Illustration of the tensor decomposition equations. Order-3 tensors are shown here as examples. (a) Block term decomposition (BTD) for terms of Kronecker tensor products of (weights) and (tensorface patterns) (see equation 2.7), where is the Kronecker product and is the number of tensorfaces in the decomposition. In general, the algorithm allows tensor size for each term to be set individually, as shown in the diagram, but in practice, all were set the same size. (b) Rank-constrained BTD decomposition illustrated for a single term. and can each be expressed as a set of matrices and (indicated by small rectangles) (see equations 2.8 and 2.9). Setting the number of columns for those matrices equal to the desired rank values, and , respectively, imposes the rank constraints of the decomposition. Rank of the decomposition determines tensorface complexity. Rank of the decomposition was always 1.
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