From 385cf672265be860a9517643514282e22f0cad42 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Wed, 21 Jun 2023 21:19:47 +0200
Subject: Some tweaks.

---
 when_annealed.tex | 46 +++++++++++++++++++++++++++-------------------
 1 file changed, 27 insertions(+), 19 deletions(-)

(limited to 'when_annealed.tex')

diff --git a/when_annealed.tex b/when_annealed.tex
index e0d9556..24b6b59 100644
--- a/when_annealed.tex
+++ b/when_annealed.tex
@@ -53,7 +53,7 @@
 \end{abstract}
 
 Random high-dimensional energies, cost functions, and interaction networks are
-important in many fields. The energy landscape of glasses, the likelihood
+important across disciplines: the energy landscape of glasses, the likelihood
 landscape of machine learning and inference, and the interactions between
 organisms in an ecosystem are just a few examples \cite{Stein_1995_Broken, Krzakala_2007_Landscape, Altieri_2021_Properties, Yang_2023_Stochastic}. A traditional tool for
 making sense of their behavior is to analyze the statistics of points where
@@ -74,7 +74,7 @@ without much reflection \cite{Wainrib_2013_Topological, Kent-Dobias_2021_Complex
 Gershenzon_2023_On-Site}. In a few cases researchers have instead made the
 better-controlled quenched average, which averages the logarithm of the number
 of stationary points, and find deviations from the annealed approximation with
-important implications for the behavior \cite{Muller_2006_Marginal,
+important implications for behavior \cite{Muller_2006_Marginal,
 Ros_2019_Complex, Kent-Dobias_2023_How, Ros_2023_Quenched, Ros_2023_Generalized}. Generically,
 the annealed approximation to the complexity is wrong when a nonvanishing
 fraction of pairs of stationary points have nontrivial correlations in their
@@ -94,8 +94,8 @@ stationary points?
 In this paper, we show that the behavior of the ground state, or \emph{any}
 equilibrium behavior, does not govern whether stationary points will have a
 correct annealed average. In a prototypical family of models of random
-functions, we calculate a condition for determining when annealed averages
-should fail and stationary points should have nontrivial correlations in their
+functions, we determine a condition for when annealed averages
+should fail and some stationary points will have nontrivial correlations in their
 mutual position. We produce examples of models whose equilibrium is guaranteed
 to never see such correlations between thermodynamic states, but where a
 population of saddle points is nevertheless correlated.
@@ -113,12 +113,13 @@ Specifying the covariance function $f$ uniquely specifies the model. The series
 coefficients of $f$ need to be nonnnegative in order for $f$ to be a
 well-defined covariance. The case where $f$ is a homogeneous polynomial has
 been extensively studied, and corresponds to the pure spherical models of glass
-physics or the spiked tensor models of statistical inference. Here we will
+physics or the spiked tensor models of statistical inference \cite{Castellani_2005_Spin-glass}. Here we will
 study cases where $f(q)=\frac12\big(\lambda q^3+(1-\lambda)q^s\big)$ for
 $\lambda\in(0,1)$, called $3+s$ models. These are examples of \emph{mixed}
 spherical models, which have been studied in the physics and statistics
 literature and host a zoo of complex orders and phase transitions
 \cite{Crisanti_2004_Spherical, Crisanti_2006_Spherical,
+Krakoviack_2007_Comment, Crisanti_2007_Amorphous-amorphous,
 Crisanti_2011_Statistical}.
 
 There are several well-established results on the equilibrium of this model.
@@ -147,8 +148,8 @@ consider, for $s>12.430...$ nontrivial ground state configurations appear in
 a range of $\lambda$. These bounds on equilibrium order are shown in
 Fig.~\ref{fig:phases}, along with our result for where the complexity has
 nontrivial correlations between some stationary points. As evidenced in that
-figure, \textsc{rsb} among saddles is possible well outside the bounds from
-equilibrium.
+figure, correlations among saddles are possible well inside regions which
+forbid them among equilibrium states.
 
 There are two important features which differentiate stationary points
 $\pmb\sigma^*$ in the spherical models: their \emph{energy density}
@@ -177,7 +178,7 @@ has support only over positive eigenvalues, and we have stable minima.\footnote{
     \eqref{eq:condition}. The yellow region shows where $\chi(q)=f''(q)^{-1/2}$
     is not convex and therefore nontrivial correlations between states are
     possible in equilibrium. The green region shows where nontrivial
-    solutions are correct at the ground state, adapted from
+    correlations exist at the ground state, adapted from
     \cite{Auffinger_2022_The}. We find that models where correlations between
     equilibrium states are forbidden can nonetheless harbor
     correlated stationary points.
@@ -193,7 +194,12 @@ logarithm: $\Sigma_\mathrm a(E,\mu)=\frac1N\log\overline{\mathcal N(E,\mu)}$.
 The annealed complexity has been computed for these models
 \cite{BenArous_2019_Geometry, Folena_2020_Rethinking}, and the quenched
 complexity has been computed for a couple examples which have nontrivial ground
-states \cite{Kent-Dobias_2023_How}.
+states \cite{Kent-Dobias_2023_How}. The annealed complexity bounds the
+complexity from above. A positive complexity indicates the presence of an
+exponentially large number of stationary points of the indicated kind, while a
+negative one means it is vanishingly unlikely they will appear. The line of
+zero complexity is significant as the transition between many stationary points
+and none.
 
