Discriminality-driven regularization framework for indefinite kernel machine

Indefinite kernel machines have attracted more and more interests in machine learning due to their better empirical classification performance than the common positive definite kernel machines in many applications. A key to implement effective kernel machine is how to use prior knowledge as sufficiently as possible to guide the appropriate construction of the kernels. However, most of existing indefinite kernel machines actually utilize the knowledge involved in data such as discriminative and structural information insufficiently and thus construct the indefinite kernels empirically. Discriminatively regularized least-squares classification (DRLSC) is a recently-proposed supervised classification method which provides a new discriminality-driven regularizer to encourage the discriminality of the classifier rather than the common smoothness. In this paper, we rigorously validate that the discriminative regularizer actually coincides with the definition on the inner product in Reproducing Kernel Kreǐn Space (RKKS) naturally. As a result, we further present a new discriminality-driven regularization framework for indefinite kernel machine based on the discriminative regularizer. According to the framework, we firstly reintroduce the original DRLSC from the viewpoint of the proper indefinite kernelization rather than the empirical kernel mapping. Then a novel semi-supervised algorithm is proposed in terms of different definition on the regularizer. The experiments on both toy and real-world datasets demonstrate the superiority of the two algorithms.