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Injective norm of random tensors with independent entries

March T. Boedihardjo

TL;DR

The paper addresses bounding the expected injective norm $\mathbb{E}\|Z\|_{\mathrm{inj}}$ of a Gaussian random tensor with independent entries and fixed coefficients $b_{i_{1},\ldots,i_{r}}$. It introduces a Gaussian-process framework via a multilinear map $\tau$, diagonal matrices $D^{(k)}$, and secondary metrics $\eta^{(k)}$, and then uses a generalized Slepian–Fernique argument together with Dudley’s entropy integral to obtain a non-asymptotic bound of the form $\mathbb{E}\|Z\|_{\mathrm{inj}} \leq \sqrt{2r}\sum_{k=1}^{r} \max_{i_{1},\ldots,i_{k-1},i_{k+1},\ldots,i_{r}}\left(\sum_{i_{k}} b_{i_{1},\ldots,i_{r}}^{2}\right)^{1/2} + C r^{3} (\ln d)^{2} \max_{i_{1},\ldots,i_{r}} |b_{i_{1},\ldots,i_{r}}|$, together with a matching lower bound and concentration results. This extends Bandeira–van Handel-type bounds from matrices to higher-order tensors, at the cost of larger logarithmic factors and constants. The approach provides dimension-aware, non-asymptotic control of the injective norm for random tensors, enabling high-dimensional tensor analysis and generalizing known matrix results to the tensor setting.

Abstract

We obtain a non-asymptotic bound for the expected injective norm of a random tensor with independent entries. This bound is similar to the bound by Bandeira and van Handel (2016) for the expected spectral norm of a random matrix with independent entries.

Injective norm of random tensors with independent entries

TL;DR

The paper addresses bounding the expected injective norm of a Gaussian random tensor with independent entries and fixed coefficients . It introduces a Gaussian-process framework via a multilinear map , diagonal matrices , and secondary metrics , and then uses a generalized Slepian–Fernique argument together with Dudley’s entropy integral to obtain a non-asymptotic bound of the form , together with a matching lower bound and concentration results. This extends Bandeira–van Handel-type bounds from matrices to higher-order tensors, at the cost of larger logarithmic factors and constants. The approach provides dimension-aware, non-asymptotic control of the injective norm for random tensors, enabling high-dimensional tensor analysis and generalizing known matrix results to the tensor setting.

Abstract

We obtain a non-asymptotic bound for the expected injective norm of a random tensor with independent entries. This bound is similar to the bound by Bandeira and van Handel (2016) for the expected spectral norm of a random matrix with independent entries.
Paper Structure (6 sections, 12 theorems, 93 equations)

This paper contains 6 sections, 12 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. Three technical objects
  3. Multilinear map and diagonal matrices
  4. The secondary metrics
  5. Summary of properties
  6. Proof of the main result

Key Result

Theorem 1.1

Let $d,r\in\mathbb{N}$. Suppose that $g_{i_{1},\ldots,i_{r}}$, for $i_{1},\ldots,i_{r}\in[d]$, are independent standard Gaussian random variables and $b_{i_{1},\ldots,i_{r}}\in\mathbb{R}$, for $i_{1},\ldots,i_{r}\in[d]$, are fixed. Let Then where $C\geq 1$ is a universal constant.

Theorems & Definitions (25)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Corollary 1.4
  • proof
  • Corollary 1.5
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • ...and 15 more