A THIRD-ORDER GENERALIZATION OF THE MATRIX SVD AS A PRODUCT OF THIRD-ORDER TENSORS ∗ MISHA E. KILMER † , CARLA D. MARTIN ‡ , AND LISA PERRONE § Abstract. Traditionally, extending the Singular Value Decomposition (SVD) to third-order tensors (multiway arrays) has involved a representation using the outer product of vectors. These outer products can be written in terms of the n-mode product, which can also be used to describe a type of multiplication between two tensors. In this paper, we present a different type of third-order generalization of the SVD where an order-3 tensor is instead decomposed as a product of order-3 tensors. In order to define this new notion, we define tensor-tensor multiplication in such a way so that it is closed under this operation. This results in new definitions for tensors such as the tensor transpose, inverse, and identity. These definitions have the advantage they can be extended, though in a non-trivial way, to the order-p (p> 3) case [31]. A major motivation for considering this new type of tensor multiplication is to devise new types of factorizations for tensors which could then be used in applications such as data compression. We therefore present two strategies for compressing third- order tensors which make use of our new SVD generalization and give some numerical comparisons to existing algorithms on synthetic data. Key words. multilinear algebra, tensor decomposition, singular value decomposition, multidi- mensional arrays AMS subject classifications. 15A69, 65F30 1. Introduction. The Singular Value Decomposition (SVD) of a matrix gives us important information about a matrix such as its rank, an orthonormal basis for the column or row space, and reduction to diagonal form. In applications, especially those involving multiway data analysis, information about the rank and reduction of tensors to have fewer nonzero entries are useful concepts to try to extend to higher dimensions. However, many of the powerful tools of linear algebra such as the SVD do not, unfortunately, extend in a straight-forward way to tensors of order three or higher. There have been several such extensions of the matrix SVD to tensors in the literature, many of which are used in a variety of applications such as chemometrics [39], psychometrics [27], signal processing [10, 38, 8], computer vision [42, 43, 44], data mining [37, 2], graph analysis [25], neuroscience [6, 33, 34], and more. The models used most often in these areas include the CANDECOMP/PARAFAC (CP) model [7, 17] and the TUCKER model [41] or Higher-Order SVD (HOSVD) [11]. A thorough treatment of these models, other SVD extensions to tensors, special cases, applications, and additional references can be found in [26]. In this paper, we present a new way of extending the matrix SVD to tensors. Specifically, we define a new type of tensor multiplication that allows a third-order tensor to be written as a product of third-order tensors (as opposed to a linear combi- nation of outer product of vectors). This new decomposition is analogous to the SVD in the matrix case (i.e. the case when the third dimension of the tensor is one). Al- though it is possible to extend our method to higher-order tensors through recursion, we do not discuss that work here, as it is beyond the scope of the present work. * This work was supported by NSF grant DMS-0552577 and by a Tufts University Summer Faculty Research Award. † Mathematics, Tufts University, 113 Bromfield-Pearson Bldg., Medford, MA 02155, misha.kilmer@tufts.edu, ‡ Mathematics and Statistics, James Madison University, 112 Roop Hall, MSC 1911, Harrisonburg, VA 22807, carlam@math.jmu.edu § Mathematics, Hawaii Pacific University, lcperrone@gmail.com 1