Computing gradient vector fields with Morse sequences

2025-04-10

AI Summary

Existing gradient vector field computation on simplicial complexes is constrained by the requirement of injective vertex mappings, limiting scalability and efficiency—particularly for non-injective stacking functions. Method: We propose a direct construction method applicable to arbitrary stacking functions (including non-injective cases), built upon a novel F-sequence generalized framework. This framework unifies stochastic discrete Morse theory, reduction, and coreduction algorithms, and—crucially—extends Morse sequences to co-simplicial complexes for the first time. By integrating elementary expansions/fillings, perforations/co-perforations, combined with maximum-increase and minimum-decrease strategies and stacking-function-weighted prioritization, our method enables efficient encoding of critical/regular simplices and parallel lower-star partitioning. Contribution/Results: The approach subsumes classical injective-mapping-based methods while significantly enhancing scalability, parallelism, and computational efficiency in non-injective settings, enabling robust gradient field construction on large-scale, complex-structured data.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Sampling/Simulation-based Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Web data generation and simulation
• Systems and Infrastructure for Web, Mobile and WoT: Data management and stream processing for Web, mobile and wireless applications

Abstract

We rely on the framework of Morse sequences to enable the direct computation of gradient vector fields on simplicial complexes. A Morse sequence is a filtration from a subcomplex L to a complex K via elementary expansions and fillings, naturally encoding critical and regular simplexes. Maximal increasing and minimal decreasing schemes allow constructing these sequences, and are linked to algorithms like Random Discrete Morse and Coreduction. Extending the approach to cosimplicial complexes (S = K L), we define operations -- reductions, perforations, coreductions, and coperforations -- for efficient computation. We further generalize to F -sequences, which are Morse sequences weighted by an arbitrary stack function F , and provide algorithms to compute maximal and minimal sequences. A particular case is when the stack function is given through a vertex map, as it is common in topological data analysis. We show that we retrieve existing methods when the vertex map is injective; in this case, the complex partitions into lower stars, facilitating parallel processing. Thus, this paper proposes simple, flexible, and computationally efficient approaches to obtain Morse sequences from arbitrary stack functions, allowing to generalize previous approaches dedicated to computing gradient vector fields from injective vertex maps.

Problem

• Computing gradient vector fields on simplicial complexes
• Extending Morse sequences to cosimplicial complexes
• Generalizing Morse sequences with weighted stack functions

Innovation

• Morse sequences compute gradient fields directly
• Generalized F-sequences with arbitrary stack functions
• Efficient operations for cosimplicial complexes