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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} |