Recovering low-rank matrices from incomplete observations is a fundamental problem with many applications, especially in recommender systems. In theory, under certain conditions, this problem can be solved by convex or nonconvex relaxation. However, most existing provable algorithms suffer from superlinear per-iteration cost, which severely limits their applicability to large-scale problems. In this paper, we propose a novel fuzzy double trace norm minimization (DTNM) method for recommender systems. We first present a tractable DTNM model, in which we can integrate both the user social relationship and the user reputation information using a fuzzy weighting way and coupling fuzzy matrix factorization. In essence, our model is a Schatten-${1/2}$ quasi-norm minimization problem. Moreover, we develop two efficient augmented Lagrangian algorithms to solve the proposed problems, and prove the convergence of our algorithms. Finally, we investigate the empirical recoverability properties of our model and its advantage over classical trace norm. Extensive experimental results on both synthetic and real-world data sets verified both the efficiency and effectiveness of our method compared with the state-of-the-art algorithms.

中文翻译：

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推荐系统的模糊双迹范数最小化

从不完整的观察中恢复低秩矩阵是许多应用程序的基本问题，尤其是在推荐系统中。理论上，在一定条件下，这个问题可以通过凸松弛或非凸松弛来解决。然而，大多数现有的可证明算法都存在超线性迭代成本，这严重限制了它们在大规模问题上的适用性。在本文中，我们为推荐系统提出了一种新颖的模糊双迹范数最小化（DTNM）方法。我们首先提出了一个易于处理的 DTNM 模型，在该模型中，我们可以使用模糊加权方式和耦合模糊矩阵分解来整合用户社会关系和用户声誉信息。本质上，我们的模型是一个 Schatten-${1/2}$ 准范数最小化问题。而且，我们开发了两种有效的增广拉格朗日算法来解决所提出的问题，并证明了我们算法的收敛性。最后，我们研究了我们模型的经验可恢复性特性及其相对于经典迹范数的优势。与最先进的算法相比，在合成和真实世界数据集上的大量实验结果验证了我们的方法的效率和有效性。