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not symmetric. To rectify this, the t-SNE algorithm symmetrizes it. \bigskip\noindent + One way to think of this is to consider the fact that the relation ``$p$ is one of the $k$ points closest to $q$'' is not symmetric. Making $P_{j|i}$ symmetric is an analytic way of creating the symmetric relation ``$p$ is one of the $k$ points closest to $q$, or vice versa.'' \bigskip\noindent + This brings outlier points into fuller consideration when constructing the low-dimensional map. \end{frame} @@ -288,6 +290,7 @@ There are both repulsive and attractive forces at work. One can efficiently find the $k$ closest points using, for example, {\it vantage points trees.} These are a data structure specifically designed for this purpose. \bigskip\noindent + This makes $P$ sparse -- there are only a few non-zero entries in each row. \end{frame} \begin{frame} @@ -298,6 +301,7 @@ For the gradient descent phase, one can use a variant of the Barnes-Hut techniqu $$ where $q_{ij}Z=(1+\|y_i-y_j\|^2)^{-1}$ takes constant time to compute. \bigskip\noindent + The first sum requires adding terms corresponding to non-zero entries in $p_{ij}$, which is sparse, so this takes time $O(N)$. \end{frame} \begin{frame} @@ -306,6 +310,7 @@ For the gradient descent phase, one can use a variant of the Barnes-Hut techniqu that if a bunch of points $y_i$ are close together, one may approximate their contribution to the force by replacing them with their center of mass. \bigskip\noindent + The BH algorithm partitions space into cubes that are small enough that the center of mass of the points in each cube are a good summary of the data. \end{frame} \begin{frame} @@ -321,6 +326,7 @@ For the gradient descent phase, one can use a variant of the Barnes-Hut techniqu Hinton and van der Maaten, Visualizing High Dimensional Data with t-SNE, Journal of Machine Learning Research, 2008 \bigskip\noindent + van der Maaten, Accelerating t-SNE using Tree-Based Algorithms, Journal of Machine Learning Reserach, 2014. \end{frame}