From 56fdd368589a446333112e3add41b31d6e982844 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Tue, 28 Nov 2023 16:42:52 +0100 Subject: Broke of much of the isolated eigenvalue to an appendix. --- 2-point.tex | 214 +++++++++++++++++++++++++++++++----------------------------- 1 file changed, 109 insertions(+), 105 deletions(-) diff --git a/2-point.tex b/2-point.tex index 238726b..9be13e9 100644 --- a/2-point.tex +++ b/2-point.tex @@ -1218,6 +1218,115 @@ $\pmb\sigma$ replicas constrained to lie at fixed overlap with \emph{all} the $\mathbf s$ replicas, and the second is the only of the $\mathbf s$ replicas at which the Hessian is evaluated. + +In this solution, we simultaneously find the smallest eigenvalue and information +about the orientation of its associated eigenvector: namely, its overlap with +the tangent vector that points directly toward the reference spin. This is +directly related to $x_0$. This tangent vector is $\mathbf x_{0\leftarrow +1}=\frac1{1-q}\big(\pmb\sigma_0-q\mathbf s_a\big)$, which is normalized and +lies strictly in the tangent plane of $\mathbf s_a$. Then +\begin{equation} + q_\textrm{min}=\frac{\mathbf x_{0\leftarrow 1}\cdot\mathbf x_\mathrm{min}}N + =\frac{x_0}{1-q} +\end{equation} +The emergence of an isolated eigenvalue and its associated eigenvector are +shown in Fig.~\ref{fig:isolated.eigenvalue}, for the same reference point +properties as in Fig.~\ref{fig:min.neighborhood}. + +\begin{figure} + \includegraphics{figs/isolated_eigenvalue.pdf} + \hfill + \includegraphics{figs/eigenvector_overlap.pdf} + + \caption{ + Properties of the isolated eigenvalue and the overlap of its associated + eigenvector with the direction of the reference point. These curves + correspond with the lower solid curve in Fig.~\ref{fig:min.neighborhood}. + \textbf{Left:} The value of the minimum eigenvalue as a function of + overlap. The dashed line shows the continuation of the bottom of the + semicircle. Where the dashed line separates from the solid line, the + isolated eigenvalue has appeared. \textbf{Right:} The overlap between the + eigenvector associated with the minimum eigenvalue and the direction of the + reference point. The overlap is zero until an isolated eigenvalue appears, + and then it grows continuously until the nearest neighbor is reached. + } \label{fig:isolated.eigenvalue} +\end{figure} + +\section{Conclusion} +\label{sec:conclusion} + +We have computed the complexity of neighboring stationary points for the mixed +spherical models. When we studied the neighborhoods of marginal minima, we +found something striking: only those at the threshold energy have other +marginal minima nearby. For the many marginal minima away from the threshold +(including the exponential majority), there is a gap in overlap between them. + +This has implications for pictures of relaxation and aging. In most $p+s$ +models studied, quenches from infinite to zero temperature (gradient descent +starting from a random point) relax towards marginal states with energies above +the threshold energy \cite{Folena_2023_On}, while at least in some models a +quench to zero temperature from a temperature around the dynamic transition +relaxes towards marginal states with energies below the threshold energy +\cite{Folena_2020_Rethinking, Folena_2021_Gradient}. We found (see especially +Figs.~\ref{fig:marginal.prop.below} and \ref{fig:marginal.prop.above}) that the +neighborhoods of marginal states above and below the threshold are quite +different, and yet the emergent aging behaviors relaxing toward states above and +below the threshold seem to be the same. Therefore, this kind of dynamics +appears to be insensitive to the neighborhood of the marginal state being +approached. To understand something better about why certain states attract the +dynamics in certain situations, nonlocal information, like the +structure of their entire basin of attraction, seems vital. + +It is possible that replica symmetry breaking among the constrained stationary +points could change the details of the two-point complexity of very nearby +states. Indeed, it is difficult to rule out \textsc{rsb} in complexity +calculations. However, such corrections would not change the overarching +conclusions of this paper, namely that