  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, ${g}_{ij}({\bf x})$,
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, $U({\bf x})$, and its gradient, $\nabla U({\bf x})$.
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.
