Related papers: Minimax bounds for sparse PCA with noisy high-dime…
Covariance matrix estimation and principal component analysis (PCA) are two cornerstones of multivariate analysis. Classic textbook solutions perform poorly when the dimension of the data is of a magnitude similar to the sample size, or…
Modern data are increasingly both high-dimensional and heteroscedastic. This paper considers the challenge of estimating underlying principal components from high-dimensional data with noise that is heteroscedastic across samples, i.e.,…
This work provides a unified analysis of the properties of the sample covariance matrix $\Sigma_n$ over the class of $p\times p$ population covariance matrices $\Sigma$ of reduced effective rank $r_e(\Sigma)$. This class includes scaled…
We consider a high-dimensional linear regression problem. Unlike many papers on the topic, we do not require sparsity of the regression coefficients; instead, our main structural assumption is a decay of eigenvalues of the covariance matrix…
We consider the observations of an unknown $s$-sparse vector ${\boldsymbol \theta}$ corrupted by Gaussian noise with zero mean and unknown covariance matrix ${\boldsymbol \Sigma}$. We propose minimax optimal methods of estimating the…
We study statistical and computational limits of clustering when the means of the centres are sparse and their dimension is possibly much larger than the sample size. Our theoretical analysis focuses on the model $X_i = z_i \theta +…
We analyze the problem of discrete distribution estimation under $\ell_1$ loss. We provide non-asymptotic upper and lower bounds on the maximum risk of the empirical distribution (the maximum likelihood estimator), and the minimax risk in…
In this paper, we study high-dimensional sparse Quadratic Discriminant Analysis (QDA) and aim to establish the optimal convergence rates for the classification error. Minimax lower bounds are established to demonstrate the necessity of…
We study estimation and testing in the Poisson regression model with noisy high dimensional covariates, which has wide applications in analyzing noisy big data. Correcting for the estimation bias due to the covariate noise leads to a…
Principal component analysis (PCA) is a widely used dimension reduction technique in machine learning and multivariate statistics. To improve the interpretability of PCA, various approaches to obtain sparse principal direction loadings have…
We study the estimation of a high dimensional approximate factor model in the presence of both cross sectional dependence and heteroskedasticity. The classical method of principal components analysis (PCA) does not efficiently estimate the…
A continuous-time regression model with a jointly strictly sub-Gaussian random noise is considered in the paper. Upper exponential bounds for probabilities of large deviations of the least squares estimator for the regression parameter are…
This paper investigates the detection and estimation of a single change in high-dimensional linear models. We derive minimax lower bounds for the detection boundary and the estimation rate, which uncover a phase transition governed by the…
We develop a dimension reduction framework for data consisting of matrices of counts. Our model is based on assuming the existence of a small amount of independent normal latent variables that drive the dependency structure of the observed…
Although a majority of the theoretical literature in high-dimensional statistics has focused on settings which involve fully-observed data, settings with missing values and corruptions are common in practice. We consider the problems of…
The current paper presents a novel machinery for studying non-asymptotic minimax estimation of high-dimensional matrices, which yields tight minimax rates for a large collection of loss functions in a variety of problems. Based on the…
In this paper, we study the system identification problem for sparse linear time-invariant systems. We propose a sparsity promoting block-regularized estimator to identify the dynamics of the system with only a limited number of input-state…
We consider two problems of estimation in high-dimensional Gaussian models. The first problem is that of estimating a linear functional of the means of $n$ independent $p$-dimensional Gaussian vectors, under the assumption that most of…
We derive tight lower bounds on the smallest eigenvalue of a sample covariance matrix of a centred isotropic random vector under weak or no assumptions on its components.
This paper studies the problem of estimating a large coefficient matrix in a multiple response linear regression model when the coefficient matrix could be both of low rank and sparse in the sense that most nonzero entries concentrate on a…