June 2021 On statistical learning of simplices: Unmixing problem revisited
Amir Najafi, Saeed Ilchi, Amir Hossein Saberi, Seyed Abolfazl Motahari, Babak H. Khalaj, Hamid R. Rabiee
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Ann. Statist. 49(3): 1626-1655 (June 2021). DOI: 10.1214/20-AOS2016


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 O(K2ϵlogKϵ), 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.


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Amir Najafi. Saeed Ilchi. Amir Hossein Saberi. Seyed Abolfazl Motahari. Babak H. Khalaj. Hamid R. Rabiee. "On statistical learning of simplices: Unmixing problem revisited." Ann. Statist. 49 (3) 1626 - 1655, June 2021. https://doi.org/10.1214/20-AOS2016


Received: 1 April 2020; Revised: 1 August 2020; Published: June 2021
First available in Project Euclid: 9 August 2021

MathSciNet: MR4298875
zbMATH: 1475.62293
Digital Object Identifier: 10.1214/20-AOS2016

Primary: 62F99

Keywords: computer simulations , high-dimensional geometry , Inference of simplices , sample complexity , statistical learning theory

Rights: Copyright © 2021 Institute of Mathematical Statistics


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Vol.49 • No. 3 • June 2021
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