Robust High-Dimensional Precision Matrix Estimation

Abstract

The dependency structure of multivariate data can be analyzed using the covariance matrix Σ. In many fields the precision matrix Σ −1 is even more informative. As the sample covariance estimator is singular in high-dimensions, it cannot be used to obtain a precision matrix estimator. A popular highdimensional estimator is the graphical lasso, but it lacks robustness. We consider the high-dimensional independent contamination model. Here, even a small percentage of contaminated cells in the data matrix may lead to a high percentage of contaminated rows. Downweighting entire observations, which is done by traditional robust procedures, would then results in a loss of information. In this paper, we formally prove that replacing the sample covariance matrix in the graphical lasso with an elementwise robust covariance matrix leads to an elementwise robust, sparse precision matrix estimator computable in high-dimensions. Examples of such elementwise robust covariance estimators are given. The final precision matrix estimator is positive definite, has a high breakdown point under elementwise contamination and can be computed fast.

Fast Robust Correlation for High-Dimensional Data

“…The precision matrix is the inverse of the covariance matrix, and allows to construct a Gaussian graphical model of the variables. Ollerer and Croux (2015) and Tarr et al (2016) estimated the covariance matrix from rank correlations, but one could also use wrapping for this step. When the dimension d is too high the estimated covariance matrix cannot be inverted, so these authors construct a sparse precision matrix by applying GLASSO.Öllerer and Croux (2015) show that the breakdown value of the resulting precision matrix, for both implosion and explosion, is as high as that of the univariate scale estimator.…”

High-dimensional robust precision matrix estimation: Cellwise corruption under $\epsilon $-contamination

“…In this paper, we have derived statistical error bounds for high-dimensional robust precision matrix estimators, when data are drawn from a multivariate normal distribution and then observed subject to cellwise contamination. We show that in such settings, the precision matrix estimators that are obtained by plugging in pairwise robust covariance estimators to the GLasso or CLIME routine, as suggested by Oellerer and Croux (2014) and Tarr et al (2015), have error bounds that match standard high-dimensional bounds for uncontaminated precision matrix estimation, up to an additive factor involving a constant multiple of the contamination fraction . Our results for precision matrix estimators are derived via estimation error bounds for robust covariance matrix estimators, which have similar deviation properties.…”

“…In this section, we review two techniques, the GLasso and CLIME, which produce a sparse precision matrix estimator based on optimizing a function of the sample covariance matrix. As proposed by Oellerer and Croux (2014) and Tarr et al (2015), these methods may easily be modified to obtained robust versions, where the sample covariance matrix estimator is simply replaced by a robust covariance estimatorΣ as described in the previous section. The graphical lasso (GLasso) estimator (Yuan and Lin, 2007;Friedman et al, 2008) is defined as the maximizer of the following 1 -penalized log-likelihood function:…”

“…We also comment briefly on another pairwise covariance estimator appearing in the robust statistics literature. Tarr et al (2015) and Oellerer and Croux (2014) propose to use the following estimator based on an idea of Gnanadesikan and Kettenring (1972): Noting that…”

“…Turning to the derivation of the breakdown point, note that by Theorem 1 of Oellerer and Croux (2014), we have n (Ω(X), X) ≥ + n (Σ(X), X).…”