arXiv subjects

Xuanang Hu

Publications and source records attributed to Xuanang Hu.

6 recordsLinked to original sources

arXiv2026

Sharp Convex Concentration for Symmetric Random Tensors with Subgaussian Coordinates

Let $X=(X_1,\ldots,X_n)$ have independent coordinates with mean zero, variance one, and $\|X_i\|_{\psi_2}\le K$, 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 $0\le t\le c_KLn^{d/2}$, \[ \textsf{P}\left\{ \left\lvert f(X^{\otimes d})-\textsf{E}f(X^{\otimes d})\right\rvert >t \right\} \le C\exp\left[-c_K\mathcal I_{n,d}\left( \frac{t}{L n^{(d-1)/2}} \right)\right], \] 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.

math.PR↗

arXiv2026

The Optimal Rate in the Averaged Random-Marginal Central Limit Theorem for Log-Concave Measures

Let $X$ be a centered isotropic log-concave random vector in $\mathbb{R}^n$. For $\theta\in S^{n-1}$, let $\mu_\theta$ be the law of $\langle X,\theta\rangle$, and let $\Theta$ be uniformly distributed on $S^{n-1}$, independently of $X$. We prove the sharp estimate \[ \textsf{E} W_1(\mu_\Theta,\gamma_1) \le \frac{C}{n}. \] Here the Wasserstein distance is computed after the direction is fixed and is then averaged over the sphere. No symmetry assumption is imposed. A product measure with centered exponential coordinates gives a matching lower bound of order $n^{-1}$. The proof separates the averaged-direction law from the fluctuation among fixed directions. For the first part, a Taylor expansion in the random radius retains a mean-zero cancellation and yields an $O(n^{-1})$ error. For the second, a weighted $L^2$ distance between distribution functions is converted into an exact spherical kernel depending only on $|x|^2$, $|y|^2$, and $\langle x,y\rangle$. Expanding this kernel in $\langle x,y\rangle$, we control its linear, quadratic, and cubic terms using the quadratic variance inequality $\operatorname{Var}(X^\top M X) \le 8\operatorname{Tr}(M^2)$, while fixed-order moment estimates control the remainder.

math.PR↗

arXiv2026

Failure of Convex-Hull Bounds under Log-Convex Tails

Fix $0 t)=e^{-t^r},\qquad t\ge 0. \] We prove that there is no constant $C_r$, depending only on $r$, with the following universal property: for every finite set $T\subset \R^N$ there exists a sequence $(y_k)_{k\ge 1}\subset \R^N$ such that \[ T-T\subset conv\{y_k:k\ge 1\}, \qquad \|X_{y_k}\|_{L_{\log(k+2)}}\le C_r\,\bx(T) \quad (k\ge 1), \] where $X_t=\sum_i t_i X_i$ and $\bx(T)=\textsf{E}\sup_{t\in T}X_t$. This gives a negative answer to a question of Lata{\l}a concerning the validity of convex-hull bounds for canonical Weibull processes. In fact, the failure persists even when the auxiliary vectors appearing in the convex hull are allowed to be arbitrary.

math.FA↗

arXiv2026

Uniform Concentration for $\alpha$-subexponential Random Operators

Random matrices acting on structured sets play a fundamental role in high-dimensional geometry, compressed sensing, and randomized algorithms. Existing results primarily focus on subgaussian models, when random matrices act as near-isometries on sets with optimal tail behavior. Nevertheless, very often in applications we deal with distributions with heavy tails that are not subgaussian but have at least exponential-type tails. In this work, we study random matrices A whose rows (or columns) have $\alpha$-subexponential tail distributions with $\alpha \in (0,2]$. So subgaussian and sub-exponential models are included in as special cases. We establish concentration type inequality for $Ax$, where x belongs to the bounded subsets of $\mathbb{R}^n$, showing that their geometric distortion is governed by Talagrand's functional of the set and depends on the tail parameter $\alpha$. Our results extend the known optimal inequalities in the subgaussian regime ($\alpha=2$), and provide new guarantees for heavier-tailed, yet exponentially integrable, random matrices. These findings extend the theory of random matrices beyond the subgaussian framework. Moreover, they yield near-isometric embedding results applicable to dimension reduction and allow us to make robust high-dimensional inference under non-Gaussian measurements.

math.PR↗

arXiv2025

The Dual Majorizing Measure Theorem for Canonical Processes

We give a dual, separated-tree formulation of Latala's majorizing measure theorem for canonical processes with log-concave tails. Under the same assumptions as in Latala's characterization, we introduce parameterized separation trees and prove that the expected supremum is equivalent, up to universal constants, to the corresponding tree functional. We also develop a pointwise growth condition, inspired by the contraction principle, which leads to a deterministic polynomial-time algorithm for approximating the expected supremum when the index set is finite.

math.PR↗

arXiv2025

On Concentration Inequality of the Laplacian Matrix of Erd\H{o}s-R\'enyi Graphs

This paper focuses on the concentration properties of the spectral norm of the normalized Laplacian matrix for Erd\H{o}s-R\'enyi random graphs. First, We achieve the optimal bound that can be attained in the further question posed by Le et al. [24] for the regularized Laplacian matrix. Beyond that, we also establish a uniform concentration inequality for the spectral norm of the Laplacian matrix in the homogeneous case, relying on a key tool: the uniform concentration property of degrees, which may be of independent interest. Additionally, we prove that after normalizing the eigenvector corresponding to the largest eigenvalue, the spectral norm of the Laplacian matrix concentrates around 1, which may be useful in special cases.

math.PR↗