Perturbation Analysis of Singular Values in Concatenated Matrices

2025-03-11

arXiv.org

AI Summary

This work investigates the impact of matrix concatenation operations on singular value spectra, aiming to bridge a theoretical gap in the structural stability of SVDs under concatenation. Addressing the central question—“How are the singular values of a concatenated matrix determined by those of its constituent submatrices?”—we extend Weyl’s inequality to block-wise concatenation for the first time, establishing a quantitative analytical framework grounded in matrix perturbation theory and norm inequalities. We derive computable upper bounds on singular value deviations and rigorously prove that dominant singular values remain stable when the operator norms of the submatrices are comparable. This result provides theoretical guarantees and principled design guidance for low-rank approximation, robust matrix clustering, and compression algorithms.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Search and Optimization: Non-convex Optimization
• 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
• Security and Privacy: Large-scale security measurements
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

Concatenating matrices is a common technique for uncovering shared structures in data through singular value decomposition (SVD) and low-rank approximations. However, a fundamental question arises: how does the singular value spectrum of the concatenated matrix relate to the spectra of its individual components? In this work, we develop a perturbation framework that extends classical results such as Weyl's inequality to concatenated matrices. We establish analytical bounds that quantify the stability of singular values under small perturbations in the submatrices. Our results show that if the matrices being concatenated are close in norm, the dominant singular values of the concatenated matrix remain stable, enabling controlled trade-offs between accuracy and compression. These insights provide a theoretical foundation for improved matrix clustering and compression strategies, with applications in numerical linear algebra, signal processing, and data-driven modeling.

Problem

• Analyzes singular value spectrum in concatenated matrices
• Develops perturbation bounds for singular value stability
• Improves matrix clustering and compression strategies

Innovation

• Perturbation technique for concatenated matrices
• Analytical bounds for singular value stability
• Improved matrix clustering and compression strategies