[Paper Review] The Compressed Annotation Matrix: an Efficient Data Structure for Computing Persistent Cohomology

Jean‐Daniel Boissonnat, Tamal K. Dey+1 more|arXiv (Cornell University)|Apr 25, 2013

Topological and Geometric Data Analysis16 references4 citations

TL;DR

This paper introduces the Compressed Annotation Matrix (CAM), a novel data structure that significantly improves the efficiency of computing persistent cohomology by decoupling the simplicial complex representation from cohomology group maintenance. By compressing the annotation matrix and employing a reordering heuristic, CAM reduces both time and space complexity, achieving up to 6.9x speedup over state-of-the-art tools like DioCoH and maintaining stable performance across high-dimensional datasets with hundreds of millions of simplices.

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.

Motivation & Objective

• To address the high computational and memory costs of persistent cohomology computation in large, high-dimensional simplicial complexes.
• To improve upon existing algorithms by decoupling the representation of the simplicial complex from the cohomology group data structure.
• To reduce the number of field operations and matrix updates during computation through a novel compression technique for the annotation matrix.
• To enhance performance across different coefficient fields, including finite fields and rationals, by minimizing matrix operations.
• To enable scalable computation of persistent cohomology on complexes with hundreds of millions of simplices using efficient data structures.

Proposed method

• The method separates the simplicial complex representation (using Hasse diagrams or simplex trees) from the cohomology group representation via a compressed annotation matrix (CAM).
• CAM stores annotation vectors in a compressed sparse format, eliminating duplicate columns and reducing the number of field operations during matrix reduction.
• A reordering heuristic is applied to iso-simplices (simplices with the same filtration value) to minimize the size of the annotation matrix and reduce computational overhead.
• The algorithm maintains cohomology groups using a dynamic data structure that supports efficient column operations and row reductions during the persistent cohomology computation.
• Field operations are optimized by precomputing small-field arithmetic and minimizing the number of operations through compression, which reduces the number of updates by up to 46 times.
• The implementation integrates with the simplex tree data structure to efficiently manage large complexes, enabling scalable computation on high-dimensional datasets.

Experimental results

Research questions

• RQ1 Can a compressed representation of the annotation matrix significantly reduce the time and space complexity of persistent cohomology computation?
• RQ2 How does separating the simplicial complex representation from the cohomology data structure improve algorithmic performance and scalability?
• RQ3 To what extent does reordering iso-simplices reduce the size of the annotation matrix and improve runtime efficiency?
• RQ4 How does the performance of the CAM method compare to state-of-the-art tools like PHAT and DioCoH across diverse datasets and coefficient fields?
• RQ5 Can the CAM approach maintain stable performance per simplex across high-dimensional complexes with complex topology?

Key findings

• CAM reduces the number of field operations by up to 46 times compared to uncompressed methods, achieving less than 1.5 field operations per simplex on average for large complexes like Nep.
• The compressed annotation matrix reduces the size of the stored matrix by a factor of 4.5, with the number of non-zero elements per column and row remaining small (average 0.79, maximum 18).
• Reordering iso-simplices leads to a 4.9x speedup in computation time, as demonstrated on the Bro dataset, significantly improving performance.
• CAM achieves up to 6.9x speedup over DioCoH and maintains stable performance across all tested examples, with a consistent 2.7–9.1×10⁻⁷ seconds per simplex.
• The method scales effectively to complexes with hundreds of millions of simplices, such as S4, L57, Kl, and L35, where it outperforms PHAT in terms of runtime growth.
• The computational time increases by only 8% when switching from the finite field ℤ₁₁ to the rational field ℚ, despite a 34% increase in field operation cost, due to effective compression reducing the number of operations.