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. From a practical standpoint, we show that our approximation results are robust to input noise, and that function-geometric multifiltrations have good statistical convergence properties. We also provide an algorithm to compute our estimators, and we use its implementation to conduct extensive experiments, on both synthetic and real biological data, in order to validate our theoretical results and to assess the practicality of our approach.

翻译：给定度量空间$X$上的未知$\mathbb{R}^n$值函数$f$，我们能否从$X$的有限采样（已知成对距离和函数值）中逼近$f$的持续同调？当$n=1$时，在假设$f$是Lipschitz连续且$X$是足够正则的测地度量空间、并使用固定尺度参数的滤波几何复形进行逼近的条件下，该问题已有解答。本文在类似假设下，利用函数-几何多重滤波回答了任意$n$值情形的这一问题。我们的分析通过聚焦这些多重滤波的逼近性质而非稳定性性质，提供了对其的新视角。我们还利用多参数设定，揭示了尺度参数的影响——该参数的选择对此类方法至关重要。从实践角度，我们证明了逼近结果对输入噪声具有鲁棒性，且函数-几何多重滤波具有良好的统计收敛性质。此外，我们给出了计算估计量的算法，并利用其实现进行了大量实验（涵盖合成数据与真实生物数据），以验证理论结果并评估方法的实用性。