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November 2020 Concentration inequalities for random tensors
Roman Vershynin
Bernoulli 26(4): 3139-3162 (November 2020). DOI: 10.3150/20-BEJ1218

Abstract

We show how to extend several basic concentration inequalities for simple random tensors $X=x_{1}\otimes\cdots\otimes x_{d}$ where all $x_{k}$ are independent random vectors in ${\mathbb{R}}^{n}$ with independent coefficients. The new results have optimal dependence on the dimension $n$ and the degree $d$. As an application, we show that random tensors are well conditioned: $(1-o(1))n^{d}$ independent copies of the simple random tensor $X\in{\mathbb{R}}^{n^{d}}$ are far from being linearly dependent with high probability. We prove this fact for any degree $d=o(\sqrt{n/\log n})$ and conjecture that it is true for any $d=O(n)$.

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Roman Vershynin. "Concentration inequalities for random tensors." Bernoulli 26 (4) 3139 - 3162, November 2020. https://doi.org/10.3150/20-BEJ1218

Information

Received: 1 August 2019; Revised: 1 March 2020; Published: November 2020
First available in Project Euclid: 27 August 2020

zbMATH: 07256171
MathSciNet: MR4140540
Digital Object Identifier: 10.3150/20-BEJ1218

Rights: Copyright © 2020 Bernoulli Society for Mathematical Statistics and Probability

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Vol.26 • No. 4 • November 2020
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