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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 |