Sharp Convex Concentration for Symmetric Random Tensors with Subgaussian Coordinates

Abstract: Let $X=(X_1,\ldots,X_n)$ have independent coordinates with mean zero, variance one, and $|X_i|<em>{ψ_2}\le K$, and let $$H_d=(\mathbb R^{n)^{\otimes_2} d}$$. Let $L&gt;0$ and let $f:H_d\to\mathbb R$ be convex and $L$-Lipschitz. We prove that, for $0\le t\le c_KLn<sup>{d/2}$, [ \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 $|X|$. 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.

• The paper presents a minimally exponential in $d$ upper tail bound for convex functions of symmetric random tensors.
• This concentration bound is achieved by summing fluctuations relative to the norm and to directions orthogonal to $x$ at typical radius $c=n^{1/2},$ rather than local approaches.
• For bounded coordinates, the paper provides exact concentration bounds mirroring Huang-Tikhomirov.
• Confirm that the upper bound of $mn$ holds when the expectation is divided by subgraph status is $m^2$
• Confirm solutions $h(x)=0$ when $u$ is $c_Kn/d^2$

This paper by Xuanang Hu resolves a problem posed by Vershynin concerning concentration of convex Lipschitz functionals of a single symmetric random tensor $X^{\otimes d}$, where $X=(X_1,\ldots,X_n)$ has independent, centered, variance-one coordinates with $\psi_2$ norm bounded by an absolute constant $K$ (2608.19832). The main theorem gives a two-scale tail bound that the paper proves is minimax sharp, scale by scale, even within the full subgaussian class. The proof combines a new entropy-to-coupling machinery for product measures with a deterministic second-order estimate of the tensor map $x\mapsto x^{\otimes d}$.

The two-scale phenomenon

The central difficulty is geometric. Writing $\Phi_d(x)=x^{\otimes d}$, the derivative satisfies

$$\|D\Phi_d(x)h\|^2 = d\|x\|^{2d-2}\|h\|^2 + d(d-1)\|x\|^{2d-4}\langle x,h\rangle^2,$$

so perturbations parallel to $x$ are amplified by a factor $d$, while perturbations orthogonal to $x$ are amplified only by $X=(X_1,\ldots,X_n)$0. At the typical radius $X=(X_1,\ldots,X_n)$1 this predicts the deviation scale

$X=(X_1,\ldots,X_n)$2

with the first term driven by fluctuations of the norm and the second by fluctuations at nearly fixed norm. This separation is essential: Vershynin's results for simple tensors (independent factors) carry the smaller variance scale $X=(X_1,\ldots,X_n)$3 in the bounded case (2608.19832), but the symmetric model forces the larger scale $X=(X_1,\ldots,X_n)$4, as witnessed already by the Euclidean functional $X=(X_1,\ldots,X_n)$5, where $X=(X_1,\ldots,X_n)$6. Vershynin explicitly identified the symmetric tensor as an open case, noting that decoupling arguments are expected to lose factors exponential in $X=(X_1,\ldots,X_n)$7; this paper avoids such losses entirely.

The main result states that for every convex $X=(X_1,\ldots,X_n)$8-Lipschitz $X=(X_1,\ldots,X_n)$9, if $\psi_2$0, then

$\psi_2$1

with matching moment bounds. A companion formulation covers the entire natural range $\psi_2$2 via the rate function

$\psi_2$3

giving tails of order $\psi_2$4. For bounded coordinates ($\psi_2$5), a separate argument based on Talagrand's convex-distance inequality removes the logarithmic factor entirely, yielding subgaussian tails with variance scale $\psi_2$6 — and the paper shows this scale is sharp even for one fixed bounded marginal law. This mirrors the Huang–Tikhomirov dichotomy between bounded product measures and the full subgaussian class.

The coupling machinery

The probabilistic core is a construction converting relative entropy into two distinct cost functionals simultaneously. For $\psi_2$7 on a product of $\psi_2$8-subgaussian marginals with $\psi_2$9, the paper builds an explicit maximal coupling $K$0 satisfying both

$K$1

and

$K$2

Both estimates must hold for a single coupling because both enter the deterministic tensor inequality. The one-dimensional building block is a maximal coupling that leaves common mass fixed and transports only the excess $K$3 against the deficit $K$4; pointwise entropy calculus converts $K$5 into control of the conditional displacements in both directions. The product version chains these couplings coordinatewise using the chain rule $K$6 and concavity of $K$7.

A notable optimality result: the $K$8 term in the mean squared distance is best possible uniformly over the subgaussian class. For biased measures on $K$9 with bias $x\mapsto x^{\otimes d}$0, one has $x\mapsto x^{\otimes d}$1 while $x\mapsto x^{\otimes d}$2. Consequently, any improvement of the final concentration rates for special subclasses must exploit additional structure beyond general coupling bounds — which is exactly what the paper does for Euclidean functionals, whose polynomial structure permits cancellations invisible to the coupling argument.

The deterministic tensor estimate

On a sphere of radius $x\mapsto x^{\otimes d}$3, for $x\mapsto x^{\otimes d}$4,

$x\mapsto x^{\otimes d}$5

The proof integrates along geodesics: since the integrand's tangent direction is orthogonal to $x\mapsto x^{\otimes d}$6 after averaging, the parallel component of the mean displacement contributes only at second order ($x\mapsto x^{\otimes d}$7), while the orthogonal part enjoys the $x\mapsto x^{\otimes d}$8 factor from the derivative formula. A covering argument extends this to an annulus $x\mapsto x^{\otimes d}$9, adding a term $\Phi_d(x)=x^{\otimes d}$0 with $\Phi_d(x)=x^{\otimes d}$1, controlled by a Bernstein estimate for $\Phi_d(x)=x^{\otimes d}$2 with probability $\Phi_d(x)=x^{\otimes d}$3.

