Modelling of Kernel Machines by Infinite and Semi-infinite Programming

In Machine Learning (ML) algorithms, one of the crucial issues is the representation of the data. As the data become heterogeneous and large-scale, single kernel methods become insufficient to classify nonlinear data. The finite combinations of kernels are limited up to a finite choice. In order to overcome this discrepancy, we propose a novel method of ”infinite” kernel combinations for learning problems with the help of infinite and semi-infinite programming regarding all elements in kernel space. Looking at all infinitesimally fine convex combinations of the kernels from the infinite kernel set, the margin is maximized subject to an infinite number of constraints with a compact index set and an additional (Riemann-Stieltjes) integral constraint due to the combinations. After a parametrization in the space of probability measures, it becomes semi-infinite. We analyze the regularity conditions which satisfy the Reduction Ansatz and discuss the type of distribution functions within the structure of the constraints and our bi-level optimization problem.