Our study addresses the inference of jumps (i.e. sets of discontinuities) within multivariate signals from noisy observations in the non-parametric regression setting. Departing from standard analytical approaches, we propose a new framework, based on geometric control over the set of discontinuities. This allows to consider larger classes of signals, of any dimension, with potentially wild discontinuities (exhibiting, for example, self-intersections and corners). We study a simple estimation procedure relying on histogram differences and show its consistency and near-optimality for the Hausdorff distance over these new classes. Furthermore, exploiting the assumptions on the geometry of jumps, we design procedures to infer consistently the homology groups of the jumps locations and the persistence diagrams from the induced offset filtration.

翻译：本研究针对非参数回归设置中从含噪观测推断多元信号内跳跃（即不连续点集）的问题。不同于标准的解析方法，我们提出了一种基于对不连续点集进行几何控制的新框架。这使得我们可以考虑更广泛的信号类别（任意维度），这些信号可能具有复杂的不连续性（例如表现出自相交和角点）。我们研究了一种基于直方图差异的简单估计程序，并证明了其在这些新类别上对于豪斯多夫距离的一致性和近似最优性。此外，通过利用关于跳跃几何形状的假设，我们设计了能够一致推断跳跃位置同调群以及由诱导偏移滤链产生的持久图的方法。