The purpose of this thesis is to develop new theories on high-dimensional structured signal recovery under a rather weak assumption on the measurements that only a finite number of moments exists. High-dimensional recovery has been one of the emerging topics in the last decade partly due to the celebrated work of Candes, Romberg and Tao (e.g. [CRT06, CRT04]). The original analysis there (and the works thereafter) necessitates a strong concentration argument (namely, the restricted isometry property), which only holds for a rather restricted class of measurements with light-tailed distributions. It had long been conjectured that high-dimensional recovery is possible even if restricted isometry type conditions do not hold, but the general theory was beyond the grasp until very recently, when the works [Men14a, KM15] propose a new small-ball method. In these two papers, the authors initiated a new analysis framework for general empirical risk minimization (ERM) problems with respect to the square loss, which is robust and can potentially allow heavy-tailed loss functions. The materials in this thesis are partly inspired by [Men14a], but are of a different mindset: rather than directly analyzing the existing ERMs for signal recovery for which it is difficult to avoid strong moment assumptions, we show that, in many circumstances, by carefully re-designing the ERMs to start with, one can still achieve the minimax optimal statistical rate of signal recovery with very high probability under much weaker assumptions than existing works.
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