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@@ -0,0 +1,601 @@ +\documentclass[aspectratio=169,dvipsnames]{beamer} + +\setbeamerfont{title}{family=\bf} +\setbeamerfont{frametitle}{family=\bf} +\setbeamerfont{normal text}{family=\rm} +\setbeamertemplate{navigation symbols}{} +\setbeamercolor{titlelike}{parent=structure,fg=cyan} + +\usepackage{enumitem} +\usepackage[utf8]{inputenc} +\usepackage[T1]{fontenc} +\usepackage{pifont} +\usepackage{graphicx} +\usepackage{xcolor} +\usepackage{tikz} + +\definecolor{ictpblue}{HTML}{0471b9} +\definecolor{ictpgreen}{HTML}{0c8636} + +\definecolor{mb}{HTML}{5e81b5} +\definecolor{my}{HTML}{e19c24} +\definecolor{mg}{HTML}{8fb032} +\definecolor{mr}{HTML}{eb6235} + +\setbeamercolor{titlelike}{parent=structure,fg=ictpblue} +\setbeamercolor{itemize item}{fg=ictpblue} + +\usepackage[ + style=phys, + eprint=true, + maxnames = 100, + terseinits=true +]{biblatex} + + +\addbibresource{zif.bib} + +\title{ + Understanding the flat parts of random landscapes +} +\author{\textbf{Jaron Kent-Dobias}} +\date{9 September 2025} + +\begin{document} + +\begin{frame} + \maketitle + + \vspace{-8pc} + \begin{minipage}[c]{10pc} + \centering + \includegraphics[height=6pc]{figs/ift-unesp.png} + + \vspace{2em} + + \includegraphics[height=1.5pc]{figs/Simons-Foundation-Logo_blue.png} + \end{minipage} + \hfill\begin{minipage}[c]{10pc} + \centering + \includegraphics[height=6pc]{figs/logo-ictp-saifr.jpg} + + \vspace{2em} + + \includegraphics[height=1.5pc]{figs/fapesp.png} + \end{minipage} + \vspace{2pc} +\end{frame} + +\begin{frame} + \frametitle{How to count: Kac--Rice} + + Number of stationary points with $\nabla H(\pmb x)=0$ 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| + \end{align*} + Note absolute value of the determinant: want to account for curvature but not add $-1$ + + \bigskip + + 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)} + \end{align*} + + \emph{How do we condition on marginal minima?} +\end{frame} + +\begin{frame} + \frametitle{Conditioning on the type of minimum: the spherical models} + \begin{columns} + \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 + \[ + \mu=\frac1N\operatorname{Tr}\operatorname{Hess} H(\pmb x) + \] + + \bigskip + + Condition on marginal minima by inserting + \[ + \delta\big(\operatorname{Tr}\operatorname{Hess}H(\pmb x)-N\mu_\text{m}\big) + \] + + \end{column} + \begin{column}{0.5\textwidth} + \begin{overlayarea}{\textwidth}{14.5em} + \only<1-2>{\includegraphics[width=\columnwidth]{figs/mu_0.75.pdf}\\\hphantom{ello}\includegraphics[width=0.8\columnwidth]{figs/land_0.75.pdf}} + \only<3>{\includegraphics[width=\columnwidth]{figs/mu_1.5.pdf}\\\hphantom{ello}\includegraphics[width=0.8\columnwidth]{figs/land_1.5.pdf}} + \only<4>{\includegraphics[width=\columnwidth]{figs/mu_2.25.pdf}\\\hphantom{ello}\includegraphics[width=0.8\columnwidth]{figs/land_2.25.pdf}} + \only<5>{\includegraphics[width=\columnwidth]{figs/mu_3.5.pdf}\\\hphantom{ello}\includegraphics[width=0.8\columnwidth]{figs/land_3.5.pdf}} + \only<6>{\includegraphics[width=\columnwidth]{figs/mu_2.pdf}\\\hphantom{ello}\includegraphics[width=0.8\columnwidth]{figs/land_2.pdf}} + \end{overlayarea} + \end{column} + \end{columns} +\end{frame} + +\begin{frame} + \frametitle{Less simple mean-field models} + \framesubtitle{The mixed spherical spin