论文标题
通过非凸放松和自适应相关学习完成矩阵完成
Matrix Completion via Non-Convex Relaxation and Adaptive Correlation Learning
论文作者
论文摘要
现有的矩阵完成方法着重于优化等级功能的放松，例如核定标准，Schatten-P Norm等。它们通常需要许多迭代才能收敛。此外，在大多数现有模型中仅利用矩阵的低排名属性，而在实践中，几种结合其他知识的方法很耗时。为了解决这些问题，我们提出了一种新型的非凸代替代物，可以通过封闭形式的解决方案进行优化，以便在数十个迭代范围内经验收敛。此外，优化是无参数的，并证明了收敛性。与等级的放松相比，替代物是通过优化上层排名来激发的。从理论上讲，我们验证它等于现有的矩阵完成模型。除了低排名的假设外，我们打算利用矩阵完成的列的相关性，从而开发了自适应相关性学习，该学习是缩放不变的。更重要的是，在结合了相关学习后，该模型仍然可以通过封闭式解决方案来解决，以使其仍然快速收敛。实验显示了非凸替代和适应性相关性学习的有效性。
The existing matrix completion methods focus on optimizing the relaxation of rank function such as nuclear norm, Schatten-p norm, etc. They usually need many iterations to converge. Moreover, only the low-rank property of matrices is utilized in most existing models and several methods that incorporate other knowledge are quite time-consuming in practice. To address these issues, we propose a novel non-convex surrogate that can be optimized by closed-form solutions, such that it empirically converges within dozens of iterations. Besides, the optimization is parameter-free and the convergence is proved. Compared with the relaxation of rank, the surrogate is motivated by optimizing an upper-bound of rank. We theoretically validate that it is equivalent to the existing matrix completion models. Besides the low-rank assumption, we intend to exploit the column-wise correlation for matrix completion, and thus an adaptive correlation learning, which is scaling-invariant, is developed. More importantly, after incorporating the correlation learning, the model can be still solved by closed-form solutions such that it still converges fast. Experiments show the effectiveness of the non-convex surrogate and adaptive correlation learning.
