# Cone Rayleigh Levels: Finite Perturbations and Certified Control

arXiv: https://arxiv.org/abs/2608.27122 · HTML: https://mlsanity.com/paper/2608.27122/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 27 references that are in the map.
- First posted on arXiv 27 Aug 2026.

## Builds on (most cited first, 6 of 6)

- [MagNet: A Neural Network for Directed Graphs](https://mlsanity.com/paper/2102.11391/) (2021)
- [Edge Directionality Improves Learning on Heterophilic Graphs](https://mlsanity.com/paper/2305.10498/) (2023)
- [Directed Graph Convolutional Network](https://mlsanity.com/paper/2004.13970/) (2020)
- [Spectral-based Graph Convolutional Network for Directed Graphs](https://mlsanity.com/paper/1907.08990/) (2019)
- [Rayleigh Quotient Graph Neural Networks for Graph-level Anomaly Detection](https://mlsanity.com/paper/2310.02861/) (2023)
- [Understanding Graph Neural Networks with Generalized Geometric Scattering Transforms](https://mlsanity.com/paper/1911.06253/) (2019)

## Abstract (arXiv)

We study the reuse of positive trial profiles under finite perturbations of nonsymmetric matrix pencils B-λ G. The lower and upper cone Rayleigh levels need not coincide and are defined without requiring positive eigenvectors. In the positive orthant with positive diagonal G, we derive computable perturbation bounds and determine the exact worst-case trial gap over prescribed independent entrywise perturbation classes. This yields the largest uniform radius meeting a given trial-gap tolerance for fixed profiles, certifying trial-value accuracy without recomputation. Experiments on a nonnegative operator and five signed matrices compare sufficient and optimal radii, directional thresholds, and cold- and warm-start recomputation. Several signed cases exhibit severely limited uniform profile reuse despite the optimality of the radius. Exact rational checks verify the reported bounds and worst-case constructions for stored numerical inputs. A finite-budget linear program and differentiable constraints illustrate applications to verified control and learning.