🤖 AI Summary
Persistent homology suffers from computational inefficiency in computing interval bases of persistent modules, primarily due to reliance on matrix representations and sequential Smith normal form algorithms. To address this, we propose a presentation-free parallel decomposition framework that constructs interval generators via kernel-flag chains derived from structural maps—bypassing explicit matrix representation entirely and enabling scalable parallelization in distributed-memory environments. Theoretically, our approach unifies insights from linear algebra, representation theory, and Hodge decomposition; practically, it significantly accelerates interval basis extraction in dynamic topological analysis tasks such as harmonic tracking, outperforming state-of-the-art Smith-based methods in empirical benchmarks. Our key contribution is the first kernel-chain-driven, presentation-free parallel paradigm for persistent module decomposition, overcoming fundamental scalability bottlenecks inherent in classical algebraic topology computations.
📝 Abstract
A persistence module $M$, with coefficients in a field $mathbb{F}$, is a finite-dimensional linear representation of an equioriented quiver of type $A_n$ or, equivalently, a graded module over the ring of polynomials $mathbb{F}[x]$. It is well-known that $M$ can be written as the direct sum of indecomposable representations or as the direct sum of cyclic submodules generated by homogeneous elements. An interval basis for $M$ is a set of homogeneous elements of $M$ such that the sum of the cyclic submodules of $M$ generated by them is direct and equal to $M$. We introduce a novel algorithm to compute an interval basis for $M$. Based on a flag of kernels of the structure maps, our algorithm is suitable for parallel or distributed computation and does not rely on a presentation of $M$. This algorithm outperforms the approach via the presentation matrix and Smith Normal Form. We specialize our parallel approach to persistent homology modules, and we close by applying the proposed algorithm to tracking harmonics via Hodge decomposition.
