On the Dimension-Free Concentration of Simple Tensors via Matrix Deviation
Pedro Abdalla, Roman Vershynin
TL;DR
This work addresses dimension-free concentration bounds for the empirical mean of simple tensors X^{⊗ p} when X is a mean-zero subgaussian vector, with p ≥ 2. It offers a simpler proof of sharp bounds previously established by Al-Ghattas, Chen and Sanz-Alonso by combining a matrix deviation inequality for ℓ^p norms with a chaining argument, avoiding coordinate-projection analyses. The resulting bound scales with the covariance’s operator norm and effective rank as E ||(1/N) ∑ X_i^{⊗ p} − E X^{⊗ p}|| ≲ C_{K,p} ||Σ||^{p/2} ( r(Σ)^{p/2}/N + sqrt{ r(Σ)/N } ), and holds in an isotropic reduction (via X = Σ^{1/2} Z) with T = Σ^{1/2} S^{d−1}. The approach extends to non-integer p and to broader function classes, contributing a more direct non-asymptotic random matrix perspective on high-order tensor concentration with potential implications for high-dimensional tensor estimation.
Abstract
We provide a simpler proof of a sharp concentration inequality for subgaussian simple tensors obtained recently by Al-Ghattas, Chen and Sanz-Alonso. Our approach uses a matrix deviation inequality for $\ell^p$ norms and a basic chaining argument.