Sparse Support Vector Machines in Reproducing Kernel Banach Spaces
We present a novel approach for support vector machines in reproducing kernel Banach spaces induced by a finite basis. In particular, we show that the support vector classification in the 1-norm reproducing kernel Banach space is mathematically equivalent to the sparse support vector machine. Finally, we develop fixed-point proximity algorithms for finding the solution of the non-smooth minimization problem that describes the sparse support vector machine. Numerical results are presented to demonstrate that the sparse support vector machine outperforms the classical support vector machine for the binary classification of simulation data.
The first author is supported in part by the Special Project on High-performance Computing under the National Key R&D Program (No. 2016YFB0200602), and by the Natural Science Foundation of China under grants 11471013 and 91530117. The third author would like to express his gratitude to the grant of the “Thousand Talents Program” for junior scholars of China, the grant of the Natural Science Foundation of China (11601162), and the grant of South China Normal University (671082, S80835, and S81031).
- 15.Sriperumbudur, B., Fukumizu, K., Lanckriet, G.: Learning in Hilbert vs.Banach spaces: a measure embedding viewpoint. In: Advances in Neural Information Processing Systems, pp. 1773–1781. MIT, Cambridge (2011)Google Scholar
- 18.Villmann, T., Haase, S., Kästner, M.: Gradient based learning in vector quantization using differentiable kernels. In: Advances in Self-Organizing Maps, pp. 193–204. Springer, Santiago (2013)Google Scholar
- 19.Xu, Y., Ye, Q.: Generalized Mercer kernels and reproducing kernel Banach spaces. Mem. AMS (accepted). arXiv:1412.8663Google Scholar
- 21.Zhu, J., Rosset, S., Hastie, T., Tibshirani, R.: 1-norm support vector machines. In: Thrun, S., Saul, L., Schölkopf, B. (eds.) The Annual Conference on Neural Information Processing Systems 16, pp. 1–8 (2004)Google Scholar