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.

中文翻译：

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一般基础下嘈杂的低秩矩阵完成

在本文中，我们考虑了通用基数下的低秩矩阵完成问题，该问题旨在通过预定基数的线性组合来恢复结构化矩阵。现有作品主要集中在正交基础上。但是，在某些实际应用中通常需要采用非正交基。因此，在这种情况下非常需要解决一些流行的估计器的可行性。我们提出了一些合理且广泛适用的假设，作为特殊情况，它们涵盖了训练有素的正交系统。在这些假设下，证明了具有核范数正则化的最小二乘估计量能够以很高的概率实现成功的恢复。建立Frobenius规范下的误差范围，以恢复精确和近似低秩矩阵。