One common function class in machine learning is the class of ReLU neural networks. ReLU neural networks induce a piecewise linear decomposition of their input space called the canonical polyhedral complex. It has previously been established that it is decidable whether a ReLU neural network is piecewise linear Morse. In order to expand computational tools for analyzing the topological properties of ReLU neural networks, and to harness the strengths of discrete Morse theory, we introduce a schematic for translating between a given piecewise linear Morse function (e.g. parameters of a ReLU neural network) on a canonical polyhedral complex and a compatible (``relatively perfect") discrete Morse function on the same complex. Our approach is constructive, producing an algorithm that can be used to determine if a given vertex in a canonical polyhedral complex corresponds to a piecewise linear Morse critical point. Furthermore we provide an algorithm for constructing a consistent discrete Morse pairing on cells in the canonical polyhedral complex which contain this vertex. We additionally provide some new realizability results with respect to sublevel set topology in the case of shallow ReLU neural networks.

翻译：ReLU神经网络是机器学习中常见的函数类别。ReLU神经网络会诱导输入空间的逐段线性分解，称为典范多面体复形。已有研究证明，判定ReLU神经网络是否为逐段线性Morse函数是可计算问题。为拓展分析ReLU神经网络拓扑性质的计算工具，并发挥离散Morse理论的优势，本文提出一种转换框架：将典范多面体复形上的给定逐段线性Morse函数（例如ReLU神经网络参数）转化为同一复形上兼容的（“相对完美”）离散Morse函数。我们的方法具有构造性，所提算法可用于判定典范多面体复形中的给定顶点是否对应逐段线性Morse临界点。此外，我们提供了在包含该顶点的典范多面体复形胞腔上构建一致离散Morse配对算法的构造方法。针对浅层ReLU神经网络情形，我们还给出了关于子水平集拓扑的新可实现性结果。