# Successive Projection Algorithm Robust to Outliers

## TL;DR

This paper introduces Robust SPA, an improved version of the successive projection algorithm, which is more resilient to outliers and considers data fitting, enhancing nonnegative matrix factorization applications like hyperspectral unmixing.

## Contribution

The paper proposes Robust SPA (RSPA), a novel variant of SPA that handles outliers effectively and incorporates data fitting into index selection, with proven robustness in low-noise conditions.

## Key findings

- RSPA outperforms SPA in the presence of outliers.
- RSPA maintains robustness in low-noise environments.
- Effective on synthetic and hyperspectral data.

## Abstract

The successive projection algorithm (SPA) is a fast algorithm to tackle separable nonnegative matrix factorization (NMF). Given a nonnegative data matrix $X$, SPA identifies an index set $\mathcal{K}$ such that there exists a nonnegative matrix $H$ with $X \approx X(:,\mathcal{K})H$. SPA has been successfully used as a pure-pixel search algorithm in hyperspectral unmixing and for anchor word selection in document classification. Moreover, SPA is provably robust in low-noise settings. The main drawbacks of SPA are that it is not robust to outliers and does not take the data fitting term into account when selecting the indices in $\mathcal{K}$. In this paper, we propose a new SPA variant, dubbed Robust SPA (RSPA), that is robust to outliers while still being provably robust in low-noise settings, and that takes into account the reconstruction error for selecting the indices in $\mathcal{K}$. We illustrate the effectiveness of RSPA on synthetic data sets and hyperspectral images.