Estimating the Persistent Homology of R^n-Valued Functions Using Functional-Geometric Multifiltrations

Explore advanced techniques for approximating persistent homology of multidimensional functions through this research lecture on functional-geometric multifiltrations in topological data analysis. Delve into the fundamental problem of estimating persistent homology for unknown $\mathbb{R}^n$-valued functions defined on metric spaces, given only finite sampling with known pairwise distances and function values. Examine how this extends previous work from the single-dimensional case ($n=1$) to arbitrary dimensions using multiparameter filtrations that have gained prominence for their stability properties. Learn about the statistical convergence proofs for persistent homology estimators, including deviation bounds and confidence intervals that validate the approximation quality. Discover computational methods for computing free presentations of homology modules induced by these multifiltrations, enabling calculation of various topological invariants. Understand the role of structure and stability in persistence modules within this applied topological data analysis framework, building upon more than a decade of research in filtered geometric complexes with fixed scale parameters.