Inference in dynamical systems and the geometry of learning group actions

Abstract: We examine consistency of the Gibbs posterior for dynamical systems using a classical idea in dynamical systems called the thermodynamic formalism in tracking dynamical systems. We state a variation formulation under which there is a unique posterior distribution of parameters as well as hidden states using using classic ideas from dynamical systems such as pressure and joinings. We use an example of consistency of hidden Markov with infinite lags as an application of our theory.

We develop a geometric framework that characterizes the synchronization problem — the problem of consistently registering or aligning a collection of objects. The theory we formulate characterizes the cohomological nature of synchronization based on the classical theory of fibre bundles and that synchronization can be characterized by trivial holonomy.  We then develop a twisted cohomology theory to quantify obstructions to synchronization, this  is a discrete version of the twisted cohomology in differential geometry.

Motivated by our geometric framework, we study the problem of learning group actions — partitioning a collection of objects based on the local synchronizability of pairwise correspondence relations. A dual interpretation is to learn finitely generated subgroups of an ambient transformation group from noisy observed group elements. A synchronization-based algorithm is also provided, and we demonstrate its efficacy in a problem in geometric morphometrics, clustering the molars of primates according to their eating habits.

Biography: Sayan Mukherjee is Professor of Statistical Science, Mathematics, Computer Science, and Biostatistics \& Bioinformatics at Duke University. He received a PhD in 2001 from the Center for Biological and Computational Learning at MI. He was a  Sloan-DOE Postdoctoral Fellow in Computational Biology 2001-2004 at the Broad Institute of MIT and Harvard. My research areas cover Bayesian methodology; computational and statistical methods in statistical genetics, quantitative genetics, cancer biology, and morphology; discrete Hodge theory, geometry and topology in statistical inference; inference in dynamical systems; machine learning; and stochastic topology. biology being cited over 11,000 times since 2004.