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We propose a new framework -- Square Root Principal Component Pursuit -- forlow-rank matrix recovery from observations corrupted with noise and outliers.Inspired by the square root Lasso, this new formulation does not require priorknowledge of the noise level. We show that a single, universal choice of theregularization parameter suffices to achieve reconstruction error proportionalto the (a priori unknown) noise level. In comparison, previous formulationssuch as stable PCP rely on noise-dependent parameters to achieve similarperformance, and are therefore challenging to deploy in applications where thenoise level is unknown. We validate the effectiveness of our new method throughexperiments on simulated and real datasets. Our simulations corroborate theclaim that a universal choice of the regularization parameter yields nearoptimal performance across a range of noise levels, indicating that theproposed method outperforms the (somewhat loose) bound proved here.

Square Root Principal Component Pursuit: Tuning-Free Noisy Robust Matrix Recovery