From 8331cf653f6ac80ebb9c96c9c844803ce0278d43 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Tue, 11 Feb 2025 15:26:41 -0300
Subject: Split up code, worked on presentation slides.

---
 ictp-saifr_colloquium.tex | 41 ++++++++++++++++++++++++++++++++++++++---
 1 file changed, 38 insertions(+), 3 deletions(-)

(limited to 'ictp-saifr_colloquium.tex')

diff --git a/ictp-saifr_colloquium.tex b/ictp-saifr_colloquium.tex
index dc152f8..1fa3a0b 100644
--- a/ictp-saifr_colloquium.tex
+++ b/ictp-saifr_colloquium.tex
@@ -377,11 +377,11 @@
 
       \medskip
 
-      Overparameterized fits are not unique: $M$ constraints
+      Overparameterized fits are not unique: $\chi^2=0$ gives $M$ constraints
       \[
         0=y_i-\hat f(x_i\mid\pmb a)\qquad\text{for all $1\leq i\leq M$}
       \]
-      plus $N$ unknowns $\pmb a=[a_1,\ldots, a_N]$ gives a manifold of $N-M$ dimensions
+      with $N$ unknowns $\pmb a=[a_1,\ldots, a_N]$ gives a manifold of $N-M$ dimensions
 
       \medskip
 
@@ -393,11 +393,46 @@
 
       \bigskip
 
-      \includegraphics[width=\columnwidth]{figs/fit_overparamfit_poly.pdf}
+      \begin{overprint}
+        \onslide<1>\includegraphics[width=\columnwidth]{figs/fit_overparamfit_poly.pdf}
+        \onslide<2>\includegraphics[width=\columnwidth]{figs/fit_overparamfit_poly_1.pdf}
+        \onslide<3>\includegraphics[width=\columnwidth]{figs/fit_overparamfit_poly_2.pdf}
+        \onslide<4>\includegraphics[width=\columnwidth]{figs/fit_overparamfit_poly_3.pdf}
+        \onslide<5>\includegraphics[width=\columnwidth]{figs/fit_overparamfit_poly_6.pdf}
+        \onslide<6>\includegraphics[width=\columnwidth]{figs/fit_overparamfit_poly_7.pdf}
+        \onslide<7>\includegraphics[width=\columnwidth]{figs/fit_overparamfit_poly_8.pdf}
+        \onslide<8>\includegraphics[width=\columnwidth]{figs/fit_overparamfit_poly_9.pdf}
+      \end{overprint}
     \end{column}
   \end{columns}
 \end{frame}
 
+\begin{frame}
+  \frametitle{Overparamaterized curve fitting and algorithms}
+  \begin{columns}
+    \begin{column}{0.5\textwidth}
+      Overparameterized fits found with gradient descent algorithm: take small
+      steps in the direction $\nabla\chi^2$ until $\|\nabla\chi^2\|<\epsilon$
+
+      \medskip
+
+      Result of descent depends on initial condition: what $\pmb a$ to start with?
+
+      \medskip
+
+      \textbf{Unexpected fact:} gradient descent with small initialization equivalent to unique optimum in \emph{regularized} problem:
+      \[
+        \chi^2_\text{eff}(\pmb a\mid\text{data})=\chi^2(\pmb a\mid\text{data})+\lambda\|\pmb a\|^2
+      \]
+    \end{column}
+    \begin{column}{0.5\textwidth}
+    \end{column}
+  \end{columns}
+\end{frame}
+
+\begin{frame}
+\end{frame}
+
 \begin{frame}
   \frametitle{The Euler characteristic \boldmath{$\chi$}}
   \begin{columns}
-- 
cgit v1.2.3-70-g09d2

