From c927abe4379b796ec67cf9cc225833a256076737 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Mon, 1 Sep 2025 08:15:24 -0300
Subject: Many changes

---
 zif.tex | 536 ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++------
 1 file changed, 486 insertions(+), 50 deletions(-)

(limited to 'zif.tex')

diff --git a/zif.tex b/zif.tex
index 323c6c0..396e908 100644
--- a/zif.tex
+++ b/zif.tex
@@ -49,7 +49,7 @@
   \vspace{-8pc}
   \begin{minipage}[c]{10pc}
     \centering
-    \includegraphics[height=6pc]{figs/ift-unesp.png}
+    \includegraphics[height=6pc]{figs/logo-ictp-saifr.jpg}
 
     \vspace{2em}
 
@@ -57,7 +57,7 @@
   \end{minipage}
   \hfill\begin{minipage}[c]{10pc}
     \centering
-    \includegraphics[height=6pc]{figs/logo-ictp-saifr.jpg}
+    \includegraphics[height=6pc]{figs/ift-unesp.png}
 
     \vspace{2em}
 
@@ -66,14 +66,95 @@
   \vspace{2pc}
 \end{frame}
 
+\begin{frame}
+  \frametitle{Understanding the flat parts of random landscapes}
+  \begin{columns}
+    \begin{column}{0.33\textwidth}
+      \textbf{Nearly flat parts}
+
+      \vspace{-0.75em}
+
+      \rule{\columnwidth}{1pt}
+
+      \medskip
+
+      Conditioning stationary point complexity on marginal optima
+
+      \bigskip
+
+      \centering
+      \includegraphics[height=10pc]{figs/msg_marg_spectra.pdf}
+
+    \end{column}
+    \begin{column}{0.66\textwidth}
+      \textbf{Really flat parts}
+
+      \vspace{-0.75em}
+
+      \rule{\columnwidth}{1pt}
+
+      \bigskip
+
+      \begin{minipage}{0.49\columnwidth}
+        \raggedright
+        Topology of solution manifolds via the average Euler characteristic
+
+        \bigskip
+
+        \centering
+        \includegraphics[height=10pc]{figs/function_2.png}
+      \end{minipage}
+      \hfill
+      \begin{minipage}{0.49\columnwidth}
+        \raggedright
+        Geometry of solution sets via the statistics of wedged and inscribed spheres
+
+        \bigskip
+
+        \centering
+        \includegraphics[height=10pc]{figs/max_size_zoom.pdf}
+      \end{minipage}
+    \end{column}
+  \end{columns}
+\end{frame}
+
+\begin{frame}
+  \frametitle{Marginal complexity}
+  \begin{columns}
+    \begin{column}{0.5\textwidth}
+      Statistics of local optima imagined to govern behavior in rugged optimisation
+
+      \bigskip
+
+      High-dimensional landscapes differ from cartoons in important ways
+
+      \bigskip
+
+      Generic rugged landscapes covered mostly by basins attached to marginal optima
+
+      \bigskip
+
+      Understanding marginal optima is more important for typical dynamics than understanding typical optima
+    \end{column}
+    \begin{column}{0.5\textwidth}
+      \begin{overprint}
+        \onslide<1>\includegraphics[width=\textwidth]{figs/complex_landscapes.png}\centering\\\tiny C Cammarota
+        \onslide<2>\includegraphics[width=\textwidth]{figs/fake_landscape.png}\\\tiny S Martiniani, YouTube
+        \onslide<3>\includegraphics[width=\textwidth]{figs/real_landscape.png}\smallskip\\\tiny\fullcite{Suryadevara_2024_The}
+        \onslide<4>\includegraphics[width=\textwidth]{figs/eigenvalue_convergence.pdf}\smallskip\\\tiny\fullcite{Kent-Dobias_2024_Algorithm-independent}
+      \end{overprint}
+    \end{column}
+  \end{columns}
+\end{frame}
+
 \begin{frame}
   \frametitle{How to count: Kac--Rice}
 
-      Number of stationary points with $\nabla H(\pmb x)=0$ given by integral
+      Number of stationary points with $\nabla H(\boldsymbol x)=0$ given by integral
       over Kac--Rice measure
       \begin{align*}
         \#_\text{points}
-        &=\int_\Omega d\pmb x\,\delta\big(\nabla H(\pmb x)\big)\,\big|\det\operatorname{Hess}H(\pmb x)\big|
+        &=\int_\Omega d\boldsymbol x\,\delta\big(\nabla H(\boldsymbol x)\big)\,\big|\det\operatorname{Hess}H(\boldsymbol x)\big|
       \end{align*}
       Note absolute value of the determinant: want to account for curvature but not add $-1$
 
