利用代数拓扑与李理论估计神经网络维度
Estimate of the Neural Network Dimension using Algebraic Topology and Lie Theory
AI总结：
本文提出通过持久同调研究数据集所在流形的拓扑不变量，确定神经网络层最小神经元数，假设数据位于光滑流形上并具有交换群结构，利用持久景观确定嵌入维度。
摘要：
本文提出一种方法，用于确定神经网络层中最小可能的神经元数量，以足够好地学习输入空间的拓扑。我们介绍一种基于持久同调的一般程序，用于研究我们怀疑数据集所在流形的拓扑不变量。我们精确指定所需维度，假设数据位于或接近光滑流形上。此外，我们要求此空间是连通的，并在数学意义上具有交换群结构。这些假设使我们能够推导出底层空间的分解，其拓扑结构已知。我们使用持久景观中的k维同调群代表来确定此分解的整数维度。此数字是能够捕捉数据流形拓扑的嵌入维度。我们推导了理论并在玩具数据集上进行了实验验证。
英文摘要：
In this paper we present an approach to determine the smallest possible number of neurons in a layer of a neural network in such a way that the topology of the input space can be learned sufficiently well. We introduce a general procedure based on persistent homology to investigate topological invariants of the manifold on which we suspect the data set. We specify the required dimensions precisely, assuming that there is a smooth manifold on or near which the data are located. Furthermore, we require that this space is connected and has a commutative group structure in the mathematical sense. These assumptions allow us to derive a decomposition of the underlying space whose topology is well known. We use the representatives of the $k$-dimensional homology groups from the persistence landscape to determine an integer dimension for this decomposition. This number is the dimension of the embedding that is capable of capturing the topology of the data manifold. We derive the theory and validate it experimentally on toy data sets.
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