Generalizing the column–row matrix decomposition to multi-way arrays - Caiafa, Cesar F.; Cichocki, Andrzej (Linear algebra and its applications, 2010)
In this paper, we provide two generalizations of the CUR matrix decomposition Y = CUR (also known as pseudo-skeleton approximation method [1]) to the case of N -way arrays (tensors). These generalizations, which we called Fiber Sampling Tensor Decomposition types 1 and 2 (FSTD1 and FSTD2), provide explicit formulas for the parameters of a rank- ( R , R , … , R ) Tucker representation (the core tensor of size R × R × ⋯ × R and the matrix factors of sizes I n × R , n = 1 , 2 , … N ) based only on some selected entries of the original tensor. FSTD1 uses P N - 1 ( P ⩾ R ) n -mode fibers of the original tensor while FSTD2 uses exactly R fibers in each mode as matrix factors, as suggested by the existence theorem provided in Oseledets et al. (2008) [2], with a core tensor defined in terms of the entries of a subtensor of size R × R × ⋯ × R . For N = 2 our results are reduced to the already known CUR matrix decomposition where the core matrix is defined as the inverse of the intersection submatrix, i.e. U = W - 1 . Additionally, we provide an adaptive type algorithm for the selection of proper fibers in the FSTD1 model which is useful for large scale applications. Several numerical results are presented showing the performance of our FSTD1 Adaptive Algorithm compared to two recently proposed approximation methods for 3-way tensors.