Geometry/topology and statistical inference
Speaker: Sayan Mukherjee , Duke University
Date: October 4 2011
Time: 2:30PM to 3:30PM
Location: Sem Rm G449 (Patil/ Kiva)
Host: Lorenzo Rosasco, Istituto Italiano di Tecnologia (IIT); CBCL, MIT
Contact: Kathleen Sullivan, 617-253-0551, kdsulliv@mit.edu
Relevant URL: http://cbcl.mit.edu/
In this talk I will illustrate two examples where geometric/topological ideas and statistical inference complement each other. In the first example, computational geometry is a central tool used to address a classic problem in statistics, inference of conditional dependence. In the second example, a classic object in topology and geometry, a Whitney stratified space, is stated as a mixture model and an algorithm for inference of mixture elements is provided as well as finite sample bounds for the algorithm.
The first part of the talk develops a parameterization of hyper-graphs based on the geometry of points in d-dimensions, the geometric tool here is the abstract simplicial complex. Informative prior distributions on hyper-graphs are induced through this parameterization by priors on point configurations via spatial processes. The approach combines tools from computational geometry and topology with spatial processes and offers greater control on the distribution of graph features than Erdos-Renyi random graphs.
In the second part of the talk, I describe the problem of stratification learning. Strata correspond to unions and intersections of arbitrary manifolds of possibly different dimension. We consider a mixture distribution on the strata and formulate the following learning problem: given n points sampled iid from the mixture model which points belong to the same strata. I will state a bound on the minimum number of sample points required to infer with high probability which points belong to the same strata. I will show results of this clustering procedure on real data. The clustering procedure uses tools from computational topology, specifically persistence homology.
No knowledge of geometry and topology is assumed in the talk.
The Brains & Machines Seminar Series 2011-2012 is being organized by the IIT@MIT lab (a joint lab between MIT and the Italian Institute of Technology.)
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