Proof architecture for concentration

The argument compares level sets around a median $\Phi_d(x)=x^{\otimes d}$4. Restricting to the shell $\Phi_d(x)=x^{\otimes d}$5, sets below and above $\Phi_d(x)=x^{\otimes d}$6 separated by a gap $\Phi_d(x)=x^{\otimes d}$7 are coupled via the two-set coupling theorem (which routes both conditional laws through one copy of $\Phi_d(x)=x^{\otimes d}$8, costing $\Phi_d(x)=x^{\otimes d}$9). For $$\|D\Phi_d(x)h\|^2 = d\|x\|^{2d-2}\|h\|^2 + d(d-1)\|x\|^{2d-4}\langle x,h\rangle^2,$$0 in the upper set, convexity gives

$$\|D\Phi_d(x)h\|^2 = d\|x\|^{2d-2}\|h\|^2 + d(d-1)\|x\|^{2d-4}\langle x,h\rangle^2,$$1

so Lipschitzness forces $$\|D\Phi_d(x)h\|^2 = d\|x\|^{2d-2}\|h\|^2 + d(d-1)\|x\|^{2d-4}\langle x,h\rangle^2,$$2, and taking expectation against both coupling costs yields the contradiction when $$\|D\Phi_d(x)h\|^2 = d\|x\|^{2d-2}\|h\|^2 + d(d-1)\|x\|^{2d-4}\langle x,h\rangle^2,$$3 exceeds a constant multiple of the predicted scale. Conversion to moments uses a quantile representation with careful handling of the exponential factor $$\|D\Phi_d(x)h\|^2 = d\|x\|^{2d-2}\|h\|^2 + d(d-1)\|x\|^{2d-4}\langle x,h\rangle^2,$$4 arising near the edge of the moderate range, valid up to $$\|D\Phi_d(x)h\|^2 = d\|x\|^{2d-2}\|h\|^2 + d(d-1)\|x\|^{2d-4}\langle x,h\rangle^2,$$5.

For large degrees, $$\|D\Phi_d(x)h\|^2 = d\|x\|^{2d-2}\|h\|^2 + d(d-1)\|x\|^{2d-4}\langle x,h\rangle^2,$$6, the logarithmic term is dominated and the bound collapses to $$\|D\Phi_d(x)h\|^2 = d\|x\|^{2d-2}\|h\|^2 + d(d-1)\|x\|^{2d-4}\langle x,h\rangle^2,$$7 — shown sharp by the Gaussian radial example. A corollary for Hilbert-valued linear images $$\|D\Phi_d(x)h\|^2 = d\|x\|^{2d-2}\|h\|^2 + d(d-1)\|x\|^{2d-4}\langle x,h\rangle^2,$$8 yields the RMS shift bound $$\|D\Phi_d(x)h\|^2 = d\|x\|^{2d-2}\|h\|^2 + d(d-1)\|x\|^{2d-4}\langle x,h\rangle^2,$$9.

Sharpness

Two complementary lower bounds establish minimax sharpness under an absolute subgaussian constraint:

• Radial lower bound: for $x$0 and $x$1, density comparisons near the mode $x$2 give $x$3, forcing the $x$4 term.
• Sparse-coordinate lower bound: a three-point law with rare large coordinates ($x$5, probability $x$6) combined with the convex functional $x$7, where $x$8 projects onto tensors with exactly one slot in $x$9, produces deviations of order $d$0 with probability $d$1, forcing the logarithmic term.

These are assembled into a rate-function sharpness statement: whenever $d$2 with $d$3, some subgaussian law and convex one-Lipschitz functional achieve $d$4. Minimax sharpness here is scale by scale: each admissible deviation level may use its own extremizing pair, and no single example is claimed to work across all scales.

Limitations and open questions

Several restrictions are explicit. The moment estimate requires $d$5; outside this range the natural range contains no nontrivial fixed-exponent moderate regime, and the analysis of very high degrees relative to $d$6 is not pursued. The logarithmic factor persists throughout for unbounded coordinates, consistent with the Huang–Tikhomirov lower bound, so removing it would require restricting beyond the full subgaussian class — and Proposition on $d$7 optimality shows no uniform improvement of the coupling costs is possible. The paper notes but does not develop an extension to base vectors with multiplicities $d$8, where parallel and orthogonal contributions scale as $d$9 and $x$0 respectively. Finally, stronger rates for structured subclasses (Euclidean functionals, bounded coordinates) rely on algebraic or isoperimetric structure beyond the coupling method; characterizing precisely which subclasses admit improvements remains open.

Conclusion

The paper settles the symmetric-tensor case of convex concentration with the correct dependence on dimension and degree, resolving an open question of Vershynin without exponential losses in $x$1. Its two principal technical contributions — a maximal-coupling construction controlling conditional displacement and squared distance simultaneously under one entropy budget, and a norm-aware second-order tensor estimate separating radial from transverse motion — are of independent interest and appear tight in generality. Together with the matching lower bounds, the results delineate exactly where the $x$2 versus $x$3 variance scales and where the logarithmic correction apply across the bounded, subgaussian, and Gaussian hierarchies.