Jinho Baik, Jack W. Silverstein
We consider a spiked population model, proposed by Johnstone, whose population eigenvalues are all unit except for a few fixed eigenvalues. The question is to determine how the sample eigenvalues depend on the non-unit population ones when both sample size and population size become large. This paper completely determines the almost sure limits for a general class of samples.
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In this paper, we study the asymptotic behavior of the extreme eigenvalues and eigenvectors of the spiked covariance matrices, in the supercritical regime. Specifically, we derive the joint distribution of the extreme eigenvalues and the generalized components of their associated eigenvectors in this regime.
We consider general high-dimensional spiked sample covariance models and show that their leading sample spiked eigenvalues and their linear spectral statistics are asymptotically independent when the sample size and dimension are proportional to each other. As a byproduct, we also establish the central limit theorem of the leading sample spiked eigenvalues by removing the block diagonal assumption on the population covariance matrix, which is commonly needed in the literature. Moreover, we propose consistent estimators of the $L_4$ norm of the spiked population eigenvectors. Based on these res…
We introduce a class of separable sample covariance matrices of the form $\widetilde{\mathcal{Q}}_1:=\widetilde A^{1/2} X \widetilde B X^* \widetilde A^{1/2}.$ Here $\widetilde{A}$ and $\widetilde{B}$ are positive definite matrices whose spectrums consist of bulk spectrums plus several spikes, i.e. larger eigenvalues that are separated from the bulks. Conceptually, we call $\widetilde{\mathcal{Q}}_1$ a \emph{spiked separable covariance matrix model}. On the one hand, this model includes the spiked covariance matrix as a special case with $\widetilde{B}=I$. On the other hand, it allows for more…
The Asymptotic Properties of the Extreme Eigenvectors of High-dimensional Generalized Spiked Covariance Model
本文针对高维广义尖峰协方差模型，去除块对角等限制假设，运用随机矩阵理论推导极端特征向量投影的渐近性质，技术稳健，还提出了协方差矩阵特征空间假设检验的强效统计量。
We study the asymptotic distributions of the spiked eigenvalues and the largest nonspiked eigenvalue of the sample covariance matrix under a general covariance matrix model with divergent spiked eigenvalues, while the other eigenvalues are bounded but otherwise arbitrary. The limiting normal distribution for the spiked sample eigenvalues is established. It has distinct features that the asymptotic mean relies on not only the population spikes but also the nonspikes and that the asymptotic variance in general depends on the population eigenvectors. In addition, the limiting Tracy-Widom law for …
Asymptotics of ultra-high-dimensional generalized spiked sample covariance matrix
本文研究了在超高维设置下尖峰协方差矩阵模型中样本协方差矩阵的特征结构渐进行为，建立了适当缩放后的样本协方差矩阵特征值位置和特征向量投影的一阶收敛极限。
We consider a more generalized spiked covariance matrix $Σ$, which is a general non-definite matrix with the spiked eigenvalues scattered into a few bulks and the largest ones allowed to tend to infinity. By relaxing the matching of the 4th moment to a tail probability decay, a {\it Generalized Four Moment Theorem} (G4MT) is proposed to show the universality of the asymptotic law for the local spectral statistics of generalized spiked covariance matrices, which implies the limiting distribution of the spiked eigenvalues of the generalized spiked covariance matrix is independent of the actual d…
This paper considers sparse spiked covariance matrix models in the high-dimensional setting and studies the minimax estimation of the covariance matrix and the principal subspace as well as the minimax rank detection. The optimal rate of convergence for estimating the spiked covariance matrix under the spectral norm is established, which requires significantly different techniques from those for estimating other structured covariance matrices such as bandable or sparse covariance matrices. We also establish the minimax rate under the spectral norm for estimating the principal subspace, the pri…
