Motivated by structures that appear in deep neural networks, we investigate nonlinear composite models alternating proximity and affine operators defined on different spaces. We first show that a wide range of activation operators used in neural networks are actually proximity operators. We then establish conditions for the averagedness of the proposed composite constructs and investigate their asymptotic properties. It is shown that the limit of the resulting process solves a variational inequality which, in general, does not derive from a minimization problem. The analysis relies on tools from monotone operator theory and sheds some light on a class of neural networks structures with so far elusive asymptotic properties.

中文翻译：

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深度神经网络结构解决变分不等式

受深层神经网络中出现的结构的影响，我们研究了在不同空间上定义接近度和仿射算子的非线性复合模型。我们首先证明神经网络中使用的多种激活算子实际上是接近算子。然后，我们为提出的复合构造的平均性建立条件，并研究其渐近性质。结果表明，所得过程的极限解决了通常不由最小化问题引起的变分不等式。该分析依赖于单调算符理论的工具，并为一类到目前为止还具有渐近性质的神经网络结构提供了一些启示。