Sliced Inner Product Gromov-Wasserstein Distances
Authors: Xiaoyun Gong, Gabriel Rioux, Ziv Goldfeld
Summary
The authors tackle the Gromov-Wasserstein (GW) problem—a framework for aligning datasets by matching their intrinsic geometry—using slicing techniques to improve scalability. Unlike standard Wasserstein distances, GW problems don't have closed-form solutions in one dimension, making slicing hard. They solve this for the inner-product version (IGW), propose a sliced IGW distance with rotational invariance, and study its properties.
Main takeaways:
- Gromov-Wasserstein provides a way to align heterogeneous datasets by matching geometry, but it's computationally expensive in high dimensions
- Slicing (projecting to one dimension and solving there) works well for Wasserstein distances but GW problems lack one-dimensional closed forms
- They resolve this for inner-product GW (IGW) and define a sliced IGW distance that's rotationally invariant
- Comprehensive theory covers structural and computational properties
- Applications include clustering text data from different sources and comparing language model representations
Relevance
Could be tangentially relevant to comparing persona representations across models or understanding how different behavioral installations relate geometrically—the language model representation comparison application is close to my domain, though the core method is about alignment rather than behavioral conditioning.
Abstract
arXiv:2605.08546v1 Announce Type: new Abstract: The Gromov-Wasserstein (GW) problem provides a framework for aligning heterogeneous datasets by matching their intrinsic geometry, but its statistical and computational scaling remains an issue for high-dimensional problems. Slicing techniques offer an appealing route to scalability, but, unlike Wasserstein distances, GW problems do not generally admit closed-form solutions in one-dimension. We resolve this problem for the GW problem with inner product cost (IGW), propose a sliced IGW distance that enjoys a natural rotational invariance property, and comprehensively study its structural and computational properties. Numerical experiments validating our theory are presented, followed by applications to heterogeneous clustering of text data and language model representation comparison.