Data Analysis using Computational Topology and Geometric Statistics

(in alphabetic order by speaker surname) Speaker: Dominique Attali (CNRS, Grenoble) Title: Persistence-sensitive simplification of functions on surfaces in linear time. Abstract: Let f be a real-valued function defined on a triangulated surface S. The persistence diagram of f encodes the homological variations in the sequence of sublevel sets St = f−1(−∞, t]. A point (x, y) in the persistence diagram of f corresponds to a homological class which appears in Sx and disappears in Sy. The distance y − x of the point (x, y) to the diagonal represents the importance of the associated homological class: the further away a point is from the diagonal, the more important the associated feature. An εsimplification of f is a map g on S whose persistence diagram consists only of those points in the diagram of f that are more than ε away from the diagonal. In this talk, we give an algorithm for constructing an ε-simplification of f which is also ε-close to f . This is a joint work with M. Glisse, S. Hornus, F. Lazarus and D. Morozov. Speaker: Peter Bubenik (Cleveland State University) Title: Persistent homology and nonparametric regression. Abstract: We estimate the persistent homology of sublevel sets of a function on a compact Riemannian manifold, from a finite noisy sample. The Stability Theorem of Cohen-Steiner, Edelsbrunner and Harer bounds the distance between the persistent homologies of the sublevel sets of two functions by the supremum norm of the difference between the two functions. This allows us to convert our topological problem to the statistical nonparametric regression problem on a compact manifold under the sup-norm loss. We calculate the sharp asymptotic minimax bound. Furthermore, the construction of the estimator in the proof is wellsuited to calculations of the persistent homology of its sublevel sets. We illustrate these techniques with an application to brain image data. This is joint work with Gunnar Carlsson, Moo Chung, Peter Kim, and Zhiming Luo. Speaker: Gunnar Carlsson (Stanford University) Title: Generalized Persistence, Noise, and Statistical Significance Abstract: Persistent homology has been shown to be a useful way to detect qualitative structure in various kinds of data sets. Recently, in joint work with V. de Silva and D. Morozov, we have shown that a generalized form of persistence, which we call ”zig-zag persistence”, can be useful both in removing noise in certain geometric problems as well as in understanding statistical significance of qualitative geometric invariants. I will describe the techniques and the various ways in which it can be applied. Speaker: Fred Chazal and David Cohen-Steiner (INRIA) Title: Geometric inference for probability distributions Abstract: Data often comes in the form of a point cloud sampled from an unknown compact subset of Euclidean space. The general goal of geometric inference is then to recover geometric and topological features (Betti numbers, curvatures,) of this subset from the approximating point cloud data. In recent years, it appeared that the study of distance functions allows to address many of these questions successfully. However, one of the main limitations of this framework is that it does not cope well with outliers nor with background noise. In this talk, we will show how to extend the framework of distance functions to overcome this problem. Replacing compact subsets by measures, we will introduce a notion of distance function to a probability distribution in Rn. These functions share many properties with classical distance functions, which makes them suitable for inference purposes. In particular, by considering appropriate level sets of