Efficient Sparsification of Simplicial Complexes via Local Densities of States

TL;DR

This paper introduces a probabilistic sparsification method for simplicial complexes that preserves spectral properties using local densities of states, significantly reducing complexity for large datasets.

Contribution

We develop a novel sparsification technique based on local densities of states that efficiently approximates the spectrum of simplicial complexes, enabling scalable analysis.

Findings

Effective approximation of generalized effective resistance

Reduces simplices while maintaining spectral properties

Demonstrated on Vietoris–Rips complexes

Abstract

Simplicial complexes (SCs) 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 often leads to dense SCs, with many higher-order simplicies, which results in prohibitive computational requirements in terms of time and memory consumption. The sparsification of such complexes is thus of broad interest, i.e., the approximation of an original SC with a sparser surrogate SC (with typically only a log-linear number of simplices) that maintains the spectrum of the original SC as closely as possible. In this work, we develop a novel method for a probabilistic sparsification of SCs that uses so-called local densities of states. Using this local densities of states, we can efficiently approximate so-called generalized effective resistance of each simplex, which is proportional to the required sampling probability for the sparsification of the SC. To avoid degenerate structures in the spectrum of the corresponding Hodge Laplacian operators, we suggest a ``kernel-ignoring'' decomposition to approximate the sampling probability. Additionally, we utilize certain error estimates to characterize the asymptotic algorithmic complexity of the developed method. We demonstrate the performance of our framework on a family of Vietoris--Rips filtered simplicial complexes.