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High-dimensional inference refers to problems of statistical estimation inwhich the ambient dimension of the data may be comparable to or possibly evenlarger than the sample size. We study an instance of high-dimensional inferencein which the goal is to estimate a matrix $\Theta^* \in \real^{k \times p}$ onthe basis of $N$ noisy observations, and the unknown matrix $\Theta^*$ isassumed to be either exactly low rank, or ``near'' low-rank, meaning that itcan be well-approximated by a matrix with low rank. We consider an$M$-estimator based on regularization by the trace or nuclear norm overmatrices, and analyze its performance under high-dimensional scaling. Weprovide non-asymptotic bounds on the Frobenius norm error that hold for ageneral class of noisy observation models, and then illustrate theirconsequences for a number of specific matrix models, including low-rankmultivariate or multi-task regression, system identification in vectorautoregressive processes, and recovery of low-rank matrices from randomprojections. Simulation results show excellent agreement with thehigh-dimensional scaling of the error predicted by our theory.

Estimation of (near) low-rank matrices with noise and high-dimensional scaling