Noisy low‐rank matrix completion under general bases

Abstract

In this paper, we consider the low‐rank matrix completion problem under general bases, which intends to recover a structured matrix via a linear combination of prespecified bases. Existing works focus primarily on orthonormal bases; however, it is often necessary to adopt nonorthonormal bases in some real applications. Thus, there is a great need to address the feasibility of some popular estimators in such situations. We present several reasonable and widely applicable assumptions that cover the well‐cultivated orthonormal systems as a special case. Under these assumptions, it is proved that the least squares estimator with the nuclear norm regularization is capable of achieving successful recovery with high probability. The error bounds under the Frobenius norm are established for recovery of both exact and approximate low‐rank matrices. Theoretical findings are further corroborated via simulations.