The problem of estimating a spiked covariance matrix in high dimensions under Frobenius loss, and the parallel problem of estimating the noise in spiked PCA is investigated. We propose an estimator of the noise parameter by minimizing an unbiased estimator of the invariant Frobenius risk using calculus of variations. The resulting estimator is shown, using random matrix theory, to be strongly consistent and essentially asymptotically normal and minimax for the noise estimation problem. We apply the construction to construct a robust spiked covariance matrix estimator with consistent eigenvalue…
In the spiked population model introduced by Johnstone (2001),the population covariance matrix has all its eigenvalues equal to unit except for a few fixed eigenvalues (spikes). The question is to quantify the effect of the perturbation caused by the spike eigenvalues. Baik and Silverstein (2006) establishes the almost sure limits of the extreme sample eigenvalues associated to the spike eigenvalues when the population and the sample sizes become large. In a recent work (Bai and Yao, 2008), we have provided the limiting distributions for these extreme sample eigenvalues. In this paper, we exte…
Spiked eigenvalues of high-dimensional sample autocovariance matrices: CLT and applications
本文建立高维样本自协方差矩阵尖峰特征值的 CLT，并据此提出尖峰特征值等价性检验及层次聚类方法，应用于多国死亡率数据。
Computational Equivalence of Spiked Covariance and Spiked Wigner Models via Gram-Schmidt Perturbation
本文提出从稀疏 Spiked Covariance Model 到稀疏 Spiked Wigner Model 的首个平均情形归约，借助 Gram-Schmidt 正交化的扰动等变性去噪保信号，证明两类高维统计模型的计算等价性。
On Convergence Rates of Spiked Eigenvalue Estimates: A General Study of Global and Local Laws in Sample Covariance Matrices
本文研究了具有一般维度增长率的样本协方差矩阵的全局与局部律，允许样本量与总体维度的比值趋于零、常数或无穷，并据此确定了尖刺特征值估计的收敛速度。
These expository notes serve as a reference for an accompanying post Morales-Jimenez et al. [2018]. In the spiked covariance model, we develop results on asymptotic normality of sample leading eigenvalues and certain projections of the corresponding sample eigenvectors. The results parallel those of Paul [2007], but are given using the non-Gaussian model of Bai and Yao [2008]. The results are not new, and citations are given, but proofs are collected and organized as a point of departure for Morales-Jimenez et al. [2018].
Edgeworth corrections for the spiked eigenvalues of non-Gaussian sample covariance matrices with applications
本文为非高斯数据下的单尖刺和多尖刺场景建立了尖刺特征值的一阶 Edgeworth 展开，借此构建更准确的置信区间并估计尖刺数量，在稳健性和准确性上优于现有方法。
The universality for the local spiked eigenvalues is a powerful tool to deal with the problems of the asymptotic law for the bulks of spiked eigenvalues of high-dimensional generalized Fisher matrices. In this paper, we focus on a more generalized spiked Fisher matrix, where $Σ_1Σ_2^{-1}$ is free of the restriction of diagonal independence, and both of the spiked eigenvalues and the population 4th moments are not necessary required to be bounded. By reducing the matching four moments constraint to a tail probability, we propose a Generalized Four Moment Theorem (G4MT) for the bulks of spiked e…
This paper aims to derive asymptotical distributions of the spiked eigenvalues of the large-dimensional spiked Fisher matrices without Gaussian assumption and the restrictive assumptions on covariance matrices. We first establish invariance principle for the spiked eigenvalues of the Fisher matrix. That is, we show that the limiting distributions of the spiked eigenvalues are invariant over a large class of population distributions satisfying certain conditions. Using the invariance principle, we further established a central limit theorem (CLT) for the spiked eigenvalues. As some interesting …
