On the post selection inference constant under restricted isometry properties

François Bachoc (1) , Gilles Blanchard (2) , Pierre Neuvial (1)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 Institut für Mathematik [Potsdam]

Uniformly valid confidence intervals post model selection in regression can be constructed based on Post-Selection Inference (PoSI) constants. PoSI constants are minimal for orthogonal design matrices, and can be upper bounded in function of the sparsity of the set of models under consideration, for generic design matrices. In order to improve on these generic sparse upper bounds, we consider design matrices satisfying a Restricted Isometry Property (RIP) condition. We provide a new upper bound on the PoSI constant in this setting. This upper bound is an explicit function of the RIP constant of the design matrix, thereby giving an interpolation between the orthogonal setting and the generic sparse setting. We show that this upper bound is asymptotically optimal in many settings by constructing a matching lower bound.

Mots clés: Inference post model-selection; confidence intervals; PoSI constants; linear regression; high-dimensional inference; sparsity; restricted isometry property

Domaines: Statistiques [math.ST]; Apprentissage [cs.LG]; Intelligence artificielle [cs.AI]

François Bachoc, Gilles Blanchard, Pierre Neuvial. On the post selection inference constant under restricted isometry properties. Electronic Journal of Statistics , 2018, 12 (2), ⟨10.1214/18-EJS1490⟩. ⟨hal-03327160v3⟩