Nonnegative Matrix Factorization for Signal and Data Analytics: Identifiability, Algorithms, and Applications

Abstract

Nonnegative matrix factorization (NMF) has become a workhorse for signal and data analytics, triggered by its model parsimony and interpretability. Perhaps a bit surprisingly, the understanding to its model identifiability---the major reason behind the interpretability in many applications such as topic mining and hyperspectral imaging---had been rather limited until recent years. Beginning from the 2010s, the identifiability research of NMF has progressed considerably: Many interesting and important results have been discovered by the signal processing (SP) and machine learning (ML) communities. NMF identifiability has a great impact on many aspects in practice, such as ill-posed formulation avoidance and performance-guaranteed algorithm design. On the other hand, there is no tutorial paper that introduces NMF from an identifiability viewpoint. In this paper, we aim at filling this gap by offering a comprehensive and deep tutorial on model identifiability of NMF as well as the connections to algorithms and applications. This tutorial will help researchers and graduate students grasp the essence and insights of NMF, thereby avoiding typical `pitfalls' that are often times due to unidentifiable NMF formulations. This paper will also help practitioners pick/design suitable factorization tools for their own problems.

accepted version, IEEE Signal Processing Magazine; supplementary materials added. Some minor revisions implemented

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• Algorithms for Nonnegative Matrix Factorization with the Kullback-Leibler Divergence
• Generalized Separable Nonnegative Matrix Factorization
• Computing Large-Scale Matrix and Tensor Decomposition with Structured Factors: A Unified Nonconvex Optimization Perspective
• A consistent and flexible framework for deep matrix factorizations
• Quaternion Non-negative Matrix Factorization: definition, uniqueness and algorithm
• Non-negative tensor factorization for vibration-based local damage detection
• Detecting Central Nodes from Low-rank Excited Graph Signals via Structured Factor Analysis
• Global Mapping of the Surface Composition on an Exo-Earth using Color Variability
• Simplex-Structured Matrix Factorization: Sparsity-based Identifiability and Provably Correct Algorithms
• Nonlinear Multiview Analysis: Identifiability and Neural Network-assisted Implementation
• Recovering Joint Probability of Discrete Random Variables from Pairwise Marginals
• Multi-Resolution Beta-Divergence NMF for Blind Spectral Unmixing
• Probabilistic Simplex Component Analysis
• Polytopic Matrix Factorization: Determinant Maximization Based Criterion and Identifiability
• Multiplicative Updates for NMF with $\beta$-Divergences under Disjoint Equality Constraints
• Majorization-minimization for Sparse Nonnegative Matrix Factorization with the $\beta$-divergence
• Accelerating Block Coordinate Descent for Nonnegative Tensor Factorization
• Learning Nonlinear Mixtures: Identifiability and Algorithm
• Memory-Efficient Convex Optimization for Self-Dictionary Separable Nonnegative Matrix Factorization: A Frank-Wolfe Approach
• Multi-block Bregman proximal alternating linearized minimization and its application to orthogonal nonnegative matrix factorization
• Partial Identifiability for Nonnegative Matrix Factorization
• Multiple Kernel Representation Learning on Networks
• Near-Convex Archetypal Analysis
• Mixed Membership Graph Clustering via Systematic Edge Query
• Identifiability-Guaranteed Simplex-Structured Post-Nonlinear Mixture Learning via Autoencoder
• Deep Nonnegative Matrix Factorization with Beta Divergences
• Least-squares methods for nonnegative matrix factorization over rational functions
• Sketching for low-rank nonnegative matrix approximation: Numerical study
• Joint Majorization-Minimization for Nonnegative Matrix Factorization with the $\beta$-divergence
• Symmetric Nonnegative Matrix Trifactorization
• Provably robust blind source separation of linear-quadratic near-separable mixtures
• Identifiability of Complete Dictionary Learning
• Bounded Simplex-Structured Matrix Factorization: Algorithms, Identifiability and Applications
• Global Mapping of Surface Composition on an Exo-Earth Using Sparse Modeling
• Refinement of Hottopixx Method for Nonnegative Matrix Factorization Under Noisy Separability
• Checking the Sufficiently Scattered Condition using a Global Non-Convex Optimization Software
• Dual Simplex Volume Maximization for Simplex-Structured Matrix Factorization
• Majorization-minimization Bregman proximal gradient algorithms for NMF with the Kullback--Leibler divergence
• Conic-Optimization Based Algorithms for Nonnegative Matrix Factorization
• Leveraging Joint-Diagonalization in Transform-Learning NMF
• Probabilistic Simplex Component Analysis by Importance Sampling
• Endmember Extraction from Hyperspectral Images Using Self-Dictionary Approach with Linear Programming