A Unified Framework for Sparse Relaxed Regularized Regression: SR3

Peng Zheng, Travis Askham, Steven L. Brunton, J. Nathan Kutz, Aleksandr Y. Aravkin

Research output: Contribution to journalArticlepeer-review

95 Scopus citations


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, essential for total variation (TV) as well as sparsity-promoting formulations using tight frames. We demonstrate the advantages of SR3 (computational efficiency, higher accuracy, faster convergence rates, and 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. Following standards of reproducible research, we also provide a companion MATLAB package that implements these examples.

Original languageEnglish (US)
Article number8573778
Pages (from-to)1404-1423
Number of pages20
JournalIEEE Access
StatePublished - 2019
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • General Computer Science
  • General Materials Science
  • General Engineering


  • Nonconvex optimization
  • compressed sensing
  • matrix completion
  • sparse regression
  • total variation regularization


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