Group Regularized Estimation Under Structural Hierarchy

References

• Bauschke, H. H., and Combettes, P. L. (2008), “A Dykstra-Like Algorithm for Two Monotone Operators,” Pacific Journal of Optimization, 4, 383–391.
   Web of Science ®Google Scholar
• Bickel, P. J., Ritov, Y., and Tsybakov, A. B. (2009), “Simultaneous Analysis of Lasso and Dantzig Selector,” The Annals of Statistics, 37, 1705–1732.
   Web of Science ®Google Scholar
• Bickel, P. J., Ritov, Y., and Tsybakov, A. B. (2010), “Hierarchical Selection of Variables in Sparse High-Dimensional Regression,” in Borrowing Strength: Theory Powering Applications–A Festschrift for Lawrence D. Brown, eds. J. O. Berger, T. T. Cai, and I. M. Johnstone, Beachwood, OH: Institute of Mathematical Statistics, pp. 56–69.
   Google Scholar
• Bien, J., Bunea, F., and Xiao, L. (2016), “Convex Banding of the Covariance Matrix,” Journal of the American Statistical Association, 111, 834–845.
   PubMed Web of Science ®Google Scholar
• Bien, J., Taylor, J., and Tibshirani, R. (2013), “A Lasso for Hierarchical Interactions,” The Annals of Statistics, 41, 1111–1141.
   PubMed Web of Science ®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
• Chipman, H. (1996), “Bayesian Variable Selection with Related Predictors,” Canadian Journal of Statistics, 24, 17–36.
   Web of Science ®Google Scholar
• Choi, N. H., Li, W., and Zhu, J. (2010), “Variable Selection With the Strong Heredity Constraint and its Oracle Property,” Journal of the American Statistical Association, 105, 354–364.
   Web of Science ®Google Scholar
• Cohen, J., Cohen, P., West, S. G., and Aiken, L. S. (2013), Applied Multiple Regression/Correlation Analysis for the Behavioral Sciences. Mahwah, NJ: Routledge.
   Google Scholar
• Davidson, E. H., and Erwin, D. H. (2006), “Gene Regulatory Networks and the Evolution of Animal Body Plans,” Science, 311, 796–800.
   PubMed Web of Science ®Google Scholar
• Hamada, M., and Wu, C. J. (1992), “Analysis of Designed Experiments With Complex Aliasing,” Journal of Quality Technology, 24, 130–137.
   Web of Science ®Google Scholar
• Hao, N., and Zhang, H. H. (2014), “Interaction Screening for Ultra-High Dimensional Data,” Journal of the American Statistical Association, 109, 1285–1301.
   PubMed Web of Science ®Google Scholar
• Hastie, T., Tibshirani, R., and Friedman, J. (2009), The Elements of Statistical Learning (2nd ed.), New York: Springer-Verlag.
   Google Scholar
• He, Y., She, Y., and Wu, D. (2013), “Stationary-Sparse Causality Network Learning,” Journal of Machine Learning Research, 14, 3073–3104.
   Web of Science ®Google Scholar
• Jenatton, R., Mairal, J., Obozinski, G., and Bach, F. (2011), “Proximal Methods for Hierarchical Sparse Coding,” Journal of Machine Learning Research, 12, 2297–2334.
   Web of Science ®Google Scholar
• Lim, M., and Hastie, T. (2015), “Learning Interactions via Hierarchical Group-Lasso Regularization,” Journal of Computational and Graphical Statistics, 24, 627–654.
   PubMed Web of Science ®Google Scholar
• Lounici, K., Pontil, M., Van De Geer, S., and Tsybakov, A. B. (2011), “Oracle Inequalities and Optimal Inference Under Group Sparsity,” The Annals of Statistics, 39, 2164–2204.
   Web of Science ®Google Scholar
• McCullagh, P., and Nelder, J. A. (1989), Generalized Linear Models, London: Chapman and Hall.
