Year: 2022
Type: preprint
Abstract: Inference by means of mathematical modeling from a collection of observations remains a crucial tool for scientific discovery and is ubiquitous in application areas such as signal compression, imaging restoration, and supervised machine learning. The inference problems may be solved using variational formulations that provide theoretically proven methods and algorithms. With ever-increasing model complexities and growing data size, new specially designed methods are urgently needed to recover meaningful quantifies of interest. We consider the broad spectrum of linear inverse problems where the aim is to reconstruct quantities with a sparse representation on some vector space; often solved using the (generalized) least absolute shrinkage and selection operator (lasso). The associated optimization problems have received significant attention, in particular in the early 2000's, because of their connection to compressed sensing and the reconstruction of solutions with favorable sparsity properties using augmented Lagrangians, alternating directions and splitting methods. We provide a new perspective on the underlying l1 regularized inverse problem by exploring the generalized lasso problem through variable projection methods. We arrive at our proposed variable projected augmented Lagrangian (vpal) method. We analyze this method and provide an approach for automatic regularization parameter selection based on a degrees of freedom argument. Further, we provide numerical examples demonstrating the computational efficiency for various imaging problems.
Source: arXiv (Cornell University)
Authors: Chung, Matthias, Renaut, Rosemary
Institutions: Emory University, Arizona State University
Language: English
Subfield: Computational Mechanics
Field: Engineering
Domain: Physical Sciences
Open Access status: green
Funder: National Science Foundation
Awards: Approximate Singular Value Expansions and Solutions of Ill-Posed Problems, Collaborative Research: Randomized Numerical Linear Algebra for Large Scale Inversion, Sparse Principal Component Analysis, and Applications, Collaborative Research: Stochastic Approximations for the Solution and Uncertainty Analysis of Data-Intensive Inverse Problems, Collaborative Research: Randomized Numerical Linear Algebra for Large Scale Inversion, Sparse Principal Component Analysis, and Applications
OpenAlex ID: W4286233342