Generative modeling of time-dependent densities via optimal transport and projection pursuit

Such processes are visible in many applications ranging from geoscience, bioscience, and engineering to computer vision. In particular, deep learning algorithms, such as neural network parameterized normalizing flows, neural ordinary differential equations, diffusion models, and generative adversarial networks, have shown remarkable advances in learning and enabling rapid sampling from these stochastic processes. Such advances are further pronounced for very high-dimensional systems where classical methods are seen to saturate their effectiveness. However, the effective use of deep learning is frequently hampered by difficulties associated with computational cost as well as optimal hyperparameter selection. In this article, we propose a novel approach based on projection-pursuit optimal transport, which learns to sample from the densities of time-varying stochastic processes. It is competitive (both in terms of computational cost and accuracy) with a state-of-the-art deep learning algorithm (given by the neural spline flow). Crucially, our proposed method requires few hyperparameter choices by the user in contrast with most neural network-based methodologies. Thus, our main contributions to this work are as follows: 1. We implement a projection-pursuit optimal transport-based method to learn maps between time-varying densities from snapshots of particles sampled from these densities.