   Google Scholar
• Negahban, S. N., Ravikumar, P., Wainwright, M. J., and Yu, B. (2012), “A Unified Framework for High-Dimensional Analysis of M-Estimators With Decomposable Regularizers,” Statistical Science, 27, 538–557.
   Web of Science ®Google Scholar
• Nelder, J. A. (1977), “A Reformulation of Linear Models,” Journal of the Royal Statistical Society, Series A, 140, 48–77.
   Google Scholar
• Nesterov, Y. (1988), “On an Approach to the Construction of Optimal Methods of Minimization of Smooth Convex Functions,” Ekonom. i. Mat. Metody (in Russian), 24, 509–517.
   Google Scholar
• Owen, A. B. (2007), “A Robust Hybrid of Lasso and Ridge Regression,” Contemporary Mathematics, 443, 59–72.
   Google Scholar
• Pace, R. K., and Barry, B. (1997), “Sparse Spatial Autoregressions,” Statistics & Probability Letters, 33, 291–297.
   Web of Science ®Google Scholar
• Peixoto, J. L. (1987), “Hierarchical Variable Selection in Polynomial Regression Models,” The American Statistician, 41, 311–313.
   Web of Science ®Google Scholar
• ——— (1990), “A Property of Well-Formulated Polynomial Regression Models,” The American Statistician, 44, 26–30.
   Web of Science ®Google Scholar
• Radchenko, P., and James, G. M. (2010), “Variable Selection Using Adaptive Nonlinear Interaction Structures in High Dimensions,” Journal of the American Statistical Association, 105, 1541–1553.
   Web of Science ®Google Scholar
• Ravikumar, P. D., Liu, H., Lafferty, J. D., and Wasserman, L. A. (2007), “SpAM: Sparse Additive Models,” in Advances in Neural Information Processing Systems (Vol. 20), Cambridge, MA: MIT Press, pp. 1201–1208.
   Google Scholar
• Schmidt, M., Roux, N. L., and Bach, F. R. (2011), “Convergence Rates of Inexact Proximal-Gradient Methods for Convex Optimization,” in Advances in Neural Information Processing Systems (Vol. 24), eds. J. Shawe-Taylor, R. Zemel, P. Bartlett, F. Pereira, and K. Weinberger, Granada, Spain: Curran Associates, Inc., pp. 1458–1466.
   Google Scholar
• She, Y. (2012), “An Iterative Algorithm for Fitting Nonconvex Penalized Generalized Linear Models With Grouped Predictors,” Computational Statistics & Data Analysis, 56, 2976–2990.
   Web of Science ®Google Scholar
• Simon, N., Friedman, J., Hastie, T., and Tibshirani, R. (2013), “A Sparse-Group Lasso,” Journal of Computational and Graphical Statistics, 22, 231–245.
   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
• van de Geer, S. (2014), “Weakly Decomposable Regularization Penalties and Structured Sparsity,” Scandinavian Journal of Statistics, 41, 72–86.
   Web of Science ®Google Scholar
• Wei, F., and Huang, J. (2010), “Consistent Group Selection in High-Dimensional Linear Regression,” Bernoulli, 16, 1369–1384.
   PubMed Web of Science ®Google Scholar
• Wu, J., Devlin, B., Ringquist, S., Trucco, M., and Roeder, K. (2010), “Screen and Clean: A Tool for Identifying Interactions in Genome-Wide Association Studies,” Genetic Epidemiology, 34, 275–285.
   PubMed Web of Science ®Google Scholar
• Yuan, M., and Lin, Y. (2006), “Model Selection and Estimation in Regression With Grouped Variables,” Journal of the Royal Statistical Society, Series B, 68, 49–67.
   Google Scholar
• Zhang, C.-H., and Huang, J. (2008), “The Sparsity and Bias of the Lasso Selection in High-Dimensional Linear Regression,” The Annals of Statistics, 36, 1567–1594.
   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