Sharp Convex Concentration for Symmetric Random Tensors with Subgaussian Coordinates

Abstract

Let $X=(X_1,\dots ,X_n)$ have independent coordinates with mean zero, variance one, and $\Vert X_i{\Vert }_{{\psi }_2}\le K$, and let $H_d=({\mathbb{R}}^n{)}^{{\otimes }_{2}d}$. Let $L>0$ and let $f:H_d\to \mathbb{R}$ be convex and $L$-Lipschitz. We prove that, for $0\le t\le c_{K}L{n}^{d/2}$, \[ \textsf{P}\left\{ \left\lvert f(X^{\otimes d})-\textsf{E}f(X^{\otimes d})\right\rvert >t \right\} \le C\exp\left[-c_K\mathcal I_{n,d}\left( \frac{t}{L n^{(d-1)/2}} \right)\right], \] where \[ \mathcal I_{n,d}(s)= \min\left\{ \frac{s^2}{d^2}, \frac{s^2}{d\log(e+nd/s^2)} \right\},\qquad s>0, \qquad \mathcal I_{n,d}(0)=0. \] The first rate is forced by changes in $\Vert X\Vert$. The second comes from changes of $X$ when its norm is nearly fixed. The proof constructs one coupling that controls both the coordinatewise conditional displacement and the mean squared Euclidean distance, and combines these bounds with a second-order estimate for $x\mapsto x^{\otimes d}$. The rate is minimax sharp, scale by scale, even when the subgaussian norms are bounded by an absolute constant. For bounded coordinates the logarithm in the second rate disappears.