From e72ee50421716fde6646afd3a444b993413d3440 Mon Sep 17 00:00:00 2001 From: Justin Bedo Date: Fri, 9 Dec 2022 10:46:11 +1100 Subject: add illustrative figures and abtract --- position.tikz | 170 ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ slides.tex | 99 ++++++++++++++++++++++++++-------- 2 files changed, 246 insertions(+), 23 deletions(-) create mode 100644 position.tikz diff --git a/position.tikz b/position.tikz new file mode 100644 index 0000000..b30b6fe --- /dev/null +++ b/position.tikz @@ -0,0 +1,170 @@ + +\begin{tikzpicture}[y=0.80pt,x=0.80pt,yscale=-1, inner sep=0pt, outer sep=0pt] +\begin{scope}[cm={{1.33333,0.0,0.0,-1.33333,(0.0,672.0)}}] + \path[draw=black,line join=round,line cap=round,miter limit=10.00,line + width=0.600pt] (74.4000,310.7400) -- (74.6100,303.9800) -- (74.8100,304.8800) + -- (75.0200,290.8100) -- (75.2300,283.5400) -- (75.4400,297.6600) -- + (75.6400,288.3800) -- (75.8500,300.0300) -- (76.0600,297.4600) -- + (76.2600,287.7100) -- (76.4700,299.9700) -- (76.6800,295.6400) -- + (76.8900,293.5500) -- (77.0900,290.4800) -- (77.3000,297.3000) -- + (77.5100,317.1600) -- (77.7100,284.8300) -- (77.9200,318.2400) -- + (78.1300,265.7600) -- (78.3400,271.2300) -- (78.5400,274.3600) -- + (78.7500,341.6700) -- (78.9600,279.3700) -- (79.1600,290.8300) -- + (79.3700,369.3200) -- (79.5800,362.9200) -- (79.7900,330.9900) -- + (79.9900,316.5500) -- (80.2000,299.1200) -- (80.4100,305.5700) -- + (80.6100,301.8000) -- (80.8200,294.4900) -- (81.0300,309.1600) -- + (81.2300,329.8600) -- (81.4400,319.0300) -- (81.6500,276.8600) -- + (81.8600,311.1800) -- (82.0600,303.3600) -- (82.2700,337.4400) -- + (82.4800,305.5700) -- (82.6800,323.0600) -- (82.8900,306.7000) -- + (83.1000,318.4400) -- (83.3100,327.1000) -- (83.5100,302.5700); + \path[draw=black,line join=round,line cap=round,miter limit=10.00,line + width=0.600pt] + (83.7200,121.6100) -- (83.9300,136.3400) -- (84.1300,115.0000) -- + (84.3400,90.4700) -- (84.5500,132.1500) -- (84.7600,124.2100) -- + (84.9600,112.6500) -- (85.1700,115.4600) -- (85.3800,148.5400) -- + (85.5800,115.9100) -- (85.7900,123.2900) -- (86.0000,117.3900) -- + (86.2100,117.4100) -- (86.4100,129.2300) -- (86.6200,117.7300) -- + (86.8300,163.6200) -- (87.0300,159.1300) -- (87.2400,128.6100) -- + (87.4500,179.7100) -- (87.6600,109.8200) -- (87.8600,87.2000) -- + (88.0700,117.2500) -- (88.2800,118.8500) -- (88.4800,107.3300) -- + (88.6900,132.7200) -- (88.9000,177.6200) -- (89.1100,134.2500) -- + (89.3100,202.0000) -- (89.5200,180.9300) -- (89.7300,104.0300) -- + (89.9300,96.1200) -- (90.1400,118.6700) -- (90.3500,109.8800) -- + (90.5600,112.8600) -- (90.7600,143.2900) -- (90.9700,195.2400) -- + (91.1800,158.8400) -- (91.3800,94.7100) -- (91.5900,115.2700) -- + (91.8000,129.8900) -- (92.0100,167.0800) -- (92.2100,137.4700) -- + (92.4200,174.1300) -- (92.6300,162.5500) -- (92.8300,184.2400) -- + (93.0400,91.6200) -- (93.2500,128.3800) -- (93.4600,195.4500) -- + (93.6600,147.8900) -- (93.8700,175.5300) -- (94.0800,192.7300) -- + (94.2800,175.1400) -- (94.4900,186.7700) -- (94.7000,126.4000) -- + (94.9000,127.3700) -- (95.1100,218.5900); + \path[draw=black,line join=round,line cap=round,miter limit=10.00,line + width=0.600pt] (412.0100,385.3000) -- + (412.2100,389.3200) -- (412.4200,355.1300) -- (412.6300,331.5300) -- + (412.8300,375.2800) -- (413.0400,381.3800) -- (413.2500,391.3700) -- + (413.4600,337.4200) -- (413.6600,354.6000) -- (413.8700,356.1700) -- + (414.0800,358.8000) -- (414.2800,385.7800) -- (414.4900,393.1800) -- + (414.7000,363.8300) -- (414.9000,354.5800) -- (415.1100,400.4400) -- + (415.3200,380.6200) -- (415.5300,363.3400) -- (415.7300,342.3300) -- + (415.9400,347.7400) -- (416.1500,369.9400) -- (416.3500,393.6700) -- + (416.5600,371.8700) -- (416.7700,368.2300) -- (416.9800,376.0700) -- + (417.1800,390.4200) -- (417.3900,361.8300) -- (417.6000,325.2500) -- + (417.8000,328.3000) -- (418.0100,359.1700) -- (418.2200,358.9500) -- + (418.4300,353.0500) -- (418.6300,322.0300) -- (418.8400,296.7000) -- + (419.0500,325.4400) -- (419.2500,275.2400) -- (419.4600,307.4800) -- + (419.6700,305.8000) -- (419.8800,301.6200) -- (420.0800,325.5300) -- + (420.2900,289.5600) -- (420.5000,293.3200) -- (420.7000,316.5700) -- + (420.9100,326.9800) -- (421.1200,312.4600) -- (421.3300,320.9300) -- + (421.5300,318.6300) -- (421.7400,331.4700) -- (421.9500,330.5300) -- + (422.1500,308.8200) -- (422.3600,333.8700) -- (422.5700,331.1800) -- + (422.7800,331.0800) -- (422.9800,330.8100) -- (423.1900,320.0100) -- + (423.4000,327.5800) -- (423.6000,340.3100) -- (423.8100,337.5100) -- + (424.0200,342.4000) -- (424.2300,306.5900) -- (424.4300,334.2100); + \path[draw=black,line join=round,line cap=round,miter limit=10.00,line + width=0.600pt] + (424.6400,246.5100) -- (424.8500,222.3300) -- (425.0500,210.7300) -- + (425.2600,217.1800) -- (425.4700,223.4900) -- (425.6800,210.7100) -- + (425.8800,226.4300) -- (426.0900,235.8000) -- (426.3000,213.5600) -- + (426.5000,220.2800) -- (426.7100,240.2700) -- (426.9200,228.0300) -- + (427.1200,233.9000) -- (427.3300,229.6300) -- (427.5400,229.3100) -- + (427.7500,215.5400) -- (427.9500,237.2800) -- (428.1600,210.3900) -- + (428.3700,236.5900) -- (428.5700,212.8100) -- (428.7800,225.0300) -- + (428.9900,220.5700) -- (429.2000,216.6800) -- (429.4000,226.2600) -- + (429.6100,210.8600) -- (429.8200,207.9100); + \path[draw=black,line join=round,line cap=round,miter limit=10.00,line + width=0.600pt] (430.0200,281.7000) -- + (430.2300,332.5200) -- (430.4400,300.3400) -- (430.6500,323.7900) -- + (430.8500,333.7900) -- (431.0600,324.6700) -- (431.2700,323.5200) -- + (431.4700,324.4100) -- (431.6800,316.5700) -- (431.8900,306.1100) -- + (432.1000,358.7400) -- (432.3000,353.0700) -- (432.5100,351.9800) -- + (432.7200,340.6200) -- (432.9200,316.0600) -- (433.1300,316.1800) -- + (433.3400,338.6200) -- (433.5500,337.6100) -- (433.7500,349.5700) -- + (433.9600,338.5200) -- (434.1700,363.3000) -- (434.3700,340.9100) -- + (434.5800,324.7200) -- (434.7900,372.3700) -- (435.0000,323.4100) -- + (435.2000,333.0500) -- (435.4100,330.9100) -- (435.6200,339.8300) -- + (435.8200,338.4700) -- (436.0300,326.4300) -- (436.2400,336.6900) -- + (436.4500,348.3700) -- (436.6500,328.8100) -- (436.8600,358.7200) -- + (437.0700,301.2700) -- (437.2700,341.2400) -- (437.4800,317.6300) -- + (437.6900,357.9600) -- (437.9000,329.9400) -- (438.1000,321.1800) -- + (438.3100,319.2500) -- (438.5200,318.3500) -- (438.7200,352.8200) -- + (438.9300,339.0100) -- (439.1400,324.8000) -- (439.3400,333.1300) -- + (439.5500,213.9600) -- (439.7600,285.0700) -- (439.9700,332.1300) -- + (440.1700,289.1900) -- (440.3800,290.6900) -- (440.5900,301.3400) -- + (440.7900,339.6700) -- (441.0000,314.6400) -- (441.2100,276.0700) -- + (441.4200,289.5500) -- (441.6200,243.7900) -- (441.8300,270.8300) -- + (442.0400,322.7300