@@ -82,8 +163,8 @@
       Can specify properties of points by inserting $\delta$-functions:
       \begin{align*}
         \#_\text{points}\alert<2>{(E)}
-        &=\int_\Omega d\pmb x\,\delta\big(\nabla H(\pmb x)\big)\,\big|\det\operatorname{Hess}H(\pmb x)\big|
-        \alert<2>{\,\delta\big(H(\pmb x)-NE\big)}
+        &=\int_\Omega d\boldsymbol x\,\delta\big(\nabla H(\boldsymbol x)\big)\,\big|\det\operatorname{Hess}H(\boldsymbol x)\big|
+        \alert<2>{\,\delta\big(H(\boldsymbol x)-NE\big)}
       \end{align*}
 
       \emph{How do we condition on marginal minima?}
@@ -95,14 +176,14 @@
     \begin{column}{0.5\textwidth}
       In spherical spin glasses, all points have the same Hessian spectral density: semicircle with radius $\mu_\text m$, but different shifts
       \[
-        \mu=\frac1N\operatorname{Tr}\operatorname{Hess} H(\pmb x)
+        \mu=\frac1N\operatorname{Tr}\operatorname{Hess} H(\boldsymbol x)
       \]
 
       \bigskip
 
       Condition on marginal minima by inserting
       \[
-        \delta\big(\operatorname{Tr}\operatorname{Hess}H(\pmb x)-N\mu_\text{m}\big)
+        \delta\big(\operatorname{Tr}\operatorname{Hess}H(\boldsymbol x)-N\mu_\text{m}\big)
       \]
 
     \end{column}
@@ -228,8 +309,8 @@
 
       \alert<2>{Example: multi-spherical model
       \[
-        H(\{\boldsymbol s_1,\boldsymbol s_2\})
-        =H_p^{(1)}(\boldsymbol s_1)+H_p^{(2)}(\boldsymbol s_2)+\epsilon\boldsymbol s_1\cdot\boldsymbol s_2
+        H(\{\boldsymbol x_1,\boldsymbol x_2\})
+        =H_p^{(1)}(\boldsymbol x_1)+H_p^{(2)}(\boldsymbol x_2)+\epsilon\boldsymbol x_1\cdot\boldsymbol x_2
       \]}
       \hspace{-0.25em}In most models we don't understand the Hessian at all
     \end{column}
@@ -252,11 +333,11 @@
       {
       \small
       \begin{align*}
-        \hspace{-1em}&\delta(\lambda_\text{min}(A)) \\
+        \hspace{0em}&\delta(\lambda_\text{min}(A)) \\
         &=\lim_{\beta\to\infty}\int
-          \frac{d\pmb s\,\delta(N-\|\pmb s\|^2)e^{-\beta\pmb s^TA\pmb s}}
-            {\int d\pmb s'\,\delta(N-\|\pmb s'\|^2)e^{-\beta\pmb s'^TA\pmb s'}}
-        \delta\left(\frac{\pmb s^TA\pmb s}N\right)
+          \frac{d\boldsymbol s\,\delta(N-\|\boldsymbol s\|^2)e^{-\beta\boldsymbol s^TA\boldsymbol s}}
+            {\int d\boldsymbol s'\,\delta(N-\|\boldsymbol s'\|^2)e^{-\beta\boldsymbol s'^TA\boldsymbol s'}}
+        \delta\left(\frac{\boldsymbol s^TA\boldsymbol s}N\right)
       \end{align*}
       }
 
@@ -291,11 +372,11 @@
       {
       \small
       \begin{align*}
-        \hspace{-3em}&\delta(\lambda_\text{min}(A)) \\
-        \hspace{-3em}&=\lim_{\beta\to\infty}\int
-          \frac{d\pmb s\,\delta(N-\|\pmb s\|^2)e^{-\beta\pmb s^TA\pmb s}}
-            {\int d\pmb s'\,\delta(N-\|\pmb s'\|^2)e^{-\beta\pmb s'^TA\pmb s'}}
-        \delta\left(\frac{\pmb s^TA\pmb s}N\right)
+        \hspace{0em}&\delta(\lambda_\text{min}(A)) \\
+        \hspace{0em}&=\lim_{\beta\to\infty}\int
+          \frac{d\boldsymbol s\,\delta(N-\|\boldsymbol s\|^2)e^{-\beta\boldsymbol s^TA\boldsymbol s}}
+            {\int d\boldsymbol s'\,\delta(N-\|\boldsymbol s'\|^2)e^{-\beta\boldsymbol s'^TA\boldsymbol s'}}
+        \delta\left(\frac{\boldsymbol s^TA\boldsymbol s}N\right)
       \end{align*}
       }
 
