From 188f305c35b46a2331027bc69c08b65a85e9516d Mon Sep 17 00:00:00 2001
From: Justin Bedo <cu@cua0.org>
Date: Mon, 12 Dec 2022 10:04:41 +1100
Subject: final edits

---
 slides.tex | 119 ++++++++++++++++++++++++++++++++++++++-----------------------
 1 file changed, 75 insertions(+), 44 deletions(-)

(limited to 'slides.tex')

diff --git a/slides.tex b/slides.tex
index 1cf5229..c908b50 100644
--- a/slides.tex
+++ b/slides.tex
@@ -5,7 +5,7 @@
 \usefonttheme{professionalfonts}
 \setbeamerfont{footnote}{size=
   \tiny}
-  \usepackage{unicode-math}
+\usepackage{unicode-math}
 
 \usepackage{microtype}
 
@@ -29,7 +29,6 @@
 \DeclareMathOperator{\alr}{alr}
 \DeclareMathOperator{\clr}{clr}
 
-
 \usepackage[natbib=true,url=false,style=verbose-ibid]{biblatex}
 \addbibresource{slides.bib}
 \AtBeginBibliography{\small}
@@ -167,20 +166,30 @@
 
   \begin{frame}{Deep Mutational Scanning: Integration issues}
     \begin{enumerate}
-      \item Scores calculated a variety of ways, e.g., Rubin et al.
-      \footfullcite{Rubin2017}:
-      \[L_{v,t}=\log\left(\frac{(c_{v,t}+\frac12)(c_{wt,0}+\frac12)}{(c_{v,0}+\frac12)(c_{wt,t}+\frac12)}\right)   \]
       \item Assays can measure different properties
       \item Numerous different experimental designs
+      \item Scores calculated a variety of ways, e.g., Rubin et al.
+      \footfullcite{Rubin2017}:
+      \[L_{v,t}=\log\left(\frac{(c_{v,t}+\frac12)(c_{wt,0}+\frac12)}{(c_{v,0}+\frac12)(c_{wt,t}+\frac12)}\right)    \]
     \end{enumerate}
   \end{frame}
 
+  \begin{frame}{Representational learning on
+      \ac{dms} data} For a given protein:
+    \begin{itemize}
+      \item Learn a representation of the available
+      \ac{dms} data
+      \item unsupervised to deal with varying designs
+      \item work on counts not scores
+    \end{itemize}
+  \end{frame}
+
   \begin{frame}{Compositional simplex}
     \begin{columns}[T]
       \begin{column}{.63
           \textwidth}
-        \begin{definition}[Compositional data] Data \(X \in \R^{n \times d}\) is compositional if rows \(\bx_i\) are in the simplex
-          \[S^d=\{\,\bx \in \R^d : \forall j,x_j > 0 ; \sum_{j=1}^d x_j = \kappa\,\}     \]
+        \begin{definition}[Compositional data] Data \(X \in \R_{\geq 0}^{n \times d}\) is compositional if rows \(\bx_i\) are in the simplex
+          \[S^d=\{\,\bx \in \R^d_{\geq 0} : \forall j,x_j > 0 ; \sum_{j=1}^d x_j = \kappa\,\}      \]
           for constant \(\kappa > 0\).
         \end{definition}
       \end{column}
@@ -201,10 +210,10 @@
   \begin{frame}{Isomorphisms to Euclidean vector spaces} The simplex forms a \(d-1\) dimensional Euclidean vector space
     \footfullcite{Aitchison1982}:
     \begin{definition}[\ac{alr}]
-      \[\alr_i(\bx) = \log \frac{x_i}{x_0}     \]
+      \[\alr_i(\bx) = \log \frac{x_i}{x_0}      \]
     \end{definition}
     \begin{definition}[\ac{clr}]
-      \[\clr_i(\bx) = \log \frac{x_i}{\left(\prod_{j=1}^d x_j\right)^{\frac 1 d}}     \]
+      \[\clr_i(\bx) = \log \frac{x_i}{\left(\prod_{j=1}^d x_j\right)^{\frac 1 d}}      \]
     \end{definition}
   \end{frame}
 
