Output-sensitive Computation of Generalized Persistence Diagrams for 2-filtrations

TL;DR

This paper introduces an output-sensitive algorithm for computing generalized persistence diagrams in 2-filtrations, significantly improving efficiency by leveraging a connection to 1-parameter persistence, reducing complexity from brute-force methods.

Contribution

The authors develop a novel output-sensitive algorithm for 2-parameter persistence diagrams, reducing computational complexity by connecting 2-parameter and 1-parameter cases.

Findings

The algorithm runs in O(n^3 + Cn) time, where C is the size of the diagram.

It generalizes persistence diagrams to multi-parameter settings while maintaining key properties.

The approach improves computational efficiency over brute-force methods.

Abstract

When persistence diagrams are formalized as the Mobius inversion of the birth-death function, they naturally generalize to the multi-parameter setting and enjoy many of the key properties, such as stability, that we expect in applications. The direct definition in the 2-parameter setting, and the corresponding brute-force algorithm to compute them, require $\Omega(n^4)$ operations. But the size of the generalized persistence diagram, $C$, can be as low as linear (and as high as cubic). We elucidate a connection between the 2-parameter and the ordinary 1-parameter settings, which allows us to design an output-sensitive algorithm, whose running time is in $O(n^3 + Cn)$.