# Formation Shape Control using the Gromov-Wasserstein Metric

arXiv: https://arxiv.org/abs/2503.21538 · HTML: https://mlsanity.com/paper/2503.21538/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 31 references that are in the map.
- First posted on arXiv 27 Mar 2025.

## Builds on (most cited first, 11 of 11)

- [Optimal Transport for structured data with application on graphs](https://mlsanity.com/paper/1805.09114/) (2018)
- [Gromov-Wasserstein Learning for Graph Matching and Node Embedding](https://mlsanity.com/paper/1901.06003/) (2019)
- [Gromov-Wasserstein Alignment of Word Embedding Spaces](https://mlsanity.com/paper/1809.00013/) (2018)
- [Scalable Gromov-Wasserstein Learning for Graph Partitioning and Matching](https://mlsanity.com/paper/1905.07645/) (2019)
- [The Unbalanced Gromov Wasserstein Distance: Conic Formulation and Relaxation](https://mlsanity.com/paper/2009.04266/) (2020)
- [First-Order Methods for Nonsmooth Nonconvex Functional Constrained Optimization with or without Slater Points](https://mlsanity.com/paper/2212.00927/) (2022)
- [Optimal transport over a linear dynamical system](https://mlsanity.com/paper/1502.01265/) (2015)
- [Semidefinite Relaxations of the Gromov-Wasserstein Distance](https://mlsanity.com/paper/2312.14572/) (2023)
- [Globally solving the Gromov-Wasserstein problem for point clouds in low dimensional Euclidean spaces](https://mlsanity.com/paper/2307.09057/) (2023)
- [Minimum energy density steering of linear systems with Gromov-Wasserstein terminal cost](https://mlsanity.com/paper/2402.15942/) (2024)
- [The Exact Projective Penalty Method for Constrained Optimization](https://mlsanity.com/paper/2312.02094/) (2023)

## Built on by (most cited first, 2 of 2)

- [Data-Driven Density Steering via the Gromov-Wasserstein Optimal Transport Distance](https://mlsanity.com/paper/2508.06052/) (2025)
- [Fixed Horizon Linear Quadratic Covariance Steering in Continuous Time with Hilbert-Schmidt Terminal Cost](https://mlsanity.com/paper/2510.21944/) (2025)

## Abstract (arXiv)

This article introduces a formation shape control algorithm, in the optimal control framework, for steering an initial population of agents to a desired configuration via employing the Gromov-Wasserstein distance. The underlying dynamical system is assumed to be a constrained linear system and the objective function is a sum of quadratic control-dependent stage cost and a Gromov-Wasserstein terminal cost. The inclusion of the Gromov-Wasserstein cost transforms the resulting optimal control problem into a well-known NP-hard problem, making it both numerically demanding and difficult to solve with high accuracy. Towards that end, we employ a recent semi-definite relaxation-driven technique to tackle the Gromov-Wasserstein distance. A numerical example is provided to illustrate our results.