Wasserstein Geometryedit

Wasserstein Geometry refers to the use of optimal-transport distances and related geometric ideas to compare probability distributions. Unlike pointwise metrics, Wasserstein distances account for the cost of moving probability mass from one distribution to another, which makes them useful for reasoning about distribution shift and generated data.

Research contextedit

Wasserstein geometry supports synthetic-data analysis, collaborative evaluation, and distributed barycenter computation. When data are split across silos, a distributional comparison can be more informative than a single scalar accuracy score. Wasserstein-style measures provide a language for describing how generated data drift across classes, modes, or visual features.

This geometry is not limited to Euclidean domains. Recent work on generative modeling over Wasserstein spaces of Riemannian manifolds illustrates how McCann displacement interpolants connect probability measures on a base manifold $(\mathcal{M}, d_g)$ to a geodesic path in the Wasserstein manifold $(\mathcal{P}_2(\mathcal{M}), W_2)$ above it—the same transport geometry used to compare and combine distributions.¹

Connection to Qiao's workedit

The ICML 2026 paper When Sample Selection Bias Precipitates Model Collapse uses collaborative Wasserstein-style signals to reason about model collapse under low-resource selection bias. The later doctoral focus on distributed Wasserstein barycenters keeps the same geometry but shifts attention toward computation: how a shared distributional reference can be obtained from several local measures. Wasserstein geometry provides tools for diagnosing distributional change when data are distributed and direct global inspection is unavailable.

See alsoedit

• When Sample Selection Bias Precipitates Model Collapse
• Collaborative Evaluation
• Distributed Wasserstein Barycenter
• Model Collapse
• Data Silos

Footnotesedit

1. Figure from Haviv et al., "When Riemann flows with Wasserstein: Generative Modeling of Probability Distributions on Manifolds", arXiv:2609.25659. ↩