ADMM for High-Dimensional Sparse Penalized Quantile Regression

References

• Bai, Z., and Wu, Y. (1994), “Limiting Behavior of M-Estimators of Regression Coefficients in High Dimensional Linear Models I. Scale Dependent Case,” Journal of Multivariate Analysis, 51, 211–239.
   Web of Science ®Google Scholar
• Beck, A., and Teboulle, M. (2009), “A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems,” SIAM Journal on Imaging Sciences, 2, 183–202.
   Web of Science ®Google Scholar
• Belloni, A., and Chernozhukov, V. (2011), “ℓ1-Penalized Quantile Regression in High-Dimensional Sparse Models,” The Annals of Statistics, 39, 82–130.
   Web of Science ®Google Scholar
• Bien, J., Taylor, J., and Tibshirani, R. (2013), “A Lasso for Hierarchical Interactions,” Annals of Statistics, 41, 1111–1141.
   PubMed Web of Science ®Google Scholar
• Bogdan, M., Berg, E. v. d., Su, W., and Candes, E. (2013), “Statistical Estimation and Testing via the Sorted ℓ1 Norm,” arXiv preprint arXiv:1310.1969v2.
   Google Scholar
• Boyd, S., Parikh, N., Chu, E., Peleato, B., and Eckstein, J. (2011), “Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers,” Foundations and Trends® in Machine Learning, 3, 1–122.
   Google Scholar
• Chen, X., Bai, Z., and Zao, L. (1990), “Asymptotic Normality of Minimum L1-Norm Estimates in Linear Models,” Science in China Series A: Mathematics, Physics, Astronomy, 33, 1311–1328.
   Google Scholar
• Chiang, A. P., Beck, J. S., Yen, H.-J., Tayeh, M. K., Scheetz, T. E., Swiderski, R. E., Nishimura, D. Y., Braun, T. A., Kim, K.-Y. A., Huang, J., Elbedour, K., Carmi, R., Slusarski, D. C., Casavant, T. L., Stone, E. M., and Sheffield, V. C. (2006), “Homozygosity Mapping With SNP Arrays Identifies Trim32, an e3 Ubiquitin Ligase, as a Bardet–Biedl Syndrome Gene (BBS11),” Proceedings of the National Academy of Sciences, 103, 6287–6292.
   PubMed Web of Science ®Google Scholar
• Eckstein, J. (1994), “Some Saddle-Function Splitting Methods for Convex Programming,” Optimization Methods and Software, 4, 75–83.
   Google Scholar
• Efron, B., Hastie, T., Johnstone, I., and Tibshirani, R. (2004), “Least Angle Regression,” The Annals of Statistics, 32, 407–499.
   Web of Science ®Google Scholar
• Fan, J., Fan, Y., and Barut, E. (2014a), “Adaptive Robust Variable Selection,” The Annals of Statistics, 42, 324–351.
   PubMed Web of Science ®Google Scholar
• Fan, J., and Li, R. (2001), “Variable Selection via Nonconcave Penalized Likelihood and Its Oracle Properties,” Journal of the American Statistical Association, 96, 1348–1360.
   Web of Science ®Google Scholar
• Fan, J., and Lv, J. (2011), “Nonconcave Penalized Likelihood With np-Dimensionality,” IEEE Transactions on Information Theory, 57, 5467–5484.
   PubMed Web of Science ®Google Scholar
• Fan, J., Xue, L., and Zou, H. (2014b), “Strong Oracle Optimality of Folded Concave Penalized Estimation,” The Annals of Statistics, 42, 819–849.
   PubMed Web of Science ®Google Scholar
• Fazel, M., Pong, T. K., Sun, D., and Tseng, P. (2013), “Hankel Matrix Rank Minimization With Applications to System Identification and Realization,” SIAM Journal on Matrix Analysis and Applications, 34, 946–977.
   Web of Science ®Google Scholar
• Frank, L. E., and Friedman, J. H. (1993), “A Statistical View of Some Chemometrics Regression Tools,” Technometrics, 35, 109–135.
   Web of Science ®Google Scholar
• Friedman, J., Hastie, T., Höfling, H., and Tibshirani, R. (2007), “Pathwise Coordinate Optimization,” The Annals of Applied Statistics, 1, 302–332.
   Web of Science ®Google Scholar
• Friedman, J., Hastie, T., and Tibshirani, R. (2010), “Regularization Paths for Generalized Linear Models via Coordinate Descent,” Journal of Statistical Software, 33, 1–22.
   PubMed Web of Science ®Google Scholar
• Goldstein, T., and Osher, S. (2009), “The Split Bregman Method for l1-Regularized Problems,” SIAM Journal on Imaging Sciences, 2, 323–343.
   Web of Science ®Google Scholar
• He, B., Liao, L.-Z., Han, D., and Yang, H. (2002), “A New Inexact Alternating Directions Method for Monotone Variational Inequalities,” Mathematical Programming, 92, 103–118.
   Web of Science ®Google Scholar
• He, X., and Shao, Q.-M. (2000), “On Parameters of Increasing Dimensions,” Journal of Multivariate Analysis, 73, 120–135.
   Web of Science ®Google Scholar
• Huang, J., Ma, S., and Zhang, C.-H. (2008), “Adaptive Lasso for Sparse High-Dimensional Regression Models,” Statistica Sinica, 18, 1603–1618.
   Web of Science ®Google Scholar
• Hunter, D. R., and Lange, K. (2000), “Quantile Regression via an MM Algorithm,” Journal of Computational and Graphical Statistics, 9, 60–77.
   Web of Science ®Google Scholar
• Koenker, R. (2005), Quantile Regression, Cambridge, United Kingdom: Cambridge University Press.
