Incorporating biological neuronal properties into Artificial Neural Networks
(ANNs) to enhance computational capabilities poses a formidable challenge in
the field of machine learning. Inspired by recent findings indicating that
dendrites adhere to quadratic integration rules for synaptic inputs, we propose
a novel ANN model, Dendritic Integration-Based Quadratic Neural Network
(DIQNN). This model shows superior performance over traditional ANNs in a
variety of classification tasks. To reduce the computational cost of DIQNN, we
introduce the Low-Rank DIQNN, while we find it can retain the performance of
the original DIQNN. We further propose a margin to characterize the
generalization error and theoretically prove this margin will increase
monotonically during training. And we show the consistency between
generalization and our margin using numerical experiments. Finally, by
integrating this margin into the loss function, the change of test accuracy is
indeed accelerated. Our work contributes a novel, brain-inspired ANN model that
surpasses traditional ANNs and provides a theoretical framework to analyze the
generalization error in classification tasks.

将生物神经元属性纳入人工神经网络以增强计算能力是机器学习领域中面临的重大挑战，我们提出了一种新颖的人工神经网络模型 DIQNN (Dendritic Integration-Based Quadratic Neural Network)，该模型在各种分类任务中表现出优于传统人工神经网络的性能。为了降低 DIQNN 的计算成本，我们介绍了一种低秩 DIQNN，发现它可以保持原始 DIQNN 的性能。我们进一步提出一个边界来描述泛化误差，并在理论上证明该边界在训练过程中会单调增加。我们通过数值实验展示了泛化性和我们的边界之间的一致性。最后，通过将该边界整合到损失函数中，我们的实验结果证明了测试精度的变化确实得到了加速。我们的工作提供了一种新颖的、受大脑启发的人工神经网络模型，优于传统人工神经网络，并提供了分析分类任务中泛化误差的理论框架。