Persistence-based Modes Inference

TL;DR

This paper presents a topological data analysis method using persistent homology to accurately estimate the number and locations of modes in multivariate densities, with theoretical guarantees on performance.

Contribution

It introduces a novel approach based on $H_0$-persistence diagrams for mode inference, including a critical separation threshold for reliable detection.

Findings

Method achieves minimax optimal rates above the separation threshold.

Identifies a critical separation threshold for mode distinguishability.

Applicable to broad classes of piecewise-continuous functions.

Abstract

We address the problem of estimating multiple modes of a multivariate density using persistent homology, a central tool in Topological Data Analysis. We introduce a method based on the preliminary estimation of the $H_0$-persistence diagram to infer the number of modes, their locations, and the corresponding local maxima. For broad classes of piecewise-continuous functions with geometric control on discontinuities loci, we identify a critical separation threshold between modes, also interpretable in our framework in terms of modes prominence, below which modes inference is impossible and above which our procedure achieves minimax optimal rates.