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We analyze linear independence of rank one tensors produced by tensor powersof randomly perturbed vectors. This enables efficient decomposition of sums ofhigh-order tensors. Our analysis builds upon [BCMV14] but allows for a widerrange of perturbation models, including discrete ones. We give an applicationto recovering assemblies of neurons.  Assemblies are large sets of neurons representing specific memories orconcepts. The size of the intersection of two assemblies has been shown inexperiments to represent the extent to which these memories co-occur or theseconcepts are related; the phenomenon is called association of assemblies. Thissuggests that an animal's memory is a complex web of associations, and posesthe problem of recovering this representation from cognitive data. Motivated bythis problem, we study the following more general question: Can we reconstructthe Venn diagram of a family of sets, given the sizes of their $\ell$-wiseintersections? We show that as long as the family of sets is randomlyperturbed, it is enough for the number of measurements to be polynomiallylarger than the number of nonempty regions of the Venn diagram to fullyreconstruct the diagram.

Smoothed Analysis of Discrete Tensor Decomposition and Assemblies of Neurons