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Manifold learning is a fundamental problem in machine learning with numerousapplications. Most of the existing methods directly learn the low-dimensionalembedding of the data in some high-dimensional space, and usually lack theflexibility of being directly applicable to down-stream applications. In thispaper, we propose the concept of implicit manifold learning, where manifoldinformation is implicitly obtained by learning the associated heat kernel. Aheat kernel is the solution of the corresponding heat equation, which describeshow "heat" transfers on the manifold, thus containing ample geometricinformation of the manifold. We provide both practical algorithm andtheoretical analysis of our framework. The learned heat kernel can be appliedto various kernel-based machine learning models, including deep generativemodels (DGM) for data generation and Stein Variational Gradient Descent forBayesian inference. Extensive experiments show that our framework can achievestate-of-the-art results compared to existing methods for the two tasks.

Learning Manifold Implicitly via Explicit Heat-Kernel Learning