Shifted Matrix-Sign Reflectors for Prescribed-Index Eigenspace Reflection
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866914597245550592 |
|---|---|
| author | Zhao, Jin |
| author_facet | Zhao, Jin |
| contents | Spectral projectors and the reflectors derived from them are basic objects in numerical linear algebra. This paper studies the prescribed-index reflector I-2P_k, where P_k is the spectral projector associated with the first k eigenvectors of a symmetric matrix. If a shift s lies in the target spectral gap, then this reflector is exactly the shifted matrix sign sign(H-sI). The exact identity is elementary, but its algorithmic consequences are not: all admissible shifts give the same exact reflector, while finite-step sign filters can have very different errors. We analyze odd sign-preserving spectral filters, prove local inheritance and discrete stability for the induced reflector iterations, derive a gap-dependent Newton-Schulz operator bound, and give deterministic admissibility certificates for inexact and reused shifts. The analysis identifies the shifted spectral margin as the quantity controlling finite-step reflector accuracy and explains why the midpoint shift is the natural default. Numerical experiments separate the matrix-function issues from the outer saddle-search dynamics: controlled spectra verify the margin predictions, low-dimensional tests distinguish shifted signs from raw signs, target-index scans probe non-small k, and Allen-Cahn and dense timing tests identify the regimes in which full-matrix sign filters are useful and the stiff regimes in which stronger sign engines are needed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_24840 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Shifted Matrix-Sign Reflectors for Prescribed-Index Eigenspace Reflection Zhao, Jin Numerical Analysis 65F60, 65F15, 65F30, 65K05 Spectral projectors and the reflectors derived from them are basic objects in numerical linear algebra. This paper studies the prescribed-index reflector I-2P_k, where P_k is the spectral projector associated with the first k eigenvectors of a symmetric matrix. If a shift s lies in the target spectral gap, then this reflector is exactly the shifted matrix sign sign(H-sI). The exact identity is elementary, but its algorithmic consequences are not: all admissible shifts give the same exact reflector, while finite-step sign filters can have very different errors. We analyze odd sign-preserving spectral filters, prove local inheritance and discrete stability for the induced reflector iterations, derive a gap-dependent Newton-Schulz operator bound, and give deterministic admissibility certificates for inexact and reused shifts. The analysis identifies the shifted spectral margin as the quantity controlling finite-step reflector accuracy and explains why the midpoint shift is the natural default. Numerical experiments separate the matrix-function issues from the outer saddle-search dynamics: controlled spectra verify the margin predictions, low-dimensional tests distinguish shifted signs from raw signs, target-index scans probe non-small k, and Allen-Cahn and dense timing tests identify the regimes in which full-matrix sign filters are useful and the stiff regimes in which stronger sign engines are needed. |
| title | Shifted Matrix-Sign Reflectors for Prescribed-Index Eigenspace Reflection |
| topic | Numerical Analysis 65F60, 65F15, 65F30, 65K05 |
| url | https://arxiv.org/abs/2605.24840 |