非负神经网络的不动点
Fixed points of nonnegative neural networks
AI总结：
本文利用非线性 Perron–Frobenius 理论分析非负神经网络，给出更弱的不动点存在条件并刻画其不动点集结构，为自编码器和深度平衡模型提供数学基础。
摘要：
我们运用不动点理论来分析非负神经网络，并将其定义为把非负向量映射为非负向量的神经网络。我们首先表明，具有非负权重和偏置的非负神经网络可在非线性 Perron–Frobenius 理论框架内被视为单调且（弱）可扩展的映射。这一事实使我们能够为输入与输出维度相同的非负神经网络给出不动点存在的条件，而且这些条件比近期利用凸分析论证所得的条件更弱。此外，我们证明了具有非负权重和偏置的非负神经网络，其不动点集的形状是一个区间，并在温和条件下退化为一个点。随后，这些结果被用于获得更一般的非负神经网络的不动点存在性。从实践角度看，我们的结果有助于理解自编码器的行为，同时也为深度平衡模型未来的发展提供了有价值的数学工具。
英文摘要：
We use fixed point theory to analyze nonnegative neural networks, which we define as neural networks that map nonnegative vectors to nonnegative vectors. We first show that nonnegative neural networks with nonnegative weights and biases can be recognized as monotonic and (weakly) scalable mappings within the framework of nonlinear Perron-Frobenius theory. This fact enables us to provide conditions for the existence of fixed points of nonnegative neural networks having inputs and outputs of the same dimension, and these conditions are weaker than those recently obtained using arguments in convex analysis. Furthermore, we prove that the shape of the fixed point set of nonnegative neural networks with nonnegative weights and biases is an interval, which under mild conditions degenerates to a point. These results are then used to obtain the existence of fixed points of more general nonnegative neural networks. From a practical perspective, our results contribute to the understanding of the behavior of autoencoders, and we also offer valuable mathematical machinery for future developments in deep equilibrium models.
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