# Linear Size Sparsifier and the Geometry of the Operator Norm Ball

## TL;DR

This paper explores the geometry of the operator norm ball related to the Matrix Spencer Conjecture, demonstrating that a certain convex body has large Gaussian measure, enabling efficient spectral sparsifier sampling.

## Contribution

It establishes that the convex body of fractional signings is measure-rich, allowing for a new discrepancy-based sampling method with fewer phases for spectral sparsification.

## Key findings

- The convex body of good fractional signings has large Gaussian measure.
- A discrepancy algorithm can efficiently sample linear size spectral sparsifiers.
- Fewer sampling phases are needed compared to previous methods.

## Abstract

The Matrix Spencer Conjecture asks whether given $n$ symmetric matrices in $\mathbb{R}^{n \times n}$ with eigenvalues in $[-1,1]$ one can always find signs so that their signed sum has singular values bounded by $O(\sqrt{n})$. The standard approach in discrepancy requires proving that the convex body of all good fractional signings is large enough. However, this question has remained wide open due to the lack of tools to certify measure lower bounds for rather small non-polyhedral convex sets. A seminal result by Batson, Spielman and Srivastava from 2008 shows that any undirected graph admits a linear size spectral sparsifier. Again, one can define a convex body of all good fractional signings. We can indeed prove that this body is close to most of the Gaussian measure. This implies that a discrepancy algorithm by the second author can be used to sample a linear size sparsifer. In contrast to previous methods, we require only a logarithmic number of sampling phases.