Laplacian Semi-Supervised Regression on a Manifold

Recently, manifold learning and semi-supervised learning are two hot topics in the field of machine learning. But there are only a few researches on semi-supervised learning from the point of manifold learning, especially for semi-supervised regression. In this paper, semi-supervised regression on manifolds is studied, which can employ the manifold structure hidden in datasets to the problem of regression estimation. Firstly the framework of Laplacian regularization presented by M. Belkin et al. is introduced. Then the framework of Laplacian semi-supervised regression with a class of generalized loss functions is deduced. Under this framework, Laplacian semi-supervised regression algorithms with linear ε-insensitive loss functions, quadric ε-insensitive loss functions and Huber loss functions are presented. Their experimental results on S-curve dataset and Boston Housing dataset are given and analyzed. The problem of semi-supervised regression on manifolds is interesting but quite difficult. The aim of this paper is only to accumulate some experience for further research in the future. There are still many hard problems on semi-supervised regression estimation on manifolds, such as constructing statistical basis of the algorithm, looking for better graph regularizer in the framework of Laplacian semi-supervised regression, designing quicker algorithms, implementing the algorithm on more datasets and so on.