Relative Error Bound Analysis for Nuclear Norm Regularized Matrix Completion
Lijun Zhang, Tianbao Yang, Rong Jin, Zhi-Hua Zhou
TL;DR
This paper establishes a relative error bound for nuclear norm regularized matrix completion in the full-rank setting, under incoherence of the top eigenspaces. It presents a general bound valid for any $oldsymbol{ extlambda}>0$ and a tighter bound when the residual $A-A_r$ is not highly spiky, with explicit dependence on the observation count $|oldsymbol{ extOmega}|$, matrix dimensions, rank, and the approximation error $oldsymbol{ extvarepsilon}=ig\|A-A_rigigig_F$. The results imply near-perfect recovery when $A$ is low-rank and improve upon prior additive bounds by scaling with $ig\|A-A_rigig_F$, offering advantages when eigenvalue distributions are skewed. The analysis relies on optimality conditions for the regularized problem and existing low-rank matrix-completion guarantees, and it points to extensions to noisy settings and universal completion frameworks. Overall, the work advances theoretical understanding of regularized matrix completion and provides practically meaningful sample-size–dependent error controls that tighten as the target deviates less from a low-rank approximation.
Abstract
In this paper, we develop a relative error bound for nuclear norm regularized matrix completion, with the focus on the completion of full-rank matrices. Under the assumption that the top eigenspaces of the target matrix are incoherent, we derive a relative upper bound for recovering the best low-rank approximation of the unknown matrix. Although multiple works have been devoted to analyzing the recovery error of full-rank matrix completion, their error bounds are usually additive, making it impossible to obtain the perfect recovery case and more generally difficult to leverage the skewed distribution of eigenvalues. Our analysis is built upon the optimality condition of the regularized formulation and existing guarantees for low-rank matrix completion. To the best of our knowledge, this is the first relative bound that has been proved for the regularized formulation of matrix completion.