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diff --git a/marginal.tex b/marginal.tex index 3a59487..21fec76 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 +1503,23 @@ 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. + +\begin{figure} + \includegraphics{figs/most_squares_stability.pdf} + \caption{ + 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} +\end{figure} + \cite{Urbani_2023_A, Kamali_2023_Dynamical, Kamali_2023_Stochastic, Urbani_2024_Statistical} \cite{Montanari_2023_Solving, Montanari_2024_On} |