Chebyshev Approximations to the Histogram Χ2 Kernel

Abstract

The random Fourier embedding methodology can be used to approximate the performance of non-linear kernel classifiers in linear time on the number of training examples. However, there still exists a non-trivial performance gap between the approximation and the nonlinear models, especially for the exponential χ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> kernel, one of the most powerful models for histograms. Based on analogies with Chebyshev polynomials, we propose an asymptotically convergent analytic series of the χ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> measure. The new series removes the need to use periodic approximations to the χ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> function, as typical in previous methods, and improves the classification accuracy when used in the random Fourier approximation of the exponential χ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> 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. The proposed approaches are tested on the PASCAL VOC 2010 segmentation and the ImageNet ILSVR-C 2010 datasets, and shown to give statistically significant improvements over alternative approximation methods.