Transfer Learning for Matrix Completion
This work addresses the challenge of enhancing matrix completion accuracy for data analysis tasks, but it is incremental as it builds on existing transfer learning frameworks.
The paper tackles the problem of improving low-rank matrix completion by transferring knowledge from auxiliary source datasets, achieving minimax optimal convergence rates and outperforming traditional single-dataset methods when sources are relevant.
In this paper, we explore the knowledge transfer under the setting of matrix completion, which aims to enhance the estimation of a low-rank target matrix with auxiliary data available. We propose a transfer learning procedure given prior information on which source datasets are favorable. We study its convergence rates and prove its minimax optimality. Our analysis reveals that with the source matrices close enough to the target matrix, out method outperforms the traditional method using the single target data. In particular, we leverage the advanced sharp concentration inequalities introduced in \cite{brailovskaya2024universality} to eliminate a logarithmic factor in the convergence rate, which is crucial for proving the minimax optimality. When the relevance of source datasets is unknown, we develop an efficient detection procedure to identify informative sources and establish its selection consistency. Simulations and real data analysis are conducted to support the validity of our methodology.