Deep Kuratowski Embedding Neural Networks for Wasserstein Metric Learning

• URL: http://arxiv.org/abs/2604.04343v1
• Date: Mon, 06 Apr 2026 01:35:27 GMT
• Title: Deep Kuratowski Embedding Neural Networks for Wasserstein Metric Learning
• Abstract summary: We propose two neural architectures for learning to approximate the Wasserstein-2 distance from data.<n>The first, DeepKENN, aggregates distances across all intermediate feature maps of a CNN using learnable weights.<n>The second, ODE-KENN, replaces the discrete layer stack with an implicit ODE, embedding each input into the infinite-dimensional Banach space.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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 ($W_2$) 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 $C^1([0,1], \mathbb{R}^d)$ and providing implicit regularization via trajectory smoothness. Experiments on MNIST with exact precomputed $W_2$ 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 $W_2$ oracle in downstream pairwise distance computations.

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