We study the expressivity and learning process for polynomial neural networks (PNNs) with monomial activation functions. The weights of the network parametrize the neuromanifold. In this paper, we study certain neuromanifolds using tools from algebraic geometry: we give explicit descriptions as semialgebraic sets and characterize their Zariski closures, called neurovarieties. We study their dimension and associate an algebraic degree, the learning degree, to the neurovariety. The dimension serves as a geometric measure for the expressivity of the network, the learning degree is a measure for the complexity of training the network and provides upper bounds on the number of learnable functions. These theoretical results are accompanied with experiments.

翻译：我们研究了具有单项激活函数的多项式神经网络（PNNs）的表达能力与学习过程。网络权重参数化神经流形。本文运用代数几何工具研究若干神经流形：给出其作为半代数集的明确描述，并刻画其Zariski闭包（称为神经簇）。我们研究了神经簇的维数，并为其赋予代数度数——学习度数。其中，维数作为网络表达能力的几何度量，学习度数则衡量网络训练的复杂度，并为可学习函数的数量提供上界。这些理论结果均辅以实验验证。