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

Persistent Laplacian-enhanced Algorithm for Scarcely Labeled Data Classification