Collaborative Research: Non-negative Matrix Factorizations for Data Mining: Foundations, Capabilities, and Applications

非负矩阵分解（NMF）将输入的非负矩阵分解为两个低秩的非负矩阵。最近发现，NMF具有解决具有挑战性的数据挖掘和机器学习问题的独特能力。与现有的无监督学习方法相比，NMF的优势在于：(1)NMF可以对变化很大的数据分布进行建模；(2)NMF可以同时进行硬聚类和软聚类。(3)许多其他数据挖掘问题，如半监督聚类问题，可以重新表述为NMF问题。在这些基础上，研究人员建议建立一个基于NMF的数据挖掘综合框架：(a)对NMF的聚类能力提供更深入的理解；(b)扩展NMF的数据挖掘能力，解决各种数据挖掘和机器学习问题；(c)发展快速数值算法，其中包括各种矩阵分解模型的数值最优化的最新发展；(d)制订新颖和严格的证明策略，以证明数值算法的正确性和收敛性；(e)在实际应用中应用和评价这些新算法。提出的工作创造了一个新的范式，分析大量的数据，并通过转换既定的矩阵计算方法从数据中发现新的知识。这项新技术可以自动将新闻文章分组为有意义的类别，在蛋白质网络中发现蛋白质模块，从气候数据中提取天气模式，将图片分割为不同的对象，在网络上检测社区，并使许多其他科学发现和新技术创造成为可能。在基本层面上，所提出的工作表明，一个简单的矩阵分解实际上解决了具有挑战性的数据挖掘问题。这项研究强调了数学在当今以数据为中心的世界中的重要性，并鼓励学生学习数学。

Nonnegative matrix factorization (NMF) factorizes an input nonnegative matrix into two nonnegative matrices of lower rank. It was recently discovered that NMF has unique ability to solve challenging data mining and machine learning problems. The advantage of NMF over existing unsupervised learning methods are (1) NMF can model widely varying data distributions, (2) NMF performs both hard and soft clustering simultaneously. (3) Many other data mining problems such as semi-supervised clustering problems can be reformulated as NMF problem. Building upon these foundations, the investigators propose to establish a NMF-based comprehensive framework for data mining: (a) Provide deeper understanding of NMF's clustering capability;(b) Extend data mining capability of NMF for solving various data mining and machine learning problems; (c) Develop fast numerical algorithms which incorporate the state-of-the-art developments from numerical optimization for various matrix factorization models; (d) Develop novel and rigorous proof strategies to prove the correctness and convergence properties of the numerical algorithms; (e) Apply and evaluate these new algorithms in real-world applications.The proposed work creates a new paradigm of analyzing vast amount of data and discovering new knowledge from the data by transforming established matrix computational methodologies. This new technology can automatically group news articles into meaningful categories, discover protein modules in protein networks, extract weather patterns in climate data, segment pictures into distinct objects, detect communities on the Web, and enable many other scientific discoveries and new technologies creation. On a fundamental level, the proposed work establishes that a simple matrix factorization in fact solves challenging data mining problems. This research reinforces the importance of mathematics in today's data centric world and encourages students to learn mathematics.