二元仿射细化迭代的精确ReLU实现：基于反射折叠与锥切换
Exact ReLU realization of binary affine refinement iterates via reflection folding and cone switching
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中文总结 AI 辅助
本文证明二元仿射细化迭代可通过反射加倍与锥切换机制实现固定宽度、线性深度的精确ReLU网络，无需分离强迫剖面，并支持阶段相关强迫。
中文摘要 AI 辅助
我们研究具有有限支撑矩阵掩模和紧支撑连续分段线性输入及强迫数据的向量值二元仿射细化算子。我们证明，每一个有限细化迭代都允许一个精确的ReLU实现，其宽度固定，深度与迭代次数成线性关系。不需要将强迫剖面与二元单元接缝分离。主要机制是普适反射加倍。将每个残差剖面与其反射配对，将两个二元转移矩阵替换为一个固定的块矩阵以及一个固定的交换对合。单元接缝恒等式使得两个分支候选在帐篷折叠处一致，而它们的交换奇分量由到折叠的距离线性界定。这允许通过固定的连续分段线性锥开关进行精确分支选择，而无需乘以可变选择器。由此产生的原始递归需要按逆序使用残差轨道。我们通过先前为仿射细化开发的残差记忆控制器获得精确的向后重放，在此通过圆加倍的反射商来解释。该构造传播完整的向量化剖面，而不是将输入和强迫分解为参考原子。我们还处理来自固定有限维族的阶段相关强迫，并表明真正的反射等变性将加倍级联减少到单一奇偶扇区。
英文摘要
We study vector-valued binary affine refinement operators with finitely supported matrix masks and compactly supported continuous piecewise linear input and forcing data. We prove that every finite refinement iterate admits an exact ReLU realization of fixed width and depth linear in the number of iterations. No separation of the forcing profile from the binary cell seams is required. The main mechanism is universal reflection doubling. Pairing each residual profile with its reflection replaces the two binary transition matrices by one fixed block matrix together with a fixed swap involution. The cell-seam identity makes the two branch candidates agree at the tent fold, while their swap-odd component is bounded linearly by the distance to the fold. This permits exact branch selection by a fixed continuous piecewise linear cone switch, without multiplication by a variable selector. The resulting primal recursion requires the residual orbit in reverse order. We obtain exact backward replay from the residual memory controller developed previously for affine refinement, interpreted here through the reflection quotient of circle doubling. The construction propagates the full vectorized profiles rather than decomposing the input and forcing into reference atoms. We also treat stage-dependent forcing from a fixed finite-dimensional family and show that genuine reflection equivariance reduces the doubled cascade to a single parity sector.
发表机构
- McGill University（麦吉尔大学）
- National University of Mongolia（蒙古国立大学）
- 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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