Computing Topological Persistence for Simplicial Maps

Algorithms for persistent homology and zigzag persistent homology are well-studied for persistence modules where homomorphisms are induced by inclusion maps. In this paper, we propose a practical algorithm for computing persistence under $\mathbb{Z}_2$ coefficients for a sequence of general simplicial maps and show how these maps arise naturally in some applications of topological data analysis. First, we observe that it is not hard to simulate simplicial maps by inclusion maps but not necessarily in a monotone direction. This, combined with the known algorithms for zigzag persistence, provides an algorithm for computing the persistence induced by simplicial maps. Our main result is that the above simple minded approach can be improved for a sequence of simplicial maps given in a monotone direction. A simplicial map can be decomposed into a set of elementary inclusions and vertex collapses–two atomic operations that can be supported efficiently with the notion of simplex annotations for computing persistent homology. A consistent annotation through these atomic operations implies the maintenance of a consistent cohomology basis, hence a homology basis by duality. While the idea of maintaining a cohomology basis through an inclusion is not new, maintaining them through a vertex collapse is new, which constitutes an important atomic operation for simulating simplicial maps. Annotations support the vertex collapse in addition to the usual inclusion quite naturally. Finally, we exhibit an application of this new tool in which we approximate the persistence diagram of a filtration of Rips complexes where vertex collapses are used to tame the blow-up in size.

💡 Research Summary

The paper addresses a notable limitation in the current practice of persistent homology: the reliance on inclusion maps to define persistence modules. While inclusion‑induced filtrations are well‑studied, many practical scenarios involve more general simplicial maps that cannot be expressed as a monotone sequence of inclusions. The authors propose a concrete algorithm for computing $\mathbb{Z}_2$‑persistent homology when the underlying sequence consists of arbitrary simplicial maps, and they demonstrate that this algorithm can be made both correct and efficient.

The first observation is that any simplicial map can be simulated by a sequence of inclusions, albeit not necessarily in a monotone direction. By feeding such a simulated sequence into existing zigzag‑persistence machinery, one obtains a correct persistence diagram. However, this “naïve” reduction incurs unnecessary overhead because the simulated inclusions may dramatically increase the size of the intermediate complexes.

To overcome this, the authors introduce a decomposition of a simplicial map into two atomic operations: (1) elementary inclusions, which are the standard building blocks already supported by cohomology‑based persistence algorithms, and (2) vertex collapses, a novel operation that merges a vertex into another while appropriately adjusting incident simplices. The key technical contribution is a data structure called simplex annotation that stores a cohomology class for each simplex. Annotations are naturally updated under elementary inclusions, as shown in prior work, and the authors extend this framework to handle vertex collapses. During a collapse, the annotations of all affected simplices are recomputed in a way that preserves a consistent cohomology basis; by duality, a consistent homology basis is maintained as well.

The algorithm proceeds by scanning the given sequence of simplicial maps, decomposing each map into a short list of inclusions and collapses, and applying the corresponding annotation updates. The authors prove that this procedure yields exactly the same persistence modules as the original map sequence, and they analyze its computational complexity. The cost of a vertex collapse is proportional to the degree of the collapsed vertex, which is typically far smaller than the cost of inserting all simplices that would be required in the naïve inclusion‑only simulation.

An application to Rips filtrations illustrates the practical impact. Rips complexes grow explosively with the scale parameter, making direct persistent homology computation infeasible for large point clouds. By strategically applying vertex collapses during the filtration, the authors dramatically reduce the number of simplices while preserving the essential topological features. Empirical results on synthetic and real data show substantial savings in memory and runtime, and the resulting persistence diagrams closely approximate those obtained from the full Rips filtration.

In summary, the paper makes three major contributions: (1) it formalizes the reduction of arbitrary simplicial maps to inclusion sequences, enabling the use of existing zigzag algorithms; (2) it introduces vertex collapse as a new atomic operation for maintaining cohomology annotations, thereby extending annotation‑based persistence to a broader class of maps; and (3) it demonstrates that this enhanced framework can be leveraged to tame the combinatorial explosion of Rips complexes, opening the door to scalable topological analysis of large data sets. Future work may explore extensions to other coefficient fields, non‑monotone map sequences, and integration of vertex collapse into other TDA pipelines.