@@ -325,7 +406,7 @@
 
       \small
       \begin{align*}
-        \hspace{-0.5em}G_0(\mu)=\frac 1N\log\Big\langle\delta\big(\lambda_\text{min}(A-\lambda\mu)\big)\Big\rangle_{A\in\text{GOE}(N)}
+        \hspace{-0.5em}G_0(\mu)=\frac 1N\log\Big\langle\delta\big(\lambda_\text{min}(A-\mu I)\big)\Big\rangle_{A\in\text{GOE}(N)}
       \end{align*}
     \end{column}
   \end{columns}
@@ -337,11 +418,11 @@
     \begin{column}{0.5\textwidth}
       Example: model of nonlinear least squares:
       \[
-        H(\pmb s)=\frac12\sum_{i=1}^{M}V_i(\pmb s)^2
+        H(\boldsymbol x)=\frac12\sum_{i=1}^{M}V_i(\boldsymbol x)^2
       \]
-      for spherical $\pmb s\in\mathbb R^N$ and $M=\alpha N$ inhomogeneous random functions $V_i$
+      for spherical $\boldsymbol x\in\mathbb R^N$ and $M=\alpha N$ inhomogeneous random functions $V_i$
       \[
-        V_i(\boldsymbol s)=H_2^{(i)}(\boldsymbol s)+H_3^{(i)}(\boldsymbol s)
+        V_i(\boldsymbol x)=H_2^{(i)}(\boldsymbol x)+H_3^{(i)}(\boldsymbol x)
       \]
 
       \bigskip
@@ -366,6 +447,72 @@
   \end{columns}
 \end{frame}
 
+\begin{frame}
+  \frametitle{Really flat parts: zero-cost sets}
+  \begin{columns}
+    \begin{column}{0.5\textwidth}
+      Diverse optimization problems exactly fit data or satisfy constraints, reach zero cost
+
+      \bigskip
+
+      Stationary point technology useless for describing zero-cost sets
+
+      \bigskip
+
+      Mostly understood by pointwise sampling or interpolation between pairs of
+      sampled points
+    \end{column}
+    \begin{column}{0.5\textwidth}
+      \begin{overprint}
+        \onslide<1>\includegraphics[width=\textwidth]{figs/neural_landscape_1.png}\smallskip\\\tiny\fullcite{Liu_2022_Loss}
+        \onslide<2>\includegraphics[width=\textwidth]{figs/neural_landscape_2.png}\smallskip\\\tiny\fullcite{Liu_2022_Loss}
+        \onslide<3>\includegraphics[width=\textwidth]{figs/draxler_2018.png}\smallskip\\\tiny\fullcite{Draxler_2018_Essentially}
+      \end{overprint}
+    \end{column}
+  \end{columns}
+\end{frame}
+
+\begin{frame}
+  \frametitle{Zero cost manifolds vs.\ zero cost non-manifolds}
+  \begin{columns}
+    \begin{column}{0.5\textwidth}
+      \textbf{Manifolds}
+
+      \medskip
+
+      Defined by sets of \emph{analytic equalities}
+
+      \medskip
+
+      Network $f_{\boldsymbol x}:\mathbb R^P\to\mathbb R$, $\boldsymbol x\in S^{N-1}\subset\mathbb R^N$, with analytic activations (e.g.\ $\tanh$):
+      \[
+        y^\mu=f_{\boldsymbol x}(\boldsymbol a^\mu)\qquad\mu=1,\ldots,M
+      \]
+
+      \smallskip
+
+      \textbf{(Usual) Non-manifolds}
+
+      \medskip
+
+      Defined by sets of \emph{inequalities}
+
+      \medskip
+
+      Jamming of disks in $d$ dimensions:
+      \[
+        \|\boldsymbol x^i-\boldsymbol x^j\|\geq 2R\qquad i,j=1,\ldots,P
+      \]
+    \end{column}
+    \begin{column}{0.5\textwidth}
+      \begin{overprint}
+        \onslide<1>\includegraphics[width=\textwidth]{figs/intersections_3.pdf}
+        \onslide<2>\includegraphics[width=\textwidth]{figs/nothing_zoom.pdf}
+      \end{overprint}
+    \end{column}
+  \end{columns}
+\end{frame}
+
 \begin{frame}
   \frametitle{The Euler characteristic \boldmath{$\chi$}}
   \begin{columns}
@@ -476,6 +623,18 @@
         +\cdots \\
         &=\sum_{i=0}^D(-1)^i\#_\text{index i}
       \end{align*}
+      \[
+        \hspace{-1em}\operatorname{sgn}\big(\det\operatorname{Hess}(\pmb x)\big)
+        =
+        \operatorname{sgn}\left(\prod_{i=1}^D\lambda_i\right)
+        =(-1)^{\text{index}}
+      \]
+      \[
+        \chi(\Omega)
+        =\int_\Omega d\pmb x\,\delta\big(\nabla h(\pmb x)\big)
+        \,\det\operatorname{Hess}h(\pmb x)
+      \]
+      \emph{Kac--Rice without the absolute value!}
     \end{column}
     \begin{column}{0.5\textwidth}
       \begin{overprint}
@@ -491,41 +650,62 @@
 \end{frame}
 
