Minimal Cycle Representatives in Persistent Homology Using Linear Programming

Explore the optimization of cycle representatives in persistent homology through this comprehensive lecture. Delve into the effectiveness and computational costs of various $\ell_1$-minimization optimization procedures for constructing homological cycle bases with rational coefficients in dimension one. Examine uniform-weighted and length-weighted edge-loss algorithms, as well as uniform-weighted and area-weighted triangle-loss algorithms. Learn how these optimizations are conducted using standard linear programming methods and general-purpose solvers. Discover key findings on the reduction of cycle representative size, computational costs, the impact of linear solver choice, and the comparison between integer and linear programming solutions. Gain insights into the qualitative differences observed in generators between Erdős-Rényi random clique complexes and real-world or synthetic point cloud data. Enhance your understanding of topological data analysis and its applications in this 50-minute presentation by Lori Ziegelmeier for the Applied Algebraic Topology Network.