标题： 限制ReLU神经网络的输出 显示英文标题

标题： Constraining the outputs of ReLU neural networks

摘要： 我们引入了一类与ReLU神经网络自然相关的代数簇，这些簇来源于输出在输入空间中的激活区域上的分段线性结构，以及参数空间中的分段多重线性结构。 通过分析每个激活区域内的网络输出的秩约束，我们推导出表征网络可表示函数的多项式方程。 我们进一步研究了这些簇达到其预期维度的条件，从而深入了解ReLU网络的表达能力和结构特性。 显示英文摘要

摘要： We introduce a class of algebraic varieties naturally associated with ReLU neural networks, arising from the piecewise linear structure of their outputs across activation regions in input space, and the piecewise multilinear structure in parameter space. By analyzing the rank constraints on the network outputs within each activation region, we derive polynomial equations that characterize the functions representable by the network. We further investigate conditions under which these varieties attain their expected dimension, providing insight into the expressive and structural properties of ReLU networks.