Low-rank matrix completion is a widely studied problem with many variants. Inductive matrix completion (IMC) incorporates row and column side information to significantly narrow the search space. Prior work falls into two regimes: methods that exploit this structure to achieve reduced sample complexity but only in noiseless settings, and methods that handle noise but require sample complexity matching the ambient matrix dimension, forfeiting the sample efficiency that side information should provide. In this paper, we close this gap by studying noisy IMC with a nonconvex projected gradient descent algorithm with spectral initialization. Our main technical contribution is establishing a regularity condition for the IMC loss function that holds at the reduced sample complexity determined by the effective problem size, scaling with the side information dimension a rather than the ambient dimension n. This directly yields linear convergence and an estimation error that both depend only on the effective problem size rather than the ambient matrix dimension. We further extend our analysis to the inexact side information setting, demonstrating that the reduced sample complexity is maintained and the estimation error is order-optimal with respect to the inexactness of the side information. Extensive simulations and real-world experiments on the MovieLens dataset validate our theoretical findings.

翻译：低秩矩阵补全是具有多种变体的广泛研究问题。归纳矩阵补全(IMC)通过引入行与列的辅助信息显著缩小搜索空间。现有工作分为两类：一类利用该结构实现降低的样本复杂度但仅限于无噪声场景，另一类处理噪声但需要匹配环境矩阵维度的样本复杂度，从而丧失了辅助信息本应提供的样本效率。本文通过研究含噪声IMC的非凸投影梯度下降算法（结合谱初始化）填补了这一空白。我们的主要技术贡献是为IMC损失函数建立了正则性条件，该条件在由有效问题规模决定的降低样本复杂度下成立——该规模取决于辅助信息维度a而非环境维度n。这直接导出了线性收敛速度与仅依赖于有效问题规模（而非环境矩阵维度）的估计误差。我们进一步将分析扩展到辅助信息不精确的场景，证明降低的样本复杂度得以保持，且估计误差在辅助信息不精确度方面达到阶数最优。在MovieLens数据集上的大量仿真与真实实验验证了我们的理论发现。