Tensor-tensor algebra for optimal representation and compression of multiway data

Significance Many real-world data are inherently multidimensional; however, often data are processed as two-dimensional arrays (matrices), even if the data are naturally represented in higher dimension. The common practice of matricizing high-dimensional data is due to the ubiquitousness and strong theoretical foundations of matrix algebra. Various tensor-based approximations have been proposed to exploit high-dimensional correlations. While these high-dimensional techniques have been effective in many applications, none have been theoretically proven to outperform matricization generically. In this study, we propose matrix-mimetic, tensor-algebraic formulations to preserve and process data in its native, multidimensional format. For a general family of tensor algebras we prove the superiority of optimal truncated tensor representations to traditional matrix-based representations with implications for other related tensorial frameworks.