
% Module 7: Summary starts here (Slide 53-54) 

\begin{frame}
    \myheading{Module 7.7 : Summary }
\end{frame}

\begin{frame}
\begin{columns}
    \column{0.5\textwidth}
    \begin{overlayarea}{\textwidth}{\textheight}
    \only<1->{
        \begin{figure}
        \input{modules/Module6/tikz_images/under-complete-pca.tex}
        \end{figure}
    }
    \end{overlayarea}

    \column{0.5\textwidth}
    \only<2->{
    \[
        \min_\theta \| X - \underbrace{H W^*}_{\substack{ U \Sigma V^T \\ \text{(SVD)}}} \|^2_F
    \]
    }
\end{columns}
\end{frame}

\begin{frame}
\begin{columns}
    \column{0.5\textwidth}
    \begin{overlayarea}{\textwidth}{\textheight}
    % \include{denoise}
    \vspace{5pt}
    \input{modules/Module7/tikz_images/denoise.tex}
    \end{overlayarea}

    \column{0.5\textwidth}
    \begin{overlayarea}{\textwidth}{\textheight}
        \vspace{1cm}
        \onslide<2->{
        \begin{center}
            \underline{Regularization}
        \end{center}
        }
        \begin{equation*}
            \footnotesize{
            \begin{split}
                \onslide<3-> {\Omega(\theta) &= \lambda \|\theta \|^2  \;\;\; \framebox{Weight decaying}} \\
                \onslide<4-> {\Omega(\theta) &= \sum_{l=1}^{k} \rho \log \frac{\rho}{\hat{\rho}_l} + (1 - \rho) \log \frac{1 - \rho}{1 - \hat{\rho}_l} \;\;\; \framebox{Sparse}} \\
                \onslide<5-> {\Omega(\theta) &= \sum_{j=1}^{n} \sum_{l=1}^{k} \Big( \frac{\partial h_l}{\partial x_j} \Big)^2 \;\;\; \framebox{Contractive}}
            \end{split}
            }
        \end{equation*}
    \end{overlayarea}
\end{columns}
\end{frame}
% Module 7: Summary ends here (Slide 53-54) 
