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

MSPA-MCS：协作研究：快速非负矩阵分解：理论、算法和应用

• 批准号: 0732318
• 负责人: Haesun Park
• 金额: $ 25.18万
• 依托单位: Georgia Tech Research Corporation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-10-01 至 2013-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0732318&HistoricalAwards=false
• 关键词: MSPA; MCS; Collaborative; Research; Fast

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.

提案 ID（s）：0732318 和 0732299PI（s）：Haesun Park 和 Moody Chu 机构：GaTech 和 NCSU 标题：协作研究：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比比皆是。为了物理可行性和可解释性，在计算和分析中必须保留非负的性质。这项工作涉及将非负矩阵分解为低阶非负矩阵的乘积。这种非负矩阵分解的概念在广泛的重要应用中发挥着重要作用，包括文本挖掘、化学信息学、因子检索、图像清晰度、生物信息学以及模式和数据分析中的降维和聚类。这项研究的发现不仅会影响矩阵计算的高级理论基础，而且还会对数据挖掘的一般领域做出贡献，例如降维、聚类和可视化。非负矩阵分解 (NMF) 背后的基本问题是将给定的非负数据矩阵最好地近似为两个较低维度（即较低秩的非负矩阵）的乘积。两个较低秩的矩阵提供了许多基本信息，否则将很难从原始矩阵中检索这些信息。文献中已经提出了许多 NMF 技术，但关于如何鲁棒且有效地求解 NMF 的理论仍然很少。在这项工作中，将通过对有前途的研究方向（包括基于活动集和基于几何的算法）进行结构化和全面的性能评估来开发新的更快的算法，并根据实际应用数据来获得有价值的见解。所提出的对 NMF 几何结构和 NMF 算法的理论特性（例如收敛性）的研究应该为任何 NMF 方法提供评估基础。还将研究 NMF 在降维和聚类方面的适用性。这项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方面有潜在的应用。