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.

翻译：我们利用拓扑数据分析的关键工具——持久同调，解决了多元密度函数多模态估计的问题。我们提出一种基于$H_{0}-$持久图初步估计的流程，用于估计模态数量、其位置及相关局部极大值。对于广泛的分段连续函数类，我们证明了这些估计量具有一致性，并达到近乎极小极大速率。这些函数类涉及对密度函数间断点集的几何控制，与模态推断中通常考虑的函数类有所不同。值得注意的是，我们并不假设模态邻域内存在正则性甚至连续性条件。