We show that in a common high-dimensional covariance model, the choice of loss function has a profound effect on optimal estimation. In an asymptotic framework based on the Spiked Covariance model and use of orthogonally invariant estimators, we show that optimal estimation of the population covariance matrix boils down to design of an optimal shrinker $η$ that acts elementwise on the sample eigenvalues. Indeed, to each loss function there corresponds a unique admissible eigenvalue shrinker $η^*$ dominating all other shrinkers. The shape of the optimal shrinker is determined by the choice of l…
We study principal components analyses in multivariate random and mixed effects linear models, assuming a spherical-plus-spikes structure for the covariance matrix of each random effect. We characterize the behavior of outlier sample eigenvalues and eigenvectors of MANOVA variance components estimators in such models under a high-dimensional asymptotic regime. Our results show that an aliasing phenomenon may occur in high dimensions, in which eigenvalues and eigenvectors of the MANOVA estimate for one variance component may be influenced by the other components. We propose an alternative proce…
This paper studies the joint limiting behavior of extreme eigenvalues and trace of large sample covariance matrix in a generalized spiked population model, where the asymptotic regime is such that the dimension and sample size grow proportionally. The form of the joint limiting distribution is applied to conduct Johnson-Graybill-type tests, a family of approaches testing for signals in a statistical model. For this, higher order correction is further made, helping alleviate the impact of finite-sample bias. The proof rests on determining the joint asymptotic behavior of two classes of spectral…
On spiked eigenvalues of a renormalized sample covariance matrix from multi-population
本文研究超高维多总体重归一化样本协方差矩阵的尖峰特征值渐近性质，建立其一二阶收敛性，应用于子群数确定与无标签聚类评估，并提出整合高低维结果的统一框架。
In this paper, we investigate the asymptotic behavior of spiked eigenvalues of the noncentral Fisher matrix defined by ${\mathbf F}_p={\mathbf C}_n(\mathbf S_N)^{-1}$, where ${\mathbf C}_n$ is a noncentral sample covariance matrix defined by $(\mathbf Ξ+\mathbf X)(\mathbf Ξ+\mathbf X)^*/n$ and $\mathbf S_N={\mathbf Y}{\mathbf Y}^*/N$. The matrices $\mathbf X$ and $\mathbf Y$ are two independent {Gaussian} arrays, with respective $p\times n$ and $p\times N$ and the Gaussian entries of them are \textit {independent and identically distributed} (i.i.d.) with mean $0$ and variance $1$. When $p$, $…
Distributed estimation of spiked eigenvalues in spiked population models
针对海量分布式数据集分析难题，本文提出尖刺总体模型中尖刺特征值的分布式估计量，证明其相合性与渐近正态性，并验证其具有与全样本估计相同的收敛阶及优异的估计精度与传输效率。
Inference for Spiked Eigenstructure under Generalized Covariance and Correlation Models
本文研究了在一般突出模型下，基于样本协方差矩阵和样本相关矩阵的样本突出特征值的推断问题，推导了样本突出特征空间的功能渐近分布，并开发了单样本基准主方向推断和双样本比较突出强度的程序。
Limiting eigen-structure of spiked sample covariance matrices under missing observations
本文针对带缺失观测的尖峰总体模型，利用随机矩阵理论研究高维PCA的渐近行为，证明其样本特征值的渐近正态性与完整数据情形存在差异，并基于此提出尖峰总体独立结构的检验方法。
Testing the number of common factors by bootstrapped sample covariance matrix in high-dimensional factor models
本文研究了高维因子模型中自助法对样本协方差矩阵特征值分布的影响，利用尖刺与非尖刺特征值的不同渐近行为，提出了检验公共因子数目的创新方法，并证明了其在弱因子和截面相关误差下的可靠性。
Generalization for Least Squares Regression With Simple Spiked Covariances
本文针对具有尖峰协方差的线性模型，在渐近比例机制下推导了两个简单模型的泛化误差，揭示了尖峰对应的特征向量和特征值对泛化性能的关键影响。
Consider the $p\times p$ matrix that is the product of a population covariance matrix and the inverse of another population covariance matrix. Suppose that their difference has a divergent rank with respect to $p$, when two samples of sizes $n$ and $T$ from the two populations are available, we construct its corresponding sample version. In the regime of high dimension where both $n$ and $T$ are proportional to $p$, we investigate the limiting laws for extreme (spiked) eigenvalues of the sample (spiked) Fisher matrix when the number of spikes is divergent and these spikes are unbounded.