 \begin{frame}
-  \frametitle{The Euler characteristic \boldmath{$\chi$}}
-  \framesubtitle{Computing the Euler characteristic}
+  \frametitle{Computing the Euler characteristic}
 
   \begin{columns}
     \begin{column}{0.6\textwidth}
+      Pick whatever height function $h:\Omega\to\mathbb R$ you like: $h(\pmb x)=\frac1N\pmb
+      x_0\cdot\pmb x$ for arbitrary $\pmb x_0$.
       \[
-        \Omega=\left\{
-          \pmb a\in\mathbb R^N\mid\|\pmb a\|^2=N, V_0=\hat f(J^i\mid\pmb a)\;\forall\;1\leq i\leq M
-        \right\}
+        \chi(\Omega)
+        =\int_\Omega d\pmb x\,\delta\big(\nabla h(\pmb x)\big)\,\det\operatorname{Hess}h(\pmb x)
       \]
-      Pick whatever height function $h:\Omega\to\mathbb R$ you like: $h(\pmb a)=\frac1N\pmb
-      a_0\cdot\pmb a$ for arbitrary $\pmb a_0$
       \[
-        \#_{\substack{\text{critical}\\\text{points}}}
-        =\int_\Omega d\pmb x\,
-        \delta\big(\nabla h(\pmb x)\big)\,
-        \big|\det\operatorname{Hess}h(\pmb x)\big|
+        \Omega=\{\boldsymbol x\in S^{N-1}\subset\mathbb R^N\mid f_{\boldsymbol x}(\boldsymbol a^\mu)=y^\mu,1\leq\mu\leq M\}
       \]
+
+      \bigskip
+
+      Lagrange multipliers replace differential geometry:
       \[
-        \hspace{-1em}\operatorname{sgn}\big(\det\operatorname{Hess}(\pmb x)\big)
-        =
-        \operatorname{sgn}\left(\prod_{i=1}^D\lambda_i\right)
-        =(-1)^{\text{index}}
+        L(\boldsymbol  x,\boldsymbol \omega)=h(\boldsymbol  x)+\omega^0(\|\boldsymbol  x\|^2-N)+\sum_{\mu=1}^M\omega^\mu(f_{\boldsymbol x}(\boldsymbol a^\mu)-y^\mu)
       \]
+    \end{column}
+    \begin{column}{0.4\textwidth}
+      \begin{overprint}
+        \onslide<1>\includegraphics[width=\textwidth]{figs/function-1.png}
+        \onslide<2>\includegraphics[width=\textwidth]{figs/function-2.png}
+        \onslide<3>\includegraphics[width=\textwidth]{figs/function-3.png}
+      \end{overprint}
+    \end{column}
+  \end{columns}
       \[
         \chi(\Omega)
-        =\sum_{i=0}^D(-1)^i\#_\text{index i}
-        =\int_\Omega d\pmb x\,\delta\big(\nabla h(\pmb x)\big)\,\det\operatorname{Hess}h(\pmb x)
+        =\int_{\mathbb R^{N+M+1}} d\boldsymbol  x\,d\boldsymbol \omega\,\delta\big(\begin{bmatrix}\frac{\partial L}{\partial\boldsymbol  x}&\frac{\partial L}{\partial\boldsymbol \omega}\end{bmatrix}\big)
+        \,\det\begin{bmatrix}\frac{\partial^2L}{\partial\boldsymbol  x^2}&\frac{\partial^2L}{\partial\boldsymbol  x\partial\boldsymbol \omega}\\\frac{\partial^2L}{\partial\boldsymbol  x\partial\boldsymbol \omega}&\frac{\partial^2L}{\partial\boldsymbol \omega^2}\end{bmatrix}
+      \]
+\end{frame}
+
+\begin{frame}
+  \frametitle{Computing the Euler characteristic}
+  \begin{columns}
+    \begin{column}{0.5\textwidth}
+      $M$ data points, $N$ parameters, $\alpha=M/N$, $\boldsymbol J^\mu\in\mathbb R^{N^p}$, $\overline{J^2}=\frac{p!}{2N^p}$
+      \[
+        V_0=f_{\boldsymbol x}(\boldsymbol J^\mu)
+        =\frac1{p!}\sum_{i_1=1}^N\cdots\sum_{i_p=1}^NJ ^\mu_{i_1,\ldots,i_p}x_{i_1}\cdots x_{i_p}
+      \]
+      Average Euler characteristic reduced to integral over $m=\frac1N\boldsymbol x\cdot\boldsymbol x_0$,
+      \[
+        \overline{\chi(\Omega)}
+        =\left(\frac N{2\pi}\right)^{\frac12}
+        \int dm\,g(m)e^{N\mathcal S_\chi(m)}
       \]
     \end{column}
-    \begin{column}{0.4\textwidth}
+    \begin{column}{0.5\textwidth}
       \begin{overprint}
-        \onslide<1>\includegraphics[width=\textwidth]{figs/function_1.png}
-        \onslide<2>\includegraphics[width=\textwidth]{figs/function_2.png}
-        \onslide<3>\includegraphics[width=\textwidth]{figs/function_3.png}
+        \onslide<1>\includegraphics[width=\textwidth]{figs/action_1.pdf}
+        \onslide<2>\includegraphics[width=\textwidth]{figs/action_3.pdf}
       \end{overprint}
     \end{column}
   \end{columns}
@@ -536,10 +716,10 @@
   \framesubtitle{Results}
   \begin{columns}
     \begin{column}{0.5\textwidth}
-      $M$ data points, $N$ parameters, $\alpha=M/N$
+      $M$ data points, $N$ parameters, $\alpha=M/N$, $\boldsymbol J^\mu\in\mathbb R^{N^p}$, $\overline{J^2}=\frac{p!}{2N^p}$
       \[
-        V_0=\hat f(J\mid \pmb a)
-        =\sum_{i_1=1}^N\cdots\sum_{i_p=1}^NJ_{i_1,\ldots,i_p}a_{i_1}\cdots a_{i_p}
+        V_0=f_{\boldsymbol x}(\boldsymbol J^\mu)
+        =\frac1{p!}\sum_{i_1=1}^N\cdots\sum_{i_p=1}^NJ ^\mu_{i_1,\ldots,i_p}x_{i_1}\cdots x_{i_p}
       \]
       Simplest case: $p=1$, $\Omega$ is intersection of random hyperplanes with the sphere
 
