Log In Sign Up

Learning Manifold Implicitly via Explicit Heat-Kernel Learning

by   Yufan Zhou, et al.

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.


page 18

page 19

page 20

page 21


Vector Diffusion Maps and the Connection Laplacian

We introduce vector diffusion maps (VDM), a new mathematical framework ...

Geodesic Distance Function Learning via Heat Flow on Vector Fields

Learning a distance function or metric on a given data manifold is of gr...

Visualization of Unsteady Flow Using Heat Kernel Signatures

We introduce a new technique to visualize complex flowing phenomena by u...

Heat kernel coupling for multiple graph analysis

In this paper, we introduce heat kernel coupling (HKC) as a method of co...

Exact heat kernel on a hypersphere and its applications in kernel SVM

Many contemporary statistical learning methods assume a Euclidean featur...

Heat Kernel analysis of Syntactic Structures

We consider two different data sets of syntactic parameters and we discu...

A kernel-based method for coarse graining complex dynamical systems

We present a novel kernel-based machine learning algorithm for identifyi...