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Stochastic Optimal Transport in Banach Spaces for Regularized Estimation of Multivariate Quantiles
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DOI：10.1137/23M1550852.png)
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Abstract
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We introduce a new stochastic algorithm for solving entropic optimal transport (EOT) between two absolutely continuous probability measure . Our work is motivated by the specific setting of Monge-Kantorovich quantiles where the source measure 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 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.
Keywords:
entropic optimal transport
Monge-Kantorovich quantiles
multivariate quantiles
stochastic optimization in a Banach space
multiple Fourier series
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S
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2.6
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17
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