用精确循环控制器实现分段线性曲线的ReLU精确表示，固定宽度下深度仅随迭代次数线性增长。
Exact Loop Controllers for ReLU Realization of Homogeneous Curve Refinements
- 设计新型循环控制器，通过多边形环路传输残差轨迹，避免传统标量代理的误差累积。
- 首次实现均匀M元向量值细分的固定宽度ReLU网络，深度仅为O(n)，且支持自相交分形构造。
- 适合研究分形生成、神经网络几何表示与可计算细分理论的学者，尤其关注高效网络架构者。
我们研究作用于紧支连续分段线性曲线γ: R→R^p的齐次细分算子 (Vγ)(t)=∑_{j∈Z} A_j γ(Mt−j)，其中M≥2，且仅有有限个矩阵A_j∈R^{p×p}非零。证明了迭代V^nγ具有固定宽度、深度O(n)的精确ReLU实现。核心创新在于构建了精确循环控制器：不传播标量残差代理，而是通过前向精确状态在多边形环路上传输残差轨道；再通过互补的分段线性读出恢复标量因子与数字选择器。环路接缝未消除，但其歧义仅保留在最终读出/选择阶段，因标量原子支撑远离接缝而无害。该方法为标量二元可细分函数构造提供了向量值扩展，具备更几何化的控制器架构。同时给出网络权重和偏置的粗略指数界。仿射强迫项通过将仿射迭代展开为有限个齐次迭代之和处理，实现深度为O(n^2)的固定宽度实现；锚定开曲线则化为具有仿射锚点不匹配的紧支缺陷。还描述了齐次多边形生成器，包括龙形例子与任意维度的自相交希尔伯特型原型。扩展版本包含依赖阶段的强迫项、有限状态堆叠简化及更多几何构造，如科赫、戈斯珀、莫顿以及基于连接器的希尔伯特型变体。
原文摘要 · Abstract (English)
We study homogeneous refinement operators \((Vγ)(t)=\sum_{j\in\mathbb Z}A_jγ(Mt-j)\), acting on compactly supported continuous piecewise linear curves \(γ:\mathbb R\to\mathbb R^p\), where \(M\ge2\) and only finitely many matrices \(A_j\in\mathbb R^{p\times p}\) are nonzero. We prove that the iterates \(V^nγ\) admit exact ReLU realizations of fixed width and depth \(O(n)\). The main new ingredient is an exact loop controller for the residual dynamics. Instead of propagating scalar residual surrogates, the construction transports the residual orbit by a forward-exact state on a polygonal loop. Scalar factors and digit selectors are then recovered from this loop state by complementary CPwL readouts. The loop seam is not removed, but its remaining ambiguity is confined to the final readout/selector stage, where it is harmless because the scalar atom is supported away from the seam. This gives a homogeneous \(M\)-ary vector-valued extension of the scalar binary refinable-function construction with a more geometric controller architecture. We also record crude exponential bounds on the network weights and biases. Affine forcing terms are handled by expanding affine iterates into finite sums of homogeneous iterates, giving exact fixed-width realizations with depth \(O(n^2)\), and anchored open curves reduce to compactly supported defects with affine anchor mismatch. We also describe homogeneous polygonal generators, including dragon-type examples and a self-intersecting Hilbert-type prototype in arbitrary dimension. The extended version includes stage-dependent forcing, finite-state stacking reductions, and further geometric constructions such as Koch-, Gosper-, Morton-, and connector-based Hilbert-type variants.
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