Sparse Volterra and Polynomial Regression Models: Recoverability and Estimation

TL;DR

This paper explores the use of compressed sampling techniques for sparse Volterra and polynomial regression models, demonstrating their potential for efficient, interpretable nonlinear system identification with fewer measurements.

Contribution

It introduces adaptive RLS-type algorithms based on (weighted) Lasso for sparse polynomial regression and generalizes RIP conditions to these models, enabling effective recovery with limited data.

Findings

Adaptive CS algorithms outperform traditional methods in sparse polynomial modeling.

Theoretical RIP bounds are established for polynomial regression settings.

Successful application to genotype-phenotype data confirms practical utility.

Abstract

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.