We develop a model-based method for evaluating heterogeneity among several p x p covariance matrices in the large p, small n setting. This is done by assuming a spiked covariance model for each group and sharing information about the space spanned by the group-level eigenvectors. We use an empirical Bayes method to identify a low-dimensional subspace which explains variation across all groups and use an MCMC algorithm to estimate the posterior uncertainty of eigenvectors and eigenvalues on this subspace. The implementation and utility of our model is illustrated with analyses of high-dimensional multivariate gene expression.
Dimension reduction for high-dimensional compositional data plays an important role in many fields, where the principal component analysis of the basis covariance matrix is of scientific interest. In practice, however, the basis variables are latent and rarely observed, and standard techniques of principal component analysis are inadequate for compositional data because of the simplex constraint. To address the challenging problem, we relate the principal subspace of the centered log-ratio compositional covariance to that of the basis covariance, and prove that the latter is approximately iden…
Random projection is widely used as a method of dimension reduction. In recent years, its combination with standard techniques of regression and classification has been explored. Here we examine its use with principal component analysis (PCA) and subspace detection methods. Specifically, we show that, under appropriate conditions, with high probability the magnitude of the residuals of a PCA analysis of randomly projected data behaves comparably to that of the residuals of a similar PCA analysis of the original data. Our results indicate the feasibility of applying subspace-based anomaly detec…
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…
Networked sensing, where the goal is to perform complex inference using a large number of inexpensive and decentralized sensors, has become an increasingly attractive research topic due to its applications in wireless sensor networks and internet-of-things. To reduce the communication, sensing and storage complexity, this paper proposes a simple sensing and estimation framework to faithfully recover the principal subspace of high-dimensional data streams using a collection of binary measurements from distributed sensors, without transmitting the whole data. The binary measurements are designed…
We study the basic problem of robust subspace recovery. That is, we assume a data set that some of its points are sampled around a fixed subspace and the rest of them are spread in the whole ambient space, and we aim to recover the fixed underlying subspace. We first estimate "robust inverse sample covariance" by solving a convex minimization procedure; we then recover the subspace by the bottom eigenvectors of this matrix (their number correspond to the number of eigenvalues close to 0). We guarantee exact subspace recovery under some conditions on the underlying data. Furthermore, we propose…
We develop an envelope model for joint mean and covariance regression in the large $p$, small $n$ setting. In contrast to existing envelope methods, which improve mean estimates by incorporating estimates of the covariance structure, we focus on identifying covariance heterogeneity by incorporating information about mean-level differences. We use a Monte Carlo EM algorithm to identify a low-dimensional subspace which explains differences in both means and covariances as a function of covariates, and then use MCMC to estimate the posterior uncertainty conditional on the inferred low-dimensional…
Principal component analysis (PCA) is a classical dimension reduction method which projects data onto the principal subspace spanned by the leading eigenvectors of the covariance matrix. However, it behaves poorly when the number of features p is comparable to, or even much larger than, the sample size n. In this paper, we propose a new iterative thresholding approach for estimating principal subspaces in the setting where the leading eigenvectors are sparse. Under a spiked covariance model, we find that the new approach recovers the principal subspace and leading eigenvectors consistently, an…
We consider the problem of joint estimation of structured covariance matrices. Assuming the structure is unknown, estimation is achieved using heterogeneous training sets. Namely, given groups of measurements coming from centered populations with different covariances, our aim is to determine the mutual structure of these covariance matrices and estimate them. Supposing that the covariances span a low dimensional affine subspace in the space of symmetric matrices, we develop a new efficient algorithm discovering the structure and using it to improve the estimation. Our technique is based on th…
A large number of algorithms in machine learning, from principal component analysis (PCA), and its non-linear (kernel) extensions, to more recent spectral embedding and support estimation methods, rely on estimating a linear subspace from samples. In this paper we introduce a general formulation of this problem and derive novel learning error estimates. Our results rely on natural assumptions on the spectral properties of the covariance operator associated to the data distribu- tion, and hold for a wide class of metrics between subspaces. As special cases, we discuss sharp error estimates for …
