Support vector machines with adaptive Lq penalty

Yufeng Liu, Hao Helen Zhang, Cheolwoo Park, Jeongyoun Ahn

Research output: Contribution to journalArticlepeer-review

67 Scopus citations


The standard support vector machine (SVM) minimizes the hinge loss function subject to the L2 penalty or the roughness penalty. Recently, the L1 SVM was suggested for variable selection by producing sparse solutions [Bradley, P., Mangasarian, O., 1998. Feature selection via concave minimization and support vector machines. In: Shavlik, J. (Ed.), ICML'98. Morgan Kaufmann, Los Altos, CA; Zhu, J., Hastie, T., Rosset, S., Tibshirani, R., 2003. 1-norm support vector machines. Neural Inform. Process. Systems 16]. These learning methods are non-adaptive since their penalty forms are pre-determined before looking at data, and they often perform well only in a certain type of situation. For instance, the L2 SVM generally works well except when there are too many noise inputs, while the L1 SVM is more preferred in the presence of many noise variables. In this article we propose and explore an adaptive learning procedure called the Lq SVM, where the best q > 0 is automatically chosen by data. Both two- and multi-class classification problems are considered. We show that the new adaptive approach combines the benefit of a class of non-adaptive procedures and gives the best performance of this class across a variety of situations. Moreover, we observe that the proposed Lq penalty is more robust to noise variables than the L1 and L2 penalties. An iterative algorithm is suggested to solve the Lq SVM efficiently. Simulations and real data applications support the effectiveness of the proposed procedure.

Original languageEnglish (US)
Pages (from-to)6380-6394
Number of pages15
JournalComputational Statistics and Data Analysis
Issue number12
StatePublished - Aug 15 2007


  • Adaptive penalty
  • Classification
  • Shrinkage
  • Support vector machine
  • Variable selection

ASJC Scopus subject areas

  • Statistics and Probability
  • Computational Mathematics
  • Computational Theory and Mathematics
  • Applied Mathematics


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