Mittagsseminar (in cooperation with J. Lengler, A. Steger, and D. Steurer)

Date and Time: Tuesday, June 12, 2012, 12:15 pm

Duration: 30 minutes

Location: HG D5.2

Speaker: Andrea Roth (INRIA Sophia-Antipolis)

On Discrete Morse-Smale Decompositions and Bifurcations Diagrams

In the realm of differential topology, Morse theory provides a powerful framework to study the topology of a manifold from a function defined on it. But the smooth concepts do not easily translate in the discrete setting, so that defining critical points and (un-)stable manifolds remains a challenge for functions sampled over a discrete domain, be it a point set or simplicial complex.

Consider the problem of approximating the Morse-Smale (MS) complex of a function sampled on a manifold. Practically, we assume that a point cloud is given, from which a nearest neighbor graph is inferred. We present novel concepts of (selected) critical points (of any index) together with the associated (un-)stable manifolds. These concepts aim at approximating thickened versions of the (un-)stable manifolds, and thus depart from strategies merely mimicking the smooth setting.

We believe that our approach will prove useful for a variety of applications including geometry processing, computational topology, scientific computing (study of vector fields in general), gradient-free non-convex optimization, and molecular modeling.On the theoretical side, we introduce the multi-scale landscape analysis (MLA) framework, an effective version of Morse theory for sampled spaces, aiming at identifying (selected) critical points of any index together with the associated (un-)stable manifolds. Our constructions approximate thickened versions of the (un-)stable manifolds of the critical points, and thus depart from strategies merely mimicking the smooth setting. We further show how topological persistence, a body of methods from computational topology, can be used to single out the most prominent critical points.

On the experimental side, illustrations will be provided on usual non-convex multivariate functions used as benchmark in optimization, and also on polynomial energy landscapes whose critical points can be certified using real algebraic geometry tools.

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