通用逼近与拓扑神经网络
Universal Approximation and the Topological Neural Network
AI总结：
本文提出拓扑神经网络和分布神经网络，将输入从有限维空间推广到吉洪诺夫空间和博雷尔测度空间，并通过强通用逼近定理证明其能任意逼近一致连续函数，从而扩展深度集合等深度学习概念。
摘要：
本文引入了一种拓扑神经网络（TNN），其数据来自吉洪诺夫拓扑空间，而非通常的有限维空间。由此，还引入了一种以博雷尔测度为数据的分布神经网络（DNN）。这些新型神经网络相结合，有助于识别随机过程路径中的长程依赖性、重尾及其他性质，或作用于由粒子滤波或隐马尔可夫模型算法产生的信念状态。随后，本文通过针对吉洪诺夫空间的强通用逼近定理及其在测度空间上的推论，确立了 TNN 和 DNN 的可靠性。这些定理表明，神经网络可以任意逼近与唯一一致性相关联的一致连续函数（关于上确界度量）。我们还提供了一些讨论，表明正有限测度上的神经网络是近期深度学习概念“深度集合”的推广。
英文摘要：
A topological neural network (TNN), which takes data from a Tychonoff topological space instead of the usual finite dimensional space, is introduced. As a consequence, a distributional neural network (DNN) that takes Borel measures as data is also introduced. Combined these new neural networks facilitate things like recognizing long range dependence, heavy tails and other properties in stochastic process paths or like acting on belief states produced by particle filtering or hidden Markov model algorithms. The veracity of the TNN and DNN are then established herein by a strong universal approximation theorem for Tychonoff spaces and its corollary for spaces of measures. These theorems show that neural networks can arbitrarily approximate uniformly continuous functions (with respect to the sup metric) associated with a unique uniformity. We also provide some discussion showing that neural networks on positive-finite measures are a generalization of the recent deep learning notion of deep sets.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
