An Efficient Algorithm for Computing the HHSVM and Its Generalizations

REFERENCES

• Alon , U. , Barkai , N. , Notterman , D. , Gish , K. , Ybarra , S. , Mack , D. and Levine , A. 1999 . Broad Patterns of Gene Expression Revealed by Clustering Analysis of Tumor and Normal Colon Tissues Probed by Oligonucleotide Arrays . Proceedings of the National Academy of Sciences , 96 : 6745 – 6750 .
   PubMed Web of Science ®Google Scholar
• Bradley, P., and Mangasarian, O. (1998), “Feature Selection via Concave Minimization and Support Vector Machines,” in Machine Learning Proceedings of the Fifteenth International Conference (ICML'98), pp. 82–90.
   Google Scholar
• Bühlmann , P. and van de Geer , S. 2011 . Statistics for High Dimensional Data , Heidelberg : Springer .
   Google Scholar
• Burges , C. 1998 . A Tutorial on Support Vector Machines for Pattern Recognition . Data Mining and Knowledge Discovery , 2 : 121 – 167 .
   Web of Science ®Google Scholar
• Daubechies , I. , Defrise , M. and De Mol , C. 2004 . An Iterative Thresholding Algorithm for Linear Inverse Problems With a Sparsity Constraint . Communications on Pure and Applied Mathematics , 57 : 1413 – 1457 .
   Web of Science ®Google Scholar
• De Leeuw, J., and Heiser, W. (1977), “Convergence of Correction Matrix Algorithms for Multidimensional Scaling,” in Geometric Representations of Relational Data, ed. J. C. Lingoes, Ann Arbor, MI: Mathesis Press, pp. 735–752.
   Google Scholar
• Dempster , A. , Laird , N. and Rubin , D. 1977 . Maximum Likelihood From Incomplete Data via the EM Algorithm . Journal of the Royal Statistical Society , 39 : 1 – 38 . Series B
   Google Scholar
• Efron , B. , Hastie , T. , Johnstone , I. and Tibshirani , R. 2004 . Least Angle Regression . The Annals of Statistics , 32 : 407 – 451 .
   Web of Science ®Google Scholar
• Evgeniou , T. , Pontil , M. and Poggio , T. 2000 . Regularization Networks and Support Vector Machines . Advances in Computational Mathematics , 13 : 1 – 50 .
   Web of Science ®Google Scholar
• Frank , A. and Asuncion , A. 2010 . Arcene Data Set: UCI” Machine Learning Repository Available at http://archive.ics.uci.edu/ml/datasets/Arcene.
   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
   PubMed Web of Science ®Google Scholar
• Fu , W. 1998 . Penalized Regressions: The Bridge Versus the Lasso . Journal of Computational and Graphical Statistics , 7 : 397 – 416 .
   Web of Science ®Google Scholar
• Genkin , A. , Lewis , D. and Madigan , D. 2007 . Large-Scale Bayesian Logistic Regression for Text Categorization . Technometrics , 49 : 291 – 304 .
   Web of Science ®Google Scholar
• Golub , T. , Slonim , D. , Tamayo , P. , Huard , C. , Gaasenbeek , M. , Mesirov , J. , Coller , H. , Loh , M. , Downing , J. , Caligiuri , M. , Bloomfield , C. and Lander , E. 1999 . Molecular Classification of Cancer: Class Discovery and Class Prediction by Gene Expression Monitoring . Science , 286 : 531 – 537 .
   PubMed Web of Science ®Google Scholar
• Graham , K. , de las Morenas , A. , Tripathi , A. , King , C. , Kavanah , M. , Mendez , J. , Stone , M. , Slama , J. , Miller , M. , Antoine , G. , Willers , H. , Sebastiani , P. and Rosenberg , C. 2010 . Gene Expression in Histologically Normal Epithelium From Breast Cancer Patients and From Cancer-Free Prophylactic Mastectomy Patients Shares a Similar Profile . British Journal of Cancer , 102 : 1284 – 1293 .
   PubMed Web of Science ®Google Scholar
• Hastie , T. , Tibshirani , R. and Friedman , J. 2001 . The Elements of Statistical Learning: Data Mining, Inference, and Prediction , New York : Springer Verlag .
   Google Scholar
• Hunter , D. and Lange , K. 2004 . A Tutorial on MM Algorithms . The American Statistician , 58 : 30 – 37 .
   Web of Science ®Google Scholar
• Lange , K. , Hunter , D. and Yang , I. 2000 . Optimization Transfer Using Surrogate Objective Functions . Journal of Computational and Graphical Statistics , 9 : 1 – 20 .
   Web of Science ®Google Scholar
• Nesterov, Y. (2007), “Gradient Methods for Minimizing Composite Objective Function,” Technical Report, Center for Operations Research and Econometrics (CORE), Catholic University of Louvain (UCL).
   Google Scholar
• Osborne , M. , Presnell , B. and Turlach , B. 2000 . A New Approach to Variable Selection in Least Squares Problems . IMA Journal of Numerical Analysis , 20 : 389 – 404 .
   Web of Science ®Google Scholar
• Rosset , S. and Zhu , J. 2007 . Piecewise Linear Regularized Solution Paths . The Annals of Statistics , 35 : 1012 – 1030 .
   Web of Science ®Google Scholar
• Singh , D. , Febbo , P. , Ross , K. , Jackson , D. , Manola , J. , Ladd , C. , Tamayo , P. , Renshaw , A. , D'Amico , A. , Richie , J. , Lander , E. S. , Loda , M. , Kantoff , P. W. , Golub , T. R. and Sellers , W. R. 2002 . Gene Expression Correlates of Clinical Prostate Cancer Behavior . Cancer Cell , 1 : 203 – 209 .
   PubMed Web of Science ®Google Scholar
• Tibshirani , R. 1996 . Regression Shrinkage and Selection via the Lasso . Journal of the Royal Statistical Society , 58 : 267 – 288 . Series B
   Google Scholar
• Tseng , P. 2001 . Convergence of a Block Coordinate Descent Method for Nondifferentiable Minimization . Journal of Optimization Theory and Applications , 109 : 475 – 494 .
   Web of Science ®Google Scholar
• Tseng , P. and Yun , S. 2009 . A coordinate Gradient Descent Method for Nonsmooth Separable Minimization . Mathematical Programming , 117 : 387 – 423 .
   Web of Science ®Google Scholar
• Van der Kooij, A. (2007), “Prediction Accuracy and Stability of Regression With Optimal Scaling Transformations,” Ph.D. thesis, Child & Family Studies and Data Theory (AGP-D), Department of Education and Child Studies, Faculty of Social and Behavioural Sciences, Leiden University.
   Google Scholar
• Vapnik , V. 1995 . The Nature of Statistical Learning Theory , New York : Springer-Verlag .
   Google Scholar
• Wang , L. , Zhu , J. and Zou , H. 2008 . Hybrid Huberized Support Vector Machines for Microarray Classification and Gene Selection . Bioinformatics , 24 : 412 – 419 .
   PubMed 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 , T. T. , Lange , K. and Lange , K. 2010 . The MM Alternative to EM . Statistical Science , 25 : 492 – 505 .
   Web of Science ®Google Scholar
• Zhu, J., Rosset, S., Hastie, T., and Tibshirani, R. (2004), “1-Norm Support Vector Machines,” The Annual Conference on Neural Information Processing Systems, 16.
   Google Scholar
• Zou , H. and Hastie , T. 2005 . Regularization and Variable Selection via the Elastic Net . Journal of the Royal Statistical Society , 67 : 301 – 320 . Series B
   Google Scholar