Efficient Sparsification of Simplicial Complexes via Local Densities of States

TL;DR

This work tackles the challenge of sparsifying dense simplicial complexes while preserving spectral properties of the higher-order Laplacians. It introduces a novel approach that expresses the sparsification probabilities in terms of local densities of states (LDoS) via a Kernel-Ignoring Decomposition (KID), enabling efficient approximation of the generalized effective resistance without full eigendecompositions. The authors prove error and complexity bounds, achieving a near log-linear sample regime and demonstrating practical performance on Vietoris–Rips filtrations and real hypergraphs. The method broadens the toolkit for scalable topological data analysis and topological signal processing by delivering spectrally faithful, computationally efficient SC sparsification. This has potential impact on spectral clustering, label propagation, and SC-based neural networks where dense higher-order structures are common.

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.

Figures (7)

• Figure 1: Overview of the proposed method. Top row: the spectral sparsification task and existing sampling approach using generalized effective resistance vector $\mathbf{r}$, spielman2011spectralosting2017spectral. Bottom row: proposed method for approximating the vector $\mathbf{r}$ using local densities of states $\mu_j(\lambda)$ and kernel-ignoring decomposition (KID), \ref{['sec:KID']}.
• Figure 1: Example of a simplicial complex with ordering and orientation: nodes from $\mathcal{V}_{0}(\mathcal{K})$ in magenta, edges from $\mathcal{V}_{1}(\mathcal{K})$ in black, and triangles from $\mathcal{V}_{2}(\mathcal{K})$ in blue. Orientation of edges and triangles is shown by arrows; the action of the $B_2$ operator is exemplified for both triangles. Adapted from savostianov2024cholesky.
• Figure 1: Convergence of the sampled simplicial complex $\mathcal{L}$ to the original complex $\mathcal{K}$ at $0$-order in terms of the spectrum of $L_0$ Laplacian operator. Left pane: convergence rate vs the number of sampled edges $q$ for various perturbed measures $\mathbf{p}^{(\delta)}$. Right pane: convergence rate for chosen values of $\delta$ in relation to the unperturbed sparsifier. $m_0 = 100$, $m_1 = \frac{m_0(m_0-1)}{3}$. All curves are averaged over $25$ random perturbations for VR-complex (see \ref{['sec:benchmark']}).
• Figure 1: Example of VR-filtration. Left pane: point cloud with $m_0 = 40$ and filtration $\epsilon = 1.5$, inter-cluster distance $c = 3$. Right pane: dynamics of the number of simplices of different orders for varying filtration parameter $\epsilon$.
• Figure 2: Dependence of the approximation error $\| \mathbf{p} - \widehat{ \mathbf{p} } \|_\infty$ on the number of moments $M$ and number of MC vectors $N_z$. Values are tested up to (scaled) theoretical bounds from \ref{['thm:error']} (in red); line colors correspond to varying $m_0$ in the point cloud. Left pane: errors vs the number of moments $M$ with fixed theoretical $N_z$; right pane: errors vs the number of MC vectors $N_z$ with fixed theoretical $M$. Errors are averaged over several generated VR-complexes; colored areas correspond to the spread of values.

Theorems & Definitions (13)

• Definition 2.1: Simplicial complex, Lim15
• Definition 2.2: Hodge Laplacian operator
• Theorem 3.1: Simplicial Sparsification, spielman2008graphosting2017spectral
• Remark 3.2
• Definition 4.1: Density of States
• Theorem 4.2: Effective resistance through Local Densities of States
• Proof 1
• Lemma 4.3
• Proof 2
• Lemma 4.4