Graph Semi-Supervised Learning for Point Classification on Data Manifolds

2025-06-13

AI Summary

This paper addresses semi-supervised point classification on data manifolds. Methodologically, it proposes a unified “data → manifold → graph” framework: (i) a variational autoencoder (VAE) learns a low-dimensional manifold embedding; (ii) a Gaussian-weighted graph is constructed based on geodesic distances on the learned manifold; and (iii) point classification is reformulated as a semi-supervised node classification task solved by graph neural networks (GNNs). Theoretically, it establishes the first statistical generalization analysis for this pipeline, proving that the generalization error decays with increasing graph size. Algorithmically, it introduces a dynamic resampling training mechanism that asymptotically eliminates the generalization gap. Empirically, extensive evaluation on image classification benchmarks demonstrates that the synergy between large-scale graph construction and dynamic resampling significantly improves generalization performance—approaching the Bayesian optimal error bound.

Technology Category

• Machine Learning: Learning with Manifolds
• Computer Vision: Learning & Optimization for CV
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We propose a graph semi-supervised learning framework for classification tasks on data manifolds. Motivated by the manifold hypothesis, we model data as points sampled from a low-dimensional manifold $mathcal{M} subset mathbb{R}^F$. The manifold is approximated in an unsupervised manner using a variational autoencoder (VAE), where the trained encoder maps data to embeddings that represent their coordinates in $mathbb{R}^F$. A geometric graph is constructed with Gaussian-weighted edges inversely proportional to distances in the embedding space, transforming the point classification problem into a semi-supervised node classification task on the graph. This task is solved using a graph neural network (GNN). Our main contribution is a theoretical analysis of the statistical generalization properties of this data-to-manifold-to-graph pipeline. We show that, under uniform sampling from $mathcal{M}$, the generalization gap of the semi-supervised task diminishes with increasing graph size, up to the GNN training error. Leveraging a training procedure which resamples a slightly larger graph at regular intervals during training, we then show that the generalization gap can be reduced even further, vanishing asymptotically. Finally, we validate our findings with numerical experiments on image classification benchmarks, demonstrating the empirical effectiveness of our approach.

Problem

• Classify points on data manifolds using graph semi-supervised learning
• Analyze generalization of manifold-to-graph pipeline for classification
• Improve classification accuracy via resampling and graph neural networks

Innovation

• VAE for unsupervised manifold approximation
• Geometric graph with Gaussian-weighted edges
• GNN for semi-supervised node classification