From da97047729a6f02a3777727e3ab61da309a2ae63 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Thu, 27 Jun 2024 09:58:39 +0200
Subject: Some writing in the sum-of-squares section.

---
 marginal.tex | 20 +++++++++++++++++++-
 1 file changed, 19 insertions(+), 1 deletion(-)

diff --git a/marginal.tex b/marginal.tex
index 3a59487..1d2d1d6 100644
--- a/marginal.tex
+++ b/marginal.tex
@@ -1478,10 +1478,20 @@ taking the zero-temperature limit, we find
     \right]
   \end{aligned}
 \end{equation}
+\end{widetext}
+We can finally write the complexity with fixed minimum eigenvalue $\lambda^*$ as
 \begin{equation}
   \Sigma_{\lambda^*}(E,\mu)=\operatorname{extremum}_{\hat\beta,r,d,g,y_0,\Delta z,\hat\lambda}\mathcal S_\infty
 \end{equation}
-\end{widetext}
+Note that unlike the previous two examples, the effective action in this case
+does not split into two largely independent pieces, one relating to the
+eigenvalue problem and one relating to the ordinary complexity. Instead, the
+order parameters related to the eigenvalue problem are mixed throughout the
+effective action with those of the ordinary complexity. This is a signal of the
+fact that the sum of squares problem is not Gaussian, while the previous two
+examples are. In all non-Gaussian problems, conditioning on properties of the
+Hessian cannot be done independently from the complexity, and the method
+introduced in this paper becomes necessary.
 
 \begin{figure}
   \includegraphics{figs/most_squares_complexity.pdf}
@@ -1493,6 +1503,14 @@ taking the zero-temperature limit, we find
   } \label{fig:ls.complexity}
 \end{figure}
 
+Fig.~\ref{fig:ls.complexity} shows the marginal complexity in a sum-of-squares
+model with $\alpha=\frac32$ and $f(q)=q^2+q^3$. Also shown is the dominant
+complexity computed in Appendix~\ref{sec:dominant.complexity}. As the figure
+demonstrates, the range of energies at which marginal minima are found can
+differ significantly from those implied by the dominant complexity, with the
+lowest energy significantly higher than the ground state and the highest energy
+significantly higher than the threshold.
+
 \cite{Urbani_2023_A, Kamali_2023_Dynamical, Kamali_2023_Stochastic, Urbani_2024_Statistical}
 \cite{Montanari_2023_Solving, Montanari_2024_On}
 
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