论文标题
使用代数拓扑和谎言理论估算神经网络维度
Estimate of the Neural Network Dimension using Algebraic Topology and Lie Theory
论文作者
论文摘要
在本文中，我们提出了一种方法，以确定神经网络层中可能数量的神经元，以使输入空间的拓扑结构充分学习。我们介绍了基于持久同源性的一般程序，以研究我们怀疑数据集的流形的拓扑不变性。假设数据位于或附近有一个光滑的歧管，我们会精确地指定所需的尺寸。此外，我们要求该空间已连接，并且具有数学意义上的交换组结构。这些假设使我们得出了拓扑众所周知的基础空间的分解。我们使用持久景观中$ k $维的同源组的代表来确定该分解的整数维度。该数字是能够捕获数据歧管拓扑的嵌入的维度。我们得出理论并在玩具数据集上实验验证它。
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.
