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Prior for Supervised Learning

\begin{flushright} {\scriptsize \citep{Urtasun:dgplvm07}} \end{flushright}

  • We introduce a prior that is based on the Fisher criteria $$p(\latentMatrix) \propto \exp \left{ -\frac{1}{\sigma_{d}^2} \tr{ {\mathbf{S}_w^{-1} \mathbf{S}_b}}\right} ~,$$ with $\mathbf{S}_b$ the between class matrix and $\mathbf{S}_w$ the within class matrix

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    \begin{column}{3cm} \begin{figure} \only<1>{\includegraphics[width=0.92\columnwidth]{../../../gplvm/tex/diagrams/classif/original2}} \only<2>{\includegraphics[width=0.92\columnwidth]{../../../gplvm/tex/diagrams/classif/between2}} \only<3>{\includegraphics[width=0.92\columnwidth]{../../../gplvm/tex/diagrams/classif/within}} \end{figure} \end{column} \begin{column}{7cm} \only<2-3>{ [ \mathbf{S}w =\sum{i=1}^L \frac{\numData_i}{\numData}(\mathbf{M}_i - \mathbf{M}_0)(\mathbf{M}_i - \mathbf{M}0)^\top ] } \only<3>{ [ \mathbf{S}b =\sum{i=1}^L\frac{\numData_i}{\numData}\left[ \frac{1}{\numData_i} \sum{k=1}^{\numData_i} (\latentVector_k^{(i)}-\mathbf{M}_i)(\latentVector_k^{(i)}-\mathbf{M}_i)^\top \right] ] } % \begin{eqnarray} % \mathbf{S}w =\sum{i=1}^L \frac{\numData_i}{\numData}(\mathbf{M}_i - \mathbf{M}_0)(\mathbf{M}_i - \mathbf{M}0)^\top\nonumber \ % \mathbf{S}b =\sum{i=1}^L\frac{\numData_i}{\numData}\left[ % \frac{1}{\numData_i} \sum{k=1}^{\numData_i} (\latentVector_k^{(i)}-\mathbf{M}_i)(\latentVector_k^{(i)}-\mathbf{M}_i)^\top % \right]\nonumber % \end{eqnarray} \end{column} columns end

    \only<2-3>{ where $\latentMatrix^{(i)} = [\latentVector_1^{(i)}, \cdots, \latentVector_{\numData_i}^{(i)}]$ are the $\numData_i$ training points of class $i$, $\mathbf{M}_i$ is the mean of the elements of class $i$, and $\mathbf{M}_0$ is the mean of all the training points of all classes. } \only<3>{ \item As before the model is learned by maximizing $p(\dataMatrix|\latentMatrix) p(\latentMatrix)$.}

\only<4->{ \begin{figure} \includegraphics[width=0.30\columnwidth]{../../../gplvm/tex/diagrams/oil_data/subset_100/d_5_kbr_01_l_1000000_a.pdf} \includegraphics[width=0.30\columnwidth]{../../../gplvm/tex/diagrams/oil_data/subset_100/d_5_kbr_01_l_10000} % \includegraphics[width=0.5\columnwidth]{../../../gplvm/tex/diagrams/oil_data/subset_100/d_5_kbr_01_l_1000} \includegraphics[width=0.30\columnwidth]{../../../gplvm/tex/diagrams/oil_data/subset_100/d_5_kbr_01_l_0} \caption{2D latent spaces learned by D-GPLVM on the oil dataset are shown, with 100 training examples and different values of $\sigma_{d}$. Note that as $1 / \sigma_{d}^2$ increases the model becomes more discriminative but has worse generalization. } \label{fig:influence} \end{figure} }

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