Bilinear generating functions in Kernel sparse modeling and learning: Towards the kernel engineering

As the cornerstone of nonlinear kernel learning algorithms, the kernel functions play a major role in sparse modeling via learning from data, as it largely determines the performance of a learning algorithm. In the literature, almost all kernels used are either dot product functions or distance functions. However, the highly challenging and complex learning tasks encountered in the real world necessitate innovative kernel functions with the capability of overcoming the fundamental limitations of conventional kernels. In this paper, the conexus between kernel functions in machine learning and generating functions of special functions is explored, by which the special functions of mathematical physics come into the picture of kernel-based machine learning, and in particular, a new trail to construct kernel function via bilinear generating function of special functions is blazed. To our best knowledge, it is the first study to investigate the use of bilinear generating functions of some special functions in constructing innovative kernel functions for linear programming support vector learning and sparse modeling.