From e4c01a637e4cb782ff9c67d9a42c54bb62a135d6 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Thu, 13 Feb 2025 17:45:27 -0300
Subject: More work, figures

---
 ictp-saifr_colloquium.tex | 105 +++++++++++++++++++++++++++++++++++++++-------
 1 file changed, 91 insertions(+), 14 deletions(-)

(limited to 'ictp-saifr_colloquium.tex')

diff --git a/ictp-saifr_colloquium.tex b/ictp-saifr_colloquium.tex
index bedb05a..d883ed8 100644
--- a/ictp-saifr_colloquium.tex
+++ b/ictp-saifr_colloquium.tex
@@ -433,6 +433,8 @@
       \[
         \chi^2_\text{eff}(\pmb a\mid\text{data})=\chi^2(\pmb a\mid\text{data})+\lambda\|\pmb a\|^2
       \]
+
+      \tiny\fullcite{Neyshabur_2017_Implicit}
     \end{column}
     \begin{column}{0.5\textwidth}
       \begin{overprint}
@@ -818,6 +820,11 @@
       Approximated gradient takes \emph{fewer} steps to find \emph{better} solutions
     \end{column}
     \begin{column}{0.5\textwidth}
+      \includegraphics[width=\textwidth]{figs/beneventano_2023.png}
+
+      \medskip\tiny
+
+      \fullcite{Beneventano_2023_On}
     \end{column}
   \end{columns}
 \end{frame}
@@ -851,17 +858,32 @@
   \frametitle{Machine learning is just \emph{non}linear least squares}
   \begin{columns}
     \begin{column}{0.5\textwidth}
-      Structure and geometry of manifold of "perfect" solutions integral to understanding overparameterized fits
+      Structure and geometry of manifold of ``perfect'' solutions integral to understanding overparameterized fits
 
       \medskip
 
-      \textbf{BUT:} extremely little is known outside of the linear case
+      \textbf{BUT:} little is known outside the linear case
 
       \medskip
 
-      State of the art: sample two points, see if the line between them are also solutions
+      State of the art: sample two points, relax an elastic line between them
+
+      \bigskip
+
+      \textbf{Can we develop better ways to understand the solution space in nonlinear problems?}
     \end{column}
     \begin{column}{0.5\textwidth}
+      \includegraphics[width=0.52\textwidth]{figs/garipov_2018.png}
+      \hfill
+      \includegraphics[width=0.46\textwidth]{figs/draxler_2018.png}
+
+      \tiny\medskip
+
+      \fullcite{Garipov_2018_Loss}
+
+      \medskip
+
+      \fullcite{Draxler_2018_Essentially}
     \end{column}
   \end{columns}
 \end{frame}
@@ -901,15 +923,71 @@
     \begin{column}{0.33\textwidth}
       \textbf{When does \boldmath{$\chi^2$} have any solutions?}
 
-      \textbf{When does \boldmath{$\chi^2$} have suboptimal minima that can interfere with optimization?}
+      \medskip
+
+      \tiny
+      \fullcite{Fyodorov_2019_A}
+
+      \smallskip
+
+      \fullcite{Fyodorov_2020_Counting}
+
+      \smallskip
+
+      \fullcite{Fyodorov_2022_Optimization}
+
+      \smallskip
+
+      \fullcite{Montanari_2024_On}
     \end{column}
     \begin{column}{0.33\textwidth}
+      \textbf{When does \boldmath{$\chi^2$} have suboptimal minima that hinder optimization?}
+
+      \medskip
+
+      \tiny\fullcite{Kent-Dobias_2024_Conditioning}
+
+      \medskip
+
+      \normalsize
       \textbf{How does gradient descent behave?}
 
-      \textbf{How does stochastic gradient descent behave?}
+      \medskip
+
+      \tiny
+      \fullcite{Urbani_2023_A}
+
+      \smallskip
+
+      \fullcite{Kamali_2023_Dynamical}
+
+      \smallskip
+
+      \fullcite{Montanari_2023_Solving}
+
+      \normalsize
     \end{column}
     \begin{column}{0.33\textwidth}
+      \textbf{How does stochastic gradient descent behave?}
+
+      \medskip
+
+      \tiny
+      \fullcite{Kamali_2023_Stochastic}
+
+      \bigskip
+
+      \normalsize
       \textbf{What does the manifold of perfect solutions look like?}
+
+      \medskip
+
+      This presentation!
+
+      \medskip
+
+      \tiny
+      \fullcite{Kent-Dobias_2024_On}
     \end{column}
   \end{columns}
 \end{frame}
@@ -917,7 +995,7 @@
 \begin{frame}
   \frametitle{A simple model of nonlinear least squares}
   \begin{columns}
-    \begin{column}{0.65\textwidth}
+    \begin{column}{0.85\textwidth}
       Solutions form $D=N-M-1$ dimensional manifold:
       \[
         \Omega=\left\{
@@ -937,15 +1015,13 @@
 
       \smallskip
 
-      \tiny\fullcite{Kent-Dobias_2022_How}
+      \tiny\fullcite{Kent-Dobias_2023_How}
 
       \normalsize
       \medskip
 
       \textbf{Today:} how to compute the \emph{Euler characteristic} of the solution manifold
     \end{column}
-    \begin{column}{0.35\textwidth}
-    \end{column}
   \end{columns}
 \end{frame}
 
@@ -1074,7 +1150,8 @@
 \end{frame}
 
 \begin{frame}
-  \frametitle{Computing the Euler characteristic of level sets}
+  \frametitle{The Euler characteristic \boldmath{$\chi$}}
+  \framesubtitle{Computing the Euler characteristic}
 
   \begin{columns}
     \begin{column}{0.6\textwidth}
@@ -1118,7 +1195,7 @@
   \framesubtitle{Results}
   \begin{columns}
     \begin{column}{0.5\textwidth}
-      $\alpha=M/N$, $\alpha<1$ is overparameterized
+      $M$ 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}
@@ -1152,7 +1229,7 @@
   \framesubtitle{Results}
   \begin{columns}
     \begin{column}{0.5\textwidth}
-      $\alpha=M/N$, $\alpha<1$ is overparameterized
+      $M$ 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}
@@ -1226,11 +1303,11 @@
     \begin{column}{0.5\textwidth}
       \textbf{How do these structures interact with dynamics?}
 
-      \medskip
+      \bigskip
 
       \textbf{What structures exist in a problem with a non-random ground truth?}
 
-      \medskip
+      \bigskip
 
       \textbf{How do topological properties of solutions correlate with their quality?}
     \end{column}
-- 
cgit v1.2.3-70-g09d2

