Francesco Tudisco

Associate Professor (Reader) in Machine Learning

School of Mathematics, The University of Edinburgh
The Maxwell Institute for Mathematical Sciences
School of Mathematics, Gran Sasso Science Institute JCMB, King’s Buildings, Edinburgh EH93FD UK
email: f dot tudisco at

Cholesky-like Preconditioner for Hodge Laplacians via Heavy Collapsible Subcomplex

Anton Savostianov, Francesco Tudisco, Nicola Guglielmi,
preprint, (2024)


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.

Please cite this paper as:

  title={Cholesky-like Preconditioner for Hodge Laplacians via Heavy Collapsible Subcomplex},
  author={Savostianov, Anton and Tudisco, Francesco and Guglielmi, Nicola},
  journal={arXiv preprint arXiv:2401.15492},

Links: arxiv doi code

Keywords: preconditioning simplicial complex higher-order networks networks spectral clustering eigenvalue optimization graph Laplacian Hodge Laplacian