From 4d16ef77a976e9bd0b7396bca507bb3f161d4d6c Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Thu, 13 Jun 2024 18:31:52 +0200
Subject: More writing.

---
 marginal.tex | 125 +++++++++++++++++++++++++++++++++++++++++++++++++++++------
 1 file changed, 114 insertions(+), 11 deletions(-)

(limited to 'marginal.tex')

diff --git a/marginal.tex b/marginal.tex
index 6514eba..0013a4f 100644
--- a/marginal.tex
+++ b/marginal.tex
@@ -224,7 +224,7 @@ of its unique coupling with $\hat\lambda$. This gives
     \tilde q_0&1&q_0&\cdots&q_0\\
     \tilde q_0&q_0&1&\ddots&q_0\\
     \vdots&\vdots&\ddots&\ddots&\vdots\\
-    \tilde q_0&q_0&q_0&\cdots&q_0
+    \tilde q_0&q_0&q_0&\cdots&1
   \end{bmatrix}
 \end{equation}
 with $\sum_{ab}Q_{ab}^2=n+2(n-1)\tilde q_0^2+(n-1)(n-2)q_0^2$, $\sum_aQ_{1a}^2=1+(n-1)\tilde q_0^2$,
@@ -690,7 +690,7 @@ m_a$ matrix with one lower and one upper index.
 After these steps, which follow identically to those more carefully outlined in
 the cited papers \cite{Folena_2020_Rethinking, Kent-Dobias_2023_How}, we arrive at a form of the integral as over an effective action
 \begin{widetext}
-\begin{equation}
+  \begin{equation} \label{eq:spherical.complexity}
   \begin{aligned}
     &\Sigma_{\lambda^*}(E,\mu)
     =\lim_{\beta\to\infty}\lim_{n\to0}\lim_{m_1\cdots m_n\to0}
@@ -852,7 +852,7 @@ These models have more often been studied with random fully connected couplings
 between the spheres, for which it is possible to also use configuration spaces
 involving spheres of different sizes \cite{Subag_2021_TAP, Subag_2023_TAP,
 Bates_2022_Crisanti-Sommers, Bates_2022_Free, Huang_2023_Strong,
-Huang_2024_Optimization}.
+Huang_2023_Algorithmic, Huang_2024_Optimization}.
 
 Because the energy is Gaussian, properties of the Hessian are once again
 statistically independent of those of the energy and gradient. However, unlike
@@ -893,14 +893,32 @@ 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.
 
