通过残差记忆和偏移帧实现仿射一维细化迭代的精确ReLU实现

中文摘要 AI 辅助

我们研究形式为[(Wγ)(t)=\sum_{j\in\mathbb{Z}} A_jγ(Mt - j)+B(t)]的向量值仿射细化算子，其具有有限支撑矩阵掩码以及紧支撑连续分段线性输入和强迫数据。基于(B≡0)的齐次实现定理，我们证明对于(M≥3)，每个有限仿射迭代(W^nγ)都允许一个精确的固定宽度ReLU实现，其深度为(O(n))。主要新要素是残差记忆控制器，它用单射斜积取代不可逆残差动力学，并允许精确反向重放仿射强迫和的Horner型评估所需的残差状态。偏移帧将强迫原子与残差接缝对齐，允许互补循环读出精确恢复其值。剩余的分支选择模糊性仅在累积仿射状态已经消失的地方出现。对于(M≥3)，该结果适用于任意紧支撑连续分段线性强迫项。对于(M = 2)，相同构造适用于普通帧接缝分离强迫。我们还证明了固定有限维连续分段线性跨度中强迫项的阶段相关扩展，并记录了开放曲线、有限状态以及希尔伯特和莫顿型递归构造产生的线性深度升级。

英文摘要

We study vector-valued affine refinement operators of the form [ (Wγ)(t)=\sum_{j\in\mathbb{Z}} A_jγ(Mt-j)+B(t), ] with finitely supported matrix mask and compactly supported continuous piecewise linear input and forcing data. Building on the homogeneous realization theorem for (B\equiv 0), we prove that, for (M\ge 3), every finite affine iterate (W^nγ) admits an exact fixed-width ReLU realization whose depth is (O(n)). The main new ingredient is a residual memory controller. It replaces the noninvertible residual dynamics by an injective skew-product and permits exact backward replay of the residual states required by a Horner-type evaluation of the affine forcing sum. Offset frames align the forcing atoms away from residual seams, allowing complementary loop readouts to recover their values exactly. The remaining branch-selection ambiguity occurs only where the accumulated affine state has already vanished. For (M\ge 3), the result applies to arbitrary compactly supported continuous piecewise linear forcing terms. For (M=2), the same construction applies to ordinary-frame seam-separated forcing. We also prove a stage-dependent extension for forcing terms in a fixed finite-dimensional continuous piecewise linear span and record the resulting linear-depth upgrade for open-curve, finite-state, and Hilbert- and Morton-type recursive constructions.

发表机构

• National University of Mongolia（蒙古国国立大学）
• McGill University（麦吉尔大学）
• Institute of Mathematics and Digital Technology, Mongolian Academy of Sciences（蒙古科学院数学与数字技术研究所）

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