CIF: Medium: Collaborative Research: Nonconvex Optimization for High-Dimensional Signal Estimation: Theory and Fast Algorithms

高维信号估计在医学成像、视频和网络监控等各种工程和科学应用中起着重要的作用。保持统计和计算效率的估计程序具有很大的实用价值，这转化为迫切需要，例如患者需要在医疗扫描仪上花费更少的时间，更快地响应网络攻击，以及处理非常大的数据集的能力。虽然许多信号估计任务自然地被公式化为非凸优化问题，但非凸方法的现有结果具有几个基本限制，并且在何时、为什么以及哪些非凸方法对于给定问题是有效的方面，现有技术仍然是有限的。本研究课题的目的是为了深化和拓宽非凸优化在高维信号估计中的理解和应用。在本课题中，研究人员将通过直接优化非凸的、潜在非光滑的损失函数来研究高维信号估计，而不需要借助凸松弛。本研究将探讨信号估计中常见的非凸函数所共有的几何结构，并研究这些结构在确定算法收敛性方面所起的基本作用。然后，这些结果将被利用作为指导方针，以开发快速和可证明正确的算法，用于估计高维信号与物理诱导的结构和流数据观测。具体而言，该研究计划包括三个主要方向：（1）理解重要类别的非凸损失曲面的几何结构，并表征它们对优化算法收敛性的影响：（2）发展结构低秩矩阵恢复的快速算法和相关理论;（3）设计新的在线算法，其在流设置下是时间和空间有效的，具有检测和跟踪感兴趣的时变信号的能力。

High-dimensional signal estimation plays fundamental roles in various engineering and science applications, such as medical imaging, video and network surveillance. Estimation procedures that maintain both statistical and computational efficacy are of great practical value, which translate into desiderata such as less time patients need to spend in a medical scanner, faster response to cyber attacks, and capabilities to handle very large datasets. While a lot of signal estimation tasks are naturally formulated as nonconvex optimization problems, existing results for nonconvex methods have several fundamental limitations, and the current state of the art is still limited in terms of when, why and which nonconvex approaches are effective for a given problem. The goal of this research program is to significantly deepen and broaden the understanding and applications of nonconvex optimization for high-dimensional signal estimation.In this project, the investigators will study high-dimensional signal estimation via direct optimization of nonconvex, and potentially nonsmooth, loss functions, without resorting to convex relaxation. This research will explore geometric structures shared by nonconvex functions commonly encountered in signal estimation, and study the fundamental roles these structures play in determining the algorithmic convergence. These results will then be exploited as guidelines to develop fast and provably correct algorithms for estimating high-dimensional signals with physically induced structures and under streaming data observations. Specifically, the research program consists of three major thrusts: (1) understanding the geometric structures of important classes of nonconvex loss surfaces, and characterizing their impact on the convergence of optimization algorithms; (2) developing fast algorithms and the associated theory for the recovery of structured low-rank matrices; (3) designing new online algorithms that are time and space efficient under a streaming setting, with the capability of detecting and tracking the time-varying signals of interest.