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@@ -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} @@ -67,13 +67,94 @@ \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 @@ -367,6 +448,72 @@ \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} \begin{column}{0.5\textwidth} @@ -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} |