Sparse Pseudospectral Shattering

[2411.19926] Sparse Pseudospectral Shattering Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Probability arXiv:2411.19926 (math) [Submitted on 29 Nov 2024 ( v1 ), last revised 10 Apr 2026 (this version, v3)] Title: Sparse Pseudospectral Shattering Authors: Rikhav Shah , Nikhil Srivastava , Edward Zeng View a PDF of the paper titled Sparse Pseudospectral Shattering, by Rikhav Shah and 2 other authors View PDF HTML (experimental) Abstract: The eigenvalues and eigenvectors of nonnormal matrices can be unstable under perturbations of their entries. This renders an obstacle to the analysis of numerical algorithms for non-Hermitian eigenvalue problems. A recent technique to handle this issue is pseudospectral shattering [BGVKS23], showing that adding a random perturbation to any matrix has a regularizing effect on the stability of the eigenvalues and eigenvectors. Prior work has analyzed the regularizing effect of dense Gaussian perturbations, where independent noise is added to every entry of a given matrix [BVKS20, BGVKS23, BKMS21, JSS21]. We show that the same effect can be achieved by adding a sparse random perturbation. In particular, we show that given any $n\times n$ matrix $M$ of polynomially bounded norm: (a) perturbing $O(n\log^2(n))$ random entries of $M$ by adding i.i.d. complex Gaussians yields $\log\kappa_V(A)=O(\text{poly}\log(n))$ and $\log (1/\eta(A))=O(\text{poly}\log(n))$ with high probability; (b) perturbing $O(n^{1+\alpha})$ random entries of $M$ for any constant $\alpha>0$ yields $\log\kappa_V(A)=O_\alpha(\log(n))$ and $\log(1/\eta(A))=O_\alpha(\log(n))$ with high probability. Here, $\kappa_V(A)$ denotes the condition number of the eigenvectors of the perturbed matrix $A$ and $\eta(A)$ denotes its minimum eigenvalue gap. A key mechanism of the proof is to reduce the study of $\kappa_V(A)$ to control of the pseudospectral area and minimum eigenvalue gap of $A$, which are further reduced to estimates on the least two singular values of shifts of $A$. We obtain the required least singular value estimates via a streamlining of an argument of Tao and Vu [TV07] specialized to the case of sparse complex Gaussian perturbations. [Rest of abstract in pdf]. Subjects: Probability (math.PR) ; Numerical Analysis (math.NA) MSC classes: 60B20, 65F22, 65F50, 68Q87 Cite as: arXiv:2411.19926 [math.PR] (or arXiv:2411.19926v3 [math.PR] for this version) https://doi.org/10.48550/arXiv.2411.19926 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Rikhav Shah [ view email ] [v1] Fri, 29 Nov 2024 18:40:16 UTC (18 KB) [v2] Tue, 17 Jun 2025 04:01:50 UTC (20 KB) [v3] Fri, 10 Apr 2026 21:40:36 UTC (23 KB) Full-text links: Access Paper: View a PDF of the paper titled Sparse Pseudospectral Shattering, by Rikhav Shah and 2 other authors View PDF HTML (experimental) TeX Source view license Current browse context: math.PR < prev | next > new | recent | 2024-11 Change to browse by: cs cs.NA math math.NA References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... 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