Robust inference for general framework of projection structures
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We develop a general framework of projection structures and study the problem of inference on the unknown parameter within this framework by using empirical Bayes and penalization methods. The main inference problem is the uncertainty quantification, but on the way we solve the estimation and posterior contraction problems as well (and a weak version of the structure recovery problem). The approach is local in that the quality of the inference procedures is measured by the local quantity, the oracle rate, which is the best trade-off between the approximation error by a projection structure and the complexity of that approximating projection structure. The approach is also robust in that the stochastic part of the general framework is assumed to satisfy only certain mild condition, the errors may be non-iid with unknown distribution. We introduce the excessive bias restriction (EBR) under which we establish the local (oracle) confidence optimality of the constructed confidence ball. As the proposed general framework unifies a very broad class of high-dimensional models interesting and important on their own right, the obtained general results deliver a whole avenue of results (many new ones and some known in the literature) for particular models and structures as consequences, including white noise model and density estimation with smoothness structure, linear regression and dictionary learning with sparsity structures, biclustering and stochastic block models with clustering structure, covariance matrix estimation with banding and sparsity structures, and many others. Various adaptive minimax results over various scales follow also from our local results.
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