## Ramp Loss Linear Programming Support Vector Machine

*Xiaolin Huang, Lei Shi, Johan A.K. Suykens*; 15(Jun):2185−2211, 2014.

### Abstract

The ramp loss is a robust but non-convex loss for
classification. Compared with other non-convex losses, a local
minimum of the ramp loss can be effectively found. The
effectiveness of local search comes from the piecewise linearity
of the ramp loss. Motivated by the fact that the
$\ell_1$-penalty is piecewise linear as well, the
$\ell_1$-penalty is applied for the ramp loss, resulting in a
ramp loss linear programming support vector machine (ramp-
LPSVM). The proposed ramp-LPSVM is a piecewise linear
minimization problem and the related optimization techniques are
applicable. Moreover, the $\ell_1$-penalty can enhance the
sparsity. In this paper, the corresponding misclassification
error and convergence behavior are discussed. Generally, the
ramp loss is a truncated hinge loss. Therefore ramp-LPSVM
possesses some similar properties as hinge loss SVMs. A local
minimization algorithm and a global search strategy are
discussed. The good optimization capability of the proposed
algorithms makes ramp-LPSVM perform well in numerical
experiments: the result of ramp-LPSVM is more robust than that
of hinge SVMs and is sparser than that of ramp-SVM, which
consists of the $\|\cdot\|_{\mathcal{K}} $-penalty and the ramp
loss.

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