揭示对比学习中数据增强图的收敛性，解决神经网络逼近性的理论难题。
Consistency of augmentation graph and network approximability in contrastive learning
- 通过分析增强图拉普拉斯的点态与谱一致性，建立理论框架。
- 证明随数据量增大，图拉普拉斯收敛到数据流形上的加权拉普拉斯-贝尔特拉米算子。
- 首次严格证明最优对比损失可被神经网络逼近，为模型有效性提供保障。
对比学习利用数据增强构建特征表示，无需依赖大规模标注数据。尽管其在实践中表现优异，但理论基础仍不完善，尤其是关于最优谱对比损失解的神经网络可逼近性这一可实现性假设尚未得到充分验证。本文通过分析增强图拉普拉斯的点态一致性和谱一致性，证明在特定数据生成条件和图连通性下，随着增强数据集规模增大，图拉普拉斯收敛至数据自然流形上的加权拉普拉斯-贝尔特拉米算子。该一致性结果确保图拉普拉斯谱能有效捕捉流形几何结构，进而为建立神经网络逼近性提供了稳健框架，直接解决了当前范式中的可实现性假设问题。
原文摘要 · Abstract (English)
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 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.
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