CENTRO DE INVESTIGACION EN MATEMATICAS A.C.

Coloquios de CIMAT en TDA

Miércoles 26 de noviembre, 4.15 pm, Salón Diego Bricio Hernández

Sufficient Statistics for Shapes and Surfaces

Sayan Mukherjee, Duke University

 Abstract

In this talk we introduce a statistic, the persistent homology transform (PHT), to model surfaces in three-dimensions and shapes in two-dimensions. This statistic is a collection of persistence diagrams- multiscale topological summaries used extensively in topological data analysis. We use the PHT to represent shapes and execute operations such as computing distances between shapes or classifying shapes. We prove the map from the space of simplicial complexes in three-dimensions into the space spanned by this statistic is injective. This implies that the statistic is a sufficient statistic for probability densities on the space of piecewise linear shapes. We also show that a variant of this statistic, the Euler Characteristic Transform (ECT), admits a simple exponential family formulation which is of use in providing likelihood based inference for shapes and surfaces. We illustrate the utility of this statistic on simulated and real data. We close with a discussion of adapting these ideas for networks and graphs.