glasses} + + \begin{columns} + \begin{column}{0.41\textwidth} + \alert<1-2>{Pure: all procedures reach $E_\text{th}$} + + \medskip + + \alert<3-4>{Mixed: different procedures reach different energies} + + \medskip + + Dynamical endpoints bounded by complexity of marginal minima + + \bigskip + + \tiny + \fullcite{Folena_2020_Rethinking} + + \smallskip + + \fullcite{Folena_2021_Gradient} + + \smallskip + + \fullcite{Kent-Dobias_2023_How} + \end{column} + \begin{column}{0.59\textwidth} + \begin{overlayarea}{\columnwidth}{0.8\textheight} + \only<1>{% + \includegraphics[width=0.8\columnwidth]{figs/complexity_3_marginal.pdf} + }% + \only<2>{% + \bigskip + + \begin{minipage}[b]{0.83\columnwidth} + \centering + \includegraphics[width=0.18\columnwidth]{figs/land_0.75.pdf} + \includegraphics[width=0.18\columnwidth]{figs/land_0.75.pdf} + \includegraphics[width=0.18\columnwidth]{figs/land_0.75.pdf} + \includegraphics[width=0.18\columnwidth]{figs/land_0.75.pdf} + \includegraphics[width=0.18\columnwidth]{figs/land_0.75.pdf} + + \includegraphics[width=0.18\columnwidth]{figs/land_1.5.pdf} + \includegraphics[width=0.18\columnwidth]{figs/land_1.5.pdf} + \includegraphics[width=0.18\columnwidth]{figs/land_1.5.pdf} + \includegraphics[width=0.18\columnwidth]{figs/land_1.5.pdf} + + \includegraphics[width=0.18\columnwidth]{figs/land_2.pdf} + \includegraphics[width=0.18\columnwidth]{figs/land_2.pdf} + \includegraphics[width=0.18\columnwidth]{figs/land_2.pdf} + + \includegraphics[width=0.18\columnwidth]{figs/land_2.25.pdf} + \includegraphics[width=0.18\columnwidth]{figs/land_2.25.pdf} + + \includegraphics[width=0.18\columnwidth]{figs/land_3.5.pdf} + \end{minipage} + \raisebox{0.75em}{\includegraphics[width=0.15\columnwidth]{figs/energy_arrow.pdf}} + } + \only<3>{% + \includegraphics[width=0.8\columnwidth]{figs/complexity_34_marginal.pdf} + }% + \only<4>{% + \bigskip + + \begin{minipage}[b]{0.83\columnwidth} + \centering + \includegraphics[width=0.18\columnwidth]{figs/land_0.75.pdf} + \includegraphics[width=0.18\columnwidth]{figs/land_0.75.pdf} + \includegraphics[width=0.18\columnwidth]{figs/land_0.75.pdf} + \includegraphics[width=0.18\columnwidth]{figs/land_1.5.pdf} + \includegraphics[width=0.18\columnwidth]{figs/land_1.5.pdf} + + \includegraphics[width=0.18\columnwidth]{figs/land_0.75.pdf} + \includegraphics[width=0.18\columnwidth]{figs/land_1.5.pdf} + \includegraphics[width=0.18\columnwidth]{figs/land_1.5.pdf} + \includegraphics[width=0.18\columnwidth]{figs/land_2.pdf} + + \includegraphics[width=0.18\columnwidth]{figs/land_1.5.pdf} + \includegraphics[width=0.18\columnwidth]{figs/land_2.pdf} + \includegraphics[width=0.18\columnwidth]{figs/land_2.25.pdf} + + \includegraphics[width=0.18\columnwidth]{figs/land_2.25.pdf} + \includegraphics[width=0.18\columnwidth]{figs/land_3.5.pdf} + + \includegraphics[width=0.18\columnwidth]{figs/land_2.25.pdf} + \end{minipage} + \raisebox{0.75em}{\includegraphics[width=0.15\columnwidth]{figs/energy_arrow_o1.pdf}} + }% + \end{overlayarea} + \end{column} + \end{columns} +\end{frame} + +\begin{frame} + \frametitle{Marginal minima in generic models} + \begin{columns} + \begin{column}{0.5\textwidth} + \alert<1>{In spherical spin glasses, all points have the same Hessian spectral density: semicircle with radius $\mu_\text m$, but different shifts} + + \bigskip + + In generic models, spectral density depends on stationarity, energy, etc! + + \bigskip + + \alert<2>{Example: multi-spherical model + \[ + H(\{\boldsymbol s_1,\boldsymbol s_2\}) + =H_p^{(1)}(\boldsymbol s_1)+H_p^{(2)}(\boldsymbol s_2)+\epsilon\boldsymbol s_1\cdot\boldsymbol s_2 + \]} + \hspace{-0.25em}In most models we don't understand the Hessian at all + \end{column} + \begin{column}{0.5\textwidth} + \begin{overprint} + \onslide<1>\includegraphics[width=\columnwidth]{figs/mu_0.75.pdf} + \onslide<2>\includegraphics[width=0.9\columnwidth]{figs/msg_marg_spectra.pdf} + \end{overprint} + \end{column} + \end{columns} +\end{frame} + +\begin{frame} + \frametitle{Towards generic marginal complexity} + \begin{columns} + \begin{column}{0.6\textwidth} + \begin{itemize}[leftmargin=4em] + \item[\color{ictpgreen}\bf Trick \#1:] condition stationary points on \emph{value of smallest eigenvalue} + \end{itemize} + { + \small + \begin{align*} + \hspace{-1em}&\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) + \end{align*} + } + + \bigskip + + Only works if you happen to have the correct shift $\mu$ + + \bigskip + + \tiny + \fullcite{Kent-Dobias_2024_Conditioning} + + \end{column} + \begin{column}{0.4\textwidth} + \begin{overprint} + \onslide<1>\includegraphics[width=\textwidth]{figs/spectrum_less.pdf} + \onslide<2>\includegraphics[width=\textwidth]{figs/spectrum_more.pdf} + \onslide<3>\includegraphics[width=\textwidth]{figs/spectrum_eq.pdf} + \end{overprint} + \end{column} + \end{columns} +\end{frame} + + +\begin{frame} + \frametitle{Towards generic marginal complexity} + \begin{columns} + \begin{column}{0.5\textwidth} + \begin{itemize}[leftmargin=4em] + \item[\color{ictpgreen}\bf Trick \#1:] condition stationary points on \emph{value of smallest eigenvalue} + \end{itemize} + { + \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) + \end{align*} + } + + \medskip + + \begin{itemize}[leftmargin=4em] + \item[\color{ictpgreen}\bf Trick \#2:] adjust $\mu\propto\operatorname{Tr}\operatorname{Hess}H$ until order-$N$ large deviation breaks + \end{itemize} + + \bigskip + + \tiny + \fullcite{Kent-Dobias_2024_Conditioning} + + \end{column} + \begin{column}{0.5\textwidth} + \hspace{0.9em} + \includegraphics[scale=0.8]{figs/spectrum_less.pdf} + \hspace{-1.6em} + \includegraphics[scale=0.8]{figs/spectrum_eq.pdf} + \hspace{-1.6em} + \includegraphics[scale=0.8]{figs/spectrum_more.pdf} + \\ + \includegraphics[scale=0.8]{figs/large_deviation.pdf} + + \vspace{-1em} + + \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)} + \end{align*} + \end{column} + \end{columns} +\end{frame} + +\begin{frame} + \frametitle{Marginal complexity: example} + \begin{columns} + \begin{column}{0.5\textwidth} + Example: model of nonlinear least squares: + \[ + H(\pmb s)=\frac12\sum_{i=1}^{M}V_i(\pmb s)^2 + \] + for spherical $\pmb s\in\mathbb R^N$ and $M=\alpha N$ inhomogeneous random functions $V_i$ + \[ + V_i(\boldsymbol s)=H_2^{(i)}(\boldsymbol s)+H_3^{(i)}(\boldsymbol s) + \] + + \bigskip + + \tiny + \fullcite{Kent-Dobias_2024_Conditioning} + + \smallskip + + \fullcite{Kent-Dobias_2024_Algorithm-independent} + \end{column} + \begin{column}{0.5\textwidth} + \begin{overprint} + \onslide<1>\vspace{4em}\includegraphics[width=\textwidth]{figs/most_squares_complex.pdf} + \onslide<2>\vspace{-1.75em}\includegraphics[width=\textwidth]{figs/most_squares_complexity.pdf} + + \vspace{-1.95em} + + \hspace{-0.25em}\colorbox{white}{\includegraphics[width=\textwidth]{figs/most_squares_stability.pdf}} + \end{overprint} + \end{column} + \end{columns} +\end{frame} + +\begin{frame} + \frametitle{The