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. Poset towers subsume multi-parameter filtrations, zigzag filtrations, and one-parameter simplicial towers, while allowing arbitrary finite posets and simplicial maps. The homology of a poset tower is a $P$-persistence module. To compute it globally over $P$, we consider 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 tower. Unlike in one-critical multi-filtrations, the chain modules $C_\ell$ need not be projective and may have a complicated structure. We address the problem of replacing this segment by projective modules and $P$-graded matrices while preserving homology. The resulting projective implicit representation (PiRep) plays the role of the graded boundary-matrix representation in the classical persistence algorithm: it converts simplicial data into algebraic input on which persistent homology can be computed globally over $P$. In particular, a PiRep can be used as input to algorithms for computing minimal presentations of persistent homology. We give an efficient algorithm to compute a PiRep from a poset tower. It 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 the resulting data into a PiRep using an additional correction term. The method is tailored to chain complexes induced by poset towers and computes the required algebraic data combinatorially, exploiting their special structure and avoiding general-purpose algebraic reduction. In the context of poset towers, it is fully general and can serve as a foundation for efficient algorithms on specific posets.