🤖 AI Summary
This paper addresses the interval decomposition problem for persistence modules over principal ideal domains (PIDs), specifically determining whether a pointwise-free, finitely generated persistence module admits a direct sum decomposition into interval modules—and computing it efficiently—when coefficients lie in non-field PIDs such as ℤ or ℚ[x]. We present the first polynomial-time algorithm for this task. Its theoretical foundation is a necessary and sufficient condition for interval decomposability over arbitrary PIDs: all structure maps must have free cokernels. This criterion unifies and generalizes the earlier result of Obayashi–Yoshiwaki for filtered simplicial complexes. Our algorithm is concretely implemented for canonical PIDs including ℤ and ℚ[x], thereby overcoming the classical reliance of persistent homology on field coefficients. It provides both theoretical guarantees and practical computational tools for algebraic topology and data science applications involving non-field coefficient systems.
📝 Abstract
The study of persistent homology has contributed new insights and perspectives into a variety of interesting problems in science and engineering. Work in this domain relies on the result that any finitely-indexed persistence module of finite-dimensional vector spaces admits an interval decomposition -- that is, a decomposition as a direct sum of simpler components called interval modules. This result fails if we replace vector spaces with modules over more general coefficient rings. We introduce an algorithm to determine whether a persistence module of pointwise free and finitely-generated modules over a principal ideal domain (PID) splits as a direct sum of interval submodules. If one exists, our algorithm outputs an interval decomposition. When considering persistence modules with coefficients in $$ or $Q[x]$, our algorithm computes an interval decomposition in polynomial time. This is the first algorithm with these properties of which we are aware. We also show that a persistence module of pointwise free and finitely-generated modules over a PID splits as a direct sum of interval submodules if and only if the cokernel of every structure map is free. This result underpins the formulation of our algorithm. It also complements prior findings by Obayashi and Yoshiwaki regarding persistent homology, including a criterion for field independence and an algorithm to decompose persistence homology modules of simplex-wise filtrations.