Approximation Theory, Computing, and Deep Learning on the Wasserstein Space

The challenge of approximating functions in infinite-dimensional spaces from finite samples is widely regarded as formidable. In this study, we delve into the challenging problem of the numerical approximation of Sobolev-smooth functions defined on probability spaces. Our particular focus centers on the Wasserstein distance function, which serves as a relevant example. In contrast to the existing body of literature focused on approximating efficiently pointwise evaluations, we chart a new course to define functional approximants by adopting three machine learning-based approaches: 1. Solving a finite number of optimal transport problems and computing the corresponding Wasserstein potentials. 2. Employing empirical risk minimization with Tikhonov regularization in Wasserstein Sobolev spaces. 3. Addressing the problem through the saddle point formulation that characterizes the weak form of the Tikhonov functional's Euler-Lagrange equation. As a theoretical contribution, we furnish explicit and quantitative bounds on generalization errors for each of these solutions. In the proofs, we leverage the theory of metric Sobolev spaces and we combine it with techniques of optimal transport, variational calculus, and large deviation bounds. In our numerical implementation, we harness appropriately designed neural networks to serve as basis functions. These networks undergo training using diverse methodologies. This approach allows us to obtain approximating functions that can be rapidly evaluated after training. Consequently, our constructive solutions significantly enhance at equal accuracy the evaluation speed, surpassing that of state-of-the-art methods by several orders of magnitude.