In this paper we characterize sharp time-data tradeoffs for optimization problems used for solving linear inverse problems. We focus on the minimization of a least-squares objective subject to a constraint defined as the sub-level set of a penalty function. We present a unified convergence analysis of the gradient projection algorithm applied to such problems. We sharply characterize the convergence rate associated with a wide variety of random measurement ensembles in terms of the number of measurements and structural complexity of the signal with respect to the chosen penalty function. The results apply to both convex and nonconvex constraints, demonstrating that a linear convergence rate is attainable even though the least squares objective is not strongly convex in these settings. When specialized to Gaussian measurements our results show that such linear convergence occurs when the number of measurements is merely 4 times the minimal number required to recover the desired signal at all (a.k.a. the phase transition). We also achieve a slower but geometric rate of convergence precisely above the phase transition point. Extensive numerical results suggest that the derived rates exactly match the empirical performance.
Learning from samples: inverse problems over measures
INRIA (PreMeDICaL & HeKA)（法国国家信息与自动化研究所（PreMeDICaL & HeKA））；CNRS and ENS, PSL Université（法国国家科学研究中心与巴黎高等师范学院，PSL大学）；Mathematics Institute, University of Warwick（华威大学数学研究所）
The Rate-Distortion-Perception-Classification Tradeoff: Joint Source Coding and Modulation via Inverse-Domain GANs
School of Informatics, Xiamen University（厦门大学信息学院）；School of Electronics and Information Engineering, Guangxi Normal University（广西师范大学电子与信息工程学院）；School of Engineering & Physical Sciences, Heriot-Watt University（赫瑞瓦特大学工程与物理科学学院）
Sharp analysis of out-of-distribution error for "importance-weighted" estimators in the overparameterized regime
Georgia Institute of Technology（佐治亚理工学院）；School of Electrical & Computer Engineering（电气与计算机工程学院）；H. Milton School of Industrial & Systems Engineering（H. Milton工业与系统工程学院）
