We introduce a flexible framework for high-dimensional matrix estimation to incorporate side information for both rows and columns. Existing approaches, such as inductive matrix completion, often impose restrictive structure-for example, an exact low-rank covariate interaction term, linear covariate effects, and limited ability to exploit components explained only by one side (row or column) or by neither-and frequently omit an explicit noise component. To address these limitations, we propose to decompose the underlying matrix as the sum of four complementary components: (possibly nonlinear) interaction between row and column characteristics; row characteristic-driven component, column characteristic-driven component, and residual low-rank structure unexplained by observed characteristics. By combining sieve-based projection with nuclear-norm penalization, each component can be estimated separately and these estimated components can then be aggregated to yield a final estimate. We derive convergence rates that highlight robustness across a range of model configurations depending on the informativeness of the side information. We further extend the method to partially observed matrices under both missing-at-random and missing-not-at-random mechanisms, including block-missing patterns motivated by causal panel data. Simulations and a real-data application to tobacco sales show that leveraging side information improves imputation accuracy and can enhance treatment-effect estimation relative to standard low-rank and spectral-based alternatives.

翻译：我们提出了一种灵活的框架，用于在高维矩阵估计中整合行和列的辅助信息。现有方法（如归纳式矩阵补全）通常施加严格的假设结构——例如精确的低秩协变量交互项、线性协变量效应，且难以利用仅由单侧（行或列）或双侧均无法解释的成分，并且常忽略显式噪声分量。为解决这些局限，我们将潜在矩阵分解为四个互补成分的和：（可能非线性的）行与列特征之间的交互项、行特征驱动成分、列特征驱动成分，以及未被观察特征解释的残差低秩结构。通过结合基于筛的投影与核范数惩罚，每个成分可被独立估计，最终合并得到整体估计量。我们推导了收敛速率，揭示了该框架在不同模型配置下的鲁棒性——其取决于辅助信息的信息量。该方法进一步扩展至部分观测矩阵，涵盖随机缺失和非随机缺失机制，包括由因果面板数据驱动的块缺失模式。模拟实验及香烟销售的真实数据应用表明，相比于标准低秩和基于谱的方法，利用辅助信息能提升插补精度并增强处理效应估计效果。