Given an unknown $\mathbb{R}^n$-valued function $f$ on a metric space $X$, can we approximate the persistent homology of $f$ from a finite sampling of $X$ with known pairwise distances and function values? This question has been answered in the case $n=1$, assuming $f$ is Lipschitz continuous and $X$ is a sufficiently regular geodesic metric space, and using filtered geometric complexes with fixed scale parameter for the approximation. In this paper we answer the question for arbitrary $n$, under similar assumptions and using function-geometric multifiltrations. Our analysis offers a different view on these multifiltrations by focusing on their approximation properties rather than on their stability properties. We also leverage the multiparameter setting to provide insight into the influence of the scale parameter, whose choice is central to this type of approach.

翻译：给定度量空间$X$上一个未知的$\mathbb{R}^n$值函数$f$，我们能否通过$X$的有限采样（已知样本间的成对距离和函数值）来逼近$f$的持续同调？对于$n=1$的情形，该问题已得到解决，其前提是假设$f$满足Lipschitz连续性且$X$是充分规则化的测地度量空间，并采用具有固定尺度参数的滤过几何复形进行逼近。本文在类似假设下，利用函数几何多重滤过方法，对任意$n$的情形给出了肯定答案。我们的分析通过聚焦于这类多重滤过的逼近性质而非稳定性性质，为其提供了新的研究视角。此外，我们借助多参数框架深入探讨了尺度参数的影响——该参数的选择正是此类方法的核心所在。