用ReLU神经网络精确实现分段线性函数的迭代运算，深度仅随迭代次数线性增长。
Exact ReLU realization of affine one-dimensional refinement iterates via residual memory and offset frames
- 设计残差记忆控制器，实现可逆状态回放，解决非可逆残差动态问题。
- 当缩放因子M≥3时，任意分段线性输入与强迫项均能实现固定宽度的精确ReLU映射。
- 适用于分形构造、曲线递归等场景，特别适合需要精确计算的数值算法设计者。
研究形如 (Wγ)(t)=∑_{j∈ℤ} A_jγ(Mt-j)+B(t) 的向量值仿射细化算子，其中矩阵掩码有限支撑，输入和强迫项为紧支撑连续分段线性函数。基于 (B≡0) 情况下的齐次实现定理，证明当 (M≥3) 时，每个有限次仿射迭代 (W^nγ) 均可被深度为 (O(n)) 的固定宽度ReLU网络精确实现。核心创新是残差记忆控制器，将不可逆的残差动态替换为可逆的斜积系统，从而支持基于Horner型求和的残差状态精确反向重播。偏移帧将强迫项原子对齐至残差接缝之外，使互补循环读出能精确恢复其值。分支选择的模糊性仅出现在累积仿射状态已消失的位置。当 (M≥3) 时，该结果适用于任意紧支撑连续分段线性强迫项；当 (M=2) 时，同样适用于普通帧接缝分离的强迫项。此外还证明了针对固定有限维连续分段线性强迫项的阶段依赖扩展，并记录了对开曲线、有限状态以及希尔伯特型和莫顿型递归构造的线性深度提升。
原文摘要 · Abstract (English)
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.
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