From 596e488214dcaaea47fa42149067fb50604ae988 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Tue, 11 Mar 2025 14:19:52 -0300
Subject: Many small wording tweaks

---
 topology.tex | 36 ++++++++++++++++--------------------
 1 file changed, 16 insertions(+), 20 deletions(-)

(limited to 'topology.tex')

diff --git a/topology.tex b/topology.tex
index cb95e01..6fb6bce 100644
--- a/topology.tex
+++ b/topology.tex
@@ -676,7 +676,7 @@ When $M=1$ the solution manifold corresponds to the energy
 level set of a spherical spin glass with energy density $E=V_0/\sqrt N$. All the
 results from the previous sections follow, and can be translated to the spin
 glasses by taking the limit $\alpha\to0$ while keeping $E=V_0\alpha^{1/2}$ fixed.\footnote{
-  It is plausible that the limits of $N\to\infty$ implicit in the saddle point expansion and the limit of $\alpha\to0$ taken here do not commute, and that $M=1$ should be taken from the beginning of the calculation. However, in this case the two procedures do commute. The $\alpha\to0$ limit accomplishes only the elimination of the first term from the effective action \eqref{eq:S.m}, while following Appendix~\ref{sec:euler} with $M=1$ from the outset results in the same term not appearing in the effective action because it is of subleading order in $N$.
+  It is plausible that the limit of $N\to\infty$ implicit in the saddle point expansion and the limit of $\alpha\to0$ taken here do not commute, and that $M=1$ should be set from the beginning of the calculation. However, in this case the two procedures do commute. The $\alpha\to0$ limit accomplishes only the elimination of the first term from the effective action \eqref{eq:S.m}, while following Appendix~\ref{sec:euler} with $M=1$ from the outset results in the same term not appearing in the effective action because it is of subleading order in $N$.
 } With a little algebra this procedure yields
 \begin{align}
   E_\text{on}=\pm\sqrt{2f(1)}
@@ -776,37 +776,33 @@ descent from a uniformly random initial condition, but other algorithms find
 minima at other energies. Optimal message passing algorithms were shown to find
 configurations at an energy level where another topological property---the
 overlap gap property---transitions, and this energy level is believed to bound from below
-all polynomial-time algorithms. On the other hand, physically inspired
+all polynomial-time algorithms \cite{ElAlaoui_2020_Algorithmic, ElAlaoui_2021_Optimization, Gamarnik_2021-10_The}. On the other hand, physically inspired
 modifications of gradient descent---notably, drawing the initial condition from
 a nonuniform distribution like the Boltzmann distribution with a finite
-temperature---can find energy configurations with energies lower than those
-found with gradient descent from a uniform initial condition. If the
+temperature---find energy configurations with energies lower than those
+found with gradient descent from a uniform initial condition \cite{Folena_2020_Rethinking, Folena_2021_Gradient}. If the
 topological transition described in this paper does predict the asymptotic
 performance of gradient descent from a uniform initial condition, then it
 provides a topological bound from above for the performance of reasonable
-algorithms that terminate in minima. Whether the performance of gradient
+algorithms that terminate in minima. It is unknown whether the performance of gradient
 descent from better initial conditions, or of other algorithms like simulated
-annealing, can be predicted with a similar method is not known.
+annealing, can be predicted with a topological property.
 
 Finally, a common extension of the spherical spin glasses is to add a
 deterministic piece to the energy, sometimes called a signal or a spike. Recent
-work argued that gradient descent can avoid being trapped in marginal minima
-and reach the vicinity of the signal if the set of trapping marginal minima has
+work argued that gradient descent can avoid being trapped by the minima that typically trap dynamics
+and reach the vicinity of the signal if the set of typically trapping minima has
 been destabilized by the presence of the signal \cite{SaraoMannelli_2019_Passed,
 SaraoMannelli_2019_Who}. The authors of Ref.~\cite{SaraoMannelli_2019_Who}
 conjecture based on \textsc{dmft} data for $2+3$ mixed spherical spin glasses that the
-trapping marginal minima are those at the traditional threshold energy
-$E_\text{th}$. However, Ref.~\cite{Folena_2023_On} demonstrated that in mixed
-$p+s$ spherical spin glasses with small $p$ and $s$, the difference between
-$E_\text{th}$ and the true trapping energy is difficult to resolve with the
-current precision of \textsc{dmft} integration schemes. Therefore,
-the authors of Ref.~\cite{SaraoMannelli_2019_Who} may have incorrectly conflated the
-threshold with the trapping marginal minima, and that the correct set of
-marginal minima that must be destabilized to reach a signal might be the same set
-that trap dynamics in the signal-free model. This paper conjectures that the important trapping minima
-are those at the shattering energy. Comparing the predictions of
+typically trapping minima are those at the threshold energy
+$E_\text{th}$. However, as discussed above, Ref.~\cite{Folena_2023_On} demonstrated that in signal-free mixed
+$p+s$ spherical spin glasses $E_\text{th}$ is not the energy of typical trapping minima, and furthermore that, when $p$ and $s$ are small, the difference between
+$E_\text{th}$ and the energy of the actual trapping minima is difficult to resolve with the
+current precision of \textsc{dmft} integration schemes. Therefore, it is plausible that the picture described in Ref.~\cite{SaraoMannelli_2019_Who} is correct except that the set of minima that must be destabilised to reach the signal is that at the typical trapping energy of the isotropic problem and not the threshold energy $E_\text{th}$. If the conjecture made in this paper is true, then this typical trapping energy is the shattering energy $E_\text{sh}$.
+Comparing the predictions of
 Ref.~\cite{SaraoMannelli_2019_Who} to \textsc{dmft} simulations of a model with
-better separation between $p$ and $s$ would help resolve this issue.
+better separation between $p$ and $s$ would help resolve this question.
 
 \section{Conclusion}
 \label{sec:conclusion}
@@ -1155,7 +1151,7 @@ $\frac12\log\det(\mathbb Q-\mathbb M\mathbb M^T)$ in the effective action from
   =1
 \end{equation}
 where the final superdeterminant is identically 1 for any superoperator $\tilde{\mathbb Q}$, not just its saddle-point value.\footnote{
-  The subscript notation in \eqref{eq:supermatrix.saddle} indicates which superindices of the four-index superoperator associated with the Hessian belong to the domain and codomain, analogous to writing $\det A=\det_{ij}A_{ij}$ for a two-index complex-valued matrix. In this case, the domain is indexed by $\{3,4\}$ and the codomain is indexed by $\{1,2\}$.
+  The subscript notation in \eqref{eq:supermatrix.saddle} indicates which superindices of the four-index superoperator associated with the Hessian belong to the domain and codomain, analogous to writing $\det A=\det_{ij}A_{ij}$ for a two-index complex-valued operator. In this case, the domain is indexed by $\{3,4\}$ and the codomain is indexed by $\{1,2\}$.
 }
 The Hubbard--Stratonovich transformation therefore contributes a factor of
 \begin{equation}
-- 
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