We give the first polynomial time algorithm for \emph{list-decodable covariance estimation}. For any $α> 0$, our algorithm takes input a sample $Y \subseteq \mathbb{R}^d$ of size $n\geq d^{\mathsf{poly}(1/α)}$ obtained by adversarially corrupting an $(1-α)n$ points in an i.i.d. sample $X$ of size $n$ from the Gaussian distribution with unknown mean $μ_*$ and covariance $Σ_*$. In $n^{\mathsf{poly}(1/α)}$ time, it outputs a constant-size list of $k = k(α)= (1/α)^{\mathsf{poly}(1/α)}$ candidate parameters that, with high probability, contains a $(\hatμ,\hatΣ)$ such that the total variation distan…
Stationary subspace analysis (SSA) searches for linear combinations of the components of nonstationary vector time series that are stationary. These linear combinations and their number defne an associated stationary subspace and its dimension. SSA is studied here for zero mean nonstationary covariance processes. We characterize stationary subspaces and their dimensions in terms of eigenvalues and eigenvectors of certain symmetric matrices. This characterization is then used to derive formal statistical tests for estimating dimensions of stationary subspaces. Eigenstructure-based techniques ar…
This paper studies the problem of estimating the covariance of a collection of vectors using only highly compressed measurements of each vector. An estimator based on back-projections of these compressive samples is proposed and analyzed. A distribution-free analysis shows that by observing just a single linear measurement of each vector, one can consistently estimate the covariance matrix, in both infinity and spectral norm, and this same analysis leads to precise rates of convergence in both norms. Via information-theoretic techniques, lower bounds showing that this estimator is minimax-opti…
Joint estimation of high-dimensional spiked covariance matrices via a partially shared subspace
KAIST（韩国科学技术院）；Institute for Data Innovation in Science, Seoul National University（首尔大学数据创新科学研究所）；Department of Statistics, Seoul National University（首尔大学统计系）
Analyzing massive spatial datasets using Gaussian process model poses computational challenges. This is a problem prevailing heavily in applications such as environmental modeling, ecology, forestry and environmental heath. We present a novel approximate inference methodology that uses profile likelihood and Krylov subspace methods to estimate the spatial covariance parameters and makes spatial predictions with uncertainty quantification. The proposed method, Kryging, applies for both observations on regular grid and irregularly-spaced observations, and for any Gaussian process with a stationa…
The classic likelihood ratio test for testing the equality of two covariance matrices breakdowns due to the singularity of the sample covariance matrices when the data dimension $p$ is larger than the sample size $n$. In this paper, we present a conceptually simple method using random projection to project the data onto the one-dimensional random subspace so that the conventional methods can be applied. Both one-sample and two-sample tests for high-dimensional covariance matrices are studied. Asymptotic results are established and numerical results are given to compare our method with state-of…
Multi-rank Subspace Change-point Detection with Application in Monitoring Robotic Swarms
Georgia Institute of Technology（佐治亚理工学院）；Johns Hopkins University（约翰斯·霍普金斯大学）；U.S. Naval Research Laboratory（美国海军研究实验室）；Department of Applied Mathematics and Statistics, Johns Hopkins University（约翰斯·霍普金斯大学应用数学与统计学系）
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…
Rank estimation is a classical model order selection problem that arises in a variety of important statistical signal and array processing systems, yet is addressed relatively infrequently in the extant literature. Here we present sample covariance asymptotics stemming from random matrix theory, and bring them to bear on the problem of optimal rank estimation in the context of the standard array observation model with additive white Gaussian noise. The most significant of these results demonstrates the existence of a phase transition threshold, below which eigenvalues and associated eigenvecto…
Estimation and hypothesis tests for the covariance matrix in high dimensions is a challenging problem as the traditional multivariate asymptotic theory is no longer valid. When the dimension is larger than or increasing with the sample size, standard likelihood based tests for the covariance matrix have poor performance. Existing high dimensional tests are either computationally expensive or have very weak control of type I error. In this paper, we propose a test procedure, CRAMP, for testing hypotheses involving one or more covariance matrices using random projections. Projecting the high dim…
A common problem in genetics is that of testing whether a set of highly dependent gene expressions differ between two populations, typically in a high-dimensional setting where the data dimension is larger than the sample size. Most high-dimensional tests for the equality of two mean vectors rely on naive diagonal or trace estimators of the covariance matrix, ignoring dependencies between variables. A test recently proposed by Lopes et al. (2012) implicitly incorporates dependencies by using random pseudo-projections to a lower-dimensional space. Their test offers higher power when the variabl…
For a high-dimensional parameter of interest, tests based on quadratic statistics are known to have low power against subsets of the parameter space (henceforth, parameter subspaces). In addition, they typically involve an inverse covariance matrix which is difficult to estimate in high-dimensional settings. I simultaneously address these two issues by proposing a novel test statistic that is large in a conic parameter subspace of interest. This test statistic generalizes the Wald statistic and nests many well-known test statistics. For a given parameter subspace, the statistic is free of tuni…
