Random weighted projections, random quadratic forms and random eigenvectors

随机矩阵的特征向量去相关

Eigenvector decorrelation for random matrices

本文研究了随机矩阵在扰动下的特征向量去相关现象，证明了不同谱族特征向量的正交性及二次型大小特性，扩展了热化假设的应用范围。

Spectral Graph Matching and Regularized Quadratic Relaxations I: The Gaussian Model

Graph matching aims at finding the vertex correspondence between two unlabeled graphs that maximizes the total edge weight correlation. This amounts to solving a computationally intractable quadratic assignment problem. In this paper we propose a new spectral method, GRAph Matching by Pairwise eigen-Alignments (GRAMPA). Departing from prior spectral approaches that only compare top eigenvectors, or eigenvectors of the same order, GRAMPA first constructs a similarity matrix as a weighted sum of outer products between all pairs of eigenvectors of the two graphs, with weights given by a Cauchy ke…

Mesoscopic Perturbations of Large Random Matrices

We consider the eigenvalues and eigenvectors of small rank perturbations of random $N\times N$ matrices. We allow the rank of perturbation $M$ increases with $N$, and the only assumption is $M=o(N)$. In both additive and multiplicative perturbation models, we prove rigidity results for the outliers of the perturbed random matrices. Based on the rigidity results we derive the empirical distribution of outliers of the perturbed random matrices. We also compute the appropriate projection of eigenvectors corresponding to the outliers of the perturbed random matrices, which are approximate eigenvec…

非对称稀疏随机矩阵有限秩加性变形的极值特征值和特征向量

Extreme eigenvalues and eigenvectors for finite rank additive deformations of non-hermitian sparse random matrices

研究非对称稀疏随机矩阵在有限秩加性变形下的极值特征值和特征向量行为，证明其与变形矩阵的特征值渐近一致，并揭示了秩一变形下的投影行为与厄密情况相同。

The eigenvalues and eigenvectors of finite, low rank perturbations of large random matrices

We consider the eigenvalues and eigenvectors of finite, low rank perturbations of random matrices. Specifically, we prove almost sure convergence of the extreme eigenvalues and appropriate projections of the corresponding eigenvectors of the perturbed matrix for additive and multiplicative perturbation models. The limiting non-random value is shown to depend explicitly on the limiting eigenvalue distribution of the unperturbed random matrix and the assumed perturbation model via integral transforms that correspond to very well known objects in free probability theory that linearize non-commuta…

Normal fluctuation in quantum ergodicity for Wigner matrices

We consider the quadratic form of a general deterministic matrix on the eigenvectors of an $N\times N$ Wigner matrix and prove that it has Gaussian fluctuation for each bulk eigenvector in the large $N$ limit. The proof is a combination of the energy method for the Dyson Brownian motion inspired by [Marcinek, Yau 2020] and our recent multi-resolvent local laws [Cipolloni, Erdős, Schröder 2020].

稀疏随机正则图中边特征向量普适性的尖锐平方根界

Sharp Square Root Bounds for Edge Eigenvector Universality in Sparse Random Regular Graphs

本文研究稀疏随机正则图特征向量投影的普适性，证明了误差为O(√d·N^{-1/6+ε})的Berry-Esseen界，并通过匹配下界证明√d缩放最优，显著改进了先前d^3缩放的结果。

Quadratic optimal functional quantization of stochastic processes and numerical applications

In this paper, we present an overview of the recent developments of functional quantization of stochastic processes, with an emphasis on the quadratic case. Functional quantization is a way to approximate a process, viewed as a Hilbert-valued random variable, using a nearest neighbour projection on a finite codebook. A special emphasis is made on the computational aspects and the numerical applications, in particular the pricing of some path-dependent European options.

The Cramer-Wold theorem on quadratic surfaces and Heisenberg uniqueness pairs

Two measurable sets $S, Λ\subseteq \mathcal{R}^d$ form a Heisenberg uniqueness pair, if every bounded measure $μ$ with support in S whose Fourier transform vanishes on Λ must be zero. We show that a quadratic hypersurface and the union of two hyperplanes in general position form a Heisenberg uniqueness pair in $\mathcal{R}^d$. As a corollary we obtain a new, surprising version of the classical Cramér-Wold theorem: a bounded measure supported on a quadratic hypersurface is uniquely determined by its projections onto two generic hyperplanes (whereas an arbitrary measure requires the knowledge of…