Tensor–tensor products with invertible linear transforms

Research in tensor representation and analysis has been rising in popularity in direct response to a) the increased ability of data collection systems to store huge volumes of multidimensional data and b) the recognition of potential modeling accuracy that can be provided by leaving the data and/or the operator in its natural, multidimensional form. In recent work [1], the authors introduced the notion of the t-product, a generalization of matrix multiplication for tensors of order three, which can be extended to multiply tensors of arbitrary order [2]. The multiplication is based on a convolution-like operation, which can be implemented efficiently using the Fast Fourier Transform (FFT). The corresponding linear algebraic framework from the original work was further developed in [3], and it allows one to elegantly generalize all classical algorithms from numerical linear algebra. In this paper, we extend this development so that tensor–tensor products can be defined in a so-called transform domain for any invertible linear transform. In order to properly motivate this transform-based approach, we begin by defining a new tensor–tensor product alternative to the t-product. We then show that it can be implemented efficiently using DCTs, and that subsequent definitions and factorizations can be formulated by appealing to the transform domain. Using this new product as our guide, we then generalize the transform-based approach to any invertible linear transform. We introduce the algebraic structures induced by each new multiplication in the family, which is that of C⁎-algebras and modules. Finally, in the spirit of [4], we give a matrix–algebra based interpretation of the new family of tensor–tensor products, and from an applied perspective, we briefly discuss how to choose a transform. We demonstrate the convenience of our new framework within the context of an image deblurring problem and we show the potential for using one of these new tensor–tensor products and resulting tensor-SVD for hyperspectral image compression.