Asymptotic Confidence Regions and Sharp Oracle Results under Structured Sparsity
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Author
Date
2017Type
- Doctoral Thesis
ETH Bibliography
yes
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Abstract
To restrict ourselves to the regime of sparse solutions has become the new paradigm for modern statistics, machine learning and in particular for high dimensional linear regression models. Sparse solutions aim at representing the information by a small core of active explanatory parameters. As a pleasant side effect the resulting models bear plain and relatively easy interpretations. This indistinctive sparsity is commonly represented by the ℓ1-norm, which is a convex relaxation of the number of active variables. Reducing complexity in this way is well understood. Nevertheless in practical applications we commonly have more knowledge about the structure of possible arrangements. Therefore the topic of structured sparsity has recently emerged as a new promising way to represent the prior knowledge of the underlying sparsity structure. In this thesis we focus on embodying the prior knowledge of potential sparsity patterns through general norm penalties. Weak decomposability is a fundamental concept in understanding the sparsity structure a norm yields. The idea of weak decomposability is further generalized to LASSO type estimators with concave penalties. We also see that sharp oracle results can be obtained in the multivariate model. The square root LASSO is generalized to all weakly decomposable norm penalties, where sharp oracle results are given. The properties of the scaling of these square root estimators have nice applications for constructing χ2 confidence regions for the LASSO. Furthermore assigning uncertainty in high dimensionality for structured sparsity estimators is tackled by means of two related frameworks. Show more
Permanent link
https://doi.org/10.3929/ethz-b-000197854Publication status
publishedExternal links
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Publisher
ETH ZurichSubject
High-dimensional regression; Sparsity; Penalized estimation; Sharp oracle inequality; Asymptotic confidence intervallsOrganisational unit
03717 - van de Geer, Sara (emeritus) / van de Geer, Sara (emeritus)
Funding
149145 - Inference in high-dimensional statistics (SNF)
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