二进神经网络表达奇偶校验问题的隐元最小数目上界

二进神经网络采用线性分类,是结构简单又易于实现的一类神经网络,在许多应用领域中都有重要研究价值.对于单隐层二进神经网络,目前隐层规模的确定问题仍然没有明确的研究结论.本文在研究隐层规模问题的过程中,提出了布尔空间的最多孤立样本问题.在二进神经网络隐层神经元各自表达一个"与"关系,所有隐层神经元通过输出元形成"或"关系的情况下,证明了实现最多孤立样本问题需2n?1个隐层神经元.更重要的是,指出了n元奇偶校验问题和最多孤立样本结构的等价性.进一步地,通过引入隐层抑制神经元将隐元数目降为n,说明了抑制神经元在二进神经网络中的重要作用.最后,在Hamming球与SP函数的基础上,揭示出抑制神经元和n元奇偶校验问题的逻辑关系,并给出了奇偶校验问题的逻辑式表达.

The upper bound of the minimal number of hidden neurons for the parity problem in binary neural networks

Binary neural networks(BNNs)have important value in many application areas.They adopt linearly separable structures,which are simple and easy to implement by hardware.For a BNN with single hidden layer, the problem of how to determine the upper bound of the number of hidden neurons has not been solved well and truly.This paper defines a special structure called most isolated samples(MIS)in the Boolean space.We prove that at least 2n-1 hidden neurons are needed to express the MIS logical relationship in the Boolean space if the hidden neurons of a BNN and its output neuron form a structure of AND/OR logic.Then the paper points out that the n-bit parity problem is just equivalent to the MIS structure.Furthermore,by proposing a new concept of restraining neuron and using it in the hidden layer,we can reduce the number of hidden neurons to n.This result explains the important role of restraining neurons in some cases.Finally,on the basis of Hamming sphere and SP function,both the restraining neuron and the n-bit parity problem are given a clear logical meaning,and can be described by a series of logical expressions.