Theory and Applications of Nonnegative Matrix Factorization

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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.