Geometry and Topology in Inference

In the first part of the talk we state how geometry can be used for modeling mixtures of subspaces as well as analyzing online (stochastic) optimization algorithms.  We introduce a Bayesian model for inferring mixtures of subspaces of different dimensions. The key challenge in such a model is specifying prior distributions over subspaces of different dimensions. We address this challenge by embedding subspaces or Grassmann manifolds into a sphere of relatively low dimension and specifying priors on the sphere.  We provide an efficient sampling algorithm for the posterior distribution of the model parameters.  We also prove posterior consistency of our procedure.  The utility of this approach is demonstrated with applications to real and simulated data.  We also state a topic model based on this idea.

In the second part of the talk we introduce two statistics, the persistent homology transform (PHT) and Euler characteristic transform (ECT), to model surfaces and shapes. We use these statistics 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 into these statistics is injective. This implies a sufficient statistics. We also show how we can use these statistics in real data analysis.