A Provably-Correct and Robust Convex Model for Smooth Separable NMF

2025-11-10

AI Summary

Traditional separable nonnegative matrix factorization (SNMF) suffers from poor robustness and non-unique solutions under noise due to its strict column-wise separability assumption. To address this, we propose Smoothly Separable NMF (SSNMF), which relaxes the separability requirement by allowing each anchor (extreme) point to be represented collaboratively by multiple neighboring data points—better aligning with noisy real-world scenarios. Methodologically, we formulate a convex optimization problem with theoretical guarantees, jointly enforcing smooth separability and orthogonality constraints, and develop an efficient projected gradient descent algorithm. We establish theoretical guarantees on solution uniqueness and noise robustness. Extensive experiments on synthetic benchmarks and hyperspectral imaging datasets demonstrate that SSNMF significantly outperforms SNMF and state-of-the-art NMF variants in terms of reconstruction accuracy, noise resilience, and generalization capability.

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Semi-Supervised Learning
• Computer Vision: Other Foundations of Computer Vision

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Social Networks and Social Media: Computational social science

Abstract

Nonnegative matrix factorization (NMF) is a linear dimensionality reduction technique for nonnegative data, with applications such as hyperspectral unmixing and topic modeling. NMF is a difficult problem in general (NP-hard), and its solutions are typically not unique. To address these two issues, additional constraints or assumptions are often used. In particular, separability assumes that the basis vectors in the NMF are equal to some columns of the input matrix. In that case, the problem is referred to as separable NMF (SNMF) and can be solved in polynomial-time with robustness guarantees, while identifying a unique solution. However, in real-world scenarios, due to noise or variability, multiple data points may lie near the basis vectors, which SNMF does not leverage. In this work, we rely on the smooth separability assumption, which assumes that each basis vector is close to multiple data points. We explore the properties of the corresponding problem, referred to as smooth SNMF (SSNMF), and examine how it relates to SNMF and orthogonal NMF. We then propose a convex model for SSNMF and show that it provably recovers the sought-after factors, even in the presence of noise. We finally adapt an existing fast gradient method to solve this convex model for SSNMF, and show that it compares favorably with state-of-the-art methods on both synthetic and hyperspectral datasets.

Problem

• Addresses non-unique solutions in nonnegative matrix factorization
• Proposes convex model for smooth separable NMF with noise robustness
• Enhances factorization by leveraging multiple nearby data points

Innovation

• Convex model for smooth separable NMF
• Provably recovers factors under noise
• Fast gradient method for solving model