T-distributed stochastic neighbor embedding: Difference between revisions

Revision as of 11:48, 26 November 2023

Revision as of 14:54, 30 November 2023

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: <math>p_{j\mid i} = \frac{\exp(-\lVert\mathbf{x}_i - \mathbf{x}_j\rVert^2 / 2\sigma_i^2)}{\sum_{k \neq i} \exp(-\lVert\mathbf{x}_i - \mathbf{x}_k\rVert^2 / 2\sigma_i^2)}</math>

and set <math>p_{i\mid i} = 0</math>.

Note thatthe above denominator ensures <math>\sum_j p_{j\mid i} = 1</math> for all <math>i</math>.

As Van der Maaten and Hinton explained: "The similarity of datapoint <math>x_j</math> to datapoint <math>x_i</math> is the conditional probability, <math>p_{j|i}</math>, that <math>x_i</math> would pick <math>x_j</math> as its neighbor if neighbors were picked in proportion to their probability density under a Gaussian centered at <math>x_i</math>."<ref name=MaatenHinton/>