Conditionally Positive Definite Kernels for SVM Based Image Recognition

Abstract

Kernel based methods such as support vector machine (SVM) has provided successful tools for solving many recognition problems. One of the reasons of this success is the use of kernels. Positive definiteness has to be checked for kernels to be suitable for most of these methods. For instance for SVM, the use of a positive definite kernel insures that the optimized problem is convex and thus the obtained solution is unique. Alternative class of kernels called conditionally positive definite have been studied for a long time from the theoretical point of view and have drawn attention from the community only in the last decade. We propose a new kernel, named log kernel, which seems particularly interesting for images. Moreover, we prove that this new kernel is a conditionally positive definite kernel as well as the power kernel. Finally, we show from experimentations that using conditionally positive definite kernels allows us to outperform classical positive definite kernels