 In these models, trivial correlations between stationary points correspond with
 zero overlap: almost all stationary points are orthogonal to each other. This
@@ -300,9 +306,11 @@ elsewhere) the constants
   z_f=f(f''-f')+f'^2
 \end{align}
 If $f$ has at least two nonzero coefficients at second order or higher, all of
-these constants are positive. We also define $E_\textrm{min}$, the minimum
-energy at which saddle points with an extensive number of downward directions
-are found, as the energy for which $\mu_0(E_\mathrm{min})=\mu_\mathrm m$.
+these constants are positive. Though in figures we focus on the lower branch of
+saddles, another set of identical solutions always exists for $E\mapsto-E$ and $\mu\mapsto-\mu$.
+We also define $E_\textrm{min}$, the minimum energy at which saddle points with
+an extensive number of downward directions are found, as the energy for which
+$\mu_0(E_\mathrm{min})=\mu_\mathrm m$.
 
 Let $M$ be the matrix of double partial derivatives of the action with
 respect to $q_1$ and $x$. We evaluate $M$ at the replica symmetric saddle point
@@ -346,9 +354,9 @@ $e$, $g$, and $h$ are given by
 
   \caption{
     Stationary point statistics as a function of energy density $E$ and
-    stability $\mu$ for a $3+5$ model with $\lambda=\frac12$. The dashed black
-    line shows the line of zero annealed complexity, where stationary points vanish, and
-    enclosed inside they are found in exponential number. The solid black line (only visible in the inset) gives the line of zero {\oldstylenums1\textsc{rsb}} complexity. The red region (blown
+    stability $\mu$ for a model with $f(q)=\frac12(\frac12q^3+\frac12q^5)$. The dashed black
+    line shows the line of zero annealed complexity and
+    enclosed inside the annealed complexity is positive. The solid black line (only visible in the inset) gives the line of zero {\oldstylenums1\textsc{rsb}} complexity. The red region (blown
     up in the inset) shows where the annealed complexity gives the wrong count
     and a {\oldstylenums1}\textsc{rsb} complexity in necessary. The red points
     show where $\det M=0$. The left point, which is only an upper bound on the
@@ -492,10 +500,10 @@ order \emph{already exist in the landscape} as unstable saddles for small
 $\lambda$, then eventually stabilize into metastable minima and finally become
 the lowest lying states. This is different from the picture of existing
 uncorrelated low-lying states splitting apart into correlated clusters. Where
-existing stationary points do appear to split apart, when $\lambda$ is
+uncorrelated stationary points do appear to split apart, when $\lambda$ is
 decreased from large values, is among saddles, not minima.
 
-A imilar analysis can be made for other mixed models, like the $2+s$, which
+A similar analysis can be made for other mixed models, like the $2+s$, which
 should see complexities with other forms of \textsc{rsb}. For instance, in
 \cite{Kent-Dobias_2023_How} we show that the complexity transitions from
 \textsc{rs} to full \textsc{rsb} (\textsc{frsb}) along the line
@@ -512,7 +520,7 @@ of the parameter space the so-called one-full \textsc{rsb}
 ({\oldstylenums1\textsc{frsb}}), rather than \textsc{frsb}, is the correct solution, as it likely is for
 large $s$ and certain $\lambda$ in the $3+s$ models studied here. Further work to find the conditions for
 transitions of the complexity to {\oldstylenums1\textsc{frsb}} and {\oldstylenums2\textsc{frsb}} is necessary. For values
-of $s$ where there is no \textsc{rsb} of any kind in the ground state, we
+of $s$ where there is trivial \textsc{rsb} in the ground state, we
 expect that the {\oldstylenums1\textsc{rsb}} complexity is correct.
 
 What are the implications for dynamics? We find that nontrivial correlations
@@ -521,7 +529,7 @@ a given energy density, which are quite atypical in the landscape. However,
 these strangely correlated saddle points must descend to uncorrelated minima,
 which raises questions about whether structure on the boundary of a basin of
 attraction is influential to the dynamics that descends into that basin. These
-saddles might act as early-time separatrices for descent trajectories.  With
+saddles might act as early-time separatrices for descent trajectories of certain algorithms.  With
 open problems in even the gradient decent dynamics on these models (itself attracted to an atypical subset of marginal minima), it
 remains to be seen whether such structures could be influential
 \cite{Folena_2020_Rethinking, Folena_2021_Gradient, Folena_2023_On}. This structure among saddles
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