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We propose an Euler particle transport (EPT) approach for generativelearning. The proposed approach is motivated by the problem of finding anoptimal transport map from a reference distribution to a target distributioncharacterized by the Monge-Ampere equation. Interpreting the infinitesimallinearization of the Monge-Ampere equation from the perspective of gradientflows in measure spaces leads to a stochastic McKean-Vlasov equation. We usethe forward Euler method to solve this equation. The resulting forward Eulermap pushes forward a reference distribution to the target. This map is thecomposition of a sequence of simple residual maps, which are computationallystable and easy to train. The key task in training is the estimation of thedensity ratios or differences that determine the residual maps. We estimate thedensity ratios (differences) based on the Bregman divergence with a gradientpenalty using deep density-ratio (difference) fitting. We show that theproposed density-ratio (difference) estimators do not suffer from the "curse ofdimensionality" if data is supported on a lower-dimensional manifold. Numericalexperiments with multi-mode synthetic datasets and comparisons with theexisting methods on real benchmark datasets support our theoretical results anddemonstrate the effectiveness of the proposed method.

Generative Learning With Euler Particle Transport