A Unified Framework for Sparse Relaxed Regularized Regression: SR3
This work addresses sparse regression for researchers in statistics and machine learning, offering incremental improvements in speed and flexibility over existing methods.
The authors tackled the problem of sparse regression in statistical modeling by proposing the SR3 framework, which solves a relaxation of regularized problems to achieve superior solutions with lower errors and false positives, faster algorithms, and applicability to composite regularizers, demonstrating computational efficiency and higher accuracy across various applications.
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 broad framework for sparse relaxed 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 advantages of SR3 (computational efficiency, higher accuracy, faster convergence rates, greater flexibility) across a range of regularized regression problems with synthetic and real data, including applications in compressed sensing, LASSO, matrix completion, TV regularization, and group sparsity. To promote reproducible research, we also provide a companion MATLAB package that implements these examples.