论文标题
熵登记的$ 2 $ - 高斯措施之间的距离
Entropy-Regularized $2$-Wasserstein Distance between Gaussian Measures
论文作者
论文摘要
高斯分布在处理不确定性定量和扩散率的应用中很丰富。此外，它们作为为概率措施提供几何形状的框架的重要特殊案例，因为在框架下通常可以在封闭形式中表达高斯的几何形状。在这项工作中，我们通过为元素之间的距离和插值提供封闭形式的溶液来研究熵调查的2-wasserstein距离下的高斯几何形状。此外，当限制在高斯人的歧管时，我们提供了种群重中心的定点表征，该高斯人允许通过固定点迭代算法进行计算。结果，结果产生了2-链接差异的闭合表达式。随着几何形状通过改变正则化幅度的变化，我们研究了消失和无限幅度的限制案例，并在sindhorn差异的范围内重新确认了众所周知的结果。最后，我们通过数值研究说明了所得的几何形状。
Gaussian distributions are plentiful in applications dealing in uncertainty quantification and diffusivity. They furthermore stand as important special cases for frameworks providing geometries for probability measures, as the resulting geometry on Gaussians is often expressible in closed-form under the frameworks. In this work, we study the Gaussian geometry under the entropy-regularized 2-Wasserstein distance, by providing closed-form solutions for the distance and interpolations between elements. Furthermore, we provide a fixed-point characterization of a population barycenter when restricted to the manifold of Gaussians, which allows computations through the fixed-point iteration algorithm. As a consequence, the results yield closed-form expressions for the 2-Sinkhorn divergence. As the geometries change by varying the regularization magnitude, we study the limiting cases of vanishing and infinite magnitudes, reconfirming well-known results on the limits of the Sinkhorn divergence. Finally, we illustrate the resulting geometries with a numerical study.
