From 4098321621e5b30b55ee4d406cf3c8bb68318914 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Tue, 11 Feb 2025 10:50:02 -0300
Subject: More standardize notation.

---
 ictp-saifr_colloquium.tex | 21 +++++++++++++--------
 1 file changed, 13 insertions(+), 8 deletions(-)

(limited to 'ictp-saifr_colloquium.tex')

diff --git a/ictp-saifr_colloquium.tex b/ictp-saifr_colloquium.tex
index 8611f15..2d621f2 100644
--- a/ictp-saifr_colloquium.tex
+++ b/ictp-saifr_colloquium.tex
@@ -85,19 +85,20 @@
 
   \begin{columns}
     \begin{column}{0.4\textwidth}
-      Pick a basis of $N$ functions $b_1(x), \ldots, b_N(x)$
+      Pick a basis of $N$ functions
+      \[b_1(x), \ldots, b_N(x)\]
 
-      \bigskip
+      \smallskip
 
       Approximate the ground truth
       \[
         \hat f(x\mid a_1,\ldots, a_N)=\sum_{j=1}^Na_jb_j(x)
       \]
 
-      Find $a_1, \ldots, a_N$ minimizing
+      Find $\pmb a=[a_1, \ldots, a_N]$ minimizing
       \[
-        \chi^2
-        =\sum_{i=1}^M\left(y_i-\sum_{j=1}^Na_jb_j(x_i)\right)^2
+        \chi^2(\pmb a\mid\pmb x,\pmb y)
+        =\sum_{i=1}^M\left(y_i-\hat f(x_i\mid\pmb a)\right)^2
       \]
     \end{column}
     \begin{column}{0.6\textwidth}
@@ -193,7 +194,7 @@
     \begin{column}{0.5\textwidth}
       Knowing the ground truth, fit error is
       \[
-        \text{MSE}=\int dx\left(f(x)-\sum_{j=1}^Na_jb_j(x)\right)^2
+        \text{MSE}=\int dx\left(f(x)-\hat f(x\mid\pmb a)\right)^2
       \]
 
       \smallskip
@@ -237,12 +238,16 @@
 
       Fit function is a neural network:
       \[
-        \hat f(\mathbf x\mid B_1,\ldots B_L)=\sigma\left(B_L \sigma\left( B_{L-1}\cdots\sigma\left(B_2\sigma (B_1\mathbf x)\right)\cdots\right)\right)
+        \hat f(\pmb x\mid B_1,\ldots B_L)=\sigma\left(B_L \sigma\left( B_{L-1}\cdots\sigma\left(B_2\sigma (B_1\pmb x)\right)\cdots\right)\right)
       \]
 
       \medskip
 
-      $\chi^2$ is called \emph{training error}
+      $\chi^2(\pmb a\mid\text{data})$ is called \emph{cost} or \emph{objective function}
+
+      \medskip
+
+      $\chi^2(\pmb a^*\mid\text{data})$ is call the \emph{training error}
 
       \medskip
 
-- 
cgit v1.2.3-70-g09d2

