基于平均场理论的非负矩阵快速降秩
Fast Rank Reduction for Non-negative Matrices via Mean Field Theory
摘要：
提出了一种有效的非负矩阵降秩方法，其时间复杂度是矩阵行数或列数的二次方。我们的关键见解是制定降秩作为一个平均场近似建模矩阵通过一个对数线性模型结构化样本空间，这使我们能够解决降秩作为凸优化。这个公式的亮点是，最小化给定矩阵的库尔贝克-莱布勒散度的最优解可以以封闭形式解析计算。我们的经验表明，我们的降秩方法比NMF及其流行的变体lraNMF更快，同时在合成和真实世界的数据集上实现了具有竞争力的低秩近似误差。
英文摘要：
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.
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