Four-dimensional image-type data can quickly become prohibitively large, and it may not be feasible to directly apply methods, such as persistent homology or convolutional neural networks, to determine the topological characteristics of these data because they can encounter complexity issues. This study aims to determine the Betti numbers of large four-dimensional image-type data. The experiments use synthetic data, and demonstrate that it is possible to circumvent these issues by applying downscaling methods to the data prior to training a convolutional neural network, even when persistent homology software indicates that downscaling can significantly alter the homology of the training data. When provided with downscaled test data, the neural network can estimate the Betti numbers of the original samples with reasonable accuracy.

翻译：四维图像类型数据可能迅速变得庞大无比，直接应用持续同调或卷积神经网络等方法确定这些数据的拓扑特征可能面临复杂性难题。本研究旨在估计大型四维图像类型数据的贝蒂数。实验采用合成数据，证明在训练卷积神经网络之前对数据应用下采样方法可以规避这些问题，即使持续同调软件表明下采样会显著改变训练数据的同调性。当提供下采样后的测试数据时，神经网络能够以合理精度估计原始样本的贝蒂数。