🤖 AI Summary
Efficient approximation of multiparameter persistent homology modules remains challenging due to computational intractability and the lack of stable, interpretable descriptors preserving barcode structure.
Method: We introduce a novel family of parameterized topological descriptors enabling controlled approximate decompositions while retaining diagonal barcode structure. We establish the first multiparameter approximation framework with provable interpolation error bounds—quantified via interleaving and bottleneck distances—and perturbation stability, overcoming limitations of single-parameter extensions. Integrating fibered barcodes with exact matching techniques, we design the MMA algorithm, which runs in polynomial time and supports filtrations of arbitrary dimension.
Results: Experiments across multiple datasets demonstrate speedups of several-fold over baselines, approximation accuracy adhering to theoretical error bounds, and robustness to input perturbations—achieving a unified balance of theoretical rigor, algorithmic feasibility, and practical scalability.
📝 Abstract
In this article, we introduce a new parameterized family of topological invariants, taking the form of candidate decompositions, for multi-parameter persistence modules. We prove that our candidate decompositions are controllable approximations: when restricting to modules that can be decomposed into interval summands, we establish theoretical results about the approximation error between our candidate decompositions and the true underlying module in terms of the standard interleaving and bottleneck distances. Moreover, even when the underlying module does not admit such a decomposition, our candidate decompositions are nonetheless stable invariants; small perturbations in the underlying module lead to small perturbations in the candidate decomposition. Then, we introduce MMA (Multipersistence Module Approximation): an algorithm for computing stable instances of such invariants, which is based on fibered barcodes and exact matchings, two constructions that stem from the theory of single-parameter persistence. By design, MMA can handle an arbitrary number of filtrations, and has bounded complexity and running time. Finally, we present empirical evidence validating the generalization capabilities and running time speed-ups of MMA on several data sets.
