From 757baebc7aaea4436e2256a95123765ce90dbec1 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Sun, 30 Jun 2024 14:43:52 +0200
Subject: Added new paragraph about new figure.

---
 marginal.tex | 26 ++++++++++++++++++++++++++
 1 file changed, 26 insertions(+)

(limited to 'marginal.tex')

diff --git a/marginal.tex b/marginal.tex
index b95872b..4ac383d 100644
--- a/marginal.tex
+++ b/marginal.tex
@@ -1577,6 +1577,32 @@ energy in a nontrivial way. The figure also shows the dominant stability,
 which is the stability associated with the dominant complexity and coincides
 with the marginal stability only at the threshold energy.
 
+\begin{figure}
+  \includegraphics{figs/most_squares_complex.pdf}
+  \caption{
+    Real and imaginary parts of the complexity $\Sigma_0(E,\mu)$ with fixed
+    minimum eigenvalue $\lambda^*=0$ as a function of $\mu$ in the nonlinear
+    sum of squares problem with $\alpha=\frac32$, $f(q)=q^2+q^3$, and
+    $E\simeq-6.47$. The vertical line depicts the value of the marginal
+    stability $\mu_\mathrm m$.
+  } \label{fig:ls.reim}
+\end{figure}
+
+Because this version of the model has no signal, we were able to use the heuristic
+\eqref{eq:marginal.stability} to fix the marginal stability. However, we could
+also have used the more general method for finding a pseudogapped Hessian
+spectrum by locating the value of $\mu$ at which the complexity develops an
+imaginary part, as described in Section \ref{sec:pseudogap} and pictured in
+Fig.~\ref{fig:large.dev}. The real and imaginary parts of the complexity
+$\Sigma_0(E,\mu)$ are plotted in Fig.~\ref{fig:ls.reim} as a function of $\mu$
+at fixed energy. The figure also shows the marginal stability $\mu_\mathrm m$
+predicted by the variational approach \eqref{eq:marginal.stability}. The
+marginal stability corresponds to precisely the point at which an imaginary
+part develops in the complexity. This demonstrates that the principles we used
+to determine the marginal stability continue to hold even in non-Gaussian cases
+where the complexity and the condition to fix the minimum eigenvalue are
+tangled together.
+
 In our companion paper, we use a sum of squared random functions model to explore the relationship between the marginal
 complexity and the performance of two generic algorithms:
 gradient descent and approximate message passing
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