Persistent Homology and Algebraic Topology of Non-Linear Manifolds in High-Dimensional Data Spaces

We present a theoretical study of persistent homology, topological invariants, and persistence diagrams for non-linear manifolds in high-dimensional data spaces under measurement noise. Moving beyond traditional geometric analysis, we investigate Čech and Vietoris-Rips complexes, prove stability theorems for persistence diagrams, and derive novel algebraic relations for spectral sequences in filtered simplicial complexes. Our analysis establishes convergence theorems for Betti number approximations of random point fields on Riemannian manifolds, as well as an original upper bound for the total error in persistence diagram generation for multi-dimensional tori. Furthermore, we outline efficient computational protocols utilizing boundary matrix reduction algorithms for sparse data structures.