Gaussian mixtures are widely used for approximating density functions in various applications such as density estimation, belief propagation, and Bayesian filtering. These applications often utilize Gaussian mixtures as initial approximations that are updated recursively. A key challenge in these recursive processes stems from the exponential increase in the mixture's order, resulting in intractable inference. To overcome the difficulty, the Gaussian mixture reduction (GMR), which approximates a high order Gaussian mixture by one with a lower order, can be used. Although existing clustering-based methods are known for their satisfactory performance and computational efficiency, their convergence properties and optimal targets remain unknown. In this paper, we propose a novel optimization-based GMR method based on composite transportation divergence (CTD). We develop a majorization-minimization algorithm for computing the reduced mixture and establish its theoretical convergence under general conditions. Furthermore, we demonstrate that many existing clustering-based methods are special cases of ours, effectively bridging the gap between optimization-based and clustering-based techniques. Our unified framework empowers users to select the most appropriate cost function in CTD to achieve superior performance in their specific applications. Through extensive empirical experiments, we demonstrate the efficiency and effectiveness of our proposed method, showcasing its potential in various domains.

翻译：摘要：高斯混合模型广泛应用于密度估计、置信传播和贝叶斯滤波等各类应用中的密度函数逼近。在这些应用中，高斯混合常被用作递归更新的初始近似。递归过程中的一个关键挑战源于混合阶数的指数增长，导致推理难以处理。为解决这一困难，可采用高斯混合约简（Gaussian Mixture Reduction, GMR）方法，即用低阶混合逼近高阶混合。尽管现有的基于聚类的方法以其令人满意的性能和计算效率而闻名，但其收敛性质和优化目标仍然未知。本文提出了一种基于复合传输散度（Composite Transportation Divergence, CTD）的新型优化型GMR方法。我们开发了一种用于计算约简混合的majorization-minimization算法，并在一般条件下建立了其理论收敛性。此外，我们证明了现有许多基于聚类的方法都是我们方法的特例，从而有效弥合了优化型与聚类型技术之间的差距。我们的统一框架使能够根据具体应用选择最合适的CTD代价函数，以实现卓越性能。通过大量实证实验，我们展示了所提出方法的效率和有效性，并彰显了其在多个领域的应用潜力。