Smoothed Analysis of Discrete Tensor Decomposition and Assemblies of Neurons

TL;DR

This paper develops a smoothed analysis framework for tensor decomposition, enabling the recovery of neural assemblies and their associations from limited measurements, even under discrete perturbations.

Contribution

It extends previous tensor decomposition analysis to discrete perturbations and applies this to reconstruct neural assemblies from intersection data.

Findings

Efficient decomposition of high-order tensors under discrete perturbations.

Successful reconstruction of neural assemblies from intersection measurements.

Polynomially many measurements suffice for full recovery of neural set relationships.

Abstract

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.