Numerical Multilinear Algebra in Signal Processing and Environmetrics

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This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).The PI and her collaborators will develop tensor-based numerical methods for applications primarily to signal processing and environmetrics, and secondarily to image processing, data mining and scientific computing. In the past, these subjects benefited from advances in numerical linear algebra. In analogy, numerical multilinear algebra aims to find useful decompositions of a given tensor into sums of simpler tensors, such as simple tensor products. This leads to efficient data compression. By using a new tensor norm called the trace-class norm for tensors, one can substantially improve de-noising, compression, and reconstruction of signals. This norm has attracted much attention in matrix rank minimization because of compressed sensing applications. Also, the PI will develop new regularization methods for tensor decomposition. Many of the problems in tensor applications are ill-posed, such as reconstruction of multi-channel signals in the presence of noise. New numerical stable methods will be studied in the framework of inverse and ill-posed problems. Mathematics has applications to environmetrics. The EPA and other agencies use air quality models, to decide how to regulate emissions. These models include receptor modeling, source apportionment and pollution identification. The PI and her collaborators will work on new computations methods to provide more accurate estimation and analysis of environmental data. Tensor-based signal processing also has many applications in biomedical imaging, such as magnetic resonance imaging diagnosis and nuclear magnetic resonance spectroscopy in tumor detection. Better data analysis tools will be facilitated by new higher-order tensor decompositions. A key component of the research project is collaboration with other scientists as well as students. Through summer research programs for undergraduates, as well as inclusion of graduate students, they will train a new generation of applied mathematicians in the methods of tensor based analysis.