Sharp Concentration of Simple Random Tensors II: Asymmetry

TL;DR

This paper derives sharp concentration inequalities for asymmetric simple random tensors of order three or higher, revealing a unique logarithmic factor phenomenon when covariance ranks cross a critical threshold.

Contribution

It introduces a new empirical process framework for analyzing products of multiple function classes evaluated at different random variables, extending existing techniques to higher-order tensors.

Findings

Asymmetric tensors exhibit a logarithmic factor in concentration bounds when covariance ranks cross a threshold.

The developed empirical process theory extends generic chaining to higher-order product processes.

Results highlight fundamental differences between symmetric and asymmetric tensor concentration behaviors.

Abstract

This paper establishes sharp concentration inequalities for simple random tensors. Our theory unveils a phenomenon that arises only for asymmetric tensors of order $p \ge 3:$ when the effective ranks of the covariances of the component random variables lie on both sides of a critical threshold, an additional logarithmic factor emerges that is not present in sharp bounds for symmetric tensors. To establish our results, we develop empirical process theory for products of $p$ different function classes evaluated at $p$ different random variables, extending generic chaining techniques for quadratic and product empirical processes to higher-order settings.