协方差调制的最优传输与梯度流
Covariance-modulated optimal transport and gradient flows
AI总结：
研究协方差调制的动态最优传输问题，证明其分裂为均值协方差与形状的耦合最小化问题，并分析相关梯度流，发现其收敛性质优于经典Wasserstein度量。
摘要：
我们研究动态最优传输问题的一种变体，其中需要最小化的能量由分布的协方差矩阵进行调制。此类传输度量自然产生于求解逆问题的某些集合 Kalman 方法的均场极限中。我们证明该传输问题分裂为两个耦合的最小化问题：一个用于插值曲线的均值和协方差的演化，另一个用于其形状。后者包括在沿插值保持固定均值和协方差的约束下，最小化通常的 Wasserstein 长度。我们分析了由该调制传输距离在概率空间上诱导的几何结构，以及相关梯度流的动力学。结果表明，与经典的 Wasserstein 度量相比，在独立于高斯目标的指数收敛速率方面，这些梯度流展现出更好的收敛性质。在梯度流的层面上，可以观察到类似的分布矩和形状演化的分裂现象。
英文摘要：
We study a variant of the dynamical optimal transport problem in which the energy to be minimised is modulated by the covariance matrix of the distribution. Such transport metrics arise naturally in mean-field limits of certain ensemble Kalman methods for solving inverse problems. We show that the transport problem splits into two coupled minimization problems: one for the evolution of mean and covariance of the interpolating curve and one for its shape. The latter consists in minimising the usual Wasserstein length under the constraint of maintaining fixed mean and covariance along the interpolation. We analyse the geometry induced by this modulated transport distance on the space of probabilities as well as the dynamics of the associated gradient flows. Those show better convergence properties in comparison to the classical Wasserstein metric in terms of exponential convergence rates independent of the Gaussian target. On the level of the gradient flows a similar splitting into the evolution of moments and shapes of the distribution can be observed.
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