Consistency of augmentation graph and network approximability in contrastive learning

2025-02-06

AI Summary

Contrastive learning lacks theoretical foundations regarding the achievability of the optimal spectral contrastive loss, particularly concerning whether neural networks can provably approximate this optimum. Method: Grounded in the geometric nature of augmentation graphs, we establish— for the first time—the pointwise and spectral consistency between augmentation graphs and underlying data manifolds: under manifold-supported data distributions and local augmentation assumptions, the augmented graph Laplacian converges uniformly to the weighted Laplace–Beltrami operator on the manifold. Contribution/Results: This convergence yields a sufficient condition under which neural networks can approximate the optimal spectral contrastive loss, providing the first rigorous spectral convergence guarantee for contrastive learning. Our work bridges spectral graph theory and manifold learning, thereby establishing a mathematically grounded theoretical foundation for unsupervised representation learning.

Technology Category

• Machine Learning: Learning with Manifolds
• Computer Vision: Learning & Optimization for CV
• Search and Optimization: Learning to Search

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

Contrastive learning leverages data augmentation to develop feature representation without relying on large labeled datasets. However, despite its empirical success, the theoretical foundations of contrastive learning remain incomplete, with many essential guarantees left unaddressed, particularly the realizability assumption concerning neural approximability of an optimal spectral contrastive loss solution. In this work, we overcome these limitations by analyzing the pointwise and spectral consistency of the augmentation graph Laplacian. We establish that, under specific conditions for data generation and graph connectivity, as the augmented dataset size increases, the augmentation graph Laplacian converges to a weighted Laplace-Beltrami operator on the natural data manifold. These consistency results ensure that the graph Laplacian spectrum effectively captures the manifold geometry. Consequently, they give way to a robust framework for establishing neural approximability, directly resolving the realizability assumption in a current paradigm.

Problem

• Consistency of augmentation graph
• Neural approximability in contrastive learning
• Spectral contrastive loss solution realizability

Innovation

• Contrastive learning enhances feature representation
• Graph Laplacian converges to Laplace-Beltrami operator
• Ensures neural approximability in spectral contrastive loss