Perturbation Analysis of Singular Values in Concatenated Matrices

TL;DR

This paper develops a perturbation analysis framework for singular values of concatenated matrices, providing bounds on their stability and implications for matrix clustering and compression.

Contribution

It extends classical perturbation results to concatenated matrices, offering analytical bounds for singular value stability under small perturbations.

Findings

Singular values of concatenated matrices are stable under small perturbations of submatrices.

The bounds enable controlled trade-offs between accuracy and compression.

Results have applications in matrix clustering, compression, and data modeling.

Abstract

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