Title : Computational information geometry on matrix manifolds
Abstract :
In this talk, I will illustrate information-theoretically flavored algorithmic methods on the space of symmetric positive definite (SPD) matrices. SPD matrices occur as tensors in many science and engineering fields such as materials science, image analysis, statistics, finance, machine learning, radar, and robotics, just to name a few. The space of SPD matrices can be endowed with a geometric structure in several ways: vector space, Riemannian space (Lie group), and more broadly under the framework of differential information geometry (Finsler, etc). We shall focus on the matrix information manifolds implied by a divergence function, representing the dissimilarity measure of matrices. Three classes of generic divergences built on top of a convex contrast function, termed Csiszar, Burbea-Rao, and Bregman, and their interactions will be discussed. We then present basic algorithmic tools for processing and characterizing efficiently finite sets of SPD matrices: center points, clustering, and Voronoi diagrams.
References:
- Total Bregman Divergence and its Applications to DTI Analysis (IEEE
Transactions on Medical Imaging, 2010)
- Simplification and hierarchical representations of mixtures of
exponential families (Elsevier Signal Processing 2010)
- The Burbea-Rao and Bhattacharyya centroids (arxiv 2010)
- Bregman Voronoi Diagrams. (Discrete & Computational Geometry 2010)
- Sided and symmetrized Bregman centroids (IEEE Transactions on
Information Theory, 2009)
- http://www.informationgeometry.org