,d\}$ are circuits.
It follows that $F\cup d$ is a twisted cube-like $3$-separator\ in $M^*$, so (iv) holds. The labelling of the twisted cube-like $3$-separator\ in the dual is illustrated in \cref{tw3}.
\end{proof}
\begin{figure}
\begin{tikzpicture}[rotate=90,scale=0.8,line width=1pt]
\tikzset{VertexStyle/.append style = {minimum height=5,minimum width=5}}
\clip (-2.5,2) rectangle (3.0,-6);
\draw (0,0) .. controls (-3,2) and (-3.5,-2) .. (0,-4);
\draw (0,0) -- (2,-2) -- (0,-4);
\draw (0,0) -- (2.5,0.5) -- (2,-2);
\draw (0,0) -- (2.25,-0.75);
\draw (2,-2) -- (1.25,0.25);
\Vertex[x=1.25,y=0.25,LabelOut=true,L=$f_5$,Lpos=180]{c1}
\Vertex[x=2.25,y=-0.75,LabelOut=true,L=$f_3$,Lpos=90]{c2}
\Vertex[x=2.5,y=0.5,LabelOut=true,L=$f_4$,Lpos=180]{c3}
\Vertex[x=1.5,y=-0.5,LabelOut=true,L=$d$,Lpos=135]{c4}
\Vertex[x=1.33,y=-2.67,LabelOut=true,L=$f_2$,Lpos=45]{c5}
\Vertex[x=0.67,y=-3.33,LabelOut=true,L=$f_1$,Lpos=45]{c6}
\draw (0,0) -- (0,-4);
\SetVertexNoLabel
\tikzset{VertexStyle/.append style = {shape=rectangle,fill=white}}
\end{tikzpicture}
\caption{The labelling of the elongated-quad $3$-separator\ when \cref{flan5structure}(iii) holds.}
\label{pss}
\end{figure}
\begin{figure}
\centering
\begin{tikzpicture}[rotate=90,scale=0.65,line width=1pt]
\tikzset{VertexStyle/.append style = {minimum height=5,minimum width=5}}
\clip (-2.78,-6) rectangle (5.0,2);
\draw (0,0) .. controls (-3,2) and (-3.5,-2) .. (0,-4);
\draw (0,0) -- (4.0,0.9);
\draw (0,-2) -- (2.5,-2.2);
\draw (0,-4) -- (3.8,-4.9);
\Vertex[x=2.0,y=0.45,LabelOut=true,Lpos=180,L=$d$]{a3}
\Vertex[x=2.5,y=-2.2,LabelOut=true,Lpos=90,L=$f_4$]{b1}
\Vertex[x=0.61,y=-2.05,LabelOut=true,Lpos=35,L=$f_5$]{b2}
\Vertex[x=3.00,y=-4.7,LabelOut=true,L=$f_1$]{c1}
\Vertex[x=2.15,y=-4.5,LabelOut=true,L=$f_2$]{c2}
\Vertex[x=4.0,y=0.9,LabelOut=true,Lpos=180,L=$f_3$]{a1}
\draw[dashed] (3.8,-4.9) .. controls (2.0,-2) .. (4.0,0.9);
\draw[dashed] (1.3,-4.3) .. controls (0.25,-2) .. (2.0,0.45);
\draw (0,0) -- (0,-4);
\SetVertexNoLabel
\tikzset{VertexStyle/.append style = {shape=rectangle,fill=white}}
\Vertex[x=3.8,y=-4.9]{c1}
\Vertex[x=1.3,y=-4.3]{c3}
\end{tikzpicture}
\caption{The labelling of the twisted cube-like $3$-separator\ of $M^*$
when \cref{flan5structure}(iv) holds.}
\label{tw3}
\end{figure}
By combining \cref{flannew,5flan,flan5structure}, we obtain the following:
\begin{corollary}
\label{flancorollary}
Let $M$ be a $3$-connected matroid with a $3$-connected matroid~$N$ as a minor, where $|E(N)| \ge 4$, and every triangle or triad of $M$ is $N$-grounded.
Let $d$ be an element of $M$ such that $M\backslash d$ is $3$-connected
and has a flan~$F$ with ordering $(f_1,f_2,\dotsc,f_t)$ where $t \ge 5$.
Suppose that $M \backslash d \backslash f_5$ has an $N$-minor with $|\{f_1,\dotsc,f_4\} \cap E(N)| \le 1$.
Then either
\begin{enumerate}
\item $M$ has an $N$-detachable pair, or
\item $F\cup d$ is either a skew-whiff $3$-separator\ of $M$, an elongated-quad $3$-separator\ of $M$, or a twisted cube-like $3$-separator\ of $M^*$.
\end{enumerate}
\end{corollary}
\begin{proof}
If $t \ge 6$, then (i) holds by \cref{flannew}.
So suppose $t=5$ and $F$ is a maximal flan.
First, suppose that $d$ fully blocks $F$. Towards an application of \cref{5flan}, we claim that $f_3$ is $N$-deletable in $M \backslash d$.
Observe that $(F-f_5, \{f_5\}, E(M \backslash d) - F)$ is a cyclic $3$-separation of $M \backslash d$. Since $F-f_5$ is a circuit, \cref{doublylabel}(i) implies that
$M \backslash d \backslash f_3$ has an $N$-minor, as claimed. Now, by \cref{5flan}, we may assume that $d$ does not fully block $F$.
Then $d \in \cl(F)$, and so, by \cref{flan5structure}, the \lcnamecref{flancorollary} follows.
\end{proof}
\section{Unveiling the $3$-separating set $X$}
\label{secunveil}
In this section, we prove our main result, \cref{foundation}.
For the entirety of the section, we work under the following hypotheses.
Let $M$ be a $3$-connected matroid and let $N$ be a $3$-connected minor of $M$ where
$|E(N)| \ge 4$, and
every triangle or triad of $M$ is $N$-grounded.
Suppose, for some $d \in E(M)$, that $M \backslash d$ is $3$-connected and has a cyclic $3$-separation $(Y, \{d'\}, Z)$ with $|Y| \ge 4$, where $M \backslash d \backslash d'$ has an $N$-minor with $|Y \cap E(N)| \le 1$.
First, we handle the cases where $Y$ contains either a $4$-element cosegment, or a particular configuration of two triads.
\begin{lemma}
\label{coseg}
If $Y$ contains a $4$-element cosegment of $M \backslash d$, then $M$ has an $N$-detachable pair.
\end{lemma}
\begin{proof}
Suppose $X$ is a $4$-element cosegment of $M \backslash d$ contained in $Y$.
If $X \subseteq \cl^*(Z)$, then $X \cup d'$ is a cosegment in $M\backslash d$.
Since $M \backslash d \backslash d'$ has an $N$-minor, neither $d$ nor $d'$ is in a triad of $M$, and any pair of elements in $\cl^*_{M \backslash d \backslash d'}(X)$ is $N$-contractible.
In particular, there are no triads of $M$ contained in $\cl^*(X \cup \{d,d'\})$, and so
$M^*|(X \cup \{d,d'\}) \cong U_{3,6}$.
Now, by \cref{6pointplane2}, $M$ has an $N$-detachable pair.
So we may assume that $|X \cap \cl^*(Z)| \le 1$.
Let $x \in X$, where $x \in \cl^*(Z)$ if such an element exists.
Since $x' \in \cl^*(X -\{x,x'\})$ for each $x' \in X-x$, we have $x' \notin \cl(\cl^*(Z))$.
Thus, by \cref{doublylabel}, each $x' \in X-x$ is
doubly $N$-labelled in $M \backslash d$.
As $d$ blocks every triad of $M \backslash d$ contained in $X$, the set $X \cup d$ is a $5$-element coplane in $M$.
If $d$ does not fully block $X$, then $d \in \cl(X)$, in which case $X \cup d$ is $3$-separating in $M$, and
$M$ has an $N$-detachable pair by the dual of \cref{basicplaneupgrade}.
So we may assume that $d$ fully blocks $X$.
Let $p' \in X-x$.
Towards an application of \cref{planeupgrade}, we claim that
for distinct elements $u,v \in \cl(X)-X$, either $M/p' /u$ or $M/p' /v$ has an $N$-minor.
Recall that $M \backslash d/p'$ has an $N$-minor.
By the dual of \cref{rank2Remove}, $M \backslash d /p'$ is $3$-connected.
Now $(Y-p', \{d'\}, Z)$ is a path of $3$-separations in $M \backslash d/p'$.
Let $Z' = \cl^*_{M\backslash d}(Z)-d'$ and $Y' = Y-Z'$.
Then $(Y'-p', \{d'\}, Z')$ is a path of $3$-separations of $M \backslash d/p'$ where $Z' \cup d'$ is coclosed, and $X-x \subseteq Y'$.
Note that $Y'-p'$ contains a circuit in $M\backslash d/p'$, since $Y'$ contains a circuit in $M \backslash d$.
In order to show that $(Y'-p', \{d'\}, Z')$ is a cyclic $3$-separation of $M\backslash d/p'$, it remains only to observe that $d' \in \cl^*_{M \backslash d}(Y'-p')$,
which follows from the fact that $p' \in \cl^*_{M \backslash d}(X-\{x,p'\})$.
We may assume there are distinct elements $u,v \in \cl(X)-X$, otherwise the claim holds trivially.
Then
$\{u,v\} \subseteq \cl(Y)$.
If $\{u,v\} \subseteq Y-p'$, then either $M/p'/u$ or $M/p'/v$ is $3$-connected by \cref{doublylabel}(ii).
Moreover, if $\{u,v\} \cap Z \neq \emptyset$, then, since $d' \in \cl^*_{M\backslash d}(Y) \cap Z$, \cref{presingle} implies that $\{u,v\} -Z \neq \emptyset$.
So suppose, without loss of generality, that $v \in Z$ and $u \in Y$. By \cref{doublylabel}(ii) again, the claim holds unless $u \in \cl_{M/p'}(Z')$.
But then it follows that $Y - \{p',u\}$ is exactly $3$-separating in $M\backslash d/p'$, with $\{u,v\} \subseteq \cl_{M/p'}(Y-\{p',u\}) \cap Z$ and $d' \in \cl^*(Y-\{p',u\}) \cap Z$, contradicting \cref{presingle}.
Now $M$ has an $N$-detachable pair by the dual of \cref{planeupgrade}.
\end{proof}
\begin{lemma}
\label{prespecifictriads}
Suppose that $Y$ contains a set $X = \{s_1, s_2, t_1, t_2, t_3\}$ such that the following hold:
\begin{enumerate}[label=\rm(\alph*)]
\item each $x \in X$ is not in a triangle or triad of $M$;
\item $\{s_1, s_2, t_3\}$ and $\{t_1, t_2, t_3\}$ are triads of $M \backslash d$;
\item for each $i \in \seq{3}$ there are elements $v_i, w_i \in \cl(X \cup d) - (X \cup d)$ such that $\{s_1, t_i, d, v_i\}$ and $\{s_2, t_i, d, w_i\}$ are circuits; and
\item $P = \{v_1,v_2,v_3,w_1,w_2,w_3\}$ is a $6$-element rank-$3$ set, and if $P$ contains a triangle~$T$, then $T$ is either $\{v_i,v_j,w_k\}$ or $\{v_i,w_j,w_k\}$ for some $\{i,j,k\}=\{1,2,3\}$.
\end{enumerate}
Then $M$ has an $N$-detachable pair.
\end{lemma}
\begin{proof}
Let $i \in [3]$.
Since $\{s_1,t_i,d,v_i\}$ and $\{s_2,t_i,d,w_i\}$ are circuits, $\{s_1,s_2,t_i,v_i,w_i\}$ contains a circuit, by circuit elimination.
But $t_i \in \cl^*_{M \backslash d}(\{t_1,t_2,t_3\}-t_i)$, so \cref{swapSepSides} and (a) imply that $\{s_1,s_2,v_i,w_i\}$ is a circuit.
\begin{sublemma}
\label{goodminors}
Either $M$ has an $N$-detachable pair, or $M \backslash d \backslash v_i$ and $M \backslash d \backslash w_i$ have $N$-minors for each $i \in \seq{3}$.
\end{sublemma}
\begin{slproof}
Note that if $d' \in \cl^*_{M \backslash d}(\{t_1,t_2,t_3\})$, then $\{d',t_1,t_2,t_3\}$ is a cosegment in $M\backslash d$ whose triads are blocked by $d$, so $\{d,d',t_1,t_2,t_3\}$ is a $5$-element plane in $M^*$. But then, by the dual of \cref{basicplaneupgrade}, $M$ has an $N$-detachable pair.
So we may assume that $d' \notin \cl^*_{M \backslash d}(\{t_1,t_2,t_3\})$.
Now $M \backslash d/s_1$ is $3$-connected by the dual of \cref{r3cocirc}, and $M \backslash d \backslash d' / s_1$ has an $N$-minor by \cref{m2.7}.
Applying \cref{doublylabel}(ii),
we deduce that $M \backslash d / s_1 / s_2$ has an $N$-minor, since $d' \notin \cl^*_{M \backslash d/s_1}(X-\{s_1,s_2\})$.
As $\{v_i,w_i\}$ is a parallel pair in $M \backslash d /s_1 /s_2$, for each $i \in \seq{3}$, the matroids $M \backslash d \backslash v_i$ and $M \backslash d \backslash w_i$ have $N$-minors.
\end{slproof}
If $\{v_i,v_j,w_k\}$ and $\{v_i,w_j,w_k\}$ are independent for all $\{i,j,k\} = \{1,2,3\}$, then $M|P \cong U_{3,6}$, and $M$ has an $N$-detachable pair by \cref{6pointplane,goodminors}.
So, without loss of generality, we may assume that $\{v_i,v_j,w_3\}$ or $\{v_i,w_j,w_3\}$ is a triangle $T$ for some $\{i,j\} = \{1,2\}$.
We claim that $w_3$ is not in a triad of $M \backslash d$.
Towards a contradiction, suppose that $T^*$ is a triad of $M \backslash d$ containing $w_3$.
By (d), $\{v_1,v_2,w_1,w_2\}$ is a circuit $C$.
By orthogonality between $T^*$ and $T$, and between $T^*$ and $C$, we deduce that $T^* -w_3\subseteq \{v_1,v_2,w_1,w_2\}$.
But then $T^*$ intersects the circuit $\{s_1,s_2,v_3,w_3\}$ in a single element; a contradiction.
Now, by Tutte's Triangle Lemma, either $M \backslash d\backslash w_3$ or $M \backslash d\backslash v_i$ is $3$-connected.
By \cref{goodminors}, it follows that $M$ has an $N$-detachable pair, thus completing the proof.
\end{proof}
\begin{lemma}
\label{specifictriads}
Suppose that $Y$ contains a set $X = \{s_1, s_2, t_1, t_2, u\}$ such that the following hold:
\begin{enumerate}[label=\rm(\alph*)]
\item $\{s_1, s_2, u\}$ and $\{t_1, t_2, u\}$ are triads of $M \backslash d$,\label{st3}
\item $X$ is closed in $M \backslash d$,\label{st1}
\item $X$ is $3$-separating in $M \backslash d$,\label{st2}
\item $X$ is not a cosegment in $M \backslash d$, and\label{st5}
\item there are no $4$-element circuits contained in $X$.\label{st4}
\end{enumerate}
Then $M$ has an $N$-detachable pair.
\end{lemma}
\begin{proof}
Since $X$ is the union of two triads that meet at $u$, but $X$ is not a cosegment, $r_{M \backslash d}^*(X) = 3$. As $X$ is a $5$-element $3$-separating set, $r_{M \backslash d}(X) = 4$.
It follows that $E(M \backslash d)-X$ is coclosed, due to \ref{st4}.
Since $x \in \cl^*_{M \backslash d}(X-x)$ for each $x \in X$, we also have that $E(M \backslash d)-X$ is closed.
Hence each element in $X$ is doubly $N$-labelled in $M \backslash d$ by \cref{doublylabel}.
It follows that each $x \in X$ is not contained in an $N$-grounded\ triangle or triad.
Assume that $M$ does not contain an $N$-detachable pair.
\begin{sublemma}
\label{sll1}
For distinct $s \in \{s_1 , s_2, u\}$ and $t \in \{t_1 , t_2, u\}$, the matroid $M \backslash d/s/t$ is $3$-connected and has an $N$-minor.
\end{sublemma}
\begin{slproof}
Let $s \in \{s_1, s_2\}$ and $t \in \{t_1, t_2, u\}$.
Since $X$ is a corank-$3$ circuit, and $s$ is not contained in a triangle,
the dual of \cref{r3cocirc} implies that $M\backslash d /s$ is $3$-connected.
Moreover, $X-s$ is a corank-$3$ circuit in $M \backslash d / s$, so $M \backslash d / s / t$ is $3$-connected unless $\{s,t\}$ is contained in a $4$-element circuit of $M \backslash d$. But, by orthogonality, such a circuit contains another element of $X$, and so, as $X$ is closed in $M \backslash d$, the circuit is contained in $X$; a contradiction. It follows by symmetry that $M \backslash d / s /t$ is $3$-connected.
It remains to show that $M \backslash d/s/t$ has an $N$-minor.
By swapping the labels on $\{s_1,s_2\}$ and $\{t_1,t_2\}$, if necessary, we may assume that $s \neq u$.
Recall that $M\backslash d/s$ has an $N$-minor.
Now $M \backslash d/s$ is $3$-connected by the dual of \cref{r3cocirc}, and $M \backslash d \backslash d' / s$ has an $N$-minor by \cref{m2.7}.
Applying \cref{doublylabel}(ii), we deduce that $M \backslash d / s / t$ has an $N$-minor, since $t \in \cl^*_{M\backslash d}(\{t_1,t_2,u\}-t)$, so $t \notin \cl(E(M) - \{t_1,t_2,u\})$.
\end{slproof}
Now, as $M$ has no $N$-detachable pairs, \cref{sll1} implies that for each distinct pair $s,t$ with $s \in \{s_1 , s_2, u\}$ and $t \in \{t_1 , t_2, u\}$, there is a circuit of $M$ containing $\{d,s,t\}$.
\begin{sublemma}
\label{sll2}
There are no $4$-element circuits of $M$ contained in $X \cup d$.
\end{sublemma}
\begin{slproof}
Suppose $X \cup d$ contains a $4$-element circuit $C$.
Then $d \in C$, by \ref{st4}.
%
Let $S = \{s,s'\} \in \{\{s_1,s_2\}, \{t_1,t_2\}\}$, and $T = \{t,t',t''\} = X-S$.
We may assume, without loss of generality, that $C=\{d,s,t,x\}$, where $x \neq t'$.
Now $\{d,s,t'\}$ is also contained in a $4$-element circuit, $\{d,s,t',y\}$ say. By circuit elimination, $\{s,t,t',x,y\}$ contains a circuit. By \ref{st4}, $\{s,t,t',x\}$ is independent, so \ref{st1} implies that $y \in X$,
and $y \notin \{s,t,t',x\}$,
and thus $\{x,y\} = \{s',t''\}$.
If $x = t''$, then $\{d,s,t,t''\}$ and $\{d,s,t',s'\}$ are circuits, but there is also a $4$-element circuit containing $\{d,s',t\}$.
So let $\{d,s',t,z\}$ be a circuit, for some $z$. Now $\{s,t,t'',s',z\}$ contains a circuit, by circuit elimination,
and it follows, by \ref{st1} and \ref{st4}, that $z \in X-\{s,s',t,t''\}$, so $z=t'$.
But then circuit elimination on the circuits $\{d,s,t',s'\}$ and $\{d,s',t,t'\}$ implies that $\{s,s',t,t'\}$ contains a circuit; a contradiction.
The argument is similar when $x=s'$.
\end{slproof}
Now, letting $t_3 = u$,
for each $i \in \seq{3}$ there are elements $v_i, w_i \in \cl(X \cup d) - (X \cup d)$ such that $\{s_1, t_i, d, v_i\}$ and $\{s_2, t_i, d, w_i\}$ are circuits.
Observe also that $d \notin \cl(X)$, since $v_i,w_i \notin \cl(X)$.
\begin{sublemma}
\label{almostU36}
Let $P = \{v_1,v_2,v_3,w_1,w_2,w_3\}$.
Then $|P|=6$ and $r(P)=3$.
Moreover, if $P$ contains a triangle~$T$, then $T$ is either $\{v_i,v_j,w_k\}$ or $\{v_i,w_j,w_k\}$ for some $\{i,j,k\}=\{1,2,3\}$.
\end{sublemma}
\begin{slproof}
If $v_i=v_{i'}$ for distinct $i,i' \in \seq{3}$, then $\{s_1, t_i,t_{i'},d\}$ contains a circuit, by the circuit elimination axiom, contradicting \ref{sll2}.
Similarly, the $w_i$ are pairwise distinct for $i \in \seq{3}$.
Say $v_i = w_j$ for some $i, j \in \seq{3}$. Then, again by circuit elimination, there is a circuit $\{s_1, s_2, t_i, t_j, v_i\}$.
But $X$ is closed in $M \backslash d$, so $v_i \notin \cl(\{s_1, s_2, t_i, t_j\})$. Hence $\{s_1, s_2, t_i, t_j\}$ is a circuit of $M$, contradicting
\ref{st4}.
Hence the elements $v_1, v_2, v_3, w_1, w_2, w_3$ are distinct.
By \ref{st2},
$\cl(X \cup d) - (X \cup d)$ has rank at most 3.
If $r(\{v_1,v_2,v_3\}) \le 2$, then $\{s_1,d,v_1,v_2,v_3\}$ has rank at most four, but spans the rank-$5$ set $X \cup d$; a contradiction.
A similar argument applies if $r(\{w_1,w_2,w_3\}) \le 2$, or, for some distinct $i,j \in [3]$ either $r(\{v_i,v_j,w_i\}) \le 2$ or $r(\{v_i,w_i,w_j\}) \le 2$.
\end{slproof}
The \lcnamecref{specifictriads} now follows from \cref{almostU36,prespecifictriads}.
\end{proof}
Finally, we come to the main result of this paper.
Recall that $d \in E(M)$,
the matroid $M \backslash d$ is $3$-connected and has a cyclic $3$-separation $(Y, \{d'\}, Z)$ with $|Y| \ge 4$, and $M \backslash d \backslash d'$ has an $N$-minor with $|Y \cap E(N)| \le 1$.
\begin{theorem}
\label{foundation}
Suppose $M$ has no $N$-detachable pairs.
Then there is a subset $X$ of $Y$ such that
\begin{enumerate}
\item $|X| \ge 4$ and
$X$ is $3$-separating in $M \backslash d$, and\label{setup}
\item either
\begin{enumerate}[label=\rm(\alph*)]
\item $X \cup \{c,d\}$ is an elongated-quad $3$-separator\ of $M$, a skew-whiff $3$-separator\ of $M$, or a twisted cube-like $3$-separator\ of $M^*$, for some $c \in \cl^*_{M \backslash d}(X)-X$;
or\label{problemseps}
\item for every $x \in X$,
\begin{enumerate}[label=\rm(\Roman*)]
\item $\co(M \backslash d \backslash x)$ is $3$-connected,\label{ond}
\item $M \backslash d / x$ is $3$-connected, and\label{com}
\item $x$ is doubly $N$-labelled in $M \backslash d$.\label{onddl}
\end{enumerate}\label{nonproblemseps}
\end{enumerate}
\end{enumerate}
\end{theorem}
\begin{proof}
Choose $X \subseteq Y$ that is minimal with respect to
\ref{setup}.
Let $W = E(M \backslash d) -X$.
Suppose that \cref{problemseps} does not hold; then, it remains to show that \cref{nonproblemseps} holds.
By \cref{coseg}, we may
assume that $X$ is not a cosegment of $M \backslash d$.
\begin{sublemma}
\label{dli}
Every element in $Y \cup d'$ is $N$-deletable in $M \backslash d$, and
every element in $X$ is doubly $N$-labelled in $M \backslash d$.
\end{sublemma}
\begin{slproof}
If there is some element $x \in X \cap \cl^*_{M \backslash d}(Z)$, then $X-x$ is $3$-separating, by \cref{extendSep}.
If $|X| > 4$, this contradicts the minimality of $X$. On the other hand, if $|X| = 4$, then $X-x$ is a triad, since $X-x$ cannot be an $N$-grounded\ triangle by \cref{doublylabel}(ii). But then $X$ is a $4$-element cosegment, contradicting \cref{coseg}.
Now we may assume that $Z \cup d'$ is coclosed in $M \backslash d$.
By \cref{doublylabel}(i), every element in $Y$ is $N$-deletable, while $d'$ is $N$-deletable by hypothesis.
If there is some element $x \in X$ that is not $N$-contractible, then $x \in \cl(Z)$
by \cref{doublylabel}(ii). Then,
the minimality of $X$ implies that $|X|=4$.
Since $X-x$ is not an $N$-grounded\ triangle, $X-x$ is a triad, and $X$ is a circuit.
Moreover, $(X-x, \{x\},W)$ is a vertical $3$-separation, so
$\co(M \backslash d \backslash x)$ is $3$-connected by Bixby's Lemma.
Since $M$ has no $N$-detachable pairs, $x$ is in a triad of $M \backslash d$ that meets $X-x$ and $W$.
Let this triad be $\{x',x,w\}$ where $x' \in X-x$.
Since $M \backslash d \backslash x'$ has an $N$-minor, and $\{x,w\}$ is a series pair in this matroid, up to an $N$-label switch the matroid $M \backslash d / x$ has an $N$-minor after all,
thus completing the proof of \cref{dli}.
\end{slproof}
Note, in particular, that no triangle meets $X$.
\begin{sublemma}
\label{dlflan}
If $|X|=4$ and $X \cup f_5$ is a flan for some $f_5 \in W$, with flan ordering $(f_1,f_2,f_3,f_4,f_5)$ for some labelling $\{f_1,f_2,f_3,f_4\}$ of $X$,
then either $M$ has an $N$-detachable pair, or \cref{problemseps} holds.
\end{sublemma}
\begin{slproof}
Suppose $f_5$ is $N$-deletable in $M \backslash d$.
Then, by \cref{flancorollary}, either $M$ has an $N$-detachable pair, or \cref{problemseps} holds.
So we may assume that $f_5$ is not $N$-deletable in $M \backslash d$.
By \cref{dli}, $f_5 \in Z$.
Now $(Y \cup f_5, \{d'\}, Z-f_5)$ is a
path of $3$-separations in $M \backslash d$, by \cref{extendSep}.
By \cref{swapSepSides,exactSeps}, $d' \in \cl^*_{M \backslash d}(Z-f_5)$.
Moreover, $Z-f_5$ contains a circuit, since $Z$ contains a circuit and $f_5 \notin \cl(Z-f_5)$, so this path of $3$-separations is a cyclic $3$-separation, and $|(Y \cup f_5) \cap E(N)| \le 1$, by \cref{m2.7} and since $|Y \cap E(N)| \le 1$ and $|E(N)| \ge 4$.
Suppose there is some $f_6 \in \cl(X \cup f_5) \cap (W-f_5)$ so that $X \cup \{f_5,f_6\}$ is a flan.
Now $(Y \cup \{f_5,f_6\}, \{d'\}, Z-\{f_5,f_6\})$ is a path of $3$-separations
where $d'$ is a coguts element,
using a similar argument as in the previous paragraph.
To show this is a cyclic $3$-separation, we now require only that $r^*_{M \backslash d}(Z-\{f_5,f_6\}) \ge 3$.
Suppose not.
Since $M \backslash d \backslash d'$ has an $N$-minor with $|(Y \cup f_5) \cap E(N)| \le 1$, \
\cref{m2.7} implies that $|(Y \cup \{f_5,f_6\}) \cap E(N)| \le 1$.
But now $r^*_{M \backslash d \backslash d'}(Z-\{f_5,f_6\}) \le 1$; a contradiction.
By \cref{doublylabel}(ii),
since $f_5$ is not $N$-deletable we have $f_5 \in \cl^*(Z - \{f_5,f_6\})$.
But $f_6 \in \cl(Z - \{f_5,f_6\})$ and $d' \in \cl^*(Z - \{f_5,f_6\})$, contradicting \cref{presingle}.
So $X \cup f_5$ is a maximal flan.
Note that $M \backslash d \backslash f_3$ has an $N$-minor, by \cref{dli}.
If $d$ fully blocks $X \cup f_5$,
then,
by \cref{5flan}, $M$ has an $N$-detachable pair.
Towards an application of \cref{flan5structure}, we show that $M /f_i/f_{i'}$ has an $N$-minor for all distinct $i,i' \in \seq{3}$.
Let $i \in \{1,2\}$.
By \cref{flanend,dli}, $M\backslash d/f_i$ is $3$-connected and has an $N$-minor.
Now $((Y-f_i) \cup f_5, \{d'\},Z-f_5)$ is a cyclic $3$-separation in $M \backslash d/f_i$. Since $\{f_3,f_4,f_5\}$ is a triad in $M \backslash d$, we have $f_3 \notin \cl(Z-f_5)$, so $M \backslash d/f_i/f_3$ has an $N$-minor by \cref{doublylabel}(ii).
Now, $\{f_5,d'\} \subseteq \cl^*_{M \backslash d/f_i}(Z-f_5)$, so no element in $(Y-f_i) \cup f_5$ is also in $\cl_{M \backslash d/f_i}(Z-f_5)$ by \cref{presingle}.
Hence $M \backslash d/f_1/f_2$ also has an $N$-minor by \cref{doublylabel}(ii).
Now, by \cref{flan5structure}, either $M$ has an $N$-detachable pair or \cref{problemseps} holds, thus completing the proof.
\end{slproof}
Next we prove that \ref{ond} holds for each $x \in X$.
Towards a contradiction,
let $x$ be an element of $X$ such that $\co(M \backslash d \backslash x)$ is not $3$-connected, and let $(P, \{x\}, Q)$ be a cyclic $3$-separation of $M \backslash d$.
\begin{sublemma}
\label{ondi}
$W \cap P \neq \emptyset$ and $W \cap Q \neq \emptyset$.
\end{sublemma}
\begin{slproof}
Suppose that $W \cap Q = \emptyset$.
Then $Q \cup x \subseteq X$ and $|Q| \ge 3$.
But $Q$ and $Q \cup x$ are $3$-separating, so the minimality of $X$ implies that $X =Q \cup x$ and $|Q|=3$. Since $Q$ contains a circuit, $Q$ is a triangle of $M\backslash d$, and hence of $M$. But,
by \cref{dli},
$Q$ is not $N$-grounded; a contradiction. So $W \cap Q$ and, by symmetry, $W \cap P$ are non-empty.
\end{slproof}
\begin{sublemma}
\label{ondii}
Up to swapping $P$ and $Q$,
$|X \cap Q| = 2$ and
$|W \cap P| \ge 2$.
\end{sublemma}
\begin{slproof}
Since $|W| \ge 3$, we may assume that $|W \cap P| \ge 2$.
By uncrossing, $X \cap Q$ and $(X \cap Q) \cup x$ are $3$-separating in $M \backslash d$.
If $|X \cap Q| \le 1$, then $|W \cap Q| \ge 2$, in which case $X \cap P$ and $(X \cap P) \cup x$ are also $3$-separating in $M \backslash d$, by uncrossing.
By the minimality of $X$, it follows that $|X|=4$, so
either $X \cap Q = \emptyset$ and $|X\cap P| = 3$, or $|X \cap Q| = 1$ and $|X \cap P| = 2$. In the first case, $X-x$ is a triad, since it cannot be an $N$-grounded\ triangle, so $X$ is a $4$-element cosegment, contradicting \cref{coseg}. In the latter case, \cref{ondii} holds after swapping $P$ and $Q$.
On the other hand, if $|X \cap Q| > 2$, then the minimality of $X$ implies that $X \cap P = \emptyset$.
But then $X-x$ is a triad, so $X$ is a $4$-element cosegment, contradicting \cref{coseg}.
\end{slproof}
Now, note that if $|W \cap Q| = 1$, then $Q$ is a triangle in $M \backslash d$,
but $Q$ is not an $N$-grounded\ triangle since, by \cref{dli}, it contains an $N$-contractible element; a contradiction.
So $|W \cap Q| \ge 2$.
\begin{sublemma}
\label{ondiii}
$|X \cap P| = 2$.
\end{sublemma}
\begin{slproof}
By uncrossing, $X \cap P$ and $(X \cap P) \cup x$ are $3$-separating.
If $|X \cap P| > 2$, then this contradicts the minimality of $X$.
So assume that $X \cap P = \{t\}$, say.
Now $X-t$ is a triad, and
$t \in \cl^{(*)}(X-t)$.
If $t \in \cl^*(X-t)$, then $X$ is a $4$-element cosegment, contradicting \cref{coseg}.
So $t \in \cl(X-t)$.
By the dual of \cref{r3cocircsi}, $\co(M \backslash d \backslash t)$ is $3$-connected, so, as $M$ has no $N$-detachable pairs, $t$ is in a triad that, by orthogonality, meets $X-t$. If this triad does not contain $x$, then, by the dual of \cref{r3cocircsi} again, $\co(M \backslash d \backslash x)$ is $3$-connected; a contradiction.
Let $f_5$ be the element of the triad in $W$, and let $X \cap Q = \{f_1,f_2\}$.
Now $X$ is contained in a $5$-element flan with ordering $(f_1,f_2,x,t,f_5)$.
Thus, by \cref{dlflan}, either $M$ has an $N$-detachable pair or \cref{problemseps} holds; a contradiction.
\end{slproof}
\begin{sublemma}
\label{ondiv}
$X$ is closed in $M\backslash d$.
\end{sublemma}
\begin{slproof}
Suppose $c \in \cl(X) - X$. We may assume that $c \in P$.
Since $|W \cap Q| \ge 2$,
both $X \cap P$ and
$(X \cap P) \cup c$ are $3$-separating, by uncrossing. So
$c \in \cl(X \cap P)$, and $(X \cap P) \cup c$ is a triangle. Since this triangle contains an $N$-contractible element, it is not $N$-grounded, which is contradictory.
\end{slproof}
\begin{sublemma}
\label{ondv}
$X$ contains no $4$-element circuits.
\end{sublemma}
\begin{slproof}
Let $X \cap P = \{p_1,p_2\}$ and $X \cap Q = \{q_1,q_2\}$.
Suppose $X$ has a $4$-element circuit. Either this circuit contains $x$ or it does not. Suppose that it does: without loss of generality, let $\{p_1,p_2,x,q_1\}$, be this circuit. Since $\{p_1,p_2,x\}$ is a triad, $\{p_1,p_2,x,q_1\}$ is $3$-separating, contradicting the minimality of $X$.
Now we may assume there is no $4$-element circuit in $X$ containing $x$.
Thus $r(\{p_1,p_2,x,q_1,q_2\})=4$, and it follows,
by \cref{swapSepSides}, that $x \in \cl^*(W)$, so $X-x$ is $3$-separating by \cref{extendSep},
again contradicting the minimality of $X$.
\end{slproof}
Now, since \cref{ondii,ondiii,ondiv,ondv} hold, we can apply \cref{specifictriads} to deduce that $M$ has an $N$-detachable pair; a contradiction. This proves that each $x \in X$ satisfies \ref{ond}.
Recall that each $x \in X$ satisfies \cref{onddl} by \cref{dli}.
It remains to consider \ref{com}.
Suppose $M \backslash d / x$ is not $3$-connected for some $x \in X$.
Since $x$ is not in a triangle, $\si(M \backslash d/x)$ is not $3$-connected, so $M \backslash d$ has a vertical $3$-separation $(P, \{x\}, Q)$.
We may assume, without loss of generality, that $|W \cap P| \ge 2$.
Thus, by uncrossing, both $X \cap Q$ and $(X \cap Q) \cup x$ are $3$-separating.
By the minimality of $X$, we have $|X \cap Q| \le 3$, and if $|X \cap Q|=3$, then $X \cap P = \emptyset$.
If $|X \cap Q| = 2$, then $(X \cap Q) \cup x$ is a triangle or a triad, but as $x \in \cl(Q)$ and $X$ contains no triangles,
this leads to a contradiction.
Suppose $|X \cap Q| \le 1$. Then $|W \cap Q| \ge 2$, in which case $X \cap P$ and $(X \cap P) \cup x$ are $3$-separating, by uncrossing.
Now $|X\cap P| \ge 2$, but, by the minimality of $X$, $|X\cap P| \le 3$ and if $|X\cap P| = 3$ then $X \cap Q=\emptyset$.
Moreover,
$|X \cap P| \neq 2$ since $x \in \cl(X \cap P)$ and $X$ does not contain any triangles.
It follows that
$X \cap Q = \emptyset$ and $|X \cap P| = 3$.
Now $\{|X \cap P|, |X \cap Q|\} = \{0,3\}$, $X-x$ is a triad, and $X$ is a circuit.
Since $\co(M \backslash d \backslash x)$ is $3$-connected, but $M \backslash d \backslash x$ is not, $x$ is in a triad $T^*$ of $M \backslash d$ that meets both $X-x$ and $W$, by orthogonality.
Let $T^* \cap W=\{f_5\}$, and observe that $X \cup f_5$ is a $5$-element flan of $M \backslash d$.
By \cref{dlflan}, either $M$ has an $N$-detachable pair or \cref{problemseps} holds; a contradiction.
So each $x \in X$ also satisfies \ref{com}, as required.
This completes the proof of \cref{foundation}.
\end{proof}
\section*{Acknowledgements}
We thank the referees for their careful reading of the paper.
\bibliographystyle{abbrv}
|
{
"timestamp": "2020-02-25T02:19:01",
"yymm": "1804",
"arxiv_id": "1804.05637",
"language": "en",
"url": "https://arxiv.org/abs/1804.05637"
}
|
\section{Introduction}
Prediction of transition events and the determination of
governing criteria has significance in many physical, chemical, and
engineering systems where rank-1 saddles are present. To name but a few,
ionization of a hydrogen atom under
electromagnetic field in atomic physics~\cite{JaFaUz2000}, transport of
defects in solid state and semiconductor physics~\cite{Eckhardt1995},
isomerization of clusters~\cite{Komatsuzaki2001}, reaction rates in
chemical physics~\cite{Komatsuzaki1999,WiWiJaUz2001}, buckling modes in
structural mechanics~\cite{Collins2012,ZhViRo2018}, ship motion and
capsize~\cite{Virgin1989,ThDe1996,NaRo2017}, escape and
recapture of comets and asteroids in celestial
mechanics~\cite{JaRoLoMaFaUz2002,DeJuLoMaPaPrRoTh2005,Ross2003}, and escape
into inflation or re-collapse to singularity in
cosmology~\cite{DeOliveira2002}.
The theoretical criteria of transition and its agreement with
laboratory experiment have been shown for 1 degree of freedom (DOF)
systems~\cite{Virgin1991, Gottwald1995, Novick2012}.
Detailed experimental validation of the geometrical framework for
predicting transition in higher dimensional phase
space ($\geqslant 4$, that is for 2 or
more DOF systems) is still lacking.
The geometric framework of phase space conduits in such systems, termed tube
dynamics\cite{KoLoMaRo2000,JaRoLoMaFaUz2002,DeJuLoMaPaPrRoTh2005,GaKoMaRoYa2006},
has not before been demonstrated in
a laboratory
experiment. It is noted that similar notions of transition were
developed for idealized microscopic systems, particularly chemical
reactions~\cite{Almeida1990,DeMeTo1991,JaFaUz2000,MaDe1989} under
the names of transition state and reactive island theory.
However, investigations of the
predicted phase space conduits of
transition
between wells in multi-well system have
stayed within the confines of numerical simulations.
In this paper, we present a direct experimental validation of the accuracy of
the phase space conduits, as well as the transition
fraction obtained as a function of energy, in a 4 dimensional phase space using
a
controlled laboratory experiment of a macroscopic system.
\\
\indent
In~\cite{Baskan_2015,Baskan_2016,Figueroa_2017}, experimental validation of
global
characteristics of 1 DOF Hamiltonian dynamics of scalar transport has been
accomplished using direct measurement of the Poincar\'e stroboscopic sections
using dye
visualization of the
fluid flow. In~\cite{Baskan_2015,Baskan_2016}, the experimental and
computational
results of chaotic mixing were compared by measuring the observed and simulated
distribution of particles, thus confirming the theory of chaotic transport
in Hamiltonian systems for such systems.
Our objective is to validate theoretical predictions of transition between
potential wells in an exemplar experimental 2 DOF system, where qualitatively
different global
dynamics can occur.
Our setup consists of a mass rolling on a multi-well surface that
is representative of potential energy underlying systems that exhibit
transition/escape behavior.
The archetypal potential energy surface chosen has implications in
transition, escape, and recapture phenomena in many of the aforementioned
physical systems.
In some of these systems, transition in the conservative case has been
understood in terms of trajectories of a given energy crossing a hypersurface
or transition state (bounded by a normally hyperbolic invariant manifold of
geometry $\mathbb{S}^{2N-3}$ in $N$ DOF).
In this paper, for $N=2$, trajectories pass inside a tube-like separatrix,
which has the advantage of accommodating the inclusion of
non-conservative forces such as stochasticity and damping~\cite{NaRo2017,ZhViRo2018}. The semi-analytical geometry-based approach for identifying transition trajectories has also been considered for periodically forced 2 DOF systems in~\cite{gawlik_lagrangian_2009,onozaki_tube_2017}. Our analytical approach here focuses on identifying separatrices from the unforced dynamics, and generalizes to higher dimensional phase space \cite{WiWiJaUz2001,GaKoMaRo2005}.
Based on the illustrative nature of our
laboratory experiment of a 2 DOF mechanical system, and the generality of the
framework to higher degrees of freedom \cite{GaKoMaRoYa2006}, we envision the geometric
approach demonstrated here can apply to experiments regarding transition across
rank-1 saddles in 3 or more
DOF systems in many physical contexts.
\vspace{-3ex}
\section{Separatrices in N DOF}
To begin the mathematical description of the invariant manifolds that partition the 2N dimensional phase space, we perform a linear transformation of the underlying conservative Hamiltonian. This transformation involves a translation of the saddle equilibrium point to the origin and a linear change of coordinates that uses the eigenvectors of the linear system. The resulting Hamiltonian near the saddle has the quadratic (normal) form
\begin{equation}
\begin{aligned}
H_2(q_1, p_1, \ldots, q_N, p_N) = \lambda q_1 p_1 + \sum\limits_{k = 2}^N \frac{\omega_k}{2} \left( q_k^2 + p_k^2\right)
\end{aligned}
\end{equation}
where $N$ is the number of degrees of freedom, $\lambda$ is the real eigenvalue corresponding to the saddle coordinates (\emph{reactive coordinates} for chemical reactions) spanned by $(q_1, p_1)$ and $\omega_k$ are the frequencies associated with the center coordinates (\emph{bath coordinates} for chemical reactions) spanned by the pair $(q_k, p_k)$ for $k \in 2, \ldots, N$.
Next, by fixing the energy level to $h \in \mathbb{R}^+$ and $c \in \mathbb{R}^+$, we can define a co-dimension 1 region $\mathcal{R} \subset \mathbb{R}^{2N}$ in the full phase space by the conditions
\begin{equation}
H_2(q_1, p_2, \ldots, q_N, p_N) = h, \quad \text{and} \quad |p_1 - q_1| \leqslant c.
\end{equation}
\begin{figure*}[!ht]
\centering
\includegraphics[width=0.95\textwidth]{projection-saddle-center.pdf}
\caption{The flow in the region $\mathcal{R}$ can be separated into saddle $\times$ center $\times \cdots \times$ center. On the left, the saddle projection is shown on the $(q_1,p_1)$-plane. The NHIM (black dot at the origin), the asymptotic orbits on the manifolds (M), two transition trajectories (T), and two non-transition trajectories (NT).
}
\label{fig:projection-saddle-center}
\end{figure*}
This implies that $\mathcal{R}$ is homeomorphic to the product of a $(2N - 2)$-sphere and an interval $I$, that is $\mathcal{R} \cong \mathcal{S}^{2N-2} \times I$ where the $\mathcal{S}^{2N-2}$ is given by
\begin{equation}
\frac{\lambda}{4} \left( q_1 + p_1 \right)^2 + \sum\limits_{k=2}^{N} \frac{\omega_k}{2}\left( q_k^2 + p_k^2 \right) = h + \frac{\lambda}{4}\left( p_1 - q_1 \right)^2.
\end{equation}
This bounding sphere of $\mathcal{R}$ at the middle of the equilibrium region where $p_1 - q_1 = 0$ is defined as follows
\begin{equation}
\mathcal{N}^{2N-2}_h = \left\{ (q,p) | \lambda p_1^2 + \sum\limits_{k=2}^N \frac{\omega_k}{2} (q_k^2 + p_k^2) = h \right\},
\end{equation}
corresponds to the transition state in chemical reactions (and other systems with similar Hamiltonian structure \cite{JaRoLoMaFaUz2002,NaRo2017,ZhViRo2018}).
The following phase space structures and their geometry are relevant for understanding transition across the saddle:
a.~{\bf NHIM:} The point $q_1 = p_1 = 0$ corresponds to an invariant $(2N - 3)$-sphere, $\mathcal{M}^{2N-3}_h$, of periodic and quasi-periodic orbits in $\mathcal{R}$, and is given by
\begin{equation}
\sum\limits_{k = 2}^N \frac{\omega_k}{2} \big( q_k^2 + p_k^2 \big) = h, \qquad q_1 = p_1 = 0.
\label{eqn:nhim_nDOF}
\end{equation}
This is known as the {\it normally hyperbolic invariant manifold} (NHIM) which has the property that the manifold has a ``saddle-like'' stability in directions transverse to the manifold and initial conditions on this surface evolve on it for $t \rightarrow \pm \infty$. The role of unstable periodic orbits the $4$ dimensional phase space (or more generally, the NHIM in the $2N$ dimensional phase space) in transition between potential wells is acting as anchor for constructing the separatrices of transit and non-transit trajectories.
b.~{\bf Separatrix:} The four half open segments on the axes, $q_1 p_1 = 0$, correspond to four high-dimensional cylinders of orbits asymptotic to this invariant $\mathbb{S}^{2N - 3}$ either as time increases ($p_1 = 0$) or as time decreases ($q_1 = 0$). These are called {\it asymptotic} orbits and they form the stable and the unstable manifolds of $\mathbb{S}^{2N - 3}$. The stable manifolds, $\mathcal{W}_{\pm}^s(\mathbb{S}^{2N - 3})$, are given by
\begin{equation}
\sum\limits_{k = 2}^N \frac{\omega_k}{2} \big( q_k^2 + p_k^2 \big) = h, \qquad q_1 = 0.
\end{equation}
where $\pm$ denotes the left and right branches of the stable manifold attached to the NHIM. Similarly, unstable manifolds are constructed and are shown in the saddle space in Fig.~\ref{fig:projection-saddle-center} as four orbits labeled M. These form the ``spherical cylinders'' of orbits asymptotic to the invariant ($2N - 3$)-sphere. Topologically, both invariant manifolds have the structure of $(2N-2)$-dimensional ``tubes'' ($\mathbb{S}^{2N-3} \times \mathbb{R}$) inside the $(2N-1)$-dimensional energy surface. Thus, they separate two distinct types of motion: transit and non-transit trajectories. While a transition, passing from one region to another, trajectory lies inside the ($2N - 2$)-dimensional manifold, the non-transition trajectories, bouncing back to their current region of motion, are those outside the manifold.
For a value of the energy just above that of the saddle, the nonlinear motion in the equilibrium region $\mathcal{R}$ is qualitatively the same as the linearized picture above~\cite{Moser1958,WiWiJaUz2001,Waalkens2010}. For example, the NHIM for the nonlinear system which corresponds to the $(2N - 3)$ sphere in~\eqref{eqn:nhim_nDOF} for the linearized system is given by
\begin{widetext}
\begin{equation}
\begin{aligned}
\mathcal{M}_h^{2N - 3} = \Big\{ (q,p) \Big| \; \sum\limits_{k = 2}^N \frac{\omega_k}{2} \big( q_k^2
+ p_k^2 \big) + f(q_2, p_2, \cdots, q_n, p_n) = h, \qquad q_1 = p_1 = 0. \Big\}
\end{aligned}
\label{eqn:NHIM_NF}
\end{equation}
\end{widetext}
where $f$ is at least of third order. Here, $(q_2, p_2, \cdots, q_N, p_N)$ are normal form coordinates and are related to the linearized coordinates via a near-identity transformation. In the neighborhood of the equilibrium point, since the higher order terms in $f$ are negligible compared to the second order terms, the $(2N - 3)$-sphere for the linear problem is a deformed sphere for the nonlinear problem. Moreover, since the NHIMs persist for higher energies, this deformed sphere $\mathcal{M}_h^{2N - 3}$ still has stable and unstable manifolds that are given by
\begin{widetext}
\begin{equation}
\begin{aligned}
\mathcal{W}_{\pm}^{S}(\mathcal{M}_h^{2N -3}) =& \Big\{ (q,p) \Big| \; \sum\limits_{k = 2}^N \frac{\omega_k}{2} \big( q_k^2 + p_k^2 \big) + f(q_2, p_2, \cdots, q_n, p_n) = h, \qquad q_1 = 0. \Big\} \\
\mathcal{W}_{\pm}^{u}(\mathcal{M}_h^{2N -3}) =& \Big\{ (q,p) \Big| \; \sum\limits_{k = 2}^N
\frac{\omega_k}{2} \big( q_k^2 + p_k^2 \big) + f(q_2, p_2, \cdots, q_n, p_n) = h, \qquad p_1 = 0.
\Big\}
\end{aligned}
\label{eqn:manifold_NF}
\end{equation}
\end{widetext}
This geometric insight is useful for developing numerical methods for {\it globalization} of the invariant manifolds using numerical continuation~\cite{Note1}.
Now, we briefly describe the techniques that can be used to quantify and visualize the high
dimensional invariant manifolds. For positive value of excess energy, one can use a normal form
computation to obtain higher order terms of~\eqref{eqn:NHIM_NF} and~\eqref{eqn:manifold_NF}. A
brief overview of this approach is given in~\cite{Wiggins2008, *Burbanks2008BackgroundAD} along
with applications and results obtained using the computational tool for the Hamiltonian normal form.
Another approach is to sample points on these manifolds since the geometry of the manifold is known
near the equilibrium point. One would start by taking Poincar\'e sections and normal form theory
that involves high-order expansions around a saddle $\times$ center $\cdots$ $\times$ center
equilibrium. For example, in 3 DOF, the NHIM has topology $\mathbb{S}^3$ and thus a tube
cross-section on a 4D Poincar\'e section will have topology $\mathbb{S}^3$ for which it is possible
to obtain an inside and outside. If $x = {\rm constant}$ defines the Poincar\'e section, then one
can project the $\mathbb{S}^3$ structure to two transverse planes, $(y,p_y)$ and $(z, p_z)$. On
each plane, the projection appears as a disk, but because of the $\mathbb{S}^3$ topology, any point
in the $(z, p_z)$ projection corresponds to a topological circle in the $(y,p_y)$ (and vice-versa)
and from this, one can determine which initial conditions are inside, and thus transit
trajectories, as has been performed previously \cite{GoKoLoMaMaRo2004,GaKoMaRo2005}.
\section{Model of the 2 DOF experimental system}
The initial mathematical model of the transition
behavior of a rolling ball on the surface, $H(x,y)$, shown in
Fig.~\ref{fig:setup_surface_traj_sos}, is described in~\cite{Virgin2010}.
\begin{figure}[!h]
\vspace{-1ex}
\centering
\subfigure{\includegraphics[height=1.8in]
{pot_surf_proj}\label{fig:typ_traj}}
%
\subfigure{\includegraphics[height=1.8in]{hills_region_below_above}}
\vspace{-2ex}
\caption{\footnotesize{\textbf{(a)} A typical experimental
trajectory, shown in white, on the potential energy surface where
the contours denote isoheights of the surface.
This instance of the trajectory was traced by the ball released
from rest, marked by a red cross. \textbf{(b)} and
\textbf{(c)}
Show energetically accessible region projected on the configuration space
in white for $\Delta E < 0$: $\Delta E = -100
\;{\rm (cm/s)^2}$ and $\Delta
E > 0$:
$\Delta E = 100 \;{\rm (cm/s)^2}$, respectively.
}}
\label{fig:setup_surface_traj_sos}
\vspace{-2ex}
\end{figure}
The equations of motion are obtained from the Hamiltonian,
${\mathcal{H}}(x,y,p_x,p_y) = T(x,y,p_x,p_y) + V(x,y)$, where mass factors out
and
where the kinetic energy (translational and rotational for a ball
rolling without slipping) is,
{\small \vspace{-1ex}
\begin{equation}
\vspace{-1ex}
T =
\frac{5}{14} \frac{(1+H_y^2)p_x^2 +
(1+H_x^2)p_y^2 - 2 H_x H_y p_x p_y}{1 + H_x^2 + H_y^2}
\label{kin_rescale}
\vspace{-1ex}
\end{equation}}
\hspace{-1ex}where $H_{(\cdot)} = \frac{\partial H}{\partial
(\cdot)}$.
The potential energy is $V(x,y)=g H(x,y)$ where
$g=981$ cm/s$^2$ is the gravitational
acceleration and the height function is
{\small
\vspace{-1ex}
\begin{align}
H = \alpha (x^2 + y^2) - \beta \left( \sqrt{x^2 + \gamma} + \sqrt{y^2 +
\gamma} \right) - \xi x y + H_0.
\label{pot_rescale}
\vspace{-3ex}
\end{align}}
\hspace{-1ex}This is the analytical function for the machined surface shown
in~Fig.~\ref{fig:setup_surface_traj_sos}(b) and the isoheights shown
in~Fig.~\ref{fig:setup_surface_traj_sos}(c).
We use parameter values $(\alpha,\beta,\gamma,\xi,H_0) =
(0.07,1.017,15.103,0.00656,12.065)$ in the appropriate units~\footnote{See the
Supplemental Material for derivation of equations
of motion and the computational approach used to obtain the invariant
manifolds.}.
Let $\mathcal{M}(E)$ be the {\it energy manifold} in the 4D phase space given
by setting
the total energy
equal to a constant, $E$, i.e., $\mathcal{M}(E)=\{(x,y,p_x,p_y)
\subset \mathbb{R}^4 \mid \mathcal{H}(x,y,p_x,p_y)=E\}$.
The projection of the energy manifold onto the $(x,y)$ configuration space
is the region of energetically possible motion for a mass with energy
$E$,
and is given by $M(E)=\{(x,y) \mid V(x,y)\leq E\}$.
The boundary of $M(E)$ is the zero velocity curve and is defined as the
locus of points in the $(x,y)$ plane where the kinetic energy is zero.
The mass is only able to move on the side of the curve where the kinetic energy
is positive, shown as white regions in Fig.~\ref{fig:setup_surface_traj_sos}(d)
and (e).
The critical energy for transition, $E_e$, is the energy of the
rank-1 saddle points in each bottleneck, which are all equal. This
energy divides the global behavior of the mass into two cases,
according to
the sign of the excess energy above the saddle, $\Delta E = E-E_e$:
{\it Case 1:} $\Delta E < 0$ \textemdash the mass is safe against transition
and remains inside the starting well since potential wells are not
energetically connected (Fig.~\ref{fig:setup_surface_traj_sos}(d)).
{\it Case 2:} $\Delta E > 0$ \textemdash the mass can transition by crossing
the
bottlenecks that open up around the saddle points, permitting transition
between the potential wells (Fig.~\ref{fig:setup_surface_traj_sos}(e) and
Fig.~\ref{fig:stable_tube_1-2_DelE100_trajs}(a) show this case).
\begin{figure}[!t]
\vspace{-1ex}
\centering
\includegraphics[width=0.98\columnwidth]{stable_tube_1-2_DelE100_trajs}
\vspace{-2ex}
\caption{\footnotesize{(\textbf{a}) For a fixed excess energy, $\Delta E$,
above the critical value $E_e$, the permissible regions (in white) are
connected by a bottleneck around the saddle equilibria. All motion from the
well in quadrant
1 to quadrant 2 must occur through the interior of a
stable manifold associated with an unstable periodic orbit in the
bottleneck between the quadrants; seen as a 2D configuration space projection
of the 3D energy manifold. We show the stable manifold (cyan) and the periodic
orbit (black) for an excess energy of $\Delta E = 100 \;{\rm (cm/s)^2}$. A
trajectory crossing the $U_1^-$ section inside the stable manifold
will transition (red) into the quadrant 2 well, while one that is
outside stays (blue) inside quadrant 1. The zoomed-in inset in the figure
shows the structure of the manifold and how precisely the separatrix divides
transition and non-transition trajectories.
\textbf{(b)}
In the $(x,y,v_y)$ projection, the phase space
conduit for imminent transition from quadrant 1 to 2 is
the stable manifold (cyan) of geometry $\mathbb{R}^1 \times \mathbb{S}^1$
(i.e., a cylinder).
The same example
trajectories (red and blue) as in (a) that exhibit transition and
non-transition behavior
starting inside and outside the stable manifold, respectively, are shown
in the 3D projection and projected on the ($x,y$) configuration space.
A movie of a nested sequence of these manifolds can be found
\href{https://youtu.be/gMqrFX2JkLU}{here}.}}
\label{fig:stable_tube_1-2_DelE100_trajs}
\vspace{-3ex}
\end{figure}
Thus, transition between wells can occur when $\Delta E > 0$ and this
constitutes a
necessary condition. The sufficient condition for transition to occur is when a
trajectory enters a codimension-1 invariant manifold associated with the
unstable
periodic orbit in the bottleneck as shown by non-transition and transition
trajectories in
Fig.\ref{fig:stable_tube_1-2_DelE100_trajs}(a)~\cite{KoLoMaRo2000}. In 2
DOF systems, the periodic orbit residing in the bottleneck has an invariant
manifold which is codimension-1 in the energy manifold and has topology
$\mathbb{R}^1 \times \mathbb{S}^1$, that is a cylinder or
tube~\cite{Note1}. This implies that the transverse intersection of these
manifolds with Poincar\'e surfaces-of-sections, $U_1$ and $U_2$, are
topologically $\mathbb{S}^1$, a closed
curve~\cite{KoLoMaRo2000,NaRo2017,ZhViRo2018}.
All the trajectories transitioning to a different potential well (or having
just transitioned into the well) are inside a tube manifold, for
example as shown
in Fig.~\ref{fig:stable_tube_1-2_DelE100_trajs}(b)~\cite{KoLoMaRo2000,
GaKoMaRoYa2006}.
For every $\Delta E > 0$, the tubes in phase space (or
more precisely, within $\mathcal{M}(E)$) that lead to transition are the
stable (and that lead to entry are the unstable) manifolds
associated with the unstable periodic orbit of energy $E$. Thus, the mass's
imminent
transition
between adjacent wells can be predicted by considering where it crosses
$U_1$
as shown in Fig.~\ref{fig:sosU1pos}, relative to
the intersection of the tube manifold. Furthermore, nested energy manifolds
have corresponding nested
stable and unstable manifolds that mediate transition.
To simplify analysis, we focus only on the
transition of trajectories
that intersect
$U_1$ in the first quadrant. This surface-of-section is best described in polar
coordinates $(r,\theta,
p_r, p_{\theta}) $;
$U_1^{\pm } = \{(r,p_r) ~|~ \theta = \frac{\pi}{4},~-{\rm sign}(p_{\theta})
=\pm 1\}$,
where $+$ and $-$ denote motion to the right and left of the
section,
respectively
~\cite{Note1}.
This Hamiltonian flow on $U_1^{\pm}$ defines a symplectic map with typical
features such as KAM tori and chaotic regions, shown
in Fig.~\ref{fig:sosU1pos} for two values of excess energy.
Based on these phase space conduits that lead to transition, we would like to
calculate what fraction of the energetically permissible
trajectories will transition from/into a given well.
This can be answered in part by calculating the transition rate of trajectories
crossing
the rank-1 saddle in the bottleneck connecting the wells.
For computing this rate---surface integral of trajectories crossing a bounded
surface per unit time---we use the geometry of the tube manifold cross-section
on the Poincar\'e section. For low excess energy, this computation is
based on the theory of flux over a rank-1 saddle \cite{MacKay1990}, which
corresponds to the action integral around the periodic orbit at energy $\Delta
E$. By the
Poincar\'e
integral invariant~\cite{Meiss1992}, this
action is preserved for symplectic maps, such as $P^\pm : U_1^\pm \rightarrow
U_1^\pm$, and is equivalent to computing the area of the tube manifold's
intersection with the surface-of-section. The transition fraction at each
energy, $p_{\rm trans}(\Delta E)$, is calculated by the fraction of
energetically permissible trajectories at a given excess energy, $\Delta E$,
that will transition. This is given by the ratio of the cross-sections on $U_1$
of the tube to
the energy surface.
The transition area,
to leading order in $\Delta E$~\cite{MacKay1990}, is given by
$A_{\rm trans} = T_{\rm po} \Delta E$, where $T_{\rm po} = 2 \pi / \omega$ is
the
period of the periodic orbits of small energy in the
bottleneck, where $\omega$ is the imaginary part of the complex conjugate pair
of eigenvalues
resulting from the linearization about the saddle equilibrium
point~\cite{MacKay1990}.
The area of the energy surface projection on $U_1$, to leading order in
$\Delta E>0$, is $A_{\rm E} = A_{0} + \tau \Delta E $, where,
{\small
\begin{align}
A_{0} =& 2 \int_{r_{\rm min}}^{r_{\rm max}} \sqrt{
\frac{14}{5}( E_e - gH(r))(1 + 4H_r^2(r)) } \;dr, \label{energy_area_U_2_crit}
\\
\text{and} \; \tau =& \int_{r_{\rm min}}^{r_{\rm max}} \sqrt{
\frac{14}{5}\frac{ (1 + 4H_r^2(r))}{ (E_e - gH(r))}} \; dr. \;
\label{energy_area_U_2}
\end{align}}
\hspace{-1ex}The transition fraction, under the well-mixed assumption mentioned
earlier, is
given in 2 DOF by
{\small
\begin{equation}\label{transit_frac}
\begin{split}
p_{\rm trans} &= \frac{A_{\rm trans}}{A_{E}} = \frac{T_{\rm
po}}{A_{0}} \Delta E \left( 1 - \frac{\tau}{A_{0}} \Delta E +
\mathcal{O}(\Delta E^2)\right).
\end{split}
\end{equation}}
\hspace{-1ex}For small positive excess energy, the predicted growth rate is
$T_{\rm po}/A_0 \approx 0.87 \times 10^{-3} \; \rm (s/cm)^2$.
For larger values of $\Delta E$, the
cross-sectional areas are computed
numerically using Green's theorem, see
Fig.~\ref{fig:energy_poincare_section}(b).
\begin{figure}[!t]
\vspace{-1ex}
\centering
\includegraphics[width=0.47\columnwidth]
{sosU1m_DelE100_300pts_400finalT_tube.pdf}
\label{sosU1m_DelE100_300pts_400finalT}
\includegraphics[width=0.47\columnwidth]
{sosU1m_DelE500_300pts_400finalT_tube.pdf}
\label{sosU1m_DelE300_300pts_400finalT}
\vspace{-3ex}
\caption{\footnotesize{Poincar\'e section, $P^-: U_1^{-}
\rightarrow U_1^{-}$, of trajectories where $U_1^- := \{ (r, p_r) | \;
\theta = \pi/4, \; p_{\theta} > 0 \}$, at excess energy
\textbf{(a)} $\Delta E =
100 \;{\rm (cm/s)^2}$ and \textbf{(b)} $\Delta E =
500 \;{\rm (cm/s)^2}$.
The blue curves
with cyan interior denote the intersection of the tube
manifold (stable) associated with the unstable periodic orbit with $U_1^-$.
It is to be noted that these manifolds act as a boundary between
transition and non-transition trajectories, and may include KAM
tori spanning more than one well. The interior of the manifolds, ${\rm
int(\cdot)}$, denote
the region of imminent transition to the quadrant 2 from quadrant 1. A
movie
showing the Poincar\'e section for a range of excess energy
can be found \href{https://youtu.be/sNvgXCrX6oo}{here}.}}
\label{fig:sosU1pos}
\vspace{-3ex}
\end{figure}
As with any physical experiment there is dissipation present, but over
the time-scale of interest, the motion approximately conserves energy.
We compare $\delta E$, the typical energy lost during a transition, with the
typical excess energy, $\Delta E>0$, when transitions are possible.
The time-scale of interest, $t_{\rm trans}$, corresponds to the time between
crossing $U_1$ and transitioning across the saddle into a neighboring well. The
energy loss
over $t_{\rm trans}$ in terms of the measured damping ratio $\zeta \approx
0.025$ is $\delta E \approx \pi \zeta v^2(\Delta E)$ where the squared-velocity
$v^2(\Delta E)$ is approximated through the total energy.
For our experimental trajectories, all starting at $\Delta E > 1000$
(cm/s)$^2$, we find $\delta E/\Delta E \ll 1$,
suggesting the appropriateness of the assumption of short-time conservative
dynamics to study
transition between wells~\cite{NaRo2017,ZhViRo2018}.
\vspace{-3ex}
\section{Experimental setup}
We designed a surface (shown in Fig.~\ref{fig:setup_surface_traj_sos}(b)) that
has 4 wells, one in each quadrant, with saddles connecting the neighboring
quadrants.
The surface has 4 stable and 5 saddle (4 rank-1 and 1 rank-2) equilibrium
points. Inter-well first order transitions are defined as crossing the rank-1
saddles between the wells.
On this high-precision machined surface, accurate to within $0.003~{\rm mm}$
and made using stock polycarbonate, a
small rubber-coated spherical steel
mass released from
rest can
roll without slipping under the influence of gravity.
The mass is released from different locations on the
machined
surface to generate experimental trajectories. The mass is tracked using a
Prosilica
GC640 digital camera mounted on a rigid frame attached to the surface as shown
in Fig.~\ref{fig:setup_surface_traj_sos}(a), with a pixel resolution of about 0.16
cm.
\begin{figure}[!h]
\vspace{-1ex}
\includegraphics[width=0.95\columnwidth]{full-setup-zoomin.pdf}
\caption{\footnotesize{\textbf{(a)},~\textbf{(b)} Experimental apparatus showing the machined surface, tracking camera, and the rubber coated steel ball.}}
\label{fig:rolling_ball_expt}
\vspace{-2ex}
\end{figure}
The tracking is done by
capturing black and white images at 50
Hz, and calculating the coordinates of the mass's geometrical center. We
recorded 120 experimental trajectories of about 10 seconds long, only using
data
after waiting at least the Lyapunov time of $\approx0.4$
seconds~\cite{Virgin2010} ensuring that the trajectories were
well-mixed in the phase space.
To analyze the fraction of trajectories that leave/enter a well, we obtain
approximately
4000 intersections with a Poincar\'e surface-of-section, $U_1$, shown as a
black
line,
for the analyzed range of energy. One such trajectory
is shown in white in Fig.~\ref{fig:setup_surface_traj_sos}(c).
These intersections are then sorted
according to energy. The intersection
points on $U_1$ are classified as a transition from quadrant 1 to 2 if the
trajectory, followed forward in time, leaves
quadrant 1. Four hundred transition events were recorded.
\vspace{-3ex}
\section{Results}
For each of the recorded trajectories, we detect
intersections with $U_1$ and
determine the instantaneous $\Delta E$.
\begin{figure}[!t
\vspace{-1ex}
\begin{center}
\begin{tabular}{c}
\includegraphics[height=1.6in]{run_plots_tubes_sos_shane_compare.pdf}
\includegraphics[height=1.6in]
{trans_frac_theory_expt_overlay_linfit.pdf}
\end{tabular}
\end{center}
\vspace{-5ex}
\caption{\footnotesize{\textbf{(a)} On the Poincar\'e section, $U_1^-$, we show
a narrow range of energy
($\Delta E \in (40,140)~{\rm (cm/s)}^2$) and label intersecting trajectories as
no transition (black) and imminent transition (red) to quadrant 2, based on
their measured behavior.
The stable invariant manifold associated with the bottleneck periodic orbit
at excess energy, $\Delta E = 140 \; {\rm (cm/s)^2}$, intersects the Poincar\'e
section, $U_1^-$, along the blue curve. Its interior is shown in cyan and
includes the experimental transition trajectories.
The outer closed curve (magenta) is the intersection of the boundary of the
energy surface $\mathcal{M}(\Delta E)$ with $U_1^-$.
\textbf{(b)} Transition fraction of trajectories as a function of excess
energy above the saddle. The theoretical result is shown
(blue curve) and experimental values are shown as filled circles (black) with
error bars. For small excess energy above critical ($\Delta E = 0$), the
transition fraction shows linear growth (see inset) with slope $1.0 \pm 0.23
\times 10^{-3}$ (s/cm)$^2$ and shows
agreement with the analytical result~\eqref{transit_frac}. A movie of
increasing transition area on the Poincar\'e section, $U_1^-$, can be found
\href{https://youtu.be/YZKYx0N9Zug}{here}.
}}
\label{fig:energy_poincare_section}
\vspace{-3ex}
\end{figure}
Grouping intersection points by energy, for example
Fig.~\ref{fig:energy_poincare_section}(a), we get an experimental transition
fraction, Fig.~\ref{fig:energy_poincare_section}(b), by dividing points which
transitioned by the total in each energy range. Despite the experimental
uncertainty from the image analysis,
agreement between observed
and predicted values is satisfactory. In fact, a linear fit of the experimental
results for small excess energy gives a slope close to that predicted by
\eqref{transit_frac} within the margin of error. Furthermore, the
clustering of observed transitioning trajectories in each energy range, as in
Fig.~\ref{fig:energy_poincare_section}(a), is consistent with the theory of
tube dynamics. The predicted transition regions in each energy range account
for more than 99\% of
the observed transition trajectories.
\section{CONCLUSIONS}
We considered a macroscopic 2 DOF experimental system showing transitions
between potential wells and a dynamical systems theory of the
conduits which mediate those
transitions~\cite{KoLoMaRo2000,NaRo2017,ZhViRo2018}. The
experimental validation presented here confirms the robustness of the conduits
between multi-stable regions, even in the presence of non-conservative forces,
providing a strong footing for predicting transitions in a wide range of
physical systems. Given the fragility
of other structures to dissipation (for example, KAM tori and periodic
orbits), these phase space
conduits of transition may be among the most robust features to be found in
experiments of autonomous multi-degree of freedom systems. Furthermore, this
study lays the groundwork for experimental validation for $N = 3$
or more degrees of freedom system, such as ship dynamics~\cite{Mccue2005, *McCue2006},
buckling of beams \cite{ZhViRo2018} and geodesic lattice domes, hanging roller pins, isomerization and roaming
reactions~\cite{Mauguiere2014,Bowman2017}.
~\\
\section*{Acknowledgments}
SDR and LNV thank the NSF for partially funding this work through grants
1537349
and 1537425.
|
{
"timestamp": "2018-08-31T02:08:56",
"yymm": "1804",
"arxiv_id": "1804.05363",
"language": "en",
"url": "https://arxiv.org/abs/1804.05363"
}
|
\section{Introduction}
Fetal ultrasound screening is an important diagnostic protocol to detect abnormal fetal development. Abnormal development is one of the leading causes for perinatal mortality world-wide. During screening examination, multiple anatomically standardised~\cite{FASP} scan planes are used to obtain biometric measurements as well as identifying abnormalities such as lesions. While 2D ultrasound is the preferred approach for examination due to its low cost and real-time capabilities, ultrasound suffers from low signal-to-noise ratio and image artefacts. As such, diagnostic accuracy and reproducibility is limited and requires a high level of expert knowledge and training. Therefore, automated scan plane detection algorithms can help training experts, facilitate non-expert examination, support consistent data acquisition and make diagnostics more robust.
Automated scan plane detection poses many challenges: Firstly, during the examination, the majority of time is spent exploring the present anatomy. As such, there are a large number of background labels and a significant class imbalance must be considered. Secondly, even if the object of interest is localised, it may not have reached the ideal scanning plane for diagnosis and hence the frame may be labelled as background; Therefore, in addition to understanding the global context, it is essential to understand the small differences in local structures to detect a correct plane. In the past, several approaches were proposed \cite{yaqub2015guided, chen2015automatic}, however, they are computationally expensive and cannot be deployed for the real-time application.
In recent years, deep learning and convolutional neural networks (CNNs) have become popular approaches for a variety of medical image classification problems, including classification of Alzheimer's disease \cite{Sarraf070441}, lung nodule in CT/X-ray \cite{DBLP:journals/corr/abs-1801-09555}, skin lesion \cite{esteva2017dermatologist, kawahara2016multi}, anatomy \cite{roth2015anatomy} and the views for echo-cardiograms \cite{madani2018fast}.An extensive list of applications can be found in \cite{DBLP:journals/corr/LitjensKBSCGLGS17,zaharchuk2018deep}. In \cite{baumgartner2016real} the authors propose a CNN architecture called \emph{Sononet} to solve the standard plane classification problem during fetal ultrasound examination. The proposed approach achieves very good performance in real-time plane detection, retrospective frame retrieval (retrieving the most relevant frame) and weakly supervised object localisation. However, despite its success, the method suffers from relatively low precision and especially struggles differentiating anatomically related cardiac views. We argue that the reason for this is that Sononet is good at aggregating global information but it cannot preserve local information well. Moreover, the heuristics employed for the object localisation requires guided backpropagation, which limits the object localisation speed that can be achieved.
In fact, we claim that the inability to exploit local information is a common problem in medical image analysis: in many of these scenarios, typically, the object of interest is very small (e.g. lesions, local deformity, etc.) compared to the size of the input image, which can be high resolution 2D, 3D or 4D data. Such situation requires to tackle the object detection and classification problem as a two-stage process. In this work, we introduce \emph{soft-attention} in the context of medical image classification. Attention is a modular mechanism that allows to efficiently exploit localised information, which also provides soft object localisation during forward pass. In this work, we demonstrate the usefulness of such attention mechanism by applying the proposed approach to improve the scan plane detection for fetal ultrasound screening.
\subsection{Related Work}
Attention mechanisms were first popularised in the context of natural language processing \cite{shen2017disan}, such as machine translation \cite{bahdanau2014neural, DBLP:journals/corr/LuongPM15}. In these settings often recurrent neural networks are employed to model a sequence of text. In particular, given a sequence of text and a current word, a task is to extract a next word in a sentence generation or translation. The idea of attention mechanisms is to generate a \emph{context} vector which assigns weights on the input sequence. Thus, the signal highlights the salient feature of the sequence conditioned on the current word while suppressing the irrelevant counter-parts, making the prediction more contextualised. Attention mechanisms can further be separated into two types: soft-attention and hard-attention. In soft-attention, continuous functions (e.g. soft-max) are used to assign the attention weight on the input, making it fully differentiable. In comparison, hard-attention models propose specific words by sampling from the weights. As the sampling operation is not differentiable, hard-attention is trained using the gradient of the likelihood term generated by Monte-Carlo sampling \cite{xu2015show}.
In computer vision, attention mechanisms are applied to a variety of problems, including image classification \cite{jetley2018learn, wang2017residual, zhao2017survey}, segmentation \cite{DBLP:journals/corr/RenZ16}, action recognition \cite{liu2017global,DBLP:journals/corr/PeiBTM16,wang2017non}, image captioning \cite{xu2015show, lu2017knowing}, and visual question answering \cite{DBLP:journals/corr/YangHGDS15, DBLP:journals/corr/NamHK16}. In the context of medical image analysis, attention models have been exploited for medical report generation \cite{zhang2017mdnet, zhang2017tandemnet} as well as joint image and text classification \cite{DBLP:journals/corr/abs-1801-04334}. However, for standard medical image classification, despite the importance of local information, only a handful of works use attention mechanisms \cite{pesce2017learning, guan2018diagnose}. In these methods, either bounding box labels are available to guide the attention, or the local context is extracted by a hard-attention model (i.e. region proposal followed by hard-cropping). In our work, we propose incorporating self-gating, a soft-attention approach that is end-to-end trainable. This also does not require any bounding box labels and backpropagation-based saliency map generation as in \cite{baumgartner2016real}.
\subsection{Contributions}
\begin{itemize}
\item We introduce a self-gated, soft-attention mechanism in the context of pure medical image classification. We apply the proposed model to real-time fetal ultrasound scan plane detection and show its superior classification performance over the baseline approach.
\item We demonstrate that the proposed attention mechanism provides fine-scale attention maps that can be visualised, with minimal computational overhead, which is a crucial step towards explainable deep learning.
\item Finally, we show that attention maps can used for fast (weakly-supervised) object localisation, demonstrating that the attended features indeed correlates to the anatomy of interest.
\end{itemize}
\section{Methodology}
\textbf{Sononet:} We will first review \emph{Sononet} \cite{baumgartner2016real}, which will be the baseline architecture for our discussion. Sononet is a CNN architecture with two components: a feature extractor module and an adaptation module. In the feature extractor module, the first 17 layers (counting max-pooling) of the VGG network \cite{simonyan2014very} is used to extract discriminant features. Note that the number of filters are doubled after each of the first three max-pooling operations. In the adaptation module, the number of channels are first reduced to the number of target classes $C$. Subsequently, the spatial information is flattened via channel-wise global average pooling. Finally, a soft-max operation is applied to the resulting vector and the entry with maximum activation is selected as the prediction. As the network is constrained to classify based on the reduced vector, the network is forced to extract the most salient features for each class. Owing to this, Sononet obtains well-localised feature maps before the pooling layer, which can also used to perform weakly-supervised localisation (WSL) with high accuracy. However, while the global average pooling is attractive as it can quickly aggregate the spatial context, it does not have the capacity to preserve local information. As such, if two frames have very similar global appearance, it cannot well distinguish them. In the case of scan plane detection, this is manifested as it results in a low accuracy for multiple cardiac views, where each view contains similar underlying anatomy but only differ by the plane orientation.
\begin{figure}[!t]
\centering
\includegraphics[width=.9\textwidth]{figures/attention_architecture.png}
\caption{The schematics of the proposed network, termed \emph{Attention-Gated Sononet}. The proposed attention units are incorporated in layer 11 and layer 14.}
\label{fig:modelschematic}
\end{figure}
\begin{figure}[!t]
\centering
\includegraphics[width=1\textwidth]{figures/grid_attention_mechanism.pdf}
\caption{The proposed grid attention block. The global gate signal $\mb{g}_I$ is shared for the region indicated in red. Tensor dimensions for the compatibility score computation are shown.}
\label{fig:attentionblock}
\end{figure}
\textbf{Attention Unit for Medical Image Analysis:} Our work is inspired by \cite{jetley2018learn}, where the authors have introduced a similar attention mechanism for a classification problem in the computer vision domain. Let $\mathcal{F}^s = \{ \mb{f}_i^{s} \}_{i=1}^n$ be the activation map of a chosen layer $s\in \{1, \dots, S\}$, where each $\mb{f}_i^{s}$ represents the pixel-wise feature vector of length $C_s$ (i.e. the number of channels). Let $\mb{g} \in \mathbb{R}^{C_g}$ be a global feature vector extracted just before the final soft-max layer of a standard CNN classifier. In this case $\mb{g}$ encodes global, discriminative, relevant information about the objects of interest. The idea is to consider each $\mb{f}_i^{s}$ and $\mb{g}$ jointly to attend the features at each scale $s$ that is relevant to the coarse scale features (i.e. object-ness) represented by $\mb{g}$. To this end, the notion of a \emph{compatibility score} $\mathcal{C}(\mathcal{F}^s, \mb{g}) = \{c_i^s\}_{i=1}^n$ is defined and is given by an \emph{additive} attention model:
\begin{equation}
c_i^s = \langle \mbb{\Psi}, \mb{f}_i^s + \mb{g} \rangle ,
\end{equation}
where $\langle\cdot,\cdot\rangle$ is the dot product and $\mbb{\Psi} \in \mathbb{R}^{C_s}$ is a learnable parameter. In the case where $\mb{f}_i^s$ and $\mb{g}$ have different dimensions, a learnable weight $\mb{W}_g \in \mathbb{R}^{C_s \times C_g}$ is used to match the dimensionality of $\mb{g}$ to $\mb{f}_i^s$. Once the compatibility scores are computed, they are passed through soft-max operation to obtain the normalised attention coefficient: $\alpha_i^l = e^{c_i^l}/\sum_i e^{c_i^l}$. Finally, at each scale $s$, a weighted sum $\mb{g}^s = \sum_{i=1}^n \alpha_i^s \mb{f}_i^s$ is computed, and the final prediction is given by fitting a fully connected layer on $\{\mb{g}^1 \dots \mb{g}^S\}$. By constraining the prediction to be done from the weighted sum, the network is forced to learn the most salient features that contribute to the class. Therefore the attention coefficients $\{\alpha_i^l\}$ identify salient image regions, amplify their influence, and suppress irrelevant information in background regions. In this work, we consider a more general attention mechanism:
\begin{gather}
c_i^s = \mbb{\Psi} \sigma_1 \left(\mb{W}_f \mb{f}_i^s + \mb{W}_g \mb{g} + \mb{b}_g \right) + \mb{b}_\psi ,
\end{gather}
where a gating unit is characterised by a set of parameters $\Theta_{att}$ containing: linear transformations $\mb{W}_f \in \mathbb{R}^{C_{int} \times C}$, $\mb{W}_g \in \mathbb{R}^{C_{int} \times C_{g}}$, $\mbb{\Psi} \in \mathbb{R}^{C_{int} }$ and bias terms $\mb{b}_{\psi} \in \mathbb{R}$ , $\mb{b}_{g} \in \mathbb{R}^{C_{int}}$. $\sigma_1$ is a nonlinearity chosen to be ReLU: $\sigma_1(\mb{x}) = \max (0, \mb{x})$. By introducing attention mechanism, the features $\mathcal{F}^s$ branches out to two paths: one to extract the global feature vector and one which passes through gating for the prediction. The benefit of the generalised approach is that firstly, we speculate that introducing $\mb{W}_f$ allows the fine-scale layer to focus less on generating a signal that is compatible to $\mb{g}$, which helps it focus on learning the discriminant features. Secondly, by introducing $\mb{W}_f$, $\mb{W}_g$ and $\sigma_1$, we allow the network to learn nonlinear relationships between the vectors, which is more expressive. This is important because in medical imaging, images are inherently noisy and the region of interest is often highly non-homogeneous. Therefore, a linear compatibility function may be too sensitive to such fluctuation. Note that $\{\mb{W}_f\}$ and $\{\mbb{\Psi}, \mb{b}_\psi\}$ are implemented as $1 \times 1$ convolution layer and $\{\mb{W}_g, \mb{b}_g\}$ is a fully connected layer.
Similarly, for normalising the compatibility coefficients, we also argue that soft-max operation may not be the optimal approach as it typically produces sparse output. This can be over-sensitive to local changes even though we want the network to attend to the region with high variability. An alternative is a sigmoid unit. However, we observe that a sigmoid often suffers from gradient saturation problems. As such, the best working strategy is to first subtract the $\min_i \{ \alpha_i^s\}_{i=1}^n$ from all attention coefficients to align the minimum value to be 0. Then, we divide each element by $\sum_i \alpha_i^s$. This is similar to soft-max but does not sparsify the attention map.
\textbf{Grid Attention:} As seen above, the global feature vector $\mb{g}$ is a 1D vector incorporating information from all spatial context. In the context of medical imaging, since most objects of interest are extremely localised, flattening may have the disadvantage of losing important spatial context. In fact in many cases, a few max-pooling operations are sufficient to infer the global context without explicitly using the global pooling. Therefore, we propose a \emph{grid attention} mechanism. The idea is to use the feature map just before the global pooling as the gridded instance of $\mb{g}$. For example, given an input size of $N_x\times N_y$, after $r$ max pooling operations, the tensor size is reduced to $N_x / (2^r) \times N_y / (2^r)$. To generate the attention map, we upsample the coarse grid to match the spatial resolution of $\mathcal{F}^s$ (see Figure \ref{fig:attentionblock}). In this way, the attention mechanism has more flexibility in terms of what to focus on a regional basis. Concretely, in terms of implementation, $\{\mb{W}_g, \mb{b}_g\}$ is now replaced by a $1 \times 1$ convolution. For upsampling, we chose to use bilinear upsampling. Note that the upsampling can be replaced by a learnable weight, however, we did not opt for this for the sake of simplicity.
\textbf{Attention Gates in Sononet:} The proposed attention mechanism is incorporated in the Sononet architecture to better exploit local information. In the modified architecture, termed Attention-Gated Sononet (\emph{AG-Sononet}), we remove the adaptation module. The final layer of the extraction module is used as gridded global feature map $\mb{g}$. We apply the proposed attention mechanism to layer 11 and 14 just before pooling, as shown in Figure \ref{fig:modelschematic}. After the attention map $\{\alpha_i^s\}$'s are obtained, the weighted average over the spatial axes is computed for each channel in the feature map, yielding a vector of length $C^s$ at scale $s$. In addition, we also perform the global average pooling on the coarsest scale representation and use it for the final classification. This is because we hypothesise that the coarsest scale representation is still useful for the classification when fine-scale features are unnecessary.
\textbf{Aggregation Strategy:} Given the attended feature vectors at different scales, we combine them for the final prediction. We highlight that the aggregation strategy is flexible and that it can be adjusted depending on the target problem. However, the aggregation strategy also influences how the network learns; simple aggregation might not enforce the network to learn the most useful gating mechanism. The simplest approach is to fit a separate fully connected (FC) layer at each scale, and make separate predictions. The final prediction is then given by either weighted mean or max operations. This approach ensures that the network learns relevant attributes of the classes at each scale, and hence the learning process is more stable. The alternative is to first concatenate the feature vectors and fit one FC layer for the prediction. In theory, this strategy should perform better as it allows the network to combine the information at different scales. However, we observe that this approach is non-trivial to train. Since the network tends to pick up coarse-scale features quickly, it quickly abandons the gating paths for finer scales and gets stuck in a local minimum. We attempted using deep-supervision \cite{lee2015deeply} to force each scale to learn a useful prediction jointly. However in this case, the network again obtains suboptimal performance. We speculate that this is because the network tries to allocate resources for individual-scale prediction and joint scale prediction simultaneously, which are conflicting in nature. The simplest and the most stable approach is to first let the network learn the prediction at each scale. After the network has converged, we fit a new FC layer on top of the predictions at each scale and let the network fine-tune itself for the joint prediction. Thus, the network discards the features that are predicted by other scales and focuses on subtle differences that can only be observed at a given scale. We denote the model which uses simple averaging of individual predictions as \emph{AG-Sononet}, the deep supervision model as \emph{AG-Sononet-DS} and the fine-tuned model as \emph{AG-Sononet-FT}.
\section{Experiments and Results}
In this section, the proposed model is compared against Sononet in terms of classification performance, model capacity, and computation time. In addition, we compare different aggregation strategies discussed above: \emph{AG-Sononet}, \emph{AG-Sononet-DS}, and \emph{AG-Sononet-FT}.
\textbf{Evaluation Datasets:} Our dataset consisted of 2694 2D ultrasound examinations of volunteers with gestational ages between 18 and 22 weeks. Image acquisition protocol is specified in \cite{baumgartner2016real}. The dataset contains 13 types of standard scan planes and background, complying the standard specified in the UK National Health Service (NHS) fetal anomaly screening programme (FASP) handbook \cite{FASP}. The standard scan planes are: Brain (Cb.), Brain (Tv.), Profile, Lips, Abdominal, Kidneys, Femur, Spine (Cor.), Spine (Sag.), 4CH, 3VV, RVOT, LVOT. The data was cropped to central $208 \times 272$ to prevent the network from learning the surrounding annotations shown in the ultrasound scan screen. The dataset was split into training (122233), validation (30553) and testing (38243) subsets. For preprocessing, we whitened our data (normalised each image by substracting the mean intensity and divide by the variance). For training, we used the following data augmentation: horizontal and vertical translation of $\pm 4$ pixels, horizontal flips, rotation of $\pm 25 \deg$ and zoom of factor $s \in [0.7, 1.3]$. This generates a dataset at least $40000\times$ bigger than the original.
For evaluation, we used accuracy, precision, recall, F1, the number of parameters and execution speed. Note that due to large class imbalance, it is important to take the macro-averaging for precision, recall and F1: e.g $\text{recall}_\text{macro} = (\text{recall}_{c_1} + \dots + \text{recall}_{c_n})/\text{\{ the number of classes \}}$. Furthermore, we also qualitatively study the attention map generated to highlight that the network indeed attends salient local regions.
\textbf{Training:} Note that due to the nature of fetal ultrasound screening, the background label dominates the dataset. Due to large class imbalance, the training is not straightforward. In addition, background frames could contain the anatomy of interest, yet it might be classified as background as the plane is not a standard plane. Therefore, an appropriate ratio between all classes and background is important. We used a weighted sampling strategy: the sampling ``probability'' of an image from class $c$ is given by $1/n_c$, where $n_c$ is the number of images in class $c$. For the background label, we used $13/n_c$, where 13 is the number of the standard scan planes. In this way, we expect to see one background image for every standard scan plane. We used cross entropy loss and the network was optimised using Stochastic Gradient Descent with Nesterov momentum ($\rho=0.9$). The initial learning rate was set to 0.1, which was subsequently reduced by a factor of 0.1 for every 100 epoch. We also used a warm-start learning rate of 0.01 for the first 5 epochs. Each network was trained for 300 epochs. The batch size was set to 64. $\ell_2$ weight regularisation was used with $\lambda = 10^{-4}$.
\textbf{Implementation Details:} We modified the baseline Sononet architecture slightly: instead of using 2 convolution layers for the first 2 feature scales and 3 convolution layers for the last 3 feature scales, we used 3 layers for the first 3 and 2 layers for the last 2 feature scales. The architecture for AG-sononet is shown in \ref{fig:modelschematic}. As discussed, training AG-sononet is slightly more tricky as the optimal gating mechanism may not be necessarily learnt. However, we observed that the simplest approach to achieve the desired gating mechanism was to initialise AG-Sononet with a partially trained Sononet. We compare our models with different capacities, with initial number of features 8, 16 and 32. Our implementation in PyTorch library is publicly available\footnote{\url{https://github.com/ozan-oktay/Attention-Gated-Networks}}.
\textbf{Results:} Table \ref{table:result1} summarises the performance of the models. In general, AG-Sononets improve the results over Sononet at all capacity levels. In particular, AG-Sononets higher precision. AG-Sononets reduces false positive examples because the gating mechanism suppresses background noise and forces the network to make the prediction based on class-specific features. As the capacity of Sononet is increased, the gap between the methods are tightened, but we note that the performance of AG-Sononets is also close to the one of Sononet with double the capacity. In addition, the advantage of AG-Sononets is that it can provide attention maps for no extra computational cost (shown below). Therefore, attention-mechanism allows the network to allocate all resources on the most salient aspect of the problem, and can achieve higher performance with minimal number of parameters. In Table \ref{table:result_class}, we show the class-wise F1, precision and recall values for AG-Sononet-FT-8. The improvement over Sononet is indicated in brackets. In \cite{baumgartner2016real}, it was highlighted that the model often confuses between cardiac views as they appear anatomically similar. The situation is notably improved, with statistically significant improvement for 4CH and 3VV ($p<0.05$) due to fine-scale aggregating differences. However, these views remained challenging. We see that the precision increased by around 5\% for kidney, profie and spines, as well as on average 3\% for cardiac views. In some cases, we see minor reduction in recall rates. We believe that this is because the network may have become slightly more conservative when predicting the class labels.
\begin{table}
\centering
\caption{Test results for standard scan plane detection. Number of initial filters is denoted by the postfix ``-$n$''. Time taken for forward (Fwd) and backward (Bwd) passes were recorded in milliseconds. }
\begin{tabular}{lrrrrrr}
\toprule
Method & Accuracy & F1 & Precision & Recall & Fwd/Bwd ($ms$) & \#parameters \\
\hline
Sononet-8 & 0.969 & 0.899 & 0.878 & 0.922 & 1.36/2.60 & 0.16M
\\
AG-Sononet-8 & 0.976 & 0.921 & 0.911 & \textbf{0.933} & 1.86/3.46 & 0.18M
\\
AG-Sononet-DS-8 & 0.975 & 0.918 & 0.907 & 0.929 & 1.92/3.51 & 0.18M
\\
AG-Sononet-FT-8 & \textbf{0.977} & \textbf{0.922} & \textbf{0.916} & 0.929 & 1.92/3.47 & 0.18M
\\
\hline
Sononet-16 & 0.977 & 0.923 & 0.916 & 0.931 & 1.45/3.92 & 0.65M
\\
AG-Sononet-16 & 0.976 & 0.925 & 0.917 & 0.932 & 1.88/5.13 & 0.70M
\\
AG-Sononet-DS-16 & \textbf{0.978} & 0.924 & 0.919 & 0.929 & 1.90/5.19 & 0.71M
\\
AG-Sononet-FT-16 & \textbf{0.978} & \textbf{0.929} & \textbf{0.924} & \textbf{0.934} & 1.94/5.13 & 0.70M
\\
\hline
Sononet-32 & 0.979 & 0.931 & 0.924 & \textbf{0.938} & 2.40/6.72 & 2.58M
\\
AG-Sononet-32 & \textbf{0.980} & 0.932 & 0.928 & 0.937 & 3.01/8.74 & 2.79M
\\
AG-Sononet-DS-32 & 0.978 & 0.929 & 0.921 & 0.937 & 2.98/8.81 & 2.80M
\\
AG-Sononet-FT-32 & \textbf{0.980} & \textbf{0.933} & \textbf{0.931} & 0.935 & 2.92/8.68 & 2.79M
\\
\bottomrule
\end{tabular}
\label{table:result1}
\end{table}
\begin{table}
\centering
\caption{Class-wise performance for AG-Sononet-FT-8. In bracket shows the improvement over Sononet-8. Bold highlights the improvement more than 0.02.}
\scalebox{0.9}{
\begin{tabular}{llll}
\toprule{} & Precision & Recall & F1 \\ \midrule
Brain (Cb.) & 0.988 (-0.002) & 0.982 (-0.002) & 0.985 (-0.002) \\
Brain (Tv.) & 0.980 ( 0.003) & 0.990 ( 0.002) & 0.985 ( 0.003) \\
Profile & 0.953 \textbf{( 0.055)} & 0.962 ( 0.009) & 0.958 \textbf{( 0.033)} \\
Lips & 0.976 \textbf{( 0.029)} & 0.956 (-0.003) & 0.966 ( 0.013) \\
Abdominal & 0.963 ( 0.011) & 0.961 ( 0.007) & 0.962 ( 0.009) \\
Kidneys & 0.863 \textbf{( 0.054)} & 0.902 ( 0.003) & 0.882 \textbf{( 0.030)} \\
Femur & 0.975 ( 0.019) & 0.976 (-0.005) & 0.975 ( 0.007) \\
Spine (Cor.) & 0.935 \textbf{( 0.049)} & 0.979 ( 0.000) & 0.957 \textbf{( 0.026)} \\
Spine (Sag.) & 0.936 \textbf{( 0.055)} & 0.979 (-0.012) & 0.957 \textbf{( 0.024)} \\
4CH & 0.943 \textbf{( 0.035)} & 0.970 ( 0.007) & 0.956 \textbf{( 0.022)} \\
3VV & 0.694 \textbf{( 0.050)} & 0.722 (-0.014) & 0.708 \textbf{( 0.021)} \\
RVOT & 0.691 \textbf{( 0.029)} & 0.705 \textbf{( 0.044)} & 0.698 \textbf{( 0.036)} \\
LVOT & 0.925 \textbf{( 0.022)} & 0.933 \textbf{( 0.027)} & 0.929 \textbf{( 0.024)} \\
Background & 0.995 (-0.001) & 0.992 ( 0.007) & 0.993 ( 0.003) \\
\bottomrule
\end{tabular}}
\label{table:result_class}
\end{table}
\textbf{Attention Map Analysis and Object Localisation} In Figure \ref{fig:attention_ag_sononet}, we show the attention map of AG-Sononet. AG-1 and AG-2 are the attention map applied at layer 11 and 14 respectively. AG-3 is the attention map of the final layer (the coarsest). In this case, we do not use attention gates, however, we use activation maps with $C \in \{64, 128, 256\}$ channels depending on the capacity. In order to visualise class-specific attention, we employed Class Activation Mapping (CAM) \cite{zhou2016learning}. AG-all is obtained by taking the mean of the attention maps which are all normalised to have the maximum value 1. Recall that AG-Sononet simply obtains mean of the predictions at each image scale. As such, the attention maps pinpoint the class-specific information at all scales. In Figure \ref{fig:attention_ag_sononet_ds}, we show the attention map of AG-Sononet-FT. In this case, the aggregation layer relearns how to optimally combine the features at different scales. Fine-scale features do not necessarily highlight the whole object of interest, but it highlights key information within it that cannot be observed by the coarse scale representation. Similarly in some cases, fine-scale features seem to not learn anything if the prediction can be done by coarser scales.
Finally, in Figure \ref{fig:attention_ag_sononet_ds_variation}, we show the attention maps of AG-Sononet-FT across different subjects, together the bounding box annotation generated using the attention maps (see Appendix for the heuristics). We see that the network consistently focuses on the object of interest, which indicates that the network indeed learnt the most important feature for each class. We note, however, attention map outlines the discriminant region; in particular, it does not necessarily coincide with the entire object. This behaviour makes sense because some part of object will appear in background label (i.e. when the ideal plane is not reached). Qualitatively, however, the bounding boxes well agree with the annotated ground truth. Most crucially, the attention map is obtained for almost no additional computational cost; In comparison, \cite{baumgartner2016real} requires guided backpropagation for localisation, which limits the localisation speed. This highlights the advantage of attention model for the real-time applications.
\begin{figure}[!t]
\centering
\includegraphics[width=1\textwidth]{figures/config_sononet_grid_att_fs8_avg_v12.jpg}
\caption{Examples of obtained attention map from AG-sononet. AG1 and AG2 are from layer 11 and 14 respectively. AG3 is obtained using CAM \cite{zhou2016learning}. AG-all is obtained by normalising the maximum attented value across all AG's and taking mean over them.}
\label{fig:attention_ag_sononet}
\end{figure}
\begin{figure}[!t]
\centering
\includegraphics[width=1\textwidth]{figures/config_sononet_grid_att_fs8_w_avg_v1.jpg}
\caption{Examples of obtained attention map from AG-sononet-FT. AG1 and AG2 are from layer 11 and 14 respectively. AG3 is obtained using CAM \cite{zhou2016learning}. AG-all is obtained by normalising the maximum attented value across all AG's and taking mean over them.}
\label{fig:attention_ag_sononet_ds}
\end{figure}
\begin{figure}[!t]
\centering
\includegraphics[width=1\textwidth]{figures/config_sononet_grid_att_fs4_w_avg_v1_variation.jpg}
\caption{Examples of the obtained attention map and genereated bounding boxes (red) from AG-Sononet-FT across different subjects. The ground truth annotation is shown in blue. The detected region highly agrees with the object of interest.}
\label{fig:attention_ag_sononet_ds_variation}
\end{figure}
\section{Discussion}
In this work, we considered soft-attention mechanism and discuss how to incorporate them into scan plane detection for fetal ultrasound to better exploit local structures. In particular, we highlighted several aspects: a normalisation strategy for the attention map, gridded attention mechanisms, and aggregation strategies. We empirically observed and reported that soft-max tends to generate a map that is over sensitive to local intensity changes, which is problematic as in medical imaging, image quality is often low. We found that dividing by sum helped attention to distribute more evenly. A Sigmoid functions is an alternative as it only normalises the range and allows more information to flow. However, we found that training is non-trivial due to the gradient saturation problem.
We noted that training the attention-mechanism was slightly more complex than the standard network architecture. In particular, we observed that the strategy employed to aggregate the attention maps at different scales affects both the learning of the attention mechanism itself and hence the performance. Having a loss term defined at each scale ensures that the network learns to attend at each scale. We observed that first training the network at each scale separately, followed by fine-tuning was the most stable approach to get the optimal performance.
The proposed network architecture resembles the one of deep-supervision in the sense that we add modules before the final layer which helps back-propagating the gradient at the early layers of the network. However, we argue that without a proper gating mechanism, we will not see any improvement. In fact, we saw that the model trained with deep supervision did not necessarily obtain the best result. Therefore, while the network certainly benefits from backpropagating through additional pathways, the improvement in performance only came in conjunction with the attention mechanism.
\section{Conclusion}
In this work we proposed generalised self-attention mechanisms that can be easily incorporated into existing classification architectures. We applied the proposed architecture to standard scan plane detection during fetal ultrasound screening and showed that it improves overall results, especially precision, with much less parameters. This was done by generating the gating signal to pinpoint local as well as global information that is useful for the classification. Furthermore, it allows one to generate fine-grained attention map that can be exploited for object localisation. We envisage that the proposed soft-attention module will have great impact for explainable deep learning, weakly supervised object detection and/or segmentation -- which are all vital research area for medical imaging analysis.
\bibliographystyle{splncs03}
|
{
"timestamp": "2018-04-17T02:10:09",
"yymm": "1804",
"arxiv_id": "1804.05338",
"language": "en",
"url": "https://arxiv.org/abs/1804.05338"
}
|
\section{Introduction}
The Hilbert metric on convex subsets of $\mathbb R^n$ is a well-known and well-studied
object. We refer the reader to \cite{HandbookHilbert} for a comprehensive introduction
to the metric aspects. A major domain of applications is the field of
divisible convex, where the invariance of the metric under projective transformations
of the convex is leveraged. Such a study originates in large part from
the series of works of Benoist begining with \cite{Benoist}. We refer to the
survey \cite{Benoist-survey} for a more precise description of these works.
We propose in this paper a generalization of the notion of Hilbert
metric to settings without convexity. The main idea to
overcome the lack of convexity is to make use of the duality between projective
spaces and dual projective spaces. A first attempt to give such
a generalization has been done by the second author in \cite{Guilloux-padic}. Whereas
the focus of the cited paper was projective spaces over local fields,
we mainly work here in real and complex
projective spaces.
We define in Section \ref{sec:definition} a notion of generalized Hilbert metric
on a set $\Omega$ in $\P(\mathbb C^{n+1})$ as long
as $\Omega$ avoids a compact set of hyperplanes. For example, any open set in
$\mathbb {CP}^1$ inherits such a metric. The metric does not, in general, separate points
but we are able to determine the conditions under which it does (Theorem \ref{thm:metric}).
For example, any open subset of $\mathbb {CP}^1$ whose complement does not lie in a circle
inherits an actual metric. We then move on to a description of the
associated infinitesimal Finsler metric
(Theorem \ref{theorem:metric}). An example of such a generalized Hilbert metric is the
usual hyperbolic metric on
complex hyperbolic space, just as the hyperbolic metric on the real hyperbolic
space is known to be an example of Hilbert metric. Note that our metric is naturally
invariant under projective transformations of $\Omega$. A more general problem is to compare the Bergman metric to ours
for bounded domains in $\mathbb C^n$ (see remark \ref{shilov}).
We then give three different directions of application to these definitions. We hope
that the given definition will prove useful in a wealth of problems and try to
convince the reader so. The first direction (Section \ref{sec:complexkleinian})
is at the origin of this work: we explore the
meaning of the definition for complex Kleinian hyperbolic groups, i.e. discrete subgroups
of $\mathrm{PU}(n,1)$. We are able to reinterpret in a geometric way
our definition and prove that it defines a natural metric on uniformized spherical CR manifolds.
The second direction (Section \ref{section:puncturedsphere}) deals with a very
simple example, akin to the polygon case of the Hilbert metric (see \cite{delaHarpe}):
the $n$-punctured projective line, where the punctures do not lie in a single circle.
We prove that our metric is quasi-isometric - but not isometric - to the
hyperbolic metric on the $n$-punctured sphere.
Eventually, we look in
Section \ref{section:RP1} at an example that may seem strange at first
glance: any open subset of the real projective line, even if not connected, inherits a metric.
We focus on complements of self-similar compact sets (such as limit sets of Fuchsian groups or
self-similar Cantor sets) and are able to prove a generalization of a formula due to
Basmajian in the setting of hyperbolic surfaces.
We thank Gilles Courtois and Pascal Dingoyan for several useful discussions. The third author
would like to thank Hugo Parlier for an enlightening discussion. Many discussions around this project
occured at the UMPA, and we thank that institution for its kind hospitality.
\section{The metric}\label{sec:definition}
We define here a metric on subsets of projective spaces avoiding a compact set of hyperplanes, under a
non degeneracy hypothesis. The idea is to redefine the usual Hilbert metric in a more suitable way
to a generalization to projective spaces on other fields than $\mathbb R$. A first attempt, with mainly the $p$-adic fields in mind, has been done in \cite{Guilloux-padic}. We are interested in this paper in the complex case, so we will always assume that we are working in complex projective spaces. Note that the real case follows, by including the real projective spaces in its complexification.
We will regularly switch between elements in projective spaces and lifts in the vector spaces. In order to be able to do it without cumbersome definitions and notations, we fix a convention throughout this paper:
points in projective spaces will be denoted by symbols like $\omega$, $\phi$, $m$, $p$... and any lift will be denoted by the bold corresponding symbol $\bm \omega$, $\bm \phi$, $\bmm$, $\bm p$...
\subsection{Setting and first examples}
We fix throughout the following section an integer $n\geq 1$. We denote by $\P$ the $n$-dimensional projective space $\P(\mathbb C^{n+1})$ and by $\P'$ its dual
$\P((\mathbb C^{n+1})^\vee)$. We consider two non-empty subsets $\Omega \subset \P$ and
$\Lambda \subset \P'$ such that
\begin{equation}\label{eq:condi-subset}\forall \omega \in \Omega,\, \forall \phi \in \Lambda \ \ \ \ \bm \phi(\bm \omega)\neq 0.\end{equation}
Geometrically, each point in $\P'$ represents a hyperplane in $\P$, and condition
\eqref{eq:condi-subset} means that
$\Omega$ is disjoint from all hyperplanes defined by points in $\Lambda$.
The following definition gives a name to such pairs.
\begin{definition}
A pair $(\Omega,\Lambda)$, where $\Omega$ is open in $\P$, $\Lambda$ closed in $\P'$ and
condition \ref{eq:condi-subset} holds, is called \emph{admissible}.
Morever, we say that $\Omega$ is \emph{saturated} if it is the set of points on which forms
of $\Lambda$ do not vanish.
\end{definition}
We will define a weak metric on the set $\Omega$ of admissible pairs. Here are the examples of
typical situations we are
thinking of. As hinted by the notation, such examples often come with considering $\Lambda$ to be a
limit set in $\P'$ for the action of a subgroup $\Gamma \subset \mathrm{PGL}(n+1,\mathbb C)$ and
$\Omega$ to be the set of points on which elements of $\Lambda$ do not vanish.
\begin{enumerate}
\item If $\Omega$ is a bounded domain of $\mathbb C^n\subset \P$, we can take
$\Lambda$ to be the set of complex lines that do not meet $\Omega$. The pseudo-metric
we will define on these examples may not be complete or even separate points. But in
the case of the unit ball, we recover the Bergman metric (see Remark \ref{rem:complexhyperbolicspace}).
\item Let $\Gamma$ be a quasi-Fuchsian subgroup of $\mathrm{PGL}(2,\mathbb C)$,
with limit set $\Lambda_\Gamma\subset \mathbb {CP}^1$. Note that for $n=1$,
then $\mathbb {CP}^1 = \P$ is naturally identified to its dual $\P'$: the
isomorphism sends a point in $\mathbb {CP}^1$ to its orthogonal i.e. the class
of forms vanishing at this point.
So $\Lambda_\Gamma$ is seen as a subset, our $\Lambda$, of $\P'$.
We thus set $\Omega=\Omega_\Gamma$, the complement in
$\mathbb {CP}^1$ of $\Lambda_\Gamma$. In this case,
the pair $\Omega,\Lambda$ satisfies \eqref{eq:condi-subset}, it is an admissible pair and $\Omega$ is saturated.
\item One may consider a discrete subgroup $\Gamma$ of $\mathrm{PU}(n,1)$ and suppose
that its limit set $\Lambda_\Gamma$ in the sphere $\S^{2n-1}$ at
infinity of complex hyperbolic space $\H^n_\mathbb C$ is not the whole sphere.
For each point $p\in \S^{2n-1}$, there exists a unique projective complex hyperplane
tangent to $\S^{2n-1}$ at $p$, which we denote by $L_p$. To each point in $\Lambda_\Gamma$ we associate the (class of) linear form
$\phi_p$ defined by $\P(\ker(\bm \phi_p))=L_p$.
Note that $\phi_p$ is actually just the class of the bracket $\langle ,\bm p\rangle$, where
$\bm p$ is a lift of $p$ to $\mathbb C^{n+1}$, and $\langle \cdot,\cdot \rangle$ is the ambient Hermitian form of signature $(n,1)$. We then define
$$\Lambda=\lbrace \phi_p, p\in\Lambda_\Gamma\rbrace,\mbox{ and } \Omega=\underset{p\in\Lambda_\Gamma}{\bigcap} L_p^c,$$
where $L_p^c$ denotes the complement. These two sets satisfy \eqref{eq:condi-subset}.
We will go into more details in Section \ref{sec:complexkleinian}. We will see
that this $\Omega$ is the complement of the Kulkarni limit set and interpret our metric in a geometric manner.
\item If $\Omega$ is an open proper convex set in $\mathbb R^n \subset \P(\mathbb R^{n+1})$, then one take for
$\Lambda$ the set of forms that do not vanish on $\Omega$. In standard terminology used for convex sets in affine
real space, $\Lambda$ is the closure of the dual convex to $\Omega$. In this case, we have the usual notion of Hilbert metric \cite{delaHarpe,HandbookHilbert} with a huge literature studying various properties of this metric.
This example is our benchmark: everything we define in a more general setting has to specialize to
the usual Hilbert metric for these pairs $(\Omega,\Lambda)$.
\end{enumerate}
\subsection{The cross-ratio}
Our main source of inspiration is the Hilbert metric, whose definition relies on the
notion of cross-ratio of four points on a projective line. We fix in this section the convention
on cross-ratios and
define a natural notion of cross-ratio between two forms and two points and gather
some well-known properties.
We take as a definition of the cross-ratio of four distinct points in $\mathbb {CP}^1$ with coordinates $(a,b,c,d)$
the element $t=[a,b,c,d]$ such that there is an element of $\mathrm{PGL}(2,\mathbb C)$ sending the
4-tuples $(a,b,c,d)$ to
$(\infty,0,1,t)$. In any affine chart, it is given by:
\begin{equation}\label{eq:usual-crossratio}[a,b,c,d]=\dfrac{(d-b)(c-a)}{(d-a)(c-b)}.\end{equation}
We now define the cross-ratio $[\phi,\phi',\omega,\omega']$ for $\phi$ and $\phi'$ in $\P'$ and
$\omega$, $\omega'$ in $\P$. For any two $\omega\neq\omega'$ in $\P$, we denote by $(\omega\omega')$ the (complex) projective line they span.
\begin{definition} \label{def:crossratio}
\begin{enumerate}
\item For any $\phi\in \Lambda$ and any pair $(\omega,\omega')$ of distinct points in $\Omega$, let
$\phi_{\omega,\omega'}$ be the point $\ker(\phi)\cap (\omega,\omega')$ in $(\omega,\omega')$.
\item We call {\it cross-ratio of $(\phi,\phi',\omega,\omega')$} the cross-ratio
\begin{equation}\label{eq:defi-crossratio}
[\phi,\phi',\omega,\omega'] := [\phi_{\omega,\omega'},\phi'_{\omega,\omega'},\omega,\omega']
\end{equation}
\end{enumerate}
\end{definition}
Note that the four points involved in \eqref{eq:defi-crossratio} lie in a common complex line by definition.
The cross-ratio of $(\phi,\phi',\omega,\omega')$ can be computed simply with lifts $(\bm \phi,\bm \phi',
\bm \omega,\bm \omega')$ as follows (compare with \cite[Lemma 3.1]{Guilloux-padic}).
\begin{lemma}
Let $(\phi,\phi',\omega,\omega')\in\Lambda^2\times\Omega^2$. The cross-ratio
$[\phi,\phi',\omega,\omega']$ satisfies
\begin{equation}
\label{eq:comput-crossratio}[\phi,\phi',\omega,\omega']=\dfrac{\bm \phi(\bm \omega)\bm \phi'(\bm \omega')}{\bm \phi(\bm \omega')\bm \phi'(\bm \omega)}
\end{equation}
\end{lemma}
\begin{figure}
\centering
\scalebox{0.4}{\includegraphics{crossratio.eps}}
\caption{The cross-ratio $[\phi,\phi',\omega,\omega']$}\label{fig:cross-ratio}
\end{figure}
\begin{proof}
The proof is classical and elementary. It is summarized by Figure \ref{fig:cross-ratio}. We include it for completeness.
If $\phi=\phi'$ the identity is obvious. If $\phi\neq \phi'$,
choose a basis $(e_k)_{1\leqslant k\leqslant n+1}$ of $\mathbb C^{n+1}$ and lifts
such that $\bm \phi$, $\bm \phi'$, $\bm \phi_{\omega,\omega'}$ and $\bm \phi'_{\omega,\omega'}$
are as follows:
$$\bm \phi = e_1^\vee, \bm \phi'= e_2^\vee, \bm \phi_{\omega,\omega'}= e_2+ \bm w,
\bm \phi'_{\omega,\omega'} = e_1+\bm w',$$
where $\bm w$ and $\bm w'$ are vectors in ${\rm Span}(e_3\cdots e_{n+1})$.
Then $\bm \omega$ and $\bm \omega'$ have the form
$$\bm \omega = \lambda \bm \phi_{\omega,\omega'} + \lambda'\bm \phi'_{\omega,\omega'}\mbox{ and } \bm \omega'= \mu \bm \phi_{\omega,\omega'} + \mu'\bm \phi'_{\omega,\omega'}.$$
Computing the right hand side of \eqref{eq:comput-crossratio}, we obtain
\begin{equation*}
\dfrac{\bm \phi(\bm \omega)\bm \phi'(\bm \omega')}{\bm \phi(\bm \omega')\bm \phi'(\bm \omega)} =
\dfrac{\mu\lambda'}{\lambda \mu'} =
[\phi_{\omega,\omega'},\phi'_{\omega,\omega'},\omega,\omega'],
\end{equation*}
where the second equality is obtain using \eqref{eq:usual-crossratio} by noting
that with the chosen coordinates,
the four points $\bm \phi_{\omega,\omega'}$, $\bm \phi'_{\omega,\omega'}$,
$\bm \omega$ and $\bm \omega'$ are given by
$$ \bm \phi_{\omega,\omega'}\sim 0,\, \bm \phi'_{\omega,\omega'}
\sim \infty,\, \bm \omega\sim\dfrac{\lambda}{\lambda'}
\mbox{ and } \bm \omega'\sim\dfrac{\mu}{\mu'}.$$
\end{proof}
The following identities follow by a direct verification.
\begin{proposition}\label{prop:birapport-identities}
Let $\omega,\omega',\omega''$ be three points in $\Omega$, and $\phi,\phi',\phi''$ be three points in $\Lambda$.
Then
\begin{enumerate}
\item $[\phi,\phi',\omega,\omega'] = [\phi',\phi,\omega',\omega]$
\item $[\phi,\phi',\omega,\omega']= [\phi,\phi',\omega,\omega''][\phi,\phi',\omega'',\omega']$
\item $[\phi,\phi',\omega,\omega']=[\phi,\phi'',\omega,\omega'][\phi'',\phi',\omega,\omega']$
\end{enumerate}
\end{proposition}
The last two equalities are known as \emph{cocycle relation} for the cross-ratio. With these properties at hand, we may proceed to the definition of our metric.
\subsection{A generalized Hilbert pseudo-metric}
From now on, we assume that the set $\Lambda$ is compact. Our Hilbert metric
is defined by the following:
\begin{definition}
Let $d_\Lambda$ be the function defined $\Omega \times \Omega$ by
\begin{equation}\label{eq:def-dist}
d_\Lambda (\omega,\omega') = \ln\left(\max \left\{
\left|\left[\phi,\phi',\omega,\omega'\right]\right| \textrm{ for }
\phi,\phi'\textrm{ in } \Lambda\right\}\right)
\end{equation}
\end{definition}
As noted in \cite[Section 3.2]{Guilloux-padic}, in the case of open proper convex
subset of $\P(\mathbb R^{n+1})$,
we recover the usual Hilbert metric up to a factor $\frac{1}{2}$. This formula
is reminiscent of a metric associated to a Funk metric \cite{PapadopoulosTroyanov}.
We will take advantage of this remark later on, by separating the contributions
of $\phi$ and $\phi'$.
In our more general setting, $d_\Lambda$ is not quite a metric but almost:
\begin{proposition}
The function $d_\Lambda$ is a pseudo-metric: it is non-negative,
symmetric and satisfies to the triangle inequality.
\end{proposition}
In the terms of \cite{PapadopoulosTroyanov-weakMinkowski}, $d_\Lambda$ is a symmetric weak metric.
\begin{proof}
Note that exchanging $\phi$ and $\phi'$ in $\left|\left[\phi,\phi',\omega,\omega'\right]\right|$ transforms it into
its inverse. This implies in particular that $d_\Lambda$ is non-negative. The other two properties follow directly
from the first two items of Proposition \ref{prop:birapport-identities}
\end{proof}
In general this metric does not separate points. Indeed, if $\Lambda$ consists of a single point, then every cross-ratio is $1$ and $d_\Lambda$ is trivial. A more interesting example is the case of $\Lambda = \{0,\infty\} \subset \mathbb {CP}^1$ and $\Omega = \mathbb C\setminus\{0\}$. Then, it is easy to compute that, for two non zero complex numbers $z$ and $z'$, we have:
$$d_{\{0,\infty\}} (z,z') = |\ln |z| - \ln |z'|\,|.$$
Thus, points of same modulus are at distance $0$. We will focus our attention to punctured spheres in section \ref{section:puncturedsphere}.
Before exploring the conditions for $d_{\Lambda}$ to be an actual metric, let us point out two
consequences of the mere definition of this function. First, we remark that $d_\Lambda$ is invariant under projective transformations:
\begin{proposition}\label{prop:invariance}
Let $(\Omega, \Lambda)$ be an admissible pair. For any $g\in \mathrm{PGL}(n+1,\mathbb C)$, the pair $(g\cdot \Omega, g\cdot \Lambda)$ is admissible, and the action of $g$ is an isometry between $(\Omega, d_\Lambda)$ and $(g\cdot \Omega, d_{g\cdot \Lambda})$
\end{proposition}
\begin{proof}
Indeed, the cross-ratio defined in Definition \ref{def:crossratio} is invariant under projective transformation.
\end{proof}
As a consequence, when $\Lambda$ is a limit set for a group $\Gamma$ and $\Omega$ its
complement, as in the first examples described, $d_\Lambda$ is $\Gamma$-invariant.
The second fact we want to point out states the pseudo-convexity of $\Omega$.
\begin{proposition}
Let $(\Omega,\Lambda)$ be an admissible pair, with $\Omega$ satured.
Then $\Omega$ is pseudo-convex and for any $\omega_0 \in \Omega$, the function
$\omega \to d_\Lambda(\omega_0,\omega)$ is a subharmonic exhaustion.
\end{proposition}
\begin{proof}
Fix a point $\omega_0$ in $\Omega$, and consider the function
$F(\omega) = d_{\Lambda}(\omega_0,\omega)$.
This function is defined as the max on a compact set of functions
$\ln(| [\phi,\phi',\omega_0,\omega]|)$. The cross-ratios are
holomorphic functions of $\omega \in \Omega$. Hence the function
$F$ is sub-harmonic. Moreover, if $\omega$ escapes any compact in
$\Omega$, then there are forms $\phi$ in $\Lambda$ such that $\phi(\omega) \to 0$.
Then, fix a form $\phi_0 \in \Lambda$: the
cross-ratio $[\phi,\phi_0,\omega_0,\omega]$ goes to $\infty$
hence $F(\omega) \to +\infty$. Conclusion: $F$ is a subharmonic exhaustion
of $\Omega$.
\end{proof}
\subsection{Separation condition}\label{subsec:separation}
We now explore when two points $\omega$ and $\omega'$ are separated
by the metric $d_\Lambda$, meaning that $d_\Lambda (\omega,\omega')>0$.
Once $\omega$ and $\omega'$ are fixed, the cross-ratios $[\phi,\phi',\omega,\omega']$
are determined by the points $\phi_{\omega,\omega'}$ as in Definition \ref{def:crossratio}.
Let us define a notation for the set of these points:
\begin{definition}\label{def:projection}
The set $\Lambda_{\omega,\omega'}$ of points $\phi_{\omega,\omega'}$
for $\phi \in \Lambda$ is called the \emph{projection} of $\Lambda$ on
the line $(\omega,\omega')$.
\end{definition}
As stated in the following proposition, $d_\Lambda$ separates $\omega$ and $\omega'$
as soon as its projection in the complex line $(\omega,\omega')$ is not included in
a real line with $\omega$ and $\omega'$ complex conjugate w.r.t. this line.
\begin{theorem}\label{thm:metric}
Let $\omega,\omega'$ be two distinct points in $\Omega$.
The following three conditions are equivalent.
\begin{enumerate}
\item $d_\Lambda$ does not separate $\omega$ and $\omega'$.
\item For all pairs $(\phi,\phi')$ in $\Lambda\times\Lambda$, $\|[\phi,\phi',\omega,\omega']\|=1$.
\item There exists an anti-holomorphic involution of the complex line
$(\omega,\omega')$ which exchange $\omega$
and $\omega'$, and fixes pointwise the projection $\Lambda{\omega,\omega'}$.
\end{enumerate}
Moreover, if $d_\Lambda$ separates each pair of distinct points, then $d_\Lambda$ is a metric.
\end{theorem}
For an admissible pair $(\Omega,\Lambda)$, if $d_\Lambda$ is a metric, we will say that $\Lambda$
is \emph{separating}.
\begin{proof}
The first two items of the equivalence are clearly equivalent. Choosing a coordinate on the
line $(\omega,\omega')$ such that
$\omega=0$ and $\omega'=\infty$, we see that the condition
$\|[\phi,\phi',\omega,\omega']\|=1$ is equivalent to
the fact that the two points $\phi_{\omega,\omega'}$ and
$\phi'_{\omega,\omega'}$ lie on a same circle centered at
$0$. So if every cross-ratio has modulus one, the whole projection
$\Lambda_{\omega,\omega'}$ is include in this circle.
We may assume, up to a change of coordinate, that this circle
is the unit circle.
The reflection $z \to \frac{1}{\bar z}$ about this circle is an anti-holomorphic
involution which fixes pointwise $\Lambda_{\omega,\omega'}$, and exchanges
$\omega = 0$ and $\omega' = \infty$.
Conversely, suppose that an anti-holomorphic involution $\sigma$ fixes the projection
$\Lambda_{\omega,\omega'}$ and $\sigma(\omega) = \omega'$. Then,
up to a change of coordinates, $\sigma$ is the
complex conjugation, $\Lambda_{\omega,\omega'}$ is included in the real line
$\mathbb {RP}^1$ and $\omega' = \bar \omega$. Then, every cross-ratio has modulus one:
\[ [\phi_{\omega,\omega'},\phi'_{\omega,\omega'},\omega,\omega'] =
\frac{(\phi_{\omega,\omega'} - \omega)(\phi'_{\omega,\omega'}-\omega')}{\phi_{\omega,\omega'} - \omega')(\phi'_{\omega,\omega'}-\omega)} =
\frac{\overline{(\phi_{\omega,\omega'} - \omega')(\phi'_{\omega,\omega'}-\omega)}}{(\phi_{\omega,\omega'} - \omega')(\phi'_{\omega,\omega'}-\omega)} \]
This proves the equivalence. The last sentence is straightforward: the separation was the
only property lacking to $d_\Lambda$ to be a metric.
\end{proof}
\begin{remark}\label{remark:zariski}We shall give examples where this condition holds. In fact, it is not as hard to check as it may seem.
Indeed, the equivalence implies that if $\Lambda$
fails to separate two points, then it is included in a $\mathbb R$-Zariski closed subset of $\P'$. Indeed, in this case, the equation $\|[\phi,\phi',\omega,\omega']\|=1$ should be valid for every $\phi$ for any fixed $\phi'$. The intersection of all solutions for varying $\phi'$ is a $\mathbb R-$Zariski closed subset of $\P'$.
In
$\mathbb {CP}^1$, any subset $\Lambda$ which is not included in a circle defines an actual metric on its
complement $\Omega$. So this metric on a $3$-punctured sphere is never separating.
Still, we may often
decide whether $\Lambda$ separates or not, even in higher dimensions, and especially in the case of limit sets.
In Section \ref{section:CR}, we will interpret this separation condition geometrically
in the context of spherical CR structures. We will then give non trivial examples of separating
sets $\Lambda$.
\end{remark}
\begin{remark}
As we have already noted, if $\Omega$ is an open proper convex subset of $\P(\mathbb R^{n+1})$, then
we take $\Lambda$ to be the closure of its dual convex set. In this case, $d_\Lambda$ is the usual
Hilbert metric, up to a factor $\frac 12$. An intriguing remark is that we may take a smaller $\Lambda$,
and define a metric on a disconnected set $\Omega$. For example, in the plane, take
$\Lambda = \{\phi_1,\phi_2,\phi_3\}$ to be three forms (not in the same line). Then each component of the complement of the three lines in the projective plane is a triangle. On each triangle,
$d_\Lambda$ is the usual Hilbert metric (see \cite{delaHarpe} for a beautiful study of these), but
then the distance between two points in different triangles is also defined. We will come
back to this remark in section \ref{section:RP1} in the seemingly trivial case of the real line $\mathbb {RP}^1$.
\end{remark}
\subsection{An infinitesimal symmetric Finsler metric}\label{Finsler metric}
We want to explore the infinitesimal behavior of the metric $d_\Lambda$. Once again,
the real case is the source of inspiration, where it is known that the Hilbert metric is
a Finsler geometry \cite{Troyanov} and this Finsler geometry is an object of study. Beware
that in this Hilbert geometry setting, the notion of Finsler metric is not as smooth
as in other parts of the literature: a Finsler metric on $\Omega$, for our purpose, is
a continuous function on $T\Omega$ which is a norm in each tangent space $T_\omega\Omega$.
In our case the situation is a bit more intricate than in the real case.
We show in this section that $d_\Lambda$ does indeed define a Finsler metric, and are able to
compute it. But, the presence of the $\max$ in the definition of $d_\Lambda$ and
the lack of regularity of our $\Lambda$ in examples we consider interesting prevent
any smoothness. Moreover, $(\Omega,d_\Lambda)$ is not in general a length space:
the metric $d_\Lambda$ is not the infimum length of a smooth path between two points.
We will nonetheless be able to understand when the unit ball for the norms are strictly convex.
\newcommand{\norm}[2]{\|#2\|_{\Lambda,#1}}
We begin by computing the infinitesimal behavior of $d_\Lambda$ to define the Finsler metric. We have to parametrize a tangent space $T_\omega\Omega$. To do that, we choose a lift $\bm \omega$ and identify
$T_\omega\Omega$ with a subspace $T\in \mathbb C^{n+1}$
transverse to the line $\omega$.
\begin{theorem}\label{theorem:metric}
Let $(\Omega,\Lambda)$ be an admissible pair, with $\Lambda$ separating.
Then the metric $d_\Lambda$ yields a symmetric Finsler metric $(\omega,v) \to \norm{\omega}{v}$ on $T\Omega$, which is given for $T_\omega\Omega$ and $v\in T$ by:
\begin{equation}\label{formula:Finsler}
\norm{\omega}{v} = \max_{\phi, \phi' \in \Lambda}
\: \Re\left(\frac{\bm \phi'(v)}{\bm \phi'(\bm \omega)} - \frac{\bm \phi(v)}{\bm \phi(\bm \omega)}\right)
\end{equation}
\end{theorem}
Note that this formula is actually independent of the choice of lifts $\bm \phi$, $\bm \phi'$. Moreover,
multiplying the lift $\bm \omega$ by some $r \in \mathbb C$ amounts to changing the parametrization of
$T_\omega\Omega$ by $T$. Hence this formula is indeed defined on $T_\omega\Omega$. Another
point worth noting is that formula actually separates the contribution of $\phi$ and $\phi'$,
as in the usual way to pass from a Funk metric to a Finsler metric \cite{PapadopoulosTroyanov}.
\begin{proof}
Let $\omega$ be a point in $\Omega$, with $\bm \omega \in \mathbb C^{n+1}$ a lift. We identify
$T_{\omega} \Omega$ with a hyperplane
$T \subset \mathbb C^{n+1}$ transverse to the line generated by $m$.
For $v\in T$ and $t>0$, we will prove that the first
order term in the Taylor expansion of $d_\Lambda(\omega,\omega+tv)$ as
$t\rightarrow 0$ defines a norm on the tangent space
at $\omega$. Let us first fix $\phi$ and $\phi'$ two points
in $\Lambda$ and $\bm \phi$ and $\bm \phi'$ corresponding linear forms.
We first expand the cross-ratio at first order:
\begin{eqnarray*}
[\phi,\phi',\omega,\omega+tv] = &
\dfrac{\bm \phi(\bm \omega)\bm \phi'(\bm \omega+tv)}{\bm \phi'(\bm \omega)\bm \phi(\bm \omega+tv)}\\
= & \dfrac{1+t\frac{\bm \phi'(v)}{\bm \phi'(\bm \omega)}}{1+t\frac{\bm \phi(v)}{\bm \phi(\bm \omega)}}\\
= & 1 + t \left(\dfrac{\bm \phi'(v)}{\bm \phi'(\bm \omega)} - \dfrac{\bm \phi(v)}{\bm \phi(\bm \omega)}\right) + o(t)
\end{eqnarray*}
Since $\ln(|1+tz|) = \frac 12\log(|1+tz|^2)=\frac 12 \log(1 + t(z+\bar z)+o(t)) = t \Re(z) +o(t)$, we may
further compute:
$$
\frac{d}{dt}_{|t=0+} \ln\left(\left|[\phi,\phi',\omega,\omega+tv]\right|\right) =
\Re\left(\frac{\bm \phi'(v)}{\bm \phi'(\bm \omega)} - \frac{\bm \phi(v)}{\bm \phi(\bm \omega)}\right).
$$
The metric is given by $d_{\Lambda} (\omega,\omega+tv)=
\max_{\phi, \phi' \in \Lambda} \ln\left(\left|[\phi,\phi',\omega,\omega+tv]\right|\right)$. Every
function appearing in the $\max$ equals $0$ for $t=0$. Lemma \ref{lem:derive} below tells us that we may swap the max
and the derivative. This gives us the announced expression for $\| \cdot \|_\Lambda$:
\begin{eqnarray}
\norm{\omega}{v} & = & \frac{d}{dt}_{|t=0+} d_\Lambda(\omega,\omega+tv)\nonumber\\
& = & \max_{\phi, \phi' \in \Lambda} \Re\left(\frac{\bm \phi'(v)}{\bm \phi'(\bm \omega)} - \frac{\bm \phi(v)}{\bm \phi(\omega)}\right).\nonumber
\end{eqnarray}
Note that since we are taking the maximum over all pairs $(\phi,\phi')$,
the quantity is positive: exchanging $\phi$ and $\phi'$ just changes the sign.
To prove that $\| \cdot \|_\Lambda$ defines a symmetric Finsler metric, we need to show that, in each tangent space, the sublevel set
$B_\Lambda^{\omega}=\{ v\in T_{\omega}\Omega,\norm{\omega}{v}\leqslant 1\}$ is compact, convex and symmetric. This sublevel set may be written as the intersection
\begin{equation}\label{eq:unit-ball}
\underset{\phi,\phi'\in\Lambda}{\bigcap}\Bigl\{ \Re\left(\frac{\bm \phi'(v)}{\bm \phi'(\bm \omega)} - \frac{\bm \phi(v)}{\bm \phi(\bm \omega)}\right)\leqslant 1\Bigr\}.
\end{equation}
We see from \eqref{eq:unit-ball} that $B_\Lambda^{\omega}$ is convex as an intersection of half-spaces, and symmetric : if the max in \eqref{formula:Finsler} for a vector $v$ is obtained for a pair $(\phi,\phi')$, then $(\phi',\phi)$ realizes the max for $-v$.
The last point to verify is the compactness. Closedness follows from \eqref{eq:unit-ball}.
For any $t>0$, $\Lambda$ is separates the points $\omega$ and $\omega+tv$.
Therefore, the module of the cross-ratio
$[\phi,\phi',\omega,\omega+tv]$ is not identically $1$ for any $\phi$, $\phi'$ in $\Lambda$. This implies the existence of
$\phi$ et $\phi'$ such that $\Re\left(\frac{\bm \phi'(v)}{\bm \phi'(\bm \omega)} - \frac{\bm \phi(v)}{\bm \phi(\bm \omega)}\right)\neq 0$. In other
terms, for any $v$ we have $\norm{\omega}{v}>0$.
We conclude by contradiction: suppose $B_\Lambda^{\omega}$ is not bounded. We would have a sequence $v_n$ of tangent
vectors at $\omega$ such that $|v_n|\rightarrow \infty$ (where $|\cdot|$ is any norm on
$T_\omega\Omega$), and
$$\forall (\phi,\phi')\in \Lambda\times\Lambda,\,
\Re\left(\frac{\bm \phi'(v_n)}{\bm \phi'(\bm \omega)} - \frac{\bm \phi(v_n)}{\bm \phi(\bm \omega)}\right)\leqslant 1.$$
In particular, up to extraction, the sequence $v_n/|v_n|$ converges to a vector $v$ that satisfies
$|v|=1$ and
$\Re\left(\frac{\bm \phi'(v)}{\bm \phi'(\bm \omega)} - \frac{\bm \phi(v)}{\bm \phi(\bm \omega)}\right) = 0$, which is a contradiction.
\end{proof}
We now state the technical lemma used in the proof.
\begin{lemma}\label{lem:derive}
Let $\mathcal F$ be a bounded set of $C^2$-functions from $\mathbb R_+$ to $\mathbb R$ vanishing at $0$.
Let $f$ be defined by $f(t) = \max_{g\in\mathcal F} g(t)$ for $t\geq 0$.
Then $f$ has a derivative at $0$ given by $f'(0) = \max_{g\in\mathcal F} g'(0)$.
\end{lemma}
\begin{proof}
Observe first that because $t>0$,
$$\frac{f(t)}{t} = \frac{\max_{g\in\mathcal F} g(t)}{t}=\max_{g\in\mathcal F} \frac{g(t)}{t}.$$
Consider next a function $g$ in $\mathcal{F}$. The second order expansion of $g$ gives
$$\dfrac{g(t)}{t}=g'(0)+t\Bigl(\dfrac{g''(0)}{2}+\varepsilon_g(t)\Bigr),$$
where $\varepsilon_g$ is a continuous function depending on $g$, such that
$\varepsilon_g(t)\underset{t\rightarrow 0^+}\longrightarrow 0$. The boundedness of $\mathcal{F}$ implies that the two
sets $\{|\varepsilon_g(t)|,t\in[0,1],g\in\mathcal{F}\}$ and $\{|g''(0)|,g\in\mathcal{F}\}$ are bounded. In turn, there
exists a constant $C>$ such that
$$
\forall g\in\mathcal{F},\, \forall t\in[0,1],\,
g'(0) -C t \leq \frac{g(t)}{t} \leq g'(0) + Ct.
$$
We obtain therefore
$$
\max_{\mathcal F}g'(0) -C t \leq \max_{\mathcal F} \frac{g(t)}{t} \leq \max_{\mathcal F} g'(0) + Ct.
$$
This implies $\max_{\mathcal F} \frac{g(t)}{t} \to \max_{\mathcal F} g'(0)$.
\end{proof}
We give here a condition for the unit balls of the Finsler metric to be strictly convex. Recall
that in the real case, it amounts to the condition that the boundary of $\Omega$ contains no segment.
The condition we give here is the same, translated by duality.
As before, for $\omega \in \Omega$ we choose a lift $\bm \omega$ and identify $T_\omega\Omega$ to $T\subset \mathbb C^{n+1}$. We denote by
$$\Psi = \left\{ \frac{\bm \phi'}{\bm \phi'(\bm \omega)} - \frac{\bm \phi}{\bm \phi(\bm \omega)}\quad \textrm{where} \quad \phi,\phi' \in \Lambda \right\}.$$
From Theorem \ref{theorem:metric} we know that for all $v\in T$,
$$\norm{\omega}{v} = \underset{\psi\in \Psi}{\max} \: \Re \left(\psi(v)\right).$$
\begin{proposition}\label{prop:Minkovski}
Let $(\Omega,\Lambda)$ be an admissible pair, with $\Lambda$ separating and $\omega \in \Omega$.
The unit ball of the Finsler metric $\norm{\omega}{\cdot}$ is strictly convex if and only if
for any pair of distinct tangent vectors $u$ and $v$ at $\omega$, the $\max$
defining $\norm{\omega}{u}$ and $\norm{\omega}{v}$ are not obtained for the same form $\psi \in \Psi$.
\end{proposition}
\begin{proof}
For simplicity, we denote by $f(v)=\norm{\omega}{v} = \max_{\psi\in \Psi} \Re(\psi(v))$.
Note that $f$ is positively homogeneous : $f(\lambda v)=\lambda f(v)$ for all $\lambda \in\mathbb R_+$
and $v\in T$.
Suppose now that for each $\psi \in\Psi$, at least one of the terms $\Re(\psi(u))$ and $\Re(\psi(v))$ is not the maximum over all $\psi$'s in $\Psi$. Then
\begin{eqnarray}
\Re\left(\psi\left(\frac{u+v}{2}\right)\right) & = & \frac{\Re(\psi(u)) + \Re(\psi(v))}{2}\nonumber\\
& < & \frac{\max_{\psi\in\Psi}(\Re(\psi(u))) + \max_{\psi\in\Psi}(\Re(\psi(v)))}{2} \nonumber\\
& = &\frac{f(u)+f(v)}{2}.\label{ineq-convex}
\end{eqnarray}
Since $\Psi$ is compact, we obtain by taking the maximum the strict inequality
$$f\left(\frac{u+v}{2}\right) < \frac{f(u)+f(v)}{2},$$
which amounts to the strict convexity of balls.
\end{proof}
\begin{remark}
Examples of separating $\Lambda$ for which the balls are not strictly convex are easily built
using this proposition: any finite subset $\Lambda \in \mathbb {CP}^1$ not included in a circle separates points
in its complement. But from the previous proposition, balls are not strictly convex: indeed, they are
polygons.
\end{remark}
\begin{remark}\label{rem:complexhyperbolicspace}
Let $\mathbb C^{n,1}$ be $\mathbb C^{n+1}$ equipped with the Hermitian form
$$
\langle Z, W \rangle=Z_0\overline{W}_{n}+Z_2\overline{W}_2+\cdots Z_{n-1}\overline{W}_{n-1}+Z_{n}\overline{W}_0.
$$
One has three subspaces:
$$
V_{+}= \{ Z\in \mathbb C^{n,1} : \langle Z, Z \rangle >0 \},
$$
$$
V_{0}= \{ Z\in \mathbb C^{n,1}- \{0\} : \langle Z, Z \rangle =0 \},
$$
$$
V_{-}= \{ Z\in \mathbb C^{n,1} : \langle Z, Z \rangle < 0 \}.
$$
Let $P : \mathbb C^{n,1}- \{0\} \rightarrow \P(\mathbb C^{n+1}) $
be the canonical projection onto complex projective space.
Then complex hyperbolic n-space is defined as $\H^n_{\mathbb C}=P(V_{-})$ equipped with the Bergman metric.
The boundary of complex hyperbolic space is defined as $\partial \H^n_{\mathbb C}=P(V_{0})$.
Using the hermitian form, we identify $\partial \H^n_{\mathbb C}$ as a subset of the dual $\P((\mathbb C^{n+1})^\vee)$.
Then the pair $(\H^n_{\mathbb C},\partial \H^n_{\mathbb C})$ is admissible and separates points. Moreover, from Proposition \ref{prop:invariance}, the Hilbert metric $d_{\partial \H^n_{\mathbb C}}$ is invariant under the action of unitary group $\mathrm{U}(n,1)$ of the Hermitian form. Hence this metric is a multiple of the Bergman metric (one may compute that the multiplication factor is $\frac 12$). We will come back to complex hyperbolic geometry in the following section.
\end{remark}
\begin{remark}\label{shilov}
More generally, let $\Omega$ be a domain in $\mathbb C^n$. There are several situations where a natural metric can be associated to it. A very general
definition is that of the Bergman metric on any bounded domain. The construction is such that the biholomorphisms group is contained in the isometry group of that metric.
The particular case where $\Omega$ is a bounded homogeneous domain has been studied for a long time. It contains the important class of non-compact hermitian symmetric spaces. These Riemannian spaces, classified by Cartan, can be embedded as
bounded domains which contain the origin, are stable under the circle action and which turn out to be convex (see \cite{wolf} for a thorough exposition).
The group of biholomorphisms of a bounded symmetric domain is transitive and can be extended to the boundary but its action on the boundary is not transitive except in the case of the complex ball. On the other hand the isotropy group at the origin acts
by linear maps of $\mathbb C^n$ and, moreover, it acts transitively on the Shilov boundary of $\Omega$.
Consider the set $\Lambda$ of all hyperplanes tangent to the boundary which touch it in at least two points. They all pass by the Shilov boundary
and therefore the action of the isotropy preserves $\Lambda$.
The distance $d_\Lambda(0,x)$ is therefore invariant under the isotropy group and, as the action of the isotropy is irreducible,
$d_\Lambda(0,x)$ coincides up to a scalar with the Bergman metric. One can then translate this distance the whole domain using the action of the automorphism group.
As an example,
consider the bidisc $\Delta\times \Delta \subset \mathbb C^2$. Its Shilov boundary is $S^1\times S^1$ and the relevant hyperplanes
passing through it are of the form $z=z_0\in S^1$ or $w=w_0\in S^1$ where $(z_0,w_0)\in S^1\times S^1$ and $(z,w)$ are coordinates of $\mathbb C^2$. We recuperate then the Bergman metric $\max \{ d_h(x_1,x_2), d_h(y_1,y_2)\}$ for any two points
$(x_1,y_1),(x_2,y_2)\in \Delta\times \Delta$.
\end{remark}
The expository and general discussion of this paper is over. We now begin to explore three different directions, illustrating the wealth of possible applications of the definition of this metric.
We first reinterpret geometrically the definitions in the situation of a
discrete subgroup of $\mathrm{PU}(n,1)$. We begin by
recalling the definition of the Kulkarni limit set and focusing to the particular case of spherical CR
geometry. We remark that the construction gives a natural metric on uniformized spherical CR structures
on manifolds.
We proceed in the last two sections with the study of two simple cases in one
dimension projetive spaces:
first the complex projective line and the punctured spheres. We then move on to the real projective line,
with attention to self-similar sets $\Lambda$. We
give a generalization of the famous Basmajian formula in hyperbolic geometry.
The three following sections are independent.
\section{Complex hyperbolic groups}\label{sec:complexkleinian}
Natural examples of the generalized Hilbert metric can be defined for
open subsets in $\P(\mathbb C^{n+1})$ which are
domains of discontinuity of discrete subgroups $\Gamma\subset \mathrm{PGL}(n+1,\mathbb C)$.
The general theory of such sets of discontinuity is not yet fully
developed but a special case has been studied which is of major
interest for us: if $\Gamma$ is a complex hyperbolic group, i.e.
a discrete subgroup of $\mathrm{PU}(n,1)$.
We show in this section that if $\Gamma$ is a non-elementary discrete
subgroup of $\mathrm{PU}(n,1)$, then its domain of discontinuity in $\P(\mathbb C^{n+1})$
inherits a generalized Hilbert metric. We then focus on the case $n=2$,
and give geometric interpretations of this metric. Recall from Remark
\ref{rem:complexhyperbolicspace} the notations for the complex
hyperbolic space $\H^n_\mathbb C \subset \P(\mathbb C^{n+1})$ and
its boundary $\partial\H^n_\mathbb C$ as well as the projective
unitary group $\mathrm{PU}(n,1)$.
\subsection{Kulkarni limit set as a union of hyperplanes}
\label{subsec:kulkarni}
In order to define an appropriate set $\Lambda$ of 1-forms we start by recalling
the definition of limit set for these groups.
The following definition transposes in this context
a definition due to Kulkarni for general actions of
groups on topological spaces. As in the general situation,
we denote by $\P$ the projective space $\P(\mathbb C^{n+1})$ and
by $\P'$ its dual.
\begin{definition} Let $\Gamma\subset \mathrm{PGL}(n+1,\mathbb C)$ be a discrete subgroup
\begin{enumerate}
\item $L_0(\Gamma)$ is the closure of the set of points in $\P$ with infinite isotropy.
\item $L_1(\Gamma)$ is the closure of the set of cluster points of the $\Gamma$-orbits of all $z\in \P\setminus L_0(\Gamma)$.
\item $L_2(\Gamma)$ is the closure of the set of cluster points of the $\Gamma$-orbits of all compact subsets $K\subset \P\setminus (L_0(\Gamma)\cup L_1(\Gamma))$.
\item The Kulkarni limit set $L(\Gamma)$ is the set $L_0(\Gamma)\cup L_1(\Gamma)\cup L_2(\Gamma)$.
\item The Kulkarni discontinuity region $\Omega_\Gamma$ is $\P\setminus L(\Gamma)$.
\item We denote by $\Lambda_\Gamma \subset \P'$ the set of forms whose kernel is included in $L(\Gamma)$.
\end{enumerate}
\end{definition}
In the case of $\Gamma$ a complex hyperbolic group, Cano Liu and Lopez
\cite[Theorem 0.1]{CanoLiuLopez} prove that $L(\Gamma)$ is the union of kernels
of $\Lambda_\Gamma$ and that $\Omega_\Gamma$ is the largest set on which $\Gamma$
acts properly and discontinuously. Note that one may
even take the a priori smaller $\Lambda_\Gamma$ consisting of form whose kernel is tangent to the sphere at infinity $\partial \H^n_\mathbb C$. In view of our definitions, this translate to:
\begin{theorem}\label{thm:complexhyperbolic}
Let $\Gamma$ be a complex hyperbolic group.
Then the pair $(\Omega_\Gamma,\Lambda_\Gamma)$ is admissible.
Moreover, if $\Gamma$ is Zariski-dense in $\mathrm{PU}(n,1)$, then $\Lambda_\Gamma$ is separating.
\end{theorem}
\begin{proof}
The fact that the pair $(\Omega_\Gamma,\Lambda_\Gamma)$ is admissible follows from definitions and \cite[Theorem 0.1]{CanoLiuLopez}. The separation property follows from the fact that, by contraction, if the metric does not separate two points,
than $\Lambda_\Gamma$ should be contained in a Zariski closed subset of $\P'$ (see remark \ref{remark:zariski}). But, in that case, the group $\Gamma$ (which
preserves $\Lambda_\Gamma$) would not be Zariski dense.
\end{proof}
A detailed study of such examples is done in \cite{CanoParkerSeade}, for complex hyperbolic subgroups
that are included in $\mathrm{SO}(n,1)$. They prove that in this case $\Omega_\Gamma$ consists
of $3$ connected components. Their groups are not Zariski-dense in $\mathrm{SU}(n,1)$.
The metric $d_{\Lambda_\Gamma}$ gives a non-separating metric in such cases.
The metric defined here seems most interesting when restricted to
$\Omega = \Omega_\Gamma \cap \partial \H^n_\mathbb C$, as we explain in the following section.
\subsection{Complex hyperbolic geometry and spherical CR geometry}\label{section:CR}
We give here a geometric reinterpretation of the separability condition (Theorem \ref{thm:metric})
in the case of complex hyperbolic groups acting on $\P(\mathbb C^{n+1})$. We then give examples of discrete
complex hyperbolic groups for which we can check this separability condition. Those groups arise
from the construction of spherical CR geometric structure on $3$-manifolds that are uniformizable. One
example is a representation of the $8$-knot complement group in $\mathrm{PU}(2,1)$ associated to a uniformisable
spherical CR structure on the $8$-knot complement, constructed in \cite{DerauxFalbel}. Another one is
constructed for the Whitehead link complement in \cite{ParkerWill}.
We begin by recalling
the definition if some geometric objects in complex hyperbolic geometry, mainly the bisectors.
A {\sl bisector} is the locus of points equidistant from two given points $p_0$ and $p_1$
in $\H^n_\mathbb C$. In homogeneous coordinates,
it is given by the
negative vectors $\mathbf{z}=(z_0,z_1,z_2)$ that satisfy the equation
\begin{equation*}\label{eq:bisector}
|\langle \mathbf{z},\tilde{p}_0\rangle|=|\langle
\mathbf{z},\tilde{p}_1\rangle|,
\end{equation*}
where $\tilde{p}_i$ are lifts satisfying $\langle \tilde{p}_0,\tilde{p}_0 \rangle=\langle
\tilde{p}_1,\tilde{p}_1 \rangle$. This equation makes sense up to the boundary $\partial \H^n_\mathbb C$ defining {\sl spinal spheres} as boundaries of bisectors.
Observe that bisectors and spinal spheres are defined by algebraic equations.
CR structures appear naturally as boundaries of complex manifolds. The local geometry
of these structures was studied by E. Cartan \cite{C} who defined, in dimension three, a curvature analogous to
curvatures of a Riemannian structure. When that curvature is zero,
Cartan called them spherical CR structures and developed their basic properties. A much later study by
Burns and Shnider \cite{BS} contains the modern setting for these structures.
\begin{definition}
A spherical CR-structure on a $(2n-1)$-dimensional manifold is a geometric structure
modeled on the homogeneous space $\S^{2n-1}:=\partial \H^{2n-1}_{\mathbb C}$ with the above
${\mathrm{PU}(n,1)}$ action.
\end{definition}
A particular class of such structures is the natural analog of complete structure for
metric geometries:
\begin{definition}
We say a spherical CR-structure on a ${2n-1}$-manifold is uniformizable if it is
equivalent to a quotient of the domain of discontinuity in $ \S^{2n-1}$ of a
discrete subgroup of ${\mathrm{PU}(2,1)}$.
\end{definition}
Here, equivalence between CR structures is defined, as usual, by
diffeomorphisms preserving the structure. A $(2n-1)$-manifold $M$ with a spherical CR structure
is said to be an
\emph{uniformized CR spherical manifold} if there is a discrete group $\Gamma$ of $\mathrm{PU}(n,1)$,
with limit set $\Lambda$ in the sphere $\partial \H^n_\mathbb C$ such that the spherical CR structure on $M$
is equivalent to the quotient of $\Omega = \partial \H^n_\mathbb C\setminus \Lambda$ by $\Gamma$.
\subsection{Invariant metric for uniformized CR spherical manifold}
Let $M$ be a uniformized CR spherical manifold. Denote as above by $\Gamma$ the discrete group
with limit set $\Lambda$ and domain of discontinuity $\Omega$ in the sphere $\partial \H^n_\mathbb C$
such that $M \simeq \Gamma \backslash \Omega$.
We identify, as in section \ref{subsec:kulkarni}, $\Lambda$ with the subset of the dual projective space
$\P'$ consisting of forms whose kernel is tangent to the sphere at points of $\Lambda$. Then, Theorem
\ref{thm:complexhyperbolic} states that $d_\Lambda$ defines a $\Gamma$-invariant, non-separating
metric on the domain of
discontinuity of $\Gamma$ in $\P(\mathbb C^{n+1})$. We look here at this metric restricted to the hypersurface
$\Omega$. We reintrepret here the definition \ref{eq:def-dist} in terms more classical
in the framework of complex hyperbolic geometry. We the proceed with a geometric interpretation of the
separation condition given in Section \ref{subsec:separation}.
Let $\phi$ and $\phi'$ be 1-forms whose kernel are, respectively, complex tangent spaces at points $p$ and $p'$ in $\Lambda\in \S^{2n-1}$. Choose lifts $\bm p$ and $\bm p'$ of $p$ and $p'$.
Then lifts of $\phi$ and $\phi'$ are given explicitly by $\bm \phi(z)=\langle z,\bm p\rangle$ and $\bm \phi'(z)=\langle z,\bm p'\rangle$. One computes, for $\omega,\omega'\in \Omega$,
$$[\phi,\phi',\omega,\omega'] =\dfrac{\langle \bm \omega,\bm p\rangle\langle\bm \omega',\bm p'\rangle}{\langle \bm \omega,\bm p'\rangle\langle\bm \omega',\bm p\rangle}$$
which is the hermitian cross-ratio of the four points $p,p',\omega,\omega'$.
Let $\omega,\omega'$ be two points in $ \S^{2n-1}$. If $p\in \S^{2n-1}$ is another point then its projection in the geodesic in the complex disc defined by
$\omega$ and $\omega'$ is
$$\pi_{(\omega\omega')}(p)=t \bm \omega + \dfrac{1}{t}\bm \omega'\mbox{ with } t=\sqrt{\dfrac{\lvert\langle { \bm p},{ \bm \omega} \rangle\rvert}{\lvert\langle { \bm p},{ \bm \omega} \rangle\vert}},$$
As a direct corollary, we obtain a condition for cross-ratios to have modulus $1$.
\begin{proposition}
Let $(a,b,c,d)$ be four pairwise distinct points in $ \S^{2n-1}$.
The cross-ratio $[a,b,c,d]$ has modulus $1$ if and only the projection of $c$ and $d$
on the geodesic $(ab)$ coincide.
\end{proposition}
One can compare with the condition in Theorem \ref{thm:metric} for the set $\Lambda$ to separate
a pair $\omega,\omega'$. Indeed the above proposition shows that $\Lambda$ separates this
pair if its projection on the complex disc defined by
$\omega$ and $\omega$ is contained in a geodesic orthogonal to the geodesic defined by $\omega,\omega'$.
Observe also that we can exchange the roles of $a,b$ and $c,d$ in the previous proposition.
\begin{corollary}
If $\Lambda$ is not contained in a bisector then $d_{\Lambda}$ is a metric.
\end{corollary}
\begin{proof}
The inverse image of a geodesic by a projection in a complex disc is a bisector. By the proposition,
$\Lambda$ does not separate two points $\omega,\omega'$ if
$\Lambda$ is included in a bisector.
\end{proof}
We may rewrite the discussion following Theorem \ref{thm:metric}. One can think of two cases where
$d_\Lambda$ will not separate points in $\Omega$: suppose $\Gamma$ is included in a subgroup
$\mathrm{PU}(n-1,1)$ or $\mathrm{PO}(n,1)$, or in other terms that it preserves a totally geodesic submanifold.
Then its limit set would be included in a bisector, which would prevent the
separation condition. The following theorem states that it is essentially the only problem. Recall that in Theorem \ref{thm:complexhyperbolic} we proved that the metric on the complement of the Kulkarni limit set is separating.
The following theorem restates this result considering only points in the boundary of complex hyperbolic space in order to stress that
the metric provides a distance on the regular set of the action of $\Gamma$ on $\S^{2n-1}$.
\begin{theorem}\label{thm:metriccomplexhyp}
Let $\Gamma$ be a discrete subgroup of $\mathrm{PU}(n,1)$
with limit set a proper subset of $\S^{2n-1}$.
Suppose that $\Gamma$ is Zariski dense in $\mathrm{PU}(n,1)$.
Then $d_{\Lambda}$ defines a $\Gamma$-invariant metric on the regular set
$\Omega$.
\end{theorem}
A practical criterium to verify that the subgroup $\Gamma$ is Zariski dense and that the distance is well defined is given in the following corollary.
\begin{corollary}
Let $\Gamma$ be a discrete subgroup of $\mathrm{PU}(n,1)$
with limit set a proper subset of $\S^{2n-1}$.
Suppose no finite index subgroup of $\Gamma$ stabilizes a totally geodesic submanifold nor a point at infinity.
Then $d_{\Lambda}$ defines a $\Gamma$-invariant metric on the regular set
$\Omega$.
\end{corollary}
\begin{proof}
Consider $G$ the connected component of the Zariski closure of $\Gamma$. It is a connected
subgroup of $\mathrm{PU}(n,1)$ which does not stabilizes a totally geodesic subspace. From
\cite[Thm 4.4.1]{ChenGreenberg}, $G$ contains $\mathrm{PU}(n,1)$. Hence $G$ does not
stabilize a proper algebraic subset of $\S^{2n-1}$.
Suppose now that some bisector $B$ contains the limit set $\Lambda$ of $\Gamma$. Then $\Gamma$ preserves the
algebraic subset $\bigcap_{\gamma\in \Gamma}\gamma B$ and $\Lambda$ is included in this intersection.
As it is algebraic, $G$ stabilizes the intersection. This is a contradiction.
\end{proof}
\begin{remark}
It is quite easy to prove that the subgroups $\Gamma$ arising in spherical CR uniformizations of a knot
or link complement $M$ as in \cite{DerauxFalbel,ParkerWill} does not virtually preserve a
totally geodesic
submanifold. Thus to such an uniformization is associated a $\Gamma$-invariant metric on
the covering $\Omega$ of $M$.
\end{remark}
\subsection{Infinitesimal metric}
We conclude this section by a reinterpretation of the formula for the infinitesimal metric defined in
section \ref{Finsler metric}.
Observe that the formula for the Finsler metric given in Theorem \ref{theorem:metric} can be written -
with choices of lifts - for
a tangent vector $v$ at $\omega \in \Omega$ by:
\begin{eqnarray*}
\norm{\omega}{v} = & \max_{\phi, \phi' \in \Lambda} \Re\left(\frac{\bm \phi'(v)}{\bm \phi'(\bm \omega)} - \frac{\bm \phi(v)}{\bm \phi(\bm \omega)}\right)\\
= & \max_{p, p' \in \Lambda} \Re\Bigl\langle\dfrac{\bm p'}{\langle \bm p',\bm \omega \rangle}-\dfrac{\bm p}{\langle \bm p,\bm \omega \rangle},v\Bigr\rangle
\end{eqnarray*}
In order to have an explicit description of the norm we shall use the Siegel model of two dimensional complex hyperbolic space.
Its boundary is $2\Re(z_1)+|z_2|^2=0$ (intersected with an affine chart $z_3=1$). Writing $z_k=x_k+iy_k$, the tangent space in $(z_1,z_2)$ is given by
$$
2dx_1+2x_2dx_2+2y_2dy_2=0.
$$
In particular the tangent space at ${\bf o}=(0,0)$ is given by $dx_1=0$, therefore consists of vectors of the form
$$v_{z,t}=\begin{bmatrix} it\\ z\\ 0 \end{bmatrix}$$
Applying the previous formula in these coordinates we have
$$
\norm{\bf o}{v_{z,t}}=\max_{p, p' \in \Lambda} \Re\Bigl\langle\dfrac{\bm p'}{\langle \bm p',\bf o\rangle}-\dfrac{\bm p}{\langle \bm p,\bf o\rangle},v_{z,t}\Bigr\rangle.
$$
\begin{remark}
\begin{enumerate}
\item The vector $w=\dfrac{\bm p'}{\langle \bm p',\bm \omega\rangle}-\dfrac{\bm p}{\langle \bm p,\bm \omega\rangle}$ is of positive type.
Indeed, one computes
$$\langle w,w\rangle= -2\Re\left(\dfrac{\langle \bm \omega,\bm p\rangle\langle \bm p,\bm p'\rangle\langle \bm p',\bm \omega\rangle}{|\langle \bm p,\bm \omega\rangle|^2|\langle \bm p',\bm \omega\rangle|^2}\right)$$
and the triple product $\arg(-\langle \bm \omega,p\rangle\langle p,p'\rangle\langle p',\bm \omega\rangle)\in [-\pi/2,\pi/2]$ is Cartan's invariant.
In particular $w$ is polar to a complex line in ${\mathbf H}_\C^2$.
\item The conditions $\langle w,\bm p\rangle=\langle w ,\bm p'\rangle$ and $\langle w, \bm \omega\rangle=0$ imply that $w$ is orthogonal to the complex line defined by $p$ and
$p'$ passing through $\omega$.
\end{enumerate}
\end{remark}
\section{Complex projective line: $n$-punctured spheres}\label{section:puncturedsphere}
We consider in this section the complex projective line $\mathbb {CP}^1$. In fact, we restrict
ourselves to a very specific case: let $\Lambda =
\{p_1,\cdots,p_n\}$ a finite set
of points in $\mathbb C$ and $\Sigma=\mathbb {CP}^1\setminus\{p_1,\cdots,p_n\}$ its complement, the $n$-punctured sphere.
The $n$ linear forms in
$\Lambda$ are given by $\phi_i=[1 : -p_i]$, so that
$$\bm \phi_i \begin{pmatrix}z\\1\end{pmatrix}=z-p_i.$$
We have already observed that when $n=1, 2$ or $3$, the metric is not separating. So
we assume here that $n\geqslant 4$ and $\Lambda$ is not included in a circle. In this
case, $d_\Lambda$ is a metric, as follows from Theorem \ref{thm:metric}. For simplicity,
we denote this metric by $d$.
On the other hand, the surface $\Sigma$ may be equipped with
a hyperbolic metric, denoted by $d_h$. We prove that the infinitesimal metric defined by $d$
is quasi-isometric to the infinitesimal hyperbolic metric defined by $d_h$. The reader can guide himself through the proof
of the following proposition keeping in mind the physical interpretation of the infinitesimal metric given in Remark \ref{physics}.
\begin{theorem}\label{thm:QI-n-punctured-sphere}
There is a quasi-isometric diffeomorphism between $\bigl(\Sigma,d_h\bigr)$ and $\bigl(\Sigma,d\bigr)$
which fixes the punctures.
\end{theorem}
We will show that the Finsler metrics are equivalent up to a diffeomorphism and this will prove the theorem.
We begin by describing the hyperbolic metric near a cusp point of $\Sigma$.
\begin{lemma}\label{lem:local-hyp}
For $k=1\cdots n$, there exists a neighbourhood of the puncture $p_k$ on which the infinitesimal hyperbolic metric defined by $d_h$ on $\Sigma$ is
given by
$$\sqrt{\dfrac{dr^2}{r^2} + r^2dt^2},$$
in the local coordinate $(r,t) \mapsto p_k+re^{i t}$.
\end{lemma}
\begin{proof}
Note first that the hyperbolic metric on the punctured unit disc is given by
\begin{equation}\label{eq:hyp-punctured-disc}\dfrac{|dz|}{|z|\log|z|}\end{equation}
This can be seen by pushing forward the hyperbolic metric of the upper half plane, which is $\frac{|dz|}{\Im(z)}$, by the
holomorphic cover map $z\longmapsto e^{iz}$. Pushing forward \eqref{eq:hyp-punctured-disc} by the diffeomorphism
given in polar coordinates by $(r,\theta)\longmapsto (-(\log r)^{-1},\theta)$ gives the result.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:QI-n-punctured-sphere}]
For any point $m\in\Sigma$, and any tangent vector $v$ at $m$, we denote by
$\|v\|_m $ the Finsler norm of $v$, and by $\|v\|_m^{h}$ its hyperbolic norm.
We will prove that any diffeomorphism $\varphi: \Sigma \to \Sigma$ which restricts to the
identity around the punctures is a quasi-isometry. First, consider $\varphi$ such a diffeomorphism, and,
for each $k=1\cdots n$, let $V_k$ be any neighbourhood of $p_k$ such that $\varphi_{|V_k}$ is the identity.
Since $\Sigma\setminus \underset{k}{\cup} V_k$ is compact, the restriction of $\varphi$ to it is quasi-isometric.
We need therefore only to consider the situation close to the punctures. The result will be proved if we show that for
each puncture $p_k$, there exists a constant $C>1$ and a neighbourhood $V_k$ of $p_k$ such that for any $m\in V_k$ and
any tangent vector $v$ at $m$
\begin{equation}
\dfrac{\|v\|_m^h}{C}\leqslant\|v\|_{m}\leqslant C\|v\|_m^{h}.
\end{equation}
We consider the situation close to a fixed puncture, which we assume to be $p_1$. We may chose our coordinates so that
$p_1=0$.
\begin{figure}[h!]
\begin{center}
\scalebox{0.5}{\includegraphics{coords.eps}}
\end{center}
\caption{Coordinates \label{fig:coords}}
\end{figure}
In the local coordinate $(r,t) \mapsto re^{it}$, we need to find $C>1$ such that for any tangent vector
$v = a\frac{\partial}{\partial r} + b\frac{\partial}{\partial t}$, we have (see Lemma \ref{lem:local-hyp}) :
\begin{equation}\label{eq:eq}
\frac{1}{C}\sqrt{\frac{a^2}{r^2} + r^2b^2} \leq \|v\|_m \leq C\sqrt{\frac{a^2}{r^2} + r^2b^2}.
\end{equation}
The rest of the proof is devoted to this local computation. From now on, all constants $C$ that appear are {\it some positive
constants}, and we remain vague about their values for the sake of readability. The point $m$ and the tangent vector $v$ are
given by
$$m=\begin{bmatrix}re^{i\theta}\\1\end{bmatrix},v=\begin{bmatrix}\rho e^{i(\theta+\alpha)}\\0\end{bmatrix}.$$
The vector $v$ decomposes as $v =\rho \cos(\alpha) \frac{\partial}{\partial r} + \frac{\rho}{r} \sin{\alpha} \frac{\partial}{\partial t}$.
Its hyperbolic norm is given by
$$\|v\|_m^h= \rho \sqrt{\frac{\cos(\alpha)^2}{r^2} + \sin(\alpha)^2}.$$
The points $p_j$ are given by $p_j=k_je^{i\beta_j}$ (with $k_1=0$ and $k_j\neq 0$ for $j\geqslant 2$). By Theorem
\ref{theorem:metric}, the Finsler norm of $v$ is given by:
\begin{equation} \label{eq:finsler-v}
\|v\|_m=\max_j \Re\Bigl(\dfrac{\phi_j(v)}{\phi_j(m)}\Bigr)-\min_j\Re\Bigl(\dfrac{\phi_j(v)}{\phi_j(m)}\Bigr).
\end{equation}
The quantities involved in \eqref{eq:finsler-v} are given by
$$\dfrac{\phi_j(v)}{\phi_j(m)}=\dfrac{\rho e^{i(\theta+\alpha)}}{re^{i\theta}-k_je^{i\beta_j}},$$
and by direct computations, we observe (see Remark \ref{physics} for a physical interpretation of the following formula).
\begin{itemize}
\item If $j=1$ ( and so $k_1=0$ ),
\begin{equation}\label{eq:expansion1}\dfrac{1}{\rho}\Re\Bigl(\dfrac{\phi_1(v)}{\phi_1(m)}\Bigr)=\dfrac{\cos(\alpha)}{r}.
\end{equation}
\item If $j\geqslant 2$ (and thus $k_j\neq 0$),
\begin{equation}\label{eq:expansionj}\dfrac{1}{\rho}\Re\Bigl(\dfrac{\phi_j(v)}{\phi_j(m)}\Bigr)=-\dfrac{\cos(\theta+\alpha-\beta_j)}{k_j} + O(r)\end{equation}
\end{itemize}
From physical considerations one expects that the most important term in the contribution to the metric near $p_1$ is the one corresponding to $j=1$ if $\alpha\neq \pm \pi/2$ and from another point which is not aligned with $m$ and $p_1$
if $\alpha=\pm \pi/2$. In the following we establish the estimates confirming this intuition.
Let us consider the right-hand side inequality in \eqref{eq:eq}. First, we note that for all $\gamma\in\mathbb R$,
$\cos(\alpha+\gamma)\leqslant |\cos\alpha| +|\sin\alpha|$. Therefore,
$$\underset{j\geqslant 2}{\max}\left(\dfrac{-\cos(\theta+\alpha-\beta_j)}{k_j}\right)\leqslant
\dfrac{|\cos\alpha| +|\sin\alpha|}{\underset{j\geqslant 2}{\min}\, k_j}.$$
Moreover, for any finite subset $F$ of $\mathbb R$, we have $\max(F)-\min(F)\leqslant 2\max|F|$. Provided that $r$ is small
enough, this implies
\begin{eqnarray}
\|v\|_m^h & \leqslant & 2\underset{j}{\max}\left|\Re\dfrac{\phi_j(v)}{\phi_j(m)}\right| \nonumber\\
& \leqslant & 2\rho\left(\dfrac{|\cos\alpha|}{r}+C(|\cos\alpha|+|\sin\alpha|)\right)\nonumber \\
& = & \rho\left(\dfrac{C}{r}|\cos\alpha| + C|\sin\alpha|\right)\nonumber\\
& \leqslant & C \rho\left(\dfrac{|\cos\alpha|}{r}+|\sin\alpha|\right) \nonumber\\
&\leqslant & C\rho \sqrt{\dfrac{\cos\alpha^2}{r^2}+\sin^2\alpha}
\end{eqnarray}
We know consider the left-hand side inequality in \eqref{eq:eq}. By symetry, we may restrict ourselves to the case where
$\alpha\in[-\pi/2,\pi/2]$. The geometric idea is the following. When $\alpha$ is close to $0$, the $\max$ in \eqref{eq:eq} is reached for $i=1$
if $r$ is small enough, due to the $\cos (\alpha)/r$ term. When $|\alpha|$ becomes larger, the $j=1$ term becomes
less influent, and the Finsler norm of $v$ is obtained from those terms where $j>1$. In the extreme case where
$\alpha=\pm\pi/2$, the $j=1$ term vanishes.
To make this idea more precise, we first compute for any $j\leqslant 2$
\begin{eqnarray}
\dfrac{1}{\rho}\Bigl(\Re(\dfrac{\phi_1(v)}{\phi_1(m)})-\Re(\dfrac{\phi_j(v)}{\phi_j(m)}) \Bigr)
& = & \dfrac{\cos(\alpha)}{r}-\dfrac{r\cos\alpha -k_j\cos{(\alpha+\theta-\beta_j)}}{|re^{i\theta}-k_je^{i\beta_j}|^2}\nonumber\\
& = & \dfrac{k_j\left(k_j\cos\alpha-r\sin{(\theta-\beta_j)}\sin\alpha\right)}{r|re^{
i\theta}-k_je^{i\beta_j}|^2} \nonumbe
\end{eqnarray}
We observe that the latter quantity has the same sign as $k_j\cos\alpha-r\sin{(\theta-\beta_j)}\sin\alpha$. A direct
resolution shows that it is non-negative if and only if $\alpha$ belongs to an interval $I_j$, which is defined by
$I_j=[-\pi/2,\alpha_j]$ if $\sin(\theta-\beta_j) >0$, by $I_j=[\alpha_j,\pi/2]$ if $\sin(\theta-\beta_j) <0$, where
$\alpha_j$ is determined by
$$\tan(\alpha_j)=\dfrac{k_j-r\cos(\theta-\beta_j)}{r\sin{(\theta-\beta_j)}},$$
and $I_j=-[\pi/2,\pi/2]$ if $\sin(\theta-\beta_j) =0$.
As a consequence, the set of values of $\alpha$ for which the max is reached for $j=1$ is a subinterval $I$ of
$[\pi/2,\pi/2]$ which contains $0$ in its interior.
\begin{itemize}
\item For $\alpha\in I$, we have thus
$$\|v\|_m \geqslant \rho \left(\dfrac{\cos\alpha}{r}-\max\dfrac{2}{k_j}\right)\geqslant\|v\|_m^h.$$
\item For $\alpha$ outside $I$, we have
$$\|v\|_m\geqslant\rho \underset{j,\ell\geqslant 2}{\max}\left(\Re\dfrac{\phi_j(v)}{\phi_j(m)}-\Re\dfrac{\phi_\ell(v)}{\phi_\ell(m)}\right).$$
In view of \eqref{eq:expansionj}, this implies (provided that $r$ is small enough)
$$\|v\|_m\geqslant\rho \underset{j,\ell\geqslant 2}{\max}\Bigl| \dfrac{\cos(\theta+\alpha-\beta_j)}{k_j}-\dfrac{\cos(\theta+\alpha-\beta_\ell)}{k_\ell}\Bigr|-C\rho r.$$
Now, the max in the right hand side of the previous inequality is not zero since the points $p_j$ do not lie on a circle.
Therefore, making $r$ smaller if necessary, we have
$$\|v\|_m\geqslant \rho C \geqslant \rho|\sin\alpha|\geqslant C \|v\|_m^h.$$
\end{itemize}
This concludes the proof.
\end{proof}
\begin{remark} \label{physics} One can gain intuition about this metric with a physical analogy. Indeed, the contribution
of each point $p_i$ in the definition of the Finsler metric $\|v\|_m$ corresponds to a magnetic field ${\bf B}_i$ induced
by a constant current passing through an infinite line perpendicular to the plane $\mathbb C$ at the point $p_i$
(see equation \ref{eq:expansion1}). The magnetic field is tangent to the circles centred at $p_i$ and decreases with the
inverse of the distance. The force ${\bf F}_i$ on a charged particle at $m$ with velocity $v$ moving on the magnetic field
${\bf B}_i$ is given by ${\bf F}_i= v\wedge {\bf B}_i$ and it can be considered a scalar (the vector ${\bf F}_i$ is
perpendicular to the plane). The Finsler metric is then
$$
\|v\|_m=\max_i{\bf F}_i-\min_j{\bf F}_j.
$$
This suggests other natural infinitesimal metrics as various combinations or means of magnetic forces but the
previous definition has the advantage of making it clear that this metric is always strictly positive if the
points are not on the same circle or line.
\end{remark}
\section{Real projective line}\label{section:RP1}
In this section, we consider the case where $\Lambda$ is a compact subset of
$\mathbb {RP}^1$ and $\Omega$ its complement. The open set $\Omega$ is a union of pairwise disjoint intervals, and
the distance $d_\Lambda$ can be computed for points in different components of $\Omega$. In that case, $d_\Lambda$
has a close connection with the hyperbolic distance on the disc $\partial \Delta$, which is hardly a surprise since
this distance is induced by the cross-ratio on $\partial\Delta$.
\subsection{Distance between intervals}
Let us consider a closed set $\Lambda \subset \mathbb {RP}^1$ and its complement $\Omega$. As above, we identify
$\Lambda$ with a subset of the dual $\mathbb R P^1$: to any point $\lambda \in \mathbb {RP}^1$ corresponds the linear map
$(x_1,x_2)\longmapsto x_1-\lambda x_2$, which corresponds to the affine map $x\longmapsto x-\lambda$ in
the affine chart $\{x_2=1\}$. Under this identification, we will sometimes refer to points in $\Lambda$ as forms,
and, for instance call them $\phi$, $\phi'$... The open subset $\Omega $ is a union of open intervals, its connected
components, to which we will refer as its \emph{components}. The metric $d_\Lambda$ is invariant under the subgroup
of $\mathrm{PGL}(2,\mathbb R)$ preserving $\Lambda$ and is particularly interesting when this group is large as, for example, when
$\Lambda$ is the limit set of a Fuchsian group.
Viewing $\mathbb {RP}^1$ as the boundary at infinty of the Poincar\'e disc $\Delta$, there is a close relation between the cross-ratio
on $\mathbb {RP}^1$ and the hyperbolic metric in $\Delta$ :
\begin{lemma}\label{lem:cross-hyp}
If $a$, $b$, $c$ and $d$ are pairwise distinct points in $\mathbb {RP}^1$, the cross ratio satisfies $|[a,b,c,d]|=e^\delta$, where
$\delta$ is the (hyperbolic) distance between the orthogonal projections of $c$ and $d$ onto the geodesic spanned by $a$
and $b$.
\end{lemma}
\begin{proof}
Applying an element of PGL(2,$\mathbb R$), which preserves the cross-ratio, we may assume that $a=\infty$, $b=0$, $c=1$ and $d=x>0$.
Then the projections of $c$ and $d$ onto the geodesic $(ab)$ are $i$ and $ix$, which are a distance $\log (x)$ apart. The cross
ratio $[a,b,c,d]$ is equal to $x$.
\end{proof}
In order to understand the metric $d_\Lambda$ in this case, we first start with the following lemma,
which shows that the metric between points in $\Omega$ depends only on the (at most) two components
containing the points.
\begin{lemma} \label{lem:dist-RP1}
\begin{enumerate}
\item The metric $d_\Lambda$ restricts to the Hilbert metric on each component of $\Omega$.
\item If $\omega$ and $\omega'$ belong to distinct components $I$ and $I'$ of $\Omega$, then the max
defining the distance $d_\Lambda(\omega,\omega')$ is realised for $\phi$ and $\phi'$ associated to boundary
points of $I'$ and $I$ respectively.
\end{enumerate}
\end{lemma}
\begin{proof}
These facts are very natural geometrically in view of Lemma \ref{lem:cross-hyp} (see Figure \ref{fig:proj}).
Choose an affine chart such that the two points in $\Omega$ are $\omega = 0$ and $\omega' = \infty$.
Then we have:
\begin{eqnarray}
d_\Lambda (\omega,\omega') &=& \underset{\lambda, \lambda'\in\Lambda} \max \log
\left|\dfrac{(\omega-\lambda)(\omega'-\lambda')}{(\omega-\lambda')(\omega'-\lambda)}\right|\nonumber\\
& = & \underset{\lambda, \lambda'\in\Lambda}\max{\log}\left|\dfrac{\lambda}{\lambda'}\right|\label{log}
\end{eqnarray}
The component $I$ of $\Omega$ containing $\omega = 0$ is a neighborhood of $0$ and contains no point of $\Lambda$.
Similarly, the component $I'$ containing $\omega' = \infty$ is a neighborhood of $\infty$ that
contains no point of $\Lambda$. In order to maximize \eqref{log}, one
should choose $\lambda$ as big as possible in absolute value,
i.e. one of the endpoints of $I'$ and $\lambda'$ as small as possible, i.e. one of the
endpoints of $I$. This proves the second item. In case $I$ and $I'$ are equal, then the max is attained
when $\lambda$ and $\lambda'$ are distinct boundary points. This means that the distance between $\omega$
and $\omega'$ is just their Hilbert distance. \end{proof}
\begin{figure}[h!]
\begin{center}
\scalebox{0.5}{\includegraphics{proj.eps}}
\end{center}
\caption{Lemma \ref{lem:dist-RP1} : $I$ and $I'$ are components of $\Omega$, points of $\Lambda$ that are not
endpoints of $I$ and $I'$ give projections onto the geodesic $(\omega,\omega')$ that are closer than those of the
endpoints of $I$ and $I'$. \label{fig:proj}}
\end{figure}
Given two components $I$ and $I'$ of which closures are disjoint, the infimum of $d_\Lambda(x,x')$
over $x\in I$ and $x'\in I'$ is not $0$: as $\Omega$ is the complement of $\Lambda$, the distance $d_\Lambda$
on $\Omega$ is proper and the infimum is attained. Hence we define the distance $d_\Lambda(I,I')$ between
$I$ and $I'$ to be this infimum. The following lemma gives a beautiful geometrical interpretation of this distance,
which is not surprise in view of Lemma \ref{lem:cross-hyp}.
\begin{lemma}
The distance between two components $I$ and $I'$ in $\Omega$ is given by the distance
between the two geodesics $\gamma$ and $\gamma'$ in the hyperbolic space where
the endpoints of $\gamma$ are the endpoints of $I$ and those of $\gamma'$ are the endpoints of $I'$.
\end{lemma}
\begin{proof}
One can arrange, up to the action of $\mathrm{PGL}(2,\mathbb R)$, the two intervals to be
$]-1,1[$ and the complement of $[-a,a]$ (with $1<a$). In this case, the hyperbolic distance
between the two geodesics is $\log(a)$ (it is the hyperbolic distance between $i$ and $ia$).
One can then compute that the minimal distance between points in both intervals is attained for
$0$ and $\infty$ and is precisely $d(0,\infty)=\ln [\infty,0,1,a]=\log a$. Again, this can be easily seen more
geometrically (see Figure \ref{fig:dist-intervals}).
\end{proof}
\begin{figure}[h!]
\begin{center}
\scalebox{0.5}{\includegraphics{dist-intervals.eps}}
\end{center}
\caption{Lemma \ref{lem:dist-RP1} : the distance between $I$ and $I'$ is realised at the common orthgonal of
the associated geodesics. \label{fig:dist-intervals}}
\end{figure}
Remark that $[a,-a,-1,1]=\left (\frac{1+a}{1-a}\right)^2$. Therefore, the distance $d$ between the two intervals
$I=]-a,a[$ and the complement of $I'=\mathbb {RP}^1\setminus[-1,1]$ is related to the cross ratio of the four endpoints by
the relation
$
[a,-a,-1,1]=\coth^{2}{\frac{d}{2}}.
$
Reading backwards and using the invariance of the distance by projective transformation, we get that
the distance $d_\Lambda(I,I')$ between two intervals $I = ]a,b[$ and $I' = ]x,y[$
(with $a<b<x<y$) is then
$$
d_\Lambda(I,I') = 2\tanh(\sqrt{[a,b,x,y]}).
$$
This distance leads to the definition of a measure on $\mathbb {RP}^1$ associated to a closed set $\Lambda \subset \mathbb {RP}^1$
and a chosen component of $\mathbb {RP}^1\setminus \Lambda$.
\begin{definition}
Let $\Lambda\subset \mathbb {RP}^1$ be a closed set and $\Omega$ its complement. We denote by $\mathcal{A}_\Lambda$ the
sigma algebra generated by components of $\Omega$ and Borel sets in $\Lambda$.
\end{definition}
In other words , a measurable set $E$ is an union
$$\left(\underset{I'\textrm{ component of }\Omega} \bigcup I'\right) \cup B
\mbox{, where $B$ is a Borel subset of $\Lambda$.}$$
For any component $I$ of $\Omega$, we denote by $\gamma_I$ the geodesic in $H^2_\mathbb R$ whose
endpoints are the same as those of $I$. Then the hyperbolic length induces a measure on $\Lambda$,
obtained by pulling back the hyperbolic length element along $\gamma_I$ by the orthogonal projection.
We denote this measure by $\nu_I$.
\begin{definition}\label{def:measure}
The measure on $\mathcal{A}_\Lambda$ associated with $I$, denoted by $\mu_{\Lambda,I}$ is defined by
\begin{itemize}
\item For any component $I'$ of $\Omega$, $\mu_{\Lambda,I}(I')=2\ln (\coth \frac{d}{2})$ where $d$ is
the distance between $I$ and $I'$.
\item For any Borel set $B$ in $\Lambda$ $\mu_{\Lambda,I}(B)=\nu_I(B)$.
\end{itemize}
\end{definition}
As an example, if the component $I$ is $(-\infty,0)$, then $\nu_I$ is the measure on Borel sets of $\mathbb R_+$ given by
$d\nu_I=\frac{dx}{x}$. In that case the geodesic $\gamma_I$ is the one connecting $0$ to $\infty$ in the upper
half-plane. In $E\in\mathcal{A}_\Lambda$ is given by $E=\left(\cup I'\right) \cup B$, then
$$\mu_{\Lambda,I}(E)= 2 \sum_{I'} \ln \left(\coth\left(\frac{d_\Lambda(I,I')}{2}\right)\right) + \int_B \frac{dx}{x}. $$
\subsection{Self-similar closed sets in $\mathbb {RP}^1$ and Basmajian formula}
In this section, we revisit the famous Basmajian formula for surfaces with boundary from the point of view
of the metric $d_\Lambda$, in the case where $\Lambda$ is a self-similar closed set.
We refer the reader to Basmajian's work \cite{Basmajian}, or to the expository articles \cite{Calegari,McShane}, for more
details. The general form of the Basmajian formula relates the orthospectrum of a hyperbolic manifold with geodesic boundary
to the area of the geodesic boundary. We restrict here to the case of surfaces. We propose here a slight generalization for
$\Lambda$ not the full limit set of a Fuchsian group, but just some closed set with self-similarity properties.
\begin{definition}
A subset $\Lambda\subset \mathbb {RP}^1$ is \emph{projectively self-similar} if
there exists a finite family of projective maps $\{f_s\}$ such that
$\Lambda=\bigcup_s f_s(\Lambda)$. The maps $f_s$ are called
\emph{self-similarities} of $\Lambda$.
\end{definition}
One has to keep in mind the fundamental example of the limit set of a Fuchsian group.
In that case, the family can be reduced to a unique map. Another example is given by the
usual triadic Cantor set in the interval $[0,1]$ to which we add the point $\infty$. In
this case, the set of self-similarities are the contractions of
ratio $\frac 13$ and center $0$ or $1$. Both these transformations fix $\infty$.
Set, as before $\Omega = \mathbb {RP}^1 \setminus\Lambda$ and suppose that there exists a
component $I = (a,b)$ of $\Omega$ which is preserved under a self-similarity $f$. The map $f$ is a
hyperbolic element fixing $a$ and $b$. Choose an interval $D = [x,f(x)]\subset \mathbb {RP}^1 \setminus I$ such
that $x \in \Lambda$. It is a fundamental domain for the action of $f$ on $\mathbb {RP}^1 \setminus I$.
The point $f(x)$ is also in $\Lambda$ by invariance. In the example of the triadic Cantor set,
the interval $I$ is $(\infty,0)$ for the contraction about $0$ and $(1,\infty)$ for the second
contraction.
The self-similarities act on the set of components of $\Omega$, and in turn, act on the set of lengthes between
components. We call {\it orthospectrum} of $\Lambda$ the set of distances between components of $\omega$ modulo
the action of self-similarities. The following result relates the ratio of contraction of the self-similarity,
or rather its translation length in the hyperbolic space, to the measure $\mu_{\Lambda,I}$ of $D$, recovering the
a version of the Basmajian formula.
\begin{theorem}\label{theorem:basmajian} Let $\Lambda\subset \mathbb {RP}^1$ be closed and preserved under a
hyperbolic element $f\in \mathrm{PGL}(2,\mathbb R)$ with translation length $l$, let $\Omega$ be its complement.
Suppose, furthermore, that $f$ preserves a component $I_f\subset\Omega$.
Let $D$ be a fundamental domain as above. Then, we have:
$$
l=\mu_{\Lambda_f}(D).
$$
where $\mu_{\Lambda_f}$ is the canonical measure defined by $I_f$.
\end{theorem}
Remark that the formula above can be written as
$$
l=S'+\int_{I_\Lambda} d\nu_I
$$
with
$$S'= \sum 2\ln \left(\coth \frac{d(I,I')}{2}\right),$$
where the sum ranges over components $I'$ of $\Omega$ inside $D$. In the case of limit
sets of Fuchsian groups the continuous measure does not appear as the limit set always
has measure zero and the above formula reduces to the Basmajian's identity for hyperbolic
surfaces. Observe also that there exists one such identity for each hyperbolic element
preserving a component of the complement $\mathbb {RP}^1\setminus \Lambda$. Note
moreover that if a self-similarity with fixed points $a\neq b$ does not fix
a component, we may split $\Omega$ and $\Lambda$ in two according to the
sides of $(a,b)$. We retrieve then two formulas for $l$.
\begin{proof}
By projective invariance, we may assume that $I_f$ is $(-\infty, 0)$, so the axis of $f$ is
the vertical geodesic above $0$ in the upper half-plane.
Let $D_\Lambda=D\cap \Lambda$ and $D'_\Lambda=D\setminus D_\Lambda = \Omega \cap D$.