Euler characteristic \boldmath{$\chi$}} + \begin{columns} + \begin{column}{0.5\textwidth} + The Euler characteristic $\chi(\Omega)$ is a topological invariant of a manifold $\Omega$ + + \medskip + + Defined by tiling the manifold, then taking the alternating sum + \begin{align*} + \chi(\Omega_{\text{cow}}) + &= + {\only<2,5->{\color{Red}}\#_\text{vertices}} + &&\hspace{-1em}- + {\only<3,5->{\color{ictpgreen}}\#_\text{edges}} + &&\hspace{-1em}+ + {\only<4,5->{\color{ictpblue}}\#_\text{faces}} + \\ + &\color{White}\only<2->{\color{Black}}= + {\only<2,5->{\color{Red}}2904} + &&\hspace{-1em}\color{White}\only<3->{\color{Black}}- + {\only<3,5->{\color{ictpgreen}}8706} + &&\hspace{-1em}\color{White}\only<4->{\color{Black}}+ + {\only<4,5->{\color{ictpblue}}5804} \\ + &\color{White}\only<5->{\color{Black}}=2 + \end{align*} + \[ + \color{White}\only<6->{\color{Black}}\chi(\Omega_\text{football}) + ={\only<6->{\color{Red}}60}-{\only<6->{\color{ictpgreen}}90}+{\only<6->{\color{ictpblue}}32}=2 + \] + + \color{White}\only<7>{\color{Black}}Cow is homeomorphic to a sphere + \end{column} + \begin{column}{0.5\textwidth} + \begin{overprint} + \onslide<1,5>\includegraphics[width=\textwidth]{figs/cow.png} + \onslide<2>\includegraphics[width=\textwidth]{figs/cow_vert.png} + \onslide<3>\includegraphics[width=\textwidth]{figs/cow_edge.png} + \onslide<4>\includegraphics[width=\textwidth]{figs/cow_face.png} + \onslide<6->\hspace{2em}\includegraphics{figs/Football_Pallo_valmiina-cropped.jpg} + \end{overprint} + \end{column} + \end{columns} +\end{frame} + +\begin{frame} + \frametitle{The Euler characteristic \boldmath{$\chi$}} + \framesubtitle{Characteristics of the characteristic} + + \begin{columns} + \begin{column}{0.5\textwidth} + For closed, connected 2-dimensional manifolds, related to genus $g$ by + $\chi=2-2g$ + + \medskip + + General properties: + \vspace{-0.5em} + \[ + \chi(\Omega)=0 \text{ for odd-dimensional $\Omega$} + \] + \vspace{-1.6em} + \[ + \chi(S^D)=2\text{ for even }D + \] + \[ + \chi(\Omega_1\sqcup\Omega_2)=\chi(\Omega_1)+\chi(\Omega_2) + \] + \[ + \chi(\Omega_1\times\Omega_2)=\chi(\Omega_1)\times\chi(\Omega_2) + \] + + \smallskip + + Examples: + \vspace{-0.5em} + \[\chi(M\text{ even-$D$ spheres})=2M\] + \vspace{-1.6em} + \[\chi(S^1\times\text{anything})=0\] + \end{column} + \begin{column}{0.5\textwidth} + \includegraphics[width=\textwidth]{figs/genus.png} + \end{column} + \end{columns} +\end{frame} + +\begin{frame} + \frametitle{The Euler characteristic \boldmath{$\chi$}} + \framesubtitle{Computing the Euler characteristic} + \begin{columns} + \begin{column}{0.5\textwidth} + Morse theory: gradient flow on an arbitrary ``height'' function $h$ makes a complex + \begin{align*} + \chi(\Omega) + &= + {\only<2,5>{\color{Red}}\#_\text{vertices}} + - + {\only<3,5>{\color{ictpgreen}}\#_\text{edges}} + + + {\only<4,5>{\color{ictpblue}}\#_\text{faces}} + +\cdots \\ + &= + {\only<6>{\color{ictpblue}}\#_\text{index 0}} + - + {\only<6>{\color{ictpgreen}}\#_\text{index 1}} + + + {\only<6>{\color{Red}}\#_\text{index 2}} + +\cdots \\ + &=\sum_{i=0}^D(-1)^i\#_\text{index