Model Selection for Simplicial Approximation

Abstract

In the computational geometry field, simplicial complexes have been used to describe an underlying geometric shape knowing a point cloud sampled on it. In this article, an adequate statistical framework is first proposed for the choice of a simplicial complex among a parametrized family. A least-squares penalized criterion is introduced to choose a complex, and a model selection theorem states how to select the best model, from a statistical point of view. This result gives the shape of the penalty, and then the slope heuristics is used to calibrate the penalty from the data. Some experimental studies on simulated and real datasets illustrate the method for the selection of graphs and simplicial complexes of dimension two.