Statistical detection of some topological and geometric features
 
 
Description:  The subject of this talk is closely related to the theories of set estimation and manifold estimation. Our object of interest is a, possibly lower-dimensional, compact set M in Rd. The general aim is to identify (via stochastic procedures) some qualitative or quantitative features of M of geometric or topological character. The available information is just a random sample of points drawn on M (noiseless case) or in a parallel set around M (noisy case). The term "to identify" means here to achieve a correct answer almost surely (a.s.) when the sample size tends to infinity. All the proposed methods are based on explicit, feasible algorithms. More specifically, the following topics will be briefly considered: (a) A method to identify (eventually a.s.) whether or not the interior of M is empty. Under some regularity conditions this amounts to decide whether or not M has a dimension smaller than that of the ambient space. Both sample models (noisy and noiseless) are considered. (b) A method to partially denoise a sample. (c) A method to estimate the measure of the boundary of M, as given by the Minkowski content. Again both sample models are studied in this problem. (d) Some simulations and graphical illustrations.

This talk is a summary of joint unpublished work with Catherine Aaron (Université Blaise Pascal, Clermont Ferrand) and Alejandro Cholaquidis (Universidad de la República, Uruguay).
Date:  2017-01-26
Start Time:   15:00
Speaker:  Antonio Cuevas (Univ. Autónoma de Madrid, Spain)
Institution:  Univ. Autónoma de Madrid
Place:  Room 2.4
Research Groups: -Probability and Statistics
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