# The Cumulative Distribution Transform and Linear Pattern Classification

arXiv: https://arxiv.org/abs/1507.05936 · HTML: https://mlsanity.com/paper/1507.05936/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Jul 2015.

## Built on by (most cited first, 19 of 19)

- [Wassmap: Wasserstein Isometric Mapping for Image Manifold Learning](https://mlsanity.com/paper/2204.06645/) (2022)
- [Supervised learning of sheared distributions using linearized optimal transport](https://mlsanity.com/paper/2201.10590/) (2022)
- [Linearized Wasserstein dimensionality reduction with approximation guarantees](https://mlsanity.com/paper/2302.07373/) (2023)
- [The Signed Cumulative Distribution Transform for 1-D Signal Analysis and Classification](https://mlsanity.com/paper/2106.02146/) (2021)
- [SLOSH: Set LOcality Sensitive Hashing via Sliced-Wasserstein Embeddings](https://mlsanity.com/paper/2112.05872/) (2021)
- [Invariance encoding in sliced-Wasserstein space for image classification with limited training data](https://mlsanity.com/paper/2201.02980/) (2022)
- [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)
- [Set Representation Learning with Generalized Sliced-Wasserstein Embeddings](https://mlsanity.com/paper/2103.03892/) (2021)
- [Local Sliced-Wasserstein Feature Sets for Illumination-invariant Face Recognition](https://mlsanity.com/paper/2202.10642/) (2022)
- [LCOT: Linear circular optimal transport](https://mlsanity.com/paper/2310.06002/) (2023)
- [Approximation properties of slice-matching operators](https://mlsanity.com/paper/2310.10869/) (2023)
- [Normalized Radon Cumulative Distribution Transforms for Invariance and Robustness in Optimal Transport Based Image Classification](https://mlsanity.com/paper/2506.08761/) (2025)
- [Resistance Distance and Linearized Optimal Transport on Graphs](https://mlsanity.com/paper/2404.15261/) (2024)
- [Signed Cumulative Distribution Transform for Parameter Estimation of 1-D Signals](https://mlsanity.com/paper/2207.07989/) (2022)
- [The Radon Signed Cumulative Distribution Transform and its applications in classification of Signed Images](https://mlsanity.com/paper/2307.15339/) (2023)
- [Point Cloud Classification via Deep Set Linearized Optimal Transport](https://mlsanity.com/paper/2401.01460/) (2024)
- [Redistributor: Transforming Empirical Data Distributions](https://mlsanity.com/paper/2210.14219/) (2022)
- [Generalizations of the Normalized Radon Cumulative Distribution Transform for Limited Data Recognition](https://mlsanity.com/paper/2512.08099/) (2025)

## Abstract (arXiv)

Discriminating data classes emanating from sensors is an important problem with many applications in science and technology. We describe a new transform for pattern identification that interprets patterns as probability density functions, and has special properties with regards to classification. The transform, which we denote as the Cumulative Distribution Transform (CDT) is invertible, with well defined forward and inverse operations. We show that it can be useful in `parsing out' variations (confounds) that are `Lagrangian' (displacement and intensity variations) by converting these to `Eulerian' (intensity variations) in transform space. This conversion is the basis for our main result that describes when the CDT can allow for linear classification to be possible in transform space. We also describe several properties of the transform and show, with computational experiments that used both real and simulated data, that the CDT can help render a variety of real world problems simpler to solve.