Geometric aspects of uncertainty quantification in high-dimensional statistics
In high-dimensional statistics, uncertainty quantification is often of its own mathematical interest and complexity. We discuss phenomena that are particular to high-dimensional inference tasks, which closely relate to the need to `adapt' to hidden lower-dimensional structures that are not directly observable but crucial for doing inference in high-dimensional models. We demonstrate information-theoretic results that show estimation and uncertainty quantification are fundamentally distinctive problems, the difficulty of which is governed by different aspects of the data and model. As it is natural to seek uncertainty quantification procedures that match the accuracy of optimal estimation, the existence of such procedures depends on the relative difficulty of performing estimation and uncertainty quantification. We show that this is critically affected by the choice of loss functions. Thus the problem can be understood from a geometric perspective. Specifically, under the sparse regression model, we layout results for foundational choices of loss functions while adding to the field by introducing the reweighted $\ell_2$ type of loss functions. The findings under the new class of loss functions not only unify some existing theories, they also show a non-trivial regime of uncertainty quantification adaptive to a sparse estimation rate, which was previously thought to be non-existent.
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