We consider the estimation of multiple modes of a (multivariate) density. We start by proposing an estimator of the $H_0$ persistence diagram. We then derive from it a procedure 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 achieve nearly minimax rates. These classes involve geometric control over the discontinuities set and differ from commonly considered function classes in mode(s) inference. Although the global regularity assumptions are stronger, we do not suppose regularity (or even continuity) in any neighborhood of the modes.

翻译：我们考虑（多元）密度函数的多个模态估计问题。首先提出一种$H_0$持久图估计器，并由此推导出估计模态数量、位置及相关局部极大值的计算流程。针对具有分段连续特性的广泛函数类，我们证明这些估计器能够达到近乎极小极大收敛速率。这些函数类通过对间断点集的几何约束进行定义，与模态推断中通常考虑的函数类存在差异。尽管全局正则性假设较强，但我们不要求模态邻域内存在任何正则性（甚至连续性）条件。