Efficient $\chi ^{2}$ Kernel Linearization via Random Feature Maps

Explicit feature mapping is an appealing way to linearize additive kernels, such as χ² kernel for training large-scale support vector machines (SVMs). Although accurate in approximation, feature mapping could pose computational challenges in high-dimensional settings as it expands the original features to a higher dimensional space. To handle this issue in the context of χ² kernel SVMs learning, we introduce a simple yet efficient method to approximately linearize χ² kernel through random feature maps. The main idea is to use sparse random projection to reduce the dimensionality of feature maps while preserving their approximation capability to the original kernel. We provide approximation error bound for the proposed method. Furthermore, we extend our method to χ² multiple kernel SVMs learning. Extensive experiments on large-scale image classification tasks confirm that the proposed approach is able to significantly speed up the training process of the χ² kernel SVMs at almost no cost of testing accuracy.