Laplace Learning in Wasserstein Space

The curation of large-scale, fully annotated training datasets remains a major bottleneck due to the high cost and expertise required for manual labelling. For example, in biomedical imaging applications such as flow cytom-etry [12, 67], gene expression microarrays [23, 24], and proteomic assays [18], modern technologies generate high-dimensional data far faster than it can be annotated. As a result, only a small fraction of samples receive reliable labels, despite their routine use in classification tasks. This motivates graph-based semi-supervised methods, which exploits the geometric structure of the data to improve predictions with limited supervision. In this paper, we focus on a special class of graph-based semi-supervised methods, namely Laplace Learning [68], to study classification in high-dimensional settings. This method exploits the geometric structure inherent in large quantities of unlabelled data to improve label predictions. However, leveraging the underlying geometry in high-dimensional datasets presents substantial challenges, including the well-known curse of dimensionality [22, 44] and poor generalization capacity [18]. In theory, a well-established trend in statistics suggests that high-dimensional data often possess an intrinsic low-dimensional structure, a concept formalized by the manifold hypothesis [25]. This hypothesis asserts that data are supported (or nearly supported) on a low-dimensional manifold with a small intrinsic dimension.