Clustering, coding, and the concept of similarity

This paper develops a theory of clustering and coding that combines a geometric model with a probabilistic model in a principled way. The geometric model is a Riemannian manifold with a Riemannian metric, \({g}_{ij}(\textbf{x})\), which we interpret as a measure of dissimilarity. The probabilistic model consists of a stochastic process with an invariant probability measure that matches the density of the sample input data. The link between the two models is a potential function, \(U(\textbf{x})\), and its gradient, \(\nabla U(\textbf{x})\). 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.