[Paper Review] Fast computation of persistent homology representatives with involuted persistent homology

Matija Čufar, Žiga Virk|arXiv (Cornell University)|May 8, 2021

Topological and Geometric Data Analysis4 citations

TL;DR

This paper introduces an efficient algorithm for computing persistent homology representatives by leveraging persistent cohomology to accelerate the extraction of homology cycles. By first reducing the coboundary matrix to identify key simplices and then restricting boundary matrix reduction to only those columns, the method achieves speedups of up to 17x over existing tools like Eirene, especially in Rips filtrations, while maintaining exact representative cycles.

ABSTRACT

Persistent homology is typically computed through persistent cohomology. While this generally improves the running time significantly, it does not facilitate extraction of homology representatives. The mentioned representatives are geometric manifestations of the corresponding holes and often carry desirable information. We propose a new method of extraction of persistent homology representatives using cohomology. In a nutshell, we first compute persistent cohomology and use the obtained information to significantly improve the running time of the direct persistent homology computations. This algorithm applied to Rips filtrations generally computes persistent homology representatives much faster than the standard methods.

Motivation & Objective

• To address the computational bottleneck in extracting persistent homology representatives, which are geometrically meaningful cycles representing topological holes.
• To overcome the limitation of standard persistent cohomology, which yields cocycles instead of cycles, by enabling cycle extraction without sacrificing performance.
• To develop a method that combines the speed of cohomology reduction with the geometric utility of homology representatives.
• To outperform existing tools like Eirene in terms of runtime for computing homology representatives across diverse datasets.

Proposed method

• The algorithm first computes persistent cohomology by reducing the coboundary matrix, identifying death simplices via column reduction.
• It then uses the cohomology reduction results to select a minimal subset of columns from the boundary matrix relevant to representative cycles.
• The boundary matrix is restricted to only these selected columns and reduced, ensuring only necessary computations are performed.
• This two-phase approach—cohomology reduction followed by targeted boundary reduction—is termed 'involuted homology' computation.
• The method exploits structural properties of Rips filtrations and matrix sparsity to minimize computational cost.
• The algorithm is implemented and compared against standard homology reduction, cohomology reduction, and Eirene on diverse datasets including random points, Celegans, and Dragon.

Experimental results

Research questions

• RQ1 Can persistent homology representatives be computed significantly faster by leveraging persistent cohomology?
• RQ2 Does restricting boundary matrix reduction to only relevant columns preserve the correctness of homology representatives?
• RQ3 How does the performance of the proposed involuted homology method compare to standard homology reduction and Eirene in terms of runtime and scalability?
• RQ4 In what settings does the involuted homology method outperform existing tools, particularly in high-dimensional or dense data?
• RQ5 What is the impact of matrix sparsity and filtration structure on the efficiency of the proposed method?

Key findings

• The involuted homology method achieves speedups of up to 17x over Eirene on datasets like dragon1000 and o3_1024, with consistent improvements across most test cases.
• For the o3_1024 dataset, the method reduced computation time from 269.3 seconds (Eirene) to 4.047 seconds, a 66x speedup.
• On the hiv dataset, the method completed in 1.375 seconds (cohomology) and 3.336 seconds (involuted), compared to 6.696 seconds for Eirene.
• The method's total runtime is consistently close to cohomology reduction, which is known to be fast, while still producing exact homology representatives.
• The algorithm is particularly effective on Rips filtrations, where cohomology reduction is significantly faster than homology reduction, enabling substantial performance gains.
• The only case where Eirene outperformed the method was for high-dimensional representatives in random16, suggesting some data-dependent performance variation.