Robust Volume Minimization-Based Matrix Factorization for Remote Sensing and Document Clustering

TL;DR

This paper advances volume minimization-based matrix factorization by providing theoretical insights and developing a robust, computationally efficient algorithm that handles outliers, demonstrated on remote sensing and document clustering tasks.

Contribution

It proves the equivalence of two key conditions for VolMin identifiability and introduces a new algorithm that improves robustness and efficiency in practical applications.

Findings

Theoretical proof of equivalence of VolMin identifiability conditions.

A new algorithm that handles volume regularization efficiently.

Effective application demonstrated on hyperspectral images and document data.

Abstract

This paper considers \emph{volume minimization} (VolMin)-based structured matrix factorization (SMF). VolMin is a factorization criterion that decomposes a given data matrix into a basis matrix times a structured coefficient matrix via finding the minimum-volume simplex that encloses all the columns of the data matrix. Recent work showed that VolMin guarantees the identifiability of the factor matrices under mild conditions that are realistic in a wide variety of applications. This paper focuses on both theoretical and practical aspects of VolMin. On the theory side, exact equivalence of two independently developed sufficient conditions for VolMin identifiability is proven here, thereby providing a more comprehensive understanding of this aspect of VolMin. On the algorithm side, computational complexity and sensitivity to outliers are two key challenges associated with real-world applications of VolMin. These are addressed here via a new VolMin algorithm that handles volume regularization in a computationally simple way, and automatically detects and {iteratively downweights} outliers, simultaneously. Simulations and real-data experiments using a remotely sensed hyperspectral image and the Reuters document corpus are employed to showcase the effectiveness of the proposed algorithm.