High-dimensional matrix regression has been studied in various aspects, such as statistical properties, computational efficiency and application to specific instances including multivariate regression, system identification and matrix compressed sensing. Current studies mainly consider the idealized case that the covariate matrix is obtained without noise, while the more realistic scenario that the covariates may always be corrupted with noise or missing data has received little attention. We consider the general errors-in-variables matrix regression model and proposed a unified framework for low-rank estimation based on nonconvex spectral regularization. Then in the statistical aspect, recovery bounds for any stationary points are provided to achieve statistical consistency. In the computational aspect, the proximal gradient method is applied to solve the nonconvex optimization problem and is proved to converge in polynomial time. Consequences for specific matrix compressed sensing models with additive noise and missing data are obtained via verifying corresponding regularity conditions. Finally, the performance of the proposed nonconvex estimation method is illustrated by numerical experiments.

翻译：高维矩阵回归已在多个方面得到研究，包括统计性质、计算效率，以及其在多元回归、系统辨识和矩阵压缩感知等具体场景中的应用。现有研究主要考虑协变量矩阵在无噪声情况下的理想化设定，而协变量可能受到噪声污染或数据缺失的更现实场景却鲜有探讨。本文考虑一般性的变量含误差矩阵回归模型，并提出一个基于非凸谱正则化的低秩估计统一框架。在统计层面，我们为任意驻点提供了实现统计一致性的恢复界。在计算层面，采用近端梯度法求解该非凸优化问题，并证明其能在多项式时间内收敛。通过验证相应的正则性条件，我们进一步得到了含加性噪声和缺失数据的特定矩阵压缩感知模型的相关结论。最后，通过数值实验验证了所提非凸估计方法的性能。