II。弱矩假设下的高维估计：结构恢复与矩阵估计
II. High Dimensional Estimation under Weak Moment Assumptions: Structured Recovery and Matrix Estimation
摘要：
本论文的目的是在一个相当弱的假设下，即只有有限个矩存在的测量，发展新的理论，高维结构化信号恢复。在过去的十年中，高维恢复一直是新兴的主题之一，部分原因是Candes，Romberg和Tao的着名工作（例如[CRT06，CRT04]）。那里的原始分析（以及此后的工作）需要一个强有力的浓度论证（即限制等距性质），这只适用于一类相当有限的具有轻尾分布的测量。长期以来，人们一直认为，即使限制等距型条件不成立，高维恢复也是可能的，但直到最近，当作品[Men14a，KM 15]提出一种新的小球方法时，一般理论才被掌握。在这两篇论文中，作者提出了一个新的分析框架，一般的经验风险最小化（ERM）问题的平方损失，这是强大的，可以潜在地允许重尾损失函数。本文中的材料部分受到[Men14a]的启发，但有不同的心态：我们不是直接分析现有的用于信号恢复的ERM，因为很难避免强矩假设，而是表明，在许多情况下，通过仔细地重新设计ERM来开始，在比现有工作弱得多的假设下，仍然可以以很高的概率获得极小极大最优信号恢复统计速率。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
