非平稳矩阵细化的时间稳定性

中文摘要 AI 辅助

矩阵级联是一种迭代细化过程，其中向量值数据通过矩阵值掩模在逐渐变细的尺度上反复滤波。此类级联出现在多通道细分、Hermite细化、多分辨率合成和指数样条构造中，在非平稳方案中掩模可能随尺度变化。我们为此设置开发了一个时间稳定性框架，允许转移算子和相关的匹配空间逐级变化。如果时间半径满足$\rho_{\rm ch}<1$且匹配缺陷可求和，则级联一致收敛，并且对于每个$q\in(\rho_{\rm ch},1)$，\\[ \\|F_n-\Phi\\|_\infty \lesssim q^n+\sum_{j=1}^n\delta_j q^{\\,n-j} +\sum_{j>n}\delta_j. \\] 时间加权缺陷项记录了每个扰动插入的级别，匹配空间不必收敛；它们甚至可能无限交替。对于紧致分段线性种子，每个有限有序矩阵级联都允许精确的固定宽度ReLU实现，具有$O(n)$深度和参数，并且这种线性深度阶在固定宽度下是最坏情况最优的。因此，指数衰减缺陷产生大小为$O(\log\varepsilon^{-1})$的认证近似。我们进一步构造了稳定的双通道类，其中时间半径和领先度量熵系数是独立不变量，表明稳定性不能确定有限精度描述复杂度。因此，该框架为非平稳多分辨率和指数样条细化提供了容差控制的有限表示。

英文摘要

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 $ρ_{\rm ch}<1$ and the matching defects are summable, then the cascade converges uniformly and, for every $q\in(ρ_{\rm ch},1)$, \[ \|F_n-Φ\|_\infty \lesssim q^n+\sum_{j=1}^nδ_j q^{\,n-j} +\sum_{j>n}δ_j . \] 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 $O(\log\varepsilon^{-1})$. 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.

发表机构

• School of Mathematical Sciences, Zhejiang University（浙江大学数学科学学院）

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