The Applied Probability Section of the Royal Statistical Society is happy to host a half-day meeting on Probability for Topological Data Analysis
Date: 13 March 2019
Venue: Royal Statistical Society, 12 Errol Street, London, EC1Y 8LX
14:00-14:45 Frederic Chazal (Inria Saclay Ile-de-France):
Title: On the density of expected persistence diagrams and its kernel based estimation
Abstract: Persistence diagrams play a fundamental role in Topological Data Analysis (TDA) where they are used as topological descriptors of data represented as point cloud. They consist in discrete multisets of points in the plane that can equivalently be seen as discrete measures. When they are built on top of random data sets, persistence diagrams become random measures. In this talk, we will show that, in many cases, the expectation of these random discrete measures has a density with respect to the Lebesgue measure in the plane. We will discuss its estimation and show that various classical representations of persistence diagrams (persistence images, Betti curves,...) can be seen as kernel-based estimates of quantities deduced from it.
This is joint work with Vincent Divol (ENS Paris / Inria DataShape team).
14.45--15.30 Florian Pausinger (Queen's University Belfast)
Title: Persistent Betti numbers of random Cech complexes
Abstract: We study the persistent homology of random \v{C}ech complexes. Generalizing a method of Penrose for studying random geometric graphs, we first describe an appropriate theoretical framework in which we can state and address our main questions. Then we define the k-th persistent Betti number of a random \v{C}ech complex and determine its asymptotic order in the subcritical regime. This extends a result of Kahle on the asymptotic order of the ordinary k-th Betti number of such complexes to the persistent setting.
Joint work with Ulrich Bauer (TU Munich).
15.30--16.00 Break
16:00--16.45 Primoz Srkaba (Queen Mary University of London)
Title: Local-to-Global Stability Results for Persistence
Abstract: One of the main reasons for the popularity of persistence (and the corresponding invariant - a persistence diagram) is stability - under a certain class of perturbations, the difference output is well-behaved (can be bounded by the size of the perturbation of the input). The algebraically most natural metric for measuring the size of the perturbation is called the bottleneck distance. Unfortunately, this metric is a sup-norm, so proving convergence in statistical settings can be difficult. I will present the classical stability results as well as new results for Wasserstein distances. Finally I will discuss how local errors relate to global errors for persistence both for bottleneck and Wasserstein distances. No background in persistence will be assumed for the talk.
16.45--17.30 Open problem and brainstorming session
To register for this meeting, please visit
https://www.statslife.org.uk/events/eventdetail/1356/14/probability-for-topological-data-analysis.
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