Distances and kernels based on cumulative distribution functions - Emerging Trends in Image Processing, Computer Vision, and Pattern Recognition - page 464

In this paper, we propose a family of distances and kernels that are defined on the cumu-
lative distribution functions, instead of densities. This work is an extension of our previous
paper in IPCV'14 ([ 2 ] ).
This paper is organized as follows. Section 2 introduces traditional kernels and distances
defined on probability distributions. In Section 3 , a new family of distance and kernel func-
tions based on cumulative distribution functions is proposed. Experimental results on Gaussi-
an mixture distributions are presented in Section 4 . In Section 5 , we discuss the generalization
of the method to higher dimensional spaces. Section 6 provides conclusions and proposals for
improvements.
2 Distance and Similarity Measures Between
Distributions
Given two probability distributions, there are well-known measures for the differences or sim-
ilarities between the two distributions.
The Bhatacharyya ainity ([ 3 ] ) is a measure of similarity between two distributions:
In [ 4 ], the probability product kernel is deined as a generalization of Bhatacharyya ainity:
The Bhatacharyya distance is a dissimilarity measure related to the Bhatacharyya ainity:
The Hellinger distance ([ 5 ]) is another metric on distributions: