# Learning Manifold Implicitly via Explicit Heat-Kernel Learning

arXiv: https://arxiv.org/abs/2010.01761 · HTML: https://mlsanity.com/paper/2010.01761/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Oct 2020.

## Built on by (most cited first, 7 of 7)

- [Learning Signal-Agnostic Manifolds of Neural Fields](https://mlsanity.com/paper/2111.06387/) (2021)
- [General Graph Random Features](https://mlsanity.com/paper/2310.04859/) (2023)
- [Quasi-Monte Carlo Graph Random Features](https://mlsanity.com/paper/2305.12470/) (2023)
- [Landing with the Score: Riemannian Optimization through Denoising](https://mlsanity.com/paper/2509.23357/) (2025)
- [Linear Transformer Topological Masking with Graph Random Features](https://mlsanity.com/paper/2410.03462/) (2024)
- [Repelling Random Walks](https://mlsanity.com/paper/2310.04854/) (2023)
- [IIKL: Isometric Immersion Kernel Learning with Riemannian Manifold for Geometric Preservation](https://mlsanity.com/paper/2505.06288/) (2025)

## Abstract (arXiv)

Manifold learning is a fundamental problem in machine learning with numerous applications. Most of the existing methods directly learn the low-dimensional embedding of the data in some high-dimensional space, and usually lack the flexibility of being directly applicable to down-stream applications. In this paper, we propose the concept of implicit manifold learning, where manifold information is implicitly obtained by learning the associated heat kernel. A heat kernel is the solution of the corresponding heat equation, which describes how "heat" transfers on the manifold, thus containing ample geometric information of the manifold. We provide both practical algorithm and theoretical analysis of our framework. The learned heat kernel can be applied to various kernel-based machine learning models, including deep generative models (DGM) for data generation and Stein Variational Gradient Descent for Bayesian inference. Extensive experiments show that our framework can achieve state-of-the-art results compared to existing methods for the two tasks.