From abab4928e793343d8d7fc917648cbeb43610b93f Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Thu, 27 Jun 2024 10:09:01 +0200
Subject: More writing in the sum-of-squares section.

---
 marginal.tex | 18 ++++++++++++++++--
 1 file changed, 16 insertions(+), 2 deletions(-)

diff --git a/marginal.tex b/marginal.tex
index 21fec76..099f603 100644
--- a/marginal.tex
+++ b/marginal.tex
@@ -552,7 +552,7 @@ In the cases studied here with zero signal-to-noise, a simpler approach is
 possible. The marginal stability $\mu=\mu_\text{m}$ can be identified by
 requiring that the complexity is stationary with respect to changes in the
 value of the minimum eigenvalue $\lambda^*$, or
-\begin{equation}
+\begin{equation} \label{eq:marginal.stability}
   0=\frac\partial{\partial\lambda^*}\Sigma_{\lambda^*}(E,\mu_\text{m}(E))\bigg|_{\lambda^*=0}
 \end{equation}
 The marginal complexity follows by evaluating the complexity conditioned on
@@ -1517,9 +1517,23 @@ significantly higher than the threshold.
     The stability, or shift of the trace, for dominant and marginal optima in
     the nonlinear sum of squares problem for $\alpha=\frac32$ and
     $f(q)=q^2+q^3$.
-  } \label{fig:ls.complexity}
+  } \label{fig:ls.stability}
 \end{figure}
 
+Fig.~\ref{fig:ls.stability} shows the associated marginal stability
+$\mu_\mathrm m(E)$ for the same model. Recall that the definition of the
+marginal stability in \eqref{eq:marginal.stability} is that which eliminates
+the variation of $\Sigma_{\lambda^*}(E,\mu)$ with respect to $\lambda^*$ at the
+point $\lambda^*=0$. Unlike the Gaussian spherical spin glass, this varies with
+energy in a nontrivial way. That figure also shows the dominant stability,
+which is the stability associated with the dominant complexity, which coincides
+with the marginal stability only at the threshold energy.
+
+In our companion paper, we explore the relationship between the marginal
+complexity and the performance of two generic algorithms on this model:
+gradient descent and approximate message passing
+\cite{Kent-Dobias_2024_Algorithm-independent}.
+
 \cite{Urbani_2023_A, Kamali_2023_Dynamical, Kamali_2023_Stochastic, Urbani_2024_Statistical}
 \cite{Montanari_2023_Solving, Montanari_2024_On}
 
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