Abstract: There are many problems in the fields of natural science which are difficult to be resolved due to continuous variation. These complex problems can be expressed as the combination of a series of simple problems which are distributed among the discrete spaces. The approximate solution of the complex problems can be obtained by solving the simple problems. In recent years, the spectral learning based on spectral mathematic theory is attracting more and more attention in machine learning. Compared with traditional learning methods, it can not only preserve the latent structure in the data, but also obtain a global optimization solution. This paper firstly introduces the basic theory of spectral learning, then shows some typical algorithms including spectral clustering, spectral learning of latent variable probabilistic model and spectral manifold learning, and finally presents some worthy perspectives according to the current researches.

Key words: spectral learning, spectral clustering, latent variable probabilistic model, spectral manifold learning

摘要： 在自然科学研究领域存在众多因连续变化而难以解决的问题。这些复杂问题可以通过谱方法表示为一系列离散空间上的简单问题的组合，通过求解这些简单问题获得其近似解。谱学习算法是近年来国际上机器学习领域的一个研究热点。谱学习算法建立在谱数学理论基础上，与传统的学习算法相比，一方面能保持数据内部潜在结构不变，另一方面能获得全局最优解。首先介绍了谱学习的基本理论，然后从谱聚类算法、概率模型谱学习算法、谱流形学习算法3个不同方面介绍了相关的典型算法，最后针对目前的研究现状，给出了谱学习几个有价值的研究方向。

关键词: 谱学习, 谱聚类, 缺失变量概率模型, 谱流形学习