Cone Extended Rayleigh Quotients for Directed Graph Learning: Minimax Spectral Certificates, Sensitivity, and Adaptive Control

arXiv:2608.27122v1 Announce Type: new Abstract: 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_heta-lambda G, ] 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 [ Dlambda_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.

Source: arXiv cs.LG | 2026-08-28

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