Two-Dimensional Solution Surface for Weighted Support Vector Machines.

The support vector machine (SVM) is a popular learning method for binary classification. Standard SVMs treat all the data points equally, but in some practical problems it is more natural to assign different weights to observations from different classes. This leads to a broader class of learning, the so-called weighted SVMs (WSVMs), and one of their important applications is to estimate class probabilities besides learning the classification boundary. There are two parameters associated with the WSVM optimization problem: one is the regularization parameter and the other is the weight parameter. In this paper we first establish that the WSVM solutions are jointly piecewise-linear with respect to both the regularization and weight parameter. We then develop a state-of-the-art algorithm that can compute the entire trajectory of the WSVM solutions for every pair of the regularization parameter and the weight parameter, at a feasible computational cost. The derived two-dimensional solution surface provides theoretical insight on the behavior of the WSVM solutions. Numerically, the algorithm can greatly facilitate the implementation of the WSVM and automate the selection process of the optimal regularization parameter. We illustrate the new algorithm on various examples.