  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.
