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A New Proof of the Abstract Random Tensor Estimate by Deng, Nahmod, and Yue

Claire Kaneshiro

TL;DR

This work provides a direct, modular proof of the abstract random tensor estimate, previously established by Deng, Nahmod, and Yue via the moment method. It leverages the non-commutative Khintchine inequality with probabilistic decoupling to bound the spectral norm of higher-order Gaussian tensors in terms of underlying deterministic tensor norms. A key innovation is the Laguerre-type renormalization that accounts for pairings, enabling removal of the square-free (tetrahedral) condition. The approach yields a scalable framework with potential applications to higher-order random tensors and PDE contexts, and builds on merging estimates and Gaussian calculus to deliver the main bound.

Abstract

We provide a new proof of the abstract random tensor estimate. This estimate was initially proven by Deng, Nahmod, and Yue (2022) using the moment method. The key new tool in our proof is the direct use of the non-commutative Khintchine inequality with the probabilistic decoupling of the product of Gaussians. Hermite and generalized Laguerre-type polynomials allow us to account for pairings in the real and complex-valued Gaussians, respectively, and remove the square-free (tetrahedral) requirement.

A New Proof of the Abstract Random Tensor Estimate by Deng, Nahmod, and Yue

TL;DR

This work provides a direct, modular proof of the abstract random tensor estimate, previously established by Deng, Nahmod, and Yue via the moment method. It leverages the non-commutative Khintchine inequality with probabilistic decoupling to bound the spectral norm of higher-order Gaussian tensors in terms of underlying deterministic tensor norms. A key innovation is the Laguerre-type renormalization that accounts for pairings, enabling removal of the square-free (tetrahedral) condition. The approach yields a scalable framework with potential applications to higher-order random tensors and PDE contexts, and builds on merging estimates and Gaussian calculus to deliver the main bound.

Abstract

We provide a new proof of the abstract random tensor estimate. This estimate was initially proven by Deng, Nahmod, and Yue (2022) using the moment method. The key new tool in our proof is the direct use of the non-commutative Khintchine inequality with the probabilistic decoupling of the product of Gaussians. Hermite and generalized Laguerre-type polynomials allow us to account for pairings in the real and complex-valued Gaussians, respectively, and remove the square-free (tetrahedral) requirement.

Paper Structure

This paper contains 6 sections, 9 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Deterministic background
  4. Probabilistic background
  5. Laguerre Polynomials
  6. The Proof of the Abstract Random Tensor Estimate

Key Result

Theorem 1.3

Fix $k \in \mathbb{N}$. Let $h = h_{n_Jn_An_B}$ be a deterministic tensor, where $A$ and $B$ are finite index sets and $J = \{1, 2, \dots, k\}$. Let $n_J = (n_1, n_2, ..., n_k) \in (\mathbb{Z}^d)^k$ and $N \in 2^{\mathbb{N}}$, such that on the support of $h$, where $|\cdot|$ is the standard $\ell_1$ norm. Functionally, the number $N$ is a rough bound on the size of the support. Furthermore, let $

Theorems & Definitions (29)

  • Definition 1.1: Tensors
  • Definition 1.2: Tensor norm
  • Theorem 1.3: Abstract random tensor estimate DNY20
  • Definition 1.4
  • Example 1.5
  • Remark 1.6
  • Lemma 2.1: Tensor norm duality
  • Lemma 2.2: Merging estimate DNY20
  • proof
  • Lemma 2.3: Non-commutative Khintchine inequality
  • ...and 19 more