  We present an effective ansatz for the wave function of correlated electrons
that brings closer the fields of machine learning parameterizations and tensor
rank decompositions. We consider a CANDECOMP/PARAFAC (CP) tensor factorization
of a general backflow transformation in second quantization for a simple,
compact and systematically improvable Fermionic state. This directly encodes
$N$-body correlations without the ordering dependence of other tensor
decompositions. We consider and explicitly demonstrate various controllable
truncations, in the rank and range of the backflow correlations or magnitude of
local energy contributions, in order to systematically affect scaling
reductions to $\mathcal{O}[N^{3-4}]$. Benchmarking against small Fermi-Hubbard
and chemical systems reveals an improvement over other NQS-like models, while
extending towards larger strongly correlated ab initio systems demonstrates
competitive accuracy with more established DMRG techniques on ab initio 2D
hydrogenic lattices with realistic long-range Coulomb interactions.
