We study the realization map of deep ReLU networks, focusing on when a function determines its parameters up to scaling and permutation. To analyze hidden redundancies beyond these standard symmetries, we introduce a framework based on weighted polyhedral complexes. Our main result shows that for every architecture whose input and hidden layers have width at least two, there exists an open set of identifiable parameters. This implies that the functional dimension of every such architecture is exactly the number of parameters minus the number of hidden neurons. We further show that minimal functional representations can still have non-trivial parameter redundancies. Finally, we establish a generic depth hierarchy, whereby for an open set of parameters the realized function cannot be represented generically by any shallower network.

翻译：我们研究了深度ReLU网络的实现映射，重点关注函数何时能确定其参数（允许缩放和置换）。为了分析这些标准对称性之外的隐藏冗余，我们引入了一个基于加权多面体复形的框架。主要结果表明，对于输入层和隐藏层宽度均至少为2的每种架构，都存在一个可识别参数的开集。这意味着每种此类架构的函数维数恰好等于参数数量减去隐藏神经元数量。我们进一步证明，最小函数表示仍可能具有非平凡的参数冗余。最后，我们建立了一个通用的深度层次结构：对于参数的开集，所实现的函数无法由任何更浅的网络通用地表示。