Dual Simplex Volume Maximization for Simplex-Structured Matrix Factorization

Abstract

Simplex-structured matrix factorization (SSMF) is a generalization of nonnegative matrix factorization, a fundamental interpretable data analysis model, and has applications in hyperspectral unmixing and topic modeling. To obtain identifiable solutions, a standard approach is to find minimum-volume solutions. By taking advantage of the duality/polarity concept for polytopes, we convert minimum-volume SSMF in the primal space to a maximum-volume problem in the dual space. We first prove the identifiability of this maximum-volume dual problem. Then, we use this dual formulation to provide a novel optimization approach which bridges the gap between two existing families of algorithms for SSMF, namely volume minimization and facet identification. Numerical experiments show that the proposed approach performs favorably compared to the state-of-the-art SSMF algorithms.

31 pages, 10 figures

References in corpus (6)

• Nonnegative Matrix Factorization for Signal and Data Analytics: Identifiability, Algorithms, and Applications
• A Fast Hyperplane-Based Minimum-Volume Enclosing Simplex Algorithm for Blind Hyperspectral Unmixing
• Computing Large-Scale Matrix and Tensor Decomposition with Structured Factors: A Unified Nonconvex Optimization Perspective
• On the Complexity of Robust PCA and ${\ell }_1$-norm Low-Rank Matrix Approximation
• Simplex-Structured Matrix Factorization: Sparsity-based Identifiability and Provably Correct Algorithms
• Polytopic Matrix Factorization: Determinant Maximization Based Criterion and Identifiability

Cited by in corpus (1)

• Adaptive Multi-Order Graph Regularized NMF with Dual Sparsity for Hyperspectral Unmixing