  We analyze linear independence of rank one tensors produced by tensor powers
of randomly perturbed vectors. This enables efficient decomposition of sums of
high-order tensors. Our analysis builds upon [BCMV14] but allows for a wider
range of perturbation models, including discrete ones. We give an application
to recovering assemblies of neurons.
  Assemblies are large sets of neurons representing specific memories or
concepts. The size of the intersection of two assemblies has been shown in
experiments to represent the extent to which these memories co-occur or these
concepts are related; the phenomenon is called association of assemblies. This
suggests that an animal's memory is a complex web of associations, and poses
the problem of recovering this representation from cognitive data. Motivated by
this problem, we study the following more general question: Can we reconstruct
the Venn diagram of a family of sets, given the sizes of their $\ell$-wise
intersections? We show that as long as the family of sets is randomly
perturbed, it is enough for the number of measurements to be polynomially
larger than the number of nonempty regions of the Venn diagram to fully
reconstruct the diagram.
