Convex concentration for symmetric random tensors

Establish sharp convex concentration inequalities for arbitrary convex Lipschitz functionals of the symmetric random tensor X^{\otimes d}, in the setting of independent subgaussian coordinates, without incurring factors exponential in the tensor degree d through direct decoupling.

Background

The paper studies concentration for a single symmetric random tensor X^{\otimes d}, where the same random vector appears in every tensor slot. This differs from the asymmetric or independently decoupled tensor models because radial changes in X are amplified by a factor proportional to d, while tangential changes are amplified by a factor proportional to \sqrt d. Consequently, a direct reduction to results for independent tensor factors does not preserve the desired dependence on the degree.

The cited prior work had identified the symmetric tensor case as unresolved and observed that a direct decoupling strategy was expected to lose factors exponential in d. The paper develops a coupling and geometric argument that resolves this case, so the passage records the motivating open problem rather than an unresolved problem at the paper’s conclusion.