Learning with Consistency between Inductive Functions and Kernels

Regularized Least Squares (RLS) algorithms have the ability to avoid over-fitting problems and to express solutions as kernel expansions. However, we observe that the current RLS algorithms cannot provide a satisfactory interpretation even on a constant function. On the other hand, while kernel-based algorithms have been developed in such a tendency that almost all learning algorithms are kernelized or being kernelized, a basic fact is often ignored: The learned function from the data and the kernel fits the data well, but may not be consistent with the kernel. Based on these considerations and on the intuition that a good kernel-based inductive function should be consistent with both the data and the kernel, a novel learning scheme is proposed. The advantages of this scheme lie in its corresponding Representer Theorem, its strong interpretation ability about what kind of functions should not be penalized, and its promising accuracy improvements shown in a number of experiments. Furthermore, we provide a detailed technical description about heat kernels, which serves as an example for the readers to apply similar techniques for other kernels. Our work provides a preliminary step in a new direction to explore the varying consistency between inductive functions and kernels under various distributions.