# On Statistical Learning of Simplices: Unmixing Problem Revisited

arXiv: https://arxiv.org/abs/1810.07845 · HTML: https://mlsanity.com/paper/1810.07845/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Oct 2018.

## Built on by (most cited first, 5 of 5)

- [Sample Complexity Bounds for Learning High-dimensional Simplices in Noisy Regimes](https://mlsanity.com/paper/2209.05953/) (2022)
- [Fundamental Limits of Learning High-dimensional Simplices in Noisy Regimes](https://mlsanity.com/paper/2506.10101/) (2025)
- [Robust Learnability of Sample-Compressible Distributions under Noisy or Adversarial Perturbations](https://mlsanity.com/paper/2506.06613/) (2025)
- [Robustness of Minimum-Volume Nonnegative Matrix Factorization under an Expanded Sufficiently Scattered Condition](https://mlsanity.com/paper/2511.04291/) (2025)
- [Scalable Minimum-Volume Simplex Estimation with Non-asymptotic Analysis](https://mlsanity.com/paper/2609.25576/) (2026)

## Abstract (arXiv)

We study the sample complexity of learning a high-dimensional simplex from a set of points uniformly sampled from its interior. Learning of simplices is a long studied problem in computer science and has applications in computational biology and remote sensing, mostly under the name of `spectral unmixing'. We theoretically show that a sufficient sample complexity for reliable learning of a K-dimensional simplex up to a total-variation error of ε is Oleft(fracK2εlogfracKεright), which yields a substantial improvement over existing bounds. Based on our new theoretical framework, we also propose a heuristic approach for the inference of simplices. Experimental results on synthetic and real-world datasets demonstrate a comparable performance for our method on noiseless samples, while we outperform the state-of-the-art in noisy cases.