   Google Scholar
• ——— (2015), Quantreg: Quantile Regression, R Package Version 5.19. Available at https://cran.r-project.org/web/packages/quantreg/index.html.
   Google Scholar
• Koenker, R., and Bassett, G. (1978), “Regression Quantiles,” Econometrica: Journal of the Econometric Society, 46, 33–50.
   Web of Science ®Google Scholar
• ——— (1982), “Robust Tests for Heteroscedasticity Based on Regression Quantiles,” Econometrica: Journal of the Econometric Society, 50, 43–61.
   Web of Science ®Google Scholar
• Koenker, R., and Ng, P. (2005), “A Frisch-Newton Algorithm for Sparse Quantile Regression,” Acta Mathematicae Applicatae Sinica, 21, 225–236.
   Google Scholar
• Li, Y., and Zhu, J. (2008), “L1-Norm Quantile Regression,” Journal of Computational and Graphical Statistics, 17, 163–185.
   Web of Science ®Google Scholar
• Lv, S., He, X., and Wang, J. (2016), “A Unified Penalized Method for Sparse Additive Quantile Models: An RKHS Approach,” Annals of the Institute of Statistical Mathematics, 69, 897–923.
   Web of Science ®Google Scholar
• O’Donoghue, B., Stathopoulos, G., and Boyd, S. (2013), “A Splitting Method for Optimal Control,” IEEE Transactions on Control Systems Technology, 21, 2432–2442.
   Web of Science ®Google Scholar
• Parikh, N., and Boyd, S. (2013), “Proximal Algorithms,” Foundations and Trends in Optimization, 1, 123–231.
   Google Scholar
• Peng, B., and Wang, L. (2015), “An Iterative Coordinate Descent Algorithm for High-Dimensional Nonconvex Penalized Quantile Regression,” Journal of Computational and Graphical Statistics, 24, 676–694.
   Web of Science ®Google Scholar
• Pollard, D. (1991), “Asymptotics for Least Absolute Deviation Regression Estimators,” Econometric Theory, 7, 186–199.
   Web of Science ®Google Scholar
• Scheetz, T. E., Kim, K.-Y. A., Swiderski, R. E., Philp, A. R., Braun, T. A., Knudtson, K. L., Dorrance, A. M., DiBona, G. F., Huang, J., Casavant, T. L., Sheffield, V. C., and Stone, E. M. (2006), “Regulation of Gene Expression in the Mammalian Eye and Its Relevance to Eye Disease,” Proceedings of the National Academy of Sciences, 103, 14429–14434.
   PubMed Web of Science ®Google Scholar
• Tibshirani, R. (1996), “Regression Shrinkage and Selection via the Lasso,” Journal of the Royal Statistical Society, Series B, 58, 267–288.
   Google Scholar
• Wang, L. (2013), “The L1 Penalized LAD Estimator for High Dimensional Linear Regression,” Journal of Multivariate Analysis, 120, 135–151.
   Web of Science ®Google Scholar
• Wang, L., Wu, Y., and Li, R. (2012), “Quantile Regression for Analyzing Heterogeneity in Ultra-High Dimension,” Journal of the American Statistical Association, 107, 214–222.
   PubMed Web of Science ®Google Scholar
• Welsh, A. (1989), “On M-Processes and M-Estimation,” The Annals of Statistics, 17, 337–361.
   Web of Science ®Google Scholar
• Wu, T. T., and Lange, K. (2008), “Coordinate Descent Algorithms for Lasso Penalized Regression,” The Annals of Applied Statistics, 2, 224–244.
   Web of Science ®Google Scholar
• Wu, Y., and Liu, Y. (2009), “Variable Selection in Quantile Regression,” Statistica Sinica, 19, 801–817.
   Web of Science ®Google Scholar
• Xue, L., Ma, S., and Zou, H. (2012), “Positive-Definite ℓ1-Penalized Estimation of Large Covariance Matrices,” Journal of the American Statistical Association, 107, 1480–1491.
   Web of Science ®Google Scholar
• Yang, J., Meng, X., and Mahoney, M. W. (2013), “Quantile Regression for Large-Scale Applications,” arXiv preprint arXiv:1305.0087v3.
   Google Scholar
• Yi, C. (2016), hqreg: Regularization Paths for Huber Loss Regression and Quantile Regression Penalized by Lasso or Elastic-Net, R Package Version 1.3. Available at https://cran.r-project.org/web/packages/hqreg/index.html.
   Google Scholar
• Yi, C., and Huang, J. (2016), “Semismooth Newton Coordinate Descent Algorithm for Elastic-Net Penalized Huber Loss Regression and Quantile Regression,” Journal of Computational and Graphical Statistics, 26, 547–557.
   Web of Science ®Google Scholar
• Yin, W., Osher, S., Goldfarb, D., and Darbon, J. (2008), “Bregman Iterative Algorithms for l1-Minimization With Applications to Compressed Sensing,” SIAM Journal on Imaging Sciences, 1, 143–168.
   Web of Science ®Google Scholar
• Zhang, T., and Zou, H. (2014), “Sparse Precision Matrix Estimation via Lasso Penalized D-Trace Loss,” Biometrika, 101, 103–120.
   Web of Science ®Google Scholar
• Zou, H. (2006), “The Adaptive Lasso and Its Oracle Properties,” Journal of the American Statistical Association, 101, 1418–1429.
   Web of Science ®Google Scholar
• Zou, H., and Hastie, T. (2005), “Regularization and Variable Selection via the Elastic Net,” Journal of the Royal Statistical Society, Series B, 67, 301–320.
   Google Scholar
• Zou, H., and Li, R. (2008), “One-Step Sparse Estimates in Nonconcave Penalized Likelihood Models,” Annals of Statistics, 36, 1509–1533.
   PubMed Web of Science ®Google Scholar