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In this paper, we study the trace regression when a matrix of parameters B*is estimated via the convex relaxation of a rank-regularized regression or viaregularized non-convex optimization. It is known that these estimators satisfynear-optimal error bounds under assumptions on the rank, coherence, andspikiness of B*. We start by introducing a general notion of spikiness for B*that provides a generic recipe to prove the restricted strong convexity of thesampling operator of the trace regression and obtain near-optimal andnon-asymptotic error bounds for the estimation error. Similar to the existingliterature, these results require the regularization parameter to be above acertain theory-inspired threshold that depends on observation noise that may beunknown in practice. Next, we extend the error bounds to cases where theregularization parameter is chosen via cross-validation. This result issignificant in that existing theoretical results on cross-validated estimators(Kale et al., 2011; Kumar et al., 2013; Abou-Moustafa and Szepesvari, 2017) donot apply to our setting since the estimators we study are not known to satisfytheir required notion of stability. Finally, using simulations on synthetic andreal data, we show that the cross-validated estimator selects a near-optimalpenalty parameter and outperforms the theory-inspired approach of selecting theparameter.

On Low-rank Trace Regression under General Sampling Distribution