For each component $I_i$ of $D'$, one computes its distance $d_i$ to $I_f=(-\infty,0)$.
This is the \emph{orthospectrum} of $\Lambda$ with respect to the
interval $(-\infty,0)$ along the fundamental domain $D$.
Each distance $d_i$ is related to a quadrilateral in the hyperbolic half plane.
The proof then is then a simple observation guided by Figure \ref{fig:basmajian}.
Indeed, the translation length $l$ of the hyperbolic element $f$
is the integral of the hyperbolic Lebesgue measure $d\nu$ over a fundamental interval for
the action of $f$ by translation along its axis. In the present case, it is the
vertical geodesic in the hyperbolic upper half plane, and so.
As in the example following Definition \ref{def:measure}, the integral is computed
as a sum of two terms: one term corresponds to
the integration of $dx/x$ along the closed set $D_\Lambda = D\cap \Lambda$ and
the other corresponds to the integration over the components in $D' = D\cap \Omega$. For each such
component $I_i = (x,y)$ (with $x<y$)the integral of $dx/x$ on $I_i$ is computable and relates to the
distance $d_i := d_\Lambda(I,I_i)$ by (compare \cite{Basmajian}):
\begin{eqnarray*}
\int_{I_i} dx/x = & \ln \dfrac yx\\
= & \log [\infty, 0, x,y]\\
= & 2\log \left(\coth \left(\frac{d_i}{2}\right)\right)\\
= & \mu_{\Lambda,I}(I_i)
\end{eqnarray*}
This proves the result.
\begin{figure}[h!]
\begin{center}
\scalebox{0.5}{\includegraphics{Basmajian.eps}}
\end{center}
\caption{Basmajian's formula. The marked points on the vertical axis bound a fundamental
interval for the action of $f$ on its axis. The translation length of $f$ is $\ell$. \label{fig:basmajian}}
\end{figure}
\end{proof}
\subsection{Quasi-M\"obius maps and quasi-isometry}
Originating from \cite{CooperPignataro}, there is a literature
about self-similar Cantor sets up to bi-Lipschitz transformations.
In the case that is relevant to us, Cooper and Pignataro \cite{CooperPignataro}
classify the self-similar Cantor subsets of $[0,1]$ up to order
preserving bi-Lipschitz maps (or in another language, quasi-isometries).
We remark here that if two closed sets $\Lambda$ and $\Lambda'$ are
quasi-M\"obius --
the right notion extending quasi-isometry as we will see --
then their complement $(\Omega,d_\Lambda)$ and $(\Omega',d_\Lambda')$
are quasi-isometric.
Let us begin by recalling a few definitions. We fix a cyclic orientation
on the real projective line $\mathbb {RP}^1$ and every affine coordinates we will
consider will respect this ordering.
\begin{definition}[\cite{Vaisala}]
An order preserving invertible map $F$ between to subsets $E$ and $E'$ of
$\mathbb {RP}^1$ is \emph{quasi-M\"obius} if there is a constant $K\geq 1$ such that for any $4$-tuples
$(e_0,e_1,e_2,e_3)$ of distinct elements of $E$, we have:
$$\frac 1K \leq \frac{[F(e_0),F(e_1),F(e_2),F(e_3)]}{[e_0,e_1,e_2,e_3]}\leq K.$$
\end{definition}
Note that this class of maps has close ties with the quasi-symmetric maps
\cite{Vaisala}, that we will not use.
Suppose you have an order-preserving map $f$ between two Cantor
subsets of $[0,1]$ -- in fact, any
two compact subsets of $\mathbb R$.
Define $\Lambda = C\cup\{\infty\}$ and $\Lambda' = C'\cup\{\infty\}$.
Let $F$ be the extension
of $f$ to the map between $\Lambda$ and $\Lambda'$ which fixes $\infty$.
Then it is easy to check
that $f$ is bi-Lipschitz if and only if $F$ is
quasi-M\"obius. The following theorem explains that such a
quasi-M\"obius map extends to a quasi-M\"obius
map of $\mathbb {RP}^1$. It implies in turn that the complements $\Omega,d_\Lambda$
and $\Omega',d_{\Lambda'}$ are
quasi-isometric.
\begin{theorem}\label{RP1:qi}
Let $\Lambda$ and $\Lambda'$ be two closed subset of $\mathbb {RP}^1$ and denote
by $\Omega$ and $\Omega'$ their complements.
Then any order-preserving quasi-M\"obius invertible map $F : \Lambda \to \Lambda'$ extends
to a quasi-M\"obius invertible map $\bar F: \mathbb {RP}^1 \to \mathbb {RP}^1$. Moreover, the restriction of $\bar F$ to
$\Omega$ realizes a quasi-isometry between $(\Omega,d_\Lambda)$ and $(\Omega',d_{\Lambda'})$.
\end{theorem}
\begin{proof}
We normalize $\Lambda$, $\Lambda'$ and $F$ in the following way: we suppose,
up to a M\"obius transformation that $\Lambda$ and $\Lambda'$ contain
$0$, $1$ and $\infty$ and that $F$ fixes $0$, $1$,
$\infty$. Note that, as the order is preserved, two points $x,x'$
in $\Lambda$ are the endpoints of
a component $(x,x')$ of $\Omega$ if and only if $(F(x),F(x'))$
is a component of $\Omega'$. Using the almost invariance of
cross-ratios of the form $[\infty,0,1,t]$
for $t\neq 0\in \Lambda$, we get that $$\frac 1K\leq \frac{F(t)}{t}\leq K.$$
Using now cross-ratios $[\infty,t,0,t'] = \frac{t-t'}{t}$ for
$t,t'\neq 0\in \Lambda$, we get that:
$$\frac 1K\leq \frac{F(t)-F(t')}{t-t'}\frac{t}{F(t)}\leq K.$$
The first inequality grants that $F$ is $K^2$-bi-Lipschitz in
restriction to $\mathbb R$.
Define now the extension $\bar F$ in the following way: $\bar F = F$ on $\Lambda$. On each bounded
component $(x,x')$,
$\bar F$ is the unique affine bijection between $(x,x')$ and $(F(x),F(x'))$.
On an unbounded component
$(x,\infty)$ or $(\infty,x')$, then $\bar F$ is the unique translation
bijection between this component and its image. Note that $\bar F_{|\mathbb R}$
is an affine extension of a $K^2$-bi-Lipschitz map: it is itself
$K^2$-bi-Lipschitz. We claim that $\bar F$ is a quasi-M\"obius map
with constant $K^8$.
Consider indeed $4$ distinct points $(x_0,x_1,x_2,x_3)$ in $\mathbb {RP}^1$.
Up to transformations of the cross-ratio,
we assume that $x_0<x_1<x_2<x_3<\infty$. If $x_0 = \infty$, we have:
$$[\infty, \bar F(x_1),\bar F(x_2),\bar F(x_3)] = \frac{\bar F(x_1) - \bar F(x_3)}{\bar F(x_1) - \bar F(x_2)} \leq K^4 \frac{x_1-x_3}{x_1-x_2} = K^4 [\infty,x_1,x_2,x_3].$$
The inequality is obtained using the $K^2$-bi-Lipschiptz property of $\bar F$. One shows similarly the
minoration leading to:
$$\frac{1}{K^4} \leq \frac{[\infty, \bar F(x_1),\bar F(x_2),\bar F(x_3)]}{[\infty,x_1,x_2,x_3]}
\leq K^4.$$
If all four points are real, a very similar computation leads to
$$\frac{1}{K^8} \leq \frac{[\bar F(x_0), \bar F(x_1),\bar F(x_2),\bar F(x_3)]}{[x_0,x_1,x_2,x_3]}
\leq K^8.$$
So $\bar F$ is $K^8$-quasi-M\"obius. This proves the first claim of
the theorem. The second claim easily follows. Indeed, for each $p$,
$p'$ in $\Lambda$, $\omega$, $\omega'$ in $\Omega$, we compute:
\begin{eqnarray*}
d_{\Lambda'} \left(\bar F(\omega),\bar F(\omega')\right)
= & \underset{q,q' \in \Lambda'}\max \ln|[q,q',\bar F(\omega),\bar F(\omega')]|\\
= &\underset{p,p' \in \Lambda}\max \ln|[\bar F(p),\bar F(p'),\bar F(\omega),\bar F(\omega')]|\\
\leq & \underset{p,p' \in \Lambda}\max \ln|K^8[p,p',\omega,\omega']|\\
\leq & d_\Lambda(\omega,\omega') + 8\ln K
\end{eqnarray*}
Conversely, a similar computation shows that:
$$d_\Lambda(\omega,\omega') - 8\ln K\leq d_{\Lambda'} \left(\bar F(\omega),\bar F(\omega')\right)
\leq d_\Lambda(\omega,\omega') + 8\ln K.$$
This proves the theorem.
\end{proof}
\bibliographystyle{plain}
|
{
"timestamp": "2018-04-17T02:09:47",
"yymm": "1804",
"arxiv_id": "1804.05317",
"language": "en",
"url": "https://arxiv.org/abs/1804.05317"
}
|
\section{Introduction}
\emph{Automatic speech recognition} (ASR) has made great strides in recent years \cite{lideng}. Deep learning, in particular, has contributed to radical transformations in the field, allowing current technology to reach unprecedented performance levels \cite{Goodfellow-et-al-2016-Book,dahl2012context}. Despite the impressive achievements of the last years, many open challenges remain in the field \cite{watanabe_book}.
One important issue is the performance drop observed when going from off-line to online speech recognition. The latter recognition modality is significantly more challenging than off-line ASR, due to the real-time/low-latency constraints which inevitably arise. To provide a speech transcription with low-latency, the speech decoding must start while acquiring the signal itself, forcing the acoustic model to perform predictions mostly based on current and past information.
Future information plays an important role to perform robust predictions, due to both phoneme co-articulations and linguistic dependencies \cite{graves}.
Despite its complexity, online speech recognition is a key component towards a more natural human-machine interaction, and extensive effort has been devoted in the last decade to improve this technology. Past online recognizers were based on the GMM-HMM framework \cite{acero_book}, while current solutions rely on deep learning \cite{Goodfellow-et-al-2016-Book}. In particular, the use of feed-forward Deep Neural Networks (DNNs), including both fully-connected and convolutional architectures, has been largely investigated in the literature \cite{acw1,bacchiani_online}, especially in the context of online ASR performed on small-footprint devices \cite{small2,small3,small4,online2,lu_smallfootprint}. Attempts have also been made to develop robust online speech recognizers based on RNNs, exploiting both the traditional RNN-HMM framework \cite{unirnn_online,peddinti_online,bidir_online1,online_zeyer,bidir_online_chen,bidir_mohamed} and, more recently, end-to-end ASR technology \cite{online_e2e,baidu}.
A common aspect of past approaches is that they often employ asymmetric context windows \cite{RAVANELLI2018,tdnn2,ravanelli15}, that embed more past than future information. Despite their effectiveness, context windows inevitably introduce a trade-off between latency (that depends on the number of look-ahead frames) and recognition accuracy. Moreover window-based approaches focus on short-term future dependencies only, while long-term information cannot be used without incurring an unacceptable latency.
In contrast to past work, this paper attempts to predict the future rather than waiting for it.
Our technique encourages the hidden representations of a unidirectional recurrent network to embed some relevant features about the future, providing useful information on the upcoming phonetic and linguistic dependencies. Inspired by a recently proposed technique called \textit{Twin Networks} \cite{twin_ref}, we add a regularization term that forces forward hidden states to be as close as possible to cotemporal backward ones, computed by a ‘‘twin” neural network running backwards in time. The twin backward network is employed at training time, when online constraints do not arise. At test time only the forward states are computed, leading to a model that (ideally) does not introduces any latency and does not add any computation compared to standard unidirectional recurrent networks.
The experiments, conducted on several datasets, recurrent architectures, input features, and acoustic conditions, have shown the effectiveness of our approach.
To summarize, the contribution of this paper is two-fold.
Firstly, we design a novel method for online ASR that predicts future states of the model and demonstrate that it is well motivated. Secondly, we evaluate the proposed method under a variety of experimental conditions, showing its effectiveness against strong baselines.
The rest of the paper is organized as follows. Twin regularization for online ASR and the related work are outlined in Sec.~\ref{sec:twin}. The experimental setup and the results are presented and discussed in Sec.~\ref{sec:setup} and Sec.~\ref{sec:exp} respectively. Finally, Sec.~\ref{sec:conc} draws our conclusions.
\section{Twin Regularization for Online ASR} \label{sec:twin}
\begin{figure}[t!]
\centering
\includegraphics[scale=0.45,trim={0cm 5.5cm 20cm 0},clip]{twin_img.pdf}
\caption{An example of a recurrent acoustic model extended with a Twin Network. The regularization term $\Omega$ encourages forward and backward hidden states to be as close as possible. Dashed lines refer to computations done at training time only.}\label{fig:twin}
\end{figure}
In the context of hybrid RNN-HMM speech recognition, the neural network processes the input speech sequence \mbox{$X=\{x_{1},...,x_{t},..,x_{N}\}$} and computes, at each time step $t$, a hidden state in the following way:
\begin{equation}
\label{eq:forward}
\overrightarrow{h_{t}}=f_{\overrightarrow{\theta}}(x_{t},\overrightarrow{h}_{t-1}),
\end{equation}
where $f_{\overrightarrow{\theta}}$ is a function that depends on trainable parameters $\overrightarrow{\theta}$ (such as an LSTM or a GRU cell), and $\rightarrow$ highlights that the network scans the sequence in the forward direction. The forward states summarize the information about current and past elements of the sequence.
A linear transformation, followed by a softmax classifier, is then employed to perform predictions $\widehat{Y}=\{\widehat{y}_{1},...,\widehat{y}_{t},..,\widehat{y}_{N}\}$ over a set of phone states.
These posterior probabilities, after being normalized by their prior, feed a HMM-based decoder that integrates acoustic and linguistic information into a search graph and estimates the sequence of words uttered by the speaker. The decoding step is normally very computationally demanding, especially for large vocabulary speech recognition. For the unidirectional RNN models described above, however, the output $\widehat{y}_{t}$ at each time step $t$ can be computed without waiting for the full speech utterance to finish. The decoding step can thus start while acquiring the speech signal from the user.
The RNN model is trained to optimize the \emph{negative log-likelihood} (NLL) cost function:
\begin{equation*}
L(X,Y;\overrightarrow{\theta})=-\frac{1}{N}\sum_{t=1}^{N}{\log P_{\overrightarrow{\theta}}\left(y_{t}| \{x_0,..,x_t\}\right)},
\end{equation*}
where $Y=\{y_{1},...,y_{t},..,y_{N}\}$ is the sequence of targeted phone labels and $P_{\overrightarrow{\theta}}(y_{t}| \{x_0,..,x_t\})$ is the output probability estimated by the neural network (that is given by reading the entry for $y_t$ from the RNN output vector $\widehat{y}_t$).
When possible, it is very convenient to also process the speech sequence in the reverse time order, and compute backward states similarly to Eq.~\ref{eq:forward}:
\begin{equation*}
\overleftarrow{h_{t}}=f_{\overleftarrow{\theta}}(x_{t},\overleftarrow{h}_{t+1}).
\end{equation*}
The backward states summarize the information about current and future elements of the sequence.
In standard Bidirectional RNNs~\cite{bidir_schuster}, forward and backward hidden states are combined to perform predictions based on the whole speech sequence. This leads to a substantial performance improvement in ASR~\cite{graves}. Differently from unidirectional RNNs, bidirectional models cannot be used for online speech recognition, since each prediction $\widehat{y}_{t}$ depends on the full input sequence $X$.
Nevertheless, even if the future is not accessible, we can try to roughly predict it, capturing some relevant features that help phone predictions. This principle can be implemented by means of a regularization term, as highlighted in Fig. \ref{fig:twin}.
The idea is to penalize forward hidden representations $\overrightarrow{h_{t}}$ that are distant from the cotemporal backward ones $\overleftarrow{h_{t}}$ ~\cite{twin_ref}. With this regard, one can add a regularization term that encourage the network to minimize the $L_{2}$ distance between forward and backward hidden states:
\begin{equation}
\Omega(\overrightarrow{\theta},\overleftarrow{\theta})=\frac{1}{N} \sum_{t=1}^N{\|\overrightarrow{h_{t}}-\overleftarrow{h_{t}}\|^2}.
\label{eq:twin}
\end{equation}
The regularization term is averaged over all time steps. In general, multiple recurrent hidden layers are stacked together to perform more robust predictions. In this case, the regularization term can be simply averaged over all the recurrent layers.
The total objective to be minimized thus becomes a weighted sum of the NLL costs plus the regularization term:
\begin{equation*}
\widetilde{L}(X, Y; \overrightarrow{\theta},\overleftarrow{\theta}) = L(X, Y; \overrightarrow{\theta}) + L(X, Y; \overleftarrow{\theta}) + \lambda \Omega(\overrightarrow{\theta},\overleftarrow{\theta}),
\end{equation*}
where $\lambda$ is an hyper-parameter controlling the importance of the penalty term, and $\widetilde{L}$ is the total loss that is averaged over all the sentences composing the mini-batch.
Note that the backward states are needed only at training time, when online constraints do not arise. During testing, the part of the model computing backward states can be omitted. This leads to an architecture particularly suitable for online ASR, since it requires exactly the same amount of computations needed for standard unidirectional RNNs (at inference time).
Another remarkable aspect of this technique is that Eq.~\ref{eq:twin} is based on the backward states $\overleftarrow{h_{t}}$, that provide a summary of the full future part of the speech sequence. This means that our method could capture not only short-term future dependencies, but also long-term ones.
\begin{table*}[t!]
\centering
\caption{PER(\%) on the TIMIT dataset obtained with various RNN architectures and input features. Our approach \emph{UniTwin} shows stable improvement in all cases. \emph{BiDir} corresponds to the off-line recognition with bidirectional networks (reported here to provide a lower bound for the error rates of the online models).}
\label{tab:timit}
\begin{tabular}{|l|c|c|c|c|c|c|c|c|c|}
\hline
\multirow{2}{*}{} & \multicolumn{3}{c|}{MFCC} & \multicolumn{3}{c|}{FBANK} & \multicolumn{3}{c|}{fMLLR} \\ \cline{2-10}
& UniDir & UniTwin & BiDir & UniDir & UniTwin & BiDir & UniDir & UniTwin & BiDir \\ \hline
LSTM & 17.3 & \textbf{17.1} & 15.7 & 17.2 & \textbf{17.0} & 15.1 & 16.9 & \textbf{16.3} & 14.7 \\ \hline
GRU & 17.9 & \textbf{17.4} & 16.0 & 18.1 & \textbf{18.0} & 15.3 & 16.9 & \textbf{16.6} & 15.3 \\ \hline
M-GRU & 17.8 & \textbf{17.5} & 16.1 & 18.0 & \textbf{17.6} & 15.4 & 16.8 & \textbf{16.4} & 15.1 \\ \hline
Li-GRU & 17.5 & \textbf{17.1} & 15.5 & 17.3 & \textbf{16.8} & 14.6 & 16.7 & \textbf{16.2} & 14.6 \\ \hline
\end{tabular}
\end{table*}
\subsection{Related Work}
Several methods have been proposed in the literature to approach online ASR with RNNs. A popular choice is to feed the RNN with a context window that embeds some future frames \cite{unirnn_online,peddinti_online}. Attempts have also been done to build low-latency bidirectional RNNs \cite{bidir_online1,online_zeyer,bidir_online_chen,bidir_mohamed}. The latter solutions are based on chunking the speech signal into several overlapping or non-overlapping windows. Each chunk embeds both past and future speech frames and is processed by an bidirectional RNN to perform phone predictions. These approaches, however, only account for a limited fraction of the future information, inheriting the same issues discussed for feed-forward DNNs.
This paper proposes the use of twin regularization to improve the way online RNNs exploit the future information. Twin regularization has been recently proposed in~\cite{twin_ref}. Its effectiveness has been proved in sequence generation tasks, such as image captioning, language modeling, and monaural singing voice separation~\cite{drossos2018mad}. Some works~\cite{goyal2017z,shabanian2017variational} take a similar approach to train stochastic recurrent models with a backward running RNN. These approaches have been applied to speech synthesis, language modeling, image generation, and demonstrate that the idea of predicting future states is well-motivated, practically sound, and worth exploring.
To the best of our knowledge, this paper is the first attempt to build a predictor of future states for an online ASR model.
\section{Experimental Setup} \label{sec:setup}
In the following sub-sections, the corpora and the RNN-HMM setting adopted for the experimental activity are described.
\subsection{Corpora and Tasks}
The first set of experiments was performed with the TIMIT corpus \cite{timit}, considering the standard phoneme recognition task (aligned with the Kaldi s5 recipe \cite{kaldi}).
To validate our model in a more challenging scenario, experiments were also conducted in distant-talking conditions with the DIRHA-English dataset \cite{dirha_asru}.
Training was based on the original WSJ-5k corpus (consisting of 7138 sentences uttered by 83 speakers) that was contaminated with a set of impulse responses measured in a real apartment \cite{rav_is16}.
The test phase was carried out with the real-part of the dataset, consisting of 409 WSJ sentences uttered in the aforementioned apartment by six native American speakers.
Additional experiments were conducted with the CHiME~4 dataset \cite{chime3}, that is based on speech data recorded in four noisy environments (on a bus, cafe, pedestrian area, and street junction). The training set is composed of 43690 noisy WSJ sentences recorded by five microphones (arranged on a tablet) and uttered by a total of 87 speakers.
The test set \textit{ET-real} considered in this work is based on 1320 real sentences uttered by four speakers, while the subset \textit{DT-real} has been used for hyperparameter tuning. The CHiME experiments were based on the single channel setting \cite{chime3}.
Finally, experiments were performed with LibriSpeech~\cite{librispeech} dataset.
We used the training subset composed of 100 hours and the \textit{dev-clean} set for the hyperparameter search. Test results are reported on the \textit{test-clean} part.
\subsection{RNN-HMM setting}
The experiments are set up considering different acoustic features, i.e., 39 MFCCs (13 static+$\Delta$+$\Delta\Delta$), 40 log-mel filter-bank features (FBANKS), as well as 40 fMLLR features (extracted as reported in the s5 recipe of Kaldi \cite{kaldi}), that were computed using windows of 25 ms with an overlap of 10 ms.
Neural acoustic models consisted of multiple recurrent layers, that were stacked together prior to the final softmax classifier. These recurrent layers were unidirectional or bidirectional RNNs.
Beyond standard LSTM~\cite{lstm} and GRU~\cite{gru1}, we also considered recently proposed architectural variations~\cite{ravanelli_is17}: M-GRU~\cite{mgru} is the minimal GRU architecture based on replacing reset gate with updated gate activations, while light GRU (Li-GRU)~\cite{li_gru} directly avoids the reset gate and exploits ReLU activations for the hidden activations.
The feed-forward connections of the architecture were initialized according to the \textit{Glorot}'s scheme \cite{xavier}, while recurrent weights were initialized with orthogonal matrices \cite{orth_init}.
Recurrent dropout was used as regularization technique \cite{drop_asru,Gal2016}.
Batch normalization was adopted for feed-forward connections only, as proposed in \cite{laurent2016batch, ravanelli_is17}.
The optimization was done using the RMSprop algorithm running for 24 epochs. The performance on the development set was monitored after each epoch, and the learning rate was halved when the relative performance improvement went below~$0.1\%$.
Back-propagation through time was not truncated, allowing the system to learn arbitrarily long time dependencies.
The main hyperparameters of the model (i.e., learning rate, number of hidden layers, hidden neurons per layer, dropout factor, as well as the twin regularization term $\lambda$) were optimized on the development datasets.
In particular, we guessed some initial values according to our experience, and starting from them we performed a grid search to progressively explore better configurations. As a result, we adopted $\lambda=0.6$ for TIMIT experiments, and $\lambda=0.1$ for the other datasets. Please refer to the github repository referenced in the footnote below for more details about the considered hyperparameters.
The labels were derived by performing a GMM-based forced alignment on the original training datasets (see the standard s5 recipe of Kaldi for more details \cite{kaldi}).
During test, the posterior probabilities generated by the RNN were normalized by their prior probabilities.
The obtained likelihoods were processed by an HMM-based decoder, that estimated the sequence of words uttered by the speaker.
The RNN part of the system was implemented with Pytorch~\cite{paszke2017automatic}, that was coupled with the Kaldi decoder \cite{kaldi} to form a context-dependent RNN-HMM speech recognizer.\footnote{\label{foot:code}The code is available at \url{http://github.com/mravanelli/pytorch-kaldi/}.}
\section{Results} \label{sec:exp}
In the following sub-sections, we report the experimental results obtained with TIMIT, DIRHA, CHiME, and LibriSpeech datasets.
\subsection{Phoneme recognition on TIMIT}
To provide a thorough assessment of our methodology, several RNNs models and features are considered. Table \ref{tab:timit} shows the results obtained with TIMIT. The results with off-line bidirectional models (\textit{BiDir} columns) are reported only to provide a lower bound for the error rates that can be achieved with an online model. Moreover, to ensure a more accurate comparison, five experiments varying the initialization seeds were conducted for each RNN model and input feature. Results of Table \ref{tab:timit} are thus reported as the average \textit{phone error rates} (PER)\footnote{Standard deviations $\sigma$ range between $0.1$ and $0.2$ for all the experiments.}.
Twin regularization (\textit{UniTwin} columns) helps to improve the recognition performance, consistently outperforming standard unidirectional models (\textit{UniDir} columns) in all the considered experimental conditions. Although our method is still far from bridging the gap with off-line bidirectional models, we observe an average relative improvement of 2.1\%, which is obtained with a simple technique that does not introduce any additional computation at test time.
Li-GRU consistently outperforms the other RNN models, as previously observed in \cite{ravanelli_is17}. A remarkable achievement is the average PER of $14.6$\% obtained with Li-GRUs using fMLLR and fbank features. To the best of our knowledge, this result yields one of the highest published performance on the TIMIT test-set.
\begin{figure}[t!]
\centering
\includegraphics[scale=0.50]{curve.pdf}
\caption{Learning curves for unidirectional and twin RNN models (fbank features, Li-GRU model).}\label{fig:te_cure}
\end{figure}
We plot the learning curves for the \textit{frame-level error rates} (FER) obtained on the development set over the duration of training in Fig.~\ref{fig:te_cure}.
Our experiments show that twin regularization converges to a better solution.
The results presented above are obtained with RNNs fed with the current frame only. Similarly to the window-based approaches described in Sec. \ref{sec:twin}, Tab. \ref{tab:cw} extends our previous results by adding a small context window that concatenates some future frames.
\begin{table}[t]
\centering
\caption{PER(\%) on TIMIT obtained with a context windows that embeds some future frames (Li-GRU model, fbank feats).}
\begin{tabular}{|l|c|c|}
\hline
\# Future Frames & UniDir & UniTwin \\ \hline
0 frames & 17.3 & \textbf{16.8 } \\ \hline
5 frames & 16.5 & \textbf{16.1 } \\ \hline
10 frames & 16.8 & \textbf{16.5} \\ \hline
15 frames & 17.5 & \textbf{16.9} \\ \hline
\end{tabular}
\label{tab:cw}
\end{table}
The table shows that a small look-ahead context window embedding 5 or 10 future frames is helpful to improve the ASR performance. Interestingly, our method outperforms standard unidirectional RNNs even under this experimental condition. This achievement confirms that twin regularization can focus on long-term future dependencies, providing useful information also when some look-ahead frames embed a short-term future context. This allows our method to be used in conjunction with previous window-based approaches.
\subsection{Word recognition on other datasets}
As a last experiment, we extend our previous achievements to more realistic ASR tasks.
To test our technique into a complex acoustic scenario, Tab. \ref{tab:dirha} reports the \emph{word error rate} (WER) obtained with the DIRHA dataset. For the sake of compactness, only the results with MFCCs are reported. It is worth mentioning that we obtained a similar experimental evidence using fbank and fMLLR features.
\begin{table}[h]
\centering
\caption{WER(\%) for the DIRHA dataset (MFCC feats).}
\label{tab:dirha}
\begin{tabular}{|l|c|c|c|}
\hline
& UniDir & UniTwin & BiDir \\ \hline
LSTM & 32.9 & \textbf{32.5} & 27.8 \\ \hline
GRU & 30.2 & \textbf{29.6} & 27.2 \\ \hline
Li-GRU & 29.2 & \textbf{28.7} & 26.9 \\ \hline
\end{tabular}
\end{table}
\begin{table}[h]
\centering
\caption{WER(\%) for CHiME (ET-Real) and Librispeech (Test-Clean) using Li-GRU models and MFCC feats.}
\label{tab:others}
\begin{tabular}{|l|c|c|c|}
\hline
& UniDir & UniTwin & BiDir \\ \hline
CHiME & 23.7 & \textbf{23.0} & 19.2 \\ \hline
LibriSpeech & 10.4 & \textbf{10.2} & 9.2 \\ \hline
\end{tabular}
\end{table}
We conclude from this experiment that twin regularization is also effective in challenging acoustic conditions characterized by the presence of both noise and reverberation.
To further test its robustness in noisy environments, we also performed some experiments with the CHiME dataset (see first row of Tab. \ref{tab:others}). Moreover, to provide evidence on a larger vocabulary task, the second row of Tab. \ref{tab:others} reports the results achieved with LibriSpeech. These results are obtained with the 100 hours subset decoded with the \textit{tgsmall} language model (see Kaldi s5 recipe \cite{kaldi}).
Our experiments target the online ASR scenario, therefore the results reported in the table do not consider complex techniques as multi-microphone processing, data-augmentation. Neither system combination nor lattice rescoring are used here.
It is however worth noting that the effectiveness of the proposed approach is one more time confirmed.
\section{Conclusions} \label{sec:conc}
This paper explored the use of twin regularization for predicting the future states of an online RNN-HMM speech recognition. The proposed technique, that encourages forward hidden representations to be predictive of the future, has shown to be effective in several experimental conditions.
An average relative performance improvement of 2\% is obtained over a standard unidirectional RNNs. The improvement is consistent across datasets, architectures, and input features.
Furthermore, our proposed technique is simple and does not add any additional computational cost at test time.
A noteworthy aspect of our method is that it also accounts for long-term future dependencies, which differs from current dominant approaches based on short-term context windows. This offers the possibility of using twin regularization in conjunction with existing techniques.
\section{Acknowledgment} \label{sec:conc}
We would like to thank Titouan Parcollet (Universit\'e d'Avignon), Kyle Kastner (Universit\'e de Montr\'eal) and Maurizio Omologo (Fondazione Bruno Kessler) for their helpful comments. This research was enabled in part by support provided by Calcul Qu\'ebec and Compute Canada (www.computecanada.ca).
\vfill \pagebreak
\bibliographystyle{IEEEtran}
|
{
"timestamp": "2018-06-13T02:04:14",
"yymm": "1804",
"arxiv_id": "1804.05374",
"language": "en",
"url": "https://arxiv.org/abs/1804.05374"
}
|
\section{Introduction}
In the last thirty years, many studies have converged on the idea that the Heisenberg uncertainty
principle (HUP)~\cite{Heisenberg} should be modified when gravitation is taken into account.
In microphysics, gravity is usually neglected on the ground of its weakness,
when compared with the other fundamental interactions.
However, this argument should not apply when one wants to address fundamental questions in Nature. In this perspective, gravity should be included, especially when we discuss the formulation of a fundamental principle like the Heisenberg's one.
And in fact, gravitation has always played a pivotal role in the generalization of the HUP,
from the early attempts~\cite{GUPearly}, to the more recent proposals, like those in string theory, loop quantum gravity, deformed special relativity, non-commutative geometry, and studies of black hole physics~\cite{VenezGrossMende,MM,FS,Adler2,CGS,SC2013}.
\par
A possible way for this generalization is to reconsider the well-known classical argument of the
Heisenberg microscope~\cite{Heisenberg}. The size $\delta x$ of the smallest detail of an object, theoretically detectable with a beam of photons of energy $E$, is roughly given by (assuming the dispersion relation $E=p$)~\footnote{We shall always
work with $c=1$, but explicitly show the Newton constant $G_{\rm N}$ and the Planck constant
$\hbar$. The Planck length is defined as $\ell_{\rm p}=\sqrt{G_{\rm N}\,\hbar/c^3}\simeq 10^{-35}\,$m, the Planck energy as $\mathcal{E}_{\rm p}\,\ell_{\rm p} = \hbar\, c/2$, and the Planck mass
as $m_{\rm p}=\mathcal{E}_{\rm p}/c^2\simeq 10^{-8}\,$kg, so that $\ell_{\rm p}=2G_{\rm N}\,m_{\rm p}$ and $2\,\ell_{\rm p}\,m_{\rm p}= \hbar$. The Boltzmann constant $k_{\rm B}$ will be shown explicitly, unless otherwise stated.}
\begin{equation}
\delta x
\simeq
\frac{\hbar}{2\, E}
\ ,
\label{HS}
\end{equation}
since increasingly large energies are required to explore decreasingly small details.
In its original formulation, Heisenberg's gedanken experiment ignores gravity.
However, gedanken experiments involving formation of gravitational instabilities
in high energy scatterings of strings~\cite{VenezGrossMende}, or gedanken experiments taking
into account the possible formation, in high energy scatterings, of micro black holes with a
gravitational radius $R_S=R_S(E)$ proportional to the (centre-of-mass) scattering energy $E$
(see Ref.\cite{FS}), suggest that the usual uncertainty relation should be modified as
\begin{equation}
\delta x
\simeq
\frac{\hbar}{2\, E}
+
\beta\, R_S(E)
\ ,
\end{equation}
where $\beta$ is a dimensionless parameter.
Recalling that $R_S\simeq 2\,G_{\rm N}\, E = 2\, \ell_{\rm p}^2\, E/\hbar$, we can write
\begin{equation}
\delta x
\simeq
\frac{\hbar}{2\, E}
+
2\beta\, \ell_{\rm p}^2\frac{E}{\hbar}
=
\ell_{\rm p}
\left(
\frac{m_{\rm p}}{E}
+
\beta\, \frac{E}{m_{\rm p}}
\right)
\ .
\label{He}
\end{equation}
This kind of modification of the uncertainty principle was also proposed in Ref.~\cite{Adler2}.
\par
The dimensionless deforming parameter $\beta$ is not (in principle) fixed by the theory, although it is generally assumed to be of order one.
This happens, in particular, in some models of string theory (see again for instance Ref.~\cite{VenezGrossMende}), and has been confirmed by an explicit calculation in Ref.~\cite{SLV}.
However, many studies have appeared in literature, with the aim to set bounds on $\beta$
(see, for instance, Refs.~\cite{brau}).
\par
The relation~(\ref{He}) can be recast in the form of an uncertainty relation, namely a deformation of the standard HUP, usually referred to as Generalized Uncertainty Principle (GUP),
\begin{equation}
\Delta x\, \Delta p
\geq
\frac{\hbar}{2}
\left[1
+\beta
\left(\frac{\Delta p}{m_{\rm p}}\right)^2
\right]
\ .
\label{gup}
\end{equation}
For mirror-symmetric states (with $\langle \hat{p} \rangle = 0$), the inequality (\ref{gup}) is equivalent to the commutator
\begin{equation}
\left[\hat{x},\hat{p}\right]
=
i \hbar \left[
1
+\beta
\left(\frac{\hat{p}}{m_{\rm p}} \right)^2 \right]
\ ,
\label{gupcomm}
\end{equation}
since $\Delta x\, \Delta p \geq (1/2)\left|\langle [\hat{x},\hat{p}] \rangle\right|$.
Vice-versa, commutator (\ref{gupcomm}) implies inequality (\ref{gup}) for any state.
The GUP is widely studied in the context of quantum mechanics~\cite{Pedram},
quantum field theory~\cite{Husain:2012im,Majhi:2013koa}, quantum gravity~\cite{Hossain:2010wy},
and for various deformations of the quantization rules~\cite{Hossain:2010wy, Jizba:2009qf}.
The above $\beta$-deformed commutator~(\ref{gupcomm}) will be the starting point of the present
investigation.
In what follows, using~(\ref{gupcomm}), we shall describe the Unruh effect (known also as Fulling-Davies-Unruh effect~\cite{Fulling:1972md,davies,Unruh:1976db}), thereby calculating corrections to the Unruh temperature to first order in $\beta$. A direct derivation of the Unruh effect from the HUP has been given in Ref.~\cite{FS9506}. On the other hand, the necessity of this effect for the internal consistency of QFT has been confirmed by arguments based both on general covariance~\cite{Matsas:1999jx3} and thermodynamic~\cite{Becattini:2017ljh}. Moreover, non-trivial modifications to the Unruh spectrum have been recently pointed out also in different contexts, for instance, it has been shown that flavor mixing does spoil its thermal character~\cite{Blasone:2017nbf, Blasone:2018byx}, thus opening new stimulating scenarios.
\section{Heuristic derivation of Unruh Effect from uncertainty relations}
In this section we derive the Unruh temperature~\cite{Unruh:1976db} starting directly
from the HUP. Simple classical physics relations will be used together with the quantum principle, following closely Ref.~\cite{FS9506} (see also the recent Ref.~\cite{gine}).
This procedure will then allow us to estimate what kind of corrections are induced by a GUP.
\par
Let us consider some elementary particles, for example electrons, kept at rest in an uniformly
accelerated frame.
The kinetic energy acquired by each of these particles while the accelerated frame moves a
distance $\delta x$ will be given by
\begin{equation}
E_k
=
m\,a\,\delta x
\ ,
\end{equation}
where $m$ is the mass of the particle and $a$ the acceleration of the frame, and therefore
of the particle.
Now, suppose this energy is sufficient to create $N$ pairs of the same kind of particles from the quantum
vacuum.
Namely, we set
\begin{equation}
E_k
\simeq
2\,N\, m
\ ,
\end{equation}
and find that the distance along which each particle must be accelerated in order to create $N$ pairs is
\begin{equation}
\delta x
\simeq
2\,\frac{N}{a}
\ .
\label{dx}
\end{equation}
The original particles and the pairs created in this way are localized inside a spatial region of width $\delta x$,
therefore the fluctuation in energy of each single particle is
\begin{equation}
\delta E
\simeq
\frac{\hbar}{2\, \delta x}
\simeq
\frac{\hbar\, a}{4\, N}
\ .
\end{equation}
If we interpret this fluctuation as a classical thermal agitation of the particles, we can write
\begin{equation}
\frac{3}{2}\,k_{\rm B}\,T
\simeq
\delta E
\simeq
\frac{\hbar\, a}{4\,N}
\ ,
\label{dE}
\end{equation}
or
\begin{equation}
T
=
\frac{\hbar\, a}{6\,N\,k_{\rm B}}
\ .
\end{equation}
On comparing with the well-known Unruh's temperature~\cite{Unruh:1976db},
\begin{equation}
T_{\rm U}
=
\frac{\hbar\, a}{2\,\pi\, k_{\rm B}}
\ ,
\label{Tu}
\end{equation}
we can set the arbitrary parameter $N$ and obtain an effective number of pairs $N=\pi/3\simeq 1$.
\par
Now we repeat the same argument using the GUP.
Upon replacing Eq.~\eqref{dx} into Eq.~\eqref{He}, and interpreting the energy fluctuation $\delta E$
in terms of a classical thermal bath, we find
\begin{equation}
2\,\frac{N}{a}
\simeq
\frac{\hbar}{3\, k_{\rm B}\, T}
+
\beta\, \ell_{\rm p}^2\, \frac{3\, k_{\rm B}\, T}{\hbar}
\ .
\label{approx}
\end{equation}
Requiring that the $T$ equals the Unruh temperature~\eqref{Tu} for $\beta \to 0$ again fixes $N=\pi/3\simeq 1$,
and we finally obtain
\begin{equation}
\frac{2\,\pi}{a}
\simeq
\frac{\hbar}{k_{\rm B}\, T}
+
9\beta\, \ell_{\rm p}^2\, \frac{k_{\rm B}\, T}{\hbar}
=
\ell_{\rm p}\left(
\frac{2m_{\rm p}}{k_{\rm B}\,T}
+
9\,\beta\,\frac{k_{\rm B}\, T}{2m_{\rm p}}
\right)
\ .
\label{acctemp}
\end{equation}
This relation can be easily inverted for $T=T(a)$.
However, it is reasonable to assume that $\beta\,k_{\rm B} T/m_{\rm p}\sim \beta \,m/m_{\rm p}$ is very small for any fundamental particle with
$m\ll m_{\rm p}$.
We can therefore expand in $\beta\,m/m_{\rm p}$ and find
\begin{equation}
T
\simeq
T_{\rm U}
\left(1
+
\frac{9\,\beta}{4}\,\frac{\ell_{\rm p}^2\,a^2}{\pi^2}
\right)
=
T_{\rm U}
\left[1
+
\frac{9\,\beta}{4}
\left(\frac{k_{\rm B}\,T_{\rm U}}{m_{\rm p}}
\right)^2
\right]
\ .
\label{newTHeuristic}
\end{equation}
We also notice an interesting physical property suggested by Eq.~(\ref{acctemp}), that is, by the GUP. In order to maintain the inverted relation $T=T(a)$ physically meaningful (i.e. the temperature must be a real number), there will be a maximal value for the acceleration, namely
\begin{equation}
a
\lesssim
\frac{\pi}{3 \, \sqrt{\beta}\, \ell_{\rm p}}
\ ,
\end{equation}
and a corresponding maximal value for the Unruh-Davies temperature,
\begin{equation}
k_{\rm B}\,T_{\rm U}
\lesssim
\frac{m_{\rm p}}{3\,\sqrt{\beta}}
\ .
\end{equation}
These ideas and estimates naturally make contact with those reported, for example, in Refs.~\cite{caianiello}.
\section{Quantization of a massive scalar field in accelerated frame}
\label{Quantization}
In this Section we briefly review the quantization of a massive scalar field for an accelerated observer. This will serve as a basis for the analysis of Section~\ref{GUP}, where the deformation of the algebra discussed above is implemented.
For the sake of simplicity, we will work in $1+1$-dimensions, using the Minkowski metric with the
conventional signature $ds^2= \eta_{\mu\nu}\,dx^\mu\,dx^\nu=dt^2-dx^2$. In this Section we set $\hbar=c=k_{\rm B}=1$, unless otherwise explicitly stated.
\subsection{Minkowski spacetime}
For an inertial observer, the scalar field in the usual plane-wave representation reads
\begin{equation}
\phi(\bx)
=
\int dk\,
\left[
a_{k}\,U_{k}(\bx)
+
a_{k}^\dagger\,U_{k}^{*}(\bx)
\right]
\ ,
\label{eqn:planewavexpans}
\end{equation}
where $\bx\equiv\{t, x\}$ denotes the set of Minkowski coordinates.
The positive frequency plane-waves of momentum $k$ are given by
\begin{equation}
U_{k}(\bx)
=
{\left(4\,\pi\,\omega_{k}\right)}^{-\frac{1}{2}}\,
e^{i\left(k\,x-\omega_{k}\, t\right)}
\ ,
\label{eqn:modes}
\end{equation}
where $\omega_{k}=\sqrt{m^2+k^2}$, $m$ being the mass of the field.
Within the framework of canonical quantum field theory (QFT), the annihilation and creation
operators for Minkowski quanta, to wit $a_{k}$ and $a^\dagger_{k}$, satisfy the standard
commutation relation
\begin{equation}
\left[a_k, a_{k'}^\dagger\right]
=
\delta(k-k')
\ ,
\label{eqn:commutcanon}
\end{equation}
with all other commutators vanishing.
The ordinary Minkowski vacuum is accordingly defined by $a_k\,|0_M\rangle=0$ for all modes $k$.
\par
As a tool for extending this quantization scheme to an accelerated observer, let us now introduce the less familiar
Lorentz-boost eigenfunctions~\cite{Blasone:2017nbf}.
Boost modes are related to the plane-waves in Eq.~(\ref{eqn:modes}) by
\begin{equation}
\widetilde{U}_{\Omega}^{(\sigma)}(\bx)
=
\int dk\, p_\Omega^{(\sigma)*}(k)\,U_{k}(\bx)
\ ,
\label{eqn:Uwidetildemodes}
\end{equation}
where
\begin{equation}
\label{eqn:p}
p_\Omega^{(\sigma)}(k)
=
\frac{1}{\sqrt{2\,\pi\,\omega_{k}}}\,
\left(\frac{\omega_{k}+k}{\omega_{k}-k}\right)^{i\,\sigma\,\Omega/2},
\qquad
\sigma
=
\pm
\ ,
\quad
0<\Omega<\infty
\ .
\end{equation}
The physical meaning of the quantum numbers $\Omega$ and $\sigma$ will be discussed in the next Section.
In terms of the modes~(\ref{eqn:Uwidetildemodes}), the spectral representation of the field operator can be written as \footnote{Note that, although the plane-wave field expansion in Eq.~(\ref{eqn:planewavexpans}) applies to the whole of the Minkowski space-time, the representation Eq.~(\ref{eqn:expansionfieldboost}) in terms of boost-modes does hold only in the Rindler manifold $x>|t|\hspace{0.4mm}\cup\hspace{0.4mm} x<-|t|$ (see Fig.1). A globally well-defined expansion can be obtained by analytically continuing the modes Eq.~(\ref{eqn:Uwidetildemodes}) across the null asymptotes $x=\pm\, t$ (see Ref.~\cite{Gerlach}). For our purposes, however, it is enough to consider the definition Eq.~(\ref{eqn:Uwidetildemodes}) of boost modes.}
\begin{equation}
\phi(\bx)
=
\int_{0}^{+\infty}
d\Omega\,
\sum_{\sigma}
\left[d_{\Omega}^{(\sigma)}\,\widetilde{U}_{\Omega}^{(\sigma)}(\bx)
+
d_{\Omega}^{(\sigma)\dagger}\,\widetilde{U}_{\Omega}^{(\sigma)*}(\bx)
\right]
\ .
\label{eqn:expansionfieldboost}
\end{equation}
\par
It is easy to prove that the two quantum constructions introduced above are equivalent to each other.
For this purpose, let us equate the field-expansions~(\ref{eqn:planewavexpans})
and~(\ref{eqn:expansionfieldboost}) on a space-like hypersurface.
By using Eq.~(\ref{eqn:Uwidetildemodes}), it follows that
\begin{equation}
\label{eqn:operat-d}
d_{\Omega}^{(\sigma)}
=
\int dk\,
p_\Omega^{(\sigma)}(k)\,a_{k}
\ .
\end{equation}
Since the operators $d_{\Omega}^{\,(\sigma)}$ are linear combinations of the Minkowski
annihilators $a_{k}$ alone, they also annihilate the Minkowski vacuum $|0_M\rangle$.
Moreover, by exploiting the completeness and orthonormality of the set of functions
$\left}\def\lrar{\leftrightarrow\{p_\Omega^{(\sigma)}\right}\def\ti{\tilde\}$ (see Ref.~\cite{Takagi:1986kn}), it can be shown
that the transformation~(\ref{eqn:operat-d}) is canonical, so that
\begin{equation}
\label{eqn:d-commut}
\left[d_{\Omega}^{(\sigma)},d_{\Omega'}^{\,(\sigma')\dagger}\right]
=
\delta_{\sigma\sigma'}\,\delta(\Omega-\Omega')
\ ,
\end{equation}
with all other commutators vanishing.
Eqs.~(\ref{eqn:operat-d}) and (\ref{eqn:d-commut}) allow us to interpret also the
$d_{\Omega}^{\,(\sigma)}$ as annihilation operators of Minkowski quanta. This implies that the
field-expansions Eqs.~(\ref{eqn:planewavexpans}) and~(\ref{eqn:expansionfieldboost}) can be used equivalently within the framework of canonical quantization in Minkowski space-time.
For our purposes, in what follows it will be convenient to employ the latter.
\subsection{Rindler space-time}
The foregoing discussion applies to inertial observers in Minkowski space-time.
In order to investigate GUP effects on the Unruh radiation~\cite{Unruh:1976db},
let us now review the Rindler-Fulling field-quantization in a uniformly accelerating frame~\cite{Fulling:1972md}.
By introducing the usual Rindler coordinates $\{\eta,\xi\}$, in place of $\{t, x\}$, we have
\begin{equation}
\label{eqn:rindlercoordinates}
t
=
\xi\,\sinh\eta
\ ,
\quad
x=
\xi\,\cosh\eta, \qquad -\infty\,<\,\eta,\xi\,<\,\infty,
\end{equation}
and the Minkowski line element takes the well-known form
\begin{equation}
ds^2
=
dt^2-dx^2
=
\xi^2\,d\eta^2
-d\xi^2
\ .
\label{eqn:lineelement}
\end{equation}
As $\xi$ and $\eta$ range from $-\infty$ to $\infty$, the Rindler coordinates cover only two sections of Minkowski space-time, specifically the right wedge $R_+=\left}\def\lrar{\leftrightarrow\{{\bf x}\,|\,x>|t|\right}\def\ti{\tilde\}$ for $\xi>0$,
and the left wedge $R_-=\left}\def\lrar{\leftrightarrow\{{\bf x}\,|\,x<-|t|\right}\def\ti{\tilde\}$ for $\xi<0$ (see Fig.~\ref{figure:Rindler}).
Since the components of the metric in these coordinates do not depend on $\eta$,
Eq.~(\ref{eqn:lineelement}) describes a static spacetime with Killing vector ${\bf B}=\partial_\eta$.
\begin{figure}[t]
\resizebox{8cm}{!}{\includegraphics{rindler.eps}}
\caption{\small{The proper coordinate system of a uniformly accelerated observer in the Minkowski spacetime.
The branch of hyperbola $\xi=a^{-1}$ represents the worldline of an observer with proper acceleration $a$.}}
\label{figure:Rindler}
\end{figure}
\par
The worldline of a uniformly accelerated (Rindler) observer with proper acceleration $|a|$ is given by
\begin{equation}
\xi(\tau)
=
\mathrm{const}
\equiv
a^{-1}
\ ,
\label{eqn:lineauniverso}
\end{equation}
where $\tau=\eta/a$ is the proper time along the accelerated trajectory (restoring $c\neq 1$, one has $\xi(\tau)=c^2/a$, and $\tau=\eta c/a$; that is, $\eta=a\tau/c$ is dimensionless).
This is a branch of hyperbola in the $(t,x)$ plane, whose null asymptotes $t=\pm x$ act respectively as future and past event horizons for the Rindler observer.
\par
Because of the non-trivial structure of Rindler space-time, the wedges $R_\pm$ are causally
disconnected from each other~\cite{Rindler:1966zz}.
The positive frequency solutions of the Klein-Gordon equation in Rindler coordinates thus take the
form~\footnote{In what follows, the set of Rindler coordinates $\{\eta, \xi\}$ will be also denoted by $\bx$; therefore, such a symbol will refer to a space-time point, rather than its representative in a particular coordinate system.}
\begin{equation}
u_\Omega^{(\sigma)}(\bx)
=
N_\Omega\,
\theta(\sigma\,\xi)\,
K_{i\,\Omega}^{(\sigma)}(m\,\xi)\,
e^{-i\,\sigma\,\Omega\,\eta}
\ ,
\label{eqn:rindlermodes}
\end{equation}
where $\Omega$ is the Rindler frequency with respect to the time $\eta$~\footnote{Using $c\neq1$ for sake of clarity, the proper frequency $\omega$ measured by a Rindler observer is obtained from $\omega\,\tau = \omega(\eta c/a)=(\omega c/a)\eta\equiv\Omega\,\eta$.}, $\sigma=\pm$ refers to the right/left wedges
$R_\pm$ and $K_{i\,\Omega}$ is the modified Bessel function of the second kind.
In this context, we do not need to specify the normalization factor $N_\Omega$
(for more details, see Ref.~\cite{Takagi:1986kn}).
Furthermore, the Heaviside function $\theta(\sigma\,\xi)$ was inserted into Eq.~(\ref{eqn:rindlermodes}) in order to constrain the Rindler modes to only one of the two disconnected wedges $R_\pm$.
\par
Using Eq.~(\ref{eqn:rindlermodes}), we can now expand the scalar field operator in the Rindler space-time as follows
\begin{equation}
\phi(\bx)
=
\int_{0}^{+\infty}
d\Omega\,
\sum_\sigma\,
\left[b^{(\sigma)}_{\Omega}\,u_{\Omega}^{(\sigma)}(\bx)
+b^{(\sigma)\dagger}_{\Omega}\,u_{\Omega}^{(\sigma)*}(\bx)
\right]
\ ,
\label{eqn:espanrind}
\end{equation}
where the ladder operators $b^{\,(\sigma)}_\Omega$ and ${b^{\,(\sigma)\dagger}_{\Omega}}$
are assumed to satisfy the canonical commutation relations
\begin{equation}
\left[b_\Omega^{(\sigma)}, b_{\Omega'}^{( \sigma')\dagger}\right]
=
\delta_{\sigma\sigma'}\,
\delta(\Omega-\Omega')
\ ,
\label{eqn:commutcanon2}
\end{equation}
with all other commutators vanishing.
The Rindler vacuum is accordingly defined by $b_\Omega^{(\sigma)}|0_{\rm R}\rangle = 0$, for all
values of $\sigma$ and $\Omega$.
\par
The connection between the two quantization schemes, for inertial and accelerated observers, can now be investigated in detail. Specifically, we compare the
field-expansions~(\ref{eqn:expansionfieldboost}) and (\ref{eqn:espanrind}) on a spacelike hypersurface $\Sigma$ lying in the Rindler manifold $R_\pm$.
A straightforward calculation leads to the following Bogoliubov
transformation~\cite{Takagi:1986kn}
\begin{equation}
b^{(\sigma)}_{\Omega}
=
\left[1+\mathcal{N}(\Omega)\right]^{1/2}\,
d_{\Omega}^{(\sigma)}
+
\mathcal{N}(\Omega)^{1/2}\,d_{\Omega}^{(-\sigma)\dagger}
\ ,
\label{eqn:bogotransform}
\end{equation}
where
\begin{equation}
\label{eqn:B-Edist}
\mathcal{N}(\Omega)
=
\frac{1}{e^{2\,\pi\,\Omega}-1}
\end{equation}
is the Bose-Einstein distribution.
Using Eq.~(\ref{eqn:bogotransform}), we can now calculate the spectrum of Rindler quanta in the Minkowski vacuum $|0_{\rm M}\rangle$,
\begin{equation}
\langle0_{\rm M}|\,b_{\Omega}^{(\sigma)\dagger}\,b_{\Omega'}^{ (\sigma')}\,|0_{\rm M}\rangle
=
\mathcal{N}(\Omega)\,\delta_{\sigma\sigma'}\,\delta(\Omega-\Omega')
\ .
\label{eqn:aspectval}
\end{equation}
It then follows that a uniformly accelerated observer perceives the Minkowski vacuum as a thermal bath
of Rindler quanta with a temperature proportional to the acceleration (Unruh effect~\cite{Unruh:1976db}).
Restoring our standard units ($c=1$, $\hbar\neq 1$, $k_{\rm B}\neq 1$), we can in fact write
\begin{equation}
2\,\pi\,\Omega
=
\frac{2\pi}{a}\,a\,\Omega
=
\frac{\hbar \,a\, \Omega}{k_{\rm B}\,T_{\rm U}}
=
\frac{\hbar\,\omega}{k_{\rm B}\,T_{\rm U}}
\ ,
\end{equation}
where $\omega=a\,\Omega$ is the frequency measured by the Rindler observer and $T_{\rm U}$
the Unruh temperature~\eqref{Tu}.
\section{GUP and modified Unruh temperature}
\label{GUP}
In the previous Section, the Unruh temperature~(\ref{Tu}) has just been re-derived within the framework of the canonical QFT. At this stage, one may wonder how such a result gets modified when starting from the GUP commutator in Eq.~(\ref{gupcomm}).
To answer this question, an intermediate step concerning the effects of GUP on a quantum one-dimensional harmonic oscillator turns out to be useful.
In this context, we note that the ladder operators $A$ and $A^\dagger$ of the deformed algebra for the one-dimensional harmonic oscillator are linked to $\hat{x}=\hat{x}^\dagger$ and $\hat{p}=\hat{p}^\dagger$ by the usual relations
\begin{equation}
A = \frac{1}{\sqrt{2m\hbar\omega}}(m\omega\hat{x} + i \hat{p}), \qquad
A^\dagger = \frac{1}{\sqrt{2m\hbar\omega}}(m\omega\hat{x} - i \hat{p})\,,
\end{equation}
and their inverses
\begin{equation}
\hat{x} = \sqrt{\frac{\hbar}{2m\omega}}(A^\dagger + A), \qquad
\hat{p} = i\sqrt{\frac{m\hbar\omega}{2}}(A^\dagger - A) \,.
\end{equation}
It is then easy to see that
\begin{equation}
[A,A^\dagger]\,=\,\frac{1}{i\hbar}\,[\hat x, \hat p]
\end{equation}
and, due to the modified commutator (\ref{gupcomm}) between
$\hat x$ and $\hat p$, the deformed algebra for the one-dimensional harmonic oscillator should be written as
\begin{equation}
\label{eqn:oscillharm}
\left[A, A^\dagger\right]
=
\frac{1}{1-\alpha}
\left[1
-\alpha
\left(A^\dagger\, A^\dagger
+A\,A
-2\,A^\dagger\, A
\right)
\right]
\ ,
\end{equation}
where
\begin{equation}
\alpha=\beta\,\frac{m\,\hbar \omega}{2\,m_{\rm p}^2}
\ ,
\end{equation}
with $m$ and $\omega$ being the mass and frequency of the harmonic oscillator, respectively.
The modified quantization rules~(\ref{eqn:oscillharm}) can be now extended in a natural way to a scalar field in the plane-wave representation, if we consider that, for a given momentum $k$, the energy $\hbar \omega_k$ of the scalar field plays the role of the mass $m$ of the harmonic oscillator. The deformation parameter $\alpha$ can be then suitably redefined as
\begin{equation}
\tilde{\alpha}=
\beta\,\frac{\hbar^2 \omega_k^2}{2\,m_{\rm p}^2}=
2\,\beta\,\ell_{\rm p}^2\,\omega_k^2
\end{equation}
and the commutator between ladder operators becomes
\begin{equation}
\label{modplanwav}
[A_k, A_{k'}^\dagger]
=
\frac{1}{1-\tilde{\alpha}}
\left[1
-
\tilde{\alpha}
\left(A_{k}^\dagger\, A_{k'}^\dagger
+A_{k}\, A_{k'}
-2\,A_{k}^\dagger\, A_{k'}
\right)
\right]
\delta(k-k')
\ .
\end{equation}
\par
In Section~\ref{Quantization}, we have seen that the scalar field for an inertial observer can be quantized
both using plane-waves and boost-modes (see Eqs.~(\ref{eqn:planewavexpans})
and~(\ref{eqn:expansionfieldboost}), respectively).
In that context, the choice between these two representations is just a matter of convenience,
since the corresponding sets of ladder operators $a_k$ and $d_{\Omega}^{(\sigma)}$
are related by the canonical transformation~(\ref{eqn:operat-d}).
With deformed quantization rules, however, Lorentz invariance is violated and such an equivalence is not guaranteed.
Nevertheless, in the limit of very small deformation (that is, $\beta p^2\ll m_{\rm p}^2$), it appears reasonable to assume the same structure of the modified algebra for the two sets of operators. According to this argument, we thus conjecture the following deformation for the commutator in the boost-mode representation:
\begin{equation}
\label{eqn:d-D}
\left[D_{\Omega}^{(\sigma)}\,D_{\Omega'}^{(\sigma')\dagger}\right]
=
\frac{1}{1-\gamma}\left[
1-\gamma\,\left(
D_{\Omega}^{(\sigma)\dagger}\,D_{\Omega'}^{(-\sigma')\dagger}
+
D_{\Omega}^{(\sigma)}\,D_{\Omega'}^{(-\sigma')}
-D_{\Omega}^{(\sigma)\dagger}\,D_{\Omega'}^{(\sigma')}
-D_{\Omega}^{(-\sigma)\dagger}\,D_{\Omega'}^{(-\sigma')}
\right)
\right]
\delta_{\sigma\sigma'}\,
\delta(\Omega-\Omega')
\ ,
\end{equation}
where $D_{\Omega}^{\,(\sigma)}$ and $D_{\Omega}^{\,(\sigma)\dagger}$ are the ladder operators in the deformed algebra and the deforming parameter $\gamma$ is defined by
\begin{equation}
\label{ips}
\gamma
=
\beta\,
\frac{\hbar^2 \omega^2}{2\,m_{\rm p}^2}
=
\beta\,
\frac{\hbar^2 a^2\,\Omega^2}{2\,m_{\rm p}^2}
= 2\,\beta\,\ell_{\rm p}^2\,a^2\,\Omega^2
\ ,
\end{equation}
being $\omega=a\Omega$ the Rindler frequency.
Some comments about Eq.~(\ref{eqn:d-D}) are needed.
First, in order to adapt the deformed commutator~(\ref{modplanwav}) to the boost operators $D$,
we have modified \emph{ad hoc} the definition of the deforming parameter $\tilde{\alpha}$
by replacing the plane-frequency $\omega_k$ with the boost-mode frequency
$\omega=a\,\Omega$ [see Eq.~(\ref{ips})].
Furthermore, the commutator~(\ref{eqn:d-D}) has been multiplied by $\delta_{\sigma\sigma'}$
to ensure that the ladder operators in the right wedge $R_+$ are still commuting with the
corresponding operators in the left wedge $R_-$.
In addition, we symmetrized it with respect to $\sigma$ and $-\sigma$, so that
\begin{equation}
\label{eqn:equalcomm}
\left[D_{\Omega}^{(\sigma)},D_{\Omega'}^{(\sigma')\dagger}\right]
=
\left[D_{\Omega}^{(-\sigma)},D_{\Omega'}^{(-\sigma')\dagger}\right]
\ .
\end{equation}
By exploiting this property and recasting the Bogoliubov transformation~(\ref{eqn:bogotransform})
in the form
\begin{equation}
B_{\Omega}^{(\sigma)}
=
{\left[1+\mathcal{N}(\Omega)\right]}^{1/2}
\,D_{\Omega}^{(\sigma)}
+
{\mathcal{N}(\Omega)}^{1/2}\,
D_{\Omega}^{(-\sigma)\dagger}
\ ,
\label{eqn:newformbogotr}
\end{equation}
one can verify that the deformation~(\ref{eqn:d-D}) induces an identical modification to the
algebra of the Rindler operators $B$.
\par
GUP effects on the Unruh temperature can now be investigated by calculating the distribution
of $B$-quanta in the Minkowski vacuum $|0_{\rm M}\rangle$.
By use of the transformation~(\ref{eqn:newformbogotr}), it can be shown that
\begin{equation}
\label{eqn:modexpecnum}
\langle0_{\rm M}|\,
B_{\Omega}^{(\sigma)\dagger}\,B_{\Omega'}^{ (\sigma')}\,
|0_{\rm M}\rangle
=
\frac{1}{\left(e^{2\pi\Omega}-1\right)\left(1-\gamma\right)}\,
\delta_{\sigma\sigma'}\,
\delta(\Omega-\Omega')
\ ,
\end{equation}
to be compared with the standard Bose-Einstein distribution Eq.~(\ref{eqn:aspectval}).
As expected, the Unruh spectrum gets non-trivially modified by the deformed
algebra~(\ref{eqn:d-D}) and loses its characteristic thermal behavior.
However, for Rindler frequencies $\Omega$ such that $\gamma\ll1$, namely (since $\beta\sim 1$) for $\hbar\omega\llm_{\rm p}$, we have $e^{-\gamma}\simeq 1-\gamma$,
and Eq.~\eqref{eqn:modexpecnum} can be approximated as
\begin{equation}
\label{eqn:approxexpecnum}
\langle0_{\rm M}|\,
B_{\Omega}^{(\sigma)\dagger}\,B_{\Omega'}^{ (\sigma')}\,
|0_{\rm M}\rangle
\simeq
\frac{1}{e^{2\pi\Omega-\gamma}-1}\,
\delta_{\sigma\sigma'}\,
\delta(\Omega-\Omega')
\ ,
\end{equation}
where we neglected the term linear in $\gamma$ in the denominator of the r.h.s.
We can interpret Eq.~(\ref{eqn:approxexpecnum}) as a shifted Bose-Einstein thermal distribution
by introducing a shifted Unruh temperature $T$ such that the term $(2\pi\Omega-\gamma)$ can be rewritten as
\begin{equation}
\label{newT}
2\pi\Omega-\gamma
\ = \
\frac{\hbar \,a\, \Omega}{k_{\rm B}\,T_{\rm U}} - \gamma
\ \equiv \
\frac{\hbar \,a\, \Omega}{k_{\rm B}\,T}
\ .
\end{equation}
We thus find for the shifted Unruh temperature
\begin{equation}
T
=
\frac{T_{\rm U}}{1-\beta\,\pi\,\Omega\,k_{\rm B}^2\,T_{\rm U}^2/m_{\rm p}^2}
\simeq
T_{\rm U}\left(1 + \beta\, \pi\, \Omega \left(\frac{k_{\rm B}T_{\rm U}}{m_{\rm p}}\right)^2 \right)
=
T_{\rm U}
\left(1+\beta\,\pi\,\Omega\,\frac{\ell_{\rm p}^2\,a^2}{\pi^2}\right)
\,.
\label{eqn:newT}
\end{equation}
We notice that such a modified temperature $T$ contains an explicit dependence on the Rindler frequency $\Omega$. This is due to the deformed structure of the
commutator (\ref{gupcomm}), which explicitly depends on $\hat{p}^2$, that is, essentially, on the energy of the considered quantum mode. So, it is not surprising to recover such an explicit dependence in the final formulae. Nevertheless, a simple thermodynamic argument allows us to get rid of this $\Omega$-dependance. In fact, for small deformations, we are still close to the thermal black body spectrum. Therefore the vast majority of the Unruh quanta will be emitted around a Rindler frequency $\omega$ such that $\hbar\,\omega\simeq k_{\rm B}\,T_{\rm U}$, which means $\Omega\approx 1/(2\pi)$.
For this typical frequency, Eq.~\eqref{eqn:newT} reproduces quite closely the heuristic
estimate~\eqref{newTHeuristic}. In fact
\begin{equation}
T
\simeq
T_{\rm U}\left(1 + \frac{\beta}{2} \left(\frac{k_{\rm B}T_{\rm U}}{m_{\rm p}}\right)^2 \right)
=
T_{\rm U}
\left(1 + \frac{\beta}{2}\frac{\ell_{\rm p}^2\,a^2}{\pi^2}\right)
\ .
\end{equation}
It is also worth noting that the deformation of the algebra Eq.~(\ref{eqn:d-D}) would affect also the Hamiltonian. Therefore, the Rindler frequency $\Omega$ in Eq.~(\ref{newT}) should in principle be modified accordingly. In the present analysis, however, since we consider only small deformations of the quantization rules, we have reasonably neglected those corrections, thus approximating the modified Rindler Hamiltonian to the original one.
Concluding, for small deviations from the canonical quantization, we have found that the Unruh distribution maintains its original thermal spectrum, provided that a new temperature $T$ is defined as in Eq.~(\ref{eqn:newT}).
\section{Conclusions}
In the context of the Generalized Uncertainty Principle, we have computed the correction induced on the Unruh temperature by a deformed fundamental commutator. This has been done following two independent paths. First, we proceeded in a heuristic way, using very general and reasonable physical considerations. Already at this stage however we have been able to point out a dependence of the deformed Unruh temperature on the cubic power of the acceleration. These considerations have been substantiated and confirmed by means of a full fledged Quantum Field Theory calculation. This has been achieved by taking into account modified commutation relations for the ladder operators compatible with the GUP of Eq.(\ref{gupcomm}). In the limit of a small deformation of the commutator, we obtained again a dependence of the first correction term on the third power of acceleration. Besides, the more refined formalism of QFT has helped us to point out an explicit dependence of the deformed Unruh temperature from the Rindler frequency
$\Omega$, which, on the other hand, was reasonably expected. A simple and effective thermodynamic argument has then been used to identify the values of most probable emission for the $\Omega$ Rindler frequency. As a consequence the QFT calculation matches in the end the heuristic estimate, indeed with almost the same numerical coefficients.
An avenue for further investigations could be the relation between the deviation from thermality of the Unruh radiation discussed in this paper and those found in different contexts (e.g. Ref.~\cite{Blasone:2017nbf}).
|
{
"timestamp": "2018-04-17T02:08:53",
"yymm": "1804",
"arxiv_id": "1804.05282",
"language": "en",
"url": "https://arxiv.org/abs/1804.05282"
}
|
\section{Introduction}
The communication networks are observing a tremendous increase in the number of devices which are predicted to go beyond 40\% (of that were active in 2012) by 2020~\cite{zhang2014sybil}. All these devices have been arranged under a common term of ``Internet of Things" (IoT). IoT allows integration of the vast variety of communication devices irrespective of their operational technology, which is also a challenging issue as a common firmware is required for all the devices. A common firmware makes it easier to control and manage various IoT devices without many overheads. Common software platforms allow easy configurations as well as easy diagnosis of faulty operations. However, a common firmware also subjects the IoT components to various types of threats which can infiltrate the operational defense of these devices~\cite{sharma2017saca}. Some of the key features required by IoT networks are remote diagnosis and management, data analytic, software upgrades, information passing and processing, and user mobility identification~\cite{sadeghi2015security}. All these form a type of application which allows access to the entire network once a particular feature is exploited.
Since there is no formal definition of IoT, same attacks which are applicable to any computing entity hold true in their case. Also, reduction in the human interventions and use of more automated systems in the IoT networks make it extremely important to secure the entire network as it may reveal critical information~\cite{kotenko2012attack}. Apart from these, IoT networks are also considered as an integral part of civilian and military expeditions focusing surveillance, navigation, localization, equipment control, and currency transfers, etc. Recent trends have focused on using RFID tags as embedded sources for IoT devices that do not connect to the network directly. Although, such strategy holds safe for the majority of application scenarios, but manipulation with RFID tags can easily make these vulnerable similar to a normal computing entity~\cite{covington2013threat}. Thus, security of IoT devices irrespective of the mode and type of connectivity is of utmost importance and has been an area of concern for a majority of the security researchers across the globe.
Considering a common platform for IoT devices, most of the business enterprises and vendors focus on making version-based IoT firmware that can be easily upgraded and controlled. Such scenarios are possible by using a software-assisted networking. However, a software-assisted networking suffers from a major issue of zero-day vulnerabilities. Considering the level of deployment and configuration of networks, zero-day vulnerabilities are extremely dangerous for IoT networks. Exploitation of a zero-day vulnerability can lead to a zero-day attack~\cite{kaur2014survey}. Control over a single unit of IoT software may expose the entire architecture.
\begin{figure}[!ht]
\centering
\includegraphics[width=230px]{fig1}
\caption{An illustration of the window of vulnerability for zero-day attacks.}\label{fig1}
\end{figure}
\section{Background: zero-day attacks}
The name ``Zero-day" is coined considering the negligible time available in mitigating these threats. The number of days for which an anomaly has been known directly affects the countermeasures and also the probability of remaining affected. It also has to do a lot with those software users who do not update security patches regularly. Once a vulnerability is publicized, it is mandatory for the particular application users to immediately switch to the stable releases. However, failure in doing so leads to various consequences in the form of cyber attacks~\cite{portokalidis2006argos}.
The effect of a zero-day vulnerability also depends on the mode of detection. If a vulnerability is identified by white hat hackers, it allows keeping it low profile until the security patches are not available; whereas identification of such vulnerabilities by a notorious group (black hat hackers) may subject the entire enterprise to failure~\cite{kaur2014survey}. The vulnerability cycle for a zero-day attack may vary from scenario to scenario. In some cases, after identification of a bug, the hackers operate covertly leading to the full zero-day attack, while in some cases, the hackers may come forward (overt) and make threat public~\cite{palani2016invisible}~\cite{wanswett2015threat}. Thus, it can be analyzed that a zero-day attack is not only because of the covert behavior of a hacker but also because of the delays in updating security patches once these are available in the public domain. This is often explained in the terms of window of vulnerability. The window of vulnerability is the time gap in which the number of vulnerable systems remaining is negligible. It is evaluated as a software timeline considering the discovery phase, security patching, intermediate exploitation phase and patch applicability phase, as shown in Fig.~\ref{fig1}~\cite{palani2016invisible}~\cite{bilge2012before}.
\begin{figure}[!ht]
\centering
\includegraphics[width=230px]{fig3}
\caption{An illustration of DDS-assisted IoT network.}\label{fig3}
\end{figure}
\begin{figure}[!ht]
\centering
\includegraphics[width=230px]{fig4}
\caption{An illustration of strategic context graph formation for an IoT device between the SDS and CDS. The decision on matching context is performed at CDS. The counter updates and firmware version decisions are also evaluated at the SDS and the CDS.}\label{fig4}
\end{figure}
\section{Proposed Approach
The network comprises various IoT devices and gadgets that operate either individually or collectively via a common gateway. The communication can be directly between the Mobile Node (MN) and the IoT device or indirectly between the MN and the IoT device via a gateway. The service providers are responsible for maintaining trust between the IoT and the MN. Currently, the proposed model emphasizes on a particular scenario in which an IoT device receives security updates that may lead to zero-day attacks; or when an attack is already launched and security updates confirm the attacks. The proposed approach uses strategic context graphs to ensure the safety of IoT devices against the zero-day attacks. The context graphs are implemented using Distributed Diagnosis System (DDS). The DDS are divided into three parts (shown in Fig.~\ref{fig3}), namely,
\begin{itemize}
\item Central Diagnosis System (CDS): CDS is installed by the service providers on the central node of the network which is responsible for generating trust as well as the updates for the entire network. CDS is responsible for managing the Access Points (APs) control, and the operations of gateways for maintaining security in the case of high possibilities of threats.
\item Local Diagnosis System (LDS): LDS is operated as a dedicated device over the gateways. Usually, these are installed with the Home Gateways (HGW). LDS interacts with the CDS and shares its context graphs with it to ensure that all the security procedures are followed by the corresponding IoT device.
\item Semi Diagnosis System (SDS): SDS is responsible for directly managing the APs trust with the CDS. It shares the context of IoT devices which directly interacts with an MN without relying on the local gateway.
\end{itemize}
\subsection{Strategic Context}
The types of devices operable in a network are considered to have valid pre-registered signatures along with a counter value. The counter value manages the count for the number of times the firmware of an IoT device is validated or encountered. The context for each IoT device is managed by its diagnosis system and periodically stored in logs and shared with the CDS. The context outline used in the proposed model is as follows:
\begin{itemize}
\item Device signatures ($S_g$): This is the unique identity for each device. The signature is the embedded information about the IoT device which is stored at the CDS once it gets activated in the network.
\item Update Counter ($U_c$): This is the firmware update counter which is randomly selected at the beginning of network registrations. These are updated using random integer values which are finalized by the CDS and change periodically without affecting the performance.
\item Traffic Type ($T_p$): This defines the context for the type of traffic to be generated for and by an IoT device. This helps the diagnosis system to analyze the content over a particular channel for its correctness.
\item Header Length ($H_l$): It defines the bit length of the header field used by the diagnosis system. It contains all the necessary context metadata which is to be shared between the LDS, SDS, and CDS.
\item Memory Range ($M_r$): It denotes the maximum and minimum size of the packets generated by the IoT device. This helps to simply analyze if the size of the initial code is affected or not. Usually, these are not mishandled by the attackers, but still, in some cases, this is very useful to identify if the binaries of the firmware are altered or not.
\item Route ($R_t$): This field is used to check whether an IoT device is operable in LDS, SDS, or CDS region. This also allows tracking the actual route for managing the context between the network entities.
\end{itemize}
\subsection{Context Graphs and Strategic Attack Detection}
The context graphs are used to generate the strategies which help in taking a decision regarding the presence of a threat amongst the IoT devices. The number of vertices in the context graphs is equal to the number of processing procedures an IoT device follows before generating an output and demanding an input. The context explained above forms the edges of the graph. After the time instance decided in the configuration of the network, the LDS and SDS evaluate these graphs for every corresponding IoT device ad share it with the CDS which also forms its own context graph for every IoT device. Along with the context graphs, the CDS also forms the context graphs for the subordinate network which includes the layers of APs, and gateways.
In order to take a strategic decision on the management of IoT devices against the zero-day attacks, the CDS follows a principle of modeling the counter and the random integer value used to manage the counter by the LDS, SDS and the device itself. Then, it performs mutual exclusion rule to trace the presence of a zero-day threat in the IoT network. The failure in the matching of the context stored and the context received from all the subordinates as well as the IoT device indicates the presence of a zero-day attack. The operational view of the proposed approach is illustrated in the Fig.~\ref{fig4}.
It is to be noted that the strategic context graphs are applicable in the network only in the deployment phase, but not in the development phase. Thus, the proposed strategy can come handy only when a vulnerability is identified by the development team at lateral stages as well as during the release of security updates as it helps in tracking the contextual behavior of every IoT device. Once a possibility of attack is found, the proposed approach utilizes the critical data sharing protocol that helps in eliminating a particular IoT device before it exploits the entire network.
\begin{figure}[!ht]
\centering
\includegraphics[width=230px]{fig5}
\caption{An illustration of critical context/data sharing protocol used after the identification of potential zero-day threat or attack in the IoT network.}\label{fig5}
\end{figure}
\subsection{Critical Data Sharing Protocol}
The proposed approach uses a critical context/data sharing protocol in the scenarios with a potential zero-day threat. The protocol, shown in Fig.~\ref{fig5}, illustrates the procedures opted by the CDS once a threat is identified amongst the IoT devices leading to a zero-day exploitation.
Once a threat policy is violated, the CDS sends alarming messages to its connected components that are its subordinates in the network. The alarming messages are followed by the patch for fixing the affected IoT device. This is followed by the reestablishment of the trust between all the connected components with the CDS. Once an alarming request is received, each subordinate's diagnosis system shares context information to revalidate the trust. By the time, these steps are performed, the affected device updates its security mechanisms, and registers itself again with the CDS leading to the elimination of the threat without eliminating the device. On the contrary, CDS shares threat information with the SDS, trust information with the HGW, device information with the LDS, and finally, leads it to eliminate the incorrect device. This allows mitigating zero-day threats in IoT networks.
\section{Performance Evaluation}
The proposed approach is evaluated by deploying 500 sensors in two modes, namely, with CDS only and with CDS, LDS, and SDS. The proposed approach is evaluated to analyze the effect of DDS on the performance of the proposed framework. The model defined in Ref.~\cite{sharma2017saca} with similar attacker scenario (20\% nodes as the attacker) is used to evaluate the formation in the proposed approach for cost of operation and communication overheads. The cost of operation is calculated as the time required by the diagnosis system to arrive at the decision of zero-day possibility. It includes the communication time including the context sharing procedures as well as the formation of the context graphs at the interacting entities of the network. Results in Fig.~\ref{g1} show that the DDS is capable of performing better in distributed mode rather than only CDS scenario, and covers 33\% less cost of operation. With critical protocol coming into play after the identification of a zero-day threat, the proposed approach utilizes series of steps to generate alert messages and reestablish the trust between the connected devices and gateways. The DDS causes 21\% lesser overheads in comparison with the scenario with a single diagnosis system, as shown in Fig.~\ref{g2}.
\begin{figure}
\centering
\begin{subfigure}[b]{0.20\textwidth}
\centering
\includegraphics[width=115px]{g1}
\caption{Cost of Operation vs. IoT devices.}
\label{g1}
\end{subfigure}
\begin{subfigure}[b]{0.20\textwidth}
\centering
\includegraphics[width=115px]{g2}
\caption{Communication Overheads vs. IoT devices.}
\label{g2}
\end{subfigure}
\caption{Simulation Results.}
\end{figure}
\section{Conclusion}
In this article, a study was presented on zero-day threats for IoT networks. A context graph based framework was presented to provide a strategy for deciding on the zero-day attacks. The proposed approach used a distributed diagnosis system for classifying the context at the central service provider as well as at the local user site. Also, once a zero-day attack was potentially identified, a critical data sharing protocol was used to transmit alert messages and reestablish the trust between the network entities and the IoT devices.
This is a progressive paper and the details on the full-fledged implementation along with critical evaluations will be presented in future reports.
\bibliographystyle{ieeetr}
|
{
"timestamp": "2018-04-17T02:14:30",
"yymm": "1804",
"arxiv_id": "1804.05549",
"language": "en",
"url": "https://arxiv.org/abs/1804.05549"
}
|
\section{Introduction}
Elliptic equations with singularity has gained a huge attention owing to its richness both from the theoretical and application point of view. Early traces of research pertaining to problems involving singularity can be found in \cite{Lazer}, where the authors have addressed the following problem.
\begin{eqnarray}\label{refeq1}
\begin{split}
-\Delta u&= \frac{f(x)}{u^{\gamma}}~\text{in}~\Omega,\\
u&= 0~\text{on}~\partial\Omega,
\end{split}
\end{eqnarray}
where $\Omega$ is a strictly convex, bounded domain in $\mathbb{R}^N$ with $C^2$ boundary. The existence of a unique solution was guaranteed iff $0<\gamma <3$. The authors in \cite{Lazer}, has also shown the existence of a solution in $C^1(\bar{\Omega})$, for $0<\gamma <1$. Haitao \cite{Hai} studied the perturbed singular problem
\begin{eqnarray}\label{refeq2}
\begin{split}
-\Delta u&= \cfrac{\lambda}{u^\gamma}+u^p~\&~u>0~\text{in}~\Omega,\\
u&= 0~\text{on}~\partial\Omega,
\end{split}
\end{eqnarray}
and guaranteed the existence of two weak solutions for $\lambda<\Lambda$, no solution for $\lambda>\Lambda$ and atleast one solution for $0<\gamma<1<p\leq\frac{N+2}{N-2}$ and some $\Lambda>0$. A further generalization to this problem can be found in \cite{gia}, where the existence of two solutions were shown for some $0<\gamma<1<p-1<q\leq p^*-1$. An important problem involving singularity in the literature can be found in the work due to Crandall et al \cite{cran}, where the authors have addressed the problem.
\begin{eqnarray}\label{cran_prob}
-\Delta u &=& f(u)~\text{in}~\Omega,\nonumber\\
u&=&0~\text{on}~\partial\Omega,
\end{eqnarray}
where $f$ is a function with singularity near $0$. The authors in \cite{cran}, have shown the existence of a unique classical solution in $C^2(\Omega)\cap C(\bar{\Omega})$. Another noteworthy work is due to Giacomoni and Sreenadh \cite{giasri}, where the authors have investigated the following quasilinear and singular problem.
\begin{align}\label{giasri_prob}
\begin{split}
-\Delta_{p} u&=\frac{\lambda}{u^\delta}+u^q~\text{in}~\Omega,\\
u&=0~\text{on}~\partial\Omega,\\
u&>0~\text{in}~\Omega,
\end{split}
\end{align}
where $\Omega$ is a bounded domain in $\mathbb{R}^N$ with smooth boundary, $1<p-1<q$ and $\lambda,\delta>0$. The authors have shown the existence of weak solutions for small $\lambda>0$ in $W_0^{1,p}(\Omega)\cap C(\bar{\Omega})$ if and only if $\delta<2+\frac{1}{p-1}$. Further they have investigated the radial symmetry case, i.e. for $\Omega=B_R(0)$, where they have proved the global multiplicity of solutions in $C(\bar{\Omega})$ with $\delta>0$, $1<p-1<q$, by using shooting method. Readers interested in `singularity involving problem' can refer to \cite{Oliva, canino2, canino3, tali} and of late Panda et al. \cite{pgc}, who have investigated a problem involving singularity and a measure. Motivated by the work due to \cite{Bal}, which stemmed out from the work due to \cite{Arcoya}, by generalizing their result for the $p$-Laplacian, we will study the following problem.
\begin{align}\label{e0}
(P)\hspace{1 cm}\left\{ \begin{aligned} -\Delta_p u&= \cfrac{\lambda}{u^\gamma}+g(u)+\mu~\text{in}~\Omega,\\
u&= 0~\text{on}~\partial\Omega,\\
u&>0~\text{in}~\Omega,
\end{aligned}\right.
\end{align}
where $\Omega$ is a strictly convex, bounded domain in $\mathbb{R}^N$ with $C^2$ boundary, $N>2$, $1<p<N$, $\Delta_pu=\text{div}\{|\nabla u|^{p-2}\nabla u\}$, $\lambda>0$, $\gamma>0$ and $\mu$ is a bounded Radon measure. The function $g$ obeys certain growth conditions, i.e there exists some constants $C>0$ such that, $$C^{-1}t^{1+q}\leq t g(t)\leq Ct^{1+q},$$ where $p-1<q<\cfrac{N(p-1)}{N-p}$.
\section{Notations and Definitions}
We will use the notations due to \cite{Evans}, to denote $W_0^{k,p}(\Omega)$, to be the space obtained by considering the closure of $C_c^{\infty}(\Omega)$ in the Sobolev space $W^{k,p}(\Omega)$ and $W_{loc}^{k,p}(\Omega)$ to be the local Sobolev space, which consists of functions $u$ such that for any compact $K\subset \Omega$, $u\in W^{k,p}(K)$. The H\"{o}lder Space is denoted by $C^{k,\beta}(\bar{\Omega})$ with $0<\beta\leq1$ (again a notation borrowed from $\cite{Evans}$), which consists of all functions $u\in C^k(\bar{\Omega})$ such that the norm $$\sum\limits_{|\alpha|\leq k}\sup|D^{\alpha}u|+\sup\limits_{x\neq y} \left\{\frac{|D^ku(x)-D^ku(y)|}{|x-y|^{\beta}}\right\}<\infty.$$ We will use the truncation functions for fixed $k > 0$,
$$T_k(t)=\max\{-k,\min\{k,t\}\} ~\text{and}~G_k(t)=(|t|-k)^+sign(t)$$with $t\in \mathbb{R}$. Observe that $T_k(t)+G_k(t)=t$ for any $t\in \mathbb{R}$ and $k>0$.\\
We denote $\mathbb{M}(\Omega)$ as the space of all finite Radon measures on $\Omega$. For every $\mu\in \mathbb{M}(\Omega)$, we define $$||\mu||_{\mathbb{M}(\Omega)}=\int_{\Omega}d|\mu|.$$
We will use the Marcinkiewicz space $\mathcal{M}^q(\Omega)$ (or weak $L^q(\Omega)$) defined for every $0 < q <\infty$, as the space of all measurable functions $f:\Omega\rightarrow\mathbb{R}$ such that the corresponding distribution functions satisfy an estimate of the form
$$m(\{x\in \Omega:|f(x)|>t\})\leq \frac{C}{t^q}\hspace{0.4cm}t>0,\,C<\infty.$$
Indeed, for bounded domain $\Omega$ we have $\mathcal{M}^q\subset \mathcal{M}^{\bar{q}}$ if $q\geq \bar{q}$, for some fixed positive $\bar{q}$. Further, the following continuous embeddings holds
\begin{equation}\label{marcin}
L^q(\Omega)\hookrightarrow \mathcal{M}^q(\Omega)\hookrightarrow L^{q-\epsilon}(\Omega),
\end{equation}
for every $1<q<\infty$ and $0<\epsilon<q-1$. We will use this embedding result to show the existence of solutions. We now give the definition of convergence in the measure space.
\begin{definition}\label{measure}
Let $(\mu_n)$ be the sequence of measurable functions in $\mathbb{M}(\Omega)$. We say $(\mu_n)$ converges to $\mu\in \mathbb{M}(\Omega)$ in the sense of measure $\cite{Folland}$ i.e. $\mu_n\rightharpoonup \mu$ in $\mathbb{M}(\Omega)$, if
$$\int_\Omega f d\mu_n\rightarrow \int_\Omega f d\mu, \hspace{0.2cm}\forall f\in C_0(\Omega).$$
\end{definition}
\noindent In order to show the existence of solutions to the problem (\ref{e0}), we will consider the following sequence of problems $(P_n)$.
\begin{align}\label{e7}
(P_n)\hspace{1 cm}\left\{ \begin{aligned} -\Delta_{p} u&=\frac{\lambda}{(u+\frac{1}{n})^\gamma}+ g(u)+\mu_n~\text{in}~\Omega,\\
u&=0~\text{on}~\partial\Omega,\\
u&>0~\text{in}~\Omega,
\end{aligned}\right.
\end{align}
whose solution will be denoted by $u_n$. The weak formulation to \eqref{e7} is defined as
\begin{equation}\label{weak}
\int_{\Omega} |\nabla u_n|^{p-2}\nabla u_n\cdot\nabla\phi dx=\lambda\int_{\Omega} \frac{\phi}{(u_n+\frac{1}{n})^\gamma}+\int_{\Omega} g(u_n)\phi dx+ \int_{\Omega} \mu_n\phi
dx, \,\forall\,\phi\in C_0^{1}(\bar{\Omega})
\end{equation}
where, ($\mu_n$) is a sequence of smooth non-negative functions bounded in $L^1(\Omega)$ and converging weakly to $\mu$ in the sense of Definition $\ref{measure}$. We now give the definition of weak solution to the problem ($P$) in \eqref{e0}.
\begin{definition}
We say a function $u\in W_{loc}^{1,p}(\Omega)\cap L^{\infty}(\Omega)$ is a weak solution to the problem (\ref{e0}) if $\cfrac{\phi}{u^\gamma}\in L^1(\Omega)$ and it satisfies
\begin{equation}\label{e1}
\int_\Omega |\nabla u|^{p-2}\cdot\nabla u\cdot\nabla \phi~ dx=\lambda\int_\Omega \frac{\phi}{u^\gamma}dx +\int_\Omega g(u)\phi dx + \int_\Omega \phi d\mu
\end{equation}
for every $\phi\in W_0^{1,p}(\Omega^{'})$ with $\Omega^{'}\subset\subset \Omega$.
\end{definition}
\noindent In the subsequent section, we will prove a few lemmas which will be required to prove our main result in Section $\ref{main}$. Note that the solution will be named as $u_n$ in multiple places for different problems.
\section{Important Lemmas}\label{lemmas}
In this section we will prove a few important lemmas, Lemma \eqref{l1} - \eqref{l6}, which are the main tools needed to prove the main result of existence of solution to the problem \eqref{e0}.
\begin{lemma}\label{l1}
The problem
\begin{eqnarray}\label{e2}
\begin{split}
-\Delta_p u&= \cfrac{\lambda}{(u+\frac{1}{n})^\gamma}~\text{in}~\Omega,\\
u&=0~\text{on}~\partial\Omega,
\end{split}
\end{eqnarray}
possesses a nonnegative weak solution in $W_{loc}^{1,p}(\Omega)\cap L^{\infty}(\Omega)$ for each $n\in \mathbb{N}$.
\end{lemma}
\begin{proof}
The idea of the proof is to apply Schauder's fixed point argument. For a fixed $n\in \mathbb{N}$ and a fixed $v\in L^p(\Omega)$, we define the map $J_\lambda:W_0^{1,p}(\Omega)\rightarrow \mathbb{R},$ as follows,
$$J_\lambda(u)=\dfrac{1}{p}\int_\Omega |\nabla u|^p dx -\lambda\int_\Omega\cfrac{u}{(|v|+\frac{1}{n})^\gamma} dx.$$
It is easy to see that, $J_\lambda$ is continuous, coercive and strictly convex in $W_0^{1,p}(\Omega).$ Therefore, the existence of a unique minimizer $w\in W_0^{1,p}(\Omega)$ corresponding to a $v\in L^p(\Omega)$ is certain.\\
We define, $H: L^p(\Omega)\rightarrow L^p(\Omega)$ by $$H(v)= (-\Delta_p)^{-1} \left[\frac{\lambda}{(|v|+\frac{1}{n})^\gamma}\right]:=w.$$
On choosing $w$ as a test function from $W_0^{1,p}(\Omega)$ in the weak formulation of \eqref{e2}, we have
\begin{align*}\label{e3}
\begin{split}
\int_\Omega|\nabla w|^p = \int_\Omega |\nabla w|^{p-2} \nabla w\cdot\nabla w&=\int_\Omega \cfrac{\lambda}{(|v|+\frac{1}{n})^\gamma} w \\
&\leq\lambda n^{\gamma} \int_\Omega |w|.
\end{split}
\end{align*}
Hence, by using the Poincar\'{e} inequality and the H\"{o}lder's inequality on the left and right hand side respectively, we get
\begin{equation}\label{e4}
\|w\|_p\leq C(n,\gamma,\lambda).
\end{equation}
Let us consider a sequence $(v_k)$ that converges to $v$ in $L^p(\Omega)$. By using the dominated convergence theorem, we have \begin{center}
$\left\|\cfrac{\lambda}{(|v_k| +\frac{1}{n})^\gamma}-\cfrac{\lambda}{(|v| +\frac{1}{n})^\gamma}\right\|_{L^p(\Omega)}\longrightarrow 0.$
\end{center}
Thus, the convergence of $w_k=H(v_k)$ to $w=H(v)$ in $L^p(\Omega)$ can be followed from the uniqueness of the weak solution. Hence, the continuity of $H$ over $L^p(\Omega)$ is followed. By the estimate in equation $\eqref{e4}$ and by the Rellich-Kondrochov theorem, we get that $H(L^p(\Omega))$ is relatively compact in $L^p(\Omega)$. We now can apply the Schauder's fixed point theorem to guarantee the existence of a fixed point say $w$. By the regularity theorem of Lieberman \cite{Lieberman}, we have $u_n\in C^1(\bar{\Omega}), \forall n\in \mathbb{N}$. Using the strong maximum principle \cite{Guedda}, we have $w>0$ in $\Omega$ and this concludes the proof.
\end{proof}
\begin{lemma}\label{l2}
The sequence $(u_n)$ is increasing w.r.t n and for every $K\subset\subset\Omega$, there exists $C_K$ (only depends on $K$) such that $u_n\geq C_K>0$, a.e. in $K$ with $||u_n||_\infty \leq R {\lambda}^{\frac{1}{\gamma+p-1}},~\forall n\in\mathbb{N}$, $R$ is independent of $n$.
\end{lemma}
\begin{proof}
Consider a sequence of problems
\begin{align}\label{e5}
\begin{split}
-\Delta_p u&= \cfrac{\lambda}{(u +\frac{1}{n})^\gamma} ~\text{in}~\Omega,\\
u&=0~\text{on}~\partial\Omega.\\
\end{split}
\end{align}
For each $n$, let $u_n$ be the solution to the problem \eqref{e5}. Consider,
$$ \int_\Omega (|\nabla u_n|^{p-2}\cdot\nabla u_n-|\nabla u_{n+1}|^{p-2}\cdot\nabla u_{n+1})\cdot\nabla\phi~ dx=\lambda\int_\Omega \left((u_n +\frac{1}{n})^{-\gamma}-(u_{n+1} +\frac{1}{n+1})^{-\gamma}\right)\phi~ dx.$$
We choose, the test function $\phi=(u_n-u_{n+1})^+$ to obtain,
$$\int_\Omega (|\nabla u_n|^{p-2}\cdot\nabla u_n-|\nabla u_{n+1}|^{p-2}\cdot\nabla u_{n+1})\cdot \nabla(u_n-u_{n+1})^+ dx$$
$$ \hspace{6cm} \leq \lambda\int_\Omega \left((u_n +\frac{1}{n+1})^{-\gamma}-(u_{n+1} +\frac{1}{n+1})^{-\gamma}\right)(u_n-u_{n+1})^+ dx.$$
Using the inequalities from \cite{grey}, we get for $p\geq2$,
\begin{align*}
\int_\Omega (|\nabla u_n|^{p-2}\nabla u_n-|\nabla u_{n+1}|^{p-2}\cdot\nabla u_{n+1})\cdot \nabla(u_n-u_{n+1})^+ dx &\geq C_p ||\nabla(u_n-u_{+n+1})^+||^p\\
&\geq 0
\end{align*}
and for $1<p<2$,
\begin{align*}
\int_\Omega (|\nabla u_n|^{p-2}\nabla u_n-|\nabla u_{n+1}|^{p-2}\nabla u_{n+1})\cdot \nabla(u_n-u_{n+1})^+ dx &\geq C_p \frac{||u_n-u_{+n+1}||^2}{(||u_n||+||u_{+n+1}||)^{2-p}}\\
&\geq 0.
\end{align*}
\noindent Therefore, we have
$$0 \leq \lambda\int_\Omega \left\{\left( u_n +\frac{1}{n+1}\right)^{-\gamma}-\left( u_{n+1} +\frac{1}{n+1}\right)^{-\gamma}\right\}(u_n-u_{n+1})^+ dx \leq 0.$$
Hence, we get $||(u_n-u_{n+1})^+||=0$. This implies $u_n$ is monotonically increasing w.r.t $n$. Now, using the Strong Maximum principle \cite{Vazquez}, we get $u_1>0$ in $\Omega$, where $u_1$ is the solution of the problem $\eqref{e5}$ with $n=1$. Since, $u_n$ is monotonically increasing with respect to $n$, we have $u_n>u_1$ in $\Omega$ and hence we conclude that $u_n>C_K>0$, for every $K\subset\subset\Omega$ with $C_K$ being independent of $n$.\\
$\textit{Claim:}$ $(u_n)$ is uniformly bounded in $\Omega.$\\
{\it Case 1:} When $\lambda=1$. Define, $M(k)=\{x\in\Omega:u_n>k\}$ and $$S_k(u_n)= { \left\{
\begin{array}{ll}
u_n-k; & \text{if}~u_n>k \\
0; &\text{if}~u_n\leq k.
\end{array}
\right. }$$
We choose, $S_k(u_n)$ as the test function in the weak formulation of \eqref{e5} to get,
\begin{align}\label{e6}
\int_{M(k)}|\nabla u_n|^{p-2}\nabla u_n.\nabla u_n & = \int_{M(k)}|\nabla u_n|^p
\nonumber\\& = \int_{M(k)}\frac{u_n-k}{(u_n+\frac{1}{n})^\gamma}\nonumber\\
&<\int_{M(k)}\frac{u_n-k}{u_n^\gamma}\nonumber
\\&\leq ||u_n-k||_{L^p(M(k))} |M(k)|^{\frac{1}{p\prime}}\nonumber\\& \leq C ||\nabla u_n||_{L^p(M(k))}|M(k)|^{\frac{1}{p\prime}},~~~~\text{( by the Poincar\'{e} inequality)}.\nonumber \end{align}
By using the Sobolev embedding theorem, we get
$$||u_n||^{p-1}_{L^{p^*}(M(k))}< \frac{C}{S^{p-1}}|M(k)|^{\frac{1}{p\prime}},~ \text{where}~ p^*=\frac{Np}{N-p}~\text{(Sobolev conjugate of}~ p).$$
It is easy to see that, for $1<k<l, ~M(l)\subset M(k)$. Hence,
$$|M(l)|\leq \left\{\frac{C}{S^{p-1}}\right\}^{\frac{p^*}{p-1}}\frac{1}{(l-k)^{p^*}}|M(k)|^{\frac{p^*}{p}} .$$
By the Lemma 4.1 of \cite{Stampacchia}, we can guarantee the existence of a $T>0$ independent of $n$ such that $|M(T)|=0$. Therefore, $||u_n||_\infty\leq T.$ \\
{\it Case 2:} Suppose $v$ is such that
\begin{eqnarray}
\int_{\Omega}|\nabla v|^{p-2}\nabla v\cdot\nabla \phi &<& \lambda\int_{\Omega}\frac{\phi}{v^{\gamma}}~\forall\lambda\in W_0^{1,p}(\Omega),\, \phi\geq 0.\label{weak_eqn1}
\end{eqnarray}
Let $\lambda>0$. Choose $v=(\frac{1}{\lambda})^{\frac{1}{\gamma+p-1}}w$. We can see that $v$ satisfies
$$\int_{\Omega}|\nabla v|^{p-2}\nabla v.\nabla\phi < \int_\Omega\frac{\phi}{v^\gamma},~\forall\phi\in W_0^{1,p}(\Omega), ~ \phi>0.$$
Therefore, using the result from {\it Case 1}, for $\lambda=1$, we have
$||v||_\infty\leq T $, which implies that $\|u_n\|_\infty \leq R {\lambda}^{\frac{1}{\gamma+p-1}}.$ Hence, $(u_n)$ is uniformly bounded in $\Omega$. Finally, on using a result due to Lieberman \cite{Lieberman}, helps us to conclude that $u_n\in C^1(\Omega)$, $\forall n\in\mathbb{N}$.
\end{proof}
\begin{lemma}\label{l3}
Every bounded nontrivial solution $v$ of the problem $-\Delta_p u=g(u) +\mu_n$ in $\Omega$, is uniformly bounded below in $L^\infty(\Omega)$, i.e. $\|v\|_{\infty}>\delta$, for some $\delta>0$.
\end{lemma}
\noindent We instead first prove the following lemma.
\begin{lemma}\label{ll3}
Every bounded nontrivial solution $u$ of the problem $-\Delta_p u=g(u)$ in $\Omega$, is uniformly bounded below in $L^\infty(\Omega)$, i.e. $\|u\|_{\infty}>\delta$, for some $\delta>0$.
\end{lemma}
\begin{proof}
Let us consider a sequence of nontrivial solutions $(u_m)$ such that $||u_m||_{\infty}\rightarrow 0$ as $m\rightarrow \infty$. Then we can define $w_m(x)=u_m(x)||u_m||_{\infty}^{-1}.$ Clearly, $\|w_m\|_{\infty}=1$. As $u_m$ satisfies $-\Delta_p u=g(u)$, we have
\begin{align}
\Delta_p w_m & = \Delta_p (u_m(x)||u_m||_{\infty}^{-1}) \nonumber\\
& = \nabla(|\nabla (u_m(x)||u_m||_{\infty}^{-1})|^{p-2} \nabla (u_m(x)||u_m||_{\infty}^{-1}))\nonumber\\
&= \Delta_p u_m||u_m||_{\infty}^{1-p}\nonumber\\
&= g(u_m)||u_m||_{\infty}^{1-p}\nonumber\\ & \leq C u_m^q||u_m||_{\infty}^{1-p} \nonumber\\
&\leq C w_m^q||u_m||_{\infty}^{1-p+q} \nonumber\\
&= f_m.\nonumber
\end{align}
Now for very large $m$, these $f_m$'s are uniformly bounded in $L^{\infty}(\Omega)$. So, $||w_m||_{C^{1,\beta}(\bar\Omega)}\leq M$ for some $\beta\in (0,1)$, by regularity results in \cite{Tolksdorf}, where $M$ is independent of $m$. Hence, by the Ascoli-Arzela theorem, the sequence $(w_m)$ converges uniformly to $w$ in $C_0^1(\Omega)$. This implies $w=0$. But with the consideration of the Lemma 1.1 of \cite{Azizieh Cle}, we have a unique solution $w$ in $C_0^1(\Omega)$, which contradicts the fact that $||w_m||_{\infty}=1$. Hence, there exists $\delta>0$ such that $\|u\|_{\infty}>\delta$.
\end{proof}
\noindent{\it{Proof of Lemma \ref{l3}.}} Since $\mu_n\geq 0$, then the solutions of the problem in Lemma \ref{l3} are supersolutions of the problem in Lemma \ref{ll3}. Therefore, if $v$ and $u$ are solutions of the problem in Lemma \ref{l3} and Lemma \ref{ll3} respectively, then $||v||_\infty\geq ||u||_\infty>\delta>0$, for some $\delta>0$.
\begin{lemma}\label{l4}
There exists a $\bar{\lambda}>0$ such that the following problem
\begin{eqnarray}\label{ne7}
-\Delta_p u&=& \cfrac{\lambda}{(u+\frac{1}{n})^\gamma} +g(u)+\mu_n~\text{in}~\Omega,\nonumber\\
u&=&0~\text{on}~\partial\Omega,\\
u&>&0~\text{in}~\Omega\nonumber
\end{eqnarray}
does not have any weak solution $u\in W_0^{1,p}(\Omega)$ for $\lambda\geq\bar{\lambda}$.
\end{lemma}
\begin{proof}
Let $\lambda_1$ be the first eigenvalue of the operator $-\Delta_p$ and its corresponding eigenfunction $\phi_1\geq 0$ be such that
\begin{eqnarray}
-\Delta_p \phi_1&=& \lambda_1\phi_1^{p-1}~\text{in}~\Omega\nonumber,\\
\phi_1&=&0~\text{on}~\partial\Omega.\nonumber
\end{eqnarray}
Its weak formulation with the test function $\phi=\phi_1$ is given by
$$\int_{\Omega}|\nabla \phi_1|^p=\lambda_1\int_{\Omega}\phi_1^p.$$
Let $u_n$ be the weak solution of $\eqref{e7}$, then by the strong maximum principle \cite{Vazquez}, we get $\frac{\phi_1^p}{u_n^{p-1}}\in W_0^{1,p}(\Omega)$. On applying the Picone's Identity (Theorem 2.1 in \cite{Bal}), we have
\begin{align*}
&\int_{\Omega} |\nabla\phi_1|^p dx-\int_{\Omega} \nabla(\frac{\phi_1^p}{u_n^{p-1}})|\nabla u_n|^{p-2} \nabla u_n dx \geq 0\\
\Rightarrow &\int_{\Omega} \lambda_1\phi_1^p- \frac{\phi_1^p}{u_n^{p-1}}\cfrac{\lambda}{(u_n+\frac{1}{n})^\gamma}-g(u_n) \frac{\phi_1^p}{u_n^{p-1}}-\mu_n \frac{\phi_1^p}{u_n^{p-1}} dx \geq 0\\
\Rightarrow&\int_{\Omega} \left(\lambda_1 u_n^{p-1}- \lambda (u_n+\frac{1}{n})^{-\gamma}-g(u_n)-\mu_n\right)\phi_1^p dx \geq 0.
\end{align*}
Consider $\bar{\lambda}$ defined as $\bar{\lambda}=\underset{x\in\Omega}{\max}\cfrac{\lambda_1 u_n^{p-1}- g(u_n)-\mu_n}{(u_n+1)^{-\gamma}}$. Now for every $\epsilon>0$, there exists a $\delta>0$ such that $v^q < \epsilon v^{p-1}, \forall v\in[0,\delta]$. Therefore, $\bar{\lambda}>0$ for some $\epsilon$ and for $\lambda\geq\bar{\lambda}$, we have
\begin{align}\lambda&\geq \max_{x\in\Omega}\frac{\lambda_1 u_n^{p-1}- g(u)-\mu_n}{(u+1)^{-\gamma}}\nonumber\\
&\geq \frac{\lambda_1 u_n^{p-1}- g(u)-\mu_n}{(u+\frac{1}{n})^{-\gamma}}\nonumber\\
&\Rightarrow\left(\lambda_1 u_n^{p-1}-\lambda \left(u+\frac{1}{n}\right)^{-\gamma}-g(u)-\mu_n\right)<0
\end{align}
which is a contradiction to our assumption. Hence, for $\lambda\geq\bar{\lambda}$, the problem $\eqref{e7}$ does not possess any solution $u\in W_0^{1,p}(\Omega)$.
\end{proof}
\begin{lemma}\label{l5}
Let $\Omega$ be a strictly convex domain and $u_n$ be a solution of problem $\eqref{e7}$. Then there exists $M>0$, which does not depend on $n$, such that $||u_n||_\infty\leq M$.
\end{lemma}
\begin{proof}
We divide the proof of this lemma into six steps.\\
\textbf{Step 1 (Uniform H\"{o}pf Lemma)}. Our aim is to show that $\frac{\partial u_n}{\partial n}(x)< c <0$ for any $n\in \mathbb{N}$, where $c$ is some constant which is independent of $n$ but depends on $x$. $\hat{n}$ is the unit outward normal to the boundary $\partial\Omega$ at the point $x$.\\
Now $\Omega$ satisfies the interior ball condition as it has a $C^2$ boundary, i.e. for some $x_0\in \partial\Omega$, there exists a $B_r(y)\subset\Omega$ such that $\partial B_r(y)\cap \partial\Omega =\{x_0\}$. Let us define $v: B_r(y)\rightarrow \mathbb{R}$ given by $$v(x)=[2^{\frac{N-p}{p-1}}-1]^{-1}r^{\frac{N-p}{p-1}}|x-y|^{\frac{p-N}{p-1}} - [2^{\frac{N-p}{p-1}}-1]^{-1}.$$
We observe that,
\begin{enumerate}
\item[(i)] $v(x)= 1$ on $\partial B_{\frac{r}{2}}(y)$ and $v(x)=0$ on $\partial B_r(y)$, and
\item[(ii)] if $x\in B_r(y)\setminus B_{\frac{r}{2}}(y)$ with $|\nabla v(x)|> c >0$ for some constant $c$ independent of $n$.
\end{enumerate}
Therefore, we have $0< v(x)<1$. Let us define $m=\inf\{u_n(x)|x\in \partial B_{\frac{r}{2}}(y)\}.$ By using the Lemma $\ref{l2}$, we can conclude that $m>0$ and is independent of $n$. on choosing $w=mv$, we see that $w$ satisfies
\begin{eqnarray}\label{e8}
-\Delta_p w&=& 0~\text{in}~ B_r(y)- \overline{B_{\frac{r}{2}}(y)} \nonumber,\\
w&=&m~\text{if}~ x\in\partial B_{\frac{r}{2}}(y),\nonumber\\
w&=&0~\text{if} x\in \partial B_r(y)\nonumber.
\end{eqnarray}
We have $u_n\geq w$ on the boundary of $B_r(y)- \overline{B_{\frac{r}{2}}(y)}$ and $-\Delta_p w\leq -\Delta_p u_n$ in $\Omega$. Hence, by the weak comparison principle, we have $u_n\geq w$ in $B_r(y)- \overline{B_{\frac{r}{2}}(y)}$. Since, $u_n(x_0)=w(x_0)=0$, then from the properties of $v$ in (i) and (ii) above, we obtain
\begin{align}
\frac{\partial u_n}{\partial \hat{n}}(x_0) & =\lim_{t\rightarrow 0} \frac{u_n(x_0-t\hat{n})}{t} \leq \lim_{t\rightarrow 0}\frac{w(x_0-t\hat{n})}{t}\nonumber\\
& = \frac{\partial w}{\partial\hat{n}}(x_0)= m \frac{\partial w}{\partial\hat{n}}< -c <0,\nonumber
~\text{where $c>0$ is independent of $n$.}
\end{align}
\textbf{Step 2 (Existence of a neighbourhood of the boundary which does not contain any critical points of $u_n$)}. Let us denote $C(u_n)=\{x\in\Omega:\nabla u_n(x)=0\}$, as the set of critical points of $u_n$. From Step 1, we have $ \frac{\partial u_n}{\partial \eta}<0$ on the boundary. Hence, $dist(\partial\Omega, C(u_n))=b_n>0,~\forall\, n\in \mathbb{N}$ as $\partial\Omega$ and $C(u_n)$ are compact subsets in $\Omega$.\\
\textbf{Claim:} There exists $\epsilon>0$, independent of $n$, such that $b_n>\epsilon>0$. In other words there exists a neighbourhood $\Omega_\epsilon=\{x\in\Omega: dist(x,\partial\Omega)<\epsilon\}$, such that $C(u_n)\cap\Omega_{\epsilon}=\phi$.\\
{\it{Proof.}} We prove this by a contrapositive argument. Let there does not exist any such $\epsilon>0$ such that $C(u_n)\cap\Omega_{\epsilon}\neq\phi$. Then there exists $x_n\in C(u_n)$ such that $dist(x_n,\partial\Omega)\rightarrow 0$ as $n\rightarrow\infty$. Therefore, upto a subsequence $x_{n_k}\rightarrow x_0$ and $x_0\in \partial\Omega$. But from Step 1, we obtain $ \frac{\partial u_n}{\partial \eta}(x_0) <c< 0$. Hence, there exists $l>0$ such that $|\nabla u_n(x)|>\frac{c}{2}$ for $x\in B_l(x_0)\cap\Omega$, where $c$ is independent of $n$. This implies that $B_l(x_0)\cap C(u_n)=\phi$. This is a contradiction, since we can find $x_{n_0}\in B_l(x_0)\cap\Omega$ such that $\nabla u_{n_0}(x_{n_0})=0$. Hence the claim.
\newpage
\noindent\textbf{Step 3 (Monotonicity of $u_n$)}. Let $e\in \mathbb{S}^{N-1}, \delta\in \mathbb{R}$, then for a fixed $n\in \mathbb{N}$, we define the following
\begin{enumerate}
\item[(i)] The hyperplane $\mathbb{L}_{\delta,e}=\{x\in \mathbb{R}^N:x.e=\delta\}$ and $\mathbb{\sigma}_{\delta,e}=\{x\in \mathbb{R}^N:x.e<\delta\}$.
\item[(ii)] $\hat{x}$ be the reflection of $x$ with respect to the hyperplane $\mathbb{L}_{\delta,e}$ i.e. $\hat{x}=x+2(\delta-x.e)e.$
\item[(iii)] $a(e)=\underset{x\in \Omega}{\inf} \{x.e\}$ and the reflected cap of $\sigma_{\delta,e}$ with respect to $\mathbb{L}_{\delta,e}$ for any $\delta>a(e)$ denoted as $\hat{\sigma}_{\delta,e}$.
\item[(iv)] $\hat{\sigma}_{\delta ,e}$ is not internally tangent to $\partial\Omega $ at some point $p\notin\mathbb{L}_{\delta ,e}$.
\item[(v)] $\hat{n}(x)$ be the unit inward normal to $\partial\Omega$ at $x$, then $\hat{n}(x).e\neq0, \forall x\in \partial\Omega\cap \mathbb{L}_{\delta ,e}$.
\item[(vi)] $\xi(e)=\{\mu_0>a(e): \forall\delta\in (a(e),\mu_0), \text{4 and 5 holds}\}.$ and $\bar{\xi}(e)=\sup\{\xi(e)\}.$
\end{enumerate}
If $\Omega$ is strictly convex, then the map $e\mapsto \bar{\xi}(e)$ is continuous by Proposition 2 of \cite{Azizieh Lema}. Let us denote $v_n(x)=u_n(\hat{x})$. Considering the strict convexity of $\Omega$ and the property (4), we see that $\hat{\sigma}_{\delta ,e}$ is contained in $\Omega$ for any $\delta\leq \delta_1$ where $\delta_1$ only depends on $\Omega$. Since, $\Delta_p$ is invariant under reflection and both $u_n$ and $v_n$ satisfy equation $\eqref{e7}$ hence both the functions take the same value on the hyperplane $\mathbb{L}_{\delta , e}$. Let us define $\delta_0=\min(\delta_1, \epsilon)$. Also for $x\in \partial\Omega\cap\partial\sigma_{\delta ,e}$, we have $u_n(x)=0$ and $v_n(x)=u_n(\hat{x})>0$ as $\hat{x}\in\Omega$. Therefore,
\begin{eqnarray}
-\Delta_p u_n+\cfrac{\lambda}{(u_n+\frac{1}{n})^\gamma}+g(u_n)+\mu_n&=& -\Delta_p v_n+\cfrac{\lambda}{(v_n+\frac{1}{n})^\gamma}+g(v_n)+\mu_n~\text{in}~\sigma_{\delta ,e}\nonumber\\
u_n&\leq&v_n~\text{on}~\partial\sigma_{\delta ,e}\cap\partial\Omega.\nonumber
\end{eqnarray}
Then $u_n\leq v_n$ in $\sigma_{\delta ,e}$ for any $\delta\in(a(e), \delta_0)$, by the comparison principle \cite{Damascelli}. Hence, $u_n$ is nondecreasing for all $x\in\sigma_{\delta_0,e}$ along the $e$-direction.\\
\noindent\textbf{Step 4} ({\bf Existence of a measurable proper subset of $\Omega$ of nonzero measure on which $u$ is nondecreasing}). For a fixed $x_0\in\partial\Omega$, let $e=e(x_0)$ be the unit outward normal to $\partial\Omega$ at $x_0$. Then by the results in Step 3, we conclude that $u_n$ is nondecreasing in the direction of $e$ for all $x\in\sigma_{\delta ,e}$ and $a(e)<\delta<\delta_0$. For any $\theta\in\mathbb{S}^{N-1}$ in a small neighbourhood of $e$, the reflection of $\sigma_{\delta ,\theta}$ w.r.t. $\mathbb{L}_{\delta,\theta}$ is a member of $\Omega$, since the domain is strictly convex and hence the sequence $u_n$ will be nondecreasing in the $\theta$ direction. Fix $\delta=\frac{\delta_0}{2}$. Since $\Omega$ is strictly convex, there exists a neighbourhood $\Theta\in\mathbb{S}^{N-1}$ such that $\sigma_{\frac{\delta_0}{2}, e}\subset\sigma_{\delta_0,\theta}$ for all $\theta\in\Theta$. Thus, we can conclude that $u_n$ is nondecreasing in every direction for $\theta\in\Theta$ and for any $x$ with $x\cdot e<\frac{\delta_0}{2}$.\\
Consider $$\sigma_0=\left\{x\in\Omega : \frac{\delta_0}{8}<x\cdot e<\frac{3\delta_0}{8}\right\}.$$ Obviously, $\sigma_0\subset\sigma_{\frac{\delta_0}{2}, e}$ and $u_n$ is nondecreasing in every direction $\theta\in\Theta$ and $x\in\sigma_0$. Choose $\epsilon=\frac{\delta_0}{8}$ and fix a point $x\in\Omega_\epsilon$. Let $x_0$ be the projection of the point $x$ onto $\partial\Omega$. We define $\mathbb{I}_x\subset\sigma_0$ to be the truncated cone having vertex at $x_0-\epsilon e$ and an opening angle $\frac{\theta}{2}$. Then $\mathbb{I}_x$ satisfies the following properties.
\begin{enumerate}
\item[(i)] $|\mathbb{I}_x|>k$ for some $k$, where $k$ depends only on $\Omega$ and $\epsilon$,
\item[(ii)] $u_n(x)\leq u_n(y)$ for all $y\in\mathbb{I}_x$ and $n\in\mathbb{N}$.
\end{enumerate}
Then, we have $u_n(x)\leq u_n(x_0-\epsilon e)\leq u_n(y)$, for all $y\in\mathbb{I}_x$.\\
\noindent\textbf{Step 5 (A boundary `estimate')}. Let us consider the first eigenfunction $\phi_1$ of the $p$-Laplacian eigenvalue problem over $\Omega$. Using the Picone's identity on $\phi_1$, $u_n$ and then applying the strong maximum principle \cite{Vazquez}, we have $\frac{\phi_1^p}{u_n^{p-1}}\in W_0^{1,p}(\Omega)$. Denote $f_n(u_n)=\frac{\lambda}{(u_n+\frac{1}{n})^\gamma}+\mu_n$. Then, we have
\begin{eqnarray}\label{e9}
\begin{split}
\int_\Omega\cfrac{[f_n(u_n)+g(u_n)]\phi_1^p}{u_n^{p-1}}&=\int_\Omega |\nabla u_n|^{p-2}\nabla u_n\cdot\nabla \left(\frac{\phi_1^p}{u_n^{p-1}}\right)\\&\leq\int_\Omega|\nabla \phi_1|^p dx\\
&\leq C(\Omega).
\end{split}
\end{eqnarray}
Let $\phi_1(z)\geq\xi>0$ for all $z\in\Omega-\Omega_{\frac{\epsilon}{2}}$. Hence, from $\eqref{e9}$, we have
$$\xi^p \int_{\Omega-\Omega_{\frac{\epsilon}{2}}} \cfrac{[f_n(u_n)+g(u_n)]}{u_n^{p-1}}\leq C(\Omega).$$ This implies
$$\int_{\mathbb{I}_x}\cfrac{[f_n(u_n)+g(u_n)]}{u_n^{p-1}}\leq\frac{C(\Omega)}{\xi^p}.$$
Now since,
\begin{eqnarray}
\int_{\mathbb{I}_x}\cfrac{[f_n(u_n)+g(u_n)]}{u_n^{p-1}}\geq\int_{\mathbb{I}_x} g(u_n) u_n^{1-p}(z)dz\geq u_n^{q-p+1}(x)|\mathbb{I}_x|
\end{eqnarray}
we have
$$u_n^{q-p+1}(x)\leq\frac{C_1(\Omega)}{\xi^p},$$ for some constant $C_1>0,$ i.e. $u_n(x)\leq C'$, for all $x\in\Omega_\epsilon$ and for all $n\in\mathbb{N}.$\\
\noindent\textbf{Step 6 (Blow-up analysis)}. We will show that for every open set, $K\subset\subset\Omega,$ there exists $C_K>0$ such that $\|u_n\|_\infty<C_K,$ for every solution $u_n$ of $\eqref{e7}.$ We will prove it by contrapositive argument. Suppose, there exist a sequence $(u_n)$ of positive solutions of the problem $\eqref{e7}$ and a sequence of points $(Z_n)\subset\Omega$ such that $M_n=u_n(Z_n)=\max\{u_n(x):x\in\bar{K}\}\rightarrow\infty$ as $n\rightarrow\infty.$ Using the boundary estimates one can assume that $Z_n\rightarrow x_0$ as $n\rightarrow\infty$, where $x_0\in \bar{K}$.
Let $dist(\bar{K},\partial\Omega) =2d$ and $\Omega_d=\{x\in\Omega : dist(x,\Omega)<d\}.$\\
Let $R_n$ be the sequence of positive real numbers with $R_n^{\frac{p}{q-p+1}}M_n=1.$ Observe that $M_n\rightarrow\infty$ iff $R_n\rightarrow 0$ as $n\rightarrow\infty.$ Define, $w_n:B_{\frac{d}{R_n}(0)\rightarrow\mathbb{R}}$ such that $$w_n(y)=R_n^{\frac{p}{q-p+1}}u_n(Z_n+R_n y).$$ Now $u_n$ has a maximum at $Z_n$, hence we have $\|w_n\|_\infty=w_n(0)=1.$ Since $R_n\rightarrow 0$ there exists $n_0 $ such that $B_R(0)\subset B_{\frac{d}{R_n}}(0)$ for fixed $R>0.$ Again, we have that $w_n$ satisfies the following
$$\nabla w_n(y)=R_n^{\frac{p}{q-p+1}+1}\nabla u_n(Z_n+R_n y)$$ and
$$-\Delta_p w_n(y)=R_n^{\frac{pq}{q-p+1}}[\lambda f_n(u_n(Z_n+R_n y))+R_n^{\frac{-pq}{q-p+1}} w_n^q(Z_n+R_n y) +R_n^{\frac{-pq}{q-p+1}}\mu_n(Z_n+R_n y)].$$
From Lemma $\eqref{l1}$ and Lemma $\eqref{l3}$, for any $y\in B_R(0)$, we have $Z_n+R_n y\in\bar{\Omega}_d\subset\Omega$ and
\begin{align}\label{inequa1}
\begin{split}
&R_n^{\frac{pq}{q-p+1}}[\lambda f_n(u_n(Z_n+R_n y))+R_n^{\frac{-pq}{q-p+1}} w_n^q(Z_n+R_n y) +R_n^{\frac{-pq}{q-p+1}}\mu_n(Z_n+R_n y)]\leq C(\bar{\Omega}_d),
\end{split}
\end{align} for every $n\geq n_0.$ Let us fix a ball $B$ such that $\bar B\subset B_{\frac{d}{R_n}}(0),~\forall~n\geq n_0.$ Then by the interior estimates of Lieberman \cite{Lieberman} and Tolksdorf \cite{Tolksdorf}, we have the existence of a constant $C=C(N,p,B)>0$ and $\beta=\beta(N,p,B)\in(0,1)$ such that
$$w_n\in C^{1,\beta}(\bar B)~ \text{and}~ ||w_n||_{1,\beta}\leq C.$$
Using the Arzela-Ascoli theorem, we guarantee the existence of a function $w\in C^1(\bar B)$ such that there exists a convergent subsequence $w_n\rightarrow w$ in $C^1(\bar B).$ On passing the limit $n\rightarrow\infty,$ we have
\begin{align*}
&\int_B|\nabla w|^{p-2}\nabla w\cdot\nabla\phi\geq C\int_B w^q\phi, ~\forall~\phi\in C_c^\infty(B),~ w\in C^1(\bar B),~ w\geq 0 ~\text{on}~\bar B,
\end{align*}
where the constant is obtained from the growth condition over $g$ and the condition in \eqref{inequa1}. Also, we have $||w||_\infty=1.$ Hence, by using the strong maximum principle \cite{Vazquez}, we have $w(x)>0,$ $\forall~x\in B$. Now for a sequence of balls with increasing radius, the Cantor diagonal subsequence converges to $w\in C^1(\mathbb{R}^N)$, on every compact subsets of $\mathbb{R}^N$ and satisfy the following
\begin{align*}
&\int_{\mathbb{R}^N}|\nabla w|^{p-2}\nabla w\cdot\nabla\phi\geq C\int_{\mathbb{R}^N} w^q\phi; ~\forall~\phi\in C_c^\infty(\mathbb{R}^N),~w\in C^1(\mathbb{R}^N),~ w>0 ~\text{on}~\mathbb{R}^N.
\end{align*}
This contradicts the Theorem $\ref{t1}$.
\end{proof}
\begin{lemma}\label{l6}
For a strictly convex domain $\Omega$, there exists $\bar\lambda>0$ such that for $0<\lambda<\bar{\lambda}$ and $\gamma>0$ atleast two solutions (say $u_n,v_n$) exist for the problem $\eqref{e7}$ in $W_{loc}^{1,p}(\Omega)$.
\end{lemma}
\begin{proof}
We define $\bar{J_\lambda}:C(\bar{\Omega})\rightarrow C(\bar{\Omega})$ by $$\bar{J_\lambda}(u)=(-\Delta_p)^{-1}\left(\frac{\lambda}{(u+\frac{1}{n})^\gamma}+g(u)+\mu_n\right), ~\lambda\geq0.$$
Now equation $\eqref{e7}$ can be written as $u=\bar{J_\lambda}(u)$. The map $\bar{J_\lambda}$ is compact since, we know $(-\Delta_p)^{-1}$ is a compact operator on $C(\bar{\Omega})$. So, we assume the map $\bar{J_\lambda}$ is also compact. For $0<\lambda<\bar{\lambda}$, we have $(u_n)$ as solutions to the problem $\eqref{e7}$ and $||u_n||_\infty\leq M$, using Lemma $\ref{l4}$ and Lemma $\ref{l5}$. Let us define, $S_1=\{u\in C(\bar{\Omega}):u\geq0~ \text{in}~ \Omega\}$, $\bar{J_0}:S_1\rightarrow S_1$ by $ \bar{J_0}(u)=(-\Delta_p)^{-1}(g(u)+\mu_n)$ and $G:\bar{B_R}\times [0,\infty)\rightarrow S_1$ such that $G(u,\lambda)=\bar{J_\lambda}$.\\
$\textit{\bf Claim 1.}$ There exists a supersolution to the problem $\eqref{e7}$.\\
{\it Proof.} Let us define, $N(r)=\frac{1}{3}\left((\frac{r}{R})^{\gamma+p-1}-Cr^{\gamma+q}\right)$, for $r\in[0,\infty)$ where $R$ is the bound used in Lemma $\ref{l2}$ and $C>0$ is the constant used in the growth condition of $g$ and $\eta=\underset{0\leq r\leq\beta_0}{\max}N(r)$, where $\beta_0=\frac{1}{2} (2q-2p+3)^{\frac{1}{p-q-1}}R^{\frac{\gamma+p-1}{p-q-1}}.$\\
Observe that, $N(r)>0$ for $r\in(0,\beta_1)$, where $\beta_1\in(0,\min{(\gamma,\beta_0)}).$ Now applying the intermediate value property of continuous functions, we get that there exists a $\beta_2\in(0,\beta_1)$ such that $N(\beta_2)=\lambda_0.$ \\
Denote $\lambda^*=\left(\frac{\beta_2}{R}\right)^{\gamma+p-1}$. So,
\begin{align}
\lambda_0=N(\beta_2) & = \frac{1}{2}\left(\lambda^*-C\beta_2^{\gamma+q}\right)\nonumber\\
\lambda^*> \lambda_0+\beta_2^{\gamma+q} & =\lambda_0+C[R(\lambda^*)^{\frac{1}{\gamma+p-1}}]^{\gamma+q}\nonumber.
\end{align}
Let $u_{n,\lambda^*}$ satisfy $\eqref{e7}$. Then for $n\geq n_0$, we have
\begin{align}
\lambda^* & >\lambda_0 + C\left(\|u_{n,\lambda^*}\|_\infty\right)^q
\left(\|u_{n,\lambda^*}\|+\frac{1}{n}\right)^\gamma \nonumber \\&
>\lambda + C\left(u_{n,\lambda^*}\right)^q
\left(u_{n,\lambda^*}+\frac{1}{n}\right)^\gamma, ~\text{for}~ \lambda\leq\lambda_0 \\&
> \lambda + g(u_{n,\lambda^*}) \left(u_{n,\lambda^*}+\frac{1}{n}\right)^\gamma.
\end{align}
Hence,
$$-\Delta_p u_{n,\lambda^*}=\frac{\lambda^*}{(u_{n,\lambda^*}+\frac{1}{n})^\gamma}+\mu_n> \frac{\lambda}{(u_{n,\lambda^*}+\frac{1}{n})^\gamma}+\mu_n+g(u_{n,\lambda^*}), ~\text{for}~ \lambda\leq\lambda^* ~\text{and}~ n\geq n_0.$$
Therefore, $u_{n,\lambda^*}\in C^{1,\alpha}(\bar{\Omega})$ is a positive supersolution for some $\alpha>0$ and $u_{n,\lambda^*}$ is a supersolution of
\begin{eqnarray}
\label{mideq1}
-\Delta_p u&=&\frac{\lambda}{(u+\frac{1}{n})^\gamma}+g(u)+\mu_n,\nonumber\\
u&=& 0~\text{on}~\partial\Omega,
\end{eqnarray}
with $\|u_{n,\lambda^*}\|_\infty\leq \beta_2$.\\
$\textit{\bf Claim 2.}$ Problem $\eqref{e7}$ possesses a unique solution.\\
To prove the Claim 2, we define, $$f_n(x,r)=\frac{\lambda(r+\frac{1}{n})^{-\gamma}+g(r)}{r^{p-1}}, ~\text{for}~ r\in [0,\infty).$$
Now the derivative of $f_n$ w.r.t $r$ is given by
\begin{align}
f_n^\prime(x,r) &=\frac{1}{r^p}\left[\frac{\lambda\{(1-p-\gamma)r+\frac{1-p}{n}\}}{(r+\frac{1}{n})^{1+\gamma}}\right]+\frac{rg^\prime(r)-g(r)(p-1)}{r^p}\nonumber\\&
<\frac{1}{r^p}\left[\frac{\lambda[(1-p-\gamma)r+\frac{1-p}{n}]}{(r+\frac{1}{n})^{1+\gamma}}\right]+ (q-p+1)r^{q-p}.\nonumber
\end{align}
As the function $r^q(r+\frac{1}{n})^{1+\gamma}$ is convex, so there exists a unique $C_n>0$, which is increasing with respect to $\lambda$ such that
$$\lambda\left[(p+\gamma-1)C_n+\frac{p-1}{n}\right]> (q-p+1)C_n^{q}(C_n+\frac{1}{n})^{1+\gamma}.$$ Now for $r\leq C_n$, we have
$$(q-p+1)r^{q}(r+\frac{1}{n})^{1+\gamma}\leq\lambda\left[(p+\gamma-1)r+\frac{p-1}{n}\right].$$
Hence, $f_n^\prime(x,r)<0$. Consider $$F_n(x,r)= \frac{\lambda(r+\frac{1}{n})^{-\gamma}+g(r)+\mu_n}{r^{p-1}}, ~\text{for}~ r\in [0,\infty).$$
Clearly, $F_n^\prime(x,r)=f_n^\prime(x,r)-\frac{\mu_n(p-1)}{r^p}<0.$ Therefore, $F_n$ is decreasing and using the result of D\'{i}az-Sa\'{a} \cite{Diaz}, we guarantee that the problem $\eqref{e7}$ has unique solution and $||u_n||_\infty\leq C_n$.\\
Thus, we have $\beta_2\leq\delta_0$. So, $$\frac{q-p+1}{\gamma+p-1}\beta_2^{\gamma+q}<\lambda_0,~\text{for}~ \gamma>1.$$
Choose $$\lambda_m=\frac{\{(q-p+1)(\beta_2+\epsilon)^{q}-\mu_m(p-1)\} (\beta_2+\epsilon+\frac{1}{m})^{1+\gamma}}{(p+\gamma-1)(\beta_2+\epsilon)+\frac{p-1}{m}}<\lambda_0,$$
then for all $n\geq m$, we have $C_n(\lambda_0)\geq C_n(\lambda_n)=\beta_2+\epsilon.$ So, $||u_n||_\infty\leq\beta_2+\epsilon.$\\
We can see that using Lemma $\ref{l3}$, Lemma $\ref{l4}$ and Lemma $\ref{l5}$, $\bar{J_0}$ and $G$ satisfy all the conditions of Lemma \eqref{figueiredo} taken from \cite{Figueiredo} for some $0<r<\beta_2<R$. Since $\beta_2<\alpha$, $(I-\bar{J_0})(u)$ has no solution on $\partial B_r$. Now considering Lemma \ref{l4} and using Lemma \ref{arcoya lemma} of \cite{Ambrosetti}, we can obtain a continuum $A_n\subset A=\{(\lambda,u)\in[0,\bar{\lambda}]\times C(\bar{\Omega}): u-\bar{J_\lambda}(u)=0\}$ such that
\begin{equation}\label{eq2}
A_n\cap (\{0\}\times B_r)\neq \phi,~ A_n\cap (\{0\}\times (B_R-B_r))\neq \phi.
\end{equation}
Next, we define $F:[0,\lambda_0]\rightarrow C_0^{1,\alpha}(\bar{\Omega})$ a continuous map such that $F(\lambda)=u_{n,\lambda^*}$. Using Lemma $\ref{l8}$, we conclude that there exists $u_n\in A_n^{\lambda_0}=\{u\in C(\bar{\Omega}):(\lambda_0,u)\in A_n\}$ such that $0<u_n<u_{n,\lambda^*}$. We have $||u_{n,\lambda^*}||_\infty\leq\beta_2$ and hence $||u_n||_\infty\leq ||u_{n,\lambda^*}||_\infty\leq \beta_2.$\\
We have $A_n\cap(\{0\}\times(B_R-B_r))\neq\phi$ by equation $\eqref{eq2}$. Hence, for $n\geq\max(n_0,m)$, there exists $v_n$ such that $||v_n||_\infty\geq \beta_2+\epsilon.$ For $\lambda=\lambda_0$ we have at least two solutions $u_n$ and $v_n$ to the problem $\eqref{e7}$. As $\lambda_0<\bar{\lambda}$ is arbitrary, it concludes the proof.
\end{proof}
\begin{theorem}
Given $\gamma>0$ there exists $\bar{\lambda}>0$ such that the problem \eqref{e0} admits atleast two solutions $u$, $v$ in $W_{loc}^{1,p}(\Omega)$, provided $\Omega$ is strictly convex with $1<p<N$, $p-1<q<\frac{p(N-1)}{N-p}-1$ and for $0<\lambda<\bar{\lambda}$.
\end{theorem}
\begin{proof} From the above Lemma $\ref{l6}$, we can conclude the existence of atleast two solutions $u_n$ and $v_n$ of the problem $\eqref{e7}.$ Also for a suitable choice of $c>0, ~~\underbar u=(c\phi_1+n^{\frac{1+p-\gamma}{p}})^{\frac{p}{\gamma +p-1}}-\frac{1}{n}$ will be a weak subsolution to the problem $\eqref{e2}$ for $\lambda=\lambda_0.$\\
Again, using $\frac{\lambda_0}{(r+\frac{1}{n})^\gamma}\leq \frac{\lambda_0}{(r+\frac{1}{n})^\gamma}+g(r)+\mu_n$ for all $r\geq 0$ we can conclude that each solutions of the problem $\eqref{e7}$ with $\lambda=\lambda_0$ is a weak supersolution of $\eqref{e2}.$ Now by the strong comparison principle \cite{Guedda}, we have
\begin{align}\label{e10}
\bar u\leq u_{n,\lambda_0}\leq u_n\leq\beta_2,~ \bar u\leq u_{n,\lambda_0}\leq v_n ~\text{and}~ ||v_n||_\infty\geq\beta_2+\epsilon.
\end{align}
Let us take $z_n=u_n$ or $v_n$, then from $\eqref{e10}$ and the Lemma $\ref{l5}$ we have, $$\bar u\leq z_n\leq M,$$ where $M$ is independent of $n.$ By using the strong comparison principle \cite{Guedda} and Lemma $\ref{l2}$, we have
\begin{align}\label{e11}
\forall\,K\subset\subset\Omega,~\exists~C_K ~\text{such that}~ z_n\geq C_K>0 ~\text{in}~ K,~\forall\, n\in\mathbb{N} .
\end{align}
\textbf{Claim.} $(z_n)$ is bounded in $W_{loc}^{1,p}(\Omega).$
{\it Proof.} Consider $z_n\phi^p$ as a test function in the equation $\eqref{e7}$ for $\phi\in C_0^1(\Omega)$, then we get
\begin{align*}
\int_\Omega|\nabla z_n|^p\phi^p=-p\int_\omega\phi^{p-1}z_n|\nabla z_n|^{p-2}\nabla\phi\cdot\nabla z_n+\int_\Omega\frac{\lambda_0 z_n\phi^p}{(z_n+\frac{1}{n})^\gamma}+\int_\Omega z_ng(z_n)\phi^p+\int_\Omega z_n\mu_n
\end{align*}
By using the modified Young's inequality we have, $\int_\Omega|\nabla z_n|^p\phi^p\leq C_\phi~\forall~n\in\mathbb{N}$, where $C_\phi$ is a constant depending only on $\phi.$ Hence, $z_n\in W_{loc}^{1,p}(\Omega)$ and there exists $z\in W_{loc}^{1,p}(\Omega)\cap L^\infty(\Omega)$ such that $z_n\rightarrow z$ a.e upto a subsequence and $z_n\rightarrow z$ weakly in $W^{1,p}(K)$ for all $K\subset\subset\Omega.$ From the Theorem 4.4 of \cite{Canino}, $\int_\Omega|\nabla u_n|^{p-2}\nabla u_n\cdot\nabla\phi$ converges to $\int_\Omega|\nabla u|^{p-2}\nabla u\cdot\nabla\phi.$\\ Again, by using dominated convergence theorem, we have $$\lim_{n\rightarrow\infty}\int_\Omega\left(\frac{\lambda_0\phi}{(z_n+\frac{1} {n})^\gamma}+\phi~ g(z_n)\right)dx=\lambda_0\int_\Omega\frac{\phi}{z^\gamma}dx+ \int_\Omega\phi~ g(z)dx$$
Since, $||u_n||_\infty\leq\beta_2,~ ||v_n||_\infty\geq\beta_2+\epsilon>\beta_2
$ and $u_n\rightarrow u, v_n\rightarrow v$, we have the existence of two distinct solutions $u$ and $v.$
\end{proof}
\noindent We will now prove the existence result of the problem $\eqref{e0}$.
\section{Existence result}\label{main}
\subsection{The case of $\gamma < 1.$}\label{sub1}
Let us consider the problem in $\eqref{e7}$ for the case of $\gamma<1$.
\begin{lemma}\label{ml1}
Let $u_n$ be a solution of $\eqref{e7}$ with $\gamma<1$. Then $(u_n)$ is bounded in $W_0^{1,r}(\Omega)$ for every $r<\frac{N(p-1)}{N-1}$.
\end{lemma}
\begin{proof}
We will prove the boundedness of ($\nabla u_n$) in the Marcinkiewicz space $\mathcal{M}^\frac{N(p-1)}{N-1}(\Omega)$. For this, let us take $\varphi =T_k(u_n)$ as a test function in the weak formulation $\eqref{weak}$ and we have
\begin{equation}\label{me1}
\int_\Omega |\nabla T_k(u_n)|^p = \int_\Omega \frac{\lambda}{(u_n+ \frac{1}{n})^\gamma}T_k(u_n) +\int_\Omega g(u_n)T_k(u_n) + \int_\Omega T_k(u_n) \mu_n.
\end{equation}
Observe \hspace{0.3cm} $$\frac{T_k(u_n)}{(u_n+\frac{1}{n})^{\gamma}}\leq\frac{u_n}{(u_n+\frac{1}{n})^{\gamma}}=\frac{u_n^{\gamma}}{(u_n+\frac{1}{n})^{\gamma}u_n^{\gamma-1}}\leq u_n^{1-\gamma}$$ and
$$\int_\Omega T_k(u_n) \mu_n \leq k||\mu_n||_{L^1(\Omega)} \leq Ck. $$
Therefore, we have,
\begin{equation}\label{me2}
\int_\Omega |\nabla T_k(u_n)|^p \leq Ck.
\end{equation}
Now consider the following set inclusion
\begin{align*}
\{|\nabla u_n|\geq t\} & = \{|\nabla u_n|\geq t,u_n< k\} \cup \{|\nabla u_n| \geq t,u_n \geq k\}
\\& \subset \{|\nabla u_n|\geq t,u_n <k\} \cup \{u_n \geq k\}\subset \Omega.
\end{align*}
With the help of the subadditivity property of Lesbegue measure $m$ we have,
\begin{equation}\label{me3}
m( \{|\nabla u_n|\geq t\}) \leq m(\{|\nabla u_n|\geq t,u_n< k\}) + m(\{u_n \geq k\}).
\end{equation}
By the Sobolev inequality, we have
\begin{equation}\label{sobo}
\frac{1}{\lambda_1}\left(\int_\Omega |T_k(u_n)|^{p^*}\right)^{\frac{p}{p^*}}\leq \int_{\Omega}|\nabla T_k(u_n)|^p \leq Ck
\end{equation}
where $\lambda_1$ is the first eigenvalue of the $p$-Laplacian operator. Now, on restricting the left hand side of the integral \eqref{sobo} on $I=\left\lbrace x\in\Omega:u_n\geq k \right\rbrace$, such that $T_k(u_n)=k$, we obtain
\begin{align*}
& k^pm(\{u_n\geq k\})^{\frac{p}{p^*}}\leq Ck\\\Rightarrow~&m(\{u_n\geq k\})\leq \frac{C}{k^\frac{N(p-1)}{N-p}},~ \forall\,k\geq1.
\end{align*}
Hence, $(u_n)$ is bounded in $\mathcal{M}^{\frac{N(p-1)}{N-p}}(\Omega)$.\\
Similarly on restricting $\eqref{sobo}$ on $I^{'}=\{|\nabla u_n|\geq t,u_n< k\}$, we have
\begin{center}
$m(\{|\nabla u_n|\geq t,u_n< k\})\leq \frac{1}{t^p}\int_\Omega |\nabla T_k(u_n)|^p\leq \frac{Ck}{t^p}, \forall k>1. $
\end{center}
\noindent Now $\eqref{me3}$ becomes
$$ m( \{|\nabla u_n|\geq t\}) \leq m(\{|\nabla u_n|\geq t,u_n< k\}) + m(\{u_n \geq k\})\leq \frac{Ck}{t^p} + \frac{C}{k^\frac{N(p-1)}{N-p}}, \forall k>1.$$
Let us choose, $k=t^{\frac{N-p}{N-1}}$ and hence we get $$ m(\{|\nabla u_n|\geq t\})\leq \frac{C}{t^\frac{N(p-1)}{N-1}}, \hspace{0.2cm} \forall\,t\geq 1.$$
We have proved that $(\nabla u_n)$ is bounded in $\mathcal{M}^{\frac{N(p-1)}{N-1}}(\Omega)$. This implies by property $\eqref{marcin}$ that $(u_n)$ is bounded in $W_0^{1,r}(\Omega)$, for every $r<\frac{N(p-1)}{N-1}$.$\vspace{0.1cm}$
\end{proof}
\begin{theorem}\label{mt1}
Let $\gamma < 1$. Then there exists a weak solution $u$ of $\eqref{e0}$ in $W_0^{1,r}(\Omega)$ for every $r<\frac{N(p-1)}{N-1}$.
\end{theorem}
\begin{proof} Lemma $\ref{ml1}$, implies that there exists $u$ such that a subsequence of $u_n$ converges weakly to $u$ in $W_0^{1,r}(\Omega)$, for every $r<\frac{N(p-1)}{N-1}$. This implies that for $\varphi$ in $C_c^1(\Omega)$
$$\lim_{n\rightarrow +\infty} \int_{\Omega} \nabla u_n . \nabla\varphi = \int_{\Omega}\nabla u .\nabla\varphi.$$
Also due to the compact embeddings we can assume that $u_n$ converges to $u$ both strongly in $L^1(\Omega)$ and a.e. in $\Omega$. Thus, taking $\varphi$ in $C_c^1(\Omega)$, we get,
\begin{align*}
0 & \leq \big|\frac{\lambda}{(u_n+\frac{1}{n})^\gamma}\varphi\big|
\\& \leq C\lambda ||\varphi||_{L^\infty(\Omega)}
\end{align*}
This is sufficient to apply the dominated convergence theorem to obtain
$$\lim_{n\rightarrow +\infty} \int_{\Omega} \frac{\lambda}{(u_n+\frac{1}{n})^\gamma}\varphi = \int_{\Omega}\frac{\lambda}{u^\gamma}\varphi.$$
Further, since $(u_n)$ is bounded in $W_0^{1,r}(\Omega)$, we have by the compact embedding that $u_n\rightarrow u$ in $L^r(\Omega)$. By the same standard argument, there exists a subsequence that converge to $u$ uniformly except on a set of arbitrarily small Lebesgue measure. Since, by the hypothesis $g$ is continuous, the limit $n\rightarrow \infty$ can be passed on. On applying a similar argument as in step 4 of the Theorem 3.2 in \cite{Oliva}, we have a.e. convergence of the $\nabla u_n$ towards $\nabla u$ that follows in a standard way by proving that $\nabla T_k (u_n)$ goes to $\nabla T_k (u)$, in $L_{loc}^{r}(\Omega)$ for $r<p$, for every $k>0$. Finally, we can pass the limit $n\rightarrow\infty$ in the last term of $\eqref{weak}$ involving $\mu_n$.
This concludes the proof of the result as it is easy to pass to the limit in $\eqref{weak}$. Therefore, we obtain a weak solution of $\eqref{e0}$ in $W_0^{1,r}(\Omega)$ for every $r<\frac{N(p-1)}{N-1}$.
\end{proof}
\subsection{The case of $\gamma\geq1.$}\label{sub2}
Due to the strong singularity we can hold some local estimates on $u_n$ in the Sobolev space. We shall give global estimates on $T_k^{\frac{\gamma+p-1}{2}}(u_n)$ in $W_0^{1,2}(\Omega)$ with the aim of giving sense, at least in a weak sense, to the boundary values of $u$.
\begin{lemma}\label{ml2}
Let $u_n$ be a solution of $\eqref{e7}$ with $\gamma\geq1$. Then $T_k^{\frac{\gamma+p-1}{p}}(u_n)$ is bounded in $W_0^{1,p}(\Omega)$ for every fixed $k>0$.
\end{lemma}
\begin{proof}
Consider $\varphi=T_k^\gamma(u_n)$ as a test function in $\eqref{weak}$. We have
\begin{align}\label{me4}
\gamma\int_\Omega |\nabla u_n|^{p-2}\nabla u_n&\cdot\nabla T_k(u_n)T_k^{\gamma-1}(u_n)\nonumber\\ &=\int_{\Omega} \frac{\lambda}{(u_n+\frac{1}{n})^\gamma}T_k^\gamma(u_n) +\int_{\Omega} g(u_n)T_k^\gamma(u_n) + \int_{\Omega}T_k^\gamma(u_n)\mu_n.
\end{align}
We can estimate the term on the left hand side of $\eqref{me4}$ as,
\begin{equation}\label{me5}
\gamma\int_\Omega |\nabla u_n|^{p-2}\nabla u_n\cdot\nabla T_k(u_n)T_k^{\gamma-1}(u_n)=\gamma\int_\Omega |\nabla T_k^{\frac{\gamma+p-1}{p}}(u_n)|^p.
\end{equation}
As $\frac{T_k^\gamma(u_n)}{(u_n+\frac{1}{n})^\gamma}\leq \frac{u_n^\gamma}{(u_n+\frac{1}{n})^\gamma}\leq 1$, the term on the right hand side of $\eqref{me4}$ can be estimated as,
\begin{align}\label{me6}
\int_{\Omega} \frac{\lambda}{(u_n+\frac{1}{n})^\gamma}T_k^\gamma(u_n) +\int_{\Omega} g(u_n)T_k^\gamma(u_n) &+ \int_{\Omega}T_k^\gamma(u_n)\mu_n\nonumber\\
& \leq C\lambda k^\gamma +C\int_{\Omega} u_n^qT_k^\gamma (u_n)+ k^\gamma \int_{\Omega} \mu_n\nonumber
\\& \leq C\lambda k^\gamma +CM k^\gamma+ k^\gamma \int_\Omega \mu_n\nonumber
\\& \leq C(k,\gamma)k^\gamma.
\end{align}
On combining the previous inequalities $\eqref{me5}$ and $\eqref{me6}$ we get
\begin{equation}\label{me7}
\int_{\Omega} |\nabla T_k^{\frac{\gamma+p-1}{p}}(u_n)|^p \leq Ck^\gamma
\end{equation}
then, $\left(T_k^{\frac{\gamma+p-1}{p}}(u_n)\right)$ is bounded in $W_0^{1,p}(\Omega)$ for every fixed $k>0$.
\end{proof}
\noindent Now, so as to pass to the limit $n\rightarrow\infty$ in the weak formulation $\eqref{weak}$, we require to prove some local estimates on $u_n$. We first prove the following.
\begin{lemma}\label{ml3}
Let $u_n$ be a solution of $\eqref{e7}$ with $\gamma\geq1$. Then ($u_n$) is bounded in $W_{loc}^{1,r}(\Omega)$ for every $r<\frac{N(p-1)}{N-1}$.
\end{lemma}
\begin{proof}
We prove the theorem in two steps.\\
$\boldmath{\text{\bf Step 1.}}$ We claim that $\left(G_1(u_n)\right)$ is bounded in $W_0^{1,r}(\Omega)$ for every $r<\frac{N(p-1)}{N-1}$.\\
We can see that $G_1(u_n)=0$ when $0\leq u_n\leq 1$, $G_1(u_n)=u_n-1$, otherwise i.e when $u_n>1$. So $\nabla G_1(u_n)=\nabla u_n$ for $u_n>1$.\\
Now, we need to show that $\left(\nabla G_1(u_n)\right)$ is bounded in $\mathcal{M}^{\frac{N(p-1)}{N-1}}(\Omega)$, where $\mathcal{M}^{\frac{N(p-1)}{N-1}}(\Omega)$ is the Marcinkiewicz space. Then we have
\begin{align*}
\{|\nabla u_n|> t, u_n>1\} & = \{|\nabla u_n|> t,1<u_n\leq k+1\} \cup \{|\nabla u_n| > t,u_n > k+1\}
\\& \subset \{|\nabla u_n|> t,1<u_n\leq k+1\} \cup \{u_n > k+1\}\subset \Omega.
\end{align*}
Hence,
\begin{equation}\label{me8}
m( \{|\nabla u_n|> t,u_n>1\}) \leq m(\{|\nabla u_n|> t,1<u_n\leq k+1\}) + m(\{u_n > k+1\}).
\end{equation}
In order to estimate $\eqref{me8}$ we take $\varphi=T_k(G_1(u_n))$, for $k>1$, as a test function in $\eqref{e7}$.\\
We observe that $\nabla T_k(G_1(u_n))= \nabla u_n$ only when $1<u_n \leq k+1$, otherwise is zero, and $T_k(G_1(u_n))=0$ on $\{ u_n\leq 1\} $, we have
\begin{align*}
\int_\Omega |\nabla T_k(G_1(u_n))|^p & = \int_\Omega \frac{\lambda}{(u+\frac{1}{n})^\gamma}T_k(G_1(u_n)) + \int_{\Omega} g(u_n)T_k(G_1(u_n))+ \int_{\Omega}T_k(G_1(u_n))\mu_n \\& \leq C\lambda k + Ck\int_\Omega u_n^q +k\int_\Omega\mu_n
\\& \leq Ck
\end{align*}
and by restricting the above integral on $I_1={\left\lbrace 1<u_n\leq k+1 \right\rbrace}$, we get
\begin{align}
\int_{\left\lbrace 1<u_n\leq k+1 \right\rbrace} |\nabla T_k(G_1(u_n))|^p \nonumber& = \int_{\left\lbrace 1<u_n\leq k+1 \right\rbrace} |\nabla u_n|^p\nonumber \\& \geq \int_{\left\lbrace |\nabla u_n|>t, 1<u_n\leq k+1 \right\rbrace} |\nabla u_n|^p\nonumber \\&\geq t^p m(\{|\nabla u_n|> t,1<u_n\leq k+1\})\nonumber
\end{align}
so that, $$m(\{|\nabla u_n|> t,1<u_n\leq k+1\})\leq \frac{Ck}{t^p} \hspace{0.4cm}\forall k \geq 1.$$
According to $\eqref{me7}$ in the proof of Lemma $\ref{l2}$, one can see that
$$\int_{\Omega} |\nabla T_k^{\frac{\gamma+p-1}{p}}(u_n)|^p \leq Ck^\gamma \hspace{0.2cm} \text{for any}\hspace{0.2cm} k>1.$$
Therefore, from the Sobolev inequality $$\frac{1}{\lambda_1}\Bigg(\int_\Omega |T_k^{\frac{\gamma+p-1}{p}}(u_n)|^{p^*}\Bigg)^{\frac{p}{p^*}}\leq \int_{\Omega}|\nabla T_k^{\frac{\gamma+p-1}{p}}(u_n)|^p \leq Ck^{\gamma},$$ where, $\lambda_1$ is the first eigenvalue of the $p$-Laplacian operator.
Now, if we restrict the integral on the left hand side on $I_2=\left\lbrace u_n> k+1 \right\rbrace_{x\in\Omega} $, on which $T_k(u_n)=k$, we then obtain $$k^{\gamma+p-1}m(\{u_n>k+1\})^{\frac{p}{p^*}}\leq Ck^{\gamma},$$
so that $$m(\{u_n>k+1\})\leq \frac{C}{k^\frac{N(p-1)}{N-p}},~\forall\, k\geq1.$$
So, $(u_n)$ is bounded in $\mathcal{M}^{\frac{N(p-1)}{N-p}}(\Omega)$, i.e. $(G_1(u_n))$ is also bounded in $\mathcal{M}^{\frac{N(p-1)}{N-p}}(\Omega)$.\\
Now $\eqref{me8}$ becomes
\begin{eqnarray} m( \{|\nabla u_n|> t,u_n>1\})& \leq & m(\{|\nabla u_n|> t,1<u_n\leq k+1\}) + m(\{u_n > k+1\})\nonumber\\&\leq &\frac{Ck}{t^p} + \frac{C}{k^\frac{N(p-1)}{N-p}}, \forall k>1.\nonumber
\end{eqnarray}
We then choose, $k=t^{\frac{N-p}{N-1}}$ and we get $$ m(\{|\nabla u_n|> t,u_n>1\})\leq \frac{C}{t^\frac{N(p-1)}{N-1}} \hspace{0.2cm} \forall t\geq 1.$$
We just proved that $(\nabla u_n)=(\nabla G_1(u_n))$ is bounded in $\mathcal{M}^{\frac{N(p-1)}{N-1}}(\Omega)$. This implies by property $\eqref{marcin}$ that $(G_1(u_n))$ is bounded in $W_0^{1,r}$ for every $r<\frac{N(p-1)}{N-1}$.$\vspace{0.1cm}$\\
$\boldmath{\text{\bf Step 2.}}$ We claim that $T_1(u_n)$ is bounded in $W_{loc}^{1,r}(\Omega)$.\\
We have to examine the behavior of $u_n$ for small values of $u_n$ for each $n$. We want to show that for every $K\subset\subset \Omega$,
\begin{equation}\label{me9}
\int_K |\nabla T_1(u_n)|^p \leq C.
\end{equation}
We have already proved that $u_n\geq C_K>0$ on $K$ in Lemma $\ref{l2}$. We will use $\varphi=T_1^\gamma(u_n)$ as a test function in $\eqref{weak}$ to get
\begin{align}\label{me10}
\gamma\int_\Omega |\nabla u_n|^{p-2}\nabla u_n&\cdot\nabla T_k(u_n)T_k^{\gamma-1}(u_n)\nonumber\\ &=\int_{\Omega} \frac{\lambda}{(u_n+\frac{1}{n})^\gamma}T_k^\gamma(u_n) +\int_{\Omega} g(u_n)T_k^\gamma(u_n) + \int_{\Omega}T_k^\gamma(u_n)\mu_n \nonumber\\ &\leq C.
\end{align}
Now observe that
\begin{align}\label{me11}
\gamma\int_\Omega |\nabla u_n|^{p-2}\nabla u_n\cdot\nabla T_1(u_n) T_1^{\gamma-1}(u_n) &\geq\int_K |\nabla T_1(u_n)|^p T_1^{\gamma-1}(u_n)\nonumber\\ &\geq C_K^{\gamma-1}\int_{K}|\nabla T_1(u_n)|^p.
\end{align}
On combining $\eqref{me10}$ and $\eqref{me11}$ we get $\eqref{me9}$.
We completed the proof as $u_n=T_1(u_n)+G_1(u_n)$. Hence, ($u_n$) is bounded in $W_{loc}^{1,r}(\Omega)$ for every $r<\frac{N(p-1)}{N-1}$.
\end{proof}
\noindent Now, we can finally state and prove the existence result for $\gamma\geq 1$.
\begin{theorem}
Let $\gamma\geq1$. Then there exists a weak solution $u$ of $\eqref{e0}$ in $W_{loc}^{1,r}(\Omega)$ for every $r<\frac{N(p-1)}{N-1}$.
\end{theorem}
\begin{proof}
The proof of this theorem is a straightforward application of the Theorem $\ref{mt1}$ and using the results in Lemma $\ref{ml2}$ and Lemma $\ref{ml3}$.
\end{proof}
\begin{center}
${\textbf{Some Important results}}$
\end{center}
Define, $X=\{u\in C_0^{1,\alpha}(\bar{\Omega}):u(x)\geq0~\text{ in}~ \bar{\Omega}\}$ and let $\xi$ is a unit outward normal at $\partial\Omega$, then define $X_0=\{u\in C_0^{1,\alpha}(\bar{\Omega}):u(x)>0 ~\text{and}~ \frac{\partial u}{\partial\xi}(x)<0, ~\forall x\in \partial\Omega\}$. Clearly $X_0$ is the interior of $X$.
\begin{lemma}\label{l7}
If $u_1,u\in C_0^{1,\alpha}(\bar{\Omega})$ with $u_1\neq u$ and
$$-\Delta_p u_1> \frac{\lambda}{(u_1+\frac{1}{n})^\gamma} +g(u_1)+\mu_n,$$ $$-\Delta_p u= \cfrac{\lambda}{(u+\frac{1}{n})^\gamma} +g(u)+\mu_n,$$
then $(u_1-u)\notin \partial X$.
\begin{proof}
We prove this Lemma by contradiction. Suppose $(u_1-u)\in \partial X$. Then $u_1(x)\geq u(x)$. By Strong maximum principle \cite{Guedda}, we can obtain $(u_1-u)\in X_0$. But $X_0\cap\partial X=\phi$, for which we get a contradiction. Therefore, $u_1-u$ does not belong to $\partial X.$
\end{proof}
\end{lemma}
\begin{lemma}\label{l8}
Assume $I$ is an interval in $\mathbb{R}$ and $A=I\times C_0^{1,\alpha}(\bar{\Omega})$ is a connected set of solutions of $\eqref{e7}$. Define $F:I\rightarrow C_0^{1,\alpha}(\bar{\Omega})$ is continuous such that $F(\lambda)$ is a supersolution to the problem $\eqref{e7}$.\\
If $u_1\leq F(\lambda_1)$ in $\Omega$, $u_1\neq F(\lambda_1)$ for some $(\lambda_1,u_1)\in A$, then $u< F(\lambda)$ in $\Omega$, $\forall(\lambda,u)\in A$.
\begin{proof}
Let $Z:A\rightarrow C_0^{1,\alpha}(\bar{\Omega})$ is a continuous map such that $Z(\lambda,u)=F(\lambda)-u$. $A$ is connected, so by continuity $Z(A)$ is connected in $C_0^{1,\alpha}(\bar{\Omega})$.\\
Using Lemma $\ref{l7}$, $F(\lambda_1)-u_1=Z(\lambda_1,u_1)\notin\partial X$. Hence, $Z(\lambda_1,u_1)\in X_0$. So, $Z(A)\subset X_0,$ as $Z(A)$ is connected.\\
Therefore, $F(\lambda)-u>0$, which implies $F(\lambda)>u$, $\forall(\lambda,u)\in A$. Hence, we get our required result.
\end{proof}
\end{lemma}
\begin{lemma}\label{arcoya lemma}[Ambrosetti-Arcoya \cite{Ambrosetti}].
Given $X$ be a real Banach space with $U\subset X$ be open, bounded set. Let $a,b\in \mathbb{R}$ such that the equation $u-T(\lambda,u)=0$ has no solution on $\partial U$ for all $\lambda\in [a,b]$ and that $u-T(\lambda,u)=0$ has no solution in $\overline{U}$ for $\lambda=b$. Also let $U_1\subset U$ be open such that $u-T(\lambda,u)=0$ has no solution in $\partial U_1$ for $\lambda=a$ and $\deg(I-K_a, U_1, 0)\neq0
.$\\
Then there exists a continuum $C$ in $\sum=\left\{(\lambda, u)\in[a, b]\times X : u-T(\lambda,u)=0 \right\}$ such that $$C\cap(\{a\}\times U_1)\neq\emptyset~\text{and}~C\cap(\{a\}\times (U-U_1))\neq\emptyset.$$
\end{lemma}
\begin{theorem}\label{t1}[Mitidieri-Pohozaev \cite{Mitidieri}].
If $p-1<q<\frac{N(p-1)}{N-p}, p<N$ and $C>0$, then the problem $$\int_{\mathbb{R}^n} |\nabla u|^{p-2}\nabla u.\nabla\phi\geq C \int_{\mathbb{R}^n} u^q\phi;~\phi\in C_c^\infty (\mathbb{R}^n)$$ does not have any positive solution in $C^1(\mathbb{R}^n)$.
\end{theorem}
\begin{theorem}\label{figueiredo}[De Figueiredo et al. \cite{Figueiredo}].
Let $C$ be a cone in a Banach space X and $\phi:C \rightarrow C$ be a compact map such that $\phi(0)=0$. Assume that there exists $0<r<R$ such that
\begin{enumerate}
\item $x\neq t\phi(x)$ for $0\leq t \leq 1$ and $\|x\|=r$
\item a compact homotopy $F:\bar{B_R}\times[0,\infty)\rightarrow C$ such that $F(x,0)=\phi(x)$ for $\|x\|=R$, $F(x,t)\neq x$ for $\|x\|=R$ and $0\leq t<\infty$ and $F(x,t)=x$ has no solution for $x\in\bar{B_R}$ for $t\geq t_0$.
\end{enumerate}
Then if, $U=\{x\in C:r<\|x\|<R\}$ and $B_{\rho}=\{x\in C:\|x\|<\rho\}$ we have $deg(I-\phi,B_R,0)=0$, $deg(I-\phi,B_r,0)=1$ and $deg(I-\phi,U,0)=-1$.x
Juha
\end{theorem}
\section*{Acknowledgement}
Two of the authors, S. Ghosh and A. Panda, thanks for the financial assistantship received to carry out this research work from the Council of Scientific and Industrial Research (C.S.I.R.), Govt. of India and Ministry of Human Resource Development(M.H.R.D.), Govt. of India, respectively. This is also to declare that there are no financial conflict of interest whatsoever.
Finally the authors thank the anonymous referee(s) and the editor(s) for the constructive comments and suggestions.
|
{
"timestamp": "2019-07-10T02:14:00",
"yymm": "1804",
"arxiv_id": "1804.05590",
"language": "en",
"url": "https://arxiv.org/abs/1804.05590"
}
|
\section{Introduction}
Stirling's original formula for the asymptotics of $\ln z!$ has been obscured by the formula popularly known as ``Stirling's formula", namely
\begin{align}\label{1}
\ln z! \sim (z+\frac{1}{2})\ln z -z + \ln \sqrt{2\pi} + z \sum_{n\geq 1}\dfrac{B_{2n}}{2n(2n-1)}\cdot\dfrac{1}{z^{2n}} \\
\sim (z+\frac{1}{2})\ln z -z + \ln \sqrt{2\pi} + \dfrac{1}{12z}-\dfrac{1}{360z^{3}}+\mathcal{O}(\dfrac{1}{z^5})\>,
\end{align}
which was actually found by De Moivre after Stirling had found his (see, \textsl{e.g.},~\cite{bellhouse2011}). Stirling's original formula is
\begin{align}\label{2}
\ln z! \sim Z\ln Z- Z +\ln\sqrt{2\pi}-Z\sum_{n\geq 1}\dfrac{(1-2^{1-2n})B_{2n}}{2n(2n-1)Z^{2n}} \\
\sim Z\ln Z- Z +\ln\sqrt{2\pi} - \dfrac{1}{24Z}+\dfrac{7}{2880Z^3}-\mathcal{O}(\dfrac{1}{Z^5})\>.
\end{align}
where $Z=z+\frac{1}{2}$.\\
\\
As you can see here, the formulae are quite similar. Stirling's original formula in equation~\eqref{2} has been rediscovered several times. Some people call it De Moivre's formula! It seems to have been known to both Gauss and to Hermite (see \textsl{e.g.}~\cite{richard}). There is a discussion in~\cite{tweddle1984} of one such rediscovery in the physics literature; for a particularly ironic case where the rediscoverer claims the formula is ``both simpler and more accurate" than ``Stirling's formula", look at~\cite{spouge1994}. For a thorough exposition of Stirling's actual work see the original, as masterfully translated and annotated by Tweddle~\cite{tweddle1984}. \\
In this present work we give a short proof of equation~\eqref{2}, which we believe to be new, by deriving an apparently new formula that is similar to the following formula of Binet:
\begin{equation}
\ln z!=(z+\frac{1}{2})\ln z-z+\ln\sqrt{2\pi}+\int_{t=0}^{\infty}\dfrac{1}{t} \left( \dfrac{1}{t}-\dfrac{1}{e^t-1}\right) e^{-tz}dt
\end{equation}
which~\cite{whittaker} claims is valid for $\Re z>0$. We will see later that this is not quite true in the modern context. This classical formula is proved in, for example,~\cite{whittaker} and in~\cite{sasvari}. The new formula is quite similar, again using $Z=z+\frac{1}{2}$:
\begin{equation}\label{6}
\ln z!=Z\ln Z-Z +\ln\sqrt{2\pi}-\int_{t=0}^{\infty}\dfrac{1}{t}\left( \dfrac{1}{t}-\dfrac{1}{2\sinh\frac{t}{2}}\right) e^{-tZ}dt,
\end{equation}
and is valid for $\Re z>-\dfrac{1}{2}$ (again, we will adjust this caveat later). Formula \eqref{6} appears as ``Theorem $2$", without proof, in~\cite{BorweinCorless1999}.\\
In the modern computational world, a new proof of an old mathematical result is rarely of interest for its own sake, but see for instance~\cite{nplus1}. Indeed Stirling's original proof of equation \eqref{2} was algorithmic in nature and, apart from the use of ``recognition" to identify $\sqrt{2\pi}$ and the lack of a ``closed formula"--- \textsl{i.e.} a relationship to other numbers, the Bernoulli numbers--- Stirling's proof was entirely satisfactory. So why record these results? \\
We believe this formula is interesting for the following reasons. First, the rediscovery was identified as such by tracing patterns and citations in Google Scholar, and now there is some hope that the obscurity of the original formula can be lifted\footnote{Of course, there is no hope of changing the popular meaning of the name ``Stirling's formula''.}. Of course the mathematics history literature has it right, owing to the work of Tweddle, but still. Second, Stirling's original proof used what is now called ``Inverse Symbolic Computation," illustrating that a modern experimental technique worth investigation has significant historical roots. As an \textsl{homage} to Stirling we use the same technique in our `new' proof below. Finally, we test Stirling's original formula in a modern computational context by trying a nonlinear sequence acceleration technique, namely Levin's $u$-transform; this gives a surprisingly viable method, comparable in cost (for a given accuracy) to the methods discussed in~\cite{schmelzer}. The separate issue of the complexity of the computation of $\Gamma(1+z)$, $z!$, or $n!$ for $n \in \mathbb{N}$, is not addressed here. See for instance~\cite{peter},~\cite{richard} for entry into that literature. See also~\cite{hare} for the computation of $\Gamma(z)$.\\
Basic references for $\Gamma$ include the DLMF (chapter $5$), the Dynamic Dictionary, and~\cite{andrews1999}.
\section{Notation}
Here we use $z!$ and $\Gamma(z+1)$ interchangeably. As mentioned in~\cite{robgamma} the ``notation wars" and the annoyance of the continual nuisance of shifting by $1$ are amusing but not possible nowadays of resolution. We use $\ln$ for the natural logarithm because it's unambiguous and ingeniously, as pointed out by David Jeffrey offers a free location for a subscript, which we use as follows
\begin{equation}
\ln_k z=\ln z + 2\pi i k \>.
\end{equation}
The unsubscripted $\ln z$ has range $-\pi < \Im \ln z \leq \pi$, the principal branch in universal usage nowadays in computers. We write $\ln z!$ for $\ln(z!)$; \textsl{i.e.} the factorial has higher precedence. We discuss the function $\ln\Gamma(z)$ in detail below as the analytic continuation of $\ln(z-1)!$. This modern notation is in contrast to Stirling's, where he used $\ell,z$ to mean $\log_{10}z$. The factorial notation $!$ was apparently invented by Christian Kramp in $1808$; the $\Gamma$ notation was invented by Legendre, and although the shift by $1$ as apparently due to Euler himself~\cite{gronau2003gamma}, Legendre gets the blame for that, too.
\section{Divergent asymptotic series}
\bigskip
For a given sequence $\lbrace \phi_i(x) \rbrace$ where the $\phi_i(x)$'s are defined over a domain, one can define a formal series
$\sum_{i= 1}^\infty a_i\phi_i(x)$. The idea of asymptotic series is to define a formal series with special property on the underlying sequence such that its partial sums approximate a given function over the same domain even more closely as $x\rightarrow x_0$.
\\
\\
Assume $R$ is a domain and $\lbrace \phi_i(x) \rbrace$ is a sequence of functions defined over $R$. The sequence $\phi_i(x)$ is called an asymptotic sequence for $x \rightarrow x_0$ in $R$ if for each $i$, $\phi_{i+1}(x)= o(\phi_i(x))$ as $x \rightarrow x_0$. A simple
example is $\lbrace (x-x_0)^i \rbrace$ for $x \rightarrow x_0$. Recall that $f(x)=o(g(x))$ as $x\rightarrow\infty$ if
\begin{equation}
\forall c > 0 \;\; \exists N>0 \;\;\; \text{s.t.} \;\;\; |f(x)|<c|g(x)| \;\;\; \text{for} \;\;\; x>N \>.
\end{equation}
or (if $x_0$ is finite)
\begin{equation}
\forall c > 0 \;\; \exists \delta>0 \;\;\; \text{s.t.} \;\;\; |f(x)|<c|g(x)| \;\;\; \text{for} \;\;\; |x-x_0|<\delta \>.
\end{equation}
Now suppose $\lbrace \phi_i(x) \rbrace$ is an asymptotic sequence which is defined over a domain $R$ and $f(x)$ is defined over
$R$ as well. The formal series $\sum_{i= 1}^\infty a_i \phi_i(x)$ is said to be an asymptotic expansion (series) to $n$ terms of
$f(x)$ as $x \rightarrow x_0$ if
\begin{equation}
f(x) = \sum_{i= 1}^n a_i \phi_i(x) + o(\phi_{n+1}(x)) \,\,\, \text{as} \,\,\, x\rightarrow x_0 \>.
\end{equation}
The formal series $\sum a_i \phi_i$ will be called an asymptotic series. An asymptotic series can be divergent or convergent itself as $n\rightarrow \infty$. For more details see \textsl{e.g.}~\cite[Chapter 1]{erdelyi}.
\section{Tools}
\bigskip We will use Fubini's theorem, which justifies the interchange of order of iterated integrals of continuous functions, and we will use Watson's Lemma. Loosely speaking, Watson's Lemma allows the interchange of order of summation of a series and of integration even though the radius of convergence of the series is violated (leaving us with a divergent asymptotic series).\\
\begin{lem}[Watson's Lemma]~\cite{bender1999} and~\cite{copson}
Assume
$\alpha > -1$, $\beta >0$ and $b>0$.
If $f(t)$ is a continuous function on $[0,b]$ such that it has asymptotic series expansion
\begin{equation}
f(t) \sim t^{\alpha} \sum_{n=0}^\infty a_nt^{\beta n}, \,\,\, t \rightarrow 0^+ \>,
\end{equation}
(and if $b=+\infty$ then $f(t) <k\cdot e^{ct}\, (t\rightarrow +\infty)$ for some positive constants $c$ and $k$), then
\begin{equation}
\int_0^b f(t)e^{-xt}dt \sim \sum_{n=0}^\infty \dfrac{a_n \Gamma (\alpha + \beta n +1)}{x^{\alpha +\beta n+1}}, \,\,\, x \rightarrow +\infty
\end{equation}
\end{lem}
For a proof of Watson's lemma, see~\cite{bender1999}.\\
We will also use Gauss' formula
\begin{equation}
\dfrac{\Gamma'(z+1)}{\Gamma(z+1)}=\int_{t=0}^{\infty}\dfrac{e^{-t}}{t}-\dfrac{e^{-tz}}{e^t-1}dt \;\;\; \text{for} \;\;\; \Re z>0
\end{equation}
a proof of which can be found for example in~\cite{whittaker}. Alternatively, a more elementary proof can be found in~\cite{sasvari}.\\
The next mathematical tool we need comes from a Laplace transform; using $\xi+\dfrac{1}{2}$ instead of the more common symbol $s$, the Laplace transform of $1$ is
\begin{equation}
\int_{t=0}^{\infty}e^{-t(\xi+\frac{1}{2})}dt=\dfrac{1}{\xi+\frac{1}{2}}
\end{equation}
by direct integration. The integral converges if $\Re(\xi)>-\frac{1}{2}$. We can then prove the following lemma:\\
\begin{lem}[The logarithm lemma]\label{lemma}
For $\Re z > -1/2,$
\begin{equation}
\ln (z+\frac{1}{2})=\int_{t=0}^{\infty}\dfrac{e^{-t}}{t}-\dfrac{e^{-t(z+\frac{1}{2})}}{t}dt \>.
\end{equation}
\end{lem}
\bigskip
\begin{proof}
Integrate the Laplace transform with respect to $\xi$ from $\xi=\frac{1}{2}$ to $\xi=z$:
\begin{equation}
\int_{\xi=\frac{1}{2}}^{z}\dfrac{d \xi}{\xi+\frac{1}{2}}=\int_{\xi=\frac{1}{2}}^{z}\int_{t=0}^{\infty}e^{-t(\xi+\frac{1}{2})}dtd\xi
\end{equation}
Interchange the order of integration---by Fubini's Theorem this is valid---and since $\int e^{-t(\xi+\frac{1}{2})}d\xi=-\dfrac{e^{-t(\xi+\frac{1}{2})}}{t}$, we have
\begin{equation}
\ln(z+\frac{1}{2})-\ln(\frac{1}{2}+\frac{1}{2})=\int_{t=0}^{\infty}-\dfrac{e^{-t(z+\frac{1}{2})}}{t}+\dfrac{e^{-t(\frac{1}{2}+\frac{1}{2})}}{t}dt
\end{equation}
which proves the lemma.
\end{proof}
\bigskip
\section{The formula like Binet's}
\begin{thm}\label{2.3}
If $z> - \frac{1}{2}$,
\begin{equation}\label{formula17}
\ln z!=(z+\frac{1}{2})\ln (z+\frac{1}{2})-(z+\frac{1}{2})+\ln\sqrt{2\pi}-\int_{t=0}^{\infty}\dfrac{1}{t}\left( \dfrac{1}{t}-\dfrac{1}{2\sinh \frac{t}{2}}\right) e^{-t(z+\frac{1}{2})}dt
\end{equation}
\end{thm}
\begin{proof}
We start with Gauss' formula and switching to $\Gamma$ notation because the derivative $d\Gamma/dz$ is easily written $\Gamma'$,\\
\begin{equation}
\dfrac{\Gamma'(z+1)}{\Gamma(z+1)}=\int_{t=0}^{\infty}\dfrac{e^{-t}}{t}-\dfrac{e^{-tz}}{e^t-1}dt
\end{equation}
(see \textsl{e.g.}~\cite{whittaker}), and Lemma \ref{lemma}.\\
Rearranging Gauss' formula using $e^{t/2}-e^{-t/2}=2\sinh\frac{t}{2}$, \\
\begin{equation}
\dfrac{\Gamma'(z+1)}{\Gamma(z+1)}=\int_{t=0}^{\infty}\dfrac{e^{-t}}{t}-\dfrac{e^{-t(z+\frac{1}{2})}}{2\sinh\frac{t}{2}}dt
\end{equation}
Subtracting Lemma \ref{lemma},\\
\begin{equation}
\dfrac{\Gamma'(\xi+1)}{\Gamma(\xi+1)}-\ln (\xi+\frac{1}{2})=\int_{t=0}^{\infty}\dfrac{e^{-t(\xi+\frac{1}{2})}}{t}-\dfrac{e^{-t(\xi+\frac{1}{2})}}{2\sinh\frac{t}{2}}dt
\end{equation}
Integrating from $\xi=\alpha> -\frac{1}{2}$ to $\xi=z> -\frac{1}{2}$ and interchanging the order of integration using Fubini's theorem, we find (except for a branch issue that we take up later) that
$$\ln \Gamma(z+1)-\ln\Gamma(\alpha+1)-(z+\frac{1}{2})\ln (z+\frac{1}{2})+(z+\frac{1}{2})+(\alpha+\frac{1}{2})\ln(\alpha+\frac{1}{2})-(\alpha+\frac{1}{2})=$$
\begin{equation}
\int_{t=0}^{\infty}\dfrac{1}{t}\left(\dfrac{1}{t}-\dfrac{1}{2\sinh\frac{t}{2}}\right)e^{-t(\alpha+\frac{1}{2})}dt-\int_{t=0}^{\infty}\dfrac{1}{t}\left(\dfrac{1}{t}-\dfrac{1}{2\sinh\frac{t}{2}}\right) e^{-t(z+\frac{1}{2})}dt\>.
\end{equation}
We now need to evaluate the $\alpha$ integral. At $\alpha=0$ Maple and Mathematica can only find a numerical approximation; likewise at $\alpha=\frac{1}{2}$. The numerical approximation can be identified by (for instance) the Inverse Symbolic Calculator at CARMA\footnote{\textbf{https://isc.carma.newcastle.edu.au}. Remark: The ISC is currently down because a security flaw was found. Discussion is under way as to how or if this can be resolved.} (a proof is supplied in Remarks \ref{rmk5.3} and \ref{rmk5.4}.)
\begin{equation}
\int_{t=0}^{\infty}\dfrac{1}{t}\left(\dfrac{1}{t}-\dfrac{1}{2\sinh\frac{t}{2}}\right)e^{-t/2}dt=\frac{1}{2}\ln(\frac{\pi}{e})
\end{equation}
Simplification then yields our formula.\\
\begin{rmk}
According to~\cite{tweddle1984}, this may have been the method Stirling used to identify $\log_{10}\sqrt{2\pi}$, except of course all calculations were done by hand. Apparently, he simply recognized the number $0.39908$. Nowadays very few people could do that unaided, but with the ISC it's easy.
\end{rmk}
\begin{rmk}\label{rmk5.3}
In~\cite{sasvari} we find a trick that could be used to do this integral analytically; we leave this as an exercise.\\
If one desires an actual proof, one can use ``Stirling's formula" (by De Moivre) and leverage the tricky identification of $\sqrt{2\pi}$, as follows.
\end{rmk}
As $z\rightarrow \infty$,
\begin{equation}
\ln \Gamma(z+1)-(z+\frac{1}{2})\ln (z+\frac{1}{2})+(z+\frac{1}{2})\sim \ln\sqrt{2\pi}+\mathcal{O}(\frac{1}{z})\>.
\end{equation}
Therefore (since the second integral goes to $0$ as $z\rightarrow \infty$)
\begin{equation}
\ln\sqrt{2\pi}-\ln \Gamma(\alpha+1)+(\alpha+\frac{1}{2})\ln(\alpha+\frac{1}{2})-(\alpha+\frac{1}{2})
\end{equation}
\begin{equation}
=\int_{t=0}^{\infty}\dfrac{1}{t}\left(\dfrac{1}{t}-\dfrac{1}{2\sinh\frac{t}{2}}\right) e^{-t(\alpha+\frac{1}{2})}dt
\end{equation}
But this is, in fact, our desired theorem with $z=\alpha$.
\end{proof}
\begin{rmk}\label{rmk5.4}
This looks like a circular argument, but it is not. We have here used the $\sqrt{2\pi}$ from the formula popularly known as Stirling's formula, for which there are many proofs analytically (see \textsl{e.g.}~\cite{whittaker}).
\end{rmk}
\begin{cor}~\cite[p.~399]{levinson1970complex}
By analytic continuation, formula \eqref{formula17} holds for $\Re z\geq -1/2$, since the integral is convergent there.
\end{cor}
\section{Evaluation of $\Gamma$ using this divergent series}
\subsection{First attempts}
It has long been known that ``Stirling's approximation'' leads to a viable method to evaluate $\ln\Gamma(z)$. The basic idea is to use the asymptotic series to evaluate $\ln\Gamma(z+n)$ for some large $n$ (large enough that the series gives some accuracy) and then work down with the recursive formula
\begin{equation}
\ln\Gamma(z+n-1)= -\ln(z+n-1)+\ln\Gamma(z+n)
\end{equation}
until we have reached $\ln\Gamma(z)$. This naive idea is surprisingly effective. The point of discussion is just how large $n$ should be, and how many terms in ``Stirling's series'' one should retain, in order to make an effective formula. \\
Given that we now have a different asymptotic formula under consideration (the original, more accurate, but certainly not ``new'' formula) all of the discussion points are necessarily changed. Just as an example, take (say), $z=11+i/2$. If we want $\ln((11+i/2)!)$ then Stirling's original series gives
\begin{gather*}
\ln\sqrt{2\pi}+(11.5+i/2)\ln(11.5+i/2)-(11.5+i/2)-\dfrac{1}{24(11.5+i/2)}+\mathcal{O}(\dfrac{1}{z^3}) \\
=17.4914469445+1.22148819106i
\end{gather*}
Wolfram Alpha confirms this, giving
$$\ln((11+i/2)!)\doteq 17.4914485209+1.22148798i \>.$$
Rather than get into the minutiae of how many terms to take, and how far to push the argument to the right, we take a different tack: we look at automatic sequence acceleration of the original divergent series. If
\begin{equation}
S=\ln\sqrt{2\pi}+Z\ln Z-Z-Z\sum_{n\geq 1}\dfrac{(1-2^{1-2n})B_{2n}}{2n(2n-1)Z^{2n}}\>,
\end{equation}
then we wonder if simple execution of the Maple command
\begin{equation}
\texttt{evalf}\texttt{(Sum(}\texttt{a(n)}\texttt{,n=1..}\texttt{infinity))};
\end{equation}
where $a(n)$ is defined as $\dfrac{(1-2^{1-2n})B_{2n}}{2n(2n-1)Z^{2n}}$ will automatically produce an accurate result.
\begin{displayquote}
``Sometimes Maple knows things that you don't know. And then you wonder just what.''
\begin{flushright}
--Jon Borwein.
