Volterra and polynomial regression models play a major role in nonlinear system identification and inference tasks. Exciting applications ranging from neuroscience to genome-wide association analysis build on these models with the additional requirement of parsimony. This requirement has high interpretative value, but unfortunately cannot be met by least-squares based or kernel regression methods. To this end, compressed sampling (CS) approaches, already successful in linear regression settings, can offer a viable alternative. The viability of CS for sparse Volterra and polynomial models is the core theme of this work. A common sparse regression task is initially posed for the two models. Building on (weighted) Lasso-based schemes, an adaptive RLS-type algorithm is developed for sparse polynomial regressions. The identifiability of polynomial models is critically challenged by dimensionality. However, following the CS principle, when these models are sparse, they could be recovered by far fewer measurements. To quantify the sufficient number of measurements for a given level of sparsity, restricted isometry properties (RIP) are investigated in commonly met polynomial regression settings, generalizing known results for their linear counterparts. The merits of the novel (weighted) adaptive CS algorithms to sparse polynomial modeling are verified through synthetic as well as real data tests for genotype-phenotype analysis.
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Manuscript received March 03, 2011; revised June 28, 2011; accepted August 08, 2011. Date of publication August 30, 2011; date of current version November 16, 2011. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Trac D. Tran. This work was supported by the Marie Curie International Outgoing Fellowship No. 234914 within the 7th European Community Framework Programme; by NSF Grants CCF-0830480, 1016605, and ECCS-0824007, 1002180; and by the QNRF-NPRP award 09-341-2-128. Part of the results of this work was presented at CAMSAP, Aruba, Dutch Antilles, December 2009 .