# Cholesky-like Preconditioner for Hodge Laplacians via Heavy Collapsible Subcomplex

arXiv: https://arxiv.org/abs/2401.15492 · HTML: https://mlsanity.com/paper/2401.15492/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Jan 2024.

## Built on by (most cited first, 2 of 2)

- [HiPoNet: A Multi-View Simplicial Complex Network for High Dimensional Point-Cloud and Single-Cell Data](https://mlsanity.com/paper/2502.07746/) (2025)
- [Efficient Sparsification of Simplicial Complexes via Local Densities of States](https://mlsanity.com/paper/2502.07558/) (2025)

## Abstract (arXiv)

Techniques based on k-th order Hodge Laplacian operators Lk 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 Lk 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 Lk 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 Lk and performs better than the standard incomplete Cholesky factorization.