High-dimensional covariance matrix estimation with missing observations

By Karim Lounici. 2012

198 citations recorded by the source corpus.

In this paper, we study the problem of high-dimensional approximately low-rank covariance matrix estimation with missing observations. We propose a simple procedure computationally tractable in high-dimension and that does not require imputation of the missing data. We establish non-asymptotic sparsity oracle inequalities for the estimation of the covariance matrix with the Frobenius and spectral norms, valid for any setting of the sample size and the dimension of the observations. We further establish minimax lower bounds showing that our rates are minimax optimal up to a logarithmic factor.

Subfields: Computational Mechanics, Statistics and Probability, Signal Processing

View Paper: https://www.semanticscholar.org/paper/cd622077fd0c05fc01b7975245cb63968854a2ca

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Mechanics

In Statistics and Probability

In Signal Processing

Citations in the corpus

Sub-Gaussian estimators of the mean of a random matrix with heavy-tailed entries (2016, methodology): “Result stated above mimics the (almost) optimal rates obtained in [Lou14] (in the situation when no data is missing) under significantly weaker assumptions on the underlying distribution.”
Analysis of elliptical copula correlation factor model with Kendall's tau (2013, methodology): “Hence, a naive application of the method of [31] to our setting amounts to seeking a low-rank approximation to a matrix that is in fact not low-rank.”
Asymptotics and Concentration Bounds for Bilinear Forms of Spectral Projectors of Sample Covariance (2014, unknown): “nt C>0 (see Theorem 5.39 and the comments after this theorem). The proof is based on an ε-net argument that does not yield an optimal bound for general (nonisotropic) subgaussian distributions. In [20, 7], similar results were derived for subgaussian distributions and low-rank covariance matrices”
Uniform Hanson-Wright type concentration inequalities for unbounded entries via the entropy method (2018, methodology): “In what follows we discuss a more general framework suggested by Lounici (2014). Let δi,j , i ≤ N, j ≤ n be independent Bernoulli random variables with the same mean δ.”
Concentration inequalities and moment bounds for sample covariance operators (2014, unknown): “For instance, Lounici [9] showed that with some constant C > 0 and with probability at least 1− e−t ‖Σ̂− Σ‖ ≤ C‖Σ‖max { √ r̃(Σ) log d+ t n , (r̃(Σ) log d+ t) logn n } .”
Learning Latent Variable Gaussian Graphical Models (2014, background): “We make the following remarks: • The error bound (13) is a family of upper bounds deﬁned by different sets of subspace pairs ( ( E ) , M ( E ) ⊥ ) and ( ( U ) , M ( U ) ⊥ ) . appropriately E and The tightest bound can be achieved by choosing U .”
Estimating structured high-dimensional covariance and precision matrices: Optimal rates and adaptive estimation (2016, background): “Two fundamental concentration inequalities have been estalished: (i) entrywise deviation bound |r̂ij − σij |, (Xue and Zou (2012) and Liu et al.”
ECA: High-Dimensional Elliptical Component Analysis in Non-Gaussian Distributions (2013, methodology): “Such an observation is consistent with the observations in the PCA theory (Lounici 2014; Bunea and Xiao 2015).”
Geometric median and robust estimation in Banach spaces (2013, methodology): “However, it is possible to use various alternatives in place of the sample covariance, such as the shrinkage estimator [27], banding/tapering estimator [3], hard thresholding estimator [4] or the nuclear norm-penalized estimator [29], to name a few.”