From 19998d539381d5e88671ebee8ce9d7b620a6f3d4 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Tue, 11 Jun 2024 10:48:14 +0200
Subject: Lots of writing.

---
 marginal.tex | 116 ++++++++++++++++++++++++++++++++++++++++++++++++++++-------
 1 file changed, 103 insertions(+), 13 deletions(-)

(limited to 'marginal.tex')

diff --git a/marginal.tex b/marginal.tex
index 95955b5..4afa2c5 100644
--- a/marginal.tex
+++ b/marginal.tex
@@ -363,15 +363,19 @@ geometry of such landscapes is studied by their complexity, or the average
 logarithm of the number of stationary points with certain properties, e.g., of
 marginal minima at a given energy.
 
-Such problems can be studied using the method of Lagrange multipliers, with one introduced for every constraint. If the configuration space is defined by $r$ constraints, then the problem is to extremize
+Such problems can be studied using the method of Lagrange multipliers, with one introduced for every constraint. If the configuration space is defined by $r$ constraints, then the problem is to extremize the Lagrangian
 \begin{equation}
-  H(\mathbf x)+\sum_{i=1}^r\omega_ig_i(\mathbf x)
+  L(\mathbf x,\pmb\omega)=H(\mathbf x)+\sum_{i=1}^r\omega_ig_i(\mathbf x)
 \end{equation}
 with respect to $\mathbf x$ and $\pmb\omega=\{\omega_1,\ldots,\omega_r\}$. The corresponding gradient and Hessian for the problem are
 \begin{align}
-  \nabla H(\mathbf x,\pmb\omega)=\partial H(\mathbf x)+\sum_{i=1}^r\omega_i\partial g_i(\mathbf x)
+  \nabla H(\mathbf x,\pmb\omega)
+  &=\partial L(\mathbf x,\pmb\omega)
+  =\partial H(\mathbf x)+\sum_{i=1}^r\omega_i\partial g_i(\mathbf x)
   \\
-  \operatorname{Hess}H(\mathbf x,\pmb\omega)=\partial\partial H(\mathbf x)+\sum_{i=1}^r\omega_i\partial\partial g_i(\mathbf x)
+  \operatorname{Hess}H(\mathbf x,\pmb\omega)
+  &=\partial\partial L(\mathbf x,\pmb\omega)
+  =\partial\partial H(\mathbf x)+\sum_{i=1}^r\omega_i\partial\partial g_i(\mathbf x)
 \end{align}
 The number of stationary points in a landscape for a particular realization $H$ is found by integrating over the Kac--Rice measure
 \begin{equation}
@@ -422,10 +426,10 @@ again to treat each of the normalizations in the numerator. This leads to the ex
     \Sigma_{\lambda^*}(E,\mu)
     &=\lim_{\beta\to\infty}\lim_{n\to0}\frac1N\frac\partial{\partial n}\int\prod_{a=1}^n\Bigg[d\mu_H(\mathbf x_a,\pmb\omega_a\mid E,\mu)\,\delta\big(N\lambda^*-(\mathbf s_a^1)^T\operatorname{Hess}H(\mathbf x_a,\pmb\omega_a)\mathbf s_a^1\big)\\
     &\hspace{12em}\times\lim_{m_a\to0}
-      \left(\prod_{b=1}^{m_a} d\mathbf s_a^b
-      \,\delta\big(N-(\mathbf s_a^b)^T\mathbf s_a^b\big)
-      \,\delta\big((\mathbf s_a^b)^T\partial\mathbf g(\mathbf x_a)\big)
-      \,e^{-\beta(\mathbf s_a^b)^T\operatorname{Hess}H(\mathbf x_a,\pmb\omega_a)\mathbf s_a^b}\right)
+      \left(\prod_{\alpha=1}^{m_a} d\mathbf s_a^\alpha
+      \,\delta\big(N-(\mathbf s_a^\alpha)^T\mathbf s_a^\alpha\big)
+      \,\delta\big((\mathbf s_a^\alpha)^T\partial\mathbf g(\mathbf x_a)\big)
+      \,e^{-\beta(\mathbf s_a^\alpha)^T\operatorname{Hess}H(\mathbf x_a,\pmb\omega_a)\mathbf s_a^\alpha}\right)
     \Bigg]
   \end{aligned}
 \end{equation}
@@ -476,6 +480,87 @@ treat the determinant keeping the absolute value signs, as in previous works
 our examples are for models where the same techniques are impossible, it is
 useful to see the fermionic method in action in this simple case.
 
