MSPA-MCS: Collaborative Research: Fast Nonnegative Matrix Factorizations: Theory, Algorithms, and Applications

提案ID：0732318和0732299 PI（s）：Haesun Park和Moody Chu机构（s）：GaTech和NCSU标题：合作研究：快速非负矩阵分解：理论，算法和应用摘要：具有非负数据值的数学模型在科学和工程中非常丰富。为了物理上的可行性和可解释性，在计算和分析中必须保留非负的性质。本文研究非负矩阵分解为低秩非负矩阵乘积的问题。这种非负矩阵分解的概念在广泛的重要应用中起着重要作用，包括文本挖掘，化学信息学，因子检索，图像清晰度，生物信息学，以及在模式和数据分析中的降维和聚类。从这个拟议的研究发现，预计不仅会影响先进的理论基础矩阵计算，但也有助于数据挖掘的一般领域，如降维，聚类和visualization.NMF背后的基本问题是最好的近似一个给定的非负数据矩阵的产品的两个较低的维，因此，较低的秩非负矩阵。这两个较低秩的矩阵提供了大量的基本信息，否则将难以从原始矩阵中检索到这些信息。许多NMF技术已被提出在文献中，但仍然很少有理论上如何NMF可以鲁棒和有效地解决。在这项工作中，新的更快的算法的开发将通过对有前途的研究方向进行结构化和全面的性能评估来进行，包括基于活动集和几何的算法，针对真实世界的应用数据，以获得有价值的见解。拟议的研究的几何结构的NMF和NMF算法的理论属性，如收敛性，应提供评估的基础上，任何NMF方法。NMF的降维和聚类的适用性也将被调查。本研究的结果也可能在数据库管理、医学检查和诊断、生化选择和生物网络中具有潜在的应用。

Proposal ID(s): 0732318 and 0732299PI(s): Haesun Park and Moody ChuInstitition(s): GaTech and NCSUTitle: Collaborative Research: Fast Nonnegative Matrix Factorizations: Theory, Algorithms, and ApplicationsABSTRACT:Mathematical models with nonnegative data values are abounding in sciences and engineering. For the sake of physical feasibility and interpretability, the nature of nonnegative must be retained in computation and analysis. This work concerns itself with the factorization of nonnegative matrix into product of lower rank nonnegative matrices. Such a notion of the nonnegative matrix factorization plays a major role in a wide range of important applications including text mining, cheminformatics, factor retrieval, image articulation, bioinformatics, and in dimension reduction and clustering in pattern and data analysis. The discoveries from this proposed research are expected to impact not only the advanced theoretical foundations of matrix computation, but also contribute to the general areas of data mining such as dimension reduction, clustering, and visualization.The basic question behind the nonnegative matrix factorization (NMF) is to best approximate a given nonnegative data matrix as the product of two lower dimensional and, hence, lower rank nonnegative matrices. The two lower rank matrices provides lot of essential information that, otherwise, would be difficult to retrieve from the original matrix. Many NMF techniques have been proposed in the literature, yet there is still little theory on how the NMF can be robustly and efficiently solved. In this work, development of new faster algorithms will be conducted through structured and comprehensive performance evaluation of promising research directions, including the active set and geometry based algorithms, against real-world application data to obtain valuable insights. The proposed study of the geometric structure of the NMF and theoretical properties of the NMF algorithms, such as convergence, should provide the basis of assessment for any NMF methods. Applicability of the NMF to dimension reduction and clustering will also be investigated. Results of this research are also likely to have potential applications in database management, medical examination and diagnosis, bio-chemical selection, and biological networks.