We consider two-sample tests for high-dimensional data under two disjoint models: the strongly spiked eigenvalue (SSE) model and the non-SSE (NSSE) model. We provide a general test statistic as a function of a positive-semidefinite matrix. We give sufficient conditions for the test statistic to satisfy a consistency property and to be asymptotically normal. We discuss an optimality of the test statistic under the NSSE model. We also investigate the test statistic under the SSE model by considering strongly spiked eigenstructures and create a new effective test procedure for the SSE model. Fina…
Sampling Spiked Wishart Eigenvalues
该研究将现有Wishart矩阵特征值高效采样方案推广到任意尖峰数的spiked Wishart及pseudo-Wishart分布，并给出采样过程的求导方法以支持随机梯度下降拟合目标分布。
A generalized spiked Fisher matrix is considered in this paper. We establish a criterion for the description of the support of the limiting spectral distribution of high-dimensional generalized Fisher matrix and study the almost sure limits of the sample spiked eigenvalues where the population covariance matrices are arbitrary which successively removed an unrealistic condition posed in the previous works, that is, the covariance matrices are assumed to be diagonal or diagonal block-wise structure. In addition, we also give a consistent estimator of the population spiked eigenvalues. A series …
Bayesian Analysis of Spiked Covariance Models: Correcting Eigenvalue Bias and Determining the Number of Spikes
本文针对尖峰协方差模型提出贝叶斯推断框架，通过逆Wishart先验推导后验分布，设计两种特征值偏差修正策略，并基于BIC型边际似然近似推断尖峰数量，在高维情形下证明后验相合性与极小极大最优性。
Asymptotic distribution of spiked eigenvalues in the large signal-plus-noise models
本文研究大型各向异性信号加噪声矩阵，在一般结构与噪声分布假设下建立超临界尖峰奇异值的渐近联合分布，揭示其非普适性并用于混合模型及均值异质性检测。
In a spiked population model, the population covariance matrix has all its eigenvalues equal to units except for a few fixed eigenvalues (spikes). Determining the number of spikes is a fundamental problem which appears in many scientific fields, including signal processing (linear mixture model) or economics (factor model). Several recent papers studied the asymptotic behavior of the eigenvalues of the sample covariance matrix (sample eigenvalues) when the dimension of the observations and the sample size both grow to infinity so that their ratio converges to a positive constant. Using these r…
High-dimensional ridgeless least squares interpolation under spiked covariance structures
该研究针对高维无岭最小二乘插值，基于广义尖峰协方差模型，揭示了其预测行为由回归系数与尖峰特征空间的对齐程度决定，刻画了尖峰特性对双下降现象的影响，为过参数化回归的泛化提供了统一认识。
Asymptotic limits of spiked eigenvalues and eigenvectors of signal-plus-noise matrices with weak signals and heteroskedastic noise
本文在高维信号加噪声模型中，针对弱信号和异方差噪声，推导了尖峰奇异向量与 Gram 矩阵尖峰特征值的渐近极限，并提出一种新的聚类簇数估计准则。
Nonlinear spiked covariance matrices and signal propagation in deep neural networks
本文研究了深度神经网络中非线性尖峰协方差模型的信号特征值与特征向量，定量描述了输入数据尖峰特征结构在随机权重网络隐藏层中的传播机制，并分析了表示学习中目标函数与 CK 尖峰特征向量的对齐情况。
Detecting Spectral Breaks in Spiked Covariance Models
本文研究高维尖峰协方差模型中序列经验协方差矩阵最大特征值的极限过程，证明其弱收敛于非高斯过程，并基于中心化尖峰特征值构造极大统计量以检测协方差结构断点，证明检验一致性并通过模拟验证其表现。
We propose a Bayesian methodology for estimating spiked covariance matrices with jointly sparse structure in high dimensions. The spiked covariance matrix is reparametrized in terms of the latent factor model, where the loading matrix is equipped with a novel matrix spike-and-slab LASSO prior, which is a continuous shrinkage prior for modeling jointly sparse matrices. We establish the rate-optimal posterior contraction for the covariance matrix with respect to the operator norm as well as that for the principal subspace with respect to the projection operator norm loss. We also study the poste…
We study improved approximations to the distribution of the largest eigenvalue $\hat{\ell}$ of the sample covariance matrix of $n$ zero-mean Gaussian observations in dimension $p+1$. We assume that one population principal component has variance $\ell > 1$ and the remaining `noise' components have common variance $1$. In the high dimensional limit $p/n \to γ> 0$, we begin study of Edgeworth corrections to the limiting Gaussian distribution of $\hat{\ell}$ in the supercritical case $\ell > 1 + \sqrt γ$. The skewness correction involves a quadratic polynomial as in classical settings, but the co…
Limiting laws and consistent estimation criteria for fixed and diverging number of spiked eigenvalues
本文研究了高维数据中极值特征值的极限定律与一致估计准则。针对固定维度p，提出了一种通用估计准则以一致估计突出特征值数量k。在允许p和k发散的情况下，利用随机矩阵理论推导了突出样本特征值的极限分布，无需假设突出特征值有上界或趋于无穷。通过模拟研究验证了所提准则的有效性。
