Low-rank matrix completion has achieved great success in many real-world data applications. A matrix factorization model that learns latent features is usually employed and, to improve prediction performance, the similarities between latent variables can be exploited by pairwise learning using the graph regularized matrix factorization (GRMF) method. However, existing GRMF approaches often use the squared loss to measure the pairwise differences, which may be overly influenced by dissimilar pairs and lead to inferior prediction. To fully empower pairwise learning for matrix completion, we propose a general optimization framework that allows a rich class of (non-)convex pairwise penalty functions. A new and efficient algorithm is developed to solve the proposed optimization problem, with a theoretical convergence guarantee under mild assumptions. In an important situation where the latent variables form a small number of subgroups, its statistical guarantee is also developed. In particular, we theoretically characterize the performance of the complexity-regularized maximum likelihood estimator, as a special case of our framework, which is shown to have smaller errors when compared to the standard matrix completion framework without pairwise penalties. We conduct extensive experiments on both synthetic and real datasets to demonstrate the superior performance of this general framework.

中文翻译：

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在低秩矩阵完成中学习具有成对惩罚的潜在特征

低秩矩阵补全在许多现实世界的数据应用中取得了巨大的成功。通常采用学习潜在特征的矩阵分解模型，为了提高预测性能，可以使用图正则化矩阵分解 (GRMF) 方法通过成对学习来利用潜在变量之间的相似性。然而，现有的 GRMF 方法通常使用平方损失来衡量成对差异，这可能会受到不同对的过度影响并导致较差的预测。为了充分授权矩阵完成的成对学习，我们提出了一个通用优化框架，该框架允许丰富的（非）凸成对惩罚函数。开发了一种新的有效算法来解决所提出的优化问题，并在温和的假设下具有理论收敛性保证。在潜在变量形成少量子组的重要情况下，其统计保证也得到了发展。特别是，我们在理论上将复杂性正则化最大似然估计器的性能表征为我们框架的一个特例，与没有成对惩罚的标准矩阵完成框架相比，它的误差更小。我们对合成数据集和真实数据集进行了大量实验，以证明该通用框架的卓越性能。与没有成对惩罚的标准矩阵完成框架相比，它显示出较小的误差。我们对合成数据集和真实数据集进行了大量实验，以证明该通用框架的卓越性能。与没有成对惩罚的标准矩阵完成框架相比，它显示出较小的误差。我们对合成数据集和真实数据集进行了大量实验，以证明该通用框架的卓越性能。