# Perturbation Analysis of Singular Values in Concatenated Matrices

arXiv: https://arxiv.org/abs/2505.01427 · HTML: https://mlsanity.com/paper/2505.01427/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 2 of its 14 references that are in the map.
- First posted on arXiv 11 Mar 2025.

## Builds on (most cited first, 2 of 2)

- [Communication-Efficient Learning of Deep Networks from Decentralized Data](https://mlsanity.com/paper/1602.05629/) (2016)
- [Advances and Open Problems in Federated Learning](https://mlsanity.com/paper/1912.04977/) (2019)

## Built on by (most cited first, 2 of 2)

- [Decom-Renorm-Merge: Model Merging on the Right Space Improves Multitasking](https://mlsanity.com/paper/2505.23117/) (2025)
- [Eigenspace-Based Clustering for Personalized System Identification](https://mlsanity.com/paper/2606.20811/) (2026)

## Abstract (arXiv)

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.