# Robust blind unmixing: A geometric approach to overcoming basis variation

arXiv: https://arxiv.org/abs/2610.04091 · HTML: https://mlsanity.com/paper/2610.04091/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 54 references that are in the map.
- First posted on arXiv 2 Oct 2026.

## Builds on (most cited first, 10 of 10)

- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Multilevel Clustering via Wasserstein Means](https://mlsanity.com/paper/1706.03883/) (2017)
- [Spectral Variability in Hyperspectral Data Unmixing: A Comprehensive Review](https://mlsanity.com/paper/2001.07307/) (2020)
- [Wasserstein Dictionary Learning: Optimal Transport-based unsupervised non-linear dictionary learning](https://mlsanity.com/paper/1708.01955/) (2017)
- [Sliced Optimal Partial Transport](https://mlsanity.com/paper/2212.08049/) (2022)
- [Optimal spectral transportation with application to music transcription](https://mlsanity.com/paper/1609.09799/) (2016)
- [Optimal Transport for Machine Learners](https://mlsanity.com/paper/2505.06589/) (2025)
- [A Gaussian mixture model representation of endmember variability in hyperspectral unmixing](https://mlsanity.com/paper/1710.00075/) (2017)
- [A laboratory-created dataset with ground-truth for hyperspectral unmixing evaluation](https://mlsanity.com/paper/1902.08347/) (2019)
- [Pattern Decomposition with Complex Combinatorial Constraints: Application to Materials Discovery](https://mlsanity.com/paper/1411.7441/) (2014)

## Abstract (arXiv)

Signal separation problems are common in science. A prominent example of this occurs during the use of diffraction or spectroscopy to identify the individual components of a mixture by measuring it. In the simplest case, the measured signal is a linear combination of basis patterns corresponding to the constituent parts. The unmixing problem is to infer all or some of these basis patterns and abundances of components from measurements of distinct mixtures. One of the core challenges of this task is the variation of the basis from mixture to mixture due to noise and the exact physics of the measurement process. This is usually addressed with tailored model-based and parametric methods that are then limited in use to specific application domains by the nature of the assumptions made. We propose a novel geometric approach to unmixing problems which views the generation of data during measurement through a metric space lens, thereby shifting the focus from parametrised models to a general relationship between basis transformations and the corresponding geometry. We take advantage of the optimal transport distances to capture commonly occurring basis variations, and use minimisation of in-class variance of candidate solutions to drive the optimisation. We pay special attention to the one-dimensional case due to its practical importance and availability of efficient distance and transport map routines. The effectiveness of our approach is demonstrated on a range of unmixing tasks using random Gaussian mixture models, simulated powder X-ray diffraction, and laboratory hyperspectral imaging datasets.