# Recovering missing features in nonnegative matrix factorization via generalized singular value decomposition

## TL;DR

This paper introduces a new method to recover missing components in nonnegative matrix factorization using generalized singular value decomposition, improving efficiency and accuracy.

## Contribution

GSVD-NMF efficiently recovers missing components in under-complete NMF without rerunning from scratch, using generalized singular value decomposition.

## Key findings

- GSVD-NMF effectively recovers multiple missing components in under-complete NMF.
- The method reaches better local optima and is compatible with various NMF algorithms.
- Efficient rank expansion is achieved by augmenting components rather than rerunning NMF.

## Abstract

Nonnegative matrix factorization (NMF) is widely used to separate mixed sources into components. Algorithms for NMF require choosing the rank in advance, and if the results are unsatisfying, one typically executes NMF again with a different rank. To make NMF more interactive, here we introduce GSVD-NMF, a method that proposes new components based on the generalized singular value decomposition (GSVD) to address discrepancies between initial under-complete NMF results and the SVD of the original matrix. Simulation and experimental results demonstrate that GSVD-NMF often effectively recovers multiple missing components in under-complete NMF, with the recovered NMF solutions frequently reaching better local optima. The results further show that GSVD-NMF is compatible with various NMF algorithms and that directly augmenting components is more efficient than rerunning NMF from scratch with additional components. Furthermore, the under-complete NMF can be computed with a relaxed convergence tolerance, greatly reducing runtime while still enabling accurate feature recovery.

•Recovers missing components from under-complete nonnegative matrix factorization•Uses generalized singular value decomposition to identify discrepant directions•Enables efficient rank expansion without rerunning NMF from scratch•Improves solution quality and efficiency across multiple NMF algorithms

Recovers missing components from under-complete nonnegative matrix factorization

Uses generalized singular value decomposition to identify discrepant directions

Enables efficient rank expansion without rerunning NMF from scratch

Improves solution quality and efficiency across multiple NMF algorithms

Applied sciences; Network