CIF: Small: Bypassing the L1 Norm: Non-Convex Regularization, Convex Optimization, and Sparse Signal Processing

CIF：小：绕过 L1 范数：非凸正则化、凸优化和稀疏信号处理

• 批准号: 1525398
• 负责人: Ivan Selesnick
• 金额: $ 31.52万
• 依托单位: New York University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-08-01 至 2019-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1525398&HistoricalAwards=false
• 关键词: CIF; Small; Bypassing; L1; Norm

Numerous problems in nonlinear signal processing and data science are successfully tackled through their formulation as ill-conditioned inverse problems. Such problems arise in medical imaging, speech and audio processing, biomedical time-series analysis, manufacturing, and remote sensing. Many ongoing advances in these fields are based on sparse regularization (i.e., the modeling of data as highly compressible when appropriately transformed). This research program aims to advance mathematical and computational tools to pose and solve ill-conditioned inverse problems by developing new techniques for sparse regularization. Convex formulations of problems arising in science and engineering are attractive because one may leverage a wealth of algorithms that are globally convergent and computationally efficient, even for large-scale non-smooth problems. While the L1 norm is a cornerstone of convex sparse regularization, it tends to under-estimate signal values. Non-convex sparse regularization is therefore a popular and valuable alternative; however, it is hampered by several complications: the optimal solution is generally a discontinuous function of the data and the objective function to be minimized generally possesses many sub-optimal local minima which can entrap optimization algorithms. This research aims to exploit the effectiveness of non-convex sparse regularization without forgoing the principles of convex optimization, by applying the principle of convex relaxation to the objective function as a whole, rather than to the regularizer alone. In particular, this program undertakes the development of new non-separable non-convex penalty functions that ensure the convexity of the objective function (comprising data fidelity and sparse regularization terms) to be minimized.

非线性信号处理和数据科学中的许多问题都是通过将其表述为病态逆问题来成功解决的。这些问题出现在医学成像、语音和音频处理、生物医学时间序列分析、制造和遥感中。这些领域中的许多正在进行的进展是基于稀疏正则化（即，当适当地变换时，将数据建模为高度可压缩的）。该研究计划旨在通过开发稀疏正则化的新技术来推进数学和计算工具，以提出和解决病态逆问题。在科学和工程中出现的问题的凸公式是有吸引力的，因为人们可以利用大量的算法，全局收敛和计算效率，即使是大规模的非光滑问题。虽然L1范数是凸稀疏正则化的基石，但它往往会低估信号值。因此，非凸稀疏正则化是一种流行且有价值的替代方案;然而，它受到几个复杂因素的阻碍：最优解通常是数据的不连续函数，并且要最小化的目标函数通常具有许多次优局部极小值，这可能会陷入优化算法。本研究的目的是利用非凸稀疏正则化的有效性，而不放弃凸优化的原则，通过将凸松弛的原则应用于目标函数作为一个整体，而不是单独的正则化。特别是，该计划负责开发新的不可分离的非凸罚函数，以确保目标函数（包括数据保真度和稀疏正则化项）的凸性最小化。