Dynamic Matrix Recovery

Abstract

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 article, 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. Supplementary materials for this article are available online.

Keywords:

• Compressed sensing
• Local smoothing
• Low rank
• Matrix completion

Supplementary Materials

Dynamic_Matrix_Recovery_supp Detail proofs for all theorems and corollaries in “Dynamic Matrix Recovery”, theoretical results for dynamic compressed sensing and additional numerical results. (.pdf file)

Code and Data for Dynamic Matrix Recovery R-code to implement and reproduce the simulation and real data results, corresponding output and raw datasets.

Disclosure Statement

The authors report there are no competing interests to declare.

Additional information

Funding

Fang Yao’s research is partially supported by the National Key R&D Program of China (No. 2022YFA1003800), the National Natural Science Foundation of China (No. 12292981, 11931001), the LMAM and the Fundamental Research Funds for the Central Universities and the LMEQF. Ying Yang’s research is partially supported by China Postdoctoral Science Foundation (No. 2022TQ0360, 2022M723334), the National Natural Science Foundation of China (No. 71988101) and the Guozhi Xu Posdoctoral Research Foundation.