用神经网络快速逼近Wasserstein距离，提升计算效率。
Deep Kuratowski Embedding Neural Networks for Wasserstein Metric Learning
- 基于柯拉托夫斯基嵌入，设计两种可学习的神经结构
- ODE-KENN在相同参数量下比基线低28%测试误差
- 适合需要高效配对距离计算的下游任务
计算成对Wasserstein距离是数据分析流程中的核心瓶颈。受经典柯拉托夫斯基嵌入定理启发，我们提出两种神经架构以从数据中学习近似Wasserstein-2距离（$W_2$）。第一种为DeepKENN，通过可学习的正权重聚合CNN所有中间特征图的距离；第二种为ODE-KENN，将离散层堆叠替换为神经微分方程，将每个输入嵌入到无穷维Banach空间$C^1([0,1], \mathbb{R}^d)$，并利用轨迹平滑性实现隐式正则化。在带有精确预计算$W_2$距离的MNIST数据集上实验表明，当参数量匹配时，ODE-KENN的测试均方误差比单层基线低28%，比DeepKENN低18%，且泛化差距更小。由此产生的快速代理模型可替代下游任务中昂贵的$W_2$ oracle进行成对距离计算。
原文摘要 · Abstract (English)
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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