On Statistical Learning of Simplices: Unmixing Problem Revisited

TL;DR

This paper investigates the sample complexity of learning high-dimensional simplices from interior points, providing improved theoretical bounds and a heuristic method that performs well especially in noisy conditions.

Contribution

It offers a new theoretical framework that reduces the sample complexity bounds and introduces a heuristic approach for simplex inference.

Findings

Sample complexity bound is $O(rac{K^2}{} ext{log}rac{K}{})$, improving previous results.

The proposed heuristic performs comparably to existing methods on noiseless data.

Our method outperforms state-of-the-art algorithms in noisy scenarios.

Abstract

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 $\epsilon$ is $O\left(\frac{K^2}{\epsilon}\log\frac{K}{\epsilon}\right)$, 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.