On the Post Selection Inference constant under Restricted Isometry Properties

TL;DR

This paper derives a new upper bound on Post-Selection Inference constants for design matrices satisfying the Restricted Isometry Property, improving understanding of confidence intervals after model selection.

Contribution

It introduces an explicit upper bound on PoSI constants based on RIP constants, bridging orthogonal and generic sparse design matrices, with asymptotic optimality proofs.

Findings

New upper bound on PoSI constants for RIP matrices

Explicit function of RIP constant interpolates between orthogonal and sparse cases

Asymptotically optimal bounds with matching lower bounds

Abstract

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.