) -- (442.2400,258.0300) -- (442.4500,331.6800) -- + (442.6600,355.6400) -- (442.8700,369.7900) -- (443.0700,341.1800) -- + (443.2800,343.7300) -- (443.4900,351.1200) -- (443.6900,357.2600) -- + (443.9000,346.3500) -- (444.1100,355.3100) -- (444.3200,350.6100) -- + (444.5200,362.3600) -- (444.7300,349.7200) -- (444.9400,352.0000) -- + (445.1400,346.6900) -- (445.3500,340.4400) -- (445.5600,336.9700) -- + (445.7700,349.8600) -- (445.9700,374.4700) -- (446.1800,357.6900) -- + (446.3900,342.8900) -- (446.5900,332.6100) -- (446.8000,347.6600) -- + (447.0100,349.8900) -- (447.2200,332.9600) -- (447.4200,344.4800); + \path[draw=black,line join=round,line cap=round,miter limit=10.00,line + width=0.600pt] (447.6300,246.2500) -- (447.8400,236.0600) -- (448.0400,244.5300) -- + (448.2500,270.1000) -- (448.4600,241.4300) -- (448.6700,353.4800) -- + (448.8700,357.3800) -- (449.0800,290.6700) -- (449.2900,252.5700) -- + (449.4900,248.9800) -- (449.7000,215.3000) -- (449.9100,302.5100) -- + (450.1200,256.4400) -- (450.3200,262.8700) -- (450.5300,213.7700) -- + (450.7400,275.9000) -- (450.9400,265.8200) -- (451.1500,268.3300) -- + (451.3600,226.8700) -- (451.5700,190.9700); + \path[draw=black,line join=round,line cap=round,miter limit=10.00,line + width=0.600pt] (451.7700,355.8800) -- + (451.9800,379.7500) -- (452.1900,380.9800) -- (452.3900,361.7900) -- + (452.6000,355.4500) -- (452.8100,351.3500) -- (453.0100,379.2500) -- + (453.2200,386.5200) -- (453.4300,390.2600) -- (453.6400,384.0700) -- + (453.8400,383.8300) -- (454.0500,410.8100) -- (454.2600,401.7100) -- + (454.4600,394.0000) -- (454.6700,411.6200) -- (454.8800,431.2000) -- + (455.0900,369.2600) -- (455.2900,393.5500) -- (455.5000,390.2600) -- + (455.7100,385.7800) -- (455.9100,398.1600) -- (456.1200,382.3700) -- + (456.3300,364.7800) -- (456.5400,393.1800) -- (456.7400,366.8600) -- + (456.9500,365.1000) -- (457.1600,370.3900) -- (457.3600,403.7900) -- + (457.5700,398.4700) -- (457.7800,377.4100) -- (457.9900,364.6900) -- + (458.1900,361.5300) -- (458.4000,370.1200); + \path[draw=black,line join=round,line cap=round,miter limit=10.00,line + width=0.600pt] (74.1900,73.4400) -- (384.8700,73.4400); + \path[draw=black,line join=round,line cap=round,miter limit=10.00,line + width=0.600pt] (74.1900,73.4400) -- (74.1900,66.2400); + \path[draw=black,line join=round,line cap=round,miter limit=10.00,line + width=0.600pt] (177.7500,73.4400) -- (177.7500,66.2400); + \path[draw=black,line join=round,line cap=round,miter limit=10.00,line + width=0.600pt] (281.3100,73.4400) -- (281.3100,66.2400); + \path[draw=black,line join=round,line cap=round,miter limit=10.00,line + width=0.600pt] (384.8700,73.4400) -- (384.8700,66.2400); + \path[cm={{1.0,0.0,0.0,-1.0,(70.86,47.52)}},fill=black,nonzero rule] (0,0) + node[above right] (text36) {0}; + \path[cm={{1.0,0.0,0.0,-1.0,(167.74,47.52)}},fill=black,nonzero rule] (0,0) + node[above right] (text40) {500}; + \path[cm={{1.0,0.0,0.0,-1.0,(267.97,47.52)}},fill=black,nonzero rule] (0,0) + node[above right] (text44) {1000}; + \path[cm={{1.0,0.0,0.0,-1.0,(371.53,47.52)}},fill=black,nonzero rule] (0,0) + node[above right] (text48) {1500}; + \path[draw=black,line join=round,line cap=round,miter limit=10.00,line + width=0.600pt] (59.0400,105.8200) -- (59.0400,399.0000); + \path[draw=black,line join=round,line cap=round,miter limit=10.00,line + width=0.600pt] (59.0400,105.8200) -- (51.8400,105.8200); + \path[draw=black,line