Compressed sensing for volterra and polynomial regression models

Volterra filtering and polynomial regression are two widely utilized tools for nonlinear system modeling and inference. They are both critically challenged by the curse of dimensionality, which is typically alleviated via kernel regression. However, exciting diverse applications ranging from neuroscience to genome-wide association (GWA) analysis call for parsimonious polynomial expansions of critical interpretative value. Unfortunately, kernel regression cannot yield sparsity in the primal domain, where compressed sampling approaches can offer a viable alternative. Following the compressed sampling principle, a sparse polynomial expansion can be recovered by far fewer measurements compared to the least squares (LS)-based approaches. But how many measurements are sufficient for a given level of sparsity? This paper is the first attempt to answer this question by analyzing the restricted isometry properties for commonly met polynomial regression settings. Additionally, the merits of compressed sampling approaches to polynomial modeling are corroborated on synthetic and real data for quantitative genotype-phenotype analysis.