Computing Projective Implicit Representations from Poset Towers

Abstract

A family of simplicial complexes, connected by simplicial maps and indexed by a finite poset \(P\), is called a poset tower. The concept of poset towers subsumes classical objects of study in the persistence literature, such as multi-parameter filtrations, zigzag filtrations, and one-parameter simplicial towers, while also allowing arbitrary finite posets 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 the chain complex segment of \(P\)-persistence modules, \(C_{\ell - 1}\xleftarrow{\partial_{\ell}}C_\ell \xleftarrow{\partial_{\ell+1}}C_{\ell+1}\), induced by the simplices of the poset tower. Contrary to the case of one-critical multi-filtrations, the chain modules \(C_\ell\) of a poset tower are not necessarily projective and can have a complicated structure. In this work, we address the problem of computing a replacement of such a chain complex segment by projective modules and \(P\)-graded matrices, resulting in what we call a projective implicit representation (PiRep), which preserves the homology. Such a representation plays the role of the graded boundary-matrix representation in the classical persistence algorithm: it converts the simplicial data of a poset tower into algebraic input on which persistent homology can be computed globally over \(P\). In particular, a PiRep can be used as input to existing algorithms for computing minimal presentations of persistent homology. We give an efficient algorithm to compute a PiRep from a poset tower. Our algorithm constructs degreewise minimal presentations and asymptotically minimal second terms of projective resolutions of the chain modules \(C_\ell\), lifts the boundary maps $\partial_\ell$ to these resolutions, and assembles this data into a PiRep using an additional correction term. The method is tailored to the chain complexes induced by poset towers and computes the required algebraic data combinatorially by exploiting their special structure, avoiding general-purpose algebraic reduction. In the context of poset towers, it is fully general and can serve as a foundation for developing efficient algorithms on specific posets.