most marginal minima are separated from +each other by a macroscopic overlap gap and high barriers. This is because the +replica symmetric complexity bounds any \textsc{rsb} complexities from above, +and so \textsc{rsb} corrections can only decrease the complexity. Therefore, +the overlap gaps, which correspond to regions of negative complexity, cannot be +removed by a more detailed saddle point ansatz. + +Our calculation studied the neighborhood of typical reference points with the +given energy and stability. However, it is possible that marginal minima with +atypical neighborhoods actually attract the dynamics. To determine this, a +different type of calculation is needed. As our calculation is akin to the +quenched Franz--Parisi potential, study of atypical neighborhoods would entail +something like the annealed Franz--Parisi approach, i.e., +\begin{equation} + \Sigma^*(E_0,\mu_0,E_1,\mu_1,q)=\frac1N\overline{\log\left( + \int d\nu_H(\pmb\sigma,\varsigma\mid E_0,\mu_0)\,d\nu_H(\mathbf s,\omega\mid E_1,\mu_1)\,\delta(Nq-\pmb\sigma\cdot\mathbf s) + \right)} +\end{equation} +which puts the two points on equal footing. This calculation and exploration of +the atypical neighborhoods it reveals is a clear future direction. + +The methods developed in this paper are straightforwardly (if not easily) +generalized to landscapes with replica symmetry broken complexities +\cite{Kent-Dobias_2023_How}. We suspect that many of the qualitative features +of this study would persist, with neighboring states being divided into +different clusters based on the \textsc{rsb} order but with the basic presence +or absence of overlap gaps and the nature of the stability of near-neighbors +remaining unchanged. Interesting structure might emerge in the arrangement of +marginal states in \textsc{frsb} systems, where the ground state itself is +marginal and coincides with the threshold. + +\paragraph{Acknowledgements} + +The author would like to thank Valentina Ros, Giampaolo Folena, Chiara +Cammarota, and Jorge Kurchan for useful discussions related to this work. + +\paragraph{Funding information} + +JK-D is supported by a \textsc{DynSysMath} Specific Initiative by the +INFN. + +\appendix + +\section{Details of calculation for the isolated eigenvalue} +\label{sec:eigenvalue-details} + Using the same methodology as above, the disorder-dependent terms are captured in the linear operator \begin{equation} \mathcal O(\mathbf t)= @@ -1588,111 +1697,6 @@ $y^2\geq f''(1)$. In practice, there is at most \emph{one} $y$ which produces a zero eigenvalue of $B-yC$ and satisfies this inequality, so the solution seems to be unique. -In this solution, we simultaneously find the smallest eigenvalue and information -about the orientation of its associated eigenvector: namely, its overlap with -the tangent vector that points directly toward the reference spin. This is -directly related to $x_0$. This tangent vector is $\mathbf x_{0\leftarrow -1}=\frac1{1-q}\big(\pmb\sigma_0-q\mathbf s_a\big)$, which is normalized and -lies strictly in the tangent plane of $\mathbf s_a$. Then -\begin{equation} - q_\textrm{min}=\frac{\mathbf x_{0\leftarrow 1}\cdot\mathbf x_\mathrm{min}}N - =\frac{x_0}{1-q} -\end{equation} -The emergence of an isolated eigenvalue and its associated eigenvector are -shown in Fig.~\ref{fig:isolated.eigenvalue}, for the same reference point -properties as in Fig.~\ref{fig:min.neighborhood}. - -\begin{figure} - \includegraphics{figs/isolated_eigenvalue.pdf} - \hfill - \includegraphics{figs/eigenvector_overlap.pdf} - - \caption{ - Properties of the isolated eigenvalue and the overlap of its associated - eigenvector with the direction of the reference point. These curves - correspond with the lower solid curve in Fig.