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, $g_{ij}(\mathbf{x})$, 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, $U(\mathbf{x})$, and its gradient, $\nabla U(\mathbf{x})$. 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.

Revised and expanded in response to referee reports. Current version: 65 pages, 18 figures

References in corpus (1)

• Principal Manifolds and Nonlinear Dimension Reduction via Local Tangent Space Alignment

Cited by in corpus (1)

• Differential Similarity in Higher Dimensional Spaces: Theory and Applications