Variance-enlarged Poisson Learning for Graph-based Semi-Supervised Learning with Extremely Sparse Labeled Data

Graph-based semi-supervised learning, particularly in the context of extremely sparse labeled data, often suffers from degenerate solutions where label functions tend to be nearly constant across unlabeled data. In this paper, we introduce Variance-enlarged Poisson Learning (VPL), a simple yet powerful framework tailored to alleviate the issues arising from the presence of degenerate solutions. VPL incorporates a variance-enlarged regularization term, which induces a Poisson equation specifically for unlabeled data. This intuitive approach increases the dispersion of labels from their average mean, effectively reducing the likelihood of degenerate solutions characterized by nearly constant label functions. We subsequently introduce two streamlined algorithms, V-Laplace and V-Poisson, each intricately designed to enhance Laplace and Poisson learning, respectively. Furthermore, we broaden the scope of VPL to encompass graph neural networks, introducing Variance-enlarged Graph Poisson Networks (V-GPN) to facilitate improved label propagation. To achieve a deeper understanding of VPL's behavior, we conduct a comprehensive theoretical exploration in both discrete and variational cases. Our findings elucidate that VPL inherently amplifies the importance of connections within the same class while concurrently tempering those between different classes. We support our claims with extensive experiments, demonstrating the effectiveness of VPL and showcasing its superiority over existing methods. The code is available at https://github.com/hitcszx/VPL.