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Estimation of Multiple Modes and Persistent Homology.
Abstract
Detecting and localizing the modes of a probability density (i.e., the points where the density attains a local maximum) is a classical problem in nonparametric statistics. Estimating the global mode, when it is unique, particularly for unimodal densities, has long attracted attention, leading both to the design of efficient algorithms and to a precise characterization of minimax rates under various assumptions on the underlying density. The more general problem of estimating the set of all modes is considerably more challenging. Several approaches have been proposed, notably mean-shift methods, which perform well in practice but whose theoretical properties remain poorly understood. In this talk, we will propose an alternative approach based on a central tool from topological data analysis (TDA): persistent homology and its practical representation through persistence diagrams. We will present several results on the consistency of this method for broad classes of densities, including those that may have discontinuities (even at the modes), as well as its minimax optimality. Beyond mode estimation, we will also discuss the problem of estimating persistence diagrams for such densities.
