# Concentration inequalities for random tensors

## TL;DR

This paper extends concentration inequalities to random tensors with independent entries, demonstrating optimal dimension dependence and showing that most such tensors are well-conditioned and far from linear dependence.

## Contribution

It introduces new concentration inequalities for random tensors with optimal dimension and degree dependence, and proves their well-conditioning properties.

## Key findings

- Random tensors satisfy concentration inequalities with optimal dependence on dimension and degree.
- Most random tensors are far from linear dependence with high probability.
- The results hold for degrees up to o(√(n/ log n)) and conjecture to extend to degrees O(n).

## 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 $d = O(n)$.