DENSITY-SENSITIVE SEMISUPERVISED INFERENCE

Semisupervised methods are techniques for using labeled data (X-1, Y-1), ..., (X-n, Y-n) together with unlabeled data Xn+1, ..., X-N to make predictions. These methods invoke some assumptions that link the marginal distribution P-X of X to the regression function f(x). For example, it is common to assume that f is very smooth over high density regions of P-X. Many of the methods are ad-hoc and have been shown to work in specific examples but are lacking a theoretical foundation. We provide a minimax framework for analyzing semisupervised methods. In particular, we study methods based on metrics that are sensitive to the distribution P-X. Our model includes a parameter alpha that controls the strength of the semisupervised assumption. We then use the data to adapt to alpha.