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% ====================
% Title
% ====================
\title{Data Association with Gaussian Processes}
\author{Markus Kaiser \inst{1,2} \and Clemens Otte \inst{1} \and Thomas Runkler \inst{1,2} \and Carl Henrik Ek \inst{3}}
\institute[shortinst]{\inst{1} Siemens AG \samelineand \inst{2} Technical University of Munich \samelineand \inst{3} University of Bristol}
\addbibresource{../zotero_export.bib}
\addbibresource{../additional.bib}
\begin{document}
\begin{frame}[t]
\begin{columns}[t]
\separatorcolumn
%
\begin{column}{\colwidth}
\begin{block}{Multimodal Data}
\begin{figure}
\centering
\includestandalonewithpath{figures/semi_bimodal}
\end{figure}
\begin{itemize}
\item AMO-GP correctly recovers the latent shared function, warping and alignment
\end{itemize}
\end{block}
\begin{block}{Noise Separation}
\begin{figure}
\centering
\begin{subfigure}{\textwidth}
\centering
\includestandalonewithpath{figures/noise_separation_data}
\caption{
Data
}
\end{subfigure}\\[2ex]
\begin{subfigure}{\textwidth}
\centering
\includestandalonewithpath{figures/noise_separation_attribution}
\caption{
Attribution
}
\end{subfigure}\\[2ex]
\begin{subfigure}{\textwidth}
\centering
\includestandalonewithpath{figures/noise_separation_joint}
\caption{
Joint
}
\end{subfigure}
\end{figure}
\begin{itemize}
\item AMO-GP correctly recovers the latent shared function, warping and alignment
\end{itemize}
\end{block}
\end{column}
%
\separatorcolumn
%
\begin{column}{\colwidth}
\begin{block}{Graphical model of DAGP}
\begin{figure}
\centering
\includestandalonewithpath{figures/graphical_model}
\end{figure}
\end{block}
\vspace{3ex}
\begin{block}{Scalable Inference}
\begin{itemize}
\item AMO-GP connects multiple deep GPs via a shared layer which is a multi-output GP
\smallskip
\item Application to non-linear time series alignment with very noisy observations
\smallskip
\item Variational distribution
\begin{align}
\Variat*{\mat{F}, \mat{\alpha}, \mat{U}}
&= \Variat*{\mat{\alpha}, \Set*{\mat{F^{\pix{k}}}, \mat{u^{\pix{k}}}, \mat{u_\alpha^{\pix{k}}}}_{k=1}^K} \\
\MoveEqLeft = \prod_{k=1}^K\prod_{n=1}^N \Prob*{\mat{\alpha_n^{\pix{k}}} \given \mat{u_\alpha^{\pix{k}}}, \mat{x_n}}\Variat*{\mat{u_\alpha^{\pix{k}}}}
\prod_{k=1}^K \prod_{n=1}^N \Prob*{\mat{f_n^{\pix{k}}} \given \mat{u^{\pix{k}}}, \mat{x_n}}\Variat*{\mat{u^{\pix{k}}}}.
\end{align}
\smallskip
\item Variational joint
\begin{align}
\Variat*{\mat{Y}, \mat{A}} &=
\int
\Prob*{\mat{Y} \given \mat{F}, \mat{A}}
\Prob*{\mat{A} \given \mat{\alpha}}
\Variat*{\mat{F}, \mat{\alpha}}
\diff \mat{F} \diff \mat{\alpha},
\end{align}
\smallskip
\item Variational bound
\begin{align}
\Ell_{\text{DAGP}}
&= \Moment*{\E_{\Variat*{\mat{F}, \mat{\alpha}, \mat{U}}}}{\log\frac{\Prob*{\mat{Y}, \mat{A}, \mat{F}, \mat{\alpha}, \mat{U} \given \mat{X}}}{\Variat*{\mat{F}, \mat{\alpha}, \mat{U}}}} \\
&= \sum_{n=1}^N \Moment*{\E_{\Variat*{\mat{f_n}}}}{\log \Prob*{\mat{y_n} \given \mat{f_n}, \mat{a_n}}}
+ \sum_{n=1}^N \Moment*{\E_{\Variat*{\mat{\alpha_n}}}}{\log \Prob*{\mat{a_n} \given \mat{\alpha_n}}} \\
&\quad - \sum_{k=1}^K \KL*{\Variat*{\mat{u^{\pix{k}}}}}{\Prob*{\mat{u^{\pix{k}}} \given \mat{Z^{\pix{k}}}}}
- \sum_{k=1}^K \KL*{\Variat*{\mat{u_\alpha^{\pix{k}}}}}{\Prob*{\mat{u_\alpha^{\pix{k}}} \given \mat{Z_\alpha^{\pix{k}}}}}
\end{align}
\smallskip
\item Predictive posterior
\begin{align}
\Variat*{\mat{f_\ast} \given \mat{x_\ast}}
&= \int \sum_{k=1}^K \Variat*{a_\ast^{\pix{k}} \given \mat{x_\ast}} \Variat*{\mat{f_\ast^{\pix{k}}} \given \mat{x_\ast}} \diff \mat{a_\ast^{\pix{k}}} \\
&\approx \sum_{k=1}^K \hat{a}_\ast^{\pix{k}} \mat{\hat{f}_\ast^{\pix{k}}}
\end{align}
\end{itemize}
\end{block}
\end{column}
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\begin{block}{Wet-Chicken RL Benchmark}
\centering
\includestandalonewithpath{figures/wetchicken}
\end{block}
% \begin{block}{Interpretable Transition Model}
% \begin{figure}
% \centering
% \includestandalonewithpath{figures/dynamics_posterior}
% \end{figure}
% \begin{itemize}
% \item AMO-GP correctly recovers the latent shared function, warping and alignment
% \end{itemize}
% \end{block}
% \begin{block}{Conservative Policy}
% \begin{figure}
% \centering
% \begin{subfigure}{.475\textwidth}
% \centering
% \includestandalonewithpath{figures/policy_quiver}
% \caption{
% $\Fun*{R}{x, y} = x$
% }
% \end{subfigure}
% \hfill
% \begin{subfigure}{.475\textwidth}
% \centering
% \includestandalonewithpath{figures/conservative_policy_quiver}
% \caption{
% $\Fun*{R^\prime}{x, y} = \Fun*{R}{x, y} - 5 \cdot \Prob{\text{drop} \given x, y}$
% }
% \end{subfigure}
% \end{figure}
% \begin{itemize}
% \item AMO-GP correctly recovers the latent shared function, warping and alignment
% \end{itemize}
% \end{block}
\end{column}
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\end{document}