From 1fb1f1c391c01006cfc62f2820178040250fae06 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Wed, 12 Feb 2025 22:26:45 -0300
Subject: Started introducing model

---
 ictp-saifr_colloquium.tex | 29 +++++++++++++++++++++++++++++
 1 file changed, 29 insertions(+)

(limited to 'ictp-saifr_colloquium.tex')

diff --git a/ictp-saifr_colloquium.tex b/ictp-saifr_colloquium.tex
index 1d823a7..e1c78bb 100644
--- a/ictp-saifr_colloquium.tex
+++ b/ictp-saifr_colloquium.tex
@@ -862,6 +862,35 @@
   \end{columns}
 \end{frame}
 
+\begin{frame}
+  \frametitle{A simple model of nonlinear least squares}
+  \begin{columns}
+    \begin{column}{0.9\textwidth}
+      Data: $x_i\mapsto J^i$, $N\times\cdots\times N$ $p$-tensors of Gaussian random numbers\\
+      \hphantom{Data: }$y_i\mapsto V_0$, a constant \\
+
+      \medskip
+
+      Parameters: $\pmb a$ is an $N$-dimensional vector constrained to live on the sphere \\
+
+      \medskip
+
+      Model: $\hat f(J\mid\pmb a)$ is the homogeneous degree-$p$ polynomial
+      \[
+        \hat f(J\mid \pmb a)
+        =\sum_{i_1=1}^N\cdots\sum_{i_p=1}^NJ_{i_1,\ldots,i_p}a_{i_1}\cdots a_{i_p}
+      \]
+      Cost function: the usual
+      \[
+        \chi^2(\pmb a\mid J^1,\ldots,J^M)
+        =\sum_{i=1}^M\left[
+          V_0-\hat f(J^i\mid\pmb a)
+        \right]^2
+      \]
+    \end{column}
+  \end{columns}
+\end{frame}
+
 \begin{frame}
   \frametitle{The Euler characteristic \boldmath{$\chi$}}
   \begin{columns}
-- 
cgit v1.2.3-70-g09d2

