# A Heat Diffusion Perspective on Geodesic Preserving Dimensionality Reduction

arXiv: https://arxiv.org/abs/2305.19043 · HTML: https://mlsanity.com/paper/2305.19043/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 36 references that are in the map.
- First posted on arXiv 30 May 2023.

## Builds on (most cited first, 7 of 7)

- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [Clustering with t-SNE, provably](https://mlsanity.com/paper/1706.02582/) (2017)
- [Attraction-Repulsion Spectrum in Neighbor Embeddings](https://mlsanity.com/paper/2007.08902/) (2020)
- [The Vector Heat Method](https://mlsanity.com/paper/1805.09170/) (2018)
- [Ground Metric Learning on Graphs](https://mlsanity.com/paper/1911.03117/) (2019)
- [Fast Polynomial Approximation of Heat Kernel Convolution on Manifolds and Its Application to Brain Sulcal and Gyral Graph Pattern Analysis](https://mlsanity.com/paper/1911.02721/) (2019)
- [Time-inhomogeneous diffusion geometry and topology](https://mlsanity.com/paper/2203.14860/) (2022)

## Built on by (most cited first, 6 of 6)

- [Geometry-Aware Generative Autoencoders for Warped Riemannian Metric Learning and Generative Modeling on Data Manifolds](https://mlsanity.com/paper/2410.12779/) (2024)
- [Enforcing Latent Euclidean Geometry in Single-Cell VAEs for Manifold Interpolation](https://mlsanity.com/paper/2507.11789/) (2025)
- [HiPoNet: A Multi-View Simplicial Complex Network for High Dimensional Point-Cloud and Single-Cell Data](https://mlsanity.com/paper/2502.07746/) (2025)
- [MS-IMAP -- A Multi-Scale Graph Embedding Approach for Interpretable Manifold Learning](https://mlsanity.com/paper/2406.02778/) (2024)
- [A Spectral Decomposition Framework for Multiscale Nonlinear Dimensionality Reduction](https://mlsanity.com/paper/2604.02535/) (2026)
- [EntroPath: Maximum Entropy Path Ensemble Embedding for Manifold Learning](https://mlsanity.com/paper/2607.06497/) (2026)

## Abstract (arXiv)

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).