Persistence-based modes inference

Laboratoire de Mathématiques d’Orsay, 307 Rue Michel Magat, 91400 Orsay, France

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Received: 23 May 2025
Accepted: 18 January 2026

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 H₀-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 discontinuity loci, we identify a critical separation threshold between modes, equiv-alently interpretable in our framework in terms of modes' prominence, below which modes inference is impossible and above which our procedure achieves minimax optimal rates.