# Fast and Robust Comparison of Probability Measures in Heterogeneous Spaces

arXiv: https://arxiv.org/abs/2002.01615 · HTML: https://mlsanity.com/paper/2002.01615/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Feb 2020.

## Built on by (most cited first, 17 of 17)

- [Linear-Time Gromov Wasserstein Distances using Low Rank Couplings and Costs](https://mlsanity.com/paper/2106.01128/) (2021)
- [Sliced Optimal Partial Transport](https://mlsanity.com/paper/2212.08049/) (2022)
- [Quantized Gromov-Wasserstein](https://mlsanity.com/paper/2104.02013/) (2021)
- [Efficient Approximation of Gromov-Wasserstein Distance Using Importance Sparsification](https://mlsanity.com/paper/2205.13573/) (2022)
- [Re-evaluating Word Mover's Distance](https://mlsanity.com/paper/2105.14403/) (2021)
- [A Novel Sliced Fused Gromov-Wasserstein Distance](https://mlsanity.com/paper/2508.02364/) (2025)
- [Unbalanced Low-rank Optimal Transport Solvers](https://mlsanity.com/paper/2305.19727/) (2023)
- [Slicing Wasserstein Over Wasserstein Via Functional Optimal Transport](https://mlsanity.com/paper/2509.22138/) (2025)
- [Feedback Schr"odinger Bridge Matching](https://mlsanity.com/paper/2410.14055/) (2024)
- [Generalized Wasserstein Flow Matching: Transport Plans, Everywhere, All at Once](https://mlsanity.com/paper/2605.08424/) (2026)
- [Partial Transport for Point-Cloud Registration](https://mlsanity.com/paper/2309.15787/) (2023)
- [Fast Gradient Computation for Gromov-Wasserstein Distance](https://mlsanity.com/paper/2404.08970/) (2024)
- [Budget-Constrained Bounds for Mini-Batch Estimation of Optimal Transport](https://mlsanity.com/paper/2210.13630/) (2022)
- [Keypoint-Guided Optimal Transport: Models, Algorithms, and Applications](https://mlsanity.com/paper/2303.13102/) (2023)
- [Shape-of-You: Fused Gromov-Wasserstein Optimal Transport for Semantic Correspondence in-the-Wild](https://mlsanity.com/paper/2603.11618/) (2026)
- [Gromov-Wasserstein and optimal transport: from assignment problems to probabilistic numeric](https://mlsanity.com/paper/2509.04089/) (2025)
- [Reliable Modeling of Distribution Shifts via Displacement-Reshaped Optimal Transport](https://mlsanity.com/paper/2605.04965/) (2026)

## Abstract (arXiv)

Comparing two probability measures supported on heterogeneous spaces is an increasingly important problem in machine learning. Such problems arise when comparing for instance two populations of biological cells, each described with its own set of features, or when looking at families of word embeddings trained across different corpora/languages. For such settings, the Gromov Wasserstein (GW) distance is often presented as the gold standard. GW is intuitive, as it quantifies whether one measure can be isomorphically mapped to the other. However, its exact computation is intractable, and most algorithms that claim to approximate it remain expensive. Building on citememoli-2011, who proposed to represent each point in each distribution as the 1D distribution of its distances to all other points, we introduce in this paper the Anchor Energy (AE) and Anchor Wasserstein (AW) distances, which are respectively the energy and Wasserstein distances instantiated on such representations. Our main contribution is to propose a sweep line algorithm to compute AE exactly in log-quadratic time, where a naive implementation would be cubic. This is quasi-linear w.r.t. the description of the problem itself. Our second contribution is the proposal of robust variants of AE and AW that uses rank statistics rather than the original distances. We show that AE and AW perform well in various experimental settings at a fraction of the computational cost of popular GW approximations. Code is available at urlhttps://github.com/joisino/anchor-energy.