In recent years, learning methods are desirable because of their reliability and efficiency in real-world problems. We propose a novel method to find infinitely many kernel combinations for learning problems with the help of infinite and semi-infinite optimization regarding all elements in kernel space. This will provide to study variations of combinations of kernels when considering heterogeneous data in real-world applications. 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 parametrisation in the space of probability measures it becomes semi-infinite. We analyze the conditions which satisfy the Reduction Ansatz and discuss the type of distribution functions of the kernel coefficients within the structure of the constraints and our bilevel optimization problem.
Sureyya Ozogur and G.W. Weber
In: International Conferece 20th Euro Mini Conferece, Continuous Optimization and Knowledge Based Technologies, 20 - 23 May 2008, Neringa, Lithuania.

Abstract

In recent years, learning methods are desirable because of their reliability and efficiency in real-world problems. We propose a novel method to find infinitely many kernel combinations for learning problems with the help of infinite and semi-infinite optimization regarding all elements in kernel space. This will provide to study variations of combinations of kernels when considering heterogeneous data in real-world applications. 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 parametrisation in the space of probability measures it becomes semi-infinite. We analyze the conditions which satisfy the Reduction Ansatz and discuss the type of distribution functions of the kernel coefficients within the structure of the constraints and our bilevel optimization problem.

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