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 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 …
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…
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…
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 …
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 $p$, $…
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…
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…
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…
Using the Coulomb Fluid method, this paper derives central limit theorems (CLTs) for linear spectral statistics of three "spiked" Hermitian random matrix ensembles. These include Johnstone's spiked model (i.e., central Wishart with spiked correlation), non-central Wishart with rank-one non-centrality, and a related class of non-central $F$ matrices. For a generic linear statistic, we derive simple and explicit CLT expressions as the matrix dimensions grow large. For all three ensembles under consideration, we find that the primary effect of the spike is to introduce an $O(1)$ correction term t…
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].
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…
This paper derives central limit theorems (CLTs) for general linear spectral statistics (LSS) of three important multi-spiked Hermitian random matrix ensembles. The first is the most common spiked scenario, proposed by Johnstone, which is a central Wishart ensemble with fixed-rank perturbation of the identity matrix, the second is a non-central Wishart ensemble with fixed-rank noncentrality parameter, and the third is a similarly defined non-central $F$ ensemble. These CLT results generalize our recent work to account for multiple spikes, which is the most common scenario met in practice. The …
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…
We propose a new statistical model, the spiked transport model, which formalizes the assumption that two probability distributions differ only on a low-dimensional subspace. We study the minimax rate of estimation for the Wasserstein distance under this model and show that this low-dimensional structure can be exploited to avoid the curse of dimensionality. As a byproduct of our minimax analysis, we establish a lower bound showing that, in the absence of such structure, the plug-in estimator is nearly rate-optimal for estimating the Wasserstein distance in high dimension. We also give evidence…
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…
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…
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 …
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.
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.
Characterizing the asymptotic distributions of eigenvectors for large random matrices poses important challenges yet can provide useful insights into a range of statistical applications. To this end, in this paper we introduce a general framework of asymptotic theory of eigenvectors (ATE) for large spiked random matrices with diverging spikes and heterogeneous variances, and establish the asymptotic properties of the spiked eigenvectors and eigenvalues for the scenario of the generalized Wigner matrix noise. Under some mild regularity conditions, we provide the asymptotic expansions for the sp…
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…
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…
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…
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…
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 present manuscript studies signal detection by likelihood ratio tests in a number of spiked random matrix models, including but not limited to Gaussian mixtures and spiked Wishart covariance matrices. We work directly with multi-spiked cases in these models and with flexible priors on the signal component that allow dependence across spikes. We derive asymptotic normality for the log-likelihood ratios when the signal-to- noise ratios are below certain thresholds. In addition, we show that the variances of the log-likelihood ratios can be asymptotically decomposed as the sums of those of a …
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…
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…
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…
In this paper, we study the power iteration algorithm for the spiked tensor model, as introduced in [44]. We give necessary and sufficient conditions for the convergence of the power iteration algorithm. When the power iteration algorithm converges, for the rank one spiked tensor model, we show the estimators for the spike strength and linear functionals of the signal are asymptotically Gaussian; for the multi-rank spiked tensor model, we show the estimators are asymptotically mixtures of Gaussian. This new phenomenon is different from the spiked matrix model. Using these asymptotic results of…
With the development of high-throughput technologies, principal component analysis (PCA) in the high-dimensional regime is of great interest. Most of the existing theoretical and methodological results for high-dimensional PCA are based on the spiked population model in which all the population eigenvalues are equal except for a few large ones. Due to the presence of local correlation among features, however, this assumption may not be satisfied in many real-world datasets. To address this issue, we investigated the asymptotic behaviors of PCA under the generalized spiked population model. Bas…
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…
We consider the problem of detecting signals in the rank-one signal-plus-noise data matrix models that generalize the spiked Wishart matrices. We show that the principal component analysis can be improved by pre-transforming the matrix entries if the noise is non-Gaussian. As an intermediate step, we prove a sharp phase transition of the largest eigenvalues of spiked rectangular matrices, which extends the Baik-Ben Arous-Péché (BBP) transition. We also propose a hypothesis test to detect the presence of signal with low computational complexity, based on the linear spectral statistics, which mi…
We study the statistical decision process of detecting the low-rank signal from various signal-plus-noise type data matrices, known as the spiked random matrix models. We first show that the principal component analysis can be improved by entrywise pre-transforming the data matrix if the noise is non-Gaussian, generalizing the known results for the spiked random matrix models with rank-1 signals. As an intermediate step, we find out sharp phase transition thresholds for the extreme eigenvalues of spiked random matrices, which generalize the Baik-Ben Arous-Péché (BBP) transition. We also prove …
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 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…
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 study the problem of detecting the presence of a single unknown spike in a rectangular data matrix, in a high-dimensional regime where the spike has fixed strength and the aspect ratio of the matrix converges to a finite limit. This setup includes Johnstone's spiked covariance model. We analyze the likelihood ratio of the spiked model against an "all noise" null model of reference, and show it has asymptotically Gaussian fluctuations in a region below---but in general not up to---the so-called BBP threshold from random matrix theory. Our result parallels earlier findings of Onatski et al.\ …
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…
