Closing the Gap Between Matrix and Tensor Computation

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A strong case can be made that tensor computation is the ``next big thing'' in numerical analysis. High-dimensional modeling is becoming commonplace and it requires the manipulation and analysis of huge multidimensional arrays. The investigator and his colleagues will enrich the interplay between matrix computations and tensor computations by pursuing four basic directions of research. They will(1) develop tensor approximation techniques based on matrices that have low Kronecker product rank, (2) implement a pair of basic tensor algebra subprograms, one that showcases a new contraction-level generalization of Strassen multiplication and one that demonstrates how to compute effectively contractions between tensors that have symmetry, (3) analyze the data sparse representation of huge vectors through tensor networks, and (4) develop a unifying framework for SVD-like tensor decompositions through an embedding idea that involves symmetric tensors. A table of data is 2-dimensional and many matrix computation techniques exist for extracting information from the numbers that appear in the rows and columns. A tensor can be thought of as a table whose entries are other tables. For example, a table having 10 rows and 8 columns has 80 ``cells''. If each of those cells is itself a 5-by-4 table, then the entire data set can be thought of as a 10-by-8-by-5-by-4 tensor. Data sets of this variety are increasingly prominent in engineering and the sciences because it is the natural way to structure the information associated with a model that depends upon many factors.The research plan is to help build an infrastructure for the scientific community that makes tensor-based computation as natural and easy as matrix-based computation. The successful problem-solving and problem-analysis tools provided by the matrix computation field will be broadened and generalized. The outreach agenda includes the production of educational materials that will help ensure the development of a tensor-savvy scientific community. These materials include an online, ten-lecture short course on tensor computation, participation in a Visiting Lecturer program that targets 4-year colleges, and the addition of a tensor computation chapter in the upcoming fourth edition of the highly-cited textbook on matrix computations by Golub and Van Loan.