Computing Minimal Persistent Cycles - Polynomial and Hard Cases

Explore the computational challenges and solutions for finding minimal persistent cycles in this 55-minute lecture from the Applied Algebraic Topology Network. Delve into the polynomial and hard cases of computing minimal persistent cycles, examining their significance in augmenting persistence diagrams. Learn about the NP-hardness of computing minimal persistent d-cycles for both finite and infinite intervals in arbitrary simplicial complexes. Discover two polynomially tractable cases involving weak pseudomanifolds and their applications in scientific data analysis. Investigate the reduction of the problem to a minimal cut problem for finite intervals and the additional constraints required for infinite intervals. Gain insights into the effectiveness of minimal persistent cycles in capturing significant data features through experimental results.

Intro
Barcode/Persistence diagram
Problem definition
Summary
Weak pseudomanifold
Duality
Correctness of Algorithm 1
Void boundary reconstruction: Orientation
Preprocessing
Correctness of Algorithm 2
Suspension: Shifting dimension for reduction
Infinite interval hardness