Towards One Model for Classical Dimensionality Reduction: A Probabilistic Perspective on UMAP and t-SNE

This paper shows that dimensionality reduction methods such as UMAP and t-SNE, can be approximately recast as MAP inference methods corresponding to a model introduced in ProbDR, that describes the graph Laplacian (an estimate for the precision/inverse covariance) matrix using a Wishart distribution, with a mean given by a non-linear covariance function evaluated on the latents. This interpretation offers deeper theoretical and semantic insights into such algorithms, by showing that variances corresponding to these covariances are low (and misspecified), and forging a connection to Gaussian process latent variable models by showing that well-known kernels can be used to describe covariances implied by graph Laplacians. We also introduce tools with which similar dimensionality reduction methods can be studied, and pose two areas of research arising from these interpretations.