Computing Projective Implicit Representations from Poset Towers

TL;DR

This paper introduces a method to compute projective implicit representations of homology chain complexes derived from poset towers, generalizing classical persistence tools to more complex filtrations.

Contribution

It develops algorithms for asymptotically minimal projective resolutions of chain modules in poset towers, enabling efficient computation of their homology representations.

Findings

Algorithms achieve asymptotic minimality of resolutions

Efficient computation tailored to poset tower structures

Potential foundation for advanced algorithms on complex filtrations

Abstract

A family of simplicial complexes, connected with simplicial maps and indexed by a poset $P$, is called a poset tower. The concept of poset towers subsumes classical objects of study in the persistence literature, as, for example, one-critical multi-filtrations and zigzag filtrations, but also allows multi-critical simplices and arbitrary simplicial maps. The homology of a poset tower gives rise to a $P$-persistence module. To compute this homology globally over $P$, in the spirit of the persistence algorithm, we consider the homology of a chain complex of $P$-persistence modules, $C_{\ell-1}\xleftarrow{}C_\ell\xleftarrow{}C_{\ell+1}$, induced by the simplices of the poset tower. Contrary to the case of one-critical filtrations, the chain-modules $C_\ell$ of a poset tower can have a complicated structure. In this work, we tackle the problem of computing a representation of such a chain complex segment by projective modules and $P$-graded matrices, which we call a projective implicit representation (PiRep). We give efficient algorithms to compute asymptotically minimal projective resolutions (up to the second term) of the chain modules and the boundary maps and compute a PiRep from these resolutions. Our algorithms are tailored to the chain complexes and resolutions coming from poset towers and take advantage of their special structure. In the context of poset towers, they are fully general and could potentially serve as a foundation for developing more efficient algorithms on specific posets.