arXiv subjects

Zhaorui Dong

Publications and source records attributed to Zhaorui Dong.

5 recordsLinked to original sources

arXiv2026

Extreme principal minors of Wishart and deformed GOE matrices

We study the laws of large numbers for the largest eigenvalues among all principal minors of Wishart matrices and deformed GOE matrices. We propose a new method based on identifying the deterministic sets to which the random sets formed by suitably normalized principal minors converge in Hausdorff distance, thereby reducing the original extreme-value problems to finite-dimensional convex optimization problems. We demonstrate the effectiveness of this method in regimes not covered by the existing second-moment arguments in \cite{cai2021asymptotic,hu2023extreme}. For deformed GOE matrices with fixed minor size $k$, we determine the limit for every diagonal variance $a>0$ and identify a phase transition at $a=2$. Above the transition, the limiting constant satisfies an explicit recursion with no close-form expression, and the optimizers exhibit a nested hierarchical structure, thereby resolving the case left open in \cite{cai2021asymptotic}. For Wishart matrices with general sub-Gaussian entries and fixed $k$, we characterize the limit through an entropy-constrained deterministic convex set. When the entries are standard Gaussian, we solve the resulting optimization problem explicitly and obtain the exact value of the limiting constant.

math.PR↗

arXiv2026

Limiting spectral distributions of large consistent rank correlation matrices

We study random matrices whose entries are obtained by applying consistent rank correlations, such as Hoeffding's $D$, pairwise to a high-dimensional random vector with mutually independent components. Prior work has shown that, in the proportional high-dimensional regime, the empirical spectral distributions of large Kendall's tau and Spearman's rho matrices converge weakly almost surely to the Marchenko--Pastur law. By contrast, we prove that for consistent rank correlations such as Hoeffding's $D$, the limiting spectral distribution is given by the semicircle law. Our result thus generalizes a recent work of Dong, Han, and Yao (2025), who considered the special case of Chatterjee's rank correlation and established the first semicircle law for a large correlation matrix in the proportional regime.

math.PR↗

arXiv2026

Quadratic form of heavy-tailed self-normalized random vector with applications in $\alpha$-heavy Mar\v{c}enko--Pastur law

Let $\mathbf{x}$ be a random vector with $n$ i.i.d.\ real-valued components in the domain attraction of an $\alpha$-stable law with $\alpha\in(0,2)$, and let $\mathbf{y}=\mathbf{x}/\|\mathbf{x}\|_2$ be the associated self-normalized vector on the unit sphere. For a (possibly random) Hermitian matrix $\mathbf{A}_n=\big(a_{ij}^{(n)}\big)$ independent of $\mathbf{y}$, we study the asymptotic law of the quadratic form $\mathbf{y}^\top \mathbf{A}_n \mathbf{y}$. Building on the sharp separation between diagonal and off-diagonal contributions in this heavy-tailed setting, we show that under a mild assumption on the Frobenius norm of the off-diagonal part of $\mathbf{A}_n$ the limiting law is solely governed by the empirical distribution of the diagonal entries and the index $\alpha$. More precisely, if $n^{-1}\sum_{i=1}^n \delta_{a^{(n)}_{ii}}$ converges weakly almost surely to a deterministic $\nu$, then $Q_n$ converges in distribution to a non-degenerate law $\mu_{\nu,\alpha}$ characterized through its Stieltjes transform. The law $\mu_{\nu,\alpha}$ is shown to be atom-free (provided that $\nu$ is non-degenerate) with an explicit density and tractable tail behavior. As an application in random matrix theory, we derive an implicit resolvent-based representation of the $\alpha$-heavy Mar\v{c}enko--Pastur law $H_{\alpha,\gamma}$ for heavy-tailed sample correlation matrices and prove that $H_{\alpha,\gamma}$ has no atoms except possibly at the origin. For comparison with the light-tailed setting, we also provide a Hanson--Wright-type concentration inequality for $\mathbf{y}^\top \mathbf{A}_n \mathbf{y}$ when the components of $\mathbf{x}$ are sub-Gaussian.

math.PR↗

arXiv2025

Spectral analysis of large dimensional Chatterjee's rank correlation matrix

This paper studies the spectral behavior of large dimensional Chatterjee's rank correlation matrix when observations are independent draws from a high-dimensional random vector with independent continuous components. Limits for the empirical spectral distributions of its two symmetrized versions are established in the proportional high-dimensional regime, one of them being the semicircle law, thereby giving a first example of a correlation matrix with a non-Marchenko--Pastur spectral limit, in contrast to the Pearson, Kendall, and Spearman cases. We further establish central limit theorems for linear spectral statistics of the symmetrized matrices. As an important application of this theory, we develop Chatterjee's rank correlation-based tests for the complete independence among the components.

math.ST↗

arXiv2025

Towards Quantifying the Hessian Structure of Neural Networks

Empirical studies reported that the Hessian matrix of neural networks (NNs) exhibits a near-block-diagonal structure, yet its theoretical foundation remains unclear. In this work, we reveal that the reported Hessian structure comes from a mixture of two forces: a ``static force'' rooted in the architecture design, and a ''dynamic force'' arisen from training. We then provide a rigorous theoretical analysis of ''static force'' at random initialization. We study linear models and 1-hidden-layer networks for classification tasks with $C$ classes. By leveraging random matrix theory, we compare the limit distributions of the diagonal and off-diagonal Hessian blocks and find that the block-diagonal structure arises as $C$ becomes large. Our findings reveal that $C$ is one primary driver of the near-block-diagonal structure. These results may shed new light on the Hessian structure of large language models (LLMs), which typically operate with a large $C$ exceeding $10^4$.

cs.LG↗