This paper addresses the following question of neural network identifiability: Does the input–output map realized by a feed-forward neural network with respect to a given nonlinearity uniquely specify the network architecture, weights, and biases? The existing literature on the subject (Sussman in Neural Netw 5(4):589–593, 1992; Albertini et al. in Artificial neural networks for speech and vision, 1993; Fefferman in Rev Mat Iberoam 10(3):507–555, 1994) suggests that the answer should be yes, up to certain symmetries induced by the nonlinearity, and provided that the networks under consideration satisfy certain “genericity conditions.” The results in Sussman (1992) and Albertini et al. (1993) apply to networks with a single hidden layer and in Fefferman (1994) the networks need to be fully connected. In an effort to answer the identifiability question in greater generality, we derive necessary genericity conditions for the identifiability of neural networks of arbitrary depth and connectivity with an arbitrary nonlinearity. Moreover, we construct a family of nonlinearities for which these genericity conditions are minimal, i.e., both necessary and sufficient. This family is large enough to approximate many commonly encountered nonlinearities to within arbitrary precision in the uniform norm.

本文解决了神经网络可识别性的以下问题：前馈神经网络针对给定的非线性实现的输入输出映射是否唯一地指定了网络体系结构，权重和偏差？有关该主题的现有文献（Sussman in Neural Netw 5（4）：589-593，1992; Albertini等人在“用于语音和视觉的人工神经网络”中，1993； Fefferman在Rev Mat Iberoam 10（3）：507-555中） （1994年）提出的答案应该是肯定的，取决于非线性所引起的某些对称性，并且要考虑的网络满足某些“一般性条件”。Sussman（1992）和Albertini等人的结果。（1993）适用于具有单个隐藏层的网络，在Fefferman（1994）中，网络需要完全连接。为了更广泛地回答可识别性问题，我们得出了任意深度和具有任意非线性连通性的神经网络的可识别性的必要通用条件。此外，我们构造了一个非线性族，对于这些非线性，这些通用性条件极小，即既必要又充分。这个族足够大，可以在统一范数内以任意精度近似许多常见的非线性。