Year: 2026
Type: preprint
Abstract: We prove that the probability that Gaussian elimination with partial pivoting on $n \times n$ Gaussian random matrices has growth $ρ$ is at least inverse quasi-polynomial: $Ω\left(\exp\left(-c \log^2 \left(\fracρ{\sqrt{n}}\right)\log(n)\right)\right)$ for some constant $c > 0$. In particular, the probability of growth $ρ\ge n^{1/2 + ε}$ is at least $\exp(- O (ε^2 \log^3 (n) ))$. This lower bound breaks standard conjectures in the smoothed and average-case analysis of Gaussian elimination. To the best of our knowledge, it is the first non-trivial lower bound on the probability of large growth for Gaussian random matrices.
Source: arXiv (Cornell University)
Authors: Spielman, Daniel A., Yu, Xifan
Institution: Yale University
Language: English
Topic: Matrix Theory and Algorithms
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
Funder: Office of Naval Research
OpenAlex ID: W7220957172