A shrinkage principle for heavy-tailed data: High-dimensional robust low-rank matrix recovery

Abstract

This paper introduces a simple principle for robust statistical inference via appropriate shrinkage on the data. This widens the scope of high-dimensional techniques, reducing the distributional conditions from subexponential or sub-Gaussian to more relaxed bounded second or fourth moment. As an illustration of this principle, we focus on robust estimation of the low-rank matrix ${\mathrm{\Theta }}^{\ast }$ from the trace regression model $\mathit{Y}=Tr({{\mathrm{\Theta }}^{\ast }}^{\top }\mathbf{X})+\mathit{\epsilon }$. It encompasses four popular problems: sparse linear model, compressed sensing, matrix completion and multitask learning. We propose to apply the penalized least-squares approach to the appropriately truncated or shrunk data. Under only bounded $2+\mathit{\delta }$ moment condition on the response, the proposed robust methodology yields an estimator that possesses the same statistical error rates as previous literature with sub-Gaussian errors. For sparse linear model and multitask regression, we further allow the design to have only bounded fourth moment and obtain the same statistical rates. As a byproduct, we give a robust covariance estimator with concentration inequality and optimal rate of convergence in terms of the spectral norm, when the samples only bear bounded fourth moment. This result is of its own interest and importance. We reveal that under high dimensions, the sample covariance matrix is not optimal whereas our proposed robust covariance can achieve optimality. Extensive simulations are carried out to support the theories.