When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2021.35
URN: urn:nbn:de:0030-drops-147281
URL: https://drops.dagstuhl.de/opus/volltexte/2021/14728/
 Go to the corresponding LIPIcs Volume Portal

### The Product of Gaussian Matrices Is Close to Gaussian

 pdf-format:

### 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).

### BibTeX - Entry

```@InProceedings{li_et_al:LIPIcs.APPROX/RANDOM.2021.35,
author =	{Li, Yi and Woodruff, David P.},
title =	{{The Product of Gaussian Matrices Is Close to Gaussian}},
booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)},
pages =	{35:1--35:22},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-207-5},
ISSN =	{1868-8969},
year =	{2021},
volume =	{207},
editor =	{Wootters, Mary and Sanit\`{a}, Laura},
publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},