On the spectral norm of Gaussian random matrices

By Ramon van Handel. 2015

59 citations recorded by the source corpus.

Let $X$ be a $d\times d$ symmetric random matrix with independent but non-identically distributed Gaussian entries. It has been conjectured by Lata\l{a} that the spectral norm of $X$ is always of the same order as the largest Euclidean norm of its rows. A positive resolution of this conjecture would provide a sharp understanding of the probabilistic mechanisms that control the spectral norm of inhomogeneous Gaussian random matrices. This paper establishes the conjecture up to a dimensional factor of order $\sqrt{\log\log d}$. Moreover, dimension-free bounds are developed that are optimal to leading order and that establish the conjecture in special cases. The proofs of these results shed significant light on the geometry of the underlying Gaussian processes.

Subfields: Applied Mathematics, Discrete Mathematics and Combinatorics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/6ab17fe5acfbbf97960fa7c8dcaadb4ef33ec721

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Discrete Mathematics and Combinatorics

In Statistics and Probability

Citations in the corpus

Matrix concentration inequalities and free probability (2021, background): “The independent entry setting is the only general model of nonhomogeneous random matrices for which satisfactory norm bounds were obtained prior to this work [5, 38, 22].”
Structured Random Matrices (2016, methodology): “While Conjectures 1 and 2 appear to arise from different mechanisms, it is observed in [30] that these conjectures are actually equivalent: it is not difficult to show that the right-hand side in both inequalities is equivalent, up to the universal constant, to the explicit expression recorded in Co”