  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.
