In this paper, we introduce a new class of functions on $\mathbb{R}$ that is closed under composition, and contains the logistic sigmoid function. We use this class to show that any 1-dimensional neural network of arbitrary depth with logistic sigmoid activation functions has at most three fixed points. While such neural networks are far from real world applications, we are able to completely understand their fixed points, providing a foundation to the much needed connection between application and theory of deep neural networks.

翻译：在本文中，我们引入了$\mathbb{R}$上的一类新函数，该类函数在复合运算下封闭，且包含逻辑斯谛Sigmoid函数。利用这一函数类，我们证明了任意深度且激活函数为逻辑斯谛Sigmoid的一维神经网络最多存在三个固定点。尽管此类神经网络远未达到实际应用水平，但我们对它们的固定点有了完全的理解，从而为深度神经网络的应用与理论之间亟需建立的联系奠定了基础。