[Paper] On the application of the Wasserstein metric to 2D curves classification

Source: arXiv - 2601.07749v1

Overview

The paper explores how to adapt the Wasserstein (optimal‑transport) distance for classifying 2‑D curves when only specific parts of the curves matter. By assigning importance weights to curve fragments through discrete probability measures, the authors show that you can steer a classifier to focus on the most relevant sections. They validate the idea with clustering experiments on archaeological shape data, demonstrating a practical way to blend geometric similarity with domain‑specific knowledge.

Key Contributions

• Fragment‑aware Wasserstein variants: Introduces several weighted versions of the Wasserstein distance that let users emphasize or de‑emphasize chosen curve segments.
• Discrete probability measure design: Proposes simple schemes for constructing probability measures that encode fragment importance directly from the data or expert input.
• Empirical validation on real‑world data: Applies the method to a collection of 2‑D archaeological outlines, showing improved clustering quality compared to unweighted baselines.
• Open‑source implementation hints: The authors release code snippets (Python/NumPy) that illustrate how to plug the weighted distance into existing pipelines.

Methodology

1. Curve representation: Each 2‑D curve is sampled uniformly to obtain a set of points ({x_i}_{i=1}^N).
2. Fragment weighting: The curve is split into predefined fragments (e.g., head, torso, tail). For each fragment a weight (w_j) is assigned, and a discrete probability measure (\mu = \sum_j w_j \delta_{x_j}) is built, where (\delta) is a Dirac mass.
3. Weighted Wasserstein distance: The classic Earth Mover’s Distance (EMD) between two measures (\mu) and (\nu) is computed, but the transport cost is multiplied by the fragment weights, effectively penalizing moves that cross “important” regions.
4. Clustering pipeline: Pairwise distances are fed to a standard hierarchical clustering algorithm (average linkage). Cluster quality is assessed with silhouette scores and domain‑specific visual inspection.
5. Baseline comparison: The same pipeline is run with the standard (unweighted) Wasserstein distance and with simple Euclidean shape descriptors to quantify the benefit of weighting.

Results & Findings

• Higher silhouette scores: Weighted Wasserstein distances consistently yielded silhouette scores 10‑15 % higher than the unweighted version, indicating tighter, more meaningful clusters.
• Domain‑aligned groups: In the archaeological dataset, clusters matched known typological categories (e.g., pottery styles) when fragment weighting emphasized the rim and handle regions, whereas unweighted clustering mixed styles that differed only in those parts.
• Robustness to noise: Adding synthetic jitter to the point clouds degraded performance of Euclidean descriptors dramatically, while the weighted Wasserstein remained stable, thanks to its global transport formulation.
• Computation time: The weighted distance incurs roughly a 20‑30 % overhead over the classic EMD due to extra weight handling, but stays within practical limits for datasets up to a few thousand curves on a modern laptop.

Practical Implications

• Shape‑aware ML pipelines: Developers can plug the weighted Wasserstein metric into any model that consumes pairwise distances (e.g., k‑NN, clustering, metric‑learning) without redesigning the whole architecture.
• Customizable similarity: In domains like CAD, GIS, or medical imaging, practitioners can encode expert knowledge (e.g., “the nose of a face is more important than the cheek”) directly as fragment weights, leading to more semantically relevant similarity scores.
• Heritage tech & digital humanities: Archaeologists can automate typology classification while still respecting the parts of artifacts that matter most to their discipline, accelerating cataloguing and comparative studies.
• Scalable to higher dimensions: The same weighting idea extends to 3‑D meshes or point clouds, opening doors for robotics (grasp‑point relevance) or autonomous driving (emphasizing road edges).

Limitations & Future Work

• Fragment definition is manual: The current approach relies on pre‑defined fragment boundaries, which may be subjective or labor‑intensive for large datasets.
• Scalability to very large corpora: While acceptable for a few thousand curves, the (O(N³)) transport solver can become a bottleneck; the authors suggest exploring entropic regularization (Sinkhorn) to speed up computation.
• Weight learning: Future research could integrate weight estimation into the learning loop (e.g., via differentiable optimal transport) so the model discovers which fragments are most discriminative automatically.
• Generalization tests: The paper focuses on a single archaeological dataset; broader benchmarks (handwritten characters, biological silhouettes) would help confirm the method’s universality.

Authors

• Agnieszka Kaliszewska
• Monika Syga

Paper Information

• arXiv ID: 2601.07749v1
• Categories: cs.CV
• Published: January 12, 2026
• PDF: Download PDF