i} + \end{align*} + \end{column} + \begin{column}{0.5\textwidth} + \begin{overprint} + \onslide<1>\includegraphics[width=\textwidth]{figs/other_sphere.png} + \onslide<2>\includegraphics[width=\textwidth]{figs/other_sphere_vert.png} + \onslide<3>\includegraphics[width=\textwidth]{figs/other_sphere_edge.png} + \onslide<4>\includegraphics[width=\textwidth]{figs/other_sphere_face.png} + \onslide<5>\includegraphics[width=\textwidth]{figs/other_sphere_all.png} + \onslide<6>\includegraphics[width=\textwidth]{figs/other_sphere_crit.png} + \end{overprint} + \end{column} + \end{columns} +\end{frame} + +\begin{frame} + \frametitle{The Euler characteristic \boldmath{$\chi$}} + \framesubtitle{Computing the Euler characteristic} + + \begin{columns} + \begin{column}{0.6\textwidth} + \[ + \Omega=\left\{ + \pmb a\in\mathbb R^N\mid\|\pmb a\|^2=N, V_0=\hat f(J^i\mid\pmb a)\;\forall\;1\leq i\leq M + \right\} + \] + Pick whatever height function $h:\Omega\to\mathbb R$ you like: $h(\pmb a)=\frac1N\pmb + a_0\cdot\pmb a$ for arbitrary $\pmb a_0$ + \[ + \#_{\substack{\text{critical}\\\text{points}}} + =\int_\Omega d\pmb x\, + \delta\big(\nabla h(\pmb x)\big)\, + \big|\det\operatorname{Hess}h(\pmb x)\big| + \] + \[ + \hspace{-1em}\operatorname{sgn}\big(\det\operatorname{Hess}(\pmb x)\big) + = + \operatorname{sgn}\left(\prod_{i=1}^D\lambda_i\right) + =(-1)^{\text{index}} + \] + \[ + \chi(\Omega) + =\sum_{i=0}^D(-1)^i\#_\text{index i} + =\int_\Omega d\pmb x\,\delta\big(\nabla h(\pmb x)\big)\,\det\operatorname{Hess}h(\pmb x) + \] + \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} +\end{frame} + +\begin{frame} + \frametitle{A simple model of nonlinear least squares} + \framesubtitle{Results} + \begin{columns} + \begin{column}{0.5\textwidth} + $M$ data points, $N$ parameters, $\alpha=M/N$ + \[ + V_0=\hat f(J\mid \pmb a) + =\sum_{i_1=1}^N\cdots\sum_{i_p=1}^NJ_{i_1,\ldots,i_p}a_{i_1}\cdots a_{i_p} + \] + Simplest case: $p=1$, $\Omega$ is intersection of random hyperplanes with the sphere + + \medskip + + Results in $\chi(\Omega)=2$ or $\chi(\Omega)=0$ depending on whether solutions exist + + \medskip + + \hspace{2em}\includegraphics[width=0.33\textwidth]{figs/connected.pdf} + \hfill + \includegraphics[width=0.33\textwidth]{figs/gone.pdf}\hspace{2em} + \end{column} + \begin{column}{0.5\textwidth} + \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} + \end{overprint} + \end{column} + \end{columns} +\end{frame} + +\begin{frame} + \frametitle{A simple model of nonlinear least squares} + \framesubtitle{Results} + \begin{columns} + \begin{column}{0.5\textwidth} + $M$ data points, $N$ parameters, $\alpha=M/N$ + \[ + V_0=\hat f(J\mid \pmb a) + =\sum_{i_1=1}^N\cdots\sum_{i_p=1}^NJ_{i_1,\ldots,i_p}a_{i_1}\cdots a_{i_p} + \] + + For $p\geq2$, different phases with $|\chi(\Omega)|\gg1$ with varying sign + + \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>\centering\includegraphics[width=0.8\textwidth]{figs/middle.pdf}\\$\chi(\Omega)\ll0$ + \onslide<2>\centering\includegraphics[width=0.8\textwidth]{figs/complex.pdf}\\$\chi(\Omega)\ll0$ + \onslide<3>\centering\includegraphics[width=0.8\textwidth]{figs/shattered.pdf}\\$\chi(\Omega)\gg0$ + \onslide<4>\includegraphics[width=\textwidth]{figs/phases_2.pdf} + \onslide<5>\includegraphics[width=\textwidth]{figs/phases_3.pdf} + \onslide<6>\includegraphics[width=\textwidth]{figs/phases_4.pdf} + \end{overprint} + \end{column} + \end{columns} +\end{frame} + +\end{document} |