将图模型从欧氏空间扩展到无穷维的Wasserstein空间，提升高维数据分类效果。
Laplace Learning in Wasserstein Space
- 在Wasserstein空间中构建图拉普拉斯学习框架，突破传统欧氏限制。
- 证明离散图p-Dirichlet能量收敛于连续对应项，理论严谨。
- 适用于高维数据分类，尤其适合流形结构明显的场景。
流形假设认为高维数据通常位于低维子空间中。本文基于该假设，研究图基半监督学习方法，特别考察了在Wasserstein空间中的拉普拉斯学习，将经典的图基半监督学习算法从有限维欧氏空间推广至无限维设定。为此，我们证明了离散图p-Dirichlet能量向其连续对应项的变分收敛性，并刻画了Wasserstein空间子流形上的拉普拉斯-贝尔特拉米算子。最后，通过基准数据集上的数值实验验证了所提理论框架的有效性，展示了在高维设置下分类性能的一致性。
原文摘要 · Abstract (English)
The manifold hypothesis posits that high-dimensional data typically resides on low-dimensional sub spaces. In this paper, we assume manifold hypothesis to investigate graph-based semi-supervised learning methods. In particular, we examine Laplace Learning in the Wasserstein space, extending the classical notion of graph-based semi-supervised learning algorithms from finite-dimensional Euclidean spaces to an infinite-dimensional setting. To achieve this, we prove variational convergence of a discrete graph p- Dirichlet energy to its continuum counterpart. In addition, we characterize the Laplace-Beltrami operator on asubmanifold of the Wasserstein space. Finally, we validate the proposed theoretical framework through numerical experiments conducted on benchmark datasets, demonstrating the consistency of our classification performance in high-dimensional settings.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。