Equilibrium Distribution for t-Distributed Stochastic Neighbor Embedding with Generalized Kernels
Abstract
We study the large-sample variational problem for t-distributed stochastic neighbor embedding with a class of input and output kernels. The input law has compact support and a density continuous on that support. An entropy equation determines the scale parameter in the input kernel, and we prove that this parameter exists and is unique at interior points of positive density. We then give sufficient conditions for solutions to exist and be uniformly bounded on the entire support. Under these conditions and a decay assumption on the output kernel, the discrete optimal values converge to a continuum minimum. Empirical measures of approximate minimizers are tight after translation; every subsequential limit is a compactly supported minimizer satisfying the equilibrium equation. The admissible output kernels include Gaussian kernels and, in output dimension two, the Cauchy kernel. Numerical examples compare the two-dimensional representations obtained with different kernels.
