Clustering, Coding, and the Concept of Similarity

Abstract

This paper develops a theory of clustering and coding which combines a geometric model with a probabilistic model in a principled way. The geometric model is a Riemannian manifold with a Riemannian metric, g_ij( x), which we interpret as a measure of dissimilarity. The probabilistic model consists of a stochastic process with an invariant probability measure which matches the density of the sample input data. The link between the two models is a potential function, U( x), and its gradient, U( x). We use the gradient to define the dissimilarity metric, which guarantees that our measure of dissimilarity will depend on the probability measure. Finally, we use the dissimilarity metric to define a coordinate system on the embedded Riemannian manifold, which gives us a low-dimensional encoding of our original data.

Tasks

Reproductions