@@ -220,22 +229,14 @@
         \ac{pca}
       \end{enumerate}
     \end{block}
+    \pause
     \begin{block}{Problems}
       \begin{itemize}
         \item Zeros:
         \begin{enumerate}
-          \item $\log(0)$ undefined
+          \item \(\log(0)\) undefined \(\Rightarrow\) can't handle unobserved components
           \item geometric mean is \(0\) \(\Rightarrow\)
           \ac{clr} is undefined
-          \item
-          \ac{alr} is undefined for unobserved components in the ref.
-        \end{enumerate}
-        \item Interpretation:
-        \begin{enumerate}
-          \item
-          \ac{alr} is not an isometry
-          \item
-          \ac{clr} is degenerate
         \end{enumerate}
       \end{itemize}
     \end{block}
@@ -243,36 +244,36 @@
 
   \begin{frame}{Traditional
       \ac{pca}} Given \(\X\in \R^{n\times d}\) minimise loss
-    \[\ell_{\textsc{pca}} \triangleq {\lVert \X - \V\A \rVert}^2_{\textrm{F}}\]
+    \[\ell_{\textsc{pca}} \triangleq {\lVert \X - \V\A \rVert}^2_{\textrm{F}} \]
     s.t.
     \(\V \in \R^{n \times k}\), \(\A \in \R^{k \times d}\), and \(\V^\intercal \V = \I\).
 
-    Has been generalised to exponential families
+    \pause Has been generalised to exponential families
     \footfullcite{collins2001generalization} via Bregman divergences
     \footfullcite{Amari2016-ua}.
   \end{frame}
 
   \begin{frame}{Exponential family
       \ac{pca}}
-    \begin{definition}[Bregman Divergence] Let \(\varphi \colon \R^d \to \R\) be a smooth ($C^1$) convex function on convex set \(\Omega\).
+    \begin{definition}[Bregman Divergence] Let \(\varphi \colon \R^d \to \R\) be a differentiable convex function.
       The Bregman divergence \(D_\varphi\) with generator \(\varphi\) is
-      \[ D_\varphi\left(\bu\,\Vert\,\bv\right) \triangleq \varphi(\bu)-\varphi(\bv)-\langle \nabla\varphi(\bv),\bu-\bv\rangle.          \]
+      \[ D_\varphi\left(\bu\,\Vert\,\bv\right) \triangleq \varphi(\bu)-\varphi(\bv)-\langle \nabla\varphi(\bv),\bu-\bv\rangle.           \]
     \end{definition}
-    Denote the convex conjugate of \(\varphi\) as \(\varphi^*(\bu) \triangleq \sup_\bv\left\{\langle \bu,\bv\rangle-\varphi(\bv)\right\}\).
+    \pause Denote the convex conjugate of \(\varphi\) as \(\varphi^*(\bu) \triangleq \sup_\bv\left\{\langle \bu,\bv\rangle-\varphi(\bv)\right\}\).
     The exponential family
     \ac{pca} is then given by minimising loss
-    \[\ell_{\varphi} \triangleq D_\varphi\left(\X\,\Vert\,\nabla\varphi^*\left(\V\A\right)\right)          \]
+    \[\ell_{\varphi} \triangleq D_\varphi\left(\X\,\Vert\,\nabla\varphi^*\left(\V\A\right)\right)           \]
     under the same constraints as previously, approximating \(\X \sim \nabla\varphi^*\left(\V\A\right)\).
   \end{frame}
 
   \begin{frame}{Aitchison's simplex and exponential
       \ac{pca}} Aitchison's log-transformation is a dual affine coordinate space made explicit with
-    \[\varphi(z) = z\log(z) - z \Leftrightarrow \varphi^*(z) = e^z,\]
+    \[\varphi(z) = z\log(z) - z \Leftrightarrow \varphi^*(z) = e^z, \]
     but what about normalisation?
 
