Summarizing distribution

Abstract

This paper considers the problem of summarizing a probability measure with a small subset of diverse samples such that representation quality defined by the probability density cover function is maximized. Many existing studies have addressed this problem based on submodularity

however, there does not seem to be a definitive measure for summarization. A probability density cover function which is both monotone and submodular is defined for measuring the sample quality in terms of coverage. The cover function is a generalization of other submodular functions such as the facility location, sum coverage, and truncated vertex cover. Maximizing the cover function is achieved by the lazy greedy algorithm, which guarantees a lower bound as a constant ratio of the optimal value. Simulation results show that the algorithms to maximize the probability density cover function to identify sample subset of high diversity and relevance can perform better than random sampling in terms of fidelity in representing the probability measure and of estimation accuracy of the moments while achieving high diversity and relevance. The proposed cover function is also applied to the data subset selection task. Experimental results show that the GMM and k-NN-based classifiers learned with the data subset selected by the proposed algorithm on TIDIGIT and MNIST datasets has better accuracy than the existing methods.