Matrix concentration inequalities and free probability

• 成果类型: Article
• 署名作者: Bandeira, Afonso S. S.; Boedihardjo, March T. T.; van Handel, Ramon
• 署名单位: Swiss Federal Institutes of Technology Domain; ETH Zurich; Princeton University
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-023-01204-6
• 发表日期: 2023
• 页码: 419-487
• 关键词: noncommutative polynomials; NORM; bounds; limit

摘要

A central tool in the study of nonhomogeneous random matrices, the noncommutative Khintchine inequality, yields a nonasymptotic bound on the spectral norm of general Gaussian random matrices X = S-i g(i)A(i) where g(i) are independent standard Gaussian variables and A(i) are matrix coefficients. This bound exhibits a logarithmic dependence on dimension that is sharp when the matrices A(i) commute, but often proves to be suboptimal in the presence of noncommutativity. In this paper, we develop nonasymptotic bounds on the spectrum of arbitrary Gaussian random matrices that can capture noncommutativity. These bounds quantify the degree to which the spectrum of X is captured by that of a noncommutative model X-free that arises from free probability theory. This intrinsic freeness phenomenon provides a powerful tool for the study of various questions that are outside the reach of classical methods of random matrix theory. Our nonasymptotic bounds are easily applicable in concrete situations, and yield sharp results in examples where the noncommutative Khintchine inequality is suboptimal. When combined with a linearization argument, our bounds imply strong asymptotic freeness for a remarkably general class of Gaussian random matrix models that may be very sparse, have dependent entries, and lack any special symmetries. When combined with a universality principle, our bounds extend beyond the Gaussian setting to general sums of independent random matrices.