Diffusion-based manifold learning methods have proven useful in representation learning and dimensionality reduction of modern high dimensional, high throughput, noisy datasets. Such datasets are especially present in fields like biology and physics. While it is thought that these methods preserve underlying manifold structure of data by learning a proxy for geodesic distances, no specific theoretical links have been established. Here, we establish such a link via results in Riemannian geometry explicitly connecting heat diffusion to manifold distances. In this process, we also formulate a more general heat kernel based manifold embedding method that we call heat geodesic embeddings. This novel perspective makes clearer the choices available in manifold learning and denoising. Results show that our method outperforms existing state of the art in preserving ground truth manifold distances, and preserving cluster structure in toy datasets. We also showcase our method on single cell RNA-sequencing datasets with both continuum and cluster structure, where our method enables interpolation of withheld timepoints of data. Finally, we show that parameters of our more general method can be configured to give results similar to PHATE (a state-of-the-art diffusion based manifold learning method) as well as SNE (an attraction/repulsion neighborhood based method that forms the basis of t-SNE).

翻译：基于扩散的流形学习方法在现代高维、高通量、噪声数据集（尤其在生物学和物理学领域）的表征学习与降维中已展现出显著效用。尽管学界普遍认为此类方法通过学习测地线距离的近似值来保留数据的潜在流形结构，但其具体理论联系尚未建立。本文通过黎曼几何中明确连接热扩散与流形距离的结论，首次建立了这种理论联系。在此过程中，我们提出了一种基于热核的广义流形嵌入方法——热测地线嵌入（heat geodesic embeddings）。这一新视角更清晰地揭示了流形学习与去噪中的可选项。实验结果表明，在保留真实流形距离和玩具数据集聚类结构方面，我们的方法优于现有最先进技术。我们还在兼具连续结构与聚类结构的单细胞RNA测序数据集上展示了该方法，并实现了对缺失时间点数据的插值。最后，我们证明通过配置该广义方法的参数，可得到与PHATE（基于扩散的先进流形学习方法）及SNE（构成t-SNE基础的吸引/排斥邻域方法）相似的结果。