A Linear Approximation to the chi^2 Kernel with Geometric Convergence

TL;DR

This paper introduces a new analytical approximation to the chi^2 kernel with geometric convergence, improving image classification and segmentation performance through efficient random Fourier features and PCA-based dimensionality reduction.

Contribution

The paper presents a novel analytical approximation to the chi^2 kernel with geometric convergence, enhancing efficiency and accuracy in kernel-based image tasks.

Findings

Improved classification and segmentation accuracy on PASCAL VOC 2010 and ImageNet datasets.

Enhanced performance using PCA for dimensionality reduction with minimal additional complexity.

Statistically significant improvements over existing approximation methods.

Abstract

We propose a new analytical approximation to the $\chi^2$ kernel that converges geometrically. The analytical approximation is derived with elementary methods and adapts to the input distribution for optimal convergence rate. Experiments show the new approximation leads to improved performance in image classification and semantic segmentation tasks using a random Fourier feature approximation of the $\exp-\chi^2$ kernel. Besides, out-of-core principal component analysis (PCA) methods are introduced to reduce the dimensionality of the approximation and achieve better performance at the expense of only an additional constant factor to the time complexity. Moreover, when PCA is performed jointly on the training and unlabeled testing data, further performance improvements can be obtained. Experiments conducted on the PASCAL VOC 2010 segmentation and the ImageNet ILSVRC 2010 datasets show statistically significant improvements over alternative approximation methods.