This paper presents a new distance metric to compare two continuous probability density functions. The main advantage of this metric is that, unlike other statistical measurements, it can provide an analytic, closed-form expression for a mixture of Gaussian distributions while satisfying all metric properties. These characteristics enable fast, stable, and efficient calculations, which are highly desirable in real-world signal processing applications. The application in mind is Gaussian Mixture Reduction (GMR), which is widely used in density estimation, recursive tracking, and belief propagation. To address this problem, we developed a novel algorithm dubbed the Optimization-based Greedy GMR (OGGMR), which employs our metric as a criterion to approximate a high-order Gaussian mixture with a lower order. Experimental results show that the OGGMR algorithm is significantly faster and more efficient than state-of-the-art GMR algorithms while retaining the geometric shape of the original mixture.

翻译：本文提出了一种新的距离度量，用于比较两个连续概率密度函数。该度量的主要优势在于，与其他统计测量不同，它能够为混合高斯分布提供解析闭式表达式，同时满足所有度量性质。这些特性使得计算快速、稳定且高效，这在现实世界的信号处理应用中极为理想。本文关注的应用是高斯混合约简（GMR），该技术广泛用于密度估计、递归跟踪和置信传播。为解决此问题，我们开发了一种名为基于优化的贪心GMR（OGGMR）的新算法，该算法采用我们所提出的度量作为准则，用低阶高斯混合逼近高阶高斯混合。实验结果表明，OGGMR算法在显著快于且更高效于现有最先进GMR算法的同时，能保持原始混合的几何形状。