# The Signed Cumulative Distribution Transform for 1-D Signal Analysis and Classification

arXiv: https://arxiv.org/abs/2106.02146 · HTML: https://mlsanity.com/paper/2106.02146/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 49 references that are in the map.
- First posted on arXiv 3 Jun 2021.

## Builds on (most cited first, 13 of 13)

- [Optimal Transport for Domain Adaptation](https://mlsanity.com/paper/1507.00504/) (2015)
- [Generalized Sliced Wasserstein Distances](https://mlsanity.com/paper/1902.00434/) (2019)
- [Sliced Wasserstein Kernels for Probability Distributions](https://mlsanity.com/paper/1511.03198/) (2015)
- [The Radon cumulative distribution transform and its application to image classification](https://mlsanity.com/paper/1511.03206/) (2015)
- [The Cumulative Distribution Transform and Linear Pattern Classification](https://mlsanity.com/paper/1507.05936/) (2015)
- [Linearized Optimal Transport for Collider Events](https://mlsanity.com/paper/2008.08604/) (2020)
- [Unnormalized Optimal Transport](https://mlsanity.com/paper/1902.03367/) (2019)
- [Partitioning signal classes using transport transforms for data analysis and machine learning](https://mlsanity.com/paper/2008.03452/) (2020)
- [Radon cumulative distribution transform subspace modeling for image classification](https://mlsanity.com/paper/2004.03669/) (2020)
- [A Transportation Lp Distance for Signal Analysis](https://mlsanity.com/paper/1609.08669/) (2016)
- [The Linearized Hellinger--Kantorovich Distance](https://mlsanity.com/paper/2102.08807/) (2021)
- [Unbalanced Optimal Transport: Dynamic and Kantorovich Formulation](https://mlsanity.com/paper/1508.05216/) (2015)
- [Discovery and visualization of structural biomarkers from MRI using transport-based morphometry](https://mlsanity.com/paper/1705.04919/) (2017)

## Built on by (most cited first, 9 of 9)

- [End-to-End Signal Classification in Signed Cumulative Distribution Transform Space](https://mlsanity.com/paper/2205.00348/) (2022)
- [LCOT: Linear circular optimal transport](https://mlsanity.com/paper/2310.06002/) (2023)
- [Normalized Radon Cumulative Distribution Transforms for Invariance and Robustness in Optimal Transport Based Image Classification](https://mlsanity.com/paper/2506.08761/) (2025)
- [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)
- [Geodesic Properties of a Generalized Wasserstein Embedding for Time Series Analysis](https://mlsanity.com/paper/2206.01984/) (2022)
- [An Efficient Transport-Based Dissimilarity Measure for Time Series Classification under Warping Distortions](https://mlsanity.com/paper/2505.05676/) (2025)
- [Improved Hyperspectral Anomaly Detection via Unsupervised Subspace Modeling in the Signed Cumulative Distribution Transform Domain](https://mlsanity.com/paper/2510.00148/) (2025)
- [Generalizations of the Normalized Radon Cumulative Distribution Transform for Limited Data Recognition](https://mlsanity.com/paper/2512.08099/) (2025)

## Abstract (arXiv)

This paper presents a new mathematical signal transform that is especially suitable for decoding information related to non-rigid signal displacements. We provide a measure theoretic framework to extend the existing Cumulative Distribution Transform [ACHA 45 (2018), no. 3, 616-641] to arbitrary (signed) signals on overlineℝ. We present both forward (analysis) and inverse (synthesis) formulas for the transform, and describe several of its properties including translation, scaling, convexity, linear separability and others. Finally, we describe a metric in transform space, and demonstrate the application of the transform in classifying (detecting) signals under random displacements.