证明二维分段线性函数迭代可用固定宽度的ReLU网络精确实现
Exact ReLU realization of tensor-product refinement iterates
- 通过张量积残差动态转移，将二维迭代映射到双多边形环路
- 任意紧支撑连续分段线性种子函数的迭代均可在O(n)深度内精确实现
- 为多变量细分迭代提供首个真正二维的ReLU精确实现框架
研究定义在R²上的标量二进制细分算子 (Vf)(x,y) = sum_{(j,k) in Z²} c_{j,k} f(2x-j, 2y-k)，其中仅有有限个掩码系数c_{j,k}非零。在固定支持窗假设下，证明对任意紧支撑连续分段线性种子函数g: R²→R，其迭代V^n g 均可实现为固定宽度、深度O(n)的ReLU网络。这是细分级联精确实现理论在二维空间的首次真正拓展。利用一维精确环路控制器框架，证明将张量积残差动态精确映射至两个多边形环路的乘积，并将剩余接缝歧义简化为最终读出与选择步骤。矩阵级联则由固定深度递归模块处理，一般紧支撑连续分段线性种子函数被分解为有限项并实现支持窗内的精确夹持拼接。该结果确立了张量积二进制情形作为环路控制器方法处理细分迭代的自然首个多变量实例。
原文摘要 · Abstract (English)
We study scalar dyadic refinement operators on R^2 of the form (Vf)(x,y) = sum_{(j,k) in Z^2} c_{j,k} f(2x-j, 2y-k), where only finitely many mask coefficients c_{j,k} are nonzero. Under a fixed support-window hypothesis, we prove that for every compactly supported continuous piecewise linear seed g:R^2->R, the iterates V^n g admit exact ReLU realizations of fixed width and depth O(n). This gives a first genuinely two-dimensional extension of the exact realization theory for refinement cascades. Using the one-dimensional exact loop-controller framework, the proof transports the tensor-product residual dynamics exactly on the product of two polygonal loops and reduces the remaining seam ambiguity to a final readout and selector step. The matrix cascade is then handled by a fixed-depth recursive block, and general compactly supported continuous piecewise linear seeds are reduced to a finite decomposition together with exact clamped gluing on the support window. This identifies the tensor-product dyadic case as a natural first multivariate instance of the loop-controller method for refinement iterates.
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