SGER: Collaborative Research: Non-negative Matrix Factorizations for Data Mining: Algorithms 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，最基本形式的NMF等价于数据挖掘中使用最广泛的模式发现算法松弛K-means聚类。数学和数据挖掘之间的这种直接联系推动了使用矩阵分解进行模式发现的大量发展。事实证明，NMF为许多基本的和新兴的数据挖掘问题提供了更加一致和数学上定义良好的优化公式。NMF算法具有众所周知的特性；它们简单且易于实现，非常适合分布式并行架构。本研究旨在正式建立一个全面的基于nmf的数据挖掘框架。特别是，我们将(1)将矩阵分解数据挖掘方法从当前关注的聚类（模式发现）扩展到新的问题：半监督聚类（将部分知识扩展到整个数据）和分类（模式预测，例如从正常组织中预测癌症肿瘤组织）；(2)开发快速数值算法，并结合最先进的数值优化技术；(3)在文本挖掘和生物信息学等不同的现实世界应用中应用和评估NMF算法。

Nonnegative matrix factorization (NMF) factorizes an input nonnegative matrix into two nonnegative matrices of lower rank. It is recently discovered that NMF in the most basic form is equivalent to a relaxed K-means clustering, the most widely used pattern discovery algorithm in data mining. This direct link between mathematics and data mining sets in motion a large number of developments on using matrix factorizations for pattern discovery. It turns out that NMF provides more consistent and mathematically well-defined optimization formulations for many fundamental and emerging data-mining problems. NMF algorithms have well-understood properties; they are simple and easy-to-implement, well suited for distributed parallel architectures. This research aims to formally establish a comprehensive NMF-based framework for data mining. In particular, we will (1) extend matrix factorization data-mining methodology from current focus on clustering (pattern discovery) to newer problems: semi-supervised clustering (extending partial knowledge to whole data) and classifications (pattern prediction, such as predicting a cancer tumor tissue from a normal one); (2) develop fast numerical algorithms and incorporate state-of-the-art numerical optimization techniques; and (3) apply and evaluate the NMF algorithms in different real-world applications including text mining and bioinformatics.