\end{flushright}
\end{displayquote}
\subsection{Levin's $u$-transform}
What Maple knows here is called Levin's $u$-transform. This is a method to accelerate convergence of the sequence of partial sums
\begin{equation}
S_n=\sum_{j=1}^{n}a_j
\end{equation}
of the series we consider. For an introduction to sequence acceleration, see~\cite{henrici1982} and~\cite{henrici1964}. For an introduction to Levin's $u$-transform, see~\cite{weniger}. \\
\\The basic idea is to replace the sequence $S_0, S_1, S_2,\cdots$ with a new one that has the same limit but which converges faster. More precisely, Levin's $u$-transform for $S_n$ is given as:
\begin{equation}
u_{k}^{(n)}(\beta , S_{n}) = \dfrac{\sum_{j=0}^{k} (-1)^{j} {k \choose j} \dfrac{(\beta + n + j)^{k-2}}{(\beta + n + k)^{k-1}}\dfrac{S_{n+j}}{a_{n+j}}}{\sum_{j=0}^{k} (-1)^{j} {k \choose j} \dfrac{(\beta + n + j)^{k-2}}{(\beta + n + k)^{k-1}}\dfrac{1}{a_{n+j}}}
\end{equation}
The parameter $\beta > 0$ is ``in principle completely arbitrary''~\cite{weniger}. In practice, Maple's routine chooses $\beta=1$.\\
For \textsl{irregular} sequence transforms such as Levin's $u$-transform, this may even transform divergent series into rapidly convergent ones. The price, however, is that it doesn't always work. It works well enough, though, that it is the default method coded in Maple~\cite{geddes1992}. It is accessed most simply by applying the ``evalf" command to an inert sum (denoted by capital-letter \texttt{Sum}). For instance,\\
\begin{equation}
\texttt{evalf}\texttt{(Sum((}\texttt{-2)}^\texttt{n}\texttt{,n=0..}\texttt{infinity))};
\end{equation}
yields $0.3333333333$\footnote{Correctly, in the sense of Euler summation, taking $1+r+r^2+\cdots=1/(1-r)$ even if $|r|>1$ by redefining what the infinite sum actually means: see \textsl{e.g.}~\cite{Hardy}, for more classical work on making sense of divergent series.}.\\
Other sequence acceleration methods or quadratures could be used (see for example chapter $28$ of~\cite{Trefethenbook}), but we wanted to show the capabilities of some (under-appreciated) off-the-shelf~tools.\\
If we issue the command (with a numerical value for $z$, say $z=11+i/2$)
\begin{lstlisting}
> evalf(-(z+1/2)*(Sum((1-2^(1-2*n))*$\mathrm{bernoulli}$(2*n)/
(2*n*(2*n-1)*(z+1/2)^(2*n)), n = 1 .. infinity))+
ln(sqrt(2*Pi))+ln(z+1/2)*(z+1/2)-(z-1/2);
\end{lstlisting}
we get $\ln((11+i/2)!)$ with full accuracy: $14$ digits if Digits $:=15$, $28$ digits if Digits $:=30$, $58$ digits if Digits $:=60$, and so on. This divergent series is being accurately, and quickly, summed by Maple's built-in sequence acceleration using the Levin $u$-transformation method above. \\
If we test this summation by looking at the error
\begin{equation}
\ln\Gamma(z+1)-\ln S(z)
\end{equation}
over a range $-20\leq \Re z\leq 20$, $-20\leq \Im z \leq 20$, we get the curious result in Figure~\ref{relerrRedBlue1}.
\begin{figure}[!h]
\centering
\includegraphics[scale=0.4]{relerrRedBlue1-eps-converted-to}
\caption{The region of utility for Levin's $u$-transform without an unwinding number.}
\label{relerrRedBlue1}
\end{figure}
\\
Everywhere in the red region (which includes the real axis for $x$ larger than about $2.1$) has full accuracy, whatever the setting of Digits. The region in white, in the middle, with its scalloped edges, is the region where Levin's $u$-transform fails and Maple returns an unevaluated Sum, as one can see in the example below:
\begin{lstlisting}
> Digits := 20:
\end{lstlisting}
\begin{maplegroup}
\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{Digits := 20}{\[\displaystyle {\it Digits}\, := \,20\]}
\end{maplelatex}
\end{maplegroup}
\begin{lstlisting}
> z := 1+.1*I:
\end{lstlisting}
\begin{maplegroup}
\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{z := 1.+.1*I}{\[\displaystyle z\, := \, 1.0+ 0.1\,i\]}
\end{maplelatex}
\end{maplegroup}
\begin{lstlisting}
> evalf(-(z+1/2)*(Sum((1-2^(1-2*n))*$\mathrm{bernoulli}$(2*n)/
(2*n*(2*n-1)*(z+1/2)^(2*n)), n = 1 .. infinity))+
ln(sqrt(2*Pi))+ln(z+1/2)*(z+1/2)-(z-1/2);
\end{lstlisting}
\begin{maplegroup}
\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{(-1.50000000000000-.1*I)*(Sum((1/2)*(1-2^(1-2*n))*bernoulli(2*n)/(n*(2*n-1)*(1.50000000000000+.1*I)^(2*n)), n = 1 .. infinity))+0.2380532679023624382e-1+0.4062048632794543180e-1*I}
\bigskip
{\[\displaystyle \left( - 1.50000000000000\\
\mbox{}- 0.1\,i\\
\mbox{} \right) \sum _{n=1}^{\infty }1/2\,{\frac { \left( 1-{2}^{1-2\,n} \right) {\it bernoulli}\left( 2\,n \right) }{n \left( 2\,n-1 \right) \\
\mbox{} \left( 1.50000000000000+ 0.1\,i\\
\mbox{} \right) ^{2\,n}}}\\
\bigskip
\bigskip
+ 0.02380532679023624382+ 0.04062048632794543180\,i\\
\mbox{}\]}
\end{maplelatex}
\end{maplegroup}
The boundary of this region is very curious, and we return to the proof of theorem \ref{2.3} to try to understand why. After staring at it for some time, we realize that the transition from
\begin{equation}
\dfrac{\Gamma'(z+1)}{\Gamma(z+1)} \;\;\; \text{to} \;\;\; \ln\Gamma(z+1)
\end{equation}
depends on the path that $\Gamma(\xi+1)$ takes as $\xi$ goes from $\xi=1/2$ to $\xi=z$ (a straight line in the $\xi$ variable). But $\Gamma(\frac{1}{2}+t(z-\frac{1}{2}))$ may cross the negative real axis (the branch cut for logarithm) several times as $t$ goes from $0$ to $1$. Writing our answers, as we do, as
\begin{equation}
\ln z! \sim Z\ln Z-Z+\ln\sqrt{2\pi}+Z\sum_{n\geq 1}\dfrac{(1-2^{1-n})B_{2n}}{2n(2n-1)(Z)^{2n}}
\end{equation}
obscures the fact that the imaginary part of the logarithm on the left is in $(-\pi,\pi]$ while the imaginary part on the right might be anything. To make this equation actually true, we must subtract a multiple of $2\pi i$. To force the imaginary part of $S$ into $(-\pi,\pi]$ there is only one choice: replace $S$ by
\begin{equation}
S-2\pi i \ensuremath{\mathcal{K}}(S)
\end{equation}
where $\ensuremath{\mathcal{K}} (z)=\left\lceil \dfrac{\Im z-\pi}{2\pi} \right\rceil$ is the unwinding number of $z$ (see~\cite{unwindHighm},~\cite{CorlessUnwind1} and~\cite{CorlessUnwind2}). This means that $\ln z! \sim S-2\pi i\ensuremath{\mathcal{K}}(S)$ not $\sim S$.\\
\begin{rmk}
As pointed out by a referee, this is because the sum $S$ is ``really" asymptotic to the analytic function $\ln\Gamma(z+1)$, obtained by analytic continuation of the function compostion $\ln(\Gamma(z+1))$ for $z>0$. See \textsl{e.g.}~\cite{hare} for details and for some simple formulae for $\ensuremath{\mathcal{K}}(S)$ in special cases.
\end{rmk}
When we plot the error $\ln z! - \ln S + 2\pi i \ensuremath{\mathcal{K}}(S)$ as in Figure~\ref{relerrRedb} we see that whenever the Levin's $u$-transform actually returns an answer, we have only roundoff error. We get essentially perfect accuracy\footnote{Except of course for rounding error. We do not attempt a numerical analysis here, which appears involved. The main difficulty is predicting the number of arithmetic operations.} everywhere to the right of the scalloped boundary in Figure~\ref{relerrRedb}. So far as we know, this result is new. Of course, the detailed accuracy needs a proof: we have only provided experimental evidence, here. What every mathematician wants is a guarantee that the acceleration will work, or a perfect description of just when it will fail. We do not have this. \\
However, when we plot the contours of the error $\ln z! - \ln S + 2\pi i\ensuremath{\mathcal{K}}(S)$ as in Figure~\ref{relerrContour2} we see that the Levin's $u$-transform works as well as could possibly be expected: the visible contours are all less than $10^{-28}$, when we work in $30$ Digits; clearly the error is zero up to roundoff. We have computed the error at ten thousand locations in the region $[0-1000i, 1000+1000i]$ and the maximum error was $10^{-27}$ (on a $100 \times 100$ grid).\\
\begin{figure}[!h]
\centering
\includegraphics[scale=0.3]{relerrRedb-eps-converted-to}
\caption{The region of utility for Levin's $u$-transform. We have essentially perfect accuracy (up to roundoff error) outside the region around the negative real axis and the ``lozenge of failure''. Curiously, the error increases gradually near the negative real axis.}
\label{relerrRedb}
\end{figure}
\begin{figure}[!h]
\centering
\includegraphics[scale=0.45]{relerrContour2-eps-converted-to}
\caption{$3D$ plot looking straight down of the error of $\ln z! - \ln S + 2\pi i\ensuremath{\mathcal{K}}(S)$. The errors are everywhere less than $10^{-27}$. We work in $30$ digits of precision.}
\label{relerrContour2}
\end{figure}
\subsection{Truncating the series without Levin's $u$-transform}
In this section, we plot the absolute estimate error of the truncated series $T$ (not using Levin's $u$-transform) $T-\ln(Z-1/2)!$ where
\begin{align}
T=u-2\pi i\ensuremath{\mathcal{K}}(u)
\end{align}
and $u=Z\ln(Z)-Z+\ln(\sqrt{2\pi})-\dfrac{1}{24Z}$. For different contours ($10^{-3}$ and $10^{-6}$), we get a very curious result as one can see in Figure~\ref{truncseries}. The error is small outside the keyhole contour. This is more the kind of error we expect from truncated asymptotic series. We see good accuracy even with very few terms. It may be surprising to see that the error is small even in parts of the left half plane, although not near the negative real axis.
\begin{figure}[!h]
\centering
\includegraphics[scale=0.45]{truncseries-eps-converted-to}
\caption{The absolute estimate error of the $T-\ln(Z-1/2)!$. The inner contour is at level $10^{-3}$, and the outer is $10^{-6}$. The truncation error is smaller outside each contour. We used Digits $=30$ and grid $=[600,600]$ in the construction of this figure. The ``bubbles'' and ``wiggles'' in this figure are unexplained.}
\label{truncseries}
\end{figure}
\section{Concluding Remarks}
The Gamma function and the factorial function, invented in the $1700$'s, have been very thoroughly studied. Richard Brent's article~\cite{brentarxiv} points out some facts, known to Hermite and to Gauss, that were not covered in the survey~\cite{robgamma}, which looked at about $100$ references. One learns therefore that it is difficult to claim a result (formula or proof) is truly new; we are worried in particular that Gauss knew of our Binet--like formula proved here.\\
Nonetheless we believe the proof and numerical experiments have some value in the modern literature. The appearance of the unwinding number in the asymptotic series (either Stirling's or De Moivre's) may also be of value for people who write programs to compute $z!$.
\bibliographystyle{line}
|
{
"timestamp": "2019-05-08T02:02:32",
"yymm": "1804",
"arxiv_id": "1804.05263",
"language": "en",
"url": "https://arxiv.org/abs/1804.05263"
}
|
\section{Introduction}
\label{sec:introduction}
Nowadays, diverse categories of sensors can be found in various wearable devices. Such devices are now being widely applied in multiple fields, such as Internet of Things \cite{kamburugamuve2015framework,zhang2017converting}. As a result, massive multimodal sensor data are being produced continuously. The question that how we can deal with these data efficiently and effectively has become a major concern.
Compared to conventional sensor data such as images and videos, these data are naturally formed as a 1-D signal, with each element representing one sensor channel accordingly. There are several challenges for such sensor data classification. First, most existing classification methods use domain-specific knowledge and thus may become ineffective or even fail in complex situations where multimodal sensor data are being collected \cite{bigdely2015prep}. For example, one approach that works well on IMU (Inertial Measurement Unit) signals may not be able to deal with EEG (Electroencephalography) brain signals. Therefore, an effective and universal sensor data classification method is highly desirable for complex situations. This framework is expected to have both efficiency and robustness over various sensor signals.
Second, the wearable sensor data carries far less information than texts and images. For example, a sample signal gathered by a 64-channel EEG equipment only contains 64 numerical elements. Hence, a more effective classifier is required to extract discriminative information from such limited raw data. However, maximizing the utilization of the given scarce data demands cautious preprocessing and a rich fund of domain knowledge.
Inspired by attention mechanism \cite{cavanagh1992attention}, we propose to concentrate on a focal zone of the signal to automatically learn the informative attention patterns for different sensor combinations. Here, the focal zone is a selection block of the signal with a certain length, sliding over the feature dimensions. Note that reinforcement learning has been shown to be capable of learning human-control level policy on a variety of tasks \cite{mnih2015human}. Then we exploit the reinforcement learning to discover the focal zone.
Moreover, considering that the signals in different categories may have different inter-dimension dependency \cite{markham1981land}, we propose to use the LSTM (Long Short-Term Memory \cite{gers1999learning,zhang2017eeg}) to exploit the latent correlation between signal dimensions. We propose a weighted average spatial LSTM (WAS-LSTM) classifier by exploring the dependency in sensor data.
The main contributions of this paper are as follows:
\begin{figure*}[!t]
\centering
\includegraphics[width=0.9\linewidth]{flowchart.pdf}
\caption{Flowchart of the proposed approach.
The focal zone $\mathbf{\bar{x}}_i$ is a selected fragment from $\mathbf{x}'_i$ to feed in the state transition and the reward model.
In each step $t$, one action is selected by the state transition to update $s_t$ based on the agent's feedback. The reward model evaluates the quality of the focal zone to the reward $r_t$.
The dueling DQN is employed to find the optimal focal zone $\mathbf{\bar{x}}^*_i$ which will be feed into the LSTM based classifier to explore the inter-dimension dependency and predict the sample's label $y'_i$. $FCL$ denotes Fully Connected Layer. The State Transition contains four actions: left shifting, right shifting, extend, and condense. The dashed line indicates the focal zone before the action while the solid line indicates the position of the focal zone after the action.}
\label{fig:workflow}
\end{figure*}
\begin{itemize}
\item We propose a selective attention mechanism for
sensor data classification using the spatial information only. The proposed method is insensitive to sensor types since it is capable of handling multimodal sensor data.
\item We apply deep reinforcement learning to automatically select the most distinguishable features, called focal zone, for multimodal sensor data of different sensor types and combinations. We design a novel objective function as the award in reinforcement learning task to optimize the focal zone. The new reward model saves more than 98\% training time of the deep reinforcement learning.
\item We propose Weighted Average Spatial LSTM classifier to capture the cross-dimensional dependency in multimodal sensor data.
\end{itemize}
\section{Proposed Method}
\label{sec:methodology}
Suppose the input sensor data can be denoted by $\mathbf{X}=\{(\mathbf{x}_i, y_i), i=1,2,\cdots I\}$ where $(\mathbf{x}_i, y_i)$ denotes the 1-D sensor signal, called one \textit{sample} in this paper, and $I$ denotes the number of samples. In each sample, the feature $\mathbf{x}_i\in \mathbb{R}^K$ contains $K$ elements and the corresponding ground truth $y_i\in \mathbb{R}$ is an integer denotes the sample's category. $\mathbf{x}_i$ can be described as a vector with $K$ elements, $\mathbf{x}_i=\{x_{ik},k=1,2,\cdots, K\}$.
The proposed algorithm is shown in Figure~\ref{fig:workflow}. The main focus of the algorithm is to exploit the latent dependency between different signal dimensions. To this end, the proposed approach contains several components: 1) the replicate and shuffle processing; 2) the selective attention learning; 3) the sequential LSTM-based classification. In the following, we will first discuss the motivations of the proposed method and then introduce the aforementioned components in details.
\subsection{Motivation}
\label{sub:motivation}
How to exploit the latent relationship between sensor signal dimensions is the main focus of the proposed approach. The signals belonging to different categories are supposed to have different inter-dimension dependent relationships which contain rich and discriminative information. This information is critical to improve the distinctive signal pattern discovery.
In practice, the sensor signal is often arranged as 1-D vector, the signal is less informative for the limited and fixed element arrangement. The elements order and the number of elements in each signal vector can affect the element dependency.
In many real-world scenarios, the multimodal sensor data are associated with the practical placement. For example, the EEG data are concatenated following the distribution of biomedical EEG channels.
Unfortunately, the practical sensor sequence, with the fixed order and number, may not be suitable for inter-dimension dependency analysis. Meanwhile, the optimal dimension sequence \cite{tan2015lstm} varies with the sensor types and combinations. Therefore, we propose the following three techniques to amend these drawbacks.
First, we replicate and shuffle the input sensor signal vector on dimension-wise in order to provide as much latent dependency as possible among feature dimensions (Section~\ref{sub:repeat_and_shuffle}).
Second, we introduce a focal zone as a selective attention mechanism, where the optimal inter-dimension dependency for each sample only depends on a small subset of features. Here, the focal zone is optimized by deep reinforcement learning which has been proved to be stable and well-performed in policy learning (Section~\ref{sub:attention_pattern_learning}).
Third, we propose the WAS-LSTM classifier by extracting the distinctive inter-dimension dependency (Section~\ref{sub:classification}).
\subsection{Data Replicate and Shuffle}
\label{sub:repeat_and_shuffle}
To provide more potential inter-dimension spatial dependencies, we propose a method called Replicate and Shuffle (RS). RS is a two-step feature transformation method which maps $\mathbf{x}_i$ to a higher dimensional space $\mathbf{x}'_i$ with more complete element combinations:
$$\mathbf{x}_i\in \mathbb{R}^K \rightarrow \mathbf{x}'_i \in \mathbb{R}^{K'}, K'>K$$
In the first step (Replicate), replicate $\mathbf{x}_i$ for $h = K'\%K+1$ times where $\%$ denotes remainder operation. Then we get a new vector with length as $h*K$ which is not less than $K'$; in the second step (Shuffle), we randomly shuffle the replicated vector in the first step and intercept the first $K'$ element to generate $\mathbf{x}'_i$. Theoretically, compared to $\mathbf{x}_i$, $\mathbf{x}'_i$ contains more diverse and complete inter-dimension dependencies.
\subsection{Selective Attention Mechanism}
\label{sub:attention_pattern_learning}
In the next process, we attempt to find the optimal dependency which includes the most distinctive information. But $K'$, the length of $\mathbf{x}'_i$, is too large and is computationally expensive. To balance the length and the information content, we introduce the attention mechanism \cite{cavanagh1992attention}.
We introduce the attention mechanism to emphasize the informative fragment in $\mathbf{x}'_i$ and denote the fragment by $\mathbf{\bar{x}}_i$, which is called \textit{focal zone}.
Suppose $\mathbf{\bar{x}}_i \in \mathbb{R}^{\bar{K}}$ and $\bar{K}$ denotes the length of the focal zone. For simplification, we continue denote the $k$-th element by $\mathbf{\bar{x}}_{ik}$ in the focal zone. To optimize the focal zone, we employ deep reinforcement learning as the optimization framework for its excellent performance in policy optimization \cite{mnih2015human}.
\textbf{Overview.} As shown in Figure~\ref{fig:workflow}, the focal zone optimization includes two key components: the environment (including state transition and reward model), and the agent. Three elements (the state $s$, the action $a$, and the reward $r$) are exchanged in the interaction between the environment and the agent. All of the three elements are customized based on the specific situation in this paper. Next, we introduce the design of the crucial components of our deep reinforcement learning structure:
\begin{itemize}
\item The \textbf{state} $\mathcal{S}=\{s_t, t=0,1,\cdots,T\}\in \mathbb{R}^2$ describes the position of the focal zone, where $t$ denotes the time stamp. In the training, $s_0$ is initialized as $s_0=[(K'-\bar{K})/2, (K'+\bar{K})/2]$. Since the focal zone is a shifting fragment on 1-D $\mathbf{x}'_i$, we design two parameters to define the state: $s_t = \{start^t_{idx},end^t_{idx}\}$, where $start^t_{idx}$ and $end^t_{idx}$ separately denote the start index and the end index of the focal zone\footnote{For example, for a random $\mathbf{x}'_i = [3,5,8,9,2,1,6,0]$, the state $\{start^t_{idx}=2, end^t_{idx}=5\}$ is sufficient to determine the focal zone as $[8,9,2,1]$.}.
\item The \textbf{action} $\mathcal{A}=\{a_t,t=0,1,\cdots,T\}\in \mathbb{R}^4$ describes which the agent could choose to act on the environment. In our case, we define 4 categories of actions for the focal zone (as described in the \textbf{State Transition} part in Figure~\ref{fig:workflow}): left shifting, right shifting, extend, and condense. Here at time stamp $t$, the state transition only choose one action to implement following the agent's policy $\pi$: $s_{t+1}=\pi(s_t,a_t)$.
\item The \textbf{reward} $\mathcal{R}=\{r_t,t=0,1,\cdots,T\}\in \mathbb{R}$ is calculated by the reward model, which will be detailed later. The reward model $\Phi$: $r_{t}=\Phi(s_t)$ receives the current state and returns an evaluation as the reward.
\item We employ the Dueling DQN (Deep Q Networks \cite{wang2015dueling}) as the optimization \textbf{policy} $\pi(s_t,a_t)$, which is enabled to learn the state-value function efficiently. Dueling DQN learns the Q value $V(s_t)$ and the advantage function $A(s_t,a_t)$ and combines them: $Q(s_t, a_t)\leftarrow V(s_t), A(s_t,a_t)$.
The primary reason we employ a dueling DQN to optimize the focal zone is that it updates all the four Q values\footnote{Since we have four actions in $a_t$, the $Q(s_t, a_t)$ contains 4 Q values. The arrangement is similar with the one-hot label.} at every step while other policy only updates one Q value at each step.
\end{itemize}
\textbf{Reward Model.} Next, we detailedly introduce the design of the reward model for it is one crucial contribution of this paper. The purpose of reward model is to evaluate how the current state impact our final target which refers to the classification performance in our case. Intuitively, the state which can lead to the better classification performance should have a higher reward: $r_t=\mathcal{F}(s_t)$. As a result, in the standard reinforcement learning framework, the original reward model regards the classification accuracy as the reward. $\mathcal{F}$ refers to the WAS-LSTM.
Note, WAS-LSTM focuses on the spatial dependency between different dimensions at the same time-point while the normal LSTM focuses on the temporal dependency between a sequence of samples collected at different time-points.
However, the WAS-LSTM requires considerable training time, which will dramatically increase the optimization time of the whole algorithm. In this section, we propose an alternative method to calculate the reward: construct a new reward function $r_t=\mathcal{G}(s_t)$ which is positively related with $r_t=\mathcal{F}(s_t)$. Therefore, we can employ $\mathcal{G}$ to replace $\mathcal{F}$. Then, the task is changed to construct a suitable $\mathcal{G}$ which can evaluate the inter-dimension dependency in the current state $s_t$ and feedback the corresponding reward $r_t$. We propose an alternative $\mathcal{G}$ composed by three components: the autoregressive model \cite{akaike1969fitting} to exploit the inter-dimension dependency in $\mathbf{x}'_i$, the Silhouette Score \cite{laurentini1994visual} to evaluate the similarity of the autoregressive coefficients, and the reward function based on the silhouette score.
The autoregressive model \cite{akaike1969fitting} receives the focal zone $\mathbf{\bar{x}}_i$ and specifies that how the last variable depends on its own previous values.
$$\bar{x}_{i\bar{K}} = \sum_{j=1}^{p}\varphi_j \bar{x}_{i(\bar{K}-j)}+ C +\bar{\varepsilon}$$
where $p$ is the order of the autoregressive model, $C$ indicates a constant, and $\bar{\varepsilon}$ indicates the withe noise. From this equation, we can infer that
the autoregressive coefficient $\boldsymbol{\varphi}=\{\varphi_j,j=1,2,\cdots,p\} \in \mathbb{R}^p$ incorporates the dependent relationship in the focal zone. Then, to evaluate how rich information is taken in the $\boldsymbol{\varphi}$, we employ silhouette score \cite{lovmar2005silhouette}
to interpret the consistence of $\boldsymbol{\varphi}$. The higher score $ss_t$ indicates the focal zone in state $s_t$ contains more inter-dimension dependency, which means $\mathbf{\bar{x}}_i$ is easier to be classified by the classifier in the next stage. At last, based on the $ss_t\in[-1,1]$, we design a \textbf{reward function}:
$$r_t = \frac{e^{ss_t+1}}{e^2-1}-\beta \frac{\bar{K}}{K'}$$
The function contains two parts, the first part is a normalized exponential function with the exponent $ss_t+1 \in[0,1]$, this part encourages the reinforcement learning algorithm to search the better $s_t$ which leads to a higher $ss_t$. The motivation of the exponential function is that: the reward growth rate is increasing with the silhouette score's increase\footnote{For example, for the same silhouette score increment 0.1, $ss_t: 0.9\rightarrow 1.0$ can earn higher reward increment than $ss_t: 0.1\rightarrow 0.2$.}. The second part is a penalty factor for the focal zone length to keep the bar shorter and the $\beta$ is the penalty coefficient.
In summary, the aim of focal zone optimization is to learn
the optimal focal zone $\mathbf{\bar{x}}^*_i$ which can lead to the maximum reward. The optimization totally iterates $N=n_e*n_s$ times where $n_e$ and $n_s$ separately denote the number of episodes and steps \cite{wang2015dueling}. $\varepsilon$-greedy method \cite{tokic2010adaptive} is employed in the state transition.
\subsection{Weighted Average Spatial LSTM Classifier}
\label{sub:classification}
In this section, we propose Weighted Average Spatial LSTM classification for two purposes. The first attempt is to capture the cross-relationship among feature dimensions in the optimized focal zone $\mathbf{\bar{x}}^*_i$. The LSTM-based classifier is widely used for its excellent sequential information extraction ability which is approved in several research areas such as natural language processing \cite{gers2001lstm,sundermeyer2012lstm}. Compared to other commonly employed spatial feature extraction methods, such as Convolutional Neural Networks, LSTM is less depends on the hyper-parameters setting. However, the traditional LSTM focuses on the temporal dependency among a sequence of samples. Technically, the input data of traditional LSTM is 3-D tensor shaped as $[n_b, n_t, \bar{K}]$ where $n_b$ and $n_s$ denote the batch size and the number of temporal sample, separately. The WAS-LSTM aims to capture the dependency among various dimensions at one temporal point, therefore, we set $n_t=1$ and transpose the input data as: $[n_b, n_t, \bar{K}]\rightarrow[n_b, \bar{K}, n_t]$.
The second advantage of WAS-LSTM is that it could stabilize the performance of LSTM via moving average method \cite{lipton2015learning}.
Specifically, we calculate the LSTM outputs $\mathbf{O}_i$ by averaging the past two outputs instead of only the final one (Figure~\ref{fig:workflow}):
$$\mathbf{O}_i = (\mathbf{O}_{i(\bar{K}-1)}+\mathbf{O}_{i\bar{K}})/2$$
The predicted label is calculated by $y'_i = \mathcal{L}(\mathbf{\bar{x}}^*_i)$ where $\mathcal{L}$ denotes the LSTM algorithm. $\ell_2$-norm (with parameter $\lambda$) is adopted as regularization to prevent overfitting. The sigmoid activation function is used on hidden layers. The loss function is cross-entropy and is optimized by the AdamOptimizer algorithm \cite{kingma2014adam}.
\section{Experiments}
\label{sec:experiments}
In this section, we evaluate the proposed approach over 3 sensor signal datasets (separately collected by EEG headset, environmental sensor, and wearable sensor) including 2 widely used public datasets and 2 limited but more practical local datasets. Firstly we describe the details of each dataset. Secondly, we demonstrate the effectiveness and robustness by comparing the performance of our approach to baselines and state-of-the-art. Lastly, we provide the efficiency of the alternative reward model designed in Section~\ref{sub:attention_pattern_learning}.
\begin{table}[]
\centering
\caption{Datasets description. PID denotes Person Identification, AR denotes Activity Recognition, and S-rate denotes Sampling rate. \#-S, \#-C, \#-D separately denote the number of subjects, classes, and dimensions.}
\label{tab:datasets}
\resizebox{\linewidth}{!}{
\begin{tabular}{llllllll}
\hline
\textbf{Datasets} & \textbf{Type} & \textbf{Task} & \textbf{\#-S} & \textbf{\#-C} & \textbf{Samples} & \textbf{\#-D} & \textbf{S-rate (Hz)} \\ \hline
\textbf{EID} & EEG & PID & 8 & 8 & 168,000 & 14 & 128 \\
\textbf{RSSI} & RFID & AR & 6 & 21 & 3,100 & 12 & 2 \\
\textbf{PAMAP2} & IMU & AR & 9 & 8 & 120,000 & 14 & 100 \\
\hline
\end{tabular}
}
\end{table}
\subsection{Datasets}
\label{sub:datasets}
More details refer to Table~\ref{tab:datasets}.
\begin{itemize}
\item \textbf{EID.} The EID (EEG ID identification) is collected in a constrained setting where 8 subjects (5 males and 3 females) aged $26\pm 2$. EEG signal monitors the electrical activity of the brain. This dataset gathers the raw EEG signals by Emotiv EPOC+ headset with 14 channels at the sampling rate of 128 Hz.
\item \textbf{RSSI.} The RSSI (Radio Signal Strength Indicator) \cite{yao2018compressive} collects the signals from passive RFID tags. 21 activities, including 18 ADLs (Activity of Daily Living) and 3 abnormal falls, are performed by 6 subject aged $25\pm 5$. RSSI measures the power present in a received radio signal, which is a convenient environmental measurement in ubiquitous computing.
\item \textbf{PAMAP2.} The PAMAP2 \cite{fida2015} is collected by 9 participants (8 males and 1 females) aged $27\pm 3$. 8 ADLs are selected as a subset of our paper. The activity is measured by 1 IMU attached to the participants' wrist. The IMU collects sensor signal with 14 dimensions including two 3-axis accelerometers, one 3-axis gyroscopes, one 3-axis magnetometers and one thermometer.
\end{itemize}
\begin{table}[t]
\centering
\caption{Comparison of EID}
\label{tab:comparison_eid}
\resizebox{\linewidth}{!}{
\begin{tabular}{cp{3cm}|llll}
\hline
\multicolumn{2}{c|}{\multirow{2}{*}{\textbf{Methods}}} & \multicolumn{4}{c}{\textbf{EID Dataset}} \\ \cline{3-6}
\multicolumn{2}{l|}{} & \textbf{Acc} & \textbf{Pre} & \textbf{Rec} & \textbf{F1} \\\hline
& \textbf{SVM} & 0.1438 & 0.1653 & 0.1545 & 0.1445 \\
& \textbf{RF} & 0.9365 & 0.9261 & 0.9142 & 0.9457 \\
& \textbf{KNN} & 0.9413 & 0.9471 & 0.9298 & 0.9511 \\
& \textbf{AB} & 0.2518 & 0.2684 & 0.2491 & 0.2911 \\
\multirow{-5}{*}{\textbf{\begin{tabular}[c]{@{}c@{}}Non-\\ DL\\\end{tabular}}} & \textbf{LDA} & 0.1485 & 0.1524 & 0.1358 & 0.1479 \\ \hline
& \textbf{LSTM} & 0.4315 & 0.5132 & 0.4278 & 0.4532 \\
& \textbf{GRU} & 0.4314 & 0.455 & 0.4288 & 0.4218 \\
\multirow{-3}{*}{\textbf{\begin{tabular}[c]{@{}c@{}}DL\\ \end{tabular}}} & \textbf{1-D CNN} & 0.8031 & 0.8127 & 0.805 & 0.8278 \\ \hline
\multicolumn{1}{l}{\textbf{}} & \textbf{WAS-LSTM} & 0.9518 & 0.9657 & \textbf{0.9631} & \textbf{0.9658} \\
\multicolumn{1}{l}{\textbf{}} & \textbf{Ours} & \textbf{0.9621} & \textbf{0.9618} & 0.9615 & 0.9615\\ \hline
\end{tabular}
}
\end{table}
\begin{table}[t]
\centering
\caption{Comparison of RSSI}
\label{tab:comparison_rssi}
\resizebox{\linewidth}{!}{
\begin{tabular}{cp{3cm}|llll}
\hline
\multicolumn{2}{c|}{\multirow{2}{*}{\textbf{Methods}}} & \multicolumn{4}{c}{\textbf{RSSI Dataset}} \\ \cline{3-6}
\multicolumn{2}{l|}{} & \textbf{Acc} & \textbf{Pre} & \textbf{Rec} & \textbf{F1} \\\hline
& \textbf{SVM} & 0.8918 & 0.8924 & 0.8908& 0.8805 \\
& \textbf{RF} & 0.9614 & 0.9713& 0.9652 & 0.9624 \\
& \textbf{KNN} & 0.9612 & 0.9628 & 0.9618 & 0.9634 \\
& \textbf{AB} & 0.4704 & 0.4125 & 0.4772 & 0.3708 \\
\multirow{-5}{*}{\textbf{\begin{tabular}[c]{@{}c@{}}Non-\\ DL\\ \end{tabular}}} & \textbf{LDA} & 0.8842 & 0.8908 & 0.8845 & 0.8802 \\ \hline
& \textbf{LSTM} & 0.7421 & 0.6505 & 0.6132 & 0.6858 \\
& \textbf{GRU} & 0.7049 & 0.7728 & 0.6584 & 0.6915 \\
\multirow{-3}{*}{\textbf{\begin{tabular}[c]{@{}c@{}}DL\\ \end{tabular}}} & \textbf{1-D CNN} & 0.9714 & 0.9676 & 0.9635 & 0.9645 \\ \hline
\multicolumn{1}{l}{\textbf{}} & \textbf{WAS-LSTM} & 0.9553 & 0.9533 & 0.9545 & 0.9592 \\
\multicolumn{1}{l}{\textbf{}} & \textbf{Ours} & \textbf{0.9838} & \textbf{0.9782} & \textbf{0.9669} & \textbf{0.9698}\\ \hline
\end{tabular}
}
\end{table}
\begin{table}[t]
\centering
\caption{Comparison of PAMAP2}
\label{tab:comparison_pamap2}
\resizebox{\linewidth}{!}{
\begin{tabular}{cp{3cm}|llll}
\hline
\multicolumn{2}{c|}{\multirow{2}{*}{\textbf{Methods}}} & \multicolumn{4}{c}{\textbf{PAMAP2 Dataset}} \\ \cline{3-6}
\multicolumn{2}{l|}{} & \textbf{Acc} & \textbf{Pre} & \textbf{Rec} & \textbf{F1} \\\hline
\multicolumn{1}{c}{} & \textbf{SVM} & 0.7492& 0.7451 & 0.7522 & 0.7486 \\
\multicolumn{1}{c}{} & \textbf{RF} & 0.9817 &0.9893 & 0.9711 & \textbf{0.9801} \\
\multicolumn{1}{c}{} & \textbf{KNN} & 0.9565 & 0.9651 & 0.9625& 0.9638 \\
\multicolumn{1}{c}{} & \textbf{AB} & 0.5776& 0.4298 & 0.5814 & 0.4942 \\
\multicolumn{1}{c}{\multirow{-5}{*}{\textbf{\begin{tabular}[c]{@{}c@{}}Non-\\ DL\end{tabular}}}} & \textbf{LDA} & 0.7127 &0.7175 &0.7298 & 0.7236 \\\hline
\multicolumn{1}{c}{} & \textbf{LSTM} & 0.7925 &0.7487 & 0.7478 &0.7482 \\
\multicolumn{1}{c}{} & \textbf{GRU} & 0.8625 & 0.8515 & 0.8349 & 0.8431 \\
\multicolumn{1}{c}{\multirow{-3}{*}{\textbf{\begin{tabular}[c]{@{}c@{}}DL\\\end{tabular}}}} & \textbf{1-D CNN} & 0.9819& 0.9715 & 0.9721& 0.9718 \\\hline
& \textbf{\cite{fida2015}} & 0.96 & - & - & - \\
& \textbf{\cite{chowd2017}} & 0.8488 & - & - & 0.841 \\
& \textbf{\cite{erfani2017}} & 0.967 & - & - & - \\
\multirow{-4}{*}{\textbf{\begin{tabular}[c]{@{}l@{}}State-\\ of-the \\ -Arts\end{tabular}}} & \textbf{\cite{zheng2014time}} & 0.9336 & - & - & - \\\hline
\textbf{} & \textbf{WAS-LSTM} & 0.9821 & \textbf{0.9981} & 0.9459 &0.9713 \\
\textbf{} & \textbf{Ours} & \textbf{0.9882} & 0.9804 & \textbf{0.9756} & 0.9780 \\\hline
\end{tabular}
}
\end{table}
\vspace{-2mm}
\subsection{Results}
\label{sub:comparision}
In this section, we compare the proposed approach with baselines and the state-of-the-art methods.
Our method focuses on the focal zone which is optimized by deep reinforcement learning and then explores the dependency between sensor signal elements by a deep learning classifier. All the three datasets are randomly split into the training set (90\%) and the testing set ($10\%$). Each sample is one sensor vector recording collected at one time point.
Through the previous experimental tuning and the Orthogonal Array based hyper-parameters tuning method \cite{zhang2017intent}, the hyper-parameters are set as following.
In the selective attention learning: the order of autoregressive is 3; $\bar{K}=128$, the Dueling DQN has 4 lays and the node number in each layer are: 2 (input layer), 32 (FCL), 4 ($A(s_t,a_t)$) + 1 ($V(s_t)$), 4 (output). The decay parameter $\gamma =0.8$, $n_e=n_s=50$, $N=2,500$, $\epsilon=0.2$, learning rate$ =0.01$, memory size $ =2000$, length penalty coefficient $\beta=0.1$, and the minimum length of focal zone is set as 10. In the deep learning classifier: the node number in the input layer equals to the number of feature dimensions, three hidden layers with 164 nodes, two layers of LSTM cells and one output layer. The learning rate $ =0.001$, $\ell_2$-norm coefficient $\lambda=0.001$, forget bias $=0.3$, batch size $ =9$, and iterate for 1000 iterations.
Tables~\ref{tab:comparison_eid} $\sim$~\ref{tab:comparison_pamap2} show the classification metrics comparison between our approach and baselines including Non-DL and DL baselines. Since the EID and RSSI are local datasets, we only compare with state-of-the-art over the public dataset PAMAP2. Table~\ref{tab:comparison_pamap2} shows that our approach achieves the highest accuracy on both datasets. DL represents deep learning. The notation and hyper-parameters of the baselines are listed here. RF denotes Random Forest, AdaB denotes Adaptive Boosting, LDA denotes Linear Discriminant Analysis. In addition, the key parameters of the baselines are listed here: Linear SVM ($C=1$), RF ($n=200$), KNN ($k=3$). In LSTM, $n_{steps}=5$, another set is the same as the WAS-LSTM classifier, along with the GRU (Gated Recurrent Unit \cite{chung2014empirical}). The CNN works on sensor data and contains 2 stacked convolutional layers (both with stride $[1,1]$, patch $[2,2]$, zero-padding, and the depth are 4 and 8, separately.) and followed by one pooling layer (stride $[1,2]$, zero-padding) and one fully connected layer (164 nodes). Relu activation function is employed in the CNN.
The results from Tables~\ref{tab:comparison_eid} $\sim$~\ref{tab:comparison_pamap2} show that:
\begin{itemize}
\item Our approach outperforms all the baselines and the state-of-the-arts over all the local and public datasets ranging from EEG, RFID to wearable IMU sensors;
\item The sensor spatial based WAS-LSTM classifier achieves a high-level performance, which indicates the method that extracting inter-dimension dependency for classification is effective;
\item Our method (WAS-LSTM with focal zone) performs better than WAS-LSTM, which illustrates that the learned informative attention is effective.
\end{itemize}
To have a closer observation, the CM (confusion matrix) and the ROC curves (including the AUC score) of the datasets are reported in Figure~\ref{fig:cm_roc}. The CMs illustrate that the robustness of the proposed approach keeps high accuracy even over few samples and numerous categories.
\begin{figure*}[t]
\hspace{5mm}
\centering
\begin{subfigure}[t]{0.28\textwidth}
\centering
\includegraphics[width=\textwidth]{EID_cm_summer.png}
\caption{EID CM}
\label{fig:eid_cm}
\end{subfigure}%
\begin{subfigure}[t]{0.28\textwidth}
\centering
\includegraphics[width=\textwidth]{RSSI_cm_summer.png}
\caption{RSSI CM}
\label{fig:rssi_cm}
\end{subfigure}%
\begin{subfigure}[t]{0.28\textwidth}
\centering
\includegraphics[width=\textwidth]{PAMAP_cm_summer.png}
\caption{PAMAP CM}
\label{fig:pamap_cm}
\end{subfigure}%
\centering
\begin{subfigure}[t]{0.28\textwidth}
\centering
\includegraphics[width=\textwidth]{EID_roc.png}
\caption{EID ROC}
\label{fig:eid_roc}
\end{subfigure}%
\begin{subfigure}[t]{0.28\textwidth}
\centering
\includegraphics[width=\textwidth]{RSSI_roc.png}
\caption{RSSI ROC}
\label{fig:rssi_roc}
\end{subfigure}%
\begin{subfigure}[t]{0.28\textwidth}
\centering
\includegraphics[width=\textwidth]{PAMAP_roc.png}
\caption{PAMAP ROC}
\label{fig:pamap_roc}
\end{subfigure}%
\caption{Confusion matrix and ROC curves of three datasets. CM denotes confusion matrix. The RSSI dataset overall contains 21 classes and we only select several representative classes.}
\label{fig:cm_roc}
\end{figure*}
\subsection{Reward Model Efficiency Demonstration}
\label{sub:reward_model_efficiency_demonstration}
In this paper, we propose a new reward model to replace the original reward model: $\mathcal{G} \rightarrow \mathcal{F}$. The original $\mathcal{F}$, in our case, refers to the WAS-LSTM classifier (Section~\ref{sub:classification}), intuitively. $\mathcal{F}$ requires a large amount of training time to find the optimal focal zone $\mathbf{\bar{x}^*}$. Take the EID dataset as an example, $\mathcal{F}$ needs around $4000$ sec on the Titan X (Pascal) GPU
for each step while the whole focal zone optimization contains $N$ ($N>2000$) iterations.
Therefore, to save training time, we attempt to employ $\mathcal{G}$
to approximate $\mathcal{G}$ to update the reward. Thus, two prerequisites are demanded: 1) $\mathcal{G}$ should have high correlation with $\mathcal{F}$ to guarantee $\mathop{\arg\max}\limits_{\mathbf{\bar{x}^*}}\mathcal{G} \approx \mathop{\arg\max}\limits_{\mathbf{\bar{x}^*}}\mathcal{F}$; 2) the training time of $\mathcal{G}$ should be shorter than $\mathcal{F}$. In this section, we demonstrate the two prerequisites by experimental analyzes.
First, on the focal zone optimization procedure on EID dataset, we conduct an experiment to measure a batch of data pairs of the reward (represents the reward of $\mathcal{G}$) and the WAS-LSTM classifier accuracy (represents the reward of $\mathcal{F}$). The relationship between the reward and the accuracy is shown in Figure~\ref{fig:reward_acc}. The figure illustrates that the accuracy has an approximately linear relationship with the reward. The correlations coefficient is 0.8258 (with p-value as 0.0115), which demonstrates the accuracy and reward are highly positive related. As a result, we can estimate $\mathop{\arg\max}\limits_{\mathbf{\bar{x}^*}}\mathcal{F}$ by $\mathop{\arg\max}\limits_{\mathbf{\bar{x}^*}}\mathcal{G}$. Moreover, another experiment is carried on to measure the single step training time of two reward models $\mathcal{G}$ and $\mathcal{F}$. The training times are marked as T1 and T2, respectively. Figure~\ref{fig:time_compare} qualitatively shows that T2 is much higher than T1 (8 states represent 8 different focal zones). Quantitatively, the sum of T1 over 8 states is $35237.41$ sec while the sum of T2 is $601.58$ sec. This results demonstrate that the proposed approach, designing a $\mathcal{G}$ to approximate and estimate the $\mathcal{F}$, saves $\mathbf{98.3\%}=1-601.58/35237.41$ training time in focal zone optimization.
\begin{figure}[t]
\centering
\begin{minipage}[b]{0.44\linewidth}
\centering
\includegraphics[width=\textwidth]{reward_acc-eps-converted-to.pdf}
\caption{The relationship between the classifier accuracy and the reward. The correlationship coefficient is 0.8258 while the p-value is 0.0115.}
\label{fig:reward_acc}
\end{minipage}
\hspace{1mm}
\begin{minipage}[b]{0.44\linewidth}
\centering
\includegraphics[width=\textwidth]{time_compare-eps-converted-to.pdf}
\caption{Reward model training time in various states. T1 and T2 separately denote the training time in reward model $\mathcal{G}$ and $\mathcal{F}$. }
\label{fig:time_compare}
\end{minipage}
\end{figure}
\subsection{Discussions}
\label{sub:discussion}
In this section, we discuss several characteristics of the proposed approach.
First, we propose a robust, universal, and adaptive classification framework which can efficiently deal with multi-modality sensor data. Specifically, our approach works better on high-dimensional feature space in that the information of inter-dimension dependency is richer.
In addition, we propose a novel idea that adopts an alternative reward model to estimate and replace the original reward model. In this way, the disadvantages of the original model, such as expensive computation, can be eliminated. The key is to keep the reward produced by the new model highly related to the original reward. The higher correlation coefficient, the better. This sheds light on the possible combination of deep learning classifier and reinforcement learning.
Nevertheless, one weakness is that the reinforcement learning policy only works well in the specific environment in which the model is trained. The dimension indices should be consistent in training and testing stages. Various policies should be trained according to different sensor combinations.
Furthermore, the proposed WAS-LSTM directly focuses on the dependency among the sensor dimensions and can produce a predicted label for each point.
This provides the foundation for the quick-reaction online detection and other applications which require instantaneous detection. However, this reuires a enough number of signal dimensions to carry sufficient information for the aim of accurately recognition.
\section{Conclusion}
\label{sec:conclusion}
In this paper, we present a robust and efficient multi-modality sensor data classification framework
which integrates selective attention mechanism, deep reinforcement learning, and WAS-LSTM classification. In order to boost the chance of inter-dimension dependency in sensor features, we replicate and shuffle the sensor data. Additionally, the optimal spatial dependency is required for high-quality classification, for which we introduce the focal zone with attention mechanism. Furthermore, we extended the LSTM to exploit the cross-relationship among spatial dimensions, which is called WAS-LSTM, for classification. The proposed approach is evaluated on three different sensor datasets, namely, EEG, RFID and wearable IMU sensors. The experimental results show that our approach outperforms the state-of-the-art baselines. Moreover, the designed reward model saves $\textbf{98.3\%}$ of the training time in reinforcement learning.
\bibliographystyle{named}
|
{
"timestamp": "2018-05-02T02:09:35",
"yymm": "1804",
"arxiv_id": "1804.05493",
"language": "en",
"url": "https://arxiv.org/abs/1804.05493"
}
|
\section{Introduction}
In this note, we complete and extend previous analyses
on dual formulations of massive and (partially) massless
spin-2 theories in (A)dS backgrounds of arbitrary dimension
$n>3\,$.
We resort to the parent-action technique employed in the papers
\cite{Curtright:1980yj,West:2001as,West:2002jj,Boulanger:2003vs,Matveev:2004ac,
Zinoviev:2005zj,Zinoviev:2005qp,Gonzalez:2008ar,Khoudeir:2008bu,Basile:2015jjd}
in order to derive equivalent, dual
actions in the sense of Fradkin and Tseytlin \cite{Fradkin:1984ai}.
In brief, in this framework one obtains two equivalent second-order
actions ---~whose field equations are related by electric-magnetic duality~---
by eliminating different sets of fields from a common ``parent''
first-order action.
In (A)dS$_n$ these techniques have been employed for massless and massive
gravitons, while the partially-massless case has been discussed recently
only in $n=3$ \cite{Galviz:2017tda}.
The same setup has also been used in the context of
Ho{\v r}ava-Lifshitz gravity \cite{Cortese:2014pfa}.
For all values of the mass and of the cosmological constant,
we furnish dual formulations at the action level
and in a manifestly Lorentz-invariant way.
The dual actions
that we built are such that the flat and massless
limits are smooth, thereby making the identification of the physical
degrees of freedom and of the helicities straightforward.
In the partially-massless case \cite{Deser:1983mm},
we obtain for the first
time a dual, manifestly covariant action principle featuring
a mixed-symmetry gauge field.
At the level of the field equations, (self-)duality symmetry,
often named pseudo (self-)duality,
has been studied in flat spacetime for linearised gravity
in \cite{Hull:2000zn,Hull:2001iu};
see also \cite{Bekaert:2002jn}.
In (A)dS$_4\,$, pseudo-duality symmetry for
partially massless spin-2 fields
was studied in \cite{Hinterbichler:2014xga,Cherney:2015jxp}.
These are first steps towards the establishment of an
equivalence between theories,
for which an off-shell duality relation is necessary.
In flat spacetime, the duality between the massless
Fierz-Pauli action and the Curtright action
\cite{Curtright:1980yk,Aulakh:1986cb}
was proven in the series of works
\cite{West:2001as,West:2002jj,Boulanger:2003vs}.
The action principles that we present feature
both the original spin-2 field and its dual, in a manifestly
Lorentz-invariant fashion. On the other hand, the pair of dual fields
does not enter the action in a duality-symmetric way.
For such a democratic appearance of electric and magnetic fields inside
the action, the price to pay is the loss
of manifest spacetime covariance, as explained
for massless spin-2 and higher-spin theories around flat spacetime
in the papers \cite{Henneaux:2004jw,Bunster:2013oaa,Henneaux:2016zlu}
and references therein. In the same, non manifestly Lorentz-covariant
framework, a double-potential formulation of linearised gravity
around (A)dS$_n$ spacetime was studied in \cite{Julia:2005ze,Leigh:2007wf}
for $n=4$ and in \cite{Hortner:2016omi} for $n>4\,$.
As for what concerns partially-massless fields of maximal depth,
the paper \cite{Deser:2013xb} provides an off-shell formulation
exhibiting a nearly manifest electric-magnetic duality symmetry.
Interestingly enough, manifest duality-invariant formulations
of linearised gravity, in the presence of
sources, have been given in \cite{Bunster:2006rt} and in an
alternative way in \cite{Barnich:2008ts}; in the partially-massless
case, see also \cite{Hinterbichler:2015nua}.
Finally, the integrability properties of duality-symmetric systems
were studied in \cite{Barnich:2008ar}.
In more details, the unified treatment of spin-2 duality presented
in this note leads to the following results:
\begin{itemize}
\item In the case of a massless graviton in (A)dS,
we complete the programme sketched in
\cite{Basile:2015jjd} by linking the dual action obtained therein
to its Stueckelberg formulation admitting a smooth flat limit;
\item For partially massless spin-2 field in (A)dS, we obtain a dual
description at the action level, thereby elevating the duality from a pseudo to
a genuine off-shell duality;
\item In the massive case in (A)dS, we clarify the flat limit of
the dual model presented in \cite{Zinoviev:2005zj} in that we
have Stueckelberg gauge fields representing the dual spin-2,
spin-1 and scalar sectors.
Therefore, our actions admit a smooth flat limit in \emph{both}
electric and magnetic formulations.
\end{itemize}
In sections~\ref{sec:parent} and \ref{sec:electric}
we recall the main features
of the first-order description of massive spin-two fields,
that we use as a parent action. Section~\ref{sec:magnetic}
collects our original results on dual formulations for
spin-2 fields in (A)dS.
\section{The parent action}\label{sec:parent}
We consider as parent action the first-order Stueckelberg action describing,
for generic values of the parameters, the propagation of a massive spin-2
field in a constant curvature background \cite{Zinoviev:2008ze}.
It is obtained by coupling the free actions for massless fields of spin two,
one and zero. It thus comprises the kinetic terms for these
fields,\footnote{We denote the background vielbein by $\bar{e}^a\,$,
while $\nabla$ is the Lorentz-covariant derivative on (A)dS$_n$.
In our conventions, it satisfies $\nabla^2 V^{c} = -\,\sigma\,\lambda^2\, \bar{e}^c \wedge \bar{e}_b \ V^b$, so that $\sigma = 1$ in AdS$_n$ and $\sigma = -1$ in dS$_n$.
We define the Levi-Civita symbol $\epsilon_{a_1\cdots a_n}$ such that
$\epsilon_{01\cdots n-1}=-1$ and we adopt the mostly-plus convention for
the metric. In the following we omit wedge products and we substitute
groups of antisymmetrised indices with a label denoting the total number
of indices. For instance, we introduce the
$k$-form $H^{a[k]} \equiv H^{a_1 \cdots a_k} = \bar{e}^{\,a_1} \cdots \bar{e}^{\,a_k}$.
Indices enclosed between square brackets are antisymmetrised, and dividing
by the number of terms involved is understood (strength-one convention).
Finally, repeated indices also denote an antisymmetrisation, e.g.,
$A_a B_a \equiv A_{[a_1} B_{a_2]}.$}
\begin{align}
{\cal L}^{(2)} & = - \frac{\epsilon_{abcp[n-3]}}{2(n-3)!}
\left( \omega^{ab} \nabla h^c + \frac{1}{n-2}\, \omega^a{}_q \omega^{qb} \bar{e}^c \right) H^{p[n-3]} \, , \label{K2} \\[5pt]
{\cal L}^{(1)} & = \frac{\epsilon_{ab c[n-2]}}{2(n-2)!}\, F^{ab} \left( \nabla A - \frac{1}{4}\, F_{kl} \, \bar{e}^k \bar{e}^l \right) H^{c[n-2]} \, , \label{K1} \\[5pt]
{\cal L}^{(0)} & = \frac{\epsilon_{ab[n-1]}}{(n-1)!}\, \pi^a \left( \nabla \varphi -\frac{1}{2} \, \pi_k \, \bar{e}^k \right) H^{b[n-1]} \, , \label{K0}
\end{align}
together with cross couplings and mass terms:
\begin{equation}
\begin{split}
&{\cal L}_{\textrm{cross}} = \frac{\epsilon_{ab c[n-2]}}{(n-1)!}\, \bigg( (n-1)m\, \omega^{ab} A + m\, F^a{}_d \, h^d \bar{e}^b + \mu\, \pi^a A\, \bar{e}^b \\
& - \frac{(n-2)\mu^2}{4}\, h^a
h^b - m\,\mu\, \varphi\, h^a \bar{e}^b -
\frac{m^2}{n-2}\, \varphi^2\, \bar{e}^a \bar{e}^b \bigg)\, H^{c[n-2]} \, .
\end{split}
\end{equation}
The full action is the integral of ${\cal L} = \sum_{s=0}^2 {\cal L}^{(s)} + {\cal L}_{\textrm{cross}}$ and it is invariant under the gauge symmetries
\begin{subequations}
\begin{align}
\delta h^{a} & = \nabla \xi^a - \Lambda^{ab} \bar{e}_b + \frac{2m}{n-2}\, \epsilon\, \bar{e}^a \, , \label{varh} \\
\delta \omega^{ab} & = \nabla \Lambda^{ab} + \frac{\mu^2}{n-1}\, \bar{e}^{[a} \xi^{b]} \, , \label{varomega}
\end{align}
\end{subequations}
and
\begin{alignat}{5}
\delta A & = \nabla \epsilon - m\, \xi^a \bar{e}_a \, , \qquad
& \delta F^{ab} & = 2m\, \Lambda^{ab} \, , \label{var1} \\[5pt]
\delta \varphi & = -\,\mu\, \epsilon \, , \qquad
& \delta \pi^a & = -\,m\,\mu\, \xi^a \, . \label{var0}
\end{alignat}
For later convenience, we introduced the constants $m$ and $\mu$, even if the action actually depends only on a single mass parameter (besides the (A)dS radius). Gauge invariance requires
\begin{equation} \label{mu-m}
\mu^2 = \frac{2(n-1)}{n-2} \left(\, 2m^2 + \sigma (n-2) \lambda^2 \,\right) .
\end{equation}
When $m=0$ the fields of spin one and zero decouple from the
spin-two sector and one recovers the usual first-order
formulation of linearised gravity in (A)dS.
At $\mu = 0$, the sole scalar sector decouples and one obtains
a first-order description of a partially-massless graviton,
propagating helicities two and one in the flat limit.
The first-order description for the spin-$s$ totally
symmetric partially-massless cases of all depths
was given in \cite{Skvortsov:2006at}.
With the manifestly unitary conventions used in \eqref{K2}--\eqref{K0}, one can set $\mu$ to zero by tuning the mass $m \in \mathbb{R}$ only in dS ($\sigma = -1$). In section \ref{sec:PM} we shall show that partially-massless fields in AdS can be described in this formalism at the price of flipping the sign of the spin-one kinetic term, which makes their lack of unitarity manifest.
Eq.~\eqref{K2} can be expressed in terms of the field \cite{West:2001as}
\begin{equation} \label{defY}
Y^{bc|a} = \omega^{a|bc} + g^{ab} \omega_{d|}{}^{cd} - g^{ac} \omega_{d|}{}^{bd} \; ,
\end{equation}
which is antisymmetric in its first two indices and transforms as
\begin{equation}
\delta Y^{bc|}{}_a = \nabla_{\!a} \Lambda^{bc} + 2\, \bar{e}_a{}^{[b} \nabla_{\!d} \Lambda^{c]d} - \frac{(n-2)\mu^2}{n-1}\,\bar{e}_a{}^{[b}\xi^{c]} \; .
\label{varY}
\end{equation}
The spin-2 kinetic term can then be cast in the form (from now on we will omit the integration measure $d^nx \sqrt{-g}$ brought by $\bar{e}^{a_1} \cdots \bar{e}^{a_n} = \det(\bar{e}) \epsilon^{a_1 \cdots a_n} d^nx$)
\begin{equation} \label{West_action}
{\cal L}^{(2)} = \nabla_{\!b\phantom{|}\!} h_{c|}{}^a Y^{bc|}{}_a
- \frac{1}{2} \left( Y^{bc|a} Y_{ab|c} + \frac{1}{n-2}\, Y^{ab|}{}_b Y_{ac|}{}^c \right)\, ,
\end{equation}
while the cross couplings and mass terms read
\begin{equation} \label{Lcross}
\begin{split}
&{\cal L}_{\textrm{cross}} = -\, \frac{2m}{n-2}\, Y^{ab|}{}_b A_a - m\, F^{ab} h_{a|b} - \mu\, \pi^a A_a \\
& - \frac{(n-2)\mu^2}{4(n-1)} \left(h_{a|b}h^{b|a} - h^2\right) + m\,\mu\, h\,\varphi + \frac{n\,m^2}{n-2}\, \varphi^2 \, ,
\end{split}
\end{equation}
where $h = h_{a|}{}^a$ denotes the trace of the linearised vielbein.
As recalled in section~\ref{sec:electric}, eliminating the auxiliary fields $Y^{bc|}{}_a$, $F^{ab}$ and $\pi^a$ from the parent action ${\cal L}$ one obtains a second-order description of a massive spin-2 field in terms of the linearised metric and the fields $A_\mu$ and $\varphi$, which reduces to the Fierz-Pauli action for $m=0$. In section~\ref{sec:magnetic} we will instead show how eliminating the fields $h_{a|}{}^b$, $A_a$ and $\varphi$ leads to its dual description, involving mixed-symmetry fields for generic values of $n$.
\section{Electric reduction}\label{sec:electric}
The equations of motion for $Y^{bc|}{}_a$, $F^{ab}$ and $\pi^a$ arising from ${\cal L}[h,Y,A,F,\varphi,\pi]$ allow to solve for them algebraically. E.g.
\begin{equation}
\begin{split}
Y_{ab|c} &= \nabla_{\!c}h_{[a|b]}-\nabla_{\!a}h_{(b|c)}+\nabla_{\!b}h_{(a|c)} \\
&\phantom{==========} +2g_{c[a} \left(\nabla^d h_{b]|d}-\nabla_{\!b]} h
+2m\, A_{b]}\right)\, .
\end{split}
\end{equation}
By plugging this and the similar expressions for $F^{ab}$ and
$\pi^a$ into the parent Lagrangian ${\cal L}\,$, the latter reduces,
modulo a total derivative, to the second-order Stueckelberg Lagrangian
for a symmetric spin-2 field \cite{Zinoviev:2001dt,Zinoviev:2006im}:
\begin{equation}
\begin{split}\label{electricLag}
& {\cal L}[h,A,\varphi]= -\tfrac{1}{2}\,\nabla_{\!a} h_{(b|c)}\nabla^a h^{(b|c)} + \nabla_{\!a} h_{(b|c)}\nabla^c h^{(b|a)} \\
& + \tfrac{1}{2}\,\nabla_{\!a} h \nabla^a h - \nabla_{\!a} h\nabla_b h^{(a|b)} -\tfrac{(n-1)\sigma\lambda^2}{2} \left(2 h_{(a|b)}h^{(a|b)}-h^2\right) \\
& -\nabla_{\![a}A_{b]} \nabla^{[a}A^{b]} -(n-1)\sigma\, \lambda^2 A_a A^a -\tfrac{1}{2}\, \nabla_{\!a}\varphi \nabla^a\varphi \\
&-2m\, A_a\left(\nabla^a h-\nabla_{\!b} h^{(a|b)}\right) +\mu\, \varphi \nabla_{\!a} A^a \\
&- m^2\left(h_{(a|b)}h^{(a|b)}- h^2\right) +\tfrac{n\,m^2}{n-2}\, \varphi^2 + m\,\mu\, h\,\varphi \; .
\end{split}
\end{equation}
The resulting action is invariant under the gauge transformations \eqref{varh}
for $h_{(a|b)}$, to be identified with the linearised metric, together
with \eqref{var1} and \eqref{var0} for the Stueckelberg fields $A_a$
and $\varphi\,$.
The antisymmetric part of the vielbein, $h_{[a|b]}\,$,
enters the reduced Lagrangian only through a total derivative,
consistently with the shift symmetry it enjoys under Lorentz transformations.
The first two lines of \eqref{electricLag} gives the Fierz-Pauli Lagrangian
for a massless spin-2 field in (A)dS.
For $\mu = 0$ one obtains a description of a partially-massless
spin-2 field in dS in terms of the Stueckelberg coupling of the
Fierz-Pauli and Proca Lagrangians.
The field $A_a$ can be gauged away using $\xi^a$, and the resulting action
is invariant under
\begin{equation}
\delta h_{(a|b)} = \tfrac{1}{\lambda} \left( \nabla_{\!(a} \nabla_{\!b)} \epsilon + \lambda^2 g_{ab} \epsilon \right) \, .
\end{equation}
In this context,
the partially-massless gauge symmetry thus follows because
gauge transformations \eqref{varh} and \eqref{var1} with
\mbox{$\nabla_{\!a} \epsilon - m\, \xi_a = 0$} preserve
the gauge fixing $A_a = 0\,$.
\section{Magnetic reduction}\label{sec:magnetic}
\subsection{Massless case}\label{sec:massless}
When $m = 0$ the fields of spin one and zero decouple and one can consider the parent Lagrangian
\begin{equation} \label{Lmassless}
\begin{split}
{\cal L}_{0}[h,Y] & = {\cal L}^{(2)}[h,Y] - \tfrac{(n-2)\sigma\lambda^2}{2} \left(h_{a|b}h^{b|a} - h^2\right) \, ,
\end{split}
\end{equation}
with ${\cal L}^{(2)}$ given in \eqref{West_action}.
Its gauge symmetries are obtained by setting $m = 0$ in \eqref{varh}
and \eqref{varY}. Contrary to flat space \cite{Boulanger:2003vs}
(where it enters the action linearly), in (A)dS the linearised vielbein
is an auxiliary field thanks to the mass term in \eqref{Lmassless}: it
can thus be eliminated through its own equation of motion \cite{Matveev:2004ac}.
This leads to an action depending only on the traceless projection of $Y^{bc|}{}_a\,$:
\begin{equation} \label{LdualY}
\hat{Y}^{bc|}{}_a = Y^{bc|}{}_a
+ \tfrac{2}{n-1}\,\bar{e}_a{}^{[b} Y^{c]d|}{}_d \; .
\end{equation}
After the elimination of $h_{a|b}\,$,
the trace of $Y^{bc|a}$ indeed contributes to the action only via
a boundary term, consistently with the shift symmetry generated by $\xi^a$
in \eqref{varY}, which is still present for $m = 0\,$.
One can cast the resulting Lagrangian in the form
\begin{equation}
{\cal L}_{0}[Y] = \tfrac{\sigma}{2(n-2)\lambda^2}\,\Big[
\nabla_{\!a\,} \hat{Y}^{cd|}{}_b\, \nabla_{\!c\,} \hat{Y}^{ab|}{}_d
+ \sigma \lambda^2 \hat{Y}_{bc|a}\,\hat{Y}^{ba|c} \Big]\;,
\end{equation}
in agreement with the result obtained by eliminating the vielbein
from the linearised McDowell-Mansouri action \cite{Basile:2015jjd}.
Introducing the Hodge dual
$T_{a[n-2]|b} = \tfrac12 \epsilon_{a[n-2]cd} \hat{Y}^{cd|}{}_b$
(which satisfies $\epsilon^{a[n-2]bc} T_{a[n-2]|b} = 0$
on account of $\hat{Y}^{ab|}{}_b = 0\,$),
one obtains a dual description of a massless graviton in (A)dS$_n\,$.
The field $T$, however, has the same structure as a massive graviton
in flat space \cite{Curtright:1980yj}; when $\lambda = 0\,$,
the dual of a massless spin-two field is instead a $GL(n)$
Young-projected\footnote{Two-column, $GL(n)$-irreducible fields
are denoted by $[p,q]$, where $p$ and $q$ stand for the lengths of
the first and second column of the corresponding Young tableau,
respectively.} $[n-3,1]$
field \cite{West:2001as,West:2002jj,Boulanger:2003vs}.
As discussed in \cite{Basile:2015jjd}, the different nature of the dual
graviton in (A)dS and flat space can be explained as follows: massless
mixed-symmetry fields display less gauge symmetries in (A)dS than in flat
space. This is the Brink-Metsaev-Vasiliev (BMV) mechanism conjectured
in \cite{Brink:2000ag}, proved for AdS$_n$
in \cite{Boulanger:2008up,Boulanger:2008kw,Alkalaev:2009vm}
and for dS$_n$ in \cite{Basile:2016aen}. It is also
discussed in \cite{Campoleoni:2012th} from the point of view of
reducibility conditions.
As a result, in the flat limit, mixed-symmetry gauge fields decompose
in multiplets of gauge fields.
In this case, in the limit $\lambda \to 0$ the field $T$ decomposes
into a ``proper'' $[n-3,1]$ dual graviton plus an additional field
of type $[n-2,1]$ that does not carry any local degrees of freedom. See \cite{Basile:2015jjd,Joung:2016naf} for further comments on the role of the field $T$.
This phenomenon can be described by introducing a suitable set
of Stueckelberg fields. In the current example, following \cite{Boulanger:2008nd}
one can introduce a new field, antisymmetric in its first three
indices and traceless, implementing the shift
\begin{equation} \label{shiftY}
\hat{Y}^{bc|}{}_a \, \to \, \hat{Y}^{bc|}{}_a
+ \tfrac{1}{\lambda}\,\nabla_{\!d} W^{bcd|}{}_a\;,\quad
W^{abc|}{}_c\equiv 0\;,
\end{equation}
either in the parent action \eqref{Lmassless} or in \eqref{LdualY}.
This leads to the Lagrangian
\begin{equation}
\label{purespin2masslessandmassive}
\begin{split}
{\cal L}_{0}[Y,W] & =
\tfrac{1}{\lambda^2} \Big[ \tfrac{1}{2}\,\nabla_{\!c} W^{abc|d} \nabla^e W_{dbe|a}
+ \lambda\,\hat{Y}^{ab|c} \nabla^e W_{cbe|a} \\
& + \tfrac{\sigma}{2(n-2)}\, \nabla_{\!b}
\hat{Y}^{ab|c} \nabla^d \hat{Y}_{cd|a} + \tfrac{\lambda^2}{2}\,
\hat{Y}^{ab|c} \hat{Y}_{ac|b} \Big] \; , \\
\end{split}
\end{equation}
that is invariant up to total derivatives
under\footnote{If one implements the Stueckelberg
shift \eqref{shiftY} already in the parent action \eqref{Lmassless},
the vielbein acquires the new transformation
$\delta_{\chi} h^{a|b} = \frac{n-3}{(n-2)\lambda}\, \nabla_{\!c} \chi^{abc}\,$.}
\begin{align}
\delta \hat{Y}^{bc|}{}_{a} &= \nabla_{\!d} \zeta^{bcd|}{}_{a}
\!+\! \nabla_{\!a} \Lambda^{bc} \!+\! \tfrac{2}{n-1}\, \bar{e}_a{}^{[b} \nabla_{\!d} \Lambda^{c]d} \!+\! \tfrac{(n-3)\lambda}{\sigma}\, \chi^{bc}{}_{a} \;,
\label{gaugevariatforYandWandhA}
\\[5pt]
\delta W^{bcd|}{}_{a} &= \nabla_{\!e} {\upsilon}^{bcde|}{}_{a}
+ \nabla_{\!a} \chi^{bcd} - \tfrac{3}{n-2}\, \bar{e}_a{}^{[b} \nabla_{\!e} \chi^{cd]e}
-\lambda\, \zeta^{bcd|}{}_{a} \; . \label{varW_gen}
\end{align}
Note that the new field can be gauged away using the shift symmetry
generated by the traceless $\zeta^{bcd|a}$, while it also brings its own
differential symmetries generated by $\upsilon^{bcde|a}$
(which is traceless and antisymmetric in the first four indices)
and by the fully antisymmetric $\chi^{abc}\,$.
Introducing the Hodge dual
$C_{a[n-3]|b} = \tfrac{1}{3!} \epsilon_{a[n-3]cde} W^{cde|}{}_b$
(that is a $GL(n)$ Young-projected $[n-3,1]$ field, since
$W^{cde|}{}_b$ is traceless) and denoting
${C'}{}_{\!a[n-4]}=C_{a[n-4]b|}{}^b$ together with
${T'}{}_{\!a[n-3]}=T_{a[n-3]b|}{}^b\,$, one obtains the dual Lagrangian
\begin{equation} \label{Ldual_massless}
{\cal L}_0[C,T] = -\tfrac{1}{2\lambda^2(n-3)!} \left[
{\cal L}[C] + \widehat{{\cal L}}_{\textrm{cross}}
+ \tfrac{\sigma}{(n-2)^2}{\cal I}[T]
\right]\, ,
\end{equation}
where (denoting antisymmetrisations with repeated indices)
\begin{align}
& {\cal L}[C] = \nabla_{\!a}C_{c[n-3]|b}\,\nabla^a C^{c[n-3]|b}
- \nabla_{\!a} C_{b[n-3]|}{}^a \,\nabla^c C^{b[n-3]|}{}_c
\label{LagC} \\&
- (n-3)\,\Big[ \nabla_{\!a}{C'}{}_{b[n-4]}\,\nabla^{a}{C'}^{b[n-4]}
+\nabla_{\!b} C_{ab[n-4]|}{}^c \,\nabla^{a}C^{b[n-3]|}{}_c
\nonumber \\
& -2(-1)^{n}\nabla^a C_{b[n-3]|a}\,\nabla^{b} {C'}^{b[n-4]}
- (n-4) \nabla_{\!b}{C'}{}_{\!cb[n-5]}\,\nabla^{c}{C'}^{b[n-4]} \Big]\;,
\nonumber
\end{align}
\begin{equation}
\begin{split}
\widehat{{\cal L}}_{\textrm{cross}} = 2\lambda\,&\Big[ T_{a[n-3]b|}{}^c\,\nabla^b C^{a[n-3]|}{}_c
- {T'}{}_{a[n-3]}\,\nabla^{b} C^{a[n-3]|}{}_b \\
&+(-1)^n(n-3)\, {T'}{}_{\!a[n-3]}\,\nabla^{a}{C'}{}^{a[n-4]}\Big]\;,
\end{split}
\end{equation}
and
\begin{equation}
\begin{split}
{\cal I}[T] =\;& {\cal L}[T] + \sigma(n-2)\lambda^2 \,\times
\\
&\,\times\, \Big[T^{a[n-2]|b}T_{a[n-2]|b} - (n-2){T'}^{a[n-3]}{T'}_{\!a[n-3]}\Big]\;.
\end{split}
\end{equation}
The expression for ${\cal L}[T]$ is obtained from ${\cal L}[C]$
in \eqref{LagC}
by replacing everywhere in the latter expression the symbols $C$ and $n$
by $T$ and $n+1\,$, respectively.
Lagrangian~\eqref{Ldual_massless} is invariant, up to total derivatives, under
\begin{align}
\delta T_{a[n-2]|b} & = (-1)^{n-1}(n-2)\, \Big[ \nabla_{\!a} \tilde{\zeta}_{a[n-3]|b} + (n-3)\sigma\lambda\, g_{ba} \tilde{\chi}_{a[n-3]} \nonumber \\
& \phantom{=} + \tfrac{(-1)^{n-1}}{n-1} \left( \nabla_{\!b} \tilde{\Lambda}_{a[n-2]} + (-1)^{n-1} \nabla_{\!a} \tilde{\Lambda}_{a[n-3]b} \right) \Big]\;, \\[5pt]
\delta C_{a[n-3]|b} & = (-1)^{n-1}(n-3)\, \nabla_{\!a} \tilde{\upsilon}_{a[n-4]|b}
- \lambda\, \tilde{\zeta}_{a[n-3]|b} \nonumber \\
& \phantom{=} + \tfrac{n-3}{n-2} \left( \nabla_{\!b} \tilde{\chi}_{a[n-3]} + (-1)^n \nabla_{\!a} \tilde{\chi}_{a[n-4]b} \right) \,,
\end{align}
where the parameters $\tilde{\zeta}_{a[n-3]|b}$, $\tilde{\Lambda}_{a[n-2]}\,$,
$\tilde{\upsilon}_{a[n-4]|b}$ and $\tilde{\chi}_{a[n-3]}$ are the Hodge duals of
those entering the transformations \eqref{gaugevariatforYandWandhA}
and \eqref{varW_gen} (the dualisation always involves only the group of
antisymmetrised indices).
In the limit $\lambda \to 0$ the field $T$ decouples and does not
propagate any degrees of freedom, while one retains the gauge field
$C_{a[n-3]|b}\,$, the dual graviton in flat space~\cite{Boulanger:2003vs}.
In a spacetime of any dimension $D > n$, the action \eqref{Ldual_massless} ---~featuring one of the two possible BMV couples of fields including $T_{a[n-2]|b}$~--- would give a non-unitary propagation in dS. This is manifested by the $\sigma$-dependent relative sign between the kinetic terms that we obtained. In this specific case, the relative sign is irrelevant because $Y$ is a topological field in flat space and, indeed, the massless theory is unitary for any value of the cosmological constant.
\subsection{Partially-massless case}\label{sec:PM}
Partially-massless spin-2 fields exist for any
non-vanishing values of the cosmological
constant, although they are unitary only in
dS \cite{Higuchi:1986wu}.
To exhibit these facts, in this subsection we slightly modify our conventions,
multiplying ${\cal L}^{(1)}$ by $-\sigma\,$.
With this choice the factor $\sigma$ in \eqref{mu-m} is replaced by $-1$,
so that one can reach the point $\mu = 0$ in both dS and AdS. This leads to the parent Lagrangian
\begin{equation} \label{LPM3}
\begin{split}
& {\cal L}_{\textrm{PM}}[h,Y,A,F] = h_{a|b}\, {\cal C}^{a|b} + \tfrac{\sigma}{\widetilde{m}}\, A_a \nabla_{\!b}\, {\cal C}^{b|a} \\
& - \tfrac{1}{2} \left( Y^{bc|a} Y_{ab|c} + \tfrac{1}{n-2}\, Y^{ab|}{}_b Y_{ac|}{}^c \right) - \tfrac{\sigma}{4}\, F_{ab} F^{ab} \, ,
\end{split}
\end{equation}
where we defined
\begin{equation} \label{mPM}
{\cal C}^{a|b} = \nabla_{\!c} Y^{ac|b} - \widetilde{m}\, F^{ab} \, , \qquad
\widetilde{m} = \pm\, \lambda \sqrt{\frac{n-2}{2}} \, .
\end{equation}
In the conventions adopted in this subsection, the gauge symmetries of the action are
\begin{align}
\delta h_{a|}{}^{b} & = \nabla_{\!a} \xi^b + \Lambda_a{}^b + \tfrac{2\widetilde{m}}{n-2}\, \bar{e}_a{}^b \, \epsilon \, , \label{varh_PM} \\[3pt]
\delta Y^{bc|}{}_{a} &= \nabla_{\!a} \Lambda^{bc} + 2\, \bar{e}_a{}^{[b} \nabla_{\!d} \Lambda^{c]d} \, , \\[3pt]
\delta A_a & = \nabla_{\!a} \epsilon + \sigma\, \widetilde{m}\, \xi_a \, , \\[3pt]
\delta F^{ab} & = -\,2\,\sigma\,\widetilde{m}\, \Lambda^{ab} \, . \label{varF_PM}
\end{align}
In \eqref{LPM3} we stressed that the fields $h_{a|b}$ and $A_a$ are both Lagrange multipliers when $\mu = 0$ (although the constraint imposed by the latter field is not independent). The analysis of the partially-massless case therefore follows closely that of a massless graviton in flat space \cite{Boulanger:2003vs}, rather than those presented in sections \ref{sec:massless} and \ref{sec:massive}. The constraint \mbox{${\cal C}_{a|b} = 0$} is solved by
\begin{equation} \label{solY}
Y^{bc|}{}_a = \frac{1}{\lambda}\, \nabla_{\!d} W^{bcd|}{}_a - \frac{\sigma}{2\widetilde{m}} \left( \nabla_{\!a} F^{bc} + 2\, \bar{e}_a{}^{[b} \nabla_{\!d} F^{c]d} \right) \,,
\end{equation}
where $W^{bcd|}{}_a$ has the same structure as the field introduced in the
Stueckelberg shift~\eqref{shiftY}. In particular, it is traceless.
Substituting \eqref{solY} in \eqref{LPM3}, one obtains
\begin{equation} \label{LPM_fin}
\begin{split}
& {\cal L}_{\textrm{PM}}[W] = -\,\frac{1}{2\lambda^2}\,\nabla_{\!d} W^{bcd|a} \nabla^{e} W_{abe|c} \\
& + \nabla_{\!a} \left( F^{ab} \nabla^{c} F_{bc} - F_{bc} \nabla^{c} F^{ab} + \tfrac{4\sigma\widetilde{m}}{\lambda}\, F_{bc} \nabla_{\!d} W^{abd|c} \right) \,.
\end{split}
\end{equation}
This Lagrangian actually depends only on the field
$W^{bcd|a}$: $F^{ab}$ contributes only via a total derivative consistently with the shift symmetry \eqref{varF_PM}. It is still invariant under
\begin{equation}
\delta W^{bcd|}{}_{a} = \nabla_{\!e} {\upsilon}^{bcde|}{}_{a} \, ,
\end{equation}
while the other differential symmetry that was present in the massless case (cf.~\eqref{varW_gen}) is absent.
All gauge symmetries that the field $W^{bcd|a}$ and, consequently,
its Hodge dual would display in flat space can be recovered by implementing the
Stueckelberg shift
\begin{equation}
W^{bcd|}{}_a \to W^{bcd|}{}_a + \frac{\widetilde{m}^{-1}}{n-3}\,
\left( \nabla_{\!a} U^{bcd} - \tfrac{3}{n-2}\, \bar{e}_a{}^{[b}
\nabla_{\!e} U^{cd]e} \right) \,.
\end{equation}
Substituting in \eqref{LPM_fin} one obtains the Lagrangian
\begin{equation}
\begin{split}
& {\cal L}_{\textrm{PM}}[W,U] = -\,\tfrac{1}{2\lambda^2}\,\nabla_{\!d} W^{bcd|a} \nabla^{e}
W_{abe|c} + \tfrac{\sigma}{\widetilde{m}} \, U_{abc} \nabla_{\!d} W^{abd|c} \\
& - \tfrac{\sigma}{2(n-2)\widetilde{m}^2}\, \nabla_{\!c} U^{abc} \nabla^d U_{abd} - \tfrac{\lambda^2}{2\widetilde{m}^2}\, U_{abc} U^{abc} \, ,
\end{split}
\end{equation}
which is invariant up to total derivatives under
\begin{align}
\delta W^{bcd|}{}_a & = \nabla_{\!e} \upsilon^{bcde|}{}_a \!+\! \nabla_{\!a} \chi^{bcd} \!-\! \tfrac{3}{n-2}\, \bar{e}_a{}^{[b} \nabla_{\!e} \chi^{cd]e} \!-\! \tfrac{\sigma\lambda^2}{\widetilde{m}}\, \rho^{bcd}{}_a \, , \label{varPMW} \\[5pt]
\delta U^{abc} & = \nabla_{\!d} \rho^{abcd} - (n-3)\,\widetilde{m}\, \chi^{abc} \, . \label{varPMU}
\end{align}
The contribution in $\rho$ in \eqref{varPMW} (that was absent in \eqref{varW_gen}) is necessary because, contrary to the massless case, the field $U$ does not enter the action only via its divergence.
As in the massless case, the sign of one of the two kinetic terms depends on $\sigma$.
This is consistent with the observation that, after Hodge dualisation,
one obtains a BMV couple of fields which is unitary only in
dS \cite{Basile:2016aen}.
However, in this case both fields propagate in the flat limit:
the $[n-3,1]$ dual of $W$ carries the spin-2 helicities,
while the $[n-3]$ dual of $U$ carries the spin-1 helicities.
Consequently, the sign flip of a kinetic terms does matter: recovering
the BMV couple of fields that is not-unitary in AdS is just another
way to see that partially-massless fields are not unitary in AdS.
Using the dual field $C_{a[n-3]|b}$ defined as in the massless case
and introducing the Hodge dual field
${A}_{a[n-3]} = \tfrac{1}{3!} \epsilon_{a[n-3]bcd} U^{bcd}\,$,
the Stueckelberg Lagrangian we obtain for the dual partially massless
spin-2 field in (A)dS$_n$ is
\begin{align}
{\cal L}_{\textrm{PM}} = -\tfrac{1}{2(n-3)!\lambda^2}\Big[
{\cal L}[C] \,-\, & \tfrac{2\sigma \lambda^2}{(n-2)\widetilde{m}^2}\,
{\cal L}[A]
+ \tfrac{4\sigma\lambda^2}{\widetilde{m}}
\widetilde{\cal L}_{\textrm{cross}} \Big]\;,
\end{align}
where ${\cal L}[C]$ is given in \eqref{LagC},
\begin{align}
{\cal L}[A] = \nabla_{\!a} A_{b[n-3]}\,\nabla^a A^{b[n-3]}
\,-\,&(n-3)\nabla^a{A}_{c[n-4]a}\,\nabla_{\!b}{A}^{c[n-4]b}
\nonumber \\
& + 3\sigma \lambda^2 \,{A}_{a[n-3]} {A}^{a[n-3]}\;,
\end{align}
and the cross terms are
\begin{equation}
\widetilde{\cal L}_{\textrm{cross}} =
{A}^{a[n-3]} \left( \nabla_{\!b}C_{a[n-3]|}{}^b
+ (-1)^{n-1} (n-3)\,\nabla_{\!a}{C'}{}_{a[n-2]} \right) .
\end{equation}
The action is invariant under
\begin{align}
\delta C_{a[n-3]|b} & = (-1)^{n-1}(n-3) \left( \nabla_{\!a} \tilde{\upsilon}_{a[n-4]|b} - \tfrac{\sigma\lambda^2}{\widetilde{m}}\, g_{ba}\, \tilde{\rho}_{a[n-4]} \right) \nonumber \\
& \phantom{=} + \tfrac{n-3}{n-2} \left( \nabla_{\!b} \tilde{\chi}_{a[n-3]} + (-1)^n \nabla_{\!a} \tilde{\chi}_{a[n-4]b} \right) , \\[5pt]
\delta A_{a[n-3]} & = (n-3) \left( (-1)^{n-1}\nabla_{\!a} \tilde{\rho}_{a[n-4]}
- \widetilde{m}\, \tilde{\chi}_{a[n-3]} \right) ,
\end{align}
where the parameters $\tilde{\upsilon}_{a[n-4]|b}$, $\tilde{\chi}_{a[n-3]}$ and $\tilde{\rho}_{a[n-4]}$ are the Hodge duals of those entering the transformations \eqref{varPMW} and \eqref{varPMU}.
\subsection{Massive case}\label{sec:massive}
We now consider the full Stueckelberg action presented in section \ref{sec:parent}. The elimination of the fields $h_{a|b}$, $A_a$ and $\varphi$ has been considered in \cite{Zinoviev:2005zj,Khoudeir:2008bu}. In the spirit of our discussion of the special points $m = 0$ and $\mu = 0$, we complement these works by exhibiting a dual description with a smooth massless and flat limit. For generic values of $m$, $h_{a|b}$ is an auxiliary field and it can be eliminated through its equation of motion as in section~\ref{sec:massless}. The field $A_a$ is instead a Lagrange multiplier enforcing the constraint
\begin{equation}
\nabla_{\!b} F^{ba} - \tfrac{2m}{n-2}\, Y^{ab|}{}_b - \mu\,\pi^a = 0 \, ,
\end{equation}
which can be solved by expressing $\pi^a$ in terms of the other fields.
The equation of motion of $\varphi$ does not bring any new information,
since it is not independent on account of the Noether identity associated
with the gauge symmetry generated by $\epsilon\,$.
Substituting the on-shell values of $h_{a|b}$ and $\pi^a$
in the Stueckelberg Lagrangian leads to \cite{Zinoviev:2005zj}
\begin{align}
& {\cal L}[\hat{Y},F] = \tfrac{1}{\mu^2} \left[ \tfrac{n-1}{n-2}\, \nabla_{\!b} \hat{Y}^{ab|c} \nabla^d \hat{Y}_{cd|a} + \tfrac{\mu^2}{2}\, \hat{Y}^{ab|c} \hat{Y}_{ac|b} \right. \label{Lag2massive} \\
& \left. + \tfrac{1}{2}\, \nabla_{\!b} F^{ab} \nabla^c F_{ac} - \tfrac{2(n-1)m}{n-2}\, F_{ab} \nabla_{\!c} \hat{Y}^{bc|a} + \left( \tfrac{\mu^2}{4} - \tfrac{(n-1)m^2}{n-2} \right) F_{ab} F^{ab} \right] , \nonumber
\end{align}
where we recall that the parameters $m$ and $\mu$ are related by \eqref{mu-m}.
One can then consider the Hodge duals of the fields $\hat{Y}$ and $F$
and obtain a dual theory for a massive graviton in terms of the Stueckelberg
coupling of a massless spin-2 field (accounted by the $[n-2,1]$ dual of $\hat{Y}$)
with a Proca field (accounted by the $[n-2]$ dual of $F$).
Its gauge symmetries are those inherited from \eqref{var1} and
\eqref{varY} after dualisation.
In order to obtain a smooth massless and flat limit, one should introduce
two additional fields: the traceless $W^{bcd|}{}_a$ that we already encountered
in section \ref{sec:massless} and a 3-form $U^{abc}\,$.
This will allow to recover all Curtright gauge symmetries for the Hodge dual
of $\hat{Y}^{bc|}{}_a$ and the usual gauge symmetry for the massless $(n-2)\,$-form
which is the Hodge dual of $F^{ab}\,$.
Due to the coupling $F^{ab} h_{a|b}$ in \eqref{Lcross},
introducing the 3-form via a Stueckelberg shift of $F^{ab}$ would modify
the equation of motion for $h_{a|b}$ and, as a result, it would introduce
second-order kinetic terms mixing $U^{abc}$ with the fully antisymmetric
projection of $Y^{ab|c}\,$. On the other hand, the shifts
\begin{subequations} \label{shift_massive}
\begin{align}
Y^{bc|a} & \to Y^{bc|a} + \tfrac{1}{\mu}\, \nabla_{\!d} W^{bcd|a} - \tfrac{m}{\mu}\, U^{abc} \, , \\
F^{ab} & \to F^{ab} + \tfrac{1}{\mu}\, \nabla_{\!c} U^{abc}
\end{align}
\end{subequations}
do not modify the equation of motion for $h_{a|b}$ and therefore they cannot introduce any mixed kinetic term. The elimination of the fields $h_{a|b}$, $A_a$ and $\varphi$ then proceeds as above and one obtains the sum of the kinetic terms
\begin{equation} \label{kinetic_massive}
\begin{split}
{\cal K} =\; & \tfrac{1}{\mu^2}
\Big[
\tfrac{1}{2}\,\nabla_{\!c} W^{abc|d} \nabla^e W_{dbe|a}
+ \tfrac{n-1}{n-2}\, \nabla_{\!b} \hat{Y}^{ab|c} \nabla^d \hat{Y}_{cd|a} \\
& + \tfrac{1}{4}\, \nabla_{\!c} U^{abc} \nabla^d U_{abd} + \tfrac{1}{2}\,
\nabla_{\!b} F^{ab} \nabla^c F_{ac} \Big]\;,
\end{split}
\end{equation}
with the cross couplings
\begin{equation} \label{Lcross_massive}
\begin{split}
{\cal L}^{(1)}_{\textrm{cross}} =\; & \tfrac{1}{\mu}\, \Big[ \hat{Y}_{ab|c}
\nabla_{\!d} W^{acd|b} + \tfrac{m}{\mu}\, U_{abc} \nabla_{\!d} W^{abd|c} \\
& - \tfrac{2(n-1)m}{(n-2)\mu}\, F_{ab} \nabla_{\!c} \hat{Y}^{bc|a}
+ \tfrac{1}{2}\, F_{ab} \nabla_{\!c} U^{abc} \Big]
\end{split}
\end{equation}
and mass-like terms
\begin{equation} \label{Lmass_massive}
\begin{split}
& {\cal L}^{(2)}_{\textrm{cross}} = \tfrac{1}{2}\,
\hat{Y}^{ab|c} \hat{Y}_{ac|b} + \tfrac{m}{\mu}\, \hat{Y}^{ab|c} U_{abc} \\
& - \tfrac{m^2}{2\mu^2}\, U^{abc} U_{abc}
+ \left( \tfrac{1}{4} - \tfrac{(n-1)m^2}{(n-2)\mu^2} \right)
F^{ab} F_{ab} \;.
\end{split}
\end{equation}
This Lagrangian is invariant up to total derivatives under the following gauge transformations:\footnote{Implementing the shift \eqref{shift_massive} before the elimination of $h_{a|b}$ etc.\ from the parent action does not modify the gauge transformations of $A_a$, $\varphi$ and $\pi^a$. The variation of $h_{a|b}$ takes instead the same form as in the massless case and acquires a contribution $\delta_{\chi} h^{a|b} = \frac{n-3}{(n-2)\mu}\, \nabla_{\!c} \chi^{abc}$.}
\begin{align}
\delta W^{bcd|}{}_a & = \nabla_{\!e} {\upsilon}^{bcde|}{}_{a}
+ \nabla_{\!a} \chi^{bcd} - \tfrac{3}{n-2}\, \bar{e}_a{}^{[b} \nabla_{\!e} \chi^{cd]e} \nonumber \\
& \phantom{=} -\mu\, \zeta^{bcd|}{}_{a} - m\, \rho^{bcd}{}_{a} \, , \label{varWmassive}\\[5pt]
\delta \hat{Y}^{bc|}{}_a & = \nabla_{\!d} \zeta^{bcd|}{}_{a} + \nabla_{\!a} \Lambda^{bc} + \tfrac{2}{n-1}\, \bar{e}_a{}^{[b} \nabla_{\!d} \Lambda^{c]d} \nonumber \\
& \phantom{=} + \tfrac{(n-3)\mu}{2(n-1)}\,\chi^{bc}{}_{a} - m\, \psi^{bc}{}_a \, , \label{varYmassive} \\[5pt]
\delta U^{abc} & = \nabla_{\!d} \rho^{abcd} - \mu\, \psi^{abc} + \tfrac{2(n-3)m}{n-2}\, \chi^{abc} \, , \label{varUmassive} \\[5pt]
\delta F^{ab} & = \nabla_{\!c} \psi^{abc} + 2m\, \Lambda^{ab} \, . \label{varFmassive}
\end{align}
The action involving the Hodge duals of the previous fields now admits a
smooth flat and massless limit, in which different helicities decouple.
The spin-two ones are carried by the $[n-3,1]$ Hodge dual of $W$ (as discussed
in section~\ref{sec:massless}), while spin-one and zero helicities are carried,
respectively, by the fully-antisymmetric Hodge duals of $U$ and $F\,$.
We refrain from displaying this action explicitly, as it can
straightforwardly be obtained by expressing all fields in terms of their Hodge
duals in \eqref{kinetic_massive}--\eqref{Lmass_massive}. One can also check
that the appropriate Curtright gauge symmetries are recovered from
\eqref{varWmassive}--\eqref{varFmassive} together with their gauge-for-gauge
symmetries.
\section*{Acknowledgments}
We are grateful to Th.~Basile for discussions and collaboration at an early
stage of this work.
We thank X.~Bekaert, J.~A.~Garc\'ia and L.~Traina for fruitful discussions.
We performed or checked several computations with the package xTras \cite{Nutma:2013zea}
of the suite of Mathematica packages xAct.
N.B.\ thanks ETH Z\"urich and the Institut Denis Poisson (Universit\'e de Tours),
while A.C.\ thanks the Universit\`a di Firenze and INFN (sezione di Firenze),
for kind hospitality during the completion of this paper.
The stay of N.B.\ at the Institut Denis Poisson in Tours was
funded by a grant of the Universit\'e de Mons (UMONS).
The work of N.B.\ has been supported in part by a FNRS PDR grant (number T.1025.14), while the work of A.C.\ has been supported in part by the NCCR SwissMAP, funded by the Swiss National Science Foundation.
|
{
"timestamp": "2018-07-13T02:10:30",
"yymm": "1804",
"arxiv_id": "1804.05588",
"language": "en",
"url": "https://arxiv.org/abs/1804.05588"
}
|
\section{Introduction}
A common task in the context of intelligent vehicles is risk assessment, where a risk score is calculated from interpreting a set of possible future paths as generated from a path prediction method.
The risk score can e.g. encode the chance of a collision between a pedestrian and a vehicle for some point in the future, where strong prediction capabilities are required in order to enhance the time horizon where a useful prediction is possible.
Basing decisions on anticipated pedestrian behavior yields the advantage of earlier decisions, especially in time critical situations.
Further, the generation of multi-modal predictions is an essential capability, since environments, like the one depicted in Fig. \ref{fig:problem} commonly offer multiple choices, which are not equal likely but in general none of the options is totally unlikely.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.375\textwidth]{problem.jpg}
\end{center}
\vspace{-0.2cm}
\caption{Multi-modal path prediction: Where might the person in the orange circle be in $N$ time steps?}
\label{fig:problem}
\vspace{-0.28cm}
\end{figure}
In contrast to prediction approaches, taking only the motion state of an object into account, path prediction in dynamic environments can strongly benefit from a complex transition or motion model, capable of capturing motion decisions of pedestrians, including:
\begin{enumerate}
\item Long-term dependencies, where decisions are influenced from past motion in a non-linear fashion
\item The inclusion of the scene geometry, e.g. implicitly included in the statistical model (\cite{huang2016deep})
\item Local interaction with dynamic objects in the scene (\cite{bartoli2017context}\cite{alahi2016social})
\item Decision making on different time scales, separating local and global path planning
\item Goal oriented motion, mainly summarized as the intuition of a pedestrian
\end{enumerate}
While the intuition of observed pedestrians is hardly measurable, the other components can either be observed or learned from data when using a statistical transition model.
A class of flexible statistical models that can be used on varying time scales are recurrent neural networks.
For multi-modal path prediction, these models can be built, such that a probability density function over the state space is generated.
Approaches dealing with path prediction can roughly be categorized by the targeted type of path prediction, which is goal-directed (\emph{what is the most likely path towards a given destination}) or undirected path prediction (\emph{where will the observed pedestrian be in $N$ time steps}).
Since the goal location is usually unknown, this paper is mainly concerned with undirected path prediction.
While recent undirected path prediction approaches are only concerned with unimodal or example-based multi-hypothesis prediction, multi-modal path prediction is only subject to goal-directed prediction approaches.
Towards this end, the goal of this paper is to introduce an undirected, multi-modal path prediction approach based on techniques known from particle filtering.
In the following, a brief summary of related work and the recurrent model used for path prediction are provided (section \ref{s:background}).
Next, the proposed path prediction approach is presented in section \ref{s:path_pred}.
The quality of the generated probability distribution is evaluated on different synthetic and real-world conditions (section \ref{sec:exp}).
Section \ref{s:conclusion} provides some concluding remarks.
\section{Background}
\label{s:background}
The path prediction problem has been addressed with a variety of approaches based on dynamic models, like recurrent neural networks (e.g. \cite{bartoli2017context}\cite{alahi2016social}) or Markov processes (e.g. \cite{kitani2012activity}), or on just a single observation (e.g. \cite{huang2016deep}).
In the case of goal-directed prediction, a policy optimization is performed after processing observations.
For undirected predictions, a state-transition model is used to project given information into the future.
Further, path prediction can be approached on a local and a global scale.
On a global scale, a path taken by a pedestrian is mainly influenced by the intended destination (indicated by observed past positions) and the scene geometry (e.g. \cite{bartoli2017context}\cite{kitani2012activity}).
On a local scale, dynamic obstacles come into play and alter the intended path (e.g. \cite{alahi2016social}).
There are also hybrid approaches, targeting both scales of path prediction (e.g. \cite{lee2017desire}).
In the context of risk assessment, global-scale path prediction should be considered for long-term decisions and complemented by local-scale path predictions for short-term decisions, respectively.
The latter is especially relevant, as pedestrian interactions may lead to risky situations.
Although being an inference problem, path prediction can also be construed as a sequence generation problem by applying recurrent neural networks (RNNs).
One of the main advantages in using RNNs is their capability of processing and generating sequences of variable lengths.
While simple RNNs struggle in capturing long-term dependencies, gated memory blocks, like the long short-term memory (LSTM, \cite{hochreiter1997long}) or gated recurrent unit (GRU, \cite{cho2014learning}), can be used.
Further, when combining the RNN with a mixture density layer (MDL, \cite{bishop2006pattern}), the model can be trained to output a continuous probability distribution.
Here, undirected, multi-modal path prediction is based on a LSTM model that is combined with a MDL, due to the flexibility of recurrent models and the probabilistic output.
\subsection{The LSTM-MDL model}
\label{s:model}
The LSTM-MDL model \cite{graves2013generating} is a recurrent neural network (using LSTM cells\footnote{It could as well be built using Gated Recurrent Units.}) that is used to parameterize a mixture density output layer (MDL).
The model is trained by minimizing the negative log-likelihood loss.
Given an input sequence $\mathcal{X}^T = \{\mathbf{x}^1, \mathbf{x}^2, ..., \mathbf{x}^T\}$ of $T$ consecutive pedestrian positions along a trajectory, the model generates a, at least $K$-modal, prediction for the next position $\mathbf{x}^{t+1}$ or position offset $\delta^t$ at each time step.
This prediction of a model $\mathcal{M}$ at time $t$ is a $K$-component Gaussian mixture model (GMM) defined by parameters $\pi_k$, $\mu_k$ and $\Sigma_k$ (weight, mean and covariance of the k'th Gaussian):
\begin{align}
\label{eq:gmm}
\begin{split}
\mathcal{M}(\mathbf{x}^t) &= (\pi^t, \mu^t, \Sigma^t) \\
&\Rightarrow p(\mathbf{\delta}|\mathbf{x}^t, \pi^t, \mu^t, \Sigma^t) \\
&\overset{\wedge}{=} p(\mathbf{\delta}^t|\mathbf{x}^t) = \sum_{k=1}^{K} \pi_k \mathcal{N}(x|\mu_k,\Sigma_k),
\end{split}
\end{align}
\subsection{Path Prediction using the LSTM-MDL model}
Several approaches base undirected path prediction on this architecture, but are limited to unimodal predictions.
For example in \cite{bartoli2017context}\cite{alahi2016social}, the LSTM-MDL model serves as an egocentric pedestrian motion model and interaction was included in order to improve prediction results.
For the purpose of this paper, however, a simpler model is used to reduce necessary amounts of training data and further to reduce side-effects while evaluating the proposed prediction approach.
Therefore, the model presented in \cite{hug2017reliability}, which does not regard any context (e.g. neighboring pedestrians or the scenery), is used.
Even though the LSTM-MDL model outputs a probability distribution, the model itself is inherently deterministic.
Because of this, the model has to be embedded in an inference scheme that explores the probability distributions generated by the model.
Due to the models property of processing each time step in succession, inference needs to follow an iterative approach and prompt new positional distribution from the model in each time step.
Further, this structure leads to a series of conditional distributions, which is why global samplers, like Gibbs sampling \cite{bishop2006pattern}, cannot be applied.
The approach commonly used to generate predictions using a trained LSTM-MDL model $\mathcal{M}$ that generates a GMM $p(\mathbf{\delta}^t|\mathbf{x}^t)$ describing the offset distribution to the next position (see (\ref{eq:gmm})) is as follows (for each time step):
\begin{enumerate}
\item Generate $p(\mathbf{\delta}^t|\mathbf{x}^t)$ by passing the current position $\mathbf{x}^t$ through $\mathcal{M}$
\item Transform offset distribution into a positional distribution $p(\mathbf{x}^{t+1}|\mathbf{x}^t)$ by adding $\mathbf{x}^t$ to every $\mu^t_k \in \mu^t$
\item Sample a position $\hat{\mathbf{x}}^{t+1}$ from $p(\mathbf{x}^{t+1}|\mathbf{x}^t)$
\item Use $\hat{\mathbf{x}}^{t+1}$ in step 1 to generate the next prediction.
\end{enumerate}
Most commonly, $\hat{\mathbf{x}}^{t+1}$ maximizes $p(\cdot|\mathbf{x}^t)$, thus generating a maximum likelihood solution, disregarding less likely paths.
To achieve a multi-modal prediction, the pdf output of the model has to be fed back into it (as opposed to a single position) in order to move forward in time.
Further, the LSTM-MDL model expects single positions as input, thus an efficient sample-based approximation of the probability distribution has to be used.
This, in turn, may lead to exponential growth in the number of samples and the number of LSTM cell states.
In total this leads to complex conditional distributions which are usually intractable in a straightforward way.
\subsection{Particle-based inference on recurrent models}
To cope with these problems, a particle-based prediction approach is proposed.
Towards this end, common particle filtering algorithms are adapted for usage on top of a LSTM-MDL model.
Thereby, different well-established techniques in the area of particle filtering were incorporated and evaluated in terms of applicability and usefulness.
Although using a particle-based approach for inference on RNNs is no novelty when looking at state-space models \cite{gu2015neural}\cite{zheng2017state} concerned with modeling latent space, there is a fundamental difference between these models and the LSTM-MDL model.
While the aforementioned state-space models are themselves stochastic in their latent space and monte carlo simulation is applied in the training phase, the LSTM-MDL model itself is deterministic and turns into a probabilistic model by embedding it into a monte carlo simulation when performing prediction.
\subsection{Problem description}
\label{s:problem}
In the following, the goal is to generate a probabilistic prediction for the next $N$ positions $\{p(\mathbf{x}^{T+1}), p(\mathbf{x}^{T+2}), ..., p(\mathbf{x}^{T+N})\}$, given an observation $\mathcal{X}^T = \{\mathbf{x}^1, \mathbf{x}^2, ..., \mathbf{x}^T\}$ of $T$ consecutive pedestrian positions along a trajectory.
In order to explore all likely paths the model is aware of, the GMMs $p(\mathbf{x}^{t+1}|\mathbf{x}^t)$ have to be fed back into $\mathcal{M}$ to progress in time at each time step.
As the model requires positions as input, $p(\mathbf{x}^{t+1}|\mathbf{x}^t)$ can be represented by a sufficiently large number of samples, which are then passed through $\mathcal{M}$.
The distributions $p_s(\mathbf{x}^{t+2}|\mathbf{x}^{t+1})$ generated by each sample $s$ can then be used to draw the next set of samples and so on.
Following this na\"ive approach, the number of samples increases exponentially with each time step, making this exploration prohibitive in terms of computation time and memory consumption.
Additionally, each sample is associated with a separate LSTM cell state, thus the number of states also grows exponentially.
\section{Particle-based path prediction}
\label{s:path_pred}
A more sophisticated approach to approximate a series of distributions using a fixed number of weighted samples is taken in particle filtering \cite{arulampalam2002tutorial}.
Here, the samples (particles) are updated in each time step considering a simple motion model and frequent measurements.
In the process of predicting the positional distribution at time $t+n$, the motion model could be applied recursively.
On a considerable high level, the particle filtering scheme can be adapted for use in multi-modal path prediction using a LSTM-MDL model as follows.
A set $\mathcal{P}^t = \{\mathbf{p}^t_1, \mathbf{p}^t_2, ..., \mathbf{p}^t_M\}$ of $M$ particles with corresponding weights $\Omega^t = \{\omega^t_1, \omega^t_2, ..., \omega^t_M\}$ is used to describe $p(\mathbf{x}_{t}|\mathbf{x}_{t-1})$ at each time step $t$.
The LSTM-MDL model serves as the motion model to propagate the set of particles forward in time.
When passing $\mathcal{P}^t$ through $\mathcal{M}$, a set of $M$ GMMs (see (\ref{eq:gmm}))
\begin{align}
\mathcal{M}(\mathcal{P}^t) \Rightarrow \mathcal{G}^{t+1} = \{p_m(\cdot|\mathbf{p}^t_m) \mid \mathbf{p}^t_m \in \mathcal{P}^t\},
\end{align}
one per particle, is generated.
To allow a fixed number of particles, all distributions in $\mathcal{G}^{t+1}$ need to be aggregated into a single $K \cdot M$ - component GMM
\begin{align}
\label{eq:comb_gmm}
p(\mathbf{x}^{t+1}|\mathcal{G}^{t+1}) = \sum_{m=1}^{M} \omega^t_m \cdot p_m.
\end{align}
Here, it is vital that $\mathcal{M}$ is an accurate model of various paths, as there are no incoming measurements to correct intermediate predictions.
Further, without new measurements, it has to be clarified how to determine the particle weights $\Omega$.
This step is of great importance, as the particle weights affect the aggregation of $\mathcal{G}^{t+1}$.
Additionally, due to the fact that the LSTM-MDL model cannot be used to update given particles, the usual particle update step known from the particle filter has to be replaced by completely re-sampling all particles in each time step given $p(\mathbf{x}^{t+1}|\mathcal{G}^{t+1})$.
\begin{algorithm}[h]
\begin{algorithmic}[1]
\Require $\mathcal{X}^T,~N,~\mathcal{M}$ \Comment{observation $\mathcal{X}^t$, model $\mathcal{M}$}
\State $p(\cdot|\mathbf{x}^T) \gets precondition(\mathcal{M}, \mathcal{X}^t)$ \Comment{$K$ components}
\State $\mathcal{P}^1 \gets draw\_samples(p(\cdot|\mathbf{x}^T))$
\State $\Omega^1 \gets weight\_particles(\mathcal{P}^1, p(\cdot|\mathbf{x}^T))$
\For{$s \gets [2..N]$} \Comment{propagate particles}
\State $\mathcal{G}^s \gets \mathcal{M}(\mathcal{P}^{s-1})$ \Comment{$\{p_m(\cdot|\mathbf{p}^{s-1}_m) \mid \mathbf{p}^{s-1}_m \in \mathcal{P}^{s-1}\}$}
\State $p(\cdot|\mathcal{G}^{s}) \gets \sum_{m=1}^{M} \omega^{s-1}_m \cdot p_m$ \Comment{$M \cdot K$ components}
\State $\mathcal{P}^{s} \gets draw\_samples(p(\cdot|\mathcal{G}^{s}))$
\State $\Omega^s \gets weight\_particles(\mathcal{P}^{s}, p(\cdot|\mathcal{G}^{s}))$
\EndFor
\State \textbf{return} $p(\cdot|\mathcal{G}^{s}),~\mathcal{P}^s$
\end{algorithmic}
\caption{Particle Propagation}\label{alg:particle_prop}
\end{algorithm}
The proposed prediction method is summarized in algorithm \ref{alg:particle_prop}.
The algorithm starts given an observation sequence $\mathcal{X}^T$, the number $N$ of time steps to predict and the trained LSTM-MDL model $\mathcal{M}$.
In the initialization stage (lines 1 to 3), the model is conditioned on the observations by consecutively passing each position $\mathbf{x}^t \in \mathcal{X}^T$ through $\mathcal{M}$, while updating the LSTM cell states.
The transformed model output at time step $T$ $p(\cdot|\mathbf{x}^T)$ (cf. (\ref{eq:gmm})) describes the first predicted positional distribution.
From here, the steps described above are performed in a loop to produce the sequence of conditional distributions $\{p(\mathbf{x}^{T+1}|\cdot), p(\mathbf{x}^{T+2}|\cdot), ..., p(\mathbf{x}^{T+N}|\cdot)\}$ (see section \ref{s:problem}).
It has to be noted, that in step 5, where the offset distribution is transformed, the particle that generated the corresponding distribution has to be used.
I.e. $p_m(\mathbf{x}^{t+1}|\mathbf{p}^t_m)$ is obtained from $p_m(\mathbf{\delta}^{t}|\mathbf{p}^t_m)$ by adding $\mathbf{p}^t_m$ to the mean vectors.
Further, new particles inherit the LSTM state from their ancestors.
The following sections break down the algorithms main parts: how to sample particles from a given GMM (section \ref{sss:sampling}) and how to weight particles (section \ref{sss:weighting}).
The presented strategies are intended to address the particle set degeneration problem common to particle filtering, where all particles collapse into a dense region, thus preventing the exploration of different paths.
As it is unclear which strategies will be adequate for the proposed approach, different techniques are investigated.
\subsection{Sampling strategies}
\label{sss:sampling}
The common approach for sampling from a GMM $p(x) = \sum_{k=1}^{K \cdot M} \pi_k \mathcal{N}(x|\mu_k,\Sigma_k)$, is to select one of the $K \cdot M$ Gaussian components and then draw a sample from that Gaussian distribution.
The strategies described in this section are concerned with the selection (i.e. the index $k$) of the Gaussian component to sample from.
Usually, $k$ is selected by performing a multinomial sampling given the component weights.
In particle filtering, there are other commonly used sampling approaches to tackle particle set degeneration.
Here, a degenerating particle set results in less likely paths vanishing from the prediction.
Prominent examples are systematic \cite{kitagawa1996monte} and stratified \cite{fishman2013monte} sampling.
Due to their similar impact on the sampling results \cite{hol2006resampling}, only stratified sampling is described and evaluated in the remainder of this paper.
\paragraph{Multinomial sampling}
When using a multinomial sampling approach, the Gaussian component to sample from is determined by drawing $k$ from a multinomial distribution, where each $k \in [1..(K \cdot M)]$ is one possible outcome and each $k$ is drawn with probability $\pi_k$.
Having chosen $k$, a sample is drawn from $\mathcal{N}(\mu_k,\Sigma_k)$.
This is repeated $M$ times.
\paragraph{Stratified sampling}
While multinomial sampling may lead to only few of the $K \cdot M$ components being selected when drawing multiple samples in succession (due to small weights on some components), stratified sampling aims at drawing $k$ more uniformly from the weights $\pi_k$.
In theory, this leads to less probable components being chosen more frequently.
This is achieved by first calculating the cumulative sum of the weights $\pi_k$.
Then, the sum is subdivided into $C$ equally sized bins.
Here, $C=M$ is the number of samples to be drawn.
Each of these sections refers to one or more values for $k$.
Lastly, for each section a component $k$ is chosen, such that $k = l^k + u$, where $u \sim \mathcal{U}(\left[0, 1\right])$ and $l^k$ and $r^k$ are the values of the left and right border of the $k$'th bin.
As before, samples are drawn from the corresponding Gaussian components.
\subsection{Particle weighting}
\label{sss:weighting}
According to common particle filtering, a weight needs to be assigned to each particle.
Here, these weights determine how $\mathcal{G}^t$ is combined into a single GMM $p(\cdot|\mathcal{G}^{t})$ (cf. (\ref{eq:comb_gmm})).
When assigning no weight to the particles, all GMMs $p_m(\cdot|\mathbf{p}^{t-1}_m) \in \mathcal{G}^t$ will be combined equally-weighted, thus
\begin{align}
\omega^t_m = \frac{1}{M},~\forall m \in [1..M].
\end{align}
As a consequence, there are more particles in regions of high probability, leading to an indirect weighting.
By assigning different weights to the particles, the probability density of different regions in $p(\cdot|\mathcal{G}^{t})$ can be increased or decreased to some degree.
Thus, more emphasis can be put on the largest mode in the distribution, eventually leveling out secondary peaks.
Conversely weak regions can be pushed in order to prevent a collapse onto a single peak.
Pushing weak regions might help in exploring less likely paths.
In the following, several weighting strategies are presented.
The first strategy, \emph{density value weighting}, yields a straightforward way to assign a weight to each particle by using $p(\cdot|\mathcal{G}^{t})$ (cf. (\ref{eq:comb_gmm})).
The strategies \emph{temperature weighting} and \emph{interpolation weighting} can be used to shift the weight distribution achieved by \emph{density value weighting}.
Depending on the parameters, these strategies are intended to force exploration of less likely paths.
\paragraph{Density value weighting}
A straightforward way of weighting the particles is to apply the pdf $p(\cdot|\mathcal{G}^{t})$ (cf. (\ref{eq:comb_gmm})) that was used to produce the particle set, by passing each particle $\mathbf{p}^t_m$ through it
\begin{align}
\omega^t_m = p(\mathbf{p}^t_m|\mathcal{G}^{t}).
\end{align}
Following that, the set of weights $\Omega^t$ has to be normalized, to conform $\sum^M_{m=1} \omega^m = 1$.
\paragraph{Temperature weighting}
First, the particle weights are determined using density value weighting.
After that, the weight distribution is shifted by applying the transformation
\begin{align}
\begin{split}
\omega^t_{m, temp} = \frac{e^{\frac{ln \omega^t_m}{\tau}}}{\sum^M_{i=1} e^{\frac{ln \omega^t_i}{\tau}}} = \frac{(\omega^t_m)^{\frac{1}{\tau}}}{\sum^M_{i=1} (\omega^t_i)^{\frac{1}{\tau}}}
\end{split}
\end{align}
to each weight $\omega^t_m$ for $m \in [1..M]$.
The parameter $\tau$ is referred to as the temperature.
For $\tau \rightarrow 0$, the new weights $\Omega^t_{temp}$ are shifted towards the strongest component, while for $\tau \rightarrow \infty$, $\Omega^t_{temp}$ approaches an uniform distribution.
This transformation is a slight variation of the \emph{temperature softmax}, sometimes used in reinforcement learning \cite{sutton1998reinforcement}.
It deviates from the original formula in that the natural logarithm is applied to each $\omega^t_m$.
This variation adds the identity transformation $\Omega^t_{temp} = \Omega^t$ for $\tau = 1$.
\paragraph{Interpolation weighting}
Temperature weighting allows to put more emphasis on the largest weights or to approach an uniform distribution.
Interpolation weighting allows to put more emphasis on smaller weights as well.
This is achieved by linearly interpolating each weight $\omega^t_m$ for $m \in [1..M]$ using an interpolation factor $\kappa$:
\begin{align}
\omega^t_{m, interp} = (1 - \kappa) * \omega^t_m + \kappa * (1 - \omega^t_m).
\end{align}
Again, the set of weights has to be re-normalized.
Using this approach, $\kappa = 0$ does not change the weights, $\kappa = 0.5$ results in an uniform distribution and $\kappa \in (0.5, 1]$ puts more emphasis on smaller weights.
For $\kappa \in (0, 0.5)$, this approach yields results similar to temperature weighting when $\tau \rightarrow \infty$.
The motivation to strengthen less probable components is to help paths that may be highly probable in later time steps to \emph{survive} early stages of the prediction.
Still, from the initial set of particles, there are more particles from high probability regions than there are from low probability regions, thus the most relevant paths are not lost, even with $\kappa = 1$.
These weights only affect how single GMMs are combined and not individual component weights, thus there is no risk that components with weights close to $0$ are enhanced.
\section{Experiments}
\label{sec:exp}
This section shows a quantitative (\ref{ss:quant}) and a qualitative (\ref{ss:quali}) evaluation for the proposed approach.
Both evaluations are concerned with verifying the overall viability of the approach in different situations given different sets of parameters.
Assuming that the LSTM-MDL model is capable of modeling all relevant paths in the training data \cite{hug2017reliability}, the quantitative evaluation is concerned with the difference between the probability distribution generated by the predictor using different configurations and the true distribution as provided by the training data.
A set of synthetic test conditions is used in order to reduce the number of factors that might cause the predictor to bias into a certain path or direction and to allow for a more systematic evaluation.
In the qualitative evaluation part, the performance of the best (in terms of the quantitative evaluation) predictor is shown on real-world scenes.
\begin{figure*}[t]
\centering
\begin{tabular}{c@{\hspace{1pt}}c@{\hspace{2pt}}c@{\hspace{1pt}}c}
\includegraphics[width=0.24\textwidth]{noisy_tmaze_heat.jpg} & \includegraphics[width=0.24\textwidth]{noisy_tmaze_dir-biased_heat.jpg} & \includegraphics[width=0.24\textwidth]{noisy_tmaze_pos-biased-gap_aligned_heat.jpg} & \includegraphics[width=0.24\textwidth]{noisy_tmaze_pos-biased-nogap_aligned_heat.jpg} \\
\small{1: \emph{tmaze-heavy-left}} & \small{2: \emph{tmaze-dirbias}} & \small{3: \emph{tmaze-posbias-gap}} & \small{4: \emph{tmaze-posbias-nogap}}
\end{tabular}
\caption{Density plot for trajectories in synthetic test conditions (all trajectories start at the bottom, red = high density).}
\label{fig:synth_data}
\vspace{-0.2cm}
\end{figure*}
\subsection{Implementation details}
The LSTM-MDL model and the prediction algorithm have been implemented using tensorflow.
The implementation strongly utilizes vectorized calculations performed on the GPU, which come at the cost of higher memory consumption, thus limiting the maximum number of particles for the prediction.
On the test system (Intel Core i7-5930K, 32GB RAM, Nvidia Titan X), the number of particles is capped at 5000.
For the quantitative evaluation, each prediction has been performed 10 times and particles have been aggregated to a total number of 50000 particles per prediction.
\subsection{Quantitative results}
\label{ss:quant}
For the quantitative evaluation, several variations of noisy trajectories along a t-shaped junction have been generated synthetically.
A t-shape can be considered as a prototypical decision scenario that is resembled in various forms in the real world (e.g. see Fig. \ref{fig:qual_res}).
Test conditions are designed to provide different difficulties, with respect to such decision points.
The test conditions are as follows:
\begin{enumerate}
\item \emph{tmaze} (cf. Fig. \ref{fig:synth_data}-1): Observations prior to the junction yield no information about which side to choose. The predictor is expected to generate a prediction evenly distributed to both sides.
\item \emph{tmaze-heavy-left} (Fig. \ref{fig:synth_data}-1): Similar to \emph{tmaze}, but the prediction to the left side should have higher weight (i.e. more particles should be located on the left side).
\item \emph{tmaze-dirbias} (Fig. \ref{fig:synth_data}-2): Observations prior to the junction yield strong information about which side to choose, as the movement direction clearly indicates it. Given the LSTM-MDL has captured this property in the training phase, the predictor should generate paths which are clearly biased to one side.
\item \emph{tmaze-posbias-gap} (Fig. \ref{fig:synth_data}-3): Similar to \emph{tmaze-dirbias} there is strong information about which side to choose in observations prior to the junction. Here, the absolute positioning of the observations indicate where to go. The gap between both directions should help the model to distinguish between the to classes.
\item \emph{tmaze-posbias-nogap} (Fig. \ref{fig:synth_data}-4): Similar to \emph{tmaze-posbias-gap}, but there is no gap between both classes. That way, the model will start to mix predictions towards both sides when looking at observations located towards the center. With respect to the distribution of particles to either side, it should shift from one side to the other when looking at trajectories ranging from left to right.
\end{enumerate}
The biased test conditions resemble real world situations, where the decision on where to go at a junction can be indicated by the positioning and the movement direction.
Further, a t-shape was chosen as it provides a case where maximum likelihood predictors fail because it provides two possible outcomes, yet it is simple enough to ensure the reliability of the LSTM-MDL model.
\paragraph{Parameter configurations}
For the prediction algorithm, a fixed set of parameter configurations is evaluated.
Here, multinomial and stratified sampling strategies are tested.
Considering the weighting strategies, the following configurations are used:
\begin{enumerate}
\item Unweighted (equal weights)
\item Density value weighting
\item Temperature weighting with $\tau \in \{0.01, 1000\}$
\item Interpolation weighting with $\kappa \in \{0.25, 0.5, 0.75, 1\}$
\end{enumerate}
The parameters of temperature and interpolation weighting were chosen to cover a wide range of possible transformations with each strategy.
Combining the sampling and weighting strategies, yields a total of $16$ configurations.
\paragraph{Setup and Metric}
In the following, the $N$-step prediction $\mathcal{P}^s$ (cf. algorithm \ref{alg:particle_prop}) given an observed trajectory fraction is evaluated.
For this, 50 trajectories are chosen from each synthetic test condition, such that their starting positions are evenly distributed from left to right.
For measuring the prediction quality, the distance between the particle distribution $\mathcal{P}^s$ and the expected distribution $D_{ex}$ is used.
$D_{ex}$ is obtained by selecting the endpoints of trajectories starting closely to the observed trajectory.
For $\mathcal{P}^s$, the first 15 positions are observed for each evaluation trajectory and the remaining $N$ positions are predicted (usually $N \in \left[50, 60\right]$).
As $\mathcal{P}^s$ and $D_{ex}$ are both sets of points, the distance is measured by calculating the euclidean distance between the corresponding centroids (average of all points):
\begin{align}
\text{CE} = ||\text{centroid}(\mathcal{P}^s) - \text{centroid}(D_{ex})||_2.
\end{align}
\begin{figure*}[t]
\begin{center}
\includegraphics[width=0.3\textwidth]{centroids_noisy_tmaze-unweighted_multinomial-00.png}
\includegraphics[width=0.3\textwidth]{centroids_noisy_tmaze-unweighted_multinomial-24.png}
\includegraphics[width=0.3\textwidth]{centroids_noisy_tmaze-unweighted_multinomial-49.png}
\includegraphics[width=0.3\textwidth]{centroids_noisy_tmaze-vanilla_stratified-00.png}
\includegraphics[width=0.3\textwidth]{centroids_noisy_tmaze-vanilla_stratified-24.png}
\includegraphics[width=0.3\textwidth]{centroids_noisy_tmaze-vanilla_stratified-49.png}
\end{center}
\vspace{-0.2cm}
\caption{Predictions for the endpoint distribution (orange) of different observed trajectories (solid green, dashed green is the true remainder), ground truth distributions (blue) and distribution centroids (stars).
First row: \emph{Unweighted multinomial} configuration.
Second row: \emph{Density stratified} configuration.}
\label{fig:centroid}
\vspace{-0.1cm}
\end{figure*}
\paragraph{Comparison of configurations}
In a first step, the performance of different predictor configurations is compared.
Therefore, the centroid error for all test conditions and respective trajectories is averaged for each configuration (mean centroid error, MCE).
Additionally, the outlier ratio is calculated.
Here, the outlier ratio is the fraction of particles that are not within the bounds of all trajectory endpoints on the left or right side.
For calculating the MCE, outlier particles were disregarded.
The results are summarized in table \ref{tab:overall_res}.
\begin{table}
\centering
\begin{tabular}{l|l||c|c}
sampling & weighting & MCE (std) & OR (std) \\
\hline
\hline
multinomial & - & \textbf{1.232} (2.484) & 0.018 (0.029) \\
stratified & - & 1.242 (2.523) & 0.018 (0.029) \\
multinomial & density & 5.303 (6.135) & 0.000 (0.006) \\
stratified & density & 5.346 (6.195) & 0.000 (0.000) \\
multinomial & $\tau = 0.01$ & 5.142 (7.295) & 0.073 (0.202) \\
stratified & $\tau = 0.01$ & 5.241 (7.491) & 0.072 (0.205) \\
multinomial & $\tau = 1000$ & 1.247 (2.522) & 0.017 (0.028) \\
stratified & $\tau = 1000$ & 1.254 (2.543) & 0.017 (0.028) \\
multinomial & $\kappa = 0.25$ & 1.239 (2.522) & 0.019 (0.030) \\
stratified & $\kappa = 0.25$ & 1.242 (2.529) & 0.018 (0.029) \\
multinomial & $\kappa = 0.5$ & 1.281 (2.589) & 0.019 (0.030) \\
stratified & $\kappa = 0.5$ & 1.241 (2.524) & 0.018 (0.029) \\
multinomial & $\kappa = 0.75$ & 1.294 (2.523) & 0.019 (0.030) \\
stratified & $\kappa = 0.75$ & 1.242 (2.521) & 0.018 (0.029) \\
multinomial & $\kappa = 1$ & 1.241 (2.556) & 0.018 (0.030) \\
stratified & $\kappa = 1$ & 1.244 (2.522) & 0.019 (0.030) \\
\hline
\end{tabular}
\caption{Mean centroid error (MCE) in meters, outlier ratio (OR) and standard deviations (std) for each configuration over all test conditions.
Weighting: none (-), density value (density), temperature ($\tau =~?$) and interpolation ($\kappa =~?$).}
\label{tab:overall_res}
\vspace{-0.8cm}
\end{table}
It can be seen that not weighting the particles and using a multinomial sampling strategy performed best and density value weighting with stratified sampling performed worst.
Further, regardless of the sampling strategy, configurations using density value or temperature (with small $\tau$) weighting perform significantly worse than other configurations.
Also, the standard deviation of the results is much higher.
These configurations perform slightly better w.r.t. the outlier ratio.
\paragraph{Unbiased test conditions}
Following this, the cause for the discrepancy between configurations and the high standard deviation of worse configurations is examined.
Due to the similarities between configurations of either group, only \emph{unweighted multinomial} (best performance) and \emph{density stratified} (worst performance) are considered in the following.
While both configurations mostly work as expected on the biased test conditions, results differ on the unbiased test conditions.
Due to similar results for \emph{tmaze} and \emph{tmaze-heavy-left}, this evaluation is restricted to the \emph{tmaze} test condition.
Fig. \ref{fig:centroid} shows the prediction results for 3 trajectories (starting from the left, center and right of the starting region) for the \emph{unweighted multinomial} (top) and the \emph{density stratified} (bottom) configurations.
Here, the orange circles represent the prediction set $\mathcal{P}^s$, the blue circles the ground truth $D_{ex}$ and the stars the respective centroids.
The \emph{unweighted multinomial} approach manages to predict both expected endpoint regions, but also causes the particles to spread more, which is the cause for a slightly higher outlier ratio.
A more important aspect visible in this figure, is that the particle set of the \emph{density stratified} configuration collapses in a dense set of particles, which often times leads to the prediction of only one of the two possible directions.
This particle set degeneration is the cause for higher centroid errors and an increased standard deviation.
\paragraph{Distribution of particles}
Another interesting aspect exclusive to the \emph{tmaze(-heavy-left)} and \emph{gap} conditions is the ratio of particles located in the left and right region for the 50 selected trajectories.
This ratio can be expressed as the fraction of particles predicted to be inside the left endpoint region (disregarding outliers).
Considering \emph{tmaze} and \emph{tmaze-heavy-left}, this fraction is expected to be $0.5$ and $0.66$ on average for each of the 50 trajectories.
Here, the fractions generated by the \emph{unweighted multinomial} predictor are very close to the expected values, with average values of $0.54$ and $0.65$.
Fig. \ref{fig:prog} depicts the fraction of particles for each of the 50 trajectories selected from the gap test conditions.
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.45\textwidth]{prog_noisy_tmaze_pos-biased-gap_aligned-unweighted_multinomial.jpg}
\includegraphics[width=0.45\textwidth]{prog_noisy_tmaze_pos-biased-nogap_aligned-unweighted_multinomial.jpg}
\end{center}
\vspace{-0.2cm}
\caption{Fraction of particles predicted to be in the left endpoint region for the \emph{unweighted multinomial} (yellow) configuration and all 50 evaluation trajectories of the \emph{tmaze-posbias-gap} test condition (top) and the \emph{tmaze-posbias-nogap} test condition (bottom). Blue plot: Ground truth.}
\label{fig:prog}
\vspace{-0.2cm}
\end{figure}
For \emph{tmaze-posbias-gap}, a jump from a fraction of $1$ to $0$ is expected when passing the trajectories closest to the center of the starting region, as indicated by the ground truth (blue).
As for \emph{tmaze-posbias-nogap}, a smooth transition from $1$ to $0$ is expected when passing the center-located trajectories.
In general, the \emph{unweighted multinomial} configuration manages to get close to these fractions (yellow).
Although the transition in \emph{tmaze-posbias-nogap} is not as smooth as desired, particles are still distributed to both sides for observations close to the center.
\paragraph{Conclusions}
The experiments show that configurations using density value or temperature (with small $\tau$) weighting suffer from particle set degeneration.
Although unsuitable for multi-modal path prediction, these configurations could be used for unimodal predictions.
These configurations, put strong emphasize on high weighted particles, thus a maximum likelihood prediction could be approximated.
As intended, given proper parameterization, the proposed sampling and weighting strategies enforce exploration of all paths known by the model.
Yet surprisingly, the simplest approach, calculating no weight and using a multinomial sampling approach, is on par with these configurations.
A possible explanation is that the LSTM-MDL model was reliable enough to properly capture motions seen in the training data.
Thus, the distribution output by $\mathcal{M}$ does not need to be manipulated artificially to put more emphasize on less likely modes.
\subsection{Qualitative results}
\label{ss:quali}
\begin{figure*}[t]
\centering
\begin{tabular}{l@{\hspace{3pt}}l@{\hspace{3pt}}r@{\hspace{3pt}}r}
\includegraphics[width=0.24\textwidth]{hyang_2_heat.jpg} & \includegraphics[width=0.24\textwidth]{hyang_6-2_heat.jpg} & \includegraphics[width=0.24\textwidth]{deathcircle_5-3_heat.jpg} & \includegraphics[width=0.24\textwidth]{deathcircle_5-4_heat.jpg} \\
\small{~~1: 30 observed, 116 predicted} & \small{~~2: 60 observed, 102 predicted} & \small{3: 40 observed, 52 predicted~~~} & \small{4: 60 observed, 32 predicted~~~}
\end{tabular}
\caption{Prediction results (heatmap) for \emph{hyang} (1, 2) and \emph{deathcircle} (3, 4).
For each trajectory, a certain number of time steps is observed (solid cyan) and the remaining time steps (ground truth: dashed cyan) are predicted.
The respective numbers are indicated in the subcaptions.}
\label{fig:qual_res}
\vspace{-0.1cm}
\end{figure*}
In this section, the performance of the \emph{unweighted multinomial} configuration is shown on different real-world scenes.
For this, it is emphasized to use only datasets, that include at least one junction, thus leading to multiple hypotheses for future progressions of an observed trajectory.
Like this, scenarios are considered, where maximum likelihood predictors would fail because of their inability to generate multiple hypotheses.
This excludes popular datasets like \emph{ETH} \cite{pellegrini2009you} or \emph{UCY} \cite{lerner2007crowds}.
Instead, two scenes, taken from the \emph{Stanford Drone Dataset} \cite{robicquet2016learning}, are used: \emph{hyang} and \emph{deathcircle}.
In order to increase the amount of training data, the combined \emph{hyang} dataset only containing pedestrian trajectories, as provided by \cite{hug2017reliability}, was used.
Likewise, available video annotations for \emph{deathcircle} were combined into a single dataset by mapping image coordinates into a common reference frame using a homography projection calculated from 4 manually selected pixels, using the video's reference images.
It has to be noted, that although there are many pedestrians in the combined \emph{deathcircle} dataset, biker trajectories were used, as these follow the roads and the roundabout, leading to visually more inspectable results.
Further, as pedestrian interactions are rather short-termed events leading to small deviations from an intended path, these only have a limited influence on the trajectories on the regarded larger time scale.
Thus, this dataset can be used for evaluation, even with the LSTM-MDL model disregarding such interactions.
\paragraph{Prediction results}
Exemplary prediction results are depicted in Fig. \ref{fig:qual_res} and \ref{fig:qual_res2}.
Here, the trajectory to be observed and predicted is illustrated in cyan (solid: observation, dashed: ground truth) and the prediction is illustrated as a heatmap calculated from the propagated particles.
Two exemplary trajectories are shown for \emph{hyang} (Fig. \ref{fig:qual_res}-1 and \ref{fig:qual_res}-2), where the first trajectory has been observed just before the junction and the second trajectory until it is extended into the junction.
Here, the predictor produces a prediction going straight over the junction and one heading towards the right side of the scene.
A prediction towards the left side is disregarded in this case due to the positioning of the observation close to the right edge of the pathway.
When the trajectory is observed for a longer period of time, the predictor disregards the prediction torwards the left or right side, and indicates that the trajectory is most likely to go downwards and eventually on the stairway.
Similar behavior can be observed for \emph{deathcircle}, where the predictor considers the correct paths to progress along.
In all cases, the ground truth lies within the predicted distribution.
\paragraph{Erroneous prediction results}
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.23\textwidth]{hyang_6_heat.jpg}
\includegraphics[width=0.23\textwidth]{deathcircle_6_heat.jpg}
\end{center}
\vspace{-0.2cm}
\caption{Erroneous prediction results for \emph{hyang} (left; 30 steps observed, 132 predicted) and \emph{deathcircle} (right; 20 steps observed, 110 predicted).}
\label{fig:qual_res2}
\vspace{-0.25cm}
\end{figure}
As stated in section \ref{s:path_pred}, the quality of the prediction also relies on the capabilities of the trained LSTM-MDL model.
Therefore, inaccuracies of the model are reproduced by the predictor, occasionally resulting in the generation of artifacts or missing predictions as depicted in Fig. \ref{fig:qual_res2}.
Looking at the prediction for a trajectory taken from \emph{hyang} (left) starting at the top, correct predictions (straight over the junction and towards the left side), but also a significant artifact (moving onto the grass) are produced.
As this artifact is located between the two correct paths, it most likely started as an outlier, hence entering a state, where less data was observed.
In such cases, the LSTM-MDL model presumably interpolates known motions, which results in a drift producing the artifact.
Besides this, when comparing this prediction to the prediction in Fig. \ref{fig:qual_res}-1, the previously indicated influence of the positioning of the observation on the prediction result is visible.
Here, the trajectory is located closer to the center of the pathway, making a prediction to the left side of the scene a viable option.
Due to the trajectory slightly heading towards the left side in the observation, a prediction to the right side is disregarded.
The right image in Fig. \ref{fig:qual_res2} shows a prediction containing artifacts for \emph{deathcircle}.
Here, supposedly due to drifting, some particles are spread over the building in the bottom right portion of the scene.
Besides these artifacts, the predictor didn't identify the first exit of the roundabout as a possible progression for the trajectory.
This most likely stems from the fact, that only few trajectories in the training dataset actually take this route, making it statistically less relevant.
Also, in the last steps of the observation, the trajectory is very close to the center of the roundabout.
This may indicate that the trajectory is going straight or wrapping around the roundabout.
\section{Conclusions and outlook}
\label{s:conclusion}
In this paper, an approach to enable LSTM-MDL models to perform multi-modal pedestrian path prediction based on modified particle filter methods has been presented.
As this predictor can be stacked on top of any recurrent model with an MDL output layer, it could also be applied in different contexts.
In the experimental section, different predictor configurations have been tested using synthetic test conditions, in order to identify the best configurations to use.
These experiments show the counter-intuitive result of the simplest configuration being the best performing.
For this configuration, prediction results have been evaluated on different real-world scenes, concluding that the predictor is capable of producing plausible hypothesis on future paths.
Given the multi-modal prediction generated by the proposed approach, future works can elaborate on determining a global solution of a maximum likelihood prediction, which is different from a greedy solution in some cases.
Further, the prediction approach will be applied to more complex LSTM-MDL models which incorporate pedestrian interaction or local scene context.
\bibliographystyle{IEEEtran}
|
{
"timestamp": "2018-08-30T02:06:14",
"yymm": "1804",
"arxiv_id": "1804.05546",
"language": "en",
"url": "https://arxiv.org/abs/1804.05546"
}
|
\section{Introduction}
One of the most surprising applications of Szemer\'edi's regularity lemma is the proof of Roth's theorem~\cite{Roth54}.
This ingenuous proof, due to Ruzsa and Szemer\'edi~\cite{RuzsaSz76}, implicitly relies on what is now known as the
{\em triangle removal lemma}. Erd\H{o}s, Frankl and R\"odl~\cite{ErdosFrRo86} asked if this lemma can be extended
to the setting of $k$-uniform hypergraphs, and Frankl and R\"odl~\cite{FrankRo02} observed that such a result would allow one to extend the Ruzsa--Szemer\'edi~\cite{RuzsaSz76} argument and thus obtain an alternative proof of Szemer\'edi's theorem~\cite{Szemeredi75} for progressions of arbitrary length (see also~\cite{Solymosi04}). Frankl and R\"odl~\cite{FrankRo02} further initiated a programme for proving such a hypergraph removal lemma
via a {\em hypergraph regularity lemma} and proved such a lemma for $3$-uniform hypergraphs. This task was completed only 10 years later when R\"odl, Skokan, Nagle and Schacht~\cite{NagleRoSc06, RodlSk04} and independently Gowers~\cite{Gowers07} obtained regularity lemmas $k$-uniform hypergraphs (from now on we will use $k$-graphs instead of $k$-uniform hypergraphs).
Shortly after, Tao~\cite{Tao06} and R\"odl and Schacht~\cite{RodlSc07,RodlSc07-B} obtained two more versions of the lemma.
The above-mentioned variants of the hypergraph regularity lemma relied on four different notions of {\em quasi-randomness}, which
are not known to be equivalent; see R\"odl's recent ICM survey~\cite{Rodl14} and our recent paper \cite{MS3} for more on this.
What all of these proofs {\em do} have in common however, is that they supply only Ackermann-type bounds for the size of a regular partition. More precisely,
if we let $\Ack_1(x)=2^x$ and then define $\Ack_k(x)$ to be the $x$-times iterated\footnote{So $\Ack_2(x)$ is a tower of exponents of height $x$, i.e.\ the tower function, $\Ack_3(x)$ is the so-called wowzer function, etc.} version of $\Ack_{k-1}$, then all the above proofs guarantee to produce a regular partition of a $k$-graph whose order can be bounded from above by an $\Ack_k$-type function.
Gowers~\cite{Gowers97} famously proved that $\Ack_2$-type upper bounds for graph regularity
are unavoidable. Tao~\cite{Tao06-h} predicted that Gowers's result can be extended to the setting
of $k$-graphs, that is, that $\Ack_k$-type bounds are unavoidable for the $k$-graph regularity lemma.
Until very recently, no analogue of Gowers's lower bound was known for any $k>2$.
In a recent paper \cite{MS3} we obtained such a result for the $3$-graph regularity lemma.
We refer the reader to Section $1$ of \cite{MS3} for a thorough discussion of this result
and some of the key ideas behind it.
In the present paper we extend this result to arbitrary $k \geq 3$, thus conforming
Tao's prediction~\cite{Tao06-h}. Our main result can be informally stated as follows.
\begin{theo}{\bf[Main result, informal statement]}\label{thm:main-informal}
The following holds for every $k\geq 2$: every regularity lemma for $k$-graphs satisfying some
mild conditions can only guarantee to produce partitions of size bounded by an $\Ack_k$-type function.
\end{theo}
Our main result, stated formally as Theorem~\ref{theo:main}, establishes an $\Ack_k$-type lower bound
for \emph{$\langle \d \rangle$-regularity} of $k$-graphs, which is a new notion we first introduced in~\cite{MS3}.
The main advantage of this notion is threefold:
$(i)$ It is much simpler to state compared to all other notions of $k$-graph regularity.
$(ii)$ It is weak enough to allow one to induct on $k$, that is, to use lower bounds for $k$-graph regularity in order to obtain lower bounds for $(k+1)$-regularity\footnote{See Section $1$ of \cite{MS3} for a more detailed discussion on this aspect of the proof.}.
$(iii)$ All known notions of regularity appear to be stronger than $\langle \d \rangle$-regularity, so a lower bound
for $\langle \d \rangle$-regularity gives a lower bound for such lemmas, that is, for any lemma
whose requirements/guarantees imply those that are needed in order to satisfy $\langle \d \rangle$-regularity.
We will demonstrate the effectiveness of item~$(iii)$ above by deriving from Theorem~\ref{thm:main-informal} a lower bound for
the $k$-graph regularity lemma of R\"odl--Schacht~\cite{RodlSc07}.
\begin{coro}[Lower bound for $k$-graph regularity]\label{coro:RS-LB}
For every $k\geq 2$, there is an $\Ack_k$-type lower bound for the $k$-graph regularity lemma of R\"odl--Schacht~\cite{RodlSc07}.
\end{coro}
As we discuss at the beginning of Section~\ref{sec:coro}, the lower bound stated in Corollary~\ref{coro:RS-LB} holds even for
a very weak/special case of the $k$-graph regularity lemma of~\cite{RodlSc07}.
As Theorem \ref{thm:main-informal} establishes a lower bound for
$\langle \d \rangle$-regularity, it is natural to ask if this notion is in fact equivalent to other notions. In particular, is this notion strong enough for ``counting'', that is,
for proving the hypergraph removal lemma, which was one of the main reasons for developing
the hypergraph regularity lemma? Our final result (see Proposition \ref{claim:example} for the formal statement)
answers both questions negatively.
This of course makes our lower bound even stronger as it already applies to a very weak notion of regularity.
\begin{prop}{\bf[Informal statement]}\label{prop:counter}
$\langle \d \rangle$-regularity is not strong enough even for proving the graph triangle removal lemma.
\end{prop}
\subsection{Paper overview}
Broadly speaking, Section \ref{sec:define} serves as the technical introduction to this paper, while
Section \ref{sec:LB} contains the main technical proofs.
More concretely, in Section \ref{sec:define} we will first define the new notion of $k$-graph regularity, which we term $\langle \d \rangle$-regularity,
for which we will prove our main lower bound. We will then give the formal version of Theorem~\ref{thm:main-informal} (see Theorem~\ref{theo:main}).
This will be followed by the statement of the main technical result we will use in this paper, Theorem~\ref{theo:core},
and an overview of how this technical result is used in the proof of Theorem~\ref{theo:main}.
The proof of Theorem~\ref{theo:main} appears in Section~\ref{sec:LB}.
In Section \ref{sec:coro} we describe how Theorem~\ref{theo:main} can be used in order to prove Corollary~\ref{coro:RS-LB}, thus establishing tight
$\Ack_k$-type lower bounds for a concrete version of the hypergraph regularity lemma.
Since at its core, the proof of Corollary~\ref{coro:RS-LB} is very similar to the
way we derived lower bounds for concrete regularity lemmas for $3$-uniform hypergraphs in \cite{MS3},
just with a much more elaborate set of notations (due to having to deal with arbitrary $k$),
we decided to put the proof of some technical claims only in the appendix of the Arxiv version of this paper.
Finally, in Section~\ref{sec:example} we prove Proposition \ref{prop:counter} by describing an example showing that even in the setting of graphs,
$\langle \d \rangle$-regularity is strictly weaker than the usual notion of graph regularity, as it does not allow one even to count triangles.
\paragraph{How is this paper related to~\cite{MS3}:}
For the reader's convenience we explain how this paper differs from~\cite{MS3}, in which we prove Theorem~\ref{theo:main} for $k=3$.
First, the definitions given in Section~\ref{sec:define} when specialized to $k=3$ are the same notions used in (Section~2 of)~\cite{MS3}.
The heart of the proof of Theorem~\ref{theo:main} is given by Lemma~\ref{lemma:ind-k} which is proved by induction on $k$.
The (base) case $k=2$ follows easily from Lemma~\ref{theo:core} which was proved in~\cite{MS3}.
Hence, the heart of the matter is the proof of Lemma~\ref{lemma:ind-k} by induction on $k$. Within this framework,
the argument given in~\cite{MS3}
is precisely the deduction of Lemma~\ref{lemma:ind-k} for $k=3$ from the case $k=2$.
Hence, the reader interested in seeing the inductive proof of Lemma~\ref{lemma:ind-k} ``in action''---without the clutter
caused by the complicated definitions related to $k$-graphs---is advised to check~\cite{MS3}.
\section{$\langle \d \rangle$-regularity and Proof Overview}\label{sec:define}
\subsection{Preliminary definitions}\label{subsec:preliminaries}
Before giving the definition of $\langle \d \rangle$-regularity, let us start with some standard definitions regarding partitions of hypergraphs.
Formally, a \emph{$k$-graph} is a pair $H=(V,E)$, where $V=V(H)$ is the vertex set and $E=E(H) \sub \binom{V}{k}$ is the edge set of $H$.
The number of edges of $H$ is denoted $e(H)$ (i.e., $e(H)=|E|$).
We denote by $K^k_\ell$ the complete $\ell$-vertex $k$-graph (i.e., containing all possible $\binom{\ell}{k}$ edges).
The $k$-graph $H$ is \emph{$\ell$-partite} $(\ell \ge k)$ on (disjoint) vertex classes $(V_1,\ldots,V_\ell)$ if every edge of $H$ has at most one vertex from each $V_i$.
We denote by $H[V'_1,\ldots,V'_\ell]$ the $\ell$-partite $k$-graph induced on vertex subsets $V_1' \sub V_1,\ldots,V_\ell'\sub V_\ell$; that is, $H[V'_1,\ldots,V'_\ell]=((V_1',\ldots,V_\ell'),\, \{e \in E(H) \,\vert\, \forall i \colon e \cap V_i \in V_i'\})$.
The \emph{density} of a $k$-partite $k$-graph $H$ is $e(H)/\prod_{i=1}^k |V_i|$.
The set of edges of $G$ between disjoint vertex subsets $A$ and $B$ is denoted by $E_G(A,B)$; the density of $G$ between $A$ and $B$ is denoted by $d_G(A,B)=e_G(A,B)/|A||B|$, where $e_G(A,B)=|E_G(A,B)|$. We use $d(A,B)$ if $G$ is clear from context.
When it is clear from context, we sometimes identify a hypergraph with its edge set. In particular, we will use $V_1 \times V_2$ for the complete bipartite graph on vertex classes $(V_1,V_2)$, and more generally, $V_1 \times\cdots\times V_k$ for the complete $k$-partite $k$-graph on vertex classes $(V_1,\ldots,V_k)$.
For a partition $\Z$ of a vertex set $V$,
the complete multipartite $k$-graph on $\Z$ is denoted by
$\Cross_k(\Z)=
\big\{ e \sub V \,\big\vert\, \forall\, V \in \Z \colon |e \cap V| \le 1 \text{ and } |e|=k\big\}$.
For partitions $\P,\Q$ of the same underlying set, we say that $\Q$ \emph{refines} $\P$, denoted $\Q \prec \P$, if every member of $\Q$ is contained in a member of $\P$.
We use the notation $x \pm \e$ for a number lying in the interval $[x-\e,\,x+\e]$.
We now define a \emph{$k$-partition},
which is a notion of a hypergraph partition\footnote{This is the standard notion, identical to the one used by R\"odl-Schacht (\cite{RodlSc07}, Definition10).}.
A $k$-partition $\P$ is of the form $\P=\P^{(1)} \cup\cdots\cup \P^{(k)}$ where $\P^{(1)}$ is a vertex partition, and for each $2 \le r \le k$, $\P^{(r)}$ is a partition of $\Cross_r(\P^{(1)})$ satisfying a condition we will state below.
First, to ease the reader in, let us describe here what a $k$-partition is for $1 \le k \le 3$.
A $1$-partition is simply a vertex partition.
A $2$-partition $\P=\P^{(1)} \cup \P^{(2)}$ consists of a vertex partition $\P^{(1)}$ and a partition $\P^{(2)}$ of $\Cross_2(\P^{(1)})$
such that the complete bipartite graph between any two distinct clusters of $\P^{(1)}$ is a union of parts of $\P^{(2)}$.
A $3$-partition $\P=\P^{(1)} \cup \P^{(2)} \cup \P^{(3)}$ consists of a $2$-partition $\P^{(1)} \cup \P^{(2)}$ and a partition $\P^{(3)}$ of $\Cross_3(\P^{(1)})$
such that for every tripartite graph $G$ whose three vertex clusters lie in $\P^{(1)}$ and three bipartite graphs lie in $\P^{(2)}$, the $3$-partite $3$-graph consisting of all triangles in $G$ is a union of parts of $\P^{(3)}$.
Before defining a $k$-partition in general, we need some terminology.
A \emph{$k$-polyad} is simply a $k$-partite $(k-1)$-graph.
Thus, a $2$-polyad is just a pair of disjoint vertex sets, and a $3$-polyad is a tripartite graph.
In the rest of this paragraph let $P$ be a $k$-polyad on vertex classes $(V_1,\ldots,V_k)$.
We
often identify $P$
with the $k$-tuple $(F_1,\ldots,F_k)$ where each $F_i$ is the induced $(k-1)$-partite $(k-1)$-graph $F_i=P[\bigcup_{j \neq i} V_j]$.
We denote by $\K(P)$ the set of $k$-element subsets of $V(P)$ that span a clique (i.e., a $K^{k-1}_{k}$) in $P$; we view $\K(P)$ as a $k$-graph on $V(P)$.
Note that $\K(P)$ is a $k$-partite $k$-graph.
For example, if $P$ is a $2$-polyad then $\K(P)$ is a complete bipartite graph (since $K^1_2$ is just a pair of vertices), and if $P$ is a $3$-polyad then $\K(P)$ is the $3$-partite $3$-graph whose edges correspond to the triangles in $P$.
For a family of hypergraphs $\P$, we say that the $k$-polyad $P=(F_1,\ldots,F_k)$ is a $k$-polyad \emph{of} $\P$ if $F_i \in \P$ for every $1 \le i \le k$.
We are now ready to define a $k$-partition for arbitrary $k$.
\begin{definition}[$k$-partition]\label{def:r-partition}
$\P$ is a \emph{$k$-partition} $(k \ge 1)$ on $V$ if $\P=\P^{(1)} \cup\cdots\cup \P^{(k)}$ with $\P^{(1)}$ a partition of $V$,
and for every $2 \le r \le k$, $\P^{(r)}$ is a partition of $\Cross_r(\P^{(1)})$ into $r$-partite $r$-graphs with $\P^{(r)} \prec \K_r(\P):=\{\K(P) \,\vert\, P \text{ is an $r$-polyad of } \P\}$.
\end{definition}
Note that, by Definition~\ref{def:r-partition}, each $r$-partite $r$-graph $F \in \P^{(r)}$ satisfies $F \sub \K(P)$ for a unique $r$-polyad $P$ of $\P$.
In this context, let $\under$ be the function mapping $F$ to $P$.
So for example, if $\P=\P^{(1)}\cup\P^{(2)}\cup\P^{(3)}$ is a $3$-partition, for every bipartite graph $F \in \P^{(2)}$ we have that $\under(F)$ is the pair of vertex classes of $F$;
similarly, for every $3$-partite $3$-graph $F \in \P^{(3)}$ we have that $\under(F)$ is the unique $3$-partite graph of $\P^{(2)}$ whose set of triangles contains the edges of $F$.
We encourage the reader to verify that Definition~\ref{def:r-partition} is indeed compatible with the explicit description of a $1$-, $2$- and $3$-partition given above.
\subsection{$\langle \d \rangle$-regularity of graphs and hypergraphs}
In this subsection we define our new notion of $\langle\d\rangle$-regularity, first for graphs and then for $k$-graphs for any $k \ge 2$ in Definition~\ref{def:k-reg} below.
\begin{definition}[graph $\langle\d\rangle$-regularity]\label{def:star-regular}
A bipartite graph $G$ on $(A,B)$ is \emph{$\langle \d \rangle$-regular} if for all subsets $A' \sub A$, $B' \sub B$ with $|A'| \ge \d|A|$, $|B'|\ge\d|B|$ we have $d_G(A',B') \ge \frac12 d_G(A,B)$.\\
A vertex partition $\P$ of a graph $G$ is \emph{$\langle \d \rangle$-regular}
if one can add/remove at most $\d \cdot e(G)$ edges so that the bipartite graph induced on each $(A,B)$ with $A \neq B \in \P$ is $\langle \d \rangle$-regular.
\end{definition}
For the reader worried that in Definition~\ref{def:star-regular} we merely replaced the $\e$
from the definition of $\e$-regularity with $\d$, we refer to the discussion following Theorem~\ref{theo:main} below.
The definition of $\langle\d\rangle$-regularity for hypergraphs involves the $\langle\d\rangle$-regularity notion for graphs, applied to certain auxiliary graphs which are defined as follows.
Henceforth, if $P$ is a $(k-1)$-graph and $H$ is $k$-graph then we say that $H$ is \emph{underlied} by $P$ if $H \sub \K(P)$.
\begin{definition}[The auxiliary graph $G_{H}^i$]\label{def:aux}
For a $k$-partite $k$-graph $H$ on vertex classes $(V_1,\ldots,V_k)$,
we define a bipartite graph $G_{H}^1$ on the vertex classes $(V_2 \times\cdots\times V_k,\,V_1)$ by
$$E(G_{H}^1) = \big\{ ((v_2,\ldots,v_k),v_1) \,\big\vert\, (v_1,\ldots,v_k) \in E(H) \big\} \;.$$
The graphs $G_{H}^i$ for $2 \le i \le k$ are defined in an analogous manner.
More generally, if $H$ is underlied by the $k$-polyad $P=(F_1,\ldots,F_k)$ then we define $G_{H,P}^i$ as the induced subgraph $G_{H,P}^i=G_{H}^i[F_i,V_i]$.
\end{definition}
As a trivial example, if $H$ is a bipartite graph then $G_H^1$ and $G_H^2$ are both isomorphic to $H$.
Importantly, for a $k$-partition (as defined in Definition~\ref{def:r-partition}) to be $\langle\d\rangle$-regular it must first satisfy a requirement on the regularity of its parts.
\begin{definition}[$\langle\d\rangle$-good partition]\label{def:k-good}
A $k$-partition $\P$ on $V$ is \emph{$\langle\d\rangle$-good} if for every $2 \le r \le k$ and every $F \in \P^{(r)}$ the following holds;
letting $P=\under(F)$ be the $r$-polyad of $\P$ underlying $F$, for every $1 \le i \le r$ the bipartite graph $G_{F,P}^i$ is $\langle \d \rangle$-regular.
\end{definition}
Note that a $1$-partition is trivially $\langle \d \rangle$-good for any $\d$.
Moreover, a $2$-partition $\P$ is $\langle \d \rangle$-good if and only if every bipartite graph in $\P^{(2)}$ (between any two distinct vertex clusters of $\P^{(1)}$) is $\langle \d \rangle$-regular (recall the remark after Definition \ref{def:aux}).
For a $(k-1)$-partition $\P$ with $\P^{(1)} \prec \{V_1,\ldots,V_k\}$ we henceforth denote, for every $1 \le i \le k$,
\begin{equation}\label{eq:partition-notation}
V_i(\P) = \Big\{Z \in \P^{(1)} \,\vert\, Z \sub V_i\Big\} \quad\text{ and }\quad E_i(\P) = \Big\{E \in \P^{(k-1)} \,\vert\, E \sub \prod_{j \neq i} V_j\Big\}.\footnote{$\prod_{j \neq i} V_j = V_1\times\cdots\times V_{i-1}\times V_{i+1}\times\cdots\times V_k$.}
\end{equation}
\begin{definition}[$\langle\d\rangle$-regular partition]\label{def:k-reg}
Let $H$ be a $k$-partite $k$-graph on vertex classes $(V_1,\ldots,V_k)$
and $\P$ be a $\langle \d \rangle$-good $(k-1)$-partition with $\P^{(1)} \prec \{V_1,\ldots,V_k\}$.
We say that $\P$ is a \emph{$\langle \d \rangle$}-regular partition of $H$ if
for every $1 \le i \le k$,
$E_i(\P) \cup V_i(\P)$ is a $\langle \d \rangle$-regular partition of $G_H^i$.
\end{definition}
Note that for $k=2$, Definition~\ref{def:k-reg} reduces to Definition~\ref{def:star-regular}.
For $k=3$, a $\langle \d \rangle$-regular partition of a $3$-partite $3$-graph $H$ on vertex classes $(V_1,V_2,V_3)$ is a $2$-partition $\P=\P^{(1)} \cup \P^{(2)}$ satisfying that: $(i)$ $\P$ is $\langle \d \rangle$-good per Definition~\ref{def:k-good};
$(ii)$ from the auxiliary graph
$G_H^1$, on $(V_2 \times V_3,\,V_1)$,
one can add/remove at most $\d$-fraction of the edges such that for every graph $F \in E_1(\P)$ (so $F \sub V_2 \times V_3$)
and every vertex cluster $V \in V_1(\P)$ (so $V \sub V_1$),
the induced bipartite graph $G_H^1[F,V]$ is $\langle \d \rangle$-regular; and~$(iii)$ the analogues of~$(ii)$ in $G_H^2$ and $G_H^3$ hold as well.
\subsection{Formal statement of the main result}
We are now ready to formally state our Ackermann-type lower bound for $k$-graph $\langle \d \rangle$-regularity (the formal version of
Theorem~\ref{thm:main-informal} above). Recall that we set $\Ack_{1}(x)=2^x$ and then define for every $k \geq 1$ the $(k+1)^{th}$
Ackermann function $\Ack_{k+1}(n)$ to be the $n$-times composition of $\Ack_{k}(n)$, that is,
$\Ack_{k+1}(n)= \Ack_{k}(\Ack_{k}(\cdots(\Ack_{k}(1))\cdots))$.
\begin{theo}[Main result]\label{theo:main}
The following holds for every $k \ge 2$, $s \in \N$.
There exists a $k$-partite $k$-graph $H$ of density $2^{-s-k}$,
and a partition $\V_0$ of $V(H)$ with $|\V_0| \le 2^{200}k$, such that for every $\langle 2^{-16^k} \rangle$-regular partition $\P$ of $H$, if $\P^{(1)} \prec \V_0$ then $|\P^{(1)}| \ge \Ack_k(s)$.
\end{theo}
Let us draw the reader's attention to an important and perhaps surprising aspect of Theorem~\ref{theo:main}.
All the known tower-type lower bounds for graph regularity depend on the error parameter $\epsilon$,
that is, they show the existence of graphs $G$ with the property that every $\epsilon$-regular partition of $G$ is of order at least $\Ack_2(\poly(1/\e))$.
This should be contrasted with the fact that our lower bounds for $\langle \d \rangle$-regularity holds for a {\em fixed} error parameter $\delta$.
Indeed, instead of the dependence on the error parameter, our lower bound depends on the {\em density} of the graph.
This delicate difference makes it possible for us to execute the inductive part of the proof of Theorem~\ref{theo:main}.
In \cite{MoshkovitzSh18} we gave a $\Ack_2(\log 1/p)$ upper bound for a notion of regularity that is
slightly stronger than $\langle \d \rangle$-regularity, where we use $p$ to denote the edge density of a graph/$k$-graph.
This allowed us to devise a new proof of Fox's upper bound
for the graph removal lemma. We believe that it should
be possible to match our lower bound stated in Theorem \ref{theo:main} with an $\Ack_k(\log 1/p)$ upper bound that applies even
for a slightly stronger notion of regularity analogous to the one used in \cite{MoshkovitzSh16}.
This should allow one to deduce an $\Ack_k(\log 1/\epsilon)$ upper bound
for the $k$-graph removal lemma. The best known bounds for this problem are (at least) $\Ack_k(\poly(1/\epsilon))$.
\subsection{The core construction and proof overview}
The graph construction in Theorem~\ref{theo:core} below is the main technical result we will need in order to prove Theorem~\ref{theo:main}.
This lemma was proved in \cite{MS3}.
We will first need to define ``approximate'' refinement (a notion that goes back to Gowers~\cite{Gowers97}).
\begin{definition}[Approximate refinements]
For sets $S,T$ we write $S \sub_\b T$ if $|S \sm T| \le \b|S|$.
For a partition $\P$ we write $S \in_\b \P$ if $S \sub_\b P$ for some $P \in \P$.
For partitions $\P,\Q$ of the same set of size $n$ we write $\Q \prec_\b \P$ if
$$\sum_{\substack{Q \in \Q\colon\\Q \notin_\b \P}} |Q| \le \b n \;.$$
\end{definition}
Note that for $\Q$ equitable, $\Q \prec_\b \P$ if and only if
all but at most $\b|\Q|$ parts $Q \in \Q$ satisfy $Q \in_\b \P$.
We note that throughout the paper we will only use approximate refinements with $\b < 1/2$, and so if $S \in_\b \P$ then $S \sub_\b P$ for a unique $P \in \P$.
We will later need the following claim.
\begin{claim}\label{claim:refinement-size}
If $\Q \prec_{1/2} \P$ and $\P$ is equitable then $|\Q| \ge \frac12|\P|$.
\end{claim}
\begin{proof}
Since $\Q \prec_{1/2} \P$, the underlying set $U$ has a subset $U^*$ of size $|U^*|\ge \frac12|U|$ such that the partitions $\Q^*=\{Q \cap U^* \colon Q \in \Q\}$ and $\P^*=\{P \cap U^* \colon P \in \P\}$ of $U^*$ satisfy $\Q^* \prec \P^*$.
Since $\P$ is equitable, $|\P^*| \ge \frac12|\P|$.
Therefore, $|\Q| \ge |\Q^*| \ge |\P^*| \ge \frac12|\P|$, as desired.
\end{proof}
We stress that in Theorem~\ref{theo:core} below we only use notions related to graphs. In particular, $\langle \d \rangle$-regularity refers to Definition~\ref{def:star-regular}.
\begin{lemma}[\cite{MS3}]\label{theo:core}
Let $\Lside,\Rside$ be disjoint sets. Let
$\L_1 \succ \cdots \succ \L_s$ and $\R_1 \succ \cdots \succ \R_s$ be two sequences of $s$ successively refined equipartitions of $\Lside$ and $\Rside$, respectively,
that satisfy for every $i \ge 1$ that:
\begin{enumerate}
\item\label{item:core-minR}
$|\R_i|$ is a power of $2$ and $|\R_1| \ge 2^{200}$,
\item\label{item:core-expR} $|\R_{i+1}| \ge 4|\R_i|$,
\item\label{item:core-expL} $|\L_i| = 2^{|\R_i|/2^{i+10}}$.
\end{enumerate}
Then there exists a sequence of $s$ successively refined edge equipartitions $\G_1 \succ \cdots \succ \G_s$ of $\Lside \times \Rside$ such that for every $1 \le j \le s$, $|\G_j|=2^j$, and
the following holds for every $G \in \G_j$ and $\d \le 2^{-20}$.
For every $\langle \d \rangle$-regular partition $\P \cup \Q$ of $G$, where $\P$, $\Q$ are partitions of $\Lside$, $\Rside$, respectively, and every $1 \le i \le j$,
if $\Q \prec_{2^{-9}} \R_{i}$ then $\P \prec_{\g} \L_{i}$ with
$\g = \max\{2^{8}\sqrt{\d},\, 32/\sqrt[6]{|\R_1|} \}$.
\end{lemma}
\begin{remark}
Every $G \in \G_j$ is a bipartite graph of density $2^{-j}$ since $\G_j$ is equitable.
\end{remark}
Let us end this section by explaining the role Theorem~\ref{theo:core} plays in the proof of Theorem~\ref{theo:main}.
\paragraph{Using graphs to construct $k$-graphs:}
It is not hard to use Theorem \ref{theo:core} in order to prove a tower-type lower bound
for graph $\langle \d \rangle$-regularity. Perhaps the most surprising aspect of the proof
of Theorem~\ref{theo:main} is that in order to construct a $k$-graph we also use the graph construction
of Theorem~\ref{theo:core} in a somewhat unexpected way.
In this case, $\Lside$ will be a complete $(k-1)$-partite $(k-1)$-graph and the $\L_i$'s will be partitions of this complete $(k-1)$-graph themselves given by another application of Theorem~\ref{theo:core}.
The size of the partitions will be of $\Ack_{k-1}$-type growth, and this application of Theorem~\ref{theo:core} will ``multiply'' the $(k-1)$-graph partitions (given by the $\L_i$'s) to
produce a partition of the complete $k$-partite $k$-graph into $k$-graphs that are hard for $\langle \d \rangle$-regularity.
We will take $H$ in Theorem~\ref{theo:main} to be an arbitrary $k$-graph in this partition.
\paragraph{Why is Theorem \ref{theo:core} one-sided?}
As is evident from the statement of Theorem~\ref{theo:core}, it is one-sided in nature; that is, under the premise that the partition $\Q$
refines $\R_i$ we may conclude that $\P$ refines $\L_i$.
It is natural to ask if one can do away with this assumption, that is, be able to show that,
under the same assumptions, $\Q$ refines $\R_i$ and $\P$ refines $\L_i$.
As we mentioned in the previous item, in order to prove an Ackermann-type lower bound for hypergraph
regularity we have to apply Theorem~\ref{theo:core} with a sequence of partitions whose size grows as an Ackermann function of the same level.
Now, in this setting, Theorem~\ref{theo:core} does not hold without the one-sided assumption, because if it did, then
one would have been able to prove an Ackermann-type lower bound for graph $\langle \d \rangle$-regularity, and hence also for Szemer\'edi's regularity lemma.
Put differently, if one wishes to have a construction that holds with arbitrarily fast growing partition sizes, then
one has to introduce the one-sided assumption.
\paragraph{How do we remove the one-sided assumption?}
The proof of Theorem \ref{theo:main} proceeds by first proving a one-sided version of Theorem \ref{theo:main},
stated as Lemma~\ref{lemma:ind-k}. In order to get a construction that does not require such a one-sided assumption,
we will need one final trick; we will take $2k$ clusters of
vertices and arrange $2k$ copies
of this one-sided construction
along the $k$-edges of a cycle. This will give us a ``circle of implications'' that will eliminate the one-sided assumption.
See Subsection~\ref{subsec:pasting}.
\section{Proof of Theorem~\ref{theo:main}}\label{sec:LB}
\renewcommand{\k}{r}
\newcommand{\w}{w}
\renewcommand{\t}{t}
\newcommand{\GG}{\mathbf{G}}
\newcommand{\FF}{\mathbf{F}}
\newcommand{\VV}{\mathbf{V}}
\renewcommand{\Hy}[1]{H_{{#1}}}
\renewcommand{\A}{A}
\newcommand{\subs}{\subset_*}
\newcommand{\pad}{P}
\renewcommand{\K}{\mathcal{K}}
\newcommand{\U}{U}
\renewcommand{\k}{k}
\renewcommand{\K}{\mathcal{K}}
\renewcommand{\r}{k}
The purpose of this section is to prove the main result, Theorem~\ref{theo:main}.
Its proof crucially relies on a subtle inductive argument (see Lemma~\ref{lemma:ind-k} below).
This section is self-contained save for the application of Theorem~\ref{theo:core}.
The key step of our lower bound proof for $k$-graph regularity, stated as Lemma~\ref{lemma:ind-k} and proved in Subsection~\ref{subsec:main-induction}, relies on a construction that applies Theorem~\ref{theo:core} $k-1$ times. This lemma only gives a ``one-sided'' lower bound, in the spirit of Theorem~\ref{theo:core}.
In Subsection~\ref{subsec:pasting} we show how to use Lemma~\ref{lemma:ind-k} in order to complete the proof of Theorem~\ref{theo:main}.
We first state some properties of $k$-partitions whose proofs are deferred to the end of this section.
The first property relates $\d$-refinements of partitions and $\langle \d \rangle$-regularity of partitions.
The reader is advised to recall the notation in~(\ref{eq:partition-notation}).
\begin{claim}\label{claim:uniform-refinement}
Let $\P$ be a $(k-1)$-partition with $\P^{(1)} \prec \{V_1,\ldots,V_k\}$,
and let $\F$ be a partition of $V_1\times\cdots\times V_{k-1}$ with $E_k(\P) \prec_\d \F$.
If $\P$ is $\langle \d \rangle$-good
then the $(k-2)$-partition $\P'$ obtained by restricting $\P$ to $\bigcup_{i=1}^{k-1} V_i$ is a $\langle 3\d \rangle$-regular partition of some $F \in \F$.
\end{claim}
The second property is given by following easy (but slightly tedious to state) claim.
\begin{claim}\label{claim:restriction}
Let $H$ be a $k$-partite $k$-graph on vertex classes $(V_1,\ldots,V_k)$, and let $H'$ be the induced $k$-partite $k$-graph on vertex classes $(V_1',\ldots,V_k')$ with $V_i' \sub V_i$ and $\b \cdot e(H)$ edges.
If $\P$ is a $\langle \d \rangle$-regular partition of $H$ with
$\P^{(1)} \prec \bigcup_{i=1}^k \{V_i,\,V_i \sm V_i'\}$
then its restriction $\P'$ to $V(H')$ is a $\langle \d/\b \rangle$-regular partition of $H'$.
\end{claim}
\renewcommand{\K}{k}
\subsection{Key inductive proof}\label{subsec:main-induction}
\paragraph*{Set-up.}
We next introduce a few more definitions that are needed for the statement of Lemma~\ref{lemma:ind-k}.
Denote $e(i) = 2^{i+10}$. We define the following tower-type function $\t\colon\N\to\N$;
\begin{equation}\label{eq:t}
\t(i+1) = \begin{cases}
2^{\t(i)/e(i)} &\text{if } i \ge 1\\
2^{200} &\text{if } i = 0
\end{cases}
\end{equation}
It is easy to prove, by induction on $i$, that $\t(i) \ge e(i)\t(i-1)$ for $i \ge 2$ (for the induction step,
$t(i+1) \ge 2^{\t(i-1)} = t(i)^{e(i-1)}$, so $t(i+1)/e(i+1) \ge \t(i)^{e(i-1)-i-11} \ge \t(i)$).
This means that $t$ is a monotone increasing function, and that it is always an integer power of $2$ (follows by induction as $t(i)/e(i) \ge 1$ is a positive power of $2$ and in particular an integer).
We record the following facts regarding $\t$ for later use:
\begin{equation}\label{eq:monotone}
\t(i) \ge 4\t(i-1) \quad\text{ and }\quad
\text{ $\t(i)$ is a power of $2$} \;.
\end{equation}
For a function $f:\N\to\N$ with $f(i) \ge i$ we denote
\begin{equation}\label{eq:f*}
f^*(i) = \t\big(f(i)\big)/e(i) \;.
\end{equation}
Note that $f^*(i)$ is indeed a positive integer (by the monotonicity of $\t$, $f^*(i) \ge \t(i)/e(i)$ is a positive power of $2$).
In fact, $f^*(i) \ge f(i)$ (as $f^*(i) \ge 4^{f(i)}/e(i)$ using~(\ref{eq:monotone})).
We recursively define the function $\A_\k\colon\N\to\N$ for any integer $\k \ge 2$ as follows: $\A_2(i)=i$, whereas for $\k \ge 3$,
\begin{equation}\label{eq:Ak}
\A_\k(i+1) = \begin{cases}
\A_{\k-1}(\A_\k^*(i)) &\text{if } i \ge 1\\
2^{2^{3\K+2}} &\text{if } i = 0
\end{cases}
\end{equation}
Note that $\A_\k$ is well defined since $\A_\k^*(i) \in \N$ for $i \ge 1$ and $k \ge 2$, and that $\A_\k^{},\A_\k^*$ are both monotone increasing.
It is evident that $\A_\k$ grows like the $k$-th level Ackermann function; in fact, one can check that for every $\k \ge 2$ we have
\begin{equation}\label{eq:A_k}
\A_\k(i) \ge \Ack_\k(i) \;.
\end{equation}
Furthermore, we denote, for $\k \ge 1$,
\begin{equation}\label{eq:delta_k}
\d_\k = 2^{-8^\k} \;.
\end{equation}
We moreover denote, for $\k \ge 2$,
\begin{equation}\label{eq:m_k}
m_\k(i):=\A_2^*(\cdots(\A^*_{\k}(i))\cdots) \;.
\end{equation}
We next record a few easy bounds for later use.
Recall that
\begin{equation}\label{eq:delta_12}
\d_1 = 2^{-8} \quad\text{ and }\quad \d_2 = 2^{-64} \;.
\end{equation}
Noting the relation $\d_{\k} = \d_{\k-1}^8$, we have for $\k \ge 3$ that
\begin{equation}\label{eq:delta_k-bound}
\d_\k^{1/4} = \d_{\k-1}^2 \le \d_2 \d_{\k-1} = 2^{-64}\d_{\k-1} \;.
\end{equation}
Noting the relation $\A_\K(1) = \d_\K^{-4}$ for $\K \ge 3$, we have for $\K \ge 3$ that
\begin{equation}\label{eq:t1-bound}
1/\sqrt[6]{\A_\K(1)} \le \d_{\K}^{1/2} = \d_{\K-1}^{4} \le \d_1^3\d_{\K-1} = 2^{-24}\d_{\K-1} \;.
\end{equation}
\paragraph*{Key inductive proof.}
The key argument in our lower bound proof for $\K$-graph regularity is the following theorem, which is proved by induction on the hypergraph's uniformity.
\begin{lemma}[$\k$-graph induction]\label{lemma:ind-k}
Let $s \in \N$, let $\Vside^1,\ldots,\Vside^\k$ be $\k \ge 2$ mutually disjoint sets of equal size $n$ and let $\V_1 \succ\cdots\succ \V_m$ be a sequence of
$m=m_\k(s)$
successive equitable refinements of $\{\Vside^1,\ldots,\Vside^\k\}$
with $|V_h(\V_i)|=\t(i)$ for every $i,h$.\footnote{Since we assume that each $\V_i$ refines $\{\Vside^1,\ldots,\Vside^k\}$ then $V_1(\V_i)$ is (by the notation mentioned before Claim \ref{claim:star-union}) the restriction of $\V_i$ to $\Vside^1$.}
Then there exists a sequence of $s$ successively refined equipartitions $\HH_1 \succ \cdots \succ \HH_s$ of $\Vside^1 \times \cdots \times \Vside^\k$ such that for every $1 \le j \le s$, $|\HH_j|=2^j$
and every $H \in \HH_j$ satisfies the following property:\\
If $\P$ is a $\langle \d_\k \rangle$-regular
partition of $H$, and for some $1 \le i \le \A_\k(j)$ we have $V_h(\P) \prec_{2^{-9}} V_h(\V_i)$ for every $2 \le h \le \k$, then $V_1(\P) \prec_{2^{-9}} V_1(\V_{i+1})$.
\end{lemma}
Note that $\V_i$ is well defined in the property described in Lemma~\ref{lemma:ind-k} since $i \le \A_\k(j) \le m$.
\begin{proof}
We proceed by induction on $\k \ge 2$.
For the induction basis $\k=2$ we are given $s \in \N$, two disjoint sets $\Vside^1,\Vside^2$
as well as $m=\A_2^*(s)$ ($\ge s+1$) successive equitable refinements $\V_1 \succ\cdots\succ \V_{m}$ of $\{\Vside^1,\Vside^2\}$.
Our goal is to find a sequence of $s$ successively refined equipartitions $\HH_1 \succ \cdots \succ \HH_s$ of $\Vside^1 \times \Vside^2$ as in the statement.
To prove the induction basis, apply Theorem~\ref{theo:core} with
$$\Lside=\Vside^1,\quad \Rside=\Vside^2 \quad\text{ and }\quad V_1(\V_2) \succ \cdots \succ V_1(\V_{s+1}) ,\quad V_2(\V_1) \succ \cdots \succ V_2(\V_{s}) \;,$$
and let
$$\G^1 \succ \cdots \succ \G^{s} \quad\text{ with }\quad |G^\ell|=2^\ell \text{ for every } 1 \le \ell \le s $$
be the resulting sequence of $s$ successively refined equipartitions of $\Vside^1 \times \Vside^2$.
These two sequences indeed satisfy assumptions~\ref{item:core-minR},~\ref{item:core-expR} in Theorem~\ref{theo:core} since $|V_2(\V_j)|=\t(j)$ and by~(\ref{eq:monotone});
moreover, they satisfy assumption~\ref{item:core-expL} since for every $1 \le j \le s$ we have
$$|V_1(\V_{j+1})| = \t(j+1) = 2^{\t(j)/e(j)} = 2^{|V_2(\V_{j})|/e(j)} \;,$$
where the second equality uses the definition of the function $\t$ in~(\ref{eq:t}).
We will show that taking $\HH_j=\G_j$ for every $1 \le j \le s$ yields a sequence as required by the statement.
Fix $1 \le j \le s$ and $G \in \G_j$; note that $G$ is a bipartite graph on the vertex classes $(\Vside^1,\Vside^2)$.
Moreover, let $1 \le i \le j$ (recall $\A_2(j)=j$) and let $\P$ be a $\langle \d_2 \rangle$-regular
partition of $G$
with $V_2(\P) \prec_{2^{-9}} V_2(\V_{i})$.
Since $\d_2 \le 2^{-20}$ by~(\ref{eq:delta_12}),
Theorem~\ref{theo:core} implies that $V_1(\P) \prec_{x} V_1(\V_{i+1})$ with $x=\max\{2^{8}\sqrt{\d_2},\, 32/\sqrt[6]{\t(1)} \}$.
Using~(\ref{eq:delta_12}) and~(\ref{eq:t})
we have $x \le 2^{-9}$, completing the proof of the induction basis.
To prove the induction step, recall that we are given $s \in \N$, disjoint sets $\Vside^1,\ldots,\Vside^\k$
and a sequence of $m=m_\k(s)$ successive equitable refinements $\V_1 \succ\cdots\succ \V_m$ of $\{\Vside^1,\ldots,\Vside^\k\}$,
and our goal is to construct a sequence of $s$ successively refined equipartitions $\HH_1 \succ \cdots \succ \HH_s$ of $\Vside^1 \times \cdots \times \Vside^\k$ as in the statement.
We begin by applying the induction hypothesis with $\k-1$ (which would imply Proposition~\ref{prop:main-k-hypo} below).
We henceforth put $c=2^{-9}$.
Now, apply the induction hypothesis with $\k-1$ on
\begin{equation}\label{eq:ind-hypo}
\text{
$s':=\A_\k^*(s)$ (in place of $s$),\, $\Vside^1,\ldots,\Vside^{\k-1}$ and $\bigcup_{h=1}^{\k-1}V_h(\V_1) \succ\cdots\succ \bigcup_{h=1}^{\k-1}V_h(\V_m)$} \;,
\end{equation}
and let
\begin{equation}\label{eq:main-k-colors}
\F_1 \succ \cdots \succ \F_{s'} \quad\text{ with }\quad |F_ \ell|=2^\ell \text{ for every } 1 \le \ell \le s'
\end{equation}
be the resulting sequence of $s'$ successively refined equipartitions of $\Vside^1 \times \cdots \times \Vside^{\k-1}$.
\begin{prop}[Induction hypothesis]\label{prop:main-k-hypo}
Let $1 \le \ell \le s'$ and $F \in \F_\ell$.
If $\P'$ is a $\langle \d_{\k-1} \rangle$-regular partition of $F$ with ${\P'}^{(1)} \prec \{\Vside^1,\ldots,\Vside^{\k-1}\}$, and for some $1 \le i \le \A_{\k-1}(\ell)$ we have
$V_h(\P') \prec_c V_h(\V_i)$ for every $2 \le h \le \k$,
then $V_1(\P) \prec_c V_1(\V_{i+1})$.
\end{prop}
\begin{proof}
It suffices to verify that the number $m$ of partitions in~(\ref{eq:ind-hypo}) is as required by the induction hypothesis. Indeed, by~(\ref{eq:m_k}),
$$m_{\k-1}(s') = \A_2^*(\cdots(\A_{\k-1}^*(s'))\cdots) = \A_2^*(\cdots(\A^*_{\k}(s))\cdots) = m_\k(s) = m \;.$$
\end{proof}
For each $1 \le j \le s$ let
\begin{equation}\label{eq:main-k-dfns}
\F_{(j)} = \F_{\A_k^*(j)}
\quad\text{ and }\quad
\V_{(j)} = V_k(\V_{\A_k(j)}) \;.
\end{equation}
All these choices are well defined since $\A_\k^*(j)$ satisfies $1 \le \A_\k^*(1) \le \A_\k^*(j) \le \A_\k^*(s) = s'$ by our choice of $s'$ in~(\ref{eq:ind-hypo}), and since $\A_\k(j)$ satisfies $1 \le \A_\k(1) \le \A_\k(j) \le \A_\k(s) \le m$.
Observe that we have thus chosen two subsequences of $\F_1,\cdots,\F_{s'}$ and $V_k(\V_1),\ldots,V_k(\V_m)$, each of length $s$.
Recalling that each $\F_{(j)}$ is a partition of $\Vside^1 \times\cdots\times \Vside^{k-1}$, apply Theorem~\ref{theo:core} with
$$
\Lside=\Vside^1 \times\cdots\times \Vside^{k-1},\quad \Rside=\Vside^k \quad\text{ and }\quad \F_{(1)} \succ \cdots \succ \F_{(s)}, \quad \V_{(1)} \succ \cdots \succ \V_{(s)} \;,
$$
and let
\begin{equation}\label{eq:ind-colors2}
\G_1 \succ \cdots \succ \G_{s} \quad\text{ with }\quad |G_\ell|=2^\ell \text{ for every } 1 \le \ell \le s
\end{equation}
be the resulting sequence of $s$ successively refined graph equipartitions of $(\Vside^1\times\cdots\times\Vside^{\k-1})\times\Vside^\k$.
\begin{prop}[Core proposition]\label{prop:ind-prop2}
Let $1 \le j \le s$ and $G \in \G_j$.
If $\E \cup \V$ is a $\langle \d_\k \rangle$-regular partition of $G$ (where $\E$ and $\V$ are partitions of $\Vside^1\times\cdots\times\Vside^{\k-1}$ and $\Vside^\k$ respectively),
and for some $1 \le j' \le j$ we have $\V \prec_{c} \V_{(j')}$,
then $\E \prec_{\frac14\d_{\k-1}} \F_{(j')}$.
\end{prop}
\begin{proof}
First we need to verify that we may apply Theorem~\ref{theo:core} as above.
Assumptions~\ref{item:core-minR} and~\ref{item:core-expR} follow from the fact that $|\V_{(j)}|=\t(\A_\k(j))$, from~(\ref{eq:monotone})
and the fact that $\A_k(1) \ge 2^{2^{11}} \ge 2^{200}$ for $\k \ge 3$ by~(\ref{eq:Ak}).
To verify that assumption~\ref{item:core-expL} holds, note that $|\F_{(j)}|=2^{\A_\k^*(j)}$
by~(\ref{eq:main-k-colors}),
and that
$|\V_{(j)}|=\t(\A_\k(j))$
by the statement's assumption that $|\V_i[\k]|=\t(i)$.
Thus, indeed,
$$
|\F_{(j)}| = 2^{\A_\k^*(j)} = 2^{\t(\A_\k(j))/e(j)} = 2^{|\V_{(j)}|/e(j)} \;,
$$
where the second equality uses the definition in~(\ref{eq:f*}).
Moreover, note that $\d_\k \le \d_2 \le 2^{-20}$ by~(\ref{eq:delta_12}).
We can thus use Theorem~\ref{theo:core} to infer that the fact that $\V \prec_{c} \V_{(j')}$ implies that $\E \prec_x \F_{(j')}$ with $x=\max\{2^{5}\sqrt{\d_\k},\, 32/\sqrt[6]{\t(\A_\K(1))} \}$.
To see that indeed $x \le \frac14\d_{\k-1}$,
apply~(\ref{eq:delta_k-bound}) as well as the fact that $\t(\A_\K(1)) \ge \A_\K(1)$ and~(\ref{eq:t1-bound}).
\end{proof}
For each $G \in \G_j$ let $\Hy{G}$ be the $k$-partite $k$-graph on vertex classes $(\Vside^1,\ldots,\Vside^k)$ with edge set
$$E(\Hy{G}) = \big\{ (v_1,\ldots,v_k) \,:\, ((v_1,\ldots,v_{k-1}),v_k) \in E(G) \big\} \;,$$
and note that we have (recall Definition \ref{def:aux})
\begin{equation}\label{eqH}
G=G_{\Hy{G}}^k\;.
\end{equation}
For every $1 \le j \le s$ let $\HH_j=\{\Hy{G} \colon G \in \G_j \}$, and note that $|\HH_j|=|\G_j|=2^j$ by~(\ref{eq:ind-colors2}),
that $H_j$ is an equipartition of $\Vside^1\times\cdots\times\Vside^k$,
and that $\HH_1 \succ\cdots\succ \HH_s$.
Our goal is to show that these partitions satisfy the property guaranteed by the statement.
Henceforth fix $1 \le j \le s$ and $H \in \HH_j$, and write $H=\Hy{G}$ with $G \in \G_j$.
To complete the proof is suffices to show that $H$ satisfies the property is the statement.
Assume now that $i$ is such that
\begin{equation}\label{eq:ind-i-assumption}
1 \le i \le \A_\k(j)
\end{equation}
and:
\begin{enumerate}
\item\label{item:ind-reg}
$\P$ is a $\langle \d_\k \rangle$-regular partition of $H$,
\item\label{item:ind-refine} $V_h(\P) \prec_{c} V_h(\V_i)$ for every $2 \le h \le \k$.
\end{enumerate}
In the remainder of the proof we will complete the induction step by showing that
\begin{equation}\label{eq:ind-goal}
V_1(\P) \prec_{c} V_1(\V_{i+1}) \;.
\end{equation}
It follows from Item~\ref{item:ind-reg},
by Definition~\ref{def:k-reg} and~(\ref{eqH}), that in particular
\begin{equation}\label{eq:ind-reg}
E_k(\P) \cup V_k(\P) \text{ is a $\langle \d_\k \rangle$-regular partition of } G.
\end{equation}
Let
\begin{equation}\label{eq:ind-j'}
1 \le j' \le s
\end{equation}
be the unique integer satisfying
\begin{equation}\label{eq:ind-sandwich}
\A_k(j') \le i < \A_k(j'+1) \;.
\end{equation}
Note that (\ref{eq:ind-j'}) holds due to~(\ref{eq:ind-i-assumption}).
Recalling~(\ref{eq:main-k-dfns}),
the lower bound in~(\ref{eq:ind-sandwich}) implies that $V_k(\V^i) \prec V_k(\V_{\A_k(j')}) = \V_{(j')}$.
Therefore, the assumption $V_k(\P) \prec_{c} V_k(\V^i)$ in Item~\ref{item:ind-refine} implies that
\begin{equation}\label{eq:ind-Zk}
V_k(\P) \prec_{c} \V_{(j')} \;.
\end{equation}
Apply Proposition~\ref{prop:ind-prop2} on $G$, using~(\ref{eq:ind-reg}),~(\ref{eq:ind-j'}) and~(\ref{eq:ind-Zk}), to deduce that
\begin{equation}\label{eq:ind-E}
E_k(\P) \prec_{\frac14\d_{k-1}} \F_{(j')} = \F_{\A_k^*(j')} \;,
\end{equation}
where for the equality again recall~(\ref{eq:main-k-dfns}).
Let $\P^*$ be the restriction of $\P$ to $\Vside^1\cup\cdots\cup\Vside^{\k-1}$,
and let $\P' = \P^* \sm \P^*[\Vside^1\times\cdots\times\Vside^{\k-1}]$.
Note that $\P^*$ is a $(\k-1)$-partition on $(\Vside^1,\ldots,\Vside^{\k-1})$ and that $\P'$ is a $(\k-2)$-partition on $(\Vside^1,\ldots,\Vside^{\k-1})$.
Since $\P$ is a $\langle \d_k \rangle$-regular partition of $H$ (by Item~\ref{item:ind-reg} above), $\P^*$
is in particular $\langle \d_k \rangle$-good. By~(\ref{eq:ind-E}) we may thus apply Claim~\ref{claim:uniform-refinement} on $\P^*$
to conclude that
\begin{equation}\label{eq:ind-reg2}
\P' \text{ is a }
\langle \d_{k-1} \rangle
\text{-regular partition of some $F\in\F_{\A_k^*(j')}$.}
\end{equation}
By~(\ref{eq:ind-reg2}) we may apply Proposition~\ref{prop:main-k-hypo} with $G$, $\P'$, $\ell=\A_k^*(j')$ and $i$, observing (crucially)
that $i \leq \ell$ by (\ref{eq:ind-sandwich}).
Note that Item~\ref{item:ind-refine} in particular implies that $V_h(\P') \prec_{c} V_h(\V_i)$ for every $2 \le h \le k$.
