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In the t-SNE paper "Visualizing Data using t-SNE" and a Deep Embedded Clustering (DEC) approach "Unsupervised Deep Embedding for Clustering Analysis", they both use the Student t-distribution to quantify the probability/similarity/distance between two data points:

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But how is this q related to the Student's t-distribution shown below?

I'm not quite familiar with the t distribution. When I check the PDF of the t distribution from Wikipedia (in the following), I could not see the link between them. enter image description here

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Note that for $\nu=1$ the PDF becomes

$$f(t)\propto C \cdot (1+t^2)^{-1}$$

With a scaling term $C=\frac{\Gamma(1)}{\sqrt{\pi}\cdot\Gamma\left(\tfrac12\right)}$ independent of t that will eventually cancel out when normalizing the values to a sum of 1.

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    $\begingroup$ So they compute t-distribution probability from Squared Euclidean distance between a data point and a centroid, then the t-distribution probabilities are normalized across all clusters to become the q probabilities. I see. Thank you! $\endgroup$
    – CyberPlayerOne
    Commented Mar 5, 2019 at 5:43
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    $\begingroup$ Neighbors, not clusters. $\endgroup$
    – Has QUIT--Anony-Mousse
    Commented Mar 5, 2019 at 7:08

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