The Compressed Annotation Matrix: an Efficient Data Structure for Computing Persistent Cohomology

TL;DR

This paper introduces the Compressed Annotation Matrix, a new data structure that enhances the efficiency of persistent cohomology computations by reducing matrix operations and space requirements, supported by theoretical and experimental analysis.

Contribution

It presents a novel, compact data structure for persistent cohomology that separates complex representation from cohomology groups, improving computational efficiency.

Findings

Significantly reduces matrix operations.

Decreases memory usage compared to existing methods.

Achieves faster computation times in experiments.

Abstract

The persistent homology with coefficients in a field F coincides with the same for cohomology because of duality. We propose an implementation of a recently introduced algorithm for persistent cohomology that attaches annotation vectors with the simplices. We separate the representation of the simplicial complex from the representation of the cohomology groups, and introduce a new data structure for maintaining the annotation matrix, which is more compact and reduces substancially the amount of matrix operations. In addition, we propose heuristics to simplify further the representation of the cohomology groups and improve both time and space complexities. The paper provides a theoretical analysis, as well as a detailed experimental study of our implementation and comparison with state-of-the-art software for persistent homology and cohomology.