The success of many machine learning (ML) methods depends crucially on having large amounts of labeled data. However, obtaining enough labeled data can be expensive, time-consuming, and subject to ethical constraints for many applications. One approach that has shown tremendous value in addressing this challenge is semi-supervised learning (SSL); this technique utilizes both labeled and unlabeled data during training, often with much less labeled data than unlabeled data, which is often relatively easy and inexpensive to obtain. In fact, SSL methods are particularly useful in applications where the cost of labeling data is especially expensive, such as medical analysis, natural language processing (NLP), or speech recognition. A subset of SSL methods that have achieved great success in various domains involves algorithms that integrate graph-based techniques. These procedures are popular due to the vast amount of information provided by the graphical framework and the versatility of their applications. In this work, we propose an algebraic topology-based semi-supervised method called persistent Laplacian-enhanced graph MBO (PL-MBO) by integrating persistent spectral graph theory with the classical Merriman-Bence- Osher (MBO) scheme. Specifically, we use a filtration procedure to generate a sequence of chain complexes and associated families of simplicial complexes, from which we construct a family of persistent Laplacians. Overall, it is a very efficient procedure that requires much less labeled data to perform well compared to many ML techniques, and it can be adapted for both small and large datasets. We evaluate the performance of the proposed method on data classification, and the results indicate that the proposed technique outperforms other existing semi-supervised algorithms.

本文提出一种称为PL-MBO的代数拓扑半监督学习方法，它将持久谱图理论与经典的MBO方法相结合，通过一系列链复杂体和关联的简单复杂体构建一族持久拉普拉斯变换，从而实现在标记数据量较少的情况下实现良好的性能。

用于稀疏标记数据分类的持久性拉普拉斯增强算法