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Robust subspace recovery (RSR) is a fundamental problem in robustrepresentation learning. Here we focus on a recently proposed RSR method termedDual Principal Component Pursuit (DPCP) approach, which aims to recover a basisof the orthogonal complement of the subspace and is amenable to handlingsubspaces of high relative dimension. Prior work has shown that DPCP canprovably recover the correct subspace in the presence of outliers, as long asthe true dimension of the subspace is known. We show that DPCP can provablysolve RSR problems in the {\it unknown} subspace dimension regime, as long asorthogonality constraints -- adopted in previous DPCP formulations -- arerelaxed and random initialization is used instead of spectral one. Namely, wepropose a very simple algorithm based on running multiple instances of aprojected sub-gradient descent method (PSGM), with each problem instanceseeking to find one vector in the null space of the subspace. We theoreticallyprove that under mild conditions this approach will succeed with highprobability. In particular, we show that 1) all of the problem instances willconverge to a vector in the nullspace of the subspace and 2) the ensemble ofproblem instance solutions will be sufficiently diverse to fully span thenullspace of the subspace thus also revealing its true unknown codimension. Weprovide empirical results that corroborate our theoretical results and showcasethe remarkable implicit rank regularization behavior of PSGM algorithm thatallows us to perform RSR without being aware of the subspace dimension.

Implicit Bias of Projected Subgradient Method Gives Provable Robust Recovery of Subspaces of Unknown Codimension