Transductive versions of the LASSO and the Dantzig Selector

TL;DR

This paper introduces transductive versions of the LASSO and Dantzig Selector that leverage both labeled and unlabeled data to improve high-dimensional prediction, especially with correlated predictors.

Contribution

It develops novel transductive estimators for LASSO and Dantzig Selector, providing theoretical guarantees under weaker assumptions than traditional methods.

Findings

Transductive estimators outperform classical methods in many scenarios.

They are particularly effective in high-dimensional, correlated predictor settings.

Theoretical analysis confirms their sparsity properties under relaxed conditions.

Abstract

Transductive methods are useful in prediction problems when the training dataset is composed of a large number of unlabeled observations and a smaller number of labeled observations. In this paper, we propose an approach for developing transductive prediction procedures that are able to take advantage of the sparsity in the high dimensional linear regression. More precisely, we define transductive versions of the LASSO and the Dantzig Selector . These procedures combine labeled and unlabeled observations of the training dataset to produce a prediction for the unlabeled observations. We propose an experimental study of the transductive estimators, that shows that they improve the LASSO and Dantzig Selector in many situations, and particularly in high dimensional problems when the predictors are correlated. We then provide non-asymptotic theoretical guarantees for these estimation methods. Interestingly, our theoretical results show that the Transductive LASSO and Dantzig Selector satisfy sparsity inequalities under weaker assumptions than those required for the "original" LASSO.