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Third Order Tensors As Operators On Matrices: A Theoretical And Computational Framework With Applications In Imaging
Authors: Kilmer, Misha E.; Braman, Karen; Hao, Ning
Date:January, 2011
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Recent work by Kilmer and Martin, and Braman provides a setting in which the familiar tools of linear algebra can be extended to better understand third-order tensors. Continuing along this vein, this paper investigates further implications including: 1) a bilinear operator on the matrices which is nearly an inner product and which leads to denitions for length of matrices, angle between two matrices and orthogonality of matrices and 2) the use of t-linear combinations to characterize the range and kernel of a mapping dened by a third-order tensor and the t-product and the quan- tication of the dimensions of those sets. These theoretical results lead to the study of orthogonal projections as well as an eces. The theoretical framework also leads us to consider the notion of tensor polynomials and their relation to tensor eigentuples dened in Braman. Implications for extending basic algorithms such as the power method, QR iteration, and Krylov subspace methods are discussed. We conclude with two examples in image processing: using the orthogonal elements generated via a Golub-Kahan iterative bidiagonalization scheme for facial recognition and solving a regularized image deblurring problem.

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