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\usepackage{acronym}
\usepackage{xspace}
\renewcommand*{\acsfont}[1]{\textsc{#1}}
\newacro{dms}{Deep Mutational Scanning}
\newacro{clr}{Centred Log-Ratio}
\newacro{alr}{Additive Log-Ratio}
\newacro{pca}{Principal Component Analysis}
\newcommand{\dms}{\ac{dms}
  \xspace}

\author{Justin Bed\H{o}}
\title{Exploration of deep mutational scanning data with unsupervised methods}
\date{December 13, 2022}

\begin{document}

  \maketitle

  \section{Deep mutational scanning data}

  \begin{frame}{\dms data} Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua.
    Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat.
    Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla pariatur.
    Excepteur sint occaecat cupidatat non proident, sunt in culpa qui officia deserunt mollit anim id est laborum.
  \end{frame}

  \section{Compositional data}

  \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).
  \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}]
      \[\alr(\bx)_i = \log \frac{x_i}{x_0} \]
    \end{definition}
    \begin{definition}[\ac{clr}]
      \[\clr(\bx)_i = \log \frac{x_i}{\left(\prod_{j=1}^d x_j\right)^{\frac 1 d}} \]
    \end{definition}
  \end{frame}

  \begin{frame}{\textsc{Pca} on
      \ac{dms} data}
    \begin{block}{Transformation approach}
      \begin{enumerate}
        \item Map
        \dms data to Euclidean space via
        \ac{alr} /
        \ac{clr}
        \item Apply standard
        \ac{pca}
      \end{enumerate}
    \end{block}
    \begin{block}{Problems}
      \begin{itemize}
        \item Zeros:
        \begin{enumerate}
          \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 isometry
          \item
          \ac{clr} is degenerate
        \end{enumerate}
      \end{itemize}
    \end{block}
  \end{frame}

  \begin{frame}{Traditional
      \ac{pca}} Given \(\X\in \R^{n\times d}\) minimise loss
    \[\ell_{\textsc{pca}} \triangleq {\lVert \X - \V\A \rVert}^2_{\textrm{F}}      \]
    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
    \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 \(\Omega\).
      The Bregman divergence \(D_\varphi\) with generator \(\varphi\) is
      \[ D_\varphi\left(\bu\,\Vert\,\bv\right) \triangleq \varphi(\bu)-\varphi(\bv)-\langle \nabla\varphi(\bv),\bu-\bv\rangle.      \]
    \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
    \[\ell_{\varphi} \triangleq D_\varphi\left(\X\,\Vert\,\nabla\varphi^*\left(\V\A\right)\right)      \]
    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
    \[\varphi(z) = z\log(z) - z \Leftrightarrow \varphi^*(z) = e^z,   \]
    but what about normalisation?

    Consider
    \ac{alr}:
    \[\alr(\bx) \triangleq x_0 \sum_{i=1}^d\varphi\left(\frac{x_i}{x_0}\right) \Leftrightarrow \alr^*(\bx) = x_0\sum_{i=1}^d e^{\frac{x_i}{x_0}} \]

  \end{frame}

  \begin{frame}
    \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
      \[g(\bx)\cdot D_\varphi\left(\frac{\bx}{g(\bx)}\,\middle\Vert\,\frac{\by}{g(\by)}\right) =  D_{\breve{\varphi}}\left(\bx\,\middle\Vert\,\by\right) \]
      where
      \[\breve{\varphi} \triangleq g(\bx) \cdot \varphi\left(\frac{x}{g(\bx)}\right)  \]
    \end{theorem}

    Avalos et al.
    \footfullcite{avalos2018representation}
    \ considered a relaxed form for \ac{clr} recently.
  \end{frame}

\end{document}