# Algorithms and Comparisons of Non-negative Matrix Factorization with Volume Regularization for Hyperspectral Unmixing

## TL;DR

This paper introduces efficient algorithms for volume-regularized non-negative matrix factorization tailored for hyperspectral unmixing, demonstrating superior performance over existing methods especially in highly mixed or less separable data scenarios.

## Contribution

The work develops and compares new algorithms for volume-regularized NMF, improving endmember recovery in hyperspectral unmixing with real-world applicability.

## Key findings

- Outperforms state-of-the-art volume-regularized NMF methods.
- Effective in scenarios with highly mixed endmembers.
- Different regularizers perform best depending on data separability.

## Abstract

In this work, we consider nonnegative matrix factorization (NMF) with a regularization that promotes small volume of the convex hull spanned by the basis matrix. We present highly efficient algorithms for three different volume regularizers, and compare them on endmember recovery in hyperspectral unmixing. The NMF algorithms developed in this work are shown to outperform the state-of-the-art volume-regularized NMF methods, and produce meaningful decompositions on real-world hyperspectral images in situations where endmembers are highly mixed (no pure pixels). Furthermore, our extensive numerical experiments show that when the data is highly separable, meaning that there are data points close to the true endmembers, and there are a few endmembers, the regularizer based on the determinant of the Gramian produces the best results in most cases. For data that is less separable and/or contains more endmembers, the regularizer based on the logarithm of the determinant of the Gramian performs best in general.