We consider problems of minimizing functionals $\mathcal{F}$ of probability measures on the Euclidean space. To propose an accelerated gradient descent algorithm for such problems, we consider gradient flow of transport maps that give push-forward measures of an initial measure. Then we propose a deterministic accelerated algorithm by extending Nesterov's acceleration technique with momentum. This algorithm do not based on the Wasserstein geometry. Furthermore, to estimate the convergence rate of the accelerated algorithm, we introduce new convexity and smoothness for $\mathcal{F}$ based on transport maps. As a result, we can show that the accelerated algorithm converges faster than a normal gradient descent algorithm. Numerical experiments support this theoretical result.

翻译：我们考虑欧几里得空间中概率测度泛函 $\mathcal{F}$ 的最小化问题。为提出此类问题的加速梯度下降算法，我们研究传输映射的梯度流，该映射给出初始测度的前推测度。随后通过扩展带有动量的Nesterov加速技术，提出一种确定性加速算法。该算法不依赖于Wasserstein几何。此外，为估计加速算法的收敛速率，我们基于传输映射引入了 $\mathcal{F}$ 的新凸性和光滑性。结果表明，加速算法比普通梯度下降算法收敛更快。数值实验支持了这一理论结果。