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I'm trying to understand t-SNE better and I was hoping someone could elaborate on how the $\sigma _i$'s are chosen.

I was also wondering why they aren't just calculated in the normal way standard deviation is calculated from the data with $x_{i} $ as the expected value for each $\sigma _i$.

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σi is basically the radius for xi, so that there are roughly the same number of neighboring data point within that radius of xi. The number of those neighboring data points is controlled by the 'perplexity' parameter of the t-SNE algorithm.

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