# Measure transfer via stochastic slicing and matching

arXiv: https://arxiv.org/abs/2307.05705 · HTML: https://mlsanity.com/paper/2307.05705/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 68 references that are in the map.
- First posted on arXiv 11 Jul 2023.

## Builds on (most cited first, 19 of 19)

- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Generalized Sliced Wasserstein Distances](https://mlsanity.com/paper/1902.00434/) (2019)
- [Learning with a Wasserstein Loss](https://mlsanity.com/paper/1506.05439/) (2015)
- [Sliced-Wasserstein Flows: Nonparametric Generative Modeling via Optimal Transport and Diffusions](https://mlsanity.com/paper/1806.08141/) (2018)
- [On the representation and learning of monotone triangular transport maps](https://mlsanity.com/paper/2009.10303/) (2020)
- [Sliced Iterative Normalizing Flows](https://mlsanity.com/paper/2007.00674/) (2020)
- [Linear Optimal Transport Embedding: Provable Wasserstein classification for certain rigid transformations and perturbations](https://mlsanity.com/paper/2008.09165/) (2020)
- [Efficient Gradient Flows in Sliced-Wasserstein Space](https://mlsanity.com/paper/2110.10972/) (2021)
- [Sliced Optimal Partial Transport](https://mlsanity.com/paper/2212.08049/) (2022)
- [The Cumulative Distribution Transform and Linear Pattern Classification](https://mlsanity.com/paper/1507.05936/) (2015)
- [Fr'echet Means and Procrustes Analysis in Wasserstein Space](https://mlsanity.com/paper/1701.06876/) (2017)
- [Large-scale optimal transport map estimation using projection pursuit](https://mlsanity.com/paper/2106.05838/) (2021)
- [Partitioning signal classes using transport transforms for data analysis and machine learning](https://mlsanity.com/paper/2008.03452/) (2020)
- [Supervised learning of sheared distributions using linearized optimal transport](https://mlsanity.com/paper/2201.10590/) (2022)
- [Slicing Unbalanced Optimal Transport](https://mlsanity.com/paper/2306.07176/) (2023)
- [Applications of No-Collision Transportation Maps in Manifold Learning](https://mlsanity.com/paper/2304.00199/) (2023)
- [No-collision Transportation Maps](https://mlsanity.com/paper/1912.02317/) (2019)
- [Fast Optimal Transport through Sliced Wasserstein Generalized Geodesics](https://mlsanity.com/paper/2307.01770/) (2023)
- [Stochastic Gradient Descent for Barycenters in Wasserstein Space](https://mlsanity.com/paper/2201.04232/) (2022)

## Built on by (most cited first, 6 of 6)

- [Conditional Optimal Transport on Function Spaces](https://mlsanity.com/paper/2311.05672/) (2023)
- [Approximation properties of slice-matching operators](https://mlsanity.com/paper/2310.10869/) (2023)
- [Towards Understanding Gradient Dynamics of the Sliced-Wasserstein Distance via Critical Point Analysis](https://mlsanity.com/paper/2502.06525/) (2025)
- [Partial Transport for Point-Cloud Registration](https://mlsanity.com/paper/2309.15787/) (2023)
- [Convergence Rates for Distribution Matching with Sliced Optimal Transport](https://mlsanity.com/paper/2602.10691/) (2026)
- [Distribution Steering via Sliced Optimal Transport Control](https://mlsanity.com/paper/2608.12828/) (2026)

## Abstract (arXiv)

This paper studies iterative schemes for measure transfer and approximation problems, which are defined through a slicing-and-matching procedure. Similar to the sliced Wasserstein distance, these schemes benefit from the availability of closed-form solutions for the one-dimensional optimal transport problem and the associated computational advantages. While such schemes have already been successfully utilized in data science applications, not too many results on their convergence are available. The main contribution of this paper is an almost sure convergence proof for stochastic slicing-and-matching schemes. The proof builds on an interpretation as a stochastic gradient descent scheme on the Wasserstein space. Numerical examples on step-wise image morphing are demonstrated as well.