We study convolutional neural networks with monomial activation functions. Specifically, we prove that their parameterization map is regular and is an isomorphism almost everywhere, up to rescaling the filters. By leveraging on tools from algebraic geometry, we explore the geometric properties of the image in function space of this map - typically referred to as neuromanifold. In particular, we compute the dimension and the degree of the neuromanifold, which measure the expressivity of the model, and describe its singularities. Moreover, for a generic large dataset, we derive an explicit formula that quantifies the number of critical points arising in the optimization of a regression loss.

翻译：我们研究了具有单项式激活函数的卷积神经网络。具体而言，我们证明了其参数化映射是正则的，并且在几乎处处（除滤波器缩放外）同构。通过利用代数几何工具，我们探索了该映射在函数空间中的像（通常称为神经流形）的几何性质。特别地，我们计算了神经流形的维数和次数，这些度量了模型的表达能力，并描述了其奇点。此外，对于通用的大规模数据集，我们推导出一个显式公式，量化了回归损失优化中出现的临界点数量。