+Because of the independence of the Hessian, the method introduced in this
+article is not necessary to characterize the marginal minima of this system.
+Rather, we could take the spectral density derived in
+Appendix~\ref{sec:multispherical.spectrum} and found the Lagrange multipliers
+$\omega_1$ and $\omega_2$ corresponding with marginality by tuning the edge of
+the spectrum to zero. In some ways the current method is more convenient than
+this, since it is a purely variational method and therefore can be reduced to a
+since root-finding exercise.
+
+The calculation of the marginal complexity in this problem follows very closely
+to that of the spherical spin glasses in the previous subsection, making
+immediately the simplifying assumptions that the soft directions of different
+stationary points are typically uncorrelated and therefore $X=\hat X=0$ and the
+overlaps $Q$ between eigenvectors are only nonzero when in the same replica.
+The result has the schematic form of \eqref{eq:spherical.complexity}, but with
+different effective actions depending now on overlaps inside each of the two
+spheres and between the two spheres. These are
 \begin{widetext}
 \begin{equation}
   \begin{aligned}
-    &\mathcal S_\mathrm{MSG}(\hat\beta,C^{11},R^{11},D^{11},G^{11},C^{22},R^{22},D^{22},G^{22})= \\
+    &\mathcal S_\mathrm{MSG}(\hat\beta,C^{11},R^{11},D^{11},G^{11},C^{22},R^{22},D^{22},G^{22},C^{12},R^{12},R^{21},D^{12},G^{12}
+    \mid E,\omega_1,\omega_2)= \\
     &\quad
     \mathcal S_\mathrm{SSG}(\hat\beta,C^{11},R^{11},D^{11},G^{11}\mid E_1,\omega_1)
     +\mathcal S_\mathrm{SSG}(\hat\beta,C^{22},R^{22},D^{22},G^{22}\mid E_2,\omega_2)
-    -\epsilon(r_{12}+r_{21})+\hat\beta(E-E_1-E_2) \\
+    -\epsilon(r^{12}_d+r^{21}_d)+\hat\beta(E-E_1-E_2-\epsilon c_d^{12}) \\
     &\quad
     +\frac12\log\det\left(
     I+
@@ -916,13 +934,13 @@ spectral density of the Hessian in these models using standard methods.
     -\log\det(I+(G^{11}G^{22})^{-1}G^{12}G^{21})
   \end{aligned}
 \end{equation}
-
+and
 \begin{equation}
   \begin{aligned}
-    \mathcal U_\mathrm{MSG}(
+    &\mathcal U_\mathrm{MSG}(\hat q,\hat\lambda,Q^{11},Q^{22},Q^{12}\mid\lambda^*,\omega_1,\omega_2,\beta) \\
     &\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]
+    +\hat\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\hat\lambda\sum_a^n(Q^{ii}_{1a})^2+\hat\lambda^2(Q^{ii}_{11})^2\right]
     +\frac12\log\det\begin{bmatrix}
       Q^{11}&Q^{12}\\
       Q^{12}&Q^{22}
@@ -930,6 +948,91 @@ spectral density of the Hessian in these models using standard methods.
   \end{aligned}
 \end{equation}
 \end{widetext}
+where again the problem of fixing marginality has completely separated from
+that of the complexity. The biggest change between this problem and the
+spherical one is that now the spherical constraint in the tangent space at each
+stationary point gives the constraint on the order parameters
+$q^{11}_d+q^{22}_d=1$. Therefore, the diagonal of the $Q$ matrices cannot be
+taken to be 1 as before. To solve the marginal problem, we take each of the
+matrices $Q^{11}$, $Q^{22}$, and $Q^{12}$ to have the planted replica symmetric
+form \eqref{eq:Q.structure}, but with the diagonal not necessarily equal to 1, so
+\begin{equation}
+ Q^{ij}=\begin{bmatrix}
+   \tilde q^{ij}_d & \tilde q^{ij}_0 & \tilde q^{ij}_0 & \cdots & \tilde q^{ij}_0 \\
+   \tilde q^{ij}_0 & q^{ij}_d & q^{ij}_0 & \cdots & q^{ij}_0 \\
+   \tilde q^{ij}_0 & q^{ij}_0 & q^{ij}_d & \ddots & q^{ij}_0 \\
+   \vdots & \vdots & \ddots & \ddots & \vdots \\
+   \tilde q^{ij}_0 & q^{ij}_0 & q^{ij}_0 & \cdots & q^{ij}_d
+ \end{bmatrix}
+\end{equation}
+
+\begin{widetext}
+  \begin{equation}
+    \begin{aligned}
+      &\sum_{i=1,2}f_i''(1)\left[
+        \beta^2\left(
+          (\tilde q^{ii}_d)^2
+          -(q^{ii}_d)^2
+          +2(q^{ii}_0)^2
