论文标题
通过明确的热 - 内核学习隐式学习歧管
Learning Manifold Implicitly via Explicit Heat-Kernel Learning
论文作者
论文摘要
多种应用程序中，多种学习是机器学习的基本问题。大多数现有方法直接在某些高维空间中直接学习数据的低维嵌入，并且通常缺乏直接适用于下游应用程序的灵活性。在本文中，我们提出了隐式多种学习的概念，其中通过学习相关的热内核可以隐式获得多种信息。热核是相应的热方程的解，它描述了歧管上的“热”如何转移，从而包含歧管的充足几何信息。我们提供实用算法和框架的理论分析。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生成模型（DGM）和贝叶斯推断的Stein变异梯度下降。广泛的实验表明，与这两个任务的现有方法相比，我们的框架可以实现最新的结果。
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.
