# Rubix: Global Correspondence-Free Point Set Alignment through Assignment Geometry

arXiv: https://arxiv.org/abs/2610.10408 · HTML: https://mlsanity.com/paper/2610.10408/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 4 of its 79 references that are in the map.
- First posted on arXiv 7 Oct 2026.

## Builds on (most cited first, 4 of 4)

- [Fast globally optimal Truncated Least Squares point cloud registration with fixed rotation axis](https://mlsanity.com/paper/2508.15613/) (2025)
- [DC-Reg: Globally Optimal Point Cloud Registration via Tight Bounding with Difference of Convex Programming](https://mlsanity.com/paper/2603.25442/) (2026)
- [A Second-Order Lower Bound for Globally Optimal 2D Registration](https://mlsanity.com/paper/1901.09641/) (2019)
- [Beyond Procrustes distances: a multilinear Gromov-Wasserstein distance capturing chirality](https://mlsanity.com/paper/2608.27774/) (2026)

## Abstract (arXiv)

Procrustes-Wasserstein alignment jointly estimates a matching and rotation without supplied correspondences, but alternating minimization can stop at suboptimal solutions. Rubix solves the equally weighted planar problem globally under squared Euclidean loss. Each matching σ of two centered n-point sets defines a complex correlation zσ=sumibar xi yσ(i). Their convex hull is the permutation polygon: supporting vertices give optimal matchings at fixed rotations, and the farthest vertex gives the global alignment. We prove the sharp bound of n(n-1) vertices for n≥2, answering Rote's rotation-assignment open problem. In exact arithmetic, assignment queries recover the polygon in mathcal O(n5) operations. Assignment-based bounds extend the approach to three-dimensional rotations and partial matching at a supplied translation through branch-and-bound. On timed MPEG-7 shape pairs, Rubix attains every numerical reference value in 12 ms on average, 50 times faster than a rotation grid at the same accuracy. Its distances improve gravity-aligned matching of real 3D scans, shape retrieval and noisy crystal classification over alternating minimization.