We thus deduce that $V_1(\P') \prec_{c} V_1(\V_{i+1})$.
Since $V_1(\P') = V_1(\P)$, this proves~(\ref{eq:ind-goal}) and thus completes the induction step and the proof.
\end{proof}
\subsection{Putting everything together}\label{subsec:pasting}
We can now prove our main theorem, Theorem~\ref{theo:main}, which we repeat here for convenience.
\addtocounter{theo}{-1}
\begin{theo}[Main theorem]\label{theo:main}
The following holds for every $k \ge 2$, $s \in \N$.
There exists a $k$-partite $k$-graph $H$ of density $2^{-s-k}$,
and a partition $\V_0$ of $V(H)$ with $|\V_0| \le 2^{200}k$, such that if $\P$ is a $\langle 2^{-16^k} \rangle$-regular partition of $H$ with $\P^{(1)} \prec \V_0$ then $|\P^{(1)}| \ge \Ack_k(s)$.
\end{theo}
\begin{remark}\label{remark:main}
As can be easily checked, the proof of Theorem~\ref{theo:main} also gives that $H$ has the same number of vertices in all vertex classes.
\end{remark}
\begin{proof}
Let the $k$-graph $B$ be the tight $2k$-cycle; that is, $B$ is the $k$-graph on vertex classes $\{0,1,\ldots,2k-1\}$ with edge set $E(B)=\{\{0,1,\ldots,k-1\},\{1,2,\ldots,k\},\ldots,\{2k-1,0,\ldots,k-2\}\}$.
Note that $B$ is $k$-partite with vertex classes $(\{0,k\},\{1,k+1\},\ldots,\{k-1,2k-1\}\}$.
Put $m=m_k(s-k)$ and let $n \ge \t(m)$.
Let $\Vside^0,\ldots,\Vside^{2k-1}$ be $2k$ mutually disjoint sets of size $n$ each.
Let $\V^1 \succ\cdots\succ \V^m$ be an arbitrary sequence of $m$ successive equitable refinements of $\{\Vside^0,\ldots,\Vside^{2k-1}\}$ with $|\V^i_h|=\t(i)$ for every $1 \le i \le m$ and $0 \le h \le 2k-1$, which exists as $n$ is large enough.
Extending the notation $\V_x$ (above Definition~\ref{def:k-reg}), for every $0 \le x \le 2k-1$ we henceforth denote the restriction of the vertex partition $\V \prec \{\Vside^0,\ldots,\Vside^{2k-1}\}$ to $\Vside^x$ by $\V_x = \{V \in \V \,\vert\, V \sub \Vside^x\}$.
For each edge $e=\{x,x+1,\ldots,x+k-1\} \in E(B)$ (here and henceforth when specifying an edge, the integers are implicitly taken modulo $2k$)
apply Lemma~\ref{lemma:ind-k} with
$$s,\,
\Vside^{x},\Vside^{x+1},\ldots,\Vside^{x+k-1}
\text{ and }
\bigcup_{j=0}^{k-1}\V^{1}_{x+j}
\succ\cdots\succ \bigcup_{j=0}^{k-1}\V^{m}_{x+j} \;.$$
Let $H_e$ denote the resulting $k$-partite $k$-graph on $(\Vside^{x},\Vside^{x+1},\ldots,\Vside^{x+k-1})$.
Note that $d(H_e) = 2^{-s}$.
Moreover, denoting
$$c = 2^{-9} \quad\text{ and }\quad K=\A_k(s)+1 \;,$$
$H_e$ has the property that for every $\langle \d_k \rangle$-regular partition $\P'$ of $H_e$ and every $1 \le i < K$,
\begin{equation}\label{eq:paste-property}
\text{If $V_{x+h}(\P') \prec_{c} V_{x+h}(\V_i)$ for every $1 \le h \le k-1$, then $V_x(\P) \prec_{c} V_x(\V_{i+1})$.}
\end{equation}
We construct our $3$-graph on the vertex set $\Vside:=\Vside^0 \cup\cdots\cup \Vside^{2k-1}$ as
$E(H) = \bigcup_{e} E(H_e)$; that is, $H$ is the edge-disjoint union of all $2k$ $k$-partite $k$-graphs $H_e$ constructed above.
Note that $H$ is a $k$-partite $k$-graph (on vertex classes $(\Vside^0 \cup \Vside^k,\, \Vside^1 \cup \Vside^{k+1},\ldots, \Vside^{k-1} \cup \Vside^{2k-1}))$ of density $\frac{2k}{2^k} 2^{-s} \ge 2^{-s-k}$, as needed.
We will later use the following fact.
\begin{prop}\label{prop:restriction}
Let $\P$ be an $\langle 2^{-16^k} \rangle$-regular partition of $H$ with $\P^{(1)} \prec \{\Vside^0,\ldots,\Vside^{2k-1}\}$, and let
$e \in E(B)$.
Then the restriction $\P'$ of $\P$ to $V(H_e)$ is a $\langle \d_k \rangle$-regular partition of $H_e$.
\end{prop}
\begin{proof}
Immediate from Claim~\ref{claim:restriction} using the fact that $e(H_e) = \frac{1}{2k}e(H)$.
\end{proof}
Now, let $\P$ be an $\langle 2^{-16^k} \rangle$-regular partition of $H$
with $\P^{(1)} \prec \V^1$.
Our goal will be to show that
\begin{equation}\label{eq:paste-goal}
\P^{(1)} \prec_{c} \V^{K} \;.
\end{equation}
Proving~(\ref{eq:paste-goal}) would complete the proof, by setting $\V_0$ in the statement to be $\V^1$ here (notice $|\V^1|=k\t(1) = k2^{200}$ by~(\ref{eq:t}));
indeed, Claim~\ref{claim:refinement-size} would imply that
$$|\P^{(1)}| \ge \frac12|\V^{K}| = \frac12 \cdot 2k \cdot \t(K)
\ge \t(K)
\ge \t(\A_k(s))
\ge \A_k(s)
\ge \Ack(s) \;,$$
where the last inequality uses~$(\ref{eq:A_k})$.
Assume towards contradiction that $\P^{(1)} \nprec_{c} \V^{K}$. By averaging,
\begin{equation}\label{eq:assumption}
\P^{(1)}_h \nprec_c \V^{K}_h \text{ for some } 0 \le h \le 2k-1.
\end{equation}
For each $0 \le h \le 2k-1$ let $1 \le \b(h) \le K$ be the largest integer satisfying $\P^{(1)}_h \prec_c \V^{\b(h)}_h$,
which is well defined since $\P^{(1)}_h \prec_c \V^1_h$ (in fact $\P^{(1)} \prec \V^1$).
Put $\b^* = \min_{0 \le h \le 2k-1} \b(h)$, and note that by~(\ref{eq:assumption}),
\begin{equation}\label{eq:paste-star}
\b^* < K \;.
\end{equation}
Let $0 \le x \le 2k-1$ minimize $\b$, that is, $\b(x)=\b^*$.
Therefore:
\begin{equation}\label{eqcontra}
\P^{(1)}_{x+k-1} \prec_c \V^{\b^*}_{x+k-1},\,\ldots,\,\P^{(1)}_{x+1} \prec_c \V^{\b^*}_{x+1} \text{ and }
\P^{(1)}_{x} \nprec_c \V^{\b^*+1}_{x}.
\end{equation}
Let $e=\{x,x+1,\ldots,x+k-1\} \in E(B)$.
Let $\P'$ be the restriction of $\P$ to $V(H_e)=\Vside^{x} \cup \Vside^{x+1} \cup\cdots\cup \Vside^{x+k-1}$, which is a $\langle \d_k \rangle$-regular partition of $H_e$ by Proposition~\ref{prop:restriction}. Since ${\P'}^{(x+h)}_{h}=\P^{(x+h)}_h$ for every $0 \le h \le k-1$ we get
from~(\ref{eqcontra}) a contradiction to~(\ref{eq:paste-property}) with $i=\beta^*$.
We have thus proved~(\ref{eq:paste-goal}) and so the proof is complete.
\end{proof}
\subsection{Deferred proofs: properties of $k$-partitions}\label{subsec:k-partitions-proofs}
\renewcommand{\K}{\mathcal{K}}
Henceforth, for a $(k-1)$-partite $(k-1)$-graph on $(V_1,\ldots,V_{k-1})$ and a disjoint vertex set $V$ we denote by $F \circ V$ the $k$-partite $k$-graph on $(V_1,\ldots,V_{k-1},V)$ given by
$$F \circ V := \{ (v_1,\ldots,v_{k}) \,\vert\, (v_1,\ldots,v_{k-1}) \in F \text{ and } v_{k} \in V \} \;.$$
We will use the following additional property of $k$-partitions.
\begin{claim}\label{claim:decomposition}
Let $\P$ be a $(k-1)$-partition
with $\P^{(1)} \prec (\Vside^1,\ldots,\Vside^k)$,
$F \in E_k(\P)$ and $V \in V_k(\P)$.
Then there is a set of $k$-polyads $\{P_i\}_i$ of $\P$ such that
\begin{equation}\label{eq:red-k-partitionP-gen}
F \circ V = \bigcup_i \K(P_i) \,\text{ is a partition,
with } P_i = (P_{i,1},\ldots,P_{i,k-1},F) \;.
\end{equation}
\end{claim}
\begin{proof
We proceed by induction on $\k \ge 2$, noting that the induction basis $\k=2$ is trivial since in this case
$F=V' \in V_2(\P)$,
hence $F \circ V = V' \times V$ is simply $\K(P)$ where $P$ is the $2$-polyad of $\P$ corresponding to the pair $(V',V)$.
For the induction step assume the statement holds for $k \ge 2$ and let us prove it for $k+1$. Let $\P$ be a $k$-partition on $(\Vside^1,\ldots,\Vside^{k+1})$, let $F \in E_{k+1}(\P)$ and let $V \in V_{k+1}(\P)$,
and denote the vertex classes of $\F$ by $(V_1,\ldots,V_k)$ with $V_j \sub \Vside^j$ for every $1 \le j \le k$.
Recall that, by Definition~\ref{def:r-partition},
$\P^{(\k)} \prec \K_{\k}(\P)$.
Thus, $F \sub \K(G_1,\ldots,G_\k)$ with $G_j \in \P^{(\k-1)}$ for every $1 \le j \le \k$, where $G_j$ is a $(\k-1)$-partite $(\k-1)$-graph on $(V_1,\ldots,V_{j-1},V_{j+1},\ldots,V_\k)$.
We have
\begin{equation}\label{eq:decompose}
F \circ V = \K(G_1 \circ V, \ldots,\, G_{\k} \circ V,\, F) \;;
\end{equation}
indeed, the inclusion~$\sub$ follows from the fact that $F \sub \K(G_1,\ldots,G_\k)$, and the reverse inclusion~$\supseteq$ is immediate.
Now, for every $1 \le j \le \k$, let $\P_j$ denote the restriction of $\P$ to the vertex classes $(\Vside^1,\ldots,\Vside^{j-1},\Vside^{j+1},\ldots,\Vside^\k,\Vside^{k+1})$
and apply the induction hypothesis with the $(\k-1)$-partition $\P_j$, the $(k-1)$-graph $G_j$ and $V$.
It follows that there is a partition $G_j \circ V = \bigcup_i \K(P_{j,i})$ where each $P_{j,i}$ is a $\k$-polyad of $\P_j$ (and thus of $\P$) on $(V_1,\ldots,V_{j-1},V_{j+1},\ldots,V_\k,V)$.
Since $\K_{\k}(\P_j) \succ \P_j^{(\k)}$, again by Definition~\ref{def:r-partition},
for each $i$ and $j$ we have a partition $\K(P_{j,i}) = \bigcup_\ell F_{j,i,\ell}$ with $F_{j,i,\ell} \in \P_j^{(\k)}$, where each $F_{j,i,\ell}$ is a $\k$-partite $\k$-graph on $(V_1,\ldots,V_{j-1},V_{j+1},\ldots,V_\k,V)$.
Summarizing, for every $1 \le j \le k$ we have the partition $G_j \circ V = \bigcup_{i,\ell} F_{j,i,\ell}$,
and so it follows using~(\ref{eq:decompose}) that we have the partition
$$
F \circ V = \K\Big(\bigcup_{i,\ell} F_{1,i,\ell}, \ldots,\, \bigcup_{i,\ell} F_{\k,i,\ell},\, F \Big)
= \bigcup_{\substack{i_1,\ldots,i_\k\\\ell_1,\ldots,\ell_\k}} \K(F_{1,i,\ell},\ldots,\,F_{\k,i,\ell},\, F) \;.
$$
As each $(\k+1)$-tuple $(F_{1,i,\ell},\ldots,\,F_{\k,i,\ell}, F)$ corresponds to a $(\k+1)$-polyad of $\P$, this completes the inductive step.
\end{proof}
Before proving Claim~\ref{claim:uniform-refinement} we will also need the following two easy claims.
\begin{claim}\label{claim:star-union
Let $G_1,\ldots,G_\ell$ be mutually edge-disjoint bipartite graphs on the same vertex classes $(Z,Z')$.
If every $G_i$ is $\langle \d \rangle$-regular then $G=\bigcup_{i=1}^\ell G_i$ is also $\langle \d \rangle$-regular.
\end{claim}
\begin{proof
Let $S \sub Z$, $S' \sub Z'$ with $|S| \ge \d|Z|$, $|S'| \ge \d|Z'|$.
Then
$$d_G(S,S') = \frac{e_G(S,S')}{|S||S'|}
= \sum_{i=1}^\ell \frac{e_{G_i}(S,S')}{|S||S'|}
= \sum_{i=1}^\ell d_{G_i}(S,S')
\ge \sum_{i=1}^\ell \frac12 d_{G_i}(Z,Z')
= \frac12 d_{G}(Z,Z') \;,$$
where the second and last equalities follows from the mutual disjointness of the $G_i$, and the inequality follows from the $\langle \d \rangle$-regularity of each $G_i$.
Thus, $G$ is $\langle \d \rangle$-regular, as claimed.
\end{proof}
\begin{claim}\label{claim:refinement-union}
If $\Q \prec_\d \P$ then there exist $P \in \P$ and $Q$ that is a union of members of $\Q$ such that $|P \triangle Q| \le 3\d|P|$.
\end{claim}
\begin{proof}
For each $P\in \P$ let $\Q(P) = \{Q \in \Q \colon Q \sub_\d P\}$,
and denote $P_\Q = \bigcup_{Q \in \Q(P)} Q$.
We have
\begin{align*}
\sum_{P \in \P} |P \triangle P_\Q|
&= \sum_{P \in \P} |P_\Q \sm P| + \sum_{P \in \P} |P \sm P_\Q|
= \sum_{P \in \P} \sum_{\substack{Q \in \Q \colon\\Q \sub_\d P}} |Q \sm P|
+ \sum_{P \in \P} \sum_{\substack{Q \in \Q \colon\\Q \nsubseteq_\d P}} |Q \cap P| \\
&\le \sum_{P \in \P} \sum_{\substack{Q \in \Q \colon\\Q \sub_\d P}} \d|Q|
+ \Big( \sum_{\substack{Q \in \Q\colon\\Q \notin_\d \P}} |Q|
+ \sum_{\substack{Q \in \Q \colon\\Q \in_\d \P}} \d|Q| \Big)
\le 3\d\sum_{Q \in \Q} |Q|
= 3\d\sum_{P \in \P} |P| \;,
\end{align*}
where the last inequality uses the statement's assumption $\Q \prec_\d \P$ to bound the middle summand.
By averaging, there exists $P \in \P$ such that $|P \triangle P_\Q| \le 3\d|P|$, thus completing the proof.
\end{proof}
\paragraph*{Proofs of properties.}
We are now ready to prove the properties of $k$-partitions stated at the beginning of Section~\ref{sec:LB}.
\renewcommand{\k}{r}
\begin{proof}[Proof of Claim~\ref{claim:uniform-refinement}]
Put $\E = E_k(\P)$, and let us henceforth use $\k=k-1$. Since $\E \prec_\d \F$, Claim~\ref{claim:refinement-union} implies that there exist $F \in \F$ (an $\k$-partite $\k$-graph on $(\Vside^1,\ldots,\Vside^{\k})$), as well as an $\k$-partite $\k$-graph $F_\E$ that is a union of members of $\E$, such that $|F \triangle F_\E| \le 3\d|F|$.
Denote by $\Q$ the $(\k-1)$-partition $\P'$ obtained by restricting $\P$ to $\bigcup_{i=1}^{\k} V_i$, and note that $\Q$ is $\langle \d \rangle$-good since $\Q \sub \P$ and, by assumption, $\P$ is $\langle \d \rangle$-good.
Our goal is to prove that $\Q$ is a $\langle 3\d \rangle$-regular partition of $F$.
Recalling Definition~\ref{def:k-reg}, note that it suffices to show, without loss of generality, that $E_{\k}(\Q) \cup V_{\k}(\Q)$ is a $\langle \d \rangle$-regular partition of the bipartite graph $G_{F}^{\k}$.
We have $|G_{F}^{\k} \triangle G_{F_\E}^\k|=|F \triangle F_\E| \le 3\d|F|=3\d|G_{F}^{\k}|$, that is, $G_{F_\E}^\k$ is obtained from $G_{F}^{\k}$ by adding/removing at most $3\d |G_{F}^{\k}|$ edges.
Therefore, to complete the proof it suffices to show that for every $Z \in E_{\k}(\Q)$ and $Z' \in V_{\k}(\Q)$, the induced bipartite graph $G_{F_\E}^\k[Z,Z']$ is $\langle \d \rangle$-regular (recall Definition~\ref{def:star-regular}).
Apply Claim~\ref{claim:decomposition} on the $(\k-1)$-partition $\Q$ with $Z$ and $Z'$. Since $\E \prec \K_{\r-1}(\Q)$ (recall Definition~\ref{def:r-partition}),
this means that $Z \circ Z'$ is a (disjoint) union of members $E$ of $\E$ all underlied by $\k$-polyads of the form $(P_1,\ldots,P_{\k-1},Z)$.
Since $\Q$ is $\langle \d \rangle$-good (recall Definition~\ref{def:k-good}), for each such $E$ we in particular have that $G^\k_E[Z,Z']$ ($=G^\k_{E,\,\U(E)}$) is $\langle \d \rangle$-regular.
It follows that $G_{F_\E}^\k[Z,Z']$ is a disjoint union of $\langle \d \rangle$-regular bipartite graphs on $(Z,Z')$.
Claim~\ref{claim:star-union} thus implies that $G_{F_\E}^\k[Z,Z']$ is a $\langle \d \rangle$-regular bipartite graph. As explained above, this completes the proof.
\end{proof}
We end this subsection with the easy proof of Claim~\ref{claim:restriction}.
\begin{proof}[Proof of Claim~\ref{claim:restriction}]
Recall Definition~\ref{def:k-reg}.
Clearly, $\P'$ is $\langle \d \rangle$-good. We will show that $E_1(\P') \cup V_1(\P')$ is a $\langle \d/\b \rangle$-regular partition of $G^1_{H'}$.
The argument for $G^i_{H'}$ for every $2 \le i \le k$ will be analogous, hence the proof would follow.
Observe that $G^1_{H'}$ is an induced subgraph of $G^1_{H}$, namely, $G^1_{H'} = G^i_{H}[V_2' \times\cdots\times V_k',\, V_1']$.
By assumption, $e(H') = \b e(H)$, and thus $e(G^1_{H'}) = \b e(G^1_{H})$.
By the statement's assumption on $\P^{(1)}$ and since $E_1(\P) \cup V_1(\P)$ is a $\langle \d \rangle$-regular partition of $G^1_{H}$, we deduce---by adding/removing at most $\d e(G^1_{H}) = (\d/\b)e(G^1_{H'})$ edges of $G^1_{H'}$---that $E_1(\P') \cup V_1(\P')$ is a $\langle \d/\b \rangle$-regular partition of $G_{H'}^1$.
As explained above, this completes the proof.
\end{proof}
\section{Ackermann-type Lower Bounds for the R\"odl-Schacht Regularity Lemma}\label{sec:coro}
\renewcommand{\K}{\mathcal{K}}
The purpose of this section is to apply Theorem~\ref{theo:main} in order to prove Corollary~\ref{coro:RS-LB}, giving level-$k$ Ackermann-type lower bounds for the $k$-graph regularity lemma of R\"odl-Schacht~\cite{RodlSc07}.
We begin with the required definitions.
The definitions we state here are essentially equivalent to (though shorter than) those in~\cite{RodlSc07}.
We will rely on the definitions in Subsection~\ref{subsec:preliminaries}, and in particular, the definition of a $k$-partition.
For a $k$-graph $H$, the \emph{density} of a $(k-1)$-graph $S$ in $H$ is
$$d_H(S) = \frac{|H \cap \K(S)|}{|\K(S)|} \;,$
where $d_H(S)=0$ if $|\K(S)|=0$.
The notion of $\e$-regularity for $k$-graphs is defined as follows.
\begin{definition}[$\e$-regular $k$-graph]\label{def:e-reg}
A $k$-partite $k$-graph $H$ is \emph{$(\e,d)$-regular}---or simply \emph{$\e$-regular}---in a $k$-polyad $P$ with $H \sub \K(P)$
if for every $S \sub P$ with $|\K(S)| \ge \e|\K(P)|$ we have $d_H(S) = d \pm \e$
\end{definition}
A \emph{partition} of a $k$-graph $H$ is simply a $(k-1)$-partition on $V(H)$.
\begin{definition}[$\e$-regular partition]\label{def:e-reg-partition}
A partition $\P$ of a $k$-graph $H$ is \emph{$\e$-regular} if
$\sum_P |\K(P)| \le \e|V(H)|^k$ where the sum is over all $k$-polyads $P$ of $\P$ for which
$H \cap \K(P)$ is not $\e$-regular in $P$.
\end{definition}
Henceforth, an \emph{$(r,a_1,\ldots,a_r)$-partition} is simply an $r$-partition $\P$ (recall Definition~\ref{def:r-partition}) where $|\P^{(1)}|=a_1$ and for every $2 \le i \le r$,
$\P^{(i)}$ subdivides each $K \in \K_i(\P)$ into $a_i$ parts
\begin{definition}[$f$-equitable partition]\label{def:r-equitable}
Let $f\colon[0,1]\to[0,1]$.
An $(r,a_1,\ldots,a_r)$-partition $\P$ is \emph{$f$-equitable} if $\P^{(1)}$ is equitable and for every $2 \le i \le r$, every $i$-graph $F \in \P^{(i)}$ is $(\e,1/a_i)$-regular in $\U(F)$, where $\e=f(d_0)$ and $d_0=\min\{1/a_2,\ldots,1/a_{r}\}$.
\end{definition}
\subsection{The lower bound}
The $k$-graph regularity of R\"odl-Schacht~\cite{RodlSc07} states, roughly, that for every $\e>0$ and every function $f\colon\N\to(0,1]$, every $k$-graph has an $\e$-regular $f$-equitable equipartition $\P$ where $|\P|$ is bounded by a level-$k$ Ackermann-type function.
In fact, R\"odl-Schacht's $k$-graph regularity lemma (Theorem~17 in~\cite{RodlSc07}) uses a considerably stronger notion of regularity of a partition that involves an additional function $r$ which we shall not discuss here (this stronger notion was crucial in~\cite{RodlSc07} for allowing them to prove a counting lemma).
Our lower bound below applies even to the weaker notion stated above, which corresponds to taking $r \equiv 1$.
The proof of Corollary~\ref{coro:RS-LB}
will follow quite easily from Theorem~\ref{theo:main} together with Claim~\ref{claim:k-reduction} below.
Claim~\ref{claim:k-reduction} basically shows that a $\langle \d \rangle$-regularity ``analogue'' of R\"odl-Schacht's notion of regularity implies graph $\langle \d \rangle$-regularity.
It is essentially a generalization of a similar claim from~\cite{MS3}.
The proof of Claim~\ref{claim:k-reduction} is deferred to the Appendix~\ref{sec:RS-appendix}.
Henceforth we say that a graph partition is \emph{perfectly $\langle \d \rangle$-regular} if all pairs of distinct clusters are $\langle \d \rangle$-regular without modifying any of the graph's edges.
\begin{claim}\label{claim:k-reduction}
Let $H$ be a $k$-partite $k$-graph on vertex classes $(\Vside^1,\ldots,\Vside^k)$, and
let $\P$ be an $f$-equitable partition of $H$ with $\P^{(1)} \prec \{\Vside^1,\ldots,\Vside^k\}$,
$f(x) = \d^4(x/2)^{2^{k+3}}$ and $|V(H)| \ge n_0(\d,|\P|)$.
Suppose that for each $k$-polyad $P$ of $\P$, every $S \sub P$ with $|\K(S)| \ge \d|\K(P)|$ has $d_{H}(S) \ge \frac23 d_{H}(P)$.
Then $E_k(\P) \cup V_k(\P)$ is a perfectly $\langle 2\sqrt{\d} \rangle$-regular
partition of $G_H^k$.
\end{claim}
We now formally restate and prove Corollary~\ref{coro:RS-LB}.
We mention that, as will be immediate from the proof, our lower bound not only applies to the hypergraph regularity lemma of R\"odl and Schacht but also to the hypergraph regular approximation lemma~\cite{RodlSc07}.
\begin{theo}[Lower bound for R\"odl-Schacht's $k$-graph regularity lemma]\label{theo:RS-LB}
Let $s \ge k \ge 2$ and
put $c = 2^{-32^k}$.
For every $s \in \N$ there exists a $k$-partite $k$-graph $H$ of density $p=2^{-s-k}$, and a partition $\V_0$ of $V(H)$ with $|\V_0| \le k 2^{200}$,
such that if $\P$ is an $\e$-regular $f$-equitable partition of $H$
with
$\e \le c p$,
$f(x) \le c^4(x/2)^{2^{k+3}}$, $|V(H)| \ge n_0(k,|\P|)$
and $\P^{(1)} \prec \V_0$, then $|\P^{(1)}| \ge \Ack_k(s)$.
\end{theo}
\begin{remark}
One can easily remove the assumption $\P^{(1)} \prec \V_0$ by replacing $\P$ with its appropriate intersection
with $\V_0$. Since $|\V_0|=O(k)$ this has only a minor effect on the parameters of $\P$ and thus
one gets essentially the same lower bound. We omit the details of this routine transformation.
\end{remark}
\begin{proof}[Proof of Theorem~\ref{theo:RS-LB}]
Put $\a_k = 2^{-16^k}$ (recall $c = 2^{-32^k}$).
The bound $|\P^{(1)}| \ge \Ack_k(s)$ would follow from Theorem~\ref{theo:main} if we show that $H$ is $\langle \a_k \rangle$-regular relative to the $(k-1)$-partition $\P$.
Henceforth put $\d=(\a_k/2)^2$. We will later use the inequalities
\begin{equation}\label{eq:RS-LB-ineq}
c \le \frac{\a_k^k}{7k^k} \le \d \;.
\end{equation}
First we claim that $\P$ is $\langle \a_k \rangle$-good (recall Definition~\ref{def:k-good}).
Let $2 \le r \le k-1$, let $F \in \P^{(r)}$ be an $r$-partite $r$-graph on $(V_1,\ldots,V_r)$, and denote by $P=\U(F)$ the $r$-polyad underlying $F$.
We will show that the bipartite graph $G_{F,P}^r$ is $\langle \a_k \rangle$-regular, and since an analogous argument will hold for $G_{F,P}^i$ for every $1 \le i \le r$, this would prove our claim.
Recalling Definition~\ref{def:aux}, we have that $G_{F,P}^r = G_F^r[E,V_r]$ with $E:=P[V_1,\ldots,V_{r-1}] \in E_r(\P)$.
Now, suppose $\P$ is a $(k-1,a_1,\ldots,a_{k-1})$-partition, put $d_r = 1/a_r$ for each $2 \le r \le k-1$, and put $d_0=\min\{1/a_2,\ldots,1/a_{k-1}\}$.
Recalling Definition~\ref{def:r-equitable}, since $\P$ is $f$-equitable we have that $F$ is $(f(d_0),d_r)$-regular in $P$. Thus, recalling Definition~\ref{def:e-reg}, for every $S \sub P$ with $|K(S)| \ge \d|K(P)|$ (note $\d \ge c \ge f(d_0)$ using~(\ref{eq:RS-LB-ineq})) we have $d_F(S) \ge d_r-f(d_0) \ge d_F(P) - 2f(d_0) \ge \frac23 d_F(P)$.
Let the $(r-1)$-partition $\P'$ be obtained by restricting $\P$ to $V_1 \cup\cdots \cup V_r$ (so in particular $V_i(\P')=\{V_i\}$), and note that $P$ is an $r$-polyad of $\P'$.
Observe that $d_F(S) \ge \frac23 d_F(P)$ trivially holds for any $r$-polyad $P'$ of $\P'$ other than $P$ as well, since $d_F(P')=0$ as $F \sub \K(P)$.
Apply Claim~\ref{claim:k-reduction}, with (the almost trivial choice of) the $r$-partite $r$-graph $F$ and the $f$-equitable $(r-1)$-partition $\P'$ of $F$, to deduce that $E_{r}(\P') \cup \{V_r\}$
is a perfectly $\langle\a_k\rangle$-regular (i.e., $\langle 2\sqrt{\d} \rangle$-regular) partition of $G_{F}^r$.
Since $E \in E_{r}(\P')$, this in particular implies that $G_F^r[E,V_r]$ is $\langle\a_k\rangle$-regular, which proves our claim as explained above.
It remains to show that $H$ is $\langle \a_k \rangle$-regular relative to the $\langle \a_k \rangle$-good $(k-1)$-partition $\P$.
Let $H'$ be obtained from $H$ by removing all its ($k$-)edges underlied by $k$-polyads of $\P$ such that either $d_H(P) \le 6\e$ or the $k$-graph $H \cap \K(P)$ is not $\e$-regular in $P$.
By Definition~\ref{def:e-reg-partition}, the number of edges removed from $H$ to obtain $H'$ is at most
$$\e|V(H)|^k + 6\e|V(H)|^k \le 7\cdot c p |V(H)|^k
\le (\a_k p/k^k)|V(H)|^k = \a_k\cdot e(H) \;,$$
where the inequalities use the statement's assumption on $\e,c$ and~(\ref{eq:RS-LB-ineq}), and the equality uses the fact that all $k$ vertex classes of $H$ are of the same size (see Remark~\ref{remark:main}).
Thus, in $H'$, every non-empty $k$-polyad of $\P$ is $\e$-regular and of density at least $6\e$.
Again by Definition~\ref{def:e-reg-partition}, for every $k$-polyad $P$ of $\P$ and every $S \sub P$ with $|\K(S)| \ge \d |\K(P)|$ ($\ge \e |\K(P)|$
by~(\ref{eq:RS-LB-ineq})) we have $d_H(S) \ge d_H(P)-2\e \ge \frac23 d_H(P)$.
Apply Claim~\ref{claim:k-reduction}, this time with $H$ and $\P$.
It follows that
$E_{k}(\P) \cup V_{k}(\P)$
is an $\langle \a_k \rangle$-regular
partition of $G_H^k$. An analogous argument applies for $G_H^i$ for every $1 \le i \le k$, thus completing the proof.
\end{proof}
\section{$\langle \d \rangle$-regularity does not suffice for triangle counting}\label{sec:example}
Here we construct for every fixed $\delta>0$ and small enough $p$, a graph of density $p$ which is $\langle \d \rangle$-regular yet does not
contain the expected number of triangles
(actually, the example is going to be triangle free).
This shows that $\langle \d \rangle$-regularity, even just for graphs, is strictly weaker than Szemer\'edi's regularity.
The precise statement is the following.
\begin{prop}\label{claim:example}
For every $0 < p \le 10^{-3}\d^5$ and large enough $n$ there is a $n$-vertex tripartite graph of density at least $p$, whose every pair of classes span a $\langle \d \rangle$-regular graph, and yet is triangle free.
\end{prop}
We use the following well-known lemma, where we denote by $\norm{v}_1$ the $\ell_1$-norm of a vector $v$ (for a proof see, e.g., Lemma 4.3 in~\cite{KaliSh13}).
\begin{lemma}\label{lemma:convex
Every vector $x \in [0,1]^n$ is a convex combination of binary vectors $y \in \{0,1\}^n$ each with $\norm{y}_1=\norm{x}_1$.
\end{lemma}
We will also apply the following version of the Chernoff bound.
\begin{lemma}[Multiplicative Chernoff bound]\label{lemma:Chernoff}
Let $X_1,\ldots,X_n$ be mutually independent Bernoulli random variables,
and put $X=\sum_{i=1}^n X_i$, $\mu = \Ex[X]$.
For every $\d \in [0,1]$ we have
$$\Pr(X \neq (1 \pm \d)\mu) \le 2\exp\Big(-\frac13\d^2\mu\Big) \;.$$
\end{lemma}
\begin{proof}[Proof of Proposition~\ref{claim:example}]
We will in fact construct a graph satisfying an even somewhat stronger property than $\langle \d \rangle$-regularity (namely, the constant $\frac12$ is Definition~\ref{def:star-regular} will be replaced by $1-\d$).
Consider a random tripartite graph on vertex classes $(V_1,V_2,V_3)$, each of size $k$, obtained by independently retaining each edge with probability
$q:=3p$ $(\le 1)$, where $k$ is any integer satisfying
\begin{equation}\label{eq:example-k}
64\d^{-2}q^{-1} \le k
\le \frac14\d^3 q^{-2} \;.
\end{equation}
Note that $k$ is well defined in~(\ref{eq:example-k}) by the statement's bound on $p$.
Denoting by $X$ the random variable counting the triangles, one can easily check that
$$\Exp[X] = k^3 q^3 \quad\text{ and }\quad
\Var[X] \le k^3 q^3 + 3\binom{k}{2}k^2 q^5 \;.$$
Chebyshev's inequality thus implies that
$$\Pr[X \ge \d^3k^2 q] \le \frac{\Var[X]}{(\d^3k^2q - \Exp[X])^2}
\le \frac{k^3q^3(1+\frac32 k q^2)}{k^4q^2(\d^3 - kq^2)^2}
\le \frac{q}{k} \cdot 8\d^{-6}
\le \frac18 q^2 \d^{-4}
= \frac98 p^2 \d^{-4}
< \frac12 \;,$$
where the third inequality uses the upper bound $kq^2 \le \frac14 \d^3$ $(\le \frac14)$ from~(\ref{eq:example-k}), the fourth inequality uses the lower bound from~(\ref{eq:example-k}), and the last inequality uses the statement's bound on $p$.
Next, by using Lemma~\ref{lemma:Chernoff}
together with the union bound we deduce that
\begin{equation}\label{eq:example-property}
\forall 1 \le a<b \le 3\,\,\,
\forall S \sub V_a,\, T \sub V_b \text{ with } |S| \ge \d|V_a|,\, |T| \ge \d|V_b| \,\,\colon\,\,
d(S,T) = \big(1\pm\frac13\d\big)q
\end{equation}
except with probability at most
$$3 \cdot 2^{2k} \cdot 2\exp\Big(-\frac{1}{27}\d^2 qk^2\Big) \le \frac12 \;,$$
where the inequality uses the lower bound $kq \ge 64\d^{-2}$ from~(\ref{eq:example-k}).
We deduce from all of the above that there exists a tripartite graph that has $k$ vertices in each vertex class, at most $\d^3 k^2 q$ triangles and satisfies~(\ref{eq:example-property}).
By removing an edge from each triangle, one-third of them from each of the three pairs of vertex classes, we obtain a triangle-free graph $G_0$ such that for every $a<b$ and subsets $S \sub V_a$, $T \sub V_b$
with $|S|\ge\d|V_a|$, $|T| \ge \d|V_b|$ we have
$$e(S,T) \ge \big(1-\frac13\d\big)q|S||T|-\frac13 \d^3k^2q \ge \big(1-\frac23\d\big)q|S||T|
\ge \frac{1-\frac23\d}{1+\frac13\d}d(V_a,V_b)|S||T|
\ge (1-\d)d(V_a,V_b)|S||T|$$
where the first and third inequalities follows from the lower and upper bound in~(\ref{eq:example-property}), respectively.
In particular, we deduce from the inequality $e(S,T) \ge (1-\frac23\d)q|S||T|$ above that $d(V_a,V_b) \ge \frac13 q=p$.
Summarizing, $G_0$ is triangle free, has density at least $p$ and satisfies
\begin{equation}\label{eq:example-property2}
\forall a<b\,\,
\forall S \sub V_a,\, T \sub V_b \text{ with } |S| \ge \d|V_a|,\, |T| \ge \d|V_b| \,\,\colon\,\, d(S,T) \ge (1-\d)d(V_a,V_b) \;.
\end{equation}
To obtain from $G_0$ a graph on a large enough number of vertices we simply take a blow-up, replacing each vertex $v$ of $G_0$ by a set $G(v)$ of $m$ new vertices, for $m \in \N$ arbitrarily large. The resulting tripartite graph $G$ is clearly triangle free and of density $d(G_0) \ge p$.
It thus remains to prove that any two vertex classes of $G$ span a bipartite graph satisfying the desired regularity property.
Let $a<b$ and let $\overline{V_a},\overline{V_b}$ denote the vertex classes of $G$ corresponding to $V_a,V_b$.
Note that $|V_a|=|V_b|=k$ and $|\overline{V_a}|=|\overline{V_b}|=mk$.
Let $S \sub \overline{V_a}$, $T \sub \overline{V_b}$ with $|S| \ge \d|\overline{V_a}|$ and $|T| \ge \d|\overline{V_b}|$.
We will show that
\begin{equation}\label{eq:example-goal}
d(S,T) \ge (1-\d)d(\overline{V_a},\overline{V_b}) \;,
\end{equation}
which would complete the proof.
Consider the two vectors $s,t \in [0,1]^k$ defined as follows:
$$s=(|S \cap G(u)|/m)_{u \in \overline{V_a}} \quad\text{ and }\quad
t=(|T \cap G(v)|/m)_{v \in \overline{V_b}} \;.$$
Note that $\norm{s}_1 = |S|/m \ge \d k$ and $\norm{t}_1 = |T|/m \ge \d k$.
By Lemma~\ref{lemma:convex} applied on $s$,
$$s = \sum_i \a_i s_i \quad\text{ with }\quad s_i \in \{0,1\}^k,\, \norm{s_i}_1 = |S|/m \text{ and } \a_i \ge 0,\, \sum_i \a_i = 1 \;.$$
By Lemma~\ref{lemma:convex} applied on $t$,
$$t = \sum_j \b_j t_j \quad\text{ with }\quad t_j \in \{0,1\}^k,\, \norm{t_j}_1 = |T|/m \text{ and } \b_j \ge 0,\, \sum_j \b_j = 1 \;.$$
Let $A$ denote the $k \times k$ bi-adjacency matrix of $G_0[V_a,V_b]$.
Observe that $e_G(S,T) = (ms)^T A (mt)$.
Moreover, observe that for every $i,j$ we have that $s_i^T A t_j$ is the number of edges of $G_0$ between the subsets of $V_a,V_b$ corresponding to $s_i,t_j$, respectively. Note that these subsets are of size $\norm{s_i}_1,\norm{t_j}_1$, respectively, which are both at least $\d k$.
Thus, by~(\ref{eq:example-property2}),
$$s_i^T A t_j \ge (1-\d)d(\overline{V_a},\overline{V_b})\norm{s_i}_1\norm{t_j}_1
= (1-\d)d(\overline{V_a},\overline{V_b})|S||T|/m^2 \;.$$
We deduce that
\begin{align*}
e(S,T) &= m^2 \cdot s^T A t
= m^2\Big(\sum_i \a_i s_i \Big)^T A \Big(\sum_j \b_j t_j\Big)
= m^2\sum_{i,j} \a_i\b_j s_i^T A t_j\\
&\ge \Big(\sum_i \a_i \Big)\Big(\sum_j \b_j \Big)(1-\d) d(\overline{V_a},\overline{V_b})|S||T| = (1-\d)d(\overline{V_a},\overline{V_b})|S||T|\;.
\end{align*}
This gives~(\ref{eq:example-goal}) and thus completes the proof.
\end{proof}
|
{
"timestamp": "2018-04-17T02:13:51",
"yymm": "1804",
"arxiv_id": "1804.05513",
"language": "en",
"url": "https://arxiv.org/abs/1804.05513"
}
|
\section{Introduction}
Over the past seven years, deep learning has transformed computer vision and has been implemented in scores of consumer-facing products. Many are excited that these approaches will continue to expand in scope and that new tools and products will be improved through the use of deep learning. One particularly exciting application area of deep learning has been in clinical medicine. There are many recent high-profile examples of deep learning achieving parity with human physicians on tasks in radiology \cite{gale2017detecting,rajpurkar2017chexnet}, pathology \cite{bejnordi2017diagnostic}, dermatology \cite{esteva2017dermatologist}, and opthalmology \cite{gulshan2016development}. In some instances, the performance of these algorithms exceed the capabilities of most individual physicians in head-to-head comparisons. This has lead some to speculate that entire specialties in medical imaging, such as radiology and pathology, may be radically reshaped \cite{jha2016adapting} or cease to exist entirely. Furthermore, on April 11, 2018, an important step was taken towards this future: the U.S. Food and Drug Administration announced the approval of the first computer vision algorithm that can be utilized for medical diagnosis without the input of a human clinician \citep{PressAnn97:online}.
In parallel to this progress in medical deep learning, the discovery of so-called `adversarial examples' has exposed vulnerabilities in even state-of-the-art learning systems \citep{goodfellow2014explaining}. Adversarial examples -- inputs engineered to cause misclassification -- have quickly become one of the most popular areas of research in the machine learning community \citep{Szegedy2013, nguyen2015deep, moosavi2016deepfool, papernot2016distillation}. While much of the interest with adversarial examples has stemmed from their ability to shed light on possible limitations of current deep learning methods, adversarial examples have also received attention because of the cybersecurity threats they may pose for deploying these algorithms in both virtual and physical settings \citep{papernot2016distillation, melis2017deep, kurakin2016adversarial, athalye2017synthesizing, brown2017adversarial, grosse2017adversarial}.
Given the enormous costs of healthcare in the US, it may seem prudent to take the expensive human `out of the loop' and replace him or her with an extremely cheap and highly accurate deep learning algorithm. This seems especially tempting given a recent study's finding that physician and nursing pay is one of the key drivers of high costs in the US relative to other developed countries \cite{papanicolas2018health}. However, there is an under-appreciated downside to widespread automation of medical imaging tasks given the current vulnerabilities of these algorithms. If we seriously consider taking the human doctor completely `out of the loop' (which now has legal sanction in at least one setting via the FDA, with many more to likely follow), we are forced to also consider how adversarial attacks may present new opportunities for fraud and harm. In fact, even with a human in the loop, any clinical system that leverages a machine learning algorithm for diagnosis, decision-making, or reimbursement could be manipulated with adversarial examples.
In this paper, we extend previous results on adversarial examples to three medical deep learning systems modeled after the state of the art medical classifiers. On the basis of these results and knowledge of the healthcare system, we argue that medical imaging is particularly vulnerable to adversarial attacks and that there are potentially enormous incentives to motivate prospective bad actors to carry out these attacks. We hope that by highlighting these vulnerabilities, more researchers will explore potential defenses against these attacks within the healthcare domain. Because the healthcare system is complex and administrative processes can appear byzantine, it may be difficult to imagine how these attacks could be operationalized. To ground these abstract potentials for harm in actual use cases, we describe realistic scenarios where clinical tasks might rely on deep learning and give specific examples of the fraud that could be mediated by adversarial attacks. Our goal is to provide background on the distinct features of the medical pipeline that make adversarial attacks a threat, and also to demonstrate the practical feasibility of these attacks on real medical deep learning systems.
\subsection{Adversarial Examples}
Adversarial examples are inputs to machine learning models that have been crafted to force the model to make a classification error. This problem extends back in time at least as far as the spam filter, where systematic modifications to email such as `good word attacks' or spelling modifications have long been employed to try to bypass the filters \citep{lowd2005adversarial, lowd2005good, dalvi2004adversarial}. More recently, adversarial examples were re-discovered and described in the context of deep computer vision systems through the work of \citet{Szegedy2013} and \citet{goodfellow2014explaining}. Particularly intriguing in these early examples was the fact that adversarial examples could be crafted to be extremely effective despite being imperceptibly different from natural images to human eyes. In the years since, adversarial examples -- visible and otherwise -- have been shown to exist for a wide variety of classic and modern learning systems \citep{papernot2016transferability, brown2017adversarial, elsayed2018adversarial}. By the same token, adversarial examples have been extended to various other domains such as text and speech processing \citep{jia2017adversarial, carlini2016hidden}. For an interesting history of adversarial examples and methods used to combat them see \citet{biggio2017wild}.
Figure ~\ref{fig:attack_taxonomy} places adversarial attacks within the broader context of machine learning development and deployment. While there are many possible intentional and inadvertent failures of real-world machine learning systems, adversarial attacks are particularly important to consider from the standpoint of model deployment, because they enable those submitting data into a running ML algorithm to subtly influence its behavior without ever achieving direct access to the model itself or the IT infrastructure that hosts it.
\begin{figure*}[!htb]
\centering
\includegraphics[width=\textwidth]{attack_taxonomy.png}
\caption{\label{fig:attack_taxonomy} Adversarial attacks within the broader taxonomy of risks facing the machine learning pipeline. Adversarial attacks pose just one of many possible risks in the development and deployment of ML systems, but are noteworthy because they enable end-users to manipulate model outputs without ever influencing the training process or gaining access to the deployed model itself.}
\end{figure*}
Adversarial examples are generally thought to arise from the piecewise linear components of complex nonlinear models \citep{goodfellow2014explaining}. They are not random, they are not due to overfitting or incomplete model training, they occupy only a comparatively small subspace of the feature landscape, they are robust to random noise, and they have been shown to transfer in many cases from one model to another \cite{papernot2016transferability, Tramer2017}. Furthermore, in addition to executing many successful attacks in purely virtual settings, researchers in the past several years have also demonstrated that adversarial attacks can generalize to physical world settings \citep{kurakin2016adversarial, evtimov2017robust, athalye2017synthesizing, brown2017adversarial}.
One natural question raised by the existence of adversarial examples is to what extent and in what forms they constitute a viable risk for harm in real-world machine learning settings. Many authors have discussed the feasibility of and possible motivations for adversarial attacks on certain real-world systems such as self-driving cars \citep{evtimov2017robust, carlini2017towards, lu2017no}. However, to our knowledge, previous machine learning literature has yet to thoroughly discuss the possibility of adversarial attacks on medical systems.
\section{Identifying factors in the U.S. healthcare system that favor adversarial attacks}
In this section, we provide a synthesis of aspects of the healthcare system that may create both the incentive and the opportunity for a bad actor to carry out an adversarial attack.
\subsection{Background on the healthcare economy and possible incentives for fraud via adversarial attacks}
\textbf{The healthcare economy is huge and fraud is already pervasive}. The United States spent approximately \$3.3 trillion (17.8\% of GDP) on healthcare in 2016 \cite{papanicolas2018health}, and healthcare is projected to represent 1/5 of the US economy by 2025. Given the vast sums of money represented by the healthcare economy, inevitably some seek to profit by committing fraud against the healthcare system. Medical fraud is estimated to cost hundreds of billions of dollars each year, and one study estimated this number to be as high as \$272 billion in 2011 \cite{jain2014corruption}. Fraud is committed both by large institutions and by individual actors. Large institutions engage in fraud by systemically inflating costs for services to increase revenue \cite{jama_fraud_def, rudman2009healthcare}. Likewise, it has been found that some individual physicians routinely bill for the highest allowable amount over 90\% of the time \cite{charles_ornstein_2018}.
\textbf{Algorithms will likely make medical reimbursement decisions in the future.} Due to the amount of money involved with the delivery of healthcare, complex book-keeping systems have been created to facilitate billing and reimbursement. In fact, most of the data generated by the healthcare system in the electronic healthcare record (EHR) is created to justify payments from `payers' (private or public insurers) to 'providers' (hospitals and physicians). In many cases, the level of monetary reimbursement for a given patient hinges on the establishment of specific diagnostic `codes', which are used to record a patient's diagnoses and treatments with high granularity \footnote{See for instance the International Classification of Disease code V97.33XD, which represents the following diagnosis: `Sucked into jet engine, subsequent encounter'}\citep{sanders2012road}. In an effort to increase revenue, some providers engage in the practice of 'upcoding' diagnoses or procedures -- selecting the codes which will allow them to bill for the highest amount. For their part, insurance companies seek to minimize total expenditure by investing millions of dollars in IT and personnel to identify unjustified billing codes. The resultant struggle between payors and providers has been extensively documented \citep{kesselheim2005overbilling, wynia2000physician}. To ensure consistency and justifiability, insurance companies will often demand specific gold standard tests as proof of diagnosis before reimbursing a given medical claim, and leverage increasingly sophisticated analytics to determine reimbursement value. Given these dynamics, it is seemingly inevitable that insurance companies will begin to require algorithmic confirmation of certain diagnoses before providing reimbursement. If and when this occurs, the ability to undetectably influence (either as a provider or as a payer) the outputs of trusted and otherwise unbiased diagnostic systems would result in the ability to influence the movement of billions of dollars through the healthcare economy. Even today, the practice of upcoding, which often entails finding subtle combinations of codes that influence reimbursement \citep{reynolds2005metabolic}, can itself be arguably considered a form of adversarial attack against reimbursement algorithms.
\textbf{Algorithms will increasingly determine pharmaceutical and device approvals.} The monetary value of a successful clinical trial is immense, with one recent study estimating the median revenue across individual cancer drugs to be as high as \$1.67 billion only four years after approval \citep{prasad2017research}. At the same time, regulatory bodies such as the FDA are increasingly allowing for the approval of new drugs based on digital \textit{surrogates} for patient response, including medical imaging \citep{pien2005using}. As algorithmic endpoints for clinical trials become increasingly accepted -- and deep learning algorithms continue to assert themselves as equal or superior to humans on well-defined visual diagnostic tasks -- we could soon reach a future where billion dollar drug decisions are made primarily by machines. In such a future, effectively executed adversarial attacks could allow trialists to imperceptibly `put their thumb on the scale,' even if images are vetted to ensure they are coming from the correct patients.
\subsection{Distinctive technical sources of vulnerability to adversarial attacks among medical machine learning systems}
\textbf{Ground truth is often ambiguous.} Compared to most common image classification tasks, the ground truth in medical imaging is often controversial, with even specialty radiologists disagreeing on well defined tasks \citep{njeh2008tumor, li2009variability, brouwer20123d}. As such, if end-users selectively perturb images for which it is difficult to establish the true diagnosis, they can make it extremely difficult to detect their influence through even expert human review. Somewhat ironically, it is these borderline cases are where deep learning is likely to be most valuable.
\textbf{Medical imaging is highly standardized.} Compared to other domains of computer vision, medical imaging is extremely standardized, with images generally captured with pre-defined and well-establisehd positioning and exposure \citep{simon1971principles}. As such, medical adversarial attacks do not need to meet the same standards of invariance to lighting and positional changes as attacks on other real-world systems such as self-driving cars. This is potentially important, as some have argued that dynamic viewing conditions imply that `there is a good prospect that adversarial examples can be reduced to a curiosity with little practical impact' \cite{lu2017no}.
\textbf{Commodity network architectures are often used.} Nearly all of the most successful published methods in medical computer vision have consisted of the same fundamental architecture: one of a small set of pretained ImageNet models that was fine-tuned to the specific task \cite{gulshan2016development,wang2017chestx, esteva2017dermatologist}. This lack of architectural diversity could make it easier for potential attackers to build transferable attacks against medical systems. By the same token, given the importance of peer review and publication in validating and approving medical diagnostics, it is likely the architectures of most medical image classification models will be public for the sake of transparency, allowing for more targeted adversarial attacks.
\textbf{Medical data interchange is limited and balkanized}. Five electronic health record (EHR) vendors constitute about half of the market and hundreds of others serve the other half. Even within an EHR vendor, data sharing is spotty and the terminologies and their semantics vary considerably from one implementation to another. On the one hand, this means that there are no universally shared mechanisms for authentication, verification of message integrity, data quality metrics, nor mechanisms for automated oversight. On the other hand, this allows healthcare providers to customize their EHR's, billing, and other information technology systems in ways that are opaque to most external auditors using one-size-fits-all tools and methods.
\textbf{Hospital infrastructure is very hard to update.} Medical software is often implemented within monolithic enterprise-wide proprietery software systems, making updates, revisions and fixes expensive and time-consuming. For context, consider the coding dictionaries used to classify patients' diseases, the International Classification of Disease (ICD) system. As recently as 2013, most hospitals were operating using the ninth edition of this coding scheme, published in 1978, despite the fact that a revised version (ICD-10) was published in 1990. All told, the conversion to the ICD-10 coding scheme has been estimated to cost major health centers up to \$20 million \textit{each} and require up to 15 years \citep{sanders2012road}. Others decided in the early 2000s that it would be preferable to skip the 1990 schema entirely and wait for ICD-11, despite the fact that its release wasn't scheduled until 2018. Thus, vulnerabilities present in medical software are likely to persist for years due to the difficulty and expense involved with update hospital infrastructure.
\textbf{Medicine contains a mix of technical and non-technical workers.} Compared to many other industries, medicine is extremely interdisciplinary and mostly comprised of members who lack a strong computational or statistical training background. For example, in the case of self-driving cars, the teams developing computer vision systems are likely to be led and staffed primarily by engineers, if not computer scientists. In contrast, since the clinical usability of medical imaging systems is also extremely important, hospitals are likely to lean heavily on physician-researchers in developing these systems, who tend to lack robust computational expertise \cite{manrai2014medicine}.
\textbf{Biomedical images carry personal signatures that could be used to defend against many simpler attacks, but not against adversarial examples.} One potential alternative to adversarial attacks on part of malicious end-users would be to simply substitute in true images of the target class. However, biomedical images -- including retinal images, X-rays, and skin photographs -- are often as unique to their owners as fingerprints \citep{qamber2012personal, angyal1998personal, miller2010personal}. This provides several advantages against substitution attacks: First, algorithms could be designed to check input images against prior images from the same patient, to ensure that their identities match. Second, algorithms could query against the database of previously submitted images and flag any inputs that are likely from the same patients as previous entries. Both strategies would make it more difficult for fraudsters to continually execute substitution attacks against ML systems, but neither would defend against adversarial attacks, which needn't change the personal identifiers in the image. By analogy, adversarial attacks applied to billing codes or medical text could serve to manipulate reimbursement algorithms by selecting combinations of codes or words that are individually truthful but in combination yield anomalously high reimbursement.
\textbf{There are many potential adversaries.} The medical imaging pipeline has many potential attackers and is thus vulnerable at many different stages. While in theory one could devise elaborate image verification schemes throughout the data processing pipeline to try to guard against attacks, this would be extremely costly and difficult to implement in practice.
\section{Attacking representative clinical deep learning systems}
To demonstrate the feasibility of medical adversarial attacks, we developed a series of medical classifiers modeled after state-of-the-art clinical deep learning systems, and launched successful white- and black-box attacks against each.
\subsection{Methods}
\subsubsection{Construction of medical classification models}
We developed baseline models to classify referable diabetic retinopathy from retinal fundoscopy (similar to \citet{gulshan2016development}), pneumothorax from chest-xray (similar to \citet{wang2017chestx} and \citet{rajpurkar2017chexnet}), and melanoma from dermoscopic photographs (similar to \citet{esteva2017dermatologist}). The decision to build models for these particular tasks was made both due to public data availability, as well as the fact that they represent three of the most highly visible successes for medical deep learning.
All of our models were trained on publicly available data. For diabetic retinopathy, this was the Kaggle Diabetic Retinopathy dataset \citep{kaggle}. The key distinction with the Kaggle dataset, however, was that we were seeking to predict \textit{referable} (grade 2 or worse) diabetic retinopathy in accordance with \citet{gulshan2016development} rather than predicting the retinopathy grade itself as was the case in the Kaggle competition. As such, the training and test sets from the kaggle competition were merged, relabeled using their provided grades, and split by patient into training and test sets with probability 0.88/0.12. For the chest x-rays, we used the ChestX-Ray14 dataset described by \citet{wang2017chestx}. We identified cases and controls by selecting images whose labeled contained `pneumothorax' or `no finding', respectively, and additionally excluded from our control group any images from patients who had received both labels. We then split by patient into training and test sets with probability 0.85/0.15. For melanoma, we downloaded images labeled as benign or malignant melanocytic lesions from the International Skin Imaging Collaboration website \citep{isic}, splitting again into training and test sets with probability 0.85/0.15.
As in the case of all three of the original papers that inspired these models, we built our classifiers by fine-tuning a pretrained ImageNet model. For convenience and consistency, we chose to build each of our networks using a pretrained ResNet-50 model, fine-tuned in Keras using stochastic gradient descent with a learning rate of 1E-3 and Momentum of 0.9. Data was augmented using 45$^{\circ}$ rotation and horizontal flipping chest x-ray images, and with 360$^{\circ}$ rotation, vertical and horizontal flipping, and mixup \cite{zhang2017mixup} for fundoscopy and dermoscopy images. In all three cases, these settings provided respectable performance and we therefore didn't perform dedicated hyperparameter optimization.
\subsubsection{Construction of adversarial attacks}
To demonstrate the vulnerability of our models to adversarial attacks under a variety of threat models, we implemented both human-imperceptible and patch attacks.
For our human-imperceptible attacks, we followed the white and black box PGD attack strategies published in \citet{madry2017towards}, which were also described as baselines in \citet{kannan2018adversarial}. The PGD attack (see \citet{madry2017towards} and also \citet{kurakin2016adversarial}) is an iterative extension of the canonical fast gradient sign method (FGSM) attack developed by \citet{goodfellow2014explaining}. In the PGD attack, given input $x\in \mathbb{R}^d$, loss function $L(\theta, x, y)$, and a set of allowed perturbations $\mathcal{S} \subseteq \mathbb{R}^d$ (most commonly the $\ell_\infty$ ball around $x$), one can perform a projected gradient descent on the negative loss function:
\begin{align*}
x^{t+1} = \Pi_{x+\mathcal{S}} (x^t + \epsilon \textrm{sgn}(\nabla_x L(\theta, x, y)))
\end{align*}
in order to identify an optimal perturbation. We implemented the PGD attack using the library Cleverhans, conducting 20 iterations with hyperparameter $\epsilon=0.02$ ($\epsilon$ corresponds to the maximum permitted $\ell_\infty$ norm of the perturbation) \citep{papernot2016cleverhans}.
For our adversarial patch attacks, we followed the approach of \citet{brown2017adversarial}. The learning process of the adversarial patch $\hat{p}$ uses a variant of the expectation over transformation algorithm (originally proposed by \citet{Athalye2017}), which can be expressed as the following maximization:
\begin{equation}
\hat{p} = \underset {p}{\operatorname {arg\,max}}\ \mathbb{E} \left[ \log p_{Y|X}\left( \left. \hat{y} \right| A\left(p, X, L, T \right) \right) \right]
\end{equation}
where $p_{Y|X}(\cdot|\cdot)$ represents the probability output from the classifier given the input image $X$, $L$ is the location of the patch, $T$ is the transformation (rotation and scaling), $\hat{y}$ is the target label (for our application binary label), and $A(\cdot, \cdot, \cdot, \cdot)$ is the deterministic mapping of given image, patch, location and transformation into the adversarially patched input image. The expectation is over the locations, transformations and input images, thus allowing the resulting patch to be `universal,' because it was trained over the entire training set and robust since training averaged over various location and transformations. We deployed adversarial patches with scaling parameter 0.4.
White box attacks for each attack were implemented by building attacks directly on the victim model itself. As in \citet{madry2017towards} and \citet{kannan2018adversarial}, black box attacks were performed by crafting the attack against an independently-trained model with the same architecture and then transferring the resultant adversarial examples to the victim.
Finally, as a control, we implemented a naive patch attack using natural images, as in \citet{brown2017adversarial}. To make this as strong a baseline as reasonably possible, we built our natural image patches using the images assigned by the model with the highest probability of the target class. Resultant patches were then applied to the test set images with random rotation and scaling factor 0.4, so match the adversarial patch.
\textbf{Availability of code:} Code to reproduce the analyses and results can be found at the first author's Github account: \\
\texttt{https://github.com/sgfin/adversarial-medicine}.
\subsection{Results}
The results of our experiments are depicted in Table~\ref{results-table} and in Figure ~\ref{fig:attack_results}.
While discrepancies in data source and train-test partitioning makes direct comparison to state-of-the-art models unfeasible, all of our baseline models achieved performance reasonably consistent with the results reported in the original manuscripts on natural images: AUROC of 0.910 for diabetic retinopathy compared with 0.936 reported in \citet{gulshan2016development}, AUROC of 0.936 for pneumothorax compared with 0.90 reported by \citet{rajpurkar2017chexnet}, and AUROC of 0.86 on melanoma compared with 0.91-0.94 reported in \citet{esteva2017dermatologist}.
Projected gradient descent attacks, targeting the incorrect answer in every case, produced effective AUROCs of 0.000 and accuracies of 0\% for all white box attacks. Black box attacks produced AUROCs of less than 0.10 for all tasks, and accuracies ranging from 0.01\% on fundoscopy to 37.9\% on dermoscopy. Qualitatively, all attacks were human-imperceptible.
Adversarial patch attacks also achieved effective AUROCs of 0.000 and accuracies of <1\% for white box attacks on all tasks. Black box adversarial patch attacks achieved AUROCs of less than 0.005 for all tasks and accurracies less than 10\%. The "natural patch" controls created by adding patches created from the most-strongly classified image of the desired class resulted in AUROCs ranging from 0.48-0.83 with accuracies ranging from 67.5\% to 92.1\%.
\begin{table*}[ht]
\centering
\scalebox{1.0}{%
\begin{tabular}{l|ccc|ccc|ccc}
\hline
& \multicolumn{3}{c|}{\textit{Fundoscopy}} & \multicolumn{3}{c|}{\textit{Chest X-Ray}} & \multicolumn{3}{c|}{\textit{Dermoscopy}} \\
\multicolumn{1}{c|}{Input Images} & \multicolumn{1}{l}{Accuracy} & \multicolumn{1}{l}{AUROC} & \multicolumn{1}{l|}{Avg. Conf.} & \multicolumn{1}{l}{Accuracy} & \multicolumn{1}{l}{AUROC} & \multicolumn{1}{l|}{Avg. Conf.} & \multicolumn{1}{l}{Accuracy} & \multicolumn{1}{l}{AUROC} & \multicolumn{1}{l|}{Avg. Conf.} \\
\midrule
Clean & 91.0\% & 0.910 & 90.4\% & 94.9\% & 0.937 & 96.1\% & 87.6\% & 0.858 & 94.1\% \\
PGD - White Box & 0.00\% & 0.000 & 100.0\% & 0.00\% & 0.000 & 100.0\% & 0.00\% & 0.000 & 100.0\% \\
PGD - Black Box & 0.01\% & 0.002 & 90.9\% & 15.1\% & 0.014 & 92.6\% & 37.9\% & 0.071 & 92.0\% \\
Patch - Natural & 78.5\% & 0.828 & 80.8\% & 92.1\% & 0.539 & 95.8\% & 67.5\% & 0.482 & 85.6\% \\
Patch - White Box & 0.3\% & 0.000 & 99.2\% & 0.00\% & 0.000 & 98.8\% & 0.00\% & 0.000 & 99.7\% \\
Patch - Black Box & 3.9\% & 0.000 & 97.5\% & 9.7\% & 0.004 & 83.3\% & 1.37\% & 0.000 & 97.6\% \\
\bottomrule
\end{tabular}%
}
\caption{Results of medical deep learning models on clean test set data, white box, and black box attacks.}
\label{results-table}
\end{table*}
\begin{figure*}[!htb]
\centering
\includegraphics[width=\textwidth]{figure_1_smaller.png}
\caption{\label{fig:attack_results} Characteristic results of adversarial manipulation. Each clean image represents the natural image to which the model assigns the highest probability for the given diagnosis. The percentage displayed on the bottom left of each image represents the probability that the model assigns that image of being diseased. Green = Model is correct on that image. Red = Model is incorrect.}
\end{figure*}
\section{Discussion}
Our experiments indicate that adversarial attacks are likely to be feasible even for extremely accurate medical classifiers, regardless of whether prospective attackers have direct access to the model or require their attacks to be human imperceptible. Of note, while the PGD attacks require digital access to the specific images to be sent into the model, adversarial patch attacks are universal in the sense that they can be applied to any image. This could open the possibility for implementation of attacks upstream from the image capture itself, rendering data processing defenses such as image hashing at point-of-capture ineffective. In addition, it is noteworthy to recognize that adversarial patches were far more potent than "photoshop"-style natural patch attacks that alter the image using the most strongly classified image from the training set.
We now discuss how someone might perform adversarial attacks against the systems developed in previous section under a realistic set of conditions. As stated above, we focus this discussion on imaging-based ML models, though similar arguments apply to adversarial examples crafted on billing code submissions or medical text. For the purposes of illustration, consider a scenario where the ML models have been subjected to extensive testing and validation and are now clinically deployed. These systems would function much like laboratory tests do now and provide confirmation of suspected diagnoses. In some instances, an insurance company may require a confirmatory diagnosis from one of these systems in order for a reimbursement to be made. Further, the insurance company or regulatory agency may have separate methods deployed to ensure the patient identity matches from prior images or has never been submitted from the same provider before. We provide the examples below to show that in many instances there is both the \emph{opportunity} and \emph{incentive} for someone to use an adversarial example to defraud the healthcare system.
\subsection{Hypothetical examples}
\textbf{Adversarial examples in dermatology}: Dermatology in the US operates under a `fee for service' model wherein a physician or practice is paid for the services or procedures they perform for the patient. Under this model, dermatologists are incentivized to perform as many procedures as possible, as their revenue is directly tied to the amount of procedures they perform. This has caused some dermatologists to perform a huge number of unnecessary procedures to increase revenue. For example, one dermatologist in Florida was recently sentenced to 22 years in prison after performing more than three thousand unnecessary surgical procedures \cite{rudman2009healthcare}. To combat fraud and unnecessary procedures such as this, an insurance company could require that a deep learning system (e.g. the one from Section 4) analyzes all dermoscopy images to confirm that surgery is necessary. In this scenario, a bad actor could add adversarial noise to images to ensure that the deep learning model always gives the diagnosis that he or she desires. Furthermore, they could add this noise to `borderline' cases, which would render the attack nearly impossible to detect by human review. Thus, a bad actor like the Florida dermatologist from \cite{rudman2009healthcare} could sidestep an insurance company's image-based fraud detector and continue to defraud the system in perpetuity.
\textbf{Adversarial examples in radiology}. Thoracic radiology images (typically CT scans, which is a 3D application of X-Ray technology) are also often used to measure tumor burden, a common secondary endpoint of cancer therapy response\citep{pien2005using}. To foster more rapid and more universally standardized clinical trials, the FDA might consider requiring that trial endpoints, such as tumor burden in chest imaging, be evaluated by a deep learning system such as the one from Section 4. By applying undetectable adversarial perturbations to the images, a company running a trial could effectively guarantee a positive trial result with respect to this endpoint, even if images are subsequently released to the public for inspection. In addition, chest X-rays provide a common screening test for dozens of diseases, and a positive chest X-ray result is often used to justify more heavily reimbursed procedures such as biopsies, CT or MR imaging, or surgical resection. As such, one could imagine many scenarios arising around chest X-rays that are directly analogous to the melanoma detection situation described above.
\textbf{Adversarial examples in ophthalmology}. As described in Section 3, providers and pharmaceutical companies are not the only organizations that could be incentivized to employ adversarial manipulation. Often, the entities who pay for healthcare (such as private or public insurers) wish to curtail the utilization rates of certain procedures to reduce costs. However, there are often guidelines from government agencies (such as the Centers for Medicare and Medicaid Services) that specify diagnostic criteria which if present dictate that certain procedures must be covered. One such criterion could be that any patient with a confirmed diabetic retinopathy diagnosis from a deep learning system such as the one from Section 4 must have the resulting vitrectomy surgery covered by their insurer. Even though the insurer has no ability to control the policy, they could still control the rate of surgeries by applying adversarial noise to mildly positive images, reducing the number of procedures. On the other end of the spectrum, an ophthalmologist could affix a universal adversarial patch to the lens of his image capture system, forcing a third party image processing system to mistake all images for positive cases without having to make an alterations to the image within the IT system itself.
\subsection{Possible areas for further research}
We hope that our discussion and demonstrations can help motivate further research into adversarial examples generally as well as within the specific context of healthcare. In particular, we consider the following areas to be of high priority:
\textbf{Algorithmic defenses} against adversarial examples remain an extremely open and challenging problem. We defer a full discussion of the extremely rapidly evolving field of adversarial defenses to the primary literature \citep{kannan2018adversarial, goodfellow2014explaining, kolter2017provable, madry2017towards, papernot2016transferability, Szegedy2013, Tramer2017, papernot2016distillation, kolter2017provable, dvijotham2018dual, raghunathan2018certified, buckman2018thermometer}. Unfortunately, despite the explosive emergence of defense strategies, there does not appear to be a simple and general algorithmic fix for the adversarial problem available in the short term. For example, one recent analysis investigated a series of promising methods that relied on gradient obfuscation, and demonstrated that they could be quickly broken \citep{athalye2018obfuscated}. Despite this, we also note that principled approaches to adversarial robustness are beginning to show promise. For example, several papers have demonstrated what appears to be both high accuracy and strong adversarial robustness on smaller datasets such as MNIST, \citep{madry2017towards,kannan2018adversarial}, and there have also been several results including theoretical \textit{guarantees} of adversarial robustness, albeit on small datasets and/or with still-insufficient accuracy \citep{kolter2017provable}. Generalized attempts at algorithmic robustness are promising, but have yet to provide methods that can demonstrate high levels of both accuracy and adversarial robustness at ImageNet scale. However, \textbf{domain-specific algorithmic} defenses such as dataset-specific image preprocessing have been shown to be highly effective on some datasets \citep{graese2016assessing}. In this light, particularly given the highly standardized image capture procedures in biomedical imaging, we feel that medical-domain-specific algorithmic defenses offer an important and promising area of future research.
\textbf{Infrastructural defenses} against clinical adversarial attacks include methods deployed to prevent potential bad actors from altering medical images -- or at least make it easier to confirm image tampering if adversarial examples are suspected. For example, imaging devices could immediately store a hashed version of any image they generate, which could subsequently be used as a reference. Likewise, raw clinical images could be processed and analyzed on a third-party system to prevent any possible systemic manipulation by payers or providers. This family of approaches to standardized best practices is reminiscent of the system of Clinical Laboratory Improvement Amendments (CLIA), a set of federal policies that regulates the process by which clinical laboratory samples are handled and analyzed in the United States \citep{us1992medicare}. Given that algorithmic defenses against adversarial attacks are still very much an area of research, we feel that infrastructural defenses should be strongly considered for all medical classifier systems that could carry incentives for adversarial attacks. However, implementing healthcare system-wide standardization to this end represents and immense challenge that will require buy-in from both the medical and CS communities.
\textbf{Ethical tradeoffs} are introduced by adversarial examples. As outlined above, several papers have demonstrated greater improvements in adversarial robustness that come at a cost of lower accuracy \citep{paschali2018generalizability, tsipras2018robustness}. However, in medical imaging, this introduces an ethical conundrum: how does one weigh protecting against adversarial examples against any inaccurate diagnosis? Quantifying and making this trade-off explicit would allow for more informed decision making and system design.
\section{Conclusion}
The prospect of improving healthcare and medicine with the use of deep learning is truly exciting. There is reasonable cause for optimism that these technologies can improve outcomes and reduce costs, if judiciously implemented \cite{beam2016translating}. In this light, it is unsurprising that dozens of private companies and large health centers have initiated efforts to deploy deep learning classifiers in clinical practice settings. As such efforts continue to develop, it seems inevitable that medical deep learning algorithms will become entrenched in the already multi-billion dollar medical information technology industry. However, the massive scale of the healthcare economy brings with it significant opportunity and incentive for fraudulent behavior and ultimately, patient harm.
In this work, we have outlined the systemic and technological reasons that cause adversarial examples to pose a disproportionately large threat in the medical domain, and provided examples of how such attacks may be executed. We hope that our results help facilitate a discussion on the threat of adversarial examples among both computer scientists and medical professionals. For machine learning researchers, we recommend research into infrastructural and algorithmic solutions designed to guarantee that attacks are infeasible or at least can be retroactively identified. For medical providers, payers, and policy makers, we hope that these practical examples can motivate a meaningful discussion into how precisely these algorithms should be incorporated into the clinical ecosystem despite their current vulnerability to such attacks.
\begin{acks}
The authors would like to thank Aleksander Madry, Dimitris Tsipras, Yun Liu, Jasper Snoek, Alex Wiltschko for a helpful review of our manuscript. In addition, SGF was supported by training grants T32GM007753 and T15LM007092; the content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institute of General Medical Sciences or the National Institutes of Health.
\end{acks}
\bibliographystyle{ACM-Reference-Format}
|
{
"timestamp": "2019-02-05T02:23:02",
"yymm": "1804",
"arxiv_id": "1804.05296",
"language": "en",
"url": "https://arxiv.org/abs/1804.05296"
}
|
\section{Introduction}\label{intro}
\indent
Though studied for decades, recent important breakthroughs in automata theory have turned it into an important field of study in discrete mathematics and theoretical computer science. For a comprehensive introduction in the theory and other related subjects, see the book of Hopcroft, Motwani and Ullman \cite{Hopcroft}.
One of the long standing famous conjectures in automata theory is the road colouring problem introduced in 1970 by Adler, Goodwyn and Weiss in \cite{Adler1}, \cite{Adler2}. The conjecture states that a strongly connected digraph in which all vertices have the same out-degree, which is aperiodic (i.e. the gcd of the lengths of all of its oriented cycles is one) has a synchronising colouring. A synchronising colouring of a strongly connected digraph $G$ of uniform out-degree $k$ is a labelling of the edges of $G$ with colours $1, \hdots, k$ such that all the vertices have out-edges of all colours and for every vertex $v$ of $G$ there exists a word $W_v$ in the alphabet of colours such that every path in $G$ corresponding to $W_v$ terminates at $v$. We note that the existence of a synchronising colouring makes it possible to reset the automaton back to its original state after the detection of an error. In fact, it is because of this important property that the road coloring problem has received so much attention over the past few decades. There have been many positive partial results published over the years, such as \cite{New1}, \cite{New2}, \cite{New3}. In 2009, Trahtman made one of the most notable advances in the field by proving this conjecture in \cite{Trahtman}.
Another famous related problem in the field is {\v C}ern{\' y}'s conjecture which appeared in \cite{CCerny} in 1964 and states that the length of the shortest synchronising word for any $n$-state deterministic finite automaton is bounded above by $(n-1)^2$ (for more details see \cite{Pin15}, \cite{Trahtman16}). For some partial results concerning {\v C}ern{\' y}'s conjecture see \cite{Grech}, \cite{Cerny}.
In this paper we introduce and study a new model of automata which turns out to be abundant in profound and intricate phenomena. This model does not seem to have occurred in literature, and is motivated by the following coffee time problem of Leader, popularised by Balister.
\begin{prob}\label{pb1}
Consider the classical $8 \times 8$ chessboard as the board of a maze, where every small square is a room, such that between any two adjacent rooms there is either a wall that prevents the transit between them, or there is no wall and transit is possible. Additionally, the boundary of the board is formed only by walls.
Say that a robot starts in one of the $64$ squares and it receives a sequence of instructions from the set of cardinal directions: \emph{north}, \emph{south}, \emph{east}, \emph{west}. Each time the robot receives such an instruction, it executes it by moving to the corresponding adjacent room, provided there is no wall to prevent it from moving as instructed; if there is such a wall, the robot simply does not move and it continues with the following instruction. The robot does not give any feedback whether it moves or not when executing an instruction.
Naturally, the board of the maze can be regarded as a subgraph of the square lattice $8 \times 8$ where there is an edge between two vertices if and only if there is no wall between the corresponding squares. Without knowing the subgraph and the starting vertex of the robot, can one write a finite sequence of instructions such that at the end the robot is guaranteed to have visited all accessible vertices?
\end{prob}
To see the existence of such an algorithm, simply enumerate all the possible mazes and solve them one by one, keeping track of the updated position of the robot when passing to a new maze. A related problem which can be solved in the same way is the following:
\begin{prob}\label{pb2}
Consider a subgraph of some finite dimensional hypercube $Q_1, Q_2, \hdots$ as the board of a maze. Say that a robot starts in one of the vertices and it receives a sequence of instructions from the set of coordinate directions $\pm e_1, \pm e_2, \hdots$. Each time the robot receives such an instruction, it executes it by moving to the corresponding adjacent vertex, provided there is an edge between these two vertices; if there is no such edge, the robot simply does not move and it continues with the following instruction. Without knowing the subgraph and the starting vertex of the robot, can one write an infinite sequence of instructions such that at the end the robot is guaranteed to have visited all accessible vertices?
\end{prob}
Problem~\ref{pb1} lead Spink and Leader to ask the following research question, which was later passed to us by Balister.
\begin{que}\label{qloha}
What happens if in Problem~\ref{pb1} we replace the (finite) $8 \times 8$ square lattice with the infinite square lattice $\mathbb{Z}^2$?
\end{que}
To our knowledge, Question~\ref{qloha} turns out to be extremely difficult to answer. In this paper we make progress towards answering this question, by establishing the following main result.
\begin{thm}\label{mthm}
There exists an infinite sequence of instructions for a robot to visit all accessible vertices in any maze for which the board is the graph $\mathbb{Z}^2$ with arbitrarily many horizontal edges removed but only finitely many vertical edges removed in consecutive columns.
\end{thm}
We note that Theorem~\ref{mthm} follows immediately from two separate beautiful results, Theorem~\ref{ch2th1} and Theorem~\ref{ch2th2} binded together by the more technical Proposition~\ref{ch2prop1}.
The structure of the paper is as follows. In Section~\S\ref{prelims} we start by developing a general set-up that encompasses a class of similar problems which we call ``solvability of mazes by blind robots''. We then return to the Leader-Spink problem and state all our main results in Section~\S\ref{results}. In Section~\S\ref{BC} we present a toy model that represents the foundation on which the general model is constructed. As part of the toy model, we prove Theorem~\ref{ch2th1}; this allows us to introduce and investigate some generic algorithms that are used as building blocks in the proof of Theorem~\ref{ch2th2}. In Section~\S\ref{fnv} we present a series of technical definitions that are used to construct a countable cover of the set of mazes in Theorem~\ref{ch2th2} with subsets of mazes that we can treat individually. In Section~\S\ref{final} we give the constructive proof of Theorem~\ref{ch2th2}. We continue with the proof of the technical result Proposition~\ref{ch2prop1} in Section~\S\ref{sectiunea1}. Finally, in Section~\S\ref{conclusions} we present several further directions of research and some of our conjectures.
\section{Preliminaries}\label{prelims}
\indent
In this section we introduce the general framework for our model giving formal definitions and some relevant examples.
A \emph{maze} is a quadruple $(M, c, o, d)$, where $M$ is a countable strongly connected digraph called the board, and $c:E(M) \longrightarrow \mathbb{N}$ is a proper colouring of the edges of $M$, i.e. one in which the out-edges from any vertex have distinct colours. Further, $o$ and $d$ are two special vertices of $M$ called the \emph{origin} and the \emph{destination}, respectively.
An \emph{instruction} $I \in \mathbb{N}$ is an element from the set of colours $\mathbb{N}$. An \emph{algorithm}
\[
A = (I_i)_{i=1}^n \text{ or } A = (I_i)_{i=1}^\infty
\]
is a finite or infinite sequence of instructions. A \emph{subalgorithm} $A'$ of an infinite algorithm $A$ as above is any truncation of $A$ of the form
\[
A' = (I_i)_{i=k}^j \text{ or } A' = (I_i)_{i=k}^\infty,
\]
for some $k \leq j$. Similarly, a \emph{subalgorithm} $A'$ of a finite algorithm $A= (I_i)_{i=1}^n$ is any truncation of $A$ of the form $A' = (I_i)_{i=k}^j$ for some $k \leq j \leq n$. In order to describe dynamically our process of visiting the graph we look at the following model.
Given an algorithm $A = (I_i)_{i=1}^\infty$ and a maze $(M, c, o, d)$, a \emph{robot} - which is just a travelling tracking object or a pointer - starts at the origin $o$ and moves in between the vertices of $M$, as it follows the instructions $I_1, I_2, \hdots$ one by one in order: for $n \in \mathbb{N}$ the robot executes the $n$-th instruction $I_n \in \mathbb{N}$ by moving from its current vertex $v$ to the next vertex $w$ if and only if there exists an oriented edge $e$ of colour $I_n$ from $v$ to $w$; if there is no such oriented edge $e$, the robot remains at $v$. In short, we say that the robot \emph{follows the algorithm $A$ in the maze $(M, c, o, d)$}. We say that an algorithm $A$ \emph{solves} the maze $(M, c, o, d)$ if the robot visits the destination $d$ at some time by following $A$ in $(M, c, o, d)$. Similarly, we say that an algorithm $A$ \emph{solves} a set $\mathcal{M}$ of mazes if it solves every maze in $\mathcal{M}$.
We remark that each connected graph can be regarded as a strongly connected digraph by doubling edges. Throughout the paper all the boards of the mazes arise in this way and hence from now on we define the board of a maze to be a graph. We can omit the condition that the graph is connected if we require that the origin and destination are in the same connected component of the graph.
In this set-up, the fundamental question that arises is the existence of algorithms that simultaneously solve certain natural sets of mazes. As we shall see from the arguments which appear in this paper, and also from our conclusions and open questions in Section~\S\ref{conclusions}, this set up is rich in very deep insights related to the phenomenon of state automata.
For example, we note that there is no algorithm that solves the set of all mazes. Indeed, let us assume for a contradiction that $A = (I_i)_{i=1}^\infty$ does the job. We construct $M$ to be the path with vertices $v_0, v_1, \hdots$ and its only edges $v_i \rightarrow v_{i-1}$ and $v_{i-1} \rightarrow v_i$ for all $i \in \mathbb{N}$. We set $o=v_1$, $d=v_0$ and colour the edge $v_i \rightarrow v_{i+1}$ with colour $I_i$ and the rest of the edges in any way that does not violate the proper colouring condition. A robot that starts in this maze and follows $A$ will visit in order $v_1, v_2, v_3, \hdots$ as it follows $I_1, I_2, \hdots$, never reaching $v_0=d$. As $M$ was constructed to be strongly connected, we have reached a contradiction.
As another example, we note that for any countable set of mazes, there exist algorithms that solve it. In particular, this solves Problem~\ref{pb1} and more importantly, it shows that there exist algorithms that solve the set of all finite mazes. Indeed, let $(M_1, c_1, o_1, d_1), (M_2, c_2, o_2, d_2) \hdots$ be an enumeration of a countable set of mazes $\mathcal{M}$. Considering the strongly connectedness property, given any maze $(M, c, o, d)$ one can write by inspection a finite algorithm that solves the maze. Then, let $A_1$ be any finite algorithm that solves $(M_1, c_1, o_1, d_1)$; let $o_2'$ be the position of the robot after it follows the algorithm $A_1$ in $(M_2, c_2, o_2, d_2)$; let $A_2$ be any finite algorithm that solves $(M_2, c_2, o_2', d_2)$ with origin $o_2'$; let $o_3'$ be the position of the robot after it follows the algorithm $A_1A_2$ in $(M_3, c_3, o_3, d_3)$, etc. Continue in this way to create algorithms $A_1, A_2, \hdots$. We claim that the algorithm $A=A_1 A_2 \hdots$ obtained by concatenating $A_1, A_2, \hdots$ solves the set of mazes $\mathcal{M}$. Indeed, consider the maze $(M_i, c_i, o_i, d_i) \in \mathcal{M}$ for some $i \geq 2$. After the robot follows the initial subalgorithm $A_1 A_2 \hdots A_{i-1}$ of $A$ it gets to the vertex $o_i'$ of $M_i$ and then after it follows $A_i$ it gets to the destination point $d_i$. Trivially, for the maze $(M_1, c_1, o_1, d_1)$, the robot gets to the destination point $d_1$ after it follows the initial subalgorithm $A_1$ of $A$. This shows that $A$ solves $\mathcal{M}$.
We can see from the two examples above that the most interesting cases of our model occur ''in between``, when we consider natural uncountable sets of mazes for which we seek to construct algorithms to solve them. We present below two uncountable sets of mazes, for which it is not hard to find such algorithms.
Firstly, let $Q=Q_1\cup Q_2 \cup \hdots$ be the nested union of all finite dimensional hypercubes i.e. the graph with vertices all possible infinite $\{ 0, 1 \}$ sequences with trailing zeros and edges between those pairs of vertices which differ in only one coordinate. Let $\mathcal{Q}$ be the set of mazes for which the board is a connected subgraph of $Q$ and the colouring assigns to each directed edge the corresponding coordinate direction $\pm e_1, \pm e_2, \hdots$.
Secondly, let $Z=\mathbb{Z}^1\cup \mathbb{Z}^2\cup \hdots$ be the nested union of all finite dimensional integer lattices i.e. the graph with vertices all possible infinite integer sequences with trailing zeros and edges between those pairs of vertices which differ in only one coordinate and the difference is one. An increasing path inside $Z$, is an infinite path that passes through the origin and always goes in a positive coordinate direction $ +e_1, +e_2, \hdots$. Let $\mathcal{P}$ be the set of mazes for which the board is an increasing path and the colouring assigns to each directed edge the corresponding coordinate direction.
The main object of study in this paper, though much more challenging, resembles Problem~\ref{pb1}. One of the most fundamental and fascinating sets of mazes is the set $\mathcal{M}$ for which the board is the square lattice $\mathbb{Z}^2$ considered as a graph with arbitrarily many edges deleted, the colouring assigns to each directed edge the corresponding cardinal direction from the set $\{N, S, E, W\} = \{S^{-1}, N^{-1}, W^{-1}, E^{-1}\}$, and the origin and destination are in the same connected component. From now on we define a maze to be a triple $(M, \textbf{o}, \textbf{d}) \in \mathcal{M}$, where $M$ is the board, $\textbf{o} = (x_o, y_o)$ is the origin, and $\textbf{d} = (x_d, y_d)$ is the destination.
We call all the vertices in the connected component of the origin \emph{accessible points}. For a vertex $\textbf{x} = (x, y)$ we refer to its coordinates $x$ and $y$ as the \emph{longitude} and \emph{latitude}, respectively. We also label the columns and rows of $\mathbb{Z}^2$ by $c_i = \{(i, y) \mid y \in \mathbb{Z} \}$ and $r_i= \{(x, i) \mid x \in \mathbb{Z} \}$ for $i\in \mathbb{Z}$, respectively. We say that $c_i$ and $c_{i+1}$ are \emph{joined} at latitude $j$ if the vertices $(i, j)$ and $(i+1, j)$ are adjacent.
We often create new algorithms by concatenations of other algorithms, and it is very convenient to use multiplication to denote concatenation. For example
\[
SNSSNS = SNS^2NS=(SNS)^2
\].
For a finite algorithm $A$, we write $|A|$ for the number of instructions in $A$; similarly we write $|A|_I$ for the number of instructions $I$ in $A$, for all $I \in \{N, S, E, W\}$.
\begin{figure}
\centering
\resizebox{0.5\textwidth}{!}{
\begin{tikzpicture}
[
dot/.style={circle,draw=black, fill,inner sep=2pt},
cross/.style={cross out, draw=black, minimum size=2*(#1-\pgflinewidth), inner sep=4pt, outer sep=4pt},
]
\foreach \x in {-4,...,4}{
\foreach \y in {-4,...,4}{
\node[dot] at (\x,\y){ };
}}
\draw[line width=0.7mm, red] (-4, -4) -- (-4, 4);
\draw[line width=0.7mm, red] (-3, -4) -- (-3, 4);
\draw[line width=0.7mm, red] (2, -4) -- (2, 4);
\draw[line width=0.7mm, red] (3, -4) -- (3, 4);
\draw[line width=0.7mm, red] (4, -4) -- (4, 4);
\draw[line width=0.7mm, red] (-2, 4) -- (-2, 2);
\draw[line width=0.7mm, red] (-2, 0) -- (-2, -1);
\draw[line width=0.7mm, red] (-2, -2) -- (-2, -3);
\draw[line width=0.7mm, red] (-1, 3) -- (-1, 1);
\draw[line width=0.7mm, red] (-1, -1) -- (-1, -2);
\draw[line width=0.7mm, red] (0, -2) -- (0, -4);
\draw[line width=0.7mm, red] (0, 0) -- (0, 3);
\draw[line width=0.7mm, red] (1, 4) -- (1, 3);
\draw[line width=0.7mm, red] (1, 2) -- (1, -4);
\draw[line width=0.7mm, red] (3, 4) -- (4, 4);
\draw[line width=0.7mm, red] (2, 4) -- (-2, 4);
\draw[line width=0.7mm, red] (-2, 3) -- (0, 3);
\draw[line width=0.7mm, red] (1, 3) -- (3, 3);
\draw[line width=0.7mm, red] (-4, 3) -- (-3, 3);
\draw[line width=0.7mm, red] (-4, 2) -- (-3, 2);
\draw[line width=0.7mm, red] (-1, 2) -- (1, 2);
\draw[line width=0.7mm, red] (2, 2) -- (3, 2);
\draw[line width=0.7mm, red] (-1, 1) -- (2, 1);
\draw[line width=0.7mm, red] (-3, 1) -- (-2, 1);
\draw[line width=0.7mm, red] (3, 1) -- (4, 1);
\draw[line width=0.7mm, red] (-2, 0) -- (-1, 0);
\draw[line width=0.7mm, red] (1, 0) -- (3, 0);
\draw[line width=0.7mm, red] (-2, -1) -- (1, -1);
\draw[line width=0.7mm, red] (2, -1) -- (3, -1);
\draw[line width=0.7mm, red] (-4, -1) -- (-3, -1);
\draw[line width=0.7mm, red] (-4, -2) -- (-2, -2);
\draw[line width=0.7mm, red] (2, -2) -- (4, -2);
\draw[line width=0.7mm, red] (-1, -3) -- (2, -3);
\draw[line width=0.7mm, red] (3, -3) -- (4, -3);
\draw[line width=0.7mm, red] (-2, -4) -- (-1, -4);
\draw[line width=0.7mm, red] (0, -4) -- (2, -4);
\draw[line width=0.7mm, red] (3, -4) -- (4, -4);
\draw[line width=0.7mm, red] (-4, -4) -- (-3, -4);
\draw (3,-2) node[cross] {};
\foreach \x in {-4,...,4}
\draw (\x,.1) -- node[below,yshift=-1mm] {\x} (\x,-.1);
\foreach \y in {-4,...,4}
\draw (.1,\y) -- node[below,xshift=-3mm, yshift=3mm] {\y} (-.1,\y);
\draw[->,line width=0.15mm] (0,-4.5) -- (0,4.5);
\draw[->,line width=0.15mm] (-4.5,0) -- (4.5,0);
\end{tikzpicture}
}
\caption{A local representation of a general maze $M$, where edges are marked by red lines. We also mark the destination point $(3, -2)$ with a cross and note that in every maze there is a path from the origin to the destination point. When the robot follows the algorithm $SNWWN$ in $M$ it gets to the point $(-1, 2)$ it follows the path $(0, 0),$ $(0, 0),$ $(0, 1),$ $(-1, 1),$ $(-1, 1),$ $(-1, 2)$; the robot does not move when it executes the first and fourth instructions, as there is no edge between $(0, 0)$ and $(0, -1)$ and between $(-1, 1)$ and $(-2, 1)$.} \label{0}
\end{figure}
\section{Our Results}\label{results}
Our main result, Theorem~\ref{mthm} follows directly from Theorem~\ref{ch2th1}, Theorem~\ref{ch2th2} and Proposition~\ref{ch2prop1}, all of which are interesting results on their own.
\begin{thm}\label{ch2th1}
Let $\mathcal{C} \subseteq \mathcal{M}$ be the set of all mazes for which the board has arbitrarily many horizontal edges removed but no vertical edges removed. There exists an algorithm that solves $\mathcal{C}$.
\end{thm}
\begin{thm}\label{ch2th2}
Let $\mathcal{F} \subseteq \mathcal{M}$ be the set of all mazes for which the board has arbitrarily many horizontal edges removed and nonzero finitely many vertical edges removed in consecutive columns. There exists an algorithm that solves $\mathcal{F}$.
\end{thm}
We should note that, as one might expect, the proof of Theorem~\ref{ch2th2} turns out to be much more difficult than the proof of Theorem~\ref{ch2th1} and that both proofs are constructive. In Section~\S\ref{BC}, in which we give the proof of Theorem~\ref{ch2th1}, we also introduce some generic algorithms which constitute the main building blocks of the algorithm which solves $\mathcal{F}$. In Lemma~\ref{l2}, which is a technical key result of the paper, we present their properties that we use multiple times in the proof of Theorem~\ref{ch2th2}.
Finally, we use the following result as a binder between Theorem~\ref{ch2th1} and Theorem~\ref{ch2th2} in order to obtain Theorem~\ref{mthm}. This result ascertains the intuitive fact that under certain technical conditions, if two sets of mazes are solvable so is there union and moreover if a set of mazes is solvable, then it is also solvable infinitely often.
\begin{prop}\label{ch2prop1}
Let $e(\mathbb{Z}^2)$ be the set of edges of $\mathbb{Z}^2$. We can regard any board of a maze as an indicator function $f: e(\mathbb{Z}^2) \longrightarrow \{ 0,1 \}$. Hence, the set of boards of mazes equipped with the product topology is a compact metrizable space. Let $\mathcal{A}_1, \mathcal{A}_2 \subseteq \mathcal{M}$ be two sets of mazes with the following properties:
\begin{enumerate}
\item for all $i\in \{1,2\}$, all origins $o\in \mathbb{Z}^2$, all destination $d\in \mathbb{Z}^2$ and all paths $P$ between $o$ and $d$, the sets of boards $B_i = \{ M \mid (M, o, d) \in \mathcal{A}_i, P\le M \}$ that contain the path $P$ are compact;
\item for all $i\in \{1,2\}$ if $(M,o,d) \in \mathcal{A}_i$, then $(M,o',d') \in \mathcal{A}_i$ for all $o',d'$ in the same connected component as $o,d$;
\item there exist algorithms $A_1$ and $A_2$ that solve the sets $\mathcal{A}_1$ and $\mathcal{A}_2$, respectively.
\end{enumerate}
Then there exists an algorithm $A$ that solves the set $\mathcal{A}=\mathcal{A}_1\cup \mathcal{A}_2$ and that furthermore guides the robot to visit the destination of any maze in the set infinitely often. Moreover, if we cut or add an initial segment to $A$, the algorithm obtained in this way has the same property.
\end{prop}
\section{The Toy Model}\label{BC}
The aim of this section is to prove Theorem~\ref{ch2th1} and to introduce the general strategy and some generic algorithms that are used in the proof of Theorem~\ref{ch2th2} as well.
For a subset of mazes $\mathcal{C} \subseteq \mathcal{M}$, in order to construct an algorithm $A$ that solves $\mathcal{C}$ we adopt the following natural strategy: we find a countable cover $\mathcal{C}=\cup_{i=1}^{\infty}\mathcal{C}_i$ such that for each $i \in \mathbb{N}$ and each finite algorithm $X$ we are able to find a finite algorithm $A_X^i$ such that the concatenated algorithm $XA_X^i$ solves $\mathcal{C}_i$. Then we are able to find an algorithm $A$ that solves $\mathcal{C}$. Indeed, we construct recursively the finite algorithms $(B_n)_{n\ge0}$ with $B_0= \emptyset$ and $B_{n}=A_{B_0B_1 \hdots B_{n-1}}^{n}$, then we take $A=B_1B_2 \hdots$.
In the toy model, let $\mathcal{C}$ be the set of mazes with no vertical edges removed. Without loss of generality, we assume that for any maze in $\mathcal{C}$ the origin is the point $(0, 0)$. The main property of this set of mazes is that at each step of the algorithm we know the robot's latitude.
\begin{proof}[Proof of Theorem~\ref{ch2th1}]
We begin the proof by defining two classes of algorithms. The aim of the first one is to move the robot eastwards in a certain organised pattern and we call it $move \_ east$; it is defined as follows for all $a, e \geq 1$: \\
$ME(a, e) := (((((E)^e NES)^e SEN)^e N^2ES^2)^e \hdots S^a E N^a)^e.$
We view $ME(a, e)$ as being composed from the multiple concatenation of $2a+1$ different building blocks which we call \emph{locomotory moves}: $E,$ $NES,$ $SEN,$ $N^2ES^2,$ $\hdots$ $N^aES^a$, $S^a E N^a.$ We constructed the class of algorithms $move \_ east$ in such a way so that the following holds:
Let $a, e$ be two natural numbers. Assume that the robot starts at the point $\textbf{x} = (x, y)$ in any maze $M \in \mathcal{C}$ with no vertical edges removed. Take the maximal $k \leq e$ such that in $M$ the columns $c_i$ and $c_{i+1}$ are joined at some latitude in $\{-a+y, \hdots, a+y \}$ for all $x \le i \le x+k-1$. Then, as the robot follows the algorithm $ME(a, e)$, it oscillates about the row $r_y$ at latitudes between $y-a$ and $y+a$. After the algorithm is followed, the robot gets to a point $\textbf{x'} = (x', y)$ with $x'\ge x+k$, in particular $x' = x+k$ if $k<e$. Moreover if we well order $\mathbb{Z}$ by $y<1+y<-1+y<2+y<-2+y< \hdots$, then for all $x \le i \le x+k-1$ the robot passes from the column $c_i$ to the column $c_{i+1}$ through the edge at the lowermost latitude with respect to this order.
This holds as a particular case of Lemma~\ref{l2}, which is a technical result used extensively, proved later in this section. One can also see how this statement follows from the construction of $ME(a, e)$, more specifically from the order in which the locomotory moves appear in the algorithm. The counterpart of $move \_ east$ is called $move \_ west$, and we have: \\
$MW(a, e) := (((((W)^e NWS)^e SWN)^e N^2WS^2)^e \hdots S^a W N^a)^e.$
The second class of algorithms that we define is called $oscillating \_ move \_ east$, which is a slight alteration of $move \_ east$ formed by inserting the \emph{oscillatory} algorithm $(N^bS^{2b}N^b)^e$ in between some locomotory moves; it is defined as follows for all $a, e \geq 1$ and $b \in \mathbb{Z}$: \\
$OME(a, e, b) := ((((((N^b S^{2b} N^b)^e E)^e NES)^e SEN)^e N^2ES^2)^e \hdots S^a E N^a)^e.$
We note that in every maze with no vertical edge removed, after the robot follows the oscillatory algorithm $(N^bS^{2b}N^b)^e$, it gets back to the starting point. Therefore, for any parameters $a, e, b$, as the robot follows $OME(a, e, b)$ in any maze $M \in \mathcal{C}$, it has the same dynamics as it follows $ME(a, e)$ in $M$ and in addition the robot visits some consecutive columns, beginning with the one which contains its starting point $\textbf{x} = (x, y)$, at all latitudes between $y-b$ and $y+b$. Finally, we use the oscillatory algorithm $(N^bS^{2b}N^b)^e$ instead of $N^bS^{2b}N^b$ which works just as well for this purpose, only because we want $OME(a, e, b)$ to be a particular case of a much more general algorithm, $SME(a, e, K)$ that is defined later in this section.
The counterpart of $oscillating \_ move \_ east$ is called $oscillating \_ move \_ west$, and we have:\\
$OMW(a, e, b) = ((((((N^b S^{2b} N^b)^e W)^e NWS)^e SWN)^e N^2WS^2)^e \hdots S^a W N^a)^e.$
We are now ready to prove the theorem using the general strategy described at the beginning of the section. In order to produce the desired countable cover, define $C_{n, \textbf{x}}$ to be the set of all mazes with no vertical edges removed, with the destination point $\textbf{x} = (x, y)$ and such that any two consecutive columns at longitude between $0$ and $x$ are joined at some latitude between $-n$ and $n$. Then $C=\cup_{n,\textbf{x}}C_{n,\textbf{x}}$ is a countable cover.
We let $X$ be any finite algorithm and we fix the values $n, \textbf{x}$. We now consider just the set of mazes $C_{n,\textbf{x}}$ and we aim to construct an algorithm $A$ such that $XA$ solves $C_{n,\textbf{x}}$, which by the discussion of our strategy at the beginning of the section is enough to conclude.
Say that the robot starts in any maze $M \in C_{n,\textbf{x}}$ (as always, it starts in the origin) and it gets to the point $(a, 0)$ after it follows some finite algorithm $Y$. Define $a:=\max \{|Y|_N, |Y|_S\}$; $e:=|Y|_W$. The following observation is crucial: for each pair $\{i,i+1\}\subset \{0, \hdots, a\}$, the columns $c_i$ and $c_{i+1}$ are joined at some latitude in $\{-|Y|_S, \hdots, |Y|_N \} \subseteq \{ -|Y|, \hdots, |Y| \}$. Therefore, after the robot follows the algorithm $Y$ $ME(a,e)$ in $M$ it gets to some point $(a', 0)$ with $a' \ge 0$.
Now we build $A$ as a concatenation of three algorithms $A:=A_1A_2A_3$.
We construct $A_1:= S^{|X|_N-|X|_S}$; then after the robot follows the algorithm $XA_1$ in any maze $M \in C_{n,\textbf{x}}$ it gets to $r_0$.
Define $a:= \max \{|XA_1|_S, |XA_1|_N, n\}$; $e:=|XA_1|_W+|x|$. Define $A_2:= ME(a,e)$. Then after the robot follows the algorithm $X A_1 A_2$ in any maze $M \in C_{n,\textbf{x}}$ it gets to some point $(x^+,0)$ with $x^+\ge x$.
Define $a:= \max \{|XA_1A_2|_S, |XA_1A_2|_N, n\}$; $w:=|XA_1A_2|_E+ |x|+1$; $b:=|y|$. Define $A_3:= OMW(a,w,b)$. Then after the robot follows the algorithm $XA_1A_2A_3$ in any maze $M \in C_{n,\textbf{x}}$, it gets to some point $(x^-,0)$ with $x^-\le x$ and it visits every intermediate column $c_i$ with $x^-\le i\le x^+$ including $c_x$ at every latitude in $\{-b, \hdots, b \}$ including $y$.
Therefore, after the robot follows $XA =XA_1A_2A_3$ in any maze $M \in C_{n,\textbf{x}}$, it visits the destination point $\textbf{x}$. Hence there exists an algorithm $A$ such that $XA$ that solves $C_{n,xy}$. This finishes the proof.
\end{proof}
We note that the missing vertical edges in the general model usually make the latitude of the robot unknown but it turns out that we can actually make use of the missing edges to regain the latitude of the robot. However, the unknown longitude and the missing edges require the robot to use a very subtle path to get to the destination point. As a result of these difficulties in the proof of Theorem~\ref{ch2th2} we need to make a much finer covering than in the proof of Theorem~\ref{ch2th1}.
In the remainder of this section we introduce an algorithm which is a generalisation of $ME(a, e)$ and $OME(a, e, b)$ called $special \_ move \_ east$ which is the main building block of the algorithms used in the general model. We then group all its properties in Lemma~\ref{l2}, which makes it one of the main results of the paper. For $a, e \geq 1$ and a finite algorithm $K$ we define:\\
$SME (a,e,K):=(((((K^eE)^eNES)^eSEN)^eN^2ES^2)^e...S^aEN^a)^e.$ We view $SME(a, e, K)$ as being composed from the multiple concatenation of $2a+2$ different building blocks: the $2a+1$ locomotory moves $E,$ $NES,$ $SEN,$ $\hdots$ $S^aEN^a$ and the special algorithm $K.$
Its counterpart, $special \_ move \_ west$ is defined as:\\
$SMW (a,e,K):=(((((K^eW)^eNWS)^eSWN)^eN^2WS^2)^e...S^aWN^a)^e.$
Recall that $\mathcal{C} \subset \mathcal{M}$ is the set of mazes with no vertical edges removed. The following result encompasses the main properties of $SME (a,e,K)$ that are used countless times in the proof of Theorem~\ref{ch2th2}.
\begin{lemma}\label{l2}
Let $a, e \geq 1$ and $K$ be a finite algorithm such that for any maze $M \in \mathcal{C}$, if the robot follows $K$ in $M$ starting from the origin, it returns on the $x$-axis and it has a non-negative longitude. Let $0\le l \le e-2$ have the following properties:
(1) for any $0\le x \le l$ the columns $c_x$ and $c_{x+1}$ are joined at some latitude between $-a$ and $a$;
(2) for any $\textbf{v}=(x_v,0)$ with $0 \le x_v \le l$, if the robot starts from $\textbf{v}$ and follows $K$ in any maze in $\mathcal{C}$ it gets at some point $\textbf{w} = (x_w,0)$ with $x_v \le x_w \le l$ without visiting any vertex on the column $c_{l+1}$, i.e. without visiting any point of longitude at least $l+1$.
Then, after the robot follows $SME(a,e,K)$ in any maze in $\mathcal{C}$, it gets to some point $\textbf{v}=(x_v,0)$ on the $x$-axis with $x_v \geq l+1$. Moreover, it does not pass from the column $c_l$ to the column $c_{l+1}$ for the first time while executing $K$; it passes from the column $c_l$ to the column $c_{l+1}$ for the first time while executing a locomotory move $N^mES^m$, where $m \in \mathbb{Z}$ is the lowermost latitude with respect to the standard well order on $\mathbb{Z}: 0<1<-1<2<-2< \hdots$ such that the columns $c_l$ and $c_{l+1}$ are joined at latitude $m$; finally, immediately after this locomotory move $N^mES^m$ is executed, the robot follows $K$.
\end{lemma}
\begin{proof}
Let $M \in \mathcal{C}$ be any maze with no vertical edge removed. We prove the result for $M$, so for brevity, we make the convention that every time we say that the robot follows an algorithm, it follows that algorithm in $M$.
By the hypothesis on $K$, if the robot is on the $x$-axis and follows $K$ (or $N^b E S^b$, $b \in \{ -a, \hdots a \}$), it returns to the $x$-axis and its longitude does not strictly decrease. We fix $x$ between $0$ and $l$, so that the columns $c_x$ and $c_{x+1}$ are joined at some latitude $b$ between $-a$ and $a$. Hence, if the robot starts from the point $(x, 0)$ and follows $N^bES^b$ it gets to the point $(x+1, 0)$. Therefore, if the robot is on the $x$-axis at some longitude between $0$ and $l$, then after each instance of the algorithm $(((((K^eE)^eNES)^eSEN)^eNNESS)^e...S^aEN^a)^1$, the longitude of the robot increases by at least one. This proves the first statement of the conclusion, that if the robot follows $SME(a,e,K)$, it gets to some point $\textbf{v}=(x_v,0)$ on the $x$-axis with $x_v \geq l+1$.
The second statement of the conclusion is that if the robot follows $SME(a,e,K)$, it does not pass from the column $c_l$ to the column $c_{l+1}$ for the first time while executing $K$. This follows directly from the hypothesis: indeed, for any $\textbf{v}=(x_v,0)$ with $0 \le x_v \le l$, if the robot starts from $\textbf{v}$ and follows $K$, it gets at some point $\textbf{w} = (x_w,0)$ with $x_v \le x_w \le l$ without visiting any vertex on the column $c_{l+1}$, i.e. without visiting any point of longitude at least $l+1$.
From the first two statements of the conclusion proved above it follows that the robot passes for the first time from the column $c_l$ to the column $c_{l+1}$ while executing some instance of the move of the form $N^bES^b$, $-a \leq b \leq a$. Assume for a contradiction that $b \neq 0$ is not the lowermost latitude with respect to the well order on $\mathbb{Z}$ at which the columns $c_l$ and $c_{l+1}$ are joined. Let $b' \in \mathbb{Z}$ be the predecessor of $b$ in the well order on $\mathbb{Z}$. Say $Y$ is the first segment of the algorithm $SME(a,e,K)$ strictly before this specific instance of this specific locomotory move, $N^bES^b$.
We define $A=(((((K^eE)^eNES)^eSEN)^eNNESS)^e...N^{b'}ES^{b'})^1$ and note that $A' = A^e = (((((K^eE)^eNES)^eSEN)^eNNESS)^e...N^{b'}ES^{b'})^e$ is a last segment of $Y$. Let $B$ be the first segment of $Y$ strictly before $A'$, i.e. $Y = BA'$. For some $0 \leq x \leq l$ we denote by $(x, 0)$ the vertex where the robot gets if it starts from the origin and follows $B$. If the robot starts from $(x, 0)$ and follows $A'$, it gets to the point $(l, 0)$. Also notice that if the robot starts from $(l, 0)$ and follows $A$, it gets to some point $(l', 0)$ with $l' \geq l+1$. Say the robot starts from the point $(x, 0)$ and follows the algorithm $A^{e+1}$. If the robot starts from the $x$- axis and follows $A$ it advances eastwards at least $0$ columns. When the robot starts from the $x$- axis and follows the $e+1$-th instance of $A$, it returns to the $x$-axis and advances eastwards at least one column. This means that if the robot starts from the $x$-axis and follows the $w$-th instance of $A$ it returns to the $x$-axis and advances eastwards at least one column for each $1 \leq w \leq e+1$.
Therefore, if the robot starts from $(x, 0)$ and follows $A' = A^e$, it gets to the point $(l, 0)$ and advances eastwards at least $e$ columns. This is a contradiction as $l+1 \leq e$. This proves the third statement of the conclusion, that if the robot follows $SME(a,e,K)$, it passes from the column $c_l$ to the column $c_{l+1}$ for the first time while executing a locomotory move $N^mES^m$, where $m \in \mathbb{Z}$ is the lowermost latitude with respect to the standard well order on $\mathbb{Z}: 0<1<-1<2<-2< \hdots$ such that the columns $c_l$ and $c_{l+1}$ are joined at latitude $m$.
By the third statement of the conclusion, we know that the robot passes for the first time from the column $c_l$ to the column $c_{l+1}$ while executing the move $N^mES^m$. Assume $K$ does not follow immediately that after this move is executed. Say $Y$ is the first segment of the algorithm $SME(a,e,K)$ before and including this specific instance of this specific locomotory move, $N^mES^m$.
We define $A=(((((K^eE)^eNES)^eSEN)^eNNESS)^e...N^{m}ES^{m})^1$ and note that $A' = A^e = (((((K^eE)^eNES)^eSEN)^eNNESS)^e...N^{m}ES^{m})^e$ is the last segment of $Y$. Let $B$ be the first segment of $Y$ strictly before $A'$, i.e. $Y = BA'$. For some $0 \leq x \leq l$ we denote by $(x, 0)$ the vertex where the robot gets if it starts from the origin and follows $B$. If the robot starts from $(x, 0)$ and follows $A'$, it gets to the point $(l+1, 0)$. Say the robot starts from the point $(x, 0)$ and it follows the algorithm $A^e$. If the robot starts from the $x$- axis and follows $A$, it advances eastwards at least $0$ columns. When the robot starts from the $x$- axis and follows the $e$-th instance of $A$, it returns to the $x$-axis and advances eastwards at least one column. This means that if the robot starts from the $x$-axis and follows the $w$-th instance of $A$ it returns to the $x$-axis and advances eastwards at least one column for each $1 \leq w \leq e$.
Therefore, if the robot starts from $(x, 0)$ and it follows $A' = A^e$, it gets to the point $(l, 0)$ and advances eastwards at least $e$ columns. This is a contradiction as $l+2 \leq e$, proving the last statement of the conclusion, that after the robot passes for the first time from $c_l$ to $c_{l+1}$ following the locomotory move $N^mES^m$, the robot follows $K$. This finishes the proof.
\end{proof}
We end this section with the following immediate corollary of Lemma~\ref{l2}.
\begin{cor}\label{c1}
Under the assumptions of Lemma~\ref{l2}, let us choose another order on $\mathbb{Z}$, say the \emph{$n$-special order on $\mathbb{Z}$}: $0<n<1<-1< \hdots$ instead of the well order on $\mathbb{Z}$ we considered in Lemma~\ref{l2}. Then if we construct
\[
SME^{(n)} (a, e, K):= (((((((K)^e N^{n} E S^{n})^e E)^e NES)^e SEN)^e NNESS)^e \hdots S^aEN^a)^e,
\]
the results in Lemma~\ref{l2} still hold, with the amendment that after the robot follows $SME^{(n)} (a, e, K)$ in any maze in $\mathcal{C}$, it passes for the first time from the column $c_l$ to the column $c_{l+1}$ while executing $N^mES^m$, where $m$ is the lowermost latitude with respect to the $n$-special order on $\mathbb{Z}$.
\end{cor}
\section{The Cover}\label{fnv}
In the general model, let $\mathcal{F} \subset \mathcal{M}$ be the set of mazes with nonzero finitely many vertical edges removed in consecutive columns. Without loss of generality, we assume that for any maze in $\mathcal{F}$ the origin is the point $(0, 0)$. In this section, we introduce a series of technical definitions that are used to classify the mazes in $\mathcal{F}$ in order to prove Theorem~\ref{ch2th2}.
\begin{figure}[h!]
\centering
\resizebox{0.5\textwidth}{!}{
\begin{tikzpicture}
[
dot/.style={circle,draw=black, fill,inner sep=2pt},
cross/.style={cross out, draw=black, minimum size=2*(#1-\pgflinewidth), inner sep=4pt, outer sep=4pt},
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\node[dot] at (\x,\y){ };
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\draw[line width=0.7mm, red] (2, -4) -- (2, 4);
\draw[line width=0.7mm, red] (3, -4) -- (3, 4);
\draw[line width=0.7mm, red] (4, -4) -- (4, 4);
\draw[line width=0.7mm, red] (-2, 4) -- (-2, 2);
\draw[line width=0.7mm, red] (-2, 0) -- (-2, -1);
\draw[line width=0.7mm, red] (-2, -2) -- (-2, -3);
\draw[line width=0.7mm, red] (-1, 3) -- (-1, 1);
\draw[line width=0.7mm, red] (-1, -1) -- (-1, -2);
\draw[line width=0.7mm, red] (0, -2) -- (0, -4);
\draw[line width=0.7mm, red] (0, 0) -- (0, 3);
\draw[line width=0.7mm, red] (1, 4) -- (1, 3);
\draw[line width=0.7mm, red] (1, 2) -- (1, -4);
\draw[line width=0.7mm, red] (3, 4) -- (4, 4);
\draw[line width=0.7mm, red] (2, 4) -- (-2, 4);
\draw[line width=0.7mm, red] (-2, 3) -- (0, 3);
\draw[line width=0.7mm, red] (1, 3) -- (3, 3);
\draw[line width=0.7mm, red] (-4, 3) -- (-3, 3);
\draw[line width=0.7mm, red] (-4, 2) -- (-3, 2);
\draw[line width=0.7mm, red] (-1, 2) -- (1, 2);
\draw[line width=0.7mm, red] (2, 2) -- (3, 2);
\draw[line width=0.7mm, red] (-1, 1) -- (2, 1);
\draw[line width=0.7mm, red] (-3, 1) -- (-2, 1);
\draw[line width=0.7mm, red] (3, 1) -- (4, 1);
\draw[line width=0.7mm, red] (-2, 0) -- (-1, 0);
\draw[line width=0.7mm, red] (1, 0) -- (3, 0);
\draw[line width=0.7mm, red] (-2, -1) -- (1, -1);
\draw[line width=0.7mm, red] (2, -1) -- (3, -1);
\draw[line width=0.7mm, red] (-4, -1) -- (-3, -1);
\draw[line width=0.7mm, red] (-4, -2) -- (-2, -2);
\draw[line width=0.7mm, red] (2, -2) -- (4, -2);
\draw[line width=0.7mm, red] (-1, -3) -- (2, -3);
\draw[line width=0.7mm, red] (3, -3) -- (4, -3);
\draw[line width=0.7mm, red] (-2, -4) -- (-1, -4);
\draw[line width=0.7mm, red] (0, -4) -- (2, -4);
\draw[line width=0.7mm, red] (-4, -4) -- (-3, -4);
\draw (3,-2) node[cross] {};
\foreach \x in {-4,...,4}
\draw (\x,.1) -- node[below,yshift=-1mm] {\x} (\x,-.1);
\foreach \y in {-4,...,4}
\draw (.1,\y) -- node[below,xshift=-3mm, yshift=3mm] {\y} (-.1,\y);
\draw[->,line width=0.15mm] (0,-4.5) -- (0,4.5);
\draw[->,line width=0.15mm] (-4.5,0) -- (4.5,0);
\end{tikzpicture}
}
\caption{A local representation of a general maze $M \in \mathcal{F}$ that we use in order to illustrate our definitions. The destination point $(3, -2)$ is marked with an 'X'. We assume that there are no vertical edges removed from $M$ other than the ones shown in the figure. For simplicity, we further assume that $M$ is connected, though this may not be true for all mazes.} \label{1}
\end{figure}
We recall that in order to construct an algorithm $A$ that solves the set of mazes $\mathcal{F} \subset \mathcal{M}$ we adopt the following strategy: we find a countable cover $\mathcal{F}=\cup_{i=1}^{\infty}F_i$ with $(F_i)_{i\ge1}\subseteq \mathcal{F}$ such that for each $i \in \mathbb{N}$ and each finite algorithm $X$ we are able to find a finite algorithm $A_X^i$ such that the concatenated algorithm $XA_X^i$ solves $F_i$.
The aim of this section is to introduce the definitions that we need to use in order to construct the cover $(F_i)_{i \geq 1}$.
For any maze $M \in \mathcal{F}$ we denote by \emph{HE}, \emph{HNE}, \emph{VE}, \emph{VNE} a horizontal edge, horizontal non edge, vertical edge and vertical non edge, respectively. For $M$ as in Figure~\ref{1}, between $(2, 2)$ to $(3, 2)$ there is a HE, between $(-1, -2)$ and $(0, -2)$ there is a HNE, between $(0, 0)$ and $(0, 1)$ there is a VE and between $(1, 2)$ and $(1, 3)$ there is a VNE.
From any maze $M \in \mathcal{F}$ we construct the maze $\overline{M} \in \mathcal{F}$ by adding all the possible VEs such that the connected component of the origin is unchanged in the process. The new maze $\overline{M}$ has the nice property that the robot can get from the origin to one vertex of every VNE in $\overline{M}$. We note that an algorithm solves $M$ if and only if it solves $\overline{M}$. Therefore, in order to prove Theorem~\ref{ch2th2} it is enough to construct an algorithm $A$ which solves $\overline{\mathcal{F}} = \{ \overline {M} \mid M \in \mathcal{F} \} \subseteq \mathcal{F}$.
The rest of the section will only address mazes in $\overline{\mathcal{F}}$, so for any maze $\overline{M} \in \overline{\mathcal{F}}$ we introduce the following definitions.
Let $S$ be the smallest vertical strip that contains all the VNEs, the origin and the destination point together with all the HEs incident to it on its left and right sides. As there is only a finite number of VNEs, $S$ contains a finite number of (consecutive) columns. For $M$ as in Figure~\ref{1}, $S$ is the subgraph formed from the columns $c_{-2}, \hdots, c_3$ together with all the HEs between $c_{-3}$ and $c_{-2}$ and all the HEs between $c_3$ and $c_4$; in particular the vertex $(-3, -2)$ and the edge between $(3, 1)$ and $(4, 1)$ are in $S$, but the vertex $(-3, 2)$ is not.
Considering the maze with all its HEs deleted, we can label the connected components obtained in this way by \emph{upper infinite columns}, \emph{lower infinite columns}, \emph{infinite columns}, and \emph{finite columns} accordingly. For $M$ as in Figure~\ref{1}, there are $4$ upper infinite columns, e.g. the infinite path $(-2,2), (-2, 3), \hdots$; there are also $4$ lower infinite columns, e.g. the infinite path $(-2, -4), (-2, -5), \hdots$; the infinite columns are $c_{-3}, c_{-4}, \hdots$ and $c_2, c_3, \hdots$; examples of finite columns are $(-2, 1)$, the path $(-2, -1), (-2, 0)$ or the path $(0, 0), (0, 1), (0, 2), (0, 3)$.
Considering only the HEs in $S$, we call a \emph{pass} any of the following edges:
\begin{enumerate}
\item the HE of smallest latitude between two upper infinite columns, or between an upper infinite column and an infinite column, e.g. the edge between $(-2, 4)$ and $(-1, 4)$ or the edge between $(1, 3)$ and $(2, 3)$, respectively in Figure~\ref{1};
\item the HE of largest latitude between two lower infinite columns, or between an lower infinite column and an infinite column, e.g. the edge between $(0, -3)$ and $(1, -3)$ or the edge between $(1, 1)$ and $(2, 1)$, respectively in Figure~\ref{1};
\item the HE of smallest latitude between two infinite columns with respect to the well order on $\mathbb{Z}: 0<1<-1<2<-2< \hdots $, e.g. the edge between $(2, 0)$ and $(3, 0)$ in Figure~\ref{1}.
\end{enumerate}
Every maze has a finite number of VNEs, so every maze has a finite number of passes. We further note that between two consecutive columns in $S$ there might not be a pass, if there is no HE between them. Finally, as a few more revealing examples, we note that in Figure~\ref{1} the edge between $(-3, 1)$ and $(-2, 1)$ is not a pass, and neither is the one between $(-4, -1)$ and $(-3, -1)$ which is not in $S$; however, the edge between $(3, 1)$ and $(4, 1)$ is in $S$ and it is also a pass.
Furthermore, we define the following regions: the \emph{obstacle strip} is the smallest vertical strip that contains all VNEs, together with all the HEs incident to it on its left and right sides. For example, in Figure~\ref{1} the obstacle strip is formed from the columns $c_{-2}, \hdots c_1$ together with all the HEs incident with any vertex on $c_{-2}$ or $c_1$. The \emph{west strip} and \emph{east strip} are the vertical strips situated at the left and right of the obstacle strip, respectively. For example, in Figure~\ref{1} is formed from the columns $c_{-3}, c_{-4}, \hdots$ and the east strip is formed from the columns $c_2, c_3, \hdots$. We note that the obstacle strip and the east or west strip may intersect only in a certain set of vertices, i.e. the eastern or western endvertices of the edges that emerge on the right or left side of the obstacle strip, respectively; they have no edges in common.
We define the \emph{primary rectangle} to be the subgraph contained in the smallest rectangle that contains the origin, the destination point, all the passes and all the VNEs. The primary rectangle is well defined, as there is a finite number of passes and VNEs. Let $p$ be the smallest positive integer such that the primary rectangle is strictly contained in the interior of the square centred at the origin with the set of vertices $\{ ( \pm p, \pm p )\}$ (see Figure~\ref{2}). We call $p$ the \emph{parameter of the primary rectangle}.
\begin{figure}[h!]
\centering
\resizebox{0.5\textwidth}{!}{
\begin{tikzpicture}
[
dot/.style={circle,draw=black, fill,inner sep=2pt},
cross/.style={cross out, draw=black, minimum size=2*(#1-\pgflinewidth), inner sep=5pt, outer sep=5pt},
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\foreach \x in {-4,...,4}{
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\node[dot] at (\x,\y){ };
}}
\foreach \x in {-2,...,4}{
\draw node[fill,circle,inner sep=5.5pt,minimum size=2pt] at (\x, 4) {};
}
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\draw node[fill,circle,inner sep=5.5pt,minimum size=2pt] at (\x, 3) {};
}
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\draw node[fill,circle,inner sep=5.5pt,minimum size=2pt] at (\x, 2) {};
}
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\draw node[fill,circle,inner sep=5.5pt,minimum size=2pt] at (\x, 1) {};
}
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}
\foreach \x in {-2,...,3}{
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}
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}
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}
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\draw[line width=0.7mm, red] (-2, 0) -- (-2, -1);
\draw[line width=0.7mm, red] (-2, -2) -- (-2, -4);
\draw[line width=0.7mm, red] (-1, 3) -- (-1, 1);
\draw[line width=0.7mm, red] (-1, -1) -- (-1, -2);
\draw[line width=0.7mm, red] (-1, -3) -- (-1, -4);
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\draw[line width=0.7mm, red] (0, 0) -- (0, 3);
\draw[line width=0.7mm, red] (1, 4) -- (1, 3);
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\draw[line width=0.7mm, red] (-1, 1) -- (1, 1);
\draw[line width=0.7mm, green] (1, 1) -- (2, 1);
\draw[line width=0.7mm, red] (-3, 1) -- (-2, 1);
\draw[line width=0.7mm, green] (3, 1) -- (4, 1);
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\draw[line width=0.7mm, green] (2, 0) -- (3, 0);
\draw[line width=0.7mm, red] (-2, -1) -- (1, -1);
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\draw[line width=0.7mm, red] (-4, -1) -- (-3, -1);
\draw[line width=0.7mm, red] (-4, -2) -- (-3, -2);
\draw[line width=0.7mm, green] (-3, -2) -- (-2, -2);
\draw[line width=0.7mm, red] (2, -2) -- (4, -2);
\draw[line width=0.7mm, green] (-1, -3) -- (1, -3);
\draw[line width=0.7mm, red] (1, -3) -- (2, -3);
\draw[line width=0.7mm, red] (3, -3) -- (4, -3);
\draw[line width=0.7mm, green] (-2, -4) -- (-1, -4);
\draw[line width=0.7mm, red] (0, -4) -- (2, -4);
\draw[line width=0.7mm, red] (-4, -4) -- (-2, -4);
\draw (3,-2) node[cross] {};
\foreach \x in {-4,...,4}
\draw (\x,.1) -- node[below,yshift=-2mm] {\x} (\x,-.1);
\foreach \y in {-4,...,4}
\draw (.1,\y) -- node[below,xshift=-4mm, yshift=3mm] {\y} (-.1,\y);
\draw[->,line width=0.15mm] (0,-4.5) -- (0,4.5);
\draw[->,line width=0.15mm] (-4.5,0) -- (4.5,0);
\end{tikzpicture}
}
\caption{We assume that there are no vertical edges removed other than the ones shown in the figure. The destination point is $(3, -2)$. All the passes are marked with green edges. The primary rectangle has vertices $(-3, 4), (4, 4), (-4, 4), (-3, -4)$ and $p=5$. The special vertices are drawn larger.} \label{2}
\end{figure}
We define the \emph{special vertices} to be all the vertices in $S$ that are connected to the destination point and have the same latitude as an endpoint of a VNE (see Figure~\ref{2}). Notice that there is a finite number of special vertices and label them $1,2, \hdots, s$. We note that there must exist a path contained in the primary rectangle between each special vertex and the destination point. Indeed, the fact that all all VNEs are contained in the primary rectangle and the way we define passes allows us to find paths contained in the primary rectangle between the accessible infinite/upper and lower infinite and finite columns of the primary rectangle; this further allows us to find paths contained in the primary rectangle from the special points to the destination point.
Let $l_i$ be the length of a shortest path contained in the primary rectangle from $i \in \{1, 2, \hdots, s\}$ to the destination point and set the following constant which depends only on the local configuration of the maze inside the primary rectangle:
$$l'=1+ ((((l_1)2+l_2)2+l_3)2+ \hdots + l_{s-1})2+l_s.$$
The \emph{secondary rectangle} is obtained from the primary rectangle by augmenting it $l'$ units in each of the four directions. Note that given the local configuration of the maze inside the secondary rectangle, we can construct a finite algorithm $L'$ such that if the robot follows $L'$ starting from any special point, it visits the destination point without leaving the secondary rectangle. Indeed, assume the robot starts at the special point labeled $1$. We construct firstly a finite algorithm $L_1$ that takes the robot to the destination point with $|L_1| = l_1$. Then assume that the robot starts at the special point labeled $2$ and that it first follows the algorithm $L_1$. We write the algorithm $L_2$ as a concatenation of two sub-algorithms. The first cancels the action of $L_1$ and brings the robot back to the special point $2$ and the second sub-algorithm takes the robot further to the destination point. This can be done with at most $l_1 + l_2$ instructions, so without loss of generality $|L_2| \le l_1 + l_2$. Moreover, if the robot starts at any of the special points labeled $1$ or $2$ and follows $L_1L_2$ it gets to the destination point. We continue in this way: given $L_1, L_2, \hdots, L_{i-1}$ and assuming that the robot starts at the special point $i$, we construct $L_i$ as a concatenation of two sub-algorithms. The first brings the robot back to the special point $i$ and the second takes the robot further to the destination point. This can be done with $|L_i| \leq (|L_1| + \hdots + |L_{i-1}|)+l_i$. Finally, take $L'=L_1L_2 \hdots L_s$ with $|L'| \leq l'$ which has the property that if the robot follows $L'$ starting from any special point, it visits the destination point. The role of adding $1$ to the sum is that to ensure that the secondary rectangle augments non-trivially the primary rectangle.
In the rest of the section we define a series of very technical configurations. We group the mazes according to these configurations and obtain the desired countable cover at the end of the section. The importance of these configurations only becomes clear in Section~\ref{final} and we will recall them where appropriate.
For simplicity we use cardinal directions in our definitions. We say that \emph{the row $r_i$ is to the north of the row $r_j$} or \emph{above row $r_j$}, provided $i>j$. By an \emph{easternmost H(N)E} $e$ with a certain property $\mathcal{P}$ we mean that $e$ has $\mathcal{P}$ and no other H(N)E with $\mathcal{P}$ has longitude greater than $e$. These definitions easily extend to the other directions: \emph{westernmost}, \emph{uppermost}, \emph{lowermost}. In pairings (e.g.``the lowermost easternmost HNE with $\mathcal{P}$'') we always give priority to the first direction and then to the second one. For example, in order to find the uppermost easternmost HNE below all VNEs in the west strip, we first look for the row of highest latitude below all VNEs on which there is a HNE in the west strip and then on this row we pick the one HNE in the west strip with the largest longitude.
Define a \emph{west bump} to be any of the easternmost HNE in the west strip or at the border between the west strip and the obstacle strip (i.e. with at least one vertex in the west strip) on a row that intersects some finite column. For example in Figure~\ref{1}, the HNE between $(-4, 1)$ and $(-3, 1)$ and the HNE between $(-3, 2)$ and $(-2, 2)$ are both west bumps with the rows $r_1$ and $r_2$ intersecting the finite column $(-1, 1), (-1, 2), (-1, 3)$. Using symmetry, define similarly an \emph{east bump}. We note that there are a finite number of west and east bumps.
If there exists a row which is a path when restricted to the west strip, but contains a HNE, then call the smallest such row with respect to the standard well order on $\mathbb{Z}$ a \emph{magical west row}; define its \emph{west cutoff} to be its westernmost HNE. Define similarly a \emph{magical east row} and its \emph{east cutoff}.
We define a \emph{west pipe} to be any of the easternmost configurations in the west strip of three vertices $(x, y), (x+1, y), (x+2, y)$ where between $(x, y)$ and $(x+1, y)$ there is a HE and between $(x+1, y)$ and $(x+2, y)$ there is a HNE, which can be at the border between the west strip and the obstacle strip. For example in Figure~\ref{1}, $(-4, 2), (-3, 2), (-2, 2)$ is a west pipe. Note that a maze may have infinitely many west pipes. We define similarly an \emph{east pipe} to be any of the westernmost configurations in the east strip of three vertices $(x, y), (x+1, y), (x+2, y)$ where between $(x+1, y)$ and $(x+2, y)$ there is a HE and between $(x, y)$ and $(x+1, y)$ there is a HNE, which can be at the border between the east strip and the obstacle strip. For example in Figure~\ref{1}, $(2, 1), (3, 1), (4, 1)$ is an east pipe.
Furthermore, we define the \emph{special west pipe} to be the west pipe on the smallest row that has a west pipe, with respect to the standard well order on $\mathbb{Z}$, if such a row exists. Note that in Figure~\ref{1} the special west pipe may not be $(-4, -1), (-3, -1), (-2, -1)$ as we do not know from the picture whether there are west pipes on $r_0$ or $r_1$, but we do know that it is the west pipe on $r_{-1}$. We define similarly the \emph{special east pipe}. Note that if a maze does not have any special west pipe, then in the west strip any row is either a path or it is the complement of an infinite path followed by a finite path.
We define an \emph{almost empty west row} to be a row that in the west strip is the complement of an infinite path followed by a non-empty finite path. Thus, in Figure~\ref{1}, both $r_0$ and $r_1$ cannot be almost empty west rows as the non-empty finite path in the west strip is missing for both of these columns; the edge between $(-3, 1)$ and $(-2, 1)$ does not belong to the west strip. We define similarly an \emph{almost empty east row}. We define the \emph{special almost empty west row} to be the smallest almost empty west row with respect to the standard well order on $\mathbb{Z}$, if such a row exists. We define the \emph{west cutoff} of a special almost empty west row to be its easternmost HNE in the west strip. We define similarly the \emph{special almost empty east row} and its \emph{east cutoff}. For example, if in Figure~\ref{1} $r_2$ was the special almost empty east row, its east cutoff would be the edge between $(3, 2)$ and $(4, 2)$. Finally, we define an \emph{empty west row} to be a row that in the west strip is empty; for the `special' label in this context, we need in addition that the latitude of the row is large in absolute value. So we define the \emph{special empty west row} to be the empty west row of smallest latitude, greater than $-3p$ (where the parameter of the primary rectangle, $p$, is defined above) with respect to the standard well order on $\mathbb{Z}$, if such a row exists. We define the \emph{natural special empty west row} to be the empty west row of smallest latitude, without the additional constraint. We define similarly the \emph{special empty east row} and the \emph{natural special empty west row}.
We define the \emph{upper west pass} to be the lowermost HE between the easternmost infinite column of the west strip and the westernmost upper infinite column with the property that its latitude $k$ is greater than that of any pass in the obstacle strip, if such a HE exists. We define similarly the \emph{upper east pass}, \emph{lower west pass} and \emph{lower east pass}. For example, in Figure~\ref{2} the edge between $(-3, -4)$ and $(-2, -4)$ is the lower west pass. Also, in Figure~\ref{3} the upper/lower west/east passes are the green edges.
Let us call the pair of columns at the border between the west strip and the obstacle strip $(c_a, c_{a+1})$, so $c_a$ is in the west strip and $c_{a+1}$ is in the obstacle strip. Let us call the pair of columns at the border between the obstacle strip and the east strip $(c_b, c_{b+1})$, so $c_b$ is in the obstacle strip and $c_{b+1}$ is in the east strip. We define the \emph{west ascending chain} (if such a structure exists) to be the finite sequence of HEs: $HE_a, HE_{a+1}, \hdots , HE_b$ such that $HE_a$ is the upper west pass and $HE_{m}$ is the lowermost HE between the pair of columns $(c_m, c_{m+1})$ at latitude at least that of $HE_{m-1}$ for $m=a+1, \hdots, b$ (see Figure~\ref{3}). Similarly, we define the \emph{east ascending chain}, \emph{west descending chain} and \emph{east descending chain}. If a west ascending chain $HE_a, \hdots , HE_b$ exists with $HE_b$ on some row $r_t$, we define the \emph{upper west constant} $c_{uw}:=t+p$, where $p$ is the parameter of the primary rectangle. We define similarly the constants \emph{lower west constant}, \emph{upper east constant} and \emph{lower east constant}.
\begin{figure}[h!]
\centering
\resizebox{0.75\textwidth}{!}{
\begin{tikzpicture}
[
dot/.style={circle,draw=black, fill,inner sep=2pt},
cross/.style={cross out, draw=black, minimum size=2*(#1-\pgflinewidth), inner sep=4pt, outer sep=4pt},
]
\foreach \x in {-6,...,6}{
\foreach \y in {-6,...,6}{
\node[dot] at (\x,\y){ };
}}
\draw[line width=0.7mm, red] (-6, -6) -- (-6, 6);
\draw[line width=0.7mm, red] (-5, -6) -- (-5, 6);
\draw[line width=0.7mm, red] (-4, -6) -- (-4, 6);
\draw[line width=0.7mm, red] (-3, -6) -- (-3, -4);
\draw[line width=0.7mm, red] (-3, -2) -- (-3, 6);
\draw[line width=0.7mm, red] (-2, -6) -- (-2, -4);
\draw[line width=0.7mm, red] (-2, -1) -- (-2, 6);
\draw[line width=0.7mm, red] (-1, -6) -- (-1, -4);
\draw[line width=0.7mm, red] (-1, -3) -- (-1, 0);
\draw[line width=0.7mm, red] (-1, 1) -- (-1, 6);
\draw[line width=0.7mm, red] (0, -6) -- (0, -4);
\draw[line width=0.7mm, red] (0, 0) -- (0, 6);
\draw[line width=0.7mm, red] (1, -6) -- (1, -4);
\draw[line width=0.7mm, red] (1, 1) -- (1, 6);
\draw[line width=0.7mm, red] (2, -6) -- (2, -2);
\draw[line width=0.7mm, red] (2, -1) -- (2, 6);
\draw[line width=0.7mm, red] (3, -6) -- (3, -5);
\draw[line width=0.7mm, red] (3, -2) -- (3, 0);
\draw[line width=0.7mm, red] (3, 1) -- (3, 6);
\draw[line width=0.7mm, red] (4, -6) -- (4, 6);
\draw[line width=0.7mm, red] (5, -6) -- (5, 6);
\draw[line width=0.7mm, red] (6, -6) -- (6, 6);
\draw[line width=0.7mm, red] (-5, 6) -- (-3, 6);
\draw[line width=0.7mm, red] (-2, 6) -- (-1, 6);
\draw[line width=0.7mm, red] (0, 6) -- (1, 6);
\draw[line width=0.7mm, red] (3, 6) -- (6, 6);
\draw[line width=0.7mm, red] (-6, 5) -- (-5, 5);
\draw[line width=1.5mm, blue] (-1, 5.2) -- (4, 5.2);
\draw[line width=0.7mm, red] (-1, 5) -- (6, 5);
\draw[line width=1.5mm, blue] (-2, 4.2) -- (-1, 4.2);
\draw[line width=0.7mm, red] (-2, 4) -- (-1, 4);
\draw[line width=0.7mm, red] (0, 4) -- (2, 4);
\draw[line width=0.7mm, red] (5, 4) -- (6, 4);
\draw[line width=0.7mm, red] (-6, 3) -- (-5, 3);
\draw[line width=1.5mm, blue] (-3, 3.2) -- (-2, 3.2);
\draw[line width=0.7mm, red] (-3, 3) -- (-2, 3);
\draw[line width=0.7mm, green] (3, 3) -- (4, 3);
\draw[line width=0.7mm, red] (2, 3) -- (3, 3);
\draw[line width=0.7mm, green] (-4, 2) -- (-3, 2);
\draw[line width=1.5mm, blue] (-4, 2.2) -- (-3, 2.2);
\draw[line width=0.7mm, red] (-3, 2) -- (-1, 2);
\draw[line width=0.7mm, red] (4, 2) -- (6, 2);
\draw[line width=0.7mm, red] (-4, 1) -- (2, 1);
\draw[line width=0.7mm, red] (-6, 1) -- (-5, 1);
\draw[line width=0.7mm, red] (2, 1) -- (3, 1);
\draw[line width=0.7mm, red] (2, 0) -- (6, 0);
\draw[line width=0.7mm, red] (-1, -1) -- (2, -1);
\draw[line width=0.7mm, red] (-6, -2) -- (-5, -2);
\draw[line width=0.7mm, red] (-4, -2) -- (-3, -2);
\draw[line width=0.7mm, red] (-2, -2) -- (-1, -2);
\draw[line width=0.7mm, red] (0, -2) -- (1, -2);
\draw[line width=0.7mm, red] (5, -2) -- (6, -2);
\draw[line width=0.7mm, red] (-3, -3) -- (1, -3);
\draw[line width=0.7mm, red] (4, -3) -- (5, -3);
\draw[line width=0.7mm, red] (-6, -4) -- (-2, -4);
\draw[line width=0.7mm, red] (2, -4) -- (5, -4);
\draw[line width=0.7mm, red] (-5, -6) -- (-4, -6);
\draw[line width=0.7mm, green] (-4, -6) -- (-3, -6);
\draw[line width=0.7mm, red] (-3, -6) -- (1, -6);
\draw[line width=0.7mm, red] (2, -6) -- (3, -6);
\draw[line width=0.7mm, green] (3, -6) -- (4, -6);
\draw[line width=0.7mm, red] (4, -6) -- (5, -6);
\draw[line width=0.7mm, red] (-2, -5) -- (4, -5);
\draw (2,-2) node[cross] {};
\foreach \x in {-6,...,6}
\draw (\x,.1) -- node[below,yshift=-1mm] {\x} (\x,-.1);
\foreach \y in {-6,...,6}
\draw (.1,\y) -- node[below,xshift=-3mm, yshift=3mm] {\y} (-.1,\y);
\draw[->,line width=0.15mm] (0,-6.5) -- (0,6.5);
\draw[->,line width=0.15mm] (-6.5,0) -- (6.5,0);
\end{tikzpicture}
}
\caption{We assume that there are no vertical edges removed other than the ones shown in the figure. The green edges are the upper/lower west/east passes. Neither the HE $(-4, -2), (-3, -2)$ nor $(-4, 1), (-3, 1)$ is the upper west pass, as they are not above all the passes in the obstacle strip. The blue coloured edges in order from left to right form the west ascending chain.} \label{3}
\end{figure}
Assume that the upper west pass is on some row $r_k$. We define an \emph{upper west paired HNEs} to be any pair of HNEs with the same longitude in the west strip such that the upper HNE is at latitude $k$ and the lower HNE is at latitude at most $k-c_{uw}$, where $c_{uw}$ defined above is the upper-west constant. We define similarly the \emph{upper east paired HNEs}, \emph{lower west paired HNEs} and \emph{lower east paired HNEs} with respect to the corresponding constants $c_{ue}$, $c_{lw}$, and $c_{le}$, respectively. We define the \emph{special upper west paired HNEs} (if such a structure exists) to be the upper west paired HNEs with the uppermost and easternmost lower HNE. We recall that in all such instances we give priority to the first condition and then the second one. We define similarly the \emph{special upper east paired HNEs}, \emph{special lower west paired HNEs} and \emph{special lower east paired HNEs}.
With $k$ being as always the latitude of the upper west pass, we define the \emph{upper west pipe} to be the west pipe on the row $r_k$, if one exists. We define similarly the \emph{lower west pipe}, the \emph{upper east pipe} and the \emph{lower east pipe}. We define the \emph{upper west cutoff} to be the easternmost HNE on the row $r_k$ in the west strip, if one exists. We define similarly the \emph{lower west cutoff}, the \emph{upper east cutoff} and the \emph{lower east cutoff}.
We define the \emph{upper west HNE} (if such a structure exists) to be the lowermost westernmost HNE at the north-east of the uppermost westernmost VNE. We define similarly the \emph{upper east HNE}, \emph{lower west HNE} and the \emph{lower east HNE}.
Being consistent with the constants $a$ and $b$ introduced in the definition of the west ascending chain, we define the parameters $h_{(m, m+1)}$ to be the latitude of the uppermost HE between two consecutive upper infinite columns or between an infinite column and an upper infinite column on $c_m$ and $c_{m+1}$ if such a HE exists and $\infty$ otherwise for $m=a, \hdots, b$.
We define similarly the parameters $l_{(m, m+1)}$ to be the latitude of the lowermost HE between two consecutive lower infinite columns or between an infinite column and a lower infinite column on $c_m$ and $c_{m+1}$ if such a HE exists and infinity otherwise for $m=a, \hdots, b$.
We define the parameters $w_1, e_1, w_2, e_2, w_3, e_3, w_4, e_4$ to be the latitude of the magical west/east row, the special almost empty west/east row, the special empty west/east row, and the natural special empty west/east row if such a configuration exists and infinity otherwise, respectively.
We finally define the \emph{tertiary rectangle} to be the subgraph contained in the smallest rectangle that contains the secondary rectangle and all the west/east bumps, upper/lower west/east cutoffs, upper/lower west/east pipes, special west/east pipes, upper/lower west/east passes, west/east ascending/descending chains, upper/lower west/east paired HNEs and the upper/lower west/east HNEs.
As in the case of the primary rectangle, let $q$ be the smallest positive integer such that the tertiary rectangle is strictly contained
in the interior of the square centred at the origin with vertices $\{ \pm \frac{q}{3}, \pm \frac{q}{3} \}$. We call $q$, together with the upper/lower west/east constants, $h_{(m, m+1)}, l_{(m, m+1)}$ for $m=a, \hdots, b$, $w_1, e_1, w_2, e_2, w_3, e_3, w_4, e_4$ the \emph{parameters of the tertiary rectangle}. Therefore, when we construct an algorithm by inspecting the tertiary rectangle, we have access to the subgraph contained in the tertiary rectangle and the values of all its parameters.
We group the mazes in $\overline{\mathcal{F}}$ according to agreeing on the destination point, the subgraph contained in the square $\{ \pm q, \pm q \}$ and the set of parameters of the tertiary rectangle. We thus obtain a countable cover $\overline{\mathcal{F}} = \cup_{i=1}^{ \infty} F_i$. It is obvious directly from the definitions that we set above that such a construction is achievable.
All of these definitions are used in the following section to prove Theorem~\ref{ch2th2} and the relevant ones will be recalled where appropriate.
\section{The General Model}\label{final}
In this section we prove Theorem~\ref{ch2th2}.
Following our strategy, we assume that we are given $F_i$ and a finite algorithm $X$ and we aim to construct a finite algorithm $A$ such that $XA$ visits the destination point of $F_i$. We construct the algorithm $A$ from several sub-algorithms treated in separate subsections, each with a specific task: in \textbf{Part I} we position the robot in the east strip, in \textbf{Part II} we position the robot in the west strip at latitude 0 and in \textbf{Part III} we guide the robot through the destination point. In each part we consider a finite number of cases for the subsets $F_i$ so that although a sub-algorithm depends quantitatively on $F_i$ and $X$, it does depend qualitatively only on the case. We treat each case separately. According to their degree of generality, we label the broader cases as ``Propositions'', and the more specific cases as ``Claims''.
\begin{proof}[Proof of Theorem~\ref{ch2th2}]
Let $F_i$ be any of the classes of mazes defined above and assume we are given a finite algorithm $X$. Let $\lambda := |X|$.
\subsection*{Part 0.}
We recall the finite algorithm $L'$ defined in Section~\S\ref{fnv} for a particular maze $M$, which had the property that if the robot starts at any special point of $M$ and follows $L'$, it visits the destination point. Take $M \in F_i$ and construct its $L'$ as described in Section~\S\ref{fnv}. We claim that for any $M' \in F_i$, the algorithm $L'$ has the same property in $M'$, i.e. if the robot starts at any special point of $M'$ and follows $L'$, it visits the destination point of $M'$. This follows from the fact that all the mazes in $F_i$ share the destination point, the secondary rectangle and in particular the set of special points. Therefore, we pick this $L'$ as a representative for the set of mazes $F_i$.
We now define the algorithm $$L=L_E=L' \text{ } N^{\varepsilon} \text{ } ME(|L' N^{\varepsilon}|, |L' N^{\varepsilon}|),$$ where the correcting constant ${\varepsilon} \in \mathbb{Z}$ is picked such that $|L' N^{\varepsilon}|_N=|L' N^{\varepsilon}|_S$ and therefore $|L|_N=|L|_S$; let $l:=|L|$. The counterpart of $L=L_E$ is $$L_W = L' \text{ } N^{\varepsilon} \text{ } MW(|L' N^{\varepsilon}|, |L' N^{\varepsilon}|),$$ and as before we have $|L_W|_N=|L_W|_S$ and also $|L_W|=l$.
The algorithm $L$ is a generic algorithm used in several other algorithms below. We remark that if the robot starts from a special point and it follows $L$ in any maze in $F_i$, it gets to the destination point; this property is inherited from $L'$. We further note that if the robot is at the origin on a maze with no VNEs and it follows $L$, then it returns to the $x$-axis and its longitude does not decrease. These properties are crucial in order to apply Lemma~\ref{l2}.
We finally note that all mazes in $F_i$ also share the same parameter of the primary rectangle $p$ and parameter of the tertiary rectangle $q$ and we keep this notation consistent for the rest of the proof.
\subsection*{Part I}
The algorithm $rough \_ positioning \_ east$ defined in this part aims to either position the robot in the east strip or to make the robot visit the destination point. We define
\[
RPE := ME (\lambda+p, \lambda+p) \text{ } N^{l+\lambda+4p} \text{ } L \text{ } S^{2(l+\lambda+4p)} \text{ } L.
\]
\begin{prop}\label{pp1}
For any maze in $F_i$, after the robot follows the algorithm $X \text{ } RPE$, it is either in the east strip or it has visited the destination point.
\end{prop}
\begin{proof}
Pick any maze in $F_i$. We claim that by our choice of parameters of $ME$, after the robot follows $X \text{ } ME (\lambda+p, \lambda+p)$, it is either in the east strip or in the obstacle strip, but not in the west strip. Indeed, assume for a contradiction that after the robot follows $X \text{ } ME (\lambda+p, \lambda+p)$, it is in the west strip. Denote by $\textbf{x} = (x, y)$ the position of the robot after it follows $X$ starting from the origin. By assumptionm $\textbf{x}$ must be in the west strip as the algorithm $ME$ has no instruction $W$. Therefore, as the robot follows $ME (\lambda+p, \lambda+p)$, it does not visit any endvertex of a VNE. We recall that all mazes in $F_i \subset \overline{\mathcal{F}}$ have the property that for every VNE at least one of its vertices is accessible, hence the westernmost column of the obstacle strip $c_{a+1}$ is accessible from $\textbf{x}$. The robot starts in the origin which is at most $p$ units in longitude away from the obstacle strip as the primary rectangle contains the origin and all the VNEs. Hence the column$c_{y+\lambda+p}$ is not in the west strip. Moreover, every pair of consecutive columns at longitude between $y$ and $a+1$ are connected by a HE at some latitude between $x+\lambda+p$ and $x-\lambda-p$, as the primary rectangle contains all the passes and the VNEs. Therefore, if the robot starts from $\textbf{x}$ and follows $ME (\lambda+p, \lambda+p)$, it gets to a longitude at least $y+\lambda+p$, which is not in the west strip. This contradiction proves the claim.
Hence, after the robot follows $X \text{ } ME (\lambda+p, \lambda+p)$, its longitude is at least $a+1$ and so it is either in the east strip or in the obstacle strip. In the former case, after the robot follows $X \text{ } RPE$, it remains in the east strip. Indeed, while the robot follows $N^{l+\lambda+4p} \text{ } L$ starting in the east strip, its latitude is too high to meet any VNE and on a maze with no VNE if the robot follows $L$ its longitude does not decrease so the robot remains in the east strip. Therefore, after the robot follows also $S^{2(l+\lambda+4p)} \text{ } L$ its latitude is too low to meet any VNE, and it remains in the east strip by a similar argument. To conclude, if the robot gets to the east strip after it follows the initial segment $X \text{ } ME (\lambda+p, \lambda+p)$, then it remains in the east strip after it follows $X \text{ } RPE$.
In the latter case, after the robot follows $X \text{ } ME (\lambda+p, \lambda+p)$, it is in the obstacle strip either in (1) a lower infinite column or a finite column or (2) an upper infinite column. In case (1), after the robot follows $X \text{ } ME (\lambda+p, \lambda+p) \text{ } N^{l+\lambda+4p}$ it gets to a special point. Therefore after the robot follows $X \text{ } ME (\lambda+p, \lambda+p) \text{ } N^{l+\lambda+4p} \text{ } L$, it gets to the destination point. In case (2), while the robot follows $N^{l+\lambda+4p} \text{ } L$ it does not meet any VNE and its longitude does not decrease, so after it follows the initial segment $X \text{ } ME (\lambda+p, \lambda+p) \text{ } N^{l+\lambda+4p} \text{ } L$, it is either in the east strip or in the obstacle strip in an upper infinite column. In both cases, it is clear that after the robot follows $X \text{ } RPE$ it is either in the east strip or it has visited the destination point.
\end{proof}
\begin{remark}
In the first part of the proof of Proposition~\ref{pp1} we argue that the parameters $(\lambda+p, \lambda+p)$ of $ME$ are large enough for the robot to have longitude at least $a+1$. The key of this argument is two-fold: firstly, all the passes are in the primary rectangle which has parameter $p$; secondly, if the robot starts from the origin and follows the algorithm $X$ with $|X|=\lambda$, it can not advance more than $\lambda$ columns east or west and any two consecutive columns between its initial and final position are connected at latitude no more than $\lambda$ in absolute value. We do not expand this argument every time we use it, but instead we use the phrase ``by our choice of parameters'' to mark that the same reasoning is used in similar instances to prove that the robot advances westwards/eastwards to the desired longitude.
\end{remark}
At the end of \textbf{Part I}, we note that although we used in the proof of Proposition~\ref{pp1} the fact there are no infinite columns in the obstacle strip, a variation of $RPE$ can be used to position the robot in the east strip, even if we drop this assumption. This note is important, because it shows that \textbf{Part I} can be generalised to improve Theorem~\ref{ch2th2} by dropping the consecutive column condition for the finite number of VNEs. To present this variation, assume that infinite columns are allowed in the obstacle strip, i.e. the (finitely many) VNEs need not be in consecutive columns.
We define now the algorithm $RPE'$ that generalises $RPE$ as described above. It is formed by $\lambda+p$ subalgorithms $S_1, \hdots S_{\lambda+p}$ concatenated in order. We define
\[
S_i = N^{\lambda_i+p+2l} \text{ } L \text{ } S^{\mu_i+p+2l} \text{ } ME (\gamma_i, 1),
\]
for $i=1, \hdots, \lambda+p$. The parameters $\lambda_i, \mu_i, \gamma_i \in \mathbb{N}$ are chosen to be at least the number of instructions written in the whole algorithm until they occur, for example we can take $\lambda_1 = |X| = \lambda$, $\mu_1 = |X \text{ } N^{\lambda_1+p+2l} \text{ } L|$, $\gamma_1 = |X \text{ } N^{\lambda_1+p+2l} \text{ } L \text{ } S^{\mu_1+p+2l}|$, etc. Finally, let
\[
RPE' = S_1 \text{ } S_2 \text{ } \hdots \text{ } S_{\lambda+p}.
\]
Note that for the mazes we consider, we first replace all the VNEs with VEs which do not change the connected component of the origin, so every pair of consecutive columns in the obstacle strip must be connected by an accessible HE. The reason why $RPE'$ indeed generalises $RPE$ is similar to the argument in the proof of Proposition~\ref{pp1}: here, after the robot follows every $S_i$ it either moves at least one column to the east or it has visited the destination point.
Moving on from this digression, by Proposition~\ref{pp1} we may assume that we are given $F_i$ and a finite algorithm $X$ with $\lambda = |X|$ such that after the robot follows $X$ in any maze in $F_i$, it is either in the east strip or it has visited the destination point. Without loss of generality, we assume that the robot is in the east strip and our aim is to build a finite algorithm $A$ such that $X A$ solves $F_i$.
\subsection*{Part II} The algorithm $reset\_latitude\_west$ defined in this part aims to either position the robot in the west strip on the $x$-axis (i.e. at latitude $0$) or to make the robot visit the destination point.
\begin{figure}[h!]
\centering
\resizebox{0.75\textwidth}{!}{
\begin{tikzpicture}
[
dot/.style={circle,draw=black, fill,inner sep=2pt},
cross/.style={cross out, draw=black, minimum size=2*(#1-\pgflinewidth), inner sep=4pt, outer sep=4pt},
]
\foreach \x in {-6,...,6}{
\foreach \y in {-6,...,6}{
\node[dot] at (\x,\y){ };
}}
\draw[line width=0.7mm, red] (-6, -6) -- (-6, 6);
\draw[line width=0.7mm, red] (-5, -6) -- (-5, 6);
\draw[line width=0.7mm, red] (-4, -6) -- (-4, 6);
\draw[line width=0.7mm, red] (-3, -6) -- (-3, 6);
\draw[line width=0.7mm, red] (-2, 6) -- (-2, 2);
\draw[line width=0.7mm, red] (-2, 0) -- (-2, -6);
\draw[line width=0.7mm, red] (-1, 6) -- (-1, 3);
\draw[line width=0.7mm, red] (-1, 1) -- (-1, -3);
\draw[line width=0.7mm, red] (-1, -4) -- (-1, -6);
\draw[line width=0.7mm, red] (0, 6) -- (0, 0);
\draw[line width=0.7mm, red] (0, -2) -- (0, -4);
\draw[line width=0.7mm, red] (0, -5) -- (0, -6);
\draw[line width=0.7mm, red] (1, 6) -- (1, 5);
\draw[line width=0.7mm, red] (1, 4) -- (1, 2);
\draw[line width=0.7mm, red] (1, 1) -- (1, -6);
\draw[line width=0.7mm, red] (2, 6) -- (2, -6);
\draw[line width=0.7mm, red] (3, 6) -- (3, -6);
\draw[line width=0.7mm, red] (4, 6) -- (4, -6);
\draw[line width=0.7mm, red] (5, 6) -- (5, -6);
\draw[line width=0.7mm, red] (6, 6) -- (6, -6);
\draw[line width=1mm, green] (5, -4) -- (5, -3);
\draw[line width=1mm, green] (4, -2) -- (4, -6);
\draw[line width=1mm, green] (3, -3) -- (3, -6);
\draw[line width=1mm, green] (2, 3) -- (2, -6);
\draw[line width=1mm, green] (1, 2) -- (1, 3);
\draw[line width=0.7mm, red] (2, -6) -- (5, -6);
\draw[line width=0.7mm, red] (-1, -5) -- (0, -5);
\draw[line width=0.7mm, red] (2, -5) -- (3, -5);
\draw[line width=0.7mm, red] (5, -5) -- (6, -5);
\draw[line width=0.7mm, red] (5, -4) -- (6, -4);
\draw[line width=0.7mm, red] (-5, -3) -- (-2, -3);
\draw[line width=0.7mm, red] (-1, -3) -- (0, -3);
\draw[line width=0.7mm, red] (4, -3) -- (6, -3);
\draw[line width=0.7mm, red] (-2, -2) -- (-1, -2);
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\draw[line width=0.7mm, red] (0, 3) -- (2, 3);
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\draw[line width=0.7mm, red] (1, 4) -- (2, 4);
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\draw[line width=0.7mm, red] (2, 5) -- (6, 5);
\draw[line width=0.7mm, red] (-3, 6) -- (-2, 6);
\draw[line width=1mm, green] (5, -4) -- (6, -4);
\draw[line width=1mm, green] (5, -3) -- (4, -3);
\draw[line width=1mm, green] (4, -6) -- (3, -6);
\draw[line width=1mm, green] (3, -5) -- (2, -5);
\draw[line width=1mm, green] (1, 3) -- (2, 3);
\draw[line width=1mm, blue] (6, 3) -- (6, -4);
\draw node[fill,circle,inner sep=5.5pt,minimum size=2pt] at (6, -4) {};
\draw node[fill,circle,inner sep=5.5pt,minimum size=2pt] at (1, 2) {};
\draw node[fill,circle,inner sep=5.5pt,minimum size=2pt] at (6, 3) {};
\draw (0,-4) node[cross] {};
\foreach \x in {-6,...,6}
\draw (\x,.1) -- node[below,yshift=-1mm] {\x} (\x,-.1);
\foreach \y in {-6,...,6}
\draw (.1,\y) -- node[below,xshift=-3mm, yshift=3mm] {\y} (-.1,\y);
\draw[->,line width=0.15mm] (0,-6.5) -- (0,6.5);
\draw[->,line width=0.15mm] (-6.5,0) -- (6.5,0);
\end{tikzpicture}
}
\caption{\textbf{Part II, Case (1).} There is no pass between the obstacle strip and the east strip. We assume that there are no VEs removed other than the ones shown in the figure and that the position of the robot after following $X$ is $(6, 3)$. The first segment $S^{\lambda+p+l}$ of $RLW$ takes the robot to some very small latitude $j$ such that if it follows $L$ starting from any point of $r_j$ in the east strip, it will always remain in the east strip. This can be done, as there is no pass between the obstacle strip and the east strip. For our example, we may assume that after the robot follows $X S^{\lambda+p+l}$ it gets to the point $(6, -4)$, though this latitude should be much smaller. The green route to the special point $(1, 2)$ is the route of the robot if it would follow the algorithm $MW$. The algorithm $SMW$ used in $RLW$ generalises $MW$ by inserting the algorithm $L$ between locomotory moves. However, $L$ is constructed in such a way that if the robot follows it while it is in the east strip, its longitude does not increase. Moreover, the latitude of the robot is so small that it will never pass from the column $c_{b+1}$ to $c_b$ while following $L$. By Lemma~\ref{l2} and the choice of parameters of $SMW$, the robot reaches the special point $(1, 2)$ while executing a locomotory move. Immediately afterwards, it executes $L$ and it gets to the destination point. Finally, we remark that when the robot reaches the obstacle strip from the east strip for the first time, it does not enter the finite column $(1, 2), (1, 3), (1, 4)$ via the HE $(1, 4), (2, 4)$ or indeed it does not enter any other finite column which is above $R$. Indeed, this follows from the order of locomotory moves in $SMW$ which prioritises smaller latitudes.} \label{4}
\end{figure}
\noindent
\textbf{Case (1). } We assume that the mazes in $F_i$ do not contain a pass between the obstacle strip and the east strip. Then from the assumptions on the mazes in $F_i$, the east strip is connected to a finite column in the easternmost column of the obstacle strip $c_b$. This follows from the fact that for every VNE of every maze in $F_i$, at least one of its vertices is accessible from the origin. Let $R$ be the lowermost finite column in $c_b$ such that there exists a HE between $R$ and the east strip. Let $\textbf{v} = (b,i)$ be the lowermost vertex of the finite column $R$. In this case, we define the algorithm
\[
RLW:= S^{\lambda+p+l} \text{ } SMW(2\lambda+2p+l,\lambda+2p,L).
\]
\textbf{Claim. } For any maze in $F_i$, after the robot follows the algorithm $X \text{ }RLW$, it visits the destination point.
\begin{proof}
After the robot follows $X \text{ } S^{\lambda+p+l}$, it is in the east strip at a certain point $\textbf{x} = (x, j)$, with $j \le i-l$. By the choice of parameters and by Lemma~\ref{l2}, while the robot follows $SMW(2\lambda+2p+l,\lambda+2p,L)$ it advances westwards in the east strip oscillating about the row $r_j$. It passes for the first time from the column $c_{b+1}$ to the column $c_b$ not while executing $L$, but while executing a locomotory move. Moreover, if we well order $\mathbb{Z}$ by $j<1+j<-1+j<2+j<-2+j< \hdots$, then the robot passes for the first time from the column $c_{b+1}$ to the column $c_b$ through the smallest HE with respect to this order and so it gets to the point $\textbf{v}$, which is a special point. Immediately afterwards, it follows $L$ and it reaches the destination point (see Figure~\ref{4}). The conclusion follows.
\end{proof}
\noindent
\begin{figure}[h!]
\centering
\resizebox{0.75\textwidth}{!}{
\begin{tikzpicture}
[
dot/.style={circle,draw=black, fill,inner sep=2pt},
cross/.style={cross out, draw=black, minimum size=2*(#1-\pgflinewidth), inner sep=4pt, outer sep=4pt},
]
\foreach \x in {-6,...,6}{
\foreach \y in {-6,...,6}{
\node[dot] at (\x,\y){ };
}}
\draw[line width=0.7mm, red] (-6, -6) -- (-6, 6);
\draw[line width=0.7mm, red] (-5, -6) -- (-5, 6);
\draw[line width=0.7mm, red] (-4, -6) -- (-4, 6);
\draw[line width=0.7mm, red] (-3, -6) -- (-3, 6);
\draw[line width=0.7mm, red] (-2, 6) -- (-2, 2);
\draw[line width=0.7mm, red] (-2, 0) -- (-2, -6);
\draw[line width=0.7mm, red] (-1, 6) -- (-1, 3);
\draw[line width=0.7mm, red] (-1, 1) -- (-1, -3);
\draw[line width=0.7mm, red] (-1, -4) -- (-1, -6);
\draw[line width=0.7mm, red] (0, 6) -- (0, 0);
\draw[line width=0.7mm, red] (0, -2) -- (0, -4);
\draw[line width=0.7mm, red] (0, -5) -- (0, -6);
\draw[line width=0.7mm, red] (1, 6) -- (1, 5);
\draw[line width=0.7mm, red] (1, 4) -- (1, 2);
\draw[line width=0.7mm, red] (1, 1) -- (1, -6);
\draw[line width=0.7mm, red] (2, 6) -- (2, -6);
\draw[line width=0.7mm, red] (3, 6) -- (3, -6);
\draw[line width=0.7mm, red] (4, 6) -- (4, -6);
\draw[line width=0.7mm, red] (5, 6) -- (5, -6);
\draw[line width=0.7mm, red] (6, 6) -- (6, -6);
\draw[line width=1mm, green] (5, -4) -- (5, -3);
\draw[line width=1mm, green] (4, -4) -- (4, -3);
\draw[line width=1mm, green] (4, -2) -- (4, -6);
\draw[line width=1mm, green] (3, -3) -- (3, -6);
\draw[line width=1mm, green] (2, -6) -- (2, -2);
\draw[line width=1mm, green] (1, -2) -- (1, -4);
\draw[line width=0.7mm, red] (2, -6) -- (5, -6);
\draw[line width=0.7mm, red] (-1, -5) -- (0, -5);
\draw[line width=0.7mm, red] (2, -5) -- (3, -5);
\draw[line width=0.7mm, red] (5, -5) -- (6, -5);
\draw[line width=0.7mm, red] (5, -4) -- (6, -4);
\draw[line width=0.7mm, red] (-5, -3) -- (-2, -3);
\draw[line width=0.7mm, red] (-1, -3) -- (0, -3);
\draw[line width=0.7mm, red] (4, -3) -- (6, -3);
\draw[line width=0.7mm, red] (-2, -2) -- (-1, -2);
\draw[line width=0.7mm, red] (1, -2) -- (2, -2);
\draw[line width=0.7mm, red] (2, -2) -- (3, -2);
\draw[line width=0.7mm, red] (-2, -1) -- (-1, -1);
\draw[line width=0.7mm, red] (2, -1) -- (3, -1);
\draw[line width=0.7mm, red] (-1, 0) -- (0, 0);
\draw[line width=0.7mm, red] (-1, 0) -- (0, 0);
\draw[line width=0.7mm, red] (1, 0) -- (2, 0);
\draw[line width=0.7mm, red] (2, 1) -- (3, 1);
\draw[line width=0.7mm, red] (5, 1) -- (6, 1);
\draw[line width=0.7mm, red] (-4, 3) -- (-3, 3);
\draw[line width=0.7mm, red] (0, 3) -- (2, 3);
\draw[line width=0.7mm, red] (-1, 4) -- (0, 4);
\draw[line width=0.7mm, red] (1, 4) -- (2, 4);
\draw[line width=0.7mm, red] (3, 4) -- (5, 4);
\draw[line width=0.7mm, red] (-6, 5) -- (-4, 5);
\draw[line width=0.7mm, red] (-1, 5) -- (0, 5);
\draw[line width=0.7mm, red] (2, 5) -- (6, 5);
\draw[line width=0.7mm, red] (-3, 6) -- (-2, 6);
\draw[line width=1mm, green] (6, -4) -- (5, -4);
\draw[line width=1mm, green] (4, -3) -- (5, -3);
\draw[line width=1mm, green] (4, -6) -- (3, -6);
\draw[line width=1mm, green] (3, -5) -- (2, -5);
\draw[line width=1mm, green] (2, -2) -- (1, -2);
\draw[line width=1mm, blue] (6, 3) -- (6, -4);
\draw node[fill,circle,inner sep=5.5pt,minimum size=2pt] at (6, 3) {};
\draw node[fill,circle,inner sep=5.5pt,minimum size=2pt] at (6, -4) {};
\draw node[fill,circle,inner sep=5.5pt,minimum size=2pt] at (1, -4) {};
\draw (0,-4) node[cross] {};
\foreach \x in {-6,...,6}
\draw (\x,.1) -- node[below,yshift=-1mm] {\x} (\x,-.1);
\foreach \y in {-6,...,6}
\draw (.1,\y) -- node[below,xshift=-3mm, yshift=3mm] {\y} (-.1,\y);
\draw[->,line width=0.15mm] (0,-6.5) -- (0,6.5);
\draw[->,line width=0.15mm] (-6.5,0) -- (6.5,0);
\end{tikzpicture}
}
\caption{\textbf{Part II, Case (2).} There is a pass $\pi$, which for this example is $(1, 0), (2, 0)$ between a lower infinite column and the east strip. We assume that there are no VEs removed other than the ones shown in the figure and that the position of the robot after following $X$ is $(6, 3)$. The first segment $S^{\lambda+p}$ of $RLW$ takes the robot at a latitude lower than that of the pass $\pi$. For our example, we may assume that after the robot follows $X S^{\lambda+p}$ it gets to the point $(6, -4)$, though this latitude should be much smaller. While the robot is in the east strip, after it executes $K = N^{2\lambda+4p} S^{2\lambda+4p}$, it returns to the starting point. By the choice of parameters, the robot enters the easternmost lower infinite column at longitude $b$ for the first time via a locomotory move (in our case, $b=1$). Ignoring, as we may, the action of $K$ in the east strip, the path of the robot to the column $c_b$ is coloured in green. Immediately after the robot enters the column $c_b$, it executes $K$ which sets it latitude so small that the parameters of $SMW$ are not large enough to make the robot visit any other configurations in the obstacle strip other than the lower infinite columns.} \label{5}
\end{figure}
\textbf{Case (2).} We assume without loss of generality that the mazes in $F_i$ contain a pass $\pi$ between the easternmost lower infinite column and the east strip. In this case, we define the algorithm
\[
RLW:= S^{\lambda+p} \text{ } SMW(2\lambda+2p,\lambda+2p,K) \text{ } N^{2 \lambda+6p+l} \text{ } L_W \text{ } S^{2p+l-k},
\]
where $K = N^{2\lambda+4p} S^{2\lambda+4p}$ and $k$ is the latitude of the lowermost special vertex.
\begin{prop}\label{pp192}
For any maze in $F_i$, after the robot follows the algorithm $X \text{ }RLW$, it is either in the west strip on the $x$-axis or it has visited the destination point.
\end{prop}
\begin{proof}
After the robot follows $X \text{ } S^{\lambda+p}$ it is in the east strip at a certain latitude say $j$, smaller than the latitude of the pass $\pi$. By the choice of parameters and by Lemma~\ref{l2}, while the robot follows $SMW(2\lambda+2p,\lambda+2p,K)$ it advances westwards in the east strip oscillating about the row $r_j$. It passes for the first time from the east strip to the obstacle strip not while executing $K$. Moreover, if we well order $\mathbb{Z}$ by $j<1+j<-1+j<2+j<\hdots,$ then the robot passes from the east strip to the easternmost lower infinite column in the obstacle strip through the smallest HE with respect to this order. Immediately afterwards, it follows $K$ and gets at latitude $2\lambda+4p$ below the easternmost lowermost special vertex. By the choice of parameters the robot advances westwards only through lower infinite columns while in the obstacle strip. Therefore, after the robot follows $X \text{ } S^{\lambda+p} \text{ } SMW(2\lambda+2p,\lambda+2p,K)$, it is either (1) in the west strip at latitude $2\lambda+4p$ below the lowermost special vertex, i.e. at latitude $k-2\lambda-4p$ or (2) in the obstacle strip in a lower infinite column $c_m$ at latitude $2\lambda+4p$ below some special vertex (see Figure~\ref{5}).
In case (1), while the robot follows $N^{2 \lambda+6p+l} \text{ } L_W$ its latitude is too large for it to hit any VNE and after it follows $N^{2 \lambda+6p+l} \text{ } L_W$ its longitude does not increase, so it remains in the west strip. Hence, after it follows $X \text{ } RLW$, the robot is in the west strip on the $x$-axis.
In case (2), after the robot follows $N^{2 \lambda+6p+l}$ it gets to a special point, more specifically to the uppermost vertex of the lower infinite column $c_m$. Immediately afterwards, it follows $L_W$ and it reaches the destination point. The conclusion follows.
\end{proof}
In \textbf{Part II} we see an example on how we divide all the sets of mazes $F_i$ in two classes in such a way that our algorithm $RLW$ depends qualitatively only on the class. This is why we treat each class in a separate case. In \textbf{Part III} the principle is the same, but we need to consider many more cases and write a different algorithm for each one of them.
At the end of \textbf{Part II}, we note that although we used in this part the fact there are no infinite columns in the obstacle strip, a variation of $RLW$ can be used to position the robot in the west strip on the $x$-axis, even if we drop this assumption. This note is important, because it shows that \textbf{Part II} can also be generalised to improve Theorem~\ref{ch2th2} by dropping the consecutive column condition for the finite number of VNEs. To present this variation, assume that infinite columns are allowed in the obstacle strip, i.e. the (finitely many) VNEs need not be in consecutive columns.
We begin with the remark that \textbf{Case (1)} considered above can be treated in the exact same way with or without infinite columns in the obstacle strip, so we may assume without loss of generality that \textbf{Case (2)} holds, i.e. that every maze in the class of mazes we consider contain a pass $\pi$ between a lower infinite column and the east strip. We recall that by the $\overline{()}$ transformation we apply on mazes, there are always passes between any two consecutive infinite columns in the obstacle strip. We now need to consider $2$ cases: (i) there exist passes between all consecutive lower infinite columns and between consecutive lower infinite columns and infinite columns; this case can be treated similarly with \textbf{Case (2)} above; (ii) there exist two entities, one of which is a lower infinite column and the other is either a lower infinite column or an infinite column with no pass between them. In this case we define the algorithm $RLW'$ which generalises $RLW$ as described above,
\[
RLW' := S^{\lambda+p} \text{ } OMW(2\lambda+2p, \lambda+p, 2\lambda+4p) \text{ } N^{2 \lambda+4p} \text{ } L.
\]
The reason why $RLW'$ indeed generalises $RLW$ in this case is that after the robot follows $X \text{ } S^{\lambda+p} \text{ } OMW(2\lambda+2p, \lambda+p, 2\lambda+4p)$, it remains trapped in the lower infinite column or infinite column in the obstacle strip with largest longitude $m$ which is not connected with the lower infinite column or infinite column at longitude $m-1$. The robot's latitude is $2\lambda+4p$ below the lowermost VNE at longitude between $m$ and $b$. Hence from this starting position, after the robot follows $N^{2 \lambda+4p}$ it gets to a special vertex (by definition) and therefore, $X \text{ } RLW'$ takes the robot to the destination point.
Moving on from this digression, by \textbf{Case (1)} and \textbf{Case (2)}, we may assume that we are given $F_i$ and a finite algorithm $X$ with $\lambda=|X|$ such that after the robot follows $X$ it is either in the west strip on the $x$-axis or it has visited the destination point. Without loss of generality, we assume that the robot is in the west strip on the $x$-axis and our aim is to build a finite algorithm $F$ such that $X F$ solves $F_i$.
\subsection*{Part III} The algorithm $finish$ defined in this part aims to make the robot visit the destination point. \\
\textbf{Case (1). } We assume that the destination point is in an infinite column in the west strip. We define the algorithm:
\[
F = MW(p, 2p) \text{ } OME (\lambda+\mu, \lambda+\mu, p),
\]
where $\mu = |MW(p, 2p)|$.
\textbf{Claim. } For any such maze in $F_i$, after the robot follows $X \text{ } F$, it visits the destination point.
\begin{proof}
After the robot follows $X \text{ } MW(p, 2p)$ it is in the west strip, to the west of the origin or it has already visited the destination point. By the choice of parameters and by the consequence of Lemma~\ref{l2} when applied to the particular algorithm $OME$, after it follows $X \text{ } F$, the robot visits the destination point.
\end{proof}
At the end of \textbf{Case (1)} we note that we note that although we used the fact there are no infinite columns in the obstacle strip, a variation of $F$ can be used in order to make the robot visit the destination point, even if we drop this assumption. To present this variation, we assume that infinite columns are allowed in the obstacle strip, i.e. the (finitely many) VNEs need not be in consecutive columns. Let us assume for now that the destination point is in an infinite column in the east strip or obstacle strip.
In this case, given any finite algorithm $A$ we will construct a finite algorithm $U(A)$ with the following $2$ properties: if the robot starts in the origin of any maze in $F_i$, it follows $A$ and it gets to the west of the destination point then (1) after the robot follows $U(A)$, either its latitude strictly increases or the robot remains stuck in a finite column, or upper/lower infinite column at some longitude $i$ with no HE connecting that column to points at longitude $i+1$; (2) as the robot follows $U(A)$, if the robot visits the infinite column which contains the destination point, then the robot visits the destination point. We will construct our algorithm $U$ from bricks of the form
\[
B(k, A) = N^{|A|+2p} S^{2|A|+4p} N^{|A|+2p} N^k E S^k
\]
where $k$ is an integer and $A$ is a finite algorithm. Every time we insert a brick $B(k, A)$ as a subalgorithm of $X \text{ } F$, we take $A$ to be the entire algorithm written until that instance of $B(k, A)$. Hence, every brick depends on the length of the algorithm written up to it in $X \text{ } F$. With this convention, from now on we shall drop the second argument from the definition of a brick and let $B(k) = B(k, A)$. We note in advance that the aim of the first segment $N^{|A|+2p} S^{2|A|+4p} N^{|A|+2p}$ of a brick $B(k, A)$ is the following: for any maze in $F_i$, if the robot is in the same column as the destination point after it follows a finite algorithm $A$, if the robot then follows $N^{|A|+2p} S^{2|A|+4p} N^{|A|+2p}$, it visits the destination point. Hence we regard the first segment $N^{|A|+2p} S^{2|A|+4p} N^{|A|+2p}$ of a brick just as an oscillation large enough to make the robot visit the destination point after it reaches the right longitude.
We note that for any $k$, if the robot follows $B(k)$ starting in any point of any maze, its longitude does not decrease. We define four types of steps by concatenating bricks, so each of the steps also have this property.
The first step $U_1$ is designed to have the following property: if the robot starts in the origin of any maze in $F_i$, it follows a finite algorithm $A$ and it gets to an infinite column strictly at the west of the destination point, if the robot then follows $U_1$, its longitude strictly increases. For instance we can take
\[
U_1 (A) := B(-|A|-p) \text{ } B(-|A|-p+1) \hdots B(|A|+p),
\]
where $A$ is always taken to be the entire algorithm written before the occurrence of this step. With this convention, we drop the argument $A$ and $U_1$ has the desired property (cf. steps $3$ and $4$ below). We also note that formally, the first brick in $U_1(A)$ is $B(-|A|-p, A)$, the second brick is $B(-|A|-p+1, A \text{ } B(-|A|-p, A))$, etc.
The second step $U_2$ is designed to have the following property: if the robot starts in the origin of any maze in $F_i$, it follows a finite algorithm $A$ and it gets to a finite column at longitude $i$ which is connected to any point at longitude $i+1$ by a HE (i.e. no HEs emerging in the east part of the finite column), if the robot then follows $U_2$, its longitude strictly increases. For instance we can take
\[
U_2 (A) := B(0) \text{ } B(1) \hdots B(2p),
\]
where the definition of $U_2(A)$ does not depend on $A$, as every finite column has at most $2p$ vertices. Therefore, let $U_2 = U_2 (A)$ have the desired property (cf. steps $3$ and $4$ below).
The third and forth step $U_3$ and $U_4$ are designed to have the following property: if the robot starts in the origin of any maze in $F_i$, it follows a finite algorithm $A$ and it gets to an upper infinite or lower infinite column at longitude $i$ which is connected to any point at longitude $i+1$ by a HE, if the robot then follows $U_3$ or $U_4$, respectively, its longitude strictly increases. For instance we can take both $U_3$ and $U_4$ to be a concatenation of $2p$ bricks in the following way
\[
U_3 (A) = B(-|A|-2p)\text{ } B(-|A|-|B(-|A|-2p)|-2p+1) \text{ }
\]
\[
B(-|A|-|B(-|A|-2p)|-|B(-|A|-|B(-|A|-2p)|-2p+1)|-2p+2) \hdots ,
\]
\[
U_4 (A) = B(|A|+2p)\text{ } B(|A|+|B(|A|+2p)|+2p-1) \text{ }
\]
\[
B(|A|+|B(|A|+2p)|+|B(|A|+|B(|A|+2p)|+2p-1)|+2p-2) \hdots ,
\]
where $A$ is always taken to be the entire algorithm written before the occurrence of this step. With this convention, we drop the argument $A$ and $U_3$, $U_4$ have the desired property. Indeed, let's assume that the robot is in an upper infinite column $c = (i, y), (i, y+1), \hdots$ at longitude $i$ which is connected to any point at longitude $i+1$ by a HE, and it follows $U_3$. Let $j$ be the smallest non-negative integer such that the vertices $(i, y+j)$ and $(i+1, y+j)$ are connected by a HE. From the definition of passes and the primary rectangle, we first note that $-p \leq y+j \leq p$ and $j \leq 2p$. As the robot follows the first brick $B(-|A|-2p) = N^{|A|+2p} S^{2|A|+4p} N^{|A|+2p} S^{|A|+2p} E N^{|A|+2p}$ in $U_3$, it oscillates in $c$, executing an $E$ instruction at the vertex $(i, y)$ in $c$. If $j=0$ we are done; otherwise, after the robot follows $B(-|A|-2p)$, it gets at the vertex $(i, y+|A|+2p)$. Therefore, we can track the position of the robot as it follows the second brick $B(-|A|-|B(-|A|-2p)|-2p+1)$ in $U_3$, and we observe that it oscillates in $c$, executing an $E$ instruction at the vertex $(i, y+1)$ in $c$. We continue in the same way; as $-p \leq y+j \leq p$, $j \leq 2p$, we are done.
Let us make one more remark regarding these steps. If the robot follows a concatenation of bricks and it reaches a finite column, an upper infinite column or a lower infinite column at longitude $i$ with no HE connecting it to points at longitude $i+1$, the robot remains stuck in that structure while it follows the rest of the algorithm. Let us define
\[
U (A) = U_1 U_2 U_3 U_4,
\]
or formally $U(A) = U_1(A) \text{ } U_2(A \text{ } U_1 (A)) \text{ } U_3 (A \text{ } U_1(A) \text{ } U_2(A \text{ } U_1 (A))) \text{ } U_4 (\hdots)$. As usual, every time we use the algorithm $U(A)$ as a subalgorithm, we take $A$ to be the entire algorithm written before the occurrence of $U(A)$, so with this convention we drop the argument of $U$. Therefore, it is clear that the algorithm $U$ has the two promised properties at the beginning of the case: if the robot starts in the origin of any maze in $F_i$, it follows a finite algorithm $A$ and it gets to the west of the destination point then (1) after the robot follows $U$, either its latitude strictly increases or the robot remains stuck in a finite column, or upper/lower infinite column at some longitude $i$ with no HE connecting that column to points at longitude $i+1$; (2) as the robot follows $U$, if the robot visits the infinite column which contains the destination point, then the robot visits the destination point. Furthermore, let $V(A) = \underbrace{UU \hdots U}_{\lambda+p},$ or formally
\[
V(A) = \underbrace{U(A) \text{ } U(A \text{ } U(A)) \text{ } U(A \text{ } U(A) \text{ } U(A \text{ } U(A))) \hdots}_{\lambda+p \text{ terms}}.
\]
We finally define the algorithm
\[
F = V(X) \text{ } N^{|V(X)|+l+p} \text{ } L \text{ } S^{2|V(X)|+l+2p} L.
\]
Let us see that indeed, after the robot follows $X \text{ } F$, it visits the destination point. We may assume without loss of generality that after the robot follows $X$ it is in the west strip on the $X$ axis, clearly to the west of the destination point (which is assumed to be in the east strip or in the obstacle strip). From property (1) of $U$, after the robot follows $X \text{ } V(X)$, it either visits the destination point or it remains stuck in a finite, lower or upper infinite column at longitude $i$, at the west of the destination point, with no HE connecting it to points at longitude $i+1$. In the first two cases, it is clear that after the robot follows $X \text V(X) \text{ } N^{|V(X)|+l+p} \text{ } L$ it visits the destination point. In the third case, the robot is stuck in an upper infinite column $c$ after it follows $X \text V(X) \text{ } N^{|V(X)|+l+p}$. We claim that the robot returns to the same vertex in $c$ after it follows $X \text V(X) \text{ } N^{|V(X)|+l+p} \text{ } L$. Indeed, as the robot follows $L$ its latitude is too large to meet any VNE, so by the construction of $L$ the latitude of the robot does not change after it follows $L$. Moreover, the fact that $c$ has no HE connecting it to points at longitude $i+1$ makes the robot return at longitude $i$ after it follows $L$. After that, the robot follows $S^{2|V(X)|+l+2p}$ and it gets to a special point, and then it follows $L$ which further takes it to the destination point.
Moving on from this digression, we have solved \textbf{Case (1)} in which the destination point is in an infinite column in the west strip. By the symmetry of this case and \textbf{Part II}, we similarly solve the case when the destination point is in an infinite column in the east strip. Likewise, the generalisation of \textbf{Case (1)} proved at the end of the section generalises to the case when the destination point is in an infinite column in the west strip. In fact, in the generalisation of \textbf{Case (1)} we could have only considered the case when the destination point is in an infinite column in the obstacle strip, as \textbf{Case (1)} itself works just as well even in the generalised set up, when the destination point is either in the west strip or east strip. That would not have simplified the argument, though.
\noindent
\textbf{Case (2). } We assume that the destination point is in the obstacle strip in a finite column, upper infinite column or lower infinite column and it is connected to the west strip via a path through a (finite) sequence of finite columns. Let $R_1, R_2, \hdots , R_k$ be a sequence of finite columns and $R$ be a finite column, upper infinite column or lower infinite column such that $R$ contains the destination point and there exists a HE between the west strip and $R_1$, between $R_m$ and $R_{m+1}$ for $1 \leq m \leq k$, where by convention $R_{k+1}=R$. Let $\textbf{w} = (a+1, u)$ be the uppermost point of the finite column $R_1$.
We consider the following sub-cases: \\
\begin{figure}[h!]
