From c8cecec1bb0b491c21d140d5253f7abd0dc5db15 Mon Sep 17 00:00:00 2001
From: Laurent Modolo <laurent.modolo@ens-lyon.fr>
Date: Thu, 19 Oct 2023 15:03:29 +0200
Subject: [PATCH] M1_biosciences_dimension_reduction: update
---
M1_biosciences_dimension_reduction/PCA.tex | 2 +-
M1_biosciences_dimension_reduction/perspectives.tex | 6 +++---
2 files changed, 4 insertions(+), 4 deletions(-)
diff --git a/M1_biosciences_dimension_reduction/PCA.tex b/M1_biosciences_dimension_reduction/PCA.tex
index f442fb3..14db8ac 100644
--- a/M1_biosciences_dimension_reduction/PCA.tex
+++ b/M1_biosciences_dimension_reduction/PCA.tex
@@ -397,7 +397,7 @@ $$
$$
\item The correlation coefficient is a distance measure between variables:
$$
- \operatorname{r}(\xbf^{j'},\xbf^{j}) = \frac{1}{2} \times \left( 1 - \frac{1}{n}\|\widetilde{\xbf}^j_c-\widetilde{\xbf}_c^{j'}\|^2 \right)
+ \operatorname{r}(\xbf^{j'},\xbf^{j}) = 1 - \frac{1}{2} \times \frac{1}{n}\|\widetilde{\xbf}^j_c-\widetilde{\xbf}_c^{j'}\|^2
$$
\end{itemize}
\end{frame}
diff --git a/M1_biosciences_dimension_reduction/perspectives.tex b/M1_biosciences_dimension_reduction/perspectives.tex
index 8c8df9e..848e7c2 100644
--- a/M1_biosciences_dimension_reduction/perspectives.tex
+++ b/M1_biosciences_dimension_reduction/perspectives.tex
@@ -95,10 +95,10 @@ $\rightarrow$ Low-rank representation of $\mb X$
\begin{columns}[c]
\begin{column}{.5\textwidth}
\begin{itemize}
- \item In many situations only the distance $d_{ij}$ between individuals is available
+ \item In many situations only the distance $d_{ii'}$ between individuals $(i,i')$ is available
\item The objective of MDS is to find new coordinates $\ubf_1,\hdots,\ubf_n$ that minimize:
$$
- \sum_{ij} \Big( d_{ij} - \|\ubf_i-\ubf_j \|^2 \Big)^2
+ \sum_{i,i'} \Big( d_{ii'} - \|\ubf_i-\ubf_{i'} \|^2 \Big)^2
$$
\item The information regarding the variables is not considered (not available)
\end{itemize}
@@ -120,7 +120,7 @@ $\rightarrow$ Low-rank representation of $\mb X$
\item These distances depend on a dot product
\item This dot product can be generalized by the so-called kernel
$$
- K(\xbf_i,\xbf_j) =\langle \phi(\xbf_i), \phi(\xbf_j)\rangle
+ K(\xbf_i,\xbf_{i'}) =\langle \phi(\xbf_i), \phi(\xbf_{i'})\rangle
$$
\item $\phi$ is called the feature map and is unknown
\item Grounds most non linear methods (kernel-PCA, kernel MDS, etc)
--
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