Cholesky-like Preconditioner for Hodge Laplacians via Heavy Collapsible Subcomplex

TL;DR

This work targets fast, scalable solutions for linear systems arising from higher-order Hodge Laplacians $L_k$ on simplicial complexes. By exploiting a combinatorial notion of weakly collapsible 2-skeletons, the authors develop HeCS, a sparse, Cholesky-like preconditioner built from a heavy collapsible subcomplex that preserves topology while enabling an exact Cholesky multiplier on the substructure. They derive conditioning bounds and present an end-to-end algorithm combining greedy 2-core reduction with heavy subsampling to produce a bijective, efficient preconditioner for $L_1^{\uparrow}$, reporting substantial improvements in convergence for sparse complexes. The approach offers a practical route to accelerate CG-based LS solves in high-order network dynamics and topology inference, outperforming standard incomplete Cholesky in sparse regimes while remaining robust to nontrivial topology.

Abstract

Techniques based on $k$-th order Hodge Laplacian operators $L_k$ are widely used to describe the topology as well as the governing dynamics of high-order systems modeled as simplicial complexes. In all of them, it is required to solve a number of least square problems with $L_k$ as coefficient matrix, for example in order to compute some portions of the spectrum or integrate the dynamical system. In this work, we introduce the notion of optimal collapsible subcomplex and we present a fast combinatorial algorithm for the computation of a sparse Cholesky-like preconditioner for $L_k$ that exploits the topological structure of the simplicial complex. The performance of the preconditioner is tested for conjugate gradient method for least square problems (CGLS) on a variety of simplicial complexes with different dimensions and edge densities. We show that, for sparse simplicial complexes, the new preconditioner reduces significantly the condition number of $L_k$ and performs better than the standard incomplete Cholesky factorization.

Figures (8)

• Figure 1: Example of the simplicial complex with ordering and orientation: nodes from $\mathcal{V} _{0}( \mathcal{K} )$ in orange, triangles from $\mathcal{V} _{2}( \mathcal{K} )$ in blue. Orientation of edges and triangles is shown by arrows; the action of $B_2$ operator is given for both triangles. \newlabelfig:orientation0
• Figure 1: $2$-Core, examples: all $3$-cliques in graphs are included in corresponding $\mathcal{V} _{2}( \mathcal{K} )$. \newlabelfig:2-core0
• Figure 1: The scheme of the simplicial complex transformation: from the original $\mathcal{K}$ to the heavy weakly collapsible subcomplex $\mathcal{L}$. \newlabelfig:scheme0
• Figure 1: Timings of HeCS-perconditioner
• Figure 2: Example of weakly collapsible but not collapsible simplicial complex \newlabelfig:weak_example0

Theorems & Definitions (26)

• Definition 2.1: Homology group and higher-order Laplacian
• Definition 2.2: Weighted and normalised boundary matrices
• Theorem 3.1: Joint $k$-Laplacian solver
• Proof 1
• Lemma 4.1: Rank-1 decomposition of $L_{k}^{\uparrow}$
• Lemma 4.2: First Schur complement $S_1$ for $L_{1}^{\uparrow}$
• Proof 2
• Example 4.3
• Definition 5.1: Collapsing sequence
• Definition 5.2: Collapsible simplicial complex