\centering
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\caption{\textbf{Part III, Case (2)(i).} We assume that there exists a row $r_i$ that intersects the finite column $R_1$ and that $r_i$ has a west bump. We assume that there are no VEs removed other than the ones shown in the figure. In this example we further take $R_1 = (1, 1), (1, 2), (1, 3)$, $R_2 = (2, 0), (2, 1), (2, 2)$, $R_3 = (3, 1), (3, 2)$, $R_4 = (4, -3), ... (4, 1)$, $R_5=R=(3, -1), (3, -2), (3, -3)$ and $r_i = r_2$ is the row that intersects $R_1$ with its west bump $(-3, 2), (-2, 2)$ and $\textbf{v}=(-2, 2)$. In general, we do not require $R$ to be a finite column. Let $H' = EEEENN^{-1}EEN^3N^{-3}E^{-1}N^2N^{-2}E$ and note that if the robot starts at $\textbf{v}$ and follows $H'$ it follows the green path and gets to the destination point. However, if the robot starts at the west of $\textbf{v}$ on $r_i$ and it follows $H'$, its longitude is always strictly smaller than that of $\textbf{v}$. Therefore, it does not hit any VNE and as $|H'|_N=|H'|_S$ its latitude does not change. In our case, $H = H' E^{20}$ which has the extra property that after the robot follows $H$ in a maze with no VNEs, its longitude does not decrease - in fact, it can be proven that there always exits a certain $H'$ that has this property itself, but we do not wish to complicate the argument.} \label{6}
\end{figure}
\textbf{2(i)} We assume that there exists a row $r_i$ that intersects the finite column $R_1$ and that it has a west bump. We recall that the west bumps are the easternmost HNEs with at least one vertex in the west strip on a row that intersects some finite column. Assume first that the eastern vertex $\textbf{v}$ of that west bump is in the west strip. By inspecting the longitude of the west bump and the primary rectangle we can construct an algorithm of the form $H' := \prod_{m=1}^h N^{k_m} N^{-k_m} E^{\varepsilon_m}$, where $\varepsilon_m \in \{ -1,1 \}$ and $k_m$ is an integer for all $1 \leq m \leq h$, such that if the robot starts at $v$ and follows $H'$, it visits the destination point. Indeed if the robot is at some specified latitude in the finite column $R_m$ and follows $N^{k_m} N^{-k_m}E$ for suitable $k_m$, it gets to some specified latitude in the finite column $R_{m+1}$. Let $H = H' E^{|H'|}$. We define the algorithm
\[
F = N^i \text{ } W^q \text{ } SME (\lambda+q, \lambda+q, H).
\]
\begin{prop}\label{plm77}
For any maze in $F_i$, after the robot follows $X \text{ } F$, it visits the destination point.
\end{prop}
\begin{proof}
We may assume without loss of generality that after the robot follows $X \text{ } N^i$, it is on the row $r_i$. Hence after the robot follows $X \text{ } N^i \text{ } W^q$ it is on the row $r_i$ at a longitude at most that of $\textbf{v}$. By the choice of parameters and by Lemma~\ref{l2}, while the robot follows $SME (\lambda+q, \lambda+q, H)$ it advances eastwards in the west strip oscillating about row $r_i$ and passing through the smallest HE with respect to the well order on $\mathbb{Z}$: $i<1+i<-1+i<2+i<\hdots$. Considering that $|H|_N=|H|_S$, while the robot is in the west strip, after it follows $H$ its latitude does not change and its longitude does not decrease. It eventually arrives at the point $\textbf{v}$ on $r_i$ not while executing $H$ (from the form of $H$ and the shape of the maze which has a HNE with its eastern vertex at $v$). Immediately after the robot reaches $\textbf{v}$, it follows $H$ and it gets to the destination point (see Figure~\ref{6}).
\end{proof}
In the case that there exists a west bump positioned at the border between the obstacle strip and the west strip on a row $r_i$ that intersects $R_1$, we consider $r_j$ to be a row on which there exists a HE between the west strip and $R_1$. As before, let $\textbf{v}$ be the eastern vertex of the west bump, $\textbf{v} \in \mathbb{R}_1$. We recall the algorithm
\[
SME^{(j-i)} (a, e, H):= (((((((H)^e N^{j-i} E S^{j-i})^e E)^e NES)^e SEN)^e NNESS)^e \hdots S^aEN^a)^e
\]
introduced in Corollary~\ref{c1}. We further define the algorithm
\[
F = N^i \text{ } W^q \text{ } SME^{(j-i)} (\lambda+q, \lambda+q, H).
\]
\textbf{Claim. } For any maze in $F_i$ after the robot follows $X \text{ } F$, it visits the destination point.
\begin{proof}
The conclusion follows by the same reasoning as in the proof of Proposition~\ref{plm77} and by Corollary~\ref{c1}.
\end{proof}
\begin{figure}[h!]
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\end{tikzpicture}
}
\caption{\textbf{Part III, Case (2)(ii).} Every row that intersects $R_1$ does not have a west bump and there exists a special west pipe on some row $r_j$. We assume that there are no VEs removed other than the ones shown in the figure. We have $r_j = r_4$ and so the special west pipe is $(-2, 4), (-1, 4), (0, 4)$ with $\textbf{v} = (-1, 4)$ and $\textbf{w} = (1, 3)$. Let us observe that after the robot follows $WPF(1, 1000)$ starting at $(-6, 4)$ it gets to $\textbf{v'} = (-4, 4)$ which is the middle vertex of the ``fake'' west pipe $(-5, 4), (-4, 4), (-3, 4)$. Further note that after the robot follows $WPF(1, 1000)$ starting at $(-3, 4)$ it gets to $\textbf{v}$. For this example we have $K=SE^2NWSW^2NE$ and after the robot follows $K$ starting from $\textbf{v}$ it gets to $\textbf{z} = (-1, 3)$, which is indeed on $r_u=r_3$ (see the blue walk). In addition, note that if the robot starts from $\textbf{v'}$ and follows $K$ it gets back to $\textbf{v'}$ (see the green circuit). We can take $H' = EENSENSEEN^3S^3E^{-1}N^2S^2E$ which has the required form and the property that after the robot starts from $\textbf{z} = (-1, 3)$ and follows $H'$ it visits the destination point. The reader may assume that the robot starts at $(-4, 0)$ and it follows $F = N^3 W^2 N (WPF(1, 1000) \text{ } K \text{ } H \text{ } SEN)^{10}$ to see how the algorithm $F$ solves the maze: after it follows $N^3 W^2 N$, the robot gets to $(-6, 4)$; further, after the first iteration of $WPF(1, 1000) \text{ } K \text{ } H \text{ } SEN$, it gets to $(-3, 4)$ as $K$ and $H$ do not change the position of the robot while it is strictly at the west of $\textbf{v}$; after the second iteration of $WPF(1, 1000) \text{ } K \text{ } H \text{ } SEN$, the robot visits the destination point.} \label{7}
\end{figure}
\noindent
\textbf{2(ii)} We assume that the previous case does not hold, so every row that intersects the column $R_1$ does not have a west bump, i.e. each such row is a path in the west strip. In addition, we assume there exists a special west pipe on some row $r_j$. We recall that the west pipes are easternmost configurations in the west strip formed by a HE followed by a HNE. Denote by $\textbf{v}$ the easternmost vertex of the HE of the special west pipe. Assume without loss of generality that $j>u$, where $\textbf{w} = (a+1, u)$ is the uppermost point of the finite column $R_1$.
We start by defining a new algorithm called $west \_ pipe \_ finder$:
\[
WPF(a,e):=(E^eWS^aEN^a)^e,
\]
with its counterpart $east \_ pipe \_ finder$. This is used directly in the final algorithm $F$ and it will be analysed later (see Figure~\ref{7}).
We then define the algorithm $$K=S^{j-u}E^dN^{j-u}WS^{j-u}W^dN^{j-u}E,$$ where $d$ is the difference in longitude between $c_{a+1}$ and $\textbf{v}$.
\textbf{Claim. } For any maze in $F_i$, if the robot starts at $\textbf{v}$ and follows $K$ it gets at a certain known point $\textbf{z}$ (given the tertiary rectangle) on the row $r_u$.
\begin{proof}
Starting at $\textbf{v}$, after the robot follows $S^{j-u}$ it gets on the row $r_u$; after it follows $S^{j-u}E^d$ it gets to $\textbf{w}$; after it follows $S^{j-u}E^dN^{j-u}$ it remains fixed at $\textbf{w}$; after it follows $S^{j-u}E^dN^{j-u}W$ it gets to $(a, u)$, on the row $r_u$ to the west of $\textbf{w}$; finally, while it executes $S^{j-u}W^dN^{j-u}E$ starting at $(a, u)$ it does not leave the square $\{ (\pm q, \: \pm q) \}$; while it executes both the subalgorithms $S^{j-u}$ and $N^{j-u}$ of $S^{j-u}W^dN^{j-u}E$ it does not hit any VNE (see Figure~\ref{7}). The conclusion follows.
\end{proof}
\noindent
\textbf{Remark. } It is easy to check that if the robot starts from the easternmost vertex $\textbf{v'}$ of a HE followed by a HNE on $r_j$ with $\textbf{v'}$ strictly at the west of $\textbf{v}$ and it follows $K$, then the robot remains in the west strip while following $K$ and after it follows $K$, it returns back to the starting point $\textbf{v'}$ (see Figure~\ref{7}). The algorithm $K$ was constructed specifically to have this property, together with the one proved in the Claim above.
By inspecting the tertiary rectangle, we construct an algorithm $H'$ of the form $H'=\prod_{i=1}^hN^{k_i}N^{-k_i}E^{\epsilon_i}$, where $\epsilon_i \in \{-1,1\}$ and $k_i$ is an integer for all $1 \le i \le h$, such that if the robot starts at $\textbf{z}$ and it follows $H'$ it visits the destination point. Let $H=H'E^{|H'|}$. We observe that if the robot starts from the easternmost vertex $\textbf{v'}$ of a HE followed by a HNE on the row $r_j$ in the west strip and follows $H$, it remains in the west strip and it returns to the same point $\textbf{v'}$. We finally define the algorithm:
\[
F = N^u \text{ } W^q \text{ } N^{j-u} \text{ } (WPF(j-u,\lambda+q) \text{ } K \text{ } H \text{ } S^{j-u} \text{ } E \text{ } N^{j-u})^{\lambda+q}.
\]
\begin{prop}
For any maze in $F_i$, after the robot follows $X \text{ } F$, it visits the destination point.
\end{prop}
\begin{proof}
We may assume without loss of generality that after the robot follows $X \text{ } N^u \text{ } W^q \text{ } N^{j-u}$ it gets on the row $r_j$, to the west of the point $\textbf{v}$. While the robot is at the west of the point $\textbf{v}$ on the row $r_j$, after each instance of $WPF (j-u,\lambda+q)$ it advances eastwards to the easternmost vertex $\textbf{v'}$ of a HE followed by a HNE on the row $r_j$. While $\textbf{v'}$ is strictly at the west of $\textbf{v}$, the robot follows the algorithm $K \text{ } H$ and returns back to $\textbf{v'}$; while the robot follows the algorithm $S^{j-u}EN^{j-u}$ it advances one unit to the east of $\textbf{v'}$ on the row $r_j$. By the choice of parameters, the robot eventually arrives at $\textbf{v'}=\textbf{v}$. Immediately afterwards, it follows $K \text{ } H$ and it visits the destination point (see Figure~\ref{7}).
\end{proof}
\noindent
\textbf{2(iii)} We assume there exists a magical west row $r_j$. We recall that a magical west row is a row which is a path when restricted to the west strip, and it contains a HNE; its west cutoff is its westernmost HNE. Denote by $\textbf{v}$ the westernmost vertex of the west cutoff of $r_j$. Then, by inspecting the tertiary rectangle, we can construct an algorithm $K$ such that if the robot starts from $\textbf{v}$ and follows $K$ it gets to the destination point. We define the algorithm
\[
F = N^jE^{\lambda+q}K.
\]
\textbf{Claim. } For any maze in $F_i$, after the robot follows $X \text{ } F$, it visits the destination point.
\begin{proof}
We may assume without loss of generality that after the robot follows $X \text{ } N^j$, it gets on the row $r_j$. Therefore, after it follows $X \text{ } N^j \text{ } E^{\lambda+q}$ the robot gets to the point $\textbf{v}$. Hence, after the robot follows $X \textbf{ } F$ it gets to the destination point.
\end{proof}
\begin{figure}[h!]
\centering
\resizebox{0.75\textwidth}{!}{
\begin{tikzpicture}
[
dot/.style={circle,draw=black, fill,inner sep=2pt},
cross/.style={cross out, draw=black, minimum size=2*(#1-\pgflinewidth), inner sep=4pt, outer sep=4pt},
]
\foreach \x in {-6,...,6}{
\foreach \y in {-4,...,6}{
\node[dot] at (\x,\y){ };
}}
\draw[line width=0.7mm, red] (-6, -4) -- (-6, 6);
\draw[line width=0.7mm, red] (-5, -4) -- (-5, 6);
\draw[line width=0.7mm, red] (-4, -4) -- (-4, 6);
\draw[line width=0.7mm, red] (-3, -4) -- (-3, 6);
\draw[line width=0.7mm, red] (-2, -4) -- (-2, 6);
\draw[line width=0.7mm, red] (-1, -4) -- (-1, 6);
\draw[line width=0.7mm, red] (0, -4) -- (0, 6);
\draw[line width=0.7mm, red] (1, 6) -- (1, 4);
\draw[line width=0.7mm, red] (1, 3) -- (1, 1);
\draw[line width=0.7mm, red] (1, 0) -- (1, -2);
\draw[line width=0.7mm, red] (1, -3) -- (1, -4);
\draw[line width=0.7mm, red] (2, 6) -- (2, 3);
\draw[line width=0.7mm, red] (2, 2) -- (2, 0);
\draw[line width=0.7mm, red] (2, -2) -- (2, -4);
\draw[line width=0.7mm, red] (3, 6) -- (3, 3);
\draw[line width=0.7mm, red] (3, 2) -- (3, 1);
\draw[line width=0.7mm, red] (3, -1) -- (3, -3);
\draw[line width=0.7mm, red] (3, -4) -- (3, -4);
\draw[line width=0.7mm, red] (4, 6) -- (4, 4);
\draw[line width=0.7mm, red] (4, 3) -- (4, 2);
\draw[line width=0.7mm, red] (4, 1) -- (4, -3);
\draw[line width=0.7mm, red] (4, -4) -- (4, -4);
\draw[line width=0.7mm, red] (5, -4) -- (5, 6);
\draw[line width=0.7mm, red] (6, -4) -- (6, 6);
\draw[line width=0.7mm, red] (-5, 5) -- (-2, 5);
\draw[line width=0.7mm, red] (1, 5) -- (3, 5);
\draw[line width=0.7mm, red] (5, 5) -- (6, 5);
\draw[line width=0.7mm, red] (-6, 4) -- (-4, 4);
\draw[line width=0.7mm, red] (-2, 4) -- (-1, 4);
\draw[line width=0.7mm, red] (3, 4) -- (5, 4);
\draw[line width=0.7mm, red] (-6, 3) -- (1, 3);
\draw[line width=0.7mm, red] (-6, 2) -- (2, 2);
\draw[line width=0.7mm, red] (-5, 2) -- (-3, 2);
\draw[line width=0.7mm, red] (2, 1) -- (4, 1);
\draw[line width=0.7mm, red] (-6, 1) -- (1, 1);
\draw[line width=0.7mm, red] (-3, -1) -- (2, -1);
\draw[line width=0.7mm, red] (-2, -2) -- (1, -2);
\draw[line width=0.7mm, red] (3, -2) -- (4, -2);
\draw[line width=0.7mm, red] (5, -2) -- (6, -2);
\draw[line width=0.7mm, red] (-1, -3) -- (1, -3);
\draw[line width=1mm, red] (-4, 4) -- (-4, 3);
\draw[line width=1mm, red] (-4, 3) -- (-2, 3);
\draw[line width=1mm, red] (-2, 3) -- (-2, 4);
\draw[line width=1mm, red] (-1, 4) -- (-1, 3);
\draw[line width=1mm, red] (-1, 3) -- (1, 3);
\draw[line width=1mm, red] (0, 3) -- (0, 2);
\draw[line width=1mm, red] (0, 2) -- (-2, 2);
\draw[line width=1mm, red] (-2, 2) -- (-2, 3);
\draw[line width=1mm, red] (-2, 3) -- (-1, 3);
\draw[line width=1mm, blue] (-5, -1) -- (-5, 3);
\draw[line width=1mm, blue] (-5, 3) -- (-4, 3);
\draw[line width=1mm, blue] (-4, -1) -- (-4, 3);
\draw[line width=1mm, green] (-3, -1) -- (-3, 3);
\draw[line width=1mm, green] (-3, 3) -- (-2, 3);
\draw[line width=1mm, green] (-3, -1) -- (-2, -1);
\draw[line width=1mm, green] (-2, -1) -- (-2, 3);
\draw node[fill,circle,inner sep=5.5pt,minimum size=2pt] at (-3, -1) {};
\draw (-2.7, -0.7) -- node {\textbf{v}} (-2.7, -0.7);
\draw (3,-3) node[cross] {};
\foreach \x in {-6,...,6}
\draw (\x,.1) -- node[below,yshift=-1mm] {\x} (\x,-.1);
\foreach \y in {-4,...,6}
\draw (.1,\y) -- node[below,xshift=-3mm, yshift=3mm] {\y} (-.1,\y);
\draw[->,line width=0.15mm] (0,-4.5) -- (0,6.5);
\draw[->,line width=0.15mm] (-6.5,0) -- (6.5,0);
\end{tikzpicture}
}
\caption{\textbf{Part III, Case (2)(iv).} Every row that intersects $R_1$ does not have a west bump, i.e. all such rows are paths in the west strip and there exists a special almost empty west row $r_j$. We assume that there are no VEs removed other than the ones shown in the figure. We have $R_1 = (1, 1), (1, 2), (1, 3)$, so $r_1$, $r_2$, $r_3$ are paths in the west strip, moreover $j = -1$, so $r_{-1}$ is the special almost empty west row. Its west cutoff is the HNE $\{ (-4, -1), \textbf{v} = (-3, -1) \}$. We construct an algorithm $K$ by inspecting the tertiary rectangle such that if the robot starts from $\textbf{v}$ and follows $K$, it gets to the destination point. For example we may take $K = N^3 E^5 S E^2 S^3 W S$. We may assume that the robot starts at $(-5, 0)$ and it follows $F = N^{-1}$ $W^{100}$ $(S^{-4} E N^{-4} W)^{100}$ $K$. After the robot follows $N^{-1}$ $W^{100}$, it gets to $(-5, -1)$ on the row $r_j=r_{-1}$ at a longitude not greater than that of $\textbf{v}$. Let us see what is the position of the robot after it follows one instance of $(S^{-4} E N^{-4} W)$, starting from $r_{-1}$: while it starts strictly at the west of $\textbf{v}$, its longitude increases by $1$ (see the blue path); if it starts at $\textbf{v}$, it comes back to $\textbf{v}$ (see the green path). The exponent of $(S^{-4} E N^{-4} W)$ is large enough for the robot to reach $\textbf{v}$ after it follows $(S^{-4} E N^{-4} W)^{100}$. After that, the robot follows $K$ and it visits the destination point.} \label{8}
\end{figure}
\noindent
\textbf{2(iv) } We assume that every row that intersects the finite column $R_1$ does not have a west bump and there exists a special almost empty west row $r_j$. We recall that a special almost empty west row is a row that in the west strip is the complement of an infinite path followed by a non-empty finite path; its west cutoff is its easternmost HNE in the west strip. We recall that $\textbf{w} = (a+1, u)$ is the uppermost point of $R_1$ and let $\textbf{v}$ be the easternmost vertex of the west cutoff of $r_j$. Then, by inspecting the tertiary rectangle, we can construct an algorithm $K$ such that if the robot starts from $\textbf{v}$ and follows $K$ it gets to the destination point. We define the algorithm
\[
F= N^j \text{ } W^{\lambda+q} \text{ } (S^{j-u}EN^{j-u}W)^{\lambda+q} \text{ } K.
\]
\textbf{Claim. } For any maze in $F_i$, after the robot follows $X \text{ } F$, it visits the destination point.
\begin{proof}
We may assume without loss of generality that after the robot follows $X \text{ } N^j \text{ } W^{\lambda+q}$ it gets on the row $r_j$ to the west of the point $\textbf{v}$. While the robot follows one instance of $S^{j-u}EN^{j-u}W$ it returns on the row $r_j$ and advances one unit eastwards if it is at the westernmost vertex of a HNE; it returns to the same point if it is at the westernmost vertex of a HE. By the choice of exponent, after the robot follows $N^j \text{ } W^{\lambda+q} \text{ } (S^{j-u}EN^{j-u}W)^{\lambda+q}$ it remains stuck at the point $\textbf{v}$. Immediately afterwards, it follows $K$ and it gets to the destination point (see Figure~\ref{8}).
\end{proof}
\noindent
\textbf{2(v) } We assume that every row that intersects the column $R_1$ does not have a west bump. In addition we assume that there exists a special empty west row $r_{w_3}$. We recall that an empty west row is a row that in the west strip is empty and the special empty west row is the empty west row of smallest latitude greater than $-3p$ with respect to the standard well order on $\mathbb{Z}$. We recall that $\textbf{w} = (a+1, u)$ is the uppermost point of the finite column $R_1$ and let $\textbf{v}$ be the easternmost vertex in the west strip on the row $r_{w_3}$. We may assume without loss of generality that $w_3>u$.
By inspecting the primary rectangle, we construct an algorithm $H'$ of the form $H' = \prod_{m=1}^hN^{k_m}N^{-k_m}E^{\epsilon_m}$, where ${\epsilon_m} \in \{-1, 1\}$ and $k_m$ is an integer with $|k_m| \le 2p$ for all $1 \le m \le h$, such that if the robot starts at $\textbf{w}$ and it follows $H'$, it visits the destination point (see $H'$ in Figure~\ref{7}). Let $H = H'W^{|H'|}$. We note that if the robot is in the origin in a maze with no VNEs and it follows $H$ it returns to the $x$-axis and its latitude does not increase. We further note that if the robot starts from $\textbf{v}$ and it follows $H$, it oscillates about latitude $w_3$ without hitting any VNE and at the end it returns back to the starting point $\textbf{v}$.
We define the algorithm
\[
F = N^{w_3} (S^{w_3-u} E N^{w_3-u} H)^{\lambda+q}.
\]
\textbf{Claim. } For any maze in $F_i$, after the robot follows $X \text{ } F$, it visits the destination point.
\begin{proof}
We may assume without loss of generality that after the robot follows $X \text{ } N^{w_3}$ it gets on the row $r_{w_3}$ at the west of the point $\textbf{v}$. While the robot follows each instance of $S^{w_3-u} E N^{w_3-u} H$ in the west strip, it advances eastwards one unit making an oscillation about the row $r_{w_3}$. By the choice of exponent, after a certain instance of $S^{w_3-u} E N^{w_3-u} H$, the robot eventually gets to the point $\textbf{v}$. Immediately afterwards, it follows another instance of $S^{w_3-u} E N^{w_3-u} H$ and it gets to the destination point. Indeed, if the robot starts at the point $\textbf{v}$ and it follows $S^{w_3-u} E N^{w_3-u}$, it gets to the point $\textbf{w}$. If the robot starts at $\textbf{w}$ and it follows $H$, it gets to the destination point. The conclusion follows.
\end{proof}
\noindent
\textbf{2(vi) } This is the final case, where we may assume all of the following: every row that intersects the column $R_1$ does not have a west bump; there does not exist a west pipe; there does not exist a magical west row; there does not exist a special almost empty west row; there does not exist a special empty west row. Then every row at latitude greater than $-3p$ with respect to the well order on $\mathbb{Z}$ is a path in the west strip and indeed a path in the maze; every row that intersects the finite column $R_1$ is a path in the west strip and indeed a path in the maze; each row at latitude at most $3p$ with respect to the standard well order on $\mathbb{Z}$ is known to be either a path or the complement of a path in the west strip. We recall that $\textbf{w} = (a+1, u)$ is the uppermost point of the finite column $R_1$.
By inspecting the primary rectangle we can construct an algorithm $H'$ of the form $H' = \prod_{m=1}^hN^{k_m}N^{-k_m}E^{\epsilon_m}$, where ${\epsilon_m} \in \{-1, 1\}$ and $k_m$ is an integer with $|k_m| \le 2p$ for all $1 \le m \le h$, such that if the robot starts at $\textbf{w}$ and follows $H'$ it visits the destination point (see $H'$ in Figure~\ref{7}). Let $H = H' E^r$, where $r$ is an integer such that if the robot follows $H$ on a maze without meeting any VNE and HNE then it returns back to its starting point. We construct the algorithm
\[
F = N^u (E N^{6p} H S^{6p})^{\lambda+p}.
\]
\textbf{Claim. } For any maze in $F_i$, after the robot follows $X \text{ } F$, it visits the destination point.
\begin{proof}
We may assume without loss of generality that after the robot follows $X \text{ } N^u$ it gets in the west strip on the row $r_u$. While the robot executes one instance of $E N^{6p} H S^{6p}$ it advances one unit eastwards in the west strip on the row $r_u$ without meeting any VNE or HNE. Indeed, every row at latitude greater than $3p$ is a path in the maze. The robot eventually eventually gets at $\textbf{w}$. Immediately afterwards, it follows $N^{6p}$, remaining at $\textbf{w}$ and then $H$, hence it gets to the destination point.
\end{proof}
This finally solves \textbf{Case (2)} in which the destination point was connected with the west strip by a finite number of finite columns. It is immediate to see that the presence of infinite columns in the obstacle strip does not affect any of the arguments made in this case. \\
\textbf{Case (3). } We assume that the destination point is in the obstacle strip and there exists some parameter $h_{(i, i+1)} < \infty$. We recall that this is equivalent to the existence of a pair of consecutive upper infinite columns (or a consecutive upper infinite column and an infinite column at the border of the obstacle strip and either the east or west strip) which are not connected by HEs at arbitrarily high latitudes. By symmetry, treating this case also solves the homologous case in which there exists some parameter $l_{(i, i+1)} < \infty$. \\
\textbf{3(i) } We assume $h_{(a, a+1)}<\infty$. We recall that the pair of columns $(c_a, c_{a+1})$ is at the border between the west strip and the obstacle strip and we also recall that the pair of columns $(c_b, c_{b+1})$ is at the border between the obstacle strip and the east strip. We assume without loss of generality that there exists a HE between the west strip and a finite column or a lower infinite column (otherwise we are done by \textbf{Part I}). Let $R$ be a finite column or a lower infinite column on the column $c_{a+1}$ such that there exists a HE between the west strip and $R$ on some row $r_c$. Let $\textbf{w}$ be the uppermost vertex of $R$. Let $j=h_{(a,a+1)}+l$ and $\textbf{v} = (a,j)$ be the easternmost point on the row $r_j$ in the west strip. We recall the generic algorithm
\[
SME^{(j-c)}(a,e,L)=((((((L)^e S^{j-c} E N^{j-c})^e E)^e N E S)^e S E N)^e N N E S S)^e...S^a E N^a)^e.
\]
We define the algorithm
\[
F = N^j SME^{(j-c)}(\lambda+j+q,\lambda+j+q,L).
\]
\textbf{Claim. } For any maze in $F_i$, after the robot follows $X \text{ } F$, it visits the destination point.
\begin{proof}
We may assume without loss of generality that after the robot follows $X \text{ } N^j$ it is in the west strip on the row $r_j$. By the choice of parameters and by Corollary~\ref{c1}, while the robot follows $SME^{(j-c)}(\lambda+j+q,\lambda+j+q,L)$ it advances eastwards in the west strip oscillating about row $r_j$. After the robot starts from some point on the row $r_j$ in the west strip and follows $L$ its longitude does not decrease and it remains in the west strip. It eventually gets to the point $\textbf{v}$. After the robot starts from $\textbf{v}$ and follows $S^{j-c}EN^{j-c}$ it gets to the point $\textbf{w}$. Immediately afterwards, it follows $L$ and gets to the destination point.
\end{proof}
\noindent
\textbf{3(ii) } Consider the pair of consecutive columns $(c_i, c_{i+1})$ which is not at the border between the west strip and the obstacle strip. Assume there are not arbitrarily high HEs between the columns $c_i$ and $c_{i+1}$, i.e. $h_{(i,i+1)}<\infty$. Assume further that there exists a pass on some row $r_c$ between the west strip and an upper infinite column $R$ (see the case \textbf{3(i)}). We define $K = S^{\lambda+2q+|h_{(i,i+1)}|+1} N^{\lambda+2q+|h_{(i,i+1)}|+1}$. We define the algorithm
\[
F = N^c \text{ } SME(\lambda+q,\lambda+q,K) \text{ } S^{\lambda+2q+|h_{(i,i+1)}|+1} \text{ } L.
\]
\textbf{Claim. } For any maze in $F_i$, after the robot follows $X \text{ } F$, it visits the destination point.
\begin{proof}
We may assume without loss of generality that after the robot follows $X \text{ } N^c$ it gets in the west strip on the row $r_c$. While the robot follows $SME(\lambda+q,\lambda+q,K)$ it advances eastwards in the west strip oscillating about the row $r_c$. It eventually enters the upper infinite columns $R$. Immediately afterwards it executes $K$ and gets to latitude at least $\lambda+q+|h_{(i,i+1)}|+1$ in $R$. While the robot is in the obstacle strip and follows $SME$ it advances eastwards through upper infinite columns at latitudes greater than $h_{(i,i+1)}+1$. Hence the robot remains stuck in some column $c_j$ with $j \le i$ at latitude $\lambda+2q+|h_{(i,i+1)}|+1$ above the highest VNE in the columns $c_r$ with $a \le r \le j$. After that the robot follows $S^{\lambda+2q+|h_{(i,i+1)}|+1}$ and gets to a special point. Therefore, after the robot follows $X \text{ } F$ it gets to the destination point.
\end{proof}
This finally solves \textbf{Case (3)} in which the destination point is in the obstacle strip in a finite or infinite column and there exists some parameter $h_{(i, i+1)} < \infty$. Moreover, the case in which there exists some parameter $l_{(i, i+1)} < \infty$ is tackled similarly by symmetry. Finally, it is immediate to see that the presence of infinite columns in the obstacle strip does not affect any of the arguments made in this case.\\
\textbf{Case (4). } This is the final case, in which we may assume that \textbf{Case (3)} does not hold and the destination point is in the obstacle strip in a finite column, an upper infinite column or lower infinite column and it is connected to the west strip by a (finite, possibly empty) sequence of finite columns followed by a (finite, non-empty) sequence of upper infinite columns, in this order starting from the destination point and advancing towards the west strip. Indeed, we may assume that the west strip is accessible by \textbf{Part II}. The case in which is the destination point is not in the obstacle strip is tackled in \textbf{Case (1)}. Furthermore, if we assume that the destination point is in the obstacle strip, it may either be reachable from the west strip through a finite sequence of finite columns tackled in \textbf{Case (2)} or otherwise it must be reachable from the west strip through a finite sequence of upper/lower infinite and finite columns which contains at least one upper or lower infinite column. Choose any such finite sequence of columns which leads to the destination point starting from the west strip and call the last upper or lower infinite column in the sequence $c$; this may either be the last element of the sequence or it might be followed by a finite sequence of finite columns. By \textbf{Case (3)} we may assume that there are horizontal edges between consecutive upper infinite columns and between consecutive lower infinite columns at latitudes arbitrarily high and low, respectively. Hence, assuming without loss of generality as we may that $c$ is an upper infinite column, $c$ can be reached from the west strip through a finite sequence of upper infinite columns. Therefore, the last case that we tackle is the one in which we assume that the destination point is connected to the west strip by a (finite, possibly empty) sequence of finite columns followed by a (finite, non-empty) sequence of upper infinite columns, in this order starting from the destination point and advancing towards the west strip.
The condition that \textbf{Case (3)} does not hold means that in this case we assume that all the parameters $h_{(i, i+1)}$ and $l_{(i, i+1)}$ are all infinity for $a \leq i \leq b$; in particular, this implies that there exists a west ascending chain. We recall that the pair of columns $(c_a, c_{a+1})$ are at the border between the west strip and the obstacle strip; we also recall that the pair of columns $(c_b, c_{b+1})$ are at the border between the obstacle strip and the east strip. We further recall that a west ascending chain is a finite sequence of HEs: $HE_a, HE_{a+1}, \hdots, HE_b$ such that $HE_a$ is the upper west pass (i.e. the lowermost HE between the west strip and the upper infinite column on $c_{a+1}$ above all passes in the obstacle strip) and $HE_m$ is the lowermost HE between the pair of columns $(c_m, c_{m+1})$ at latitude at least that of $HE_{m-1}$ for $m=a+1, \hdots, b$. In this case, we take $R_{a+1}, R_{a+2}, \hdots , R_n$ to be a finite non-empty sequence of upper infinite columns and $R_{n+1}, \hdots , R_k$ to be a finite possibly empty sequence of finite columns and finally we take $R$ to be a finite, upper infinite or lower infinite column such that $R$ contains the destination point and there exists a HE between the west strip and $R_{a+1}$, between $R_m$ and $R_{m+1}$ for $a+1 \leq m \leq k -1$ and between $R_k$ and $R$. By the discussion at the beginning of the case, we may assume that such a sequence has the extra property that $R_m$ is on the column $c_m$ for $a+1 \leq m \leq n$. Moreover, if $R_{n+1}$ exists we may assume that $R_{n+1} \in c_{n+1}$; indeed, $R_{n+1} \in c_{n+1}$ or $R_{n+1} \in c_{n-1}$ and if $R_{n+1} \in c_{n-1}$ then we can use the symmetry of the argument in \textbf{Part II} to assume that the robot is in the east strip on the $x$-axis. From that perspective, we can use the arguments from the case that we are treating with $R_{n+1} \in c_{n+1}$. Obviously, if $R_{n+1}$ does not exist, by the same argument we may assume that $R$ is in $c_{n+1}$. Finally, say that the row $r_i$ contains the upper west pass and note that the upper west pass is above all passes in the obstacle strip and therefore, as \textbf{Case (3)} does not hold, it is above all special vertices. \\
\textbf{4(i) }We assume there exists a magical west row $r_j$. We recall that a magical west row is a row which is a path when restricted to the west strip, and it contains a HNE; its west cutoff is its westernmost HNE. We see in the end that our argument also solves the case when there exists a magical east row. Denote by $\textbf{v}$ the westernmost vertex of the west cutoff of $r_j$. Then, by inspecting the tertiary rectangle, we can construct an algorithm $K$ such that if the robot starts from $\textbf{v}$ and follows $K$ it gets to the destination point. We construct the algorithm
\[
F = N^jE^{\lambda+q}K.
\]
\textbf{Claim. } For any maze in $F_i$, after the robot follows $X \text{ } F$, it visits the destination point.
\begin{proof}
We may assume without loss of generality that after the robot follows $X \text{ } N^j$, it gets on the row $r_j$. Therefore, after it follows $X \text{ } N^j \text{ } E^{\lambda+q}$ the robot gets to the point $\textbf{v}$. Hence, after the robot follows $X \textbf{ } F$ it gets to the destination point.
\end{proof}
Clearly, in this case we may easily drop the general assumption that $R_{n+1} \in c_{n+1}$. Therefore, this argument also solves the case when there exists a magical east row.
\begin{figure}
\centering
\resizebox{0.5\textwidth}{!}{
\begin{tikzpicture}
[
dot/.style={circle,draw=black, fill,inner sep=2pt},
cross/.style={cross out, draw=black, minimum size=2*(#1-\pgflinewidth), inner sep=4pt, outer sep=4pt},
]
\foreach \x in {-4,...,4}{
\foreach \y in {-4,...,4}{
\node[dot] at (\x,\y){ };
}}
\draw[line width=0.7mm, red] (-4, -4) -- (-4, 4);
\draw[line width=0.7mm, red] (-3, -4) -- (-3, 4);
\draw[line width=0.7mm, red] (-2, -4) -- (-2, 4);
\draw[line width=0.7mm, red] (-1, -4) -- (-1, 4);
\draw[line width=0.7mm, red] (0, -4) -- (0, 4);
\draw[line width=0.7mm, red] (1, -4) -- (1, 4);
\draw[line width=0.7mm, red] (2, 4) -- (2, 2);
\draw[line width=0.7mm, red] (2, 0) -- (2, -1);
\draw[line width=0.7mm, red] (2, -2) -- (2, -3);
\draw[line width=0.7mm, red] (3, 3) -- (3, 1);
\draw[line width=0.7mm, red] (3, -1) -- (3, -2);
\draw[line width=0.7mm, red] (4, -2) -- (4, -4);
\draw[line width=0.7mm, red] (4, 0) -- (4, 3);
\draw[line width=0.7mm, red] (3, 4) -- (4, 4);
\draw[line width=0.7mm, red] (2, 4) -- (-2, 4);
\draw[line width=0.7mm, red] (-2, 3) -- (0, 3);
\draw[line width=0.7mm, red] (1, 3) -- (3, 3);
\draw[line width=0.7mm, red] (-4, 3) -- (-3, 3);
\draw[line width=0.7mm, red] (-4, 2) -- (-3, 2);
\draw[line width=0.7mm, red] (-1, 2) -- (1, 2);
\draw[line width=0.7mm, red] (2, 2) -- (3, 2);
\draw[line width=0.7mm, red] (-1, 1) -- (2, 1);
\draw[line width=0.7mm, red] (-3, 1) -- (-2, 1);
\draw[line width=0.7mm, red] (3, 1) -- (4, 1);
\draw[line width=0.7mm, red] (-2, 0) -- (-1, 0);
\draw[line width=0.7mm, red] (1, 0) -- (3, 0);
\draw[line width=0.7mm, red] (-2, -1) -- (1, -1);
\draw[line width=0.7mm, red] (2, -1) -- (3, -1);
\draw[line width=0.7mm, red] (-4, -1) -- (-3, -1);
\draw[line width=0.7mm, red] (1, -2) -- (-1, -2);
\draw[line width=0.7mm, red] (2, -2) -- (4, -2);
\draw[line width=0.7mm, red] (-1, -3) -- (2, -3);
\draw[line width=0.7mm, red] (3, -3) -- (4, -3);
\draw[line width=0.7mm, red] (-2, -4) -- (-1, -4);
\draw[line width=0.7mm, red] (0, -4) -- (2, -4);
\draw[line width=0.7mm, red] (3, -4) -- (4, -4);
\draw[line width=0.7mm, red] (-4, -4) -- (-3, -4);
\draw[line width=0.7mm, green] (-3, -2) -- (-3, 1);
\draw[line width=0.7mm, green] (-2, -2) -- (-2, 1);
\draw[line width=0.7mm, green] (-3, 1) -- (-2, 1);
\draw[line width=0.7mm, green] (-2, -1) -- (-1, -1);
\draw[line width=0.7mm, green] (-1, -2) -- (-1, -1);
\draw node[fill,circle,inner sep=5.5pt,minimum size=2pt] at (-1, -2) {};
\draw (-0.7, -1.7) -- node {\textbf{v}} (-0.7, -1.7);
\draw (4,0) node[cross] {};
\foreach \x in {-4,...,4}
\draw (\x,.1) -- node[below,yshift=-1mm] {\x} (\x,-.1);
\foreach \y in {-4,...,4}
\draw (.1,\y) -- node[below,xshift=-3mm, yshift=3mm] {\y} (-.1,\y);
\draw[->,line width=0.15mm] (0,-4.5) -- (0,4.5);
\draw[->,line width=0.15mm] (-4.5,0) -- (4.5,0);
\end{tikzpicture}
}
\caption{\textbf{Part III, Case (4)(ii).} There exists a special almost empty west row $r_j$ and let $\textbf{v}$ be the easternmost vertex of the west cutoff of $r_j$. We assume that there are no VEs removed other than the ones shown in the figure. In this example, let $r_j = r_{-2}$ and so $\textbf{v} = (-1, -2)$. Let us see how the robot gets to $\textbf{v}$ after it follows $AME (5,5)$ starting from $(-3, -2)$. As long as the robot is at the west of $\textbf{v}$, the $W$ instructions do not decrease the longitude of the robot, as there are no HEs on $r_j$ at the west of $\textbf{v}$. Therefore, the robot takes the green path to $\textbf{v}$. Once the robot gets at $\textbf{v}$, every subsequent subalgorithm of the form $N^iES^iW$ takes it back to $\textbf{v}$: after $N^iES^i$, the robot is either at $\textbf{v} = (-1, -2)$ or $(0, -2)$; immediately afterwards, the robot follows $W$ and the presence of a HE between $(-1, -2)$ and $(0, -2)$ guarantees that the robot returns back to $\textbf{v}$. Immediately after the robot follows $AME$ and it gets to $\textbf{v}$, it follows $K$ and it visits the destination point. For our example we can take $K = N^3 E^2 N^2 E^2 S^2 E S$.} \label{10}
\end{figure}
\noindent
\textbf{4(ii) } We assume there exists a special almost empty west row $r_j$ and we call $\textbf{v}$ the easternmost vertex of the west cutoff of $r_j$. We recall that an almost empty west row is a row that in the west strip is the complement of an infinite path followed by a non-empty finite path; its west cutoff is its easternmost HNE in the west strip. Then, by inspecting the tertiary rectangle, we can construct an algorithm $K$ such that if the robot starts from $\textbf{v}$ and follows $K$ it gets to the destination point (see Figure~\ref{10}).
We then define the algorithm $auxiliary \_ move \_ east$,
\[
AME (a, e) = ((NESW)(SENW)(N^2ES^2W)(S^2EN^2W) \hdots (S^aEN^aW))^e.
\]
We finally define the algorithm
\[
F:= N^j \text{ } W^q \text{ } AME (\lambda+q, \lambda+q) \text{ } K.
\]
\textbf{Claim. } For any maze in $F_i$, after the robot follows $X \text{ } F$, it visits the destination point.
\begin{proof}
We may assume without loss of generality that after the robot follows $X \text{ } N^j \text{ } W^q$ it gets on the row $r_j$ at a longitude at most that of $\textbf{v}$. By the choice of parameters, while the robot follows $AME (\lambda+q, \lambda+q)$ it advances eastwards in the west strip oscillating about the row $r_j$ and it remains stuck at the point $\textbf{v}$. Hence, after the robot follows $X \text{ } F$, it reaches the destination point (see Figure~\ref{10}).
\end{proof}
\begin{figure}[h!]
\centering
\resizebox{0.75\textwidth}{!}{
\begin{tikzpicture}
[
dot/.style={circle,draw=black, fill,inner sep=2pt},
cross/.style={cross out, draw=black, minimum size=2*(#1-\pgflinewidth), inner sep=4pt, outer sep=4pt},
]
\foreach \x in {-6,...,6}{
\foreach \y in {-6,...,6}{
\node[dot] at (\x,\y){ };
}}
\draw[line width=0.7mm, red] (-6, -6) -- (-6, 6);
\draw[line width=0.7mm, red] (-5, -6) -- (-5, 6);
\draw[line width=0.7mm, red] (-4, -6) -- (-4, 6);
\draw[line width=0.7mm, red] (-3, -6) -- (-3, 6);
\draw[line width=0.7mm, red] (-2, -6) -- (-2, 6);
\draw[line width=0.7mm, red] (-1, -6) -- (-1, 6);
\draw[line width=0.7mm, red] (0, -6) -- (0, 6);
\draw[line width=0.7mm, red] (1, -6) -- (1, 6);
\draw[line width=0.7mm, red] (2, -6) -- (2, -4);
\draw[line width=0.7mm, red] (2, -3) -- (2, 6);
\draw[line width=0.7mm, red] (3, -5) -- (3, -4);
\draw[line width=0.7mm, red] (3, -2) -- (3, 6);
\draw[line width=0.7mm, red] (4, -5) -- (4, -4);
\draw[line width=0.7mm, red] (4, -3) -- (4, 6);
\draw[line width=0.7mm, red] (5, -6) -- (5, -5);
\draw[line width=0.7mm, red] (5, -4) -- (5, -1);
\draw[line width=0.7mm, red] (5, 0) -- (5, 6);
\draw[line width=0.7mm, red] (6, -6) -- (6, 6);
\draw[line width=0.7mm, red] (-2, 5) -- (1, 5);
\draw[line width=0.7mm, red] (3, 5) -- (6, 5);
\draw[line width=0.7mm, red] (4, 4) -- (6, 4);
\draw[line width=0.7mm, red] (-5, 4) -- (-4, 4);
\draw[line width=0.7mm, red] (-1, 3) -- (0, 3);
\draw[line width=0.7mm, red] (2, 3) -- (4, 3);
\draw[line width=0.7mm, red] (5, 3) -- (6, 3);
\draw[line width=0.7mm, red] (-2, 2) -- (-3, 2);
\draw[line width=0.7mm, red] (0, 2) -- (1, 2);
\draw[line width=0.7mm, red] (2, 2) -- (3, 2);
\draw[line width=0.7mm, red] (-5, 1) -- (-4, 1);
\draw[line width=0.7mm, red] (-2, 1) -- (-1, 1);
\draw[line width=0.7mm, red] (0, 1) -- (2, 1);
\draw[line width=0.7mm, red] (4, 1) -- (5, 1);
\draw[line width=0.7mm, red] (0, 0) -- (1, 0);
\draw[line width=0.7mm, red] (4, 0) -- (5, 0);
\draw[line width=0.7mm, red] (-1, -1) -- (0, -1);
\draw[line width=0.7mm, red] (2, -1) -- (4, -1);
\draw[line width=0.7mm, red] (-5, -2) -- (-3, -2);
\draw[line width=0.7mm, red] (-1, -2) -- (0, -2);
\draw[line width=0.7mm, red] (1, -2) -- (2, -2);
\draw[line width=0.7mm, red] (4, -2) -- (5, -2);
\draw[line width=0.7mm, red] (-2, -3) -- (0, -3);
\draw[line width=0.7mm, red] (-6, -4) -- (-3, -4);
\draw[line width=0.7mm, red] (-2, -4) -- (-1, -4);
\draw[line width=0.7mm, red] (4, -4) -- (5, -4);
\draw[line width=0.7mm, red] (-5, -5) -- (-3, -5);
\draw[line width=0.7mm, red] (-1, -5) -- (0, -5);
\draw[line width=0.7mm, red] (3, -5) -- (4, -5);
\draw[line width=0.7mm, red] (-1, -6) -- (0, -6);
\draw[line width=0.7mm, red] (4, -6) -- (5, -6);
\draw[line width=1mm, green] (1, 1) -- (2, 1);
\draw[line width=1mm, green] (2, 2) -- (3, 2);
\draw[line width=1mm, green] (3, 3) -- (4, 3);
\draw[line width=1mm, green] (4, 4) -- (6, 4);
\draw node[fill,circle,inner sep=5.5pt,minimum size=2pt] at (-2, 1) {};
\draw node[fill,circle,inner sep=5.5pt,minimum size=2pt] at (2, 1) {};
\draw node[fill,circle,inner sep=5.5pt,minimum size=2pt] at (4, 4) {};
\draw node[fill,circle,inner sep=5.5pt,minimum size=2pt] at (5, -1) {};
\draw (-1.7,1.4) -- node {\textbf{v}} (-1.7,1.4);
\draw (2.4,1) -- node {\textbf{t}} (2.4,1);
\draw (5.4, -1) -- node {\textbf{z}} (5.4, -1);
\draw (4.3, 4.4) -- node {\textbf{w}} (4.3, 4.4);
\draw (3,-4) node[cross] {};
\foreach \x in {-6,...,6}
\draw (\x,.1) -- node[below,yshift=-1mm] {\x} (\x,-.1);
\foreach \y in {-6,...,6}
\draw (.1,\y) -- node[below,xshift=-3mm, yshift=3mm] {\y} (-.1,\y);
\draw[->,line width=0.15mm] (0,-6.5) -- (0,6.5);
\draw[->,line width=0.15mm] (-6.5,0) -- (6.5,0);
\end{tikzpicture}
}
\caption{\textbf{Part III, Case (4)(iii).} We assume there exist the special upper west paired HNEs. We assume that there are no VEs removed other than the ones shown in the figure, so $a=1$ and $b= 5$. The upper west pass between $c_1$ and $c_2$ is $HE_1 = (1, 1), (2, 1)$, above all the passes in the obstacle strip; the west ascending chain is coloured green. The chosen path from the west strip to the destination point goes through $R_2 = \{ (2, -3), (2, -2), \hdots \}$, then $R_3$, $R_n = R_4$, $R_5 = \{ (5, -1), (5, -2), (5, -3), (5, -4) \}$, $R_6 = \{ (4, -4), (4, -5) \}$, $R = R_7 = \{ (3, -4), (3, -5) \}$. The point $\textbf{z} = (5, -1)$ is the uppermost vertex of $R_5$. For the purpose of this example, let us assume $c_{uw} = 6$, although this should be larger. To find the upper west paired HNEs, the set of HNEs on $r_1$ is the set of all possible candidates for the upper HNE in the pair. To find the second HNE in the pair, we look on $r_{1-6} = r_{-5}$ (i.e. at latitude $i-c_{uw}$) to find a matching HNE at the same longitude with one on $r_1$ and we choose the easternmost one. If none such HNE exists, we repeat the same process on $r_{-6}$, then on $r_{-7}$ and so on. In this example, we find the upper west paired HNEs to be $(-3, 1), (-2, 1)$ and $(-3, -5), (-2, -5)$. Therefore $\textbf{v} = (-2, 1)$ and $\textbf{t} = (2, 1)$. The only HE between $R_4$ and $R_5$ is $(4, -2), (5, -2)$ at latitude $\nu = -2$, so $\textbf{w} = (4, 4)$. Then, if the robot follows $K_1 = E S^6 E N^6 EE$ starting from $\textbf{v}$ it gets to $\textbf{t}$; if the robot follows $K_2 = S^5 N^5 E S^5 N^5 E S^7 N^7$ starting from $\textbf{t}$ it gets to $\textbf{w}$, passing through the green edges from $R_2$ to $R_3$ and from $R_3$ to $R_4$; if the robot follows $K_3 = S^6 E N^6$ starting from $\textbf{w}$ it gets to $\textbf{z}$; if the robot follows $K_4 = N^3 S^3 W NS W NS E$ starting from $\textbf{z}$ it gets to the destination point; $K_5 = E^{13}$. Finally, we remark that if the robot follows $K= K_1 K_2 K_3 K_4 K_5$ starting on any point of $r_1$ strictly at the west of $\textbf{v}$, then it returns on $r_1$ strictly at the west of $\textbf{v}$.} \label{11}
\end{figure}
\noindent
\textbf{4(iii) } We assume there exist the special upper west paired HNEs. We recall the following definitions: let $HE_a, \hdots, HE_b$ be the west ascending chain with $HE_a$ being the upper west pass say on some row $r_i$ and also say that $HE_b$ is on some row $r_t$. Then $c_{uw} = t+p$ is the upper west constant, where $p$ is the parameter of the primary rectangle. The upper west paired HNEs are any pair of HNEs with the same longitude, in the west strip, such that the upper HNE is at latitude $i$, on the same row as the upper west pass, and the lower HNE is at latitude at most $i-c_{uw}$. For the special upper west paired HNEs, we choose the upper west paired HNEs with the uppermost easternmost lower HNE. In this subcase, we assume that there exist the special upper west paired HNEs, with the upper HNE on the row $r_i$ and the lower HNE on the row $r_j$, $j \leq i-c_{uw}$.
Let the point $\textbf{v}$ be the easternmost vertex of the upper HNE of the pair and let the point $\textbf{t}$ be the easternmost vertex of the upper west pass. We pick any HE between the upper infinite column $R_n$ and the finite column $R_{n+1}$ at latitude say $\nu$. In the case that $R_{n+1}$ does not exist, we pick the lowermost HE between the upper infinite column $R_n$ and $R$ at latitude say $\nu$. Let the point $\textbf{w}$ be the vertex in the infinite column $R_n$ at latitude $\nu+i-j$. Then the eastern vertex of $HE_{n-1}$ which has a latitude of at most $t$ by definition is in the column $c_n$ below $\textbf{w}$; indeed, $\nu+i-j \geq \nu + t + p$ and $\nu+p \geq 0$. Finally, let the point $\textbf{z}$ be the uppermost vertex of the finite column $R_{n+1}$ if $R_{n+1}$ exists. In the following argument, we assume that $R_{n+1}$ exists and it will be clear how this also naturally treats the case when $R_n$ is connected to $R$ which contains the destination point. For an illustration of all these definitions in a concrete example, see Figure~\ref{11}.
In what follows, we will construct $5$ algorithms $K_1, \hdots, K_5$, by inspecting the tertiary rectangle.
We start by constructing a finite algorithm $K_1$ of the form $K_1=\prod_{m=1}^{h_1} S^{\epsilon_m}EN^{\epsilon_m}$, where $\epsilon_m \in \{0, i-j \}$ for all $1 \le m \le h_1$, such that after the robot follows $K_1$ starting from the point $\textbf{v}$ it gets to the point $\textbf{t}$. We make use of the fact that in the west strip at the east of the special upper west paired HNEs at each given longitude at least one of the rows $r_i$ and $r_j$ contains a HE. Clearly, $\epsilon_{h_1}=0$.
We construct a finite algorithm $K_2$ of the form $K_2=(\prod_{m=a+1}^{n-1} S^{k_m}N^{k_m}E)S^{k_n}N^{k_n} $, where $k_m$ is a positive integer for all $a+1 \le m \le n$, such that if the robot starts from the point $\textbf{t}$ and it follows $K_2$, it gets to the point $\textbf{w}$. More specifically, if the robot is in the upper infinite column $R_m$ in the column $c_m$ at the easternmost end of $HE_{m-1}$ and it follows $S^{k_m}N^{k_m}E$ it gets in the upper infinite column $R_{m+1}$ in the column $c_{m+1}$ at the easternmost end of $HE_{m}$, for $a+1 \le m \le n-1$; if the robot is in the upper infinite column $R_n$ in the column $c_n$ at the easternmost end of $HE_{n-1}$ and it follows $S^{k_n}N^{k_n}$ it gets to the point $\textbf{w}$.
We construct an algorithm $K_3=S^{i-j} E N^{i-j}$, such that if the robot starts from $\textbf{w}$ and it follows $K_3$, it gets to the point $\textbf{z}$.
We construct an algorithm $K_4$ of the form $K_4=(\prod_{m=n+1}^{k} N^{k_m}S^{k_m}E^{\epsilon_m})$ $N^{k_{k+1}} \text{ } N^{-k_{k+1}}$, where $\epsilon_m \in \{-1, 1\}$ and $k_m$ is an integer for all $n+1 \le m \le k+1$, such that if the robot starts from the point $\textbf{z}$ and it follows $K_4$, it visits the destination point. More specifically, if the robot is at some specified latitude in the finite column $R_m$ and it follows $N^{k_m}N^{-k_m}E^{\epsilon_m}$, it gets to some specified latitude in the finite column $R_{m+1}$ for $n+1 \le m \le k$, where by convention we write $R_{k+1}$ for $R$. If the robot is at some specified latitude inside $R$ and follows $N^{k_{k+1}}N^{-k_{k+1}}$ it visits the destination point.
We define the algorithm $K_5=E^{|K_4|}.$
We define the algorithm $K =K_1 K_2 K_3 K_4 K_5.$ Note that if the robot is on the row $r_i$ strictly at the west of the point $\textbf{v}$ and it follows $K$ it returns on the row $r_i$ strictly at the west of $\textbf{v}$. Indeed, by examining $K_1, \hdots, K_5$ one by one, we conclude that if the robot starts strictly at the west of $\textbf{v}$, while executing $K$ it can only change its longitude at latitudes $i$ or $j$. Thus, the existence of the special west paired HNEs prevents the robot from reaching a longitude at least that of $\textbf{v}$. In particular, if the robot is on the row $r_i$ strictly at the west of the point $\textbf{v}$ and it follows $K$ it does not meet any VNE, so it is easy to see that it returns back to the row $r_i$. Finally, the only $W$ instructions in $K$ could appear as part of $K_4$, which is followed by $K_5=E^{|K_4|}$ in $K$; therefore if the robot is on the row $r_i$ strictly at the west of the point $\textbf{v}$ and it follows $K$ its longitude does not decrease. If the robot starts at the point $\textbf{v}$ and it follows $K$, then it visits the destination point; this follows directly from the definitions of $K_1, \hdots, K_5$ (see Figure~\ref{11}).
Finally, we construct the algorithm
\[
F = N^i \text{ } MW (i-j, q) \text{ } SME(\mu+\lambda+2q,\mu+\lambda+2q,K),
\]
where $\mu = |MW (i-j, q)|.$
\begin{prop}
For any maze in $F_i$, after the robot follows $X \text{ } F$, it visits the destination point.
\end{prop}
\begin{proof}
We may assume without loss of generality that after the robot follows the algorithm $X \text{ } N^i \text{ } MW (i-j, q)$ it gets on the row $r_i$ at a longitude at most that of the point $\textbf{v}$. By the choice of parameters and by Lemma~\ref{l2}, while the robot follows $SME (\mu+\lambda+2q,\mu+\lambda+2q,K)$, it advances eastwards in the west strip oscillating about the row $r_i$. After defining $K$, we checked that it satisfies the conditions required in order to apply Lemma~\ref{l2}. Therefore, by Lemma~\ref{l2}, the robot gets for the first time to the point $\textbf{v}$ not while executing $K$, but while executing a locomotory move. Immediately afterwards, it follows $K$ and it gets to the destination point. The conclusion follows.
\end{proof}
\begin{figure}[h!]
\centering
\resizebox{0.75\textwidth}{!}{
\begin{tikzpicture}
[
dot/.style={circle,draw=black, fill,inner sep=2pt},
cross/.style={cross out, draw=black, minimum size=2*(#1-\pgflinewidth), inner sep=4pt, outer sep=4pt},
]
\foreach \x in {-6,...,6}{
\foreach \y in {-6,...,6}{
\node[dot] at (\x,\y){ };
}}
\draw[line width=0.7mm, red] (-6, -6) -- (-6, 6);
\draw[line width=0.7mm, red] (-5, -6) -- (-5, 6);
\draw[line width=0.7mm, red] (-4, -6) -- (-4, 6);
\draw[line width=0.7mm, red] (-3, -6) -- (-3, 6);
\draw[line width=0.7mm, red] (-2, -6) -- (-2, 6);
\draw[line width=0.7mm, red] (-1, -6) -- (-1, 6);
\draw[line width=0.7mm, red] (0, -6) -- (0, 6);
\draw[line width=0.7mm, red] (1, -6) -- (1, 6);
\draw[line width=0.7mm, red] (2, -6) -- (2, -4);
\draw[line width=0.7mm, red] (2, -3) -- (2, 6);
\draw[line width=0.7mm, red] (3, -5) -- (3, -4);
\draw[line width=0.7mm, red] (3, -2) -- (3, 6);
\draw[line width=0.7mm, red] (4, -5) -- (4, -4);
\draw[line width=0.7mm, red] (4, -3) -- (4, 6);
\draw[line width=0.7mm, red] (5, -6) -- (5, -5);
\draw[line width=0.7mm, red] (5, -4) -- (5, -1);
\draw[line width=0.7mm, red] (5, 0) -- (5, 6);
\draw[line width=0.7mm, red] (6, -6) -- (6, 6);
\draw[line width=0.7mm, red] (-2, 5) -- (1, 5);
\draw[line width=0.7mm, red] (3, 5) -- (6, 5);
\draw[line width=0.7mm, red] (4, 4) -- (6, 4);
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\draw[line width=0.7mm, red] (-1, 3) -- (0, 3);
\draw[line width=0.7mm, red] (2, 3) -- (4, 3);
\draw[line width=0.7mm, red] (5, 3) -- (6, 3);
\draw[line width=0.7mm, red] (-2, 2) -- (-3, 2);
\draw[line width=0.7mm, red] (0, 2) -- (1, 2);
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\draw[line width=0.7mm, red] (-5, 1) -- (-4, 1);
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\draw[line width=0.7mm, red] (-3, 1) -- (2, 1);
\draw[line width=0.7mm, red] (4, 1) -- (5, 1);
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\draw[line width=0.7mm, red] (-1, -2) -- (0, -2);
\draw[line width=0.7mm, red] (1, -2) -- (2, -2);
\draw[line width=0.7mm, red] (4, -2) -- (5, -2);
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\draw[line width=0.7mm, red] (4, -4) -- (5, -4);
\draw[line width=0.7mm, red] (-5, -5) -- (-3, -5);
\draw[line width=0.7mm, red] (-1, -5) -- (0, -5);
\draw[line width=0.7mm, red] (3, -5) -- (4, -5);
\draw[line width=0.7mm, red] (-1, -6) -- (0, -6);
\draw[line width=0.7mm, red] (4, -6) -- (5, -6);
\draw[line width=0.7mm, red] (-4, -6) -- (-3, -6);
\draw[line width=1mm, green] (1, 1) -- (2, 1);
\draw[line width=1mm, green] (2, 2) -- (3, 2);
\draw[line width=1mm, green] (3, 3) -- (4, 3);
\draw[line width=1mm, green] (4, 4) -- (6, 4);
\draw node[fill,circle,inner sep=5.5pt,minimum size=2pt] at (-3, 1) {};
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\draw node[fill,circle,inner sep=5.5pt,minimum size=2pt] at (5, -1) {};
\draw (-2.7,1.4) -- node {\textbf{v}} (-2.7,1.4);
\draw (2.4,1) -- node {\textbf{t}} (2.4,1);
\draw (5.4, -1) -- node {\textbf{z}} (5.4, -1);
\draw (4.3, 4.4) -- node {\textbf{w}} (4.3, 4.4);
\draw (3,-4) node[cross] {};
\foreach \x in {-6,...,6}
\draw (\x,.1) -- node[below,yshift=-1mm] {\x} (\x,-.1);
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\draw (.1,\y) -- node[below,xshift=-3mm, yshift=3mm] {\y} (-.1,\y);
\draw[->,line width=0.15mm] (0,-6.5) -- (0,6.5);
\draw[->,line width=0.15mm] (-6.5,0) -- (6.5,0);
\end{tikzpicture}
}
\caption{\textbf{Part III, Case (4)(iv).} We assume that there do not exist some special upper west paired HNEs and there exists an upper west pipe on the row $r_i$. We assume that there are no VEs removed other than the ones shown in the figure, so $a=1$ and $b= 5$. We have $r_i = r_1$ with the upper west pipe $\{ (-5, 1), (-4, 1), (-3, 1) \}$. The points $\textbf{v}$, $\textbf{t}$, $\textbf{w}$ and $\textbf{z}$ are marked on the figure and the west ascending chain is coloured green. We also have $HE_{special} = \{ (4, -2), (5, -2) \}$, $d=6$ and let us assume for this example that $c_{uw} = 6$, though this value should be larger. Then, if the robot follows $K_1 = E^5$ starting from $\textbf{v}$ it gets to $\textbf{t}$; if the robot follows $K_2 = (S^5N^5E)^2 \text{ } (S^7N^7E) \text{ } W$ starting from $\textbf{t}$ it gets to $\textbf{w}$; if the robot follows $K_3 = S^6 E N^6$ starting from $\textbf{w}$ it gets to $\textbf{z}$; if the robot follows $K_4 = (N^4 S^4 W) (NSW) NS$ starting from $\textbf{z}$ it gets to the destination point; $K_5 = E^{15}$. We define $K=K_1 K_2 K_3 K_4 K_5$ and note that if the robot follows $K$ starting from $\textbf{v}$ it visits the destination point, but if the robot follows $K$ starting on $r_i = r_1$ strictly at the west of $\textbf{v}$, it returns on $r_i$ strictly at the west of $\textbf{v}$.} \label{12}
\end{figure}
\noindent
\textbf{4(iv) } We assume that there do not exist some special upper west paired HNEs and there exists an upper west pipe on the row $r_i$ which contains the upper west pass. The upper west pipe is the west pipe (the easternmost configuration of a HE followed by a HNE) on the row $r_i$. Let the point $\textbf{v}$ in the west strip be the easternmost vertex of the HNE of the upper west pipe. Let the point $\textbf{t}$ be the easternmost vertex of the upper west pass. Consider the finite sequence of HEs in the west ascending chain $HE_a, HE_{a+1}, \hdots , HE_b$. Let the point $\textbf{w}$ in the upper infinite column $R_n$ in $c_n$ be the westernmost vertex of $HE_n$. Let $HE_{special}$ be a HE between the upper infinite column $R_n$ and the finite column $R_{n+1}$. As in case \textbf{4(iii)}, if $R_{n+1}$ does not exist, let $HE_{special}$ be the lowermost HE between $R_n$ and $R$. Let the constant $d$ be the difference in latitude between $HE_n$ and $HE_{special}$, with $d \geq 0$ from the definition of the upper west pass which is above all passes and special vertices in the obstacle strip. Let $\textbf{z}$ be the uppermost point in the finite column $R_{n+1}$ if $R_{n+1}$ exists. In the following argument, we assume that $R_{n+1}$ exists and it will be clear how this also naturally treats the case when $R_n$ is connected to $R$ which contains the destination point.
In what follows, we will construct $5$ algorithms $K_1, \hdots, K_5$, by inspecting the tertiary rectangle.
We start by constructing the algorithm $K_1=(WS^{c_{uw}}EN^{c_{uw}})^{h_1} E^{h_2}$, where $h_1$ and $h_2$ are positive integers, such that if the robot starts from the point $\textbf{v}$ and follows $K_1$ it gets to the point $\textbf{t}$. We make use of the fact that in the west strip at each given longitude at least one of the rows $r_i$ and $r_j$, $j=i-c_{uw}$ contains a HE. We also make use of the fact that in the west strip the section of the row $r_i$ at the east of the upper west pipe is the complement of a path, followed by a path (which is nonempty from the existence of the upper west pass). However, we remark that if the robot starts on $r_i$ strictly at the west of $\textbf{v}$ and it follows $K_1$, it always remains strictly at the west of $\textbf{v}$, due to the HNE of the west pipe and the fact that there are no VNEs at the west of $\textbf{v}$.
We construct the algorithm $K_2=(\prod_{m=a+1}^{n} S^{k_m}N^{k_m}E)W $, where $k_m$ is a positive integer for all $a+1 \le m \le n$, such that if the robot starts from the point $\textbf{t}$ and follows $K_2$ it gets to the point $\textbf{w}$. More specifically, if the robot is in the upper infinite column $R_m$ in the column $c_m$ at the easternmost point of $HE_{m-1}$ and follows $S^{k_m}N^{k_m}E$, it gets in the upper infinite column $R_{m+1}$ in the column $c_{m+1}$ at the easternmost end of the $HE_{m}$, for $a+1 \le m \le n$. After the robot follows the last instruction in the product, $S^{k_m}N^{k_m}E$, it gets to $c_{n+1}$ at the easternmost point of $HE_n$ and so after it follows the last instruction in $K_2$, that is $W$, the robot gets to the point $\textbf{w}$.
We define the algorithm $K_3=S^d E N^d$, such that if the robot starts from $\textbf{w}$ and it follows $K_3$, it gets to the point $\textbf{z}$. However, we remark that if the robot starts on $r_i$ strictly at the west of $\textbf{v}$ and it follows $K_2 \text{ } K_3$, it always remains strictly at the west of $\textbf{v}$. Indeed, while the robot follows $K_2$ starting strictly at the west of $\textbf{v}$, the HNE of the west pipe prevents it from visiting longitudes greater than that of $\textbf{v}$. Hence, the robot could only potentially get to a large longitude by reaching $\textbf{v}$ after it follows $K_3$; however, this is impossible as the last instruction in $K_2$ is $W$.
We construct the algorithm $K_4=(\prod_{m=n+1}^{k} N^{k_m}N^{-k_m}E^{\epsilon_m})N^{k_{k+1}}N^{-k_{k+1}}$, where $\epsilon_m \in \{-1, 1\}$ and $k_m$ is an integer for all $n+1 \le m \le k+1$, such that if the robot starts from the point $\textbf{z}$ and follows $K_4$ it passes through the destination point. More specifically if the robot is at some specified latitude in the finite column $R_i$ and it follows $N^{k_i}N^{-k_i}E^{\epsilon_i}$, it gets to some specified latitude in the finite column $R_{i+1}$ for $n+1 \le i \le k$, where by convention we write $R_{k+1}$ for $R$. If the robot is at some specified latitude in $R$ and it follows $N^{k_{k+1}}N^{-k_{k+1}}$ it passes through the destination point. However, we remark that if the robot starts on $r_i$ strictly at the west of $\textbf{v}$ and it follows $K_4$ it always remains strictly at the west of $\textbf{v}$, as the robot follows the $E$ instructions at latitude $i$ and the HNE of the west pipe prevents it from visiting longitudes greater than that of $\textbf{v}$.
We finally construct the algorithm $K_5=E^{|K_4|+1}$ and note that if the robot starts on $r_i$ strictly at the west of $\textbf{v}$ and it follows $K_5$ it always remains strictly at the west of $\textbf{v}$.
We define the algorithm $K =K_1 K_2 K_3 K_4 K_5.$ Note that if the robot starts on the row $r_i$ strictly at the west of the point $\textbf{v}$ and it follows $K$ then it returns on the row $r_i$ strictly at the west of $\textbf{v}$. Indeed, the last part follows by the remarks we made on $K_1, \hdots, K_5$ individually and the first part follows from the fact that the robot does not meet any VNEs if it starts on the row $r_i$ strictly at the west of $\textbf{v}$ and it follows $K$. If the robot starts at the point $\textbf{v}$ and it follows $K$, then it visits the destination point; this follows directly from the definitions of $K_1, \hdots, K_5$. Finally, we claim that if the robot starts on the row $r_i$ strictly at the west of the point $\textbf{v}$ and it follows $K$, its longitude does not decrease. Indeed, the only $W$ instructions in $K$ occur either in $K_4$, which is followed by $K_5$ specifically designed to negate them or as the last instruction in $K_2$, which is preceded by an $E$ instruction. Therefore, the claim holds (see Figure~\ref{12}).
Finally, we define the algorithm
\[
F = N^i \text{ } MW (c_{uw}, q) \text{ } SME (\mu+\lambda+2q,\mu+\lambda+2q,K),
\]
where $\mu = |MW (c_{uw}, q)|.$
\begin{prop}
For any maze in $F_i$, after the robot follows $X \text{ } F$, it visits the destination point.
\end{prop}
\begin{proof}
We may assume without loss of generality that after the robot follows $X N^i$ it gets in the west strip on the row $r_i$. While the robot follows the algorithm $MW (c_{uw}, q)$ it gets on the row $r_i$ at $\textbf{v}$ or to the west of $\textbf{v}$. By the choice of parameters and by Lemma~\ref{l2}, if the robot is in the west strip on the row $r_i$ at the west of the point $\textbf{v}$ and it follows $SME (\mu+\lambda+2q,\mu+\lambda+2q,K)$, it advances eastwards oscillating about the row $r_i$. While the robot is on the row $r_i$ strictly at the west of $\textbf{v}$ and it follows $K$, it remains on the row $r_i$ strictly at the west of $\textbf{v}$. After defining $K$, we checked that it satisfies the conditions required in order to apply Lemma~\ref{l2}. Finally, the robot reaches the point $\textbf{v}$ not while executing $K$, but while executing a locomotory move in $SME$. Immediately afterwards, the robot follows $K$ and it gets to the destination point. The conclusion follows.
\end{proof}
\begin{figure}
\centering
\resizebox{0.5\textwidth}{!}{
\begin{tikzpicture}
[
dot/.style={circle,draw=black, fill,inner sep=2pt},
cross/.style={cross out, draw=black, minimum size=2*(#1-\pgflinewidth), inner sep=4pt, outer sep=4pt},
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\node[dot] at (\x,\y){ };
}}
\draw[line width=0.7mm, red] (-4, -4) -- (-4, 4);
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\draw[line width=0.7mm, red] (-2, -4) -- (-2, 4);
\draw[line width=0.7mm, red] (-1, -4) -- (-1, -3);
\draw[line width=0.7mm, red] (-1, -2) -- (-1, 4);
\draw[line width=0.7mm, red] (0, -4) -- (0, -2);
\draw[line width=0.7mm, red] (0, -1) -- (0, 4);
\draw[line width=0.7mm, red] (1, -2) -- (1, 4);
\draw[line width=0.7mm, red] (2, -3) -- (2, -1);
\draw[line width=0.7mm, red] (2, 0) -- (2, 4);
\draw[line width=0.7mm, red] (3, -4) -- (3, 4);
\draw[line width=0.7mm, red] (4, -4) -- (4, 4);
\draw[line width=0.7mm, red] (0, 0) -- (2, 0);
\draw[line width=0.7mm, red] (1, 1) -- (3, 1);
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\draw[line width=0.7mm, green] (-2, 2) -- (0, 2);
\draw[line width=0.7mm, green] (0, 3) -- (3, 3);
\draw[line width=0.7mm, red] (-1, 1) -- (0, 1);
\draw[line width=0.7mm, red] (1, -2) -- (2, -2);
\draw[line width=0.7mm, red] (3, -1) -- (4, -1);
\draw[line width=0.7mm, red] (-4, -3) -- (-2, -3);
\draw[line width=0.7mm, red] (-4, -4) -- (-1, -4);
\draw node[fill,circle,inner sep=5.5pt,minimum size=2pt] at (-2, 2) {};
\draw node[fill,circle,inner sep=5.5pt,minimum size=2pt] at (-4, 0) {};
\draw node[fill,circle,inner sep=5.5pt,minimum size=2pt] at (-1, -3) {};
\draw (-1.7,2.4) -- node {\textbf{v}} (-1.7,2.4);
\draw (-1, -2.6) -- node {\textbf{z}} (-1, -2.6);
\draw (2,-3) node[cross] {};
\foreach \x in {-4,...,4}
\draw (\x,.1) -- node[below,yshift=-1mm] {\x} (\x,-.1);
\foreach \y in {-4,...,4}
\draw (.1,\y) -- node[below,xshift=-3mm, yshift=3mm] {\y} (-.1,\y);
\draw[->,line width=0.15mm] (0,-4.5) -- (0,4.5);
\draw[->,line width=0.15mm] (-4.5,0) -- (4.5,0);
\end{tikzpicture}
}
\caption{\textbf{Part III, Case (4)(v).} There does not exist a magical west row, there does not exist a special almost empty west row, there does not exist an upper west pipe, there do not exist the special upper west paired HNEs, but there exists an upper west cutoff. We assume that there are no VEs removed other than the ones shown in the figure. Then the row $r_i = r_2$ is the complement of a path in the west strip and all the rows $r_k$ with $k\le j=i - c_{uw} \le -p$ are paths in the west strip and indeed paths in the entire maze. For the purpose of this example, we can take $c_{uw}$ to be any large constant, say $c_{uw}=100$. The points $\textbf{v}$ and $\textbf{z}$ are marked on the figure, $z= -3$, $j=i-c_{uw}=-98$ and $\textbf{w} = (-2, -103)$. Let us see what is the path of the robot as it follows $F = N^2 (S^{100} E N^{200} S^{100} W)^4 N^{105} K$ starting from $(-4, 0)$, where $K$ is any algorithm that takes the robot from $\textbf{v}$ to the destination point. When the robot follows $S^{100} E N^{200} S^{100} W$ starting from $(-4, 2)$, it first reaches a row which is a path after it executes $S^{100}$, so its longitude increases by $1$ after it executes $S^{100} E$; so after the robot executes $S^{100} E N^{200} S^{100}$ it is back on $r_2=r_i$ with its latitude increased by one, at $(-3, 2)$; the $W$ instruction at the end does not change the longitude of the robot, as $r_2$ is the complement of a path in the west strip. Similarly, after the robot follows $S^{100} E N^{200} S^{100} W$ starting from $(-3, 2)$ it gets to $\textbf{v} = (-2, 2)$. After the robot follows $S^{100} E N^{200} S^{100} W$ starting from $\textbf{v}$, it enters the lower infinite column on $c_{a+1}$: after $S^{100} E$ it is at $(a+1, j) = (-1, -98)$; after the robot follows $S^{100} E N^{200} S^{100}$, it is at $(-1, -103)$; finally, after the robot follows $S^{100} E N^{200} S^{100} W$, it is at $\textbf{w} = (-2, -103)$. Similarly, we can see that after the robot follows each subsequent instance of $S^{100} E N^{200} S^{100} W$ starting at $\textbf{w}$, it returns to $\textbf{w}$. After the robot follows enough instances of $S^{100} E N^{200} S^{100} W$ to reach $\textbf{w}$, it follows $N^{i+c_{uw}-z} = N^{105}$ and it reaches $\textbf{v}$; immediately afterwards, the robot follows $K$ and it reaches the destination point.} \label{13}
\end{figure}
\noindent
\textbf{4(v) } We assume that there does not exist a magical west row, there does not exist a special almost empty west row, there does not exist an upper west pipe, there do not exist the special upper west paired HNEs, but there exists an upper west cutoff. We recall that the upper west cutoff is the easternmost HNE in the west strip on the row $r_i$ which contains the upper west pass. Then the row $r_i$ is the complement of a path in the west strip and all the rows $r_k$ with $k\le j=i - c_{uw} \le -p$ are paths in the west strip and indeed paths in the entire maze (from the non existence of the special upper west paired HNEs). Let $\textbf{v} = (a, i)$ be the easternmost vertex of the row $r_i$ in the west strip. Let $\textbf{z} = (a+1, z)$ be the uppermost vertex of the westernmost lower infinite column in the column $c_{a+1}$. Let $\textbf{w} = (a, z-c_{uw})$. By inspecting the tertiary rectangle, we can construct an algorithm $K$ that takes the robot from $\textbf{v}$ to the destination point.
We define the algorithm
\[
F := N^i \text{ } (S^{i-j} E N^{2i-2j} S^{i-j} W)^{\lambda+q} \text{ } N^{i+c_{uw}-z} \text{ } K.
\]
\textbf{Claim. } For any maze in $F_i$, after the robot follows $X \text{ } F$, it visits the destination point.
\begin{proof}
We may assume without loss of generality that after the robot follows $X \text{ } N^i$ it gets in the west strip on the row $r_i$. By the choice of exponents, while the robot follows $(S^{i-j} E N^{2i-2j} S^{i-j} W)^{\lambda+q}$ it gets to the point $\textbf{w}$ and remains stuck there. Indeed, while the robot follows each instance of $S^{i-j} E N^{2i-2j} S^{i-j} W$, it advances one unit to the east, oscillating about the row $r_i$ until it gets to $\textbf{v}$. Immediately afterwards, it follows $S^{i-j} E N^{2i-2j} S^{i-j} W$ and gets to $\textbf{w}$. After the robot gets to $\textbf{w}$, after each other instance of $S^{i-j} E N^{2i-2j} S^{i-j} W$, the robot gets back to $\textbf{w}$. If the robot starts at $\textbf{w}$ and it follows $N^{i+c_{uw}-z}$, it gets to $\textbf{v}$. Therefore, after the robot follows $X \text{ } F$, it gets to the destination point. The conclusion follows.
\end{proof}
\begin{figure}
\centering
\resizebox{0.5\textwidth}{!}{
\begin{tikzpicture}
[
dot/.style={circle,draw=black, fill,inner sep=2pt},
cross/.style={cross out, draw=black, minimum size=2*(#1-\pgflinewidth), inner sep=4pt, outer sep=4pt},
]
\foreach \x in {-4,...,4}{
\foreach \y in {-4,...,4}{
\node[dot] at (\x,\y){ };
}}
\draw[line width=0.7mm, red] (-4, -4) -- (-4, 4);
\draw[line width=0.7mm, red] (-3, -4) -- (-3, 4);
\draw[line width=0.7mm, red] (-2, -4) -- (-2, 4);
\draw[line width=0.7mm, red] (-1, -4) -- (-1, 4);
\draw[line width=0.7mm, red] (0, -2) -- (0, 4);
\draw[line width=0.7mm, red] (1, -1) -- (1, 4);
\draw[line width=0.7mm, red] (1, -2) -- (1, -3);
\draw[line width=0.7mm, red] (2, -2) -- (2, 4);
\draw[line width=0.7mm, red] (2, -3) -- (2, -4);
\draw[line width=0.7mm, red] (3, -1) -- (3, 4);
\draw[line width=0.7mm, red] (3, -2) -- (3, -4);
\draw[line width=0.7mm, red] (4, -4) -- (4, 4);
\draw[line width=0.7mm, red] (3, 3) -- (4, 3);
\draw[line width=0.7mm, red] (1, 2) -- (3, 2);
\draw[line width=0.7mm, red] (-4, 1) -- (4, 1);
\draw[line width=0.7mm, red] (-4, 1) -- (4, 1);
\draw[line width=0.7mm, red] (-4, 0) -- (4, 0);
\draw[line width=0.7mm, red] (-3, -1) -- (-2, -1);
\draw[line width=0.7mm, red] (0, -1) -- (3, -1);
\draw[line width=0.7mm, red] (-4, -2) -- (-3, -2);
\draw[line width=0.7mm, red] (-2, -2) -- (-1, -2);
\draw[line width=0.7mm, red] (0, -2) -- (2, -2);
\draw[line width=0.7mm, red] (-3, -3) -- (-2, -3);
\draw[line width=0.7mm, red] (3, -3) -- (4, -3);
\draw[line width=0.7mm, red] (-3, -4) -- (-1, -4);
\draw[line width=0.7mm, green] (-1, 0) -- (0, 0);
\draw node[fill,circle,inner sep=5.5pt,minimum size=2pt] at (3, 2) {};
\draw node[fill,circle,inner sep=5.5pt,minimum size=2pt] at (1, -1) {};
\draw node[fill,circle,inner sep=5.5pt,minimum size=2pt] at (-3, 0) {};
\draw (3.3, 2.3) -- node {\textbf{v}} (3.3, 2.3);
\draw (1.3, -0.7) -- node {\textbf{w}} (1.3, -0.7);
\draw (1,-3) node[cross] {};
\foreach \x in {-4,...,4}
\draw (\x,.1) -- node[below,yshift=-1mm] {\x} (\x,-.1);
\foreach \y in {-4,...,4}
\draw (.1,\y) -- node[below,xshift=-3mm, yshift=3mm] {\y} (-.1,\y);
\draw[->,line width=0.15mm] (0,-4.5) -- (0,4.5);
\draw[->,line width=0.15mm] (-4.5,0) -- (4.5,0);
\end{tikzpicture}
}
\caption{\textbf{Part III, Case (4)(vi).} We assume there exists an upper west HNE on some row $r_j$. We further assume there does not exist a magical west row, there does not exist a magical east row and there does not exist an upper west cutoff and that there are no VEs removed other than the ones shown in the figure. Then all the rows $r_m$ with $i\le m < j$ are paths in the maze. In this example, the upper west pass is coloured green, the uppermost westernmost VNE is $\{ (1, -2), (1, -1) \}$, the upper west HNE is $\{(3, 2), (4, 2) \}$ and so $j=2$. The vertices $\textbf{v}$ and $\textbf{w}$ are marked on the figure. We can take $K = S^3 (WS)^2$, so if the robot follows $K$ starting from $\textbf{v}$ it visits the destination point. Let us observe how the robot follows $F = (ES^3 N^3)^{10}K$ starting from $(-3, 0)$. As long as the robot is in the west strip, each instance of $ES^3 N^3$ increases its longitude by one. Eventually, the robot gets to $(0, 0)$. After that, the robot follows $S^3 N^3$ and it gets to $(0,1)$. Considering that every row at latitude between $i=0$ and $j=2$ is a path in the maze, every further instance of $ES^3 N^3$ increases the longitude of the robot by one, until it arrives at $\textbf{v} = (3, 2)$, as its latitude is determined by the uppermost VNEs at the west of $\textbf{v}$. Once the robot reaches $\textbf{v}$, we can see that after each instance of $ES^3 N^3$, the robot returns to $\textbf{v}$. Finally, the robot follows $K$ and it visits the destination point.} \label{14}
\end{figure}
\noindent
\textbf{4(vi) } We assume there exists an upper west HNE on some row $r_j$. We recall that the upper west HNE is the lowermost westernmost HNE at the north-east of the uppermost westernmost VNE. We further assume there does not exist a magical west row, there does not exist a magical east row and there does not exist an upper west cutoff. Then all the rows $r_m$ with $i\le m < j$ are paths in the maze (from the minimality of $j$ and the non-existence of a magical east row). Let $\textbf{v}$ be the western vertex of the upper west HNE. Let $\textbf{w} = (x_w, y_w)$ be the upper vertex of the uppermost westernmost VNE. Then $\textbf{v}$ is at the east of $\textbf{w}$. By inspecting the tertiary rectangle, we construct an algorithm $K$ which takes the robot from $\textbf{v}$ to the destination point (see Figure~\ref{14}).
We define the algorithm
\[
F= N^i (ES^{j-y_w}N^{j-y_w})^{\lambda+q}K.
\]
\textbf{Claim. } For any maze in $F_i$, after the robot follows $X \text{ } F$, it visits the destination point.
\begin{proof}
We may assume without loss of generality that after the robot follows $X N^i$ it gets in the west strip on the row $r_i$. In the west strip, while the robot follows $ES^{j-y_w}N^{j-y_w}$ it advances eastwards oscillating about the row $r_i$. In the obstacle strip, while the robot follows $ES^{j-y_w}N^{j-y_w}$ it advances eastwards, potentially increasing its latitude as it meets VNEs. It eventually gets on the row $r_j$ and remains stuck at the point $\textbf{v}$. Therefore, after the robot follows $X \text{ } F$ it gets to the destination point. The conclusion follows (see Figure~\ref{14}).
\end{proof}
\begin{figure}
\centering
\resizebox{0.5\textwidth}{!}{
\begin{tikzpicture}
[
dot/.style={circle,draw=black, fill,inner sep=2pt},
cross/.style={cross out, draw=black, minimum size=2*(#1-\pgflinewidth), inner sep=4pt, outer sep=4pt},
]
\foreach \x in {-5,...,4}{
\foreach \y in {-4,...,4}{
\node[dot] at (\x,\y){ };
}}
\draw[line width=0.7mm, red] (-5, -4) -- (-5, 4);
\draw[line width=0.7mm, red] (-4, -4) -- (-4, 4);
\draw[line width=0.7mm, red] (-3, -4) -- (-3, 4);
\draw[line width=0.7mm, red] (-2, -4) -- (-2, 4);
\draw[line width=0.7mm, red] (-1, -4) -- (-1, 4);
\draw[line width=0.7mm, red] (0, -2) -- (0, 4);
\draw[line width=0.7mm, red] (1, -1) -- (1, 4);
\draw[line width=0.7mm, red] (1, -2) -- (1, -3);
\draw[line width=0.7mm, red] (2, -2) -- (2, 4);
\draw[line width=0.7mm, red] (2, -3) -- (2, -4);
\draw[line width=0.7mm, red] (3, -1) -- (3, 4);
\draw[line width=0.7mm, red] (3, -2) -- (3, -4);
\draw[line width=0.7mm, red] (4, -4) -- (4, 4);
\draw[line width=0.7mm, red] (-5, 4) -- (4, 4);
\draw[line width=0.7mm, red] (-5, 3) -- (4, 3);
\draw[line width=0.7mm, red] (-5, 2) -- (4, 2);
\draw[line width=0.7mm, red] (-5, 1) -- (4, 1);
\draw[line width=0.7mm, red] (-5, 0) -- (4, 0);
\draw[line width=0.7mm, red] (-5, -1) -- (-4, -1);
\draw[line width=0.7mm, red] (-3, -1) -- (-2, -1);
\draw[line width=0.7mm, red] (-1, -1) -- (3, -1);
\draw[line width=0.7mm, red] (-4, -2) -- (-3, -2);
\draw[line width=0.7mm, red] (-2, -2) -- (-1, -2);
\draw[line width=0.7mm, red] (0, -2) -- (2, -2);
\draw[line width=0.7mm, red] (-5, -3) -- (-4, -3);
\draw[line width=0.7mm, red] (-3, -3) -- (-2, -3);
\draw[line width=0.7mm, red] (3, -3) -- (4, -3);
\draw[line width=0.7mm, red] (-3, -4) -- (-1, -4);
\draw[line width=0.7mm, green] (-1, 0) -- (0, 0);
\draw node[fill,circle,inner sep=5.5pt,minimum size=2pt] at (-2, -1) {};
\draw node[fill,circle,inner sep=5.5pt,minimum size=2pt] at (0, -2) {};
\draw node[fill,circle,inner sep=5.5pt,minimum size=2pt] at (-2, 0) {};
\draw node[fill,circle,inner sep=5.5pt,minimum size=2pt] at (1, -2) {};
\draw node[fill,circle,inner sep=5.5pt,minimum size=2pt] at (-4, -1) {};
\draw (-1.7, -0.7) -- node {\textbf{v}} (-1.7, -0.7);
\draw (-3.7, -0.7) -- node {\textbf{v'}} (-3.7, -0.7);
\draw (0.3, -1.7) -- node {\textbf{w}} (0.3, -1.7);
\draw (-1.7, 0.3) -- node {\textbf{t}} (-1.7, 0.3);
\draw (1.3, -1.7) -- node {\textbf{z}} (1.3, -1.7);
\draw (1,-3) node[cross] {};
\foreach \x in {-5,...,4}
\draw (\x,.1) -- node[below,yshift=-1mm] {\x} (\x,-.1);
\foreach \y in {-4,...,4}
\draw (.1,\y) -- node[below,xshift=-4mm, yshift=3mm] {\y} (-.1,\y);
\draw[->,line width=0.15mm] (0,-4.5) -- (0,4.5);
\draw[->,line width=0.15mm] (-5.5,0) -- (4.5,0);
\end{tikzpicture}
}
\caption{\textbf{Part III, Case (4)(vii).} We assume there does not exist a magical west row, there does not exist a magical east row, there does not exist an upper west cutoff, there does not exist an upper west HNE, but there does exist a special west pipe on some row $r_j$. We assume that there are no VEs removed other than the ones shown in the figure. The upper west pass is coloured green and it is on the row $r_i=r_0$. From the assumptions, it follows that for every $m \geq i$, the row $r_m$ is a path in the maze. The special west pipe is $\{ (-3, -1), (-2, -1), (-1, -1) \}$ on $r_j = r_{-1}$. We take $R_{n+1}$ to be $\{(1, -2), (1, -3)\}$, accessible from $R_n = \{(0, -2), (0, -1), \hdots \}$ via $HE_{special} = \{(0, -2), (1, -2)\}$ on $r_{\gamma} = r_{-2}$. Then, if the robot follows $K_1 = NE^2S^3N^3W^2S$ starting from $\textbf{v}$, it gets to $\textbf{t}$ passing from $\textbf{w}$; however, note that if the robot follows $K_1$ starting from $\textbf{v'}$ (which is the eastern vertex of the HE of the ``fake west pipe'' $\{(-5, -1), (-4, -1), (-3, -1) \}$ on $r_j$ strictly at the west of $\textbf{v}$), it returns to $\textbf{v'}$. If the robot follows $K_2 = E^3 W S^2 E N^2$ starting from $\textbf{t}$, it gets to $\textbf{z}$; however, if the robot follows $K_2$ starting from $\textbf{v'}$ it gets back to $\textbf{v'}$; in general, we are certain that if the robot follows $K_2$ starting from $\textbf{v'}$ it either gets back to $\textbf{v'}$ or to the western neighbour of $\textbf{v'}$. If the robot follows $K_3 = NSW$ starting from $\textbf{z}$ it visits the destination point. In this case, $K_4 = E^4$. Therefore, if the robot follows $K_3 K_4$ starting either from $\textbf{v'}$ or from the western neighbour of $\textbf{v'}$, it gets to $\textbf{v'}$.} \label{15}
\end{figure}
\noindent
\textbf{4(vii) } We assume there does not exist a magical west row, there does not exist a magical east row, there does not exist an upper west cutoff, there does not exist an upper west HNE, but there does exist a special west pipe on some row $r_j$. We recall that the special west pipe is the west pipe (the easternmost configuration in the west strip of a HE followed by a HNE) on the smallest row that has a west pipe with respect to the standard well order on $\mathbb{Z}$. Then all the rows $r_m$ with $m \geq i$ are paths in the maze (from the non existence of an upper west HNE and the non existence of a magical east row).
Let $\textbf{v} = (x_v,j)$ be the eastern vertex of the HE of the special west pipe. Let $\textbf{w} = (a+1,y_w)$ be the lowermost vertex of the westernmost upper infinite column $R_{a+1}$. Let $\textbf{t} = (x_v,i)$ be the vertex at the intersection between the column $c_{x_v}$ and the row $r_i$. Let $\textbf{z} = (n+1, y_z)$ be the uppermost vertex of the finite column $R_{n+1}$ or the uppermost vertex of the lower infinite column $R_{n+1} = R$ that contains the destination point. The special case that the destination point is in the upper infinite column $R_{n+1} = R$ is much more easy and we will make a note on how to solve it before defining the finish algorithm $F$. Let $HE_{special}$ be a HE on some row $r_{\gamma}$ between the upper infinite column $R_n$ and the finite column $R_{n+1}$. Let $\textbf{v'}$ be the eastern vertex of the HE of any ``fake west pipe'', i.e. a configuration in the west strip on $r_j$ that is formed by a HE followed by a HNE, strictly at the west of the special west pipe (see Figure~\ref{15}).
We define the algorithm $K_1=N^{i-j}E^{a+1-x_v}S^{2i-j-y_w}N^{2i-j-y_w}W^{a+1-x_v}S^{i-j}$ with the property that if the robot starts from $\textbf{v}$ and follows $K_1$ it passes through the point $\textbf{w}$ and gets to the point $\textbf{t}$. However, if the robot starts at $\textbf{v'}$ and it follows $K_1$ then it returns at $\textbf{v'}$. The second statement follows from the fact that the robot moves at every instruction in $K_1$: indeed, while the robot executes $N^{i-j}$ starting from $\textbf{v'}$, it is in the west strip which contains no VNEs, so it changes its latitude to $i$; considering that $r_i$ is a path in the maze, when the robot continues to follow $E^{a+1-x_v}$, its longitude increases by exactly $a+1-x_v$ which is the exact difference in longitude between $\textbf{v}$ and the westernmost column in the obstacle strip, $c_{a+1}$; as $\textbf{v'}$ is strictly at the west of $\textbf{v}$, we conclude that after the robot follows $N^{i-j}E^{a+1-x_v}$ starting from $\textbf{v'}$, it is still in the west strip on the row $r_i$ which is a path in the maze; hence, if the robot follows $K_1$ starting from $\textbf{v'}$, it gets back to $\textbf{v'}$. Similarly, we can show the first statement about $K_1$, that if the robot starts from $\textbf{v}$ and follows $K_1$ it gets to the point $\textbf{t}$; in this case, we note that the only instructions in $K_1$ that do not change the position of the robot are instructions of type $S$ from the group $S^{2i-j-y_w}$ that occur immediately after the robot reaches $\textbf{w}$ (see Figure~\ref{15}).
We define the algorithm $K_2=E^{n+1-x_v}WS^{i-\gamma}EN^{i-\gamma}$ such that if the robot starts from $\textbf{t}$ and follows $K_2$ it gets to the point $\textbf{z}$. This is clear as the robot starts on $r_i$ which is a path, so after it follows $E^{n+1-x_v}W$ it gets at the point $(n, i)$ and so after it follows $K_2$ it is in $R_{n+1}$; moreover, as the upper west pass at latitude $i$ is above all the passes in the obstacle strip and so, in this case, also above all the VNEs, the robot actually gets to $\textbf{z}$ in $R_{n+1}$ after it follows $K_2$ starting from $\textbf{t}$. However, if the robot follows $K_2$ starting from $\textbf{v'}$, it does not move after it follows $E^{n+1-x_v}$ and its longitude decreases by $1$ after it follows $E^{n+1-x_v}W$. Hence, if the robot follows $K_2$ starting from $\textbf{v'}$, it either gets back to $\textbf{v'}$ or it gets to the western neighbour of $\textbf{v'}$ (see Figure~\ref{15}).
By inspecting the tertiary rectangle, we construct the algorithm $K_3$ of the form $K_3=(\prod_{m=n+1}^{k} N^{k_m}N^{-k_m}E^{\epsilon_m})N^{k_{k+1}}N^{-k_{k+1}}$, where $\epsilon_m \in \{-1, 1\}$ and $k_m$ is an integer for all $n+1 \le m \le k+1$, such that if the robot starts from the point $\textbf{z}$ and follows $K_3$ it passes through the destination point. More specifically, if the robot is at some specified latitude in the finite column $R_m$ and follows $N^{k_m}N^{-k_m}E^{\epsilon_m}$ it gets to some specified latitude in the finite column $R_{m+1}$ for $n+1 \le m \le k$, where by convention we write $R_{k+1}$ for $R$. If the robot is at some specified latitude inside $R$ and it follows $N^{k_{k+1}}N^{-k_{k+1}}$, it visits the destination point.
We construct the algorithm $K_4=E^{|K_3|+1}$. We note that from the structure of a fake west pipe and its position in the west strip, if the robot starts either at $\textbf{v'}$ or at the western neighbour of $\textbf{v'}$ and it follows $K_3 K_4$, it gets to $\textbf{v'}$.
We define the algorithm $K=K_1 K_2 K_3 K_4$ with the property that if the robot starts at $\textbf{v}$ and it follows $K$, it passes through the destination point. However, if the robot starts at $\textbf{v'}$ and it follows $K$, it gets back to $\textbf{v'}$. In the special case when $\textbf{z}$ does not exist and so the destination point $(n+1, \delta)$ is in the upper infinite column $R_{n+1}=R$ we define $K_2' = E^{n+1 - x_v} N^{\delta - i} S^{\delta - i}$. In this case we define $K = K_1 K_2'$ instead and we note that, as before, if the robot starts at $\textbf{v}$ and it follows $K$, it passes through the destination point; moreover, if the robot starts at $\textbf{v'}$ and it follows $K$, it gets back to $\textbf{v'}$.
We recall the algorithm $WPF(a,e):=(E^eWS^aEN^a)^e$, defined in the case \textbf{2(ii)}. Finally, we define the algorithm
\[
F= N^i \text{ } W^{\lambda-x_v} \text{ } S^{i-j} \text{ } (WPF(j-i, 2\lambda+q)KN^{i-j}ES^{i-j})^{2 \lambda+q}.
\]
\textbf{Claim. } For any maze in $F_i$, after the robot follows $X \text{ } F$, it visits the destination point.
\begin{proof}
We may assume without loss of generality that after the robot follows $X \text{ } N^i \text{ } W^{\lambda - x_v} \text{ } S^{i-j}$ it gets in the west strip on the row $r_j$ at the west of the point $\textbf{v}$. While the robot follows each instance of $WPF(j-i, 2\lambda+q)$ it advances eastwards to the easternmost vertex $\textbf{v'}$ of a HE of a fake west pipe on the row $r_j$. If $\textbf{v'}$ is strictly at the west of $\textbf{v}$, after the robot follows the algorithm $K$ it returns to the point $\textbf{v'}$; after the robot follows the algorithm $N^{i-j}ES^{i-j}$ starting from $\textbf{v'}$, it advances to the east of $\textbf{v'}$ on the row $r_j$. By the choice of parameters, the robot eventually gets to the point $\textbf{v'}=\textbf{v}$. Immediately afterwards, it follows $K$ and it gets to the destination point. The conclusion follows.
\end{proof}
\begin{figure}
\centering
\resizebox{0.5\textwidth}{!}{
\begin{tikzpicture}
[
dot/.style={circle,draw=black, fill,inner sep=2pt},
cross/.style={cross out, draw=black, minimum size=2*(#1-\pgflinewidth), inner sep=4pt, outer sep=4pt},
]
\foreach \x in {-5,...,4}{
\foreach \y in {-4,...,4}{
\node[dot] at (\x,\y){ };
}}
\draw[line width=0.7mm, red] (-5, -4) -- (-5, 4);
\draw[line width=0.7mm, red] (-4, -4) -- (-4, 4);
\draw[line width=0.7mm, red] (-3, -4) -- (-3, 4);
\draw[line width=0.7mm, red] (-2, -4) -- (-2, 4);
\draw[line width=0.7mm, red] (-1, -4) -- (-1, 4);
\draw[line width=0.7mm, red] (0, -2) -- (0, 4);
\draw[line width=0.7mm, red] (1, -1) -- (1, 4);
\draw[line width=0.7mm, red] (1, -2) -- (1, -3);
\draw[line width=0.7mm, red] (2, -2) -- (2, 4);
\draw[line width=0.7mm, red] (2, -3) -- (2, -4);
\draw[line width=0.7mm, red] (3, -1) -- (3, 4);
\draw[line width=0.7mm, red] (3, -2) -- (3, -4);
\draw[line width=0.7mm, red] (4, -4) -- (4, 4);
\draw[line width=0.7mm, red] (-5, 4) -- (4, 4);
\draw[line width=0.7mm, red] (-5, 3) -- (4, 3);
\draw[line width=0.7mm, red] (-5, 2) -- (4, 2);
\draw[line width=0.7mm, red] (-5, 1) -- (4, 1);
\draw[line width=0.7mm, red] (-5, 0) -- (4, 0);
\draw[line width=0.7mm, red] (-5, -1) -- (4, -1);
\draw[line width=0.7mm, red] (0, -2) -- (2, -2);
\draw[line width=0.7mm, red] (3, -3) -- (4, -3);
\draw[line width=0.7mm, green] (-1, 0) -- (0, 0);
\draw node[fill,circle,inner sep=5.5pt,minimum size=2pt] at (-1, -2) {};
\draw node[fill,circle,inner sep=5.5pt,minimum size=2pt] at (0, -2) {};
\draw node[fill,circle,inner sep=5.5pt,minimum size=2pt] at (-1, 2) {};
\draw node[fill,circle,inner sep=5.5pt,minimum size=2pt] at (-3, 0) {};
\draw (-0.7, -1.7) -- node {\textbf{v}} (-0.7, -1.7);
\draw (0.3, -1.7) -- node {\textbf{z}} (0.3, -1.7);
\draw (-0.7, 2.3) -- node {\textbf{w}} (-0.7, 2.3);
\draw (1,-3) node[cross] {};
\foreach \x in {-5,...,4}
\draw (\x,.1) -- node[below,yshift=-1mm] {\x} (\x,-.1);
\foreach \y in {-4,...,4}
\draw (.1,\y) -- node[below,xshift=-4mm, yshift=3mm] {\y} (-.1,\y);
\draw[->,line width=0.15mm] (0,-4.5) -- (0,4.5);
\draw[->,line width=0.15mm] (-5.5,0) -- (4.5,0);
\end{tikzpicture}
}
\caption{\textbf{Part III, Case (4)(viii).} We assume that there does not exist a magical west row, there does not exist a magical east row, there does not exist an upper west cutoff, there does not exist an upper west HNE, but there exists a natural special empty west row on $r_{-2} = r_j$. We assume that there are no VEs removed other than the ones shown in the figure. Let us assume that the robot starts at $(-3, 0)$ and it follows $F = N^{-2} (N^2 E S^6 N^4 W)^{10} S^4 K$, where $K = N(ES)^2$ is an algorithm with the property that if the robot follows it starting from $\textbf{v}$ it reaches the destination point. While the robot is on $r_j = r_{-2}$ strictly at the west of $\textbf{v}$, its longitude increases by one after each instance of $N^2 E S^6 N^4 W$. After the robot reaches $\textbf{v}$ and it follows $N^2 E S^6 N^4 W$, it gets to $\textbf{w}$. If the robot follows $N^2 E S^6 N^4 W$ starting from $\textbf{w}$ it gets back to $\textbf{w}$.} \label{16}
\end{figure}
\noindent
\textbf{4(viii) } We assume that there does not exist a magical west row, there does not exist a magical east row, there does not exist an upper west cutoff, there does not exist an upper west HNE, but there exists a natural special empty west row on $r_j$. Then, as in \textbf{4(vii)}, all the rows $r_m$ for $m \geq i$ are paths in the maze. Let $\textbf{v} = (a,j)$ be the easternmost vertex of the row $r_j$ in the west strip. Let $\textbf{z} = (a+1, \gamma)$ be the lowermost vertex of the westernmost upper infinite column $R_{a+1}$. Let $\textbf{w} = (a, 2i-\gamma)$. By inspecting the tertiary rectangle, we construct an algorithm $K$ that takes the robot from $\textbf{v}$ to the destination point.
We define the algorithm
\[
F=N^j \text{ } (N^{i-j}ES^{3i-2 \gamma-j}N^{2i-2 \gamma}W)^{\lambda+q} \text{ } S^{2i-\gamma-j} \text{ } K.
\]
\textbf{Claim. } For any maze in $F_i$, after the robot follows $X \text{ } F$, it visits the destination point.
\begin{proof}
We may assume without loss of generality that after the robot follows $X \text{ }N^j$, it gets in the west strip on the row $r_j$. While the robot follows each instance of $N^{i-j}ES^{3i-2\gamma-j}N^{2i-2\gamma}W$, it advances eastwards one unit making an oscillation about the row $r_j$. By the choice of exponent, the robot eventually gets to the point $\textbf{v}$. Immediately afterwards, it follows $N^{i-j}ES^{3i-2\gamma-j}N^{2i-2\gamma}W$ and gets to the point $\textbf{w}$. The robot remains stuck at $\textbf{w}$, i.e. while it follows each instance of $N^{i-j}ES^{3i-2\gamma-j}N^{2i-2\gamma}W$, it gets back to $\textbf{w}$ (see Figure~\ref{16}). Hence after the robot follows $X \text{ } N^i \text{ } W^{\lambda-a} \text{ } S^{i-j} \text{ } (N^{i-j}ES^{3i-2\gamma-j}N^{2i-2\gamma}W)^{\lambda+q}$ $S^{2i-\gamma-j}$, it gets to $\textbf{v}$. Hence, after the robot follows $X \text{ } F$, it gets to the destination point. The conclusion follows.
\end{proof}
\noindent
\textbf{4(ix) } As a final case, we may assume that there does not exist a magical west/east row, there does not exist a special west pipe, there does not exist a natural special empty west row, there does not exist a special almost empty west row. Then all the rows are paths in the maze and hence the maze does not contain any HNE. Therefore, both the latitude and the longitude of the robot are known and, by inspecting the primary rectangle, we can write an algorithm $F$ that takes the robot from its known position to the destination point. The conclusion follows.
This finally solves \textbf{Case (4)} in which the destination point is connected to the west strip by a (finite, possibly empty) sequence of finite columns followed by a (finite, non-empty) sequence of upper infinite columns.
We have therefore treated all possible cases, as detailed in the arguments above. This completes the proof of Theorem~\ref{ch2th2}.
\end{proof}
\section{Proof of Proposition~\ref{ch2prop1}}\label{sectiunea1}
In this short section we present a proof of the slightly technical but easy Proposition~\ref{ch2prop1}.
The following observation represents the main idea of the proof.
\begin{obs}\label{ch7obs1}
Let $o,d$ be fixed vertices in $\mathbb{Z}^2$ and let $\mathbb{B}$ be a set of subgraphs of $\mathbb{Z}^2$ which is compact in the product topology. Let $A$ be a possibly infinite algorithm that solves the set of mazes $\mathbb{A}=\{(B,o,d)\mid B\in \mathbb{B}\}$. Then there exists a finite initial segment $A_0$ of $A$ that solves $\mathbb{A}$.
\end{obs}
\begin{proof}
Assume for a contradiction that there does not exists such an initial segment $A_0$. For each $i\ge 1$, let $A_i$ be the initial segment of $A$ with the first $i$ instructions. By assumption, for each $i\ge 1$ there exists a board $B_i \in \mathbb{B}$ such that $A_i$ does not solve $B_i$. By compactness there exists a subsequence $(B_{i_j})_{j\ge 1}$ such that $\displaystyle \lim_{j\rightarrow \infty} B_{i_j}=B_0 \in \mathbb{B}$ in the product topology. As $A$ solves $B_0$, there exists an initial segment $A_0$ of $A$ which solves $(B_0,o,d)$. As $\displaystyle \lim_{j\rightarrow \infty} B_{i_j}=B_0 \in \mathbb{B}$, $A_0$ solves $(B_{i_j})_{j\ge 1}$ for all $j\ge |A_0|$ sufficiently large. This gives the desired contradiction.
\end{proof}
We are now ready to prove Proposition~\ref{ch2prop1}.
\begin{proof}[Proof of Proposition~\ref{ch2prop1}]
By hypothesis (1) and (3) and by Observation~\ref{ch7obs1},for all $i\in \{1,2\}$, all origins $o\in \mathbb{Z}^2$, all destination $d\in \mathbb{Z}^2$ and all paths $P$ between $o$ and $d$, there exists a finite initial segment $A_{i,P}$ of $A_i$ that solves the set of mazes $\{(M, o, d)\mid (M, o, d)\in \mathcal{A}_i, P\le M \}$ that contain the path $P$ (this set of mazes might be empty). By hypothesis (2), for all $i\in \{1,2\}$, all origins $o\in \mathbb{Z}^2$ and all $j\in \mathbb{N}$, there exists a finite initial segment $A_{i,o,j}$ of $A_i$ that guides the robot to visit all accessible points at distance at most $j$ from the origin $o$ in the set of mazes $\{(M, o, d)\mid (M, o, d)\in \mathcal{A}_i\}$ that have origin $o$ (notice that here the destination $d$ plays no role so we might as well drop it). But then for all $i\in \{1,2\}$ and all $j, k\in \mathbb{N}$, there exists a finite initial segment $A_{i,j,k}$ of $A_i$ such that for any origin $o$ at distance at most $k$ from $\mathbf{0}$ in the graph $\mathbb{Z}^2$, the algorithm guides the robot to visit all accessible points at distance at most $j$ from the origin $o$ in the set of mazes $\{(M, o)\mid (M, o)\in \mathcal{A}_i\}$ that have origin $o$.
In order to construct the algorithm $A$, we define the algorithms $B_i$ recursively to be $B_i=A_{\lfloor\frac{i}{2}\rfloor,(2|B_1\hdots B_{i-1}|+1),(2|B_1\hdots B_{i-1}|+1)}$ and take $A:=B_1B_2\hdots$. Clearly, the algorithm $A$ has the desired properties.
\end{proof}
\section{Open Problems}\label{conclusions}
As we emphasised in the proof of Theorem~\ref{mthm}, we strongly believe that there exists an algorithm which solves the set of all mazes with arbitrarily many HNEs and finitely many VNEs. The only case in our proof where an argument for this result breaks down is \textbf{Case 4} of \textbf{Part III}. We believe that this problem, together with Conjecture~\ref{plmconj} below could be solved using similar techniques with those developed in this paper.
\begin{conj}\label{plmconj}
There exists an algorithm that solves the set of all mazes with arbitrarily many HNEs and arbitrarily many VNEs in one column.
\end{conj}
Furthermore, we believe the following positive result to hold.
\begin{conj}
Consider the subset $\mathcal{N}$ of mazes in which the connected component of the origin is a simple (possibly infinite) path. Then there exists an algorithm that solves $\mathcal{N}$.
\end{conj}
In the opposite direction, we believe the following to be true.
\begin{conj}\label{mconj}
There is no algorithm that solves the class $\mathcal{M}$ of all mazes.
\end{conj}
From another perspective, let us call $\mathcal{M}_k \subseteq \mathcal{M}$ the set of mazes for which the destination is at distance $k$ from the origin. From Proposition~\ref{ch2prop1}, the following conjecture is equivalent to Conjecture~\ref{mconj}.
\begin{conj}\label{ojlk}
There exists a $k$ for which $\mathcal{M}_k$ is not solvable.
\end{conj}
Perhaps the following stronger results also hold.
\begin{conj}\label{pbneb}
Let $\mathcal{N}_3 \subset \mathcal{M}$ be the set of all mazes for which there are only HNEs between the pairs of columns $(c_{-4}, c_{-3})$ and $(c_3, c_4)$. Then there is no algorithm that solves $\mathcal{N}_3$.
\end{conj}
\begin{conj}
Conjecture~\ref{ojlk} holds for $k=10$.
\end{conj}
Conjecture~\ref{pbneb} is one of the main reasons why we think Conjecture~\ref{mconj} holds.
Finally, we strongly believe that the classes of mazes in higher dimensions arising from the lattice $\mathbb{Z}^k$ with suitable mild restrictions should represent a captivating further study.
|
{
"timestamp": "2018-04-17T02:12:09",
"yymm": "1804",
"arxiv_id": "1804.05439",
"language": "en",
"url": "https://arxiv.org/abs/1804.05439"
}
|
\section{Introduction}
Let $k$ be a perfect field of characteristic $p$ and let
$K$ be a local field of characteristic $p$ with residue
field $k$. Let $\mathrm{Aut}_k(K)$ denote the group of
continuous $k$-automorphisms of $K$ and let
$\sigma\in\mathrm{Aut}_k(K)$. Let $v_K$ denote the normalized
valuation of $K$ and let $\pi_K$ be a uniformizer for
$K$. The (lower) ramification number of $\sigma$ is
defined to be $i(\sigma)=v_K(\sigma(\pi_K)-\pi_K)-1$;
this value does not depend on the choice of $\pi_K$. We
say that $\sigma$ is a wild automorphism of $K$ if
$i(\sigma)\ge1$. In this paper we study the
ramification numbers of the $p^n$-powers of wild
automorphisms of $K$. Our most important tool is the
field of norms construction of Fontaine and
Wintenberger, which allows us to interpret $p$-adic Lie
subgroups of $\mathrm{Aut}_k(K)$ in terms of totally ramified
$p$-adic Lie extensions of local fields.
Automorphisms of local fields of characteristic $p$
are closely connected to power series over the residue
field $k$. Let $k((t))$ denote the field of formal
Laurent series in one variable over $k$. There is a
continuous $k$-isomorphism from $k((t))$ to $K$ which
carries $t$ to $\pi_K$; in particular, there is
$\eta_{\sigma}\in k[[t]]$ such that $\sigma(\pi_K)
=\eta_{\sigma}(\pi_K)$. The set of power series
\[\mathcal{A}(k)=\{a_0t+a_1t^2+a_2t^3+\cdots:
a_i\in k,\,a_0\not=0\}\]
forms a group with the operation
$\phi(t)\cdot\psi(t)=\phi(\psi(t))$, and the map
$\theta:\mathrm{Aut}_k(K)\rightarrow\mathcal{A}(k)$ defined by
$\theta(\sigma)=\eta_{\sigma^{-1}}(t)$ is a group
isomorphism. The results of this paper are phrased in
terms of elements and subgroups of $\mathrm{Aut}_k(K)$, but they
can be interpreted as statements about $\mathcal{A}(k)$. We work
with $\mathrm{Aut}_k(K)$ rather than $\mathcal{A}(k)$ because the functor
$\mathcal{F}$ described in Section~\ref{norms} maps Galois groups
to subgroups of $\mathrm{Aut}_k(K)$.
\section{The field of norms} \label{norms}
Let $k$ be a perfect field of characteristic $p$.
We define a category $\mathcal{L}$ whose objects are totally
ramified Galois extensions $E/F$, where $F$ is a local
field with residue field $k$, and $\mathrm{Gal}(E/F)$ is a
$p$-adic Lie group of dimension $d\ge1$. An
$\mathcal{L}$-morphism from $E/F$ to $E'/F'$ is defined to be a
continuous embedding $\rho:E\rightarrow E'$ such that
\begin{enum1}
\item $\rho$ induces the identity on $k$.
\item $E'$ is a finite separable extension of $\rho(E)$.
\item $F'$ is a finite separable extension of $\rho(F)$.
\end{enum1}
Let $\rho^*:\mathrm{Gal}(E'/F')\rightarrow\mathrm{Gal}(E/F)$ be the homomorphism
induced by $\rho$. It follows from conditions (2) and
(3) in the definition that $\rho^*$ has finite kernel
and finite cokernel.
Let $K$ be a local field with residue field $k$ and
let $\mathrm{Aut}_k(K)$ denote the group of continuous
automorphisms of $K$ which induce the identity map on
$k$. Define a metric on $\mathrm{Aut}_k(K)$ by setting
$d(\sigma,\tau)=2^{-a}$, where
$a=v_K(\sigma(\pi_K)-\tau(\pi_K))$ and $\pi_K$ is a
uniformizer for $K$. We define a category $\mathcal{C}$ whose
objects are pairs $(K,G)$, where $K$ is a local field of
characteristic $p$ with residue field $k$, and $G$ is a
closed subgroup of $\mathrm{Aut}_k(K)$ which is a compact
$p$-adic Lie group of dimension $d\ge1$. A
$\mathcal{C}$-morphism from $(K,G)$ to $(K',G')$ is defined to be
a continuous field embedding $\gamma:K\rightarrow K'$ such that
\begin{enum1}
\item $\gamma$ induces the identity on $k$.
\item $K'$ is a finite separable extension of
$\gamma(K)$.
\item $G'$ stabilizes $\gamma(K)$, and the image of $G'$
in $\mathrm{Aut}_k(\gamma(K))\cong\mathrm{Aut}_k(K)$ is an open subgroup
of $G$.
\end{enum1}
Let $\gamma^*:G'\rightarrow G$ be the map induced by $\gamma$.
It follows from conditions (2) and (3) in the definition
that $\gamma^*$ has finite kernel and finite cokernel.
Let $E/F$ be a totally ramified $p$-adic Lie
extension. Then the field of norms of $E/F$ is defined
\cite[Th.\,1.2]{wlie}. The field of norms $X_F(E)$ is a
local field of characteristic $p$ with residue field
$k$, and there is a faithful continuous $k$-linear
action of $\mathrm{Gal}(E/F)$ on $X_F(E)$. It follows from the
properties of the field of norms construction that there
is a functor $\mathcal{F}:\mathcal{L}\rightarrow\mathcal{C}$ defined by
\[\mathcal{F}(E/F)=(X_F(E),\mathrm{Gal}(E/F)).\]
Let $\mathcal{L}^{ab}$ denote the full subcategory of $\mathcal{L}$
consisting of extensions $E/F\in\mathcal{L}$ such that
$\mathrm{Gal}(E/F)$ is abelian, and let $\mathcal{C}^{ab}$ denote the
full subcategory of $\mathcal{C}$ consisting of pairs $(K,G)$
such that $G$ is abelian. Wintenberger proved the
following:
\begin{theorem} \label{equiv}
$\mathcal{F}$ induces an equivalence of categories from
$\mathcal{L}^{ab}$ to $\mathcal{C}^{ab}$.
\end{theorem}
\begin{proof}
See \cite{Wab} for an outline of the proof; see
\cite{WZp} and \cite{Llie} for the details.
\end{proof}
Let $(K,G)\in\mathcal{C}$. The lower ramification number of
$\sigma\in\mathrm{Aut}_k(K)$ is defined to be
$i(\sigma)=v_K(\sigma(\pi_K)-\pi_K)-1$, where $\pi_K$ is
any uniformizer for $K$. If $\sigma\not=\mathrm{id}_K$ then
$i(\sigma)$ is a nonnegative integer. For
$x\in\mathbb{R}_{\ge0}$ we define the $x$th lower ramification
subgroup of $G$ by $G_x=\{\sigma\in G:i(\sigma)\ge x\}$.
We also define the Hasse-Herbrand function
$\phi_G:\mathbb{R}_{\ge0}\rightarrow\mathbb{R}_{\ge0}$ by
\[\phi_G(x)=\int_0^x\frac{dt}{|G:G_t|}.\]
Since $G$ is compact the open subgroup $G_x$ of $G$ has
finite index. Hence $|G:G_x|<\infty$ for all $x\ge0$,
so $\phi_G$ is one-to-one.
For $\sigma\in G$ we define the upper ramification
number of $\sigma$ by $u^G(\sigma)=\phi_G(i(\sigma))$.
Note that while the lower ramification number
$i(\sigma)$ depends only on $\sigma$, the upper
ramification number $u^G(\sigma)$ depends on $G$ as
well. If $\sigma\not=\mathrm{id}_K$ then $u^G(\sigma)$ is a
nonnegative rational number, but not necessarily an
integer. For $x\ge0$ we define the $x$th upper
ramification subgroup of $G$ to be
$G^x=\{\sigma\in G:u^G(\sigma)\ge x\}$. Suppose
$\displaystyle\lim_{x\rightarrow\infty}\phi_G(x)=\infty$. Then $\phi_G$
is a bijection, so we may define
$\psi_G:\mathbb{R}_{\ge0}\rightarrow\mathbb{R}_{\ge0}$ by
$\psi_G(x)=\phi_G^{-1}(x)$. We have then
\[\psi_G(x)=\int_0^x|G:G^t|\,dt.\]
We say that $\sigma$ is a wild automorphism of $K$
if $i(\sigma)\ge1$. In this case we define
$i_n(\sigma)=i(\sigma^{p^n})$ and $u_n^G(\sigma)
=u^G(\sigma^{p^n})$ for $n\ge0$. Then
$(i_n(\sigma))_{n\ge0}$ and $(u_n(\sigma))_{n\ge0}$ are
increasing sequences. Let $H$ be the closure of the
subgroup of $\mathrm{Aut}_k(K)$ generated by $\sigma$. Then
$\sigma$ is a wild automorphism of $K$ if and only if
either $H$ is a cyclic $p$-group or $H\cong{\mathbb Z}_p$. Hence
if $G\le\mathrm{Aut}_k(K)$ is a pro-$p$ group then every
$\sigma\in G$ is a wild automorphism of $K$.
Suppose $E/F\in\mathcal{L}$. Then $\mathrm{Gal}(E/F)$ has a
filtration by upper ramification groups $\mathrm{Gal}(E/F)^x$
for $x\ge0$ (see \cite[IV]{cl} or \cite[III\,\S3]{FV}).
Since $E/F$ is a totally ramified $p$-adic Lie
extension, $E/F$ is arithmetically profinite
\cite[Th.\,1.2]{wlie}. In other words, for every
$x\ge0$ the upper ramification group $\mathrm{Gal}(E/F)^x$ has
finite index in $\mathrm{Gal}(E/F)$. Therefore we may define
Hasse-Herbrand functions
\[\psi_{E/F}(x)=\int_0^x|\mathrm{Gal}(E/F):\mathrm{Gal}(E/F)^t|\,dt\]
and $\phi_{E/F}(x)=\psi_{E/F}^{-1}(x)$. We define the
lower numbering for the ramification subgroups of
$\mathrm{Gal}(E/F)$ by setting
$\mathrm{Gal}(E/F)_x=\mathrm{Gal}(E/F)^{\phi_{E/F}(x)}$. The crucial
fact for our purposes is that the functor $\mathcal{F}$ respects
the ramification filtrations:
\begin{theorem} \label{ram}
Let $E/F\in\mathcal{L}$ and suppose $\mathcal{F}(E/F)\cong(K,G)$. Then
$\phi_G$ is onto, so $\psi_G=\phi_G^{-1}$ is defined.
Furthermore, for all $x\ge0$ the following hold:
\begin{enuma}
\item The isomorphism $\mathrm{Gal}(E/F)\cong G$ induces
isomorphisms $\mathrm{Gal}(E/F)_x\cong G_x$ and
$\mathrm{Gal}(E/F)^x\cong G^x$.
\item $\phi_{E/F}(x)=\phi_G(x)$ and
$\psi_{E/F}(x)=\psi_G(x)$.
\end{enuma}
\end{theorem}
\begin{proof} See \cite[Cor.\,3.3.4]{cn}. \end{proof}
Say that $a\ge0$ is an upper ramification break for
$E/F$ if $\mathrm{Gal}(E/F)^a\not=\mathrm{Gal}(E/F)^{a+\epsilon}$ for
all $\epsilon>0$. Say that $b\ge0$ is a lower
ramification break for $E/F$ if
$\mathrm{Gal}(E/F)_b\not=\mathrm{Gal}(E/F)_{b+\epsilon}$ for all
$\epsilon>0$. Let $E/F\in\mathcal{L}$ and let $(K,G)=\mathcal{F}(E/F)$.
Let $\boldsymbol{\sigma}\in\mathrm{Gal}(E/F)$ and let $\sigma$ be the
automorphism of $X_F(E)$ induced by $\boldsymbol{\sigma}$. We
define the upper and lower ramification numbers of
$\boldsymbol{\sigma}$ by $u(\boldsymbol{\sigma})=u^G(\sigma)$ and
$i(\boldsymbol{\sigma})=i(\sigma)$. If $\boldsymbol{\sigma}\in
\mathrm{Gal}(E/F)_1$ then for $n\ge0$ we set
$u_n(\boldsymbol{\sigma})=u(\boldsymbol{\sigma}^{p^n})$ and
$i_n(\boldsymbol{\sigma})=i(\boldsymbol{\sigma}^{p^n})$. It follows from
Theorem~\ref{ram} that
$i_n(\boldsymbol{\sigma})=\psi_{E/F}(u_n(\boldsymbol{\sigma}))$. Suppose
$\boldsymbol{\sigma},\boldsymbol{\tau}\in\mathrm{Gal}(E/F)$. Since
$i(\sigma\tau)\ge\min\{i(\sigma),i(\tau)\}$ we get
$i(\boldsymbol{\sigma}\boldsymbol{\tau})\ge\min\{i(\boldsymbol{\sigma}),i(\boldsymbol{\tau})\}$.
\section{$p$-adic Lie subgroups of $\mathrm{Aut}_k(K)$}
Suppose we have an isomorphism $\mathcal{F}(E/F)\cong(K,G)$
in the category $\mathcal{C}$. We wish to compute the absolute
ramification index $e_F$ of $F$ using the ramification
data of $G$. In Section~4 of \cite{liht} $e_F$ is
computed in the cases where $G\cong{\mathbb Z}_p$ or
$G\cong{\mathbb Z}_p\times{\mathbb Z}_p$. In \cite[Th.\,1.2]{wlie} it is
proved that $e_F=\infty$ if and only if
$\displaystyle\lim_{x\rightarrow\infty} \frac{|G:G_x|}{x}=0$. The
following theorem gives a general method for computing
$e_F$ in terms of the ramification data of $G$.
\begin{theorem}
Let $K$ be a local field of characteristic $p$ with
residue field $k$ and let $G$ be a $p$-adic Lie subgroup
of $\mathrm{Aut}_k(K)$ of dimension $d\ge1$. Assume there is
$E/F\in\mathcal{L}$ such that $\mathcal{F}(E/F)\cong(K,G)$. Set
$e_F=v_F(p)$, so that $e_F=\infty$ if $\char(F)=p$.
Then
\[\lim_{x\rightarrow\infty}\frac{\log_p(|G:G^x|)}{x}
=\frac{d}{e_F}.\]
\end{theorem}
\begin{proof} Suppose $\char(F)=p$. Set
$\displaystyle\lambda(x)=\frac{|G:G^x|}{\psi_G(x)}=
\frac{\psi_G'(x)}{\psi_G(x)}$, where $\psi_G'(x)$
denotes the left derivative of $\psi_G(x)$. Then
$\Lambda(x):=\log\psi_G(x)$ is an antiderivative of
$\lambda(x)$, and $\psi_G(x)=e^{\Lambda(x)}$. It
follows that
\begin{align} \nonumber
|G:G^x|&=\psi_G'(x)=e^{\Lambda(x)}\cdot\lambda(x) \\
\frac{\log_p(|G:G^x|)}{x}
&=\log_pe\cdot\frac{\Lambda(x)+\log\lambda(x)}{x}.
\label{nonneg}
\end{align}
By \cite[Th.\,1.2]{wlie} we have
$\displaystyle\lim_{x\rightarrow\infty}\lambda(x)=0$. It follows that
$\log\lambda(x)<0$ for sufficiently large $x$. Using
(\ref{nonneg}) we see that for sufficiently large $x$ we
have
\[0\le\frac{\log_p(|G:G^x|)}{x}
\le\log_pe\cdot\frac{\Lambda(x)}{x}.\]
Since $\Lambda'(x)=\lambda(x)$ goes to 0 as
$x\rightarrow\infty$, we have
$\displaystyle\lim_{x\rightarrow\infty}\frac{\Lambda(x)}{x}=0$.
Therefore
\[\lim_{x\rightarrow\infty}\frac{\log_p(|G:G^x|)}{x}=0
=\frac{d}{\infty}.\]
Now suppose $\char(F)=0$. By
\cite[III,\,Prop.\,3.1.3]{laz} there exists a sequence
$G\ge G(0)\ge G(1)\ge\dots$ of open normal subgroups of
the $p$-adic Lie group $G$ such that $G(n)/G(n+1)$ is an
elementary abelian $p$-group of rank $d$ for every
$n\ge0$. Since $G$ is compact, $G(0)$ has finite index
in $G$. Hence by setting $A=|G:G(0)|$ we get
$|G:G(n)|=Ap^{dn}$ for all $n\ge 0$. By Sen's theorem
\cite{sen} and Theorem~\ref{ram} there is $c>0$ such
that $G^{ne_F+c}\le G(n)\le G^{ne_F-c}$ for all $n\ge0$,
where we define $G^x=G$ for $x<0$. It follows that for
$x\ge c$ we have $G(a)\le G^x\le G(b)$ with
$\displaystyle a=\left\lceil\frac{x+c}{e_F}\right\rceil$ and
$\displaystyle b=\left\lfloor\frac{x-c}{e_F}\right\rfloor$. Hence
\[\log_pA+d\left\lceil\frac{x+c}{e_F}\right\rceil
\ge\log_p(|G:G^x|)\ge
\log_pA+d\left\lfloor\frac{x-c}{e_F}\right\rfloor.\]
Dividing these inequalities by $x$ and taking the limit
as $x\rightarrow\infty$ gives the theorem.
\end{proof}
For the rest of this section we restrict our
attention to the cases where $G\cong{\mathbb Z}_p^d$ for some
$d\ge1$. For groups $G$ of this form we set
$G(n)=\{\sigma^{p^n}:\sigma\in G\}$ for $n\ge0$. In
order to prove our next theorem we need some preliminary
results.
\begin{lemma} \label{Gx}
Let $G$ be a subgroup of $\mathrm{Aut}_k(G)$ such that
$G\cong{\mathbb Z}_p^d$ for some $d\ge1$. Assume that there
exist positive real numbers $C$ and $\lambda$ such that
for every $n\ge0$ and $\sigma\in G\smallsetminus G(1)$
we have $i_n(\sigma)<Cp^{\lambda n}$. Then for every
$x\ge0$ we have $|G:G_x|>(x/C)^{d/\lambda}$.
\end{lemma}
\begin{proof}
Let $n\ge0$ and let $\tau\in G_{Cp^{\lambda n}}$. If
$\tau\not\in G(n+1)$ then $\tau=\sigma^{p^m}$ for
some $m\le n$ and $\sigma\in G\smallsetminus G(1)$.
It follows that $i(\tau)=i_m(\sigma)\le i_n(\sigma)
<Cp^{\lambda n}$. This is a contradiction, so we have
$\tau\in G(n+1)$. It follows that
$G_{Cp^{\lambda n}}\le G(n+1)$, and hence that
\[|G:G_{Cp^{\lambda n}}|\ge|G:G(n+1)|=p^{d(n+1)}.\]
If $0\le x<C$ then the conclusion of the lemma certainly
holds. Suppose $x\ge C$. Then there is $n\ge0$ such
that $Cp^{\lambda n}\le x<Cp^{\lambda(n+1)}$. Therefore
\[|G:G_x|\ge|G:G_{Cp^{\lambda n}}|\ge p^{d(n+1)}
>(x/C)^{d/\lambda}.\qedhere\]
\end{proof}
\begin{prop} \label{breaks}
Let $F$ be a local field of characteristic $p$ with
perfect residue field $k$ and let $(a_n)_{n\ge0}$ be a
sequence of positive integers. Then the following
statements are equivalent:
\begin{enum1}
\item There exists a totally ramified ${\mathbb Z}_p$-extension
$E/F$ whose upper ramification sequence is
$(a_n)_{n\ge0}$.
\item $p\nmid a_0$, and for all $n\ge0$ we have
$a_{n+1}\ge pa_n$, with $p\nmid a_{n+1}$ if
$a_{n+1}>pa_n$.
\end{enum1}
\end{prop}
\begin{proof} This follows from \cite[Th.\,3]{ls}. Note
that the hypothesis in \cite{ls} that the Galois group
of the maximal abelian pro-$p$ extension of $F$ is a
free abelian pro-$p$ group is automatically satisfied
when $\char(F)=p$ (see Theorem~8 and Remark~5 of
\cite{mar}).
\end{proof}
\begin{cor} \label{any}
Let $E/F\in\mathcal{L}$, with $\char(F)=p$. Set $(K,G)=\mathcal{F}(E/F)$
and let $\sigma$ be a nontorsion element of $G_1$. Then
$u_n^G(\sigma)\ge p^n$ for all $n\ge0$.
\end{cor}
\begin{proof}
Let $H\cong{\mathbb Z}_p$ be the closure of the subgroup of $G$
generated by $\sigma$, and let $\boldsymbol{H}$ be the subgroup
of $\mathrm{Gal}(E/F)$ that corresponds to $H$. Set
$D=E^{\boldsymbol{H}}$. If $|G:H|$ is finite then
$(K,H)\cong\mathcal{F}(E'/F')$, with $F'=D$ and $E'=E$. If
$|G:H|$ is infinite then it follows from
\cite[Prop.\,3.4.1]{cn} that $(K,H)\cong\mathcal{F}(E'/F')$,
where $F'=X_F(D)$ and $E'=X_{D/F}(E)$ is the injective
limit of $X_F(C)$ over all finite subextensions $C/D$ of
$E/D$. In either case we get $(K,H)\cong\mathcal{F}(E'/F')$ with
$\char(F')=p$. Hence by Proposition~\ref{breaks} and
Theorem~\ref{ram} we have $u_n^G(\sigma)\ge p^n$ for all
$n\ge0$.
\end{proof}
The case $d=1$ of the following theorem can be
deduced from \cite[Th.\,3]{ls}; the case $d=2$ follows
from \cite[Th.\,3.1]{liht}.
\begin{theorem} \label{htd}
Let $K$ be a local field of characteristic $p$ with
residue field $k$. Let $G$ be a subgroup of $\mathrm{Aut}_k(K)$
such that $G\cong{\mathbb Z}_p^d$ for some $d\ge1$. Then the
following statements are equivalent:
\begin{enum1}
\item There is $M\in\mathbb{N}$ such that for every $n\ge M$ and
every $\sigma\in G\smallsetminus\{\mathrm{id}_K\}$ we
\vspace{1mm} have
$\displaystyle\frac{i_{n+1}(\sigma)-i_n(\sigma)}{i_n(\sigma)
-i_{n-1}(\sigma)}=p^d$.
\item For every $\sigma\in G\smallsetminus\{\mathrm{id}_K\}$ the
limit $\displaystyle\lim_{n\rightarrow\infty}\frac{i_n(\sigma)}{p^{dn}}$
exists and is nonzero.
\item There is $E/F\in\mathcal{L}$ such that $\char(F)=0$ and
$\mathcal{F}(E/F)\cong(K,G)$.
\end{enum1}
\end{theorem}
\begin{proof} The equivalence of statements (1) and (3)
is proved in Theorem~1 of \cite{hl}. We will prove
$(1)\Rightarrow(2)\Rightarrow(3)$.
\\[\smallskipamount]
$(1)\Rightarrow(2)$: Let $\sigma\in
G\smallsetminus\{\mathrm{id}_K\}$ and set
\[A=\frac{p^di_M(\sigma)-i_{M+1}(\sigma)}{p^d-1},
\hspace{1cm}
B=\frac{i_{M+1}(\sigma)-i_M(\sigma)}{p^d-1}.\]
Then for $n\ge M$ we
have $i_n(\sigma)=A+Bp^{d(n-M)}$. It follows that
\[\lim_{n\rightarrow\infty}\frac{i_n(\sigma)}{p^{dn}}
=\lim_{n\rightarrow\infty}\frac{A+Bp^{d(n-M)}}{p^{dn}}
=\frac{B}{p^{dM}}\not=0.\]
$(2)\Rightarrow(3)$: By Theorem~\ref{equiv} there is
$E/F\in\mathcal{L}$ such that $(K,G)\cong\mathcal{F}(E/F)$. Let
$S=G\smallsetminus G(1)$ and define $f:S\rightarrow\mathbb{R}$ by
$\displaystyle f(\sigma)=\sup_{n\ge0}\frac{i_n(\sigma)}{p^{dn}}$.
Since we are assuming that statement (2) holds, the
function $f$ is well-defined. We claim that $f$ is
locally constant. Let $\sigma\in S$ and set
$A=f(\sigma)$. Then $i_n(\sigma)\le Ap^{dn}$ for all
$n\ge0$. Let $\{\tau_1,\ldots,\tau_d\}$ be a
generating set for the ${\mathbb Z}_p$-module $G$. Since
$\displaystyle\lim_{n\rightarrow\infty}\frac{i_n(\tau_i)}{p^{dn}}>0$
there is $B_i>0$ such that $i_n(\tau_i)\ge B_ip^{dn}$
for all $n\ge0$. Choose $r\ge0$ such that $B_ip^{rn}>A$
for $1\le i\le d$. Then $i_{r+n}(\tau_i)>Ap^{dn}$ for
all $n\ge0$ and $1\le i\le d$. It follows that for
every $\gamma\in G(r)$ and $n\ge0$ we have
$i_n(\gamma)>Ap^{dn}$. Since $Ap^{dn}\ge i_n(\sigma)$
we get $i_n(\sigma\gamma)=i_n(\sigma)$. Therefore
$f(\sigma\gamma)=f(\sigma)$ for all $\gamma\in G(r)$,
which shows that $f$ is locally constant.
Since $S$ is compact there is $\sigma_0\in S$ such
that $f(\sigma_0)$ is the maximum value of $f$. Setting
$C=f(\sigma_0)+1$ we get $i_n(\sigma)<Cp^{dn}$ for all
$\sigma\in S$ and $n\ge0$. Hence by Lemma~\ref{Gx} we
have $|G:G_x|>x/C$ for all $x>0$. Therefore for $x\ge1$
we get
\[\phi_G(x)=1+\int_1^x\frac{dt}{|G:G_t|}
<1+\int_1^xCt^{-1}\,dt=1+C\log x.\]
Since $i_n(\sigma_0)<Cp^{dn}$ this implies
\[u_n^G(\sigma_0)=\phi_G(i_n(\sigma_0))<
1+C(\log C+(\log p)\cdot dn)\]
for all $n\ge0$. If $\char(F)=p$ then by
Corollary~\ref{any} we have $u_n^G(\sigma_0)\ge p^n$ for
all $n$, which gives a contradiction. It follows that
$\char(F)=0$.
\end{proof}
\begin{remark}
We easily see that statement (2) of Theorem~\ref{htd}
implies
\begin{enumerate}[(1)]
\setcounter{enumi}{3}
\item $\displaystyle\lim_{n\rightarrow\infty}
\frac{i_{n+1}(\sigma)}{i_n(\sigma)}=p^h$ for all
$\sigma\in G\smallsetminus\{\mathrm{id}_K\}$
\end{enumerate}
(cf.\ equation (\ref{Ht23}) below).
It follows from \cite[Th.\,3.1]{liht} that when $d=2$
statement (4) is equivalent to statements (1)--(3) of
Theorem~\ref{htd}. It would be interesting to know
whether this holds when $d\ge3$.
\end{remark}
\section{Heights of elements of $\mathrm{Aut}_k(K)$} \label{hts}
Let $K$ be a local field of characteristic $p$ with
residue field $k$ and let $\sigma$ be a wild
automorphism of $K$. There are several possible
definitions for the height of $\sigma$. (In all three
cases we set $\mathrm{Ht}_j(\sigma)=\infty$ if $\sigma$ has
finite order.)
\begin{enumerate}[(1)]
\setlength{\itemsep}{0cm}
\setlength{\parskip}{0cm}
\setlength{\parsep}{0cm}
\item Say $\mathrm{Ht}_1(\sigma)=h$ if there is $M\in\mathbb{N}$ such
that $\displaystyle\frac{i_{n+1}(\sigma)-i_n(\sigma)}{i_n(\sigma)
-i_{n-1}(\sigma)}=p^h$ for all $n\ge M$.
\item Say $\mathrm{Ht}_2(\sigma)=h$ if $\displaystyle\lim_{n\rightarrow\infty}
\frac{i_n(\sigma)}{p^{hn}}$ exists and is nonzero.
\item Say $\mathrm{Ht}_3(\sigma)=h$ if $\displaystyle\lim_{n\rightarrow\infty}
\frac{i_{n+1}(\sigma)}{i_n(\sigma)}=p^h$.
\end{enumerate}
Let $1\le j\le3$. Then $\mathrm{Ht}_j(\sigma)=h$ for at most
one $h>0$. If there is no $h>0$ with $\mathrm{Ht}_j(\sigma)=h$
we say that $\mathrm{Ht}_j(\sigma)$ is undefined.
The definition of $\mathrm{Ht}_1(\sigma)$ is implicit in
the definition of the height of an invertible stable
series given in \cite[Def.\,1.2]{lidynam}. The
definition of $\mathrm{Ht}_2(\sigma)$ is motivated by statement
(2) of Theorem~\ref{htd}, and also by statement (2) of
\cite[Th.\,3.1]{liht}. The definition of
$\mathrm{Ht}_3(\sigma)$ agrees with the definition of height
given in \cite[Def.\,1.1]{liht}.
It follows from the
proof of $(1)\Rightarrow(2)$ in Theorem~\ref{htd} that
if $\mathrm{Ht}_1(\sigma)=h$ then $\mathrm{Ht}_2(\sigma)=h$. Suppose
$\mathrm{Ht}_2(\sigma)=h$. Then
$\displaystyle\lim_{n\rightarrow\infty}\frac{i_n(\sigma)}{p^{hn}}=L$ for
some $L\not=0$. Hence
\begin{equation} \label{Ht23}
\lim_{n\rightarrow\infty}\frac{i_{n+1}(\sigma)}{i_n(\sigma)}
=\lim_{n\rightarrow\infty}
\frac{i_{n+1}(\sigma)/p^{h(n+1)}}{i_n(\sigma)/p^{hn}}
\cdot p^h=\frac{L}{L}\cdot p^h=p^h,
\end{equation}
so we have $\mathrm{Ht}_3(\sigma)=h$. In Example~\ref{3not2} we
will construct a wild automorphism $\sigma$ such that
$\mathrm{Ht}_3(\sigma)$ is defined but $\mathrm{Ht}_2(\sigma)$ is
undefined. In Example~\ref{2not1} we will construct a
wild automorphism $\tau$ such that $\mathrm{Ht}_2(\tau)$ is
defined but $\mathrm{Ht}_1(\tau)$ is undefined.
\begin{prop}
Let $\sigma,\sigma'$ be wild automorphisms of $K$ such
that $\mathrm{Ht}_j(\sigma)=h$, $\mathrm{Ht}_j(\sigma')=h'$, and
$\sigma\sigma'=\sigma'\sigma$. Let $1\le j\le3$ and
$\alpha\in{\mathbb Z}_p\smallsetminus\{0\}$. Then
\begin{enuma}
\item $\mathrm{Ht}_j(\sigma^{\alpha})=h$.
\item If $h<h'$ then $\mathrm{Ht}_j(\sigma\sigma')=h$.
\end{enuma}
\end{prop}
\begin{proof}
Let $w=v_p(\alpha)$. Then $i_n(\sigma^{\alpha})
=i_{n+w}(\sigma)$, so we get $\mathrm{Ht}_j(\sigma^{\alpha})
=\mathrm{Ht}_j(\sigma)=h$. If $h<h'$ then for $n$ sufficiently
large we have $i_n(\sigma)<i_n(\sigma')$, and hence
\[i_n(\sigma\sigma')=i(\sigma^{p^n}(\sigma')^{p^n})
=i(\sigma^{p^n})=i_n(\sigma).\]
Therefore $\mathrm{Ht}_j(\sigma\sigma')=\mathrm{Ht}_j(\sigma)=h$.
\end{proof}
Let $G$ be a closed subgroup of $\mathrm{Aut}_k(K)$ such
that $G\cong{\mathbb Z}_p^d$ and let $1\le j\le3$. It follows
from the proposition that $\mathrm{Ht}_j(\sigma)$ takes on at
most $d$ distinct values for $\sigma\in
G\smallsetminus\{\mathrm{id}_K\}$. Suppose that $\mathrm{Ht}_j(\sigma)$
is defined for every $\sigma\in G$. Then
$\mathrm{Ht}_3(\sigma)=\mathrm{Ht}_j(\sigma)$ for every $\sigma\in G$,
and for every $h>0$ the set
\[G[h]=\{\sigma\in G:\mathrm{Ht}_3(\sigma)\ge h\}\]
is a ${\mathbb Z}_p$-submodule of $G$ such that $G/G[h]$ is a
free ${\mathbb Z}_p$-module. We define the multiplicity of $h>0$
to be
\[m(h)=\mathrm{rank}_{{\mathbb Z}_p}(G[h])-\mathrm{rank}_{{\mathbb Z}_p}(G[h+\epsilon])\]
for sufficiently small $\epsilon>0$. Then $G$ has $d$
heights when they are counted with multiplicities.
Let $\sigma$ be a wild automorphism of $K$. The
possibilities for $\mathrm{Ht}_1(\sigma)$ are quite limited,
since if $\mathrm{Ht}_1(\sigma)=h$ then $p^h$ must be
rational. On the other hand, Proposition~\ref{exists}
below shows that if $h=1$ or $h\ge2$ then there exists a
wild automorphism $\sigma$ with
$\mathrm{Ht}_3(\sigma)=\mathrm{Ht}_2(\sigma)=h$. To prove the
proposition we need a lemma.
\begin{lemma} \label{H3}
Let $\sigma$ be a wild automorphism of $K$ such that
$\mathrm{Ht}_3(\sigma)=h$ for some $h>0$. Then for every
$\epsilon>0$ there exists $N\ge1$ such that for all
$n\ge N$ we have $p^{(h-\epsilon)n}\le i_n(\sigma)\le
p^{(h+\epsilon)n}$.
\end{lemma}
\begin{proof}
Let $\epsilon>0$. Since $\mathrm{Ht}_3(\sigma)=h$ there is
$M\ge1$ such that for all $n\ge M$ we have
\begin{align*}
|\log_p(i_{n+1}(\sigma))-\log_p(i_n(\sigma))-h|
&\le\textstyle\frac12\epsilon \\[.1cm]
|\log_p(i_n(\sigma))-\log_p(i_M(\sigma))-(n-M)h|
&\le\textstyle\frac12(n-M)\epsilon.
\end{align*}
Let $C=|\log_p(i_M(\sigma))-Mh|$ and
$N=\max\{M,\lceil2C/\epsilon\rceil\}$. Then for
$n\ge N$ we have
\[|\log_p(i_n(\sigma))-nh|\le\textstyle\frac12(n-M)\epsilon+C\le
\frac12n\epsilon+\frac12n\epsilon=n\epsilon.\]
It follows that $p^{(h-\epsilon)n}\le i_n(\sigma)\le
p^{(h+\epsilon)n}$ for $n\ge N$.
\end{proof}
\begin{prop} \label{exists}
Let $K$ be a local field of characteristic $p$ with
residue field $k$ and let $h>0$ be a real number. Then
the following statements are equivalent:
\begin{enum1}
\item There exists a wild automorphism $\sigma$ of $K$
such that $\mathrm{Ht}_2(\sigma)=h$.
\item There exists a wild automorphism $\sigma$ of $K$
such that $\mathrm{Ht}_3(\sigma)=h$.
\item Either $h=1$ or $h\ge2$.
\end{enum1}
\end{prop}
\begin{proof}
$(1)\Rightarrow(2)$: If $\mathrm{Ht}_2(\sigma)=h$ then
$\mathrm{Ht}_3(\sigma)=h$.
\\[\smallskipamount]
$(2)\Rightarrow(3)$: Let $\sigma$ be a wild automorphism
of $K$ such that $\mathrm{Ht}_3(\sigma)=h$, and let
$G=\sigma^{{\mathbb Z}_p}$ be the closure of the subgroup of
$\mathrm{Aut}_k(K)$ generated by $\sigma$. By
Theorem~\ref{equiv} there is a local field with residue
field $k$ and a totally ramified ${\mathbb Z}_p$-extension $E/F$
such that $\mathcal{F}(E/F)\cong(K,G)$. If $\char(F)=0$ then by
Theorem~\ref{htd} we get $h=1$. Suppose $\char(F)=p$
and $h<2$. Then there is $\delta$ such that
$0<\delta<1$ and $h<2-\delta$. Let
$\epsilon=2-\delta-h>0$. Then by Lemma~\ref{H3}
there is $N\ge1$ such that for $n\ge N$ we have
$i_n(\sigma)\le p^{(h+\epsilon)n}=p^{(2-\delta)n}$.
Hence there is $C>0$ such that
$i_n(\sigma)<Cp^{(2-\delta)n}$ for all $n\ge0$.
It follows from Lemma~\ref{Gx} that
$|G:G_x|>(x/C)^{1/(2-\delta)}$ for all $x\ge0$.
Therefore we have
\[\phi_G(x)=\int_0^x\frac{dt}{|G:G_t|}<
\int_0^xC^{1/(2-\delta)}t^{-1/(2-\delta)}\,dt
=Dx^{(1-\delta)/(2-\delta)}\]
with $D=\displaystyle C^{1/(2-\delta)}\frac{2-\delta}{1-\delta}$.
It follows that
\begin{align*}
u_n^G(\sigma)&=\phi_G(i_n(\sigma)) \\
&<D\cdot i_n(\sigma)^{(1-\delta)/(2-\delta)} \\
&<D\cdot(Cp^{(2-\delta)n})^{(1-\delta)/(2-\delta)} \\
&=DC^{(1-\delta)/(2-\delta)}p^{(1-\delta)n}.
\end{align*}
By Corollary~\ref{any} we have $u_n^G(\sigma)\ge p^n$.
This gives a contradiction, so we have $h\ge2$ if
$\char(F)=p$.
\\[\smallskipamount]
$(3)\Rightarrow(1)$: Let $h\in\{1\}\cup[2,\infty)$. To
prove statement (1) for $h$ it suffices by
Theorem~\ref{ram} to prove the following: There exists a
local field $F$ with residue field $k$ and a totally
ramified ${\mathbb Z}_p$-extension $E/F$ such that for any
generator $\boldsymbol{\sigma}$ of the ${\mathbb Z}_p$-module $\mathrm{Gal}(E/F)$,
the limit $\displaystyle\lim_{n\rightarrow\infty}
\frac{i_n(\boldsymbol{\sigma})}{p^{hn}}$ exists and is nonzero.
If $h=1$ we let $F$ be the field of fractions of
the ring of Witt vectors of $k$. Using
Artin-Schreier-Witt theory we construct a totally
ramified ${\mathbb Z}_p$-extension $E/F$. Let $\boldsymbol{\sigma}$ be a
generator for the ${\mathbb Z}_p$-module $\mathrm{Gal}(E/F)$. Then by
Theorem~\ref{htd} the limit
$\displaystyle\lim_{n\rightarrow\infty}\frac{i_n(\boldsymbol{\sigma})}{p^n}$
exists and is nonzero. If $h=2$ we let $F=k((t))$.
By Proposition~\ref{breaks} there exists a ${\mathbb Z}_p$-extension
$E/F$ such that for all $n\ge0$ the $n$th upper
ramification break of $E/F$ is $a_n=p^n$. The $n$th
lower ramification break of $E/F$ is then
$b_n=(p^{2n+1}+1)/(p+1)$. Hence if $\boldsymbol{\sigma}$ is a
generator for $\mathrm{Gal}(E/F)$ then
\[\lim_{n\rightarrow\infty}\frac{i_n(\boldsymbol{\sigma})}{p^{2n}}
=\lim_{n\rightarrow\infty}\frac{(p^{2n+1}+1)/(p+1)}{p^{2n}}
=\frac{p}{p+1}.\]
This proves (1) in the cases $h=1$ and $h=2$.
Let $h>2$. Then there is $n_0\ge1$ such that
\begin{equation} \label{assume}
\frac{p^{h-1}-1}{p-1}\ge1+\frac{1}{p^{(h-1)n}-2}
=\frac{p^{(h-1)n}-1}{p^{(h-1)n}-2}
\end{equation}
for all $n\ge n_0$. For $0\le n\le n_0$ let
\begin{align*}
a_n=\frac{p^{n+1}-1}{p-1} \hspace{1cm}
b_n=\frac{p^{2(n+1)}-1}{p^2-1}.
\end{align*}
Then $a_0=b_0=1$, and for $1\le n\le n_0$ we have
$b_n=b_{n-1}+p^n(a_n-a_{n-1})$. For $n>n_0$ we define
$a_n,b_n$ recursively by letting $a_n$ be the largest
integer such that $p\nmid a_n$ and
\begin{equation} \label{largest}
a_n\le a_{n-1}
+p^{-n}\left(\frac{p^{h(n+1)}-1}{p^h-1}-b_{n-1}\right)
\end{equation}
and setting $b_n=b_{n-1}+p^n(a_n-a_{n-1})$. Then
for $n>n_0$ we have
\begin{equation} \label{ange}
a_n>a_{n-1}
+p^{-n}\left(\frac{p^{h(n+1)}-1}{p^h-1}-b_{n-1}\right)-2.
\end{equation}
It follows that
\begin{equation} \label{ineq}
\frac{p^{h(n+1)}-1}{p^h-1}-2p^n<b_n
\le\frac{p^{h(n+1)}-1}{p^h-1}.
\end{equation}
We claim that
$\displaystyle a_n\le\frac{p^{(h-1)(n+1)}-1}{p^{h-1}-1}$ for all
$n\ge0$. Since $h>2$ this holds for $n\le n_0$. To
prove the claim for $n_0+1$ we note that since $h>2$ we
have
\[p^{-n_0-1}\sum_{i=0}^{n_0}(p^{hi}-p^{2i})
\le\sum_{i=0}^{n_0}p^{-i}(p^{hi}-p^{2i})
=\sum_{i=0}^{n_0}(p^{(h-1)i}-p^{i}).\]
It follows that
\begin{align*}
p^{-n_0-1}\left(\frac{p^{h(n_0+1)}-1}{p^h-1}
-\frac{p^{2(n_0+1)}-1}{p^2-1}\right)
&\le\frac{p^{(h-1)(n_0+1)}-1}{p^{h-1}-1}
-\frac{p^{n_0+1}-1}{p-1} \\
p^{-n_0-1}\left(p^{h(n_0+1)}
+\frac{p^{h(n_0+1)}-1}{p^h-1}-b_{n_0}\right)
&\le p^{(h-1)(n_0+1)}
+\frac{p^{(h-1)(n_0+1)}-1}{p^{h-1}-1}-a_{n_0} \\
a_{n_0}+p^{-n_0-1}\left(
\frac{p^{h(n_0+2)}-1}{p^h-1}-b_{n_0}\right)
&\le\frac{p^{(h-1)(n_0+2)}-1}{p^{h-1}-1}.
\end{align*}
Hence by (\ref{largest}) we get $\displaystyle a_{n_0+1}\le
\frac{p^{(h-1)(n_0+2)}-1}{p^{h-1}-1}$. Let $n\ge n_0+2$
and assume the claim holds for $n-1$. Since $n-1>n_0$
it follows from (\ref{largest}) and (\ref{ineq}) that
\begin{align*}
a_n&<a_{n-1}+p^{-n}\left(\frac{p^{h(n+1)}-1}{p^h-1}
-\frac{p^{hn}-1}{p^h-1}+2p^{n-1}\right)\\
&\le\frac{p^{(h-1)n}-1}{p^{h-1}-1}+p^{(h-1)n}+2p^{-1} \\
&=\frac{p^{(h-1)(n+1)}-1}{p^{h-1}-1}+2p^{-1}.
\end{align*}
Since $a_n$ is an integer the claim holds for $n$.
Hence by induction the claim holds for all $n\ge0$.
We claim that $a_{n+1}>pa_n$ for all $n\ge0$. For
$0\le n\le n_0-1$ we have $a_{n+1}=pa_n+1>pa_n$. Let
$n\ge n_0$. Then by the preceding paragraph and
(\ref{assume}) we get
\begin{align*}
(p-1)a_n&\le(p-1)\frac{p^{(h-1)(n+1)}-1}{p^{h-1}-1} \\
&\le p^{(h-1)(n+1)}-2.
\end{align*}
Hence by (\ref{ineq}) and (\ref{ange}) we get
\begin{align*}
pa_n&\le a_n+p^{(h-1)(n+1)}-2 \\
&\le a_n+p^{(h-1)(n+1)}+p^{-n-1}
\left(\frac{p^{h(n+1)}-1}{p^h-1}-b_n\right)-2 \\
&=a_n+p^{-n-1}
\left(\frac{p^{h(n+2)}-1}{p^h-1}-b_n\right)-2 \\
&<a_{n+1}.
\end{align*}
Set $F=k((t))$. Since $p\nmid a_n$ and
$a_{n+1}>pa_n$ for all $n\ge0$ it follows from
Proposition~\ref{breaks} that there is a totally
ramified ${\mathbb Z}_p$-extension $E/F$ whose upper ramification
breaks are $a_0,a_1,a_2,\ldots$. Hence by construction
the lower ramification breaks of $E/F$ are
$b_0,b_1,b_2,\ldots$. Let $\boldsymbol{\sigma}$ be a generator
for the ${\mathbb Z}_p$-module $\mathrm{Gal}(E/F)$. Then
$i_n(\boldsymbol{\sigma})=b_n$. It follows from (\ref{ineq})
that
\[\lim_{n\rightarrow\infty}\frac{i_n(\boldsymbol{\sigma})}{p^{hn}}
=\lim_{n\rightarrow\infty}\frac{b_n}{p^{hn}}
=\frac{p^h}{p^h-1}.\]
This proves (1) for the cases with $h>2$.
\end{proof}
\section{Some examples} \label{examples}
In this section we construct several examples which
illustrate and limit the results of the previous
section. We begin with an example of a subgroup $G$
of $\mathrm{Aut}_k(K)$ such that
\begin{enum1}
\item $G\cong{\mathbb Z}_p\times{\mathbb Z}_p$.
\item $\mathrm{Ht}_2(\gamma)$ is defined for all $\gamma\in G$.
\item There are $\sigma_1,\sigma_2\in
G\smallsetminus\{\mathrm{id}_K\}$ such that
$\mathrm{Ht}_2(\sigma_1)\not=\mathrm{Ht}_2(\sigma_2)$.
\end{enum1}
\begin{example}
Let $p>2$, set $F=\mathbb{F}_{\!p^2}((t))$, and let $F^{sep}$ be a
separable closure of $F$. Let $E_1/F$ be a totally
ramified ${\mathbb Z}_p$-subextension of $F^{sep}/F$ such that
\begin{enum1}
\item The sequence of upper ramification breaks of
$E_1/F$ is $p^4+1,p^8+1,p^{12}+1,\ldots$.
\item The automorphism $\boldsymbol{\phi}$ of $F$ which fixes $t$
and acts as the Frobenius on $\mathbb{F}_{\!p^2}$ extends to an
automorphism of $E_1$ which induces
$\boldsymbol{\gamma}\mapsto\boldsymbol{\gamma}^{-1}$ on $\mathrm{Gal}(E_1/F)$.
\end{enum1}
Such an extension can be constructed as follows. Set
\[\theta(X)=\frac{1+X}{1-X}=1+2X+2X^2+2X^3+\cdots
\in\mathbb{F}_p[[X]],\]
let $b\in\mathbb{F}_{\!p^2}$ satisfy $\boldsymbol{\phi}(b)=-b$, and set
$S=\{n\ge1:p\nmid n\}$. Then every element $v$ in the
group $1+t\mathbb{F}_{\!p^2}[[t]]$ of 1-units in $F$ can be expressed
uniquely in the form
\[v=\prod_{n\in S}\theta(t^n)^{\lambda_n}\cdot
\prod_{n\in S}\theta(bt^n)^{\mu_n}\]
with $\lambda_n,\mu_n\in{\mathbb Z}_p$. Let
$S_1=\{p^{4i+4}+1:i\ge0\}$ and
$S_2=S\smallsetminus S_1$. We define a continuous
homomorphism $\chi:F^{\times}\rightarrow{\mathbb Z}_p$ by setting
\begin{align*}
\chi(t)&=0 \\
\chi(r)&=0\text{ for }r\in\mathbb{F}_{\!p^2}^{\times} \\
\chi(\theta(t^n))&=0\text{ for }n\in S \\
\chi(\theta(bt^n))&=0\text{ for }n\in S_2 \\
\chi(\theta(bt^n))&=p^i\text{ for }n=p^{4i+4}+1.
\end{align*}
Let $E_1/F$ be the abelian extension associated to
$\chi$ by local class field theory. Then
$\mathrm{Gal}(E_1/F)\cong\chi(F^{\times})={\mathbb Z}_p$. Since
$p^{4(i+1)+4}+1>p(p^{4i+4}+1)$ for all $i\ge0$, the
sequence of upper ramification breaks of $E_1/F$ is
${p^4+1,p^8+1,p^{12}+1,\ldots}$. Since
$\theta(-X)=\theta(X)^{-1}$ we get
$\chi(\boldsymbol{\phi}(v))=-\chi(v)$ for all $v\in F^{\times}$.
Hence $\boldsymbol{\phi}$ stabilizes $E_1$ and induces
$\boldsymbol{\gamma}\mapsto\boldsymbol{\gamma}^{-1}$ on $\mathrm{Gal}(E_1/F)$.
We now construct another totally ramified
${\mathbb Z}_p$-subextension $E_2/F$ of $F^{sep}/F$. Let
$\alpha\in\mathbb{R}$ be the solution to the linear equation
\begin{equation} \label{alpha}
p^6-p-p^2\alpha+p\alpha
=p^{9/2}\alpha-p^{9/2}+p^{5/2}-p^{-3/2}\alpha.
\end{equation}
Then $\displaystyle\alpha=\frac{p^{5/2}+p}{p^{5/2}+1}\cdot p^{3/2}$
satisfies $p^{3/2}<\alpha<p^2$. For $n\ge0$ let $c_n$
be the smallest integer such that $c_n\ge p^{4n}\alpha$
and $p\nmid c_n$. Then $|c_n-(p^{4n}\alpha+1)|<1$.
Combining this inequality with the bounds on $\alpha$ we
get $c_n>p(p^{4n}+1)$ and $p^{4(n+1)}+1>pc_n$. Hence by
Proposition~\ref{breaks} there is a totally ramified
${\mathbb Z}_p$-extension $E_2'/\mathbb{F}_p((t))$ whose sequence of
upper ramification breaks is
\[p^0+1,c_0,p^4+1,c_1,p^8+1,c_2,p^{12}+1,c_3,\ldots.\]
Let $E_2=FE_2'$. Then $E_2/F$ is a totally ramified
${\mathbb Z}_p$-extension with the same upper ramification
sequence as $E_2'/\mathbb{F}_p((t))$. The automorphism
$\boldsymbol{\phi}$ of $F$ stabilizes $E_2$ and induces the
identity on $\mathrm{Gal}(E_2/F)$.
Viewing $E_1$ and $E_2$ as subfields of $F^{sep}$
we define $E=E_1E_2$. For $i=1,2$ let $E_i^{(1)}/F$ be
the unique $({\mathbb Z}/p{\mathbb Z})$-subextension of the
${\mathbb Z}_p$-extension $E_i/F$. Then $E_1^{(1)}/F$ has
ramification break $p^4+1$ and $E_2^{(1)}/F$ has
ramification break 2. Therefore
$E_1^{(1)}\cap E_2^{(1)}=F$ and $E_1^{(1)}E_2^{(1)}/F$
is a totally ramified $({\mathbb Z}/p{\mathbb Z})^2$-extension. It
follows that $E_1\cap E_2=F$, and that $E/F$ is totally
ramified. Set $\boldsymbol{G}=\mathrm{Gal}(E/F)$,
$\boldsymbol{H}_1=\mathrm{Gal}(E/E_2)$, and $\boldsymbol{H}_2=\mathrm{Gal}(E/E_1)$.
Then $\boldsymbol{H}_1\cap\boldsymbol{H}_2=\{\mathrm{id}_E\}$ and
$\boldsymbol{H}_1\boldsymbol{H}_2=\boldsymbol{G}$, so we get
\[\boldsymbol{G}\cong\boldsymbol{H}_1\times\boldsymbol{H}_2\cong{\mathbb Z}_p\times{\mathbb Z}_p.\]
Let $\boldsymbol{\sigma}_1$ be a generator for the
${\mathbb Z}_p$-module $\boldsymbol{H}_1$ and let $\boldsymbol{\sigma}_2$ be a
generator for the ${\mathbb Z}_p$-module $\boldsymbol{H}_2$. Since
$\boldsymbol{\phi}\in\mathrm{Aut}(F)$ can be extended to automorphisms of
both $E_1$ and $E_2$, there is an automorphism $\widetilde{\boldsymbol{\phi}}$
of $E$ such that $\widetilde{\boldsymbol{\phi}}|_F=\boldsymbol{\phi}$,
$\widetilde{\boldsymbol{\phi}}(E_1)=E_1$ and $\widetilde{\boldsymbol{\phi}}(E_2)=E_2$. It follows
from the constructions of $E_1/F$ and $E_2/F$ that
\begin{align*}
(\widetilde{\boldsymbol{\phi}}\circ\boldsymbol{\sigma}_1\circ\widetilde{\boldsymbol{\phi}}^{-1})|_{E_1}
&=\boldsymbol{\sigma}_1^{-1}|_{E_1} \\
(\widetilde{\boldsymbol{\phi}}\circ\boldsymbol{\sigma}_2\circ\widetilde{\boldsymbol{\phi}}^{-1})|_{E_2}
&=\boldsymbol{\sigma}_2|_{E_1}.
\end{align*}
We also have
\begin{align*}
(\widetilde{\boldsymbol{\phi}}\circ\boldsymbol{\sigma}_1\circ\widetilde{\boldsymbol{\phi}}^{-1})|_{E_2}
&=\mathrm{id}_{E_2}=\boldsymbol{\sigma}_1^{-1}|_{E_2} \\
(\widetilde{\boldsymbol{\phi}}\circ\boldsymbol{\sigma}_2\circ\widetilde{\boldsymbol{\phi}}^{-1})|_{E_1}
&=\mathrm{id}_{E_1}=\boldsymbol{\sigma}_2|_{E_1}.
\end{align*}
It follows that
\begin{align*}
\boldsymbol{\sigma}_1^{\boldsymbol{\phi}}
&=\widetilde{\boldsymbol{\phi}}\circ\boldsymbol{\sigma}_1\circ\widetilde{\boldsymbol{\phi}}^{-1}
=\boldsymbol{\sigma}_1^{-1} \\
\boldsymbol{\sigma}_2^{\boldsymbol{\phi}}
&=\widetilde{\boldsymbol{\phi}}\circ\boldsymbol{\sigma}_2\circ\widetilde{\boldsymbol{\phi}}^{-1}
=\boldsymbol{\sigma}_2.
\end{align*}
Let $\widetilde{v}_F$ denote the valuation on $E$
which extends $v_F$. Then
\[(\widetilde{v}_F\circ\widetilde{\boldsymbol{\phi}})|_F=v_F\circ\boldsymbol{\phi}
=v_F.\]
Since $v_F$ extends uniquely to a valuation on $E$
we get $\widetilde{v}_F\circ\widetilde{\boldsymbol{\phi}}=\widetilde{v}_F$.
It follows that for $\boldsymbol{\gamma}\in\boldsymbol{G}$ we have
$i(\boldsymbol{\gamma})=i(\boldsymbol{\gamma}^{\boldsymbol{\phi}})$. Writing
$\boldsymbol{\gamma}=\boldsymbol{\sigma}_1^a\boldsymbol{\sigma}_2^b$ with
$a,b\in{\mathbb Z}_p$ we get $\boldsymbol{\gamma}^{\boldsymbol{\phi}}
=\boldsymbol{\sigma}_1^{-a}\boldsymbol{\sigma}_2^b$, and hence
\[i(\boldsymbol{\gamma})\le i(\boldsymbol{\gamma}\gammabold^{\boldsymbol{\phi}})
=i(\boldsymbol{\sigma}_2^{2b})=i(\boldsymbol{\sigma}_2^b).\]
A similar argument gives $i(\boldsymbol{\gamma})\le
i(\boldsymbol{\sigma}_1^a)$. Since we also have $i(\boldsymbol{\gamma})\ge
\min\{i(\boldsymbol{\sigma}_1^a),i(\boldsymbol{\sigma}_2^b)\}$ we get
$i(\boldsymbol{\gamma})=\min\{i(\boldsymbol{\sigma}_1^a),i(\boldsymbol{\sigma}_2^b)\}$.
Hence for $x\ge0$ we have
$\boldsymbol{\sigma}_1^a\boldsymbol{\sigma}_2^b\in\boldsymbol{G}_x$ if and only if
$\boldsymbol{\sigma}_1^a\in\boldsymbol{G}_x$ and $\boldsymbol{\sigma}_2^b\in \boldsymbol{G}_x$.
For $n\ge0$ set $a_n=u(\boldsymbol{\sigma}_1^{p^n})$. Then
$\boldsymbol{\sigma}_1^{p^n}\in \boldsymbol{G}^{a_n}$, so we have
$\boldsymbol{\sigma}_1^{p^n}\boldsymbol{H}_2\in\boldsymbol{G}^{a_n}\boldsymbol{H}_2/\boldsymbol{H}_2$.
In addition, for $\epsilon>0$ we have
$\boldsymbol{\sigma}_1^{p^n}\not\in\boldsymbol{G}^{a_n+\epsilon}$.
Therefore $\boldsymbol{\sigma}_1^{p^n}\boldsymbol{\sigma}_2^b\not\in
\boldsymbol{G}^{a_n+\epsilon}$ for all $b\in{\mathbb Z}_p$, so we have
$\boldsymbol{\sigma}_1^{p^n}\boldsymbol{H}_2\not\in
\boldsymbol{G}^{a_n+\epsilon}\boldsymbol{H}_2/\boldsymbol{H}_2$. Let
$\overline{\boldsymbol{\sigma}}_1=\boldsymbol{\sigma}_1|_{E_1}$ denote the image of
$\boldsymbol{\sigma}_1$ in $\mathrm{Gal}(E_1/F)\cong\boldsymbol{G}/\boldsymbol{H}_2$.
Then $\overline{\boldsymbol{\sigma}}_1^{p^n}\in(\boldsymbol{G}/\boldsymbol{H}_2)^{a_n}$ and
for $\epsilon>0$ we have
$\overline{\boldsymbol{\sigma}}_1^{p^n}\not\in(\boldsymbol{G}/\boldsymbol{H}_2)^{a_n+\epsilon}$.
Thus $a_n$ is the $n$th upper ramification
break of $E_1/F$, so we have
$u(\boldsymbol{\sigma}_1^{p^n})=a_n=p^{4n+4}+1$ for $n\ge0$. A
similar argument shows that $u(\boldsymbol{\sigma}_2^{p^n})$ is
equal to the $n$th upper ramification break of $E_2/F$.
Therefore for $n\ge0$ we have
$u_{2n}(\boldsymbol{\sigma}_2)=p^{4n}+1$ and
$u_{2n+1}(\boldsymbol{\sigma}_2)=c_n$. Combining these facts we
get
\begin{equation} \label{index}
|\boldsymbol{G}:\boldsymbol{G}^x|=\begin{cases}
1&(0\le x\le2) \\
p^{3i+1}&(p^{4i}+1<x\le c_i) \\
p^{3i+2}&(c_i<x\le p^{4i+4}+1).
\end{cases}
\end{equation}
For sequences $(d_n)$ and $(e_n)$ we write
$d_n\sim e_n$ to indicate that $(d_n)$ is asymptotic
to $(e_n)$ as $n\rightarrow\infty$. Using (\ref{index}) we get
\begin{align} \nonumber
\psi_{E/F}(p^{4n}+1)
&=\int_0^{p^{4n}+1}|\boldsymbol{G}:\boldsymbol{G}^x|\,dx \\
&=2+\sum_{i=0}^{n-1}(p^{3i+1}(c_i-p^{4i}-1)
+p^{3i+2}(p^{4i+4}+1-c_i)) \nonumber \\
&=2+\sum_{i=0}^{n-1}(p^{7i+6}-p^{7i+1}+p^{3i+2}-p^{3i+1})
-\sum_{i=0}^{n-1}(p^{3i+2}-p^{3i+1})c_i \nonumber \\
&=2+(p^6-p)\frac{p^{7n}-1}{p^7-1}
+(p^2-p)\frac{p^{3n}-1}{p^3-1}
-(p^2-p)\sum_{i=0}^{n-1}p^{3i}c_i \nonumber \\
&\sim\frac{p^6-p}{p^7-1}p^{7n}
-(p^2-p)\sum_{i=0}^{n-1}p^{3i}\cdot p^{4i}\alpha
\nonumber \\
&=\frac{p^6-p}{p^7-1}p^{7n}
-(p^2-p)\frac{p^{7n}-1}{p^7-1}\alpha \nonumber \\
&\sim\frac{p^6-p-p^2\alpha+p\alpha}{p^7-1}p^{7n}.
\label{p4n}
\end{align}
\begin{align}
\psi_{E/F}(c_n)&=\psi_{E/F}(p^{4n}+1)
+p^{3n+1}(c_n-p^{4n}-1) \nonumber \\
&\sim\frac{p^6-p-p^2\alpha+p\alpha}{p^7-1}p^{7n}
+p^{3n+1}(p^{4n}\alpha-p^{4n}) \nonumber \\
&=\frac{p^8\alpha-p^8+p^6-p^2\alpha}{p^7-1}p^{7n}
\nonumber \\
&=\frac{p^{9/2}\alpha-p^{9/2}+p^{5/2}-p^{-3/2}\alpha}{p^7-1}
(p^{7/2})^{2n+1}. \label{an}
\end{align}
Let $(K,G)=\mathcal{F}(E/F)$ and let
$\sigma_1,\sigma_2\in G$ be the automorphisms of $K$
induced by $\boldsymbol{\sigma}_1,\boldsymbol{\sigma}_2\in\mathrm{Gal}(E/F)$.
Then the ${\mathbb Z}_p$-module $G\cong\boldsymbol{G}\cong{\mathbb Z}_p\times{\mathbb Z}_p$
is generated by $\sigma_1$ and $\sigma_2$. Since
\[i_n(\sigma_1)=i_n(\boldsymbol{\sigma}_1)=\psi_{E/F}(u_n(\boldsymbol{\sigma}_1))
=\psi_{E/F}(p^{4(n+1)}+1)\]
it follows from (\ref{p4n}) that $\mathrm{Ht}_2(\sigma_1)=7$.
Similarly, we have
\begin{align*}
i_{2n}(\sigma_2)&=i_{2n}(\boldsymbol{\sigma}_2)=\psi_{E/F}(u_{2n}(\boldsymbol{\sigma}_2))
=\psi_{E/F}(p^{4n}+1) \\
i_{2n+1}(\sigma_2)&=i_{2n+1}(\boldsymbol{\sigma}_2)
=\psi_{E/F}(u_{2n+1}(\boldsymbol{\sigma}_2))=\psi_{E/F}(c_n).
\end{align*}
Since $\alpha$ was chosen to satisfy (\ref{alpha})
it follows from (\ref{p4n}) and (\ref{an}) that
$\mathrm{Ht}_2(\sigma_2)=7/2$.
Let $\gamma\in G$. Then
$\gamma=\sigma_1^a\sigma_2^b$ for some $a,b\in{\mathbb Z}_p$.
Therefore for $n\ge0$ we have $\gamma^{p^n}
=\sigma_1^{ap^n}\sigma_2^{bp^n}$, and hence
$i(\gamma^{p^n})=
\min\{i(\sigma_1^{ap^n}),i(\sigma_2^{bp^n})\}$. Since
$\mathrm{Ht}_2(\sigma_1)=7$ and $\mathrm{Ht}_2(\sigma_2)=7/2$ there are
$L_1,L_2>0$ such that $i(\sigma_1^{p^n})\sim L_1p^{7n}$ and
$i(\sigma_2^{p^n})\sim L_2p^{\frac72n}$. Hence if
$b\not=0$ then $i(\gamma^{p^n})\sim L_2'p^{\frac72n}$ with
$L_2'=L_2p^{\frac72v_p(b)}$, while if $a\not=0$ and $b=0$
then $i(\gamma^{p^n})\sim L_1'p^{7n}$ with
$L_1'=L_1p^{7v_p(a)}$. Therefore $\mathrm{Ht}_2(\gamma)$ is
defined in all cases, and we have
\[\mathrm{Ht}_2(\sigma_1^a\sigma_2^b)=\begin{cases}
7/2&(b\not=0) \\
7&(a\not=0,\,b=0) \\
\infty&(a=b=0).
\end{cases}\]
It follows that
\[G[h]=\begin{cases}
G&(0<h\le7/2) \\
\sigma_1^{{\mathbb Z}_p}&(7/2<h\le7) \\
\{\mathrm{id}_K\}&h>7.
\end{cases}\]
\qed
\end{example}
Let $\sigma$ be a wild automorphism of $K$ and let
$h\ge1$. In Section~\ref{hts} we showed that if
$\mathrm{Ht}_1(\sigma)=h$ then $\mathrm{Ht}_2(\sigma)=h$, and if
$\mathrm{Ht}_2(\sigma)=h$ then $\mathrm{Ht}_3(\sigma)=h$. In order to
give examples which show that the converses of these
implications do not hold, we describe a method for
producing wild automorphisms of $K$ with specified
ramification data. Let $F$ be a local field of
characteristic $p$ with residue field $k$ and let
$(\nu_n)_{n\ge0}$ be a sequence such that
$\nu_0\in\mathbb{N}\smallsetminus p\mathbb{N}$ and
$\nu_n\in(\mathbb{N}\smallsetminus p\mathbb{N})\cup\{0\}$ for $n\ge1$.
Define a sequence $(a_n)_{n\ge0}$ by $a_0=\nu_0$ and
$a_n=pa_{n-1}+\nu_n$ for $n\ge1$. Then $p\nmid a_0$,
and for $n\ge0$ we have $a_{n+1}\ge pa_n$ with $p\nmid
a_{n+1}$ if $a_{n+1}\not=pa_n$. Therefore by
Proposition~\ref{breaks} there is a totally ramified
${\mathbb Z}_p$-extension $E/F$ whose upper ramification sequence
is $(a_n)_{n\ge0}$.
Let $\mathcal{F}(E/F)\cong(K,G)$ and let $\sigma$ be a
generator for $G$. Then $u_n^G(\sigma)=a_n
=\displaystyle\sum_{j=0}^n p^{n-j}\nu_j$ for $n\ge0$. In
addition, the lower ramification numbers of $\sigma$ are
given by
\begin{align}
i_n(\sigma)&=a_0+\sum_{h=1}^np^h(a_h-a_{h-1})
\nonumber \\
&=\sum_{h=0}^np^ha_h-\sum_{h=0}^{n-1}p^{h+1}a_h
\nonumber \\
&=\sum_{h=0}^n\sum_{j=0}^hp^{2h-j}\nu_j
-\sum_{h=0}^{n-1}\sum_{j=0}^hp^{2h-j+1}\nu_j
\nonumber \\
&=\sum_{j=0}^n\sum_{h=j}^np^{2h-j}\nu_j
-\sum_{j=0}^{n-1}\sum_{h=j}^{n-1}p^{2h-j+1}\nu_j
\nonumber \\
&=\sum_{j=0}^n\frac{p^{2n+2-j}-p^j}{p^2-1}\nu_j
-\sum_{j=0}^{n-1}\frac{p^{2n+1-j}-p^{j+1}}{p^2-1}
\nu_j \nonumber \\
&=\sum_{j=0}^n\frac{p^{2n+1-j}+p^j}{p+1}\nu_j.
\label{innu}
\end{align}
It follows that
\begin{align}
i_{n+1}(\sigma)-p^2i_n(\sigma)
&=p^{n+1}\nu_{n+1}+\sum_{j=0}^n
\frac{p^j-p^{j+2}}{p+1}\nu_j \nonumber \\
&=p^{n+1}\nu_{n+1}-(p-1)\sum_{j=0}^np^j\nu_j
\nonumber \\
\frac{i_{n+1}(\sigma)}{i_n(\sigma)}
&=p^2+\frac{1}{i_n(\sigma)}\left(p^{n+1}\nu_{n+1}
-(p-1)\sum_{j=0}^np^j\nu_j\right). \label{irat}
\end{align}
\begin{example} \label{3not2}
Let $\nu_0=a_0=b_0=1$, and for $n\ge1$ define
$\nu_n,a_n,b_n$ recursively by
\begin{align} \label{nun}
\nu_n&=\left\lfloor\frac{b_{n-1}}{np^n}
+(p-1)\sum_{j=0}^{n-1}p^{j-n}\nu_j\right\rfloor
+\gamma_n \\
a_n&=\sum_{j=0}^n p^{n-j}\nu_j \nonumber \\
b_n&=\sum_{j=0}^n\frac{p^{2n+1-j}+p^j}{p+1}\nu_j,
\label{bn}
\end{align}
where $\gamma_n\in\{0,1\}$ is chosen so that
$p\nmid\nu_n$. Then the construction above gives
$\sigma\in G$ such that $u_n^G(\sigma)=a_n$ and
$i_n(\sigma)=b_n$ for $n\ge0$. We claim that
$\mathrm{Ht}_3(\sigma)=2$ but $\mathrm{Ht}_2(\sigma)$ is undefined.
Using (\ref{nun}) we get
\[\nu_n=\frac{b_{n-1}}{np^n}
+(p-1)\sum_{j=0}^{n-1}p^{j-n}\nu_j+\epsilon_n,\]
with $|\epsilon_n|\le1$. Therefore by (\ref{irat}) we
have
\begin{equation} \label{quot}
\frac{i_n(\sigma)}{i_{n-1}(\sigma)}=p^2+\frac{1}{n}
+\frac{p^n\epsilon_n}{b_{n-1}}.
\end{equation}
Since $\nu_0=1$ it follows from (\ref{bn}) that
$b_{n-1}\ge\displaystyle\frac{p^{2n-1}+1}{p+1}$. Therefore
$\displaystyle\lim_{n\rightarrow\infty}
\frac{p^n\epsilon_n}{b_{n-1}}=0$, so we get
$\displaystyle\lim_{n\rightarrow\infty}
\frac{i_n(\sigma)}{i_{n-1}(\sigma)}=p^2$. Hence
$\mathrm{Ht}_3(\sigma)=2$.
On the other hand, since $i_0(\sigma)=b_0=1$ it
follows from (\ref{quot}) that
\[\frac{i_N(\sigma)}{p^{2N}}=\prod_{n=1}^N
\left(1+\frac{1}{np^2}+\frac{p^{n-2}\epsilon_n}{b_{n-1}}
\right).\]
Since $b_{n-1}>p^{2n-1}/(p+1)$ and $\epsilon_n\ge-1$ we
get $p^{n-2}\epsilon_n/b_{n-1}>-(p+1)p^{-n-1}$. Hence
\[\frac{i_N(\sigma)}{p^{2N}}>\prod_{n=1}^N
\left(1+\frac{1}{np^2}-(p+1)p^{-n-1}\right).\]
Since the product on the right diverges,
$\displaystyle\lim_{N\rightarrow\infty}\frac{i_N(\sigma)}{p^{2N}}$ is
undefined. Hence $\mathrm{Ht}_2(\sigma)$ and $\mathrm{Ht}_1(\sigma)$
are undefined. \qed
\end{example}
\begin{example} \label{2not1}
For $n\ge0$ let $\nu_n=\displaystyle\frac{p^{2n+1}+1}{p+1}+p$.
Then $\nu_n\in\mathbb{N}\smallsetminus p\mathbb{N}$, and the element
$\tau\in\mathrm{Aut}_k(K)$ associated to the sequence
$(\nu_n)_{n\ge0}$ satisfies
\begin{align*}
i_n(\tau)&=\sum_{j=0}^n\frac{p^{2n+1-j}+p^j}{p+1}
\left(\frac{p^{2j+1}+p^2+p+1}{p+1}\right) \\
&=\sum_{j=0}^n\frac{p^{2n+2+j}+p^{3j+1}
+(p^2+p+1)(p^{2n+1-j}+p^j)}{(p+1)^2} \\
&=\frac{1}{(p+1)^2}\left(\frac{p^{3n+3}-p^{2n+2}}{p-1}
+\frac{p^{3n+4}-p}{p^3-1}+(p^2+p+1)\frac{p^{2n+2}-1}{p-1}
\right) \\
&=\frac{1}{(p+1)^2}\left(\frac{p^{3n+3}-1}{p-1}
+p\cdot\frac{p^{3n+3}-1}{p^3-1}
+(p^2+p)\frac{p^{2n+2}-1}{p-1}\right) \\
&=\frac{1}{(p+1)^2}\left(
\frac{(p^2+2p+1)(p^{3n+3}-1)}{p^3-1}
+(p^2+p)\frac{p^{2n+2}-1}{p-1}\right) \\
&=\frac{p^{3n+3}-1}{p^3-1}+p\cdot
\frac{p^{2n+2}-1}{p^2-1}.
\end{align*}
It follows that $\displaystyle\lim_{n\rightarrow\infty}
\frac{i_n(\tau)}{p^{3n}}=\frac{p^3}{p^3-1}$, so
$\mathrm{Ht}_3(\tau)=\mathrm{Ht}_2(\tau)=3$. In addition, we get
\begin{align*}
i_{n+1}(\tau)-i_n(\tau)&=
\frac{p^{3n+6}-p^{3n+3}}{p^3-1}+p\cdot
\frac{p^{2n+4}-p^{2n+2}}{p^2-1} \\
&=p^{3n+3}+p^{2n+3} \\[.2cm]
\frac{i_{n+1}(\tau)-i_n(\tau)}
{i_n(\tau)-i_{n-1}(\tau)}
&=\frac{p^{3n+3}+p^{2n+3}}{p^{3n}+p^{2n+1}} \\
&=\frac{p^n+1}{p^{n-1}+1}\cdot p^2.
\end{align*}
Therefore $\mathrm{Ht}_1(\tau)$ is undefined.
\end{example}
|
{
"timestamp": "2018-04-17T02:06:26",
"yymm": "1804",
"arxiv_id": "1804.05222",
"language": "en",
"url": "https://arxiv.org/abs/1804.05222"
}
|
\section{Introduction }\label{secintro}
We consider the problem of minimizing a polynomial $f:{\mathbb R}^n\to{\mathbb R}$ over a compact set $\mathbf{K}\subseteq {\mathbb R}^n$.
That is, we consider the problem of computing the parameter:
\begin{equation*}\label{fmink}
f_{\min,\mathbf{K}}:= \min_{x\in \mathbf{K}}f(x).
\end{equation*}
We recall the following reformulation for $f_{\min,\mathbf{K}}$, established by Lasserre \cite{Las11}:
\begin{equation*}\label{fminkreform2}
f_{\min,\mathbf{K}}=\inf_{\sigma\in\Sigma[x]}\int_{\mathbf{K}}\sigma(x)f(x)d\mu(x) \ \ \mbox{s.t. $\int_{\mathbf{K}}\sigma(x)d\mu(x)=1$,}
\end{equation*}
where $\Sigma[x]$ denotes the set of sums of squares of polynomials, and $\mu$ is a signed Borel measure supported on $\mathbf{K}$.
