The Spectrum of the Singular Values of Z-Shaped Graph Matrices

TL;DR

This paper investigates the spectral distribution of Z-shaped graph matrices, providing new insights into their singular values and extending results to m-layer variants, which enhances understanding of their mathematical properties.

Contribution

It determines the limiting distribution of singular values for Z-shaped graph matrices and generalizes these results to m-layer configurations, advancing theoretical knowledge.

Findings

Derived the limiting spectral distribution of Z-shaped graph matrices

Extended results to m-layer Z-shaped graph matrices

Improved understanding of graph matrix spectral properties

Abstract

Graph matrices are a type of matrix which has played a crucial role in analyzing the sum of squares hierarchy on average case problems. However, except for rough norm bounds, little is known about graph matrices. In this paper, we take a step towards better understanding graph matrices by determining the limiting distribution of the spectrum of the singular values of Z-shaped graph matrices. We then give a partial generalization of our results for $m$-layer Z-shaped graph matrices.