Motivated by the computational difficulties incurred by popular deep learning algorithms for the generative modeling of temporal densities, we propose a cheap alternative that requires minimal hyperparameter tuning and scales favorably to high-dimensional problems. In particular, we use a projection-based optimal transport solver [Meng et al.,Advances in Neural Information Processing Systems (Curran Associates, 2019), Vol. 32] to join successive samples and, subsequently, use transport splines (Chewi et al., 2020) to interpolate the evolving density. When the sampling frequency is sufficiently high, the optimal maps are close to the identity and are, thus, computationally efficient to compute. Moreover, the training process is highly parallelizable as all optimal maps are independent and can, thus, be learned simultaneously. Finally, the approach is based solely on numerical linear algebra rather than minimizing a nonconvex objective function, allowing us to easily analyze and control the algorithm. We present several numerical experiments on both synthetic and real-world datasets to demonstrate the efficiency of our method. In particular, these experiments show that the proposed approach is highly competitive compared with state-of-the-art normalizing flows conditioned on time across a wide range of dimensionalities.
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