Sparse interactions: Identifying high-dimensional multilinear systems via compressed sensing

This paper investigates the problem of identifying sparse multilinear systems. Such systems are characterized by multiplicative interactions between the input variables with sparsity meaning that relatively few of all conceivable interactions are present. This problem is motivated by the study of interactions among genes and proteins in living cells. The goal is to develop a sampling/sensing scheme to identify sparse multilinear systems using as few measurements as possible. We derive bounds on the number of measurements required for perfect reconstruction as a function of the sparsity level. Our results extend the notion of compressed sensing from the traditional notion of (linear) sparsity to more refined notions of sparsity encountered in nonlinear systems. In contrast to the linear sparsity models, in the multilinear case the pattern of sparsity may play a role in the sensing requirements.