# Deep Kuratowski Embedding Neural Networks for Wasserstein Metric Learning

arXiv: https://arxiv.org/abs/2604.04343 · HTML: https://mlsanity.com/paper/2604.04343/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 9 references that are in the map.
- First posted on arXiv 6 Apr 2026.

## Builds on (most cited first, 5 of 5)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Perceptual Losses for Real-Time Style Transfer and Super-Resolution](https://mlsanity.com/paper/1603.08155/) (2016)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Wassmap: Wasserstein Isometric Mapping for Image Manifold Learning](https://mlsanity.com/paper/2204.06645/) (2022)
- [Linearized Wasserstein dimensionality reduction with approximation guarantees](https://mlsanity.com/paper/2302.07373/) (2023)

## Abstract (arXiv)

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.