A Survey on Nonconvex Regularization Based Sparse and Low-Rank Recovery in Signal Processing, Statistics, and Machine Learning (1808.05403v3)

Abstract: In the past decade, sparse and low-rank recovery have drawn much attention in many areas such as signal/image processing, statistics, bioinformatics and machine learning. To achieve sparsity and/or low-rankness inducing, the $\ell_1$ norm and nuclear norm are of the most popular regularization penalties due to their convexity. While the $\ell_1$ and nuclear norm are convenient as the related convex optimization problems are usually tractable, it has been shown in many applications that a nonconvex penalty can yield significantly better performance. In recent, nonconvex regularization based sparse and low-rank recovery is of considerable interest and it in fact is a main driver of the recent progress in nonconvex and nonsmooth optimization. This paper gives an overview of this topic in various fields in signal processing, statistics and machine learning, including compressive sensing (CS), sparse regression and variable selection, sparse signals separation, sparse principal component analysis (PCA), large covariance and inverse covariance matrices estimation, matrix completion, and robust PCA. We present recent developments of nonconvex regularization based sparse and low-rank recovery in these fields, addressing the issues of penalty selection, applications and the convergence of nonconvex algorithms. Code is available at https://github.com/FWen/ncreg.git.

• The paper demonstrates that nonconvex regularization techniques reduce bias and require fewer measurements for accurate sparse signal recovery compared to convex approaches.
• The methodology improves high-dimensional variable selection and enhances image processing and matrix completion by leveraging penalties like SCAD and MCP.
• The paper’s comprehensive analysis paves the way for developing scalable algorithms with global convergence guarantees in complex nonconvex optimization problems.

A Survey on Nonconvex Regularization Based Sparse and Low-Rank Recovery in Signal Processing, Statistics, and Machine Learning

The paper provides an exhaustive examination of nonconvex regularization techniques for sparse and low-rank recovery, which have garnered significant interest due to their efficacy in various domains like signal processing, statistics, and machine learning. The research investigates the ways in which nonconvex penalties outperform traditional convex ones usually employed in tasks such as compressive sensing (CS), sparse regression, and matrix completion, offering solutions to overcome problems related to bias and sample efficiency.

The traditional approach to inducing sparsity and low-rank structure—using convex penalties such as the $\ell_1$ norm and nuclear norm—facilitates algorithmic tractability. While convexity simplifies convergence, these methods exhibit limitations, such as introducing bias and necessitating more observations than theoretically possible for optimal recovery. The paper critically evaluates nonconvex penalties like $\ell_0$ norm, $\ell_q$ norm, Smoothly Clipped Absolute Deviation (SCAD), and Minimax Concave Penalty (MCP), which address these issues.

Key Findings and Contributions

1. Compressive Sensing (CS): For sparse signal recovery, nonconvex regularization requires fewer measurements than convex methods. Algorithms using nonconvex penalties can achieve more precise and efficient reconstructions, thanks to the alleviation of the problems associated with bias.
2. Sparse Regression and Variable Selection: Nonconvex penalties significantly enhance the accuracy of variable selection and coefficient estimation in high-dimensional statistical models. SCAD and MCP, among others, have shown effectiveness over the traditional LASSO method.
3. Image Processing Applications: In tasks like image inpainting and robust recovery under impulsive noise, nonconvex penalties outperform convex counterparts, offering superior restoration quality and robustness.
4. Principal Component Analysis (PCA): In sparse PCA, which seeks interpretable loading vectors, nonconvex penalties promote sparsity effectively, thereby offering meaningful reductions in dimensionality while enhancing interpretability.
5. Covariance and Precision Matrix Estimation: Large sample covariance and sparse inverse covariance matrices benefit from nonconvex regularization, contributing to accurate high-dimensional data analyses such as in finance and bioinformatics.
6. Matrix Completion and Robust PCA: For low-rank matrix recovery tasks, nonconvex approaches such as Schatten-$q$ norm outperform nuclear-norm based methods, which translates into more accurate recommendations and computer vision tasks.

Practical Implications and Future Directions

This comprehensive survey underscores the potential and flexibility of nonconvex regularization in numerous applications, establishing a marked improvement over traditional convex methods. The paper discusses the need for caution in penalty selection, highlighting that the optimal penalty might depend on specific application conditions like data noise and intrinsic sparsity.

Theoretical advancements in optimization suggest more scalable and efficient algorithms for nonconvex problems. Proximal algorithms and the Alternating Direction Method of Multipliers (ADMM) have shown global convergence under certain conditions, emphasizing their practicality for real-world applications. Future work should focus on advancing algorithmic convergence guarantees for complex nonconvex optimization tasks, further enhancing practical efficacy across broader domains.

In conclusion, this paper provides a detailed and systematic exploration of nonconvex regularization techniques, situating them as crucial tools for current and future advancements in signal processing, statistics, and machine learning. The articulated results and methodologies serve as a valuable resource for researchers and practitioners seeking to leverage the advantages of nonconvex regularization in complex data recovery tasks.