\smallskip
\noindent
Given an integer $d\in {\mathbb N}$, by bounding the degree of the polynomial $\sigma\in \Sigma[x]$ by $2d$, Lasserre \cite{Las11} defined the parameter:
\begin{eqnarray}\label{fundr}
\underline{f}^{(d)}_{\mathbf{K}}:=\inf_{\sigma\in\Sigma[x]_d}\int_{\mathbf{K}}\sigma(x)f(x)d\mu(x) \ \ \mbox{s.t. $\int_{\mathbf{K}}\sigma(x)d\mu(x)=1$,}
\end{eqnarray}
where $\Sigma[x]_d$ consists of the polynomials in $\Sigma[x]$
with degree at most $2d$.
The inequality \smash{$f_{\min,\mathbf{K}}\le\underline{f}^{(d)}_{\mathbf{K}}$} holds for all $d\in{\mathbb N}$ and, in view of the identity (\ref{fminkreform2}), it follows that the sequence \smash{$\underline{f}^{(d)}_{\mathbf{K}}$} converges to $f_{\min,\mathbf{K}}$ as $d\rightarrow \infty$.
De Klerk and Laurent \cite{DKL MOR}
established the following rate of convergence for the sequence
$\underline{f}^{(d)}_{\mathbf{K}}$, when $\mu$ is the Lebesgue measure and ${\mathbf K}$ is a convex body.
\begin{theorem}\cite{DKL MOR}
\label{thm:dKLS}
Let $f\in {\mathbb R}[x]$, $\mathbf{K}$ a convex body, and $\mu$ the Lebesgue measure on $\mathbf{K}$. There exist constants $C_{f,{\mathbf K}}$ (depending only on $f$ and ${\mathbf K}$) and $d_{\mathbf K}\in {\mathbb N}$ (depending only on ${\mathbf K}$) such that
\begin{equation}\label{thmmaineq2}
\underline{f}^{(d)}_{\mathbf{K}}-f_{\min,\mathbf{K}} \le {C_{f,{\mathbf K}} \over {d}}\ \ \text{ for all } d\ge d_{\mathbf K}.
\end{equation}
That is, the following asymptotic convergence rate holds: $\underline{f}^{(d)}_{\mathbf{K}}-f_{\min,\mathbf{K}} = O\left( {1\over d}\right).$
\end{theorem}
This result was an improvement on an earlier result by De Klerk, Laurent and Sun \cite[Theorem~3]{KLS MPA}, who showed a convergence rate in \smash{$O(1/\sqrt d)$} (for ${\mathbf K}$ convex body or, more generallly, compact under a mild assumption).
As explained in \cite{Las11} the parameter $\underline{f}^{(d)}_{\mathbf{K}}$ can be computed using semidefinite programming,
assuming one knows the (generalised) moments of the measure $\mu$ on ${\mathbf K}$ with respect to some polynomial basis. Set
\begin{equation*}\label{mack}
m_{\alpha}(\mathbf{K}):=\int_{\mathbf{K}}b_{\alpha}(x)d\mu(x), \;
m_{\alpha,\beta}(\mathbf{K}):=\int_{\mathbf{K}}b_{\alpha}(x)b_{\beta}(x)d\mu(x) \ \ \ \mbox{ for } \alpha, \beta\in {\mathbb N}^n,
\end{equation*}
where the polynomials $\{b_\alpha\}$ form a basis for the space ${\mathbb R}[x_1,\ldots,x_n]_{2d}$ of polynomials of degree at most $2d$, indexed by
$N(n,2d)=\{\alpha\in {\mathbb N}^n: \sum_{i=1}^n\alpha_i\le 2d\}$.
For example, the standard monomial basis in ${\mathbb R}[x_1,\ldots,x_n]_{2d}$ is $b_\alpha(x) = x^\alpha := \prod_{i=1}^n x_i^{\alpha_i}$ for $\alpha\in N(n,2d)$, and then
$m_{\alpha,\beta}(\mathbf{K})= m_{\alpha+\beta}(\mathbf{K})$.
\smallskip
If $f(x)=\sum_{\beta\in N(n,d_0)}f_{\beta}b_{\beta}(x)$ has degree $d_0$, and
writing $\sigma\in\Sigma[x]_{d}$ as $\sigma(x)=\sum_{\alpha\in N(n,2d)}\sigma_{\alpha}b_{\alpha}(x)$, then the parameter $\underline{f}^{(d)}_{\mathbf{K}}$ in
(\ref{fundr}) can be computed as follows:
\begin{eqnarray}\label{eqSDP}
\underline{f}^{(d)}_{\mathbf{K}}&=&\min\sum_{\beta\in N(n,d_0)}f_{\beta}\sum_{\alpha\in N(n,2d)}\sigma_{\alpha}m_{\alpha,\beta}(\mathbf{K})\label{fundr2}\\
& &\mbox{ s.t. } \ \ \sum_{\alpha\in N(n,2d)}\sigma_{\alpha}m_{\alpha}(\mathbf{K})=1,\nonumber\\
&&\ \ \ \ \ \ \ \sum_{\alpha\in N(n,2d)}\sigma_{\alpha}b_{\alpha}(x)\in\Sigma[x]_d.\nonumber
\end{eqnarray}
Since the sum-of-squares condition on $\sigma$ may be written as a linear matrix inequality, this is a semidefinite program.
In fact, since the program (\ref{eqSDP}) has only one linear equality constraint, using semidefinite programming duality it can be rewritten as a generalised eigenvalue problem.
In particular,
\smash{ $\underline{f}_{\mathbf{K}}^{(d)}$} {is equal to the the smallest generalised eigenvalue} of the system:
\[
Ax = \lambda Bx \quad \quad\quad (x \neq 0),
\]
where the symmetric matrices $A$ and $B$ are of order ${n + d \choose d}$ with rows and columns indexed by $N(n,d)$,
and
\begin{equation}
\label{matrices A and B}
A_{\alpha, \beta} = \sum_{\delta \in N(n,d_0)} f_\delta \int_{\mathbf{K}} b_{\alpha}(x) b_{\beta}(x) b_{\delta}(x) d\mu(x),
\quad B_{\alpha, \beta} = \int_{\mathbf{K}} b_\alpha(x)b_{\beta}(x)d\mu(x) \quad \text{ for } \alpha, \beta \in {N}(n,d).
\end{equation}
For more details, see \cite{Las11,KLS MPA}.
In particular, if the basis $\{b_\alpha\}$ is orthonormal
with respect to the measure $\mu$, then $B$ is the identity matrix, and \smash{$\underline{f}_{\mathbf{K}}^{(d)}$} is
the smallest eigenvalue of the above matrix $A$. For further reference we summarize this result, {which will play a central role in our approach.}
\begin{lemma}\label{lemsummarize}
Assume $\{b_\alpha:\alpha\in N(n,2d)\}$ is a basis of the space ${\mathbb R}[x_1,\ldots,x_n]_{2d}$, which is orthonormal w.r.t. the measure $\mu $ on $K$, i.e.,
$\int_K b_\alpha(x)b_{\beta}(x)d\mu(x)=\delta_{\alpha,\beta}$. Then the parameter \smash{$\underline{f}^{(d)}_{\mathbf{K}}$} is equal to the smallest eigenvalue of the matrix $A$ in (\ref{matrices A and B}).
\end{lemma}
Under the conditions of the lemma, note in addition that, if the vector $u=(u_\alpha)_{\alpha\in N(n,d)}$ is an eigenvector of the matrix $A$ in (\ref{matrices A and B}) for its smallest eigenvalue, then the (square) polynomial
$\sigma(x)=(\sum_{\alpha\in N(n,d)}u_\alpha x^\alpha)^2$ is an optimal density function for the parameter \smash{$\underline{f}^{(d)}_{\mathbf{K}}$}.
\subsubsection*{Related hierarchy by De Klerk, Hess and Laurent}
For the hypercube ${\mathbf K} = [-1,1]^n$, De Klerk, Hess and Laurent \cite{DHL SIOPT} considered a variant on the Lasserre hierarchy \eqref{fundr},
where the density function $\sigma$ is allowed to take the more general form
\begin{equation}\label{eqSch}
\sigma(x) = \sum_{I \subseteq \{1,\ldots,n\}} \sigma_I(x) \prod_{i\in I} (1-x_i^2)
\end{equation}
and the polynomials $\sigma_I$ are sum-of-squares polynomials with degree at most $2d-2|I|$ (to ensure that the degree of $\sigma$ is at most $2d$), and $I = \emptyset$ is included in the summation.
Moreover the measure $\mu$ is fixed to be
\begin{equation}\label{eqmu}
d\mu(x) = \left(\prod_{i=1}^n \sqrt{1-x_i^2} \right)^{-1}dx_1\cdots dx_n.
\end{equation}
As we will recall below, this measure is associated with the Chebyshev orthogonal polynomials.
We let \smash{$f^{(d)}$} denote the parameter\footnote{We drop the dependence on ${\mathbf K}$ which is implictly selected to be the box $[-1,1]^n$.} obtained by using in (\ref{fundr}) these choices (\ref{eqSch}) of density functions $\sigma(x)$ and (\ref{eqmu}) of measure $\mu$.
By construction, we have
$$f_{\min,{\mathbf K}} \le f^{(d)}\le \underline{f}^{(d)}_{\mathbf{K}}.$$
De Klerk, Hess and Laurent \cite{DHL SIOPT} proved a stronger convergence rate for the bounds $f^{(d)}$.
\begin{theorem}\cite{DHL SIOPT}
\label{theoDKHL}
Let $f\in{\mathbb R}[x]$ be a polynomial and ${\mathbf K}=[-1,1]^n.$ We have
$$f^{(d)}-f_{\min,{\mathbf K}} = O\left({1\over d^2}\right).$$
\end{theorem}
\subsubsection*{Contribution of this paper}
{In this paper we investigate the rate of convergence of the hierarchies \smash{$\underline{f}^{(d)}_{\mathbf{K}}$} and
\smash{$f^{(d)}$} to $f_{\min,{\mathbf K}}$ for the case of the box ${\mathbf K}=[-1,1]^n$. The above discussion raises naturally the following questions:
\begin{itemize}
\item[$\bullet$] Is the sublinear convergence rate $f^{(d)}-f_{\min,{\mathbf K}} = O\left({1\over d^2}\right)$ tight, or can this result be improved?
\item[$\bullet$] Does this convergence rate extend to the Lasserre bounds, where we restrict to sums-of-squares density functions?
\end{itemize}
We give a positive answer to both questions.
Regarding the first question we show that the convergence rate is $\Omega(1/d^2)$ when $f$ is a linear polynomial,
which implies that the convergence analysis in Theorem \ref{theoDKHL} for the bounds \smash{$f^{(d)}$} is tight.
This relies on the eigenvalue reformulation of the bounds (from Lemma \ref{lemsummarize}) and an additional link to the extremal zeros of the associated Chebyshev
polynomials. We also show that the same lower bound holds for the convergence rate of the Lasserre bounds
\smash{$\underline{f}^{(d)}_{\mathbf{K}}$} when considering measures on the hypercube corresponding to general Jacobi polynomials.}
{Regarding the second question we show that also the Lasserre bounds have a $O(1/d^2)$ convergence rate when using the Chebyshev type measure from (\ref{eqmu}). The starting point is again the reformulation from Lemma \ref{lemsummarize} in terms of eigenvalues, combined with some further analytical arguments.}
{The paper is organised as follows. In Section \ref{secprel} we group preliminary results about orthogonal polynomials and their extremal roots. Then, in Section \ref{secLas} we analyse the convergence rate of the Lasserre bounds
\smash{$\underline{f}^{(d)}_{\mathbf{K}}$} when $f$ is a linear polynomial
and, in Section \ref{secDKHL}, we analyse the bounds \smash{$f^{(d)}$}.
In both cases we show a $\Omega(1/d^2)$ lower bound. In Section \ref{secupper} we show a $O(1/d^2)$
upper bound for the convergence rate of the Lasserre bounds \smash{$\underline{f}^{(d)}_{\mathbf{K}}$}, and this analysis is tight in view of the previously shown lower bounds.}
\subsubsection*{Notation}
We recap here some notation that is used throughout.
For an integer $d\in{\mathbb N}$, ${\mathbb R}[x]_d$ denotes the set of $n$-variate polynomials in the variables $x=(x_1,\ldots,x_n)$ with degree at most $d$ and $\Sigma[x]_d$ denotes the set of polynomials with degree at most $2d$ that can be written as a sum of squares of polynomials.
We use the classical Landau notation. For two functions $f,g:{\mathbb N}\rightarrow {\mathbb R}_+$, the notation $f(n)=O(g(n))$ (resp., $f(n)=\Omega(g(n))$, $f(n)=o(g(n))$) means
$\limsup_{n\to\infty} f(n)/g(n) <\infty$ (resp., $\liminf_{n\to\infty} f(n)/g(n)>\infty$, $\lim_{n\to\infty}f(n)/g(n) =0$), and $f(n)=\Theta(g(n))$
means $f(n)=O(g(n))$ and $f(n)=\Omega(g(n))$.
We also use this notation when $f,g$ are functions of a continuous variable $x$ and we want to indicate the behavior of $f(x)$ and $g(x)$ in the neighbourhood of a given scalar $x_0$ when $x\to x_0$.
So, $f(x)=O(g(x))$ as $x\to x_0$ means $\limsup_{x\to x_0} f(x)/g(x) <\infty$, etc.
\section{Preliminaries on orthogonal polynomials}\label{secprel}
In what follows we review some known facts on classical orthogonal polynomials that we need for our treatment.
Unless we give detailed references, the relevant results may be found in the classical text by Szeg\"o \cite{Szego_1975}
(see also \cite{Gautsch}).
We consider families of univariate polynomials $\{p_k(x)\}$ $(k=0,1,\ldots,d)$, that satisfy a three-term
recursive relation of the form:
\begin{equation}
\label{eq:recursion}
xp_k(x) = a_kp_{k+1}(x) + b_kp_k(x) + c_kp_{k-1}(x) \quad\quad (k=1,\ldots,d-1),
\end{equation}
where $p_0$ is a constant,
$p_1(x) = (x-b_0)p_0/a_0$, and $a_k$, $b_k$ and $c_k$ are
real values that satisfy $a_{k-1}c_k >0$ for $k=1,\ldots,d-1$. If we set $c_0=0$ then relation (\ref{eq:recursion}) also holds for $k=0$).
\noindent Defining the $k\times k$ tri-diagonal matrix
\begin{equation}
\label{matrix_Ak}
A_k :=
\left(
\begin{array}{ccccc}
b_0 & a_0 & 0 & \cdots & 0 \\
c_1 & b_1 & a_1 & & 0 \\
0 & \ddots & \ddots & \ddots & \\
\vdots & & c_{k-2} & b_{k-2} &a_{k-2} \\
0 & 0 & \cdots & c_{k-1} &b_{k-1} \\
\end{array}
\right),
\end{equation}
one has the classical relation:
\begin{equation}
\label{eq:det_roots}
\left(\prod_{j=0}^{k-1} a_j\right) p_k(x) = \det(xI_k - A_k){p_0} \quad \text{ for } k=1,\ldots,d,
\end{equation}
which can be easily verified using induction on $k\ge 1$ and the relation (\ref{eq:recursion}) (see, e.g., \cite{Ismail_Li_1992}).
Therefore, the roots of the polynomial $p_k$ are precisely the eigenvalues of the matrix $A_k$ in (\ref{matrix_Ak}).
\medskip
Recall that the polynomials $p_k$ $(k=0,1,\ldots,d)$ are {\em orthogonal with respect to a
weight function} $w:[-1,1]\rightarrow \mathbb{R}$, that is continuous and positive on $(-1,1)$,
if
\[
\langle p_i,p_j\rangle := \int_{-1}^1 p_i(x)p_j(x) w(x)dx = 0 \quad \mbox{ for all \ $i \neq j$}.
\]
We denote by $\hat p_k:=p_k/\sqrt{\langle p_k,p_k\rangle}$ the corresponding normalized polynomial, so that $\langle \hat p_k,\hat p_k\rangle =1$.
As is well known, if the polynomials $p_k$ are degree $k$ polynomials that are pairwise orthogonal with respect to such a weight function then they satisfy a three-terms recurrence relation of the form (\ref{eq:recursion}) (see, e.g., \cite[\S 1.3]{Gautsch}). Of course, the corresponding orthonormal polynomials $\hat p_k$ also satisfy such a three-terms recurrence relation (for different scaled parameters $a_k,b_k,c_k$).
By taking the inner product of both sides in (\ref{eq:recursion}) with $p_{k-1}$ and $p_{k+1}$ one gets the relations $c_k \langle p_{k-1},p_{k-1}\rangle =\langle p_k,xp_{k-1}\rangle$ and
$a_k \langle p_{k+1},p_{k+1}\rangle= \langle p_{k+1},xp_k\rangle$, which imply $c_k \langle p_{k-1},p_{k-1}\rangle
=a_{k-1}\langle p_k,p_k\rangle$ and thus $a_{k-1}c_k>0$. Moreover, when considering the recurrence relations associated with the orthonormal polynomials $\hat p_k$, we have $a_{k-1}=c_k$ for any $k\ge 1$, i.e., the matrix $A_k$ in (\ref{matrix_Ak}) is symmetric.
We will use later the following fact.
\begin{lemma}\label{lem_Ak}
Let $\{\hat p_k\}$ be orthonormal polynomials for the measure $d\mu(x)=w(x)dx$ on $[-1,1]$, where $w(x)$ is continuous and positive on $(-1,1)$, and assume they satisfy the three-terms recurrence relation (\ref{eq:recursion}).
Then, the matrix
\begin{equation}\label{eqhatA}
\left( \langle x\hat p_i,\hat p_j\rangle= \int_{-1}^1 x\hat p_i(x)\hat p_j(x)w(x)dx\right)_{i,j=0}^{k-1}
\end{equation}
is equal to the matrix $A_k$ in (\ref{matrix_Ak}). In particular, its smallest eigenvalue is the smallest root of the polynomial $p_k$.
\end{lemma}
\begin{proof}
Using the recurrence relation (\ref{eq:recursion}) we obtain
\begin{eqnarray*}
\langle x \hat p_i,\hat p_j\rangle &=& \langle a_i\hat p_{i+1}+b_i \hat p_i+c_i \hat p_{i-1},\hat p_j\rangle
\\
&=& \left\{
\begin{array}{ll}
a_i & \mbox{if $j=i+1$} \\
b_i & \mbox{if $j=i$} \\
c_i & \mbox{if $j=i-1$} \\
0 & \mbox{otherwise}.
\end{array}\right.
\end{eqnarray*}
Hence the matrix in (\ref{eqhatA}) is equal to $A_k$ and the last claim follows from (\ref{eq:det_roots}).
\end{proof}
It is also known that the roots of $p_k$ are all real, simple and lie in $(-1,1)$, and that they interlace the roots of $p_{k+1}$ (see, e.g., \cite[\S 1.2]{Gautsch}). In what follows we will use the smallest (and largest) roots to give closed-form expressions for the bounds \smash{$ \underline f^{(d)}_{\mathbf K}$ and $f^{(d)}$} in some examples.
\medskip
We now recall several classical univariate orthogonal polynomials on the interval $[-1,1]$ and some information on their smallest roots.
\subsection*{Chebyshev polynomials}
We will use the univariate Chebyshev polynomials (of the first kind),
defined by:
\begin{equation}\label{eqTUn}
T_k(x)= \cos(k\arccos (x)),\ \ \text{ for } \ x\in [-1,1],\ k = 0,1,\ldots.
\end{equation}
They satisfy the following three-terms recurrence relationships:
\begin{equation}\label{eqTnrec}
T_0(x)=1,\
T_1(x)=x, \ T_{k+1}(x)= 2xT_k(x)-T_{k-1}(x) \ \ \text{ for } \ k\ge 1.
\end{equation}
The Chebyshev polynomials are orthogonal with respect to the weight function
$w(x) = \frac{1}{ \sqrt{1-x^2}}$
and the roots of $T_k$ are given by
\begin{equation}\label{rootTk}
\cos\left( \frac{2i-1}{2k}\pi \right) \quad \text{ for } i = 1,\ldots,k.
\end{equation}
\subsection*{Jacobi polynomials}
The Jacobi polynomials, denoted by $\{P_k^{\alpha,\beta}\}$ $(k=0,1,\ldots)$, are orthogonal with respect to the weight function
\begin{equation}
\label{Jacobi_weight_fn}
w_{\alpha,\beta}(x) := (1-x)^\alpha(1+x)^\beta, \quad x \in (-1,1)
\end{equation}
where $\alpha > -1$ and $\beta >-1$ are given parameters.
The normalized Jacobi polynomials are denoted by $\hat P^{\alpha,\beta}_k$, so that
$\int_{-1}^1 (\hat P^{\alpha,\beta}_k(x))^2 w_{\alpha,\beta}(x)dx =1$.
Thus the Chebyshev polynomials may be seen as the special case corresponding to $\alpha = \beta = -\frac{1}{2}$.
Likewise, the Legendre polynomials are the orthogonal polynomials w.r.t.\ the constant weight function ($w(x)=1$), so they correspond to the special case $\alpha = \beta = 0$.
There is no closed-form expression for the roots of Jacobi polynomials in general. But some bounds are known for the smallest root of $P_k^{\alpha,\beta}$, denoted by $\xi^{\alpha,\beta}_k$, that we recall in the next theorem.
\begin{theorem}\label{thm_root}
The smallest root, denoted $\xi^{\alpha,\beta}_k$, of the Jacobi polynomial $P^{\alpha,\beta}_k$ satisfies the following inequalities:
\begin{itemize}
\item[(i)] (\cite{Driver_Jordaan_2012})
$\xi^{\alpha,\beta}_k\le -1 + \frac{2(\beta+1)(\beta+3)}{2(k-1)(k+\alpha+\beta+2)+(\beta+3)(\alpha+\beta+2)}.$
\item[(ii)] (\cite{Dimitrov_Nikolov_2010})
$\xi^{\alpha,\beta}_k\ge \frac{F-4(k-1)\sqrt{\Delta}}{E},$ where
\begin{eqnarray*}
F &=& (\beta - \alpha)\left((\alpha + \beta + 6)k + 2(\alpha + \beta)\right), \\
E &=& (2k + \alpha + \beta)\left(k(2k + \alpha + \beta) + 2(\alpha + \beta + 2)\right) \\
\Delta &=& k^2(k + \alpha + \beta + 1)^2 +(\alpha + 1)(\beta + 1)(k^2 + (\alpha + \beta + 4)k + 2(\alpha + \beta)).
\end{eqnarray*}
\end{itemize}
\end{theorem}
\ignore{
may upper bounded by (see \cite{Driver_Jordaan_2012}):
\begin{equation}
\label{smallest_Jacobi_root_bound}
\end{equation}
In the case of Legendre polynomials ($\alpha = \beta = 0$) this bound becomes:
\[
-1 + \frac{3}{(k-1)(k+2)+3}.
\]
Moreover, the smallest root of $P_k^{\alpha,\beta}$ is lower bounded by (see \cite{Dimitrov_Nikolov_2010}):
\begin{equation}
\label{Jacobi_root_upper_bound}
\frac{F-4(k-1)\sqrt{\Delta}}{A},
\end{equation}
where
\begin{eqnarray*}
F &=& (\beta - \alpha)\left((\alpha + \beta + 6)k + 2(\alpha + \beta)\right), \\
A &=& (2k + \alpha + \beta)\left(k(2k + \alpha + \beta) + 2(\alpha + \beta + 2)\right) \\
\Delta &=& k^2(k + \alpha + \beta + 1)^2 +(\alpha + 1)(\beta + 1)(k^2 + (\alpha + \beta + 4)k + 2(\alpha + \beta)).
\end{eqnarray*}
Note that $\frac{B-4(k-1)\sqrt{\Delta}}{A} = -1 + \Omega\left(\frac{1}{k^2} \right)$.
For example, if $\alpha = \beta = 0$, one has
\begin{eqnarray*}
\frac{B-4(k-1)\sqrt{\Delta}}{A} &=& \frac{-4(k-1)\sqrt{k^2(k + 1)^2 +k^2 + 4k }}{2k\left(2k^2 + 4\right)}\\
&\ge& \frac{-4\sqrt{k^2(k^2 - 1)^2 +(k-1)^2(k^2 + 4k) }}{4k^3 } \\
&=& -\sqrt{1 - \frac{1}{k^2} + \frac{2}{k^3} - \frac{6}{k^4} + \frac{4}{k^5}} \\
&=& -1 + \Omega\left(\frac{1}{k^2}\right),
\end{eqnarray*}
where the last equality follows from the expansion
\[
\sqrt{1+x} = 1 +\frac{1}{2}x - \frac{1}{8}x^2 + \frac{1}{16}x^3 - \ldots
\]
}
The smallest roots $\xi^{\alpha,\beta}_k$ of the Jacobi polynomials $P^{\alpha,\beta}_k$ converge to $-1$ as $k\to \infty$. Using the above bounds we see that the rate of convergence is
$O(1/k^2)$.
\begin{corollary}
\label{cor_root}
The smallest roots of the Jacobi polynomials $P^{\alpha,\beta}_k$ satisfy
$$\xi^{\alpha,\beta}_k= -1 + \Theta\left({1\over k^2}\right) \ \ \text{ as } \ k\rightarrow \infty.$$
\end{corollary}
\begin{proof}
The upper bound in Theorem \ref{thm_root}(i) gives directly $\xi^{\alpha,\beta}_k= -1 + O\left({1\over k^2}\right)$.
We now use the lower bound in Theorem \ref{thm_root}(ii) to show $\xi^{\alpha,\beta}_k= -1 + \Omega\left({1\over k^2}\right)$. For this we give asymptotic estimates for the quantities $E,F,\Delta$. First, using the expansion $\sqrt{1+x}= 1+{x\over 2}-{x^2\over 8}+o(x^2)$ as $x\rightarrow 0$ we obtain
$$\sqrt \Delta = k^2\left(1+{\alpha+\beta+1\over k} +{(\alpha+1)(\beta+1)\over 2k^2} +o\left({1\over k^2}\right)\right).$$
Second, using the expansion ${1\over 1+x}=1-x+x^2+o(x^2)$ as $x\rightarrow 0$ we obtain
$${1\over E}={1\over 4k^3}\left(1-{\alpha+\beta\over k} -{4(\alpha+\beta+2)\over k^2} +o\left({1\over k^2}\right)\right).$$
Combining these two relations gives
$$\begin{array}{lll}
{4(k-1)\sqrt\Delta\over E} &=
& \left(1-{1\over k}\right)
\left(1+{\alpha+\beta+1\over k} +{(\alpha+1)(\beta+1)\over 2k^2} +o\left({1\over k^2}\right)\right)
\left(1-{\alpha+\beta\over k} -{4(\alpha+\beta+2)\over k^2} +o\left({1\over k^2}\right)\right)\\
&=& 1 +{C\over 2k^2} +o\left({1\over k^2}\right),
\end{array}$$
where we set $C= (\alpha+1)(\beta+1) -8(\alpha+\beta+2) -2(\alpha+\beta)(\alpha+\beta+1)-2$.
Finally, using
$${F\over E}= {(\beta-\alpha)(\beta+\alpha+6)\over 4k^2} +o\left({1\over k^2}\right),$$
we obtain
$${F-4(k-1)\sqrt\Delta\over E}
= -1 + {1\over k^2}\left( {(\beta-\alpha)(\beta+\alpha+6)\over 4} -{C\over 2} \right)+o\left({1\over k^2}\right),$$
where the coefficient of $1/k^2$ can be verified to be strictly positive, which thus implies the estimate $\xi^{\alpha,\beta}_k=-1 +\Omega(1/k^2)$.
\end{proof}
It is also known that $P^{\alpha,\beta}_k(x)=(-1)^k P^{\beta,\alpha}_k(-x)$. Therefore the largest root of $P^{\alpha,\beta}_k(x)$ is equal to $-\xi^{\beta,\alpha}_k= 1-\Theta(1/ k^2)$.
\section{{Tight lower bounds for a class of examples}}
In this section we consider the following simple examples
\begin{equation}
\label{example}
\min \left\{\sum_{i=1}^n c_ix_i: \; \; x \in [-1,1]^n\right\},
\end{equation}
asking to minimize the linear polynomial $f(x)=\sum_{i=1}^n c_ix_i$ over the box ${\mathbf K}=[-1,1]^n$. Here $c_i\in {\mathbb R}$ are given scalars for $i\in [n]$. Hence, $f_{\min,{\mathbf K}}=-\sum_{i=1}^n |c_i|$.
For these examples we can obtain explicit closed-form expressions for the Lasserre bounds \smash{$\underline f^{(d)}_{\mathbf K}$} when using product measures with weight functions $w_{\alpha,\beta}$ on $[-1,1]$, and also for the strengthened bounds \smash{$f^{(d)}$} considered by De Klerk, Hess and Laurent, which use product measures with weight functions $w_{-1/2,-1/2}$. These closed-form expressions are in terms of extremal roots of Jacobi polynomials.
\subsection{Tight lower bound for the Lasserre hierarchy}\label{secLas}
Here we consider the bounds \smash{$\underline{f}^{(d)}_{\mathbf{K}}$} for the example (\ref{example}), when the measure $\mu$ on ${\mathbf K}=[-1,1]^n$ is a product of univariate measures given by weight functions.
First we consider
the univariate case $n=1$.
When the measure $\mu$ on $\mathbf{K} = [-1,1]$
is given by a continuous positive weight function $w$ on $(-1,1)$,
one can obtain a closed form expression for $\underline{f}^{(d)}_{\mathbf{K}}$ in terms of the smallest root of the corresponding orthogonal polynomials.
\begin{theorem}\label{thm:main1}
Consider the measure $d\mu(x) = w(x)dx$ on ${\mathbf K}=[-1,1]$, where $w$ is a positive, continuous weight function on $(-1,1)$, and let $p_k$ be univariate degree $k$ polynomials that are orthogonal
with respect to this measure.
For the univariate polynomial $f(x)=x$ {(resp., $f(x)=-x$),}
the parameter \smash{$\underline{f}^{(d)}_{\mathbf{K}}$} is equal to the smallest root {(resp., the opposite of the largest root)} of the polynomial $p_{d+1}$.
\end{theorem}
\begin{proof}
Let $\hat p_0,\ldots,\hat p_{d+1}$ denote the corresponding orthonormal polynomials, with $\hat p_i=p_i/\sqrt{\langle p_i,p_i\rangle}$.
Consider first $f(x)=x$. Using Lemma \ref{lemsummarize}, we see that \smash{$\underline{f}^{(d)}_{\mathbf{K}}$} is equal to the smallest eigenvalue of the matrix $A$ in (\ref{eqhatA}) (for $k=d+1$),
which coincides with the matrix $A_{d+1}$ in (\ref{matrix_Ak}), so that its smallest eigenvalue is equal to the smallest root of $p_{d+1}$.\\
{Assume now $f(x)=-x$. Then \smash{$\underline{f}^{(d)}_{\mathbf{K}}$} is equal to $\lambda_{\min}(-A)=-\lambda_{\max}(A)$, which in turn is equal to the opposite of the largest root of $p_{d+1}$.}
\ignore{
As recalled earlier, we may assume that $\hat p_0,\ldots,\hat p_{d+1}$ satisfy three-terms recurrence relations of the form
\eqref{eq:recursion}. This permits to compute the entries of the matrix $A$ as follows:
\begin{eqnarray*}
A_{ij} &=& \int_{-1}^1 x\hat p_i(x)\hat p_j(x) w(x)dx \\
&=& \int_{-1}^1\left(\alpha_i\hat p_{i+1}(x) + \beta_i\hat p_i(x) + \gamma_i\hat p_{i-1}(x)\right) \hat p_j(x) w(x)dx \\
&=& \left\{
\begin{array}{rl}
\alpha_i & \mbox{if $j=i+1$} \\
\beta_i & \mbox{if $j=i$} \\
\gamma_i & \mbox{if $j=i-1$} \\
0 & \mbox{otherwise.}
\end{array}\right.
\end{eqnarray*}
Comparing with the matrix in \eqref{matrix_Ak}, one finds $A = A_{d+1}$. Hence, the smallest eigenvalue of
$A$ is the smallest root of $p_{d+1}$, by \eqref{eq:det_roots}.}
\end{proof}
Recall that $\xi^{\alpha,\beta}_{d+1}$ denotes the smallest root of the Jacobi polynomial $P^{\alpha,\beta}_{d+1}$ and that the largest root of $P^{\alpha,\beta}_{d+1}$ is equal to $-\xi^{\beta,\alpha}_{d+1}$.
\begin{corollary}\label{corboundn1}
Consider the measure $d\mu(x)=w_{\alpha,\beta}(x)dx$ on ${\mathbf K}=[-1,1]$ with the weight function $w_{\alpha,\beta}(x) = (1-x)^\alpha(1+x)^\beta$ and $\alpha,\beta>-1$.
For the univariate polynomial $f(x)=x$ {(resp., $f(x)=-x$)}, the parameter
$\smash{\underline{f}^{(d)}_{\mathbf{K}}}$ is equal to \smash{$ \xi^{\alpha,\beta}_{d+1}$}
{(resp., to \smash{$\xi^{\beta,\alpha}_{d+1}$)}} and thus we have
$$\underline{f}^{(d)}_{\mathbf{K}}- f_{\min,{\mathbf K}}=\Theta(1/d^2).$$
In particular, \smash{$\underline{f}^{(d)}_{\mathbf{K}}=-\cos\left( \frac{\pi}{2d+2} \right)$}
when $\alpha=\beta=-1/2$.
\end{corollary}
\begin{proof}
This follows directly using Theorem \ref{thm:main1}, Corollary \ref{cor_root}, the fact that the largest root of $P^{\alpha,\beta}_{d+1}$ is equal to \smash{$-\xi^{\beta,\alpha}_{d+1}$,} and the closed form expression (\ref{rootTk}) for the roots of the Chebyshev polynomials of the first kind.
\end{proof}
We now use the above result to show $\underline{f}^{(d)}_{\mathbf{K}}- f_{\min,{\mathbf K}} = \Omega(1/d^2)$ for the example (\ref{example}) in the multivariate case $n\ge 2$.
\begin{corollary}\label{corboundn}
Consider the measure $d\mu(x) =\prod_{i=1}^n w_{\alpha_i,\beta_i}(x_i) dx_i$ on the hypercube ${\mathbf K}=[-1,1]^n$, with the weight functions $w_{\alpha_i,\beta_i}(x_i)=(1-x_i)^{\alpha_i}(1+x_i)^{\beta_i}$ and $\alpha_i,\beta_i>-1$ for $i\in [n]$.
For the polynomial $f(x)=\sum_{l=1}^n c_l x_l$, we have
$$\underline{f}^{(d)}_{{\mathbf K}}\ge \sum_{l: c_l>0} c_l\xi^{\alpha_l,\beta_l}_{d+1} +\sum_{l:c_l<0}|c_l| \xi^{\beta_l,\alpha_l}_{d+1},$$
and thus $\underline{f}^{(d)}_{\mathbf{K}} -f_{\min,{\mathbf K}}= \Omega(1/d^2)$.
\end{corollary}
\begin{proof}
Assume $\underline{f}^{(d)}_{\mathbf{K}}=\int_{\mathbf K} (\sum_{l=1}^n {c_l} x_l)\sigma(x)d\mu(x)$, where $\sigma\in {\mathbb R}[x_1,\ldots,x_n]_{2d}$ is a sum of squares of polynomials and $\int_{\mathbf K} \sigma(x)d\mu(x)=1$.
For each $l\in [n]$ consider the univariate polynomial
$$\sigma_l(x_l):= \int_{[-1,1]^{n-1}} \sigma(x_1,\ldots,x_n)\prod_{i\in [n]\setminus \{l\}} w_{\alpha_i,\beta_i}(x_i) dx_i,$$
where we integrate over all variables $x_i$ with $i\in[n]\setminus \{l\}$. Then we have $\int_{-1}^1\sigma_l(x_l)w_{\alpha_l,\beta_l}(x_l)dx_l=1$. Moreover, $\sigma_l$ has degree at most $2d$ and, as it is a univariate polynomial which is nonnegative on ${\mathbb R}$, it is a sum of squares of polynomials.
Hence, using Corollary \ref{corboundn1}, we can conclude that
$$\int_{-1}^1 x_l\sigma_l(x_l)w_{\alpha_l,\beta_l}(x_l)dx_l \ge \xi^{\alpha_l,\beta_l}_{d+1},\quad
{\int_{-1}^1 (-x_l)\sigma_l(x_l)w_{\alpha_l,\beta_l}(x_l)dx_l \ge \xi^{\beta_l,\alpha_l}_{d+1}.}
$$
Combining with the definition of $\underline{f}^{(d)}_{\mathbf{K}}$ we obtain
$$\underline{f}^{(d)}_{\mathbf{K}} =\sum_{l=1}^n {c_l}\int_{-1}^1 {x_l}\sigma_l(x_l) w_{\alpha_l,\beta_l}(x_l)dx_l \ge
{\sum_{l:c_l>0} c_l \xi^{\alpha_l,\beta_l}_{d+1}+\sum_{l: c_l<0} |c_l| \xi^{\beta_l,\alpha_l}_{d+1}}$$
and thus $\underline{f}^{(d)}_{\mathbf{K}} -f_{\min,{\mathbf K}}
\ge { \sum_{l: c_l>0} c_l(\xi^{\alpha_l,\beta_l}_{d+1}+1) +\sum_{l: c_l<0} |c_l| (\xi^{\beta_l,\alpha_l}_{d+1}+1)}=\Omega(1/d^2).$
\end{proof}
\ignore{
\begin{proof}
By definition, \smash{$\underline{f}^{(d)}_{\mathbf{K}}$} is the smallest value taken by $\int_{\mathbf K} f(x)\sigma(x) d\mu (x)$, where $\sigma \in \Sigma[x]_{2d}$ satisfies the constraint $\int_{\mathbf K} \sigma(x)d\mu(x)=1$.
We use the polynomial basis $\{\prod_{i=1}^n \hat P^{\alpha_i,\beta_i}_{k_i} (x_i)\}$ to express the property that $\sigma$ is a sum of squares of degree $2d$. For this we write
$$\sigma(x)=\sum_{\mathbf{h},\mathbf{k}\in N(n,d)} M^{}_{\mathbf{h},\mathbf{k}} \prod_{i=1}^n \hat P^{\alpha_i,\beta_i}_{h_i} (x_i)\hat P^{\alpha_i,\beta_i}_{k_i} (x_i),$$
where $M$ is a matrix indexed by $N(n,d)$ constrained to be positive semidefinite.
Then, the constraint
$\int_{\mathbf K} \sigma(x)d\mu(x)=1$ can be rewritten as $\text{\rm Tr}(M^{})=1.$
We can express the objective $\int_{\mathbf K} (\sum_{l=1}^n x_l)\sigma(x)d\mu(x)$ using the matrices in (\ref{eqAab}), namely,
$$\int_{\mathbf K} \left(\sum_{l=1}^n x_l\right)\sigma(x)d\mu(x)= \sum_{\mathbf{h},\mathbf{k}\in N(n,d)} M^{}_{\mathbf{h},\mathbf{k}} \sum_{l=1}^n \prod_{i\ne l} \delta_{h_i,k_i}
\underbrace{ \left(\int_{-1}^1 x_l \hat P^{\alpha_l,\beta_l}_{h_l}(x_l) \hat P^{\alpha_l,\beta_l}_{k_l}(x_l)w_{\alpha_l,\beta_l}(x_l)dx_l\right)}_{= (A^{\alpha_l,\beta_l}_d)_{h_l,k_l}}.$$
In other words we have
$$\int_{\mathbf K} \left(\sum_{l=1}^n x_l\right)\sigma(x)d\mu(x)= \langle \Lambda,M\rangle,$$
after defining the matrix $\Lambda= \sum_{l=1}^n I^{\otimes (l-1)}\otimes A^{\alpha_l,\beta_l}_d\otimes I^{\otimes (n-l)}$.
Therefore, we obtain
$$\underline{f}^{(d)}_{\mathbf{K}} =\min\{\langle \Lambda,M\rangle: \text{\rm Tr}(M)=1, M\succeq 0\}= \lambda_{\min}(\Lambda).$$
By the definition of the matrix $\Lambda$ it follows that $\lambda_{\min}(\Lambda)= \sum_{l=1}^n \lambda_{\min}(A^{\alpha_l,\beta_l}_d)= \sum _{l=1}^n \xi^{\alpha_l,\beta_l}_{d+1}$. The rest of the claim follows using Corollary \ref{cor_root}.
\end{proof}
}
\subsection{Tight lower bound for the De Klerk, Hess and Laurent hierarchy}\label{secDKHL}
In this section we consider the hierarchy of bounds \smash{$f^{(d)}$} studied by De Klerk, Hess and Laurent \cite{DHL SIOPT},
which are potentially stronger than the bounds \smash{$\underline f^{(d)}_{\mathbf K}$} since they involve the wider class of density functions in (\ref{eqSch}). Their convergence rate is known to be $O(1/d^2)$ (\cite{DHL SIOPT}, recall Theorem \ref{theoDKHL}).
For the example \eqref{example} we can also give an explicit expression for the bounds \smash{$f^{(d)}$} and we will show that their convergence rate to $f_{\min,{\mathbf K}}$ is also in the order $\Omega(1/d^2)$, which shows that the analysis in \cite{DHL SIOPT} is tight.
We first treat the univariate case, in order to introduce the main ideas, and then we extend to the multivariate case.
\begin{theorem}\label{thm:rateDKHL}
For the univariate polynomial $f(x)={\pm} x$, we have
$$f^{(d)} =\min\{\xi^{-1/2,-1/2}_{d+1},\xi^{1/2,1/2}_d\},$$ the smallest value among the smallest roots of the Jacobi polynomials $P^{-1/2,-1/2}_{d+1}$ and $P^{1/2,1/2}_d$. In particular, we have
$f^{(d)}-f_{\min,{\mathbf K}}= \Theta(1/d^2)$.
\end{theorem}
\begin{proof}
Consider first $f(x)=x.$ We first recall how to compute \smash{$f^{(d)}$} as an eigenvalue problem.
By definition,
it is the minimum value of $\int_{-1}^1 x (\sigma_0(x)+\sigma_1(x)(1-x^2))w_{-1/2,-1/2}(x)dx$,
where $\sigma_0\in \Sigma[x]_{2d}$, $\sigma_1\in \Sigma[x]_{2d-2}$ and
$\int_{-1}^1 (\sigma_0(x)+\sigma_1(x)(1-x^2))w_{-1/2,-1/2}(x)dx=1$.
We express the polynomial $\sigma_0$ in the normalized Jacobi (Chebychev) basis {$\{\hat P^{-1/2,-1/2}_k\}$} as
$$\sigma_0=\sum_{i,j=0}^d M^{(0)}_{ij} \hat P^{-1/2,-1/2}_i \hat P^{-1/2,-1/2}_j$$ for some matrix $M^{(0)}$ of order $d+1$, constrained to be positive semidefinite.
Based on the observation that
$(1-x^2) w_{-1/2,-1/2}(x)=w_{1/2,1/2}(x)$, we express the polynomial $\sigma_1$ in the normalized Jacobi basis {$\{\hat P^{1/2,1/2}_k\}$} as
$$\sigma_1=\sum_{i,j=0}^{d-1} M^{(1)}_{ij} \hat P^{1/2,1/2}_i \hat P^{1/2,1/2}_j$$
for some matrix $M^{(1)}$ of order $d$, also constrained to be positive semidefinite.
Then, we obtain
$$f^{(d)}= \min \{\langle A^{-1/2,-1/2}_d,M^{(0)}\rangle +\langle A^{1/2,1/2}_{d-1},M^{(1)}\rangle: \text{\rm Tr}(M^{(0)})+\text{\rm Tr}(M^{(1)})=1,\ M^{(0)}\succeq 0, M^{(1)}\succeq 0\},$$
{where $A^{1/2,1/2}_d$ and $A^{-1/2,-1/2}_{d-1}$ are instances of (\ref{eqhatA}) defined as follows: }
\begin{equation*}\label{eqAab}
A^{\alpha,\beta}_d:=\left( \int_{-1}^1x \hat P^{\alpha,\beta}_h(x) \hat P^{\alpha,\beta}_k(x) w_{\alpha,\beta}(x) dx\right)_{h,k=0}^d
\end{equation*}
{for any $\alpha,\beta>-1$ and $d\in{\mathbb N}$.}
Since strong duality holds we obtain
$$f^{(d)} =\max\{t: A^{-1/2,-1/2}_d-tI\succeq 0,\ A^{1/2,1/2}_{d-1} -tI\succeq 0\}= \min \{\lambda_{\min}(A^{-1/2,-1/2}_d),\lambda_{\min}(A^{1/2,1/2}_{d-1})\}.$$
By Lemma \ref{lem_Ak}, we have $\lambda_{\min}(A^{-1/2,-1/2}_d)=\xi^{-1/2,-1/2}_{d+1}$
and $\lambda_{\min}(A^{1/2,1/2}_{d-1})=\xi^{1/2,1/2}_d$ and thus
$f^{(d)} = \min\{\xi^{-1/2,-1/2}_{d+1},\xi^{1/2,1/2}_d\}$. {The same result holds when $f(x)=-x$.}
Finally, by Corollary \ref{cor_root}, these two smallest roots are both equal to $-1+\Theta(1/d^2)$, which concludes the proof.
\end{proof}
We now extend this result to the multivariate case of example \eqref{example}:
\begin{corollary}\label{corbounddKHLn
For the linear polynomial $f(x)=\sum_{l=1}^n c_lx_l$, we have $$f^{(d)} \ge {\left(\sum_{l=1}^n|c_l|\right)} \min\{\xi^{-1/2,-1/2}_{d+1},\xi^{1/2,1/2}_d\}$$ and thus
$f^{(d)} -f_{\min,{\mathbf K}}= \Omega(1/d^2).$
\end{corollary}
\begin{proof}
The proof is analogous to that of Corollary \ref{corboundn}, with some more technical details.
Assume $f^{(d)}=\int_{\mathbf K} (\sum_{l=1}^n x_l)\sigma(x)d\mu(x)$, where $\sigma(x)=\sum_{I\subseteq [n]}\sigma_I(x)\prod_{i\in I}(1-x_i^2)$, $\sigma_I(x)$ is a sum of squares of degree at most $2d-2|I|$ and $\int_{\mathbf K} \sigma(x)d\mu(x)=1$.
Fix $l\in [n]$. Then we can write
$$\sigma(x)= \sum_{I\subseteq [n]\setminus\{l\}} \sigma_I(x)\prod_{i\in I}(1-x_i^2)
\ + \ (1-x_l^2)
\sum_{I\subseteq [n]: l\in I} \sigma_I(x)\prod_{i\in I\setminus\{l\}}(1-x_i^2).
$$
Next, define the univariate polynomials in the variable $x_l$:
$$\sigma_{l,0}(x_l):=\sum_{I\subseteq [n]\setminus\{l\}} \int_{[-1,1]^{n-1}} \sigma_I(x)\prod_{i\in I}(1-x_i^2) \prod_{i\in [n]\setminus \{l\}} w_{-1/2,-1/2}(x_i)dx_i,$$
$$\sigma_{l,1}(x_l):=\sum_{I\subseteq [n]: l\in I} \int_{[-1,1]^{n-1}} \sigma_I(x)\prod_{i\in I{\setminus\{l\}}}(1-x_i^2) \prod_{i\in [n]\setminus \{l\}} w_{-1/2,-1/2}(x_i)dx_i,$$
$$\sigma_l(x_l):= \int_{[-1,1]^{n-1}} \sigma(x) \prod_{i\in [n]\setminus\{l\}} w_{-1/2,-1/2}(x_i)dx_i
= \sigma_{l,0}(x_l)+ (1-x_l^2)\sigma_{l,1}(x_l).$$
By construction, we have
$$\int_{\mathbf K} x_l\sigma(x)d\mu(x)= \int_{-1}^1 x_l\sigma_l(x_l)w_{-1/2,-1/2}(x_l)dx_l,\ \
\int_{-1}^1 \sigma_l(x_l) w_{-1/2,-1/2}(x_l)dx_l= \int_{\mathbf K} \sigma(x)d\mu(x)=1.$$
Moreover, the polynomial $\sigma_{l,0}$ is a sum of squares (since it is univariate and nonnegative on ${\mathbb R}$) and its degree is at most $2d$,
and the polynomial $\sigma_{l,1}$ is a sum of squares of degree at most $2d-2$.
Hence, using Theorem \ref{thm:rateDKHL}, we can conclude that
$$\int_{-1}^1 ({\pm} x_l)\sigma_l(x_l)w_{-1/2,-1/2}(x_l)dx_l \ge \min\{\xi^{-1/2,-1/2}_{d+1},\xi^{1/2,1/2}_d\}.$$
This implies that
$$
f^{(d)} =\int_{\mathbf K} (\sum_{l=1}^nc_l x_l)\sigma(x)d\mu(x)
= \sum_{l=1}^n c_l \int_{-1}^1 x_l\sigma_l(x_l)w_{-1/2,-1/2}(x_l)dx_l $$ is at least ${ (\sum_l|c_l|)} \min\{\xi^{-1/2,-1/2}_{d+1},\xi^{1/2,1/2}_d\}$
and the proof is complete.
\end{proof}
\ignore{
\begin{proof}
By definition, the parameter {$f^{(d)}$} is the smallest value of $\int_{\mathbf K} f(x)\sigma(x)d\mu(x)$, where $\mu$ is the measure on
${\mathbf K}=[-1,1]^n$ with $d\mu(x)=\prod_{i=1}^n w_{-1/2,-1/2}(x_i)dx_i$, $\sigma(x)=\sum_{I\subseteq [n]} \sigma_I(x) \prod_{i\in I}(1-x_i^2)$ with $\sigma_I\in \Sigma[x]_{2d-2|I|}$, and $\int_{\mathbf K} \sigma(x) d\mu(x)=1$.
As in the previous proof we exploit the observation that $(1-x_i^2)w_{-1/2,-1/2}(x_i)=w_{1/2,1/2}(x_i)$. Then, given $I\subseteq [n]$, we express the polynomial $\sigma_I$ using the polynomial basis
$\{\prod_{i\in I}\hat P^{1/2,1/2}_{k_i} \prod_{i\in \overline I} \hat P^{-1/2,-1/2}_{k_i}: \mathbf{k}=(k_1,\ldots,k_n)\in {\mathbb N}^n\}$.
So we write
$$\sigma_I(x)= \sum_{\mathbf{h},\mathbf{k}\in N(n,d-|I|)} M^{(I)}_{\mathbf{h},\mathbf{k}} \prod_{i\in I}\hat P^{1/2,1/2}_{h_i}(x)\hat P^{1/2,1/2}_{k_i} (x) \prod_{i\in [n]\setminus I} \hat P^{-1/2,-1/2}_{h_i}(x) \hat P^{-1/2,-1/2}_{k_i}(x)
$$
for some matrix $M^{(I)}$ indexed by $N(n,d-|I|)$, constrained to be positive semidefinite. Then we have
$$\int_{\mathbf K} \sigma(x)d\mu(x)=\sum_{I\subseteq [n] }\int_{\mathbf K} \sigma_I(x) \prod_{i\in I}(1-x_i^2)d\mu(x)= \sum_{I\subseteq [n]} \sum_{\mathbf{h},\mathbf{k}\in N(n,d-|I|)} M^{(I)}_{\mathbf{h},\mathbf{k}}\delta_{\mathbf{h},\mathbf{k}}= \sum_{I\subseteq [n]} \text{\rm Tr}(M^{(I)}).$$
Next, the objective function reads:
$$\int_{\mathbf K} (\sum_{l=1}^n x_l)\sigma(x)d\mu(x)=
\sum_{I\subseteq [n]}\sum_{l=1}^n \int_{\mathbf K} x_l \sigma_I(x)\prod_{i\in I}(1-x_i^2)d\mu(x).$$
Given a subset $I\subseteq [n]$ and $l\in [n]$, one can check that
$$ \int_{\mathbf K} x_l \sigma_I(x)\prod_{i\in I}(1-x_i^2)d\mu(x)=
\sum_{\mathbf{h},\mathbf{k}\in N(n,d-|I|)} M^{(I)}_{\mathbf{h},\mathbf{k}} (A^{\epsilon,\epsilon}_{d-|I|})_{h_lk_l} \prod_{i\ne l}\delta_{h_i,k_i} ,$$
where $\epsilon=1/2$ if $l\in I$ and $\epsilon=-1/2$ if $l\not\in I$, and $A^{\epsilon,\epsilon}_{d-|I|}$ is as in (\ref{eqAab}).
Therefore we obtain
$$\int_{\mathbf K} (\sum_{l=1}^n x_l)\sigma(x)d\mu(x)=\sum_{I\subseteq [n]} \langle \Lambda^{(I)},M^{(I)}\rangle,$$
after defining the matrices
$$\Lambda^{(I)}= \sum_{l\in I} I^{\otimes (l-1)}\otimes A^{1/2,1/2}_{d-|I|} \otimes I^{\otimes (n-l)}
+\sum_{l\in [n]\setminus I} I^{\otimes (l-1)}\otimes A^{-1/2,-1/2}_{d-|I|} \otimes I^{\otimes (n-l)}.$$
Summarizing we have shown
$$f^{(d)}=\min\left\{\sum_{I\subseteq [n]}\langle \Lambda^{(I)},M^{(I)}\rangle: \sum_{I\subseteq [n]} \text{\rm Tr}(M^{(I)}=1, M^{(I)}\succeq 0\right\}
=\min\{\lambda_{\min}(\Lambda^{(I)}): I\subseteq [n]\}.$$
By the definition of the matrix $\Lambda^{(I)}$ it follows that
$$\lambda_{\min}(\Lambda^{(I)})= |I|\lambda_{\min}(A^{1/2,1/2}_{d-|I|})+(n-|I|)\lambda_{\min}(A^{-1/2,-1/2}_{d-|I|}) =
|I|\xi^{1/2,1/2}_{d-|I|+1}+(n-|I|)\xi^{-1/2,-1/2}_{d-|I|+1}$$ is equal to $-n +\Theta(1/d^2)$.
From this follows that $f^{(d)}-f_{\min,{\mathbf K}}= \Theta(1/d^2).$
\end{proof}
}
\section{Tight upper bounds for the Lasserre hierarchy} \label{secupper}
In this section we analyze the rate of convergence of the Lasserre bounds $\underline{f}^{(d)}_{\mathbf{K}}$
when using the measure $d\mu(x)=\prod_{i=1}^n w_{-1/2,-1/2}(x_i)dx_i$ on the box ${\mathbf K}=[-1,1]^n$ (corresponding to the Chebyshev orthogonal polynomials). For this measure, it is known that the stronger bounds $f^{(d)}$ - that use a much richer class of density functions -
enjoy a $O(1/d^2)$ rate of convergence (\cite{DHL SIOPT}, see Theorem \ref{theoDKHL}). We show that the convergence rate remains $O(1/d^2)$ for the weaker bounds $\underline{f}^{(d)}_{\mathbf{K}}$, which thus also implies Thoerem \ref{theoDKHL}.
\begin{theorem}\label{theoLasrate}
Consider the measure $d\mu(x)=\prod_{i=1}^n w_{-1/2,-1/2}(x_i)dx_i$ on the hypercube ${\mathbf K}=[-1,1]^n$, with the weight function $w_{-1/2,-1/2}(x_i)=(1-x_i^2)^{-1/2}$ for $i\in [n]$. For any polynomial $f$ we have
$$\underline{f}^{(d)}_{\mathbf{K}} -f_{\min,{\mathbf K}} =O(1/d^2).$$
\end{theorem}
\noindent
It turns out that we can reduce the general result to the univariate quadratic case.
In what follows we consider first the special case when $f$ is univariate and quadratic (see Lemma \ref{lemquaduni}) and then we indicate how to derive the result for an arbitrary multivariate polynomial $f$. A key tool we use for this reduction is the existence of a quadratic upper estimator for $f$ having the same minimum as $f$ over ${\mathbf K}$.
In the quadratic univariate case we exploit again the formulation of $\underline{f}^{(d)}_{\mathbf{K}}$ in terms of the smallest eigenvalue of the associated matrix $A_d$ in (\ref{eqAdLas}) (recall Lemma \ref{lemsummarize}). This matrix $A_d$ is now 5-diagonal, but a key feature is that it contains a large Toeplitz submatrix, whose eigenvalues can be estimated by embedding it into a circulant matrix for which closed form expressions exist for the eigenvalues. This nice structure, which allows a simple analysis, follows from the choice of the Chebyshev type measure. We expect that a similar
convergence rate should hold when selecting any measure of Jacobi type, but the analysis seems more complicated.
\subsection{The quadratic univariate case}
Here we consider the case when ${\mathbf K}=[-1,1]$ and $f$ is a univariate quadratic polynomial of the form
$f(x)=x^2+\alpha x$, for some scalar $\alpha\in{\mathbb R}$.
We can first easily deal with the case when $\alpha\not\in (-2,2)$. Indeed then we have
$$f(x)\le g(x):= \alpha x+1 \quad \text{ for all } x\in [-1,1],$$
and both $f$ and $g$ have the same minimum value on $[-1,1]$. Namely, $f_{\min,{\mathbf K}}=g_{\min,{\mathbf K}}$ is equal to $1-\alpha$ if $\alpha \ge 2$, and to $1+\alpha$ if $\alpha \le -2$. Therefore we have
$$\underline{f}^{(d)}_{\mathbf{K}} -f_{\min,{\mathbf K}}\le \underline{g}^{(d)}_{\mathbf{K}} - g_{\min,{\mathbf K}} =O(1/d^2),$$
where we use Corollary \ref{corboundn} for the last estimate.
\medskip We may now assume that $f(x)=x^2+\alpha x$, where $\alpha\in [-2,2]$. Then, $f_{\min,{\mathbf K}}= -\alpha^2/4$, which is attained at $x=-\alpha /2$.
After scaling the measure $\mu$ by $2/\pi$,
the Chebyshev polynomials $T_i$ satisfy
$$\int_{-1}^1 T_i(x)T_j(x) {2\over \pi \sqrt {1-x^2}} dx= 0 \text{ if } i\ne j, \ 2 \text{ if } i=j=0,\ 1 \text{ if } i=j\ge 1.$$
So with respect to this scaled measure the normalized Chebyshev polynomials are $\hat T_0=1/\sqrt 2$ and $\hat T_i=T_i$ for $i\ge 1$, and they satisfy the 3-terms relation:
$$x\hat T_1= {1\over 2}\hat T_2+{1\over \sqrt 2}\hat T_0 \quad \text{ and } \quad x \hat T_{k} ={1\over 2}\hat T_{k+1} +{1\over 2}\hat T_{k-1} \ \text{ for } k\ge 2.$$
In view of Lemma \ref{lemsummarize} we know that the parameter $\underline{f}^{(d)}_{\mathbf{K}}$ is equal to the smallest eigenvalue of the following matrix
$$A_d =\left(\int_{-1}^1 (x^2+\alpha x) \hat T_i(x)\hat T_j(x) {2\over \pi \sqrt {1-x^2}} dx\right)_{i,j=0}^d.$$
Using the above 3-terms relations one can verify that the matrix $A_d$ has the following form:
\begin{equation}\label{eqAdLas}
A_d=\left(\begin{matrix}
{1\over 2} & {\alpha\over \sqrt 2} & {1\over 2\sqrt 2} & &&&&& \cr
{\alpha\over \sqrt 2} & {3\over 4} & {\alpha\over 2} & {1\over 4} & &&&& \cr
{1\over 2\sqrt 2} & {\alpha\over 2} & a& b & c & &&&\cr
& {1\over 4} & b & a& b& c & && \cr
& & c &b & \ddots & \ddots & \ddots & &\cr
&&& c & \ddots & \ddots & \ddots & \ddots & \cr
& &&&\ddots & \ddots & \ddots & \ddots & c\cr
& &&&&\ddots& \ddots & \ddots & b \cr
&&&&&& c& b & a
\end{matrix}\right),
\end{equation}
where we set $a=1/2$, $b=\alpha/2$ and $c=1/4$.
Observe that if we remove the first two rows and columns of $A$ then we obtain a principal submatrix, denoted $B$, which
is a symmetric 5-diagonal Toeplitz matrix.
Now we may embed $B$ into a symmetric circulant matrix of size $d+1$,
denoted $C_d$, by suitably defining the first two rows and columns. Namely,
$$
C_d=\left(\begin{matrix}
a & b & c & &&&&c& b\cr
b & a & b & c & &&&&c \cr
c & b & a & b & c & &&&\cr
& c & b & a & b &c & && \cr
& & c & b & \ddots & \ddots & \ddots & &\cr
&&& c & \ddots & \ddots & \ddots & \ddots & \cr
& &&&\ddots & \ddots & \ddots & \ddots & c\cr
c& &&&&\ddots& \ddots & \ddots & b \cr
b& c &&&&& c & b & a
\end{matrix}\right).
$$
Recall that the eigenvalues of a circulant matrix are known in closed form, see, e.g.,\ \cite{circulant matrices}.
In particular, the eigenvalues of $C_d$ are given by
\begin{equation}
\label{eigs symmetric circulant}
a + 2b\cos(2\pi j/(d+1)) + 2c\cos(2\pi 2j/(d+1), \quad j = 0,\ldots,d, \quad (d \ge 5).
\end{equation}
By the Cauchy interlacing theorem for eigenvalues (see, e.g.,\ Corollary 2.2 in \cite{Haemers95}), we have
$$\underline{f}^{(d)}_{\mathbf{K}}=\lambda_{\min}(A_d)\le \lambda_{\min}(B) \le \lambda_3(C_d),$$ where $\lambda_3(C_d)$ is the third smallest eigenvalue of $C_d$.
{As noted above the eigenvalues of $C_d$ are known in closed form as in (\ref{eigs symmetric circulant}) and this is the key fact which enables us to conclude the analysis.}
\begin{lemma}\label{lemquaduni}
For any $\alpha\in [-2,2]$, the third smallest eigenvalue of the matrix $C_d$ satisfies
$$\lambda_3(C_d)= -{\alpha^2\over 4} +O\left({1\over d^2}\right).$$
Therefore, if $f(x)=x^2+\alpha x$ with $\alpha \in [-2,2]$ then $\underline{f}^{(d)}_{\mathbf{K}}-f_{\min,{\mathbf K}}=O(1/d^2)$.
\end{lemma}
\begin{proof}
Setting $\vartheta_j= {2\pi j \over d+1}$ for $j\in{\mathbb N}$, then by \eqref{eigs symmetric circulant} the eigenvalues of the matrix $C_d$ are
the scalars
$${1\over 2} +\alpha \cos (\vartheta_j) +{1\over 2} \cos(2\vartheta_j) = \cos^2(\vartheta_j)+\alpha \cos( \vartheta_j )\quad \text{ for } 0\le j\le d.$$
Consider the function $f(\vartheta)= \cos^2(\vartheta)+\alpha \cos (\vartheta)$ for $\vartheta\in [0,2\pi]$. Then $f$ satisfies: $f(\vartheta)=f(2\pi-\vartheta)$, and its minimum value is equal to $-\alpha^2/4$, which is attained at $\vartheta= \arccos(-\alpha/2)\in [0,\pi]$ and $2\pi-\vartheta$.
Let $j$ be the integer such that $\vartheta_j \le \vartheta < \vartheta_{j+1}.$
Then the smallest eigenvalue of $C_d$ is $\lambda_{\min}(C_d)=\min\{f(\vartheta_j), f(\vartheta_{j+1})\}$ and its third smallest eigenvalue is given by $\lambda_3(C_d)= \min\{f(\vartheta_{j-1}), f(\vartheta_{j+1})\}$ if $\lambda_{\min}(C_d)= f(\vartheta_j)$,
and
$\lambda_3(C_d)= \min\{f(\vartheta_{j}), f(\vartheta_{j+2})\}$ if $\lambda_{\min}(C_d)= f(\vartheta_{j+1})$.
Therefore, $\lambda_3(C_d)=f(\vartheta_k)$ for some $k\in \{ j-1,j,j+1,j+2\}$.
Using Taylor theorem (and the fact that $f'(\vartheta)=0$) we can conclude that
$$\lambda_3(C_d)+{\alpha^2\over 4}= f(\vartheta_k)-f(\vartheta)= {1\over 2}f''(\xi)(\vartheta-\vartheta_k)^2,$$
for some scalar $\xi \in (\vartheta,\vartheta_k)$ (or $(\vartheta_k,\vartheta)$).
Finally, $f''(\xi)= -2\cos(\xi) -\alpha \cos (\xi)$ and thus we have $|f''(\xi)|\le 2+|\alpha|$.
Also $|\vartheta-\vartheta_k|\le |\vartheta_{j+2}-\vartheta_{j-1}| = {6\pi\over d+1}.$ The claimed result now follows directly. \end{proof}
\subsection{The general case}
As a direct application we can also deal with the case when $f$ is multivariate quadratic and separable.
\begin{corollary} \label{corquadsep}
Consider the box ${\mathbf K}=[-1,1]^n$ and a multivariate polynomial of the form
$f(x)=\sum_{i=1}^n x_i^2 +\alpha_i x_i$ for some scalars $\alpha_i\in {\mathbb R}$.
Then we have
$\underline{f}^{(d)}_{\mathbf{K}}-f_{\min,{\mathbf K}}=O(1/d^2)$.
\end{corollary}
\begin{proof}
The polynomial $f$ is separable: $f(x)=\sum_{i=1}^n f_i(x_i)$, after setting $f_i(x_i)= x_i^2+\alpha_i x_i.$
Hence its minimum over the box ${\mathbf K}$ is $f_{\min,{\mathbf K}}=\sum_{i=1}^n (f_i)_{\min,[-1,1]}$.
Suppose $\sigma_i\in \Sigma[x_i]_d$ is an optimal density function for the bound $\underline{f_i}^{(d)}_{[-1,1]}$ and consider the polynomial
$\sigma(x)=\prod_{i=1}^n \sigma_i(x_i) \in \Sigma[x]_{nd},$ which is a density function over ${\mathbf K}$. Then we have
$$\underline{f}^{(nd)}_{\mathbf{K}} -f_{\min,{\mathbf K}} \le \int_{{\mathbf K}} f(x)\sigma(x)d\mu(x) =\sum_{i=1}^n \left(\int_{-1}^1 f_i(x_i) d\mu(x_i) -(f_i)_{\min,[-1,1]}\right) =O(1/d^2),$$
where we use Lemma \ref{lemquaduni} for the last estimate.
This implies the claimed convergence rate for the bounds $\underline{f}^{(d)}_{\mathbf{K}}$.
\end{proof}
Assume now $f$ is an arbitrary polynomial and let $a\in {\mathbf K}=[-1,1]^n$ be a minimizer of $f$ over ${\mathbf K}$.
Consider the following quadratic polynomial
$$g(x) = f(a)+ \nabla f(a)^T (x-a) + C_f \|x-a\|_2^2,$$
where we set $C_f= \max_{x\in {\mathbf K}} \|\nabla^2 f(x)\|_2$. By Taylor's theorem we know that $f(x)\le g(x)$ for all $x\in {\mathbf K}$ and that the minimum value of $g(x)$ over ${\mathbf K}$ is $g_{\min,{\mathbf K}}=f(a)=f_{\min,{\mathbf K}}$.
This implies
$$\underline{f}^{(d)}_{\mathbf{K}}-f_{\min,{\mathbf K}}\le \underline{g}^{(d)}_{\mathbf{K}}-g_{\min,{\mathbf K}} =O(1/d^2),$$
where we use Corollary \ref{corquadsep} for the last estimate.
This concludes the proof of Theorem \ref{theoLasrate}.
\section{Concluding remarks}
Some other hierarchical upper bounds for polynomial optimization over the hypercube have been
investigated in the literature. In particular, bounds are proposed in \cite{KLLS MOR}, that rely on selecting
density functions arising from beta distributions:
\[
f^H_d:=\,\displaystyle\min_{(\alpha,\beta)\in {N}(2n,d)}\:\frac{\displaystyle\int_{\mathbf K} f(x)\,x^\alpha(1-x)^\beta\,dx}
{\displaystyle\int_{\mathbf K} x^\alpha(1-x)^\beta\,dx},
\]
where, ${\mathbf K} = [-1,1]^n$, and $(1-x)^\beta = \prod_{i=1}^n (1-x_i)^{\beta_i}$ for $\beta \in \mathbb{N}^n$.
These bounds can be computed via elementary operations only and
their rate of convergence is $f^H_d-f_{\min,{\mathbf K}}= O(1/\sqrt d)$ (or $O(1/d)$ for quadratic polynomials with rational data).
Other hierarchies involve
selecting discrete measures. They rely on polynomial evaluations at rational grid points \cite{KL SIOPT} or at polynomial
meshes like Chebyshev grids \cite{PV OL}.
The grids in \cite{PV OL} are given by the Chebyshev-Lobatto points:
\[
C_d := \left\{\cos\left(\frac{j\pi}{d}\right) \right\} \quad j = 0,\ldots,d.
\]
In particular the authors of \cite{PV OL} show that $\min_{x \in C_d^n} f(x) -f_{\min,{\mathbf K}} = O\left(\frac{1}{d^2}\right)$, where
$$ C_d^n= C_d \times \cdots \times C_d \subset [-1,1]^n.$$
Note that $|C_d^n| = (d+1)^n$, which is of course exponential in $n$ even for fixed $d$.
The same {$O\left(\frac{1}{d^2}\right)$} rate of convergence was shown in \cite{KL SIOPT} for the regular grid ({using} $d+1$ evenly spaced points).
{We also refer to the recent work \cite{PV18} where polynomial meshes are investigated for polynomial optimization over general convex bodies.}
Thus the Lasserre bound $\underline{f}^{(d)}_{\mathbf{K}}$ has the same $O\left(\frac{1}{d^2}\right)$ asymptotic rate of convergence as the grid searches, but with the advantage
that the computation may be done in polynomial time for fixed $d$.
Of course, the problem studied in this paper falls in the general framework of bound-constrained global optimization problems, and many other
algorithms are available for such problems; a recent survey is given in the thesis \cite{Pal thesis}. The point is that the
methods we studied in this paper allow analysis of the convergence rate to the global minimum.
We conclude with some unresolved questions:
\begin{itemize}
\item
Does the $O\left(\frac{1}{d^2}\right)$ rate of convergence still hold for the Lasserre bounds if ${\mathbf K}$ is a general convex body?
(The best known result is the $O(1/d)$ rate from \cite{DKL MOR}.)
\item
What is the precise influence of the choice of reference measure $\mu$ in \eqref{fminkreform2} on the convergence rate?
\item
Is is possible to show a `saturation' result for the Lasserre bounds of the type:
\[
\underline{f}^{(d)}_{\mathbf{K}} - f_{\min,{\mathbf K}} = o\left( \frac{1}{d^2}\right) \Longleftrightarrow \mbox{ $f$ is a constant polynomial?}
\]
In other words, is $O(1/d^2)$ the fastest possible convergence rate for nonconstant polynomials?
\end{itemize}
\vspace*{0.5cm}\noindent
{\bf Acknowledgements.}
The authors would like to thank Jean-Bernard Lasserre for useful discussions.
|
{
"timestamp": "2018-04-17T02:14:01",
"yymm": "1804",
"arxiv_id": "1804.05524",
"language": "en",
"url": "https://arxiv.org/abs/1804.05524"
}
|
\subsection{protocol details}
Now, we discuss the details of our method. Carry on the Equation.\ref{SDD}, defining the $\mathbf{D}$ operator
\begin{equation}
\mathbf{D} =: \sum_{\alpha} \sum_{j} \big(\prod_{i\neq j}\mathbf{X}^{T} \mathbf{A}^{\alpha}_i \mathbf{X} \big) \mathbf{A}_{j}^{\alpha}
\label{SDD}
\end{equation}
define $ Z= \mathbf{X} \mathbf{X}^{T} $, $b_{j}^{\alpha} =\mathbf{X}^{T} \mathbf{A}^{\alpha}_j \mathbf{X} $ and $ Q^{\alpha}_{j}=\mathbf{A}^{\alpha}_1Z\mathbf{A}^{\alpha}_2Z \cdots\mathbf{A}^{\alpha}_p $. Then, $$ M^{\alpha}=\prod_{i}\mathbf{X}^{T} \mathbf{A}^{\alpha}_i \mathbf{X}= \mathbf{X}^{T} Q^{\alpha} \mathbf{X}$$
The operator $\mathbf{D} $ can be rewritten as
\begin{eqnarray}
\mathbf{D} &=&\sum_{\alpha} \sum_{j} \big(\prod_{i}\mathbf{X}^{T} \mathbf{A}^{\alpha}_i \mathbf{X} \big) \dfrac{\mathbf{A}_{j}^{\alpha}}{\mathbf{X}^{T} \mathbf{A}^{\alpha}_j \mathbf{X} }\\
&=& \sum_{\alpha} \sum_{j} M^{\alpha} \dfrac{\mathbf{A}_{j}^{\alpha}}{b_{j}^{\alpha}}\\
&=& \sum_{m=1}^{Kp} c_{m}\mathbf{A}_{m}
\end{eqnarray}
where $ c_{m}= M^{\alpha}/b_{j}^{\alpha} $.
The method to obtain the expected value $ M^{\alpha} $ of $ Q_{j}^{\alpha} $ and expected value $b_{j}^{\alpha} $ of $\mathbf{A}^{\alpha}_j $ are as follows.
\begin{figure}[!h]
\centering
\includegraphics[width=0.6\textwidth]{Sfig1.pdf}
\caption{ Quantum circuit for obtaining the expected value $b_{j}^{\alpha} $. $|x\rangle$ denotes the initial state of work system, and ancillary system consists of $ n $ qubits in the $\|0\rangle_{n}$ state, where $n=T_{1}$. The squares represent unitary operations and the circles represent the state of the controlling qubit. } \label{ap1}
\end{figure}
In Fig.\ref{ap1}, the unitary matrix $ H $ represents the tenser products of $ T_{1} $ number Hadamard gates.
The process can be expressed as:
\begin{eqnarray}
|0\rangle^{T_{1}}|X\rangle &\rightarrow &H|0\rangle^{T_{1}}|X\rangle \nonumber\\
&\rightarrow &\frac{1}{\sqrt{2^{T_{1}}}}(|0\rangle A_{1}|X\rangle +|1\rangle A_{2}|X\rangle + \cdots ).\nonumber\\
& = & \frac{1}{\sqrt{2^{T_{1}}}}(\sum_{m=1}^{Kp}|m\rangle A_{m}|X\rangle)
\label{eq6}
\end{eqnarray}
When the ancillary system in state $ |m\rangle $, we measure the work system via $|X\rangle\langle X| $ basis, we can obtain the expected value $ b_{j}^{\alpha} $. The expected value $ M^{\alpha}$ can be calculated by $ M^{\alpha}=\prod_{i}b_{j}^{\alpha} $.
Now, we go to the iteration part. We conbined the minus sign and the operator $\mathbf{D}$, into a unitary operator~$\mathbf{A}_{m}$. The iteration process can be expressed as
\begin{eqnarray}
|\mathbf{X}^{(t+1)}\rangle & = & |\mathbf{X}^{(t)}\rangle - \mathbf{D} |\mathbf{X}^{(t)}\rangle \\\nonumber
& = & |\mathbf{X}^{(t)}\rangle +\sum_{m=0}^{Kp-1}c_{m}\mathbf{A}_{m} |\mathbf{X}^{(t)}\rangle .
\end{eqnarray}\label{eq7}
We can prepare the following initial state firstly
\begin{eqnarray}
|0\rangle |0\rangle^{T_{1}} |0\rangle &\rightarrow & ( \sqrt{\beta}|0\rangle|0\rangle^{T_{1}} + \sum_{m=0}^{Kp-1}c_{m}|1\rangle|m\rangle)|\mathbf{X}^{(t)}\rangle.
\end{eqnarray}
Then, we perform the first ancillary qubit controlled operation $|0\rangle \langle 0|\otimes A_{0}^{\dagger} $ and the second ancillary system controlled operations $|0\rangle \langle 0| \otimes A_{0} ,|1\rangle \langle 1|\otimes A_{1} ,\ldots ,|Kp-1\rangle \langle Kp-1| \otimes A_{Kp-1} $ on the work system $|\mathbf{x}^{(t)}\rangle$. This step transforms the initial state into
\begin{eqnarray}
( \sqrt{\beta}|0\rangle|0\rangle^{T_{1}} + \sum_{m=0}^{Kp-1}c_{m}|1\rangle|m\rangle)|\mathbf{X}^{(t)}\rangle
& = & \frac{1}{\sqrt{(\sum_{m=0}^{Kp-1} c_{m}+1)}} |0\rangle|0\rangle^{T_{1}}|\mathbf{X}^{(t)}\rangle +\frac{\sqrt{\sum_{m=0}^{Kp-1} c_{m}}}{\sqrt{(\sum_{m=0}^{Kp-1} c_{m}+1)}}|1\rangle|0\rangle^{T_{1}}|\mathbf{X}^{(t)}\rangle\\\nonumber
&\rightarrow &
\frac{1}{\sqrt{(\sum_{m=0}^{Kp-1} c_{m}+1)}} |0\rangle|0\rangle^{T_{1}}|\mathbf{X}^{(t)}\rangle +\frac{\sqrt{\sum_{m=0}^{Kp-1} c_{m}}}{\sqrt{(\sum_{m=0}^{Kp} c_{m}+1)}}|1\rangle | c_{m}\rangle A_{m}|\mathbf{X}^{(t)}\rangle\\\nonumber.
\end{eqnarray}
Finally, applying the gate sequence of preparing initial state in reverse and measuring the two registers.
An output of $ |0\rangle |0\rangle^{T_{1}} $ will result in the following state
\begin{eqnarray}
\frac{1}{\sqrt{(\sum_{m=0}^{Kp-1} c_{m}+1)}} |0\rangle|0\rangle^{T_{1}}|\mathbf{X}^{(t)}\rangle +\frac{\sqrt{\sum_{m=0}^{Kp-1} c_{m}}}{\sqrt{(\sum_{m=0}^{Kp} c_{m}+1)}}|1\rangle | c_{m}\rangle A_{m}|\mathbf{X}^{(t)}\rangle\\\nonumber
\rightarrow \frac{1}{\sum_{m=1}^{Kp} c_{m}+1} \left ( |\mathbf{X}^{(t)}\rangle +\sum_{m=1}^{Kp} c_{m}\mathbf{A}_{m} |\mathbf{X}^{(t)}\rangle \right),
\label{NewP}
\end{eqnarray}
which is proportional to $ \mathbf{X}^{(t+1)} $. When we get the final result, we can multiply $ \sum_ {m=1}^{Kp} c_{m}+1 $ to obtain $|\mathbf{X}^{(t+1)}\rangle $.
\subsection{experimental process}
\subsubsection{molecule}
The demonstration of the whole algorithm were conducted on a four-qubit nuclear magnetic resonance (NMR) system. The four-qubit sample is $^{13}$C-labeled crotonic acid dissolved in d6-acetone.
\begin{figure}[htb]
\begin{center}
\includegraphics[width= 0.6\columnwidth]{Sfig2_molecule.pdf}
\end{center}
\setlength{\abovecaptionskip}{-0.00cm}
\caption{\footnotesize{ Molecular structure of $^{13}$C-labeled crotonic acid. C$_1$ is denoted as the working system while C$_2$, C$_3$ and C$_4$ are denoted as ancillary qubits. The chemical shifts and J-couplings (in Hz) are listed by the diagonal and off-diagonal elements, respectively. T$_{2}$ (in Seconds) are also shown at bottom. All parameters were obtained on this Bruker DRX 400MHz spectrometer at room temperature (296.5K). }}\label{molecule}
\end{figure}
Fig. \ref{molecule} shows the structure of this molecule, where C$_1$ to C$_4$ are denoted as four qubits, representing the ancillary and working system. Throughout the entire experiments, M, H$_1$ and H$_2$ nuclei in the methyl group were decoupled. With the weak coupling assumption, the internal Hamiltonian of this liquid sample are written as:
\begin{align}\label{Hamiltonian}
\mathcal{H}=\sum\limits_{j=1}^4 {\frac{1}{2} \omega _j } \sigma_z^j + \sum\limits_{j < k}^4 {\frac{\pi}{2}} J_{jk} \sigma_z^j \sigma_z^k,
\end{align}
where $\nu_j$ is the chemical shift and $\emph{J}_{jk}$ is the J-coupling strength. All parameters were obtained on this Bruker DRX 400MHz spectrometer at room temperature (296.5K).
\subsubsection{Preparation of the pseudo-pure state}
In our four-qubit NMR system, the thermal equilibrium state $\rho_{eq}$ is,
\begin{equation}
\rho_{eq}=\frac{1-\epsilon}{16} \mathbb{I} + \epsilon (\gamma_{\text{C1}} \sigma_{z}^{1}+\gamma_{\text{C2}} \sigma_{z}^{2}+\gamma_{\text{C3}} \sigma_{z}^{3}+\gamma_{\text{C4}} \sigma_{z}^{4}),
\end{equation}
where $\epsilon \approx 10^{-5}$ polarization coefficient, $ \mathbb{I}$ is a $16\times 16$ identity matrix, and $\gamma_{\text{Ci}}(i=1...4)$ are the gyromagnetic ratios for each carbon nuclei, respectively. As the identity part does not influence the unitary operations or measurements in NMR experiments, the original density matrix $\rho_{eq}$ can be replaced by the deviation matrix, i.e,
\begin{equation}
\label{drho}
\rho_{eq}= \sigma_{z}^{1} + \sigma_{z}^{2} + \sigma_{z}^{3}+ \sigma_{z}^{4}.
\end{equation}
Our purpose is to create the pseudo-pure state
\begin{equation}
\rho_{0000}=\frac{1-\epsilon}{16} \mathbb{I}+ \epsilon |0000\rangle\langle0000|,
\end{equation}
Spatial average technique\cite{Cory04031997} was used here, as the pulse sequences are shown in Fig. \ref{Sfig3}. It includes local rotation and four $z$-gradient fields for destroying the unnecessary coherent terms. The entire procedure of the pseudo-pure state creation takes about 70ms with the simulated fidelity 99.8\%. In experiment, the fidelity is around 99.01\%.
\begin{figure}
\centering
\includegraphics[width=0.7\textwidth]{Sfig3.pdf}
\caption{ Circuit for creating the $4$-quibit pseudo-pure state: via spatial average technique, the entire procedure includes local operations, five J-coupling evolutions, and four z-gradient pulses which is to destroy the unnecessary coherent terms. The time of each free evolutions is several ms, determined by 1/2J, i.e. the J-coupling strengths of the relevant spins. }
\label{Sfig3}
\end{figure}
\subsubsection{operators in expeimrnt}
We depicts the details of the experimental circuits in this section, which implements one iteration to realize gradient descent algorithm. During the experiment, the unitary operators were arranged as follows.
$V^{0}$ is a sigle qubit rotation and $ W^{0}={V^{0}}^{\dagger} $ and
\begin{equation}
V^{0}=\left(
\begin{array}{cc}
\frac{1}{\sqrt{(\sum_{m=1}^{Kp} c_{m}+1)}} & \frac{\sqrt{\sum_{m=1}^{Kp} c_{m}}}{\sqrt{(\sum_{m=1}^{Kp} c_{m}+1)}} \\
\frac{\sqrt{\sum_{m=1}^{Kp} c_{m}}}{\sqrt{(\sum_{m=1}^{Kp} c_{m}+1)}} & -\frac{1}{\sqrt{(\sum_{m=1}^{Kp} c_{m}+1)}} \\
\end{array}
\right).
\end{equation}
V is a $Kp\times Kp$ unitary matrix, which maps $V|0\rangle^T=\frac{1}{\sqrt{\sum_{m=1}^{Kp} c_{m}}}(\sqrt{c_{1}}|0\rangle)+\sqrt{c_{2}}|1\rangle+...+\sqrt{c_{Kp}}|Kp-1\rangle$, where $T=2$ and $Kp=4$ in our experiment.. Using the Schmidt orthogonalization, we could find a unitary matrix implementing it. Meanwhile, $ W={V}^{\dagger} $. As for the series of $A_i$, since the decomposition $\widehat{A}=-\sigma_I\otimes \sigma_X+\sigma_X\otimes \sigma_Z$, $A_i (i=1...4)$ equals to $-\sigma_I$,$\sigma_X$,$\sigma_X$ and $\sigma_Z$, respectively.
\subsubsection{the results of each iteration}
Each time we measure the output of the gradient descent implement circuit, the 4-qubit tomography was also employed and 4-qubit state $\rho_3$ was obtained. Theoretical states $\rho_{th}$ are listed by Eq.\ref{NewP}, where $|\mathbf{X}^{(t)}\rangle$ is the input state, i.e. the current point. we calculate the fidelity between the theoretical and experimental results\cite{coryfidelity}:
\begin{equation}
F=\operatorname{tr}(\rho_{\text{th}}\rho_{\text{exp}})/\sqrt{\operatorname{tr}(\rho_{\text{th}}^{2})\operatorname{tr}(\rho_{\text{exp}}^{2})}.
\end{equation}
Here, we write the states in terms of density matrices because the experimentally prepared state is mixed due to the experimental imperfections. For those 4-qubit states, since the decoherence effect introduced by the long algorithm operation time about 70ms and the inaccurate pulse from arbitrary wave generator, they have the average of 94\% fidelity. Fig.\ref{fig4} has shown this consequence.
\begin{figure}
\centering
\includegraphics[width=0.6\textwidth]{Sfig4.pdf}
\caption{ Quantum circuit for realising gradient descent algorithm. $|X\rangle$ denotes the initial state of work system, and ancillary system are$ T_{1}+1 $ qubits in the $|0\rangle|0\rangle^{T_{1}}$ state, where$ T_{1}=log_{2}(Kp) $. The squares represent unitary operations and the circles represent the state of the controlling qubit. } \label{fig4}
\end{figure}
|
{
"timestamp": "2018-04-17T02:06:58",
"yymm": "1804",
"arxiv_id": "1804.05231",
"language": "en",
"url": "https://arxiv.org/abs/1804.05231"
}
|
\subsection{Stability Analysis}
The dynamics for autonomous vehicles can be modeled with Euler
equations to study the behavior of these systems. In a continuous time
setting, the state of the system $\vx \in \reals^n$ updates w.r.t.\ an ordinary differential equation. Formally, $\dot{\vx} = f(\vx, \vu)$,
where $\dot{\vx}$ is the derivative of $\vx$ w.r.t.\ time and $\vu \in \reals^m$ is the control input.
Stability is a fundamental property of dynamical systems. Numerous
control problems can be viewed as controlling a given system to
stabilize to a given equilibrium state $\vx_r$ or an ``equilibrium''
reference trajectory $\vx_r(t)$. Control Lyapunov functions (CLF) are
a powerful tool for designing such stabilizing
controls~\cite{sontag1983lyapunov,artstein1983stabilization}. We first describe CLFs for
stabilizing to an equilibrium state.
\begin{definition}[Control Lyapunov Functions]\label{def:clf}
A CLF $V$ is a smooth and radially unbounded function that maps each
state to a real non-negative value, such that \textbf{(a)}
$V(\vx_r) = 0$ and $V(\vx) > 0$ for all $\vx \not=\vx_r$, and
\textbf{(b)}
$(\forall\ \vx \not=\vx_r)\ (\exists \vu) \ \dot{V}(\vx, \vu) < 0$,
wherein $\dot{V}$ is the Lie-derivative of $V$ w.r.t. $f$:
$\dot{V}(\vx, \vu) = \nabla V \cdot
f(\vx, \vu)$.
\end{definition}
Condition (a) ensures that the value of $V$ is zero at
the equilibrium and strictly positive everywhere else. Condition (b)
ensures that for any state, we can find a control input that can achieve
an instantaneous decrease to the value of $V$. In this sense, CLFs are
equivalent to \emph{artificial potential functions} over the state
space~\cite{lopez1995autonomous}. Having a CLF, one can design a
feedback law which always decreases the value of $V$ and therefore
stabilizes the system to the equilibrium point. For instance, Sontag
provides a simple means to extract a feedback law from a given
CLF~\cite{sontag1989universal}.
\subsection{Trajectory Tracking vs.\ Path Following}
Another form of stability appears in trajectory tracking, wherein the
goal is to stabilize to a reference trajectory $\vx_r(t)$. Formally,
let $\vx_d(t):\ \vx(t) - \vx_r(t)$ describe the ``deviation'' from the
reference trajectory state at time $t$. The goal is to stabilize
$\vx_d$ to the equilibrium $\vzero$ under the time-varying reference frame
that places $\vx_r(t)$ as the origin at time $t$. The dynamics for
$\vx_d$ is defined as: $\dot{\vx}_d = f(\vx, \vu) - \vr(t)$, wherein
$\dot{\vx}_r = \frac{d\vx_r}{dt} = \vr(t)$.
One of the key drawbacks of trajectory tracking is that it specifies
the reference trajectory $\vx_r(t)$ along with the \emph{reference
timing}, wherein the state $\vx_r(t)$ must ideally be achieved
at time $t$. This poses a challenge for control design unless the
timing is designed very carefully. Imagine, a reference trajectory
that traverses a winding hilly road at constant speeds. This compels
the control to constantly accelerate the vehicle on upslopes only
to ``slam the brakes'' on downhill sections~\cite{Hauser+Saccon/2006/Motorcycle}.
Path following, on the other hand, separates these concerns by
allowing the user to specify a reference (feasible) path parameterized
with a scalar, $\theta$ (instead of time), $\vx_r(\theta)$ yields a
state for each $\theta$ and $\frac{d\vx_r}{d\theta} = \vr(\theta)$.
As proposed by Hauser et al., one could define $\Pi$ as a function
that maps a state $\vx$ to the closest state on the reference
trajectory
$\vx_r(\cdot)$, using an auxiliary map $\pi$ ~\cite{HAUSER1995,Saccon+Hauser+Beghi/2013/Virtual}:
\[ \pi(\vx):\ \underset{\theta}{\mbox{argmin}} \ ||\vx -
\vx_r(\theta)||_P^2 \ \,, \ \Pi(\vx) : \vx_r(\pi(\vx)) \] where $||\vx||_P^2 :\ \vx^t P \vx$ is a
Lyapunov function for the linearized dynamics around the reference
trajectory. In order to stabilize the system to the reference path,
Hauser et al.\ propose to decrease the value of
$||\vx - \Pi(\vx)||_P^2$. However, as the projection function $\pi$
can get complicated, they use local approximations of $\pi$.
Following this, others have proposed to design a control law for a
virtual input $u_0$ that controls $\theta$ as a function of time
(called the \emph{timing feedback law}), or in other words, the
progress (or sometimes regress) along the
reference~\cite{SKJETNE2004,AGUIAR2004}. Therefore, the deviation is
now defined as $\vx_d(t):\ \vx(t) - \vx_r(\theta(t))$ wherein
$\diff{\theta}{t} = u_0$. As depicted in Fig.~\ref{fig:new-system},
$\theta$ is mapped to a state on the path $\vx_r(\theta)$.
For example Faulwasser et al.~\cite{Faulwasser2014} design
the timing law as a function of $\theta$, the deviation ($\vx_d$ for
state feedback systems), and their higher derivatives:
\[
g(\theta^{(k)},\vx_d^{(k)},\ldots,\theta,\vx_d,u_0) = 0\,,
\]
wherein $\theta^{(k)}$ is the $k^{th}$ derivative of
$\theta$. However, defining the function $g$ is a nontrivial problem.
\begin{figure}[t]
\begin{center}
\vspace{0.2cm}
\includegraphics[width=0.25\textwidth]{pics/new-system}
\end{center}
\caption{Schematic View of a System along the Parameterized Path.}\label{fig:new-system}
\end{figure}
\subsection{Related Work}
Stability for autonomous vehicles is a challenging problem. Brockett~\cite{brockett1983} showed that even a simple unicycle model cannot be stabilized using continuous feedback laws. However, continuous feedback laws exist for stabilization to non-stationary trajectories; these feedback laws are usually obtained through linearization~\cite{Walsh1994}.
While trajectory tracking has been widely used to solve plan execution, it has several shortcomings which are addressed using path-following.
In pioneering work ~\cite{Nelson1990,Sampei1991,Canudas1991,Samson1992,Sordalen1993} the velocity of the vehicle tracked a desired reference velocity and the controller is designed to steer the vehicle to the path. These path-following methods have been shown to yield a smoother convergence to the trajectory while avoiding input saturation. Beside these works, a wide diversity of approaches are used to study the path-following problem. One line of work is based on designing vector fields surrounding the path to guarantee reaching and following the path~\cite{Nelson2007,lawrence2007}. Another approach is to use model predictive control~\cite{Faulwasser2009,Faulwasser2013}. In this article, we consider a line of effort distinct from these others.
Hauser et al.~\cite{HAUSER1995} proposed the conversion of the trajectory
tracking strategy to the so-called maneuver regulation strategy. The
main idea is to decrease the distance between the state and the
reference trajectory, not a specific point on the trajectory. The
reference trajectory $\vx_r(\cdot)$ is parameterized using a variable
$\theta$ (instead of time) and distance is defined as a function of
$\vx - \vx_r(\theta)$. $\theta$ and treated as a variable. An update
law (timing law) is then applied to ensure proper change of
$\theta$. Hauser and Hindman showed that this maneuver regulation
trick would yield a system that avoids input saturation. Similarly,
Pappas~\cite{Pappas1996} showed that by re-parameterizing the
trajectory, one could avoid input saturation. Subsequently, Encarnacao
et al.~\cite{Encarnacao2001} extended the technique for the output
maneuvering problem on a restricted set of dynamics. m
Following Hauser et al.~\cite{HAUSER1995}, others have divided the
task into two parts. The first task is to reach and follow the
reference trajectory using the variable $\theta$ (instead of time), and the
second task is to \textit{improve} the solution using an extra control input
$\theta$. For example, in Skjetne et al.~\cite{SKJETNE2004}, first the
system output is stabilized, and then a control law for $\theta$ is
used to adjust the velocity. In this work, we use the extra
freedom to control $\theta$ for increasing robustness. More
specifically, this extra degree of freedom allows us to design more
robust control Lyapunov functions (CLFs) from which we extract
the feedback as well as the timing law.
Control Lyapunov functions were originally \hadi{introduced by
Sontag~\cite{sontag1989universal,sontag1983lyapunov}}. Synthesis
of CLFs is hard, involving bilinear matrix inequalities
(BMIs)~\cite{tan2004searching,majumdar2013control}.
Standard approaches such as alternating minimization result often do
not converge to a solution. To combat this, Majumdar et al.\ use
LQR controllers and their associated Lyapunov functions for the
linearization of the dynamics as good initial seed
solutions~\cite{majumdar2013control}. In contrast, recent work by some
of the authors remove the bilinearity by using a demonstrator in the form of a MPC
controller~\cite{Ravanbakhsh-RSS-17}. Furthermore, this approach
avoids local saddle points and has a fast convergence guarantee.
Aguiar et al.~\cite{AGUIAR2004} argue that there are performance
limitations for systems with unstable zero dynamics if one uses
trajectory tracking. However, using an extra control input
$\theta$, this restriction vanishes. The timing law in this work is
designed as a function of $\theta$ and its higher derivatives.
\hadi{Egerstedt et al.~\cite{Egerstedt2001virtual} develop a method where the reference point dynamics are governed by tracking error feedback.
Similarly}, Faulwasser et al.~\cite{Faulwasser2014} proposed designing the timing
law as a function of $\theta$, tracking error, and their higher derivatives. For example, one
can design a timing law which slows down the progress of $\theta$ when
the distance between the state and the reference is large.
They also combine the idea of path-following with control funnels. They Similarly, we use control funnels to provide formal guarantees. However, the funnel is constructed using a CLF. Besides this, the timing law in our work is a function of the state $\vx$ and depends on the structure of the CLF.
\section{INTRODUCTION}~\label{sec:intro}
\input{intro}
\section{BACKGROUND}~\label{sec:background}
\input{background}
\section{PATH-FOLLOWING USING CLF}~\label{sec:main}
\input{path}
\section{PATH SEGMENT FOLLOWING PROBLEM}~\label{sec:bounded}
\input{local}
\section{EXPERIMENTS}~\label{sec:expr}
\input{expr}
\section{CONCLUSIONS}~\label{sec:conclusion}
In this work, we investigate the use of control Lyapunov functions for
path following and provide a characterization of control funnel
functions for tracking a trajectory segment. Our approach lends itself
to an efficient synthesis technique presented previously. We implement
the resulting controller on the Parkour car and show its effectiveness
through a set of tracking problems. Future work will focus on integrating
our approach more closely with planning approaches to augment existing
approaches to the synthesis of control funnels~\cite{majumdar2013robust}.
\section*{ACKNOWLEDGMENT}
This work was funded in part by NSF under award numbers SHF 1527075 and CPS 1646556. All opinions expressed
are those of the authors and not necessarily of the NSF.
\bibliographystyle{IEEEtran}
|
{
"timestamp": "2018-08-06T02:00:45",
"yymm": "1804",
"arxiv_id": "1804.05288",
"language": "en",
"url": "https://arxiv.org/abs/1804.05288"
}
|
\section{Introduction}
\label{sec:Intro}
Since its development in the 1980s, diffusion tensor imaging (DTI) has become an essential tool to study white matter connectivity in the human brain. Its ability to infer the orientation of white matter fibers, in-vivo and non-invasively, is key to understanding brain connectivity and associated neurological diseases \cite{hagmann2006understanding,de2011atlasing}. Since the macroscopic inference of underlying fibers from dMRI data, known as tractography, typically produces a large number of streamlines, it is common to group these streamlines into anatomically meaningful clusters called \emph{bundles} \cite{o2013fiber}. Clustering streamlines is also essential for the creation of white matter atlases, visualization, and statistical analysis of microstructure measures along tracts \cite{guevara2012automatic,Odonnell07automatictractography,siless2018anatomicuts}.
Furthermore, clinical applications of tractography analysis are also numerous and include identifying major bundles for neurological planning in patients with tumors \cite{o2017automated}, understanding difference between white matter connectivity in typically developing controls versus children with autism \cite{zhang2017whole}, and uncovering white matter bundles as bio-markers for the diagnosis of Parkinson's disease \cite{cousineau2017test}.
Clustering streamlines into anatomically meaningful bundles is a challenging task in part due to lack of gold standard. There can be several hundreds of thousands of streamlines to consider, making the clustering problem computationally complex. As illustrated in Fig. \ref{fig:corpus-callosum_KSC}, streamlines within the same bundle can have different lengths and endpoints. Thus, using standard geometric distance measures often leads to poor results. Another challenge comes from the weak separability of certain bundles, which can result in low-quality (e.g., too small or too large) clusters. Also, while many clustering approaches assume a crisp membership of streamlines to bundles, as shown in Fig. \ref{fig:corpus-callosum_KSC}, such a separation of streamlines into hard clusters is often arbitrary. In practice, streamline bundles may overlap and intersect each other, making their extraction and analysis difficult.
Moreover, when used to label the streamlines of a new subject, the clusters learned using crisp methods often give unsatisfactory results due to the variability across individual brains.
\begin{figure}[h]
{\centering
\includegraphics{Figs/Fig1_CC_hard_and_Soft_visualization_cropped.jpg}
}
\caption{Illustrative example. Clustering of the corpus callosum by our method: hard clustering (\textbf{left}), and membership of each streamline to two bundles (\textbf{center} and \textbf{right}). Dark green represents a zero membership and bright red a maximum membership to the bundles.}
\label{fig:corpus-callosum_KSC}
\end{figure}
In this paper, we propose a set of flexible and efficient streamline clustering approaches based on kernel dictionary learning and sparsity priors. The general idea of these approaches is to learn a compact dictionary of training streamlines capable of describing the whole dataset, and to encode bundles as a sparse non-negative combination of multiple dictionary prototypes. In contrast to spectral embedding methods (e.g., \cite{brun2004clustering,o2005white}) which perform the embedding and clustering in two separate steps, our approaches find clusters in the kernel space without having to explicitly compute an embedding.
The proposed streamline clustering approaches have several advantages over existing methods for this task. First, they do not require an explicit representation of the streamlines and can extend to any streamline representation or distance/similarity measure. Second, they use a non-linear kernel mapping which facilitates the separation of clusters in a manifold space. Third, unlike hard-clustering methods like the k-means algorithm and its variants (e.g. spectral clustering), they can distribute the membership of streamlines across multiple bundles, making them more robust to overlapping bundles and outliers, as well as to variability across subjects.
Our specific contributions include:
\begin{enumerate}
\setlength{\itemsep}{2pt}%
\setlength{\parskip}{2pt}
\item We propose three different streamline clustering models based on kernel k-means, non-negative factorization and sparse coding, and demonstrate the advantages of these models with respect to the state of the art;
\item We provide a flexible platform to integrate and evaluate streamline distance measures, and compare the performance of three popular measures using two different datasets;
\item Whereas dictionary learning and sparsity have shown promise in various pattern recognition and neuroimaging applications, to our knowledge, the present article is the first account of their use for streamline clustering in a peer-reviewed indexed publication.
Our results on the streamline clustering problem show the potential of this approach for other imaging applications.
\end{enumerate}
The rest of the paper is structured as follows. Section \ref{sec:related-works} provides a brief survey of relevant literature on streamline clustering. In Section \ref{sec:MaterialsAndMethods}, we present our kernel dictionary learning based methods. Section \ref{sec:Results} evaluates the methods on the task of clustering streamlines using real data. Finally, we conclude with a summary of our main contributions, and discuss potential extensions.
\section{Related works}\label{sec:related-works}
Our presentation of relevant work is divided into two parts, focusing respectively on the various approaches for representation and analysis of streamlines, and the application of sparse coding techniques in neuroimaging.
\subsection{White matter fiber analysis}
Over the years, several approaches have been proposed to cluster streamlines and provide a simplified quantitative description of white matter connections, including cross-population inferences \cite{guevara2012automatic,jin2014automatic,o2007automatic,prasad2014automatic}. These studies could be vaguely classified into two categories: representation of streamlines or streamline similarity, and clustering approaches. Features proposed to represent streamlines include the distribution parameters (mean and covariance) of points along the streamline \cite{brun2004clustering} and B-splines \cite{maddah2006statistical}. Approaches using such explicit features typically suffer from two problems: they are sensitive to the length and endpoint positions of the streamlines and/or are unable to capture their full shape. Instead of using explicit features, streamlines can also be compared using specialized distance measures. Popular distance measures for this task include the Hausdorff distance, the Minimum Direct Flip (MDF) distance and the Mean Closest Points (MCP) distance \cite{corouge2004towards,moberts2005evaluation}.
Fiber clustering approaches include manifold embedding techniques such as spectral clustering and normalized cuts \cite{brun2004clustering}, agglomerative approaches like hierarchical clustering \cite{Odonnell07automatictractography,corouge2004towards}, k-means \cite{li2010hybrid}, and Dirichlet processes
\cite{wassermann2010unsupervised,wang2011tractography}. Several studies have also focused on incorporating anatomical features into the clustering \cite{siless2018anatomicuts,o2007automatic}, or on clustering large multi-subject datasets \cite{guevara2012automatic}. A detailed description and comparison of several distances and clustering approaches can be found in \cite{moberts2005evaluation,olivetti2017comparison,siless2013comparison}.
Various studies have also focused on the segmentation of streamlines, toward the goal of drawing cross-population inferences \cite{guevara2012automatic,jin2014automatic,o2007automatic,prasad2014automatic}. These studies either follow an atlas based approach \cite{guevara2012automatic,jin2014automatic,o2007automatic} or align specific tracts directly across subjects \cite{garyfallidis2015robust,o2012unbiased}. Multi-step or multi-level approaches have also been proposed to segment streamlines, for example, by combining both voxel and streamline groupings \cite{guevara2012automatic}, fusing labels from multiple hand-labeled atlases \cite{jin2014automatic}, or using a bundle representation based on maximum density paths \cite{prasad2014automatic}. A few studies have also investigated the representation of specific streamline bundles using different techniques such as gamma mixture models \cite{maddah2008unified}, the computational model of rectifiable currents \cite{durrleman2009statistical,gori2016parsimonious}, and functional varifolds \cite{kumar2017white}. For detailed review of white matter clustering approaches, we refer the reader to \cite{o2013fiber}.
\subsection{Sparse coding for neuroimaging}
Sparse coding, with an objective of encoding a signal as a sparse combination of prototypes in a data-driven dictionary, has been applied in various domains of computer vision and pattern recognition \cite{elad2010role,wright2009robust,wright2010sparse,yang2009linear}. Various neuroimaging applications have also utilized concepts from this technique, such as the reconstruction \cite{lustig2008compressed} or segmentation \cite{tong2013segmentation} of MRI data, and for functional connectivity analysis \cite{lee2016spark,lee2016sparse}. For diffusion data, sparse coding has been used successfully for clustering white matter voxels from Orientation Density Function (ODF) data \cite{ccetingul2014segmentation}, and for finding a population-level dictionary of key white matter tracts \cite{zhu2016population}.
Recently, several studies have outlined the connection between clustering and factorization problems, such as dictionary learning \cite{aharon2006svd,sprechmann2010dictionary} and non-negative matrix factorization \cite{kim2007sparse}. Thus, dictionary learning can be seen as a soft clustering, where objects can be linked to more than one cluster. Researchers have also recognized the advantages of applying kernels to existing clustering methods, like the k-means algorithm \cite{dhillon2004kernel}, as well as dictionary learning approaches \cite{nguyen2012kernel}. Such ``kernel'' methods have been shown to better learn the non-linear relations in the data \cite{hofmann2008kernel}.
Sparse coding and dictionary learning were used in \cite{moreno2016sparse,alexandroni2017white} to obtain a compressed representation of streamlines. In our previous work \cite{kumar2015brain,kumar2016sparse}, we applied these concepts to learn an multi-subject streamline atlas for labelling the streamlines of a new subject. In recent studies, we showed how this idea can be used to derive a brain fingerprint capturing genetically-related information on streamline geometry \cite{kumar2017fiberprint}, and to incorporate along-tract measures of micro-structure in the representation \cite{kumar2017white}.
The present study extends our preliminary work in \cite{kumar2017white,kumar2015brain,kumar2016sparse,kumar2017fiberprint} by providing an in-depth analysis that compares different sparsity priors and evaluates the impact of various parameters. As algorithmic contributions, we present two extensions of the model in \cite{kumar2016sparse}, based on group sparsity and manifold regularization, that provide more meaningful bundles and can incorporate information on streamline geometry, such as the proximity of streamline endpoints, to constrain the clustering process.
\section{Kernel dictionary learning for streamline clustering}
\label{sec:MaterialsAndMethods}
In this section we propose kernel dictionary learning and sparsity priors based frameworks for white matter fiber analysis. We start with a brief review of dictionary learning and the k-means algorithm, followed by proposed methods based on various sparsity priors, and algorithm complexity analysis.
\subsection{Dictionary learning and the k-means algorithm}
\label{sec:DL_and_KM}
Let $\vec X$ be the set of $n$ streamlines, each represented as a set of 3D coordinates. For the purpose of explanation, we suppose that each streamline $i$ is encoded as a feature vector $\vec x_i \in \mathbb{R}^d$, and that $\vec X$ is a $d \! \times \! n$ feature matrix. Since our dictionary learning method is based on kernels, a fixed set of features is however not required, and streamlines having a different number of 3D coordinates could be compared with a suitable similarity measure (i.e., the kernel function).
The traditional (hard) clustering problem can be defined as assigning each streamline to a bundle from a set of $m$ bundles, such that streamlines are as close as possible to their assigned bundle's prototype (i.e., cluster center). Let $\Psi^{m \times n}$ be the set of all $m \! \times \! n$ cluster assignment matrices (i.e., matrices in which each row has a single non-zero value equal to one), this problem can be expressed as finding the matrix $\vec D$ of $m$ bundle prototypes and the streamline-to-bundle assignment matrix $\vec W$ that minimize $\|\vec X - \vec D\vec W\|_F^2$.
This formulation of the clustering problem can be seen as a special case of dictionary learning, where $\vec D$ is the dictionary and $\vec W$ is constrained to be a cluster assignment matrix, instead of enforcing its sparsity.
While solving this clustering problem is NP-hard, optimizing $\vec W$ or $\vec D$ individually is easy. For a given dictionary $\vec D$, the optimal $\vec W$ assigns each streamline $i$ to the prototype $m$ closest to its feature vector. Likewise, for a fixed $\vec W$, the optimal dictionary is found by solving a simple linear regression problem. This simple heuristic correspond to the well-known k-means algorithm.
\subsection{Kernel k-means}
\label{sec:KKM}
In our streamline clustering problem, the k-means approach described in the previous section has two important disadvantages. First, it requires to encode streamlines as a set of features, which is problematic due to the variation in their length and endpoints. Also, it assumes linear relations between the streamlines and bundle prototypes, while these relations could be better defined in a non-linear subspace (\emph{manifold}).
These problems can be avoided by using a kernel version of k-means for the streamline clustering problem. In this approach, each streamline is projected to a $q$-dimensional space using a mapping function ${\vec \phi} : \mathbb{R}^d \to \mathbb{R}^q$, where $q \gg d$. We denote by $\vec \Phi$ the $\mathbb{R}^{q \times n}$ matrix containing the tracts of $\vec X$ mapped with ${\vec \phi}$. The inner product of two streamlines in this space corresponds to a kernel function $k$, i.e. $ k(\vec x_i, \vec x_j) \ = \ \tr{{\vec \phi}(\vec x_i)}{\vec \phi}(\vec x_j)$. With $\vec K \ = \ \tr{\vec \Phi}\vec \Phi$, the kernel matrix, the \emph{kernel} clustering problem can be expressed as:
\begin{equation}\label{eqn:KKM_cost}
\argmin_{\substack{\vec D \, \in \, \mathbb{R}^{q \times k} \\
\vec W \, \in \, \{0,1\}^{m \times n}}} \
\|\vec \Phi - \vec D\vec W\|_F^2 \quad \tx{subject to} \ \ \tr{\vec W}\vec 1_m \ = \ \vec 1_n.
\end{equation}
Since the dictionary prototypes are defined in the kernel space, $\vec D$ cannot be computed explicitly. To overcome this problem, we follow the strategy proposed in \cite{nguyen2012kernel,rubinstein2010double} and define the dictionary as $\vec D = \vec \Phi A$, where $A \in \mathbb{R}^{n \times m}$.
Using a similar optimization approach as in k-means, we alternate between updating matrix $\vec W$ and $A$. Thus, we update $\vec W$ by assigning each streamline $i$ to the prototype $m$ whose features in the kernel space are the closest:
\begin{equation}\label{eqn:W-KKM}
w_{mi} \ = \ \left\{\begin{array}{ll}
1 : & \tx{if } m = \arg\min_{m'} \ [\tr{A} \vec K A]_{m'm'} \ - \ 2[\tr{A} \vec k_i]_{m'}, \\
0 : & \tx{otherwise}.
\end{array}\right.,
\end{equation}
where $\vec k_i$ corresponds to the $i$-th column of $\vec K$. Recomputing $A$ corresponds once again to solving a linear regression problem with optimal solution:
\begin{equation}\label{eqn:A-KKM}
A \ = \ \tr{\vec W} \big(\vec W \tr{\vec W}\big)^{-1}.
\end{equation}
We initialize matrix $A$ as a random selection matrix (i.e., random subset of columns in the identity matrix). This is equivalent to using a random subset of the transformed streamlines (i.e., subset of columns in $\vec \Phi$) as the initial dictionary. This optimization process is known as \emph{kernel k-means} \cite{dhillon2004kernel}.
\subsection{Non-negative kernel sparse clustering}
\label{sec:KSC}
Because they map each streamline to a single bundle, hard clustering approaches like (kernel) k-means can be sensitive to poorly separated bundles and streamlines which do not fit in any bundle (outliers). This section describes a new clustering model that allows one to control the hardness or softness of the clustering.
In the proposed model, the hard assignment constraints are replaced with non-negativity and $L_0$-norm constraints on the columns of $\vec W$. Imposing non-negativity is necessary because the values of $\vec W$ represent the membership level of streamlines to bundles. Moreover, since the $L_0$-norm counts the number of non-zero elements, streamlines can be expressed as a combination of a small number of prototypes, instead of a single one.
When updating the streamline-to-bundle assignments, the columns $\vec w_i$ of $\vec W$ can be optimized independently, by solving the following sub-problem:
\begin{equation}\label{eqn:KSC_cost}
\argmin_{\vec w_i \, \in \, \mathbb{R}_+^m} \ \|\phi(\vec x_i) - \vec \Phi A \vec w_i\|^2_2 \quad
\tx{subject to} \ \ \|\vec w_i\|_0 \leq S_\mr{max}.
\end{equation}
Parameter $S_\mr{max}$ defines the maximum number of non-zero elements in $\vec w_i$ (i.e., the sparsity level), and is provided by the user as input to the clustering method.
The algorithm summary and computational complexity is reported in Supplement material, Algorithm 1. To compute non-negative weights $\vec w_i$, we modify the kernel orthogonal matching pursuit (kOMP) approach of \cite{nguyen2012kernel} to include non-negativity constrains of sparse weights (Supplement material, Algorithm 2). Unlike kOMP, the most \emph{positively} correlated atom is selected at each iteration, and the sparse weights $\vec w_s$ are obtained by solving a non-negative regression problem. Note that, since the size of $\vec w_s$ is bounded by $S_\mr{max}$, computing $\vec w_s$ is fast.
In the case of a soft clustering (i.e., when $S_\mr{max} \geq 2$), updating $A$ with (\ref{eqn:A-KKM}) can lead to negative values in the matrix. As a result, the bundle prototypes may lie outside the convex hull of their respective streamlines. To overcome this problem, we adapt a strategy proposed for non-negative tri-factorization \cite{ding2006orthogonal} to our kernel model. In this strategy, $A$ is recomputed by applying the following update scheme, until convergence:
\begin{equation}\label{eqn:A-KSC}
[A]_{ij} \ \gets \ [A]_{ij} \cdot \dfrac {\left[ \vec K \tr{\vec W}\right]_{ij}} {\left[\vec K A \vec W \tr{\vec W}\right]_{ij}}, \quad i=1,\ldots,n, \quad j=1,\ldots,m.
\end{equation}
The above update scheme produces small positive values instead of zero entries in $A$. To resolve this problem, we apply a small threshold in post-processing.
In terms of computational complexity, the bottleneck of the method lies in computing the kernel matrix. For large datasets, we could reduce this computational complexity by approximating the kernel matrix with the Nystr\"om method \cite{fowlkes2004spectral,Odonnell07automatictractography} (Supplement material, Section 1.5).
\subsection{Extension 1: group sparse kernel dictionary learning}
\label{sec:GKSC_L1_L21}
The methods proposed above may find insignificant bundles (e.g., bundles containing only a few streamlines) when the parameter controlling the number of clusters is not properly set. Due to the lack of gold standard in tractography analysis, finding a suitable value for this parameter is challenging.
To overcome this problem, we present a new clustering method based on group sparse kernel dictionary learning. We reformulate the clustering problem as finding the dictionary $\vec D$ and non-negative weight matrix $\vec W$ minimizing the following problem:
\begin{equation}\label{eqn:GKSC_cost}
\argmin_{\substack{A \in \mathbb{R}^{n \times m} \\
\vec W \, \in \, \mathbb{R}_+^{m \times n}}} \
\frac{1}{2}\|\vec \Phi - \vec \PhiA\vec W\|_F^2 \, + \, \lambda_1 \|\vec W\|_1 \, + \, \lambda_2 \|\vec W\|_{2,1}.
\end{equation}
In this formulation, $\|\vec W\|_1 = \sum_{i=1}^K\sum_{j=1}^N |w_{ij}|$ is an $L_1$ norm prior which enforces global sparsity of $\vec W$, and $\|\vec W\|_{2,1} = \sum_{i=1}^K \|\ww_{i\LargerCdot}\|_2$ is a mixed $L_{2,1}$ norm prior imposing the vector of row norms to be sparse. Concretely, the $L_1$ norm prior limits the ``membership'' of streamlines to a small number of bundles, while the $L_{2,1}$ prior penalizes the clusters containing only a few streamlines.
%
Parameters $\lambda_1, \lambda_2 \geq 0$ control the trade-off between these three properties and the reconstruction error (i.e., the first term of the cost function).
We solve this problem using an Alternating Direction Method of Multipliers (ADMM) algorithm \cite{boyd2011distributed}. First, we introduce ancillary matrix $\vec Z$ and reformulate the problem as:
\begin{equation}\label{eqn:W-GKSC_L1_L21_cost}
\argmin_{\substack{A \in \mathbb{R}_+^{n \times m} \\ \vec W, \vec Z \, \in \, \mathbb{R}_+^{m \times n}}} \
\frac{1}{2}\|\vec \Phi - \vec \Phi A \vec W\|^2_F
\, + \, \lambda_1 \|\vec Z\|_1 \, + \, \lambda_2 \|\vec Z\|_{2,1} \ \ \tx{subject to} \ \vec W = \vec Z.
\end{equation}
We then convert this an unconstrained problem using an Augmented Lagrangian formulation with multipliers $\vec U$:
\begin{equation}\label{eqn:W-GKSC_L1_L21_aug_lag}
\argmin_{\substack{A \in \mathbb{R}_+^{n \times m} \\ \vec W, \vec Z \, \in \, \mathbb{R}_+^{m \times n}}} \
\frac{1}{2}\|\vec \Phi - \vec \Phi A \vec W\|^2_F
\, + \, \lambda_1 \|\vec Z\|_1 \, + \, \lambda_2 \|\vec Z\|_{2,1}
\, + \, \frac{\mu}{2}\|\vec W - \vec Z + \vec U\|_F^2.
\end{equation}
Parameters $\vec W$, $\vec Z$ and $\vec U$ are updated alternatively until convergence. In this work, we use primal feasibility as convergence criteria and stop the optimization once $\|\vec W - \vec Z\|_F^2$ is below a small epsilon.
Dictionary matrix $\AA$ is updated as (\ref{eqn:A-KSC}). To update $\vec W$, we derive the objective function with respect to this matrix and set the result to $0$, yielding:
\begin{equation}\label{eqn:W-GKSC_L1_L21_W}
\vec W \ = \ \big(\tr{A}\KKA + \mu \vec I\big)^{-1}\big(\tr{A}\vec K + \mu(\vec Z-\vec U)\big).
\end{equation}
Note that imposing non-negativity on $\vec W$ is not required since we ensure this property for $\vec Z$ and have $\vec W \approx \vec Z$ at convergence.
Optimizing $\vec Z$ corresponds to solving a group sparse proximal problem \cite{friedman2010note}. This can be done in two steps. First, we do a $L_1$-norm shrinkage by applying the non-negative soft-thresholding operator to each element of $\vec W+\vec U$:
\begin{equation}\label{eqn:W-GKSC-l1thres}
\hat{z}_{ij} \ = \ S^+_{\nicefrac{\lambda_1}{\mu}}\big(w_{ij} + u_{ij}\big) \ = \
\max\Big\{w_{ij} + u_{ij} -
\nicefrac{\lambda_1}{\mu}, \, 0\Big\}, \quad i \leq K, \ j \leq N.
\end{equation}
Then, $\vec Z$ is obtained by applying a group shrinkage on each row of $\hat{\vec Z}$:
\begin{equation}\label{eqn:W-GKSC-l2l1thres}
\zz_{i\LargerCdot} \ = \ \max\Big\{\|\hat{\zz}_{i\LargerCdot}\|_2 - \nicefrac{\lambda_2}{\mu}, \, 0\Big\} \cdot \frac{\hat{\zz}_{i\LargerCdot}}{\|\hat{\zz}_{i\LargerCdot}\|_2}, \quad i \leq K.
\end{equation}
Finally, the Lagrangian multipliers are updated as in standard ADMM methods: $\vec U \ := \ \vec U + (\vec W - \vec Z)$.
The overall optimization procedure and its computational complexity are reported in Supplement material, Algorithm 3.
\subsection{Extension 2: kernel dictionary learning with manifold prior}
\label{sec:GKSC_L1_Lap}
Another challenge in streamline clustering is to generate anatomically meaningful groupings. This may require incorporating prior information into the clustering process, for example, to impose streamlines ending in the same anatomical region to be grouped together. In this work, we address this challenge by proposing a manifold-regularized kernel dictionary learning method.
In the proposed method, we define the manifold as a graph with adjacency matrix $\vec G \in \mathbb{R}^{n \times n}$. In this matrix, $g_{i,i'} = 1$ if streamlines $i$ and $i'$ should be grouped in the same bundle, otherwise $g_{i,i'} = 0$. The manifold regularization prior on the streamline-to-bundle assignments can be formulated as
\begin{align}\label{eqn:manifold-reg}
\mathcal{R}_\mr{man}(\vec W) & \ = \ \lambda_L \! \sum_{i=1}^n \sum_{i'=1}^n g_{i,i'} \, \|\vec w_i - \vec w_{i'}\|_2^2 \nonumber\\
& \ = \ \lambda_L \, \mr{tr}(\vec W\vec L\tr{\vec W}),
\end{align}
where $\mathcal{L} \in \mathbb{R}^{n \times n}$ is the Laplacian of $\vec G$ and $\lambda_L$ is a parameter controlling the strength of constraints on streamlines.
Our manifold-regularized formulation is obtained by replacing the $L_{2,1}$ prior on $\vec W$ with $\mathcal{R}_\mr{man}(\vec W)$. This new formulation can be solved, as the previous one, with an ADMM algorithm. The main difference occurs when updating $\vec W$, which corresponds to the following problem:
\begin{equation}\label{eqn:W-GKSC_Lap_aug_lag}
\begin{split}
\argmin_{\vec W \, \in \, \mathbb{R}^{k \times n}} \
\|\vec \Phi - \vec \Phi A \vec W\|^2_F \, + \, \lambda_L\,\mr{tr}(\vec W\vec L\tr{\vec W})
\, + \, \mu\|\vec W - \vec Z + \vec U\|_F^2 .
\end{split}
\end{equation}
Derive this objective function with respect to $\vec W$ and setting the result to $0$ gives a Sylvester equation of the form $\vec P \vec W + \vec W \vec Q = \vec R$ where, $\vec P = \tr{A}\KKA + \mu\vec I$, $\vec Q = \lambda_L \vec L$, and $\vec R = \tr{A}\vec K + \mu(\vec Z-\vec U)$. This equation can be solved using Bartels-Stewart algorithm \cite{bartels1972solution}, which requires transforming $\vec P$ and $\vec Q$ into Schur form with a QR algorithm, and solving the resulting triangular system via back-substitution (Supplement material, Algorithm 4). The computational complexity is $O(n^3)$, $n$ being the size of $\vec Q$. However, this can be drastically reduced by pre-computing once the Schur form of $\vec Q$.
\section{Experimental results and analysis}
\label{sec:Results}
In this section, we evaluate our proposed methods on a labeled dataset, followed by parameter impact analysis, and concluding with Human Connectome Project data results on clustering and automated segmentation of new subjects.
\subsection{Data and pre-processing}
\label{subsec:DataAndPreProc}
In the first experiment, we compared the proposed methods on a dataset of manually/expert labeled streamline bundles provided by the Sherbrooke Connectivity Imaging Laboratory (SCIL). The source dMRI data was acquired from a 25 year old healthy right-handed volunteer and is described in \cite{fortin2012tractography}. We used 10 of the largest bundles, consisting of 4449 streamlines identified from the cingulum, corticospinal tract, superior cerebellar penduncle and other prominent regions. Figure \ref{fig:Fiber_bundles_GT_vis_compare} (left) shows the coronal and sagittal plane view of the ground truth set. Fibernavigator tool \cite{chamberland2014real} was used for visualizations of this dataset.
To evaluate the performance of our method across a population of subjects, we two datasets. First, consisting of 12 healthy volunteers (6 males and 6 females, between 19 to 35 years of age) from the freely available MIDAS dataset \cite{bullitt2005vessel}. For streamline tractography, we used the tensor deflection method \cite{lazar2003white} with the following parameters: minimum fractional anisotropy of 0.1, minimum streamline length of 100 mm, threshold for streamline deviation angle of 70 degrees. A mean number of 9124 streamlines was generated for the 12 subjects.
Second, used the pre-processed dMRI data of $10$ unrelated subjects (age 22--35) from the Q3 release of the Human Connectome Project \cite{glasser2013minimal,van2012human,van2013wu}, henceforth referred to as HCP data. All HCP data measure diffusivity along 270 directions distributed equally over 3 shells with b-values of 1000, 2000 and 3000 $\nicefrac{\tx{s}}{\tx{mm}^2}$, and were acquired on a Siemens Skyra 3T scanner with the following parameters: sequence = Spin-echo EPI; repetition time (TR) = 5520 ms; echo time (TE) = 89.5 ms; resolution = $1.25 \times 1.25 \times 1.25$ $\tx{mm}^3$ voxels. Further details can be obtained from HCP Q3 data release manual\footnote{\url{http://www.humanconnectome.org/documentation/Q3/}}.
For signal reconstruction and tractography, we used the freely available DSI Studio toolbox \cite{yeh2010generalized}. All subjects were reconstructed in MNI space using the Q-space diffeomorphic reconstruction (QSDR) \cite{yeh2011ntu} option in DSI Studio. We set output resolution to $1$ mm. For skull stripping, we used the masks provided with pre-processed diffusion HCP data. Other parameters were set to the default DSI Studio values. Deterministic tractography was performed with the Runge-Kutta method of DSI Studio \cite{basser2000vivo,yeh2013deterministic}, using the following parameters: minimum length of $40$ mm, turning angle criteria of $60$ degrees, and trlinear interpolation. The termination criteria was based on the quantitative anisotropy (QA) value, which is determined automatically by DSI Studio. As in the reconstruction step, the other parameters were set to the default DSI Studio values. Using this technique, we obtained a total of $50\,000$ streamlines for each subject.
As a note, whether the streamlines, generated from tractography, represent the actual white matter pathways remains a topic of debate \cite{jones2013white,thomas2014anatomical}. Streamlines derived from DSI studio are hypothetical curves in space that represent, at best, the major axonal directions suggested by the orientation distribution functions of each voxel, which may contain tens of thousands of actual axonal streamlines.
\subsection{Experimental methodology}
\label{subsec:methodology}
We tested three distance measures used in the literature for the streamline clustering problem: 1) the Hausdorff distance (Haus) \cite{corouge2004towards,o2005white} which measures the maximum distance between any point on a streamline and its closest point on the other streamline, 2) the mean of closest points distance (MCP) \cite{corouge2004towards} that computes the mean distance between any point on a streamline and its closest point on the other streamline, and 3) the end points distance (EP) \cite{moberts2005evaluation} measuring the mean distance between the endpoints of a streamline and the closest endpoint on the other streamline.
Fiber distances were converted into similarities by applying a radial basis function (RBF) kernel: $k_{i,i'} = \exp\big(\!-\!\gamma \, \mr{dist}_{i,i'}^2\big)$. Parameter $\gamma$ was adjusted separately for each distance measure, using the distribution of values in the corresponding distance matrix. Since the tested distance measures are not all metrics, we applied spectrum shift to make kernels positive semi-definite: $\vec K_\mr{psd} = \vec K + |\lambda_\mr{min}| \, \vec I$, where $\lambda_\mr{min}$ is the minimum eigenvalue of $\vec K$. This technique only modifies self similarities and is well adapted to clustering \cite{chen2009learning}.
We initialized $\vec W$ using the output of a spectral clustering method \cite{o2005white}, which applies the k-means algorithm on the $10$ first eigenvectors of the normalized Laplacian matrix of $\vec K$. To avoid inversion problems when $\vec W \tr{\vec W}$ is close to singular, we used a small regularization value of 1e-8. Finally, to compare our method with hard clustering approaches, we converted its soft clustering output to a hard clustering by mapping each streamline $i$ to the bundle $j$ for which $w_{ji}$ is maximum.
We compared our kernel sparse clustering (KSC) approach to four other methods: kernel k-means (KKM) using the same $\vec K$ and initial clustering, the spectral clustering (Spect) approach described above, single linkage hierarchical clustering (HSL) \cite{moberts2005evaluation}, and QuickBundles (QB) \cite{garyfallidis2012quickbundles}. The performance of these methods was evaluated using four clustering metrics: the Rand Index (RI) which measures the consistency of the clustering output with respect to the ground truth, the Adjusted Rand Index (ARI) adjusting ARI values by removing the chance agreement, the Normalized Adjusted Rand Index (NARI) that further normalizes the values by considering the cluster sizes, and the Silhouette (SI) measure which does not use the ground truth and measures the ratio between the intra-cluster and inter-cluster distances \cite{rousseeuw1987silhouettes}. While RI, ARI and NARI values range from $0.0$ to $1.0$, SI values are between $-1.0$ and $1.0$. In practice, SI values are generally much lower than $1.0$ due to the intrinsic intra-cluster variance. More information about these metrics can be found in \cite{moberts2005evaluation,siless2013comparison}.
\begin{table}[ht]
\caption{Clustering accuracy of our KSC method ($S_\mr{max}\!=\!3$), kernel k-means (KKM), spectral clustering (Spect), and hierarchical clustering (HSL), using the Hausdorff, MCP and EP distances, on the SCIL dataset. For KSC, KKM and Spect, the mean accuracy over $10$ initializations with $m$=10 is reported. The best results for a distance and accuracy metric are shown in boldface type.
}
{
\centering
\fontsize{8}{8}\selectfont
\begin{tabular}{ l l c c c c}
\toprule
\multirow{2}{*}{\textbf{Dist}} & \multirow{2}{*}{\textbf{Method}} & \textbf{RI} & \textbf{ARI} & \textbf{NARI} & \textbf{SI}\\
& & mean (std) & mean (std) & mean (std) & mean (std) \\
\midrule\midrule
\multirow{5}{*}{MCP} & KSC & \textbf{0.948 (0.012)} & \textbf{0.780 (0.051)} & \textbf{0.716 (0.047)} & \textbf{0.543 (0.032)} \\
& KKM & 0.947 (0.011) & 0.777 (0.049) & \textbf{0.716 (0.046)} & 0.541 (0.028)\\
& Spect & 0.942 (0.014) & 0.752 (0.058) & 0.701 (0.047) & 0.515 (0.059)\\
& HSL & 0.915 (0.000) & 0.704 (0.000) & 0.612 (0.000) & 0.474 (0.000)\\
& QB & 0.943 (0.000) & \textbf{0.780 (0.000)} & 0.696 (0.000) & 0.486 (0.000)\\
\midrule
\multirow{5}{*}{Haus} & KSC & \textbf{0.924 (0.013)} & \textbf{0.658 (0.068)} & \textbf{0.634 (0.030)} & \textbf{0.425 (0.022)}\\
& KKM & 0.904 (0.020) & 0.589 (0.082) & 0.573 (0.068) & 0.365 (0.054)\\
& Spect & 0.884 (0.018) & 0.517 (0.041) & 0.538 (0.054) & 0.317 (0.069)\\
& HSL & 0.891 (0.000) & 0.640 (0.000) & 0.609 (0.000) & 0.221 (0.000)\\
& QB & 0.851 (0.000) & 0.468 (0.000) & 0.485 (0.000) & 0.143 (0.000) \\
\midrule
\multirow{5}{*}{EP} & KSC & \textbf{0.919 (0.005)} & \textbf{0.634 (0.026)} & \textbf{0.641 (0.006)} & \textbf{0.422 (0.020)}\\
& KKM & 0.915 (0.013) & 0.621 (0.052) & 0.634 (0.034) & 0.410 (0.032)\\
& Spect & 0.911 (0.014) & 0.603 (0.053) & 0.616 (0.040) & 0.408 (0.031)\\
& HSL & 0.842 (0.000) & 0.539 (0.000) & 0.445 (0.000) & 0.197 (0.000)\\
& QB & 0.885 (0.000) & 0.534 (0.000) & 0.550 (0.000) & 0.129 (0.000)\\
\bottomrule
\end{tabular}
}
\label{tab:Table_method_distance_comp}
\end{table}
\subsection{Comparison of methods and distance measures}
Table \ref{tab:Table_method_distance_comp} gives the accuracy obtained by KSC ($S_\mr{max}$=3) and the four other tested methods on the SCIL dataset, for the same number of clusters as the ground truth ($m$=10). Since the output of spectral clustering depends on the initialization of its k-means clustering step, for Spect, KSC and KKM, we report the mean performance and standard deviation obtained using $10$ different random seeds. We see that our KSC method improves the initial solution provided by spectral clustering, and gives in most cases a higher accuracy than other clustering methods. We also observe that KSC is more robust to the choice of distance measure than other methods and, as reported in \cite{moberts2005evaluation}, that MCP is consistently better than other distance measures.
\begin{figure}
{\centering
\includegraphics{Figs/Fig4_vis_GT_KSC_cropped.jpg}
\vspace{-3mm}
\caption{Right sagittal (\textbf{top}) and inferior axial (\textbf{bottom}) views of the ground truth, and bundles obtained by KSC ($S_\mr{max}=3$) using the Haus, MCP and EP.}
\label{fig:Fiber_bundles_GT_vis_compare}
}
\end{figure}
Figure \ref{fig:Fiber_bundles_GT_vis_compare} compares the ground truth clustering of the SCIL dataset with the outputs of KSC ($S_\mr{max}$=3) using the Haus, MCP and EP distances. Except for the superior cerebellar peduncle bundle (cyan and green colors in the ground truth), the bundles obtained by KSC+MCP and KSC+Haus are similar to those of the ground truth clustering. Also, we observe that the differences between KSC+MCP and KSC+Haus occur mostly in the right inferior fronto-occipital fasciculus and inferior longitudinal fasciculus bundles (yellow and purple colors in the ground truth). Possibly due to the large variance of endpoint distances in individual bundles, KSC+EP gives poor clustering results.
\subsection{Impact of sparsity}
Figure \ref{fig:GT_KSC_KKM_Spect_comp_MCP} reports the mean ARI (over $10$ runs) obtained on the SCIL dataset by our KSC approach, using $S_\mr{max}$=1,2,3, for an increasing number of clusters (i.e., dictionary size $m$). For comparison, the performance of KKM and Spect is also shown. When the Spectral Clustering initialization is near optimal (i.e., when $m$ is near the true number of clusters and using MCP), both methods find similar solutions. However, when the initial spectral clustering is poor (e.g., Haus and EP distance or small number of clusters) the improvement obtained by KSC is more significant than KKM. Hence, KSC ($S_\mr{max}\!\geq\!2$) is more robust than hard clustering approaches (i.e., Spect, KKM or KSC with $S_\mr{max}$=1) to the number of clusters and distance measures.
\begin{figure}[t]
{\centering
\mbox{
\includegraphics[width=11cm]{Figs/Fig3_mean_ARI_plots_Hauss_MCP_EP_cropped.jpg}
}
\vspace{-3mm}
\caption{Mean ARI obtained on the SCIL dataset by KSC ($S_\mr{max} = 1,2,3$), KKM and Spect, using Haus (\textbf{left}), MCP (\textbf{center}), EP (\textbf{right}); for varying $m$.}
\label{fig:GT_KSC_KKM_Spect_comp_MCP}
}
\end{figure}
\begin{figure}[h]
{\centering
\includegraphics[width=11cm]{Figs/Fig3_vis_Wsoft_GT_cropped.jpg}
}
\vspace{-3mm}
\caption{Membership level of streamlines to two different bundles (\textbf{left} and \textbf{center}), and importance of each streamline in defining the prototype of a bundle (\textbf{right}). Blue means a null membership/importance, while non-zero values are represented by a color ranging from green (lowest value) to red (highest value).}
\label{fig:A_W_sparsity_KSC_MCP}
\end{figure}
To illustrate the soft clustering of KSC, Fig. \ref{fig:A_W_sparsity_KSC_MCP} (\textbf{left}) and (\textbf{center}) show the membership level of streamlines to two different bundles. Streamline colors in each figure correspond to the values of a row in $\vec W$ normalized so that the minimum is 0 (blue) and the maximum is $1$ (red). We observe streamlines having a membership to both bundles (e.g., orange-colored streamlines in the left image), reflecting the uncertainty of this part of the clustering. In Fig. \ref{fig:A_W_sparsity_KSC_MCP} (\textbf{right}), we show the importance of each streamline in defining the prototype of a bundle, using the normalized value of a column in $A$ as colors. It can be seen that only a few streamlines are used to define this bundle, confirming the sparsity of $A$.
\subsection{Group sparsity prior}
\begin{figure}[h]
\centering
\begin{small}
\includegraphics[width=11cm]{Figs/Fig5_mean_ARI_GKSC_hist_cropped.jpg}
\vspace{-3mm}
\caption{\textbf{(a)} Mean ARI obtained on the SCIL dataset by GKSC, MCP+L1, MCP+Manifold and Spect, using MCP; for varying $m$; \textbf{(b)} mean and standard deviation of final $m$ for varying input $m$; \textbf{(c)} Distribution of bundle sizes for a sample run using $m=20$.}
\label{fig:GKSC_analysis_final_m}
\end{small}
\end{figure}
\begin{table
\caption{Clustering accuracy of proposed methods
using MCP distances and three types of priors: $L_1$ norm sparsity alone (L1), with group sparsity (L1+L21), and with manifold regularization (L1+Manifold). Reported values are the mean accuracy over $10$ initializations with (input) $m$=10 clusters. The best result for each accuracy metric is shown in boldface type.
}
{
\centering
\fontsize{8.5}{8.5}\selectfont
\begin{tabular}{l c c c c}
\toprule
\multirow{2}{*}{\textbf{Prior}} & \textbf{RI} & \textbf{ARI} & \textbf{NARI} & \textbf{SI}\\
& mean (std) & mean (std) & mean (std) & mean (std) \\
\midrule\midrule
L1 & 0.947 (0.011) & 0.775 (0.049) & 0.714 (0.045) & 0.543 (0.029)\\
L1+Manifold & 0.948 (0.010) & 0.780 (0.044) & 0.717 (0.046) & 0.546 (0.033)\\
L1+L21 & \textbf{0.949 (0.006)} & \textbf{0.791 (0.025)} & \textbf{0.721 (0.035)} & \textbf{0.563 (0.039)}\\
\bottomrule
\end{tabular}
}
\label{tab:Table_method_distance_comp_2}
\end{table}
Figure \ref{fig:GKSC_analysis_final_m}(a) plots the mean Adjusted Rand Index (ARI) obtained by our group sparse model (MCP+L1+L21) for various cluster numbers ($m$), over $10$ runs with different spectral clustering initializations. As baseline, we also report the ARI of spectral clustering and our method without group sparsity (MCP+L1), i.e. using $\lambda_2$=0. We see that employing group sparsity improves clustering quality and provides a greater robustness to the input value of $m$. The advantages of using a group sparse prior are further confirmed in Table \ref{tab:Table_method_distance_comp_2}, which gives the mean ARI, RI, NARI and average SI for $m$=10. Results show that MCP+L1+L21 outperforms MCP+L1 for all performance metrics. In a t-test, these improvements are statistically significant with $p<$0.01.
As described in Section \ref{sec:GKSC_L1_L21}, group sparsity has the benefit of providing meaningful bundles, regardless of the number of clusters $m$ given as input. In Fig. 5(a), we see that the ARI of MCP+L1+L21 increases monotonically until reaching the ground-truth number of bundles $m^*$=10. While the clustering accuracy of other methods drops for $m>$10, the performance of MCP+L1+L21 remains stable. This is explained in Fig. (b) which plots the number of non-empty clusters found by MCP+L1+L21 as a function of $m$: the number of output clusters stays near to $m^*$=10, even for large values of $m$. As additional confirmation, Fig. \ref{fig:GKSC_analysis_final_m}(c) shows the number of streamlines per cluster for a sample run of MCP+L1+L21 with $m=20$. In this example, the output clustering contains $m^*$=10 non-empty clusters.
In Fig. \ref{fig:Param_var_GKSC}, we measure the impact of sparse regularization parameters $\lambda_1$ and $\lambda_2$ for a fixed ADMM parameter of $\mu=0.01$. As shown in (a), $\lambda_1/\mu$ controls the mean number of non-zero weights per streamline (i.e., how soft or hard is the clustering). Likewise, as illustrated in (b), $\lambda_2/\mu$ defines the size of bundles in the output. These results are consistent with the use $L_1$-norm and $L_{2,1}$-norm sparsity in (\ref{eqn:W-GKSC-l1thres}). Finally, the optimization stability of the MCP+L1+L21 model is illustrated in Fig. \ref{fig:Param_var_GKSC}(c), where convergence is reached around 20 iterations.
\begin{figure}[h]
\centering
\begin{small}
\includegraphics[width=11cm]{Figs/Fig6_lambda_imapct_GKSC_cropped.jpg}
\vspace{-3mm}
\caption{\textbf{(a)} Mean number of non-zero assignment weights per streamline, for $\lambda_2/\mu = 80$ and increasing $\lambda_1/\mu$. \textbf{(b)} Mean number of streamlines per bundle, for $\lambda_1/\mu=0.1$ and increasing $\lambda_2/\mu$. \textbf{(c)} Cost function value at each of a sample run for MCP+L1+L21.}
\label{fig:Param_var_GKSC}
\end{small}
\end{figure}
\subsection{Manifold regularization prior}
We apply the proposed manifold regularization prior to enforce the grouping of streamlines with similar end-points. The idea is to obtain bundles that correspond to localized regions of the cortex. To generate the Laplacian matrix in (\ref{eqn:manifold-reg}), we constructed a graph where the nodes are streamlines and two nodes are connected if the distance between their nearest endpoints is below some threshold. Following \cite{gori2016parsimonious}, we used a distance threshold of 7mm, giving a Laplacian matrix with overall sparsity near 15\%.
\begin{figure}[h]
\centering
\begin{small}
\includegraphics[width=11cm]{Figs/Fig7_Lap_and_HCP_SI_cropped.jpg}
\vspace{-4mm}
\caption{\textbf{(a)} Percentage overlap with EP based Laplacian prior matrix, compared with baseline initialization of spect, for varying $m$. \textbf{(b)} Mean of avg SI for KSC+MCP clustering of $10$ unrelated HCP subjects for varying $m$.}
\label{fig:Lap_prior_and_HCP_10sub}
\end{small}
\end{figure}
In Fig. \ref{fig:GKSC_analysis_final_m}(a), we see that the manifold regularization prior (MCP+Manifold) improves performance compared to spectral clustering baseline and $L_1$ norm sparsity (MCP+L1). This improvement is particularly important when the input number of clusters is below that of the ground truth (i.e., $m<$10). Conversely, for $m\!>\!10$, MCP+Manifold is outperformed by group sparsity (MCP+L1+L21) due to the over-segmentation of streamlines. Fig. \ref{fig:Lap_prior_and_HCP_10sub}(a) measures the the percentage of streamlines with nearby endpoints (i.e., edges in the graph) that are assigned to the same cluster, denoted as overlap in the figure. As expected, the prior helps preserve anatomical information defined by streamline endpoints in the clustering.
\subsection{Validation on HCP data}
We evaluated the performance of our kernel sparse clustering (KSC) method on a population of subjects from the Human Connectome Project (HCP). For this experiment, we used two datasets: $10$ unrelated HCP subjects, and subjects from the freely available MIDAS dataset \cite{bullitt2005vessel} (results in Supplement material). The objective here is to show applicability of our method across population-subjects, and analyse the impact of inter-subject variability.
Figure \ref{fig:Lap_prior_and_HCP_10sub}(b) shows the mean of average SI obtained for the $10$ unrelated subjects, using a varying number $m$ of clusters and $3$ runs for each $m$ value. This plot was generated by sampling $5000$ streamlines uniformly over the full tractography (\cite{o2007automatic,kumar2017fiberprint}) and computing their pairwise MCP distance. We observe that clustering quality decreases with higher values of $m$, and that this quality varies across subjects. A similar trend is observed for MIDAS dataset (Supplement material, Fig. 2). Comparing HCP and MIDAS datasets, a greater average SI is obtained for HCP possibly due to the higher resolution of images in this dataset. Full clustering visualization for $10$ subjects ($m=50$) and subject 1 for $m=25,50,75,100,125,150$ are shown in Supplement material, Figure 3,4). Note the optimal number of streamline clusters is still an open challenge \cite{o2007automatic}, we used $m=50$ in this study for ease of visualization and interpretation.
\begin{figure}[htb!]
{\centering
\includegraphics[scale=0.90]{Figs/Fig15_Vis_WSoft_m25_and_m50_cropped.jpg}
\vspace{-3mm}
\caption{Color coded visualization of sparse code memberships of streamlines in Corpus Callosum (row-1,2); left Inferior Occipitofrontal Fasciculus (IOF) and Cortico-Spinal-Tract (CST) (row-3,4); and right IOF and CST (row-5,6).}
\label{fig:Wsoft_HCP10sub}
%
%
}
\end{figure}
Figure \ref{fig:Wsoft_HCP10sub} shows sparse code memberships of streamlines in six different bundles: Corpus Callosum - anterior body (row 1) and central body (row 2), left Inferior Occipitofrontal Fasciculus (IOF) (row 3), left Cortico-Spinal-Tract (CST) (row 4), right IOF (row 5), and right CST (row 6). Results are reported for subject 1 ($m$=25 and $m$=50), subjects $2$ ($m$=50) and subject $3$ ($m$=50). Sparse code values are represented by a color ranging from green (lowest value) to red (highest value). While variations are observed across values of $m$ and subjects, the general shape of bundles recovered by our method is similar.
\begin{figure}[htb!]
{\centering
\includegraphics[scale=0.90]{Figs/Fig9_Vis_D1_to_D4_cropped.jpg}
\vspace{-3mm}
\caption{Unsupervised multi-subject dictionary visualization. Four different dictionaries and corresponding bundles. Top row: Axial view of full dictionary with a unique color assigned to each bundle; Second row: Anterior Body, and Central Body bundles in Corpus Callosum; Third row: Left CST, and Left IOF bundles; Last row: Right CST, and Right IOF bundles. Each dictionary has a different color code, while the bundles respect that dictionary color-code. (m=50 bundles).}
\label{fig:Multi-sub-dictionary-sets}
%
}
\end{figure}
\subsection{Application to automated tractography segmentation}
In this section, we apply the proposed KSC method for the automated segmentation of new subject streamlines. Again, the focus of our analysis is on inter-subject variability and its effect on results. To label streamlines, we used as bundle atlas the dictionaries obtained from 40 unrelated HCP subjects ($4$ dictionaries, each one learned from $10$ subjects. Dictionaries were generated by sampling $5000$ streamlines in each subject and employing MCP as distance measure. Note that expert-labeled streamlines could also be used as dictionary.
\begin{figure}[htb!]
{\centering
\includegraphics[scale=0.70]{Figs/Fig10_Vis_sub1_to_4_D1_cropped.jpg}
\vspace{-3mm}
\caption{Automated segmentation visualization. Top row: full segmentation of 4 HCP subjects using dictionary D1, with a unique color assigned to each cluster, and same color code as D1. Rows 2-7: sparse code (bundle membership) visualization for the posterior body CC, anterior body CC, left IOF, left CST, right IOF, and right CST bundles. Membership values are represented by a color ranging from green (no membership) to red (highest membership).}
\label{fig:Auto_Seg_4subs_D4}
%
%
}
\end{figure}
The bundles encoded by these dictionaries are depicted in Figure \ref{fig:Multi-sub-dictionary-sets}. Moreover, segmentation results obtained for 4 different subjects using dictionary D1 are shown in Fig. \ref{fig:Auto_Seg_4subs_D4}. For each subject, we give the full segmentation as well as membership values for CC, left/right IOF, and left/right CST bundles. Additionally, to analyze the impact of sampling streamlines from a subject, segmentation results for $5$ instances of subject 1 using D1 are provided in Supplement material. Once more, while we observe variability across segmented streamlines from different subjects, the results obtained by our method are globally consistent across subjects. Similar consistency is found across multiple instances of the subject 1 (see Supplement material, Fig. 5).
\section{Discussion}
\label{sec:Discussion}
We now summarize and discuss the findings related to proposed approaches, impact of various priors, and their applications. We then highlight limitations and additional considerations of this study.
\subsection{Main findings}
Our experiments have demonstrated the usefulness of our kernel sparse clustering (KSC) and various sparsity priors. The soft assignment provided by KSC ($S_\mr{max}\!\geq\!2$) improved performance for all measures of clustering quality compared to a hard clustering approaches like kernel k-means. This improvement was most significant when the input number of clusters (parameter $m$) is not set close to the ground truth value. In such cases, soft assignment offers a greater robustness to the ambiguous membership of streamlines to bundles.
Comparing the different streamline distances, we found that mean of closest points (MCP) performed the best. Hausdorff distance measures the maximum distance between any point on a streamline and its closest point on another streamline, and thus fails to capture bundles with branching or diverging streamlines. Likewise, end points distances may be more affected by outlier streamlines or issues in diffusion tractography output. These observations are in line with previous analyses on streamline distances \cite{moberts2005evaluation,siless2013comparison}.
Results revealed the input number of clusters to have a high impact on results. The true value of this parameter is largely unknown \cite{o2007automatic}, and even in expert labeled set could be off the mark due to labeling errors \cite{moberts2005evaluation}. Our analysis showed that group sparsity provides robustness to this confound, and recovers meaningful bundles when it is set far from the ground-truth value. Likewise, the proposed manifold regularization prior helped the clustering by enforcing related pairs of streamlines to be grouped together. This could be useful in a wide range of applications where anatomical information (e.g., cortical parcellation atlas) is available.
Unsupervised clustering of subjects from HCP and MIDAS datasets showed that our KSC method can be employed for data driven analyses, our method finding plausible clusters corresponding to well known bundles. Moreover, the visualization of clusters and membership values demonstrates that KSC can effectively capture inter-subject variability. Experiments on automated streamline segmentation also revealed that KSC can accurately recover major bundles in new subjects, and that this segmentation is robust to the number of clusters, inter-individual variations, and the sampling of streamlines from the same subject.
\subsection{Limitations and additional considerations}
Due to the lack of gold standard clustering, as well as the various challenges in diffusion tractography \cite{maier2017challenge} and its interpretation \cite{jones2013white}, validating streamline clustering approaches is difficult. A large scale and data-driven analysis, for example using data from over $1000$ HCP subjects, could lead to interesting observations on number of bundles and their population-wise variability.
An important aspect of our dictionary learning method is its initialization. While we employed spectral clustering for this task, considering other techniques could possibly lead to better results. For the automated segmentation of streamlines in new subjects, we learned the dictionary in an unsupervised setting, however expert-labeled streamlines set or atlas/clustering from other approaches can also be utilized.
One the main advantages of the proposed kernel-based framework is that it alleviates the need for an explicit streamline representation. Previous attempts in utilizing dictionary learning and sparse coding for streamline clustering might have been hindered by this. Employing kernels also provides flexibility and enables the extension to other streamline similarity measures, which can incorporate a richer set of characteristics such as along-tract diffusivity \cite{kumar2017white,charon2013varifold,charlier2014fshape}.
Another key element of our study is the anatomical interpretation of clustering results. The streamlines generated from diffusion tractography provide a macro-scale inference of the underlying fibers\cite{jones2013white,maier2017challenge}. As such, the clustering for a given distance/similarity measures focuses primarily on the geometric aspect of streamlines. Although we considered end points proximity in our manifold regularization prior, additional information such as structural parcellation could be incorporated to improve the anatomic plausibility of the final clustering \cite{o2013fiber,siless2018anatomicuts}.
The sparse code representation of streamlines conveys a wealth of information on inter-individual variability in terms of streamline geometry. Extension of this study could leverage this information for additional tasks, such as identifying noisy/spurious streamlines, discovering tract-based biomarkers to discriminate between healthy and diseased subjects \cite{o2017automated}, or establishing bundle-to-bundle correspondences across subjects.
\section{Conclusion}
\label{sec:Conclusion}
We presented a novel framework using kernel dictionary learning with various sparsity priors for the unsupervised segmentation of white matter streamlines. The proposed framework does not require explicit streamline representation and enables using any streamline similarity measure. Dictionary bundles are encoded as a non-negative combination of training streamlines, and the kernel trick is used to model non-linear relationships between streamlines and bundles.
We compared our method against state-of-the-art streamline clustering approaches using expert-labeled data, as well as subjects from the HCP and MIDAS dataset. Results demonstrate the usefulness of having a soft assignment, and that our method is suitable for scenarios where streamlines are not clearly separated, bundles overlap, or when there is important inter-individual variability. Experiments using group sparsity ($L_{2,1}$ norm) and manifold regularization show that these priors can improve clustering quality by adding robustness to the input number of clustering or incorporating anatomical constraints in the clustering.
The benefits of the proposed approach in cases of inter-individual variability was showcased for the automated segmentation of streamlines from new subjects. In future work, we will investigate the usefulness of our approach for identifying and comparing major bundles in healthy vs diseased subjects, and for incorporating along-tract measures in the clustering process.
\section*{Acknowledgements}
Data were provided in part by the Human Connectome Project, WU-Minn Consortium (Principal Investigators: David Van Essen and Kamil Ugurbil; 1U54MH091657) funded by the 16 NIH Institutes and Centers that support the NIH Blueprint for Neuroscience Research; and by the McDonnell Center for Systems Neuroscience at Washington University. We thank the Sherbrooke Connectivity Imaging Laboratory for generously providing the labeled dataset used in this work.
\section{Algorithms}
\label{sec:SupAlgo}
\subsection{Non-negative kernelized sparse clustering}
\subsubsection{Algorithm summary}
\begin{center}
\begin{small}
\begin{algorithm2e}[H]
\KwIn{Pairwise streamline distance matrix $S_{dist} \in \vec R^{n \times n}$;}
\KwIn{The desired number of streamline bundles $m$;}
\KwIn{The RBF kernel parameter $\gamma$;}
\KwIn{The sparsity level $S_\mr{max}$ and maximum number of iterations $T_\mr{max}$;}
\KwOut{The sparse assignment matrix $\vec W \in \vec R^{n \times m}$ and
hard assignment vector $\vec c \in \{1,\ldots,m\}^n$;}
\caption{Algorithm 2: Kernelized sparse clustering method}
\BlankLine
Initialize the kernel matrix: $k_{ij} \ = \ \exp(-\gamma \! \cdot \! dist_{ij}^2)$ \;
Initialize $A$ as a random selection matrix\;
\BlankLine
\For{$t = 1, \ldots, T_\mr{max}$}{
Update each column $\vec w_i$ of $\vec W$ using NNKOMP (Algorithm 2)\;
Update dictionary until convergence: $ A_{ij} \ \gets \ A_{ij}\dfrac {\left( \vec K \tr{\vec W}\right) _{ij}} {\left( \vec K A \vec W \tr{\vec W}\right) _{ij}}, \quad i=1,\ldots,n, \quad j=1,\ldots,m.$\;
$t_\mr{out} \ \gets \ t_\mr{out} + 1$\;
}
\BlankLine
Compute hard assignment: $c_i \ = \ \arg\max _{k'} w_{im'}, \ \ i = 1,\ldots,n$ \;
\BlankLine
\Return{$\{\vec W,c\}$} \;
\end{algorithm2e}
\end{small}
\end{center}
\subsubsection{Algorithm complexity}
In Algorithm 2, the user provides a matrix $S_{dist}$ of pairwise streamline distances, as well as the desired number of bundles (clusters), and obtains in return the soft (matrix $\vec W$) and hard (vector $\vec c$) streamline clusterings. Various distance measures, suitable for streamlines, are described in experiments. The distances are converted into similarities by using a Gaussian (RBF) kernel of parameter $\gamma$. Note that the obtained kernel is
semi-definite positive only if the distance is a metric. However, non-metric distances, such as the Hausdorff distance (see experiments), have been shown to be quite useful in practice \cite{laub2004feature}. In the main loop, the dictionary matrix $A$ and sparse streamline-to-bundle assignment matrix $\vec W$ are optimized alternatively, until convergence or $T_\mr{max}$ iterations have been reached. The soft clustering of $\vec W$ is converted to a hard clustering by assigning each streamline $i$ to the bundle $m$ for which $w_{im}$ is maximum.
\subsection{Non-negative kernelized orthogonal matching pursuit (NNKOMP)}
{
\fontsize{7}{7}\selectfont
\begin{algorithm2e}[htb!]
\KwIn{The dictionary matrix $A \in \vec R_+^{n \times m}$ and kernel matrix $\vec K \in \vec R^{n \times n}$;}
\KwIn{The streamline index $i$ and sparsity level $S_\mr{max}$;}
\KwOut{The set of non-zero weights $I_s$ and corresponding weight values $\vec w_s$;}
\caption{Non-negative kernelized orthogonal matching pursuit}
\BlankLine
Initialize set of selected atoms and weights: $I_0 \ = \ \emptyset, \ \vec w_0 \ = \ \emptyset$\;
\BlankLine
\For{$s = 1, \ldots, S_\mr{max}$}{
\BlankLine
$\tau_j \ = \ \big[\tr{A}\big(\vec k_i - \vec K \, A_{[I_s]} \vec w_s\big)\big]_j \, / \,
\big[\tr{A}\KKA\big]_{jj}, \ \ j = 1, \ldots, m$\;
$ j_\mr{max} \ = \ \argmax_{j \, \not \in \, I_{s-1}} \ \tau_j, \ \
I_s \ = \ I_{s-1} \, \cup \, j_\mr{max}$\;
$\vec w_s \ = \ \argmin_{\vec w \, \in \, \vec R_+^s} \
\tr{\vec w}\tr{A}_{[I_s]} \vec K A_{[I_s]}\vec w \ - \ 2\, \tr{\vec k_i}A_{[I_s]}\vec w$\;
}
\BlankLine
\Return{$\{I_s, \, \vec w_s\}$} \;
\BlankLine
\textbf{Note:} $A_{[I_s]}$ contains the columns of $A$ whose index is in $I_s$ \;
\end{algorithm2e}
}
\subsection{Group sparse kernelized dictionary learning}
\subsubsection{Algorithm summary}
\begin{small}
\centering
\begin{algorithm2e}[!htb]
\KwIn{Pairwise streamline distance matrix $S_{dist} \in \vec R^{n \times n}$;}
\KwIn{The maximum number of streamline bundles $m$;}
\KwIn{The RBF kernel parameter $\gamma$;}
\KwIn{The cost trade-off parameters $\lambda_1, \lambda_2$ and Lagrangian parameter $\mu$;}
\KwIn{The maximum number of inner and outer loop iterations $T_\mr{in}, T_\mr{out}$;}
\KwOut{\mbox{The dictionary $A \in \vec R^{n \times m}$ and assignment weights
$\vec W \in \vec R_+^{n \times m}$;}}
\caption{ADMM method for group sparse kernelized clustering}
\BlankLine
Initialize the kernel matrix: $k_{ij} \ = \ \exp(-\gamma \! \cdot \! dist_{ij}^2)$\;
Initialize $A$ as a random selection matrix and $t_\mr{out}$ to 0\;
\BlankLine
\While{$f(\vec D,\vec W)$ not converged and $t_\mr{out} \leq T_\mr{out}$}{
\BlankLine
Initialize $\vec U$ and $\vec Z$ to all zeros and $t_\mr{in}$ to zero\;
\BlankLine
\While{$||\vec W-\vec Z||_F^2$ not converged and $t_\mr{in} \leq T_\mr{in}$}{
\BlankLine
Update $\vec W$, $\vec Z$ and $\vec U$: \\
\begin{fleqn} \begin{flalign*}
\qquad \vec W & \ \gets \
\big(\tr{A}\KKA + \mu \vec I\big)^{-1}\big(\tr{A}\vec K + \mu(\vec Z-\vec U)\big); \\
\qquad \hat{z}_{ij} & \ \gets \ \max\Big\{w_{ij} + u_{ij} - \frac{\lambda_1}{\mu}, \, 0\Big\},
\quad i \leq m, \ j \leq n; \\
\qquad \zz_{i\LargerCdot} & \ \gets \ \max\Big\{||\hat{\zz}_{i\LargerCdot}||_2 - \frac{\lambda_2}{\mu}, \, 0\Big\} \cdot
\frac{\hat{\zz}_{i\LargerCdot}}{||\hat{\zz}_{i\LargerCdot}||_2}, \quad i \leq m; \\
\qquad \vec U \ & \ \gets \ \vec U \ + \ \big(\vec W - \vec Z\big);
\end{flalign*}\end{fleqn}
$t_\mr{in} \ \gets \ t_\mr{in} + 1$\;
}
\BlankLine
Update dictionary until convergence: $ A_{ij} \ \gets \ A_{ij}\dfrac {\left( \vec K \tr{\vec W}\right) _{ij}} {\left( \vec K A \vec W \tr{\vec W}\right) _{ij}}, \quad i=1,\ldots,n, \quad j=1,\ldots,m.$\;
$t_\mr{out} \ \gets \ t_\mr{out} + 1$\;
}
\BlankLine
\Return{$\{A, \vec W\}$} \;
\end{algorithm2e}
\centering
\end{small}
\subsubsection{Algorithm complexity}
The clustering process of our proposed method is summarized in Algorithm 1. In this algorithm, the user provides a matrix $S_{dist}$ of pairwise streamline distances (see experiments for more details), the maximum number of clusters $m$, as well as the trade-off parameters $\lambda_1, \lambda_2$, and obtains as output the dictionary matrix $A$ and the cluster assignment weights $\vec W$. At each iteration, $\vec W$, $\vec Z$ and $\vec U$ are updated by running at most $T_\mr{in}$ ADMM loops, and are then used to update $A$. This process is repeated until $T_\mr{out}$ iterations have been completed or the cost function $f(\vec D,\vec W)$ converged. The soft assignment of $\vec W$ can be converted to a hard clustering by assigning each streamline $i$ to the bundle $m$ for which $w_{im}$ is maximum.
The complexity of this algorithm is mainly determined by the initial kernel computation, which takes $O(n^2)$ operations, and updating the assignment weights in each ADMM loop, which has a total complexity in $O(T_\mr{out} \cdot T_\mr{in} \cdot m^2 \cdot n)$. Since $T_\mr{out}$, $T_\mr{in}$ and $m$ are typically much smaller than $n$, the main bottleneck of the method lies in computing the pairwise distances $S_{dist}$ used as input. For datasets having a large number of streamlines (e.g., more than $n=100,000$ streamlines), this matrix could be computed using an approximation strategy such as the the Nystr\"om method \cite{fowlkes2004spectral}, described later in the paper.
\subsection{Kernelized dictionary learning with Laplacian prior}
\subsubsection{Update W: Bartels-Stewart Algorithm summary }
\begin{small}
\centering
\begin{algorithm2e}[!htb]
\KwIn{$\vec P$, $\vec Q$, and $\vec R$;}
\KwOut{$\vec W$}
\caption{Bartels-Stewart Algorithm summary}
\BlankLine
Step 1: Transfrom $\vec P$ and $\vec Q$ into Schur form \\
\BlankLine
$\vec C_c \ = \ \vec R;$ \\
$[\vec Q_a,\vec T_a] \ = \ schur(\vec P); \quad \vec C_c \ = \ \tr{\vec Q_a}\vec C_c$\\
$[\vec Q_b,\vec T_b] \ = \ schur(\vec Q); \quad \vec C_c \ = \ \vec C_c\vec Q_b$\\
\BlankLine
Step 2: Solve following Simplified Sylvester equation using back substitution
$\vec T_a \vec W + \vec W \vec T_b \ = \ \vec C_c$
\BlankLine
Step 3: Recover $\vec W$:\\
$\vec W \ = \ \vec Q_a \vec W \tr{\vec Q_b}$
\Return{$\{ \vec W\}$} \;
\end{algorithm2e}
\centering
\end{small}
\subsubsection{Algorithm summary and complexity}
The algorithm below provides summary of the methods, while complexity can be computed similar to previous section, with only difference being update of W.
\begin{small}
\centering
\begin{algorithm2e}[!htb]
\KwIn{Pairwise streamline distance matrix $S_{dist} \in \vec R^{n \times n}$;}
\KwIn{The maximum number of streamline bundles $m$;}
\KwIn{The RBF kernel parameter $\gamma$;}
\KwIn{The cost trade-off parameters $\lambda_1, \lambda_L$ ;}
\KwIn{Lagrangian parameter $\mu_1$;}
\KwIn{The maximum number of inner and outer loop iterations $T_\mr{in}, T_\mr{out}$;}
\KwOut{\mbox{The dictionary $A \in \vec R^{n \times m}$ and assignment weights
$\vec W \in \vec R_+^{n \times m}$;}}
\caption{ADMM method for kernelized dictionary learning with Laplacian prior}
\BlankLine
Initialize the kernel matrix: $k_{ij} \ = \ \exp(-\gamma \! \cdot \! dist_{ij}^2)$\;
Initialize $A$ as a random selection matrix and $t_\mr{out}$ to 0\;
Precompute $ schur(\lambda_L \vec L)$;
\BlankLine
\While{$f(\vec D,\vec W)$ not converged and $t_\mr{out} \leq T_\mr{out}$}{
\BlankLine
Initialize $\vec U$ and $\vec Z$ to all zeros and $t_\mr{in}$ to zero\;
\BlankLine
\While{$||\vec W-\vec Z||_F^2$ not converged and $t_\mr{in} \leq T_\mr{in}$}{
\BlankLine
Update $\vec W$, $\vec Z$, and $\vec U$: \\
\begin{fleqn} \begin{flalign*}
\qquad \vec W & \ \gets \
Sylvester\big(\big(\tr{A}\KKA + \mu_1\vec I\big),\lambda_L \vec L,\big(\tr{A}\vec K + \mu_1(\vec Z-\vec U)\big)\big) \\
\qquad \hat{z}_{ij} & \ \gets \ \max\Big\{w_{ij} + u_{ij} - \frac{\lambda_1}{\mu_1}, \, 0\Big\},
\quad i \leq m, \ j \leq n; \\
\qquad \vec U \ & \ \gets \ \vec U \ + \ \big(\vec W - \vec Z\big);
\end{flalign*}\end{fleqn}
$t_\mr{in} \ \gets \ t_\mr{in} + 1$\;
}
\BlankLine
Update dictionary until convergence: $ A_{ij} \ \gets \ A_{ij}\dfrac {\left( \vec K \tr{\vec W}\right) _{ij}} {\left( \vec K A \vec W \tr{\vec W}\right) _{ij}}, \quad i=1,\ldots,n, \quad j=1,\ldots,m.$\;
$t_\mr{out} \ \gets \ t_\mr{out} + 1$\;
}
\BlankLine
\Return{$\{A, \vec W\}$} \;
\end{algorithm2e}
\centering
\end{small}
\subsection{Group sparsity and manifold prior visualization}
The bundles obtained by group sparsity (MCP+L1+L21) for the input number of clusters $m=20$ are presented in Figure \ref{fig:Fiber_bundles_GT_vis_compare_MCP_L21_and_Lap} (middle). We observe that the clustering is similar to the ground truth clustering, except for small differences in left/right inferior longitudinal fasciculus bundles (purple and blue colors in the ground truth). Also, we observe that superior cerebellar peduncle bundles (cyan and green colors in the ground truth) are well separated.
Figure \ref{fig:Fiber_bundles_GT_vis_compare_MCP_L21_and_Lap}(right) shows clustering output using this method for $m=10$ and MCP for a sample run. We observe that the clustering is similar to the ground truth clustering, except for small differences in left/right inferior longitudinal fasciculus bundles (purple and blue colors in the ground truth).
\begin{figure}
{\centering
\includegraphics{Figs/Fig4b_plot_GKSC_L21_Lap_vis_cropped.jpg
\vspace{-3mm}
\caption{Right sagittal (\textbf{top}) and inferior axial (\textbf{bottom}) views of the ground truth (left), and bundles obtained by MCP+L1+L21 (middle, m=20, final m=10), and MCP+L1+Lap (right, m=10).}
\label{fig:Fiber_bundles_GT_vis_compare_MCP_L21_and_Lap}
}
\end{figure}
\subsection{Results on multi-subject MIDAS dataset (KSC+MCP)}
\textbf{Data:} To evaluate the performance of our method on multiple subjects, we also used the data of 12 healthy volunteers (6 males and 6 females, 19 to 35 years of age) from the freely available MIDAS dataset \cite{bullitt2005vessel}. For fiber tracking, we used the tensor deflection method \cite{lazar2003white} with the following parameters: minimum fractional anisotropy of 0.1, minimum streamline length of 100 mm, threshold for streamline deviation angle of 70 degrees. A mean number of 9124 streamlines was generated for the 12 subjects.
\textbf{Results:} Figure \ref{fig:MIDAS_Conv_Plots} (left) shows the mean SI (averaged over all clusters) obtained by KSC ($S_\mr{max}\!=\!3$), KKM and Spect with MCP, on 12 subjects of the MIDAS dataset. We see that our soft clustering method outperforms the hard clustering approaches, especially for a small number of clusters. In Figure \ref{fig:MIDAS_Conv_Plots} (right), the results obtained for $m \ = \ 35$ are detailed for each subject. Error bars in the plot show the mean and variance of SI values obtained over 10 different initializations. As can be seen, our method shows a greater accuracy and less variance across subjects.
\begin{figure}[t]
{\centering
\mbox{
\includegraphics[width=11cm]{Figs/Sup_Fig2_MIDAS_cropped.jpg
}
\vspace{-3mm}
\caption{ MIDAS: Mean of average SI using MCP: varying $m$ mean over 12 subjects (\textbf{left}); for $m \ = \ 35$ for each subject (\textbf{middle}); Convergence plot (\textbf{right} for KSC+MCP )}
\label{fig:MIDAS_Conv_Plots}
}
\end{figure}
\subsection{Results on Human Connectome Project subjects}
Figure \ref{fig:HCP10sub_hard_clustering_vis} shows clustering output for $10$ HCP subjects for $m=50$, with an unique color assigned to each cluster. For this simplified visualization each streamline is assigned to a single cluster by taking the maximum for each column of the matrix $\vec W$. Note, we have used a unique color code for each subject, as establishing a cluster correspondence across subjects is itself a challenging problem. We observe that the overall pattern of clustering across subjects looks similar. However, there are subtle variations for clusters across subjects.
\begin{figure}[h]
{\centering
\includegraphics{Figs/Fig14_Vis_W_HCP10_m50_sub1_to_sub10_cropped.jpg
\vspace{-3mm}
\caption{Visualization of clustering output for $10$ unrelated HCP subjects using KSC, for m = $50$.}
\label{fig:HCP10sub_hard_clustering_vis}
%
%
}
\end{figure}
Similarly, Figure \ref{fig:Impact_m_HCP_sub1} shows simplified visualization of clustering output for subject 1, for varying $m$. As expected, going from $m=25$ to $m=150$ the clusters split into smaller ones, for example, observe the clusters in corpus callosum.
\begin{figure}[h]
{\centering
\includegraphics{Figs/Fig13_Vis_W_HCP10_m25_m150_cropped.jpg
\vspace{-3mm}
\caption{Visualization of clustering output for subject 1 using KSC, for varying $m$.}
\label{fig:Impact_m_HCP_sub1}
%
%
}
\end{figure}
\subsection{Multi-subject clustering as dictionary}
\subsubsection{Computing the kernel matrix using the Nystr\"om method}
When there can be multiple subjects, each subject having several thousands of streamlines, computing the similarity between all pairs of training streamlines in $\vec K$ is impossible. To alleviate this problem, we approximate the kernel matrix using the Nystr\"om method \cite{fowlkes2004spectral,Odonnell07automatictractography}. In this method, a set of $p$ representative streamlines are sampled from while set of training streamlines, where $p \ll |\vec X|$. The pairwise similarities between all selected streamlines are then computed in a reduced kernel matrix $\vec K_a \in \vec R^{p \times p}$. Likewise, the similarity between the selected and non-selected ones are obtained in a matrix $\vec K_b \in \vec R^{p \times (|\vec X|-p)}$. The whole kernel matrix is then reconstructed using a low-rank approximation $\vec K = \vec G\tr{\vec G}$, where $\tr{\vec G} = \vec K_a^{-\frac{1}{2}} \big[\tr{\vec K}_a \, \tr{\vec K}_b\big]$. In practice, the most computationally expensive step of this method is the SVD decomposition of $\vec K_a$.
\subsubsection{HCP multi-subject clustering}
When there are multiple subjects, each subject having several thousands of streamlines, computing the similarity between all pairs of training streamlines in $\vec K$ is impossible. To alleviate this problem, we approximate the kernel matrix using the Nystr\"om method \cite{fowlkes2004spectral,Odonnell07automatictractography}. Figure 9 (manuscript) shows simplified visualization of $A$ matrix of 4 sets of $10$ unrelated HCP subjects. (These subjects are utilized as dictionary in next section). We utilized $50,000$ streamlines for each set, sampling $5,000$ streamlines from each subject.
For simplification, we show full clusterings and select bundles including Anterior Body, and Central Body bundles in Corpus Callosum; Third row: Left Cortico-Spinal-Tract, and Left Arcuate Fasciculus bundles; Last row: Right Cortico-Spinal Tract, and Right Inferior Occipitofrontal Fasciculus bundles. The objective here is to show that the multi-subject clustering output provides plausible clusters, corresponding to well-known anatomical bundles. Also, comparing multi-subject clustering with single subject clustering, we observe overall similarity in terms of clusters, while also reflecting variation. We also observe subtle variations across multi-subject clustering sets, for example, within IOF or CST bundles.
\subsection{Application: automated segmentation of new subject streamlines}
To analyze the impact of sampling streamlines from a subject, Figure \ref{fig:Auto_seg_sub2_D4_instance1_to5} shows segmentation output for 5 instances of subject 1 using dictionary D1.
\begin{figure}[htb!]
{\centering
\includegraphics[scale=0.90]{Figs/Fig12_Vis_Seg_sub1_instance_1_to_5_cropped.jpg
\vspace{-3mm}
\caption{Automated segmentation of 5 instances of subject 1 using dictionary D1.}
\label{fig:Auto_seg_sub2_D4_instance1_to5}
%
%
}
\end{figure}
|
{
"timestamp": "2018-04-17T02:11:57",
"yymm": "1804",
"arxiv_id": "1804.05427",
"language": "en",
"url": "https://arxiv.org/abs/1804.05427"
}
|
\section{Introduction and Main Results}
Composition operator is a branch of modern operator theory \cite{CM1995}, \cite{S1993}, \cite{SM1993}, \cite{zhu2007} and a typical object of study involves an analytic self-map $\varphi: \mathbb{D}\to \mathbb{D}$ of the unit disk. Let $H$ be a Hilbert space of analytic functions, such as the Hardy space $H^2(\mathbb{D})$ or the Bergman space $L^2_a(\mathbb{D})$. Then one considers $C_\varphi$ defined by
$$C_\varphi(f)=f\circ \varphi, \quad f \in H.$$
The boundedness of $C_\varphi$ follows from the Littlewood subordination principle. Once the boundedness is in hand, one can ask various operator-theoretic questions such as compactness, Schatten class membership, Fredholm theory, etc.
\bigskip \noindent
{In this paper we seek to extend the symbol $\varphi$ to non-analytic self-maps of $\mathbb{D}$. This is certainly not a new idea, although the theory is far from being mature.
%
There exist quite a few references about composition operators on nonanalytic function spaces, such as Sobolve-type spaces or BMO-type spaces. In these situations, the symbols are usually not analytic. Of particular interests to us is quasiconformal composition. Reimann's theorem on composing BMO functions with quasiconformal symbols \cite{AIM2008} is perhaps the best known result along this line.
Other places where quasiconformal composition appears include
\cite{GGR1995}, \cite{HKM2014}, \cite{Hu}, \cite{KKSS1014}, \cite{KXZZ2017}, \cite{Nag}, \cite{Schippers}, etc.
In particular, in view of the similiarity of the titles, it appears enlightening to compare the present article with \cite{KXZZ2017} which studies composition on $Q$-spaces $Q_\alpha(\mathbb{R}^n)(0<\alpha<1)$ with quasiconformal symbols whose Jacobians are $A_1$ weights. The specific problems treated in these two papers are different.
The methodology in \cite{KXZZ2017} is an interplay between quasiconformal mapping and harmonic analysis, whereas ours has an additional flavor of complex analysis.
\bigskip \noindent
{In this paper we study quasiconformal composition operators on analytic function spaces and we choose to work on the Bergman space $L_a^2(\mathbb{D})$. Our construction clearly makes sense on other analytic spaces as well, but such a task is not pursued in this paper.
In this setting, the composition operator is initially defined as a map from $H^\infty(\mathbb{D})$ to $L^\infty(\mathbb{D})$. The idea of such a study can be traced back to R. Rochberg's work in 1994 \cite{rochberg1994}. To the best of our knowledge, there is little progress along this line since then. In general, when acting on analytic spaces, non-analytic symbols form conceivably a rather wild world for which the boundedness of $C_\varphi$ presents a serious challenge, not to mention other fine properties.} One contribution of this paper is to manifest that a decent theory can be established for quasiconformal composition on analytic function spaces, although the methodology is, as expected, quite different from that of analytic symbols
\bigskip \noindent
Our first motivation for considering quasiconformal composition starts with the observation that if one is set to consider non-analytic $\varphi$, then $f\circ \varphi$ is not necessarily in the Bergman/Hardy space we start with. So we consider a Toeplitz-composition type operator:
$$Q_\varphi(f)=P(f\circ \varphi),$$
where $P$ denotes the Bergman/Szego projection. If we look at the extreme situation, i.e., we ask $\varphi$ to be anti-analytic, such as $\varphi(z)=\bar{z}$, then $Q_\varphi$ kills the analytic function $f$ entirely, except for the constant term. So a reasonable idea is to allow $\varphi$ to have a controlled degree of analyticity, which points precisely to quasiconformal mappings. For any fixed $K \in [1, \infty)$, we say that a homeomorphism $\varphi:\mathbb{D}\to\mathbb{D}$ is a $K$-quasiconformal mapping if it satisfies
\begin{itemize}
\item[\emph{(1)}] $\varphi\in W^{1, 2}_{\emph{loc}}(\mathbb{D})$ \emph{(}that is, $\varphi, \partial_z \varphi, \partial_{\bar{z}} \varphi \in L^2_{\emph{loc}}(\mathbb{D})$\emph{)};
\item[\emph{(2)}] $|\partial_{\bar{z}} \varphi(z)| \leq \frac{K-1}{K+1}|\partial_z \varphi(z)|$ for almost every $z\in\mathbb{D}$.
\end{itemize}
\bignobf{Two Simplifications.} (1) It is attempting to consider general quasiregular mappings instead of the quasiconformal symbols used in this paper, which are homeomorphisms of the unit disk; this will simplify many arguments.
But given that this is the first paper along this line and quasiconformal mappings illuminate the ideas better, and moreover, they present a theory which is already interesting and complicated, it appears reasonable to us to refrain to homeomorphisms at this moment. General symbols are certainly good problems for further work.
\bigskip \noindent (2) After some initial study of the $Q_\varphi$ operator, it becomes clear to us that, when $\varphi$ is not analytic, the two maps
$$f \mapsto f\circ \varphi \quad \text{and} \quad f\mapsto P(f\circ \varphi)$$
differ significantly because the latter involves the extra complicacy of the Szego projection or the Bergman projection. Because the first map is rich and difficult enough, in this paper we focus on it. We plan to treat the second map in a separate work. {In \cite{rochberg1994}, using probabilistic methods, Rochberg considered certain special $\varphi$ and obtained several sufficient conditions for the boundedness of $Q_\varphi$, where $P$ is the Szego projection.}
\bigskip \noindent The second motivation of this paper comes from the theory of harmonic quasiconformal mapping. In 1968, Martio \cite{Ma1968} asked whether the Possion extension of a $K$-quasisymmetric mapping on the unit circle must be a quasiconformal mapping on $\mathbb{D}$. This problem is negative answered by \cite{Pa2002}. We study this question via operator theory on the analytic function spaces. With the help of Heinz's inequality \cite{Heinz1959}, if $\varphi$ is a harmonic quasiconformal mapping, then both $C_\varphi$ and $C_{\varphi^{-1}}$ are bounded from $L^2_a(\mathbb{D})$ into $L^2(\mathbb{D})$.
As an application of Theorem \ref{T:boundforC}, a harmonic quasiconformal map on $\mathbb{D}$ must be bi-Lipschitz on $\mathbb{T}$.
\bigskip \noindent The third motivation of this paper is to study twisted Bergman projections.
When $\varphi:\mathbb{D}\to\mathbb{D}$ is measurable, the study of composition operators is in a sense equivalent to the study of the following singular integral operator
\begin{equation}\label{E:soperator}
P_\varphi f(z)
=\int_{\mathbb{D}}\frac{f(w)}{(1-\varphi(z)\overline{\varphi(w)})^2}dA(w),
\end{equation}
where $dA(z)$ is the normalized Lebesgue measure on $\mathbb{D}$. Note that $P_\varphi=C_\varphi C_\varphi^*$
\bigskip \noindent Similar formulas have been used by a number of researchers. In \cite{CG2006}, Cowen and Gallardo-Gutierrez obtained a description of the adjoints of composition operators. Muhly and Xia used them to study automorphisms of the Toeplitz algebra in \cite{MX1995}.
\bigskip \noindent In contrast to the standard Bergman projection, in general, $P_\varphi$ is not a Calderon-Zygmund-type operator (CZO) over a space of homogeneous type.
When $\varphi$ is analytic, the $L^2$-boundedness of $P_\varphi$ on $L^2(\mathbb{D})$ is equivalent to that on $L^2_a(\mathbb{D})$. Zhu \cite{zhu20071}, \cite{zhu2007} obtained $L^p$ boundedness of $P_\varphi$ for $1<p<\infty$.
\bigskip \noindent In this paper we obtain a rather thorough analysis of the boundedness of $P_\varphi$ for quasiconformal $\varphi$, including both weak-$(1, 1)$ and $L^p$-estimates ($1<p\leq\infty$). The proofs form a mixture of quasiconformal mapping, harmonic analysis, and operator theory.
\bigskip \noindent Our first task is to characterize the $L^2$-boundedness of $C_\varphi$. Next are some needed notations. Let $\varphi$ be a $K$-quasiconformal mapping over the unit disk $\mathbb{D}$ with $\varphi(0)=0$. It extends uniquely to a homeomorphism from the closed disk $\bar{\mathbb{D}}$ to itself (Theorem 8.2, \cite{lv}). We let $\tilde{\varphi}: \mathbb{T} \to \mathbb{T}$ denote the boundary mapping.
Let $m_\varphi(z)=\frac{1-|z|}{1-|\varphi(z)|}$.
We also recall
the so-called extremal distortion function $\psi_K(r)$.
\begin{definition} \textsc{(\cite{AVM1988}, \cite{AVM1997})} For each $K\in[1,\infty), r \in [0, 1]$, we define
\begin{equation}\psi_K(r)=\sup_{|z|=r}\{|\varphi(z)|: \varphi \text{ is $K$-quasiconformal over } \mathbb{D} \text{ with } \varphi(0)=0\}.
\end{equation}
\end{definition}
\noindent It is easy to see that $\psi_K(r)$ is a strictly increasing function from $[0, 1]$ to $[0, 1]$, with $\psi_K(0)=0$ and $\psi_K(1)=1$. It satisfies the following remarkable semigroup property \cite{lv}: For any $K_1, K_2 \in [1, \infty)$,
\begin{equation}\label{E:semigroup}\psi_{K_1K_2}=\psi_{K_1}\circ\psi_{K_2}.\end{equation} Moreover, when $K=1$, $\psi_1(r)=r$ by the Schwarz lemma.
\begin{theorem}\label{T:boundforC}
The operator $C_\varphi$ extends to be a bounded operator $C_\varphi: L^2_a(\mathbb{D})\to L^2(\mathbb{D})$ if and only if $\widetilde{\varphi}^{-1}$ is Lipschitz on $\mathbb{T}$. Moreover, in case of boundedness, if $\varphi(0)=0$, then
$$c_2\sup_{z\in\mathbb{D}} m_\varphi(z) \leq \|C_\varphi\|_{L_a^2(\mathbb{D})\to L^2(\mathbb{D})}\leq \frac{c_1}{1-\psi_K^2(\frac{1}{2})} \sup_{z\in\mathbb{D}} m_\varphi(z),$$
where $c_1$ is a universal constant and $c_2=c_2(K)$ is a positive constant depending only on $K$.
\end{theorem}
\bigskip \noindent Koo-Smith \cite{KS2007}, Koo-Wang \cite{KW2009}, and Wogen \cite{wogen1988} contain some general boundedness criteria for smooth symbols on analytic function spaces on the higher dimensions.
\bigskip \noindent Next we consider essential boundedness. {Let $\varphi: \mathbb{D} \to \mathbb{D}$ be an analytic mapping. For each $w\in\mathbb{D}-\varphi(0)$, define the Nevanlinna counting function by
$N_{\varphi,\gamma}(w)=\mathop{\sum}\limits_{z\in \varphi^{-1}(w)}(\log\frac{1}{|z|})^\gamma, $
where $\gamma=1, 2$.
Shapiro's beautiful essential norm formula \cite{S1987} for composition operator $C_\varphi$ on $H^2(\mathbb{T})$ states
$\|C_\varphi\|_e^2=\mathop{\limsup}\limits_{{|w|\to 1}}\frac{N_{\varphi,1}(w)}{\log(\frac{1}{|w|})}.
$
On the Bergman space Poggi-Corradini \cite{Pietro1998} obtained
$\|C_\varphi\|_e^2=\mathop{\limsup}\limits_{{|w|\to 1}}\frac{N_{\varphi,2}(w)}{\log(\frac{1}{|w|})^2}.
$ For a quasiconformal mapping $\varphi$, $
N_\varphi(z)=\log\frac{1}{|\varphi^{-1}(z)|}, z\not=0.$ } This is a good example that although the quasiconformal assumption simplifies $N_\varphi$ considerably, the proof of the following theorem is still rather technical, hence better presented for quasiconformal only. The notation $N_\varphi$ serves mainly to illustrate the similiarity.
\begin{theorem}\label{T:compactforC}
Let $\varphi$ be a quasiconformal mapping over $\mathbb{D}$ with $\varphi(0)=0$. If $\widetilde{\varphi}^{-1}$ is Lipschitz on $\mathbb{T}$, then
\begin{equation}\label{E:essentialnorm}
c_2 \mathop{\lim}\limits_{{t\to 1}}\mathop{\sup}\limits_{{|z|>t}}\frac{N_\varphi(z)}{\log\frac{1}{|z|}}\leq \|C_\varphi\|_e\leq \frac{c_1 \psi_K(\frac{1}{2})}{1-\psi_K^2(\frac{1}{2})} \mathop{\lim}\limits_{{t\to 1}}\mathop{\sup}\limits_{{|z|>t}}\frac{N_\varphi(z)}{\log\frac{1}{|z|}}.
\end{equation}
where $c_1$ is a universal constant and $c_2=c_2(K)$ is a positive constant depending only on $K$.
