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.

翻译：我们引入了一类自然与ReLU神经网络相关的代数簇，这些代数簇源于网络输出在输入空间激活区域中的分段线性结构，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网络输出的秩约束，我们推导出刻画网络可表示函数的多项式方程。我们进一步研究了这些代数簇达到期望维度的条件，从而为ReLU网络的表达能力与结构特性提供见解。