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Non-asymptotic Results for Singular Values of Gaussian Matrix Products

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Abstract

This article provides a non-asymptotic analysis of the singular values (and Lyapunov exponents) of Gaussian matrix products in the regime where N,  the number of terms in the product, is large and n,  the size of the matrices, may be large or small and may depend on N. We obtain concentration estimates for sums of Lyapunov exponents, a quantitative rate for convergence of the empirical measure of the squared singular values to the uniform distribution on [0, 1], and results on the joint normality of Lyapunov exponents when N is sufficiently large as a function of n. Our technique consists of non-asymptotic versions of the ergodic theory approach at \(N=\infty \) due originally to Furstenberg and Kesten (Ann Math Stat 31(2):457–469, 1960) in the 1960s, which were then further developed by Newman (Commun Math Phys 103(1):121–126, 1986) and Isopi and Newman (Commun Math Phys 143(3):591–598, 1992) as well as by a number of other authors in the 1980s. Our key technical idea is that small ball probabilities for volumes of random projections gives a way to quantify convergence in the multiplicative ergodic theorem for random matrices.

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Acknowledgements

We are grateful to Vadim Gorin, Maurice Duits, and Gernot Akemann for pointing us to a number of interesting references. We would also like to thank a referee for a very careful and helpful reading of an earlier draft that pointed out a number of inaccuracies and ultimately lead to a substantially improved exposition.

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  1. Department of Mathematics, Texas A&M University, College Station, TX, 77843, USA

    Boris Hanin & Grigoris Paouris

  2. Google, Mountain View, CA, 94043, USA

    Boris Hanin

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  1. Boris Hanin
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Correspondence to Boris Hanin.

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BH is supported by NSF Grants DMS-1855684 and CCF-1934904.

GP is supported by NSF Grants DMS-1812240 and CCF-1900929.

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Hanin, B., Paouris, G. Non-asymptotic Results for Singular Values of Gaussian Matrix Products. Geom. Funct. Anal. 31, 268–324 (2021). https://doi.org/10.1007/s00039-021-00560-w

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