Chronological stability for nonstationary matrix refinement
Abstract
A matrix cascade is an iterative refinement process in which vector-valued data are repeatedly filtered by matrix-valued masks across successively finer scales. Such cascades arise in multichannel subdivision, Hermite refinement, multiresolution synthesis, and exponential-spline constructions, and in nonstationary schemes the masks may vary with scale. We develop a chronological stability framework for this setting, allowing both the transition operators and the associated matching spaces to change from level to level. If the chronological radius satisfies ρch<1 and the matching defects are summable, then the cascade converges uniformly and, for every q∈(ρch,1), |Fn−Φ|∞ ≲ qn+∑j=1nδj q n−j +∑j>nδj . The time-weighted defect term records the level at which each perturbation is inserted, and the matching spaces need not converge; they may even alternate indefinitely. For compact piecewise-linear seeds, every finite ordered matrix cascade admits an exact fixed-width ReLU realization with O(n) depth and parameters, and this linear depth order is worst-case optimal at fixed width. Consequently, exponentially decaying defects yield certified approximants of size O(logε−1). We further construct stable two-channel classes in which the chronological radius and the leading metric-entropy coefficient are independent invariants, showing that stability does not determine finite-accuracy description complexity. The framework therefore provides tolerance-controlled finite representations for nonstationary multiresolution and exponential-spline refinement.
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BibTeX
@misc{li2026chronological,
title = {{Chronological stability for nonstationary matrix refinement}},
author = {Yuwen Li and Yupeng Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13981},
eprint = {2609.13981},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13981},
}
RIS
TY - UNPB
TI - Chronological stability for nonstationary matrix refinement
AU - Yuwen Li
AU - Yupeng Wang
PY - 2026
DA - 2026/09/12
PB - arXiv
DO - 10.48550/arxiv.2609.13981
UR - https://arxiv.org/abs/2609.13981
AB - A matrix cascade is an iterative refinement process in which vector-valued data are repeatedly filtered by matrix-valued masks across successively finer scales. Such cascades arise in multichannel subdivision, Hermite refinement, multiresolution synthesis, and exponential-spline constructions, and in nonstationary schemes the masks may vary with scale. We develop a chronological stability framework for this setting, allowing both the transition operators and the associated matching spaces to change from level to level. If the chronological radius satisfies ρ_ch<1 and the matching defects are summable, then the cascade converges uniformly and, for every q∈(ρ_ch,1), |F_n−Φ|_∞ ≲ q^n+∑_j=1^nδ_j q^ n−j +∑_j>nδ_j . The time-weighted defect term records the level at which each perturbation is inserted, and the matching spaces need not converge; they may even alternate indefinitely. For compact piecewise-linear seeds, every finite ordered matrix cascade admits an exact fixed-width ReLU realization with O(n) depth and parameters, and this linear depth order is worst-case optimal at fixed width. Consequently, exponentially decaying defects yield certified approximants of size O(logε^−1). We further construct stable two-channel classes in which the chronological radius and the leading metric-entropy coefficient are independent invariants, showing that stability does not determine finite-accuracy description complexity. The framework therefore provides tolerance-controlled finite representations for nonstationary multiresolution and exponential-spline refinement.
ER -