Sparse precision matrix estimation under elementwise contamination

The dependency structure of multivariate data can be analyzed using the covariance matrix. In many fields the precision matrix, this is the inverse of the covariance matrix, is even more informative (e.g. gaussian graphical model, linear discriminant analysis). As the sample covariance estimator is singular in high-dimensions, it cannot be used to obtain a precision matrix estimator. In this scenario, the graphical lasso is one of the most popular estimators, but it lacks robustness. Most robust procedures assume that at least half of the observations are absolutely clean. However, often only a few variables of an observation are contaminated. An example is the high-dimensional independent contamination model, where small amounts of contamination lead to a large number of contaminated observations. Downweighting entire observations would then result in a loss of information, creating problems if the number of observations is small. We formally prove that initializing the graphical lasso with an elementwise robust covariance estimator leads to an elementwise robust, sparse precision matrix estimator computable in high-dimensions. Clever choice of the initial covariance estimator leads to a high breakdown point and positive definiteness of the final precision matrix estimator. The performance of the estimator is analyzed in a simulation study.status: accepte