The inner representation of deep neural networks (DNNs) is indecipherable, which makes it difficult to tune DNN models, control their training process, and interpret their outputs. In this paper, we propose a novel approach to investigate the inner representation of DNNs through topological data analysis (TDA). Persistent homology (PH), one of the outstanding methods in TDA, was employed for investigating the complexities of trained DNNs. We constructed clique complexes on trained DNNs and calculated the one-dimensional PH of DNNs. The PH reveals the combinational effects of multiple neurons in DNNs at different resolutions, which is difficult to be captured without using PH. Evaluations were conducted using fully connected networks (FCNs) and networks combining FCNs and convolutional neural networks (CNNs) trained on the MNIST and CIFAR-10 data sets. Evaluation results demonstrate that the PH of DNNs reflects both the excess of neurons and problem difficulty, making PH one of the prominent methods for investigating the inner representation of DNNs.

深度神经网络 (DNN) 的内部表示无法解读，这使得调整 DNN 模型、控制其训练过程和解释其输出变得困难。在本文中，我们提出了一种通过拓扑数据分析 (TDA) 研究 DNN 内部表示的新方法。持久同源性 (PH) 是 TDA 中的杰出方法之一，用于研究训练过的 DNN 的复杂性。我们在经过训练的 DNN 上构建了 clique complexes 并计算了 DNN 的一维 PH。PH 揭示了 DNN 中多个神经元在不同分辨率下的组合效应，这在不使用 PH 的情况下很难被捕获。使用全连接网络 (FCN) 以及结合 FCN 和卷积神经网络 (CNN) 的网络在 MNIST 和 CIFAR-10 数据集上进行了评估。评估结果表明，DNN 的 PH 反映了神经元的过剩和问题的难度，使 PH 成为研究 DNN 内部表示的主要方法之一。