Null models and statistical validation for Gromov-Wasserstein distance in latent space comparisons

I am using Gromov-Wasserstein (GW) distance to quantify the structural similarity between latent representations of two non-isomorphic physical systems. Given two metric spaces ( X , d X ) (X,d X ​ ) and ( Y , d Y ) (Y,d Y ​ ), the GW distance aims to find the optimal coupling that minimizes the distortion of pairwise distances. However, I've encountered a critical issue: the GW loss can be lower for a comparison between a structured dataset and a random noise cloud than for a comparison between two structured but distinct datasets. I suspect this happens because the latent geometries in my datasets (reduced via PCA to dimensions d=3 to 10) are nearly spherical, and GW is simply aligning these trivial geometric structures. I've read that GW is notoriously fragile to structural noise and geometric artifacts. My current pipeline uses a baseline of ETT (transformer) vs Random, and ETT vs UCI (electricity). The GW loss for ETT vs Random is often significantly lower than for ETT vs UCI, which makes me question the validity of using GW as a standalone metric here. Questions: What is the appropriate null hypothesis for GW? Should I be using permutations (shuffling the points within the spaces) or a Gaussian random geometric graph? Is Unbalanced Gromov-Wasserstein (UGW) statistically more robust for this specific problem (where one space might have a different mass/distribution), or should I move to Fused GW (FGW) with a semantically meaningful prior? What are the recommended non-pa…