@@ -557,9 +737,48 @@
       \begin{overprint}
         \onslide<1>\includegraphics[width=\textwidth]{figs/intersections_1.pdf}
         \onslide<2>\includegraphics[width=\textwidth]{figs/intersections_2.pdf}
-        \onslide<3>\includegraphics[width=\textwidth]{figs/intersections_3.pdf}
-        \onslide<4>\includegraphics[width=\textwidth]{figs/intersections_4.pdf}
-        \onslide<5>\includegraphics[width=\textwidth]{figs/phases_1.pdf}
+        \onslide<3>\includegraphics[width=\textwidth]{figs/action_phase_1.pdf}
+        \onslide<4>\includegraphics[width=\textwidth]{figs/intersections_3.pdf}
+        \onslide<5>\includegraphics[width=\textwidth]{figs/intersections_4.pdf}
+        \onslide<6>\includegraphics[width=\textwidth]{figs/action_phase_2.pdf}
+        \onslide<7>\includegraphics[width=\textwidth]{figs/phases_1.pdf}
+      \end{overprint}
+    \end{column}
+  \end{columns}
+\end{frame}
+
+\begin{frame}
+  \begin{columns}
+    \begin{column}{0.5\textwidth}
+      For $p>1$, new phases possible
+
+      \medskip
+
+      \alert<2>{\textbf{\boldmath{$m^*=0$, $\mathcal S_\chi(m^*)$ complex, $\overline{\chi(\Omega)}\ll0$, local maximum exists}}}
+
+      \medskip
+
+      \alert<3>{\textbf{\boldmath{$m^*=0$, $\mathcal S_\chi(m^*)$ complex, $\overline{\chi(\Omega)}\ll0$, no local maximum}}}
+
+      \medskip
+
+      \alert<4>{\textbf{\boldmath{$m^*=0$, $\mathcal S_\chi(m^*)$ real, $\overline{\chi(\Omega)}\gg0$}}}
+
+      \medskip
+
+      \includegraphics[width=0.23\textwidth]{figs/middle.pdf}
+      \includegraphics[width=0.23\textwidth]{figs/complex.pdf}
+      \includegraphics[width=0.23\textwidth]{figs/shattered.pdf}
+      \includegraphics[width=0.23\textwidth]{figs/gone.pdf}
+    \end{column}
+    \begin{column}{0.5\textwidth}
+      \begin{overprint}
+        \onslide<1-2>\includegraphics[width=\textwidth]{figs/action_phase_3.pdf}
+        \onslide<3>\includegraphics[width=\textwidth]{figs/action_phase_4.pdf}
+        \onslide<4>\includegraphics[width=\textwidth]{figs/action_phase_5.pdf}
+        \onslide<5>\includegraphics[width=\textwidth]{figs/phases_2.pdf}
+        \onslide<6>\includegraphics[width=\textwidth]{figs/phases_3.pdf}
+        \onslide<7>\includegraphics[width=\textwidth]{figs/phases_4.pdf}
       \end{overprint}
     \end{column}
   \end{columns}
@@ -598,4 +817,221 @@
   \end{columns}
 \end{frame}
 
