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.

翻译：本文研究从含噪声的部分观测条目中恢复底层矩阵的问题，该问题通常被称为矩阵补全。我们深入探究一种称为变换$L_1$（TL1）的非凸正则化方法，该方法通过超参数$a \in (0, \infty)$在矩阵的秩与核范数之间进行插值。尽管已有文献采用此类正则化进行矩阵补全，但其主要针对条目均匀缺失的情形，并侧重于算法改进。为填补当前文献的空白，我们对一般采样分布下TL1正则化恢复模型所得到的估计量进行了全面的统计分析。具体而言，我们证明当$a$充分大时，尽管TL1正则化的非凸性带来了挑战，但基于TL1的模型所恢复的矩阵在Frobenius范数度量下仍能达到与基于核范数的模型相当的收敛速率。当$a$足够小时，我们证明在真实矩阵为精确低秩的情况下，估计矩阵的秩将保持为常数阶。通过不同调节参数的选择，我们在误差控制与秩估计之间建立了权衡关系。一项涵盖多种采样机制的仿真研究以及两个实际应用案例展示了TL1正则化在实际性能上的优势。此外，我们通过实验探究了超参数$a$在基于TL1的模型中的作用，为实际应用提供了指导。