In particular, $C_\varphi$ is compact if and only if $\lim_{|z|\to 1} m_\varphi(z)=0.$
\end{theorem}
\bigskip \noindent The choice of $\frac{1}{2}$ in Theorem \ref{T:boundforC} and Theorem \ref{T:compactforC} can be replaced by any number in $(0, 1)$. It appears that optimizing on this choice still won't give us the best constant, so we refrain from doing so. Say, a crude estimate shows that one can take $c_1=147$ in the above, which is certainly not sharp. Next it comes to the weak $(1, 1)$ and $L^p$-boundedness of $P_\varphi$.
\begin{theorem}\label{T:extraforShplus}Let $\varphi$ be a quasiconformal mapping over $\mathbb{D}$ such that $\varphi(0)=0$. If $\widetilde{\varphi}$ is bilipschitz on $\mathbb{T}$, then
$P_\varphi$ satisfies the weak-$(1, 1)$ inequality: if $f\in L^1(\mathbb{D})$, then for any $\alpha>0$,
\begin{equation*}\label{E:weak11}|\{z\in\mathbb{D}: |P_\varphi f(z)|>\alpha\}|\leq \frac{c}{\alpha}\int_{\mathbb{D}}|f(z)|dA(z),
\end{equation*}
where $c$ is a constant independent of $f$, and the leftmost $|\cdot|$ denotes the normalized Lebesgue area.
\end{theorem}
\begin{theorem}\label{T:Shexpra}
Let $\varphi:\mathbb{D}\to\mathbb{D}$ be a quasiconformal mapping such that $\varphi(0)=0$ and $P_\varphi$ is bounded on $L^2(\mathbb{D})$. Then
\begin{itemize} \item[\emph{(i)}] $P_\varphi$ is bounded on $L^p(\mathbb{D})$ for $1<p<\infty$,
\item[\emph{(ii)}] {$P_\varphi$ is bounded from $L^p(\mathbb{D})$ to $L^1(\mathbb{D})$ for $0<p<1$,}
\item[\emph{(iii)}] $P_\varphi$ is bounded from $L^\infty(\mathbb{D})$ to $\emph{BMO}(\mathbb{D})$.
\end{itemize}
\end{theorem}
\section{Proofs of Theorem \ref{T:boundforC} and Theorem \ref{T:compactforC}
We first introduce Hersch-Pfluger's distortion theorem which serves as a quasiconformal version of the Schwarz lemma. To do this, we extend the definition of the
extremal distortion function from $K\geq 1$ to $K> 0$ by letting $\psi_K(r)$ be the inverse function of $\psi_{\frac{1}{K}}$ when $0<K<1$.
It is easy to see that $\psi_K(r)$ is a strictly increasing function from $[0, 1]$ to $[0, 1]$, with $\psi_K(0)=0$ and $\psi_K(1)=1$.
\begin{lemma} \emph{(\cite{lv}, p.64)}\label{L:HPD}
Let $\varphi$ be a $K$-quasiconformal mapping over $\mathbb{D}$ with $\varphi(0)=0$. Then
\begin{equation}
\psi_{\frac{1}{K}}(|z|)\leq |\varphi(z)|\leq \psi_{K}(|z|), \quad\quad z\in\mathbb{D}.
\end{equation}
\end{lemma}
\bigskip \noindent Now we analyze the area distortion of a pseudohyperbolic disk, where is typical in operator theory, under a quasiconformal mapping. Recall that
a pseudohyperbolic disk $D^{ph}(a,r)$ centered at $a\in\mathbb{D}$ with radius $0<r<1$
is given by
$D^{ph}(a,r)=\{z\in\mathbb{D}:|\tau_{a}(z)|<r\},$
where
$\tau_a(z)=\frac{a-z}{1-\bar{a}z}.$
It is a Euclidean disk with center
$C=\frac{1-r^2}{1-r^2|a|^2}a $ and
radius $R=\frac{1-|a|^2}{1-r^2|a|^2}r.$
\bigskip \noindent
The next lemma is of independent interests and will be used repeatedly.
\begin{lemma}\label{L:areachangephd} Let $\varphi$ be a $K$-quasiconformal mapping over $\mathbb{D}$ such that $\varphi(0)=0$ and $D^{ph}(z,r)$ be a pseudohyperbolic disk for $z\in\mathbb{D}$ and $0<r<1$. Then\begin{equation}\label{s4eq05}
\psi_{\frac{1}{K}}^2(r)\frac{(1-|\varphi(z)|^2)^2}
{(1-\psi_{\frac{1}{K}}(r)^2|\varphi(z)|^2)^2}\leq |\varphi(D^{ph}(z,r))|\leq \psi_K^2(r)\frac{(1-|\varphi(z)|^2)^2}{(1-\psi_K(r)^2|\varphi(z)|^2)^2},
\end{equation}
where $|\cdot|$ denotes the normalized Lebesgue area on $\mathbb{D}$.
\end{lemma}
\begin{proof} It suffices to show that for any $z\in\mathbb{D}$ and $0<r<1$,
\begin{equation}\label{E:changephd}
D^{ph}(\varphi(z),\psi_{\frac{1}{K}}(r))\subset \varphi[D^{ph}(z,r)]\subset D^{ph}(\varphi(z),\psi_K(r)).
\end{equation}
Define $\Phi(u)=\tau_{\varphi(z)}\circ \varphi\circ\tau_z(u),$ which is a $K$-quasiconformal mapping with $\Phi(0)=0$.
By Lemma \ref{L:HPD},
$|\Phi(u)|\leq\psi_K(|u|).$
Set $w=\tau_z(u)$, then $|\tau_{\varphi(z)}
(\varphi(w))|\leq\psi_K(|\tau_z^{-1}(w)|).$
Since $w\inD^{ph}(z, r)$
and $\psi_K$ is increasing, it follows $\bigg|\frac{\varphi(z)-\varphi (w)}{1-\overline{\varphi(z)}\varphi(w)}\bigg|\leq\psi_K(|\frac{z-w}{1-\bar{z}w}|)\leq\psi_K(r)$,
which yields the second inclusion
\bigskip \noindent For the first inclusion, it is sufficient to show
$\varphi^{-1}[D^{ph}(\varphi(z),\psi_{\frac{1}{K}}(r)]\subset D^{ph}(z,r).$
Since $\varphi^{-1}$ is $K$-quasiconformal, by the second inclusion of
(\ref{E:changephd}),
\begin{equation*}
\varphi^{-1}[D^{ph}(\varphi(z),\psi_{\frac{1}{K}}(r)]\subset D^{ph}(z,\psi_{K}\circ\psi_{\frac{1}{K}}(r)).
\end{equation*}Using the semigroup property of $\psi_K$, we can conclude that
$\varphi^{-1}(D^{ph}(\varphi(z),\psi_{\frac{1}{K}}(r))\subset D^{ph}(z,r)$.
The proof of Lemma \ref{L:areachangephd} is complete now.
\end{proof}
\noindent Next we introduce another gadget needed for our proof. It will also play a role
in Section \ref{S:Pplus}.
\begin{definition}\label{D:aphi}\emph{(\cite{AK2011}, p. 22)}
Let $\varphi$ be a quasiconformal mapping over $\mathbb{D}$ such that $\varphi(0)=0$. For each $z\in\mathbb{D}$, let $$B_z=\bigg\{w\in\mathbb{D}:|z-w|\leq\frac{1-|z|}{2}\bigg\}.$$ We define
\begin{equation*}\label{s4eq07}
a_\varphi(z) = \exp\bigg[\frac{1}{2|B_z|}\int_{B_z}\log J(w,\varphi)dA(w)\bigg].
\end{equation*}
\end{definition}
\noindent
Two theorems of Astala and Koskela \cite{AK2011} will be needed.
\begin{lemma}\emph{(\cite{AK2011}, Lemma 2.3)}\label{F:AK01}
Let $\varphi$ be a $K$-quasiconformal mapping over $\mathbb{D}$ with $\varphi(0)=0$. Then there exist positive constants $c_1$ and $c_2$ which depend only on $K$ such that for any $z\in\mathbb{D}$,
\begin{equation}\label{s4eq08}
c_1 \frac{1-|\varphi(z)|}{1-|z|}\leq a_\varphi(z)\leq c_2
\frac{1-|\varphi(z)|}{1-|z|}.
\end{equation}
\end{lemma}
\begin{lemma}\emph{(\cite{AK2011}, Theorem 7.3)} \label{F:AK02} Let $\varphi$ be a quasiconformal mapping over $\mathbb{D}$ with $\varphi(0)=0$. Then the boundary mapping
$\widetilde{\varphi}$ is a Lipschitz function on $\mathbb{T}$ if and only if there exists a constant $c>0$ such that $a_\varphi(z)<c$ for every $z\in\mathbb{D}$.
\end{lemma}
\noindent Now we are ready to give the proof of Theorem \ref{T:boundforC}. We actually show the following.
\begin{theorem}\label{T:boundforCC}
Let $\varphi$ be a $K$-quasiconformal mapping over the unit disk $\mathbb{D}$ with $\varphi(0)=0$. Let $C_\varphi(f)=f\circ \varphi$, initially defined as a map from $H^\infty(\mathbb{D})$ to $L^\infty(\mathbb{D})$.
\begin{itemize} \item[\emph{(1)}.] The following are equivalent:
\begin{itemize}
\item[\emph{(i)}] The map $f \mapsto f\circ \varphi$ extends to a bounded operator $C_\varphi: L^2_a(\mathbb{D})\to L^2(\mathbb{D})$;
\item[\emph{(ii)}] The boundary mapping $\widetilde{\varphi}^{-1}: \mathbb{T} \to \mathbb{T}$ is Lipschitz;
\item[\emph{(iii)}] $
\mathop{\sup}\limits_{z\in\mathbb{D}} m_\varphi(z)<\infty$, where
$m_\varphi(z)=\frac{1-|z|}{1-|\varphi(z)|}, \quad z \in \mathbb{D}.$
\end{itemize}
\item[\emph{(2)}.] When $C_\varphi: L^2_a(\mathbb{D}) \to L^2(\mathbb{D})$ is bounded, we have the following estimate of its norm:
$$ \|C_\varphi\|_{L_a^2(\mathbb{D})\to L^2(\mathbb{D})}\leq \frac{c}{1-\psi_K^2(\frac{1}{2})} \sup_{z\in\mathbb{D}} m_\varphi(z)$$
where $c$ is a universal constant.
\item[\emph{(3)}.] We also have a lower estimate $ c_1\sup_{z\in\mathbb{D}} m_\varphi(z)\leq \|C_\varphi\|_{L_a^2(\mathbb{D})\to L^2(\mathbb{D})},$
where $c_1=c_1(K)$ is a positive constant depending only on $K$.
\end{itemize}
\end{theorem}
\bigskip \noindent As hinted in the introduction, the sharp constants are unknown but certainly attractive.
\begin{proof}
For $0<r<1$, $z\in \mathbb{D}$, we introduce a new function
$
\widehat{\varphi}_r(z)=\frac{|\varphi^{-1}(D^{ph}(z,r))|}{|D^{ph}(z,r)|}
$
and consider the Toeplitz operator $T_{\widehat{\varphi}_r}$. By Lemma \ref{L:areachangephd}, $\|T_{\widehat{\varphi}_r}\|_{L_a^2(\mathbb{D})\to L_a^2(\mathbb{D})}\leq 4c_1\sup_{z\in\mathbb{D}}m_\varphi(z)^2,$ where $c_1=\frac{\psi_K^2(r)}{r^2(1-\psi_K(r)^2)^2}.$
We recall a basic estimate for $f\inL_a^2(\mathbb{D})$ (\cite{DS2003}, Lemma 13).
For $0<t<1$, set $r=\frac{1+t}{2}$. Then for any $z\in\mathbb{D}$, we have
$$
|f(z)|^2\leq\frac{4(1-r)^{-4}}{|D^{ph}(z,r)|}\int_{D^{ph}(z,r)}|f(w)|^2dA(w).
$$
Moreover, for each $w\in D^{ph}(z,r)$,
$$
\frac{(1-r|z|)^2}{(1-|z|^2)^2}\leq |\frac{1}{(1-\bar{z}w)^2}|\leq\frac{(1+r|z|)^2}{(1-|z|^2)^2}.
$$
Let $J(w)=J(w,\varphi^{-1})$ be the Jacobian of $\varphi^{-1}$ and $T_J$ be the Toeplitz operator on $L_a^2(\mathbb{D})$ with symbol $J$. We set $c_{2}=4(1-r)^{-4}$ and $c_{3}=(1-r)^2$. Then
\begin{eqnarray*}
\langle T_{J}f,f\rangle
&\leq& \frac{4c_{2}}{c_{3}r^2}\int_{\mathbb{D}}\int_{D^{ph}(z,r)}\frac{|f(w)|^2}{|1-z\bar{w}|^2}
dA(w)J(z, \varphi^{-1})dA(z)\\
&\leq&\frac{16c_{2}}{c_{3}(1-r^2)^2}\int_{\mathbb{D}}\int_{D^{ph}(z,r)}
\frac{|f(w)|^2}{|D^{ph}(w,r)|}dA(w)J(z, \varphi^{-1})dA(z)\\
&=& \frac{16c_{2}}{c_{3}(1-r^2)^2}\langle T_{\widehat{\varphi}_r}f, f\rangle.
\end{eqnarray*}
\medskip \noindent It follows that
\begin{equation*}
\|T_J\|_{L_a^2(\mathbb{D})\to L_a^2(\mathbb{D})}\leq \frac{16c_{2}}{c_{3}(1-r^2)^2}\|T_{\widehat{\varphi}_r}\|_{L_a^2(\mathbb{D})\to L_a^2(\mathbb{D})}
\leq \frac{64c_{1}c_{2}}{c_{3}(1-r^2)^2}\sup_{z\in\mathbb{D}}m_\varphi(z)^2,
\end{equation*}
for $\frac{1}{2}<r<1$. Since $C^*_\varphi C_\varphi=T_J$,
\begin{equation*}
\|C_\varphi\|_{L_a^2(\mathbb{D})\to L^2(\mathbb{D})} \leq\frac{c_4}{r(1-r)^3(1-r^2)(1-\psi_{K}^2(r))}
\sup_{z\in\mathbb{D}}m_\varphi(z).
\end{equation*}
Let $r$ approach $\frac{1}{2}$, there is a constant $c$ such that
$\|C_\varphi\|_{L_a^2(\mathbb{D})\to L^2(\mathbb{D})}\leq\frac{c}{1-\psi_K^2(\frac{1}{2})} \mathop{\sup}\limits_{z\in\mathbb{D}
m_\varphi}(z).$
\bigskip \noindent Next, for the lower bound, we need a subharmonic property of quasiregular mappings, due to Iwaniec and Nolder \cite{IN1985}, \cite{Nolder1990}, which we record below for the convenience of the reader.
\begin{lemma}\label{L:INthm01}
Let $\Omega$ be a domain in the plane, and let $z$ be the center of a cube $Q$ such that the closure $\overline{Q}$ is contained in $\Omega$. Let $\varphi:\Omega\to \mathbb{C}$ be a $K$-quasiregular mapping. Then for any $0<p<\infty$,
\begin{equation*}
|\varphi(z)|\leq c\bigg(\frac{1}{|Q|}\int_{Q}|\varphi(w)|^pdA(w)\bigg)^{\frac{1}{p}},
\end{equation*}
where the constant $c$ depends on $p$ and $K$.
\end{lemma}
\noindent For $f\in L_a^2(\mathbb{D})$, $f\circ \varphi$ is $K$-quasiregular. By Lemma \ref{L:INthm01}, there exists a constant $c=c(K)$ such that for all $z\in\mathbb{D}$,
$
|f\circ \varphi(z)|\leq\frac{c}{{1-|z|}}\bigg(\int_{\mathbb{D}}|f\circ \varphi(w)|^2dA(w)\bigg)^{\frac{1}{2}}.
$
Now fix $z_0\in\mathbb{D}$ and choose
$f(w)=\frac{1-|\varphi(z_0)|^2}{(1-\overline{\varphi(z_0)}w)^2} $ for $w\in\mathbb{D}.$
It satisfies $\|f\|_{L^2(\mathbb{D})}=1$.
Also $C_\varphi f(z_0)=\frac{1}{1-|\varphi(z_0)|^2}$.
Thus,
\begin{equation*}
\frac{c}{1-|z_0|}\bigg(\int_{\mathbb{D}}|f\circ \varphi(w)|^2dA(w)\bigg)^{\frac{1}{2}}\geq \frac{1}{1-|\varphi(z_0)|^2}.
\end{equation*}
Thus
\begin{equation*}
\|C_\varphi\|_{L_a^2(\mathbb{D})\to L^2(\mathbb{D})}\geq\frac{1}{2c}\frac{1-|z_0|}{1-|\varphi(z_0)|}.
\end{equation*}
It follows that
$\|C_\varphi\| \geq \frac{1}{2c} \mathop{\sup}\limits_{z\in\mathbb{D}}\frac{1-|z|}{1-|\varphi(z)|}$. Now we know that $C_\varphi$ is bounded from the Bergman space $L_a^2(\mathbb{D})$ into $L^2(\mathbb{D})$ if and only if $\mathop{\sup}\limits_{z\in\mathbb{D}}m_\varphi(z)<\infty$. It follows $\textrm{(i)} \Leftrightarrow \textrm{(iii)}$ in Part (1).
Observe that
$\mathop{\sup}\limits_{z\in \mathbb{D}} m_\varphi(z)<\infty$
if and only if
$
\mathop{\sup}\limits_{z\in \mathbb{D}}\frac{1-|\varphi^{-1}(z)|}{1-|z|}<\infty.
$
By Lemma \ref{F:AK01}, $\mathop{\sup}\limits_{z\in \mathbb{D}} m_\varphi(z)<\infty$ if and only if $\mathop{\sup}\limits_{z\in\mathbb{D}}a_{\varphi^{-1}}<\infty$. From Lemma \ref{F:AK02}, it follows that $\mathop{\sup}\limits_{z\in\mathbb{D}} m_\varphi(z)<\infty$ if and only if $\tilde{\varphi}^{-1}$ is Lipschitz on $\mathbb{T}$. The proof of Theorem \ref{T:boundforCC} is complete now.
\end{proof}
{\bignobf{Remarks.} (1) By classical results (\cite{zhu2007}, p. 172) on Carleson measures on $L^p_a(\mathbb{D})$ for $0<p<\infty$, we now know that $C_\varphi:L^p_a(\mathbb{C})\to L^p(\mathbb{D})$ is bounded if and only if $\tilde{\varphi}^{-1}$ is Lipschitz on $\mathbb{T}$.
\bigskip \noindent (2) By Harnack's inequality and Lemma \ref{L:HPD}, one can obtain the following quasiconformal version of the Littlewood subordination principle, which is unfortunately not enough to yield the desired $L^2$-boundedness of $C_\varphi$. We record it below, without proof, for interested readers.
Let $\varphi$ be a $K$-quasiconformal over $\mathbb{D}$ such that $\varphi(0)=0$, and let $f\geq 0$ be a subharmonic function on $\mathbb{D}$. Then for any $r\in(0, 1)$,
\begin{equation}
\frac{1}{2\pi}\int_0^{2\pi}f(\varphi(re^{i\theta}))d\theta\leq \frac{4}{1-\psi_K(r)} \sup_{0<t<1}\frac{1}{2\pi}\int_0^{2\pi}f(te^{i\theta})d\theta,
\end{equation}
where $\psi_K$ is the extremal distortion function.}
\bigskip \noindent Next we give the proof of Theorem \ref{T:compactforC}.
\begin{proof} Recall the essential norm $\|C_\varphi\|_e= \inf_K \|C_\varphi-\textit{K}\|_{L_a^2(\mathbb{D})\to L^2(\mathbb{D})},$ where $\mathit{K}: L_a^2(\mathbb{D})\to L^2(\mathbb{D})$ is compact.
We shall need an integrability criteria for the Jacobians of quasiconformal mappings (\cite{astala1994}, Corollary 1.2). Again we record it for the convenience of the reader.
\begin{lemma}\label{L:JacInt}
Let $\varphi$ be a $K$-quasiconformal mapping over $\mathbb{D}$ with $\varphi(0)=0$. Then $J(z,\varphi)\in L^q(\mathbb{D})$ for $ q\in [0, \frac{K}{K-1})$.
\end{lemma}
\noindent We shall take $q=\frac{K+1}{K}$.
Next we introduce a family of compact operators associated with compact subsets of $\mathbb{D}$. Let $E$ be a compact subset of $\mathbb{D}$, and let $M_E f= f\chi_{E}$ be a multiplication operator on $L^2(\mathbb{D})$. We define $X_E= M_E\cdot C_\varphi.$
\mednobf{Claim A. } $X_E$ is compact on $L_a^2(\mathbb{D})$ for each compact subset $E$ of $\mathbb{D}$.
\bigskip \noindent
It is sufficient to show that if
$\{f_n\}\subset L_a^2(\mathbb{D})$ uniformly converges to $0$ on each compact subset in $\mathbb{D}$, then $\|X_E f_n\|_{L^2(\mathbb{D})}$ converges to 0 as $n \to \infty$ \cite{zhu2007}. By Lemma \ref{L:JacInt}, there is a constant $c$ depending only on $\varphi$ such that
\begin{equation*}
\int_{\mathbb{D}}|f_n\circ \varphi(z)|^2\chi_{E}(z)dA(z)\leq c \bigg [\int_{\varphi(E)}|f_n(w)|^{2(K+1)}dA(w)\bigg ]^\frac{1}{K+1}.
\end{equation*}
\bigskip \noindent Then, we have $\lim_{n\to\infty}\|X_Ef_n\|^2_{L^2(\mathbb{D})}=0$.
\medskip \noindent Next we introduce another family of operators $\{T_r\}_{r \in (0, 1)}$. For any $r\in(0, 1)$, let $F_r\subset\mathbb{D}$ be such that $\varphi(F_r)=A_r \triangleq \{z\in\mathbb{D}:|z|>r\}.$
Let $E_r=\mathbb{D}-F_r$ and consider the compact operator $X_{E_r}$. Then we define $T_r$ on $L_a^2(\mathbb{D})$ by $(T_rf)=C_\varphi f-X_{E_r} f$.
\bignobf{Claim B.} There is a constant $c$ such that $$\mathop{\inf}\limits_{0<r<1}\|T_r\|\leq \frac{c\psi_K(\frac{1}{2})}{1-\psi_K^2(\frac{1}{2})}\mathop{\lim}\limits_{t\to 1}\mathop{\sup}\limits_{|z|>t}m_\varphi(z).$$
\bigskip \noindent
By a known covering lemma (\cite{axlers}, Lemma 3.5), there exists a sequence $\{\xi_n\}_{n=0}^\infty \subset\mathbb{D}$ such that $\mathop{\lim}\limits_{n\to \infty}|\xi_n|=1$ and
$\mathbb{D}=\mathop{\cup}\limits_{n=0}^\infty D^{ph}(\xi_n,\frac{1}{2}).$
Moreover, each $z\in\mathbb{D}$ is contained in at most $c_1$ pseudohyperbolic disks among $\{D^{ph}(\xi_n,\frac{3}{4})\}$.
Owing to a standard subharmonic estimate \cite{DS2003}, for $f\in L_a^2(\mathbb{D})$ and any $z\in D^{ph}(z_0,\frac{1}{2}), z_0\in\mathbb{D}$, we have
\begin{equation}\label{s7eq01}
|f(z)|^2\leq \frac{c_2}{|D^{ph}(z_0,\frac{3}{4})|}
\int_{D^{ph}(z_0,\frac{3}{4})}|f(w)|^2dA(w).
\end{equation}
In order to proceed with the proof of Claim B, we need to verify the following
\bignobf{Claim C.} For any $r \in (0, 1)$ close to $1$, we can select a subsequence $\{\xi_{n_m^{(r)}}\}_{m=0}^\infty\subset\{\xi_n\}_{n=0}^\infty$ such that for any $m\geq 0$
\begin{itemize}
\item [(1)] $D^{ph}(\xi_{n_m^{(r)}},\frac{1}{2})\cap A_r\not=\emptyset$;
\item [(2)] $A_r\subset \mathop{\cup}\limits_{m=0}^\infty D^{ph}(\xi_{n_m^{(r)}},\frac{1}{2})$;
\item [(3)] $|\xi_{n_{m^{(r)}}}|\geq t_r$, where $t_r=\min\{\frac{-2+3r+2r^2}{4-r^2}, r\}$.
\end{itemize}
\noindent We look at those $\xi_n$ such that $D^{ph}(\xi_n,\frac{1}{2})\cap A_r\not=\emptyset$. We re-label these $\xi_n$ as $\{\xi_{n^{(r)}_m}\}_{m=0}^\infty$. Then (1) and (2) are clearly satisfied. The main point is to verify (3). Clearly, if $|\xi_{n^{(r)}_m}| \ge r$, then there is nothing to prove. Next we re-label the rest of the sequence as
$\{\xi_{n^{(r)}_{m_j}}\}_{j=0}^\infty $ with $|\xi_{n^{(r)}_{m_j}}| \le r.$
For any $j\geq 0$, since the pseudohyperbolic distance between each $\xi_{n_{m_j}}^{(r)}$ and $\{z:|z|=r\}$ is at most $\frac{1}{2}$, there exists a point $z_{n_{m_j}^{(r)}}'\in \mathbb{T}_r$
such that $\xi_{n_{m_j}^{(r)}}\in D^{ph}(z_{n_{m_j}^{(r)}}', \frac{1}{2}),$ where $\mathbb{T}_r\doteq \{z\in\mathbb{D}:|z|=r\}$.
Observe that for any $j\geq 0$ the modulus of the Euclidean center of $D^{ph}(\xi_{n_{m_j}^{(r)}},\frac{1}{2})$ is equal to $\frac{3r}{4-r^2}$, and its
Euclidean radius is $\frac{2(1-r^2)}{4-r^2}.$
Then $|\xi_{n_{m_j}^{(r)}} |\geq \frac{-2+3r+2r^2}{4-r^2}$ for any $j\geq 0$. We have
for any $m\geq 0$, $|\xi_{n_m^{(r)}}|\geq t_r \triangleq \min\{\frac{-2+3r+2r^2}{4-r^2}, r\}$.
\bigskip \noindent
Take $f\inL_a^2(\mathbb{D}),\|f\|_{L_a^2(\mathbb{D})}\leq 1$, by (\ref{s7eq01}) and Lemma \ref{L:areachangephd}.
then
\begin{eqnarray*}
\int_{|z|>r}|f(z)|^2J(z,\varphi^{-1})dA(z)&\leq&\sum_{m=0}^\infty\int_{D^{ph}
(\xi_{n_m^{(r)}},\frac{1}{2})}|f(z)|^2J(z,\varphi^{-1})dA(z)\\
&\leq&\frac{c_3\psi_K^2(\frac{1}{2})}{(1-\psi_K^2(\frac{1}{2}))^2} \\
&\cdot&\sup_{|z|>t_r}\bigg[\frac{1-|\varphi^{-1}(z)|}{1-|z|}\bigg]^2
\sum_{m=0}^\infty \int_{D^{ph}(\xi_{n_m^{(r)}},\frac{3}{4})}|f_n(z)|^2dA(z)\\
&\leq& \frac{c_4 \psi_K^2(\frac{1}{2})}{(1-\psi_K^2(\frac{1}{2}))^2} \sup_{|z|>t_r}\bigg[\frac{1-|\varphi^{-1}(z)|}{1-|z|}\bigg]^2.
\end{eqnarray*}
\bigskip \noindent It follows that $\|T_r\|_{L_a^2(\mathbb{D})\to L^2(\mathbb{D})}\leq \frac{c\psi_K(\frac{1}{2})}{1-\psi_K^2(\frac{1}{2})}
\sup_{\varphi^{-1}(A_{t_r})}\frac{1-|w|}{1-|\varphi(w)|}.$
By Lemma \ref{L:HPD},
$$\|T_r\|_{L_a^2(\mathbb{D})\to L^2(\mathbb{D})}\leq \frac{c\psi_K(\frac{1}{2})}{1-\psi_K^2(\frac{1}{2})}
\sup_{|z|>\psi_{\frac{1}{K}}(t_r)}m_\varphi(z).$$
Observe that $t_r$ converges to 1 as $r\to 1$, so does $\varphi_{\frac{1}{K}}(t_r)\to 1$.
%
In summary so far, we have
\begin{equation*}\label{s6eq01}
\|C_\varphi\|_e\leq \frac{c\psi_K(\frac{1}{2})}{1-\psi_K^2(\frac{1}{2})}\lim_{t\to 1}\sup_{|z|>t}\frac{1-|z|}{1-|\varphi(z)|}.
\end{equation*}
\bigskip \noindent Next let $\{z_n\in\mathbb{D}\}_{n=1}^\infty$ be any sequence such that $|z_n|\to 1$. Then we define a sequence of functions on $\mathbb{D}$ by
$
f_n(z)=\frac{1-|\varphi(z_n)|^2}{(1-\overline{\varphi(z_n)}z)^2}.
$
This is a sequence of unit vectors in $L_a^2(\mathbb{D})$ and it weakly converges to $0$.
For any compact operator $\textit{K}$,
$$ \mathop{\limsup}\limits_{n\to\infty}\|C_\varphi f_n-\textit{K} f_n\|_{L^2(\mathbb{D})} = \mathop{\limsup}\limits_{n\to\infty}\|C_\varphi f_n\|_{L^2(\mathbb{D})}.$$
\medskip \noindent {For any $n\geq 1$, $f_n\circ \varphi$ is $K$-quasiregular on $\mathbb{D}$. By Lemma \ref{L:INthm01}, there is a constant $c$ depending only on $K$ such that}
\begin{equation*}
\frac{1-|z_n|}{1-|\varphi(z_n)|}\leq c\bigg[\int_{\mathbb{D}}|f_n\circ \varphi(w)|^2dA(w)\bigg]^{\frac{1}{2}}=
c\bigg[\int_{\mathbb{D}}|f_n(z)|^2J(z,\varphi^{-1})dA(z)\bigg]^{\frac{1}{2}}.
\end{equation*}
Therefore
$\|C_\varphi\|_e\geq c_{1} \mathop{\lim}\limits_{r\to 1}\mathop{\sup}\limits_{|z|>r}m_\varphi(z)$ for some constant
$c_{1}=c_{1}(K)$ that depends only on $K$. The proof of Theorem \ref{T:compactforC} is complete now.
\end{proof}
\section{Proof of Theorem \ref{T:extraforShplus}}\label{S:Pplus}
In the first subsection below we prove a crucial technical lemma (Lemma \ref{L:Key01}) which is clearly of independent interests. Then we prove Theorem \ref{T:extraforShplus}. Actually, we prove
\begin{theorem}\label{T:extraforShpluss}Let $\varphi$ be a quasiconformal mapping over $\mathbb{D}$ such that $\varphi(0)=0$. Let $P^+_\varphi=\int_{\mathbb{D}}\frac{f(w)}{|1-\varphi(z)\overline{\varphi(w)}|^2}dA(w),$
If $\widetilde{\varphi}$ is bilipschitz on $\mathbb{T}$, then
$P_\varphi$ satisfies the weak-$(1, 1)$ inequality: if $f\in L^1(\mathbb{D})$, then for any $\alpha>0$,
\begin{equation}\label{E:weak11}|\{z\in\mathbb{D}: |P_\varphi^+ f(z)|>\alpha\}|\leq \frac{c}{\alpha}\int_{\mathbb{D}}|f(z)|dA(z),
\end{equation}
where $c$ is a constant independent of $f$, and the leftmost $|\cdot|$ denotes the normalized Lebesgue area.
\end{theorem}
\subsection{Carleson Boxes and Quasiconformal Mappings}
This subsection is devoted to investigating the behaviors of Carlesons boxes under a quasiconformal mapping. The main result is Lemma \ref{L:Key01}. It plays a key role in the proof of Theorem \ref{T:extraforShplus}.
Let $I\subset\mathbb{T}$ be an interval. Then it induces a Carleson box $Q_I$ in $\mathbb{D}$:
\begin{equation*}
Q_I=\{z\in\mathbb{D}:1-|I|\leq |z|<1,\frac{z}{|z|}\in I\},
\end{equation*}
where $|I|$ is the normalized arc length of $I$. We call the point $z_I=r_Ie^{i\theta_I}$ the center of $Q_I$, where $r_I=1-\frac{|I|}{2}$ and $\theta_I$ is the midpoint of $I$.
\begin{lemma}\label{L:Key01}
Let $\varphi$ be a quasiconformal mapping over $\mathbb{D}$ such that $\varphi(0)=0$. If the boundary mapping $\widetilde{\varphi}$ is bilipschitz on $\mathbb{T}$, then there is a constant $0<t_0<1$ such that for each Carleson box $Q_I$ with $|I|<t_0$, there exists a Carleson box $Q_J$ such that
\begin{itemize}
\item [\emph{(1)}] $\varphi^{-1}(Q_I)\subset Q_J$, and
\item [\emph{(2)}] $\frac{|Q_J|}{|\varphi^{-1}(Q_I)|}\leq c.$
\end{itemize}
Here the constant $c$ does not depend on the choice of $Q_I$.
\end{lemma}
\begin{proof} Since $\widetilde{\varphi}$ is bilipschitz, by Theorem \ref{T:boundforC}, $C_\varphi$ and $C_{\varphi^{-1}}$ are bounded. So, there exist constants $c_{1}>0$ and $c_{2}>0$ such that
for any $z\in\mathbb{D}$,
\begin{equation}\label{1207eq03}
c_{1}<\frac{1-|\varphi^{-1}(z)|}{1-|z|}<c_{2}.
\end{equation}
Next we further divide the proof of Lemma \ref{L:Key01} into two subsections.
\subsubsection{Area Change of Carleson Boxes--the Upper Bound}\label{S:areaupper}
\smallskip \noindent The main task of this subsection is to verify the next claim.
\bigskip \noindent \textbf{Claim.} There exists a positive number $t_0\in(0,1)$ such that for any Carleson box $Q_I$ with $|I|\leq t_0$, there is a Carleson box $Q_J$ such that
\begin{itemize}
\item [(i)] $\varphi^{-1}(Q_I)\subset Q_J,$
\item [(ii)] $|Q_J|\leq c|Q_I|$.
\end{itemize}
Here the constant $c$ depends only on the function $\varphi$.
\bigskip \noindent To verify this claim, we need a theorem of Koslela \cite{Koskela1994}. Recall that $a_\varphi(\cdot)$ is given by Definition \ref{D:aphi}.
\begin{lemma}\emph{(\cite{Koskela1994}, Lemma 2.6)}\label{L:koskelath01}
Let $\varphi$ be a $K$-quasiconformal mapping over $\mathbb{D}$, and $\gamma\subset\mathbb{D}$ be a rectifiable arc with length $l(\gamma)\geq \emph{dist}(\gamma,\mathbb{T})$. Then
\begin{equation*}
\emph{diam}(\varphi(\gamma))\leq c\int_{\gamma}a_\varphi(z)|dz|,
\end{equation*}
where the constant $c$ depends only on $K$, and \emph{diam} denotes the diameter of the set $\varphi(\gamma)$.
\end{lemma}
\begin{proof}[Proof of the Claim] Assume that $Q_I$ is a Carleson box with $|I|=t$ and the center $z_I$. Let $z_0$ be the center of the arc $l_1 = \{z: |z|=1-t\}\cap \overline{Q}_I$. Let $\gamma_0$ be the radial segment connecting $z_I$ and $z_0$. Consider any point $\tilde{w}\in\partial \varphi^{-1}(Q_I)$.
Let $\tilde{z}\in\partial Q_I$ be such that $\varphi^{-1}(\tilde{z})=\tilde{w}$.
Now we distinguish two cases.
If $\tilde{z}\in\partial Q_I-\mathbb{T}$, then let $
\gamma_{\tilde{z}}\subset\partial Q_I-\mathbb{T}$ be the path connecting $z_0$ and $\tilde{z}$. More precisely, let $l_2$ be the radial segment in $\partial Q_I$ that contains $\tilde{z}$, and let $z_{1}=l_1\cap l_2$. Then $\gamma_{\tilde{z}}$ is the arc in $l_1\cup l_2$ passing through $z_1$.
Next, set $$\gamma=\gamma_0+\gamma_{\tilde{z}}.$$ Observe that $l(\gamma_0)=\frac{t}{2}$ and $l(\gamma_{\tilde{z}})|\leq (\pi+1)t$, so we have $$\frac{t}{2}\leq l(\gamma)\leq (\frac{3}{2}+\pi) t<5t.$$
Since $\textrm{dist}(\gamma, \mathbb{T})\leq \textrm{dist}(z_I, \mathbb{T})=\frac{t}{2}$,
we conclude that
\begin{equation*}
5t>l(\gamma)\geq \frac{t}{2}\geq \textrm{dist}(\gamma,\mathbb{T}).
\end{equation*}
If $\tilde{z}\in\partial Q_I\cap\mathbb{T}$, then link $\tilde{z}$ and $z_I$ by a segment contained in $Q_I$ whose arc length is at most $(\pi+\frac{1}{2})t$.
By Lemma \ref{L:koskelath01}, we have
\begin{equation*}
\text{diam}(\varphi^{-1}(\gamma))\leq c\int_{\gamma}a_{\varphi^{-1}}(z)|dz|\leq 5c c_{1}t \triangleq \delta t,
\end{equation*}
where $c_{1}=\mathop{\sup}\limits_{z\in\mathbb{D}}a_{\varphi^{-1}}(z)$. Since $\varphi^{-1}$ is Lipschitz on $\mathbb{T}$, by Lemma \ref{F:AK02}, $c_{1}<\infty$. It follows that $$\textrm{dist}(\varphi^{-1}(z_I), \partial \varphi^{-1}(Q_I))\leq \delta t<\infty.$$
\bigskip \noindent Note that the constant $\delta$ depends only on $\varphi^{-1}$. Let us choose a constant $t_0\in(0, 1)$ such that $$\delta t_0<\frac{1}{100}.$$ For any $0<t<t_0$,
\begin{equation}\label{E:carlesonboxQC} \varphi^{-1}(Q_I)\subset \{w\in\mathbb{D}:|w-\varphi^{-1}(z_I)|<\delta t\}.\end{equation}
\noindent Consider a Carleson box $Q_J$ such that $|J|=10\pi\delta t$ and the middle points of $J$ and $\varphi^{-1}(I)$ coincide.
Next, we show \begin{equation}\varphi^{-1}(Q_I)\subset Q_J.\end{equation}
\smallskip \noindent If $w'\in\partial \varphi^{-1}(Q_I)$ is the center of $\varphi^{-1}(I)$, then let $z'\in\mathbb{T}$ such that $\varphi^{-1}(z')=w'$.
By (\ref{E:carlesonboxQC}), for each $z\in Q_I$,
$$|\varphi^{-1}(z)-\varphi^{-1}(z')|\leq
|\varphi^{-1}(z)-\varphi^{-1}(z_I)|+|\varphi^{-1}(z_I)-\varphi^{-1}(z')|\leq 2\delta t.$$
Then
$$\varphi^{-1}(Q_I)\subset \{w\in\mathbb{D}:|w-w'|<2\delta t\}.$$
Since $\{w\in\mathbb{D}:|w-w'|<2\delta t\}\subset Q_J$, we have
$\varphi^{-1}(Q_I)\subset Q_J$.
\bigskip \noindent Then it follows from
$|Q_J|= (10\pi\delta t)^2(2-10\pi\delta t)$ and
$|Q_I|=t^2(2-t)$ that we have
$$|Q_J|\leq \frac{2(10\pi\delta)^2}{2-t}|Q_I|\leq 200(\pi\delta)^2|Q_I|.$$
Now we have verified the claim and obtained an upper bound for the area change of a Carleson box under a quasiconformal mapping.
\end{proof}
\subsubsection {Area Change of Carleson Boxes--the Lower Bound}
For the lower bound, we shall need to work with the so-called geometric definition of quasiconformal mappings. It is based on conformal modules of quadrilaterals.
\bigskip \noindent
A quadrilateral $Q(z_1,z_2,z_3,z_4)$ is a Jordan domain $Q$ with four consecutive boundary points $\{z_1,z_2,z_3,z_4\}$ specified and with a positive orientation. The boundary arcs $\overrightarrow{z_1z_2}$ and $\overrightarrow{z_3z_4}$ are called $a$-sides and the other two are called $b$-sides. By the Riemann mapping theorem, there is a unique conformal mapping $\phi: Q\to R$ , where $R$ is a rectangle with vertices $\{0, a, a+ib, ib: a>0, b>0\}$, such that $\phi(z_1)=0, \phi(z_2)=a, \phi(z_3)=a+ib$ and $\phi(z_4)=ib$. Then the conformal module $\textrm{Mod} (Q) $ of $Q$ is defined to be $\frac{b}{a}$. We call an orientation-preserving homeomorphism on $\mathbb{D}$ $K$-quasiconformal if
$$\sup_{Q}\frac{\textrm{Mod}(\varphi(Q))}{\textrm{Mod}(Q)}\leq K<\infty,
$$
where $Q$ runs over all quadrilaterals such that $\overline{Q}\subset \mathbb{D}$.
It is a fundamental and remarkable result in the theory of quasiconformal mapping that the geometric definition is equivalent to the analytic definition. A proof of this equivalence can be found in \cite{ah1}, \cite{lv}.
\bigskip \noindent Now we continue our estimation of Carleson boxes. We view a Carleson box $Q_I$ as a quadrilateral by
$Q_I=Q_I(z_1,z_2,z_3,z_4)$ such that
the vertices satisfy $|z_1|=|z_4| =1-t$ and $|z_2|=|z_3|=1$. In particular, the $a$-sides of $Q_I$ are the radial segments and the other two are the $b$-sides.
Let $\Gamma_a$ be the set of rectifiable arcs $\gamma\subset Q_I$ which connect the $a$-sides of $Q_I$. We define $$s_a(Q_I)=\mathop{\inf}\limits_{\gamma\in \Gamma_a}l(\gamma),$$
where $l(\gamma)$ is the arc length of $\gamma$.
Similarly, we define $s_b(Q_I)$. A direct calculation shows that
$$s_a(Q_I)=2\pi t(1-t) \quad \text{and} \quad s_b(Q_I)=t.$$
\bigskip \noindent Next we shall need Rengel's inequality \cite{lv} which says that the conformal module of a quadrilateral $Q_I$ satisfies the double inequality
\begin{equation}
\frac{(s_b(Q_I))^2}{\pi|Q_I|}\leq \textrm{Mod}(Q_I)\leq \frac{\pi |Q_I|}{(s_a(Q_I))^2}.
\end{equation}
\medskip \noindent Since $|Q_I|=t^2(2-t)$, we get
\begin{equation*}
\frac{1}{\pi(2-t)}\leq \textrm{Mod}(Q_I)\leq \frac{2-t}{4\pi(1-t)^2}.
\end{equation*}
By the geometric definition,
$$
\frac{1}{K\pi(2-t)}\leq \textrm{Mod}(\varphi^{-1}(Q_I))\leq K\frac{2-t}{4\pi(1-t)^2}.
$$
By Rengel's inequality,
$$|\varphi^{-1}(Q_I)|\geq \frac{1}{\pi} \frac{(s_b( \varphi^{-1}(Q_I) ))^2}{\textrm{Mod}( \varphi^{-1}(Q_I) )}.$$
Let $l_{Q_I}=\partial Q_I\cap\{z:|z|=1-t\}$. For any $z'\in\varphi^{-1}(l_{Q_I})$, let $z\in l_{Q_I}$ be such that $z'=\varphi^{-1}(z)$.
For any $z''\in\partial\varphi^{-1}(Q_I)\cap\mathbb{T}$, let $\gamma\subset \varphi^{-1}(Q_I)$ be a curve connecting $z'$ and $z''$. Then
\begin{equation*}
l(\gamma)\geq |z'-z''|\geq \textrm{dist}(z',\mathbb{T})=1-|z'|=1-|\varphi^{-1}(z)|\geq c_{1}(1-|z|)=c_{1}t.
\end{equation*}
The last inequality is due to the left side of (\ref{1207eq03}).
Thus $$s_b(\varphi^{-1}(Q_I))\geq c_{1}t,$$ and we deduce that
\begin{equation*}
|\varphi^{-1}(Q_I)|\geq \frac{1}{\pi}c_{1}^2\frac{4\pi(1-t)^2}{K(2-t)}t^2.
\end{equation*}
It follows that for any $t_1\in(0,1)$, there is a constant $c_{2}=c(\varphi, t_1)>0$ such that for any Carleson box $Q_I$ with $|I|\leq t_1$,
\begin{equation}\label{E:arealower}
|\varphi^{-1}(Q_I)|\geq c_{2}|Q_I|.
\end{equation}
\noindent Let $0<t_0<1$ be the constant in the upper estimate. Then for any Carleson box $Q_I$ with $|I|<t_0$, there exists a
Carleson box $Q_J$ such that $\varphi^{-1}(Q_I)\subset Q_J.$
Furthermore, by the Claim in Subsection \ref{S:areaupper} and (\ref{E:arealower}),
\begin{equation*}
\frac{|Q_J|}{|\varphi^{-1}(Q_I)|}=\frac{|Q_J|}{|Q_I|}
\frac{|Q_I|}{|\varphi^{-1}(Q_I)|}\leq c.
\end{equation*}
The proof of Lemma \ref{L:Key01} is complete now.
\end{proof}
\bigskip \noindent Next, we introduce some dyadic systems over $\mathbb{D}$.
Let $\mathbb{Z}_+=\mathbb{N} \cup \{0\}$. Consider the following two dyadic grids on $\mathbb{T}$,
\begin{equation*}
\mathcal{D}^0=\{[\frac{2\pi m}{2^j},\frac{2\pi (m+1)}{2^j}):m\in\mathbb{Z}_+, j\in\mathbb{Z}_+, 0\leq m<2^j\}
\end{equation*}
and
\begin{equation*}
\mathcal{D}^{\frac{1}{3}}=\{[\frac{2\pi m}{2^j}+\frac{2\pi}{3},\frac{2\pi (m+1)}{2^j}+\frac{2\pi}{3}):m\in\mathbb{Z}_+, j\in\mathbb{Z}_+, 0\leq m<2^j\}.
\end{equation*}
The first appearance of shifted dyadic grids in print is probably in page 30 of \cite{Christ}. A quick way to appreciate why shifted dyadic grids are powerful is to look at \cite{APR2013}, \cite{GJ1982} and \cite{Mei2003}. In particular, \cite{APR2013} contains a nice application to Sarason's problem on Toeplitz products.
For each $\beta\in\{0, \frac{1}{3}\}$, let $\mathcal{Q}^\beta$ denote the collection of Carleson boxes $Q_I$ with $I\in\mathcal{D}^\beta$ and we call $\mathcal{Q}^\beta$ a Carleson box system.
\subsection{Proof of Theorem \ref{T:extraforShpluss}}
\noindent
Let $r_0 < {t_0^2}/{16},$ where
$t_0$ is the constant in Lemma \ref{L:Key01}. Then we write
$
P_\varphi^+=(P_{\varphi}^+)_{r_0}+(P_\varphi^+)_{1-r_0},
$
where
\begin{equation*}
(P_{\varphi}^+)_{r_0}f(z)=\int_{|1-\varphi(z)\overline{\varphi(w)}|\geq r_0}\frac{f(w)}{|1-\varphi(z)\overline{\varphi(w)}|^2}dA(w)
\end{equation*}
and
\begin{equation*}
(P_{\varphi}^+)_{1-r_0}f(z)=\int_{|1-\varphi(z)\overline{\varphi(w)}|<r_0}\frac{f(w)}
{|1-\varphi(z)\overline{\varphi(w)}|^2}dA(w).
\end{equation*}
Since the integral kernel of $(P_{\varphi}^+)_{r_0}$ is bounded, $(P_{\varphi}^+)_{r_0}$
is bounded on $L^p(\mathbb{D})$ for $1\leq p\leq\infty$. Next let us focus on the $L^2$-boundedness of $(P_\varphi^+)_{1-r_0}$ under the assumption that the boundary mapping $\widetilde{\varphi}$ is bilipschitz on $\mathbb{T}$.
The next lemma is elementary, hence proof skipped.
\begin{lemma}\label{L:Key02} For any $z, w \in \mathbb{D}$, there exists a Carleson box $Q_I$ such that $z,w\in Q_I$ and
$\frac{1}{16}|Q_I|\leq |1-z\bar{w}|^2\leq 8 |Q_I|$.
\end{lemma}
\bigskip \noindent
For any $z, w \in \mathbb{D}$ such that $|1-\varphi(z)\overline{\varphi(w)}|^2\leq r_0$, by Lemma \ref{L:Key02}, there exists a Carleson box $Q_I$ such that $\varphi(z), \varphi(w)\in Q_I$ and $$|I|^2\leq 16 |1-\varphi(z)\overline{\varphi(w)}|^2 \leq t_0^2.$$
By Lemma \ref{L:Key01}, there exists a Carleson box $Q_J$ such that $z, w\in Q_J$ and
\begin{eqnarray*}
\frac{1}{|1-\varphi(z)\overline{\varphi(w)}|^2}\leq \frac{1}{|Q_I|}
\leq\frac{16 c_{1}}{|\varphi^{-1}(Q_I)|} \leq \frac{c_2}{|Q_J|}.
\end{eqnarray*}
Here we have used the Claim in Subsection \ref{S:areaupper} and Lemma \ref{L:Key01}.
Now, a few known facts will help conclude the proof. By \cite{Mei2003}, for any interval $J\subset\mathbb{T}$, there exists an interval $I'\in\mathcal{D}^0\cup\mathcal{D}^\frac{1}{3}$ such that
$J\subset I'$ and $|I'|\leq 6|J|$.
%
So we can find
an interval $I'\in\mathcal{D}^0\cup\mathcal{D}^\frac{1}{3}$ such that
\begin{eqnarray}
\frac{1}{|1-\varphi(z)\overline{\varphi(w)}|^2} \leq \frac{c_3\chi_{Q_{I'}}(z)\chi_{Q_{I'}}(w)}{|Q_{I'}|}. \label{E:dyadiccontrol03}
\end{eqnarray}
\smallskip \noindent Now we write the integral kernel of $(P_\varphi^+)_{1-r_0}$ as
$
\widetilde{K}(z, w)=\frac{\chi_{S}(z,w)}{|1-\varphi(z)\overline{\varphi(w)}|^2},
$
where $S=\{(z,w)\in\mathbb{D}\times\mathbb{D}:|1-\varphi(z)\overline{\varphi(w)}|<r_0\}$.
It follows that
\begin{equation}\label{E:dyadiccontrol}
\widetilde{K}(z,w)\leq c_{3}\sum_{ I \in \{\mathcal{D}^0\cup\mathcal{D}^{\frac{1}{3}}\} }\frac{\chi_{Q_{I}}(z)\chi_{Q_{I}}(w)}{|Q_{I}|}.
\end{equation}
Now we write
$K_{\beta}(z,w)=\sum_{I\in\mathcal{D}^\beta}\frac{\chi_{Q_I}(z)\chi_{Q_I}(w)}{|Q_I|}
$
and
$\mathcal{Q}^\beta f(z)=\int_{\mathbb{D}}K_\beta (z, w)f(w)dA(w).
$
Then the proof follows from the fact \cite{APR2013} that the weak-\emph{(1, 1)} type inequality holds for
$\mathcal{Q}^\beta$, $\beta\in\{0, \frac{1}{3}\}$.
\section{Proof of Theorem \ref{T:Shexpra}}
We shall need three different definitions of BMO functions over the unit disk, and to show that they are all equivalent. This is probably
well known to experts. Since we cannot locate a reference and it is clearly of independent interests, we present a proof for completeness. Readers familiar with BMO may skip the next subsection.
\subsection{Three Definitions of BMO on $\mathbb{D}$}
We begin with the definition of a space of homogeneous type which plays an important role in our proofs.
\begin{definition}
\emph{(\cite{CW1971}, \cite{DH2009})}\label{0828de1}
A triple $(X, \rho, \mu)$ is a space of homogeneous type if
\begin{itemize}
\item [\emph{(i)}] $X$ is a set,
\item [\emph{(ii)}] $\rho$ is a quasi-metric on $X$, that is,
$\rho:X\times X\to[0,\infty]$ is such that for any $x, y, z\in X$,
\emph{(1)} $\rho(x, y)=0 \Leftrightarrow x=y, $
\emph{(2)} $\rho(x, y) = \rho(y, x),$ and
\emph{(3)} $ \rho(x, y) \leq c(\rho(x, z)+\rho(z, y))$
for some constant $c$, and
\item [\emph{(iii)}] $\mu$ is a positive measure on $X$ such that for all $x\in X$ and $r>0$, there exists a constant $c$ such that
\emph{(a)} $ \mu(B(x,r)) < \infty, $ and
\emph{(b)} $ \mu(B(x,2r)) \leq c\mu(B(x,r))\label{0828eq1},$
where $B(x,r)=\{y\in X:\rho(x,y)<r\}$.
\end{itemize}
\end{definition}
\bigskip \noindent Let $d(z,w)=|z-w|$ be the Euclidean distance on $\mathbb{D}$. Let $|E|$ be the normalized Lebesgue area of $E \subset \mathbb{D}$. Then the triple $(\mathbb{D},d,|\cdot|)$ is a space of homogeneous type $\cite{CRW1976}$.
\bigskip \noindent Let $D^h(=D^h(z, r)) \triangleq \{w \in \mathbb{D}: |z-w|<r\}$ be a homogeneous ball in $(\mathbb{D}, d, |\cdot|)$. For a locally integrable function $f$ on $\mathbb{D}$, define its average over $E_{D^h}$,
$E_{D^h}(f)=\frac{1}{|D^h|}\int_{D^h}f(z)dA(z).
$
Now we introduce the needed BMO spaces over $\mathbb{D}$. First, we say that $f\in \textrm{BMO}_{\textrm{H}}(\mathbb{D})$ if
$$\|f\|_{\textrm{BMO}_\textrm{H}} \triangleq \sup_{D^h}E_{D^h}(|f-E_{D^h}(f)|)<\infty,$$
where $D^h$ runs over all homogeneous balls in $(\mathbb{D}, d, |\cdot|)$.
If we replace $D^h$ in the above definition by Euclidean balls and Euclidean cubes contained in $\mathbb{D}$, respectively, then we get $\textrm{BMO}_\textrm{B}(\mathbb{D})$ and $\textrm{BMO}_{\textrm{C}}(\mathbb{D})$.
\bigskip \noindent The purpose of this subsection is to show
\begin{theorem}\label{T:BMO}There are positive constants $c_1, c_{2}, c_{3}$ and $c_{4}$ such that
$$
c_{1}\|f\|_{\emph{BMO}_\emph{H}}\leq c_{2}\|f\|_{\emph{BMO}_{\emph{C}}} \leq c_{3}\|f\|_{\emph{BMO}_{\emph{B}}}\leq c_{4}\|f\|_{\emph{BMO}_\emph{H}}.
$$
for all $f\in \emph{BMO}_{\emph{H}}(\mathbb{D})$.
\end{theorem}
\noindent Theorem \ref{T:BMO} is an immediate consequence of Lemma \ref{L:BMO1} and Lemma
\ref{L:BMO2}.
\begin{lemma}\label{L:BMO1}
The identity map
$i: \emph{BMO}_\emph{H}(\mathbb{D})\to \emph{BMO}_{\emph{C}}(\mathbb{D})$
is an isomorphism between Banach spaces.
\end{lemma}
\begin{lemma}\label{L:BMO2}
The identity map
$i: \emph{BMO}_\emph{H}(\mathbb{D})\to \emph{BMO}_{\emph{B}}(\mathbb{D})$
is an isomorphism between Banach spaces.
\end{lemma}
\smallskip \noindent Since the ideas in the proofs of the above two lemmas are similar, we only prove Lemma \ref{L:BMO1}. We need the following basic fact whose proof is easy and will be skipped.
\begin{lemma}\label{L:BMO}
Fix $f \in L^1_{\emph{loc}}(\mathbb{D})$. If for any homogeneous ball $D^h \subset \mathbb{D}$, there exists a number $c$ such that
$\frac{1}{|D^h|}\int_{D^h}|f(z)-c|dA(z)\leq c,$
then $\|f\|_{\emph{BMO}_\emph{H}(\mathbb{D})}\leq 2c.$
Moreover, a similar statement holds for $\emph{BMO}_\emph{C}(\mathbb{D})$ and $\emph{BMO}_\emph{B}(\mathbb{D})$.
\end{lemma}
\begin{proof}[Proof of Lemma \ref{L:BMO1}] Let $Q=Q(z_0,2r)\subset\mathbb{D}$ be a cube with center $z_0$ and with side length $2r$. Let $B=D^h(z_0, \sqrt{2}r)$ so that $Q\subset B$. Then
\begin{eqnarray*}\label{1004eq6}
\frac{1}{|Q|}\int_{Q}|f(z)-E_Bf|dA(z)&\leq& \frac{|B|}{|Q|}\frac{1}{|B|}\int_{B}|f(z)-E_Bf|dA(z)\\
&\leq& \frac{\pi}{2}\|f\|_{\textrm{BMO}_\textrm{H}(\mathbb{D})}.
\end{eqnarray*}
By Lemma \ref{L:BMO}, $\|f\|_{\textrm{BMO}_\textrm{C}(\mathbb{D})}\leq \pi\|f\|_{\textrm{BMO}_\textrm{H}(\mathbb{D})}.$
Next, we show that the mapping $i: \textrm{BMO}_{\textrm{H}}(\mathbb{D})\to \textrm{BMO}_\textrm{C}(\mathbb{D})$ is onto. To do so, we need the following extension theorem of Jones which involves quasicircles. A Jordan curve $\gamma$ in the plane is called a $K$-quasicircle if there is a $K$-quasiconformal mapping $\varphi: \mathbb{C}\to\mathbb{C}$ such that $\varphi(\mathbb{T})=\gamma$.
In \cite{Jones1980}, Jones proved
\begin{lemma}\label{F:Jones}
Let $\Omega$ be a bounded planar domain whose boundary is a $K$-quasicircle.
If $f\in \emph{BMO}_\emph{C}(\Omega)$, then there exists an $\tilde{f}\in \emph{BMO}_\emph{C}(\mathbb{C})$ such that
\begin{itemize}
\item[\emph{(i)}] $\tilde{f}(z)=f(z), z\in\Omega$, and
\item[\emph{(ii)}] $\|\tilde{f}\|_{\emph{BMO}_\emph{C}(\mathbb{C})}\leq c\|f\|_{\emph{BMO}_\emph{C}(\Omega)}$, where $c=c(K)$ is a constant depending only on $K$.
\end{itemize}
\end{lemma}
\noindent Let us continue with the proof of Lemma \ref{L:BMO1}.
If $0<r<1+|z_0|$, observe that $D^h(z_0,r)\subset Q(z_0, 2r)$, then we set $Q(z_0)=Q(z_0, 2r)$. Otherwise, since $D^h(z_0, r)=\mathbb{D}$, we take $Q(z_0)=Q(0, 2)$. Then
\begin{eqnarray*}
\frac{1}{|D^h(z_0,r)|}\int_{D^h(z_0,r)}|f(z)-
E_{Q(z_0)}\tilde{f}|dA(z)
&\leq&\frac{1}{|D^h(z_0,r)|}\int_{Q(z_0)}|\tilde{f}(z)-E_{Q(z_0)}\tilde{f}|dA(z)\\
&\leq& \frac{32}{\pi}\|\tilde{f}\|_{\textrm{BMO}(\mathbb{C})}\\
&\leq&\frac{32c}{\pi}\|f\|_{\textrm{BMO}_\textrm{C}(\mathbb{D})}.
\end{eqnarray*}
So by Lemma \ref{L:BMO}, it follows $\|f\|_{\textrm{BMO}_\textrm{H}(\mathbb{D})}\leq \frac{64c}{\pi}\|f\|_{\textrm{BMO}_\textrm{C}(\mathbb{D})}$. The proof of Lemma \ref{L:BMO1} is complete now.
\end{proof}
\subsection{Proof of Theorem \ref{T:Shexpra}}
\begin{proof}
Part (ii) follows from Theorem \ref{T:boundforC} and the so-called Kolmogorov inquality.
It is based on the factorization \begin{equation}\label{E:factorizarionP}
P_\varphi=C_\varphi\circ I_\varphi,
\end{equation} where
$I_\varphi f(z)=\int_{\mathbb{D}}\frac{f(w)}{(1-z\overline{\varphi(w)})^2}dA(w).$
Since $\widetilde{\varphi}^{-1}$ is Lipschitz on $\mathbb{T}$, by the remark after the proof of Theorem \ref{T:boundforCC}, $C_\varphi: L_a^p(\mathbb{D})\to L^p(\mathbb{D})$ is bounded for all $0<p<\infty$.
Thus $\|P_\varphi (f)\|_{L^p(\mathbb{D})} \lesssim \|I_\varphi (f)\|_{L^p(\mathbb{D})}.$
Let $g(z)=f\circ\varphi^{-1}(z)J(z, \varphi^{-1})$, so $I_\varphi f(z)=P(g)(z).$ Now the Kolmogorov inequality \cite{DHZZ} for $P$ completes the proof.
\bigskip \noindent Next, we prove Part (i) and Part (iii). We shall use the interpolation theorem regarding BMO functions on homogeneous spaces.
\begin{lemma}\textsc{(\cite{DY2005})} \label{F:interpolation}
Let $\mathcal{X}$ be a space of homogeneous type. Let $1<p<\infty$.
If a linear operator $T$ is bounded on $L^p(\mathcal{X})$, and if it is bounded from $L^\infty(\mathcal{X})$ to $\emph{BMO}_\emph{H}(\mathcal{X})$, then $T:L^q(\mathcal{X})\to L^q(\mathcal{X})$ is bounded for all $p<q<\infty$.
\end{lemma}
\smallskip \noindent By Theorem $\ref{T:boundforC}$, $C_\varphi$ is bounded from $L_a^2(\mathbb{D})$ into $L^2(\mathbb{D})$, hence $P_\varphi$ bounded on $L^2(\mathbb{D})$. Now by standard interpolation and duality arguments, it is sufficient to prove that $P_\varphi: L^\infty(\mathbb{D})\to \textrm{BMO}_\textrm{H}(\mathbb{D})$ is bounded.
We need the following special case of Reimann's theorem \cite{Reimann1974}.
\begin{lemma} \label{L:Reimann}Let $\varphi$ be a quasiconformal mapping over $\mathbb{D}$ with $\varphi(0)=0$. If $f\in \emph{BMO}_{\emph{C}}(\mathbb{D})$, then
$$\|f\circ \varphi\|_{\emph{BMO}_\emph{C}}\leq c\|f\|_{\emph{BMO}_\emph{C}},$$
where $c=c(K)$ depends only on $K$.
\end{lemma}
\noindent So by Reimann's theorem Theorem \ref{T:BMO}, we only need to show that $I_\varphi: L^\infty(\mathbb{D})\to \textrm{BMO}_\textrm{H}(\mathbb{D})$ is bounded. We recall the Bloch space consists of holomorphic functions on $\mathbb{D}$ such that $\|f\|_{\beta}=\sup_{z\in\mathbb{D}} (1-|z|^2)|f'(z)|<\infty.$
Let $\textrm{BMOA}(\mathbb{D})=\textrm{Hol}(\mathbb{D})\cap \textrm{BMO}_{\textrm{H}}(\mathbb{D})$.
A theorem of Coifman-Rochberg-Weiss (\cite{CRW1976}, p. 632) states that for each $f\in \mathcal{B},\|f\|_{\beta}\approx\|f\|_{\emph{BMOA}(\mathbb{D})}$.
So the proof of Part (i) is reduced to show that $\textrm{I}_\varphi:L^\infty(\mathbb{D})\to \mathcal{B}$ is bounded.
\bigskip \noindent Observe that for any $f\in L^\infty(\mathbb{D})$, $I_\varphi f$ is analytic on $\mathbb{D}$. For any $z\in\mathbb{D}$,
\begin{eqnarray*}\label{s5eq01}
|(I_\varphi f)'(z)|
&\leq&2\int_{\mathbb{D}}\frac{|f(w)\varphi (w)|}{|1-\overline{z}\varphi(w)|^3}dA(w)\\
& \leq &
2\|f\|_{L^\infty} \|C_\varphi\|_{L_a^3(\mathbb{D})\to L^3(\mathbb{D})}^3 \int_{\mathbb{D}}\frac{1}{|1-z\bar{w}|^3}dA(w).
\end{eqnarray*}
Observe that $\|C_\varphi\|_{L_a^3(\mathbb{D})\to L^3(\mathbb{D})}^3<\infty$ by the remark after the proof of Theorem \ref{T:boundforCC}.
Now we can conclude the proof by a standard estimate (\cite{zhu2007}, Lemma 3.10), which is recorded below for the reader's convenience.
\begin{lemma}\label{L:integralestimates}
Let $z\in \mathbb{D}, c>0$, $t>-1$, and
$
I_{c,t}(z)=\int_{\mathbb{D}}\frac{(1-|w|^2)^t}{|1-z\bar{w}|^{2+t+c}}dA(w).
$
Then
$
I_{c,t}(z)\sim\frac{1}{(1-|z|^2)^c} \ \textrm{as} \ |z|\to 1^{-}.
$
\end{lemma}
\end{proof}
\section{Concluding Remarks: Operator-theoretic Questions}\label{Subs:OTquestions}
In this section we discuss some questions which are part of the focus for composition operators when the symbol $\varphi: \mathbb{D} \to \mathbb{D}$ is analytic; in particular, we discuss the characterization of compactness, Schatten class membership, and conditions for closed ranges.
They are relatively easy now after we have laid the ground work in the previous sections.
The real challenge on these operator-theoretic properties probably only arises when one deals with quasiregular symbols in the future.
For this reason the present section discusses some results, without proofs, to illustrate the parallelism between quasiconformal and analytic symbols, which suggests that the quasiconformal extension is a feasible idea since a decent theory can be expected.
The proofs should present no serious difficulty for anyone reasonably skilled in analytic composition operator theory.
{By Theorem \ref{T:compactforC}, $C_\varphi:L^p_a(\mathbb{D})\to L^p(\mathbb{D})$ is compact for some $0<p<\infty$ if and only if $\mathop{\lim}\limits_{|z|\to 1} m_\varphi(z)=0.$ Now we let $\mathcal{S}_p$ ($0<p<\infty$) denote the Schatten class of bounded operators between $L_a^2(\mathbb{D})$ and $L^2(\mathbb{D})$. Using ideas of Luecking and Zhu (Theorem 3, \cite{LZ1992}) in analytic settings, one can prove that the quasiconformal composition operator $C_\varphi$ on $L_a^2$ is in $\mathcal{S}_p$ if and only if for some $r \in (0, 1)$, we have
$$
\int_{r<|z|<1}\bigg(\frac{N_\varphi(z)}{\log\frac{1}{|z|}}\bigg)^pd\lambda(z)<\infty,
$$ where $d\lambda(z)$ is the Mobius-invariant measure given by
$
d\lambda(z)=\frac{1}{(1-|z|^2)^2}dA(z).
$
Next we consider when $C_\varphi$ has a closed range. In other words, we are concerned with when $C_\varphi$ is bounded below. With an application of reverse Carleson measures on the Bergman space and Lemma \ref{L:areachangephd}, one can show that $C_\varphi$ is bounded below on $L^p_a(\mathbb{D})$ if and only if $\widetilde{\varphi}$ is Lipschitz on $\mathbb{T}$. This should be compared with the classical result of Akeroyd and Ghatage \cite{AG}.} Moreover, Luecking-type conditions \cite{Luecking1981}, \cite{Luecking1985} can be used to provide sufficient conditions for closed ranges.
\bigskip \noindent \textbf{Acknowledgement}
\bigskip \noindent X. Fang is supported by MOST of Taiwan (106-2115-M-008 -001 -MY2) and NSFC 11571248 during his visit to Soochow University in China.
K. Guo is supported by NSFC (11371096).
Z. Wang is supported by NSFC (11601296
, NSF of Shaanxi (2017JQ1008) and MOST 105-2811-M-008-029 during his visit to National Central University in Taiwan.
\bigskip \noindent
The project of using quasiconformal symbols for composition operators on the Hardy or Bergman spaces was first conceived (for us) in 2011 when the first author was still at Kansas State University, after several discussions with his colleague (Pietro Poggi-Corradini), to whom we are most deeply grateful. The first draft of this paper was completed and reported in the 2014 Chongqing Analysis Meeting. We are grateful to Richard Rochberg for bringing his paper \cite{rochberg1994} to our attention after our talk and giving us other suggestions. Several drafts of this paper have been circulated in the last few years and the current version is considerably shortened.
|
{
"timestamp": "2018-04-17T02:10:21",
"yymm": "1804",
"arxiv_id": "1804.05352",
"language": "en",
"url": "https://arxiv.org/abs/1804.05352"
}
|
\section{Introduction}
Childhood obesity prevention has become increasingly important to control the global obesity pandemic. Granular-level surveillance of childhood obesity that identifies and tracks obesity trends is needed to
help design interventions and guide policy solutions when monetary resources are limited. \citep{Longjohn2010}.
Several efforts have been made to construct local surveillance systems \citep{Hoelscher2017}, which are primarily school-based \citep{Blondin2016}.
Although school-based surveillance can be effective for data collection and implementation, there are concerns about privacy, stigmatization, and dysfunctional behavioral responses \citep{Mass2014}. Alternatively, routinely collected massive health databases such as Electronic Health Records (EHRs) are gaining attention as a platform for assessing trends and local childhood obesity risk \citep{Friedman2013}.
Statistical methods for geospatial surveillance may include two aspects: i) monitoring regional trends in prevalence (also known as ``disease mapping'' or ``risk mapping'') and ii) identifying unexpected variation in the prevalence of different locations (also known as ``hot spot detection'').
Traditionally, these two tasks have been accomplished separately.
For task i),
obesity literature mainly utilized the standard generalized linear mixed effect model (GLMM) to account for individual factors and community environments \citep{Zhang2011, Davila-Payan2015}.
Those approaches assumed the regional random effects to be independent, although a spatial dependency exists even after adjusting for covariates \citep{Panczak2016}.
To account for the spatial dependence, methods for smooth disease mapping have been proposed from both frequentist and Bayesian perspectives.
Under Poisson log-linear models or multilevel logistic models, the region-specific effects were smoothed by kernels \citep{Ghosh1999} or splines \citep{Ugarte2010,Maiti2016}, or were modeled as a dependent random vector by conditional autoregressive (CAR) priors \citep{Waller1997,Pascutto2000,Lee2013a,Mercer2015}.
These strategies resulted in ``clustered'' risk maps, which enhanced interpretability, but did not explore identification of aberrant {regions}.
For task ii),
the most popular approach for detecting locations with outbreaking incidence is the spatial scan statistics approach \citep{Kulldorff1995, Kulldorff1997, Jung2009}.
The scan statistic methods search over a pre-specified set of geographical districts and conduct a generalized likelihood ratio test for testing whether the proportions of events are homogeneous across, inside, and outside the district.
However, it may not be suitable for identifying multiple locations with heterogeneous sizes. Residuals generated from regression approaches can also be used to detect regional outbreaks \citep{Kafadar1992, Farrington1996, Zhao2011}.
However, residual-based outlier detection is known to fail when an outlier is a leverage point or there are multiple outliers \citep{She2011}.
Use of the fusion penalty for smoothing was first proposed in a least squares setup \citep{Tibshirani2005}. The resulting fit from the fusion penalty appears to be piecewise constant, yielding a natural clustering of fitted values.
Smoothing by the fusion penalty enables an additional regularization using a different penalty, such as a sparse penalty, which may not be straightforward in other smooth disease mapping methods.
Sparse penalty for outlier detection was used with the squared error loss \citep{Kim2009, Tibshirani2011a, She2011}. \cite{Kim2009} and \cite{Tibshirani2011a} considered the $\ell_1$ penalty, and \cite{She2011} reported that nonconvex penalties outperformed both $\ell_1$ penalty and residual-based approaches for detection in standard multiple linear regression.
In this paper, we develop a new method that simultaneously produces an interpretable disease map and {detects outlier regions}.
We formulate a multilevel logistic model to naturally incorporate risk factors.
A novel hybrid regularization is introduced, where the region-specific effect is represented by the summation of a smooth signal and a sparse signal.
The smooth signal is regularized by a fusion penalty so that adjacent locations tend to have similar fitted baseline obesity rates.
A nonconvex sparse penalty is enforced for the sparse signals so that nonzero fitted coefficients signify potential outliers.
It is worthwhile mentioning that
estimating population health metrics from EHRs can be particularly challenging. First, subjects in EHRs systems are not randomly sampled. EHRs only capture people seeking healthcare.
Second, recorded data often suffers from missingness,
because for each patient, EHRs only collect data on the tests and conditions that clinicians order and diagnose.
Thus, statistical models utilizing EHRs should deal with missingness and non-representativeness.
Following \cite{Flood2015}, we adopt a two-step weighting procedure to account for missing data and to adjust the covariate distribution for a nationally representative sample.
We develop an alternating minimization algorithm to optimize a nonconvex objective function, which is computationally efficient and can leverage off-the-shelf software packages.
Our original contributions are twofold. First, while the hybrid regularization of the fusion and $\ell_1$ penalties has been considered in linear models \citep{Kim2009,Tibshirani2011a}, to the best of our knowledge, we are the first to incorporate a fusion penalty and a nonconvex penalty to identify outliers.
{Second, we provide an efficient optimization algorithm that guarantees convergence for the hybrid regularization model.}
{Although our algorithm is described in a Bernoulli likelihood, it can be easily extended to handle other convex loss functions.}
In Section \ref{sec:data}, we introduce the University of Wisconsin Electronic Health Record Public Health Information Exchange (PHINEX) database that motivated our study.
We introduce our model and formalize the objective function in Section \ref{sec:method}. We also develop a computational algorithm and discuss tuning parameter selection in this section.
Simulation studies are presented in Section \ref{sec:simul}, which demonstrate the superior performance of our proposed method.
We apply our method to PHINEX on childhood obesity surveillance in Section \ref{sec:application}. We provide concluding remarks in Section \ref{sec:conclusion}.
\section{Data}\label{sec:data}
The University of Wisconsin Electronic Health Record Public Health Information Exchange (UW eHealth PHINEX) database contains EHR data from a south-central Wisconsin academic healthcare system. It consists of patient records with documented primary care encounters at family medicine, pediatric, and internal medicine clinics occurring from 2007 to 2012. All PHINEX data were derived from the Epic EHR Clarity Database (EpicCare Electronic Medical Record, Epic Systems Corp., Verona WI). Furthermore, the program geocodes to the census blockgroup and links EHRs with community-level social determinants of health. It was created to improve clinical practice and population health by understanding local variations in disease risk, patients, and communities \citep{guilbert2012}.
In this paper, we focused on 93,130 patients aged 2--19 years during 2011--2012. Body mass index (BMI) values (in $kg/m^2$) were calculated from a subject's height and weight, measured at the same visit.
Any subject with a BMI at or above the 95th percentile was categorized as obese.
Among all the patients, 34,852 (37.4\%) were missing a valid BMI.
Subject covariates included sex, age, race/ethnicity, health service payer (i.e., insurance), and the 2010 census blockgroup information on subject residence. Economic hardship index (EHI) \citep{Nathan1989} was used as a measure of blockgroup socioeconomic status, and was calculated from a blockgroup's: \% of housing units with more than one person per room; \% of households below the federal poverty level; \% of people \textgreater 16 years of age who are unemployed; \% of people \textgreater 25 years of age without a high school education; \% of people \textless 18 or \textgreater 64 years of age; and per capita income. EHI was normalized for all Wisconsin census blockgroups.
The values were continuous, ranging from 0 to 100, with larger values indicating greater hardship.
Urbanicity of a census blockgroup was based on its 11 Urbanization Summary Groups, according to \cite{ESRI2012}.
These groups were derived from data on census blockgroup population density, city size, proximity to metropolitan areas, and economic/social centrality.
Urbanicity integer values ranged from 1 (the most urban) to 11 (the most rural).
\section{Method}\label{sec:method}
\subsection{Model setup}\label{subsec:model}
We use a double subscript, $ij$ ($j=1,\ldots,n_i$, $i=1,\ldots,K$) to indicate the $j$-th subject in the $i$-th region. Let $\boldsymbol{S}_i$ be the location of the $i$-th region. Let $\boldsymbol{X}_i$ denote the region-level covariates such as urbanicity and EHI.
Let $Y_{ij}$ be the obese indicator of the $(ij)$-th subject, with $Y_{ij} = 1$ indicating obese. Lastly, let $\boldsymbol{Z}_{ij}$ be a vector of the covariates of the $(ij)$-th subject such as gender, age, race/ethnicity and insurance payor.
Let $p_{ij} = \mathbb{P}(Y_{ij} = 1| \boldsymbol{Z}_{ij}, \boldsymbol{X}_i)$. We formalize our model for the $p_{ij}$ as
\begin{gather}
{\rm logit}(p_{ij}) =
\boldsymbol{Z}_{ij}^T \boldsymbol{\alpha}_1 + \boldsymbol{X}_{i}^T \boldsymbol{\alpha}_2 + \beta_i + \gamma_i,
\label{eqn:model} \\
\mbox{subject~to} \qquad \sum_{i_1 < i_2} \rho_{i_1,i_2} |\beta_{i_1} - \beta_{i_2}| \leq c_1 ;
\label{eqn:const1} \\
\qquad\qquad \sum_{i=1}^K I(\gamma_i \neq 0 ) \leq c_2,
\label{eqn:const2}
\end{gather}
where $c_1, c_2 \geq 0$, ${\rm logit}(t) = \log \{t / (1-t)\}$, and $I(\cdot)$ is the indicator function. Here, $\beta_i$ represents a regional-specific effect for the $i$-th region that is not explained by $\boldsymbol{X}_i$. Since the probability of a child being obese might be affected by the community environment he or she resides in, we expect the regional contribution on the obesity prevalence to be similar for individuals in closer locations \citep{Panczak2016}, and thus a smoothness constraint $\eqref{eqn:const1}$ is imposed on $\beta_i$.
The fusion weight $\rho_{i_1,i_2}$ ($\rho_{i_1,i_2} \geq 0$) represents the strength of the ``fusion'' for each pair of $i_1$ and $i_2$. A higher value of $\rho_{i_1,i_2}$ will lead to a more similar pair of the fitted $\beta_{i_1}$ and $\beta_{i_2}$.
With an appropriate choice of tuning parameter, the values that $\beta_i$ could take are limited, where similar locations are grouped together.
We may interpret the distinct levels of $\beta_i$ as segmentation or clustering of the regions. We also note that \eqref{eqn:const1} can be seen as a two-dimensional generalization of the total variation constraint used in the fused lasso \citep{Tibshirani2005}.
$\gamma_i$ is introduced to capture potential aberrant regions, where the $i$-th region is an outlier with unusual obesity prevalence if $\gamma_i \neq 0$. Given the sparsity constraint \eqref{eqn:const2}, we expect $\gamma_i$ will be zero (non-outlier) for most regions, but a few might be nonzero (outliers).
\subsection{Estimation with complete data}
Denote by $N = \sum_{i=1}^K n_i$ and define $\boldsymbol{\alpha} = (\boldsymbol{\alpha}_1^T, \boldsymbol{\alpha}_2^T)^T$, $\boldsymbol{\beta} = (\beta_1, \ldots, \beta_K)^T$ and $\boldsymbol{\gamma} = (\gamma_1, \ldots, \gamma_K)^T$. If all patients had complete records, the parameters could be estimated by a penalized logistic likelihood, where $(\widehat{\boldsymbol{\alpha}}, \widehat{\boldsymbol{\beta}}, \widehat{\boldsymbol{\gamma}}) = \operatornamewithlimits{argmin}_{\boldsymbol{\alpha}, \boldsymbol{\beta}, \boldsymbol{\gamma}} \phi(\boldsymbol{\alpha}, \boldsymbol{\beta}, \boldsymbol{\gamma})$, and the objective function $\phi$ is defined by
\begin{equation}\label{eqn:objFn}
\phi(\boldsymbol{\alpha}, \boldsymbol{\beta}, \boldsymbol{\gamma}) =
-{\rm loglik} (\boldsymbol{\alpha}, \boldsymbol{\beta}, \boldsymbol{\gamma}) +
P_{\lambda_1}(\boldsymbol{\beta}) + Q_{\lambda_2}(\boldsymbol{\gamma}).
\end{equation}
The normalized negative log-likelihood function is
\begin{eqnarray}
- {\rm loglik} (\boldsymbol{\alpha}, \boldsymbol{\beta}, \boldsymbol{\gamma})
&=&
\frac{1}{N} \sum_{i=1}^K \sum_{j=1}^{n_i} \bigg[
\log\{1 + \exp({\boldsymbol Z}_{ij}^T \boldsymbol{\alpha}_1 +
{\boldsymbol X}_i^T \boldsymbol{\alpha}_2 + \beta_i + \gamma_i)\} \nonumber \\
&& \qquad - Y_{ij} ({\boldsymbol Z}_{ij}^T \boldsymbol{\alpha}_1 +
{\boldsymbol X}_i^T \boldsymbol{\alpha}_2 + \beta_i + \gamma_i) \bigg]. \label{eqn:loglik}
\end{eqnarray}
The second term $P_{\lambda_1}(\boldsymbol{\beta})$ is a fusion penalty that stems from the Lagrangian of \eqref{eqn:const1}, where
$
P_{\lambda_1}(\boldsymbol{\beta}) = \lambda_1 \sum_{i_1 < i_2} \rho_{i_1,i_2} |\beta_{i_1} - \beta_{i_2}|.
$
We use $\rho_{i_1,i_2} = 1/ d(\boldsymbol{S}_{i_1}, \boldsymbol{S}_{i_2})$, where the $d(\boldsymbol{S}_{i_1}, \boldsymbol{S}_{i_2})$ denotes a distance between $\boldsymbol{S}_{i_1}$ and $\boldsymbol{S}_{i_2}$. Here, geodistance is used to define {$d(\cdot,\cdot)$} but other measures of similarity can be employed.
Without loss of generality, we assume $\max_{i_1, i_2} \rho_{i_1, i_2} = 1$, otherwise we can normalize it by redefining $\rho_{i_1, i_2}$ with $\rho_{i_1, i_2} / \max_{i_1, i_2} \rho_{i_1, i_2}$.
Since the computational cost of the optimization involving fusion penalty increases quadratically in the number of nonzero $\rho_{i_1,i_2}$'s,
one may want to retain a few $\rho_{i_1, i_2}$s with large values and truncate the others at zero for ease of computation.
The third term, $Q_{\lambda_2}(\boldsymbol{\gamma}) = \sum_{i=1}^K n_{i} q_{\lambda_2}(\gamma_i)/N$, is a sparse penalty that is a relaxation of the Lagrangian of \eqref{eqn:const2}, where $q_{\lambda}(\cdot)$ is a univariate penalty function. In particular, we consider the hard penalty function as proposed in \cite{She2011},
$
q_{\lambda} (t) = (\lambda |t| - t^2/2 ) I(t < \lambda) + {\lambda^2}/{2} I(t \geq \lambda).
$
The hard penalty results in a nonconvex formulation on \eqref{eqn:objFn}, which guarantees convergence to a local minima.
We weigh the $i$-th penalty in $Q_{\lambda_2}(\boldsymbol{\gamma}) $ by $n_i$ such that subjects across different regions are penalized with the same amount.
\subsection{Optimization algorithm}\label{subsec:optim}
We developed an alternating minimization algorithm. It alternately updates $\boldsymbol{\alpha}$, $\boldsymbol{\beta}$, and $\boldsymbol{\gamma}$, each time minimizing one of them while keeping the others fixed. Denote the current iterates by $\boldsymbol{\alpha}^{(t)}$, $\boldsymbol{\beta}^{(t)}$, and $\boldsymbol{\gamma}^{(t)}$. In addition, we denote $\boldsymbol{Q}_{ij} = (\boldsymbol{Z}_{ij}^T, \boldsymbol{X}_i^T)$. Then $\boldsymbol{Q}_{ij}^T \boldsymbol{\alpha} = \boldsymbol{Z}_{ij}^T \boldsymbol{\alpha}_1 + {\boldsymbol X}_i^T \boldsymbol{\alpha}_2$.
\paragraph{\bf Updating $\boldsymbol{\alpha}$} Fix $\boldsymbol{\beta} = \boldsymbol{\beta}^{(t)}$ and $\boldsymbol{\gamma} = \boldsymbol{\gamma}^{(t)}$. The objective function is equivalent to
\[
\phi \left(\boldsymbol{\alpha}, \boldsymbol{\beta}^{(t)}, \boldsymbol{\gamma}^{(t)}\right) = \frac{1}{N} \sum_{i=1}^K \sum_{j=1}^{n_i}
\left[ \log\{1 + \exp \left(\boldsymbol{Q}_{ij}^T \boldsymbol{\alpha} + \mu_{ij}^{(t)} \right)\}
- Y_{ij} \left(\boldsymbol{Q}_{ij}^T \boldsymbol{\alpha} + \mu_{ij}^{(t)} \right) \right]
\]
with $\mu_{ij}^{(t)} = \beta_i^{(t)} + \gamma_i^{(t)}$, which corresponds to a classical logistic regression on $N$ individuals. One can run standard packages (such as \texttt{glm} in \texttt{R}) to obtain $\boldsymbol{\alpha}^{(t+1)}$.
\paragraph{\bf Updating $\boldsymbol{\beta}$} Fix $\boldsymbol{\alpha} = \boldsymbol{\alpha}^{(t+1)}$ and $\boldsymbol{\gamma} = \boldsymbol{\gamma}^{(t)}$, then
\begin{gather}
\phi \left(\boldsymbol{\alpha}^{(t+1)}, \boldsymbol{\beta}, \boldsymbol{\gamma}^{(t)}\right)
=
\underbrace{\frac{1}{N} \sum_{i=1}^K \sum_{j=1}^{n_i} \left[
\log \left\{1 + \exp \left(\beta_i + \theta_{ij}^{(t)}\right)\right\}
- Y_{ij} \left(\beta_i + \theta_{ij}^{(t)} \right) \right]}_{=: ~l(\boldsymbol{\beta})} \nonumber \\
\qquad + \, \lambda_1 \sum_{i_1 < i_2} \rho_{i_1,i_2} |\beta_{i_1} - \beta_{i_2}|, \nonumber
\end{gather}
where $\theta_{ij}^{(t)} = \boldsymbol{Q}_{ij}^T \boldsymbol{\alpha}^{(t+1)} + \gamma_i^{(t)}$ for each $i$ and $j$.
For simplicity, define $\psi(\boldsymbol{\beta}) = \phi \left(\boldsymbol{\alpha}^{(t+1)}, \boldsymbol{\beta}, \boldsymbol{\gamma}^{(t)}\right)$, which is convex in $\boldsymbol{\beta}$. To update $\boldsymbol{\beta}^{(t)}$, we propose to minimize a surrogate objective function in which $l(\boldsymbol{\beta})$ is replaced by its local quadratic approximation around $\boldsymbol{\beta}^{(t)}$.
The same strategy was applied in implementing \texttt{R} package \texttt{glmnet} to iteratively descend the objective function of the generalized linear model with the elastic-net penalty \citep{Friedman2010a}.
Write the second-order Taylor expansion of $l(\boldsymbol{\beta})$ at $\boldsymbol{\beta}^{(t)}$ as
\[
\tilde{l}(\boldsymbol{\beta};\boldsymbol{\beta}^{(t)}) = l(\boldsymbol{\beta}^{(t)}) + \nabla_{\boldsymbol{\beta}} l(\boldsymbol{\beta}^{(t)})^T (\boldsymbol{\beta} - \boldsymbol{\beta}^{(t)}) + \frac{1}{2} (\boldsymbol{\beta} - \boldsymbol{\beta}^{(t)})^T \nabla_{\boldsymbol{\beta}\boldsymbol{\beta}}^2 l(\boldsymbol{\beta}^{(t)}) (\boldsymbol{\beta} - \boldsymbol{\beta}^{(t)}),
\]
where $\nabla_{\boldsymbol{\beta}}$ and $\nabla_{\boldsymbol{\beta}\boldsymbol{\beta}}^2$ are the first and the second derivative operators with respect to $\boldsymbol{\beta}$. Define the surrogate objective function as
$
\tilde{\psi}(\boldsymbol{\beta};\boldsymbol{\beta}^{(t)}) = \tilde{l}(\boldsymbol{\beta};\boldsymbol{\beta}^{(t)}) + P_{\lambda_1}(\boldsymbol{\beta}).
$
We calculate $\tilde {\boldsymbol \beta} = \operatornamewithlimits{argmin}_{\boldsymbol{\beta}} \tilde{\psi}(\boldsymbol{\beta}; \boldsymbol \beta^{(t)})$, where
\[
\tilde {\boldsymbol \beta} = \operatornamewithlimits{argmin}_{\boldsymbol{\beta}} \left[
\frac{1}{2} \sum_{i=1}^K a_i^{(t)} \left(\beta_i - b_i^{(t)} \right)^2
+ \lambda_1 \sum_{i_1 < i_2} \rho_{i_1,i_2} |\beta_{i_1} - \beta_{i_2}|
\right],
\]
with
\begin{gather}
a_i^{(t)} = \sum_{j=1}^{n_i}
\frac{ \exp\left( \beta_i^{(t)} + \theta_{ij}^{(t)} \right)}
{ \left\{ 1 + \exp\left( \beta_i^{(t)} + \theta_{ij}^{(t)} \right) \right\}^2 }; \quad
b_i^{(t)} = \beta_i^{(t)} - \frac{1}{a_i^{(t)}} \sum_{j=1}^{n_i} \left[
\frac{ \exp\left( \beta_i^{(t)} + \theta_{ij}^{(t)} \right)}
{ 1 + \exp\left( \beta_i^{(t)} + \theta_{ij}^{(t)} \right) }
- Y_{ij} \right]. \nonumber
\end{gather}
{For the calculation of $\tilde {\boldsymbol \beta}$, w}e applied the majorization-minimization algorithm proposed by \cite{Yu2015}, which yields a stable solution and can be easily implemented.
To ensure $\psi (\boldsymbol{\beta}^{(t)}) \geq \psi (\tilde{\boldsymbol{\beta}})$, we adopt \cite{Lee2016d}'s one-step modification of $\tilde{\boldsymbol{\beta}}$: if $\psi (\boldsymbol{\beta}^{(t)}) \geq \psi (\tilde{\boldsymbol{\beta}})$, let $\boldsymbol{\beta}^{(t+1)} = \tilde{\boldsymbol{\beta}}$; otherwise, $\boldsymbol{\beta}^{(t+1)} = \tilde h\tilde{\boldsymbol{\beta}} + (1-\tilde h)\boldsymbol{\beta}^{(t)},$
where
$
\tilde{h} = \operatornamewithlimits{argmin}_{h \in [0,1]} \psi\left(h\tilde{\boldsymbol{\beta}} + (1-h)\boldsymbol{\beta}^{(t)}\right).
$
We will show in Proposition \ref{prop:1} that $\tilde{h}$ always exists and {$\psi (\boldsymbol{\beta}^{(t)}) \geq \psi (\boldsymbol{\beta}^{(t+1)})$} holds over iterations.
\paragraph{\bf Updating $\boldsymbol{\gamma}$} Given that $\boldsymbol{\alpha} = \boldsymbol{\alpha}^{(t+1)}$ and $\boldsymbol{\beta} = \boldsymbol{\beta}^{(t+1)}$,
\begin{gather}
\phi \left(\boldsymbol{\alpha}^{(t+1)}, \boldsymbol{\beta}^{(t+1)}, \boldsymbol{\gamma}\right)
= \frac{1}{N}
\sum_{i=1}^K \sum_{j=1}^{n_i} \left[
\log \left\{1 + \exp\left(\gamma_i + \nu_{ij}^{(t)}\right) \right\}
- Y_{ij} \left(\gamma_i + \nu_{ij}^{(t)}\right) \right] \nonumber \\
\qquad + \frac{1}{N} \sum_{i=1}^K n_i q_{\lambda_2}(\gamma_i), \nonumber
\end{gather}
where $\nu_{ij}^{(t)} = \boldsymbol{Q}_{ij}^T \boldsymbol{\alpha}^{(t+1)} + \beta_i^{(t+1)}$. With a slight abuse of notation, we define a univariate objective function $\phi_i (\gamma)$ and a loss function $l_i (\gamma)$ ($i = 1, \ldots, K$) as
\[
\phi_i (\gamma) = \underbrace{\sum_{j=1}^{n_i} \left[
\log \left\{1 + \exp\left(\gamma + \nu_{ij}^{(t)}\right) \right\}
- Y_{ij} \left(\gamma + \nu_{ij}^{(t)}\right) \right]}_{l_i (\gamma)}
+ \, n_i q_{\lambda_2}(\gamma).
\]
Clearly $\phi \left(\boldsymbol{\alpha}^{(t+1)}, \boldsymbol{\beta}^{(t+1)}, \boldsymbol{\gamma}\right) = N^{-1} \sum_{i=1}^K \phi_i (\gamma_i)$.
Thus, it suffices to optimize $K$ univariate functions $\phi_i (\cdot)$, $i=1, \ldots, K$.
Although each $\phi_i (\gamma)$ is nonconvex, we can find a global optimum of $\phi_i$ as follows.
Let $\tilde{t} = \operatornamewithlimits{argmin}_{t \in \mathbb{R}} l_i (t)$.
Since $q_{\lambda_2}(\cdot)$ is constant outside $[-\lambda_2, \lambda_2]$,
a minimizer of $\phi_i (\cdot)$ either lies on $[-\lambda_2, \lambda_2]$ or equals to $\tilde{t}$.
Hence, we propose a grid search: let $\{t_1, \ldots, t_T\} \subseteq [-\lambda_2, \lambda_2]$ and put $\widehat{\gamma}_i^{(t+1)} = \operatornamewithlimits{argmin}_{\gamma \in \{\tilde{t}, t_1, \ldots, t_T \}} \phi_i (t)$.
Details of the algorithm are provided in the Web Supplementary Materials. The following property is guaranteed by the proposed algorithm.
\begin{proposition}\label{prop:1} Assume that for each $i$, there exist $j_1, j_2$ such that $Y_{ij_1} = 0$ and $Y_{ij_2} = 1$.
For any choice of $\boldsymbol{\alpha}^{(t)}$, $\boldsymbol{\beta}^{(t)}$ and $\boldsymbol{\gamma}^{(t)}$, the updated iterates $\boldsymbol{\alpha}^{(t+1)}$, $\boldsymbol{\beta}^{(t+1)}$ and $\boldsymbol{\gamma}^{(t+1)}$ by Algorithm 1 in the Web Supplementary Materials satisfy a monotone decreasing property:
$
\phi \left(\boldsymbol{\alpha}^{(t)}, \boldsymbol{\beta}^{(t)}, \boldsymbol{\gamma}^{(t)}\right)
\geq \phi \left(\boldsymbol{\alpha}^{(t+1)}, \boldsymbol{\beta}^{(t)}, \boldsymbol{\gamma}^{(t)}\right)
\geq \phi \left(\boldsymbol{\alpha}^{(t+1)}, \boldsymbol{\beta}^{(t+1)}, \boldsymbol{\gamma}^{(t)}\right)
\geq \phi \left(\boldsymbol{\alpha}^{(t+1)}, \boldsymbol{\beta}^{(t+1)}, \boldsymbol{\gamma}^{(t+1)}\right)
$.
\end{proposition}
The proof is deferred to the Web Supplementary Materials. The assumption indicates that the naive prevalence rate $\sum_{j=1}^{n_i} Y_{ij} / n_i$ lies on $(0,1)$ for each $i$, which is crucial to guarantee the existence of the optima at each step.
By Proposition \ref{prop:1}, any limit point of $\{(\boldsymbol{\alpha}^{(t)}, \boldsymbol{\beta}^{(t)}, \boldsymbol{\gamma}^{(t)})\}$ is a stationary point if $\phi$ is continuous. Since the objective function $\phi$ is nonconvex due to the nonconvexity of the hard penalty function, the proposed algorithm can only guarantee the convergence to a local optimum and requires a careful choice of the initial point.
We could use the \emph{warm start strategy}, where the solution under the previous tuning parameter is used as the initial point for the next choice of tuning parameter. We applied this strategy in our simulation and data analysis, which showed a satisfactory performance.
\noindent {\bf Remark.} Although the described algorithm handles a binomial likelihood function, it can be easily extended to other (multilevel) generalized linear models. We can still solve $\boldsymbol{\alpha}$-step using an off-the-shelf package (e.g. \texttt{glm} in \texttt{R}), and $\boldsymbol{\beta}$- and $\boldsymbol{\gamma}$-steps using the same strategies as outlined in the paper.
\subsection{Choice of tuning parameter}
The proposed procedure involves the choice of $\lambda_1$ and $\lambda_2$. We implemented a model selection procedure to tune those parameters. Particularly, we used the modified Bayesian information criterion (BIC) proposed in \cite{She2011},
$
{\rm BIC}^*(\lambda_1, \lambda_2) = - 2 N \cdot {\rm loglik}(\widehat{\boldsymbol{\alpha}}, \widehat{\boldsymbol{\beta}}, \widehat{\boldsymbol{\gamma}}) + {\rm DF} \cdot \left(1 + \log N \right).
$
Here, ${\rm loglik}(\widehat{\boldsymbol{\alpha}}, \widehat{\boldsymbol{\beta}}, \widehat{\boldsymbol{\gamma}})$ is defined in \eqref{eqn:loglik}, and
the degrees of freedom (DF) is calculated by combining the DF calculated in the lasso and fused lasso regressions \citep{Tibshirani2005, Zou2007,Tibshirani2011a}, where
\begin{eqnarray}
{\rm DF} = ({\rm dimension~of~ }\widehat{\boldsymbol{\alpha}}) + ({\rm \#~of~distinct~values~of~}\widehat{\boldsymbol{\beta}}) +({\rm \#~of~nonzero~values~of~} \widehat{\boldsymbol{\gamma}}).\label{eqn:DF}
\end{eqnarray}
We searched for the $(\lambda_1, \lambda_2)$ among a candidate set that minimizes the ${\rm BIC}^*(\lambda_1, \lambda_2)$. An alternative tuning method is cross-validation, which however, is much more computationally demanding. We use the ${\rm BIC}^*(\lambda_1, \lambda_2)$ throughout.
\subsection{Weighting to account for missingness and selection bias}
As indicated in the previous sections, our dataset involves a large number of missing values for the obese indicators ($Y_{ij}$). Furthermore, the data may not be directly comparable to a national sample. For example, in geographic areas and population groups that have traditionally experienced disparities in healthcare access and outcomes, the adoption of EHRs may not be as widespread. These locations could be less represented.
We consider a two-step weighting procedure to adjust for both missing BMI values and selection bias.
The first step is to account for the missingness of BMI. We assumed the missing at random (MAR), where the probability of missing BMI is independent of its response conditional on the covariates \citep{little2014statistical}. Let $R_{ij} = 1$ if $Y_{ij}$ is observed and $R_{ij} = 0$ otherwise. The weight was defined as the inverse probability of observing BMI, $\mathbb{P}(R_{ij}=1 | \boldsymbol Z_{ij}, \boldsymbol X_i)$. This was unknown in practice. Hence we estimated it with a logistic regression using the observed data.
The second step is to adjust for the population distribution of age, sex, and race/ethnicity. We applied a post-stratification correction using the 2012 national census data. The final weight for each subject was the product of the inverse probability weight and the post-stratification weight. The objective function and subsequent procedures are modified subsequently. Details can be found in the Web Supplementary Materials.
\section{Simulation studies}\label{sec:simul}
We compared the proposed method with classic generalized mixed effect model (GLMM) and
the covariate-adjusted spatial scan statistic proposed by \cite{Jung2009} (Scan Statistic).
The GLMM assumes ${\rm logit}(p_{ij}) = \boldsymbol{Z}_{ij}^T \boldsymbol{\alpha}_1 + \boldsymbol{X}_i \boldsymbol{\alpha}_2 + b_i + \delta$ where $b_i \sim \mathcal{N}(0, \sigma^2)$ {and $\delta$ is the global intercept}.
To implement GLMM, we used the function \texttt{glmer()} of \texttt{R} package \texttt{lme4}.
Let $\widehat{b}_i$ be the predicted random effect of the $i$-th region from the fitted model. The $i$-th region was declared as an outlier if $|\widehat{b}_i| > 2.5 \widehat{\sigma}$.
The scan statistic assumes ${\rm logit}(p_{ij}) = \boldsymbol{Z}_{ij}^T \boldsymbol{\alpha}_1 + \boldsymbol{X}_i \boldsymbol{\alpha}_2 + I(i \in S) \theta + \delta$ and searches for $S$, a cluster of regions, such that the null hypothesis of $H_{0}: \theta=0$ is rejected {with the largest likelihood ratio statistic}. We also included three ``oracle'' versions of our methods, where $\phi$ is minimized with respect to one of $\boldsymbol{\alpha}$, $\boldsymbol{\beta}$, or $\boldsymbol{\gamma}$ while the other two are set to their true values: with respect to $\boldsymbol{\alpha}$ (Oracle $\boldsymbol{\alpha}$); with respect to $\boldsymbol{\beta}$ (Oracle $\boldsymbol{\beta}$); and with respect to $\boldsymbol{\gamma}$ (Oracle $\boldsymbol{\gamma}$).
We considered $K$ $(K=20, 40)$ regions where the number of subjects in each region was $n$ $(n=50, 100)$.
We generated $\boldsymbol{Q}_{ij} = (Z_{ij}, X_i)^T$, where $Z_{ij}$ and $X_i$ were drawn from Bernoulli distributions with a probability of 0.5. We set $\boldsymbol{\alpha} = (\alpha_1, \alpha_2) = (-0.2, 0.2)$. For simplicity, we simulated $K$ locations on a one-dimensional line with $S_i \sim {\rm Unif}(5, 95)$, $i=1,\ldots,K$.
$\beta_i$ was set to ${\rm logit}(0.4)$ if $5 \leq S_i < 35$, ${\rm logit}(0.5)$ if $35 \leq S_i < 65$, and ${\rm logit}(0.6)$ if $65 \leq S_i \leq 95$. We randomly chose $K_O$ regions, where $\gamma_i = 2$ for $\lfloor K_0 / 2 \rfloor$ regions ($\lfloor t \rfloor$ is the maximum integer no larger than $t$) and $\gamma_i = -2$ for the remaining. Thus, those regions with $\gamma_i \neq 0$ should be regarded as the true outlier regions.
We varied the number of outliers so that $K_O/K = 0\%, 5\%, 10\%, 15\%$.
For each scenario, we repeatedly generated 1000 datasets. We applied different methods on each dataset and evaluated the performance metrics. The performance measures were averaged over the 1000 replications. The tuning parameters were selected using the proposed modified BIC, among a pre-defined candidate set with $\lambda_1 \in [2^{-2}, 2^{12}]$ and $\lambda_2 \in [2^{-5}, 2^{2}]$.
We compared the proposed method with GLMM and Oracle $\boldsymbol{\alpha}$ in terms of the bias of the individual-level covariate effect, $\widehat{\alpha}_1 - \alpha_1$, community-level covariate effect, $\widehat{\alpha}_2 - \alpha_2$, and the root mean squared error (RMSE) of the region-level prevalence rates, $\sqrt{\sum_{i=1}^{K}(\widehat{p}_i - p_i)^2 / K}$. Here, $p_i = \mathbb{E}(Y_{ij}|\boldsymbol{X}_i)$ and $\widehat{p}_i$ is the empirical average of the estimated individual-level prevalence estimates, $\widehat{p}_{ij}$, taken over $j=1,\ldots, n_i$. The results are presented in Table \ref{tbl:trad}. The biases of estimating $\alpha_1$, the individual-level covariate effect, were close to zero in the proposed method, especially when both $n$ and $K$ increase. The confidence intervals from the proposed and GLMM were comparable.
The biases of $\widehat{\alpha}_2$, the region-level covariate effect, were reduced as $K$ increased in the proposed. The performances of the proposed and the method Oracle $\boldsymbol{\alpha}$ were similar in terms of the biases. In addition, they were barely affected by the increased proportion of outliers.
On the other hand, $\widehat\alpha_2$ in the GLMM
had increased variability with a larger proportion of outliers.
The RMSEs of $\widehat{p}_i$ were smaller in the proposed than in the GLMM,
although as anticipated, larger than the case where only $\boldsymbol{\alpha}$ was unknown.
\begin{table}[t]
\caption{95\% confidence intervals of biases of $\widehat{\alpha}_1$ and $\widehat{\alpha}_2$, and RMSE of $\{\widehat{p}_i\}_{i=1}^K$, over 1000 replications.}
\label{tbl:trad}
\begin{center}
{\scriptsize
\begin{tabular}{c|l|ccc|ccc}
\hline
& & \multicolumn{3}{c|}{$n=50$ per region} & \multicolumn{3}{c}{$n=100$ per region} \\ \cline{3-8}
$K_O/K$ & Method & Bias & Bias & RMSE & Bias & Bias & RMSE \\
& & of $\widehat{\alpha}_1$ & of $\widehat{\alpha}_2$ & of $\{\widehat{p}_{i}\}$ & of $\widehat{\alpha}_1$ & of $\widehat{\alpha}_2$ & of $\{\widehat{p}_{i}\}$ \\\hline
\multicolumn{8}{c}{$K=20$ regions} \\ \hline
& Proposed & -.004 (-.012, .004) & -.002 (-.014, .009) & .053 & .002 (-.004, .008) & -.016 (-.025, -.007) & .040 \\
0\% & GLMM & -.004 (-.012, .004) & -.015 (-.027, -.002) & .057 & .002 (-.004, .007) & -.019 (-.030, -.008) & .044 \\
& Oracle $\boldsymbol{\alpha}$& -.002 (-.009, .005) & .002 (-.004, .008) & .016 & .002 (-.003, .007) & -.004 (-.009, .000) & .011 \\ \hline
& Proposed & -.003 (-.011, .006) & -.003 (-.015, .009) & .053 & .002 (-.004, .008) & -.016 (-.025, -.006) & .040 \\
5\% & GLMM & -.003 (-.011, .005) & -.016 (-.033, .001) & .062 & .001 (-.005, .007) & -.020 (-.036, -.004) & .046 \\
& Oracle $\boldsymbol{\alpha}$& -.001 (-.008, .006) & .002 (-.004, .009) & .016 & .002 (-.002, .007) & -.003 (-.008, .002) & .011 \\ \hline
& Proposed & -.003 (-.012, .005) & -.004 (-.016, .009) & .055 & .002 (-.003, .008) & -.017 (-.027, -.007) & .040 \\
10\% & GLMM & -.004 (-.013, .004) & -.001 (-.021, .019) & .063 & .002 (-.004, .007) & -.006 (-.026, .014) & .046 \\
& Oracle $\boldsymbol{\alpha}$& -.002 (-.009, .005) & .003 (-.003, .010) & .016 & .003 (-.002, .008) & -.003 (-.008, .002) & .011 \\ \hline
& Proposed & -.004 (-.012, .005) & -.002 (-.016, .011) & .055 & .002 (-.004, .008) & -.016 (-.026, -.006) & .041 \\
15\% & GLMM & -.005 (-.013, .003) & .009 (-.014, .033) & .063 & .001 (-.005, .007) & .002 (-.021, .026) & .046 \\
& Oracle $\boldsymbol{\alpha}$& -.003 (-.010, .004) & .005 (-.002, .011) & .016 & .002 (-.003, .007) & -.004 (-.009, .001) & .011 \\ \hline
\multicolumn{8}{c}{$K=40$ regions} \\ \hline
& Proposed & -.002 (-.008, .004) & .002 (-.005, .009) & .043 & -.000 (-.004, .004) & -.003 (-.008, .003) & .033 \\
0\% & GLMM & -.002 (-.008, .003) & .001 (-.008, .009) & .055 & -.001 (-.005, .003) & -.005 (-.012, .003) & .043 \\
& Oracle $\boldsymbol{\alpha}$& -.002 (-.007, .002) & .003 (-.001, .008) & .012 & .001 (-.002, .004) & -.001 (-.004, .002) & .008 \\ \hline
& Proposed & -.003 (-.009, .003) & .004 (-.004, .011) & .044 & .000 (-.004, .004) & -.002 (-.008, .003) & .034 \\
5\% & GLMM & -.004 (-.010, .002) & .003 (-.008, .014) & .060 & -.000 (-.005, .004) & -.003 (-.014, .008) & .045 \\
& Oracle $\boldsymbol{\alpha}$& -.002 (-.007, .002) & .004 (-.001, .009) & .012 & .001 (-.002, .005) & -.001 (-.005, .002) & .008 \\ \hline
& Proposed & -.004 (-.010, .002) & .003 (-.004, .011) & .045 & -.001 (-.005, .003) & -.001 (-.006, .005) & .034 \\
10\% & GLMM & -.004 (-.010, .002) & -.003 (-.017, .011) & .062 & -.002 (-.006, .002) & -.007 (-.021, .007) & .046 \\
& Oracle $\boldsymbol{\alpha}$& -.003 (-.008, .001) & .004 (-.001, .008) & .012 & .001 (-.003, .004) & .000 (-.003, .004) & .008 \\ \hline
& Proposed & -.004 (-.010, .002) & .005 (-.003, .013) & .046 & -.000 (-.005, .004) & -.001 (-.007, .004) & .034 \\
15\% & GLMM & -.005 (-.011, .001) & -.002 (-.018, .014) & .063 & -.001 (-.005, .003) & -.006 (-.022, .010) & .046 \\
& Oracle $\boldsymbol{\alpha}$& -.003 (-.008, .001) & .004 (-.001, .009) & .012 & .001 (-.002, .005) & .000 (-.003, .004) & .008 \\ \hline
\end{tabular}
}
\end{center}
\end{table}
We further compared the RMSE of $\boldsymbol{\beta}$, $\sqrt{\sum_{i=1}^K (\widehat{\beta}_i - \beta_i)^2/K}$, from the proposed method and Oracle $\boldsymbol\beta$. {Figure \ref{fig:RMSE}} shows that the RMSE of $\widehat{\boldsymbol{\beta}}$ decreased when $n$ or $K$ increased, and slightly increased with a larger proportion of outliers. Compared with Oracle $\boldsymbol{\beta}$, the difference between two methods became smaller with an increasing $K$. This indicates that the proposed method provides a good estimate of the baseline obesity rate.
\begin{figure}[t!]
\caption{ RMSE of $\widehat{\boldsymbol{\beta}}$, varying the number of outliers over 1000 replications. }
\label{fig:RMSE}
\centering{
\includegraphics[scale=0.8]{PHINEX_FN19_outprop_3msebeta.pdf}
}
\end{figure}
Finally, to compare the performances in outlier detection, we used Matthews Correlation Coefficient (MCC), defined by
\[
{\rm MCC = \frac{TP \cdot TN - FP \cdot FN}{\sqrt{(TP+FP)(TP+FN)(TN+FP)(TN+FN)}} }.
\]
Here, TP stands for true positive, where the detected outlier region is indeed an outlier; TN stands for true negative, where the labeled normal region is normal; FP stands for false positive, where the detected outlier region is actually normal; and FN stands for false negative, where the labelled normal region is actually an outlier. A higher value of MCC is preferred, where ${\rm MCC} =1$ indicates a perfect classifier and ${\rm MCC} =0$ indicates a random guess. We evaluated the MCC on the proposed, GLMM, Scan Statistic, and Oracle $\boldsymbol\gamma$. The MCCs are presented in Figure \ref{fig:MCC}.
The MCC of the Scan Statistic was around zero, even when $n$ and $K$ were increased, indicating that the Scan Statistic failed to detect multiple outlier regions.
The MCC of the GLMM
decreased when either the proportion of outlier or the $K$ increased. This is consistent with the literature suggesting that residual-based outlier detection may not operate well with multiple outliers. The MCC of the proposed method was comparable to its oracle counterpart, Oracle $\boldsymbol{\beta}$. It improved over increasing $K$ and stabilized over increasing proportion of outliers. In summary, our method showed promising performance in identifying outliers, especially when the proportion of outliers was increased.
\begin{figure}[t!]
\caption{ MCC, varying the number of outliers over 1000 replications. }
\label{fig:MCC}
\centering{
\includegraphics[scale=0.8]{PHINEX_FN19_outprop_10MCC.pdf}
}
\end{figure}
\section{Application to the PHINEX database}\label{sec:application}
We considered census blockgroup as the geographic unit, and we excluded certain blockgroups with small sample sizes, following the guidelines of Behavioral Risk Factor Surveillance System \citep{CDC2016}.
The individual covariates $\boldsymbol{Z}_{ij}$ included sex, age as of 2012, race/ethnicity and insurance status,
and the region-level covariates $\boldsymbol{X}_i$ included urbanicity and EHI. Age was categorized into 3 groups: 2--4 years, 5--9 years, and 10--14 years.
Race and ethnicity were combined into a single covariate, and categorized into 4 groups: Hispanic, non-Hispanic white, non-Hispanic black, and non-Hispanic other.
Patients with health service payor as commercial or Medicaid were included, where a few subjects with no insurance were excluded. Urbanicity, ranging from 1 to 11, was categorized into 3 groups: urban (1-4), suburban (5-8), or rural (9-11).
We standardized EHI for numerical stability of the proposed algorithm.
The location $\boldsymbol{S}_i$ was defined by a vector of the longitude and latitude of the centroid of the $i$-th block group. We constructed $\rho_{i_1, i_2}$ as the inverse of geodesic distances, $\rho_{i_1, i_2} = 1 / d^{{\rm geo}}(\boldsymbol{S}_{i_1}, \boldsymbol{S}_{i_2})$, where $d^{{\rm geo}}(\boldsymbol{S}_{i_1}, \boldsymbol{S}_{i_2})$ denotes the greater circle distance between $\boldsymbol{S}_{i_1}$ and $\boldsymbol{S}_{i_2}$.
For the $i_1$-th region, we retained the $L$ largest $\rho_{i_1, i_2}$s and truncated the others at zero, where we treated $L$ as a tuning parameter.
A grid search on $\lambda_1 \in [2^{-1}, 2^{17}]$, $\lambda_2 \in [2^{-5}, 2^2]$ and $L \in \{3, 5, 7\}$ was conducted to find the best combination of tuning parameters.
The confidence interval of each parameter was constructed using bootstrap over 1000 replications.
\begin{table}[t]
\centering
\caption{Fitted coefficients and {confidence intervals (in parentheses)} for covariate effects.}
\bigskip
\label{tbl:alpha}
{
\begin{tabular}{l|cc}
\hline
& \multicolumn{2}{c}{Model} \\
& Proposed & GLMM \\ \hline
\emph{Individual-level covariates} & \\
~~~~~ Sex (Base: Female) & \\
~~~~~$\sim$ Male & .235 (.144, .305) & .226 (.180, .271)\\
~~~~~Age at 2012 (Base: Pre-school) & \\
~~~~~$\sim$ School-aged & .568 (.429, .657) & .562 (.491, .632)\\
~~~~~$\sim$ Adolescent & .875 (.740, .967) & .869 (.802, .936)\\
~~~~~Race/Ethnicity (Base: White, non-Hispanic) & & \\
~~~~~$\sim$ Black, non-Hispanic & .437 (.313, .559) & .434 (.360, .508)\\
~~~~~$\sim$ Other, non-Hispanic & .035 (-.133, .180) & .042 (-.068, .153)\\
~~~~~$\sim$ Hispanic & .680 (.569, .787) & .667 (.606, .728)\\
~~~~~Insurance status (Base: Commercial) & \\
~~~~~$\sim$ Medicaid & .522 (.428, .636) & .509 (.453, .565)\\
\emph{Community-level covariates} & \\
~~~~~Urbanicity (Base: Urban) & \\
~~~~~$\sim$ Suburban & -.134 {(-.305, -.034)} & .037 (-.078, .152)\\
~~~~~$\sim$ Rural & .125 {(-.163, .302)} & .237 (.070, .405)\\
~~~~~Economic Hardship Index (standardized) & .120 {(-.001, .147)} & .143 (.095, .191)\\ \hline
\end{tabular}
}
\end{table}
\begin{figure}[t!]
\caption{Estimated baseline prevalence rates and the identified outliers in childhood obesity surveillance. Each polygon represents a census blockgroup. Top-left: Result from the proposed method. Outliers are marked as black (yellow) as above the trend, $\widehat{\gamma}_i > 0$ (below the trend, $\widehat{\gamma}_i < 0$). Top-right: Result from the GLMM. Outliers are marked as black (yellow) as above the trend, $\widehat{b}_i > 2.5 \widehat{\sigma}$ (below the trend, $\widehat{b}_i < 2.5 \widehat{\sigma}$). Bottom-left: Discovered cluster by the Scan Statistic with the highest likelihood ratio, which was {below} the trend.}
\label{fig:chol}
\centering{
\includegraphics[scale=0.62]{choroplethAll_real_impl16_1.pdf}
}
\end{figure}
The estimated $\widehat{\boldsymbol{\alpha}}$s are summarized in Table \ref{tbl:alpha}. Overall, the proposed method had wider confidence intervals than the GLMM. This result was anticipated, given that our method had more parameters to estimate, which could lead to higher variabilities. The estimated coefficients from our model and the GLMM were comparable in general, except for the suburban effect. The obesity rate in females was lower compared to males, and younger children had lower obesity rates. Obesity rates in both non-Hispanic white and non-Hispanic other were lower than those in non-Hispanic black and Hispanic patients.
The obesity prevalence was higher in subjects with Medicaid compared to those with commercial insurance. The EHI was positively associated with the estimated obesity rate.
The fitted baseline obesity rates are displayed in Figure \ref{fig:chol}, which appear to coincide with empirical knowledge of the greater Madison area.
The lowest prevalence areas included the western portion of the Madison, Middleton, and Verona areas. It is known that these areas are recently developed and expanded, and include people who are generally younger and more socioeconomically advantaged compared to the surrounding areas. The intermediate prevalence areas, comprising the greater central and eastern Madison region, are more established, historic areas of the region, and are known
to contain more stereotypical middle-class citizens. The highest prevalence areas are clearly the most geographically distant from the center of Madison, and are also all outside of Dane county, which contains Madison.
\begin{table}[t!]
\centering
\caption{The anonymized IDs of the outlier blockgroups, their sample sizes, crude obesity rates, fitted baseline obesity rates, adjusted obesity rates, {and frequencies of detections over $B=1000$ bootstrap replications}. }
\label{tbl:gamma}
{\footnotesize
\begin{tabular}{c|cc|cc|c}
\hline
blockgroup & Unweighted & Crude & Fitted & Fitted & Frequency \\
ID & sample & obesity rate & baseline & adjusted & of \\
& size & & obesity rate & obesity rate & detection \\
& ($n_i$) & ($\widehat{p}_i^{\rm crude}$) & ($\widehat{p}_i^{\rm bsl}$) & ($\widehat{p}_i^{\rm adj}$) & ($\sum_i I(\widehat{\gamma}_i \neq 0)/B$) \\
\hline
\multicolumn{6}{c}{Above the trend}\\
\hline
7 & 60 & .432 & .097 (.061, .150) & .264 (.153, .351) & .701 \\
23 & 91 & .234 & .048 (.042, .057) & .129 (.110, .167) & .483 \\
24 & 93 & .206 & .048 (.042, .057) & .113 (.095, .128) & .515 \\
25 & 104 & .207 & .048 (.042, .057) & .115 (.094, .139) & .496 \\
83 & 96 & .291 & .058 (.047, .064) & .174 (.133, .179) & .689 \\
85 & 91 & .356 & .058 (.047, .064) & .187 (.139, .196) & .895 \\
100 & 62 & .288 & .058 (.047, .064) & .149 (.117, .155) & .763 \\
102 & 53 & .335 & .058 (.047, .064) & .217 (.147, .246) & .579 \\
124 & 66 & .207 & .058 (.047, .064) & .117 (.090, .136) & .535 \\
200 & 93 & .218 & .058 (.047, .064) & .133 (.102, .151) & .474 \\
212 & 100 & .180 & .048 (.042, .057) & .085 (.076, .094) & .634 \\
244 & 71 & .203 & .053 (.044, .064) & .121 (.096, .135) & .446 \\
245 & 94 & .248 & .053 (.044, .064) & .148 (.105, .184) & .573 \\
252 & 68 & .278 & .056 (.046, .073) & .148 (.108, .179) & .706 \\
254 & 82 & .204 & .058 (.048, .071) & .103 (.080, .111) & .698 \\
264 & 74 & .257 & .048 (.042, .057) & .130 (.092, .157) & .726 \\
\hline
\multicolumn{6}{c}{Below the trend} \\
\hline
22 & 110 & .063 & .048 (.042, .057) & .123 (.109, .151) & .363 \\
32 & 221 & .045 & .048 (.042, .057) & .092 (.080, .101) & .079 \\
35 & 146 & .067 & .048 (.042, .057) & .126 (.109, .151) & .300 \\
70 & 67 & .168 & .058 (.047, .064) & .283 (.203, .292) & .269 \\
73 & 109 & .175 & .058 (.047, .064) & .273 (.202, .279) & .136 \\
82 & 259 & .173 & .058 (.047, .064) & .329 (.209, .342) & .106 \\
94 & 134 & .105 & .058 (.047, .064) & .198 (.159, .210) & .383 \\
115 & 68 & .061 & .058 (.047, .064) & .157 (.120, .161) & .468 \\
118 & 192 & .141 & .058 (.047, .064) & .248 (.155, .253) & .062 \\
125 & 81 & .047 & .058 (.047, .064) & .115 (.091, .115) & .277 \\
127 & 295 & .051 & .058 (.047, .064) & .114 (.094, .121) & .195 \\
128 & 125 & .054 & .058 (.047, .064) & .154 (.122, .175) & .698 \\
136 & 466 & .038 & .048 (.042, .057) & .114 (.100, .132) & .684 \\
153 & 469 & .045 & .048 (.042, .057) & .095 (.085, .103) & .047 \\
159 & 202 & .081 & .058 (.047, .064) & .158 (.130, .176) & .345 \\
168 & 139 & .117 & .056 (.046, .071) & .221 (.169, .247) & .418 \\
186 & 66 & .097 & .058 (.047, .064) & .167 (.118, .184) & .204 \\
229 & 133 & .139 & .077 (.048, .127) & .299 (.189, .395) & .599 \\
235 & 61 & .139 & .077 (.048, .127) & .235 (.139, .312) & .365 \\
246 & 210 & .060 & .053 (.044, .064) & .138 (.106, .158) & .283 \\
262 & 90 & .076 & .081 (.050, .114) & .189 (.116, .237) & .459 \\
\hline
\end{tabular}
}
\end{table}
The proposed method identified several outliers. Aberrant locations with obesity rates above the trend ($\widehat{\gamma}_i > 0$) and below the trend ($\widehat{\gamma}_i < 0$) were shown as black and yellow, respectively, in Figure \ref{fig:chol}. We identified 6\% of blockgroups as outliers above the trend, and 8\% as below the trend. Results are presented in Table \ref{tbl:gamma}, including:
\begin{itemize}
\item crude obesity rates, $\widehat{p}_i^{\rm crude} = \frac{1}{\sum_j w_{ij}I(R_{ij}=1)} \sum_j w_{ij} I(R_{ij}=1) Y_{ij}$;
\item baseline obesity rates, $\widehat{p}_i^{\rm bsl} = {\rm logit}^{-1}(\widehat{\beta}_i)$;
\item obesity rates adjusted for covariates and outliers,
$$
\widehat{p}_i^{\rm adj} = \widehat{\mathbb{E}}_ {\gamma_i = 0}(Y_{ij} | \boldsymbol{X}_i) = \frac{1}{\sum_j w_{ij}I(R_{ij}=1)} \sum_{j=1}^{n_i} w_{ij}I(R_{ij}=1) \cdot {\rm logit}^{-1} \left(
\boldsymbol{Z}_{ij}^T \widehat{\boldsymbol{\alpha}}_1 + \boldsymbol{X}_{i}^T \widehat{\boldsymbol{\alpha}}_2 + \widehat{\beta}_i \right),
$$
\end{itemize}
and frequencies of detections over $B=1000$ bootstrap replications. We note that the outlier identification is \emph{relative} to the fitted trend. For example, blockgroup 212 had an ordinary level of the estimated crude obesity rate (0.180). However, the crude rate was much higher than the fitted value of expected obesity prevalence (0.085). There existed unexplained information that could contribute to the elevated rate. Hence, it was declared as an outlier above the trend. The frequency of detection based on the bootstrap provides a glimpse of the uncertainty of aberrance. The outlier regions above the trend tended to have higher frequencies than those below the trend.
The localized outbreak from our model may enable comparative investigations at granular level. For example, what potential factors explain the outlier blockgroups that are significantly above or below the trend?
Obesity prevalence is determined by the interplay of patient demographic characteristics, behaviors, and community environmental factors. Our model has accounted for only a subset of them.
Thus outliers could represent communities with meaningfully different environments than expected (e.g. much better or worse than average access to grocery stores, parks, and recreational facilities), and/or it could represent community members with behaviors that are substantially different than expected (e.g. much greater or less physical activity, substantially better or worse dietary habits, etc.).
Based on our results, healthcare professionals could look into the risk factors within the outlier and compare these factors with its adjacent blockgroups.
We also applied the GLMM and the Scan Statistic for comparison.
As shown in Figure \ref{fig:chol}, the Scan Statistic identified midwestern Madison area as abnormal below trend, which appears to coincide with the lowest prevalence area identified by our method. However, it failed to inform outbreaks in obesity rates. These outbreaks could be captured by $\widehat{\boldsymbol{\gamma}}$ in our model.
The GLMM does not smooth the obesity pattern over the state, which might be difficult to explain and investigate. Thus, the proposed method could provide more informative and interpretable results.
\section{Concluding remarks}\label{sec:conclusion}
Motivated by childhood obesity surveillance using routinely collected EHR data, we developed a multilevel penalized logistic regression model. We incorporated the fusion and the nonconvex sparsity penalty in the likelihood function, which enabled us to conduct regional smoothing and outlier detection simultaneously. While in this paper we only considered spatial surveillance, we are interested in generalizing the method to longitudinal data setup that conducts spatiotemporal surveillance.
In our paper, we assume that BMI is MAR, which is not testable in observational data. The feasibility of MAR for EHRs data is an active area of research in recent years \citep{Snyder2018}. In the future, we can develop sensitivity analysis techniques that investigate sensitivity of the results to uncontrolled confounding \citep{Greenland2004}.
Another future direction is to develop principled inferential procedures for the proposed work. We could potentially use ideas from significance testing and confidence regions for penalized procedures. For example, \cite{Hyun2016} developed a post-selection inference for generalized lasso applied to linear models. A general selective inference procedure for penalized likelihood has been developed for $\ell_1$ penalty \citep{Taylor2016}.
\section*{Acknowledgement}
This work is supported by R21HD086754 awarded by the National Institutes of Health. It is also supported in part by NIH grant P30CA015704 and S10OD020069.
The authors thank Donghyeon Yu (Inha University, Incheon, South Korea) for sharing R codes on the majorization-minimization algorithm, and Albert Y. Kim (Amherst College, Amherst, MA) for sharing the source codes for the \texttt{R} package \texttt{SpatialEpi}.
|
{
"timestamp": "2019-04-16T02:18:30",
"yymm": "1804",
"arxiv_id": "1804.05430",
"language": "en",
"url": "https://arxiv.org/abs/1804.05430"
}
|
\subsection{Battery and Reserve Options in Oahu}
In 2015, the State of Hawaii adopted a 100\% renewable power target by 2045. Here we use Switch 2.0 and datasets prepared for a forthcoming study of that transition, to assess the potential benefits of obtaining operating reserves from several different types of battery and/or demand-side response. This is an important question for engineers and policymakers planning the transition to a 100\% renewable power system, and can be addressed directly using the combination of long-term modeling, chronological daily modeling, unit commitment, operating reserves, storage and customizability offered in Switch 2.0. We are not aware of any other published models that can address these questions directly.
This study uses assumptions from the Hawaiian Electric Company's latest integrated resource plan \cite{HECOHawaiianElectricCompanies2016b}; model configuration and data are available from ref. \cite{HawaiiReservesStudyRepository}. There are four types of lithium-ion batteries: contingency-oriented batteries can make 10 deep cycles per year and regulation-oriented batteries can make up to 15,000 shallow cycles per year. Neither of these can provide bulk load-shifting from hour to hour. However, load-shifting batteries can provide 4 or 6 hours of energy storage and complete up to 365 cycles per year. Peak demand could be reduced up to 10\% via demand response.
In this case study, we use Switch 2.0 to answer several questions about these resources.
(1) If load-shifting batteries could provide contingency and/or regulating reserves while charging or discharging, how much money would that save?
(2) How much money could be saved per household by implementing demand response as a simple load-shifting service?
(3) How much more could be saved if demand response also provided contingency and regulating reserves?
To address these questions, we ran Switch using these standard modules:
\begin{itemize}
\itemsep-0.4em
\item \verb|timescales|,
\item \verb|financials|,
\item \verb|balancing.load_zones|,
\item \verb|energy_sources.properties|,
\item \verb|generators.core.build|,
\item \verb|generators.core.dispatch|,
\item \verb|reporting|,
\item \verb|energy_sources.fuel_costs.markets|,
\item \verb|generators.core.proj_discrete_build|,
\item \verb|generators.core.commit.operate|,
\item \verb|generators.core.commit.fuel_use|,
\item \verb|generators.core.commit.discrete|,
\item \verb|generators.extensions.storage|,
\item \verb|balancing.operating_reserves.areas|, and
\item \verb|balancing.operating_reserves|\linebreak[0]{}\verb|.spinning_reserves_advanced|.
\end{itemize}
These comprise Switch's core formulation plus the following elements: discrete-sized generators, detailed unit-commitment, two-way flow for batteries, regional fuel markets, and spinning reserve targets. The formulation of these modules and the Hawaii reserve rule are detailed in the Supplementary Material. We also used several modules from a \verb|hawaii| subpackage, to define options for fuel market expansion, demand-response, electric-vehicle charging and operating rules for some individual generators.
We used Switch's scenario-solving system to define five scenarios: (1) "battery bulk": load-shifting batteries only provide bulk inter-hour load-shifting, not reserves; there is no demand response (DR), and electric vehicles (EVs) charge at business-as-usual times; (2) "battery bulk and conting": same as "battery bulk", but load-shifting batteries can also provide contingency reserves; (3) "battery bulk and reg": same as "battery bulk and conting", but load-shifting batteries can also provide regulating reserves; (4) "DR bulk": same as "battery bulk and reg" plus DR can provide bulk load-shifting (i.e., the model can move up to 10\% of demand from each hour to any other hour, provided it doesn't raise demand by more than 80\% in any hour), and EVs charge at optimal times each day; no reserves from DR or EVs; (5) "DR bulk and reg": same as "DR bulk", but DR and EVs can also provide up and down contingency and regulation reserves equal to the difference between the amount of load scheduled each hour and the minimum and maximum allowed loads.
We then used Switch to select the optimal investment plan for each of the five scenarios, with investment decisions made every five years from 2020 through 2045. For these scenarios, we considered 12 sample days during each period, using weather from the 23rd day of each month in 2008. Costs are scaled as if each sample day were repeated 152 times, filling out the five-year period.
\begin{figure*}[t]
\centering
\includegraphics[width=7in]{Figures/hawaii_capacity_by_scenario.pdf}
\caption{Generation and storage capacities in five scenarios.}
\label{fig:hawaii_capacity_by_scenario}
\end{figure*}
Figure \ref{fig:hawaii_capacity_by_scenario} shows the optimal generation and storage portfolios selected in the five scenarios.
From this brief study, we can draw several useful conclusions:
(1) Obtaining contingency reserves from load-shifting batteries is not likely to provide large savings (only about \$225 per customer on an NPV basis); this is because contingency batteries make up only a small share of the system's assets, and are relatively inexpensive per MW of capacity.
(2) Obtaining regulating reserves from load-shifting batteries is financially attractive (additional \$615 savings per customer). These savings occur because this control strategy reduces the need to build dedicated batteries for regulation.
(3) Inter-hour load shifting via demand response (e.g., in response to real-time pricing) could save an additional \$1,850 per customer, an attractive option.
(4) Providing regulation and contingency reserves from demand response saves only an additional \$159 per customer. The savings come mainly from a small reduction in the need for regulating batteries. This may not be cost-effective, unless this service can ``piggyback'' on the price-based response infrastructure.
\subsection{Software Architecture}
\label{architecture}
Switch 2.0 is a Python package that can be installed via standard Python package managers or directly from its Github repository. Installation instructions are at \url{http://switch-model.org}. Switch uses the open-source Pyomo \cite{hart2017pyomo} optimization framework to define models, load data and solve instances. Models can be solved using any optimization software compatible with Pyomo, which includes most commercial and open-source solvers.
\begin{figure*}[t]
\centering
\includegraphics[width=.99\textwidth]{Figures/switch_module_structure_v3.pdf}
\caption{Package and module structure of Switch 2.0. Blue boxes are subpackages, green boxes are modules.}
\label{fig:modules}
\end{figure*}
Switch 2.0 uses a fine-grained, modular approach to define power system models, which allows the formulation to be easily customized for the needs of each study. This modular architecture reflects the modularity of actual power systems, where individual elements operate independently but contribute to the system's total costs and power balance. Core modules in Switch define spatially and temporally resolved balancing constraints for energy and reserves, and an overall system cost. Separate modules represent components such as generators, batteries or transmission links. These modules interact with the overall optimization model by adding terms to the shared energy and reserve balances and the overall cost expression. They can also define additional decision variables and constraints to govern operation of each technology or subsystem. This allows technologies to be packaged in plug-and-play modules that participate as fully integrated components of the overall model.
Each built-in or user-supplied Switch module is implemented by creating a Python module file that defines one or more standard callback functions that will be called at each stage of generating and solving a model: defining and parsing command-line arguments, defining model components, defining costs or energy balance equations that are shared between modules, loading data from an input directory,
and performing post-solve functions.
These functions are detailed in the Supplementary Material.
Users configure the model by creating a text file containing a list of modules to be used. At runtime, Switch loads each module and runs through definition, compilation, solution, and export stages, calling callback functions of each module in turn, starting with core modules to define the basic framework, followed by specialized modules to define custom technologies or policies.
This system is highly flexible, making it easy to add or subtract from the codebase, typically without having to modify the built-in modules. Basic extensions are simple to write, and advanced modules have an unlimited ability to extend the model.
By changing the choice of modules, users can also switch easily between distinct modeling modes, such as running a sparse capacity-expansion model, followed by a more detailed production-cost assessment of the proposed portfolio.
\subsection{Software Functionalities}
\label{functionalities}
Figure \ref{fig:modules} presents the subpackages and modules that are included in Switch 2.0. The key modules and subpackages are summarized below. A complete mathematical formulation of the model is provided in the Supplementary Material, and thoroughly documented source code may be inspected in the model repository.
\emph{Timescales}.
This module defines three timescales for decision making: \emph{periods} of one or more years where investment decisions are made, \emph{timepoints} within each period when operational decisions are made, and \emph{timeseries} that group timepoints into chronological sequences. Timepoints within each timeseries have a fixed duration specified in hours, and timeseries have a fixed weight that denotes how many times this type of series is expected to occur in the corresponding period. This approach can represent any standard time structure: a load duration curve (many one-hour timeseries per planning period), a collection of sample days during each period (several one-day timeseries), or an 8760-hour timeseries as typically used by production cost models (a single, year-long timeseries).
\emph{Financials}.
This module defines the objective function and financial parameters for the model. Other modules may register investment or operational cost components with this module. Switch then minimizes the net present value of all costs over the entire study.
\emph{Balancing}.
This subpackage defines \emph{load zones}, geographic regions with load timeseries in which energy supply and demand must be balanced in all timepoints. Other modules may register power injections or withdrawals with the power balance constraints. An optional \emph{Unserved Load} module allows imbalances with a user-specified penalty per unit of energy. This package also includes subpackages and modules for \emph{Planning Reserves}, \emph{Operating Reserves} and \emph{Demand Response}, which are described in more detail in the Supplemental Material.
\emph{Generators}.
This subpackage defines all possible generation projects. The \emph{Core} subpackage describes construction and operation constraints and decisions for all projects standard thermal generators or renewables without storage. A simple economic dispatch may be chosen via the \emph{No Commit} module, or a full unit commitment formulation can be incorporated through the \emph{Commit} subpackage. Generators may be assigned a single energy source or allowed to switch optimally between fuels in order to meet targets for emissions or renewables.
Additional power sources may be implemented via optional modules in the \emph{Extensions} subpackage within the \emph{Generators} package. The \emph{Storage} module defines a generic framework for storage technologies, such as pumped hydro, batteries, flywheels, and others. This formulation permits independent sizing of energy and power elements. The \emph{Hydro System} module represents a cascading water network operating in parallel with the energy network, whereas the \textit{Hydro Simple} module merely enforces average water availability for each sampled timeseries. Switch 2.0 offers both linearized unit commitment and mixed-integer unit-commitment via the \emph{Commit} subpackage and \emph{Commit.Discrete} module. When using the \emph{Commit} subpackage, users may also optionally provide multi-segment heat rate curves, minimum up and down times and startup costs and energy.
Switch aggregates generators into generation ``projects''. These are stacks of one or more similar generating units in the same transmission zone, but not necessarily at the same site. Capacity can be added to each generation project in different years, and then portions of the available capacity are committed or dispatched as needed. This approach significantly reduces model size.
\emph{Transmission}.
Switch offers several approaches for representing transmission network capabilities. The most basic strategy is to use a single-zone (\textit{copperplate}) formulation, which ignores restrictions on spatial transfer of power. Alternatively, a \textit{transport model} can be used to represent transmission capabilities in a simplified manner \cite{fripp2012switch,loulou2004documentation, schaber_transmission_2012}. In a transport model, the study region is divided into zones which are internally well-connected but have constrained connections (flowgates) to neighboring zones. The size of each flowgate and the amount of power transfer are decision variables. Transport models are designed to approximate the capabilities of the network and the cost of improvements without modeling the electrical behavior of the network directly. They provide an attractive balance between granularity and tractability in expansion models, because they use only linear terms, even when network expansion is considered. Switch uses a copperplate model by default, or a transport model can be adopted by using the \emph{Transport} subpackage of the \emph{Transmission} subpackage.
The optional \emph{Local T\&D} module within the \emph{Transmission} subpackage represents power transfers from the zonal node to customers, as well as distributed energy resources. This module enables a simplified consideration of the impact of distributed generation, efficiency or demand response on distribution network investments or losses.
\emph{Energy Sources}
This subpackage defines fuel and non-fuel energy sources. Fuel costs can be either represented by a \emph{Simple} flat cost per period or through a \emph{Markets} module which supports supply curves and regional markets which may or may not be interconnected. These are important for power systems that may need to switch to new energy sources such as biofuels or LNG in order to meet strict targets for renewables or emissions
\emph{Policies}
This subpackage defines investment and operational policies. Current modules include enforcing a \emph{Simple RPS} (Renewable Portfolio Standard) and \emph{Carbon Policies}, such as carbon taxes and caps. Investment or production cost credits can be modeled by adjusting fixed or variable costs.
\emph{Testing}.
Switch uses an automated testing framework to support quality assurance and control as users develop code. The testing framework includes a mix of unit, integration, and regression tests.
This helps ensure that changes to the model formulation don't introduce bugs that break existing models, allowing faster and more robust evolution of the codebase.
\section{Motivation and significance}
\label{motivation}
\input{Sections/motivation.tex}
\section{Software description}
\label{software}
\input{Sections/software.tex}
\section{Illustrative Example}
\label{studies}
\input{Sections/case_studies.tex}
\section{Impact and Conclusions}
\label{conclusions}
\input{Sections/conclusion.tex}
\section*{Acknowledgements}
\label{acknowledgments}
The authors thank researchers who have used prior versions of Switch, whose discussions, collaborations and support have inspired continued work on this platform. Portions of the work reported here were funded by grants from the US Department of Transportation’s University Transportation Centers Program, Research and Innovative Technology Administration (PO\#291166), the National Science Foundation (\#1310634), and the Ulupono Initiative.
\bibliographystyle{elsarticle-num}
|
{
"timestamp": "2018-10-19T02:03:53",
"yymm": "1804",
"arxiv_id": "1804.05481",
"language": "en",
"url": "https://arxiv.org/abs/1804.05481"
}
|
\section{Introduction}
\label{sec:intro}
We consider optimal stopping problems of the form $ \sup_{ \tau } \mathbb{E} \, g( \tau, X_{ \tau })$,
where $X= (X_n)_{n=0}^N$ is an $\mathbb{R}^d$-valued discrete-time Markov process
and the supremum is over all stopping times $\tau$ based on observations of $X$.
Formally, this just covers situations where the stopping decision can only be made at finitely many times.
But practically all relevant continuous-time stopping problems can be approximated with time-discretized versions.
The Markov assumption means no loss of generality. We make it because it simplifies the presentation and many
important problems already are in Markovian form. But every optimal stopping problem can
be made Markov by including all relevant information from the past in the current state of $X$
(albeit at the cost of increasing the dimension of the problem).
In theory, optimal stopping problems with finitely many stopping opportunities can be solved exactly.
The optimal value is given by the smallest supermartingale that dominates the reward process -- the
so-called Snell envelope -- and the smallest (largest) optimal stopping time is the first time the
immediate reward dominates (exceeds) the continuation value; see, e.g., \cite{PS06, LL08}.
However, traditional numerical methods suffer from the curse of dimensionality. For instance, the
complexity of standard tree- or lattice-based methods increases exponentially in the dimension.
For typical problems they yield good results for up to three dimensions.
To treat higher-dimensional problems, various Monte Carlo based methods have been developed over the last
years. A common approach consists in estimating continuation values to either derive stopping rules or
recursively approximate the Snell envelope; see e.g.,
\cite{Till93, BM95, Ca96, LS01, TVR01, BKT03, BG04, BPP05, KS06, EKT07, BS08, JO15, BSST} or
\cite{HK04, KKT10}, which use neural networks with one hidden layer to do this.
A different strand of the literature has focused on approximating optimal exercise boundaries;
see, e.g., \cite{A00, G03, Be11AA}. Based on an idea of \cite{DK94}, a dual approach was
developed by \cite{R02, HK04}; see \cite{J07, CG07} for a multiplicative version and
\cite{AB04, BC08, BBS09, R10, DFM12, Be13, BSD13, Le16} for extensions and primal-dual methods.
In \cite{SS18} optimal stopping problems in continuous time are treated by approximating the
solutions of the corresponding free boundary PDEs with deep neural networks.
In this paper we use deep learning to approximate an optimal stopping time. Our approach is related to policy
optimization methods used in reinforcement learning \cite{SB}, deep reinforcement learning \cite{SLM, MKS, SHM, LHP}
and the deep learning method for stochastic control problems proposed by \cite{HE}.
However, optimal stopping differs from the typical control problems studied in this literature.
The challenge of our approach lies in the implementation of a deep learning method that can efficiently
learn optimal stopping times. We do this by decomposing an optimal stopping time into a sequence of 0-1
stopping decisions and approximating them recursively with a sequence of multilayer feedforward neural
networks. We show that our neural network policies can approximate optimal stopping times to any degree of
desired accuracy. A candidate optimal stopping time $\hat{\tau}$ can be obtained by
running a stochastic gradient ascent. The corresponding expectation
$\mathbb{E} \, g(\hat{\tau},X_{\hat{\tau}})$ provides a lower bound
for the optimal value $\sup_{\tau} \mathbb{E} \, g(\tau,X_{\tau})$.
Using a version of the dual method of \cite{R02,HK04}, we also derive an upper bound.
In all our examples, both bounds can be computed with short run times and lie close together.
The rest of the paper is organized as follows: In Section \ref{sec:dl} we introduce the
setup and explain our method of approximating optimal stopping times with neural networks.
In Section \ref{sec:bci} we construct lower bounds, upper bounds, point estimates and confidence intervals
for the optimal value. In Section \ref{sec:ex} we test the approach on three examples: the pricing of a Bermudan
max-call option on different underlying assets, the pricing of a callable multi barrier reverse convertible
and the problem of optimally stopping a fractional Brownian motion. In the first two examples, we use a
multi-dimensional Black--Scholes model to describe the dynamics of the underlying assets. Then the pricing of
a Bermudan max-call option amounts to solving a $d$-dimensional optimal stopping problem,
where $d$ is the number of assets. We provide numerical results for $d = 2,3,5, 10, 20, 30, 50, 100, 200$ and 500.
In the case of a callable MBRC, it becomes a $d+1$-dimensional stopping problem since
one also needs to keep track of the barrier event. We present results for
$d = 2,3,5,10,15$ and $30$. In the third example we only consider a one-dimensional fractional Brownian
motion. But fractional Brownian motion is not Markov. In fact, all of its increments are correlated. So,
to optimally stop it, one has to keep track of all past movements. To make it tractable, we approximate
the continuous-time problem with a time-discretized version, which if formulated as a Markovian problem,
has as many dimensions as there are time-steps. We compute a solution for 100 time-steps.
\section{Deep learning optimal stopping rules}
\label{sec:dl}
Let $X = (X_n)_{n=0}^N$ be an $\mathbb{R}^d$-valued discrete-time Markov process
on a probability space $(\Omega, {\cal F} , \mathbb{P})$, where $N$ and $d$ are
positive integers. We denote by ${\cal F}_n$ the $\sigma$-algebra generated by
$X_0, X_1, \dots, X_n$ and call a random variable $\tau \colon \Omega \to \crl{0,1, \dots, N}$
an $X$-stopping time if the event $\crl{\tau = n}$ belongs to $ {\cal F}_n $ for all
$n \in \{ 0, 1, \dots, N \}$.
Our aim is to develop a deep learning method that can efficiently learn an
optimal policy for stopping problems of the form
\begin{equation}
\label{os}
\sup_{ \tau \in {\cal T} } \mathbb{E}\, g(\tau,X_{\tau}),
\end{equation}
where $g \colon \crl{0,1,\dots, N} \times \mathbb{R}^d \to \mathbb{R}$ is a measurable function
and ${\cal T}$ denotes the set of all $X$-stopping times. To make sure that problem \eqref{os}
is well-defined and admits an optimal solution, we assume that $g$ satisfies the integrability condition
\begin{equation} \label{ic}
\mathbb{E}\, | g(n,X_n) | < \infty \quad \mbox{for all } n \in \crl{0,1,\dots,N};
\end{equation}
see, e.g., \cite{PS06, LL08}. To be able to derive confidence intervals for the optimal value \eqref{os},
we will have to make the slightly stronger assumption
\begin{equation} \label{ic2}
\mathbb{E} \edg{ g(n,X_n)^2} < \infty \quad \mbox{for all } n \in \crl{0,1,\dots,N}
\end{equation}
in Subsection \ref{subsec:ci} below. This is satisfied in all our examples in Section \ref{sec:ex}.
\subsection{Expressing stopping times in terms of stopping decisions}
Any $X$-stopping time can be decomposed into a sequence of 0-1 stopping decisions.
In principle, the decision whether to stop the process at time $n$ if it has not
been stopped before, can be made based on the whole evolution of $X$ from time $0$ until $n$.
But to optimally stop the Markov process $X$, it is enough to make stopping decisions according to
$f_n(X_n)$ for measurable functions $f_n \colon \mathbb{R}^d \to \crl{0,1}$, $ n = 0, 1, \dots, N$.
Theorem~\ref{thm:rep} below extends this well-known fact and serves as the theoretical basis of our method.
Consider the auxiliary stopping problems
\begin{equation} \label{nos}
V_n = \sup_{\tau \in {\cal T}_n} \mathbb{E} \, g(\tau,X_{\tau})
\end{equation}
for $n = 0, 1, \dots, N$, where ${\cal T}_n$ is the set of all $X$-stopping times satisfying $n \le \tau \le N$.
Obviously, ${\cal T}_N$ consists of the unique element $\tau_N \equiv N$,
and one can write $\tau_N = N f_N(X_N)$ for the constant function $f_N \equiv 1$.
Moreover, for given $n \in \crl{0, 1, \dots, N} $ and a sequence of measurable functions
$ f_n, f_{ n + 1 }, \dots, f_N \colon \mathbb{R}^d \to \crl{0,1}$ with $f_N \equiv 1$,
\begin{equation} \label{taun}
\tau_n = \sum_{m = n}^{N} m f_m(X_m) \prod_{j=n}^{m - 1} \left( 1 - f_j(X_j) \right)
\end{equation}
defines\footnote{In expressions of the form \eqref{taun}, we understand the empty product
$\prod_{j=n}^{n-1} \left( 1 - f_j(X_j) \right)$ as $1$.} a stopping time in ${\cal T}_n$.
The following result shows that, for our method of recursively computing
an approximate solution to the optimal stopping problem \eqref{os}, it will be sufficient to consider stopping
times of the form \eqref{taun}.
\begin{theorem} \label{thm:rep}
For a given $n \in \crl{0,1,\dots, N-1}$, let $\tau_{n+1}$ be a stopping time in ${\cal T}_{n+1}$ of the form
\begin{equation} \label{taun1}
\tau_{n+1} = \sum_{m =n+1}^N mf_m(X_m) \prod_{j=n+1}^{m-1} (1-f_j(X_j))
\end{equation}
for measurable functions $f_{n+1}, \dots, f_N \colon \mathbb{R}^d \to \crl{0,1}$ with $f_N \equiv 1$.
Then there exists a measurable function $f_n \colon \mathbb{R}^d \to \crl{0,1}$ such that
the stopping time $\tau_n \in {\cal T}_n$ given by \eqref{taun} satisfies
\[
\mathbb{E} \, g( \tau_n, X_{ \tau_n }) \ge V_n - \left(V_{ n + 1 } - \mathbb{E} \, g( \tau_{n+1}, X_{ \tau_{n+1}}) \right),
\]
where $V_n$ and $V_{n+1}$ are the optimal values defined in \eqref{nos}.
\end{theorem}
\begin{proof}
Denote $\varepsilon = V_{ n + 1 } - \mathbb{E} \, g( \tau_{ n + 1 } , X_{ \tau_{ n + 1 } })$,
and consider a stopping time $ \tau \in {\cal T}_n$. By the Doob--Dynkin lemma (see, e.g., Theorem 4.41 in \cite{AB}),
there exists a measurable function $h_n \colon \mathbb{R}^d \to \mathbb{R}$ such that
$h_n(X_n)$ is a version of the conditional expectation $\mathbb{E} \edg{g(\tau_{n+1},X_{\tau_{n+1}}) \mid X_n}$.
Moreover, due to the special form \eqref{taun1} of $\tau_{n+1}$,
\[
g(\tau_{n+1},X_{\tau_{n+1}}) = \sum_{m =n+1}^N g(m,X_m) 1_{\crl{\tau_{n+1} = m}}
= \sum_{m =n+1}^N g(m,X_m) 1_{\crl{f_m(X_m) \prod_{j=n+1}^{m-1} (1-f_j(X_j))=1 }}
\]
is a measurable function of $X_{n+1}, \dots, X_N$. So it follows from the Markov property of $X$ that
$h_n(X_n)$ is also a version of the conditional expectation
$\mathbb{E} \edg{g(\tau_{n+1},X_{\tau_{n+1}}) \mid {\cal F}_n}$. Since the events
\[
D = \crl{ g(n,X_n) \ge h_n(X_n)}\quad \mbox{and} \quad E = \crl{\tau = n}
\]
are in ${\cal F}_n$, $\tau_n = n 1_D + \tau_{n+1} 1_{D^c }$ belongs to
${\cal T}_n $ and $\tilde{\tau} = \tau_{n+1} 1_E + \tau 1_{E^c}$ to ${\cal T}_{n+1}$.
It follows from the definitions of $ V_{n+1} $ and $ \varepsilon $ that
$\mathbb{E} \, g( \tau_{ n + 1 }, X_{ \tau_{ n + 1 } }) = V_{n+1} - \varepsilon
\ge \mathbb{E} \, g( \tilde{ \tau }, X_{ \tilde{\tau} } )- \varepsilon$. Hence,
\[
\mathbb{E} \edg{ g( \tau_{ n + 1 }, X_{ \tau_{ n + 1 } } ) 1_{E^c}}
\ge \mathbb{E} \edg{g( \tilde{\tau} , X_{ \tilde{ \tau } } ) 1_{E^c}} - \varepsilon =
\mathbb{E} \edg{g( \tau, X_{ \tau } ) 1_{E^c}} - \varepsilon,
\]
from which one obtains
\[
\begin{split}
& \mathbb{E} \, g( \tau_n, X_{ \tau_n } ) = \mathbb{E} \edg{g(n,X_n ) I_D + g(\tau_{n+1}, X_{\tau_{n+1}}) I_{D^c}}
= \mathbb{E} \edg{g(n,X_n ) I_D + h_n(X_n) I_{D^c}}\\
& \ge \mathbb{E} \edg{g( n, X_n ) I_E + h_n(X_n) I_{E^c}} =
\mathbb{E} \edg{g( n, X_n ) I_E + g( \tau_{ n + 1 } , X_{ \tau_{ n + 1 } } ) I_{E^c}}\\
& \ge \mathbb{E} \edg{g( n, X_n ) I_E + g( \tau, X_{ \tau } ) I_{E^c}} - \varepsilon =
\mathbb{E} \, g( \tau, X_{ \tau }) - \varepsilon .
\end{split}
\]
Since $\tau \in {\cal T}_n $ was arbitrary, this shows that
$\mathbb{E} \, g(\tau_n, X_{ \tau_n } ) \ge V_n - \varepsilon$. Moreover, one has
$1_D = f_n(X_n)$ for the function $f_n \colon \mathbb{R}^d \to \crl{0,1}$ given by
\[
f_n(x) =
\begin{cases}
1 & \mbox{ if } g(n,x) \ge h_n(x)
\\
0 & \mbox{ if } g(n,x) < h_n(x)
\end{cases}.
\]
Therefore,
$$
\tau_n = n f_n(X_n) + \tau_{n+1} (1-f_n(X_n)) = \sum_{m =n}^{N} mf_m(X_m) \prod_{j=n}^{m-1} (1-f_j(X_j)),
$$
which concludes the proof.
\end{proof}
\begin{Remark}
Since for $f_N \equiv 1$, the stopping time $\tau_N = f_N(X_N)$ is optimal in ${\cal T}_N$,
Theorem \ref{thm:rep} inductively yields measurable functions $f_n \colon \mathbb{R}^d \to \crl{0,1}$ such that
for all $n \in \crl{0,1,\dots, N-1}$, the stopping time $\tau_n$ given by \eqref{taun} is optimal among ${\cal T}_n$.
In particular,
\begin{equation} \label{ost}
\tau = \sum_{n = 1}^{N} nf_n(X_n) \prod_{j=0}^{n-1} (1-f_j(X_j))
\end{equation}
is an optimal stopping time for problem \eqref{os}.
\end{Remark}
\begin{Remark}
In many applications, the Markov process $X$ starts from a deterministic initial value $x_0 \in \mathbb{R}^d$.
Then the function $f_0$ enters the representation \eqref{ost} only through the value $f_0(x_0)\in \crl{0,1}$;
that is, at time $0$, only a constant and not a whole function has to be learned.
\end{Remark}
\subsection{Neural network approximation}
Our numerical method for problem \eqref{os} consists in iteratively approximating optimal stopping
decisions
$
f_n \colon \mathbb{R}^d \to \crl{0,1}
$,
$
n = 0, 1, \dots, N - 1,
$
by a neural network
$f^{ \theta } \colon \mathbb{R}^d \to \crl{ 0, 1 }$
with parameter
$
\theta \in \mathbb{R}^q
$.
We do this by starting with the terminal stopping decision $ f_N \equiv 1 $
and proceeding by backward induction.
More precisely, let $ n \in \crl{ 0, 1, \dots, N - 1 } $, and assume parameter values
$
\theta_{ n + 1 }, \theta_{ n + 2 }, \dots, \theta_N \in \mathbb{R}^q
$
have been found such that $f^{\theta_N} \equiv 1$ and the stopping time
\[
\tau_{n+1} = \sum_{m =n+1}^{N} m f^{\theta_m}(X_m) \prod_{j=n+1}^{m-1} (1-f^{\theta_j}(X_j))
\]
produces an expected value
$
\mathbb{E} \, g(\tau_{n+1},X_{\tau_{n+1}})
$
close to the optimum $ V_{ n + 1 } $. Since $f^{\theta}$ takes values in $\crl{0,1}$, it does not directly
lend itself to a gradient-based optimization method. So, as an intermediate step, we
introduce a feedforward neural network $F^{\theta} \colon \mathbb{R}^d \to (0,1)$
of the form
\[
F^{\theta} = \psi \circ a^{\theta}_I \circ \varphi_{q_{I-1}} \circ a^{\theta}_{I-1} \circ \dots \circ \varphi_{q_1} \circ a^{\theta}_1,
\]
where
\begin{itemize}
\item
$I, q_1, q_2, \dots, q_{I-1}$ are positive integers specifying the depth of the network and
the number of nodes in the hidden layers (if there are any),
\item
$a^{\theta}_1 \colon \mathbb{R}^d \to \mathbb{R}^{q_1}, \dots,
a^{\theta}_{I-1} \colon \mathbb{R}^{q_{I-2}} \to \mathbb{R}^{q_{I-1}} $ and
$a^{\theta}_I \colon \mathbb{R}^{q_{I-1}} \to \mathbb{R}$ are affine functions,
\item
for $j \in \mathbb{N}$, $\varphi_j \colon \mathbb{R}^j \to \mathbb{R}^j$ is the
component-wise ReLU activation function given by \linebreak $\varphi_j(x_1, \dots, x_j) = (x^+_1 , \dots, x^+_j)$
\item
$\psi \colon \mathbb{R} \to (0,1)$ is the standard logistic function
$\psi(x) = e^x/(1+ e^x) = 1 / ( 1 + e^{ - x } )$.
\end{itemize}
The components of the parameter $\theta \in \mathbb{R}^q$ of $F^{\theta}$ consist of the
entries of the matrices $A_1 \in \mathbb{R}^{q_1 \times d}, \dots,$ $A_{I-1} \in \mathbb{R}^{q_{I-1} \times q_{I-2}},
A_I \in \mathbb{R}^{1 \times q_{I-1}}$ and the vectors $b_1 \in \mathbb{R}^{q_1}, \dots, b_{I-1} \in \mathbb{R}^{q_{I-1}},
b_I \in \mathbb{R}$ given by the representation of the affine functions
\[
a^{\theta}_i(x) = A_i x + b_i, \quad i = 1, \dots, I.
\]
So the dimension of the parameter space is
\[
q =
\begin{cases}
d+1 & \mbox{ if } I =1 \\
1 + q_1 + \dots + q_{I-1} + d q_1 + \dots + q_{I-2} q_{I-1} + q_{I-1} & \mbox{ if } I \ge 2,
\end{cases}
\]
and for given $x \in \mathbb{R}^d$, $F^{\theta}(x)$ is continuous as well as almost everywhere smooth in $\theta$.
Our aim is to determine $\theta_n \in \mathbb{R}^q$ so that
\[
\mathbb{E} \edg{g(n,X_n) F^{\theta_n}(X_n) + g(\tau_{n+1}, X_{\tau_{n+1}}) (1- F^{\theta_n}(X_n))}
\]
is close to the supremum
$\sup_{\theta \in \mathbb{R}^q} \mathbb{E} \edg{g(n,X_n) F^{\theta}(X_n) + g(\tau_{n+1}, X_{\tau_{n+1}}) (1- F^{\theta}(X_n))}$.
Once this has been achieved, we define the function
$f^{\theta_n} \colon \mathbb{R}^d \to \crl{0,1}$ by
\begin{equation} \label{ftheta}
f^{\theta_n} = 1_{[0,\infty)} \circ a^{\theta_n}_I
\circ \varphi_{q_{I-1}} \circ a^{\theta_n}_{I-1} \circ \dots \circ \varphi_{q_1} \circ a^{\theta_n}_1,
\end{equation}
where $1_{[0,\infty)} \colon \mathbb{R} \to \crl{0,1}$ is the indicator function of $[0,\infty)$.
The only difference between $F^{\theta_n}$ and $f^{\theta_n}$ is the final nonlinearity. While
$F^{\theta_n}$ produces a stopping probability in $(0,1)$, the output of $f^{\theta_n}$ is a hard
stopping decision given by $0$ or $1$, depending on whether $F^{\theta_n}$ takes a value below or above $1/2$.
The following result shows that for any depth $I \ge 2$, a neural network of the form \eqref{ftheta}
is flexible enough to make almost optimal stopping decisions provided it has sufficiently many nodes.
\begin{proposition} \label{prop:appr}
Let $n \in \crl{0,1,\dots, N-1}$ and fix a stopping time $\tau_{n+1} \in {\cal T}_{n+1}$. Then, for
every depth $I \ge 2$ and constant $\varepsilon > 0$, there exist positive integers $q_1, \dots, q_{I-1}$ such that
\begin{eqnarray*}
&& \sup_{\theta \in \mathbb{R}^q} \mathbb{E} \edg{g(n,X_n) f^{\theta}(X_n)
+ g(\tau_{n+1}, X_{\tau_{n+1}}) (1- f^{\theta}(X_n))}\\
&&\ge \sup_{f \in {\cal D}} \mathbb{E} \edg{g(n,X_n) f(X_n) + g(\tau_{n+1}, X_{\tau_{n+1}}) (1- f(X_n))} - \varepsilon,
\end{eqnarray*}
where ${\cal D}$ is the set of all measurable functions $f \colon \mathbb{R}^d \to \crl{0,1}$.
\end{proposition}
\begin{proof}
Fix $\varepsilon > 0$. It follows from the integrability condition \eqref{ic}
that there exists a measurable function $\tilde{f} \colon \mathbb{R}^d \to \crl{0,1}$ such that
\begin{equation} \label{fhat}
\begin{aligned}
& \mathbb{E} \edg{g(n,X_n) \tilde{f}(X_n) + g(\tau_{n+1}, X_{\tau_{n+1}}) (1- \tilde{f}(X_n))}\\
& \ge \sup_{f \in {\cal D}} \mathbb{E} \edg{g(n,X_n) f(X_n) + g(\tau_{n+1}, X_{\tau_{n+1}}) (1- f(X_n))} - \varepsilon/4.
\end{aligned}
\end{equation}
$\tilde{f}$ can be written as $\tilde{f} = 1_A$ for the Borel set $A = \{x \in \mathbb{R}^d : \tilde{f}(x) =1\}$.
Moreover, by \eqref{ic},
\[
B \mapsto \mathbb{E} \edg{|g(n,X_n)| 1_B(X_n)} \quad \mbox{and} \quad
B \mapsto \mathbb{E} \edg{|g(\tau_{n+1}, X_{\tau_{n+1}})| 1_B(X_n)}
\]
define finite Borel measures on $\mathbb{R}^d$. Since every finite Borel measure on $\mathbb{R}^d$ is tight
(see e.g., \cite{AB}), there exists a compact (possibly empty) subset $K \subseteq A$ such that
\begin{equation} \label{K}
\begin{aligned}
& \mathbb{E} \edg{g(n,X_n) 1_K(X_n) + g(\tau_{n+1}, X_{\tau_{n+1}}) (1- 1_K(X_n))}\\
&\ge \mathbb{E} \edg{g(n,X_n) \tilde{f}(X_n) + g(\tau_{n+1}, X_{\tau_{n+1}}) (1- \tilde{f}(X_n))} - \varepsilon/4.
\end{aligned}
\end{equation}
Let $\rho_K \colon \mathbb{R}^d \to [0,\infty]$ be the distance function given by
$\rho_K(x) = \inf_{y \in K} \|x-y\|_2$. Then
\[
k_j(x) = \max \crl{1 - j\rho_K(x), -1}, \quad j \in \mathbb{N},
\]
defines a sequence of continuous functions $k_j \colon \mathbb{R}^d \to [-1,1]$ that converge pointwise to
$1_K - 1_{K^c}$. So it follows from Lebesgue's dominated convergence theorem that there exists a $j \in \mathbb{N}$
such that
\begin{equation} \label{kj}
\begin{aligned}
& \mathbb{E} \edg{g(n,X_n) \, 1_{\crl{k_j(X_n) \ge 0}}
+ g(\tau_{n+1}, X_{\tau_{n+1}}) (1- 1_{\crl{k_j (X_n) \ge 0}})}\\
&\ge \mathbb{E} \edg{g(n,X_n) 1_K(X_n) + g(\tau_{n+1}, X_{\tau_{n+1}}) (1- 1_K(X_n))} - \varepsilon/4.
\end{aligned}
\end{equation}
By Theorem 1 of \cite{LLPS}, $k_j$ can be approximated uniformly on compacts by functions of the form
\begin{equation} \label{h}
\sum_{i=1}^r (v_i^T x + c_i)^+ - \sum_{i=1}^s (w_i^T x + d_i)^+
\end{equation}
for $r,s \in \mathbb{N}$, $v_1, \dots, v_r, w_1, \dots, w_s \in \mathbb{R}^d$ and $c_1, \dots, c_r,
d_1, \dots, d_s \in \mathbb{R}$. So there exists a function $h \colon \mathbb{R}^d \to \mathbb{R}$
expressible as in \eqref{h} such that
\begin{equation}
\begin{aligned} \label{hk}
&\mathbb{E} \edg{g(n,X_n) \, 1_{\crl{h(X_n) \ge 0}}
+ g(\tau_{n+1}, X_{\tau_{n+1}}) (1- 1_{\crl{h(X_n)\ge 0}})}\\
&\ge \mathbb{E} \edg{g(n,X_n) \, 1_{\crl{k_j(X_n) \ge 0}}
+ g(\tau_{n+1}, X_{\tau_{n+1}}) (1- 1_{\crl{k_j (X_n) \ge 0}})}- \varepsilon/4.
\end{aligned}
\end{equation}
Now note that for any integer $I \ge 2$, the composite mapping $1_{[0,\infty)} \circ h$ can
be written as a neural net $f^{\theta}$ of the form \eqref{ftheta} with depth $I$ for suitable integers
$q_1, \dots, q_{I-1}$ and parameter value $\theta \in \mathbb{R}^q$. Hence, one obtains
from \eqref{fhat}, \eqref{K}, \eqref{kj} and \eqref{hk} that
\[
\begin{aligned}
&\mathbb{E} \edg{g(n,X_n) \, f^{\theta}(X_n) + g(\tau_{n+1}, X_{\tau_{n+1}}) (1- f^{\theta}(X_n))}\\
&\ge \sup_{f \in {\cal D}} \mathbb{E} \edg{g(n,X_n) f(X_n)
+ g(\tau_{n+1}, X_{\tau_{n+1}}) (1- f (X_n))} - \varepsilon,
\end{aligned}
\]
and the proof is complete.
\end{proof}
We always choose $\theta_N \in \mathbb{R}^q$ such that\footnote{It is easy to see
that this is possible.} $f^{\theta_N} \equiv 1$. Then our candidate optimal stopping time
\begin{equation} \label{tautheta}
\tau^{\Theta} = \sum_{n=1}^{N} n f^{\theta_n}(X_n) \prod_{j=0}^{n-1} (1-f^{\theta_j}(X_j))
\end{equation}
is specified by the vector $\Theta = (\theta_0, \theta_1, \dots, \theta_{N-1}) \in \mathbb{R}^{Nq}$.
The following is an immediate consequence of Theorem \ref{thm:rep} and Proposition \ref{prop:appr}:
\begin{corollary}
For a given optimal stopping problem of the form \eqref{os}, a depth $I \ge 2$ and a constant $\varepsilon > 0$, there
exist positive integers $q_1, \dots, q_{I-1}$ and a vector $\Theta \in \mathbb{R}^{Nq}$ such that the
corresponding stopping time \eqref{tautheta} satisfies
$
\mathbb{E} \, g(\tau^{\Theta}, X_{\tau^{\Theta}}) \ge \sup_{\tau \in {\cal T}} \mathbb{E} \, g(\tau, X_{\tau}) - \varepsilon.
$
\end{corollary}
\subsection{Parameter optimization}
\label{ss:parameter}
We train neural networks of the form \eqref{ftheta} with fixed depth $I \ge 2$ and
given numbers $q_1, \dots, q_{I-1}$ of nodes in the hidden
layers\footnote{For a given application, one can try out different choices of $I$ and
$q_1, \dots, q_{I-1}$ to find a suitable trade-off between accuracy and efficiency. Alternatively,
the determination of $I$ and $q_1, \dots, q_{I-1}$ could be built into the training algorithm.}.
To numerically find parameters $\theta_n \in \mathbb{R}^q$ yielding good stopping decisions $f^{\theta_n}$
for all times $n \in \crl{0,1, \dots, N-1}$, we approximate expected values with averages of
Monte Carlo samples calculated from simulated paths of the process $(X_n)_{n=0}^N$.
Let $(x^k_n)_{n=0}^N$, $k =1,2,\dots$ be independent realizations of such paths.
We choose $\theta_N \in \mathbb{R}^q$ such that $f^{\theta_N} \equiv 1$ and determine
determine $\theta_n \in \mathbb{R}^q$ for $n \le N-1$ recursively. So, suppose that for a given
$n \in \crl{0,1, \dots, N-1}$, parameters $\theta_{n+1}, \dots, \theta_{N} \in \mathbb{R}^q$, have been found so that
the stopping decisions $f^{\theta_{n+1}}, \dots, f^{\theta_N}$ generate a stopping time
\[
\tau_{n+1} = \sum_{m =n+1}^{N} mf^{\theta_m}(X_m) \prod_{j=n+1}^{m-1} (1-f^{\theta_j}(X_j))
\]
with corresponding expectation
$
\mathbb{E}\, g( \tau_{ n + 1 } , X_{ \tau_{ n + 1 } } )
$
close to the optimal value $V_{n+1}$. If $n = N-1$, one has
$\tau_{n+1} \equiv N$, and if $n \le N-2$, $\tau_{n+1}$ can be written as
\[
\tau_{n+1} = l_{n+1}(X_{n+1}, \dots, X_{N-1})
\]
for a measurable function $l_{n+1} \colon \mathbb{R}^{d(N-n-1)} \to \crl{n+1, n+2, \dots, N}$. Accordingly, denote
\[
l^k_{n+1} =
\begin{cases}
N & \mbox{if } n = N - 1
\\
l_{n+1}(x^k_{n+1}, \dots, x^k_{N-1})
& \mbox{if } n \le N-2
\end{cases}
.
\]
If at time $n$, one applies the soft stopping decision $F^{\theta}$ and afterward behaves according to
$f^{\theta_{n+1}}, \dots, f^{\theta_N}$, the realized reward along the $k$-th simulated path of $X$ is
\[
r^k_n(\theta) = g(n,x^k_n)F^{\theta}(x^k_n) + g(l^k_{n+1}, x^k_{l^k_{n+1}}) (1- F^{\theta}(x^k_n)).
\]
For large $K \in \mathbb{N}$,
\begin{equation} \label{Msum}
\frac{1}{K} \sum_{k =1}^{K} r^k_n(\theta)
\end{equation} approximates the expected value
\[
\mathbb{E} \edg{g(n,X_n) F^{\theta}(X_n) + g(\tau_{n+1}, X_{\tau_{n+1}}) (1- F^{\theta}(X_n))}.
\]
Since $r^k_n(\theta)$ is almost everywhere differentiable in $\theta$, a stochastic gradient ascent method
can be applied to find an approximate optimizer $\theta_n \in \mathbb{R}^q$ of
\eqref{Msum}. The same simulations $(x^k_n)_{n=0}^N$, $k = 1,2,\dots$ can be used to train the stopping decisions
$f^{\theta_n}$ at all times $n \in \crl{ 0, 1, \dots, N-1}$. In the numerical examples in Section \ref{sec:ex} below, we
employed mini-batch gradient ascent with Xavier initialization \cite{GB10}, batch normalization
\cite{IS15} and Adam updating \cite{KiB15}.
\begin{Remark} \label{Rem:I0}
If the Markov process $X$ starts from a deterministic initial value $x_0 \in \mathbb{R}^d$, the initial
stopping decision is given by a constant $f_0 \in \crl{0,1}$. To learn $f_0$ from simulated paths
of $X$, it is enough to compare the initial reward $g(0,x_0)$ to a Monte Carlo estimate
$\hat{C}$ of $\mathbb{E} \, g(\tau_1,X_{\tau_1})$, where $\tau_1 \in {\cal T}_1$ is of the form
\[
\tau_1 = \sum_{n=1}^{N} n f^{\theta_n}(X_n) \prod_{j=1}^{n-1} (1-f^{\theta_j}(X_j))
\]
for $f^{\theta_N} \equiv 1$ and trained parameters $\theta_1, \dots, \theta_{N-1} \in \mathbb{R}^q$.
Then one sets $f_0 = 1$ (that is, stop immediately) if $g(0,x_0) \ge \hat{C}$ and
$f_0 = 0$ (continue) otherwise. The resulting stopping time is of the form
\[
\tau^{\Theta} =
\begin{cases}
0 & \mbox{if } f_0 = 1
\\
\tau_1 & \mbox{if } f_0 = 0.
\end{cases}
\]
\end{Remark}
\section{Bounds, point estimates and confidence intervals}
\label{sec:bci}
In this section we derive lower and upper bounds as well as point estimates and confidence intervals
for the optimal value $V_0 = \sup_{\tau \in {\cal T}} \, \mathbb{E} \, g(\tau, X_{\tau})$.
\subsection{Lower bound}
\label{subsec:lb}
Once the stopping decisions $f^{\theta_n}$ have been trained, the stopping time
$\tau^{\Theta}$ given by \eqref{tautheta} yields a lower bound $L = \mathbb{E} \, g(\tau^{\Theta}, X_{\tau^{\Theta}})$
for the optimal value $V_0 = \sup_{\tau \in {\cal T}} \, \mathbb{E} \, g(\tau, X_{\tau})$.
To estimate it, we simulate a new set\footnote{In particular, we assume that the samples
$(y^k_n)_{n=0}^N$, $k = 1, \dots, K_L$, are drawn independently from the
realizations $(x^k_n)_{n=0}^N$, $k = 1, \dots, K$, used in the training of the stopping decisions.}
of independent realizations $(y^k_n)_{n=0}^N$, $k = 1,2,\dots, K_L,$
of $(X_n)_{n=0}^N$. $\tau^{\Theta}$ is of the form $\tau^{\Theta} = l(X_0,\dots, X_{N-1})$ for
a measurable function $l \colon \mathbb{R}^{d N} \to \crl{0, 1, \dots, N}$. Denote $l^k = l(y^k_0, \dots, y^k_{N-1})$.
The Monte Carlo approximation
\[
\hat{L} = \frac{1}{K_L} \sum_{k=1}^{K_L} g(l^k, y^k_{l^k})
\]
gives an unbiased estimate of the lower bound $L$, and by the law of large numbers, $\hat{L}$ converges to
$L$ for $K_L \to \infty$.
\subsection{Upper bound}
The Snell envelope of the reward process $(g(n,X_n))_{n=0}^N$ is the smallest\footnote{in the $\mathbb{P}$-almost sure order}
supermartingale with respect to $({\cal F}_n)_{n=0}^N$ that dominates $(g(n,X_n))_{n=0}^N$.
It is given\footnote{\label{fn}up to $\mathbb{P}$-almost sure equality} by
\[
H_n = {\rm ess\,sup}_{\tau \in {\cal T}_n} \mathbb{E}[g(\tau) \mid {\cal F}_n], \quad n = 0, 1, \dots, N;
\]
see, e.g., \cite{PS06, LL08}. Its Doob--Meyer decomposition is
\[
H_n = H_0 + M^H_n - A^H_n,
\]
where $M^H$ is the $({\cal F}_n)$-martingale given\footref{fn} by
\[
M^H_0 = 0 \quad \mbox{and} \quad M^H_n - M^H_{n-1} = H_n - \mathbb{E}[H_n \mid {\cal F}_{n-1}],
\quad n = 1, \dots, N,
\]
and $A^H$ is the nondecreasing $({\cal F}_n)$-predictable process given\footref{fn} by
\[ A^H_0 = 0 \quad \mbox{and} \quad
A^H_n - A^H_{n-1} = H_{n-1} - \mathbb{E}[H_n \mid {\cal F}_{n-1}], \quad n =1, \dots, N.
\]
Our estimate of an upper bound for the optimal value $V_0$ is based on the following
variant\footnote{See also the discussion on noisy estimates in \cite{AB04}.
}
of the dual formulation of optimal stopping problems introduced by \cite{R02} and \cite{HK04}.
\begin{proposition}
Let $(\varepsilon_n)_{n=0}^N$ be a sequence of integrable random variables on
$(\Omega, {\cal F}, \mathbb{P})$. Then
\begin{equation} \label{V0est1}
V_0 \ge \mathbb{E} \edg{ \max_{0 \le n \le N} \brak{g(n,X_n) - M^H_n - \varepsilon_n}}
+ \mathbb{E} \edg{ \min_{0 \le n \le N} \brak{A^H_n + \varepsilon_n}}.
\end{equation}
Moreover, if $\mathbb{E} \edg{\varepsilon_n \mid {\cal F}_n} = 0$ for all $n \in \crl{0,1, \dots, N}$, one has
\begin{equation} \label{V0est2}
V_0 \le \mathbb{E} \edg{ \max_{0 \le n \le N} \brak{g(n,X_n) - M_n - \varepsilon_n}}
\end{equation}
for every $({\cal F}_n)$-martingale $(M_n)_{n=0}^N$ starting from $0$.
\end{proposition}
\begin{proof}
First, note that
\begin{eqnarray*}
&& \mathbb{E} \edg{ \max_{0 \le n \le N} \brak{g(n,X_n) - M^H_n - \varepsilon_n}}
\le \mathbb{E} \edg{ \max_{0 \le n \le N} \brak{H_n- M^H_n - \varepsilon_n}}\\
&&= \mathbb{E} \edg{ \max_{0 \le n \le N} \brak{H_0 - A^H_n - \varepsilon_n}}
= V_0 - \mathbb{E} \edg{ \min_{0 \le n \le N} \brak{A^H_n + \varepsilon_n }},
\end{eqnarray*}
which shows \eqref{V0est1}.
Now, assume that $\mathbb{E} \edg{\varepsilon_n \mid {\cal F}_n} = 0$ for all $n \in \crl{0,1, \dots, N}$,
and let $\tau$ be an $X$-stopping time.
Then
\[
\mathbb{E} \, \varepsilon_{\tau} = \mathbb{E} \edg{\sum_{n=0}^N 1_{\crl{\tau = n}} \varepsilon_n}
= \mathbb{E} \edg{\sum_{n=0}^N 1_{\crl{\tau = n}} \mathbb{E}[\varepsilon_n \mid {\cal F}_n]} = 0.
\]
So one obtains from the optional stopping theorem (see, e.g., \cite{GS01}) that
\[
\mathbb{E} \, g(\tau,X_{\tau})
= \mathbb{E} \edg{g(\tau,X_{\tau}) - M_{\tau} - \varepsilon_{\tau}}
\le \mathbb{E} \edg{\max_{0 \le n \le N} \brak{g(n,X_n) - M_n - \varepsilon_n}}
\]
for every $({\cal F}_n)$-martingale $(M_n)_{n=0}^N$ starting from $0$.
Since $V_0 = \sup_{\tau \in {\cal T}} \mathbb{E} \, g(\tau,X_{\tau})$, this implies \eqref{V0est2}.
\end{proof}
For every $({\cal F}_n)$-martingale $(M_n)_{n=0}^N$ starting from $0$ and each sequence of integrable
error terms $(\varepsilon_n)_{n=0}^N$ satisfying $\mathbb{E}\edg{\varepsilon_n \mid {\cal F}_n} = 0$
for all $n$, the right side of \eqref{V0est2} provides an upper bound\footnote{Note that
for the right side of \eqref{V0est2} to be a valid upper bound, it is sufficient that
$\mathbb{E}\edg{\varepsilon_n \mid {\cal F}_n} = 0$ for all $n$. In particular,
$\varepsilon_0, \varepsilon_1, \dots, \varepsilon_N$ can have any arbitrary dependence structure.} for $V_0$, and by
\eqref{V0est1}, this upper bound is tight if $M = M^H$ and $\varepsilon \equiv 0$.
So we try to use our candidate optimal stopping time $\tau^{\Theta}$
to construct a martingale close to $M^H$. The closer $\tau^{\Theta}$ is to an optimal stopping time,
the better the value process\footnote{Again, since $H^{\Theta}_n$, $M^{\Theta}_n$ and $C^{\Theta}_n$
are given by conditional expectations, they are only specified up to $\mathbb{P}$-almost sure equality.}
\[
H^{\Theta}_n = \mathbb{E} \edg{g(\tau^{\Theta}_n,X_{\tau^{\Theta}_n}) \mid {\cal F}_n},
\quad n = 0, 1, \dots, N,
\]
corresponding to
\[
\tau^{\Theta}_{n} = \sum_{m =n}^{N} mf^{\theta_m}(X_m) \prod_{j=n}^{m-1} (1-f^{\theta_j}(X_j)),
\quad n = 0,1,\dots, N,
\]
approximates the Snell envelope $(H_n)_{n=0}^N$. The martingale part of $(H^{\Theta}_n)_{n=0}^N$
is given by $M^{\Theta}_0 = 0$ and
\begin{equation} \label{MTheta}
M^{\Theta}_n - M^{\Theta}_{n-1} = H^{\Theta}_n - \mathbb{E} \edg{H^{\Theta}_n \mid {\cal F}_{n-1}}
= f^{\theta_n}(X_n) g(n,X_n) + (1-f^{\theta_n}(X_n)) C^{\Theta}_n - C^{\Theta}_{n-1}, \;
n\ge 1,
\end{equation}
for the continuation values\footnote{The two conditional expectations are equal since $(X_n)_{n=0}^N$
is Markov and $\tau^{\Theta}_{n+1}$ only depends on $(X_{n+1}, \dots, X_{N-1})$.}
\[
C^{\Theta}_n = \mathbb{E}[g(\tau^{\Theta}_{n+1}, X_{\tau^{\Theta}_{n+1}}) \mid {\cal F}_n] =
\mathbb{E}[g(\tau^{\Theta}_{n+1}, X_{\tau^{\Theta}_{n+1}}) \mid X_n], \quad n = 0,1, \dots, N-1.
\]
Note that $C^{\Theta}_N$ does not have to be specified. It formally appears in \eqref{MTheta} for $n = N$.
But $(1-f^{\theta_N}(X_N))$ is always $0$.
To estimate $M^{\Theta}$, we generate a third set\footnote{The realizations $(z^k_n)_{n=0}^N$, $k = 1, \dots, K_U$,
must be drawn independently of $(x^k_n)_{n=0}^N$, $k = 1, \dots, K$, so that our estimate of the upper bound
does not depend on the samples used to train the stopping decisions. But theoretically, they can depend on
$(y^k_n)_{n=0}^N$, $k = 1, \dots, K_L$, without affecting the unbiasedness of the estimate $\hat{U}$ or
the validity of the confidence interval derived in Subsection \ref{subsec:ci} below.} of independent realizations
$(z^k_n)_{n=0}^N$, $k = 1,2,\dots, K_U,$ of $(X_n)_{n=0}^N$. In addition, for every $z^k_n$, we simulate
$J$ continuation paths $\tilde{z}^{k,j}_{n+1}, \dots, \tilde{z}^{k,j}_N$, $j =1, \dots, J$,
that are conditionally independent\footnote{More precisely, the tuples
$(\tilde{z}^{k,j}_{n+1}, \dots, \tilde{z}^{k,j}_N)$, $j = 1, \dots, J$, are simulated according to $p_n(z_n^k, \cdot)$, where
$p_n$ is a transition kernel from $\mathbb{R}^d$ to $\mathbb{R}^{(N-n)d}$
such that $p_n(X_n, B) = \mathbb{P}[(X_{n+1}, \dots, X_N) \in B \mid X_n]$
$\mathbb{P}$-almost surely for all Borel sets $B \subseteq \mathbb{R}^{(N-n)d}$.
We generate them independently of each other across $j$ and $k$.
On the other hand, the continuation paths starting from $z^k_n$ do not have to be drawn independently
of those starting from $z^k_{n'}$ for $n \neq n'$.
}
of each other and of $z^{k}_{n+1}, \dots, z^{k}_N$.
Let us denote by $\tau^{k,j}_{n+1}$ the value of $\tau^{\Theta}_{n+1}$ along
$\tilde{z}^{k,j}_{n+1}, \dots, \tilde{z}^{k,j}_N$. Estimating the continuation values as
\[
C^k_n =
\frac{1}{J} \sum_{j = 1}^{J} g \brak{\tau^{k,j}_{n+1}, \tilde{z}^{k,j}_{\tau^{k,j}_{n+1}}}, \quad
n = 0, 1, \dots, N-1,\]
yields the noisy estimates
\[
\Delta M^k_n = f^{\theta_n}(z^k_n) g(n,z^k_n) + (1- f^{\theta_n}(z^k_n)) C^k_n - C^k_{n-1}
\]
of the increments $M^{\Theta}_n - M^{\Theta}_{n-1}$ along the $k$-th simulated path $z^k_0, \dots, z^k_N$.
So
\[
M^k_n = \begin{cases}
0 & \mbox{ if } n = 0\\
\sum_{m=1}^n \Delta M^k_m & \mbox{ if } n \ge 1
\end{cases}
\]
can be viewed as realizations of $M^{\Theta}_n + \varepsilon_n$
for estimation errors $\varepsilon_n$ with standard deviations proportional to
$1/\sqrt{J}$ such that $\mathbb{E}\edg{\varepsilon_n \mid {\cal F}_n} = 0$ for all $n$. Accordingly,
\[
\hat{U} = \frac{1}{K_U} \sum_{k =1}^{K_U} \max_{0 \le n \le N} \brak{g \brak{n,z^k_n} - M^k_n},
\]
is an unbiased estimate of the upper bound
\[
U = \mathbb{E} \edg{\max_{0 \le n \le N} \brak{g(n,X_n) - M^{\Theta}_n - \varepsilon_n}},
\]
which, by the law of large numbers, converges to $U$ for $K_U \to \infty$.
\subsection{Point estimate and confidence intervals}
\label{subsec:ci}
Our point estimate of $V_0$ is the average
\[
\frac{\hat{L} + \hat{U}}{2}.
\]
To derive confidence intervals, we assume that $g(n,X_n)$ is square-integrable\footnote{See condition \eqref{ic2}.}
for all $n$. Then
\[
g(\tau^{\theta}, X_{\tau^{\Theta}}) \quad \mbox{and} \quad
\max_{0 \le n \le N} \brak{g(n,X_n) - M^{\Theta}_n - \varepsilon_n}
\]
are square-integrable too. Hence, one obtains from the central limit theorem that
for large $K_L$, $\hat{L}$ is approximately normally
distributed with mean $L$ and variance $\hat{\sigma}^2_L/K_L$ for
\[
\hat{\sigma}^2_L = \frac{1}{K_L-1} \sum_{k=1}^{K_L} \brak{g(l^k,y^k_{l^k})-\hat{L}}^2.
\]
So, for every $\alpha \in (0,1]$,
\[
\left[ \hat{L} - z_{\alpha/2} \frac{\hat{\sigma}_L}{\sqrt{K}_L} \, , \, \infty \right)
\]
is an asymptotically valid $1-\alpha/2$ confidence interval for $L$, where
$z_{\alpha/2}$ is the $1-\alpha/2$ quantile of the standard normal distribution.
