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| Carla Martin |
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| Escobar Assistant Professor of Mathematics |
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Web Site N/A Contact Information
Courses & Office Hours |
Education
Ph.D. (2005), Cornell University
Research Area: Numerical linear algebra
My research involves computations with tensors, or multi-dimensional arrays. Specifically, I design fast algorithms to compute SVD-like tensor decompositions. Many of the powerful tools of linear algebra such as the Singular Value Decomposition (SVD) do not, unfortunately, extend easily to tensors of order three and higher. For second-order tensors (i.e., matrices) the SVD is particularly illuminating because it reduces a matrix to diagonal form. In applications (e.g., scientific computing and engineering), this reduction is useful because the SVD compresses the two-dimensional data and allows one to describe more easily interactions and relationships that exist. Extending the SVD to higher-order tensors is nontrivial; even familiar matrix concepts such as diagonalization and rank become ambiguous and complicated.
For my dissertation, I designed an algorithm to compute an orthogonal tensor decomposition such that the resulting decomposition is compressed. The algorithm is based on the Jacobi SVD algorithm for matrices and can be extended to p-dimensional tensors. While computing the tensor rank is still an open problem, both theoretically and computationally, I also developed an algorithm to compute the rank of nxnx2 tensors. The algorithm is based on certain eigendecompositions. I was able to prove that the rank of this special subclass of tensors depends on the generalized eigenvalues of the faces of the tensor cube. The results regarding this special subclass provide insight into the general rank problem.
Selected Publications
Shifted Kronecker product systems (with C. F. Van Loan) SIAM J. Matrix Anal. (accepted).
Solving real systems with the complex Schur decomposition (with C. F. Van Loan), SIAM J. Matrix Anal. (accepted).
Decomposing a tensor (with M. E. Kilmer), SIAM News 37 no. 9 (2004).
Product triangular systems with shift (with C. F. Van Loan), SIAM J. Matrix Anal. 24 no. 1 (2001), 292–301.
Mathematician at work, Math Horizons 5 (1997), Mathematical Association of America, p. 9.
A Jacobi-type method for computing orthogonal tensor decompositions (with C.F. Van Loan), SIAM J. Matrix Anal. (submitted).
Last modified: May 18, 2006