Title: Noisy Low-Rank Matrix Completion via Transformed $L_1$ Regularization and its Theoretical Properties Show Chinese title

Title: 通过变换的$L_1$正则化进行噪声低秩矩阵补全及其理论性质

Abstract: This paper focuses on recovering an underlying matrix from its noisy partial entries, a problem commonly known as matrix completion. We delve into the investigation of a non-convex regularization, referred to as transformed $L_1$ (TL1), which interpolates between the rank and the nuclear norm of matrices through a hyper-parameter $a \in (0, \infty)$. While some literature adopts such regularization for matrix completion, it primarily addresses scenarios with uniformly missing entries and focuses on algorithmic advances. To fill in the gap in the current literature, we provide a comprehensive statistical analysis for the estimator from a TL1-regularized recovery model under general sampling distribution. In particular, we show that when $a$ is sufficiently large, the matrix recovered by the TL1-based model enjoys a convergence rate measured by the Frobenius norm, comparable to that of the model based on the nuclear norm, despite the challenges posed by the non-convexity of the TL1 regularization. When $a$ is small enough, we show that the rank of the estimated matrix remains a constant order when the true matrix is exactly low-rank. A trade-off between controlling the error and the rank is established through different choices of tuning parameters. The appealing practical performance of TL1 regularization is demonstrated through a simulation study that encompasses various sampling mechanisms, as well as two real-world applications. Additionally, the role of the hyper-parameter $a$ on the TL1-based model is explored via experiments to offer guidance in practical scenarios. Show Chinese abstract

Abstract: 本文专注于从其噪声部分条目中恢复底层矩阵，这一问题通常被称为矩阵补全。 我们深入研究了一种非凸正则化方法，称为变换的$L_1$（TL1），它通过一个超参数$a \in (0, \infty)$在矩阵的秩和核范数之间进行插值。 虽然一些文献采用了这种正则化方法进行矩阵补全，但主要针对的是缺失条目均匀的情况，并关注算法的进步。 为了填补当前文献中的空白，我们在一般抽样分布下对TL1正则化恢复模型的估计量进行了全面的统计分析。 特别是，我们证明当$a$足够大时，基于TL1的模型恢复的矩阵在Frobenius范数下具有收敛率，与基于核范数的模型相当，尽管TL1正则化存在非凸性的挑战。 当$a$足够小时，我们证明当真实矩阵是精确低秩时，估计矩阵的秩保持为常数阶。 通过选择不同的调参方式，建立了控制误差和秩之间的权衡。 通过涵盖各种抽样机制的模拟研究以及两个实际应用，展示了TL1正则化的出色实际性能。 此外，通过实验探索了超参数$a$对基于TL1的模型的影响，以在实际场景中提供指导。