We propose an efficient matrix rank reduction method for non-negative matrices, whose time complexity is quadratic in the number of rows or columns of a matrix. Our key insight is to formulate rank reduction as a mean-field approximation by modeling matrices via a log-linear model on structured sample space, which allows us to solve the rank reduction as convex optimization. The highlight of this formulation is that the optimal solution that minimizes the KL divergence from a given matrix can be analytically computed in a closed form. We empirically show that our rank reduction method is faster than NMF and its popular variant, lraNMF, while achieving competitive low rank approximation error on synthetic and real-world datasets.
翻译:我们建议对非负矩阵采用有效的矩阵排位削减方法,其时间复杂性在矩阵的行数或列数上是四倍的。我们的主要见解是通过结构化样本空间的日志线性模型模型将排位降为中位近似值,从而通过结构化样本空间的对数-线性模型将排位降为中位近似值,从而使我们能够解决排位降为曲线优化的问题。这一提法的亮点是,能够以封闭的形式分析计算出最大限度地减少与特定矩阵之间的 KL 差异的最佳解决方案。我们的经验显示,我们的排位降方法比NMF及其流行变量(lRANMF)更快,同时在合成和真实世界数据集上实现竞争性低位近似误差。