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A Unified Framework for Sparse Relaxed Regularized Regression: SR3

Peng Zheng (zhengp***at***uw.edu)
Travis Askham (askhamwhat***at***gmail.com )
Steve Brunton (sbrunton***at***uw.edu )
Nathan Kutz (kutz***at***uw.edu )
Aleksandr Aravkin (saravkin***at***uw.edu)

Abstract: 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, vari- able 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.

Keywords: Sparse regression, Nonconvex Regularizers, Compressed Sensing, Total Variation, Matrix Completion

Category 1: Applications -- Science and Engineering (Statistics )

Category 2: Convex and Nonsmooth Optimization (Nonsmooth Optimization )

Category 3: Nonlinear Optimization (Nonlinear Systems and Least-Squares )


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Entry Submitted: 11/08/2018
Entry Accepted: 11/08/2018
Entry Last Modified: 11/08/2018

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