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Diffstat (limited to 'marginal.tex')
| -rw-r--r-- | marginal.tex | 28 |
1 files changed, 20 insertions, 8 deletions
diff --git a/marginal.tex b/marginal.tex index 1ece03b..fdb7377 100644 --- a/marginal.tex +++ b/marginal.tex @@ -883,7 +883,9 @@ gradient. This means that the form of the Hessian is parameterized solely by the values of the Lagrange multipliers $\omega^{(1)}$ and $\omega^{(2)}$, just as $\mu=\omega$ alone parameterized the Hessian in the spherical spin glasses. Unlike that case, however, the Hessian takes different shapes with different -spectral widths depending on their precise combination. +spectral widths depending on their precise combination. In +Appendix~\ref{sec:multispherical.spectrum} we derive a variational form for the +spectral density of the Hessian in these models using standard methods. \begin{widetext} \begin{equation} @@ -911,6 +913,7 @@ spectral widths depending on their precise combination. \begin{equation} \begin{aligned} + \mathcal U_\mathrm{MSG}( &\sum_a^n\left[\hat q_a(Q^{11}_{aa}+Q^{22}_{aa}-1)-\beta(\omega_1Q^{11}_{aa}+\omega_2Q^{22}_{aa}+2\epsilon Q^{12}_{aa})\right] +\lambda(\omega_1Q^{11}_{11}+\omega_2Q^{22}_{11}+2\epsilon Q^{12}_{11}) \\ &+\sum_{i=1,2}f_i''(1)\left[\beta^2\sum_{ab}^n(Q^{ii}_{ab})^2-2\beta\lambda\sum_a^n(Q^{ii}_{1a})^2+\lambda^2(Q^{ii}_{11})^2\right] @@ -921,13 +924,22 @@ spectral widths depending on their precise combination. \end{aligned} \end{equation} \end{widetext} -\begin{equation} - \log\det\begin{bmatrix} - Q^{11}&Q^{12}\\ - Q^{12}&Q^{22} - \end{bmatrix} - +\log\det(Q^{11}Q^{22}-Q^{12}Q^{12}) -\end{equation} + +\begin{figure} + \includegraphics{figs/msg_marg_legend.pdf} + + \includegraphics{figs/msg_marg_params.pdf} + \hfill + \includegraphics{figs/msg_marg_spectra.pdf} + + \caption{ + \textsc{Left}: Values of the Lagrange multipliers $\omega_1$ and $\omega_2$ + corresponding to a marginal spectrum for multispherical spin glasses with + $\sigma_1^2=f_1''(1)=1$, $\sigma_2^2=f_2''(1)=1$, and various $\epsilon$. + \textsc{Right}: Spectra corresponding to the parameters $\omega_1$ and + $\omega_2$ marked by the circles on the lefthand plot. + } \label{fig:msg.marg} +\end{figure} \subsection{Random nonlinear least squares} \label{sec:least.squares} |