The dimension-free structure of nonhomogeneous random matrices

By Rafal Latala, Ramon van Handel, Pierre Youssef. Inventiones Mathematicae, 2017

93 citations recorded by the source corpus.

Let X be a symmetric random matrix with independent but non-identically distributed centered Gaussian entries. We show that $$\begin{aligned} \mathbf {E}\Vert X\Vert _{S_p} \asymp \mathbf {E}\Bigg [ \Bigg (\sum _i\Bigg (\sum _j X_{ij}^2\Bigg )^{p/2}\Bigg )^{1/p} \Bigg ] \end{aligned}$$E‖X‖Sp≍E[(∑i(∑jXij2)p/2)1/p]for any $$2\le p\le \infty $$2≤p≤∞, where $$S_p$$Sp denotes the p-Schatten class and the constants are universal. The right-hand side admits an explicit expression in terms of the variances of the matrix entries. This settles, in the case $$p=\infty $$p=∞, a conjecture of the first author, and provides a complete characterization of the class of infinite matrices with independent Gaussian entries that define bounded operators on $$\ell _2$$ℓ2. Along the way, we obtain optimal dimension-free bounds on the moments $$(\mathbf {E}\Vert X\Vert _{S_p}^p)^{1/p}$$(E‖X‖Spp)1/p that are of independent interest. We develop further extensions to non-symmetric matrices and to nonasymptotic moment and norm estimates for matrices with non-Gaussian entries that arise, for example, in the study of random graphs and in applied mathematics.

Subfields: Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/96da11f7ff52f0126590e864f88b18d92f2293bb

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Citations in the corpus

Matrix concentration inequalities and free probability (2021, background): “5] (a completely sharp dimension-free bound, but without the optimal constant in the leading term, was proved in [24]).”
Spectral radii of sparse random matrices (2017, background): “are grateful to Sasha Sodin for pointing out and sharing with us a simpli- cation in the proof of Proposition4.2. We would like to thank Ramon van Handel for many helpful discussions on the articles [2,22]. C. B. is supported by grants ANR-14-CE25-0014 and ANR-16- CE40-0024-01. A. K. is supported by”
Largest eigenvalues of sparse inhomogeneous Erdős–Rényi graphs (2017, result): “A consequence of our results, combined with those from the companion paper [4] (see also the recent preprint [14]), where we analyse the extreme eigenvalues in the complementary regime d log n, is a crossover in the behaviour of the extreme eigenvalues around d ∼ log n (the same threshold as for the”