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

• URL: http://arxiv.org/abs/2608.27122v1
• Date: Thu, 27 Aug 2026 13:37:02 GMT
• Title: Cone Extended Rayleigh Quotients for Directed Graph Learning: Minimax Spectral Certificates, Sensitivity, and Adaptive Control
• Abstract summary: We develop a learning-oriented methodology for spectral certification, sensitivity analysis, and control without requiring symmetry, nonnegativity, or cone preservation.<n>For a simple interior level, the right and left modes satisfy [ D_C(B)[H]=v_CT H u_C, ] yielding first-order optimal graph-supported interventions.<n> Numerical experiments demonstrate the applicability of the approach beyond cone-preserving operators and in directed learning settings.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 \[ B_θ-λG, \] 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.

Related papers

• The Role of Gradient Modification in Heavy-Tailed Nonconvex Stochastic Min-Max Optimization [50.59470683896735]
  min-max optimization has attracted increasing attention due to its applications in modern machine learning.<n>We provide a comprehensive theoretical of gradient descent (SGDA)<n>We develop new clipping-free algorithms, i.e., Stoc-GDAM and Stoc-TRmax, and they both can achieve the optimal dependence on the target accuracy without using gradient clipping.
  arXiv  Detail & Related papers  (2026-09-05T12:49:28Z)
• Spectrally Deconfounded Gradient Boosting [10.803532361460741]
  We develop a nonlinear spectral deconfounding framework for gradient boosting.<n>We show that deconfounding is not achieved by the spectral loss alone.<n>We also extend the method to general likelihoods and nonlinear confounding.
  arXiv  Detail & Related papers  (2026-07-10T12:51:42Z)
• Need We Teach Foundation Models What is a Generative Image? Gradient-Free Generative Artifact Detection via Analytic Spectral Adaptation [3.881211374324378]
  Adapting foundation models to detect generative artifacts via gradient-based updates compromises their intrinsic representations.<n>We propose a gradient-free methodology reframing detection from binary classification to an out-of-distribution (OOD) anomaly measurement problem.
  arXiv  Detail & Related papers  (2026-06-03T13:04:23Z)
• Correcting Neural Operator Spectral Bias via Diffusion Posterior Sampling with Sparse Observations [0.0]
  Neural operator surrogates approximate PDE solutions orders of magnitude faster than numerical solvers.<n>Sparse sensor measurements of the field are often available too, offering pointwise accuracy without spectral distortion.<n>We address this by treating NO predictions as auxiliary observations in a diffusion posterior sampling framework.
  arXiv  Detail & Related papers  (2026-06-02T17:26:15Z)
• Emulating Non-Differentiable Metrics via Knowledge-Guided Learning: Introducing the Minkowski Image Loss [0.05997422707234518]
  We develop a framework to bridge the differentiability gap'' in Earth system deep learning.<n>First, we approximate the original non-differentiable function into a differentiable equivalent one; the second is to learn differentiable surrogates for scientific functionals.<n>We demonstrate this framework's utility by developing the Minkowski image loss, a differentiable equivalent for the integral-geometric measures of surface precipitation fields.
  arXiv  Detail & Related papers  (2026-04-13T13:04:57Z)
• Stability and Generalization of Push-Sum Based Decentralized Optimization over Directed Graphs [55.77845440440496]
  Push-based decentralized communication enables optimization over communication networks, where information exchange may be asymmetric.<n>We develop a unified uniform-stability framework for the Gradient Push (SGP) algorithm.<n>A key technical ingredient is an imbalance-aware generalization bound through two quantities.
  arXiv  Detail & Related papers  (2026-02-24T05:32:03Z)
• Direct Learning of Calibration-Aware Uncertainty for Neural PDE Surrogates [0.0]
  Cross-regularized uncertainty learns uncertainty parameters during training using routed through a held-out regularization split.<n>The predictor is optimized on the training split for fit, while low-dimensional uncertainty controls are optimized on the regularization split to reduce train-test mismatch.<n>We instantiate the approach in Fourier Neural Operators and evaluate on APEBench sweeps over observed fraction and training-set size.
  arXiv  Detail & Related papers  (2026-02-11T17:57:20Z)
• Verifying Closed-Loop Contractivity of Learning-Based Controllers via Partitioning [52.23804865017831]
  We address the problem of verifying closed-loop contraction in nonlinear control systems whose controller and contraction metric are both parameterized by neural networks.<n>We derive a tractable and scalable sufficient condition for closed-loop contractivity that reduces to checking that the dominant eigenvalue of a symmetric Metzler matrix is nonpositive.
  arXiv  Detail & Related papers  (2025-12-01T23:06:56Z)
• Revisiting Zeroth-Order Optimization: Minimum-Variance Two-Point Estimators and Directionally Aligned Perturbations [57.179679246370114]
  We identify the distribution of random perturbations that minimizes the estimator's variance as the perturbation stepsize tends to zero.<n>Our findings reveal that such desired perturbations can align directionally with the true gradient, instead of maintaining a fixed length.
  arXiv  Detail & Related papers  (2025-10-22T19:06:39Z)
• Stable Nonconvex-Nonconcave Training via Linear Interpolation [51.668052890249726]
  This paper presents a theoretical analysis of linearahead as a principled method for stabilizing (large-scale) neural network training. We argue that instabilities in the optimization process are often caused by the nonmonotonicity of the loss landscape and show how linear can help by leveraging the theory of nonexpansive operators.
  arXiv  Detail & Related papers  (2023-10-20T12:45:12Z)
• Beyond the Edge of Stability via Two-step Gradient Updates [49.03389279816152]
  Gradient Descent (GD) is a powerful workhorse of modern machine learning. GD's ability to find local minimisers is only guaranteed for losses with Lipschitz gradients. This work focuses on simple, yet representative, learning problems via analysis of two-step gradient updates.
  arXiv  Detail & Related papers  (2022-06-08T21:32:50Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.