A Geometric Approach for Multivariate Jumps Detection

TL;DR

This paper introduces a geometric framework for detecting jumps in multivariate signals from noisy data, enabling analysis of complex discontinuities and topological features with proven consistency and near-optimality.

Contribution

It proposes a novel geometric approach for jump detection that handles complex discontinuities and infers topological features, extending beyond traditional methods.

Findings

Consistent estimation of jump sets using histogram differences.

Near-optimal detection in Hausdorff distance.

Ability to infer homology groups and persistence diagrams.

Abstract

Our study addresses the inference of jumps (i.e. sets of discontinuities) within multivariate signals from noisy observations in the non-parametric regression setting. Departing from standard analytical approaches, we propose a new framework, based on geometric control over the set of discontinuities. This allows to consider larger classes of signals, of any dimension, with potentially wild discontinuities (exhibiting, for example, self-intersections and corners). We study a simple estimation procedure relying on histogram differences and show its consistency and near-optimality for the Hausdorff distance over these new classes. Furthermore, exploiting the assumptions on the geometry of jumps, we design procedures to infer consistently the homology groups of the jumps locations and the persistence diagrams from the induced offset filtration.