A Randomized Algorithm for the Exact Solution of Transductive Support Vector Machines

Random sampling is an efficient method to deal with constrained optimization problems. In computational geometry, this method has been successfully applied, through Clarkson’s algorithm (Clarkson, 1996), to solve a general class of problems called violator spaces. In machine learning, TSVM is a learning method used when only a small fraction of labeled data is available, which implies to solve a non convex optimization problem. Several approximation methods have been proposed to solve it, but they usually find suboptimal solutions. However, global optimal solution may be obtained using exact techniques, but at the cost of suffering an exponential time complexity with respect to the number of instances. In this paper, an interpretation of TSVM in terms of violator space is given. Hence, a randomized method is presented extending the use of exact methods now reducing the time complexity to exponential w.r.t. the number of support vectors of the optimal solution instead of exponential w.r.t. the number of instances.