有限域上浅层神经网络的表达能力
Expressivity of Shallow Neural Networks Over Finite Fields
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中文总结 AI 辅助
研究有限域上浅层多项式神经网络的表达能力，通过定义神经流形并量化其基数得出上下界，涉及计算有限域上有理点，还展示一种架构说明域特征对表达能力概念的关键作用。
中文摘要 AI 辅助
我们研究了具有单项式激活函数的浅层多项式神经网络（PNN）在有限域上的表达能力。对于给定架构，我们将神经流形定义为从所有可能网络权重到多项式环乘积的映射的像。通过神经流形的基数量化表达能力，得出自然的上下界。这涉及到计算有限域上的有理点，与韦伊猜想密切相关。最后，我们展示了一种架构，其在零特征域和有限特征域上考虑时神经流形有显著差异，说明了域特征在表达能力概念中的关键作用。
英文摘要
We study the expressivity of shallow polynomial neural networks (PNNs) with monomial activation functions over finite fields. For a given architecture, we define a neuromanifold as the image of the map from all possible network weights into the product of polynomial rings. We quantify the expressivity by the cardinality of the neuromanifold, and derive a natural lower and upper bound. This leads to counting rational points over finite fields, a problem closely linked to the Weil conjectures. Finally, we present an architecture that exhibits a striking difference in the neuromanifolds when considered over a characteristic zero versus a finite-characteristic field, illustrating the critical role of field characteristic in the notion of expressivity.
