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DASSO: connections between the Dantzig selector and lasso

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  • Gareth M. James
  • Peter Radchenko
  • Jinchi Lv

Abstract

Summary. We propose a new algorithm, DASSO, for fitting the entire coefficient path of the Dantzig selector with a similar computational cost to the least angle regression algorithm that is used to compute the lasso. DASSO efficiently constructs a piecewise linear path through a sequential simplex‐like algorithm, which is remarkably similar to the least angle regression algorithm. Comparison of the two algorithms sheds new light on the question of how the lasso and Dantzig selector are related. In addition, we provide theoretical conditions on the design matrix X under which the lasso and Dantzig selector coefficient estimates will be identical for certain tuning parameters. As a consequence, in many instances, we can extend the powerful non‐asymptotic bounds that have been developed for the Dantzig selector to the lasso. Finally, through empirical studies of simulated and real world data sets we show that in practice, when the bounds hold for the Dantzig selector, they almost always also hold for the lasso.

Suggested Citation

  • Gareth M. James & Peter Radchenko & Jinchi Lv, 2009. "DASSO: connections between the Dantzig selector and lasso," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 71(1), pages 127-142, January.
    • Handle: RePEc:bla:jorssb:v:71:y:2009:i:1:p:127-142
    DOI: 10.1111/j.1467-9868.2008.00668.x
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    References listed on IDEAS

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    1. Zou, Hui, 2006. "The Adaptive Lasso and Its Oracle Properties," Journal of the American Statistical Association, American Statistical Association, vol. 101, pages 1418-1429, December.
    2. Fan J. & Li R., 2001. "Variable Selection via Nonconcave Penalized Likelihood and its Oracle Properties," Journal of the American Statistical Association, American Statistical Association, vol. 96, pages 1348-1360, December.
    3. Hui Zou & Trevor Hastie, 2005. "Addendum: Regularization and variable selection via the elastic net," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 67(5), pages 768-768, November.
    4. Hui Zou & Trevor Hastie, 2005. "Regularization and variable selection via the elastic net," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 67(2), pages 301-320, April.
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    3. Gerda Claeskens, 2012. "Focused estimation and model averaging with penalization methods: an overview," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 66(3), pages 272-287, August.
    4. Christis Katsouris, 2023. "High Dimensional Time Series Regression Models: Applications to Statistical Learning Methods," Papers 2308.16192, arXiv.org.
    5. Lu, Zhaosong & Pong, Ting Kei & Zhang, Yong, 2012. "An alternating direction method for finding Dantzig selectors," Computational Statistics & Data Analysis, Elsevier, vol. 56(12), pages 4037-4046.
    6. Chun-Wei Zheng & Zong-Feng Qi & Qiao-Zhen Zhang & Min-Qian Liu, 2022. "A Method for Augmenting Supersaturated Designs with Newly Added Factors," Mathematics, MDPI, vol. 11(1), pages 1-17, December.
    7. Hongjin He & Xingju Cai & Deren Han, 2015. "A fast splitting method tailored for Dantzig selector," Computational Optimization and Applications, Springer, vol. 62(2), pages 347-372, November.
    8. Feng Li & Lu Lin & Yuxia Su, 2013. "Variable selection and parameter estimation for partially linear models via Dantzig selector," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 76(2), pages 225-238, February.
    9. Pun, Chi Seng & Wong, Hoi Ying, 2019. "A linear programming model for selection of sparse high-dimensional multiperiod portfolios," European Journal of Operational Research, Elsevier, vol. 273(2), pages 754-771.
    10. Howard D. Bondell & Brian J. Reich, 2012. "Consistent High-Dimensional Bayesian Variable Selection via Penalized Credible Regions," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 107(500), pages 1610-1624, December.
    11. Keith Knight, 2016. "The Penalized Analytic Center Estimator," Econometric Reviews, Taylor & Francis Journals, vol. 35(8-10), pages 1471-1484, December.
    12. Brown Andrew Anand & Richardson Sylvia & Whittaker John, 2011. "Application of the Lasso to Expression Quantitative Trait Loci Mapping," Statistical Applications in Genetics and Molecular Biology, De Gruyter, vol. 10(1), pages 1-35, March.
    13. Prater, Ashley & Shen, Lixin & Suter, Bruce W., 2015. "Finding Dantzig selectors with a proximity operator based fixed-point algorithm," Computational Statistics & Data Analysis, Elsevier, vol. 90(C), pages 36-46.
    14. Christoph Brauer & Dirk A. Lorenz & Andreas M. Tillmann, 2018. "A primal-dual homotopy algorithm for $$\ell _{1}$$ ℓ 1 -minimization with $$\ell _{\infty }$$ ℓ ∞ -constraints," Computational Optimization and Applications, Springer, vol. 70(2), pages 443-478, June.
    15. E. Androulakis & C. Koukouvinos, 2013. "A new variable selection method for uniform designs," Journal of Applied Statistics, Taylor & Francis Journals, vol. 40(12), pages 2564-2578, December.
    16. Luigi Augugliaro & Angelo M. Mineo & Ernst C. Wit, 2013. "Differential geometric least angle regression: a differential geometric approach to sparse generalized linear models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 75(3), pages 471-498, June.

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