+          -2(\tilde q^{ii}_0)^2
+        \right)
+        -2\beta\hat\lambda\left(
+          (\tilde q^{ii}_d)^2-(\tilde q^{ii}_0))^2
+        \right)
+        +\hat\lambda^2(\tilde q^{ii}_d)^2
+      \right]
+      +\hat\lambda\left(
+        \tilde q^{11}_d\omega_1+\tilde q^{22}_d\omega_2+2\tilde q^{12}_d
+      \right) \\
+      &-\beta\left(
+        (\tilde q^{11}_d-q^{11}_d)\omega_1
+        +(\tilde q^{22}_d-q^{22}_d)\omega_2
+        -2\epsilon(\tilde q^{12}_d-q^{12}_d)
+      \right) \\
+      &+\frac12\log\bigg[
+        \left(
+          2q^{12}_0\tilde q^{12}_0-\tilde q^{12}_0(\tilde q^{12}_d+q^{12}_d)
+          -2\tilde q^{11}_0q^{22}_0+\tilde q^{11}_d\tilde q^{22}_0+\tilde q^{11}_0q^{22}_d
+        \right)
+        \left(
+          2q^{12}_0\tilde q^{12}_0-\tilde q^{12}_0(\tilde q^{12}_d+q^{12}_d)
+          -2q^{11}_0\tilde q^{22}_0+q^{11}_d\tilde q^{22}_0+\tilde q^{11}_0\tilde q^{22}_d
+        \right) \\
+      &\qquad\qquad+2\left(3(q^{12}_0)^2-(\tilde q^{12}_0)^2-2q^{12}_0q^{12}_d-3q^{11}_0q^{22}_0+q^{11}_dq^{22}_0+\tilde q^{11}_0\tilde q^{22}_0+q^{11}_0q^{22}_d
+        \right)\left(
+          (\tilde q^{12}_0)^2-(\tilde q^{12}_d)^2-\tilde q^{11}_0\tilde q^{22}_0+\tilde q^{11}_d\tilde q^{22}_d
+        \right) \\
+      &\qquad\qquad+\left(
+        2(q^{12}_0)^2-(\tilde q^{12}_0)^2-(q^{12}_d)^2-2q^{11}_0q^{22}_0+\tilde q^{11}_0\tilde q^{22}_0+q^{11}_dq^{22}_d
+      \right)\left(
+        (\tilde q^{12}_0)^2-(\tilde q^{12}_d)^2-\tilde q^{11}_0\tilde q^{22}_0+\tilde q^{11}_d\tilde q^{22}_d
+      \right)
+      \bigg]
+      \\
+      &-\log\left[(q^{11}_d-q^{11}_0)(q^{22}_d-q^{22}_0)-(q^{12}_d-q^{12}_0)^2\right]
+    \end{aligned}
+  \end{equation}
+\end{widetext}
+To make the limit to zero temperature, we once again need an ansatz for the
+asymptotic behavior of the overlaps. These take the form
+$q^{ij}_0=q^{ij}_d-y^{ij}_0\beta^{-1}-z^{ij}_0\beta^{-2}$, with the same for
+the tilde variables. Notice that in this case, the asymptotic behavior of the
+off diagonal elements is to approach the value of the diagonal rather than one.
+We also require $\tilde q^{ij}_d=q^{ij}_d-\tilde y^{ij}_d\beta^{-1}-\tilde
+z^{ij}_d\beta^{-2}$, i.e., that the tilde diagonal term also approaches the
+same diagonal value.
+
+As before, in order for the volume term to stay finite, there are necessary
+constraints on the values $y$. These are
+\begin{align}
+  \frac12(y^{11}_d-\tilde y^{11}_d)=y^{11}_0-\tilde y^{11}_0 \\
+  \frac12(y^{22}_d-\tilde y^{22}_d)=y^{22}_0-\tilde y^{22}_0 \\
+  \frac12(y^{12}_d-\tilde y^{12}_d)=y^{12}_0-\tilde y^{12}_0
+\end{align}
+One can see that when the diagonal elements are all equal, this requires the
+$y$s for the off-diagonal elements to be equal, as in the GOE case. Here, since
+the diagonal elements are not necessarily equal, we have a more general
+relationship.
 
 \begin{figure}
   \includegraphics{figs/msg_marg_legend.pdf}
@@ -1137,8 +1240,8 @@ Given these simplifying forms of the ansatz, taking the superdeterminant yields
 where once again $\odot$ is the Hadamard product and $A^{\circ n}$ gives the
 Hadamard power of $A$. We can already see one substantive difference between
 the structure of this problem and that of the spherical models: the effective
-action in this case mixes the order parameters $G$ due to the fermions with the
-ones $C$, $R$, and $D$ due to the other variables. This is the realization of
+action in this case mixes the order parameters $G$ due to the Grassmann variables with the
+ones $C$, $R$, and $D$ due to the other variables. Notice further that the dependence on $Q$ due to the marginal constraint is likewise no longer separable. This is the realization of
 the fact that the Hessian properties are no longer independent of the energy
 and gradient.
 
-- 
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