In this paper, a particular discrete-time nonlinear and time-invariant system represented as a vector difference equation is analyzed for its stability properties. The motivation for analyzing this particular system is that it models gated recurrent unit neural networks commonly used and well known in machine learning applications. From the technical perspective, the analyses exploit the systems similarities to a convex combination of discrete-time systems, where one of the systems is trivial, and thus, the overall properties are mostly dependent on the other one. Stability results are formulated for the nonlinear system and its linearization with respect to the systems, in general, multiple equilibria. To motivate and illustrate the potential of these results in applications, some particular results are derived for the gated recurrent unit neural network models and a connection between local stability analysis and learning is provided.

中文翻译：

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门控循环单元神经网络的稳定性：凸组合公式方法

在本文中，一个特定的离散时间非线性和时不变系统表示为向量差分方程分析其稳定性属性。分析这个特定系统的动机是它对机器学习应用中常用和众所周知的门控循环单元神经网络进行建模。从技术角度来看，分析利用系统与离散时间系统的凸组合的相似性，其中一个系统是微不足道的，因此，整体属性主要依赖于另一个。稳定性结果是针对非线性系统及其相对于系统的线性化而制定的，通常是多重平衡。为了激发和说明这些结果在应用中的潜力，