Persistence-based Modes Inference

We address the problem of estimating multiple modes of a multivariate density, using persistent homology, a key tool from Topological Data Analysis. We propose a procedure, based on a preliminary estimation of the $H_{0}-$persistence diagram, to estimate the number of modes, their locations, and the associated local maxima. For large classes of piecewise-continuous functions, we show that these estimators are consistent and achieve nearly minimax rates. These classes involve geometric control over the set of discontinuities of the density, and differ from commonly considered function classes in mode(s) inference. Interestingly, we do not suppose regularity or even continuity in any neighborhood of the modes.