Adversarial Optimal Transport Through The Convolution Of Kernels With Evolving Measures
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A novel algorithm is proposed to solve the sample-based optimal transport problem. An adversarial formulation of the push-forward condition uses a test function built as a convolution between an adaptive kernel and an evolving probability distribution ν over a latent variable b. Approximating this convolution by its simulation over evolving samples b^i(t) of ν, the parameterization of the test function reduces to determining the flow of these samples. This flow, discretized over discrete time steps t_n, is built from the composition of elementary maps. The optimal transport also follows a flow that, by duality, must follow the gradient of the test function. The representation of the test function as the Monte Carlo simulation of a distribution makes the algorithm robust to dimensionality, and its evolution under a memory-less flow produces rich, complex maps from simple parametric transformations. The algorithm is illustrated with numerical examples.
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