On the post selection inference constant under restricted isometry properties
- 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.
| Author details: | Francois Bachoc, Gilles BlanchardGND, Pierre Neuvial |
|---|---|
| DOI: | https://doi.org/10.1214/18-EJS1490 |
| ISSN: | 1935-7524 |
| Title of parent work (English): | Electronic journal of statistics |
| Publisher: | Institute of Mathematical Statistics |
| Place of publishing: | Cleveland |
| Publication type: | Article |
| Language: | English |
| Date of first publication: | 2018/11/20 |
| Publication year: | 2018 |
| Release date: | 2022/02/24 |
| Tag: | Inference post model-selection; PoSI constants; confidence intervals; high-dimensional inference; linear regression; restricted isometry property; sparsity |
| Volume: | 12 |
| Issue: | 2 |
| Number of pages: | 22 |
| First page: | 3736 |
| Last Page: | 3757 |
| Funding institution: | german DFGGerman Research Foundation (DFG) [FOR-1735]; german DFG, under Collaborative Research Center [SFB-1294]; [ANR-16-CE40-0019] |
| Organizational units: | Mathematisch-Naturwissenschaftliche Fakultät / Institut für Mathematik |
| DDC classification: | 5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik |
| Peer review: | Referiert |
| Publishing method: | Open Access / Gold Open-Access |