-    Consider
+    \pause Consider
     \ac{alr}:
-    \[\alr(\bx) \triangleq x_0 \sum_{i=1}^d\varphi\left(\frac{x_i}{x_0}\right) \Leftrightarrow \alr^*(\bx) = x_0\sum_{i=1}^d e^{\frac{x_i}{x_0}}     \]
+    \[\alr(\bx) \triangleq x_0 \sum_{i=1}^d\varphi\left(\frac{x_i}{x_0}\right) \Leftrightarrow \alr^*(\bx) = x_0\sum_{i=1}^d e^{\frac{x_i}{x_0}}      \]
 
   \end{frame}
 
@@ -280,9 +281,9 @@
     \begin{theorem}[Scaled Bregman
         \footfullcite{nock2016scaled}] Let \(\varphi \colon \mathcal{X} \to \R\) be convex differentiable and \(g \colon \mathcal{X} \to \R\) be differentiable.
       Then
-      \[g(\bx)\cdot D_\varphi\left(\frac{\bx}{g(\bx)}\,\middle\Vert\,\frac{\by}{g(\by)}\right) =  D_{\breve{\varphi}}\left(\bx\,\middle\Vert\,\by\right)     \]
+      \[D_{\breve{\varphi}}\left(\bx\,\middle\Vert\,\by\right) = g(\bx)\cdot D_\varphi\left(\frac{\bx}{g(\bx)}\,\middle\Vert\,\frac{\by}{g(\by)}\right) \]
       where
-      \[\breve{\varphi} \triangleq g(\bx) \cdot \varphi\left(\frac{x}{g(\bx)}\right)\]
+      \[\breve{\varphi} \triangleq g(\bx) \cdot \varphi\left(\frac{x}{g(\bx)}\right) \]
     \end{theorem}
 
     Avalos et al.
@@ -291,11 +292,16 @@
     \ac{clr} recently.
   \end{frame}
 
-  \begin{frame}{Medians instead of means}
-    Zeros still a problem, as geometric mean is $0$. Instead, use median as gague
-    function.
+  \begin{frame}{\textsc{Clr} undefined if any component is unobserved}
+    \begin{itemize}
+      \item Zeros still a problem for
+      \ac{clr} as geometric mean is \(0\).
+      \item[\(\Rightarrow\)] use median as gague function.
+    \end{itemize}
   \end{frame}
 
+  \section{Experiments}
+
   \begin{frame}{Activation-Induced Deaminase
       \footfullcite{Gajula2014}}
     \begin{tikzpicture}[remember picture,overlay]
@@ -356,17 +362,42 @@
   \end{frame}
 
   \begin{frame}{\textsc{Brca1}: Positional effects}
-  \begin{columns}[T]
-  \begin{column}{.4\textwidth}
-  \[\V\A+\U^\intercal\Q\PP\]
-    where $\U \in \R^n$, $\Q \in \R^l$, $\PP \in \mathbb{2}^{l\times d}$
-    \end{column}\hfill
-    \begin{column}{.58\textwidth}
-  \begin{tikzpicture}
-    \node[scale=.45]{\input{position.tikz}};
-  \end{tikzpicture}
-  \end{column}
-  \end{columns}
+    \begin{columns}[T]
+      \begin{column}{.4
+          \textwidth}
+        \vspace{1cm}
+        \[\V\A+\U^\intercal\Q\PP \]
+        where \(\U \in \R^n\), \(\Q \in \R^l\), \(\PP \in \mathbb{2}^{l\times d}\)
+      \end{column}
+      \hfill
+      \begin{column}{.58
+          \textwidth}
+        \begin{tikzpicture}
+          \node[scale=.45]{\input{position.tikz}};
+        \end{tikzpicture}
+      \end{column}
+    \end{columns}
+  \end{frame}
+
+  \begin{frame}{Acknowledgements}
+    \begin{columns}[T]
+      \begin{column}{.4
+          \textwidth}
+        \textbf{Papenfuss lab}
+        \begin{itemize}
+          \item Tony Papenfuss
+          \item
+          \textit{Alan Rubin}
+          \item
+          \textit{Matthew Wakefield}
+        \end{itemize}
+      \end{column}
+      \hfill
+      \begin{column}{.4
+          \textwidth}
+        \textbf{Stafford Fox medical research foundation}
+      \end{column}
+    \end{columns}
   \end{frame}
 
 \end{document}
-- 
cgit v1.2.3

