# A Penalty Function Promoting Sparsity Within and Across Groups

arXiv: https://arxiv.org/abs/1603.03650 · HTML: https://mlsanity.com/paper/1603.03650/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Mar 2016.

## Built on by (most cited first, 2 of 2)

- [Exclusive Group Lasso for Structured Variable Selection](https://mlsanity.com/paper/2108.10284/) (2021)
- [A General Framework for Group Sparsity in Hyperspectral Unmixing Using Endmember Bundles](https://mlsanity.com/paper/2505.14634/) (2025)

## Abstract (arXiv)

We introduce a new weakly-convex penalty function for signals with a group behavior. The penalty promotes signals with a few number of active groups, where within each group, only a few high magnitude coefficients are active. We derive the threshold function associated with the proposed penalty and study its properties. We discuss how the proposed penalty/threshold function can be useful for signals with isolated non-zeros, such as audio with isolated harmonics along the frequency axis, or reflection functions in exploration seismology where the non-zeros occur on the boundaries of subsoil layers. We demonstrate the use of the proposed penalty/threshold functions in a convex denoising and a non-convex deconvolution formulation. We provide convergent algorithms for both formulations and compare the performance with state-of-the-art methods.