# A unified approach for projections onto the intersection of ℓ1 and ℓ2 balls or spheres

arXiv: https://arxiv.org/abs/1911.03946 · HTML: https://mlsanity.com/paper/1911.03946/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Nov 2019.

## Built on by (most cited first, 1 of 1)

- [Gradient Preconditioning for Efficient and Reliable Reward-Guided Generation](https://mlsanity.com/paper/2602.08646/) (2026)

## Abstract (arXiv)

This paper focuses on designing a unified approach for computing the projection onto the intersection of an ℓ1 ball/sphere and an ℓ2 ball/sphere. We show that the major computational efforts of solving these problems all rely on finding the root of the same piecewisely quadratic function, and then propose a unified numerical method to compute the root. In particular, we design breakpoint search methods with/without sorting incorporated with bisection, secant and Newton methods to find the interval containing the root, on which the root has a closed form. It can be shown that our proposed algorithms without sorting possess O(n log n) worst-case complexity and O(n) in practice. The efficiency of our proposed algorithms are demonstrated in numerical experiments.