Structured Spectral Step-Sizes and Rebound-Aware Ordering for Gradient Methods
The performance of gradient methods depends critically on step-size selection. Spectral step-size design involves constructing candidates from finite history and ordering them during iteration. We address both through a finite-moment spectral framework. For candidate construction, we relate Huang--Dai--Liu determinant pencils, limited-memory steepest-descent Krylov--Ritz extraction, pseudo-memory moment realization, and Gu--Du recovery of spectral nodes and component weights. The framework distinguishes generalized-eigenvalue recoverability, finite-window implementability, and component-weight interpretability. Under explicit positivity, rank-one, regularity, and exact-recovery assumptions, these properties become compatible in a common finite-moment regime, with the long, short, and mean LMSD constructions as representatives. For ordering, we identify spectral rebound: a step targeting a lower spectral scale may reactivate previously reduced high-frequency components. Idealized one-target, block, and recomputed memory-\(m\) models yield phase-count recurrences motivating a high-frequency-first strategy with spectral refreshes. We propose a rebound-aware spectral gradient method (RASG) that ranks Ritz candidates by frequency and estimated strength, and sets phase duration via a Ritz-uncertainty estimator. Two variants, RASG-KT and RASG-LMSD, use a Kato--Temple-inspired estimator and a long/short LMSD discrepancy. For strictly convex quadratics, global \(R\)-linear convergence holds under an admissibility safeguard. Experiments on ill-conditioned quadratic problems and two large-scale convex problems illustrate the behavior.