通过数据集的Polytope结构定义神经网络架构
Defining Neural Network Architecture through Polytope Structures of Dataset
AI总结：
本文通过数据集的polytope结构定义神经网络宽度的上下界，探讨网络宽度与数据集几何复杂性的关系，并开发算法从训练好的网络中推断数据集的polytope结构，证明MNIST等数据集可用少量polytope高效封装。
摘要：
当前神经网络的理论和实证研究表明，复杂的数据集需要大型网络架构才能进行彻底的分类，但这种关系的确切本质仍不清楚。本文通过定义神经网络宽度的上下界来解决这一问题，这些界限基于所讨论数据集的polytope结构。我们还探讨了这些原则在simplicial complexes和特定流形形状上的应用，解释了网络宽度的需求如何根据数据集的几何复杂性而变化。此外，我们开发了一种算法来研究相反的情况，即如何从相应的训练好的神经网络中推断出数据集的polytope结构。通过我们的算法，证明了MNIST、Fashion-MNIST和CIFAR10等流行数据集可以使用不超过两个具有少量面的polytope来高效封装。
英文摘要：
Current theoretical and empirical research in neural networks suggests that complex datasets require large network architectures for thorough classification, yet the precise nature of this relationship remains unclear. This paper tackles this issue by defining upper and lower bounds for neural network widths, which are informed by the polytope structure of the dataset in question. We also delve into the application of these principles to simplicial complexes and specific manifold shapes, explaining how the requirement for network width varies in accordance with the geometric complexity of the dataset. Moreover, we develop an algorithm to investigate a converse situation where the polytope structure of a dataset can be inferred from its corresponding trained neural networks. Through our algorithm, it is established that popular datasets such as MNIST, Fashion-MNIST, and CIFAR10 can be efficiently encapsulated using no more than two polytopes with a small number of faces.
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。