+For the cases studied here, fixing the trace results in a relationship
+between $\mu$ and the Lagrange multipliers enforcing the constraints. This is
+because the trace of $\partial\partial H$ is typically an order of $N$ smaller
+than that of the constraint functions $\partial\partial g_i$. The result is that
+\begin{equation}
+  \mu
+  =\frac1N\operatorname{Tr}\operatorname{Hess}H(\mathbf x)
+  =\frac1N\sum_{i=1}^r\omega_i\partial\partial g_i(\mathbf x)
+  +O(N^{-1})
+\end{equation}
+In particular, here we study only cases with quadratic $g_i$, which results in
+an expression relating $\mu$ and the $\omega_i$ that is independent of $\mathbf
+x$. Since $H$ contains the disorder of the problem, this simplification means
+that the effect of fixing the trace is independent of the disorder and only
+depends on properties of the constraint manifold.
+
+\subsection{Superspace representation}
+
+The ordinary Kac--Rice calculation involves many moving parts, and this method
+for incorporating marginality adds even more. It is therefore convenient to
+introduce compact and simplifying notation through a superspace representation.
+The use of superspace in the Kac--Rice calculation is well established, as well
+as the deep connections with BRST symmetry that is implied.
+Appendix~\ref{sec:superspace} introduces the notation and methods of
+superspace. Here we describe how it can be used to simplify the complexity
+calculation in the marginal case.
+
+We consider the $\mathbb R^{N|4}$ superspace whose Grassmann indices are
+$\bar\theta_1,\theta_1,\bar\theta_2,\theta_2$. Consider the supervector defined
+by
+\begin{equation}
+  \pmb\phi_a^\alpha(1,2)
+  =\mathbf x_a
+  +\bar\theta_1\pmb\eta_a+\bar{\pmb\eta}_a\theta_1
+  +i\hat{\mathbf x}_a\bar\theta_1\theta_1
+  +\mathbf s_a^\alpha(\bar\theta_1\theta_2+\bar\theta_2\theta_1)
+\end{equation}
+Note that this supervector does not span the whole superspace: only a couple
+terms from the $\bar\theta_2,\theta_2$ sector are present, since the rest are
+unnecessary for our representation. With this supervector so defined, the
+replicated count of stationary points with energy $E$, trace $\mu$, and
+smallest eigenvalue $\lambda^*$ can be written as
+\begin{widetext}
+\begin{equation}
+  \begin{aligned}
+    \mathcal N_H(E,\mu,\lambda^*)^n
+    &=\lim_{\beta\to\infty}\int\prod_{a=1}^nd\pmb\omega_a\lim_{m_a\to0}\prod_{\alpha=1}^{m_a}d\pmb\phi_a^\alpha
+    \exp\left\{
+      \delta^{\alpha1}N(\hat\beta_aE+\hat\lambda_a\lambda^*)
+      +\int d1\,d2\,B_a^\alpha(1,2)L(\pmb\phi_a^\alpha(1,2),\pmb\omega_a)
+    \right\}
+  \end{aligned}
+\end{equation}
+Here we have also defined the operator
+\begin{equation}
+  B_a^\alpha(1,2)=\delta^{\alpha1}\bar\theta_2\theta_2
+        (1-\hat\beta_a\bar\theta_1\theta_1)
+        -\delta^{\alpha1}\hat\lambda_a-\beta
+\end{equation}
+which encodes various aspects of the complexity problem, and the measures
+\begin{align}
+  d\pmb\phi_a^\alpha
+  &=\left[
+    d\mathbf x_a\,\delta\big(\mathbf g(\mathbf x_a)\big)\,
+    \frac{d\hat{\mathbf x}_a}{(2\pi)^N}\,
+    d\pmb\eta_a\,d\bar{\pmb\eta}_a\,
+    \delta^{\alpha1}+(1-\delta^{\alpha1})
+  \right]\,
+  d\mathbf s_a^\alpha\,\delta(\|\mathbf s_a^\alpha\|^2-N)\,
+  \delta\big((\mathbf s_a^\alpha)^T\partial\mathbf g(\mathbf x_a)\big)
+  \\
+  d\pmb\omega_a&=\prod_{i=1}^rd\omega_{ai}\,\delta\big(N\mu-\omega_{ai}\partial\partial g_i(\mathbf x_a)\big)
+\end{align}
+\end{widetext}
+With this way of writing the replicated count, the problem of marginal
+complexity temporarily takes the schematic form of an equilibrium calculation
+with configurations $\pmb\phi$, inverse temperature $B$, and energy $L$. This
+makes the intermediate pieces of the calculation dramatically simpler. Of
+course the complexity of the underlying problem is not banished: near the end
+of the calculation, terms involving the superspace must be expanded.
+
 \section{Examples}
 
