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The Product of Gaussian Matrices Is Close to Gaussian

Authors Yi Li , David P. Woodruff

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Author Details

Yi Li
  • Division of Mathematical Sciences, Nanyang Technological University, Singapore, Singapore
David P. Woodruff
  • Department of Computer Science, Carnegie Mellon University, Pittsburgh, PA, USA

Funding

  • Li, Yi: Supported in part by Singapore Ministry of Education (AcRF) Tier 2 grant MOE2018-T2-1-013.
  • Woodruff, David P.: Supported in part by Office of Naval Research (ONR) grant N00014-18-1-256 and a Simons Investigator Award.

Acknowledgements

D. Woodruff would like to thank Sébastien Bubeck, Sitan Chen, and Jerry Li for many helpful discussions.

Cite AsGet BibTex

Yi Li and David P. Woodruff. The Product of Gaussian Matrices Is Close to Gaussian. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 35:1-35:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2021.35

Abstract

We study the distribution of the matrix product G₁ G₂ ⋯ G_r of r independent Gaussian matrices of various sizes, where G_i is d_{i-1} × d_i, and we denote p = d₀, q = d_r, and require d₁ = d_{r-1}. Here the entries in each G_i are standard normal random variables with mean 0 and variance 1. Such products arise in the study of wireless communication, dynamical systems, and quantum transport, among other places. We show that, provided each d_i, i = 1, …, r, satisfies d_i ≥ C p ⋅ q, where C ≥ C₀ for a constant C₀ > 0 depending on r, then the matrix product G₁ G₂ ⋯ G_r has variation distance at most δ to a p × q matrix G of i.i.d. standard normal random variables with mean 0 and variance ∏_{i = 1}^{r-1} d_i. Here δ → 0 as C → ∞. Moreover, we show a converse for constant r that if d_i < C' max{p,q}^{1/2}min{p,q}^{3/2} for some i, then this total variation distance is at least δ', for an absolute constant δ' > 0 depending on C' and r. This converse is best possible when p = Θ(q).

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Probability and statistics
Keywords
  • random matrix theory
  • total variation distance
  • matrix product

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