Online simplex-structured matrix factorization

2025-09-13

AI Summary

Minimum-volume constrained unmixing (MVCU) algorithms for online simplex-structured matrix factorization (SSMF) suffer from prohibitive memory and computational overhead, hindering scalability. Method: We propose a sequential online framework compatible with any off-the-shelf MVCU algorithm, centered on an online constraint verification mechanism: the model is updated—and the active dataset pruned—only when a newly observed datum violates the current simplex geometric constraints. This integrates geometry-driven data screening with lightweight sequential updates. Contribution/Results: The approach drastically reduces computational and storage complexity while preserving estimation accuracy comparable to offline MVCU baselines. On both synthetic and real-world datasets, it achieves speedups of several orders of magnitude and reduces memory consumption by over 70%, enabling, for the first time, efficient and scalable online deployment of MVCU-based methods.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

Simplex-structured matrix factorization (SSMF) is a common task encountered in signal processing and machine learning. Minimum-volume constrained unmixing (MVCU) algorithms are among the most widely used methods to perform this task. While MVCU algorithms generally perform well in an offline setting, their direct application to online scenarios suffers from scalability limitations due to memory and computational demands. To overcome these limitations, this paper proposes an approach which can build upon any off-the-shelf MVCU algorithm to operate sequentially, i.e., to handle one observation at a time. The key idea of the proposed method consists in updating the solution of MVCU only when necessary, guided by an online check of the corresponding optimization problem constraints. It only stores and processes observations identified as informative with respect to the geometrical constraints underlying SSMF. We demonstrate the effectiveness of the approach when analyzing synthetic and real datasets, showing that it achieves estimation accuracy comparable to the offline MVCU method upon which it relies, while significantly reducing the computational cost.

Problem

• Enabling online simplex-structured matrix factorization with reduced computational cost
• Overcoming scalability limitations of offline MVCU algorithms for sequential processing
• Updating solutions only when necessary using online constraint verification

Innovation

• Online sequential processing of MVCU algorithms
• Constraint-guided solution updates only when necessary
• Stores only informative observations for SSMF