A growing number of modern statistical learning problems involve estimating a large number of parameters from a (smaller) number of noisy observations. In a subset of these problems (matrix completion, matrix compressed sensing, and multi-task learning) the unknown parameters form a high-dimensional matrix B*, and two popular approaches for the estimation are convex relaxation of rank-penalized regression or non-convex optimization. It is also known that these estimators satisfy near optimal error bounds under assumptions on rank, coherence, or spikiness of the unknown matrix. In this paper, we introduce a unifying technique for analyzing all of these problems via both estimators that leads to short proofs for the existing results as well as new results. Specifically, first we introduce a general notion of spikiness for B* and consider a general family of estimators and prove non-asymptotic error bounds for the their estimation error. Our approach relies on a generic recipe to prove restricted strong convexity for the sampling operator of the trace regression. Second, and most notably, we prove similar error bounds when the regularization parameter is chosen via K-fold cross-validation. This result is significant in that existing theory on cross-validated estimators do not apply to our setting since our estimators are not known to satisfy their required notion of stability. Third, we study applications of our general results to four subproblems of (1) matrix completion, (2) multi-task learning, (3) compressed sensing with Gaussian ensembles, and (4) compressed sensing with factored measurements. For (1), (3), and (4) we recover matching error bounds as those found in the literature, and for (2) we obtain (to the best of our knowledge) the first such error bound. We also demonstrate how our frameworks applies to the exact recovery problem in (3) and (4).
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