The k-PDTM : a coreset for robust geometric inference

TL;DR

This paper introduces the k-PDTM, a coreset-based approximation of the distance to measure, enabling robust and computationally efficient topological inference of shapes from noisy point clouds.

Contribution

The paper proposes the k-PDTM, a novel coreset-based approximation of the DTM, with theoretical guarantees and an algorithm for efficient computation in high-dimensional noisy data.

Findings

k-PDTM is robust to noise and reduces computational complexity.

Optimality of k = n^{1/3} for 2D shapes is established.

Algorithm for computing k-PDTM is provided.

Abstract

Analyzing the sub-level sets of the distance to a compact sub-manifold of R d is a common method in TDA to understand its topology. The distance to measure (DTM) was introduced by Chazal, Cohen-Steiner and M{\'e}rigot in [7] to face the non-robustness of the distance to a compact set to noise and outliers. This function makes possible the inference of the topology of a compact subset of R d from a noisy cloud of n points lying nearby in the Wasserstein sense. In practice, these sub-level sets may be computed using approximations of the DTM such as the q-witnessed distance [10] or other power distance [6]. These approaches lead eventually to compute the homology of unions of n growing balls, that might become intractable whenever n is large. To simultaneously face the two problems of large number of points and noise, we introduce the k-power distance to measure (k-PDTM). This new approximation of the distance to measure may be thought of as a k-coreset based approximation of the DTM. Its sublevel sets consist in union of k-balls, k << n, and this distance is also proved robust to noise. We assess the quality of this approximation for k possibly dramatically smaller than n, for instance k = n 1 3 is proved to be optimal for 2-dimensional shapes. We also provide an algorithm to compute this k-PDTM.