Density-sensitive semisupervised inference

TL;DR

This paper develops a theoretical framework for density-sensitive semisupervised inference, analyzing methods that leverage the distribution of features to improve prediction, and introduces an adaptive approach to the strength of these assumptions.

Contribution

It provides a minimax analysis of density-sensitive semisupervised methods and proposes an adaptive procedure for tuning the assumption strength parameter.

Findings

Framework unifies various density-sensitive methods

Adaptive procedure effectively tunes the assumption strength

Theoretical guarantees established for the proposed approach

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 $\alpha$.