# Revisiting Lq(0≤ q<1) Norm Regularized Optimization

arXiv: https://arxiv.org/abs/2306.14394 · HTML: https://mlsanity.com/paper/2306.14394/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 26 Jun 2023.

## Built on by (most cited first, 3 of 3)

- [Bi-Sparse Unsupervised Feature Selection](https://mlsanity.com/paper/2412.16819/) (2024)
- [Federated Structured Sparse PCA for Anomaly Detection in IoT Networks](https://mlsanity.com/paper/2503.23981/) (2025)
- [GoPrune: Accelerated Structured Pruning with ℓ2,p-Norm Optimization](https://mlsanity.com/paper/2511.22120/) (2025)

## Abstract (arXiv)

Sparse optimization has seen its advances in recent decades. For scenarios where the true sparsity is unknown, regularization turns out to be a promising solution. Two popular non-convex regularizations are the so-called L0 norm and Lq norm with q∈(0,1), giving rise to extensive research on their induced optimization. However, the majority of these work centered around the main function that is twice continuously differentiable and the best convergence rate for an algorithm solving the optimization with q∈(0,1) is superlinear. This paper explores the Lq norm regularized optimization in a unified way for any q∈[0,1), where the main function has a semismooth gradient. In particular, we establish the first-order and the second-order optimality conditions under mild assumptions and then integrate the proximal operator and semismooth Newton method to develop a proximal semismooth Newton pursuit algorithm. Under the second sufficient condition, the whole sequence generated by the algorithm converges to a unique local minimizer. Moreover, the convergence is superlinear and quadratic if the gradient of the main function is semismooth and strongly semismooth at the local minimizer, respectively. Hence, this paper accomplishes the quadratic rate for an algorithm designed to solve the Lq norm regularization problem for any q∈(0,1). Finally, some numerical experiments have showcased its nice performance when compared with several existing solvers.