Theory and Applications of Nonnegative Matrix Factorization

该研究项目将推动非负矩阵分解(NMF)算法的发展。NMF是一种数学运算，能够自动分解和分类非结构化数据集中的项。例如，NMF可以仅基于文章中出现的单词来自动识别报纸文章的大型数据集中的各种重复主题。作为第二个例子，NMF可以分析卫星的高光谱图像，以确定哪些部分是铝的，哪些是塑料的，等等。NMF还有许多其他应用，包括分析微阵列生化实验的结果，以及在流行病学中跟踪易感人群。它甚至被用来分析音乐作品。这项研究将为NMF开发新的算法，既要更快，又能更好地确保能够找到正确的分解。作为研究的一部分，将开发所谓的生成性模型，这是捕捉真实数据集的本质特征的数学模型。生成性模型的使用可以让我们深入了解为什么一些算法在真实数据上比其他算法更有效。这项研究最直接的影响将在数学家、计算机科学家和统计学家的社区内，他们开发和分析NMF和其他寻找数据结构的算法。拟议研究的更广泛影响将体现在使用NMF的许多应用中；NMF效率和准确性的提高将意味着更强的理解和分析科学和医学许多领域数据的能力。

The research program will advance the state of the art in algorithms for nonnegative matrix factorization (NMF). NMF is a mathematical operation that is able to automatically decompose and classify items in an unstructured dataset. For example, NMF can automatically identify various recurring topics in a large data set of newspaper articles based solely on the words appearing in the articles. As a second example, NMF can analyze a hyperspectral image of a satellite to determine which portions are aluminum, which are plastic and so on. NMF has many other applications including analysis of the results of microarray biochemical experiments and in tracking susceptible populations in epidemiology. It has even been used to analyze musical compositions. The research will develop new algorithms for NMF that are both faster and come with better assurances that they are able to find the correct decomposition. As part of the research, so-called generative models will be developed, which are mathematical models that capture essential qualities of real datasets. The use of generative models can give insight into why some algorithms work better on real data than others. The most direct impact of the research will be within the community of mathematicians, computer scientists and statisticians who develop and analyze NMF and other algorithms for finding structure in data. The broader impact of the proposed research will be in the many applications where NMF is used; improved efficiency and accuracy of NMF will mean greater ability to understand and analyze data across many fields of science and medicine.