Approximation Theory, Computing, and Deep Learning on the Wasserstein Space
Massimo Fornasier, Pascal Heid, Giacomo Enrico Sodini
TL;DR
This work develops and analyzes three machine-learning–driven frameworks for approximating Sobolev-smooth functionals on Wasserstein spaces, using the Wasserstein distance as a central example. It combines cylinder-function approximations, empirical risk minimization with pre-Cheeger regularization, and Euler–Lagrange–based saddle-point formulations to construct fast, trainable function classes that approximate Wasserstein-Sobolev maps, notably $F( u)=W_p( u, heta)$. The authors establish convergence guarantees via Gamma-convergence, provide explicit generalization-error bounds under data noise, and demonstrate substantial speedups over classical OT solvers on large datasets (MNIST, CIFAR-10) while achieving competitive accuracy with CNNs in some settings. The work also contributes concrete neural-network constructions for cylinder-function realizations (e.g., max-of-arguments via ReLU nets) and an adversarial training protocol to solve the corresponding saddle-point problems. Collectively, the results advance scalable, ML-based numerical procedures for function approximation on spaces of probability measures with rigorous guarantees and practical performance benefits.
Abstract
The challenge of approximating functions in infinite-dimensional spaces from finite samples is widely regarded as formidable. 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. We furnish explicit and quantitative bounds on generalization errors for each of these solutions. 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. Our constructive solutions significantly enhance at equal accuracy the evaluation speed, surpassing that of state-of-the-art methods by several orders of magnitude. This allows evaluations over large datasets several times faster, including training, than traditional optimal transport algorithms. Our analytically designed deep learning architecture slightly outperforms the test error of state-of-the-art CNN architectures on datasets of images.