Learning Manifold Implicitly via Explicit Heat-Kernel Learning

TL;DR

This paper introduces implicit manifold learning by learning heat kernels, providing a flexible approach that enhances various kernel-based models and achieves state-of-the-art results in data generation and Bayesian inference.

Contribution

It proposes a novel implicit manifold learning framework through heat kernel learning, with theoretical analysis and practical algorithms for improved downstream applications.

Findings

Achieves state-of-the-art results in data generation.

Enhances Bayesian inference with heat kernel-based methods.

Provides a flexible approach applicable to multiple kernel-based models.

Abstract

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.