Rotationally symmetric distributions on the p-dimensional unit hypersphere, extremely popular in directional statistics, involve a location parameter theta that indicates the direction of the symmetry axis. The most classical way of addressing the spherical location problem H_0:theta=theta_0, with theta_0 a fixed location, is the so-called Watson test, which is based on the sample mean of the observations. This test enjoys many desirable properties, but its implementation requires the sample size n to be large compared to the dimension p. This is a severe limitation, since more and more proble…
Consider two $p$-variate populations, not necessarily Gaussian, with covariance matrices $Σ_1$ and $Σ_2$, respectively, and let $S_1$ and $S_2$ be the sample covariances matrices from samples of the populations with degrees of freedom $T$ and $n$, respectively. When the difference $Δ$ between $Σ_1$ and $Σ_2$ is of small rank compared to $p,T$ and $n$, the Fisher matrix $F=S_2^{-1}S_1$ is called a {\em spiked Fisher matrix}. When $p,T$ and $n$ grow to infinity proportionally, we establish a phase transition for the extreme eigenvalues of $F$: when the eigenvalues of $Δ$ ({\em spikes}) are above…
The classical methods of multivariate analysis are based on the eigenvalues of one or two sample covariance matrices. In many applications of these methods, for example to high dimensional data, it is natural to consider alternative hypotheses which are a low rank departure from the null hypothesis. For rank one alternatives, this note provides a representation for the joint eigenvalue density in terms of a single contour integral. This will be of use for deriving approximate distributions for likelihood ratios and linear statistics used in testing.
We consider two types of spiked multivariate F distributions: a scaled distribution with the scale matrix equal to a rank-one perturbation of the identity, and a distribution with trivial scale, but rank-one non-centrality. The norm of the rank-one matrix (spike) parameterizes the joint distribution of the eigenvalues of the corresponding F matrix. We show that, for a spike located above a phase transition threshold, the asymptotic behavior of the log ratio of the joint density of the eigenvalues of the F matrix to their joint density under a local deviation from this value depends only on the…
We derive the asymptotic distributions of the spiked eigenvalues and eigenvectors under a generalized and unified asymptotic regime, which takes into account the spike magnitude of leading eigenvalues, sample size, and dimensionality. This new regime allows high dimensionality and diverging eigenvalue spikes and provides new insights into the roles the leading eigenvalues, sample size, and dimensionality play in principal component analysis. The results are proven by a technical device, which swaps the role of rows and columns and converts the high-dimensional problems into low-dimensional one…
In this paper, we establish some new central limit theorems for certain spectral statistics of a high-dimensional sample covariance matrix under a divergent spectral norm population model. This model covers the divergent spiked population model as a special case. Meanwhile, the number of the spiked eigenvalues can either be fixed or grow to infinity. It is seen from our theorems that the divergence of population spectral norm affects the fluctuations of the linear spectral statistics in a fickle way, depending on the divergence rate.
We study estimation of the covariance matrix under relative condition number loss $κ(Σ^{-1/2} \hatΣ Σ^{-1/2})$, where $κ(Δ)$ is the condition number of matrix $Δ$, and $\hatΣ$ and $Σ$ are the estimated and theoretical covariance matrices. Optimality in $κ$-loss provides optimal guarantees in two stylized applications: Multi-User Covariance Estimation and Multi-Task Linear Discriminant Analysis. We assume the so-called spiked covariance model for $Σ$, and exploit recent advances in understanding that model, to derive a nonlinear shrinker which is asymptotically optimal among orthogonally-equiva…
Self-Distillation is Optimal Among Spectral Shrinkage Estimators in Spiked Covariance Models
本文研究了自蒸馏在带噪协方差模型中的表现，证明了在谱收缩估计器中，s步自蒸馏在性能上最优，并展示了其在统计和机器学习中的优势。
Optimal Differentially Private PCA and Estimation for Spiked Covariance Matrices
