Sparse Relaxed Regularized Regression: SR3.

Regularized regression problems are ubiquitous in statistical modeling, signal processing, and machine learning. Sparse regression in particular has been instrumental in scientific model discovery, including compressed sensing applications, variable selection, and high-dimensional analysis. We propose a new and highly effective approach for regularized regression, called SR3. The key idea is to solve a relaxation of the regularized problem, which has three advantages over the state-of-the-art: (1) solutions of the relaxed problem are superior with respect to errors, false positives, and conditioning, (2) relaxation allows extremely fast algorithms for both convex and nonconvex formulations, and (3) the methods apply to composite regularizers such as total variation (TV) and its nonconvex variants. We demonstrate the improved performance of SR3 across a range of regularized regression problems with synthetic and real data, including compressed sensing, LASSO, matrix completion and TV regularization. To promote reproducible research, we include a companion MATLAB package that implements these popular applications.