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In this paper, we develop a relative error bound for nuclear norm regularizedmatrix completion, with the focus on the completion of full-rank matrices.Under the assumption that the top eigenspaces of the target matrix areincoherent, we derive a relative upper bound for recovering the best low-rankapproximation of the unknown matrix. Although multiple works have been devotedto analyzing the recovery error of full-rank matrix completion, their errorbounds are usually additive, making it impossible to obtain the perfectrecovery case and more generally difficult to leverage the skewed distributionof eigenvalues. Our analysis is built upon the optimality condition of theregularized formulation and existing guarantees for low-rank matrix completion.To the best of our knowledge, this is the first relative bound that has beenproved for the regularized formulation of matrix completion.

Relative Error Bound Analysis for Nuclear Norm Regularized Matrix Completion