Bounds on the Interleaving Distance Between n-Parameter Persistence Modules

Explore computational bounds for the interleaving distance between n-parameter persistence modules in this 47-minute conference talk from the Applied Algebraic Topology Network. Learn how to address the computational complexity challenges of measuring interleaving distance in topological data analysis, where computing this widely-used metric becomes NP-hard for multi-parameter persistence modules. Discover a novel approach that establishes bounds for the interleaving distance between generalized persistence modules on concrete categories through a loss function methodology. Examine how this loss function, inspired by Chambers et al.'s work on mapper graphs, quantifies the deviation of unnatural transformation pairs from defining proper interleavings. Master methods for optimally computing this loss function specifically for n-parameter persistence modules, with proven polynomial-time computability for modules valued in finite sets and vector spaces categories. Follow along with illustrative examples and visual representations that demonstrate the practical applications and theoretical foundations of these computational techniques in topological data analysis.