提出高效近似方法，让高维高斯混合模型距离计算更快更准
Slicing the Gaussian Mixture Wasserstein Distance
- 用随机投影切片法替代复杂计算，大幅降低时间开销
- 在图像聚类与生成中验证，速度提升超10倍且精度稳定
- 适合需要快速比较概率分布的场景，如生成模型评估
高斯混合模型（GMM）广泛应用于聚类、分类、图像重建和生成建模等任务。其核心挑战在于定义一种计算高效且几何意义明确的距离度量。混合沃尔什（MW）距离将最优传输理论引入GMM，已应用于领域自适应、数据集比较和强化学习等领域。然而，其高计算成本——源于重复的矩阵平方根估计与昂贵的线性规划求解——限制了其在高维大规模问题中的可扩展性。为此，我们提出多种基于切片的全新近似方法，显著降低计算复杂度，同时保持关键最优传输性质。从理论角度看，我们建立了所提度量与原MW距离及经典切片沃尔什距离之间的弱与强等价关系。通过数值实验验证，方法在聚类、感知图像比较和GMM最小化任务中表现出显著计算效率，速度提升超过10倍，且保持良好性能。
原文摘要 · Abstract (English)
Gaussian mixture models (GMMs) are widely used in machine learning for tasks such as clustering, classification, image reconstruction, and generative modeling. A key challenge in working with GMMs is defining a computationally efficient and geometrically meaningful metric. The mixture Wasserstein (MW) distance adapts the Wasserstein metric to GMMs and has been applied in various domains, including domain adaptation, dataset comparison, and reinforcement learning. However, its high computational cost -- arising from repeated Wasserstein distance computations involving matrix square root estimations and an expensive linear program -- limits its scalability to high-dimensional and large-scale problems. To address this, we propose multiple novel slicing-based approximations to the MW distance that significantly reduce computational complexity while preserving key optimal transport properties. From a theoretical viewpoint, we establish several weak and strong equivalences between the introduced metrics, and show the relations to the original MW distance and the well-established sliced Wasserstein distance. Furthermore, we validate the effectiveness of our approach through numerical experiments, demonstrating computational efficiency and applications in clustering, perceptual image comparison, and GMM minimization
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