Illustration of the tensor decomposition equations. Order-3 tensors $Y∈RI1×I2×I3$ are shown here as examples. (a) Block term decomposition (BTD) for $P$ terms of Kronecker tensor products of $Ap$ (weights) and $Xp$ (tensorface patterns) (see equation 2.7), where $⊗$ is the Kronecker product and $P$ is the number of tensorfaces in the decomposition. In general, the algorithm allows tensor size for each term $P$ 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. $Ap$ and $Xp$ can each be expressed as a set of matrices $Um$ and $V(l)$ (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, $Sp$ and $Rp$⁠, respectively, imposes the rank constraints of the decomposition. Rank of the $Xp$ decomposition determines tensorface complexity. Rank of the $Ap$ decomposition was always 1.