Similarly,
\[
\left(-\infty \, , \, \hat{U} + z_{\alpha/2} \frac{ \hat{\sigma}_U}{\sqrt{K}_U} \right] \quad \mbox{with} \quad
\hat{\sigma}^2_U = \frac{1}{K_U -1} \sum_{k=1}^{K_U} \brak{\max_{0 \le n \le N} \brak{g \brak{n,z^k_n} - M^k_n} - \hat{U}}^2,
\]
is an asymptotically valid $1- \alpha/2$ confidence interval for $U$. It follows that for every constant $\varepsilon > 0$,
one has
\begin{eqnarray*}
&& \mathbb{P} \edg{V_0 < \hat{L} - z_{\alpha/2} \frac{\hat{\sigma}_L}{\sqrt{K}_L} \; \; \mbox{ or } \;\;
V_0 > \hat{U} + z_{\alpha/2} \frac{\hat{\sigma}_U}{\sqrt{K}_U}}\\
&& \le \mathbb{P} \edg{L < \hat{L} - z_{\alpha/2} \frac{\hat{\sigma}_L}{\sqrt{K}_L}}
+ \mathbb{P} \edg{
U > \hat{U} + z_{\alpha/2} \frac{\hat{\sigma}_U}{\sqrt{K}_U}}
\le \alpha + \varepsilon
\end{eqnarray*}
as soon as $K_L$ and $K_U$ are large enough. In particular,
\begin{equation} \label{ci}
\left[\hat{L} - z_{\alpha/2} \frac{\hat{\sigma}_L}{\sqrt{K}_L} \, , \,
\hat{U} + z_{\alpha/2} \frac{\hat{\sigma}_U}{\sqrt{K}_U} \right]
\end{equation}
is an asymptotically valid $1- \alpha$ confidence interval for $V_0$.
\section{Examples}
\label{sec:ex}
In this section we test\footnote{All computations were performed in single precision (float32)
on a NVIDIA GeForce GTX 1080 GPU with 1974 MHz core clock and 8 GB GDDR5X memory
with 1809.5 MHz clock rate. The underlying system consisted of an Intel Core i7-6800K 3.4
GHz CPU with 64 GB DDR4-2133 memory running
Tensorflow 1.11 on Ubuntu 16.04.}
our method on three examples: the pricing of a Bermudan max-call option,
the pricing of a callable multi barrier reverse convertible and
the problem of optimally stopping a fractional Brownian motion.
\subsection{Bermudan max-call options}
\label{subsec:maxcall}
Bermudan max-call options are one of the most studied examples in the numerics literature on
optimal stopping problems; see, e.g., \cite{LS01,R02, G03, BKT03, HK04, BG04, AB04, BC08,
BS08, Be11AA, Be13, JO15, Le16}. Their payoff depends on the maximum of $d$ underlying assets.
Assume the risk-neutral dynamics of the assets are given by a multi-dimensional Black--Scholes
model\footnote{We make this assumption so that we can compare our results to those obtained
with different methods in the literature. But our approach works for any asset dynamics as long
as it can efficiently be simulated.}
\begin{equation} \label{BS}
S^i_t = s^i_0 \exp\!\brak{[r-\delta_i - \sigma^2_i/2] t + \sigma_i W^i_t}, \quad
i = 1, 2, \dots, d,
\end{equation}
for initial values $ s^i_0 \in (0,\infty)$, a risk-free interest rate $r \in \mathbb{R}$,
dividend yields $\delta_i \in [0,\infty)$, volatilities $\sigma_i \in (0,\infty)$ and
a $d$-dimensional Brownian motion $W$ with constant instantaneous correlations\footnote{That is,
$\mathbb{E}[(W^i_t-W^i_s)( W^j_t- W^i_s)] = \rho_{ij}(t-s)$ for all $i \neq j$ and $s < t$.} $\rho_{ij} \in \mathbb{R}$
between different components $W^i$ and $W^j$.
A Bermudan max-call option on $ S^1, S^2, \dots, S^d$ has payoff
$\brak{\max_{1 \le i \le d} S^i_t - K}^+
$
and can be exercised at any point of a time grid $0=t_0 <t_1< \dots < t_N$. Its price is given by
\[
\sup_{\tau} \mathbb{E}\!\left[ e^{ - r \tau } \left(
\max_{ 1 \le i \le d } S^i_{ \tau } - K \right)^{ \! + } \right] ,
\]
where the supremum is over all $S$-stopping times taking values in $\crl{t_0,t_1, \dots, t_N}$; see, e.g., \cite{S02}.
Denote $X^i_n = S^i_{t_n}$, $n = 0,1,\dots, N$, and let ${\cal T}$ be the set of $X$-stopping times. Then
the price can be written as $\sup_{\tau \in {\cal T}} \mathbb{E} \, g(\tau, X_{\tau} )$ for
\[
g(n,x) = e^{-r t_n} \brak{\max_{1 \le i \le d} x^i - K}^+,
\]
and it is straight-forward to simulate $(X_n)_{n=0}^N$.
In the following we assume the time grid to be of the form $t_n = nT/N$, $n =0,1,\dots,N$, for a maturity
$T > 0$ and $N+1$ equidistant exercise dates. Even though $g(n,X_n)$ does not carry any information
that is not already contained in $X_n$, our method worked more efficiently when we trained the optimal
stopping decisions on Monte Carlo simulations of the $d+1$-dimensional Markov process
$(Y_n)_{n=0}^N = (X_n,g(n,X_n))_{n=0}^N$ instead of $(X_n)_{n=0}^N$.
Since $Y_0$ is deterministic, we first trained stopping times $\tau_1 \in {\cal T}_1$ of the form
\[
\tau_1 = \sum_{n=1}^{N} n f^{\theta_n}(Y_n) \prod_{j=1}^{n-1} (1-f^{\theta_j}(Y_k))
\]
for $f^{\theta_N} \equiv 1$ and $f^{\theta_1}, \dots, f^{\theta_{N-1}} \colon \mathbb{R}^{d+1} \to \crl{0,1}$
given by \eqref{ftheta} with $I=3$ and $q_1 = q_2 = d+40$. Then we determined our
candidate optimal stopping times as
\[
\tau^{\Theta} =
\begin{cases}
0 & \mbox{if } f_0 = 1\\
\tau_1 & \mbox{if } f_0 = 0
\end{cases}
\]
for a constant $f_0 \in \crl{0,1}$ depending\footnote{In fact, in none of the examples
in this paper it is optimal to stop at time $0$. So $\tau^{\Theta} = \tau_1$ in all these cases.}
on whether it was optimal to stop immediately at time $0$ or not (see Remark \ref{Rem:I0} above).
It is straight-forward to simulate from model \eqref{BS}.
We conducted 3,00$0+d$ training steps, in each of which we generated a batch of
8,192 paths of $(X_n)_{n=0}^N$. To estimate the lower bound $L$ we simulated $K_L = 4,$096,000 trial paths.
For our estimate of the upper bound $U$, we produced $K_U = 1$,024 paths $(z^k_n)_{n = 0}^N$,
$k=1, \dots, K_U$, of $(X_n)_{n=0}^N$ and $K_U \times J$ realizations
$(v^{k,j}_n)_{n=1}^N$, $k = 1, \dots, K_U$, $j =1, \dots, J$, of $(W_{t_n}- W_{t_{n-1}})_{n=1}^N$
with $J = 16,$384. Then for all $n$ and $k$, we generated the $i$-th component of the $j$-th continuation
path departing from $z^k_n$ according to
\[
\tilde{z}^{i,k,j}_m = z^{i,k}_n
\exp \brak{[r- \delta_i - \sigma^2_i/2] (m-n) \Delta t + \sigma_i[v^{i,k,j}_{n+1} + \dots + v^{i,k,j}_m]}, \quad m = n+1, \dots, N.
\]
\medskip
{\bf Symmetric case}\\
We first considered the special case, where $s^i_0 = s_0$, $\delta_i = \delta$, $\sigma_i = \sigma$
for all $i =1, \dots, d,$ and $\rho_{ij} = \rho$ for all $i \neq j$. Our results are reported in
Table~\ref{table:symm}.
\medskip
{\bf Asymmetric case}\\
As a second example, we studied model \eqref{BS} with
$s^i_0 = s_0$, $\delta_i = \delta$ for all $i = 1, 2, \dots, d,$ and $\rho_{ij} = \rho$ for
all $i \neq j$, but different volatilities $\sigma_1 < \sigma_2 < \dots < \sigma_d$.
For $d \le 5$, we chose the specification $\sigma_i = 0.08 + 0.32 \times (i-1)/(d-1)$, $i=1,2, \dots, d$.
For $d > 5$, we set $\sigma_i = 0.1 + i/(2d)$, $i = 1,2, \dots, d$. The results are given in
Table~\ref{table:asymm}.
\begin{table}
\centering
\begin{small}
\begin{tabular}{c c c c c c c c c c}
\hline \\[-3mm]
$d$ & $s_0$ & $\hat{L}$ & $t_L$ & $\hat{U}$ & $t_U$ & Point est.\ & $95\%$ CI & Binomial & BC $95\%$ CI\\[1mm]
\hline \\[-3mm]
$2$ & $90$ & $8.072$ & $28.7$ & $8.075$ & $25.4$ & $8.074$ & $[8.060, 8.081]$ & $8.075$ &\\
$2$ & $100$ & $13.895$ & $28.7$ & $13.903$ & $25.3$ & $13.899$ & $[13.880, 13.910]$ & $13.902$ &\\
$2$ & $110$ & $21.353$ & $28.4$ & $21.346$ & $25.3$ & $21.349$ & $[21.336, 21.354]$ & $21.345$ &\\[1mm]
$3$ & $90$ & $11.290$ & $28.8$ & $11.283$ & $26.3$ & $11.287$ & $[11.276, 11.290]$ & $11.29$ &\\
$3$ & $100$ & $18.690$ & $28.9$ & $18.691$ & $26.4$ & $18.690$ & $[18.673, 18.699]$ & $18.69$ &\\
$3$ & $110$ & $27.564$ & $27.6$ & $27.581$ & $26.3$ & $27.573$ & $[27.545, 27.591]$ & $27.58$&\\[1mm]
$5$ & $90$ & $16.648$ & $27.6$ & $16.640$ & $28.4$ & $16.644$ & $[16.633, 16.648]$ & & $[16.620, 16.653]$\\
$5$ & $100$ & $26.156$ & $28.1$ & $26.162$ & $28.3$ & $26.159$ & $[26.138, 26.174]$ & & $[26.115, 26.164]$\\
$5$ & $110$ & $36.766$ & $27.7$ & $36.777$ & $28.4$ & $36.772$ & $[36.745, 36.789]$ & & $[36.710, 36.798]$\\[1mm]
$10$ & $90$ & $26.208$ & $30.4$ & $26.272$ & $33.9$ & $26.240$ & $[26.189, 26.289]$ & &\\
$10$ & $100$ & $38.321$ & $30.5$ & $38.353$ & $34.0$ & $38.337$ & $[38.300, 38.367]$ & &\\
$10$ & $110$ & $50.857$ & $30.8$ & $50.914$ & $34.0$ & $50.886$ & $[50.834, 50.937]$ & &\\[1mm]
$20$ & $90$ & $37.701$ & $37.2$ & $37.903$ & $44.5$ & $37.802$ & $[37.681, 37.942]$ & &\\
$20$ & $100$ & $51.571$ & $37.5$ & $51.765$ & $44.3$ & $51.668$ & $[51.549, 51.803]$ & &\\
$20$ & $110$ & $65.494$ & $37.3$ & $65.762$ & $44.4$ & $65.628$ & $[65.470, 65.812]$ & &\\[1mm]
$30$ & $90$ & $44.797$ & $45.1$ & $45.110$ & $56.2$ & $44.953$ & $[44.777, 45.161]$ & &\\
$30$ & $100$ & $59.498$ & $45.5$ & $59.820$ & $56.3$ & $59.659$ & $[59.476, 59.872]$ & &\\
$30$ & $110$ & $74.221$ & $45.3$ & $74.515$ & $56.2$ & $74.368$ & $[74.196, 74.566]$ & &\\[1mm]
$50$ & $90$ & $53.903$ & $58.7$ & $54.211$ & $79.3$ & $54.057$ & $[53.883, 54.266]$ & &\\
$50$ & $100$ & $69.582$ & $59.1$ & $69.889$ & $79.3$ & $69.736$ & $[69.560, 69.945]$ & &\\
$50$ & $110$ & $85.229$ & $59.0$ & $85.697$ & $79.3$ & $85.463$ & $[85.204, 85.763]$ & &\\[1mm]
$100$ & $90$ & $66.342$ & $95.5$ & $66.771$ & $147.7$ & $66.556$ & $[66.321, 66.842]$ & &\\
$100$ & $100$ & $83.380$ & $95.9$ & $83.787$ & $147.7$ & $83.584$ & $[83.357, 83.862]$ & &\\
$100$ & $110$ & $100.420$ & $95.4$ & $100.906$ & $147.7$ & $100.663$ & $[100.394, 100.989]$ & &\\[1mm]
$200$ & $90$ & $78.993$ & $170.9$ & $79.355$ & $274.6$ & $79.174$ & $[78.971, 79.416]$ & &\\
$200$ & $100$ & $97.405$ & $170.1$ & $97.819$ & $274.3$ & $97.612$ & $[97.381, 97.889]$ & &\\
$200$ & $110$ & $115.800$ & $170.6$ & $116.377$ & $274.5$ & $116.088$ & $[115.774, 116.472]$ & &\\[1mm]
$500$ & $90$ & $95.956$ & $493.4$ & $96.337$ & $761.2$ & $96.147$ & $[95.934, 96.407]$ & &\\
$500$ & $100$ & $116.235$ & $493.5$ & $116.616$ & $761.7$ & $116.425$ & $[116.210, 116.685]$ & &\\
$500$ & $110$ & $136.547$ & $493.7$ & $136.983$ & $761.4$ & $136.765$ & $[136.521, 137.064]$ & &\\[1mm]
\hline
\end{tabular}
\caption{\label{table:symm}Summary results for max-call options on $d$ symmetric assets for parameter values of
$r = 5\%$, $\delta = 10\%$, $\sigma = 20\%$, $\rho = 0$, $K=100$, $T=3$, $N=9$.
$t_L$ is the number of seconds it took to train $\tau^{\Theta}$ and compute $\hat{L}$.
$t_U$ is the computation time for $\hat{U}$ in seconds.
95\% CI is the 95\% confidence interval \eqref{ci}. The binomial values were calculated
with a binomial lattice method in \cite{AB04}. BC 95\% CI is the 95\% confidence interval computed in \cite{BC08}.}
\end{small}
\end{table}
\begin{table}
\centering
\begin{small}
\begin{tabular}{c c c c c c c c c}
\hline \\[-3mm]
$d$ & $s_0$ & $\hat{L}$ & $t_L$ & $\hat{U}$ & $t_U$ & Point est.\ & $95\%$ CI & BC $95\%$ CI\\[1mm]
\hline \\[-3mm]
$2$ & $90$ & $14.325$ & $26.8$ & $14.352$ & $25.4$ & $14.339$ & $[14.299, 14.367]$ &\\
$2$ & $100$ & $19.802$ & $27.0$ & $19.813$ & $25.5$ & $19.808$ & $[19.772, 19.829]$ &\\
$2$ & $110$ & $27.170$ & $26.5$ & $27.147$ & $25.4$ & $27.158$ & $[27.138, 27.163]$ &\\[1mm]
$3$ & $90$ & $19.093$ & $26.8$ & $19.089$ & $26.5$ & $19.091$ & $[19.065, 19.104]$ &\\
$3$ & $100$ & $26.680$ & $27.5$ & $26.684$ & $26.4$ & $26.682$ & $[26.648, 26.701]$ &\\
$3$ & $110$ & $35.842$ & $26.5$ & $35.817$ & $26.5$ & $35.829$ & $[35.806, 35.835]$ &\\[1mm]
$5$ & $90$ & $27.662$ & $28.0$ & $27.662$ & $28.6$ & $27.662$ & $[27.630, 27.680]$ & $[27.468, 27.686]$\\
$5$ & $100$ & $37.976$ & $27.5$ & $37.995$ & $28.6$ & $37.985$ & $[37.940, 38.014]$ & $[37.730, 38.020]$\\
$5$ & $110$ & $49.485$ & $28.2$ & $49.513$ & $28.5$ & $49.499$ & $[49.445, 49.533]$ & $[49.155, 49.531]$\\[1mm]
$10$ & $90$ & $85.937$ & $31.8$ & $86.037$ & $34.4$ & $85.987$ & $[85.857, 86.087]$ &\\
$10$ & $100$ & $104.692$ & $30.9$ & $104.791$ & $34.2$ & $104.741$ & $[104.603, 104.864]$ &\\
$10$ & $110$ & $123.668$ & $31.0$ & $123.823$ & $34.4$ & $123.745$ & $[123.570, 123.904]$ &\\[1mm]
$20$ & $90$ & $125.916$ & $38.4$ & $126.275$ & $45.6$ & $126.095$ & $[125.819, 126.383]$ &\\
$20$ & $100$ & $149.587$ & $38.2$ & $149.970$ & $45.2$ & $149.779$ & $[149.480, 150.053]$ &\\
$20$ & $110$ & $173.262$ & $38.4$ & $173.809$ & $45.3$ & $173.536$ & $[173.144, 173.937]$ &\\[1mm]
$30$ & $90$ & $154.486$ & $46.5$ & $154.913$ & $57.5$ & $154.699$ & $[154.378, 155.039]$ &\\
$30$ & $100$ & $181.275$ & $46.4$ & $181.898$ & $57.5$ & $181.586$ & $[181.155, 182.033]$ &\\
$30$ & $110$ & $208.223$ & $46.4$ & $208.891$ & $57.4$ & $208.557$ & $[208.091, 209.086]$ &\\[1mm]
$50$ & $90$ & $195.918$ & $60.7$ & $196.724$ & $81.1$ & $196.321$ & $[195.793, 196.963]$ &\\
$50$ & $100$ & $227.386$ & $60.7$ & $228.386$ & $81.0$ & $227.886$ & $[227.247, 228.605]$ &\\
$50$ & $110$ & $258.813$ & $60.7$ & $259.830$ & $81.1$ & $259.321$ & $[258.661, 260.092]$ &\\[1mm]
$100$ & $90$ & $263.193$ & $98.5$ & $264.164$ & $151.2$ & $263.679$ & $[263.043, 264.425]$ &\\
$100$ & $100$ & $302.090$ & $98.2$ & $303.441$ & $151.2$ & $302.765$ & $[301.924, 303.843]$ &\\
$100$ & $110$ & $340.763$ & $97.8$ & $342.387$ & $151.1$ & $341.575$ & $[340.580, 342.781]$ &\\[1mm]
$200$ & $90$ & $344.575$ & $175.4$ & $345.717$ & $281.0$ & $345.146$ & $[344.397, 346.134]$ &\\
$200$ & $100$ & $392.193$ & $175.1$ & $393.723$ & $280.7$ & $392.958$ & $[391.996, 394.052]$ &\\
$200$ & $110$ & $440.037$ & $175.1$ & $441.594$ & $280.8$ & $440.815$ & $[439.819, 441.990]$ &\\[1mm]
$500$ & $90$ & $476.293$ & $504.5$ & $477.911$ & $760.7$ & $477.102$ & $[476.069, 478.481]$ &\\
$500$ & $100$ & $538.748$ & $504.6$ & $540.407$ & $761.6$ & $539.577$ & $[538.499, 540.817]$ &\\
$500$ & $110$ & $601.261$ & $504.9$ & $603.243$ & $760.8$ & $602.252$ & $[600.988, 603.707]$ &\\[1mm]
\hline
\end{tabular}
\caption{\label{table:asymm}Summary results for max-call options on $d$ asymmetric assets for parameter values of
$r = 5\%$, $\delta = 10\%$, $\rho = 0$, $K=100$, $T=3$, $N=9$.
$t_L$ is the number of seconds it took to train $\tau^{\Theta}$ and compute $\hat{L}$.
$t_U$ is the computation time for $\hat{U}$ in seconds.
95\% CI is the 95\% confidence interval \eqref{ci}. BC 95\% CI is
the 95\% confidence interval computed in \cite{BC08}.}
\end{small}
\end{table}
\subsection{Callable multi barrier reverse convertibles}
\label{subsec:callable}
A MBRC is a coupon paying security that converts into shares of the worst-performing
of $d$ underlying assets if a prespecified trigger event occurs.
Let us assume that the price of the $i$-th underlying asset in percent of its starting value
follows the risk-neutral dynamics
\begin{equation} \label{BS2}
S^i_t =
\begin{cases}100 \exp\!\brak{[r - \sigma^2_i/2] t + \sigma_i W^i_t} & \mbox{for } t \in [0 , T_i)\\
100 (1-\delta_i) \exp\!\brak{[r - \sigma^2_i/2] t + \sigma_i W^i_t} & \mbox{for } t \in [T_i,T]
\end{cases}
\end{equation}
for a risk-free interest rate $r \in \mathbb{R}$, volatility $\sigma_i \in (0,\infty)$, maturity $T \in (0,\infty)$,
dividend payment time $T_i \in (0,T)$, dividend rate $\delta_i \in [0,\infty)$
and a $d$-dimensional Brownian motion $W$ with constant instantaneous
correlations $\rho_{ij} \in \mathbb{R}$ between different components $W^i$ and $W^j$.
Let us consider a MBRC that pays a coupon $c$ at each of $N$ time points $t_n = nT/N$, $n = 1, 2, \dots, N$,
and makes a time-$T$ payment of
\[
G = \begin{cases}
F & \mbox{ if } \min_{1 \le i \le d} \min_{1 \le m \le M} S^i_{u_m} > B \mbox{ or } \min_{1 \le i \le d} S^i_T > K\\
\min_{1 \le i \le d} S^i_T & \mbox{ if } \min_{1 \le i \le d} \min_{1 \le m \le M} S^i_{u_m} \le B \mbox{ and } \min_{1 \le i \le d} S^i_T \le K,
\end{cases}
\]
where $F \in [0,\infty)$ is the nominal amount, $B \in [0,\infty)$ a barrier,
$K \in [0,\infty)$ a strike price and $u_m$ the end of the $m$-th trading day.
Its value is
\begin{equation} \label{payoff}
\sum_{n=1}^N e^{-rt_n} c + e^{-rT} \mathbb{E} \, G
\end{equation}
and can easily be estimated with a standard Monte Carlo approximation.
A callable MBRC can be redeemed by the issuer at any of the times
$t_1, t_2, \dots, t_{N-1}$ by paying back the notional. To minimize costs, the issuer will
try to find a $\crl{t_1, t_2, \dots, T}$-valued stopping time such that
\[
\mathbb{E} \edg{\sum_{n=1}^{\tau} e^{-r t_n} c + 1_{\crl{\tau < T}} e^{-r \tau} F + 1_{\crl{\tau = T}} e^{-r T} G}
\]
is minimal.
Let $(X_n)_{n=1}^N$ be the $d+1$-dimensional Markov process given by
$X^i_n = S^i_{t_n}$ for $i = 1, \dots, d$, and
\[
X^{d+1}_n := \begin{cases}
1 & \mbox{ if the barrier has been breached before or at time $t_n$}\\
0 & \mbox{ else}.
\end{cases}
\]
Then the issuer's minimization problem can be written as
\begin{equation} \label{min}
\inf_{\tau \in {\cal T}} \mathbb{E} \, g(\tau, X_{\tau}),
\end{equation}
where ${\cal T}$ is the set of all $X$-stopping times and
\[
g(n,x) = \begin{cases}
\sum_{m=1}^n e^{-r t_m} c + e^{-rt_n} F & \mbox{ if } 1 \le n \le N-1 \mbox{ or } x^{d+1}=0\\
\sum_{m=1}^N e^{-r t_m} c + e^{-r t_N} h(x) & \mbox{ if } n = N \mbox{ and } x^{d+1} = 1,
\end{cases}
\]
where
\[
h(x) = \begin{cases}
F & \mbox{ if } \min_{1 \le i \le d} x^i > K\\
\min_{1 \le i \le d} x^i & \mbox{ if } \min_{1 \le i \le d} x^i \le K.
\end{cases}
\]
Since the issuer cannot redeem at time $0$, we trained stopping times of the form
\[
\tau^{\Theta} = \sum_{n=1}^{N} n f^{\theta_n}(Y_n) \prod_{j=1}^{n-1} (1-f^{\theta_j}(Y_k)) \in {\cal T}_1
\]
for $f^{\theta_N} \equiv 1$ and $f^{\theta_1}, \dots, f^{\theta_{N-1}} \colon \mathbb{R}^{d+1} \to \crl{0,1}$
given by \eqref{ftheta} with $I=3$ and $q_1 = q_2 = d+ 40$. Since \eqref{min} is a minimization problem,
$\tau^{\Theta}$ yields an upper bound and the dual method a lower bound.
We simulated the model \eqref{BS2} like \eqref{BS} in Subsection \ref{subsec:maxcall} with the same
number of trials except that here we used the lower number $J = 1,$024 to estimate the dual bound.
Numerical results are reported in Table~\ref{table:rbc}.
\begin{table}
\centering
\begin{small}
\begin{tabular}{c c c c c c c c c}
\hline \\[-3mm]
$d$ & $\rho$ & $\hat{L}$ & $t_L$ & $\hat{U}$ & $t_U$ & Point est.\ & $95\%$ CI & Non-callable\ \\[1mm]
\hline \\[-3mm]
$2$ & $0.6$ & $98.235$ & $24.9$ & $98.252$ & $204.1$ & $98.243$ & $[98.213,98.263]$ & $106.285$ \\
$2$ & $0.1$ & $97.634$ & $24.9$ & $97.634$ & $198.8$ & $97.634$ & $[97.609,97.646]$ & $106.112$ \\
$3$ & $0.6$ & $96.930$ & $26.0$ & $96.936$ & $212.9$ & $96.933$ & $[96.906,96.948]$ & $105.994$ \\
$3$ & $0.1$ & $95.244$ & $26.2$ & $95.244$ & $211.4$ & $95.244$ & $[95.216,95.258]$ & $105.553$ \\
$5$ & $0.6$ & $94.865$ & $41.0$ & $94.880$ & $239.2$ & $94.872$ & $[94.837,94.894]$ & $105.530$ \\
$5$ & $0.1$ & $90.807$ & $41.1$ & $90.812$ & $238.4$ & $90.810$ & $[90.775,90.828]$ & $104.496$ \\
$10$ & $0.6$ & $91.568$ & $71.3$ & $91.629$ & $300.9$ & $91.599$ & $[91.536,91.645]$ & $104.772$ \\
$10$ & $0.1$ & $83.110$ & $71.7$ & $83.137$ & $301.8$ & $83.123$ & $[83.078,83.153]$ & $102.495$ \\
$15$ & $0.6$ & $89.558$ & $94.9$ & $89.653$ & $359.8$ & $89.606$ & $[89.521,89.670]$ & $104.279$ \\
$15$ & $0.1$ & $78.495$ & $94.7$ & $78.557$ & $360.5$ & $78.526$ & $[78.459,78.571]$ & $101.209$ \\
$30$ & $0.6$ & $86.089$ & $158.5$ & $86.163$ & $534.1$ & $86.126$ & $[86.041,86.180]$ & $103.385$\\
$30$ & $0.1$ & $72.037$ & $159.3$ & $72.749$ & $535.6$ & $72.393$ & $[71.830,72.760]$ & $99.279$\\[1mm]
\hline
\end{tabular}
\caption{\label{table:rbc}Summary results for callable MBRCs with $d$ underlying assets for
$F = K = 100$, $B = 70$, $T = 1$ year ($= 252$ trading days), $N = 12$, $c = 7/12$, $\delta_i = 5\%$,
$T_i = 1/2$, $r = 0$, $\sigma_i = 0.2$ and $\rho_{ij} = \rho$ for $i \neq j$.
$t_U$ is the number of seconds it took to train $\tau^{\Theta}$ and compute $\hat{U}$.
$t_L$ is the number of seconds it took to compute $\hat{L}$. The last column lists
fair values of the same MBRCs without the callable feature. We estimated them
by averaging 4,096,000 Monte Carlo samples of the payoff. This took between 5 (for $d=2$) and
44 (for $d$ = 30) seconds.}
\end{small}
\end{table}
\subsection{Optimally stopping a fractional Brownian motion}
\label{subsec:fBm}
A fractional Brownian motion with Hurst parameter $ H \in (0,1] $
is a continuous centered Gaussian process $ ( W^H_t )_{ t \ge 0 } $ with covariance structure
\[
\mathbb{E}[W^H_t W^H_s] = \frac{1}{2} \brak{t^{2H} + s^{2H} - |t-s|^{2H}};
\]
see, e.g., \cite{MVN,ST}. For $ H = 1/2$, $W^H$ is a standard Brownian motion. So, by the
optional stopping theorem, one has $\mathbb{E}\, W^{1/2}_{ \tau }= 0$ for every $W^{1/2}$-stopping time $\tau$
bounded above by a constant; see, e.g., \cite{GS01}. However,
for $ H \neq 1/2$, the increments of $W^H$ are correlated --
positively for $ H \in (1/2, 1 ]$ and negatively for $ H \in ( 0, 1/2) $.
In both cases, $W^H$ is neither a martingale nor a Markov process,
and there exist bounded $W^H $-stopping times $\tau$ such that $ \mathbb{E}\, W^H_{ \tau }> 0$;
see, e.g., \cite{KG16} for two classes of simple stopping rules $ 0 \leq \tau \le 1 $
and estimates of the corresponding expected values $\mathbb{E} \, W^H_{ \tau }$.
To approximate the supremum
\begin{equation} \label{stfBm}
\sup_{ 0 \leq \tau \le 1 }
\mathbb{E}\, W^H_{\tau}
\end{equation} over all $W^H$-stopping times $ 0 \leq \tau \le 1 $,
we denote $ t_n = n/100$, $n = 0, 1, 2, \dots, 100$, and introduce the
$100$-dimensional Markov process $(X_n)_{n=0}^{100}$ given by
\[
\begin{split}
X_0 &= (0,0,\dots, 0)\\
X_1 &= (W^H_{t_1},0, \dots, 0)\\
X_2 &= (W^H_{t_2}, W^H_{t_1}, 0, \dots, 0)\\
\vdots & \\
X_{100} &= (W^H_{t_{100}}, W^H_{t_{99}}, \dots, W^H_{t_1}).
\end{split}
\]
The discretized stopping problem
\begin{equation} \label{disfBm}
\sup_{ \tau \in {\cal T} } \mathbb{E} \, g(X_{\tau}),
\end{equation}
where ${\cal T}$ is the set of all $X$-stopping times and $g \colon \mathbb{R}^{100} \to \mathbb{R}$
the projection $(x^1, \dots, x^{100}) \mapsto x^1$, approximates \eqref{stfBm} from below.
We computed estimates of \eqref{disfBm} for $H \in \{ 0.01, 0.05, 0.1, 0.15, \dots, 1 \}$ by
training networks of the form \eqref{ftheta} with depth $I = 3$, $d = 100 $ and $q_1 = q_2 = 140$.
To simulate the vector $Y = (W^H_{t_n})_{n=0}^{100}$, we used the representation $Y = B Z$,
where $BB^T$ is the Cholesky decomposition of the covariance matrix
of $Y$ and $Z$ a $100$-dimensional random vector with independent standard normal components.
We carried out 6,000 training steps with a batch size of 2,048. To estimate the lower bound $L$ we
generated $K_L = 4,$096,000 simulations of $Z$. For our estimate of the upper bound $U$, we first
simulated $K_U = 1,$024 realizations $v^k$, $k = 1, \dots, K_U$ of $Z$ and set $w^k = B v^k$.
Then we produced another $K_U \times J$ simulations $\tilde{v}^{k,j}$,
$k=1, \dots, K_U$, $j=1, \dots, J$, of $Z$, and generated for all $n$ and $k$, continuation paths starting from
\[
z^k_n = (w^k_n, \dots, w^k_1, 0 , \dots, 0)
\]
according to
\[
\tilde{z}^{k,j}_m
= (\tilde{w}^{k,j}_m, \dots, \tilde{w}^{k,j}_{n+1}, w^k_n, \dots, w^k_1, 0 \dots,0), \quad
m = n+1, \dots, 100,
\]
with
\[
\tilde{w}^{k,j}_l = \sum_{i=1}^n B_{li} v^k_i + \sum_{i=n+1}^l B_{li} \tilde{v}^{k,j}_i, \quad
l = n+1, \dots, m.
\]
For $H \in \crl{0.01,...,0.4} \cup \crl{0.6,...,1.0}$, we chose $J = 16,$384, and for
$H \in \crl{0.45,0.5,0.55}$, $J = 32,$768. The results are listed in Table~\ref{table:fbm}
and depicted in graphical form in Figure~\ref{fig:fbm}.
Note that for $H = 1/2$ and $H=1$, our 95\% confidence intervals contain the true values, which
in these two cases, can be calculated exactly. As mentioned above, $W^{1/2}$ is a Brownian motion, and therefore,
$\mathbb{E} \, W^{1/2}_{\tau} = 0$ for every $(W^{1/2}_{t_n})_{n = 0}^{100}$-stopping time $\tau$.
On the other hand, one has\footnote{\label{fn2}up to $\mathbb{P}$-almost sure equality} $W^1_t = t W^1_1$, $t \ge 0$.
So, in this case, the optimal stopping time is
given\footref{fn2} by
\[
\tau = \begin{cases}
1 & \mbox{ if } W^1_{t_1} > 0\\
t_1 & \mbox{ if } W^1_{t_1} \le 0,
\end{cases}
\]
and the corresponding expectation by
\[
\mathbb{E} \, W^1_{\tau} = \mathbb{E} \edg{W^1_1 1_{\crl{W^1_{t_1} > 0}} - W^1_{t_1} 1_{\crl{W^1_{t_1} \le 0}}}
= 0.99 \, \mathbb{E} \edg{W^1_1 1_{\crl{W^1_1 > 0}}} = 0.99/\sqrt{2 \pi} = 0.39495...
\]
Moreover, it can be seen that for $H \in (1/2,1)$, our estimates are up to
three times higher than the expected payoffs generated by the heuristic stopping rules of \cite{KG16}.
For $H \in(0, 1/2)$, they are up to five times higher.
\begin{table}
\centering
\begin{small}
\begin{tabular}{c c c c c c c}
\hline \\[-3mm]
$H$ & $\hat{L}$ & $\hat{U}$ & Point est.\ & $95\%$ CI\\[1mm]
\hline \\[-3mm]
$0.01$ & $1.518$ & $1.519$ & $1.519$ & $[1.517,1.520]$\\
$0.05$ & $1.293$ & $1.293$ & $1.293$ & $[1.292,1.294]$\\
$0.10$ & $1.048$ & $1.049$ & $1.049$ & $[1.048,1.050]$\\
$0.15$ & $0.838$ & $0.839$ & $0.839$ & $[0.838,0.840]$\\
$0.20$ & $0.658$ & $0.659$ & $0.658$ & $[0.657,0.659]$\\
$0.25$ & $0.501$ & $0.504$ & $0.503$ & $[0.501,0.505]$\\
$0.30$ & $0.369$ & $0.370$ & $0.370$ & $[0.368,0.371]$\\
$0.35$ & $0.255$ & $0.256$ & $0.255$ & $[0.254,0.257]$\\
$0.40$ & $0.155$ & $0.158$ & $0.156$ & $[0.154,0.158]$\\
$0.45$ & $0.067$ & $0.075$ & $0.071$ & $[0.066,0.075]$\\
$0.50$ & $0.000$ & $0.005$ & $0.002$ & $[0.000,0.005]$\\
$0.55$ & $0.057$ & $0.065$ & $0.061$ & $[0.057,0.065]$\\
$0.60$ & $0.115$ & $0.118$ & $0.117$ & $[0.115,0.119]$\\
$0.65$ & $0.163$ & $0.165$ & $0.164$ & $[0.163,0.166]$\\
$0.70$ & $0.206$ & $0.207$ & $0.207$ & $[0.205,0.208]$\\
$0.75$ & $0.242$ & $0.245$ & $0.244$ & $[0.242,0.245]$\\
$0.80$ & $0.276$ & $0.278$ & $0.277$ & $[0.276,0.279]$\\
$0.85$ & $0.308$ & $0.309$ & $0.308$ & $[0.307,0.310]$\\
$0.90$ & $0.336$ & $0.339$ & $0.337$ & $[0.335,0.339]$\\
$0.95$ & $0.365$ & $0.367$ & $0.366$ & $[0.365,0.367]$\\
$1.00$ & $0.395$ & $0.395$ & $0.395$ & $[0.394,0.395]$\\[1mm]
\hline
\end{tabular}
\caption{ \label{table:fbm}Estimates of $\sup_{\tau \in \crl{0,t_1, \dots, 1}} \mathbb{E}\, W^H_{ \tau }$. For all
$H \in \crl{0.01, 0.05, \dots, 1}$, it took about 430 seconds to train $\tau^{\Theta}$ and compute $\hat{L}$.
The computation of $\hat{U}$ took about 17,000 seconds for $H \in \crl{0.01, \dots, 0.4} \cup \crl{0.6, \dots, 1}$ and
about 34,000 seconds for $H \in \crl{0.45,0.5,0.55}$.}
\end{small}
\end{table}
\begin{figure}
\centering
\includegraphics{figure.pdf}
\caption{\label{fig:fbm}Estimates of $\sup_{\tau \in \crl{0,t_1, \dots, 1}} \mathbb{E}\, W^H_{ \tau }$ for
different values of $H$.}
\end{figure}
|
{
"timestamp": "2020-01-07T02:11:59",
"yymm": "1804",
"arxiv_id": "1804.05394",
"language": "en",
"url": "https://arxiv.org/abs/1804.05394"
}
|
\section{Introduction}
Semantic and functional scene understanding is a crucial capability of
manipulation robots. In the Computer Vision community, this
challenging problem is often approached given only a single
image. However, a robot is able to physically interact with the
environment and thereby autonomously induce motion in the scene. This
motion creates a rich, visual sensory signal that would otherwise not
be present, thus facilitating better scene understanding. Methods that
exploit physical interaction to ease perception are often referred to as
performing \textit{Interactive Perception} (IP)~\citep{Bohg:IP:17}. In this
paper, we are providing the robot with a model to process the visual
effect of its interaction. Given two consecutive RGB-D images, we are
interested in estimating a dense 3D motion field of the environment,
also known as {\em scene flow}. We show how this result helps to
segment the finite, but unknown number of moving objects in the
scene. This can provide input to tasks such as for example grasp
planning or 3D object reconstruction.
\begin{figure}[t!]
\centering
\includegraphics[width=0.94\linewidth]{imgs/teaser_final3.PNG}
\caption{We present a neural network which learns to estimate object
segmentation and scene flow given a pair of RGB-D images. The data
undergoes spatial compression, correlation, and refinement to propose
object segmentations and transformations.}
\label{fig:teaser}
\end{figure}
We propose a model that takes advantage of the fact that in a common
household scenario, scenes often consist of a set of rigidly moving
objects. Our model jointly estimates (i) the segmentation of a scene
into a finite number of rigidly moving object, (ii) the motion
trajectories of these objects and (iii) the resulting {\em object
scene flow\/}~\citep{menze2015object}. We propose to use
a deep neural network architecture that takes as input a pair of
consecutive RGB-D images. See Fig.~\ref{fig:teaser} for an overview of
the approach. In a first stage, features are extracted from each of
the four input images. The RGB features are then correlated and the
resulting values are used to weight the feature encoding of the depth
data. Intuitively, this favors correspondences between points in the
depth data that also have a strong similarity in the RGB images. The
result is then decoded to produce three images containing the object
positions, their translation, and their rotation. From this, we can
infer the object scene flow and segmentation.
Our primary contributions are: (1) generating a
challenging, large-scale dataset for scene flow estimation with
ground-truth annotated RGB-D images, (2) treating rotational symmetry
of objects in scene flow prediction, (3) estimating object scene flow
with a deep neural network architecture, and (4) predicting rigid body
transformations to segment a finite, but unknown number of moving
objects.
\section{Related Work}
Estimating scene flow has a long-standing history in the research
community starting with \citet{vedula2005three}.
We briefly review the most recent approaches that are
related to our work in terms of several aspects: input sensor, data
sets, learning-based methods and motion segmentation.
\subsection{Scene Flow based on RGB-D or Stereo Images}
\citet{gottfried2011computing} were the first to use an RGB-D sensor
for scene flow estimation. Their work also addresses the necessary
calibration process. \citet{herbst2013rgb} generalize the two-frame
variational optical flow algorithm (2D) to scene flow (3D). The
resulting dense scene flow is then used for rigid motion
segmentation. \citet{jaimez2015primal} present the first real-time
method for computing dense scene flow from RGB-D images. Their method
is based on a variational formulation that imposes brightness and
geometric consistencies. The minimization problem is efficiently
solved with a GPU and a primal-dual algorithm. \citet{6751281} were
the first to propose the estimation of piecewise rigid scene flow where
oversegmentation into superpixels constrains the scene flow
estimation. The authors obtain a new level of accuracy that may run in
real-time. Inspired by this
work, \citet{DBLP:journals/corr/abs-1710-02124} propose a multi-frame
scene flow approach which jointly optimizes the consistency of the
patch appearances and their local motions from RGB-D image
sequences. However the reliance on bottom up cues for segmentation may
lead to oversegmentation of objects. \citet{menze2015object} defined
{\em object scene flow\/} as the 3D motion associated with a set of
pixels that constitute a rigidly-moving object. By assuming that the
scene consists of a set of such objects and encouraging superpixels in
the same region to have similar 3D motion, the authors constrain the
solution space for estimating scene flow. The inference process is
computationally very expensive, taking 2-50 minutes per image pair.
For computing a matching score
between pixels across stereo frames and over time, traditional
approaches often rely on assumptions like brightness constancy and
motion smoothness within a small region. In real scenes, these
assumptions are often broken for example with non-Lambertian surfaces,
occlusions or large displacements. These effects are prevalent when
multiple objects are moving fast and simultaneously over
time. Therefore, matching pixel positions over time is the most
vulnerable component in traditional methods. Our hypothesis is that
these challenges can be mitigated by using
methods that learn powerful features of the raw input
data over multiple spatial scales. Evidence comes from successful
learning-based approaches towards optical flow as detailed in
Sec.~\ref{sec:learning_flow}.
\subsection{Datasets}
Several large scale datasets exist for benchmarking and learning
optical and scene flow. Different from our data set, they are all
under a binocular setting with flow and disparity ground
truth. KITTI~\citep{Geiger2013IJRR} consists of 194 training and 195
test scenes recorded from a calibrated pair of cameras mounted on a
car. Ground truth annotations are obtained by combining data from a 3D
laser scanner with the car's ego motion. \citet{menze2015object}
annotated the dynamic scenes with 3D CAD models for all moving
vehicles and modified the dataset with 200 training scenes and 200
test scenes. KITTI contains valuable real world data. However, the ground
truth contains some approximation error.
\citet{mayer2016large} created a synthetic dataset called
FlyingThings3D containing over 35000 stereo frames with ground truth
scene flow annotations. When using data from stereo cameras,
insufficient texture can result in matching errors across frames and over
time. RGB-D cameras deliver
dense depth measurements despite a lack of texture. This data can
support the matching process. Therefore, our data set contains pairs
of consecutive RGB-D images and is of similar size as FlyingThings3D.
Different from the aforementioned datasets, objects in our dataset
are falling onto a surface, colliding with each others, and even sliding
on the surface. It is important for a manipulation robot to understand this type of non-smooth, physically-realistic motion
due to contact. The
objects in our scenes are also much closer to each other, leading to more
challenging occlusions and motion. And lastly, we use a new annotation
method to coherently label objects with rotational symmetry. See
Section~\ref{sec:dataset} for more details on our dataset.
\subsection{Learning-Based Flow Prediction}\label{sec:learning_flow}
Learning-based methods have up till now been mainly applied to optical
flow estimation. \citet{7410673} posed this problem as a supervised
learning problem and were the first to solve it with {\em
Convolutional Neural Networks\/} (CNNs). They compare two
architectures called FlownetS and FlownetC: a generic architecture and
an architecture that includes a layer that correlates feature vectors
at different image locations. These two FlowNets were tested on
datasets like Sintel~\citep{butler2012naturalistic} and
KITTI~\citep{Geiger2013IJRR} achieving competitive accuracy at frame
rates of 5-10 fps. \citet{DBLP:journals/corr/IlgMSKDB16} extend
FlowNet by developing a stacked architecture. It includes warping of
the second images with intermediate optical flow. The authors also propose a
subnetwork specializing in small displacements resulting in
state-of-the-art results while running at real-time. For
learning-based scene flow estimation, \citet{hadfield2014scene}
introduced a novel cost function. In this new formulation, only a
limited portion of the parameters from the entire pipeline are
learned, leading to limited improvements.
\citet{mayer2016large} utilized a CNN to estimate scene flow based on
stereo images. They embed a disparity estimation network called
DispNet into FlowNet~\citep{7410673}.
We propose an hourglass deep architecture that uses two RGB-D frames
as input. It adopts the correlation layer of FlowNetC for the
RGB encoding and uses this to associate encoded point cloud features. One of our main contributions is the decoder which directly predicts object position, translation and rotation. From this we can infer object scene flow and motion-based, rigid object segmentation.
\subsection{Motion-based Segmentation}
\citet{Bohg:IP:17} extensively review the variety of work towards
motion-based segmentation within robotics. Here, we discuss a few
representative examples. Many works use over-segmentations and connect
superpixels over time using clustering
methods~\citep{brendel2009video,grundmann2010efficient}. However, the
reliance on bottom-up cues often results in some remaining
oversegmentation.
The authors of \citep{cheriyadat2009non, brox2010object,zografos2014fast}
formulate the problem as clustering of point trajectories across
different frames and solve it based on spectral clustering
methods. Instead, \citet{rahmati2014motion} utilize multi-label graph
cuts. \citet{ji2014robust} define an unbalanced energy to model both,
motion segmentation and point matching. \citet{KB15b} formulate
motion-based segmentation based on point trajectories as a minimum
cost, multi-cut problem. The minimum cost multi-cut formulation allows
for varying cluster sizes.
We propose a model where each pixel directly predicts the center and
trajectory of the object that it is associated with. We achieve
accurate motion-based segmentation by clustering in this space. This
in turn helps to refine the scene flow estimate.
\section{Problem Formulation \& Notation}
The input
to the proposed model are two consecutive RGB-D images. We assume that
the environment consists of a finite, but unknown, number of rigidly
moving objects. The network outputs (i) a pixel-wise segmentation of
each object, (ii) the rigid body motion of each object, and (iii) the
scene flow of each pixel in a reference frame.
More formally, let $\mathcal{I}^{t}$ and $\mathcal{P}^{t}$ denote an RGB image and a point cloud from a single RGB-D image at time $t$. Time $t$ and $t-1$ refer to the current and previous frames, respectively.
To calculate scene flow of each point $P^t_i \in \mathcal{P}^{t}$ in a reference frame, we predict its 3D displacement by estimating its corresponding position $P^{t-1}_i$ in the previous frame. This estimate is denoted by $\hat{P}^{t-1}_i$.
Let $\mathcal{O}$ denote the set of rigidly moving objects in the scene. The rigid body motion between two consecutive frames for $\mathcal{O}_k$ is described by an SE(3) transform consisting of a rotation $R_k$ and translation $T_k$.
Our model directly outputs three images $\mathcal{Q}$, $\mathcal{T}$ and $\mathcal{X}$ where each pixel contains an estimate of the rotation, translation and center of the object that the pixel belongs to.
Therefore, if point $P^t_i$ is generated by $\mathcal{O}_k$ then the correct value at the projected image coordinates $(u,v)$ in the respective output images will contain the ground truth rotation, translation and center of object $\mathcal{O}_k$.
We denote the rotation of a point $P_i$ based on the axis-angle representation $Q_k$ as $r(P_i,Q_k) = R_kP_i$. Therefore, the corresponding point in frame $t-1$ can be computed by
\begin{equation}\label{eq:sceneflow}
P^{t-1}_i=r(P^{t}_i-X_k,Q_k)+X_k+T_{k}
\end{equation} with per-pixel scene flow $S_i=P^{t-1}_i-P^{t}_i$.
Note, that our model outputs an estimate of the ground truth variables $Q_k, T_k$ and $X_k$ which results in $\hat{P}^{t-1}_i$ instead of $P^{t-1}_i$ and therefore only in an estimate $\hat{S}_i$ of the ground truth scene flow. During training, we aim to minimize the error between these estimates and the ground truth.
Let $\xi_k = [X_k,X_k+T_k]$ be the trajectory feature of an object
$\mathcal{O}_k$. $X_k$ and $X_k+T_k$ are the object centers at frame
$t$ and $t-1$, respectively.
Unless two objects have exactly the same object center and move with exactly the same translation trajectory, each $\xi_k$ is unique per object. Therefore, we can use it as a cue for motion-based, object segmentation.
\section{Technical Approach}
\begin{figure*}[!htb]
\centering
\includegraphics[width=0.94\linewidth]{imgs/lin_full.png}
\caption{Network architecture utilized in this paper. The RGB-D input
is split into two components, RGB and XYZ, before being passed into
Siamese neural networks. A correlation is performed on the output of
the RGB Siamese network and applied to the XYZ features from time
$t-1$. After a max pooling layer, the newly combined features undergo
upconvolutions. The output of the upconvolutions is fed into 3
different layers that predict the center of the object, translation,
and rotation. Thereafter, the segmentation ID is determined using
the center of the object and its predicted translation. For
predicting scene flow, the translation, rotation, and input XYZ data
is utilized. The final output is presented as a segmentation mask
and scene flow predictions. Note that the blue, red, and green
arrows do not have gradient flow.}
\label{fig:sceneflownet}
\end{figure*}
\subsection{Rigid Motion and Object Scene Flow}
The first stage of the proposed model, displayed in
Fig.~\ref{fig:sceneflownet}, consists of two Siamese networks that
takes RGB images $\mathcal{I}^{t-1}$, $\mathcal{I}^{t}$ and point
clouds $\mathcal{P}^{t-1}$, $\mathcal{P}^{t}$ as inputs, each with
resolution $(W,H,3)$. The pair of point clouds is fed into the first
of these networks that outputs a new feature encoding denoted by
$\mathcal{P}f^{t-1}$ and $\mathcal{P}f^{t}$, respectively. We use the
VGG architecture~\cite{simonyan2014very} for this purpose. The shape
of the output feature is $(W/8,H/8,64).$
The pair of RGB images is fed into the second Siamese network that
outputs a new feature encoding denoted by $\mathcal{I}f^{t-1}$ and
$\mathcal{I}f^{t}$, respectively. We use the ResNet50 architecture and
its weights for initialization~\cite{he2016deep}. The shape of the
output feature tensor is $(W/8,H/8,256)$.
The RGB image features are fed into a correlation layer similar to the
one used in FlowNetC~\cite{7410673}. A high correlation between
patches in consecutive RGB images indicates that they contain a
projection of the same physical object part. This correlation layer
parallels the brightness constancy assumption in traditional optical
and scene flow methods.
\begin{figure}[htb]
\centering
\includegraphics[width=0.88\linewidth]{imgs/cc.png}
\caption{Process of correlation and max-pooling. After two RGB feature
maps (black) are generated, each cell in the feature map
$\mathcal{I}f^{t}$ is correlated with every cell within a patch of
the feature map $\mathcal{I}f^{t-1}$. Let us assume that the yellow
cells $F$ and $l$ contain features corresponding to the same
object. Therefore, their correlation $c(\mathcal{I}f^{t}_{F},
\mathcal{I}f^{t-1}_{l})$ will be high. These correlation values are
used to weight corresponding cells of the XYZ feature map
$\mathcal{P}f^{t-1}$ (gray). The result is fed into a max-pooling
layer which in this example will output $c(\mathcal{I}f^{t}_{F},
\mathcal{I}f^{t-1}_{l}) \mathcal{P}f^{t-1}_{l'}$. The final feature
$l^{\prime\prime}$ containing object XYZ information at frame $t-1$
will be placed at the same location as feature $F$ at frame $t$.}
\label{fig:cc}
\end{figure}
Fig.~\ref{fig:cc} visualizes the correlation process encoded in the
layer. Let $\mathcal{I}f^{t}_{uv}$ denote a feature of RGB image
$\mathcal{I}^{t}$ at pixel $(u,v)$. Each feature is correlated with a
patch of features denoted by $\mathcal{P}\mathcal{I}f^{t-1}_{uv}$. The
patch is centered at $\mathcal{I}f^{t-1}_{uv}$ and has a side length
of $2L+1$, i.e. the dimension of the patch encoding is
$(2L+1,2L+1,256)$. The correlation operation between features
$\mathcal{I}f_{uv}^{t}$ and $\mathcal{I}f_{kl}^{t-1}$ inside the patch
$\mathcal{P}\mathcal{I}f^{t-1}_{uv}$ is defined as
\begin{equation}
c(\mathcal{I}f^{t}_{uv}, \mathcal{I}f^{t-1}_{kl}) = \langle \mathcal{I}f^{t}_{uv},\mathcal{I}f^{t-1}_{kl} \rangle \text{ if } |u-k| \le L, |v -l| \le L
\end{equation}
The output vector of correlation between the single feature
$\mathcal{I}f_{uv}^{t}$ and corresponding patch
$\mathcal{P}\mathcal{I}f^{t-1}_{uv}$ has a dimension of
$(2L+1)^2$. The correlation is performed at each pixel within
$\mathcal{I}f^{t-1}$ with a stride of $(W/8,H/8)$. The final output
shape of the correlation layer is $(W/8,H/8,(2L+1)^2)$.
Highly correlated RGB patches also indicate which parts in consecutive point clouds correspond to each other.
We therefore multiply the correlation value tensor with the
corresponding $\mathcal{P}f^{t-1}$ features to get a weighted XYZ
feature encoding $\widehat{\mathcal{P}f}^{t-1}$. Then we apply max
pooling to this result along the feature dimension as follows:
\begin{equation}
\widehat{{\mathcal{P}f}}^{t-1}_{uv} = \max_{\substack{|u-k| \le L \\ |v-l| \le L}}(c(\mathcal{I}f^{t}_{uv}, \mathcal{I}f^{t-1}_{kl})\mathcal{P}f^{t-1}_{kl} )
\end{equation}
We concatenate $[Pf^{t},Pf^{t-1},\widehat{Pf}^{t-1}]$ and feed this
into another encoder until reaching a feature map with size
(W/60,H/60,512) before feeding it into a decoder. Skip links are
created between encoder and decoder. The decoder generates three
images $\mathcal{Q}$, $\mathcal{T}$ and $\mathcal{X}$ representing
per-pixel estimates of rotation, translation and center position of
the object projected to that pixel. Per-pixel scene flow can then be
computed through Eq.~\ref{eq:sceneflow}.
\subsection{Motion-based Segmentation}\label{sec:motion-based}
As defined previously, let $\xi_k = [X_k,X_k+T_k]$ represent the start
and end point of the object trajectory of $O_k$. Pixels
belonging to the same object $\mathcal{O}_k$ will have the same value
$\xi_k$. We assume that pixels belonging to different objects
have different values. Based on this we perform object segmentation.
Our model makes a pixel-wise prediction $\hat{\xi}_{uv}$ of the
trajectory feature at pixel coordinates $(u,v)$. This is only an
approximation of the ground truth value. Therefore, each pixel
$(u,v)$ that corresponds to the same object $\mathcal O_k$ will predict
feature values $\hat{\xi}_{uv}$ that differ from ground
truth by some $\epsilon_{uv}$ such that $\hat{\xi}_{uv} = \xi_{uv} +
\epsilon_{uv}$.
To segment moving objects, we propose the
following inference process. Let $\mathcal{B}$ be an additional output
image of our model. A pixel at $(u,v)$ contains a scalar value
$B_{uv}$. This value is a radius estimate of the sphere that encloses
all pixels which belong to the
same moving object, i.e. have a similar trajectory. The sphere is
centered at $\hat{\xi}_{uv}$. Any pixel at coordinates $(o,p)$ whose
$\hat{\xi}_{op}$ falls inside the sphere centered around
$\hat{\xi}_{uv}$ will be segmented as the same
object $\mathcal{O}_k$. Any pixel at $(m,n)$ whose $\hat{\xi}_{mn}$
falls outside the sphere will be part of the background or a different
object. In addition to $\mathcal{B}$, we also learn a mask
layer to discard pixel in this segmentation process that belong to the
background.
To generate the ground truth of $\mathcal{B}^{gt}$, each pixel (u,v)
representing object $\mathcal{O}_k$ is annotated by half of the
minimum distance between $\xi_k$ and the trajectories
$\xi_l$ of all the other objects in the image pair:
\begin{equation}
B^{gt}_{uv} = \frac{1}{2} \min_{k \ne l}\|\xi_k - \xi_l\|_2
\end{equation}
Inspired by region proposals~\citep{DBLP:journals/corr/RenHG015}, our
model also outputs an image denoted by $\eta$. Each pixel in this
image at $(u,v)$ contains the probability $\eta_{uv}$ that it is the
projection of the object centroid. To generate the ground truth of $\eta$, we sort pixels representing object $\mathcal{O}_k$ by their distance to the object's centroid in ascending order. The top $D$ pixels per object in the input image
$\mathcal{I}$ will be labeled as 1, the rest will be labeled as
0. If the total number of pixels representing object $\mathcal{O}_k$
is less than $D$, all of them are labeled as 1. We found that $D=300$
worked well. This corresponds to approximately 10\% - 30\% of the
ground truth object pixels. The final performance is not very sensitive
to this parameter
Given the predicted $\hat{\mathcal{B}}$ and $\hat{\eta}$, we can now
perform multi-object segmentation as visualized in
Fig.~\ref{fig:cluster}. Pixel $(u,v)$ with the maximum predicted probability
$\hat{\eta}_{uv}$ is proposed first. Given a sphere centered at
$\hat{\xi}_{uv}$ with radius $\hat{B}_{uv}$, all pixels $(m,n)$ with a
trajectory $\hat{\xi}_{mn}$ enclosed by this sphere are assigned to
object $O_1$. All pixel assigned to $O_1$ are removed from the set of
unsegmented pixels before segmenting the next object. The remaining
pixel at $(o,p)$ with the highest $\hat{\eta}_{op}$ is used as the seed for
segmenting $O_2$. This process is repeated until all foreground
pixels are assigned an object id $k$. The final object translation
$T_k$ and rotation $R_k$
is computed by averaging over all pixels with the same id. Based on
this, also the scene flow can be recomputed.
\begin{figure}[t]
\centering
\includegraphics[width=0.9\linewidth]{imgs/cluster.png}
\caption{Object segmentation process. Left: Points represent points in
a point cloud. Stars represent ground-truth object centers. Same
color indicates same object. Middle: Each square represents the
trajectory features $\hat{\xi}$ in trajectory feature space each
associated with a point on the left. The size of the
squares represents the corresponding point's probability $\hat{\eta}$
of being an
object centroid. Right: The segmentation process cycles through the
squares starting with those having the highest probability to be an
object centroid. A sphere centered at one of those squares with
radius $\hat{\mathcal{B}}$ then segments trajectories and corresponding points.}
\label{fig:cluster}
\end{figure}
\subsection{Loss Function} \label{sec:loss}
We use the following training loss:
\begin{align}
L = & \lambda_{m}L_{m} +\lambda_{center}L_{center} + L_{p} \nonumber\\
& +\lambda_{var}L_{var}+\lambda_{vio}L_{vio}.
\end{align}
In the following, we define each term. Note that all pixel-wise loss
terms $L_p$, $L_{center}$, $L_{var}$ and $L_{vio}$ are only computed
on the ground truth foreground pixel.
\subsubsection{Mask Loss}
$L_{m}$ is the cross entropy loss between the ground truth and
estimated foreground/background segmentation. If a pixel is the
projection of an object point, we assign 1 as ground truth; otherwise
0.
\subsubsection{Cluster Center Loss}
Cross-entropy loss $L_{center}$ is used to learn the probability
$\eta_{uv}$ of a pixel $(u,v)$ to be the object center as described in
Sec.~\ref{sec:motion-based}.
\subsubsection{Pixel-wise Loss}
We use a pixel-wise loss $L_{p}$ on the predicted object rotation
$Q_{uv}$, translation $T_{uv}$, scene flow
$S_{uv}$, enclosing sphere radius $B_{uv}$ and trajectory $\xi =
[X_{uv},X_{uv}+T_{uv}]$. For each attribute, we use the L2-norm to measure
and minimize the error between predictions and ground truth. Note that
the loss on each attribute is also differently weighted. We denote
their corresponding weights $\lambda_{Q}$, $\lambda_{T}$, $\lambda_X$,
$\lambda_{S}$, $\lambda_B$ and $\lambda_{\xi}$.
\subsubsection{Variance Loss}
We use $L_{var}$ to encourage pixels $(u,v)$ belonging to the same
object $\mathcal{O}_{k}$ to have similar trajectories $\xi_{uv}$ and
thereby to reduce their variance.
\begin{equation}
L_{var} = \sum_{k}\frac{1}{N_{k}}\sum_{(u,v) \in \mathcal{O}_k}\|
\hat{\xi}_{uv} - \bar{\hat{\xi}}_{uv} \|^2
\end{equation}
where $\bar{\hat{\xi}}_{uv}$ is the mean value of $\hat{\xi}_{uv}$
over all $N_k$ pixels belonging to $\mathcal{O}_k$.
\subsubsection{Violation Loss}
$L_{vio}$ penalizes pixels $(u,v)$ that are not correctly
segmented. Any predicted trajectory $\hat{\xi}_{uv}$ that is more than
$\frac{1}{5}\mathcal{B}_{uv}$ away from the ground truth $\xi_{uv}$
will be pushed towards the ground truth trajectory by the violation
loss. Note that $\mathcal{B}_{uv}$ refers to the radius of enclosing sphere.
\begin{equation}
L_{vio} =\sum_{k}\sum_{(u,v)\in \mathcal{O}_k}
\mathbf{1}\{\|\hat{\xi}_{uv} -\xi_{uv}\|_2 >
\frac{1}{5}\mathcal{B}_{uv}\} \|\hat{\xi}_{uv} -\xi_{uv}\|_2 \nonumber
\\
\end{equation}
The variance and violation loss are designed to train the clustering framework described in
~\ref{sec:motion-based}.
\section{Dataset}\label{sec:dataset}
We generated a new dataset that consists of RGB-D image pairs showing
dynamic scenes. These scenes contain a large variety of rigidly-moving objects.
See Fig.~\ref{fig::synthetic_table} for some example frames.
To ensure a diverse data set, we used 31594 3D object mesh models from
ShapeNet~\citep{shapenet2015} covering 28 categories. We split these
models into a training, validation and test set with 21899, 3186 and
6509 objects respectively. Model sizes are adjusted to simulate their real
world sizes~\cite{lin2017crossmodal}. For each scene, 1-30 object
models are randomly selected. For simulating realistic object motion,
we use Bullet~\citep{Coumans:2015:BPS:2776880.2792704} as physics
engine. The objects are put close to each other at 0.2 meter above a
surface. After simulation begins, they start to fall down to the surface
and collide with each other in the process. The RGB-D camera is static and the simulation runs at 60 Hz. We extract frame 20 and 80 from the simulated image sequence as RGB-D image pair. They are 1 second apart with
an average object displacement of 0.085 meters. We synthesize 24994,
3360 and 7186 frame pairs for training, validation and test set. Note
that the object models are not re-used across these data sets. In
total, we generated 35540 pairs of
consecutive RGB-D frames using Blensor~\cite{gschwandtner2011blensor}
to ensure realistic depth data. For each rendered RGB image pair, we
randomly sample an image from the SUN397 dataset~\cite{xiao2010sun} to
simulate textured floor or we use a single color. We also randomly
change the lighting conditions (number of light sources, their
positions and energies) and camera viewpoint. We do not add artificial
noise in the raw dataset for two reasons. Different sensors like
time-of-flight or structured light have different noise
patterns. Adding one type of noise pattern into the dataset might
increase the simulation-to-reality gap when other sensors are
used. Extra noise can be dynamically added into the neural network
training procedure as data augmentation procedure.
\subsection*{Annotating Objects with Rotational Symmetry}
Some of the objects in ShapeNet~\citep{shapenet2015} are rotationally symmetric, e.g. bottles and bowls. Rotational symmetry is a common object attribute especially for human-made objects. However, the rotation of such an object around its symmetry axis is not observable in an image pair (especially when uniformly colored) as there might be multiple or even infinite solutions.
There are different orders of rotational symmetry denoted by $C_2$, $\cdots$, $C_n$, $\cdots$,$C_{\infty}$. An object with $C_n$ means that it will remain the same after rotating about the rotation axis by $\pm 360/n$ degrees. An object might contain several different rotational symmetries. Fig.~\ref{fig:rotational symmetry} illustrates an example.
\begin{figure}[t]
\centering
\includegraphics[width=0.8\linewidth]{imgs/multiple.png}
\caption{A rotationally symmetric object for which two different rotations yield the same RGB-D data. Therefore, multiple solutions for scene flow exist.}
\label{fig:rotational symmetry}
\end{figure}
This has implications for the ground truth annotation of our dataset. If we directly use the ground truth rotation provided by the simulator, the network might not converge during training as more than one rotation might lead to the same RGB-D data.
In the following, we describe a procedure to map the ground truth rotation of an object about its symmetry axis to the rotation with minimum angular displacement.
Consider an object with $C_n$ rotational symmetry. Let $\bar r_{t-1}$ and $\bar r_{t}$ denote this axis of symmetry at frame $t-1$ and frame $t$, respectively. Let the rotation provided by the simulator be given as a quaternion $q = [q_0, q_x, q_y, q_z]^T$. We decompose the rotation $q$ into a rotation $\alpha$ about $C_n (\bar{r}_{t-1})$ and a rotation $\theta$ perpendicular to $C_n: ~(\bar{r}_{\bot} = \bar{r}_{t-1} \times \bar{r}_t)$.
\begin{align}
\alpha = & 2\tan^{-1}(\frac{r_{t-1,x}q_x+r_{t-1,z}q_z+r_{t-1,z}q_z}{q_0})
\label{eq::quat_decomp1}\\
\theta = & 2\cos^{-1}(q_0/cos(\frac{\alpha}{2}))
\label{eq::quat_decomp2}
\end{align}
$\alpha$ is then adjusted to be $\hat\alpha \in (-\pi/n,\pi/n]$.
This corresponds to the minimum angular displacement leading to the same observation as the original angle. From this, we can construct a new quaternion $\hat q$ which corresponds to the rotation of $\hat \alpha$ about $\bar{r}_{t-1}$ and rotation of $\theta$ about $\bar{r}_{\bot}$. Note that if $\alpha = \hat\alpha \in (-\pi/n,\pi/n]$ then $q = \hat q$. This operation is performed on all the rotational axis of symmetry.
With this procedure, we reduced the ambiguous cases to a very small number, e.g. to uniformly-colored objects with non-orthogonal axes of symmetry (of which there exists one among our models) or rotations as shown in Fig.~\ref{fig:rotational symmetry} where the minimum angular displacement can either refer to a rotation in the positive or negative direction.
\section{Experiments}
We report the performance of the proposed model quantitatively on the synthesized dataset and qualitatively on real
data. We evaluate accuracy in scene flow prediction by comparing to
PD-Flow~\citep{jaimez2015primal}, semi-rigid scene flow (SR-Flow)~\cite{quiroga2014dense} and \citet{7989459}. We evaluate motion-based
segmentation performance by comparing to Higher-Order Minimum Cost
Lifted Multicuts (HOMC)~\citep{keuper2017higher}.
Furthermore, we compare to variants of the proposed architecture. We refer to the network in Fig.~\ref{fig:sceneflownet} as $\textbf{OurC}$ and propose a simpler neural net architecture denoted by $\textbf{OurS}$. It concatenates all four input images and feeds it into the encoder. Most importantly, it drops the correlation and max pooling layer. The remaining model architecture is the same.
$\textbf{OurC+vL}$ denotes added variance and violation loss compared
to training $\textbf{OurC}$. Our model $\textbf{OurC+vL}$
simultaneously predicts pixel-level segmentation IDs and scene
flow. Given all pixels with the same, predicted object ID, we compute
the mean object center $\bar{X}_k$, translation $\bar{T}_k$ and
rotation $\bar{Q}_k$. $\textbf{OurC+vL+Rig}$ denotes the model with
added rigidity constraints for improved scene flow estimation. After we infer the segmentation mask per object, we average the rigid transformations from pixels predicted to represent the same object. The rigid transformation is then applied to pixels to recalculate scene flow.
We conduct our experiments on an NVIDIA P100 with TensorFlow. For
training, we use the Adam optimizer~\citep{kingma2014adam} with its
suggested default parameters of $\beta1=0.9$ and
$\beta2=0.999$ along with a learning rate $\alpha=0.0001$~\cite{kingma2014adam}. We use a batch size of 12 image pairs. The input RGB-D images have a resolution of $240\times 320$. The loss weights, as defined in Sec.~\ref{sec:loss}, are set to $\lambda_m=1.0$, $\lambda_{var}=0.1$,
$\lambda_{vio}=0.1$, $\lambda_{Q}$=0.1, $\lambda_{T}$=100.0,
$\lambda_X$=10.0, $\lambda_{S}$=10.0, $\lambda_B$=1.0 and
$\lambda_{\xi}$=1.0.
\subsection{Evaluation of Scene Flow Performance}
We compare the proposed method with the aforementioned approaches
using standard evaluation metrics as defined in~\citep{survey}: end point error (EPE) and $4D$ average angular error (AAE) error. Each metric is calculated as averages over the entire
image and is reported in cm and degrees, respectively. Because it
is impossible to calculate the scene flow for an object that is only
present in one of the two frames, we also report masked EPE and masked
AAE which calculates the desired metrics only on objects that are in
both frames. The results are presented in
Fig.~\ref{fig::res_scene_flow}.
\begin{figure}[t]
\tiny
\renewcommand\arraystretch{1.7}
\centering
\begin{tabular}{p{0.15\linewidth} ||P{0.11\linewidth} P{0.11\linewidth} ||P{0.11\linewidth} P{0.11\linewidth}
|| P{0.08\linewidth}}
\hline
\hline
\scriptsize Method & \multicolumn{2}{c||}{\scriptsize EPE in cm} & \multicolumn{2}{c||}{\scriptsize AAE in degrees} &\scriptsize Runtime \\
&\scriptsize all & \scriptsize masked &\scriptsize all &\scriptsize masked &\scriptsize seconds\\
\hline
PD-Flow\cite{jaimez2015primal}& 2.830 $\pm$4.23& 8.041 $\pm$5.10& 1.607 $\pm$2.38& 4.572 $\pm$2.87&\textbf{0.046} \\
SR-Flow \cite{quiroga2014dense}& 2.040 $\pm$3.77 & 6.859 $\pm$4.69 &1.155 $\pm$2.12 & 3.898 $\pm$2.63& $\ge$1\\
Jaimez et al. \cite{7989459}& 2.330 $\pm$3.94 & 6.431 $\pm$5.20& 1.317 $\pm$2.20& 3.643 $\pm$2.91& 0.083 \\
\hline
OurS & 1.643 $\pm$3.07 & 5.324 $\pm$3.56 & 0.928 $\pm$1.72 & 3.020 $\pm$2.00& 0.059\\
OurC & 1.330 $\pm$2.60 & 4.333 $\pm$2.92 & 0.750 $\pm$1.45 &2.457 $\pm$1.64& 0.078\\
OurC+vL & 1.315 $\pm$2.57 & 4.333 $\pm$2.95 & 0.742 $\pm$1.44 & 2.457 $\pm$1.66& 0.078\\
OurC + vL + Rig & \textbf{1.303} $\pm$2.55 & \textbf{4.290} $\pm$2.93 & \textbf{0.734} $\pm$1.43 & \textbf{2.432} $\pm$1.64& 0.121\\
\hline
\hline
\end{tabular}
\caption{Performance of scene flow prediction measured in endpoint error (EPE) and average angular error (AAE) with standard deviation. {\em Masked\/} only contains data from objects that appear in both frames. The learned models outperform the baselines in terms of mean error and std \textbf{OurC} and its variants perform better than the simple model version \textbf{OurS} without the correlation layer.}
\label{fig::res_scene_flow}
\end{figure}
All our proposed models outperform the aforementioned approaches both in mean and standard deviation. Furthermore, \textbf{OurC} and its variants perform better than the simple
model version \textbf{OurS} without the correlation layer. This comes
at the expense of a higher processing time. However, the most complex
model $\textbf{OurC+vL+Rig}$ can still run at 8.3 frames per
second.
\subsection{Evaluation of Motion-based Segmentation}
We evaluate our model's ability to perform motion-based segmentation
by comparing to HOMC, the state-of-the-art technique
by \citet{keuper2017higher}. This method requires a sequence of RGB
images. To satisfy the input requirement of HOMC, we generate an additional test dataset called \textbf{TestSeq} following the procedure outlined in Sec.~\ref{sec:dataset}. In \textbf{TestSeq} there are 1302 image sequences each consisting of 8 frames with indices 20, 30, 40, 50, 60, 70, 80 and 90. The original data set only has 2 frames, frame 20 and 80. From this we create the data set \textbf{Test} in which these two frames are repeated 5 times (20, 80, 20, 80, 20, 80, 20, 80, 20, 80) such that it can serve as input to HOMC. \citet{keuper2017higher} provided an executable file upon
request. We run HOMC\cite{keuper2017higher} on the sequences with a subsampling of 4 and a prior cut probability of 0.5. For our proposed models, frames 20 and 80 compose the input image pair for all experiments.
To evaluate the segmentation results produced by HOMC and our three network
variants, we rely on four metrics that are frequently used in
segmentation papers: precision, recall, F-measure, and extracted
objects~\citep{KB15b}. We compute the metrics on the segmentation of frame 80 by following the
convention in~\citep{ochs2014segmentation}. We use an F-measure threshold of
0.75. The results are reported
in Fig.~\ref{tab::seq_motion_seg}.
\begin{figure}[t]
\tiny
\renewcommand\arraystretch{1.7}
\centering
\begin{tabular}{p{0.08\linewidth} ||P{0.03\linewidth} P{0.055\linewidth} ||P{0.03\linewidth} P{0.055\linewidth}
||P{0.03\linewidth} P{0.055\linewidth}
||P{0.07\linewidth} P{0.08\linewidth}
}
\hline
\hline
\scriptsize Method & \multicolumn{2}{c||}{\scriptsize Precision} & \multicolumn{2}{c||}{\scriptsize Recall} & \multicolumn{2}{c||}{\scriptsize F-measure} & \multicolumn{2}{c}{\scriptsize Extracted Objects}\\
&\fontsize{5.5}{7.2}\selectfont Test &\fontsize{5.5}{7.2}\selectfont TestSeq &\fontsize{5.5}{7.2}\selectfont Test &\fontsize{5.5}{7.2}\selectfont TestSeq &\fontsize{5.5}{7.2}\selectfont Test &\fontsize{5.5}{7.2}\selectfont TestSeq &\fontsize{5.5}{7.2}\selectfont Test &\fontsize{5.5}{7.2}\selectfont TestSeq\\
\hline
HOMC\cite{keuper2017higher}& \textbf{0.833} &\textbf{0.797} & 0.195 & 0.282 & 0.111& 0.186 & 3909/64556 & 1172/11782 \\
\hline
OurS & 0.697 & 0.714 & 0.735 & 0.749 & 0.671 & 0.683& 36369/64556 & 7077/11782 \\
OurC & 0.756 & 0.763 & 0.757 & 0.771 &0.696 & 0.711& 41274/64556 & 7839/11782 \\
OurC+vL & 0.766 & 0.768 & \textbf{0.787} & \textbf{0.783} & \textbf{0.730}& \textbf{0.725} & \textbf{43719/64556} & \textbf{8015/11782}\\
\hline
\end{tabular}
\caption{Performance of motion-based segmentation. \textbf{Test} refers to the dataset containing repetitions of 2 image frames. \textbf{TestSeq} refers to the dataset containing 8 image frames. The proposed models significantly outperform HOMC that requires longer image sequences but cannot rely on the strong depth cue. The performance increase of $\textbf{OurC+vL}$ over the simpler models highlights the importance of the correlation layer and the variance and violation loss.}
\label{tab::seq_motion_seg}
\end{figure}
On both datasets, HOMC~\citep{keuper2017higher} achieves high precision and low recall values indicating undersegmentation. In \textbf{TestSeq}, HOMC extracts more objects with
higher recall and F-measure scores than \textbf{Test}, emphasizing the dependence on an actual image sequence. All our proposed methods show a significant improvement on the recall, F-measure, and extracted
objects metrics while retaining a high precision score. While
HOMC relies on a longer sequence of images and takes more than 30 seconds to process the sequence, it does not require depth information.
A few example results are displayed in
Fig.~\ref{fig::synthetic_table}. These results highlight another
advantage of our approach, that the resulting segmentation is dense.
\subsection{Architecture Design Analysis}
\subsubsection{Effects of correlation layer}
We report the training and validation loss curve in
Fig.~\ref{fig::loss}. $\textbf{OurC}$ has a much lower
training and validation loss than $\textbf{OurS}$. We also showed that
$\textbf{OurC}$ outperforms $\textbf{OurS}$ both in scene flow
prediction and motion-based segmentation. This demonstrates the impact of adding a correlation layer in $\textbf{OurC}$. It forces our model to learn the similarity between consecutive RGB features which makes $\textbf{OurC}$ more robust to changes such as lighting conditions or viewpoints.
\begin{figure}[t]
\centering\includegraphics[width=0.5\linewidth]{imgs/loss_curve.png}
\caption{Loss curve at each epoch during training and validation for
our model with and without the correlation layer: $\textbf{OurC}$ and
$\textbf{OurS}$}
\label{fig::loss}
\end{figure}
\subsubsection{Effects of using variance and violation loss}
We utilize the variance loss to reduce the statistical variance of predicted trajectory features. The violation loss penalizes outliers in the training process. Compared to $\textbf{OurC}$, $\textbf{OurC+vL}$ improves motion-based segmentation, but only leads to small improvements on scene flow prediction.
\subsubsection{Effects of using rigid motion cues}
The best scene flow prediction performance is achieved by adding rigid
constraints ($\textbf{OurC+vL+Rig}$). However the improvement over
$\textbf{OurC}$ is only marginal. The difference to $\textbf{OurS}$
remains significant, underlining the importance of correlation
layer.
\subsection{Results and Analysis on Real World Data}
Finally, we demonstrate the networks ability to perform in a real
world setting. We recorded real RGB-D data with the Intel RealSense
SR300 Camera. The data includes large displacements, occlusions, and
collisions. It was captured using a diverse set of objects with
varying geometries, textures, and colors. Note that we do not have
any ground truth annotations and that the model is not fine-tuned to
transfer from synthetic to real data.
We apply HOMC~\citep{keuper2017higher} on the stream of real data as one long sequence. We use our $\textbf{OurC+vL+Rig}$
model to process real data sequences. Every pair of consecutive images forms one image pair which are fed into our neural network.
Some example images and corresponding outputs are displayed in
Fig.~\ref{fig::realworld_table}. The accurate real world segmentation
and scene flow prediction results strongly indicate the small
sim-to-real transfer gap of the proposed model.
There are still some failure cases including inaccurate object boundaries due to noisy sensor data and false positive segmentations due to varied lighting conditions. Also if two objects are moving along extremely similar trajectories, it is difficult to segment them.
This could be potentially alleviated by concatentating rotational motion to the trajectory feature.
Other limitations of our method include: inability to generalize to non-flat surfaces or non-rigid objects. The generalization problem of learning-based methods could be mitigated by transfer-learning techniques e.g.~\cite{ganin2015unsupervised}
\renewcommand{\tabcolsep}{0pt}
\renewcommand{\tabcolsep}{0pt}
\newcommand*\rot{\rotatebox{90}}
\begin{figure*}[htb!]
\centering\includegraphics[width=0.86\linewidth]{imgs/synthetic6.png}
\caption{Performance comparison of the proposed method on our synthetic data set. First two columns show RGB inputs, next columns are Scene Flow from Jaimez et al.~\cite{7989459}, our method, and the ground truth, the final three columns correspond to segmentation results from HOMC \cite{keuper2017higher}, our method, and the ground truth. In scene flow images, green, blue, red intensities are proportional to the velocities along $X,~Y,~Z$ respectively. In the HOMC segmentation, colored pixels (not gray or white) have been successfully clustered with longer trajectories to produce valid segmentations.}
\label{fig::synthetic_table}
\end{figure*}
\begin{figure*}[htb!]
\centering\includegraphics[width=0.86\linewidth]{imgs/realworld6.png}
\caption{Performance comparison of the proposed method on the real-world data set. First two columns show RGB inputs, next columns are Scene Flow from Jaimez et al.~\cite{7989459} and our methods, last two columns correspond to segmentation results from HOMC \cite{keuper2017higher} and our method.}
\label{fig::realworld_table}
\end{figure*}
\section{Conclusion}
We proposed a deep neural network architecture that given two consecutive RGB-D images can accurately estimate object scene flow and motion-based object segmentation. We demonstrated this on a new and challenging, synthetic data set that contains a large variety of graspable objects moving simultaneously. We showed that the correlation layer makes a crucial difference to training time and accuracy and outperforms state of the art baselines in scene flow prediction and motion-based segmentation. Additionally, we showed how our approach performs on real RGB-D data when only trained on synthetic data. The results look qualitatively more accurate than baseline methods. Overall, we demonstrated the power of learning based methods over traditional methods in situations of large displacements and strong occlusions.
In future work, we will explore how this approach enables agile, robotic manipulation in cluttered scenes.
\footnotesize{
\bibliographystyle{plainnat}
|
{
"timestamp": "2018-07-25T02:07:20",
"yymm": "1804",
"arxiv_id": "1804.05195",
"language": "en",
"url": "https://arxiv.org/abs/1804.05195"
}
|
\section{Introduction}
In the paper we consider a kinetic-type evolution equation for a time dependent family of probability measures $(\rho_t)_{t\ge 0}$. Let
$$\phi(t,\xi)=\int_{\mathbb{R}} e^{i\xi v}\rho_t({\rm d}v),\quad t\geq 0,\quad \xi\in{\mathbb{R}},
$$
be the Fourier--Stieltjes transform (the characteristic function) of $\rho_t$. We are interested in the solution of the following Cauchy problem
\begin{equation} \label{eqboltzivp}
\frac{\partial }{\partial t} \phi(t,\xi)+\phi(t,\xi)=\widehat{Q}(\phi(t,\cdot),\ldots,\phi(t,\cdot))(\xi),\quad t>0,\quad \phi(0,\xi)=\phi_0(\xi),\quad \xi\in{\mathbb{R}},
\end{equation} where $\widehat{Q}$ is a smoothing transform. The smoothing transform $\widehat{Q}$ is defined by the equality
$$\widehat{Q}(\phi_1,\ldots,\phi_N)(\xi):={\mathbb{E}}(\phi_1(A_1\xi)\cdot \ldots \cdot \phi_N(A_N\xi)), \quad \xi\in{\mathbb{R}},$$
where $\phi_1,\ldots,\phi_N$ are characteristic functions, $N$ is a fixed positive integer, and a random vector ${\mathbb{A}} = (A_1, \ldots , A_N)$ consists of positive real-valued random variables defined on a common probability space $(\Omega,\mathcal{F},\mathbb{P})$.
The initial condition $\phi_0$ is the characteristic function of some random variable $X_0$ defined on $(\Omega,\mathcal{F},\mathbb{P})$.
The equation of the form \eqref{eqboltzivp} with $N=2$ and ${\mathbb{A}} = (\sin\theta,\cos \theta)$, where $\theta$ is a random angle uniformly distributed on $[0,2\pi)$, was introduced and investigated by Kac \cite{Kac} as a model of behavior of a particle in a homogeneous gas. In subsequent works the Kac model was generalized in various directions including one dimensional dissipative Maxwell models for colliding molecules \cite{Pareschi}, models describing economical dynamics \cite{Matthes} and the inelastic Boltzmann equation \cite{bobylev,bobylev2}. We refer to \cite{BasettiLadelli2012,BasettiLadelliMatthes,BassettiPerversi:2013}
for other examples and a comprehensive bibliography.
In this paper we study asymptotic behavior of the solution $\phi$ to equation \eqref{eqboltzivp} from probabilistic point of view and prove related limit theorems. This problem was recently addressed in \cite{BasettiLadelli2012} where it was shown that under mild assumptions, which we discuss later, there exists a parameter $\mu$, depending on the initial condition $\phi_0$ and the law of ${\mathbb{A}}$, such that the rescaled solution to \eqref{eqboltzivp}, namely
\begin{equation}\label{eq:ss1}
w(t,\xi) = \phi(t, e^{-\mu t}\xi),\quad t\geq 0,\quad \xi\in{\mathbb{R}},
\end{equation}
converges to a nondegenerate limit as $t\to\infty$ and the limit is a fixed point of a smoothing transform pertained to $\widehat Q$. The main goal of our paper is to present a class of solutions to~\eqref{eqboltzivp} which, after rescaling as in \eqref{eq:ss1}, converge to a degenerate limit, yet it is possible to~find a different normalization ensuring a nondegenerate limit possessing some self-similarity properties. To achieve our aims we propose a refinement of the probabilistic construction of the solution $\phi$ presented in \cite{BasettiLadelliMatthes} and express $\phi$ via a continuous-time branching random walk.
Firstly, we state assumptions concerning the initial condition $\phi_0$. We suppose, similarly as in \cite{BasettiLadelli2012} and \cite{BassettiPerversi:2013}, that
the distribution function $F_0$ of $X_0$ satisfies one of the following hypotheses $(H_{\gamma})$ for some $\gamma\in(0,2]$:
\begin{itemize}
\item[$(H_1)$]
either
\subitem{(a)} $\int_{\mathbb{R}}|v| \,{\rm d}F_0(v) <+\infty$ \textit{and then we set}
$m_0:=\int_{\mathbb{R}} v\,{\rm d}F_0(v)$\\
\textit{or}
\subitem{(b)}
$F_0$ \textit{ satisfies the conditions}
$$
\lim_{x \to+\infty} x \bigl(1-F_0(x)\bigr) =
\lim_{x \to-\infty} |x| F_0(x) =c_0^+\in (0,\infty),
$$
and
$$
\lim_{R\to+\infty}\int_{-R}^{R}v{\rm d}F_0(v)=:m_0\in(-\infty,\infty).
$$
\item[$(H_2)$]
$0<\sigma_0^2:=\int_{\mathbb{R}}|v|^2 \,{\rm d}F_0(v) <+\infty$ \textit{and}
$\int_{\mathbb{R}} v\,{\rm d}F_0(v)=0$.
\item[$(H_\gamma)$]
\textit{If} $\gamma\in(0,1) \cup(1,2)$,
$F_0$ satisfies the conditions
$
\lim_{x \to+\infty} x^\gamma\bigl(1-F_0(x)\bigr) =c_0^+<+\infty,\qquad
\lim_{x \to-\infty} |x|^\gamma F_0(x) =c_0^-<+\infty
$$
\textit{with} $c_0^++ c_0^->0$
\textit{and}, \textit{in addition}, $\int_{\mathbb{R}} v\,{\rm d}F_0(v)=0$ \textit{if $\gamma\in (1,2)$.}
\end{itemize}
Further, we define the function $\hat g_\gamma:{\mathbb{R}}\mapsto \mathbb{C}$ by
\begin{equation}
\label{chaSta}
\hat g_\gamma(\xi):=
\begin{cases}
e^{ i m_0 \xi}, & \text{if } \gamma=1 \text{ and (a) of }(H_1) \text{ holds},\\
e^{ i m_0 \xi- \pi c_0^+ |\xi| }, & \text{if } \gamma=1 \text{ and (b) of }(H_1) \text{ holds},\\
e^{ - \sigma_0^2 |\xi|^2/2 }, &\text{if } \gamma=2 \text{ and }(H_2)\text{ holds},\\
e^{ -k_0 |\xi|^\gamma
(1-i \eta_0 \tan({\pi\gamma}/{2} )\operatorname{sign}\xi) }, &
\text{if }\gamma\in(0,1) \cup(1,2) \text{ and }(H_\gamma)\text{ holds},
\end{cases}
\end{equation}
where
\begin{equation*}
k_0 = (c_0^{+}+c_0^{-}) \frac{\pi}{2\Gamma(\gamma)\sin(\pi\gamma/2)},
\qquad \eta_0 = \frac{c_0^{+}-c_0^{-} }{c_0^{+}+c_0^{-}}.
\end{equation*}
Observe that the condition $(H_\gamma)$ is equivalent to the fact that the law of $X_0$ (the law of $X_0-m_0$ in case $H_1(b)$) is centered and (except case $H_1(a)$) belongs to the domain of normal attraction of a $\gamma$-stable law with the characteristic function $\hat g_\gamma$, see a concluding remark on p.~581 in \cite[Chapter XVIII.5]{Feller:1971}.
Now we formulate our hypotheses on the smoothing transform $\widehat Q$. Our first assumption is that the weights $(A_i)_{i=1,\ldots,N}$ are a.s.~positive. Next we define the function $\Phi:[0,\infty)\mapsto
{\mathbb{R}}\cup\{+\infty\}$ via
$$
\Phi(s) = {\mathbb{E}}\bigg[\sum_{i=1}^N A_i^s\bigg] - 1,\quad s\geq 0,
$$
and assume that $s_{\infty}>0$ where $s_{\infty} := \sup\{s\geq 0: \Phi(s) < \infty\}$. Note that the function $\Phi$ is smooth and convex on $(0,s_{\infty})$. The function
$$
\mu(s) = \frac{\Phi(s)}{s},\quad s > 0,
$$ is called spectral function, see \cite{bobylev}. Observe that $\mu(s)$ is equal to the tangent of the angle between the vector joining $(0,0)$ and $(s,\Phi(s))$ and the positive horizontal half-axis. Since $\Phi$ is strictly convex and smooth there exists exactly one point $\gamma^{\ast}$ minimizing the spectral function, and then the corresponding line is just tangent to the function $\Phi$ at point $(\gamma^{\ast}, \Phi(\gamma^{\ast}))$, see Fig.~\ref{fig_1}. Moreover, $\mu(\gamma^{\ast}) = \Phi'(\gamma^{\ast})$.
\begin{figure}[!hbtp]
\centering
\scalebox{1.5}{\begin{tikzpicture}
\draw[->] (-0.2,0) -- (3.2,0) node[right,scale=0.5] {$s$};
\draw[->] (0,-1.6) -- (0,1.8) node[above,scale=0.5] {$\Phi(s)$};
\draw[-] (0,0) -- (2,-1) ;
\draw [densely dotted] (0.95,0) -- (0.95,-0.45) ;
\node[left,scale=0.5] (c) at (0,1.5) {$N-1$};
\node[left,scale=0.5] (c) at (0,-1) {$-1$};
\node[above,scale=0.5] (c) at (0.95,0) {$\gamma^{\ast}$};
\node[xshift=-12.0, yshift=-4.3,scale=0.5,radius=5.0] (c) at (1,-0.5) {$(\gamma^{\ast},\Phi(\gamma^{\ast}))$};
\draw (0.95,-0.475) circle (0.02);
\draw [thick,red] plot [smooth, tension=1] coordinates {(0,1.5) (1.0,-0.5) (3.0,0.3)};
\draw[fill=black] (0.95,-0.475) circle (0.03);
\draw (0.2,0) arc (0:-28:0.2) node[xshift=5.0,yshift=0.5,scale=0.5] {$\alpha$};;
\end{tikzpicture}}
\caption{Plot of the function $s\mapsto \Phi(s)$ (solid red) with $\tan\alpha=\mu(\gamma^{\ast})=\Phi^{\prime}(\gamma^{\ast})=\Phi(\gamma^{\ast})/\gamma^{\ast}$.}
\label{fig_1}
\end{figure}
In the series of papers \cite{BasettiLadelli2012,BasettiLadelliMatthes} Bassetti, Ladelli and Matthes found a probabilistic interpretation of the solution $\phi$ via labelled random trees. Assuming that $(H_{\gamma})$ holds for some $\gamma\in(0,2]$ and there exists $\delta>\gamma$ such that $\mu(\delta)<\mu(\gamma)<\infty$, it is shown in \cite[Theorem 2.2]{BasettiLadelli2012} that $\phi(t, e^{-\mu(\gamma)t}\xi)$ converges to a nondegenerate limit being the characteristic function of the law of the limit of some positive martingale related to a family of random labelled trees. Clearly, if $\gamma=\argmin\mu(s)$ no such $\delta$ exists and, moreover, it can be checked that the corresponding martingale converges to $0$. As manifested in the title of the paper and motivated by Biggins and Kyprianou \cite{Biggins:Kyprianou:2005}, who considered the smoothing transform in the case $\gamma^{\ast}=1$ and $\mu(\gamma^{\ast}) = 0$, we call this situation the {\it boundary case}.
\medskip
The main result of our paper is given by Theorem \ref{thm:main} below, and provides the correct normalization in the boundary case leading to a non-degenerate limit. As we will see, the right normalization involves a subexponential term and the limit is a fixed point of some smoothing transform.
\begin{thm}\label{thm:main}
Assume that for some $\gamma\in (0,2]$ the hypothesis $(H_{\gamma})$ is satisfied and
$$
\gamma=\argmin_{s\in(0,s_\infty)}\mu(s)=\gamma^{\ast}\in(0,s_\infty).
$$
Then there exists a probability measure $\rho_{\infty}$ such that the function $\phi$, the unique solution to \eqref{eqboltzivp}, satisfies
$$
\lim_{t\to\infty}\phi\big(t,t^{\frac{1}{2\gamma}} e^{-\mu(\gamma)t}\xi\big) = w_{\infty}(\xi),\quad \xi\in{\mathbb{R}},
$$ where
$w_{\infty}$ is the Fourier-Stieltjes transform of $\rho_{\infty}$. Moreover, $w_{\infty}$ has the following representation
$w_{\infty}(\xi)={\mathbb{E}} \widehat{g}_{\gamma}(\xi c_{\gamma} D^{1/\gamma}_{\infty})$,
where $c_{\gamma}:=\left(\frac{2}{\pi\gamma^2\Phi^{\prime\prime}(\gamma)}\right)^{\frac{1}{2\gamma}}$ and $D_{\infty}$ is a.s.~positive random variable defined in Proposition \ref{prop: as} below and which satisfies the following stochastic fixed-point equation
\begin{equation}\label{eq:d_fixed_point}
D_{\infty}\overset{d}{=}\mathcal{U}^{\Phi(\gamma)}\sum_{k=1}^{N}A_k^{\gamma}D_{\infty}^{(k)},
\end{equation}
where $(D_{\infty}^{(k)})_{k=1}^N$ are independent copies of $D_{\infty}$; $\mathcal{U}$ has a uniform distribution on $(0,1)$ and $(D_{\infty}^{(k)})_{k=1}^N$, $\mathcal{U}$ and $(A_1,\ldots,A_N)$ are independent.
\end{thm}
The rest of the paper is organized as follows. In Section \ref{sec:prob_interpretation} we describe a probabilistic representation
of the solution $\phi$, which essentially reminds the construction in \cite{BasettiLadelli2012} but is more transparent and convenient for the analysis. Moreover, we reveal some further probabilistic structure behind this construction by pointing out a connection to Yule processes and branching random walks in continuous time. We strongly believe that the representation proposed in Section \ref{sec:prob_interpretation} is the most accurate probabilistic interpretation of the solution to equation \eqref{eqboltzivp}. In Section \ref{sec:biggins_convergence} we prove a convergence result for the Biggins martingale in continuous time branching random walk and explain the construction of the limiting measure $\rho_{\infty}$. The proof of Theorem \ref{thm:main} is given in Section \ref{sec:proof}.
\section{Probabilistic representation of the solution}\label{sec:prob_interpretation}
The solution to the equation \eqref{eqboltzivp} can be derived analytically in terms of the Wild series \cite{Wild}, see also Kielek \cite{Kielek}. However, based on McKean's \cite{McKean} ideas, Bassetti, Ladelli and Matthes \cite{BasettiLadelli2012,BasettiLadelliMatthes} expressed the solution in a convenient probabilistic way. Ealier results on probabilistic representation can be found in \cite{Carlen+Carvalho+Gabetta:2000,Gabetta+Regazzini:2006,Gabetta+Regazzini:2008}.
The probabilistic construction of the solution $\phi$ using labelled $N$-ary random trees is given on pp.~1938--1939 of \cite{BasettiLadelli2012} see Proposition 3.2 therein, where the authors use among other a stochastic process called $(\nu_t)_{t\geq 0}$. However, it is defined as an arbitrary stochastic process with specified marginal distributions, see the top of p.~1939 in \cite{BasettiLadelli2012}. Even though such specification is sufficient for the asymptotic analysis of $\phi$, it leaves an open and interesting question of finding a correct interpretation and pathwise construction of $(\nu_t)_{t\geq 0}$. The main purpose of this subsection is to propose a natural representation of $(\nu_t)_{t\geq 0}$ and to provide an alternative form of Proposition 3.2 of \cite{BasettiLadelli2012} revealing the complete probabilistic structure of the solution $\phi$. As we will see, $\phi(t,\cdot)$ is nothing else but the characteristic function of a smoothing transform associated with a certain continuous-time branching random walk and applied to the distribution of $X_0$, see Proposition \ref{prop:prob_solution} below.
\subsection{Representation of the solutions and connection with branching random walks in continuous time.}
Let us recall that a Yule process $(\mathcal{Y}_t)_{t\geq 0}$ is a pure birth process which starts with one particle. After exponential time with parameter $1$ the original particle dies out and produces $N$ new particles. Every particle behaves as the original one, and the particles reproduce independently. The quantity $\mathcal{Y}_t$ is the number of particles at time $t\geq 0$. Denote by $F(s,t)$ the probability generating function of $\mathcal{Y}_t$, that is
$$
F(s,t)={\mathbb{E}} s^{\mathcal{Y}_t},\quad t\geq 0,\quad |s|\leq 1.
$$
Using equations (5) and (6) on p.~106 in \cite{AthreyaNey}, see also example on p.~109 in the same reference, we obtain
$$
\frac{\partial F(s,t)}{\partial t} = F^{N}(s,t)-F(s,t),\quad t>0,\quad F(s,0)=s.
$$
By solving this differential equation, we get the explicit solution
\begin{equation}\label{eq:yule_process_gf}
F(s,t)=s\left(\frac{e^{-(N-1)t}}{1-s^{N-1}(1-e^{-(N-1)t})}\right)^{\frac{1}{N-1}},\quad t\geq 0,\quad |s|\leq 1.
\end{equation}
The full genealogical tree $\mathcal{T}_{\infty}$ of the Yule process $(\mathcal{Y}_t)_{t\geq 0}$ is an infinite $N$-ary random tree. For every fixed $T\geq 0$ the genealogical tree of $(\mathcal{Y}_t)_{t\in[0,T]}$ is a finite $N$-ary random tree with leaves representing the particles alive at time $T$ and internal nodes being the particles which have died out during $[0,T]$. Denote the number of latter particles by $\nu_T$. We have the following identity
\begin{equation}\label{eq:y_and_nu_connection}
\mathcal{Y}_t=(N-1)\nu_t+1,\quad t\geq 0.
\end{equation}
From this representation and formula \eqref{eq:yule_process_gf} we get
\begin{multline}\label{eq:nu_gen_func}
{\mathbb{E}} s^{\nu_t}=e^{-t} \left(1-s\left(1-e^{-(N-1)t}\right)\right)^{- \frac{1}{N-1}}\\
=\sum_{k\geq 0}\frac{\Gamma(\frac{1}{N-1}+k)}{k!\Gamma(\frac{1}{N-1})}e^{-t}(1-e^{-(N-1)t})^k s^k,\quad t\geq 0,\quad |s|\leq 1,
\end{multline}
in full agreement with formula (3.4) in \cite{BasettiLadelli2012}. That is to say, the process $(\nu_t)_{t\geq 0}$ introduced in \cite{BasettiLadelli2012}, should be interpreted as the number of splits during the time interval $[0,t]$ in the Yule process $(\mathcal{Y}_t)_{t\geq 0}$. This interpretation of the distribution of $\nu_t$ is the starting point of our probabilistic construction of the solution $\phi$.
By adding to the definition of a Yule process the control over positions of particles, we obtain a continuous-time branching random walk introduced in \cite{Uchiyama:1982}. More precisely, let $\zeta=\sum_{k=1}^{N}\delta_{Z_k}$ be an arbitrary point process on ${\mathbb{R}}$, where $\delta_x$ denotes the Dirac point measure at $x\in{\mathbb{R}}$. In the continuous-time branching random walk the initial single particle is located at $0$. After an exponential time with parameter $1$ it dies out and gives birth to $N$ new particles which are placed at positions $(Z_1,\ldots, Z_N)$. These particles reproduce independently in the same fashion as their mother. In particular, if at any time a particle $v$ located at some $x\in{\mathbb{R}}$ splits, its children are placed at $x+Z_1(v),\ldots,x+Z_N(v)$, where $\zeta^{(v)}:=\sum_{k=1}^{N}\delta_{Z_k(v)}$ is an independent copy of $\zeta$. In what follows we only consider branching random walks with deterministic number of children of every particle. Clearly, the number of particles in such a continuous-time branching random walk at time $t\geq 0$ is just $\mathcal{Y}_t$. Denote the locations of particles present at time $t$ by $z_{1,t},z_{2,t},\ldots,z_{\mathcal{Y}_t,t}$. The continuous-time branching random walk is formally defined as the measure-valued stochastic process
$$
\mathcal{Z}_t:=\sum_{k=1}^{\mathcal{Y}_t}\delta_{z_{k,t}},\quad t\geq 0.
$$
It will be important that the process $(\mathcal{Z}_t)_{t\geq 0}$ satisfies the following branching relation:
\begin{equation}\label{eq:brw_branching_property}
\mathcal{Z}_{t+s}(\cdot)\overset{d}{=}\sum_{k=1}^{\mathcal{Y}_t}\mathcal{Z}_{s}^{(k)}(\cdot - z_{k,t}),\quad t,s\geq 0,
\end{equation}
where $(\mathcal{Z}_{t}^{(k)})_{t\geq 0}$ for $k\in{\mathbb{N}}$ are independent copies of $(\mathcal{Z}_t)_{t\geq 0}$.
Finally, given a continuous-time branching random walk $(\mathcal{Z}_t)_{t\geq 0}$, the associated family of smoothing transforms $(\mathcal{L}^{(\gamma)}_t)_{t\geq 0}$ on the space of probability distributions on ${\mathbb{R}}$ is defined by
$$
\mathcal{L}^{(\gamma)}_t({\rm distr}(U))={\rm distr}\left(\sum_{k=1}^{\mathcal{Y}_t}e^{\gamma z_{k,t}}U_k\right),
$$
where $(U_k)_{k\geq 1}$ are independent copies of a random variable $U$ and $\gamma\in\mathbb{C}$ is a parameter. By slightly abusing notation we write $\mathcal{L}^{(\gamma)}_t(U)$ instead of $\mathcal{L}^{(\gamma)}_t({\rm distr}(U))$. We also suppress the index $\gamma$ if it is equal to $1$ by writing $\mathcal{L}_t$ instead of $\mathcal{L}^{(1)}_t$.
We are ready to state the main result of this subsection, namely the probabilistic representation of the solution $\phi$ to kinetic-type equation \eqref{eqboltzivp}. Assume that on the probability space $(\Omega,\mathcal{F},{\mathbb{P}})$ the following two objects are defined:
\begin{itemize}
\item the continuous time branching random walk $(\mathcal{Z}_{t})_{t\geq 0}$ with the displacement process $\zeta:=\sum_{k=1}^{N}\delta_{\log A_k}$:
$$
\mathcal{Z}_t:=\sum_{k=1}^{\mathcal{Y}_t}\delta_{z_{k,t}},\quad t\geq 0.
$$
\item the sequence $(X_k)_{k\geq 1}$ of independent random variables with common distribution function $F_0$, which is also independent of $(\mathcal{Z}_{t})_{t\geq 0}$.
\end{itemize}
\begin{prop}\label{prop:prob_solution}
Equation \eqref{eqboltzivp} has a unique solution $\phi(t,\cdot)$ which is given by
\begin{equation}\label{eq:prob_solution_main}
\phi(t,\xi)={\mathbb{E}}\exp\left(i\xi \left(\sum_{k=1}^{\mathcal{Y}_t}e^{z_{k,t}}X_k\right)\right),\quad t\geq 0,\quad \xi\in{\mathbb{R}},
\end{equation}
that is $\phi(t,\cdot)$ is the characteristic function of the random variable $\mathcal{L}_t(X_0)$, where $\mathcal{L}_t$ is the smoothing transform associated with the continuous-time branching random walk $(\mathcal{Z}_{t})_{t\geq 0}$.
\end{prop}
\begin{proof}
Uniqueness of the solution follows from a standard use of the Picard-Lindel\"of theorem, see e.g. proof of Proposition 2.2 in \cite{BasettiLadelliMatthes2}. Thus it is enough to show that the right-hand side of \eqref{eq:prob_solution_main} satisfies \eqref{eqboltzivp}.
To this end, denote the the right-hand side of \eqref{eq:prob_solution_main} by $\psi(t,\xi)$ and write
$$
\psi(t,\xi)={\mathbb{E}}\left[\prod_{k=1}^{\mathcal{Y}_t}
\phi_0(\xi e^{z_{k,t}})\right]={\mathbb{E}}\exp\left(\int_{{\mathbb{R}}}\log \phi_0(\xi e^y)\mathcal{Z}_t({\rm d}y)\right),\quad t\geq 0,\quad \xi\in{\mathbb{R}}.
$$
Firstly, let us show that $t\mapsto \psi(t,\xi)$ is continuous for every fixed $\xi$. For $t,s\geq 0$ we can write
\begin{multline*}
|\psi(t,\xi)-\psi(s,\xi)|\leq 2 {\mathbb{P}}\{\text{there are splits during } [t\wedge s,t\vee s]\}=2 {\mathbb{E}} (1-e^{-\mathcal{Y}_{t\wedge s}|t-s|})\to 0,
\end{multline*}
as $s\to t$, by the dominated convergence theorem and the observation $\mathcal{Y}_{t}<\infty$ a.s.
Further, for $t\geq 0$, let $\mathcal{F}_t\subset \mathcal{F}$ be the $\sigma$-algebra generated by $(\mathcal{Z}_s)_{s\in[0,t]}$. For $h\geq 0$, using formula \eqref{eq:brw_branching_property}, we obtain
\begin{align*}
\psi(t+h,\xi)&={\mathbb{E}}\left({\mathbb{E}}\left(\exp\left(\int_{{\mathbb{R}}}\log \phi_0(\xi e^y)\mathcal{Z}_{t+h}({\rm d}y)\right)\Big|\mathcal{F}_h\right)\right)\\
&={\mathbb{E}}\left({\mathbb{E}}\left(\prod_{k=1}^{\mathcal{Y}_h}\exp\left(\int_{{\mathbb{R}}}\log \phi_0(\xi e^y e^{z_{k,h}})\mathcal{Z}^{(k)}_{t}({\rm d}y)\right)\Big|\mathcal{F}_h\right)\right)\\
&={\mathbb{E}}\left(\prod_{k=1}^{\mathcal{Y}_h}{\mathbb{E}}\left(\exp\left(\int_{{\mathbb{R}}}\log \phi_0(\xi e^y e^{z_{k,h}})\mathcal{Z}^{(k)}_{t}({\rm d}y)\right)\Big|\mathcal{F}_h\right)\right)={\mathbb{E}}\left(\prod_{k=1}^{\mathcal{Y}_h}\psi(t,\xi e^{z_{k,h}})\right).
\end{align*}
The probability of having two or more splits in the branching random walk $(\mathcal{Z}_{t})_{t\geq 0}$ during $[0,h]$ is $o(h)$ as $h\to +0$, whence
\begin{align*}
\psi(t+h,\xi)&={\mathbb{E}}\left(\prod_{k=1}^{\mathcal{Y}_h}\psi(t,\xi e^{z_{k,h}})\right)\\
&=\psi(t,\xi){\mathbb{P}}\{\text{there are no splits during } [0,h]\}\\
&+{\mathbb{E}}\left(\prod_{j=1}^{N}\psi(t,\xi A_j)\right){\mathbb{P}}\{\text{there is exactly one split during } [0,h]\}+o(h)\\
&=\psi(t,\xi)e^{-h}+{\mathbb{E}}\left(\prod_{j=1}^{N}\psi(t,\xi A_j)\right)h+o(h).
\end{align*}
Likewise, we can write for $h\geq 0$ and $t\geq h$:
\begin{align*}
\psi(t,\xi)&={\mathbb{E}}\left(\prod_{k=1}^{\mathcal{Y}_h}\psi(t-h,\xi e^{z_{k,h}})\right)\\
&=\psi(t-h,\xi){\mathbb{P}}\{\text{there are no splits during } [0,h]\}\\
&+{\mathbb{E}}\left(\prod_{j=1}^{N}\psi(t-h,\xi A_j)\right){\mathbb{P}}\{\text{there is exactly one split during } [0,h]\}+o(h)\\
&=\psi(t-h,\xi)e^{-h}+{\mathbb{E}}\left(\prod_{j=1}^{N}\psi(t-h,\xi A_j)\right)h+o(h).
\end{align*}
Rearranging the terms and sending $h\to +0$ shows that
$$
\frac{\partial \psi(t,\xi)}{\partial t}+\psi(t,\xi)={\mathbb{E}}\left(\prod_{j=1}^{N}\psi(t,\xi A_j)\right)=\widehat{Q}(\psi(t,\cdot),\ldots,\psi(t,\cdot)),\quad t>0,
$$
by the dominated convergence theorem and continuity of $t\mapsto \psi(t,\xi)$ (this is required for the left derivative). Therefore, $\psi$ is a solution to \eqref{eqboltzivp}. Since the solution is unique and $\psi(0,\xi)=\phi_0(\xi)=\phi(0,\xi)$, we infer $\psi(t,\xi)\equiv\phi(t,\xi)$. The proof is complete.
\end{proof}
\begin{rem}
Let us now compare our Proposition \ref{prop:prob_solution} with Proposition 3.2 in \cite{BasettiLadelli2012} in more details. Proposition 3.2 in \cite{BasettiLadelli2012}
states that the unique solution $\phi$ to \eqref{eqboltzivp} is
$$
\phi(t,\xi)=\int_{\mathbb{R}} e^{i\xi v}\rho_t({\rm d}v)=\int_{\mathbb{R}} e^{i\xi v}{\mathbb{P}}\{W_{\nu_t}\in {\rm d}v\},
$$
where $\nu_t$ is an integer-valued random variable with the generating function \eqref{eq:nu_gen_func} and which is independent of $(W_n)_{n\geq 0}$. For $n=0,1,2,\ldots$, $W_n$ is defined by a sum
$$
W_n:=\sum_{i=1}^{(N-1)n+1} \omega(v_{i,n}) X_{v_{i,n}},
$$
where $v_{i,n}$, $i=1,\ldots,(N-1)n+1$ are the leaves of a random labelled $N$-ary recursive tree $\mathcal{T}_n$ after $n$ steps, $\omega(v)$ is the weight of a leaf $v\in\mathcal{T}_n$ (the product of all labels on the unique path from the root to $v$), and $(X_{v})$ is a family of independent random variables with common distribution function $F_0$ which is also independent of the random labelled tree $\mathcal{T}_n$.
Our construction described above unifies and reinterprets all the aforementioned ingredients: the sequence of random $N$-ary trees $(\mathcal{T}_n)$, the labels of their nodes and the subordination time $\nu_t$ via a single object, the continuous-time branching random walk $(\mathcal{Z}_t)_{t\geq 0}$. We summarize the above observations in Table \ref{tab1}.
\begin{table}[!htbp]
\caption{Comparison of two probabilistic constructions of $\phi$}
\begin{tabular}{p{7cm}p{7cm}}
\hline
The construction in \cite{BasettiLadelli2012,BasettiLadelliMatthes} & A counterpart in our construction\\
\hline
the sequence of random $N$-ary recursive trees $(\mathcal{T}_n)_{n\geq 0}$ & the skeleton of the Yule process $(\mathcal{Y}_{t})_{t\geq 0}$ pertained to $(\mathcal{Z}_{t})_{t\geq 0}$ and observed at splitting times\\
\hline
labels of the nodes in the trees $(\mathcal{T}_n)_{n\geq 0}$ & relative displacements of the particles in $(\mathcal{Z}_{t})_{t\geq 0}$\\
\hline
random variables $\nu_t$, $t\geq 0$ & random process $(\nu_t)_{t\geq 0}$, the number of splits in
$(\mathcal{Z}_{s})_{s\geq 0}$ (or $(\mathcal{Y}_{t})_{t\geq 0}$) on $[0,t]$\\
\hline
\end{tabular}
\label{tab1}
\end{table}
The connection between random $N$-ary trees, Yule processes and branching random walks, is by no means new and have already been observed in probabilistic literature, see for example \cite{Chauvin+Klein+Marckert+Rouault:2004} for the case of binary search trees. Recently this connection has been extensively exploited in the analysis of profiles of random trees in \cite{Kabluchko+Marynych+Sulzbach:2017}.
\end{rem}
\begin{rem}
As has been pointed out by the referee our Proposition \ref{prop:prob_solution} remains valid also with random $N$ such that ${\mathbb{E}} N\in (1,\infty)$. The latter condition guarantees that $(\mathcal{Z}_t)_{t\geq 0}$ does not explode and has a positive survival probability. On the other hand, probabilistic construction used in \cite{BasettiLadelli2012,BasettiLadelliMatthes} does not seem to have a direct analogue for random $N$ due to a lack of explicit distribution for $\nu_t$ for a fixed $t>0$.
\end{rem}
Last but not least, we would like to emphasize that the main advantage of Proposition \ref{prop:prob_solution} is its generality. It allows one to translate limit theorems for the smoothing transform $\mathcal{L}^{(\gamma)}_t(X_0)$, as $t\to\infty$, to the corresponding asymptotics for the solution of \eqref{eqboltzivp}, as $t\to\infty$. In particular, Proposition \ref{prop:prob_solution} is useful not only in the case considered in our paper~--~the boundary case~--~but also in other situations. As we will see in the next sections, limit theorems for $\mathcal{L}^{(\gamma)}_t(X_0)$ are intimately connected with convergence in probability of a so-called Biggins martingale for the continuous-time branching random walk $(\mathcal{Z}_t)_{t\geq 0}$.
\section{Convergence of the continuous-time Biggins martingale in the boundary case.}\label{sec:biggins_convergence}
For every $\gamma\in [0,s_{\infty})$, put
$$
\mathcal{M}_t(\gamma):= e^{-\Phi(\gamma)t}\sum_{k=1}^{\mathcal{Y}_t}e^{\gamma z_{k,t}},\quad t\geq 0,
$$
and note that by formula (5.1) in \cite{Biggins:1992} we have
\begin{equation}\label{eq:biggins_martingale_exp}
{\mathbb{E}} \mathcal{M}_t(\gamma)=1.
\end{equation}
The stochastic process $(\mathcal{M}_t(\gamma))_{t\geq 0}$ is a martingale and is called {\it continuous-time Biggins martingale}.
If $\gamma=\gamma^{\ast}=\argmin_{s\in [0,s_{\infty})}\mu(s)$ and $\gamma^{\ast}<s_{\infty}$, then the Biggins martingale $(\mathcal{M}_t(\gamma^{\ast}))_{t\geq 0}$ converges to zero a.s. For the discrete-time Biggins martingale this fact is well-known, see, for example, Lemma 5 in \cite{Biggins:1977}, and for the continuous-time Biggins martingale it follows from Theorem 1.1 of the recent paper \cite{Bertoin+Mallein:2018} as well as from Proposition \ref{prop: as}(i) below.
\begin{prop}\label{prop: as} Assume that $\gamma^{\ast}\in (0,s_\infty)$. The following limit relations hold true.
\begin{itemize}
\item[(i)] As $t\to\infty$ we have
\begin{equation}\label{eq:pp1}
\sqrt{t} \mathcal{M}_t(\gamma^{\ast})
=\sqrt{t} \sum_{k=1}^{\mathcal{Y}_t}e^{\gamma^{\ast}z^{\circ}_{k,t}}
\overset{{\mathbb{P}}}{\to}\sqrt{\frac{2}{\pi (\gamma^{\ast})^2\Phi^{\prime\prime}(\gamma^{\ast})}}D_{\infty},
\end{equation}
where $z^{\circ}_{k,t} = z_{k,t} - t \mu(\gamma^*)$, $D_{\infty}$ is the a.s. limit of the derivative martingale
\begin{equation}\label{eq:prop1_derivative_martingale}
\mathcal{D}_t(\gamma^{\ast}):=\sum_{k=1}^{\mathcal{Y}_t}e^{\gamma^{\ast}z^{\circ}_{k,t}}z^{\circ}_{k,t},\quad t\geq 0.
\end{equation}
The random variable $D_{\infty}$ is a.s.~positive and satisfies \eqref{eq:d_fixed_point}.
\item[(ii)] Moreover,
\begin{equation}\label{eq:pp2}
\sqrt{t} \max_{k=1,\ldots,\mathcal{Y}_t}e^{\gamma^{\ast}z^{\circ}_{k,t}} \overset{{\mathbb{P}}}{\to} 0,\quad t\to\infty.
\end{equation}
\end{itemize}
\end{prop}
The derivation of Proposition \ref{prop: as} utilizes ideas borrowed from \cite{Dadoun:2017}, where part (i) has been stated without a proof in Remark 2.11(iii). Firstly, we obtain two auxiliary lemmas which show that the Biggins martingale $(\mathcal{M}_t(\gamma^{\ast}))_{t\geq 0}$ is in the {\it boundary case}. In particular, this implies that every $\theta$-skeleton, that is the discrete-time Biggins martingale $(\mathcal{M}_{n\theta}(\gamma^{\ast}))_{n\ge 0}$, $\theta>0$, is also in the boundary case. Thereafter, we apply the corresponding theorem by A\"id\'ekon and Shi \cite{AidekonShi2014}, who found the appropriate normalization for the discrete-time Biggins martingales in the boundary case, to our $\theta$-skeletons and then pass to the continuous parameter with the aid of the Croft--Kingman lemma.
\begin{lem}\label{lem:1} Assume that $\gamma^{\ast}\in (0,s_\infty)$. For every $t\geq 0$ we have
\begin{equation}\label{eq:lem1_claims}
{\mathbb{E}}\bigg[ \sum_{k=1}^{\mathcal{Y}_t} e^{\gamma^{\ast} z^{\circ}_{k,t}} z^{\circ}_{k,t} \bigg] = 0\quad\text{and}\quad {\mathbb{E}}\bigg[ \sum_{k=1}^{\mathcal{Y}_t}e^{\gamma^{\ast} z^{\circ}_{k,t}}(z^{\circ}_{k,t})^2 \bigg] = t\Phi^{\prime\prime}(\gamma^{\ast}).
\end{equation}
\end{lem}
\begin{proof}
Fix $\varepsilon\in (0,\gamma^{\ast})$ such that $\gamma^{\ast}+\varepsilon<s_{\infty}$. Let us show that for every fixed $t\geq 0$ the following holds:
$$
{\mathbb{E}}\bigg[ \sum_{k=1}^{\mathcal{Y}_t}e^{\gamma^{\ast} z_{k,t}}z_{k,t} \bigg]=\frac{\partial }{\partial \gamma}\left[{\mathbb{E}}\int_{{\mathbb{R}}} e^{\gamma y}\mathcal{Z}_t({\rm d}y)\right]\Bigg|_{\gamma=\gamma^{\ast}}.
$$
To this end, it is enough to check that the partial derivative on the right-hand side can be moved inside the expectation and the integration signs. But this is a simple consequence of the dominated convergence theorem, since
$$
\lim_{\Delta \to 0}\int_{\Omega}\int_{{\mathbb{R}}} \frac{e^{(\gamma^{\ast}+\Delta) y}-e^{\gamma^{\ast} y}}{\Delta}\mathcal{Z}_t({\rm d}y){\rm d}{\mathbb{P}}=\lim_{\Delta \to 0}\int_{\Omega}\int_{{\mathbb{R}}} \frac{e^{\Delta y}-1}{\Delta}e^{\gamma^{\ast} y}\mathcal{Z}_t({\rm d}y){\rm d}{\mathbb{P}},
$$
and the absolute value of the integrand is bounded by the integrable function
$$
y\mapsto e^{(\gamma^{\ast}+\varepsilon) y}{\bf 1}_{\{y\geq 0\}}+e^{(\gamma^{\ast}-\varepsilon) y}{\bf 1}_{\{y<0\}}
$$
for sufficiently small $\Delta$ and all $y\in{\mathbb{R}}$.
Using formula \eqref{eq:biggins_martingale_exp} we derive
$$
{\mathbb{E}}\bigg[ \sum_{k=1}^{\mathcal{Y}_t}e^{\gamma^{\ast} z_{k,t}} z_{k,t} \bigg]=\frac{\partial}{\partial \gamma}\left(e^{t\Phi(\gamma)}\right)\Bigg|_{\gamma=\gamma^{\ast}}=t\Phi^{\prime}(\gamma^{\ast})e^{t\Phi(\gamma^{\ast})},\quad t\geq 0.
$$
This immediately yields
$$
{\mathbb{E}}\bigg[ \sum_{k=1}^{\mathcal{Y}_t}e^{\gamma^{\ast} z^{\circ}_{k,t}}z^{\circ}_{k,t} \bigg]=e^{-t\Phi(\gamma^{\ast})}\left(t\Phi^{\prime}(\gamma^{\ast})e^{t\Phi(\gamma^{\ast})}\right)
-t \mu(\gamma^{\ast}){\mathbb{E}} \mathcal{M}_t(\gamma^{\ast})=t\left(\Phi^{\prime}(\gamma^{\ast})- \mu(\gamma^{\ast})\right) = 0.
$$
The second claim in \eqref{eq:lem1_claims} follows from the formula
$$
{\mathbb{E}}\bigg[ \sum_{k=1}^{\mathcal{Y}_t}e^{\gamma^{\ast} z_{k,t}} z^2_{k,t} \bigg]=\frac{\partial^2 }{\partial \gamma^2}\left[{\mathbb{E}}\int_{{\mathbb{R}}} e^{\gamma y}\mathcal{Z}_t({\rm d}y)\right]\Bigg|_{\gamma=\gamma^{\ast}},
$$
which can be proved similarly. The proof is complete.
\end{proof}
\begin{rem}
As has been pointed out by the referee Lemma 3.6 also follows from the many-to-one lemma for continuous-time branching random walks.
\end{rem}
\begin{lem}\label{lem:2}
Assume that $\gamma^{\ast}<s_{\infty}$. Then for every fixed $t\geq 0$ and $\delta>0$ such that $(1+\delta)\gamma^{\ast} < s_{\infty}$ we have
\begin{equation*}
{\mathbb{E}}\bigg[ \bigg( \sum_{k=1}^{\mathcal{Y}_t} e^{\gamma^{\ast} z^{\circ}_{k,t}}\bigg)^{1+\delta} \bigg] <\infty\quad\text{and}\quad {\mathbb{E}}\bigg[ \bigg( \sum_{k=1}^{\mathcal{Y}_t} e^{\gamma^{\ast} z^{\circ}_{k,t}}( z_{k,t})_{+} \bigg)^{1+\delta} \bigg]<\infty,
\end{equation*}
where $x_{+}:=\max(x,0)$.
\end{lem}
\begin{proof}
Let us prove the first claim. Using the inequality
$$
\left(\sum_{k=1}^{n}x_k\right)^{1+\delta}\leq n^{\delta}\left(\sum_{k=1}^{n}x_k^{1+\delta}\right)
$$
which holds for $n\in{\mathbb{N}}$ and arbitrary nonnegative reals $x_1,x_2,\ldots,x_n$, we infer
$$
{\mathbb{E}}\bigg( \sum_{k=1}^{\mathcal{Y}_t} e^{\gamma^{\ast} z_{k,t}} \bigg)^{1+\delta}\leq
{\mathbb{E}}\left( \mathcal{Y}_t^{\delta} \sum_{k=1}^{ \mathcal{Y}_t}e^{(1+\delta)\gamma^{\ast} z_{k,t}}\right)=\sum_{k=1}^{\infty}{\mathbb{E}}\left( \mathcal{Y}_t^{\delta} e^{(1+\delta)\gamma^{\ast} z_{k,t}}{\bf 1}_{\{k\leq \mathcal{Y}_t\}}\right).
$$
Pick $p>1$ such that $p(1+\delta)\gamma^{\ast}<s_{\infty}$ and $q>1$ such that $1/p+1/q=1$. By H\"{o}lder's inequality we obtain
$$
{\mathbb{E}}\left( \mathcal{Y}_t^{\delta} e^{(1+\delta)\gamma^{\ast} z_{k,t}}{\bf 1}_{\{k\leq \mathcal{Y}_t\}}\right)\leq \left({\mathbb{E}} e^{p(1+\delta)\gamma^{\ast} z_{k,t}}{\bf 1}_{\{k\leq \mathcal{Y}_t\}}\right)^{1/p}\left({\mathbb{E}} \mathcal{Y}_t^{q \delta}{\bf 1}_{\{k\leq \mathcal{Y}_t\}}\right)^{1/q},
$$
and thereupon
\begin{multline}\label{eq:lem2_proof1}
{\mathbb{E}}\bigg( \sum_{k=1}^{\mathcal{Y}_t} e^{\gamma^{\ast}z_{k,t}} \bigg)^{1+\delta}\leq \sum_{k=1}^{\infty}\left({\mathbb{E}} e^{p(1+\delta)\gamma^{\ast}z_{k,t}}{\bf 1}_{\{k\leq \mathcal{Y}_t\}}\right)^{1/p}\left({\mathbb{E}}\mathcal{Y}_t^{q \delta}{\bf 1}_{\{k\leq \mathcal{Y}_t\}}\right)^{1/q}\\
\leq \left(\sum_{k=1}^{\infty}{\mathbb{E}} e^{p(1+\delta)\gamma^{\ast} z_{k,t}}{\bf 1}_{\{k\leq \mathcal{Y}_t\}}\right)^{1/p}\left(\sum_{k=1}^{\infty}{\mathbb{E}} \mathcal{Y}_t^{q \delta}{\bf 1}_{\{k\leq \mathcal{Y}_t\}}\right)^{1/q},\end{multline}
where the last passage is a consequence of H\"{o}lder's inequality for series. The first factor on the right-hand side is finite because $p(1+\delta)\gamma^{\ast}<s_{\infty}$ and
\begin{multline*}
\sum_{k=1}^{\infty}{\mathbb{E}} e^{p(1+\delta)\gamma^{\ast} z_{k,t}}{\bf 1}_{\{k\leq \mathcal{Y}_t\}}={\mathbb{E}} \left(\sum_{k=1}^{\mathcal{Y}_t}e^{p(1+\delta)\gamma^{\ast} z_{k,t}}\right)={\mathbb{E}} \int_{{\mathbb{R}}}e^{p(1+\delta)\gamma^{\ast}y}\mathcal{Z}_t({\rm d}y)\\
=e^{\Phi(p(1+\delta)\gamma^{\ast})t}{\mathbb{E}}\mathcal{M}_t(p(1+\delta)\gamma^{\ast})=e^{\Phi(p(1+\delta)\gamma^{\ast})t}<\infty.
\end{multline*}
Formulae \eqref{eq:y_and_nu_connection} and \eqref{eq:nu_gen_func} imply that $\mathcal{Y}_t$ has exponential moment of some positive order for every fixed $t$. Therefore,
$$
\sum_{k=1}^{\infty}{\mathbb{E}}\mathcal{Y}_t^{q \delta}{\bf 1}_{\{k\leq \mathcal{Y}_t\}}={\mathbb{E}}\mathcal{Y}_t^{q \delta+1}<\infty
$$
and the proof of the first claim is complete.
To prove the second inequality we use exactly the same arguments to get the upper bound
\begin{multline*}
{\mathbb{E}}\bigg( \sum_{k=1}^{\mathcal{Y}_t} e^{\gamma^{\ast}z_{k,t}}(z_{k,t})_{+} \bigg)^{1+\delta}\\
\leq \left(\sum_{k=1}^{\infty}{\mathbb{E}} e^{p(1+\delta)\gamma^{\ast}z_{k,t}}(z_{k,t})^{p(1+\delta)}_{+}{\bf 1}_{\{k \leq \mathcal{Y}_t\}}\right)^{1/p}\left(\sum_{k=1}^{\infty}{\mathbb{E}}\mathcal{Y}_t^{q \delta}{\bf 1}_{\{k\leq \mathcal{Y}_t\}}\right)^{1/q}.
\end{multline*}
It remains to note that
$$
\sum_{k=1}^{\infty}{\mathbb{E}} e^{p(1+\delta)\gamma^{\ast}z_{k,t}}(z_{k,t})^{p(1+\delta)}_{+}{\bf 1}_{\{k \leq \mathcal{Y}_t\}}={\mathbb{E}} \int_{{\mathbb{R}}}e^{p(1+\delta)\gamma^{\ast}y}y^{p(1+\delta)}_+\mathcal{Z}_t({\rm d}y)<\infty,
$$
since $p(1+\delta)\gamma^{\ast}<s_{\infty}$. The proof is complete.
\end{proof}
\begin{proof}[Proof of Proposition \ref{prop: as}]
{\sc Proof of part (i).} Fix $\theta>0$. Define a point process
$$
\Xi:=\sum_{k=1}^{\mathcal{Y}_{\theta}}\delta_{-\gamma^{\ast}z^{\circ}_{k,\theta}},
$$
and consider a discrete-time branching random walk $(\mathcal{Z}_n(\theta))_{n=0,1,2,\ldots}$, where
$$
\mathcal{Z}_n(\theta):=\sum_{k=1}^{\mathcal{Y}_{n\theta}}\delta_{ -\gamma^{\ast} z^{\circ}_{k,n\theta}},\quad n=0,1,2,\ldots.
$$
The discrete-time branching random walk $(\mathcal{Z}_k(\theta))$ has the displacement process $\Xi$ and satisfies the following three conditions:
\begin{multline}\label{eq:boundary_case_skeleton}
{\mathbb{E}}\left(\int_{{\mathbb{R}}}e^{-y}\mathcal{Z}_1(\theta)({\rm d}y)\right)=1,\quad {\mathbb{E}}\left(\int_{{\mathbb{R}}}e^{-y}y\mathcal{Z}_1(\theta)({\rm d}y)\right)=0\quad \text{and}\\
{\mathbb{E}}\left(\int_{{\mathbb{R}}}e^{-y}y^2\mathcal{Z}_1(\theta)({\rm d}y)\right)=\theta(\gamma^{\ast})^2\Phi^{\prime\prime}(\gamma^{\ast})<\infty,
\end{multline}
where the last two relations are secured by Lemma \ref{lem:1}. Moreover, Lemma \ref{lem:2} yields
$$
{\mathbb{E}}\left(\int_{{\mathbb{R}}}e^{-y}\mathcal{Z}_1(\theta)({\rm d}y)\right)^{1+\delta}<\infty\quad\text{and}\quad {\mathbb{E}}\left(\int_{{\mathbb{R}}}e^{-y}y_{+}\mathcal{Z}_1(\theta)({\rm d}y)\right)^{1+\delta}<\infty,
$$
whence conditions (5.3) in \cite{Shi:2015} hold. Therefore, Assumption (H) in the same reference holds for the discrete-time branching random walk $(\mathcal{Z}_n(\theta))_{n=0,1,2,\ldots}$ for every fixed $\theta>0$. In particular, by Theorem 5.29 in \cite{Shi:2015}, see also Theorem 1.1 in \cite{AidekonShi2014}, we have
\begin{multline}\label{eq:conv_along_integers}
\sqrt{n} \mathcal{M}_{n}(\gamma^{\ast})=\sqrt{n} \sum_{k=1}^{\mathcal{Y}_{n}}e^{\gamma^{\ast}z^{\circ}_{k,n}}
=\sqrt{n} \int_{{\mathbb{R}}}e^{-y}\mathcal{Z}_{n}(1)({\rm d}y)\\
\overset{{\mathbb{P}}}{\to} \sqrt{\frac{2}{\pi (\gamma^{\ast})^2 \Phi^{\prime\prime}(\gamma^{\ast})}}D_{\infty}=:D,\quad n\to\infty,
\end{multline}
where $D$ is a.s. positive, because in our settings the process does not extinct with probability one. It remains to show the convergence in probability to $D$ along $t\to\infty$, $t\in{\mathbb{R}}$. This can be accomplished by adopting the argument given on p.~47 in \cite{Biggins+Kyprianou:1996} as follows. From \eqref{eq:conv_along_integers} we know that for every fixed $x>0$
$$
\left(\sqrt{n+1} \mathcal{M}_{n+1}(\gamma^{\ast})-\sqrt{n} \mathcal{M}_{n}(\gamma^{\ast})\right)\mathbbm{1}_{\{\sqrt{n} \mathcal{M}_{n}(\gamma^{\ast})\leq x\}}\overset{{\mathbb{P}}}{\to} 0,\quad n\to\infty,
$$
and therefore by the dominated convergence theorem we have for every $u\geq 0$
\begin{equation*}
{\mathbb{E}} \exp\left(-u\left(\left(\sqrt{n+1} \mathcal{M}_{n+1}(\gamma^{\ast})-\sqrt{n} \mathcal{M}_{n}(\gamma^{\ast})\right)\mathbbm{1}_{\{\sqrt{n} \mathcal{M}_{n}(\gamma^{\ast})\leq x\}}\right)\right)\to 1,\quad n\to\infty.
\end{equation*}
Further, by the martingale property of $(\mathcal{M}_t(\gamma^{\ast}))$ and applying Jensen's inequality twice to the convex function $x\mapsto \exp(-ux)$ we obtain for every $t\geq 0$
\begin{align*}
& {\mathbb{E}} \exp\left(-u\left(\left(\sqrt{[t]+1} \mathcal{M}_{[t]+1}(\gamma^{\ast})-\sqrt{[t]} \mathcal{M}_{[t]}(\gamma^{\ast})\right)\mathbbm{1}_{\{\sqrt{[t]} \mathcal{M}_{[t]}(\gamma^{\ast})\leq x\}}\right)\right)\\
&={\mathbb{E}} \left[{\mathbb{E}} \left\{\exp\left(-u\left(\left(\sqrt{[t]+1} \mathcal{M}_{[t]+1}(\gamma^{\ast})-\sqrt{[t]} \mathcal{M}_{[t]}(\gamma^{\ast})\right)\mathbbm{1}_{\{\sqrt{[t]} \mathcal{M}_{[t]}(\gamma^{\ast})\leq x\}}\right)\right)\Big|\mathcal{F}_t \right\}\right]\\
&\geq {\mathbb{E}} \left\{\exp\left(-u\left(\left(\sqrt{[t]+1} \mathcal{M}_{t}(\gamma^{\ast})-\sqrt{[t]} \mathcal{M}_{[t]}(\gamma^{\ast})\right)\mathbbm{1}_{\{\sqrt{[t]} \mathcal{M}_{[t]}(\gamma^{\ast})\leq x\}}\right)\right)\right\}\\
&= {\mathbb{E}} \left[{\mathbb{E}} \left\{\exp\left(-u\left(\left(\sqrt{[t]+1} \mathcal{M}_{t}(\gamma^{\ast})-\sqrt{[t]} \mathcal{M}_{[t]}(\gamma^{\ast})\right)\mathbbm{1}_{\{\sqrt{[t]} \mathcal{M}_{[t]}(\gamma^{\ast})\leq x\}}\right)\right)\Big|\mathcal{F}_{[t]}\right\}\right]\\
&\geq {\mathbb{E}} \left\{\exp\left(-u\left(\left(\sqrt{[t]+1} \mathcal{M}_{[t]}(\gamma^{\ast})-\sqrt{[t]} \mathcal{M}_{[t]}(\gamma^{\ast})\right)\mathbbm{1}_{\{\sqrt{[t]} \mathcal{M}_{[t]}(\gamma^{\ast})\leq x\}}\right)\right)\right\}.
\end{align*}
Sending $t\to\infty$ in the above inequalities we obtain
\begin{equation}\label{eq:conv_along_integers2}
\left(\sqrt{[t]+1} \mathcal{M}_{t}(\gamma^{\ast})-\sqrt{[t]} \mathcal{M}_{[t]}(\gamma^{\ast})\right)\mathbbm{1}_{\{\sqrt{[t]} \mathcal{M}_{[t]}(\gamma^{\ast})\leq x\}}\overset{{\mathbb{P}}}{\to} 0,\quad t\to\infty.
\end{equation}
By the triangle inequality
\begin{align*}
\left|\sqrt{t}\mathcal{M}_{t}(\gamma^{\ast})-D\right|&\leq \left|\sqrt{t} \mathcal{M}_{t}(\gamma^{\ast})-\sqrt{[t]+1} \mathcal{M}_{t}(\gamma^{\ast})\right|\\
& + \left|\sqrt{[t]+1} \mathcal{M}_{t}(\gamma^{\ast})-\sqrt{[t]} \mathcal{M}_{[t]}(\gamma^{\ast})\right|\mathbbm{1}_{\{\sqrt{[t]} \mathcal{M}_{[t]}(\gamma^{\ast})\leq x\}}\\
&+ \left|\sqrt{[t]+1} \mathcal{M}_{t}(\gamma^{\ast})-\sqrt{[t]} \mathcal{M}_{[t]}(\gamma^{\ast})\right|\mathbbm{1}_{\{\sqrt{[t]} \mathcal{M}_{[t]}(\gamma^{\ast})> x\}}\\
&+ |\sqrt{[t]} \mathcal{M}_{[t]}(\gamma^{\ast})-D|.
\end{align*}
The second and fourth summands converge to zero in probability as $t\to\infty$ by \eqref{eq:conv_along_integers2} and \eqref{eq:conv_along_integers}, respectively. The first summand does this by Markov's inequality since $\sqrt{t+1}-\sqrt{t}\to 0$ as $t\to\infty$. The probability that the third summand is larger than some $\delta>0$ is bounded from above by ${\mathbb{P}}\{\sqrt{[t]} \mathcal{M}_{[t]}(\gamma^{\ast})> x\}$ which can be made arbitrarily small by choosing $x$ large enough in view of \eqref{eq:conv_along_integers} and a.s. finiteness of $D$. This completes the proof of convergence in part (i).
Let us show that $D_{\infty}$ (and also $D$) satisfies \eqref{eq:d_fixed_point}. Let $\tau_1$ be the time of the first split in $(\mathcal{Z}_t)_{t\geq 0}$, then
$$
\mathcal{Z}_t(\cdot)\overset{d}{=}{\bf 1}_{\{\tau_1>t\}}\delta_0(\cdot)+{\bf 1}_{\{\tau_1\leq t\}}\sum_{k=1}^{N}\mathcal{Z}^{(k)}_{t-\tau_1}(\cdot - z_{k,\tau_1}),
$$
and therefore
\begin{align*}
\sqrt{t}\mathcal{M}_t(\gamma^{\ast})&=\sqrt{t}e^{-\Phi(\gamma^{\ast})t}\int_{{\mathbb{R}}}e^{\gamma^{\ast}y}\mathcal{Z}_t({\rm d}y)\overset{d}{=}{\bf 1}_{\{\tau_1>t\}}\sqrt{t}e^{-\Phi(\gamma^{\ast})t}\\
&+{\bf 1}_{\{\tau_1\leq t\}}e^{-\Phi(\gamma^{\ast})\tau_1}\sum_{k=1}^{N}\sqrt{t}e^{-\Phi(\gamma^{\ast})(t-\tau_1)}A_k^{\gamma^{\ast}}\int_{{\mathbb{R}}}e^{\gamma^{\ast}y}
\mathcal{Z}^{(k)}_{t-\tau_1}({\rm d}y).
\end{align*}
Sending $t\to\infty$ yields \eqref{eq:d_fixed_point} because $\tau_1$ has the standard exponential law and is independent of $(\mathcal{Z}^{(k)}_t)_{t\geq 0}$, $k\in{\mathbb{N}}$. The proof of part (i) is complete.
\vspace{0.5mm}
\noindent
{\sc Proof of part (ii).} The claim of part (ii) can be reformulated as follows:
$$
\min_{k=1,\ldots,\mathcal{Y}_t}\left( -\gamma^{\ast}z^{\circ}_{k,t}\right)-\frac{1}{2}\log t\overset{{\mathbb{P}}}{\to}+\infty,\quad t\to\infty.
$$
Fix arbitrary $M>0$ and define a function
$$
p_{M}(t):={\mathbb{P}}\bigg\{\min_{k=1,\ldots,\mathcal{Y}_t}\left(-\gamma^{\ast}z^{\circ}_{k,t}\right)-\frac{1}{2}\log t<M\bigg\},\quad t\geq 0.
$$
By Theorem 5.12 in \cite{Shi:2015} applied to the discrete-time branching random walk $(\mathcal{Z}_n^{(\theta)})_{n=0,1,2,\ldots}$, see also \cite{AidekonShi2010,HuShi2009}, we already know that
$$
\lim_{n\to\infty}p_{M}(n\theta)=0
$$
for every fixed $\theta>0$. In order to finish the proof of part (ii) it remains to show that
$$
\lim_{t\to\infty,t\in{\mathbb{R}}}p_{M}(t)=0.
$$
According to the Croft-Kingman lemma, see Corollary 2 in \cite{Kingman:1963}, it is enough to check that $t\mapsto p_{M}(t)$ is right-continuous. To prove the latter statement, note that for $0\leq s\leq t$ we have
\begin{multline*}
|p_{M}(t)-p_{M}(s)|\leq {\mathbb{P}}\{\text{there are splits during } [s,t]\}\\
+{\mathbb{P}}\left\{\min_{k=1,\ldots,\mathcal{Y}_t}\left(-\gamma^{\ast}z^{\circ}_{k,t}\right)\in \Big[M+\frac{1}{2}\log s, M+\frac{1}{2}\log t\Big) \right\},
\end{multline*}
where we have used the equality $\min_{k=1,\ldots,\mathcal{Y}_t}\left(-\gamma^{\ast}z^{\circ}_{k,t}\right)=\min_{k=1,\ldots,\mathcal{Y}_s}\left(-\gamma^{\ast}z^{\circ}_{k,s}\right)$ which holds if there are no splits in $[s,t]$.
The right-hand side of the last display converges to $0$ as $s\to t+$. The proof of part (ii) is complete.
\end{proof}
\section{Proof of Theorem \ref{thm:main}}\label{sec:proof}
The key ingredient in the proof is Propostion \ref{prop: as} and the following lemma.
\begin{lem}\label{lem:3}
Assume that $(r_t)_{t\geq 0}$ is an integer-valued random process such that $r_t\overset{{\mathbb{P}}}{\to}\infty$, as $t\to\infty$. Further, suppose that for every $t\geq 0$ there is an array $(a_{k,t})_{k=1,\ldots,r_t}$ of a.s.~positive random weights such that
$$
\sum_{k=1}^{r_t}a_{k,t}^{\gamma}\overset{{\mathbb{P}}}{\to}a_{\infty}\quad\text{and}\quad \max_{k=1,\ldots,r_t}a_{k,t}\overset{{\mathbb{P}}}{\to} 0,\quad t\to\infty,
$$
for some a.s. positive random variable $a_{\infty}$ and $\gamma\in(0,2]$. Let $(X_k)_{k\in{\mathbb{N}}}$ be a sequence of independent random variables with common distribution function $F_0$ satisfying $(H_{\gamma})$ and which are independent of $(a_{k,t})_{k=1,\ldots,r_t}$ and $r_t$ for every fixed $t\geq 0$. Put
$$
S_t:=\sum_{k=1}^{r_t} a_{k,t}X_k,\quad t\geq 0.
$$
Then
$$
\lim_{t\to\infty}{\mathbb{E}}\exp(i\xi S_t)={\mathbb{E}} \widehat{g}_{\gamma}(\xi a_{\infty}^{1/\gamma}),\quad \xi\in{\mathbb{R}},
$$
where $\widehat{g}_{\gamma}$ is defined by \eqref{chaSta}.
\end{lem}
\begin{proof}
The proof is based on the following asymptotic expansions of the characteristic function $\phi_0$ of $X_0$, that are equivalent to the corresponding assumptions of the distribution function $F_0$, see Theorem 2.6.5 in \cite{Ibragimov+Linnik:1971}:
\begin{itemize}
\item if the case (a) of $(H_1)$ holds, then $\log \phi_0(\xi)=i m_0\xi+o(\xi)$ as $\xi\to 0$;
\item if the case (b) of $(H_1)$ holds, then $\log \phi_0(\xi)=i m_0\xi-\pi c_0^{+}|\xi|+o(\xi)$ as $\xi\to 0$;
\item if $(H_2)$ holds, then $\log \phi_0(\xi)=-\frac{\sigma_0^2}{2}\xi^2+o(\xi^2)$ as $\xi\to 0$;
\item if $(H_{\gamma})$ holds with $\gamma\in(0,1)\cup(1,2)$, then
$$
\log \phi_0(\xi)=-k_0|\xi|^{\gamma}(1-i \eta_0 \tan({\pi\gamma}/{2} )\operatorname{sign}\xi)+o(|\xi|^{\gamma}), \quad \xi\to 0.
$$
\end{itemize}
Using the above expansions the rest of the proof is standard and relies on the formula
$$
{\mathbb{E}}\exp(i\xi S_t)={\mathbb{E}} \left(\exp\left(\sum_{k=1}^{r_t}\log \phi_0(a_{k,t}\xi)\right)\right),\quad \xi\in{\mathbb{R}},\quad t\geq 0.
$$
We will give full details in the case (b) of $(H_1)$. The other cases can be checked similarly. From the equality
\begin{multline*}
{\mathbb{E}}\exp(i\xi S_t)={\mathbb{E}} \Bigg(\exp\Bigg(\sum_{k=1}^{r_t}\left(\log \phi_0(a_{k,t}\xi)-i m_0 \xi a_{k,t}+\pi c_0^{+}|\xi |a_{k,t}\right)\\
+i m_0 \xi\sum_{k=1}^{r_t}a_{k,t}-\pi c_0^{+}|\xi | \sum_{k=1}^{r_t}a_{k,t}\Bigg)\Bigg),\quad \xi\in{\mathbb{R}},\quad t\geq 0,
\end{multline*}
we see that it is enough to check that for every fixed $\xi\in{\mathbb{R}}$
\begin{equation}\label{eq:lem3}
\sum_{k=1}^{r_t}\left(\log \phi_0(a_{k,t}\xi)-i m_0 \xi a_{k,t}+\pi c_0^{+}|\xi | \sum_{k=1}^{r_t}a_{k,t}\right)\overset{{\mathbb{P}}}{\to} 0,\quad t\to\infty.
\end{equation}
Fix $\varepsilon>0$. There exists $x_0(\varepsilon)>0$ such that
$$
|\log\phi_0(x)-i m_0 x+\pi c_0^{+}|x||\leq \varepsilon |x|,\quad |x|\leq x_0(\varepsilon).
$$
Therefore, for every fixed $\varepsilon_0>0$
\begin{align*}
&\hspace{-1cm}{\mathbb{P}}\left\{\sum_{k=1}^{r_t}|\log \phi_0(a_{k,t}\xi)-i m_0 \xi a_{k,t}+\pi c_0^{+}|\xi |a_{k,t}|>\varepsilon_0\right\}\\
&\leq {\mathbb{P}}\left\{\varepsilon|\xi| \sum_{k=1}^{r_t}a_{k,t}>\varepsilon_0\right\}+{\mathbb{P}}\left\{|\xi|a_{k,t}>x_0(\varepsilon)\text { for some }k=1,\ldots,r_t\right\}\\
&={\mathbb{P}}\left\{\varepsilon|\xi| \sum_{k=1}^{r_t}a_{k,t}>\varepsilon_0\right\}+{\mathbb{P}}\left\{|\xi|\max_{k=1,\ldots,r_t}a_{k,t}>x_0(\varepsilon)\right\}.
\end{align*}
Sending $t\to\infty$ and then $\varepsilon\to +0$ yields \eqref{eq:lem3}. The proof is complete.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:main}]
Put $a_{k,t}:=t^{\frac{1}{2\gamma^{\ast}}}e^{z_{k,t}-t\frac{\Phi(\gamma^{\ast})}{\gamma^{\ast}}}$, $r_t:=\mathcal{Y}_t$, $a_{\infty}:=D'=\sqrt{\frac{2}{\pi (\gamma^{\ast})^2\Phi^{\prime\prime}(\gamma^{\ast})}}D_{\infty}$, $\gamma=\gamma^{\ast}$, and finally
$$
S_t=t^{\frac{1}{2\gamma^{\ast}}}e^{-\mu(\gamma^{\ast})t}\sum_{k=1}^{\mathcal{Y}_t}e^{z_{k,t}}X_k,\quad t\geq 0.
$$
From Proposition \ref{prop: as} we know that all the assumptions of Lemma \ref{lem:3} hold and therefore
$$
\lim_{t\to\infty}{\mathbb{E}}\exp\left(i\xi S_t\right)={\mathbb{E}} \widehat{g}_{\gamma}\left(\xi \left(\sqrt{\frac{2}{\pi (\gamma^{\ast})^2\Phi^{\prime\prime}(\gamma^{\ast})}}D_{\infty}\right)^{1/\gamma}\right),\quad \xi\in{\mathbb{R}}.
$$
By Proposition \ref{prop:prob_solution}
$$
{\mathbb{E}}\exp\left(i\xi S_t\right)=\phi(t,t^{\frac{1}{2\gamma^{\ast}}}e^{-\mu(\gamma^{\ast})t}\xi)
$$
which proves convergence. The fact that $D_{\infty}$ satisfies \eqref{eq:d_fixed_point} has already been proved above. The proof of Theorem \ref{thm:main} is complete.
\end{proof}
\section*{Acknowledgment} We thank two anonymous referees for the detailed and useful reports containing numerous remarks and suggestions.
|
{
"timestamp": "2019-03-07T02:07:23",
"yymm": "1804",
"arxiv_id": "1804.05418",
"language": "en",
"url": "https://arxiv.org/abs/1804.05418"
}
|
\section{}
Magnon computing is an emerging field of study which aims to advance information processing beyond current CMOS technology \cite{Chumak2015,Csaba2014,Csaba2017,Toedt2016,Lenk2011,Khitun2013,Sadovnikov2018,Cornelissen2018,Nikitov2015}. The limits of Moore's law are rapidly being approached \cite{Waldrop2016}, and a new computing paradigm is required to help stave off the impending computational crisis caused by the overheating of progressively smaller silicon-based transistors. Magnonics is one such technology. Magnonic computers utilize spin waves, whose quanta are called magnons, to perform complex computational processes using a fraction of the energy and space required by conventional circuitry and with potentially enormous clock speeds \cite{Chumak2015}. With its current trajectory, magnonics looks to augment this conventional circuitry with specialist designed-for-purpose micro-circuits that can outperform standard devices. Several key elements of these circuits have so far been developed \cite{Chumak2014,Chumak2010,doi:10.1063/1.4898042,Chumak2017,Chumak2015}, some of which are discussed further below.
One aspect of these device elements whose significance is often neglected is the method by which data is encoded in the magnonic signal. For some devices, such as the majority gate \cite{doi:10.1063/1.4898042}, the digital data is encoded in the phase of the magnons, whereas for the all-magnon transistor \cite{Chumak2014}, the amplitude is the relevant quantity. In a real magnonic computer where both types of components will be used, it will be necessary to convert between phase modulated and amplitude modulated signals. Some conversion methods have already been proposed \cite{Bracher2016}, however, these require external electronics. In this paper, we propose an all-magnon based method for PM to AM (P2A) and AM to PM (A2P) conversion.
As we will be discussing the use of two different binary data encoding methods, it is convenient to shift away from the nomenclature of purely 1s and 0s, and move to a system that provides the flexibility to discuss both modulation techniques simultaneously. The phase and amplitude of a magnon with respect to a reference source can be recorded in the form $\zeta(\textrm{R},\theta,\phi)=\textrm{R}\textrm{e}^{i(\theta+\phi)}$, where $\theta$ encodes the phase modulated binary state, which for one such implementation takes the values of 0 or $\pi$, and $\phi$ is a controllable phase offset. R represents the relative magnitude of the signal. If $\phi$ is set to zero, the phase encoded binary states can be written as $\zeta(\textrm{R},\{0,\pi\},0)=\{-\textrm{R},\textrm{R}\}$. For amplitude modulation, the signal is either zero or non-zero, and can take any phase, and can be written as $\zeta(\{0,\textrm{R}\},\theta,\phi)=\{0, \textrm{Re}^{i(\theta+\phi)}\}$. By combining phase and amplitude modulation and setting the amplitude to one, a simple example of a three-level mixed AM/PM system with $\zeta(\{0,1\},\{0,\pi\},0)=\{-1,0,1\}$ is obtained, which is a useful starting point for the discussion below.
A method of converting between phase and amplitude modulated signals is proposed as follows: both types of converters work by interfering two magnon channels at a junction, with the incoming data signal entering from one of the input branches (Figure \ref{fig:AMPMEquality} a and b). The other input comes from a pure tone continuous magnon source with a specific phase and amplitude.
\begin{figure*}
\centering
\includegraphics[width=0.9\textwidth]{AMPMEquality.eps}
\caption{(Color online) a) Schematic of the Phase to Amplitude modulation (P2A) converter with example waveforms. Binary 1's and 0's are indicated in red (light gray) and blue (dark gray). b) The Amplitude to Phase modulation (A2P) converter, again with example waveforms. c) Schematic of the Equality Gate, which takes a similar form to the majority gate \cite{doi:10.1063/1.4898042}, followed by the all-magnon transistor \cite{Chumak2014}. The three input channels are each separated by $\frac{2\pi}{3}$, and the output is 1 when all three inputs are in the same binary state.}
\label{fig:AMPMEquality}
\end{figure*}
For the phase to amplitude conversion (P2A) where the signal input takes the values of $\zeta=\{-1,1\}$, the continuous-wave input is set to $\zeta=+1$. When the two pulse trains combine and interfere (constructively for $+1$ and destructively for $-1$), the output will be an amplitude modulated signal with $\zeta={0,2}$, which can be attenuated if necessary (Figure \ref{fig:PM2PM}a). The reverse process (A2P) works by setting the continuous input to -0.5. In this case, an AM signal input of $\zeta=\{0,1\}$ results in a phase modulated output of $\zeta=\{-0.5,0.5\}$, which can be amplified if required.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.45\textwidth]{PM2PM.eps}
\caption{(Color online) a) Constellation plot for the P2A converter. The $\zeta=\{-1,1\}$ PM signal values shift to a $\zeta=\{0,2\}$ AM signal with the application of the continuous wave. b) Example of a Phase to Phase modulation (P2P) transformation, in which a PM signal with $\zeta(1,\{0,\frac{\pi}{2}\},0)$, (which has a $\Delta\theta=\frac{\pi}{2}$) is converted to a new PM' signal with $\zeta(\frac{\sqrt{2}}{2},\{0,\pi\},\frac{-\pi}{4})$, which has a $\Delta\theta=\pi$.}
\label{fig:PM2PM}
\end{figure}
This concept has been experimentally verified using a linear YIG waveguide in the backward volume mode in the presence of three equally spaced antennae (Figure \ref{fig:YIGexperiment}). Pulsed AM or PM signals of $3.2\ensuremath{\, \mathrm{GHz}}$ microwaves were inserted into antenna B, while the pure-tone signal was applied to antenna A. The electrical signal collected by antenna C was homodyned with a reference signal and then passed through a low-pass filter to remove the $2f$ component. The trace of the (modulated) DC component was then recorded on an oscilloscope. When the amplitude of the pure-tone signal was set correctly, both the A2P and P2A arrangements functioned as expected.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.45\textwidth]{YIG.eps}
\caption{The experimental arrangement used to test the A2P and P2A converters.}
\label{fig:YIGexperiment}
\end{figure}
By controlling the phase and amplitude of the continuous pure-tone magnon source, additional functionality is available. For example, this method can be used to convert between different types of phase modulated signals, such as the PM signals used in \cite{Bracher2016} which use a modulation of $\theta = \{0,\frac{\pi}{2}\}$. By adding a continuous signal with $\zeta(\frac{\sqrt{2}}{2},0,\frac{5\pi}{4})$, a new PM signal with $\Delta\theta = \pi$ can be obtained (Figure \ref{fig:PM2PM}b). This technique can also be used as an AM NOT gate by simply adding a wave with $\zeta=-1$ to an AM signal with $\zeta=\{0,1\}$.
By using these conversion techniques, it is possible to interface PM driven devices such as the majority gate with AM driven devices like the magnon transistor or amplifier \cite{Chumak2014}. By considering devices with mixed AM/PM encodings, it is suggested that this technique opens new possibilities for magnonic circuit design, such as the equality gate.
The equality gate has the same appearance as the majority gate followed by an all-magnon transistor (Figure \ref{fig:AMPMEquality}c), and uses PM inputs with $\theta = \{0,\pi\}$. The key difference in this device is that the three input channels are each separated by a fixed phase of $\Delta\phi=\frac{2\pi}{3}$. The eight possible output states from the majority-like section can be calculated and plotted in a constellation plot on the Argand diagram. (Figure \ref{fig:constelationequality}a).
\begin{figure*}
\centering
\includegraphics[width=0.7\textwidth]{constellation.eps}
\caption{a) Constellation plot of the 8 possible intermediary outcomes of the equality gate, prior to rectification in the magnon transistor. The two `equality' states lie at the origin, whilst the other states lie on a circle of constant radius (amplitude). b) Constellation plot of an XNOR gate (the two channel analogue of the equality gate), in which two $\pi$-separated PM signals interfere at a junction. The same method of rectification (the magnon transistor) can again be used to extract the digital information.}
\label{fig:constelationequality}
\end{figure*}
In Figure \ref{fig:constelationequality}a, the two states for which all three inputs take equal logic values (111 , 000) lie on the origin, whereas the other six states lie on a circle with a constant radius. By extracting the AM information from this mixed AM/PM output state, this device is able to function as an `equality' gate. This operation is performed by the subsequent all-magnon transistor \cite{Chumak2014}.
\begin{table}[btp]
\centering
\label{tab:truthtable}
\begin{tabular}{ccc|c}
\toprule
Input A &Input B &Input C &Output \\ \hline
0 & 0 & 0 & 1 \\
0 & 0 & 1 & 0 \\
0 & 1 & 0 & 0 \\
0 & 1 & 1 & 0 \\
1 & 0 & 0 & 0 \\
1 & 0 & 1 & 0 \\
1 & 1 & 0 & 0 \\
1 & 1 & 1 & 1 \\
\bottomrule
\end{tabular}
\caption{Truth table for the equality gate.}
\end{table}
As the transistor only relies on the presence of the gating magnons within the magnonic crystal, the phase information of the input signal is irrelevant. As the 111 and 000 states have no amplitude and thus no magnons, the transistor will be open and produce an output value of $\zeta=+1$. The other six states will close the transistor (by the same amount, as they all have the same amplitude) and produce an output of $\zeta=0$. These values represent AM encoded logic which agrees with the truth table for the equality gate shown in table \ref{tab:truthtable}.
It is worthy of note that this equality gate arrangement of multiple converging phase-separated magnon signals followed by an all-magnon transistor also functions as an XNOR gate when only two inputs separated by $\pi$ radians are used (Figure \ref{fig:constelationequality}b). We also highlight that by combining an equality gate with a majority, NOT, and XOR gate, it is possible to construct a full adder circuit.
In conclusion, we have presented a simple method for converting between amplitude and phase modulated magnonic signals using wave interference at a waveguide junction. We have shown that this technique can be used to convert between different types of phase modulation, as well as provide the basis for other simple logic operations such as an AM NOT gate. We have suggested that magnonic circuit elements can be designed that utilize both AM and PM encoding simultaneously, and have proposed the schematic of two such devices: the equality gate and an XNOR gate. We believe these new circuit elements complement the plethora of already existing magnonic components, and provides a missing piece of the puzzle that enables the sequentialization of magnonic devices.
|
{
"timestamp": "2018-04-17T02:15:04",
"yymm": "1804",
"arxiv_id": "1804.05575",
"language": "en",
"url": "https://arxiv.org/abs/1804.05575"
}
|
\section{Introduction}
The parity operation results in the inversion of the spatial coordinates of the object it acts on. Although many physical systems are symmetric under parity operations, some give rise to different physics under the inversion of spatial coordinates. The violation of symmetry under a parity operation is known as parity non-conservation (PNC). PNC measurements within the $^{133}$Cs atom~\cite{Wood:97}, which are dominated by nuclear-spin-independent (NSI) PNC effects, are in outstanding agreement with predictions from the standard model~\cite{Ginges2002,porsev09prl,Dzuba:2012}. These have placed bounds on the energy at which new physics may be discovered from this process at greater than 0.7~TeV/$c^2$ (see e.g.\cite{Dzuba:2012,Roberts:2014}). Experimental investigation has consequently shifted towards nuclear-spin-dependent (NSD) PNC effects with the aim of testing low energy quantum chromodynamics (QCD) and nuclear theory~\cite{Ginges:2004}.
The nuclear anapole moment is one example of a manifestation of NSD PNC \cite{Khriplovich1980,Sushkov1984}
and is the main mechanism behind the PNC considered in this letter. Zel'dovich developed the notion of the anapole moment of an elementary particle in 1957 \cite{Zeldovich:57}. Subsequently, Flambaum and Khriplovich proposed the existence of the nuclear anapole moment, which was found to be the dominant NSD PNC effect in heavy atoms and molecules
\cite{Sushkov:78,Khriplovich1980}.
The observable NSD PNC effects of the nuclear anapole moment include manifestations of the parity violating electric dipole transition (E1$_{PNC}$) in atoms and molecules. PNC effects have small amplitudes compared to molecular and atomic electromagnetic processes and are difficult to detect \cite{Kozlov:95}.
The nuclear anapole moment has been detected only once within the $^{133}$Cs atom \cite{Wood:97} (where NSD PNC is sub-dominant) as experimental techniques have lacked the sensitivity to detect NSD PNC effects with certainty. NSD PNC calculations in molecules provides a new window of opportunity to study parity violating nuclear forces which create the nuclear anapole moment.
PNC effects are enhanced within diatomic molecules due to closely spaced rotational levels of opposite parity \cite{Sushkov:78,Labzowsky:78}. In this Letter we show that
mercury hydride (HgH) in particular is a promising choice for the study of PNC effects, not only because it gives an enhanced, pure NSD PNC signal but also because it is easy to make at room temperature. These effects manifest as $E1_{PNC}$ transitions that violate the parity selection rules of dipole transitions. The transition can be detected via interference of the $E1_{PNC}$ amplitude with an allowed $M1$ transition amplitude between the same states. This results in the rotation of the polarisation plane of light passing through a gas of HgH molecules, which is referred to as PNC optical rotation \cite{Khriplovich:91}
\begin{equation}
\phi_{PNC} = -\frac{4 \pi l}{\lambda}(n(\omega) - 1) \frac{\textrm{Im}(E1_{PNC})}{M1}
\label{eq:rota}
\end{equation}
which depends on the experimental parameters $\omega$, $n(\omega)$, $l$ and $\lambda$ which are the optical frequency, refractive index due to the absorption line, the path length of light, and the optical wavelength, respectively.
The experimental techniques developed in~\cite{Bougas:2012} promise greater sensitivity in NSD PNC measurements of this type. The experimental set-up includes a cavity in which four mirrors are placed in a ``bow-tie'' configuration, allowing polarised light to make multiple passes through the cavity before detection.
By increasing the path length of light passing through the sample within the optical cavity, the experiment is expected to enhance optical rotation signals by up to 4 orders of magnitude.
For small optical paths $l<2L$, where $L$ is absorption length at a given frequency off-resonance, the optical rotation $\phi_{PNC}$ increases linearly with the sample density (or the number of cavity passes); it reaches $\phi_{PNC} \sim P = 2 \frac{\textrm{Im}(E1_{PNC})}{M1}$ at $\omega-\omega_r =\Delta_D$ when $l=2L$ (where the maximum signal-to-noise ratio is achieved), i.e. at transmission $1/e^2 = 13.5\%$. Here, $\omega_r$ is the resonant frequency and $\Delta_D$ is the Doppler width which is much larger than the natural width. However for $l>2L$ in the resonance, larger values of $\phi_{PNC}$ can be found by tuning the wavelength further off resonance: absorption falls rapidly as $1/(\omega - \omega_r)^2$ while $\phi_{PNC}$ falls slower as $1/(\omega - \omega_r)$. Therefore to suppress absorption, one must go to the tail of the resonance which will result in large $L$ and $\phi_{PNC}$ much larger than $P$. In order to achieve this we must have sufficiently large effective $l$ after many reflections of light in the cavity \cite{Khriplovich:91}.
The advantage of the HgH molecule for the PNC experiment is the large rotational constant, which allows optical transitions to be resolvable for levels of opposite parity.
In this Letter we have performed relativistic coupled cluster calculations of the weak interaction (anapole) matrix elements in the $A_1~^{2}\Pi_{\frac{1}{2}}$ excited state and the ground $X~^{2}\Sigma$ state of $^{199}$HgH, as well as the corresponding E1 and M1 transition amplitudes. These calculations allow for a complete extraction of the nuclear anapole moment of $^{199}$Hg from the proposed experiment.
\section{Spin-rotational Hamiltonian}
$^{199}$HgH is a heteronuclear diatomic molecule with one valence electron. The total valence electronic angular momentum can be expressed as $\vect{J}_e = \vect{S} + \vect{L}$ where \vect{S} is the electron spin and \vect{L} is the orbital angular momentum. HgH has electronic ground state of $X~^2\Sigma_{1/2}$ and first electronic excited state $A_1~^2\Pi_{1/2}$. We assign the laboratory frame coordinates \vect{x}, \vect{y} and \vect{z}, in which the molecule rotates with angular momentum \vect{N}. The magnitude of the separation between discrete rotational levels in HgH is governed by the state-specific rotational constant $B$. Rotational angular momenta can couple to the electronic angular momentum to form a vector \vect{J}:
\begin{equation}
\vect{J} = \vect{N} + \vect{J}_e.
\label{eq:J}
\end{equation}
\vect{J} has a projection along the inter-nuclear axis $\Omega$. Furthermore, both $^{199}$Hg and H have nuclear spin, denoted by \vect{I}$_1$ and \vect{I}$_2$ respectively. A general spin-rotational Hamiltonian $H_{sr}$ can be written for both the $X~^2\Sigma_{1/2}$ and $A_1~^2\Pi_{1/2}$ terms \cite{Kozlov:91,Kozlov:95}:
\begin{equation}
\label{SROT}
H_{sr}=B\vect{J}^2+\Delta\vect{J}\cdot\vect{S}^{\prime}+\vect{I}_1\cdot \hat{\vect{A}}_1 \cdot \vect{S}^{\prime} + \vect{I}_2\cdot \hat{\vect{A}}_2 \cdot \vect{S}^{\prime}.
\end{equation}
Here $\hat{\textbf{A}}_1$ and $\hat{\textbf{A}}_2$ are second rank axial tensors describing the spin-spin interaction between electrons and the nucleus, and $\Delta$ is the $\Omega$-doubling constant. In the rotating molecular frame described by $\boldsymbol{\xi}$, $\boldsymbol{\eta}$ and $\boldsymbol{\zeta}$, the tensor contractions
\begin{equation}
\label{HFSr}
\vect{I}\cdot\hat{\vect{A}}\cdot \vect{S}^{\prime}=A_{||}\,\vect{I}_{0}\vect{S}_{0}^{\prime}-
A_{\perp}\,\left(\vect{I}_{1}\vect{S}_{-1}^{\prime}+\vect{I}_{-1}\vect{S}_{1}^{\prime}\right),
\end{equation}
are determined by the parallel and perpendicular hyperfine parameters $A_{||}$ and $A_{\perp}$. $\vect{S}^{\prime}$ is the effective spin whose components act on the projection $\Omega$. If we express the tensor components of \vect{S} in the rotating molecular frame we get \cite{Kozlov:87, Dmitriev:92}
\begin{align}
\vect{S}^{\prime}_{\hat{n}}\ket{\Omega} &= \Omega\ket{\Omega}, \nonumber \\
\vect{S}^{\prime}_{\pm}\ket{\Omega=\mp 1/2} &= \ket{\Omega=\pm 1/2}, \nonumber \\
\vect{S}^{\prime}_{\pm}\ket{\Omega=\pm 1/2} &= 0.
\end{align}
The angular momenta coupling scheme in the case of $X~^2\Sigma_{1/2}$ ground state follows that known as Hund's case \textit{b}. The total electronic angular momentum $\vect{J}_e \approx \vect{S}$ since for this state $\Lambda = 0$ where $\Lambda$ is the projection of electronic orbital angular momentum on the molecular axis. Therefore, we can use $\vect{J}_e \approx \vect{S}$ and the substitution $\vect{J}=\vect{N}+\vect{S}$~\cite{Kozlov:91,Kozlov:95}.
Next, the first and second nuclear spin couple in succession \cite{Kozlov:91}:
\begin{gather}
\textbf{F}_1=\textbf{J}+\textbf{I}_1,
\label{eq:couple1} \\
\textbf{F}=\textbf{F}_1+\textbf{I}_2,
\label{eq:couple2}
\end{gather}
Furthermore, the $\Omega$ doubling constant is defined in this scheme to be
$$\Delta=-2B+\gamma,$$
where $\gamma$ is the spin-doubling constant.
Conversely, the $A_1~^2\Pi_{1/2}$ state follows the coupling scheme described by Hund's case \textit{a}. The projection of total angular momentum $\Omega$ in the direction of a unit vector along the internuclear axis $\hat{\mathbf{n}}$ couples to $\mathbf{N}$ to give
\begin{equation*}
\vect{J} = \vect{N} + \Omega\, \hat{\vect{n}}.
\end{equation*}
$\textbf{I}_1$ and $\textbf{I}_2$ couple to $\mathbf{J}$ to form $\textbf{F}_1$ and $\textbf{F}$ in turn, as in Equations~(\ref{eq:couple1}) and (\ref{eq:couple2}).
The basis states that will be used in this work are defined by quantum numbers \ket{J p F_1 F M}, where $p$ is the parity and $M$ is the projection of the total angular momentum $\textbf{F}$ on the lab axis.
\section{Weak Interaction Constants}
The nuclear anapole moment can interact (via its magnetic field) with an electron wavefunction with non-zero total angular momentum \cite{Flambaum:85b}; this is one mechanism behind NSD PNC interactions in atoms and molecules and can be described using a Hamiltonian of the form
\begin{equation}
H_{P} = \kappa\frac{G_\mathrm{F}}{\sqrt{2}}\,
\boldsymbol{\alpha}\cdot\vect{I}\, \rho(\vect{r}),
\label{HP}
\end{equation}
where \mbox{$G_\mathrm{F}~=~2.22249 \cdot 10^{-14}$~a.u.} is the Fermi coupling constant in atomic units and $\rho (\vect{r})$ is the normalised nuclear density. $\kappa$ is the dimensionless constant determined by nuclear anapole moment to be extracted from experiment. It has been estimated as \cite{Flambaum:85b}
\begin{equation}
\kappa\approx\frac{9}{10} g \left(\frac{\alpha \mu}{m r_{0}} \right) A^{\frac{2}{3}}.
\end{equation}
where $A$ is the number of nucleons in the nucleus, $m$ is the mass of the proton, $\mu$ is the magnetic moment of the external nucleon, $g$ is a dimensionless constant describing the strength of the weak nucleon-nucleus interaction, $\alpha=\frac{1}{137}$ is the fine-structure constant, and $r_{0}=1.2\times 10^{-13}$ cm is the internuclear distance.
It is possible to average over fast electron motion to obtain the effective weak interaction coefficient $W_{a}$, which will be constant for a given molecular state. An effective P-odd Hamiltonian can be written as a T-even pseudoscalar formed from the products of the vectors in the system, namely \vect{I}, the effective electron spin $\vect{S}^{\prime}$ and direction of the internuclear axis $\vect{n}$ \cite{Flambaum:85b}. Therefore, in the presence of anapole moment within the Hg nucleus the total Hamiltonian of HgH will also include the following term:
\begin{equation}
H_{\rm eff} =
W_{a}\kappa\left(\vect{n}\times\vect{S}^{\prime}\right)\cdot \vect{I},
\label{HEFF}
\end{equation}
where $W_{a}$ can be written as
\begin{equation}
W_{a}=\frac{G_\mathrm{F}}{\sqrt{2}} \bra{\Psi_{\Omega=1/2}}
\rho(\textbf{r}) {\alpha_+}
\ket{\Psi_{\Omega=-1/2}}.
\label{W_a}
\end{equation}
$\Psi_{\Omega=1/2}$ can be the $^{2}\Pi_{1/2}$ or $^{2}\Sigma_{1/2}$ state,
$\alpha_+$ is defined as
\begin{eqnarray*}
\alpha_+=\alpha_\xi+\mathrm{i}\alpha_\eta,
\end{eqnarray*}
and $\alpha_\xi, \alpha_\eta$ are the Dirac matrices in the molecular coordinate system. An approximate expression for $W_{a}$ can be used to check the corresponding calculations and has been found in \cite{Flambaum:85b} to have the form
\begin{equation}
W_{a} \approx \frac{\epsilon_{s}}{\nu_{s}^{\frac{3}{2}}} \frac{\epsilon_{p}}{\nu_{p}^{\frac{3}{2}}} Ry \frac{2\sqrt{2}}{\sqrt{3} \pi} G_{F} m_e^{2} \alpha^{2} Z^{2} R_{W} \frac{(-1)^{I+\frac{1}{2}-l}(I+\frac{1}{2})}{I(I+1)},
\label{op} \end{equation}
where the relativistic correction term $R_{W}$ can be written as
\begin{equation}
R_{W}=\frac{2 \gamma + 1}{3} \left(\frac{a_{B}}{2 Z r_{0} A^{\frac{1}{3}}}\right)^{2-2\gamma}.
\label{eq:rel}
\end{equation}
$\nu_{s}$ and $\nu_{p}$ are the effective quantum numbers for the $s$ and $p$ atomic Hg orbitals respectively, $\epsilon_{s}$ and $\epsilon_{p}$ are weighting coefficients for the contributions of each atomic orbital, $m_e$ is the mass of the electron, $Ry=13.6$ eV is the Rydberg constant, $l$ is the orbital angular momentum of an external unpaired nucleon, $Z$ is the atomic number, and $a_{B}$ is the Bohr radius.
We have calculated the $W_a(^2\Sigma_{1/2})$ and $W_a(^2\Pi_{1/2})$ constants for HgH using two different methods: the first was a Dirac-Hartree-Fock (DHF) calculation performed as a way of checking the scaling relation between $W_a$ and Z; and the second was an accurate coupled-cluster (CC) calculation which we use in our subsequent calculation of the circular polarization parameter $P$.
In the first method, all $W_a(^2\Sigma_{1/2})$ and $W_a(^2\Pi_{1/2})$ constants were calculated with the relativistic program package {\sc dirac15} \cite{DIRAC15} using the DHF method. The DHF method employs the relativistic, multi-electron Dirac Hamiltonian in conjunction with the Hartree-Fock wavefunction. The $W_a$ constants for ZnF and CdH were calculated within the same approach and used to verify that the
$W_a$ values scale as expected with the square of the atomic number Z.
The final values are displayed in the first column of Table~\ref{table:Wa}.
The calculations were carried out at the experimental bond lengths of both the ground and the excited states of the three molecules \cite{NIST}. The heavy Zn, Cd and Hg atoms were described using Dyall's cc-pvqz basis sets \cite{Dyall04,Dyall:07} and for the H atom we used the uncontracted aug-cc-pVTZ basis set \cite{Dunning89}. Finally, we multiplied the output by a core polarisation scaling factor used in other works \cite{Borschevsky:13}.
\begin{figure}[htb]
\includegraphics[width=0.45\textwidth]{logWa.pdf}
\caption{\label{fig:logWa} Ratio of weak interaction constant $W_a$ to relativistic factor $R_W$ plotted against $Z$ for $Z = 30$, 48, and 80. Circles, solid line of best fit: ground state $^2\Sigma_{1/2}$; diamonds, dashed line of best fit: excited state $^2\Pi_{1/2}$. Calculated using the DHF method (see text).}
\end{figure}
The ratio $W_a (^2\Sigma_{1/2})/R_W$ should scale linearly with $Z^{2}$, where $R_{W}$ is the relativistic factor defined by (\ref{eq:rel}). However, we observe (see Figure~\ref{fig:logWa}) a gradient of 2.5 instead of the expected gradient of 2. Similarly, we find $W_a(^2\Pi_{1/2})/R_W\sim Z^{5.3}$ rather than the expected $Z^4$. Both cases can be
understood by the filling of the Hg atomic $d$ orbital close to the nucleus. Upon filling, the $d$ orbital expands relativistically, hence increasing effective nuclear charge and enhancing relativistic and NSD PNC effects. A similar trend is also seen in the $W_a (^2\Sigma_{1/2})$ constants for HgF, ZnF and CdF in \cite{Borschevsky:13}.
Furthermore, $W_a$ constants for the HgH $^2\Sigma_{1/2}$ and $^2\Pi_{1/2}$ electronic states have been calculated within the relativistic Fock-Space coupled cluster with single and double cluster amplitudes method. 35 outer-core and valence electrons were included in correlation treatment. For Hg and H atoms Dyall's uncontracted core-valence triple zeta (cv3z) basis sets \cite{Dyall:07,Dyall:12} were used. The $W_a$ constants were calculated at the equilibrium internuclear distance for the corresponding electronic states and are presented in the second column of Table~\ref{table:Wa}. There is good agreement between the DHF and CC methods of calculating $W_a$ constants in HgH with the two methods varying by 14\% for $W_a(^2\Sigma_{1/2})$ and 13\% for $W_a(^2\Pi_{1/2})$.
\begin{table}[htb]
\caption{\label{table:Wa}Values for the effective weak interaction coefficients $W_a(^2\Sigma_{1/2})$ (ground state) and $W_a(^2\Pi_{1/2})$ (first excited state) calculated for group 12 hydrides. The $W_a(^2\Sigma_{1/2})$ for HgH is in good agreement with the semi-empirical estimates presented in \cite{Kozlov1985}}
\begin{ruledtabular}
\begin{tabular}
{p{1cm} p{1.7cm} p{1.7cm} p{1.7cm} p{1.7cm}}
Mol. & DHF (Hz) & & CC (Hz) & \\
& $W_a(^2\Sigma_{1/2})$ & $W_a(^2\Pi_{1/2})$ & $W_a(^2\Sigma_{1/2})$& $W_a(^2\Pi_{1/2})$ \\
\hline
ZnH & 61 & -0.42 &- &- \\
CdH & 284 & -6.71 &- &- \\
HgH & 3882 & -372 & 3335 &-419 \\
\end{tabular}
\end{ruledtabular}
\end{table}
\section{PNC E1 amplitude}
The parity violating dipole transition amplitude $E1_{PNC}$ can be expressed as
\begin{multline}
\label{amp}
\bra{i} E1_{PNC} \ket{k} = \\
\sum_{j} \frac{\bra{i} \vect{d}\cdot\vect{E}_0 \ket{j}\bra{j} H_{\rm eff} \ket{k}}{E_{k}-E_{j}} +
\frac{\bra{i} H_{\rm eff}\ket{j} \bra{j} \vect{d}\cdot\vect{E}_0 \ket{k}}{E_{i}-E_{j}},
\end{multline}
where $\vect{E}_0$ is the external electric field and $\vect{d}$ is the dipole moment operator.
In the first term of (\ref{amp}) \ket{j} and \ket{k} are sublevels of opposite parity situated in the ground $^2\Sigma_{1/2}$ electronic state.
In the second term of (\ref{amp}) \ket{i} and \ket{j} are sublevels of opposite parity corresponding to the $^2\Pi_{1/2}$ electronic state.
Expressions that appear in the decoupling of Eq. (\ref{amp}) can be found in the appendix of Ref.~\cite{Kozlov:91}, e.g.
for the present case ($I_1=I_2=1/2$)
\begin{multline}
\bra{JpF_1 F M} H_{\rm eff}\ket{J^\prime(-p)F_1^\prime F^\prime M} =\\
-\frac{1}{4}iW_a \kappa\,\delta_{F_1F_1^\prime}\delta_{FF^\prime}
\sqrt{\frac{3}{2}}
\left\{
\begin{array}{ccc}
J & J^\prime & 1 \\
1/2 & 1/2 & F_1
\end{array}
\right\}\\
.(-1)^{F_1+J^\prime+1/2}\chi_JX_{JJ'},
\end{multline}
where
\begin{gather}
\chi_J= \pm p\,(-1)^{J-1/2},\nonumber \\
X_{JJ}=(2J+1)\sqrt{\frac{2J+1}{J(J+1)}},\nonumber\\
X_{JJ-1}=X_{J-1J}=\sqrt{\frac{(2J+1)(2J-1)}{J}}; \nonumber
\end{gather}
the plus sign is taken for the $^2\Pi_{1/2}$ state and the minus sign is taken for the $^2\Sigma_{1/2}$ state according to \cite{Kozlov:91}.
To obtain E1 and M1 amplitudes between $^2\Sigma_{1/2}$ and $^2\Pi_{1/2}$ electronic states we require the following matrix elements which we have calculated:
\begin{align}
\label{Dperp}
D_{+} &= \bra{\Psi_{^2\Pi_{1/2}}}\vect{d}_{+} \ket{\Psi_{^2\Sigma_{-1/2}}} = 0.7~{\rm a.u.} \\
\label{Gperp}
G_{+} &= \bra{\Psi_{^2\Pi_{1/2}}} \vect{L}_{+} - g_{S} \vect{S}_{+} \ket{\Psi_{^2\Sigma_{-1/2}}} = 1.4~{\rm a.u.}
\end{align}
Here $\vect{d}_{+}=\vect{d}_{\xi} + i\vect{d}_{\eta}$ is the dipole moment operator, $\vect{L}$ and $\vect{S}$ are the electronic orbital angular momentum and spin operators, and $g_{S} = -2.0023$ is the free-electron $g$-factor. Corresponding parallel components are small due to electronic configuration and are neglected here.
Matrix elements (\ref{Dperp}) and (\ref{Gperp}) have been calculated using the relativistic linear-response coupled cluster with single and double cluster amplitudes method \cite{Kallay:5} within the Dirac-Coulomb Hamiltonian. These calculations were performed at the internuclear distance which is the average of the $^2\Sigma_{1/2}$ and $^2\Pi_{1/2}$ equilibrium distances ($R=3.14$ Bohr \cite{Dufayard:88,Mosyagin:01b}).
For the correlation calculation we used the {\sc mrcc} \cite{MRCC2013} and {\sc dirac15} \cite{DIRAC15} codes. For calculation of matrix elements over molecular bispinors the code developed in Refs.~\cite{Skripnikov:15b,Skripnikov:16b,Petrov:17b} was used.
\section{Results and discussion}
We have chosen to study a transition that occurs between the zeroth vibrational levels of the HgH molecule; this is because the ($\nu_X=0\to\nu_{A_1}=0$) transition has the maximal value of the square of vibration wave functions overlap (Frank-Condon factor) which is 0.5 \cite{Nedelec:88} and should result in a stronger transition compared to other vibrational states.
To calculate the circular polarization parameter $P=2\,\textrm{Im}(E1_{PNC})/{M1}$ we consider the ground rotational levels in both electronic states, set $F_1=0$ and use the following estimates for the energy separation between levels of opposite parity, $\Delta E$:
\begin{gather*}
\Delta E(^2\Sigma_{1/2})=2B(^2\Sigma_{1/2})-\gamma=8.64~\textrm{cm}^{-1},\\
\Delta E(^2\Pi_{1/2})=\Delta=3.36~\textrm{cm}^{-1},
\end{gather*}
where the experimental constants $B(^2\Sigma_{1/2})= 5.3888~{\rm cm}^{-1}$, $\gamma=2.14~{\rm cm}^{-1}$ and $\Delta=3.36~{\rm cm}^{-1}$ were taken from Ref.~\cite{Huber:79}.
It should be noted that the hyperfine splitting is considerably smaller than the rotational constant $B$ for both electronic states under consideration, e.g.~$A_{1, ||}(^2\Sigma_{1/2})$ is about 20 times smaller than the rotational constant for the $^2\Sigma_{1/2}$ state. Therefore, we neglect it below. For more accurate estimates one should numerically diagonalize the spin-rotational Hamiltonian (\ref{SROT}).
Furthermore, the estimated uncertainty of the calculated $W_a$ parameters are 15-20\%. This can be minimized considerably by applying combined technique developed in Refs.\cite{Skripnikov:16b,Skripnikov:17a,Skripnikov:17b}, but for our current purposes it is enough.
Using the aforementioned energy separations, the matrix elements (\ref{Gperp}, \ref{Dperp}), coupled-cluster $W_a$ constants for HgH from Table~\ref{table:Wa}, and neglecting possible phase difference in the terms in Eq.~(\ref{amp}) we obtain our final result
\begin{equation}
P=3 \cdot 10^{-6} \kappa.
\end{equation}
The leading contribution comes from the mixing of opposite parity levels of $^2\Sigma_{1/2}$ state, which is about 3 times larger than the term due to the mixing of opposite parity levels of $^2\Pi_{1/2}$ state.
\section{Conclusion}
The $^{199}$HgH molecule is a good candidate for PNC optical rotation experiments as it has closely spaced levels of opposite parity as well as a rotational constant large enough to resolve optical transitions between those levels. The circular polarization parameter was calculated to be $P=3 \cdot 10^{-6} \kappa $ which is 2 to 3 orders of magnitude larger than the estimated value for NSD PNC effects in atomic Xe, Hg, Tl, Pb and Bi \cite{Khriplovich:91,Dzuba2012}.
Furthermore, HgH gives a pure NSD PNC signal needing a single transition for measurement;
in contrast atomic experiments also give a much larger NSI PNC signal, requiring measurements on at least two different hyperfine transitions to isolate the small NSD PNC effect, which increases noise and possibly systematic effects.
\section{Acknowledgments}
L.S. is grateful to Saint-Petersburg State University for a travel grant 11.42.700.2017 and RFBR, according to the research project No.~16-32-60013 mol\_a\_dk. A. G. is grateful for the support of the Australian Government Research Training Program Scholarship. L.S. and A.B. acknowledge the support of the Gordon Godfrey Visiting Fellowship. V.F. is grateful to the Australian Research Council for support.
|
{
"timestamp": "2018-04-17T02:12:56",
"yymm": "1804",
"arxiv_id": "1804.05475",
"language": "en",
"url": "https://arxiv.org/abs/1804.05475"
}
|
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