Matrix recovery from sparse observations is an extensively studied topic emerging in various applications, such as recommendation system and signal processing, which includes the matrix completion and compressed sensing models as special cases. In this work we propose a general framework for dynamic matrix recovery of low-rank matrices that evolve smoothly over time. We start from the setting that the observations are independent across time, then extend to the setting that both the design matrix and noise possess certain temporal correlation via modified concentration inequalities. By pooling neighboring observations, we obtain sharp estimation error bounds of both settings, showing the influence of the underlying smoothness, the dependence and effective samples. We propose a dynamic fast iterative shrinkage thresholding algorithm that is computationally efficient, and characterize the interplay between algorithmic and statistical convergence. Simulated and real data examples are provided to support such findings.

翻译：从稀疏观测中恢复矩阵是推荐系统、信号处理等多种应用场景中的广泛研究课题，其特殊情形包括矩阵补全与压缩感知模型。本文提出一个通用框架，用于恢复随时间平滑演化的低秩动态矩阵。我们首先考虑观测值在时间上独立的情形，进而通过修正浓度不等式，将框架扩展至设计矩阵与噪声均具有特定时间相关性的设定。通过整合邻近时刻的观测数据，我们获得了两种设定下的尖锐估计误差界，揭示了潜在平滑性、依赖关系及有效样本量的影响。我们提出一种计算高效的动态快速迭代收缩阈值算法，并刻画了算法收敛与统计收敛之间的相互作用。模拟与真实数据案例验证了上述发现。