Sparse coding with earth mover's distance

Sparse coding (Sc) has been studied very well as a powerful data representation method. It attempts to represent the feature vector of a data sample by reconstructing it as the sparse linear combination of some basic elements, and a $l_2$ norm distance function is usually used as the loss function for the reconstruction error. In this paper, we investigate using Sc as the representation method within multi-instance learning framework, where a sample is given as a bag of instances, and further represented as a histogram of the quantized instances. We argue that for the data type of histogram, using $l_2$ norm distance is not suitable, and propose to use the earth mover's distance (EMD) instead of $l_2$ norm distance as a measure of the reconstruction error. By minimizing the EMD between the histogram of a sample and the its reconstruction from some basic histograms, a novel sparse coding method is developed, which is refereed as Sc-EMD. We evaluate its performances as a histogram representation method in tow multi-instance learning problems --- abnormal image detection in wireless capsule endoscopy videos, and protein binding site retrieval. The encouraging results demonstrate the advantages of the new method over the traditional method using $l_2$ norm distance.