Bilinear operations with tensors in the Tucker format

We consider approximate computation of a bilinear tensor function $D \approx C = F(A,B)$ of tensors $A$ and $B$. It is assumed that $A, B$ are given in the Tucker format with mode ranks $r$. In this case the same format for $C$ is easily available with mode ranks $r2$.

However, $D$ is obtained in the Tucker format with mode ranks as small as possible within a prescribed approximation accuracy. Generally we assume that these ranks are about the same value $r$. Thus, the input and output data contain an array with $r3$ elements. The main difficulty of existing approaches is a call for squared memory of $r6$ elements needed only in between of input and output. We propose how to avoid the need in squared memory and improve the performance of tensor recompression algorithms.

The new findings include a variable-rank Tucker-ALS procedure without a priori knowledge of mode ranks and a cheap initialization procedure using an intrinsic tensor structure in $C=F(A,B)$. We also investigate the merits and drawbacks of the Tucker approximation via Krylov subspaces and the pre-filtering of individual mode factors. Numerical examples include tensor operations in the Hartree-Fock and Kohn-Sham models.