论文标题
Bures-Wasserstein Barycenters的梯度下降算法
Gradient descent algorithms for Bures-Wasserstein barycenters
论文作者
论文摘要
我们研究了一阶方法，以计算有限第二刻的概率度量的概率分布$ p $的重中心。我们开发了一个框架，以得出梯度下降和随机梯度下降的全球收敛速率，尽管Barycenter功能不是地理上的凸起。我们的分析通过采用Polyak-Lojasiewicz（PL）不平等来克服这一技术障碍，并依赖于最佳运输和度量几何形状的工具。反过来，当高斯概率措施的Bures-Wasserstein歧管支持$ P $时，我们会确定PL不平等。在这种情况下，它导致了一阶方法的首个全球收敛速率。
We study first order methods to compute the barycenter of a probability distribution $P$ over the space of probability measures with finite second moment. We develop a framework to derive global rates of convergence for both gradient descent and stochastic gradient descent despite the fact that the barycenter functional is not geodesically convex. Our analysis overcomes this technical hurdle by employing a Polyak-Lojasiewicz (PL) inequality and relies on tools from optimal transport and metric geometry. In turn, we establish a PL inequality when $P$ is supported on the Bures-Wasserstein manifold of Gaussian probability measures. It leads to the first global rates of convergence for first order methods in this context.
