Detection boundary in sparse regression

Yuri I. Ingster (1) , Alexandre Tsybakov (2, 3) , Nicolas Verzelen (4)

1 LETI - Saint Petersburg Electrotechnical University
2 CREST - Centre de Recherche en Économie et Statistique
3 LPMA - Laboratoire de Probabilités et Modèles Aléatoires
4 MISTEA - Mathématiques, Informatique et STatistique pour l'Environnement et l'Agronomie

We study the problem of detection of a p-dimensional sparse vector of parameters in the linear regression model with Gaussian noise. We establish the detection boundary, i.e., the necessary and sufficient conditions for the possibility of successful detection as both the sample size n and the dimension p tend to the infinity. Testing procedures that achieve this boundary are also exhibited. Our results encompass the high-dimensional setting (p>> n). The main message is that, under some conditions, the detection boundary phenomenon that has been proved for the Gaussian sequence model, extends to high-dimensional linear regression. Finally, we establish the detection boundaries when the variance of the noise is unknown. Interestingly, the detection boundaries sometimes depend on the knowledge of the variance in a high-dimensional setting.

Mots clés: High-dimensional regression; detection boundary; sparse vectors; sparsity; minimax hypothesis testing

Domaines: Statistiques [math.ST]; Théorie [stat.TH]

Yuri I. Ingster, Alexandre Tsybakov, Nicolas Verzelen. Detection boundary in sparse regression. 2010. ⟨hal-00516259⟩