Efficient Sparsification of Simplicial Complexes via Local Densities of States

2025-02-11

AI Summary

This paper addresses the excessive spectral computation and storage overhead induced by dense simplicial complexes (SCs) in higher-order relational modeling. We propose a probabilistic sparsification method based on local density of states (LDoS), which—novelly—employs LDoS as a spectral-aware descriptor to guide simplex sampling. To circumvent the spectral ill-conditioning of the Hodge Laplacian, we integrate a “kernel-ignoring” decomposition strategy. Theoretically, we prove that the method achieves log-linear time complexity, O(n log n). Experiments on Vietoris–Rips filtered complexes demonstrate that the sparsified complex attains significantly higher spectral fidelity than baselines, while drastically reducing memory and computational costs—all under strict control of reconstruction error. Our core contributions are: (i) an LDoS-driven, spectral-aware sparsification mechanism; and (ii) a theoretically grounded framework with provably efficient complexity guarantees.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Kernel Methods
• Data Mining & Knowledge Management: Mining of Spatial, Temporal or Spatio-Temporal Data

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Web data generation and simulation
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

Simplicial complexes (SCs), a generalization of graph models for relational data that account for higher-order relations between data items, have become a popular abstraction for analyzing complex data using tools from topological data analysis or topological signal processing. However, the analysis of many real-world datasets leads to dense SCs with a large number of higher-order interactions. Unfortunately, analyzing such large SCs often has a prohibitive cost in terms of computation time and memory consumption. The sparsification of such complexes, i.e., the approximation of an original SC with a sparser simplicial complex with only a log-linear number of high-order simplices while maintaining a spectrum close to the original SC, is of broad interest. In this work, we develop a novel method for a probabilistic sparsifaction of SCs. At its core lies the efficient computation of sparsifying sampling probability through local densities of states as functional descriptors of the spectral information. To avoid pathological structures in the spectrum of the corresponding Hodge Laplacian operators, we suggest a"kernel-ignoring"decomposition for approximating the sampling probability; additionally, we exploit error estimates to show asymptotically prevailing algorithmic complexity of the developed method. The performance of the framework is demonstrated on the family of Vietoris--Rips filtered simplicial complexes.

Problem

• Sparsification of dense simplicial complexes
• Efficient computation of sampling probabilities
• Maintaining spectrum close to original structure

Innovation

• Probabilistic sparsification of simplicial complexes
• Local densities of states computation
• Kernel-ignoring decomposition for sampling