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The purpose of this thesis is to develop new theories on high-dimensionalstructured signal recovery under a rather weak assumption on the measurementsthat only a finite number of moments exists. High-dimensional recovery has beenone of the emerging topics in the last decade partly due to the celebrated workof 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 ratherrestricted class of measurements with light-tailed distributions. It had longbeen conjectured that high-dimensional recovery is possible even if restrictedisometry type conditions do not hold, but the general theory was beyond thegrasp until very recently, when the works [Men14a, KM15] propose a newsmall-ball method. In these two papers, the authors initiated a new analysisframework for general empirical risk minimization (ERM) problems with respectto the square loss, which is robust and can potentially allow heavy-tailed lossfunctions. The materials in this thesis are partly inspired by [Men14a], butare of a different mindset: rather than directly analyzing the existing ERMsfor signal recovery for which it is difficult to avoid strong momentassumptions, we show that, in many circumstances, by carefully re-designing theERMs to start with, one can still achieve the minimax optimal statistical rateof signal recovery with very high probability under much weaker assumptionsthan existing works.

II. High Dimensional Estimation under Weak Moment Assumptions: Structured Recovery and Matrix Estimation