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一个被广泛研究的问题，出现在推荐系统与信号处理等多种应用中，其中包含矩阵补全和压缩感知模型作为特例。本文提出了一个通用框架，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相互独立的设定，随后通过修正的浓度不等式，将研究扩展到设计矩阵与噪声均具有特定时间相关性的设定。通过合并相邻观测值，我们获得了这两种设定下的精确估计误差界，揭示了潜在平滑性、依赖性和有效样本量的影响。我们提出了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。模拟与真实数据实验验证了上述发现。