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In the first part, we develop a general framework for projection structuresand study several inference problems within this framework. We proposeprocedures based on data dependent measures (DDM) and make connections withempirical Bayes and penalization methods. The main inference problem is theuncertainty quantification (UQ), but on the way we solve the estimation,DDM-contraction problems, and a weak version of the structure recovery problem.The approach is local in that the quality of the inference procedures ismeasured by the local quantity, the oracle rate, which is the best trade-offbetween the approximation error by a projection structure and the complexity ofthat approximating projection structure. Like in statistical learning settings,we develop distribution-free theory as no particular model is imposed, we onlyassume certain mild condition on the stochastic part of the projectionpredictor. We introduce the excessive bias restriction (EBR) under which weestablish the local confidence optimality of the constructed confidence ball.  The proposed general framework unifies a very broad class of high-dimensionalmodels and structures, interesting and important on their own right. In thesecond part, we apply the developed theory and demonstrate how the generalresults deliver a whole avenue of local and global minimax results (many newones, some known results from the literature are improved) for particularmodels and structures as consequences, including white noise model and densityestimation with smoothness structure, linear regression and dictionary learningwith sparsity structures, biclustering and stochastic block models withclustering structure, covariance matrix estimation with banding and sparsitystructures, and many others. Various adaptive minimax results over variousscales follow also from our local results.

General framework for projection structures