[2302.00982] Stochastic optimum transport in Banach Areas for regularized estimation of multivariate quantiles

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Summary:We introduce a brand new stochastic algorithm for fixing entropic optimum transport (EOT) between two completely steady likelihood measures $mu$ and $nu$. Our work is motivated by the precise setting of Monge-Kantorovich quantiles the place the supply measure $mu$ is both the uniform distribution on the unit hypercube or the spherical uniform distribution. Utilizing the information of the supply measure, we suggest to parametrize a Kantorovich twin potential by its Fourier coefficients. On this approach, every iteration of our stochastic algorithm reduces to 2 Fourier transforms that permits us to utilize the Quick Fourier Remodel (FFT) with a purpose to implement a quick numerical technique to resolve EOT. We research the virtually positive convergence of our stochastic algorithm that takes its values in an infinite-dimensional Banach area. Then, utilizing numerical experiments, we illustrate the performances of our method on the computation of regularized Monge-Kantorovich quantiles. Particularly, we examine the potential advantages of entropic regularization for the graceful estimation of multivariate quantiles utilizing knowledge sampled from the goal measure $nu$.