~\ref{fig:min.neighborhood}. - \textbf{Left:} The value of the minimum eigenvalue as a function of - overlap. The dashed line shows the continuation of the bottom of the - semicircle. Where the dashed line separates from the solid line, the - isolated eigenvalue has appeared. \textbf{Right:} The overlap between the - eigenvector associated with the minimum eigenvalue and the direction of the - reference point. The overlap is zero until an isolated eigenvalue appears, - and then it grows continuously until the nearest neighbor is reached. - } \label{fig:isolated.eigenvalue} -\end{figure} - -\section{Conclusion} -\label{sec:conclusion} - -We have computed the complexity of neighboring stationary points for the mixed -spherical models. When we studied the neighborhoods of marginal minima, we -found something striking: only those at the threshold energy have other -marginal minima nearby. For the many marginal minima away from the threshold -(including the exponential majority), there is a gap in overlap between them. - -This has implications for pictures of relaxation and aging. In most $p+s$ -models studied, quenches from infinite to zero temperature (gradient descent -starting from a random point) relax towards marginal states with energies above -the threshold energy \cite{Folena_2023_On}, while at least in some models a -quench to zero temperature from a temperature around the dynamic transition -relaxes towards marginal states with energies below the threshold energy -\cite{Folena_2020_Rethinking, Folena_2021_Gradient}. We found (see especially -Figs.~\ref{fig:marginal.prop.below} and \ref{fig:marginal.prop.above}) that the -neighborhoods of marginal states above and below the threshold are quite -different, and yet the emergent aging behaviors relaxing toward states above and -below the threshold seem to be the same. Therefore, this kind of dynamics -appears to be insensitive to the neighborhood of the marginal state being -approached. To understand something better about why certain states attract the -dynamics in certain situations, nonlocal information, like the -structure of their entire basin of attraction, seems vital. - -It is possible that replica symmetry breaking among the constrained stationary -points could change the details of the two-point complexity of very nearby -states. Indeed, it is difficult to rule out \textsc{rsb} in complexity -calculations. However, such corrections would not change the overarching -conclusions of this paper, namely that most marginal minima are separated from -each other by a macroscopic overlap gap and high barriers. This is because the -replica symmetric complexity bounds any \textsc{rsb} complexities from above, -and so \textsc{rsb} corrections can only decrease the complexity. Therefore, -the overlap gaps, which correspond to regions of negative complexity, cannot be -removed by a more detailed saddle point ansatz. - -Our calculation studied the neighborhood of typical reference points with the -given energy and stability. However, it is possible that marginal minima with -atypical neighborhoods actually attract the dynamics. To determine this, a -different type of calculation is needed. As our calculation is akin to the -quenched Franz--Parisi potential, study of atypical neighborhoods would entail -something like the annealed Franz--Parisi approach, i.e., -\begin{equation} - \Sigma^*(E_0,\mu_0,E_1,\mu_1,q)=\frac1N\overline{\log\left( - \int d\nu_H(\pmb\sigma,\varsigma\mid E_0,\mu_0)\,d\nu_H(\mathbf s,\omega\mid E_1,\mu_1)\,\delta(Nq-\pmb\sigma\cdot\mathbf s) - \right)} -\end{equation} -which puts the two points on equal footing. This calculation and exploration of -the atypical neighborhoods it reveals is a clear future direction. - -The methods developed in this paper are straightforwardly (if not easily) -generalized to landscapes with replica symmetry broken complexities -\cite{Kent-Dobias_2023_How}. We suspect that many of the qualitative features -of this study would persist, with neighboring states being divided into -different clusters based on the \textsc{rsb} order but with the basic presence -or absence of overlap gaps and the nature of the stability of near-neighbors -remaining unchanged. Interesting structure might emerge in the arrangement of -marginal states in \textsc{frsb} systems, where the ground state itself is -marginal and coincides with the threshold. - -\paragraph{Acknowledgements} - -The author would like to thank Valentina Ros, Giampaolo Folena, Chiara -Cammarota, and Jorge Kurchan for useful discussions related to this work. - -\paragraph{Funding information} - -JK-D is supported by a \textsc{DynSysMath} Specific Initiative by the -INFN. - -\appendix - \section{Franz--Parisi potential} \label{sec:franz-parisi} -- cgit v1.2.3-70-g09d2