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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 sub-exponential 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 Θ* from the trace regression model Y= Tr (Θ*⊤ X)+ ϵ. It encompasses four popular problems: sparse linear model, compressed sensing, matrix completion and multi-task learning. We propose to apply the penalized least-squares approach to the appropriately truncated or shrunk data. Under only bounded 2+ δ 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 multi-task …