# Cluster-Aware Matching via Laplacian Optimal Transport

arXiv: https://arxiv.org/abs/2607.16178 · HTML: https://mlsanity.com/paper/2607.16178/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 22 of its 53 references that are in the map.
- First posted on arXiv 17 Jul 2026.

## Builds on (most cited first, 22 of 22)

- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Optimal Transport for Domain Adaptation](https://mlsanity.com/paper/1507.00504/) (2015)
- [Gromov-Wasserstein Alignment of Word Embedding Spaces](https://mlsanity.com/paper/1809.00013/) (2018)
- [Unsupervised Alignment of Embeddings with Wasserstein Procrustes](https://mlsanity.com/paper/1805.11222/) (2018)
- [Low-Rank Sinkhorn Factorization](https://mlsanity.com/paper/2103.04737/) (2021)
- [Regularized Discrete Optimal Transport](https://mlsanity.com/paper/1307.5551/) (2013)
- [Learning Generative Models across Incomparable Spaces](https://mlsanity.com/paper/1905.05461/) (2019)
- [Efficient random graph matching via degree profiles](https://mlsanity.com/paper/1811.07821/) (2018)
- [Statistical Optimal Transport via Factored Couplings](https://mlsanity.com/paper/1806.07348/) (2018)
- [Low-rank Optimal Transport: Approximation, Statistics and Debiasing](https://mlsanity.com/paper/2205.12365/) (2022)
- [Bayesian cluster analysis: Point estimation and credible balls](https://mlsanity.com/paper/1505.03339/) (2015)
- [On the existence of Monge maps for the Gromov-Wasserstein problem](https://mlsanity.com/paper/2210.11945/) (2022)
- [MREC: a fast and versatile framework for aligning and matching point clouds with applications to single cell molecular data](https://mlsanity.com/paper/2001.01666/) (2020)
- [Reversible Gromov-Monge Sampler for Simulation-Based Inference](https://mlsanity.com/paper/2109.14090/) (2021)
- [Alignment of Density Maps in Wasserstein Distance](https://mlsanity.com/paper/2305.12310/) (2023)
- [AlignOT: An optimal transport based algorithm for fast 3D alignment with applications to cryogenic electron microscopy density maps](https://mlsanity.com/paper/2210.09361/) (2022)
- [Statistical Significance of Clustering using Soft Thresholding](https://mlsanity.com/paper/1305.5879/) (2013)
- [Inference for Clustering: Conformal Sets for Cluster Labels](https://mlsanity.com/paper/2604.03488/) (2026)
- [A Convexified Matching Approach to Imputation and Individualized Inference](https://mlsanity.com/paper/2407.05372/) (2024)
- [Optimal Transport for Brain-Image Alignment: Unveiling Redundancy and Synergy in Neural Information Processing](https://mlsanity.com/paper/2503.10663/) (2025)
- [Weighted Conformal Clustering](https://mlsanity.com/paper/2606.00436/) (2026)
- [Robust Point Matching with Distance Profiles](https://mlsanity.com/paper/2312.12641/) (2023)

## Abstract (arXiv)

In many applications of matching, the point clouds to be matched are not merely unstructured sets of points but rather samples from distributions with an intrinsic cluster structure. In such cases, as individual points are often interchangeable within a coherent region, finding a robust region-to-region alignment is more desirable than establishing a precise point-to-point correspondence. To this end, we propose a novel approach for cluster-aware matching based on Laplacian Optimal Transport (LapOT). The key idea is to regularize the optimal transport problem with quadratic Laplacian terms constructed from similarity graphs of the point clouds, which encourages the optimal coupling to respect the cluster structure of both point sets. We also introduce Refined Simultaneous Clustering (RSC), a method that leverages the cluster-aware coupling obtained from LapOT to produce consistent partitions across the point sets, which can overcome the limitations of independent clustering and yield more stable and interpretable results. We demonstrate the effectiveness of our approach through theoretical analysis and empirical experiments, showing that LapOT indeed produces cluster-aware matching that leads to more consistent and meaningful alignments between point clouds.