In this work we use the persistent homology method, a technique in topological data analysis (TDA), to extract essential topological features from the data space and combine them with deep learning features for classification tasks. In TDA, the concepts of complexes and filtration are building blocks. Firstly, a filtration is constructed from some complex. Then, persistent homology classes are computed, and their evolution along the filtration is visualized through the persistence diagram. Additionally, we applied vectorization techniques to the persistence diagram to make this topological information compatible with machine learning algorithms. This was carried out with the aim of classifying images from multiple classes in the MNIST dataset. Our approach inserts topological features into deep learning approaches composed by single and two-streams neural networks architectures based on a multi-layer perceptron (MLP) and a convolutional neral network (CNN) taylored for multi-class classification in the MNIST dataset. In our analysis, we evaluated the obtained results and compared them with the outcomes achieved through the baselines that are available in the TensorFlow library. The main conclusion is that topological information may increase neural network accuracy in multi-class classification tasks with the price of computational complexity of persistent homology calculation. Up to the best of our knowledge, it is the first work that combines deep learning features and the combination of topological features for multi-class classification tasks.

翻译：本研究采用持续同调方法（一种拓扑数据分析技术），从数据空间中提取关键拓扑特征，并将其与深度学习特征融合以完成分类任务。在拓扑数据分析中，复形和过滤是基础构建单元。首先基于特定复形构建过滤，随后计算持续同调类，并通过持续图可视化其沿过滤的演化过程。此外，我们对持续图应用向量化技术，使拓扑信息能够与机器学习算法兼容。该工作的目标是实现MNIST数据集中多类别图像的分类。我们的方法将拓扑特征引入基于多层感知机（MLP）和卷积神经网络（CNN）的单流与双流神经网络架构中，这些架构专门针对MNIST数据集的多类别分类任务进行了优化。在分析中，我们评估了模型结果，并与TensorFlow库中提供的基线方法进行了对比。主要结论是：在付出持续同调计算复杂度的代价下，拓扑信息能够提升神经网络在多类别分类任务中的准确率。据我们所知，这是首个将深度学习特征与拓扑特征相结合用于多类别分类任务的研究工作。