A First Course in Sparse Optimization

TL;DR

This paper provides a comprehensive overview of sparse optimization, covering foundational concepts, algorithms, and applications in signal recovery and regularization, serving as an entry point and reference for researchers and practitioners.

Contribution

It offers a thorough, self-contained introduction to sparse optimization, combining intuitive explanations with rigorous mathematical details, and covers key algorithms and applications.

Findings

Detailed explanation of sparse signal recovery methods

Overview of sparse regularization techniques like LASSO and elastic net

Discussion of algorithms such as basis pursuit and matching pursuit

Abstract

This article aims to provide a comprehensive overview of sparse optimization, with a focus on both sparse signal recovery and sparse regularization techniques. We will begin by exploring the foundations of sparse optimization, delving into the mathematical tools and models that underpin sparse signal recovery and LASSO. We will then discuss key algorithms for both sparse recovery (e.g., basis pursuit, matching pursuit) and sparse regularization (e.g., LASSO, elastic net), along with their applications in real-world problems. Throughout the text, we balance intuitive explanations with rigorous mathematical formulations to provide a comprehensive resource for both newcomers and experts in the field. Our aim is twofold: to provide a self-contained entry point for students and researchers new to the field, and to offer a rigorous reference for practitioners seeking to apply sparse optimization in science and engineering.