Abstract

A vast number of methods have been proposed in literature for point estimation of eigenstructure of covariance matrices in high-dimensional settings. In this work, we study uncertainty quantification and propose methodology for inference and hypothesis testing for individual loadings of the covariance matrix. We base our methodology on a Lasso-penalized M-estimator which, despite non-convexity, may be solved by a polynomial-time algorithm such as coordinate or gradient descent. Our results provide theoretical guarantees on asymptotic normality of the new estimators and may be used for valid hypothesis testing and variable selection. These results are achieved under a sparsity condition relating the number of non-zero loadings, sample size, dimensionality of the covariance matrix and spectrum of the covariance matrix. This talk is based on joint work with Sara van de Geer.