# Supervised learning of sheared distributions using linearized optimal transport

arXiv: https://arxiv.org/abs/2201.10590 · HTML: https://mlsanity.com/paper/2201.10590/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 38 references that are in the map.
- First posted on arXiv 25 Jan 2022.

## Builds on (most cited first, 8 of 8)

- [Kernel Mean Embedding of Distributions: A Review and Beyond](https://mlsanity.com/paper/1605.09522/) (2016)
- [Linear Optimal Transport Embedding: Provable Wasserstein classification for certain rigid transformations and perturbations](https://mlsanity.com/paper/2008.09165/) (2020)
- [The Radon cumulative distribution transform and its application to image classification](https://mlsanity.com/paper/1511.03206/) (2015)
- [Quantitative stability of optimal transport maps and linearization of the 2-Wasserstein space](https://mlsanity.com/paper/1910.05954/) (2019)
- [The Cumulative Distribution Transform and Linear Pattern Classification](https://mlsanity.com/paper/1507.05936/) (2015)
- [Partitioning signal classes using transport transforms for data analysis and machine learning](https://mlsanity.com/paper/2008.03452/) (2020)
- [Two-sample Statistics Based on Anisotropic Kernels](https://mlsanity.com/paper/1709.05006/) (2017)
- [People Mover's Distance: Class level geometry using fast pairwise data adaptive transportation costs](https://mlsanity.com/paper/1707.00514/) (2017)

## Built on by (most cited first, 14 of 14)

- [Algorithms for mean-field variational inference via polyhedral optimization in the Wasserstein space](https://mlsanity.com/paper/2312.02849/) (2023)
- [Wassmap: Wasserstein Isometric Mapping for Image Manifold Learning](https://mlsanity.com/paper/2204.06645/) (2022)
- [Linearized Wasserstein dimensionality reduction with approximation guarantees](https://mlsanity.com/paper/2302.07373/) (2023)
- [Applications of No-Collision Transportation Maps in Manifold Learning](https://mlsanity.com/paper/2304.00199/) (2023)
- [Measure transfer via stochastic slicing and matching](https://mlsanity.com/paper/2307.05705/) (2023)
- [Manifold learning in Wasserstein space](https://mlsanity.com/paper/2311.08549/) (2023)
- [Statistical learning on measures: an application to persistence diagrams](https://mlsanity.com/paper/2303.08456/) (2023)
- [Approximation properties of slice-matching operators](https://mlsanity.com/paper/2310.10869/) (2023)
- [Discrete transforms of quantized persistence diagrams](https://mlsanity.com/paper/2312.17093/) (2023)
- [Point Cloud Classification via Deep Set Linearized Optimal Transport](https://mlsanity.com/paper/2401.01460/) (2024)
- [Low dimensional representation of multi-patient flow cytometry datasets using optimal transport for minimal residual disease detection in leukemia](https://mlsanity.com/paper/2407.17329/) (2024)
- [Linearized Optimal Transport for Analysis of High-Dimensional Point-Cloud and Single-Cell Data](https://mlsanity.com/paper/2510.22033/) (2025)
- [Structured Approximations of Measures](https://mlsanity.com/paper/2310.09149/) (2023)
- [Recovering Wasserstein Distance Matrices from Few Measurements](https://mlsanity.com/paper/2509.19250/) (2025)

## Abstract (arXiv)

In this paper we study supervised learning tasks on the space of probability measures. We approach this problem by embedding the space of probability measures into L2 spaces using the optimal transport framework. In the embedding spaces, regular machine learning techniques are used to achieve linear separability. This idea has proved successful in applications and when the classes to be separated are generated by shifts and scalings of a fixed measure. This paper extends the class of elementary transformations suitable for the framework to families of shearings, describing conditions under which two classes of sheared distributions can be linearly separated. We furthermore give necessary bounds on the transformations to achieve a pre-specified separation level, and show how multiple embeddings can be used to allow for larger families of transformations. We demonstrate our results on image classification tasks.