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.

研究使用单项式激活函数的多项式神经网络 (PNNs) 的表达能力和学习过程。探讨了使用代数几何工具对某些神经流形进行研究：给出了半代数集的显式描述，并表征了其 Zariski 闭包，称之为神经多样性。研究了神经多样性的维度，并将一个代数度量，即学习度，与神经多样性相关联。维度用作网络表达能力的几何度量，学习度用作训练网络的复杂度度量，并提供了可学习函数数量的上限。这些理论结果与实验证明相伴。