Nowadays, Machine Learning and Deep Learning methods have become the state-of-the-art approach to solve data classification tasks. In order to use those methods, it is necessary to acquire and label a considerable amount of data; however, this is not straightforward in some fields, since data annotation is time consuming and might require expert knowledge. This challenge can be tackled by means of semi-supervised learning methods that take advantage of both labelled and unlabelled data. In this work, we present new semi-supervised learning methods based on techniques from Topological Data Analysis (TDA), a field that is gaining importance for analysing large amounts of data with high variety and dimensionality. In particular, we have created two semi-supervised learning methods following two different topological approaches. In the former, we have used a homological approach that consists in studying the persistence diagrams associated with the data using the Bottleneck and Wasserstein distances. In the latter, we have taken into account the connectivity of the data. In addition, we have carried out a thorough analysis of the developed methods using 3 synthetic datasets, 5 structured datasets, and 2 datasets of images. The results show that the semi-supervised methods developed in this work outperform both the results obtained with models trained with only manually labelled data, and those obtained with classical semi-supervised learning methods, reaching improvements of up to a 16%.

本文基于拓扑数据分析 (Topological Data Analysis,TDA) 技术提出了两种半监督学习方法，一种基于骨架結构和水斗距离，另一种考虑数据的连通性， 并将其应用于三个综合数据集和七个各色各样的结构体和图像数据集，实验结果表明，这些半监督方法能够有效地提高数据分类准确性达到16%。

半监督学习的拓扑方法