A unified approach for projections onto the intersection of $\ell_1$ and $\ell_2$ balls or spheres

TL;DR

This paper introduces a unified numerical approach for efficiently projecting onto the intersection of $\\ell_1$ and $\\ell_2$ balls or spheres, leveraging root-finding of a piecewise quadratic function.

Contribution

It develops a general method that simplifies and accelerates projections onto these intersections by solving a common root-finding problem with optimized algorithms.

Findings

Algorithms without sorting have $O(n \log n)$ worst-case complexity.

Proposed methods are efficient in practice, demonstrated through numerical experiments.

Unified approach reduces computational effort across different projection problems.

Abstract

This paper focuses on designing a unified approach for computing the projection onto the intersection of an $\ell_1$ ball/sphere and an $\ell_2$ 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.