# A Scale Invariant Approach for Sparse Signal Recovery

arXiv: https://arxiv.org/abs/1812.08852 · HTML: https://mlsanity.com/paper/1812.08852/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Dec 2018.

## Built on by (most cited first, 13 of 13)

- [Minimizing L1 over L2 norms on the gradient](https://mlsanity.com/paper/2101.00809/) (2021)
- [An Efficient Smoothing and Thresholding Image Segmentation Framework with Weighted Anisotropic-Isotropic Total Variation](https://mlsanity.com/paper/2202.10115/) (2022)
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- [Quaternion Nuclear Norms Over Frobenius Norms Minimization for Robust Matrix Completion](https://mlsanity.com/paper/2504.21468/) (2025)
- [Difference of Anisotropic and Isotropic TV for Segmentation under Blur and Poisson Noise](https://mlsanity.com/paper/2301.03393/) (2023)
- [Robust Autoencoders for Collective Corruption Removal](https://mlsanity.com/paper/2303.02828/) (2023)
- [Orthogonally weighted ℓ2,1 regularization for rank-aware joint sparse recovery: algorithm and analysis](https://mlsanity.com/paper/2311.12282/) (2023)
- [Transformed ℓ1 Gradient Regularization for Image Denoising](https://mlsanity.com/paper/2511.15060/) (2025)
- [The Effective Number of Nonzeros: Theory and Regularization for Sparse Recovery](https://mlsanity.com/paper/2603.13826/) (2026)
- [A Unified Fractional Regularization Framework for Sparse Recovery](https://mlsanity.com/paper/2604.23184/) (2026)
- [Low-Rank Tensor Completion Based on Fractional Regularization with Ky Fan p-k Norm](https://mlsanity.com/paper/2606.19046/) (2026)
- [Separable Nonnegative Matrix Factorization Using Powered Ratio-of-Norms Regularization](https://mlsanity.com/paper/2608.28799/) (2026)

## Abstract (arXiv)

In this paper, we study the ratio of the L1  and L2  norms, denoted as L1/L2, to promote sparsity. Due to the non-convexity and non-linearity, there has been little attention to this scale-invariant model. Compared to popular models in the literature such as the Lp model for p∈(0,1) and the transformed L1 (TL1), this ratio model is parameter free. Theoretically, we present a strong null space property (sNSP) and prove that any sparse vector is a local minimizer of the L1 /L2  model provided with this sNSP condition. Computationally, we focus on a constrained formulation that can be solved via the alternating direction method of multipliers (ADMM). Experiments show that the proposed approach is comparable to the state-of-the-art methods in sparse recovery. In addition, a variant of the L1/L2 model to apply on the gradient is also discussed with a proof-of-concept example of the MRI reconstruction.