本文研究了尖峰协方差模型下的最优差分隐私主成分分析和协方差估计，刻画了特征量敏感度并建立了极小极大收敛速度，提出了计算高效且在对数因子范围内最优的差分隐私估计器，适用于发散秩、更广信号强度及小样本高维场景。
We consider matrices formed by a random $N\times N$ matrix drawn from the Gaussian Orthogonal Ensemble (or Gaussian Unitary Ensemble) plus a rank-one perturbation of strength $θ$, and focus on the largest eigenvalue, $x$, and the component, $u$, of the corresponding eigenvector in the direction associated to the rank-one perturbation. We obtain the large deviation principle governing the atypical joint fluctuations of $x$ and $u$. Interestingly, for $θ>1$, in large deviations characterized by a small value of $u$, i.e. $u<1-1/θ$, the second-largest eigenvalue pops out from the Wigner semi-circ…
Recovery Limits for Eigenvector Alignment in HDLSS Spiked Covariance Models
该研究证明在高维低样本量尖峰协方差模型中，尽管行格拉姆矩阵可逐模型一致估计，但完整的特征向量对齐矩阵无法在嵌套路径上实现均匀可靠的几乎必然恢复。
Sample correlation matrices are employed ubiquitously in statistics. However, quite surprisingly, little is known about their asymptotic spectral properties for high-dimensional data, particularly beyond the case of "null models" for which the data is assumed independent. Here, considering the popular class of spiked models, we apply random matrix theory to derive asymptotic first-order and distributional results for both the leading eigenvalues and eigenvectors of sample correlation matrices. These results are obtained under high-dimensional settings for which the number of samples n and vari…
Spectrally-Corrected and Regularized QDA Classifier for Spiked Covariance Model
针对高维场景下QDA分类失效的问题，提出结合谱校正与正则化、通过最大化Fisher判别比选参的SR-QDA方法，实验证明其在中高维场景下表现优于多种对比方法。
In this paper, we study the asymptotic behavior of the extreme eigenvalues and eigenvectors of the high dimensional spiked sample covariance matrices, in the supercritical case when a reliable detection of spikes is possible. Especially, we derive the joint distribution of the extreme eigenvalues and the generalized components of the associated eigenvectors, i.e., the projections of the eigenvectors onto arbitrary given direction, assuming that the dimension and sample size are comparably large. In general, the joint distribution is given in terms of linear combinations of finitely many Gaussi…
In this paper, we establish the central limit theorem (CLT) for linear spectral statistics (LSS) of large-dimensional sample covariance matrix when the population covariance matrices are not uniformly bounded, which is a nontrivial extension of the Bai-Silverstein theorem (BST) (2004). The latter has strongly stimulated the development of high-dimensional statistics, especially the application of random matrix theory to statistics. However, the assumption of uniform boundedness of the population covariance matrices is found strongly limited to the applications of BST. The aim of this paper is …
We consider the five classes of multivariate statistical problems identified by James (1964), which together cover much of classical multivariate analysis, plus a simpler limiting case, symmetric matrix denoising. Each of James' problems involves the eigenvalues of $E^{-1}H$ where $H$ and $E$ are proportional to high dimensional Wishart matrices. Under the null hypothesis, both Wisharts are central with identity covariance. Under the alternative, the non-centrality or the covariance parameter of $H$ has a single eigenvalue, a spike, that stands alone. When the spike is smaller than a case-spec…
Federated PCA and Estimation for Spiked Covariance Matrices: Optimal Rates and Efficient Algorithm
本文研究差分隐私约束下尖峰协方差矩阵的联邦PCA与估计，证明中心服务器最优速率为各客户端极小极大速率的调和平均，并推导矩阵版van Trees不等式作为下界工具，提出达到近最优速率的高效隐私保护算法。
Distributed Online Estimation of Spiked Eigenvalues with Adaptive Weighting under Persistent Aspect Ratio Heterogeneity
针对多节点异质性有效样本量的尖峰协方差特征值在线估计问题，提出先校正再聚合的分布式框架，实现高效低通信成本的在线推断与系统性风险实时跟踪。
Principal component analysis (PCA) is a widely used method for dimension reduction. In high dimensional data, the "signal" eigenvalues corresponding to weak principal components (PCs) do not necessarily separate from the bulk of the "noise" eigenvalues. Therefore, popular tests based on the largest eigenvalue have little power to detect weak PCs. In the special case of the spiked model, certain tests asymptotically equivalent to linear spectral statistics (LSS)---averaging effects over all eigenvalues---were recently shown to achieve some power. We consider a nonparametric, non-Gaussian genera…
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