Learning Manifold Implicitly via Explicit Heat-Kernel Learning
Learning Manifold Implicitly via Explicit Heat-Kernel Learning
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发表时间：
2020-10
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通讯作者：
Yufan Zhou;Changyou Chen;Jinhui Xu
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作者：
Yufan Zhou;Changyou Chen;Jinhui Xu
流形学习是机器学习中的一个基本问题，有许多应用。现有的方法大多直接学习数据在某些高维空间的低维嵌入，通常缺乏直接适用于下游应用的灵活性。在本文中，我们提出了隐式流形学习的概念，其中流形信息是通过学习相关的热核隐式地获得。热核是相应热方程的解，它描述了“热”如何在流形上传递，因此包含了流形的丰富几何信息。我们提供了我们的框架的实际算法和理论分析。学习的热核可以应用于各种基于内核的机器学习模型，包括用于数据生成的深度生成模型（DGM）和用于贝叶斯推理的Stein变分梯度下降。大量的实验表明，我们的框架可以实现国家的最先进的结果相比，现有的方法，这两个任务。
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.