Spectral Relaxations of the Persistent Rank Invariant - Applied Algebraic Topology Network

Explore spectral relaxations of the persistent rank invariant in this 53-minute lecture from the Applied Algebraic Topology Network. Delve into a framework for constructing continuous relaxations of the persistent rank invariant for parametrized families of persistence vector spaces indexed over the real line. Discover how these families, derived from simplicial boundary operators, obey inclusion-exclusion and encode all necessary information for constructing persistence diagrams. Learn about their unique stability and continuity properties, including smoothness and differentiability over the positive semi-definite cone. Investigate the connection between stochastic Lanczos quadrature and implicit trace estimation, revealing how to iteratively approximate persistence invariants such as Betti numbers, persistent pairs, and cycle representatives in a linear space and "matrix-free" manner suitable for GPU parallelization.