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：高斯混合广泛应用于密度估计、信念传播和贝叶斯滤波等任务中的密度函数逼近。这些应用通常将高斯混合作为初始近似，并通过递归更新进行优化。此类递归过程的核心挑战在于混合阶数的指数增长，导致推理难以处理。为克服该困难，可采用高斯混合约简（GMR）方法，通过低阶混合近似高阶混合。尽管现有基于聚类的方法因其性能优良且计算高效而广为人知，但其收敛性质与最优目标仍不明确。本文提出一种基于复合传输散度（CTD）的新型优化型GMR方法。我们开发了用于计算约简混合的极大极小化算法，并在一般条件下建立了其理论收敛性。此外，我们证明许多现有基于聚类的方法均为本方法的特例，有效弥合了优化型方法与聚类型方法之间的鸿沟。该统一框架使用户能够根据具体需求选择CTD中最合适的代价函数，在特定应用中实现更优性能。通过广泛实证实验，我们验证了所提方法的效率与有效性，展示了其在多个领域的应用潜力。