Uncertainty quantification and confidence sets in high-dimensional models

While much attention has been paid recently to the construction of optimal algorithms that adaptively estimate low-dimensional parameters (described by sparsity, low-rank, or smoothness) in high-dimensional models, the theory of statistical inference and uncertainty quantification (in particular hypothesis tests & confidence sets) is much less well-developed. We will discuss some perhaps surprising impossibility results in the basic high-dimensional compressed sensing model, and some of the recently remerging positive results in the area.