+\begin{frame}
+  \frametitle{Non manifolds: constraint satisfaction}
+  \begin{columns}
+    \begin{column}{0.5\textwidth}
+      Continuous constraint satisfaction problems with $\boldsymbol x\in\Omega\subseteq\mathbb R^N$, with $\Omega$ $D$-dimensional
+      \[
+        h^\mu(\boldsymbol x)\geq0 \qquad 1\leq\mu\leq M
+      \]
+      Spherical perceptron: $M$ patterns $\boldsymbol\xi^\mu\in\mathbb R^N$,
+      \[
+        h^\mu(\boldsymbol x)=\boldsymbol\xi^\mu\cdot\boldsymbol x-\kappa
+      \]
+    \end{column}
+    \begin{column}{0.5\textwidth}
+      \begin{overprint}
+        \onslide<1>\includegraphics[width=\textwidth]{figs/nothing_zoom.pdf}
+        \onslide<2>\includegraphics[width=\textwidth]{figs/fixed_size_zoom_points.pdf}
+        \onslide<3>\includegraphics[width=\textwidth]{figs/max_size_zoom.pdf}
+      \end{overprint}
+    \end{column}
+  \end{columns}
+\end{frame}
+
+\begin{frame}
+  \frametitle{Wedged spheres}
+  \begin{columns}
+    \begin{column}{0.5\textwidth}
+      Sphere of radius $r$ uniquely defined by
+      $h^\mu(\boldsymbol x)=r$ for $D$ constraints,
+      $h^\mu(\boldsymbol x)\geq r$ for other $M-D$ constraints
+      \[
+        \begin{aligned}
+          \#_r
+          =
+          \int_{\mathbb R^D} d\boldsymbol x
+          \sum_{\substack{\sigma\subset[M]\\|\sigma|=D}}
+          \bigg(\prod_{\mu\in[M]\backslash\sigma}\theta\big(h^\mu(\boldsymbol x)-r\big)\bigg)
+          \\
+          \times\bigg(\prod_{\mu\in\sigma}\delta\big(h^\mu(\boldsymbol x)-r\big)\bigg)
+          \\
+          \times\left|
+            \det\begin{bmatrix}
+              \frac\partial{\partial\boldsymbol x}h^{\sigma_1}(\boldsymbol x)
+              &
+              \cdots
+              &
+              \frac\partial{\partial\boldsymbol x}h^{\sigma_D}(\boldsymbol x)
+            \end{bmatrix}
+          \right|
+        \end{aligned}
+      \]
+      With margin, $\#_r(\kappa)=\#_0(r+\kappa)$
+    \end{column}
+    \begin{column}{0.5\textwidth}
+      \begin{overprint}
+        \onslide<1>\includegraphics[width=\textwidth]{figs/nothing_zoom.pdf}
+        \onslide<2>\includegraphics[width=\textwidth]{figs/fixed_size_zoom_r.pdf}
+        \onslide<3>\includegraphics[width=\textwidth]{figs/fixed_size_zoom_0.pdf}
+      \end{overprint}
+    \end{column}
+  \end{columns}
+\end{frame}
+
+\begin{frame}
+  \frametitle{Inscribed spheres}
+  \begin{columns}
+    \begin{column}{0.5\textwidth}
+      Sphere of maximal radius uniquely defined by
+      $h^\mu(\boldsymbol x)=r$ for $D+1$ constraints,
+      $h^\mu(\boldsymbol x)\geq r$ for other $M-D-1$ constraints
+      \[
+        \begin{aligned}
+          \#_\text{insc}
+          =
+          \int_0^\infty dr\int_{\mathbb R^D} d\boldsymbol x
+          \sum_{\substack{\sigma\subset[M]\\|\sigma|=D+1}}
