Wasserstein-Distance-Based Gaussian Mixture Reduction

Gaussian mixtures (GMs) are widely used in signal processing applications to capture the multimodal behavior of dynamic systems. Due to an exponential increase in the number of GM components in such applications, Gaussian mixture reduction (GMR) approaches are deemed necessary. Traditionally, the Kullback-Leibler divergence (KLD) is used for GMR along with a moment-matching merging approach to minimize the information loss. However, in certain applications such as image retrieval, preserving the geometric shape of the GM is more appealing. For such applications, this work prescribes the use of the Wasserstein distance (WD), which quantifies the minimum cost of converting one density into another and, therefore, is mostly concerned with the shape difference between the densities. To minimize the change in the shape of the GM, first, similar GM components are identified utilizing the WD. Next, these components are merged by proposing a novel WD-based averaging method. The simulation results confirm the success of the proposed WD-based GMR techniques in providing a better approximation of the original GM in the WD sense as compared to KLD-based methods.