We apply a generalized piecewise-linear (PL) version of Morse theory due to Grunert-Kuhnel-Rote to define and study new local and global notions of topological complexity for fully-connected feedforward ReLU neural network functions, F: R^n -> R. Along the way, we show how to construct, for each such F, a canonical polytopal complex K(F) and a deformation retract of the domain onto K(F), yielding a convenient compact model for performing calculations. We also give a construction showing that local complexity can be arbitrarily high.

翻译：我们应用Grunert-Kuhnel-Rote提出的广义分段线性（PL）Morse理论，定义并研究了全连接前馈ReLU神经网络函数F: R^n → R的局部与全局拓扑复杂度新概念。在此过程中，我们展示了如何为每个此类函数F构造典范多面体复形K(F)，并给出定义域到K(F)的形变收缩，从而获得便于计算的紧致模型。此外，我们还通过构造证明了局部复杂度可达到任意高阶。