Equilibrium Distribution for t-Distributed Stochastic Neighbor Embedding with Generalized Kernels
Read the original on arXiv Machine Learning →The paper investigates the large‑sample variational problem for t‑Distributed Stochastic Neighbor Embedding (t‑SNE) using a broad class of input and output kernels. It shows that an entropy equation uniquely determines the scale parameter of the input kernel at interior points of positive density, and provides sufficient conditions for the existence and uniform boundedness of solutions across the entire support. Under these conditions, together with a decay assumption on the output kernel, the discrete optimal values converge to a continuum minimum, and empirical measures of approximate minimizers are tight after translation, yielding compactly supported minimizers that satisfy the equilibrium equation. The study includes Gaussian and, in two dimensions, Cauchy output kernels, and presents numerical comparisons of two‑dimensional representations obtained with different kernels.
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