Rank minimization is of interest in machine learning applications such as recommender systems and robust principal component analysis. Minimizing the convex relaxation to the rank minimization problem, the nuclear norm, is an effective technique to solve the problem with strong performance guarantees. However, nonconvex relaxations have less estimation bias than the nuclear norm and can more accurately reduce the effect of noise on the measurements. We develop efficient algorithms based on iteratively reweighted nuclear norm schemes, while also utilizing the low rank factorization for semidefinite programs put forth by Burer and Monteiro. We prove convergence and computationally show the advantages over convex relaxations and alternating minimization methods. Additionally, the computational complexity of each iteration of our algorithm is on par with other state of the art algorithms, allowing us to quickly find solutions to the rank minimization problem for large matrices.

在推荐器系统和可靠的主成分分析之类的机器学习应用程序中，排名最小化是令人感兴趣的。最小化对秩最小化问题的凸松弛，即核范数，是一种具有强大性能保证的有效解决问题的技术。但是，非凸弛豫比核标准具有较小的估计偏差，并且可以更准确地减少噪声对测量的影响。我们基于迭代加权核规范方案开发了有效的算法，同时还将低秩分解用于Burer和Monteiro提出的半确定程序。我们证明了收敛性并通过计算显示了优于凸松弛和交替最小化方法的优势。此外，