Computing Projective Implicit Representations from Poset Towers

TL;DR

This work develops a framework to compute the persistent homology of poset towers K: P -> SCpx by introducing a projective implicit representation (PiRep) built from chain modules represented as $P$-graded projective modules. It provides an algorithm (Presentation) to obtain a minimal two-term projective resolution (p^1 and p^2) and a lift f of the boundary maps, exploiting the special structure of poset towers through simplex generators and edge events. From these resolutions, the paper shows how to assemble a PiRep of $H_ u(K)$ and, via graph-based lifts ($\gamma$, $\vartheta$), to compute a minimal presentation of the homology. The results yield practical, scalable complexity bounds for computing PiReps on general poset towers, laying the groundwork for efficient persistence computations over arbitrary indexing posets and maps.

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.

Figures (11)

• Figure 1: Common indexing posets: (lower left) one-parameter persistence, (upper left) zigzag persistence, (center) two-parameter persistence, (right) two-parameter zigzag persistence.
• Figure 2: One-parameter persistent homology pipeline. Upper left: a filtered simplicial complex together with its algebraic representation by the simplicial chain complex. Right: The simplicial chain spaces as projective interval modules representing the simplices. Lower left: The representation of the boundary maps by graded matrices.
• Figure 3: Left: a simplicial tower with its algebraic representation by a simplicial chain complex. Right: The chain spaces represented by interval modules corresponding to the simplices. Contrary to the case of filtrations, an interval can end if the corresponding simplex dies in the tower.
• Figure 4: Left: a bifiltered simplicial complex with simplex labels shown at their generating grades. Right: The interval modules corresponding to some of the simplices.
• Figure 5: Left: a poset tower over a $2\times 3$-grid. The orange arrow denotes the collapse of the vertex $v_3$ into the vertex $v_2$ when moving to the right. Right: The indecomposable persistence module representing $C_1(K)$.

Theorems & Definitions (45)

• Definition 1: Poset tower
• Definition 2: Simplex generators
• Definition 3: Edge events
• Proposition 4
• Definition 5: $P$-Persistence module
• Definition 6: Projective modules
• Proposition 7
• Definition 8: Projective resolution
• Definition 9
• Theorem 10