We introduce a new stochastic algorithm for solving entropic optimal transport (EOT) between two absolutely continuous probability measures $\mu$ and $\nu$. Our work is motivated by the specific setting of Monge-Kantorovich quantiles where the source measure $\mu$ is either the uniform distribution on the unit hypercube or the spherical uniform distribution. Using the knowledge of the source measure, we propose to parametrize a Kantorovich dual potential by its Fourier coefficients. In this way, each iteration of our stochastic algorithm reduces to two Fourier transforms that enables us to make use of the Fast Fourier Transform (FFT) in order to implement a fast numerical method to solve EOT. We study the almost sure convergence of our stochastic algorithm that takes its values in an infinite-dimensional Banach space. Then, using numerical experiments, we illustrate the performances of our approach on the computation of regularized Monge-Kantorovich quantiles. In particular, we investigate the potential benefits of entropic regularization for the smooth estimation of multivariate quantiles using data sampled from the target measure $\nu$.

翻译：我们提出了一种新的随机算法，用于求解两个绝对连续概率测度μ和ν之间的熵正则化最优输运问题（EOT）。本研究源于Monge-Kantorovich分位数的特定场景，其中源测度μ为单位超立方体上的均匀分布或球面均匀分布。利用源测度的已知信息，我们提出将Kantorovich对偶势函数通过其傅里叶系数进行参数化。由此，算法每次迭代简化为两次傅里叶变换，这使得我们能够借助快速傅里叶变换来实现求解EOT的快速数值方法。我们研究了该取值于无穷维Banach空间的随机算法的几乎必然收敛性。随后通过数值实验，展示了该方法在计算正则化Monge-Kantorovich分位数方面的性能。特别地，我们探讨了熵正则化对利用目标测度ν采样数据进行多元分位数平滑估计的潜在优势。