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An in depth look at the Procrustes-Wasserstein distance: properties and barycenters
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Shows that Procrustes-Wasserstein is a distance on a quotient space of discrete measures, defines PW barycenters with a Newton-style point-location update, and tests initializations, k-means clustering on MNIST, and an archaeological sheep astragalus application.
Problem
Wasserstein distance is not invariant to rigid transformations, and Gromov-Wasserstein barycenters are pairwise distance matrices that are hard to map back to feature space; PW had not been proven to be a metric and PW barycenters had not been defined or learned from data.
Approach
Define a quotient space of discrete probability measures (modulo permutation and orthogonal transformation) on which the Procrustes-Wasserstein cost is a metric, and define a PW barycenter whose support locations are updated via an alternating scheme combining OT plans with orthogonal alignments.
Key results
- On MNIST digits 0-4 (50 point clouds), PW-based k-means reaches ARI 0.7669 and NMI 0.8361, versus 0.5500/0.6815 for Euc-GW and 0.4069/0.5652 for EMD.
- PW clustering took 130.11 s versus 675.19 s for Euc-GW and 378.82 s for Geo-GW (EMD was fastest at 9.18 s).
- Fiedler-vector Wasserstein initialization gave consistently robust convergence in the synthetic matching tests, while UPCA-based initialization was ineffective especially in 3D.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use PW-based rather than EMD-based clustering | k-means clustering on MNIST point clouds · MNIST | ARI: 0.4069 -> 0.7669 | Helped |
| Use PW-based rather than Euc-GW-based clustering | k-means clustering on MNIST point clouds · MNIST | ARI: 0.5500 -> 0.7669 | Helped |
| Use PW-based rather than Geo-GW-based clustering | k-means clustering on MNIST point clouds · MNIST | ARI: 0.3797 -> 0.7669 | Helped |
| Use PW-based rather than EMD-based clustering | k-means clustering on MNIST point clouds · MNIST | NMI: 0.5652 -> 0.8361 | Helped |
| Use PW-based rather than Euc-GW-based clustering | k-means clustering on MNIST point clouds · MNIST | NMI: 0.6815 -> 0.8361 | Helped |
| Use PW-based rather than Geo-GW-based clustering | k-means clustering on MNIST point clouds · MNIST | NMI: 0.5724 -> 0.8361 | Helped |
| Initialize PW matching with Fiedler-W instead of other tested strategies (ablation) | PW point-cloud matching · Randomly perturbed 2D/3D shapes | Matching convergence | Helped |
| Initialize PW matching with UPCA-W (ablation) | PW point-cloud matching · Randomly perturbed 2D/3D shapes | Matching convergence | Hurt |
| Initialize PW matching with Euc-GW or Geo-GW (ablation) | PW point-cloud matching · Randomly perturbed 2D/3D shapes | Matching convergence | Mixed |
Surprising findings
- UPCA-W initialization did not work effectively, particularly in the 3D case.
- EMD-based clustering consistently failed to identify digit 3, despite being a simpler baseline.
Limitations
- Non-convex problem; barycenter optimization only finds local minima via Newton-type updates and depends on initialization.
- Translations and scaling are not handled; point clouds are assumed centered and scaled to the unit ball.
- Experiments are small-scale (2D/3D shapes, 50 MNIST clouds, one pair of bones).
Experimental setup
Datasets: MNIST synthetic perturbed point clouds (2D dog, 3D dragon) archaeological and modern sheep astragalus 3D scans
Benchmarks: MNIST k-means clustering (ARI, NMI) initialization convergence on synthetic point cloud matching
Metrics: ARI NMI computational time (s) convergence success rate
Baselines: EMD / discrete Wasserstein Euc-GW Geo-GW Gromov-Wasserstein with MDS Gromov-Wasserstein with TSNE
Builds on: Wasserstein barycenter Gromov-Wasserstein Procrustes-Wasserstein k-means++ Sinkhorn algorithm
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