T-distributed stochastic neighbor embedding: Difference between revisions

Revision as of 14:12, 5 July 2016

Revision as of 18:23, 15 July 2016

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t-SNE aims to learn a <math>d</math>-dimensional map <math>\mathbf{y}_1, \dots, \mathbf{y}_N</math> (with <math>\mathbf{y}_i \in \mathbb{R}^d</math>) that reflects the similarities <math>p_{ij}</math> as well as possible. To this end, it measures similarities <math>q_{ij}</math> between two points in the map <math>\mathbf{y}_i</math> and <math>\mathbf{y}_j</math>, using a very similar approach. Specifically, <math>q_{ij}</math> is defined as:

: <math>q_{ij} = \frac{(1 + \lVert \mathbf{y}_i - \mathbf{y}_j\rVert^2)^{-1}}{\sum_{k \neq il} (1 + \lVert \mathbf{y}_k - \mathbf{y}_i_l\rVert^2)^{-1}}</math>

Herein a heavy-tailed [[Student-t distribution]] (with one-degree of freedom, which is the same as a [[Cauchy distribution]]) is used to measure similarities between low-dimensional points in order to allow dissimilar objects to be modeled far apart in the map.