Computing Multidimensional Persistence

TL;DR

This paper introduces a polynomial time algorithm for computing multidimensional persistence, enabling efficient topological analysis of complex scientific data through innovative algebraic geometry techniques.

Contribution

It presents the first polynomial time algorithm for multidimensional persistence computation, leveraging algebraic geometry to improve practicality.

Findings

Algorithms are implemented and tested successfully.

Practical algorithms outperform standard doubly-exponential methods.

Feasibility demonstrated through statistical experiments.

Abstract

The theory of multidimensional persistence captures the topology of a multifiltration -- a multiparameter family of increasing spaces. Multifiltrations arise naturally in the topological analysis of scientific data. In this paper, we give a polynomial time algorithm for computing multidimensional persistence. We recast this computation as a problem within computational algebraic geometry and utilize algorithms from this area to solve it. While the resulting problem is Expspace-complete and the standard algorithms take doubly-exponential time, we exploit the structure inherent withing multifiltrations to yield practical algorithms. We implement all algorithms in the paper and provide statistical experiments to demonstrate their feasibility.