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

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而成功解决的。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、制造和遥感等领域。这些领域中许多正在进行的进展都是基于稀疏正则化（即，在适当转换时将数据建模为高度可压缩的）。本研究计划旨在通过开发稀疏正则化新技术来推进数学和计算工具来提出和解决病态逆问题。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，这些算法是全局收敛的，计算效率高，甚至对于大规模的非光滑问题也是如此。虽然L1范数是凸稀疏正则化的基础，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，它受到一些复杂性的阻碍：最优解通常是数据的不连续函数，而要最小化的目标函数通常具有许多次优的局部最小值，这可能会使优化算法陷入困境。本研究旨在利用非凸稀疏正则化的有效性，同时不放弃凸优化原理，将凸松弛原理应用于整个目标函数，而不是单独应用于正则化器。特别是，该程序开发了新的不可分离的非凸惩罚函数，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。

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.