Deep Kuratowski Embedding Neural Networks for Wasserstein Metric Learning

Computing pairwise Wasserstein distances is a fundamental bottleneck in data analysis pipelines. Motivated by the classical Kuratowski embedding theorem, we propose two neural architectures for learning to approximate the Wasserstein-2 distance (W2) from data. The first, DeepKENN, aggregates distances across all intermediate feature maps of a CNN using learnable positive weights. The second, ODE-KENN, replaces the discrete layer stack with a Neural ODE, embedding each input into the infinite-dimensional Banach space C1([0,1], ℝd) and providing implicit regularization via trajectory smoothness. Experiments on MNIST with exact precomputed W2 distances show that ODE-KENN achieves a 28% lower test MSE than the single-layer baseline and 18% lower than DeepKENN under matched parameter counts, while exhibiting a smaller generalization gap. The resulting fast surrogate can replace the expensive W2 oracle in downstream pairwise distance computations.