# Beyond Procrustes distances: a multilinear Gromov-Wasserstein distance capturing chirality

arXiv: https://arxiv.org/abs/2608.27774 · HTML: https://mlsanity.com/paper/2608.27774/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 48 references that are in the map.
- First posted on arXiv 27 Aug 2026.

## Builds on (most cited first, 6 of 6)

- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Optimal Transport Tools (OTT): A JAX Toolbox for all things Wasserstein](https://mlsanity.com/paper/2201.12324/) (2022)
- [Convergence Rate of Frank-Wolfe for Non-Convex Objectives](https://mlsanity.com/paper/1607.00345/) (2016)
- [A contribution to Optimal Transport on incomparable spaces](https://mlsanity.com/paper/2011.04447/) (2020)
- [An in depth look at the Procrustes-Wasserstein distance: properties and barycenters](https://mlsanity.com/paper/2507.00894/) (2025)
- [On the Wasserstein alignment problem](https://mlsanity.com/paper/2503.06838/) (2025)

## Built on by (most cited first, 1 of 1)

- [Rubix: Global Correspondence-Free Point Set Alignment through Assignment Geometry](https://mlsanity.com/paper/2610.10408/) (2026)

## Abstract (arXiv)

Efficiently and robustly analyzing shape data is critical across many scientific disciplines. While chirality is a fundamental property in numerous applications - most notably in molecular science - existing shape analysis metrics fail to distinguish between a shape and its mirror image. To address this gap, we introduce a multilinear generalization of the Gromov-Wasserstein objective. Under mild assumptions, this objective yields a distance between shapes, represented as probability distributions quotiented by a symmetry group G. In particular, for G = SO(d), we introduce the Chiral Gromov-Wasserstein (CGW) distance, sensitive to chirality. We establish robustness properties for the multilinear Gromov-Wasserstein distances and develop efficient algorithms to compute them, reformulating the underlying optimization problem by projecting couplings onto a low-dimensional space. We derive algorithms for both local and approximate global solutions, yielding a fully polynomial-time approximation scheme for these problems. We validate the framework through numerical experiments that demonstrate the effectiveness of CGW as a shape metric for chiral objects.