# Fast L1-L2 minimization via a proximal operator

arXiv: https://arxiv.org/abs/1609.09530 · HTML: https://mlsanity.com/paper/1609.09530/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Sep 2016.

## Built on by (most cited first, 16 of 16)

- [Quaternion Nuclear Norm minus Frobenius Norm Minimization for color image reconstruction](https://mlsanity.com/paper/2409.07797/) (2024)
- [Multi-channel Nuclear Norm Minus Frobenius Norm Minimization for Color Image Denoising](https://mlsanity.com/paper/2209.08094/) (2022)
- [Minimizing L1 over L2 norms on the gradient](https://mlsanity.com/paper/2101.00809/) (2021)
- [Weighted Anisotropic-Isotropic Total Variation for Poisson Denoising](https://mlsanity.com/paper/2307.00439/) (2023)
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- [An Efficient Smoothing and Thresholding Image Segmentation Framework with Weighted Anisotropic-Isotropic Total Variation](https://mlsanity.com/paper/2202.10115/) (2022)
- [DC-LA: Difference-of-Convex Langevin Algorithm](https://mlsanity.com/paper/2601.22932/) (2026)
- [Sparse recovery based on the generalized error function](https://mlsanity.com/paper/2105.13189/) (2021)
- [Non-convex approaches for low-rank tensor completion under tubal sampling](https://mlsanity.com/paper/2303.12721/) (2023)
- [Multi-modal and frequency-weighted tensor nuclear norm for hyperspectral image denoising](https://mlsanity.com/paper/2106.12489/) (2021)
- [Difference of Anisotropic and Isotropic TV for Segmentation under Blur and Poisson Noise](https://mlsanity.com/paper/2301.03393/) (2023)
- [Untrained neural network embedded Fourier phase retrieval from few measurements](https://mlsanity.com/paper/2307.08717/) (2023)
- [Alternating Direction Method of Multipliers for Negative Binomial Model with The Weighted Difference of Anisotropic and Isotropic Total Variation](https://mlsanity.com/paper/2408.16117/) (2024)
- [Robust Tensor Completion via Gradient Tensor Nulclear L1-L2 Norm for Traffic Data Recovery](https://mlsanity.com/paper/2506.22732/) (2025)
- [Geometry Denoising with Preferred Normal Vectors](https://mlsanity.com/paper/2511.04848/) (2025)
- [Transformed ℓ1 Gradient Regularization for Image Denoising](https://mlsanity.com/paper/2511.15060/) (2025)

## Abstract (arXiv)

This paper aims to develop new and fast algorithms for recovering a sparse vector from a small number of measurements, which is a fundamental problem in the field of compressive sensing (CS). Currently, CS favors incoherent systems, in which any two measurements are as little correlated as possible. In reality, however, many problems are coherent, and conventional methods such as L1 minimization do not work well. Recently, the difference of the L1 and L2 norms, denoted as L1-L2, is shown to have superior performance over the classic L1 method, but it is computationally expensive. We derive an analytical solution for the proximal operator of the L1-L2 metric, and it makes some fast L1 solvers such as forward-backward splitting (FBS) and alternating direction method of multipliers (ADMM) applicable for L1-L2. We describe in details how to incorporate the proximal operator into FBS and ADMM and show that the resulting algorithms are convergent under mild conditions. Both algorithms are shown to be much more efficient than the original implementation of L1-L2 based on a difference-of-convex approach in the numerical experiments.