Computing Projective Implicit Representations from Poset Towers

2025-05-13

AI Summary

This paper addresses the computational challenge of computing global homology for simplicial towers indexed by arbitrary posets—extending beyond the classical limitations of persistent homology, which applies only to single-critical or multi-filtered chain complexes. We introduce **Projection Implicit Representation (PiRep)**, the first unified algebraic framework capable of modeling arbitrary simplicial maps and multi-critical simplices. To exploit the poset structure, we design an **asymptotically minimal projection resolution algorithm (up to the second term)**, grounded in *P*-persistent module chain complexes, *P*-graded matrices, and projective module representations. Furthermore, we develop specialized algebraic algorithms to efficiently handle intricate chain module structures. PiRep establishes the first general, scalable algebraic foundation for persistent homology over arbitrary posets, thereby enabling theoretically sound and computationally efficient custom algorithm design.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Learning with Manifolds
• Humans and AI: Other Foundations of Human Computation & AI

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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, $C_{ell-1}xleftarrow{}C_ellxleftarrow{}C_{ell+1}$, 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.

Problem

• Computing projective implicit representations for poset towers
• Analyzing homology of chain complexes in P-persistence modules
• Developing efficient algorithms for projective resolutions

Innovation

• Computing homology via projective implicit representations
• Efficient algorithms for minimal projective resolutions
• Tailored solutions for poset tower chain complexes