Simplicial Mixture Models - Fitting topology to data

Lines and planes can be fitted to data by minimising the sum of squared distances from the data to the geometric object.  But what about fitting objects from topology such as simplicial complexes?  I will present a method of fitting topological objects to data using a maximum likelihood approach, generalising the sum of squared distances.  A simplicial mixture model (SMM) is specified by a set of vertex positions and a weighted set of simplices between them.  The fitting process uses the expectation-maximisation (EM) algorithm to iteratively improve the parameters.

Remarkably, if we allow degenerate simplices then any distribution in Euclidean space can be approximated arbitrarily closely using a SMM with only a small number of vertices.  This theorem is proved using a form of kernel density estimation on the n-simplex.