join=round,line cap=round,miter limit=10.00,line + width=0.600pt] (59.0400,203.5400) -- (51.8400,203.5400); + \path[draw=black,line join=round,line cap=round,miter limit=10.00,line + width=0.600pt] (59.0400,301.2700) -- (51.8400,301.2700); + \path[draw=black,line join=round,line cap=round,miter limit=10.00,line + width=0.600pt] (59.0400,399.0000) -- (51.8400,399.0000); + \path[cm={{0.0,1.0,1.0,0.0,(41.76,95.64)}},fill=black,nonzero rule] (0,0) + node[above right] (text62) {−10}; + \path[cm={{0.0,1.0,1.0,0.0,(41.76,196.7)}},fill=black,nonzero rule] (0,0) + node[above right] (text66) {−5}; + \path[cm={{0.0,1.0,1.0,0.0,(41.76,297.94)}},fill=black,nonzero rule] (0,0) + node[above right] (text70) {0}; + \path[cm={{0.0,1.0,1.0,0.0,(41.76,395.66)}},fill=black,nonzero rule] (0,0) + node[above right] (text74) {5}; + \path[draw=black,line join=round,line cap=round,miter limit=10.00,line + width=0.600pt] (59.0400,73.4400) -- (473.7600,73.4400) -- (473.7600,444.9600) + -- (59.0400,444.9600) -- cycle; + \path[cm={{1.0,0.0,0.0,-1.0,(210.44,18.72)}},fill=black,nonzero rule] (0,0) + node[above right,scale=2] (text82) {Position}; + \path[cm={{0.0,1.0,1.0,0.0,(12.96,248.53)}},fill=black,nonzero rule] (0,0) + node[above right,scale=2] (text86) {$\Q$}; +\end{scope} + +\end{tikzpicture} diff --git a/slides.tex b/slides.tex index f8121d9..1cf5229 100644 --- a/slides.tex +++ b/slides.tex @@ -5,12 +5,13 @@ \usefonttheme{professionalfonts} \setbeamerfont{footnote}{size= \tiny} + \usepackage{unicode-math} \usepackage{microtype} \usepackage{tikz} -\usetikzlibrary{shapes} -\usetikzlibrary{bayesnet} +\usepackage{pgfplots} +\usepgfplotslibrary{ternary} \usepackage{stmaryrd} \newcommand{\R}{\mathbb{R}} @@ -22,9 +23,13 @@ \newcommand{\V}{\mathbf{V}} \newcommand{\A}{\mathbf{A}} \newcommand{\I}{\mathbf{I}} +\newcommand{\U}{\mathbf{u}} +\newcommand{\Q}{\mathbf{q}} +\newcommand{\PP}{\mathbf{P}} \DeclareMathOperator{\alr}{alr} \DeclareMathOperator{\clr}{clr} + \usepackage[natbib=true,url=false,style=verbose-ibid]{biblatex} \addbibresource{slides.bib} \AtBeginBibliography{\small} @@ -84,9 +89,19 @@ \definecolor{cb3}{HTML}{7570b3} \author{Justin Bed\H{o}} -\title{Representation learning of compositional counts: exploration of deep mutational scanning data} +\title{Representation learning of compositional counts: an exploration of deep mutational scanning data} \date{December 13, 2022} +% Abstract: + +% Deep mutational scanning data provides important functional information on the % effects of protein variants. Many different aspects of proteins can be assayed, % many different experimental designs are possible, and many different scores are % computed leading to very heterogeneous data that is difficult to integrate. + +% In this talk I will explore a representational learning approach on raw count % data. This technique uses recent methods combining compositional data analysis % with a generalised form of principal component analysis to infer protein % representations without specific knowledge of the experimental design or assay % type. + +% Bio + +% Dr Justin Bedő is the Stafford Fox Centenary Fellow in Bioinformatics and % Computational Biology at the Walter and Eliza Hall Institute. He studied % computer science followed by a PhD in machine learning at the Australian % National University and was awarded his doctorate in 2009. He subsequently % worked as a researcher