Abstract: The low rank matrix approximation model based on nuclear norm can not exactly reflect the properties of the original matrix because of imposing the same punishments to all singular values. To solve this problem，Weighted Nuclear Norm Minimization（WNNM）model with predicted values for missed items was proposed . Firstly，the approximation matrix could be well approximated to the original one by constructing the weights opposite to the size of the singular values. Then，a new algorithm named APGL-WNNM was proposed based on improved APGL（Accelerated Proximal Gradient Algorithm）. Moreover，an appropriate prediction method to construct the initial matrix was used to accelerate the convergence of the new algorithm. At last，the convergence of the proposed algorithm was proved. The comparison results between the new proposed algorithm and APGL algorithm on MovieLens dataset show that the improved weighted nuclear norm algorithm works better than the mainstream optimization algorithm.

Key words: low rank matrix approximation, weighted nuclear norm, collaborative filtering, personalized recommendation

摘要： 基于核范数的低秩矩阵近似模型，由于对所有奇异值的惩罚力度一样，导致不能很好地反映原矩阵的特性，针对此问题提出了带初始值引导的加权核范数最小模型。首先，通过构造和奇异值的大小相反的权值，使得近似矩阵能够很好地逼近原矩阵；其次，改进线性搜索加速近端梯度算法（APGL），提出了求解加权核范数最小模型的APGL-WNNM 算法；然后，使用适当的预估方法，构造初始引导矩阵，来提高算法的收敛速度；最后，证明了新提出算法的收敛性，使用 MoiveLens数据集，对所提出的 APGL-WNNM算法和 APGL算法进行比较。基于同样的数据集，改进的加权核范数算法比主流的优化算法效果更好。

关键词: 低秩矩阵近似, 加权核范数, 协同过滤, 个性化推荐