# Sparsified Cholesky Solvers for SDD linear systems

arXiv: https://arxiv.org/abs/1506.08204 · HTML: https://mlsanity.com/paper/1506.08204/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 26 Jun 2015.

## Built on by (most cited first, 1 of 1)

- [Almost linear time differentially private release of synthetic graphs](https://mlsanity.com/paper/2406.02156/) (2024)

## Abstract (arXiv)

We show that Laplacian and symmetric diagonally dominant (SDD) matrices can be well approximated by linear-sized sparse Cholesky factorizations. We show that these matrices have constant-factor approximations of the form L LT, where L is a lower-triangular matrix with a number of nonzero entries linear in its dimension. Furthermore linear systems in L and LT can be solved in O (n) work and O(lognlog2logn) depth, where n is the dimension of the matrix. We present nearly linear time algorithms that construct solvers that are almost this efficient. In doing so, we give the first nearly-linear work routine for constructing spectral vertex sparsifiers---that is, spectral approximations of Schur complements of Laplacian matrices.