We study inductive matrix completion (matrix completion with side information) under an i.i.d. subgaussian noise assumption at a low noise regime, with uniform sampling of the entries. We obtain for the first time generalization bounds with the following three properties: (1) they scale like the standard deviation of the noise and in particular approach zero in the exact recovery case; (2) even in the presence of noise, they converge to zero when the sample size approaches infinity; and (3) for a fixed dimension of the side information, they only have a logarithmic dependence on the size of the matrix. Differently from many works in approximate recovery, we present results both for bounded Lipschitz losses and for the absolute loss, with the latter relying on Talagrand-type inequalities. The proofs create a bridge between two approaches to the theoretical analysis of matrix completion, since they consist in a combination of techniques from both the exact recovery literature and the approximate recovery literature.

本文研究了在低噪声环境下使用iid子高斯噪声的归纳矩阵填充问题（带侧面信息的矩阵填充），首次获得了普适性界限，并呈现出标准差与零误差恢复情况下的规模趋近，结果表明：在样本大小趋近于无穷大时，噪声即使存在也会趋近于零，对于侧面信息的固定维度而言，它们只有对矩阵大小的对数依赖性。

低噪声情况下归纳矩阵补全的泛化界