Sliced Inner Product Gromov-Wasserstein Distances
This work addresses the computational bottleneck of Gromov-Wasserstein distances for high-dimensional alignment tasks, offering a more scalable alternative with theoretical guarantees.
The authors propose a sliced version of the Gromov-Wasserstein distance with inner product cost (IGW) to improve scalability for high-dimensional data, achieving rotational invariance and demonstrating effectiveness in heterogeneous clustering and language model comparison.
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.