  Causal Optimal Transport (COT) results from imposing a temporal causality
constraint on classic optimal transport problems, which naturally generates a
new concept of distances between distributions on path spaces. The first
application of the COT theory for sequential learning was given in Xu et al.
(2020), where COT-GAN was introduced as an adversarial algorithm to train
implicit generative models optimized for producing sequential data. Relying on
(Xu et al., 2020), the contribution of the present paper is twofold. First, we
develop a conditional version of COT-GAN suitable for sequence prediction. This
means that the dataset is now used in order to learn how a sequence will evolve
given the observation of its past evolution. Second, we improve on the
convergence results by working with modifications of the empirical measures via
kernel smoothing due to (Pflug and Pichler (2016)). The resulting kernel
conditional COT-GAN algorithm is illustrated with an application for video
prediction.
