正交非负矩阵分解的算法、理论与应用

项目名称： 正交非负矩阵分解的算法、理论与应用

项目编号： No.11726618

项目类型： 专项基金项目

立项/批准年度： 2018

项目学科： 数理科学和化学

项目作者： 申远

作者单位： 南京财经大学

项目金额： 10万元

中文摘要： 正交非负矩阵分解（ONMF）是一种用于矩阵近似的数学模型，来源于线性代数和多元统计分析。该问题可以被描述为：将给定矩阵近似分解为两个规模较小的非负矩阵的乘积，且其中一个矩阵满足正交性约束。该模型源于非负矩阵分解（NMF），已被成功应用于计算机视觉、基因表达、文档聚类、统计学习、化学计量学、图像及音频信号处理、文本挖掘、推荐系统等诸多科学与工程领域。ONMF和NMF最大的区别在于添加了正交性约束，这个非凸约束使得求解该问题的难度大大提升。截至目前，关于ONMF的研究较为零散，求解ONMF的算法更为稀少。我们希望提出求解ONMF问题的新型高效算法，新算法应该具有框架简单、计算效率高、尺度可扩展性高（即问题规模增长时计算效率不会出现显著下降）等优点。

中文关键词： 低秩优化；一阶方法；交替方向乘子法；邻近点算法；计算复杂性

英文摘要： Orthogonal nonnegative matrix factorization (ONMF) is a mathematical model for matrix approximation, deriving from linear algebra and multivariate statistical analysis. The problem can be described as: the given matrix is approximately decomposed into the product of two smaller nonnegative matrices, and one of the matrices satisfies the orthogonality constraint. The model is derived from nonnegative matrix factorization (NMF), which has been successfully used in computer vision, gene expression, document clustering, statistical learning, chemometrics, image and audio signal processing, text mining, recommendation systems and other fields of science and engineering. The biggest difference between ONMF and NMF is the addition of orthogonality constraints, which makes the problem more difficult to solve. Up to now, the study on ONMF is fragmented, and algorithms for solving ONMF is still scarce. We hope to propose a new efficient algorithm for solving the ONMF problem. The new algorithm should have the advantages of simple frame, high computational efficiency and high dimensional scalability (i.e., the computational efficiency can hardly be affected as the dimension increases).

英文关键词： low-rank optimization;first-order method;alternating Direction Method of Multipliers;proximal point algorithm;computational complexity