Type-2 constant for random tensors (injective norm bound)

Establish that for any p ≥ 2, tensor order r ≥ 2, dimension d, and symmetric tensors T1,…,Tn ∈ (R^d)^{⊗ r}, the expected injective ℓ_p norm satisfies E ||∑_{i=1}^n g_i T_i||_{I_p} ≤ ˜O_{r,p}( d^{1/2−1/p} √(∑_{i=1}^n ||T_i||_{I_p}^2) ), where g_i are i.i.d. standard Gaussian variables and ˜O_{r,p} hides constants depending on r and p and polylogarithmic factors in d and n.

Background

The problem asks for sharp concentration bounds for the injective norm of Gaussian sums of symmetric tensors, generalizing classical matrix (r = p = 2) bounds such as Ahlswede–Winter and the noncommutative Khintchine inequality. The conjecture asserts a dimension scaling d^{1/2−1/p}, up to polylogarithmic factors, uniform over the family of tensors.

Existing techniques yield the conjectured bound when p ≥ 2r using geometric covering arguments, but a volumetric barrier prevents extending these arguments to p < 2r. Even the matrix case (r = p = 2) has no known proof by purely geometric entropy/chaining methods. A proof would unify and sharpen tensor concentration inequalities with broad implications across analysis, probability, and theoretical computer science.