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Sharp Concentration of Simple Random Tensors II: Asymmetry

Jiaheng Chen, Daniel Sanz-Alonso

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.

Sharp Concentration of Simple Random Tensors II: Asymmetry

Abstract

This paper establishes sharp concentration inequalities for simple random tensors. Our theory unveils a phenomenon that arises only for asymmetric tensors of order 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 different function classes evaluated at different random variables, extending generic chaining techniques for quadratic and product empirical processes to higher-order settings.

Paper Structure

This paper contains 14 sections, 13 theorems, 177 equations.

Table of Contents

  1. Introduction
  2. Main contributions and outline
  3. Notation
  4. Main results
  5. Concentration of simple random tensors
  6. Multi-product empirical processes
  7. Concentration of simple random tensors
  8. Multi-product empirical processes
  9. Background and auxiliary results
  10. Proof of Theorem \ref{['thm:main2']}
  11. Proof of Theorem \ref{['thm:key_lemma']}
  12. Log-concave ensembles
  13. Proofs of auxiliary results
  14. Optimality of expectation bound \ref{['eq:aux_key_lemma']}

Key Result

Theorem 2.1

For any integer $p\ge 2$ and $1 \leq k \leq p$, let $X^{(k)}, X_1^{(k)}, \ldots, X_N^{(k)}$ be i.i.d. centered sub-Gaussian and pre-Gaussian random variables in $H^{(k)}$ with covariance operator $\Sigma^{(k)}$. Then, for any $q\ge 1$, where Moreover, if $X^{(1)},\ldots, X^{(p)}, (X^{(1)}_i)_{i=1}^{N},\ldots,(X^{(p)}_i)_{i=1}^N$ are independent Gaussian, then

Theorems & Definitions (30)

  • Theorem 2.1
  • Remark 2.1
  • Remark 2.2
  • Definition 2.1: Talagrand's $\gamma$ functional
  • Theorem 2.2
  • Remark 2.3
  • proof : Proof of Theorem \ref{['thm:main1']}
  • Proposition 3.1
  • proof : Proof of Proposition \ref{['prop:lowerbound']}
  • Remark 3.1: Discussion on Proposition \ref{['prop:lowerbound']}
  • ...and 20 more