+          \\
+          \bigg(\prod_{\mu\in[M]\backslash\sigma}\theta\big(h^\mu(\boldsymbol x)-r\big)\bigg)
+          \bigg(\prod_{\mu\in\sigma}\delta\big(h^\mu(\boldsymbol x)-r\big)\bigg)
+          \\
+          \times\left|
+            \det\begin{bmatrix}
+              \frac\partial{\partial\boldsymbol x}h^{\sigma_1}(\boldsymbol x)
+              &
+              \cdots
+              &
+              \frac\partial{\partial\boldsymbol x}h^{\sigma_{D+1}}(\boldsymbol x)
+              \\
+              -1 & \cdots & -1
+            \end{bmatrix}
+          \right|
+        \end{aligned}
+      \]
+    \end{column}
+    \begin{column}{0.5\textwidth}
+      \begin{overprint}
+        \onslide<1>\includegraphics[width=\textwidth]{figs/max_size_zoom.pdf}
+      \end{overprint}
+    \end{column}
+  \end{columns}
+\end{frame}
+
+\begin{frame}
+  \frametitle{Treating the determinant}
+  \begin{columns}
+    \begin{column}{0.9\textwidth}
+      \[
+        |\det M|=\sqrt{\det MM^T}
+        =\int\frac{d\boldsymbol s}{(2\pi)^{D/2}}\,d\bar{\boldsymbol\eta}\,d\boldsymbol\eta\,
+        e^{-\frac12\boldsymbol s^TMM^T\boldsymbol s-\bar{\boldsymbol\eta}^TMM^T\boldsymbol\eta}
+      \]
+      \begin{align*}
+        &\#_r
+        =
+        \int\frac{d\boldsymbol x\,d\boldsymbol s}{(2\pi)^{D/2}}d\bar{\boldsymbol\eta}\,d\boldsymbol\eta
+        \sum_{\substack{\sigma\subset[M]\\|\sigma|=D}}
+        \bigg(\prod_{\mu\in[M]\backslash\sigma}\theta\big(h^\mu(\boldsymbol x)-r\big)\bigg)
+        \\
+        &\hspace{8em}
+        \times
+        \bigg(
+          \prod_{\mu\in\sigma}
+          \delta\big(h^\mu(\boldsymbol x)-r\big)
+          e^{
+            -\frac12[\boldsymbol s\cdot\frac\partial{\partial\boldsymbol x}h^\mu(\boldsymbol x)]^2
+            -[\bar{\boldsymbol\eta}\cdot\frac\partial{\partial\boldsymbol x}h^\mu(\boldsymbol x)]
+            [\boldsymbol\eta\cdot\frac\partial{\partial\boldsymbol x}h^\mu(\boldsymbol x)]
+          }
+        \bigg)
+      \end{align*}
+    \end{column}
+  \end{columns}
+\end{frame}
+
+\begin{frame}
+  \frametitle{Treating the sum over subsets}
+  \begin{columns}
+    \begin{column}{0.9\textwidth}
+      \begin{align*}
+        &\#_r
+        =
+        \int\frac{d\boldsymbol x\,d\boldsymbol s}{(2\pi)^{D/2}}d\bar{\boldsymbol\eta}\,d\boldsymbol\eta
+        \\
+        &\times
+        \sum_{\substack{\sigma\subset[M]\\|\sigma|=D}}
+        \bigg(\prod_{\mu\in[M]\backslash\sigma}\theta\big(h^\mu(\boldsymbol x)-r\big)\bigg)
+        \bigg(
+          \prod_{\mu\in\sigma}
+          \delta\big(h^\mu(\boldsymbol x)-r\big)
+          e^{
+            -\frac12[\boldsymbol s\cdot\frac\partial{\partial\boldsymbol x}h^\mu(\boldsymbol x)]^2
+            -[\bar{\boldsymbol\eta}\cdot\frac\partial{\partial\boldsymbol x}h^\mu(\boldsymbol x)]
