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Fixed-Point Continuation for $\ell_1$-Minimization: Methodology and Convergence
Abstract
We present a framework for solving the large-scale $\ell_1$-regularized convex minimization problem:\[ \min\|x\|_1+\mu f(x). \] Our approach is based on two powerful algorithmic ideas: operator-splitting and continuation. Operator-splitting results in a fixed-point algorithm for any given scalar $\mu$; continuation refers to approximately following the path traced by the optimal value of x as $\mu$ increases. In this paper, we study the structure of optimal solution sets, prove finite convergence for important quantities, and establish q-linear convergence rates for the fixed-point algorithm applied to problems with $f(x)$ convex, but not necessarily strictly convex. The continuation framework, motivated by our convergence results, is demonstrated to facilitate the construction of practical algorithms.
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Cites
Regression shrinkage and selection via the lasso
Tibshirani, R. (1996) · Journal of the Royal Statistical Society Series B: Statistical Methodology · doi:10.1111/j.2517-6161.1996.tb02080.x · Cited by 51,482
Analog-to-Information Conversion via Random Demodulation
Kirolos, S., Laska, J., Wakin, M., Duarte, M., Baron, D., Ragheb, T., Massoud, Y., & Baraniuk, R. (2006) · 2006 IEEE Dallas/CAS Workshop on Design, Applications, Integration and Software · doi:10.1109/dcas.2006.321036 · Cited by 393
Cited by
Parikh, N., & Boyd, S. (2014) · Foundations and Trends® in Optimization · doi:10.1561/2400000003 · Cited by 4,102
Beyond Nyquist: Efficient Sampling of Sparse Bandlimited Signals
Tropp, J. A., Laska, J. N., Duarte, M. F., Romberg, J. K., & Baraniuk, R. G. (2010) · IEEE Transactions on Information Theory · doi:10.1109/tit.2009.2034811 · Cited by 1,089
Power Quality Disturbance Classification based on Adaptive Compressed Sensing and Machine Learning
Cheng, L., Wu, Z., Xuanyuan, S., & Chang, H. (2020) · 2020 IEEE Green Technologies Conference(GreenTech) · doi:10.1109/greentech46478.2020.9289735 · Cited by 11
Metadata
- Authors
- Hale, E. T., Yin, W., & Zhang, Y.
- Venue
- SIAM Journal on Optimization
- Published
- 2008-01
- Volume
- 19
- Issue
- 3
- Pages
- 1107-1130
- ISSN
- 1052-6234 (ISSN-L)
- 1095-7189
- OpenAlex
- W2009702064
- Cited by
- 909
- References
- 2
- FWCI
- 49.01
- Type
- journal article
- Language
- en
- Open access
- closed
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