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\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}
