Probabilistic Simplex Component Analysis

TL;DR

This paper introduces PRISM, a probabilistic method for identifying simplex vertices from data, with applications in hyperspectral unmixing and non-negative matrix factorization, demonstrating robustness to noise and strong theoretical foundations.

Contribution

The paper proposes PRISM, a novel probabilistic approach for simplex component analysis, with provable vertex identifiability and connections to geometric and matrix factorization methods.

Findings

PRISM effectively identifies vertices in noisy data.

The variational inference scheme resembles a regularized matrix factorization.

Numerical results show PRISM's potential in practical applications.

Abstract

This study presents PRISM, a probabilistic simplex component analysis approach to identifying the vertices of a data-circumscribing simplex from data. The problem has a rich variety of applications, the most notable being hyperspectral unmixing in remote sensing and non-negative matrix factorization in machine learning. PRISM uses a simple probabilistic model, namely, uniform simplex data distribution and additive Gaussian noise, and it carries out inference by maximum likelihood. The inference model is sound in the sense that the vertices are provably identifiable under some assumptions, and it suggests that PRISM can be effective in combating noise when the number of data points is large. PRISM has strong, but hidden, relationships with simplex volume minimization, a powerful geometric approach for the same problem. We study these fundamental aspects, and we also consider algorithmic schemes based on importance sampling and variational inference. In particular, the variational inference scheme is shown to resemble a matrix factorization problem with a special regularizer, which draws an interesting connection to the matrix factorization approach. Numerical results are provided to demonstrate the potential of PRISM.