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A very interesting blocked version of the CP decomposition which under the hood uses ALS with fixed block sizes. Very similar to the paneled ALS method however, the difference comes from the following: After optimizing a block of size
r<=R_{target}
you compute the gradient of the target tensor. The following block of sizer'
now tries to approximate the gradient. This continues until there are no blocks left. ifr = R
, i.e. one block, one recovers the exact CP-ALS decomposition.It is very interesting for two reasons:
r * number of blocks
.r < R_{target}
. This implies that some tensors (using tensors in general may be too strong assumption) are made up of a fixed number of component tensorsT_{i,j,k} = A_{i,j,k} + B_{i,j,k} + C_{i,j,k} + \dots
which themselves have relatively low CP rank. These component tensors are "discovered" as the gradient ofT
.I have done limited testing with this method beyond getting it to work using a small DF coulomb integral tensors (water in the basis aug-cc-pVDZ/aug-cc-pVDZ-RI) and the random test tensors.