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Diffstat (limited to 'slides.tex')
| -rw-r--r-- | slides.tex | 124 |
1 files changed, 112 insertions, 12 deletions
@@ -29,6 +29,46 @@ \addbibresource{slides.bib} \AtBeginBibliography{\small} +%% Tikz relative positioning https://tex.stackexchange.com/questions/89588/positioning-relative-to-page-in-tikz +\makeatletter +\def +\parsecomma#1,#2 +\endparsecomma{\def + \page@x{#1} + \def + \page@y{#2}} +\tikzdeclarecoordinatesystem{page}{\parsecomma#1 + \endparsecomma + \pgfpointanchor{current page}{north east} + \pgf@xc= + \pgf@x + \pgf@yc= + \pgf@y + \pgfpointanchor{current page}{south west} + \pgf@xb= + \pgf@x + \pgf@yb= + \pgf@y + \pgfmathparse{(\pgf@xc- + \pgf@xb)/2.* + \page@x+( + \pgf@xc+ + \pgf@xb)/2.} + \expandafter + \pgf@x + \expandafter= + \pgfmathresult pt + \pgfmathparse{(\pgf@yc- + \pgf@yb)/2.* + \page@y+( + \pgf@yc+ + \pgf@yb)/2.} + \expandafter + \pgf@y + \expandafter= + \pgfmathresult pt} +\makeatother + \usepackage{acronym} \usepackage{xspace} \renewcommand*{\acsfont}[1]{\textsc{#1}} @@ -59,7 +99,7 @@ \begin{frame}{Basics} \begin{definition}[Compositional data] Data \(X \in \R^{n \times d}\) is compositional if rows \(\bx_i\) are in the simplex - \[S^d=\{\,\bx \in \R^d : \forall j,x_j > 0 ; \sum_{j=1}^d x_j = \kappa\,\} \] + \[S^d=\{\,\bx \in \R^d : \forall j,x_j > 0 ; \sum_{j=1}^d x_j = \kappa\,\} \] for constant \(\kappa > 0\). \end{definition} Information is therefore given only by the ratios of components and any composition can be normalised to the standard simplex where \(\kappa = 1\) (c.f., dividing by library size). \end{frame} @@ -67,10 +107,10 @@ \begin{frame}{Isomorphisms to Euclidean vector spaces} The simplex forms a \(d-1\) dimensional Euclidean vector space \footfullcite{Aitchison1982}: \begin{definition}[\ac{alr}] - \[\alr(\bx)_i = \log \frac{x_i}{x_0} \] + \[\alr(\bx)_i = \log \frac{x_i}{x_0} \] \end{definition} \begin{definition}[\ac{clr}] - \[\clr(\bx)_i = \log \frac{x_i}{\left(\prod_{j=1}^d x_j\right)^{\frac 1 d}} \] + \[\clr(\bx)_i = \log \frac{x_i}{\left(\prod_{j=1}^d x_j\right)^{\frac 1 d}} \] \end{definition} \end{frame} @@ -108,7 +148,7 @@ \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 \(\V^\intercal \V = \I\). @@ -121,39 +161,99 @@ \ac{pca}} \begin{definition}{Bregman Divergence} Let \(\varphi \colon \R^d \to \R\) be a smooth ($C^1$) convex function on convex set \(\Omega\). 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 \(\varphi^*(\bu) \triangleq \sup_\bv\left\{\langle \bu,\bv\rangle-\varphi(\bv)\right\}\). 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 \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} - \begin{frame} + \begin{frame}{Scaled Bregman} \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 - \[g(\bx)\cdot D_\varphi\left(\frac{\bx}{g(\bx)}\,\middle\Vert\,\frac{\by}{g(\by)}\right) = D_{\breve{\varphi}}\left(\bx\,\middle\Vert\,\by\right) \] + \[g(\bx)\cdot D_\varphi\left(\frac{\bx}{g(\bx)}\,\middle\Vert\,\frac{\by}{g(\by)}\right) = D_{\breve{\varphi}}\left(\bx\,\middle\Vert\,\by\right) \] 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. \footfullcite{avalos2018representation} - \ considered a relaxed form for \ac{clr} recently. + \ considered a relaxed form for + \ac{clr} recently. + \end{frame} + + \begin{frame}{Activation-Induced Deaminase + \footfullcite{Gajula2014}} + \begin{tikzpicture}[remember picture,overlay] + \node[scale=0.85] at (page cs:0,0.08){\input{106-samples.tikz}}; + \end{tikzpicture} + \end{frame} + + \begin{frame}{Activation-Induced Deaminase} + \begin{tikzpicture} + \node at (page cs:-0.5,0.08){\input{106-Leu113.tikz}}; + \node at (page cs:0.5,0.5){\includegraphics{gku689fig3-a.pdf}}; + \node at (page cs:0.5,-0.25){\includegraphics{gku689fig3-key.pdf}}; + \end{tikzpicture} + \end{frame} + + \begin{frame}{Activation-Induced Deaminase} + \begin{tikzpicture} + \node at (page cs:-0.5,0.08){\input{106-Phe115.tikz}}; + \node at (page cs:0.5,0.5){\includegraphics{gku689fig3-b.pdf}}; + \node at (page cs:0.5,-0.25){\includegraphics{gku689fig3-key.pdf}}; + \end{tikzpicture} + \end{frame} + + \begin{frame}{Activation-Induced Deaminase} + \begin{tikzpicture} + \node at (page cs:-0.5,0.08){\input{106-Glu117.tikz}}; + \node at (page cs:0.5,0.5){\includegraphics{gku689fig3-c.pdf}}; + \node at (page cs:0.5,-0.25){\includegraphics{gku689fig3-key.pdf}}; + \end{tikzpicture} + \end{frame} + + \begin{frame}{Activation-Induced Deaminase} + \begin{tikzpicture} + \node at (page cs:-0.7,0.9){\textbf{Bregman}}; + \node at (page cs:0.3,0.9){\textbf{+1-log + \ac{pca}}}; + \node[scale=0.9] at (page cs:-0.5,0.08){\input{106-samples.tikz}}; + \node[scale=0.9] at (page cs:0.5,0.08){\input{106-samples-log.tikz}}; + \end{tikzpicture} + \end{frame} + + \begin{frame}{\textsc{Erbb2} + \footfullcite{Elazar2016}} + \begin{tikzpicture} + \node[scale=0.8] at (page cs: -0.5,0){\input{helix-erbb2.tikz}}; + \node at (page cs: 0.5,0.07){\includegraphics[width=0.4 + \textwidth]{helix-erbb2-pub.jpg}}; + \end{tikzpicture} + \end{frame} + + \begin{frame}{\textsc{Brca1} + \footfullcite{Findlay2018}} + \begin{tikzpicture}[remember picture,overlay] + \node[inner sep=0pt] at (5,0.5){\input{brca1-density.tikz}}; + + \node[inner sep=0pt] at (11,1.25){\includegraphics{brca1-hist-pub.jpg}}; + \end{tikzpicture} \end{frame} \end{document} |