发表机构
McGill University; National University of Mongolia; Mongolian Academy of Sciences（麦吉尔大学; 蒙古国立大学; 蒙古科学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对二元细化迭代的神经网络实现问题，提出双图册选择器构造，通过双坐标系切换解决不连续与连续映射的矛盾，实现ReLU网络对相关函数的精确表达。
AI 中文摘要
细化算子生成小波构造、细分格式与几何建模中使用的众多函数，其有限迭代会产生数量快速增长的线性片段，是深度神经网络表达能力的自然测试案例。早期研究表明，对于具有有限支撑掩码的标量二元细化，每个紧支撑连续分段线性种子的有限细化迭代都可由固定宽度、深度随细化步数线性增长的ReLU网络精确实现。本文对该已知定理给出新构造。难点在于细化级联由不连续的二元数字选择驱动，而ReLU网络生成连续分段线性映射。我们在圆的多边形模型上表示残差动力学，用两个重叠坐标系描述每个残差位置：一个为普通坐标系，一个偏移1/2。两者的不连续点不同，网络仅在两者均有效且对应固定线性级联更新一致的位置切换两种描述，该切换是精确的，无需乘以可变选择器。该构造还可精确读出满足自然端点相容条件的每个连续分段线性圆函数。局部种子由双通路网络处理，平移协方差、有限分解与粘合操作将结果扩展到任意紧支撑连续分段线性种子，且支撑窗口保持不变。
英文摘要
Refinement operators generate many functions used in wavelet constructions, subdivision schemes, and geometric modeling. Their finite iterates can develop rapidly increasing numbers of linear pieces, making them a natural test case for the expressive power of deep neural networks. Earlier work showed that, for scalar binary refinement with a finitely supported mask, every compactly supported continuous piecewise linear seed has finite refinement iterates that admit exact ReLU realizations of fixed width and depth growing linearly with the number of refinement steps. The present paper gives a new construction of this known theorem. The difficulty is that the refinement cascade is driven by discontinuous binary digit choices, whereas ReLU networks produce continuous piecewise linear maps. We represent the residual dynamics on a polygonal model of the circle and describe each residual position in two overlapping coordinate systems, one ordinary and one shifted by one half. Their discontinuities occur at different points. The network switches between the two descriptions only where both are valid and the corresponding fixed linear cascade updates agree, so the switch is exact and requires no multiplication by a variable selector. The construction also gives exact readout of every continuous piecewise linear circle function satisfying the natural endpoint compatibility condition. Localized seeds are handled by a two-pass network, and translation covariance, finite decomposition, and gluing extend the result to arbitrary compactly supported continuous piecewise linear seeds in a preserved support window.
Comments30 pages
