Efficient Semi-Discrete Optimal Transport Using the Maximum Relative Error between Distributions

Semi-Discrete Optimal Transport (SDOT) transforms a continuous distribution to a discrete distribution. However, existing SDOT algorithms for high dimensional distributions have two limitations. 1) It is difficult to evaluate the quality of the transport maps produced by SDOT algorithms, because computing a high-dimensional Wasserstein distance for SDOT is intractable and 2) The transport map cannot guarantee that all target points have the corresponding source points that are mapped to them. To address these limitations, we introduce the Maximum Relative Error (\texttt{MRE}) between the target distribution and the transported distribution computed by an SDOT map. If the \texttt{MRE} is smaller than 1, then every target point is guaranteed to have an area in the source distribution that is mapped to it. We propose a statistical method to compute the lower and upper bounds of the \texttt{MRE} given a confidence threshold and a precision. The gap between the lower bound and the upper bound approaches 0 as the number of samples goes to infinity. We present an efficient Epoch Gradient Descent algorithm for SDOT (SDOT-EGD) that computes the learning rate, number of iterations, and number of epochs in order to guarantee an arbitrarily small \texttt{MRE} in expectation. Experiments on both low and high-dimensional data show that SDOT-EGD is much faster and converges much better than state-of-the-art SDOT algorithms. We also show our method's potential to improve GAN training by avoiding the oscillation caused by randomly changing the association between noise and the real images.