Training Data Selection for Optimal Generalization in Trigonometric Polynomial Networks

In this paper, we consider the problem of active learning in trigonometric polynomialnetworks and give a necessary and sufficient condition of sample points to provide the optimal generalization capability. By analyzing thecondition from the functional analytic point of view, we clarify the mechanism of achieving the optimal generalization capability. We also show that a set of training examples satisfying the condition does not only provide the optimal generalization but also reduces the computational complexityand memory required for the calculation of learning results. Finally, examples of sample points satisfying the condition are given and computer simulations are performed to demonstrate the effectiveness ofthe proposed active learning method. 1 Introduction Supervised learning is obtaining an underlying rule from training examples, and can be formulated as a function approximation problem. If sample points are actively designed, then learning can be performed more efficiently. In this paper, we discuss the problem of designing sample points, referred to as active learning, for optimal generalization. Active learning is classified into two categories depending on the optimality. One is global optimal, where a set of all training examples is optimal (e.g.