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