  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).
