Density-sensitive semisupervised inference
Martin Azizyan, Aarti Singh, Larry Wasserman
The Annals of Statistics · 2013 · Volume 41 · Issue 2
Abstract
Semisupervised methods are techniques for using labeled data $(X_{1},Y_{1}),\ldots,(X_{n},Y_{n})$ together with unlabeled data $X_{n+1},\ldots,X_{N}$ 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 $\alpha$.
Keywords
62G07, 62G15, efficiency, kernel density, nonparametric inference, semisupervised
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