 \subsection{Spherical spin glasses}
@@ -1031,10 +1116,14 @@ the Grassmann indices $\bar\theta_1,\theta_1,\bar\theta_2,\theta_2$. Such a supe
 \begin{equation}
   (M\pmb\phi)(1)=\int d1\,M(1,2)\pmb\phi(2)
 \end{equation}
+The identity supermatrix is given by
+\begin{equation}
+  \delta(1,2)=(\bar\theta_1-\bar\theta_2)(\theta_1-\theta_2)I
+\end{equation}
 Integrals involving superfields contracted into such operators result in schematically familiar expressions, like that of the standard Gaussian:
 \begin{equation}
   \int d\pmb\phi\,e^{\int\,d1\,d2\,\pmb\phi(1)^TM(1,2)\pmb\phi(2)}
-  =(\operatorname{sdet}M)^{-N/2}
+  =(\operatorname{sdet}M)^{-1/2}
 \end{equation}
 where the usual role of the determinant is replaced by the superdeterminant.
 The superdeterminant can be defined using the ordinary determinant by writing a
@@ -1056,12 +1145,15 @@ block representation of $M$ in analogy to the matrix form of an operator in quan
     A & B \\ C & D
   \end{bmatrix}
 \end{equation}
-Then the superdeterminant of $M$ is given by
+where each of the blocks is a $2N\times 2N$ real matrix. Then the
+superdeterminant of $M$ is given by
 \begin{equation}
   \operatorname{sdet}M=\det(A-BD^{-1}C)\det(D)^{-1}
 \end{equation}
 which is the same for the normal equation for the determinant of a block matrix
-save for the inverse of $\det D$.
+save for the inverse of $\det D$. The same method can be used to calculate the
+superdeterminant in arbitrary superspaces, where for $\mathbb R^{N|2D}$ each
+basis has $2^{2D-1}$ elements. For instance, for $\mathbb R^{N|4}$ we have $\mathbf e(1,2)=\{1,\bar\theta_1\theta_1,\bar\theta_2\theta_2,\bar\theta_1\theta_2,\bar\theta_2\theta_1,\bar\theta_1\bar\theta_2,\theta_1\theta_2,\bar\theta_1\theta_1\bar\theta_2\theta_2\}$ and $\mathbf f(1,2)=\{\bar\theta_1,\theta_1,\bar\theta_2,\theta_2,\bar\theta_1\theta_1\bar\theta_2,\bar\theta_2\theta_2\theta_1,\bar\theta_1\theta_1\theta_2,\bar\theta_2\theta_2\theta_1\}$.
 
 \section{Complexity of dominant optima in the least-squares problem}
 \label{sec:dominant.complexity}
@@ -1122,8 +1214,6 @@ and $\hat v$ with kernel
     i\delta_{ab}\delta(1,2) & f\left(\frac{\pmb\phi_a(1)^T\pmb\phi_b(2)}N\right)
   \end{bmatrix}
 \end{equation}
-where $\delta(1,2)=(\bar\theta_1-\bar\theta_2)(\theta_1-\theta_2)$ is the
-identity operator for convolutions with $d1$ or $d2$.
 Making the $M$ independent Gaussian integrals, we therefore have
 \begin{equation}
   \begin{aligned}
-- 
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