An Order-$p$ Tensor Factorization with Applications in Imaging

• Published in: SIAM journal on scientific computing Vol. 35; no. 1; pp. A474 - A490
• Main Authors: Martin, Carla D., Shafer, Richard, LaRue, Betsy
• Format: Journal Article
• Language: English
• Published: Philadelphia Society for Industrial and Applied Mathematics 01.01.2013
• Subjects: Algorithms; Applied mathematics; Arrays; Computation; Data analysis; Decomposition; Face recognition; Factorization; Fourier transforms; Imaging; Linear algebra; Mathematical analysis; Strategy; Tensors
• ISSN: 1064-8275; 1095-7197
• DOI: 10.1137/110841229

Operations with tensors, or multiway arrays, are increasingly prevalent in many applications involving multiway data analysis. This paper extends a third-order factorization strategy and tensor operations defined in a recent paper [M. E. Kilmer and C. D. Martin, Linear Algebra Appl., 435 (2011), pp. 641--658] to order-$p$ tensors. The extension to order-$p$ tensors is explained in a recursive way but for computational speed is implemented directly using the fast Fourier transform. A major motivation for considering factorization strategies for order-$p$ tensors is to devise new types of algorithms for general order-$p$ tensors which can be used in applications. We conclude with two applications in imaging. The first application is image deblurring, and the second application is video facial recognition. Both applications involve order-4 tensors. [PUBLICATION ABSTRACT]