Computing Homomorphisms of Poset Representations with Applications to Multiparameter Persistence

TL;DR

This paper introduces algorithms for computing homomorphisms between finitely generated representations of posets, with applications to multiparameter persistence, improving efficiency and implementing in a C++ library.

Contribution

It generalizes homomorphism computation algorithms to any poset, introduces a new theoretical uniqueness result, and benchmarks practical performance in persistent homology.

Findings

Algorithms improve on naive bounds when thick(Y) is small.

Classical approach achieves better asymptotic runtime, especially for decompositions.

Lifting algorithm is fastest in practice for 2-parameter modules.

Abstract

We present algorithms to compute the vector space of homomorphisms Hom(X,Y) between finitely generated representations of the partially ordered set Z^d. Our results generalise to any partially ordered set. Our main theoretical contribution is a uniqueness result for lifts of homomorphisms along free resolutions, which we use to obtain an algorithm running in O(n^4 (thick(Y) + thick(Omega^1 Y))^2 + T_ker(d,n)) time, where thick(Y) denotes the maximal pointwise dimension of Y and T_ker is the time it takes to compute the kernel of a map between projective Z^d-modules. We also apply and analyse a classical approach due to Green, Heath, and Struble (J. Symbolic Comput., 2001), achieving O(n^3 thick(Y)^3 + n^4). Both improve on the naive O(n^6) bound when thick(Y) is small. Applied to the decomposition algorithm AIDA (Dey-J-Kerber, SoCG '25), the classical approach improves the asymptotic runtime the most, strengthening the result of Dey and Xin (J. Appl. Comput. Topology, 2022) for uniquely graded modules. We implement all algorithms in the Persistence Algebra C++ library and benchmark them on the persistent homology of density-alpha bi-filtrations of immune-cell locations. The classical approach has the best worst-case complexity, yet for 2-parameter modules, the lifting algorithm is fastest in practice.