across both academia and industry at NICTA, IBISC % (Informatique, BioInformatique, Systèmes Complexes) CNRS, and IBM Research on % machine learning methods development and applications to biology before joining % the WEHI in 2016. + \begin{document} \maketitle @@ -105,6 +120,7 @@ \item Growing resource of functional data \item MaveDB \footfullcite{Esposito2019} + \unskip \footnote{\url{https://www.mavedb.org}} catalogs a number of datasets and provides easy access \end{enumerate} \end{frame} @@ -153,24 +169,42 @@ \begin{enumerate} \item Scores calculated a variety of ways, e.g., Rubin et al. \footfullcite{Rubin2017}: - \[L_{v,t}=\log\left(\frac{(c_{v,t}+\frac12)(c_{wt,0}+\frac12)}{(c_{v,0}+\frac12)(c_{wt,t}+\frac12)}\right) \] + \[L_{v,t}=\log\left(\frac{(c_{v,t}+\frac12)(c_{wt,0}+\frac12)}{(c_{v,0}+\frac12)(c_{wt,t}+\frac12)}\right) \] + \item Assays can measure different properties + \item Numerous different experimental designs \end{enumerate} \end{frame} - \begin{frame}{Basics} - \begin{definition}[Compositional data] Data \(X \in \R^{n \times d}\) is compositional if rows \(\bx_i\) are in the simplex - \[S^d=\{\,\bx \in \R^d : \forall j,x_j > 0 ; \sum_{j=1}^d x_j = \kappa\,\} \] - 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). + \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 + \[S^d=\{\,\bx \in \R^d : \forall j,x_j > 0 ; \sum_{j=1}^d x_j = \kappa\,\} \] + for constant \(\kappa > 0\). + \end{definition} + \end{column} + \hfill + \begin{column}{.26 + \textwidth} + \begin{tikzpicture}[scale=0.5] + \begin{ternaryaxis} + \addplot3 coordinates{(0.25,0.5,0.25)}; + \path (0.25,0.5,0.25) coordinate (M) (1,0,0) coordinate (C) (0,1,0) coordinate (A) (0,0,1) coordinate (B); + \end{ternaryaxis} + \end{tikzpicture} + \end{column} + \end{columns} + \vspace{10pt} \(\Rightarrow\) Information is 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} \begin{frame}{Isomorphisms to Euclidean vector spaces} The simplex forms a \(d-1\) dimensional Euclidean vector space \footfullcite{Aitchison1982}: \begin{definition}[\ac{alr}] - \[\alr(\bx)_i = \log \frac{x_i}{x_0} \] + \[\alr_i(\bx) = \log \frac{x_i}{x_0} \] \end{definition} \begin{definition}[\ac{clr}] - \[\clr(\bx)_i = \log \frac{x_i}{\left(\prod_{j=1}^d x_j\right)^{\frac 1 d}} \] + \[\clr_i(\bx) = \log \frac{x_i}{\left(\prod_{j=1}^d x_j\right)^{\frac 1 d}} \] \end{definition} \end{frame} @@ -190,6 +224,7 @@ \begin{itemize} \item Zeros: \begin{enumerate} + \item $\log(0)$ undefined \item geometric mean is \(0\) \(\Rightarrow\) \ac{clr} is undefined \item @@ -198,7 +233,7 @@ \item Interpretation: \begin{enumerate} \item - \ac{alr} is not isometry + \ac{alr} is not an isometry \item \ac{clr} is degenerate \end{enumerate} @@ -208,7 +243,7 @@ \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 \(\V^\intercal \V = \I\). @@ -219,36 +254,35 @@ \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 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}{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. + \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. @@ -257,6 +291,11 @@ \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. + \end{frame} + \begin{frame}{Activation-Induced Deaminase \footfullcite{Gajula2014}} \begin{tikzpicture}[remember picture,overlay] @@ -316,4 +355,18 @@ \end{tikzpicture} \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} + \end{frame} + \end{document} -- cgit v1.2.3