  We propose a new framework -- Square Root Principal Component Pursuit -- for
low-rank matrix recovery from observations corrupted with noise and outliers.
Inspired by the square root Lasso, this new formulation does not require prior
knowledge of the noise level. We show that a single, universal choice of the
regularization parameter suffices to achieve reconstruction error proportional
to the (a priori unknown) noise level. In comparison, previous formulations
such as stable PCP rely on noise-dependent parameters to achieve similar
performance, and are therefore challenging to deploy in applications where the
noise level is unknown. We validate the effectiveness of our new method through
experiments on simulated and real datasets. Our simulations corroborate the
claim that a universal choice of the regularization parameter yields near
optimal performance across a range of noise levels, indicating that the
proposed method outperforms the (somewhat loose) bound proved here.