+            [\boldsymbol\eta\cdot\frac\partial{\partial\boldsymbol x}h^\mu(\boldsymbol x)]
+          }
+        \bigg)
+      \end{align*}
+      \begin{align*}
+        &\#_r
+        =
+        \lim_{\omega\to\infty}\int\frac{d\boldsymbol x\,d\boldsymbol s}{(2\pi)^{D/2}}d\bar{\boldsymbol\eta}\,d\boldsymbol\eta
+        \\
+        &
+        \times\prod_{\mu=1}^M\bigg(\omega\theta\big(h^\mu(\boldsymbol x)-r\big)
+          +
+          \omega^{1-\frac MD}\delta\big(h^\mu(\boldsymbol x)-r\big)
+          e^{
+            -\frac12[\boldsymbol s\cdot\frac\partial{\partial\boldsymbol x}h^\mu(\boldsymbol x)]^2
+            -[\bar{\boldsymbol\eta}\cdot\frac\partial{\partial\boldsymbol x}h^\mu(\boldsymbol x)]
+            [\boldsymbol\eta\cdot\frac\partial{\partial\boldsymbol x}h^\mu(\boldsymbol x)]
+          }
+        \bigg)
+      \end{align*}
+    \end{column}
+  \end{columns}
+\end{frame}
+\begin{frame}
+  \frametitle{Treating the sum over subsets}
+  \begin{columns}
+    \begin{column}{0.9\textwidth}
+      Why does this work? Expand the product:
+      \begin{align*}
+        &\prod_{\mu=1}^M
+        \bigg(
+          \omega\theta\big(h^\mu(\boldsymbol x)-r\big)
+          +
+          \omega^{1-\frac MD}\delta\big(h^\mu(\boldsymbol x)-r\big)
+          e^{
+            -\frac12[\boldsymbol s\cdot\frac\partial{\partial\boldsymbol x}h^\mu(\boldsymbol x)]^2
+            -[\bar{\boldsymbol\eta}\cdot\frac\partial{\partial\boldsymbol x}h^\mu(\boldsymbol x)]
+            [\boldsymbol\eta\cdot\frac\partial{\partial\boldsymbol x}h^\mu(\boldsymbol x)]
+          }
+        \bigg)
+        \\
+        &=\sum_{d=0}^M\omega^{M(1-\frac dD)}
+        \sum_{\substack{\sigma\subset[M]\\|\sigma|=d}}
+        \bigg(\prod_{\mu\in[M]\backslash\sigma}\theta\big(h^\mu(\boldsymbol x)-r\big)\bigg)
+        \\
+        &
+        \hspace{8em}\times\bigg(
+          \prod_{\mu\in\sigma}
+          \delta\big(h^\mu(\boldsymbol x)-r\big)
+          e^{
+            -\frac12[\boldsymbol s\cdot\frac\partial{\partial\boldsymbol x}h^\mu(\boldsymbol x)]^2
+            -[\bar{\boldsymbol\eta}\cdot\frac\partial{\partial\boldsymbol x}h^\mu(\boldsymbol x)]
+            [\boldsymbol\eta\cdot\frac\partial{\partial\boldsymbol x}h^\mu(\boldsymbol x)]
+          }
+        \bigg)
+      \end{align*}
+      Our term is $d=D$, giving $\omega^0$. For the undesired terms:
+      \begin{itemize}
+        \item $d>D$: $\omega$ raised to a negative power, limit kills term
+        \item $d<D$: integral over $\boldsymbol s$, $\bar{\boldsymbol\eta}$, $\boldsymbol\eta$ gives determinant of matrix with rank less than dimension
+      \end{itemize}
+    \end{column}
+  \end{columns}
+\end{frame}
+
 \end{document}
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