Cone Extended Rayleigh Quotients for Directed Graph Learning
Cone Extended Rayleigh Quotients for Directed Graph Learning: Minimax Spectral Certificates, Sensitivity, and Adaptive Control
The method improves spectral control in directed graphs while preserving predictive information, outperforming symmetrization approaches that lose directional insights.
The paper develops a methodology for spectral certification and control in directed graph learning using nonsymmetric propagation operators.
It introduces computable cone bounds for a posteriori enclosure of spectral levels and differentiable surrogates with explicit approximation errors.
Numerical experiments on the Cora citation network show a 21.5% reduction in spectral level under 0.
5% edge-weight reduction without test accuracy loss.
Cone Extended Rayleigh Quotients for Directed Graph Learning: Minimax Spectral Certificates, Sensitivity, and Adaptive Control
Directed graph learning naturally leads to trainable nonsymmetric propagation operators with distinct right and left spectral structures. Building on the two-sided cone Rayleigh framework for generalized pencils \[ B_θ-λG, \] we develop a learning-oriented methodology for spectral certification, sensitivity analysis, and control without requiring symmetry, nonnegativity, or cone preservation. In the positive-orthant setting, computable lower and upper cone bounds provide an a posteriori enclosure of a distinguished cone level, while smooth soft-min/max surrogates preserve rigorous one-sided bounds with explicit approximation errors and remain differentiable with respect to the trainable parameters. For a simple interior level, the right and left modes satisfy \[ Dλ_C(B)[H]=v_C^T H u_C, \] yielding first-order optimal graph-supported interventions under prescribed perturbation budgets and motivating adaptive spectral control. Numerical experiments demonstrate the applicability of the approach beyond cone-preserving operators and in directed learning settings. Signed nonsymmetric perturbations reveal a transition from interior eigenpairs to boundary complementary quasi-pairs, including non-spectral cone levels, while controlled experiments show that symmetrization can remove predictive information carried solely by edge direction. On the directed Cora citation network, adaptive recomputation of the right--left sensitivity reduces the distinguished spectral level by approximately $21.5\%$ under a cumulative edge-weight reduction budget of $0.5\%$, with no observed change in test accuracy for the trained model and data split considered.
This report is categorized under AITOP's "Papers" desk, focusing on China's AI industry and domestic foundation model ecosystem. Key entities and technical topics under continuous tracking include directed graph learning, spectral certification, Cora citation network, adaptive control.
Source attribution: curated from arXiv cs.LG (@Yavdat Sh. Il'yasov, Nur F. Valeev).