论文标题
矩阵完成的数值集成
Splitting numerical integration for matrix completion
论文作者
论文摘要
低等级矩阵近似是机器学习中的一个流行话题。在本文中，我们通过最大程度地减少对固定级矩阵的riemannian歧管的最小二乘估计来提出一种针对此主题的新算法。该算法是对歧管优化框架内经典梯度下降的适应。特别是，我们将低级别多种多样的不受约束的优化问题重新制定为差异动态系统。我们通过将分裂集成方案应用于动态系统来开发分裂数值集成方法。我们对分裂数值整合算法进行收敛分析。可以保证，恢复的矩阵和真实结果之间的误差在弗罗贝尼乌斯规范中单调减少。此外，我们的拆分数值集成可以调整为矩阵完成方案。实验结果表明，我们的方法对大规模问题具有良好的可扩展性，精度令人满意
Low rank matrix approximation is a popular topic in machine learning. In this paper, we propose a new algorithm for this topic by minimizing the least-squares estimation over the Riemannian manifold of fixed-rank matrices. The algorithm is an adaptation of classical gradient descent within the framework of optimization on manifolds. In particular, we reformulate an unconstrained optimization problem on a low-rank manifold into a differential dynamic system. We develop a splitting numerical integration method by applying a splitting integration scheme to the dynamic system. We conduct the convergence analysis of our splitting numerical integration algorithm. It can be guaranteed that the error between the recovered matrix and true result is monotonically decreasing in the Frobenius norm. Moreover, our splitting numerical integration can be adapted into matrix completion scenarios. Experimental results show that our approach has good scalability for large-scale problems with satisfactory accuracy
