Added new figure for multispherical case.

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}