Neural Optimal Transport in Hilbert Spaces

Functional data such as time series, trajectories, and fields are increasingly central in machine learning and are naturally modeled in infinite-dimensional Hilbert spaces. We study Semi-dual Neural Optimal Transport (SNOT) for learning transport maps between probability measures on separable Hilbert spaces. A critical challenge in this setting is the spurious solution problem: a neural map can globally optimize the SNOT objective while failing to recover an optimal transport map. We show that this failure arises from non-regular source measures, which make the inner minimization in the semi-dual objective non-identifiable. We prove that regular source measures, defined via Gaussian null sets, restore inner-minimizer uniqueness and recover the Monge map. For singular sources, we introduce Gaussian smoothing and establish a necessary and sufficient kernel condition for smoothing to restore regularity. We further prove plan-level consistency: as smoothing vanishes, every Wasserstein accumulation point of the induced optimal plans solves the original Kantorovich problem. Experiments on synthetic singular transports and real-world unpaired time-series imputation benchmarks validate the theory, achieving the best performance on several benchmarks and competitive performance on the remaining ones.