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This paper primarily focuses on computing the Euclidean projection of avector onto the $\ell_{p}$ ball in which $p\in(0,1)$. Such a problem emerges asthe core building block in statistical machine learning and signal processingtasks because of its ability to promote the sparsity of the desired solution.However, efficient numerical algorithms for finding the projections are stillnot available, particularly in large-scale optimization. To meet thischallenge, we first derive the first-order necessary optimality conditions ofthis problem. Based on this characterization, we develop a novel numericalapproach for computing the stationary point by solving a sequence ofprojections onto the reweighted $\ell_{1}$-balls. This method is practicallysimple to implement and computationally efficient. Moreover, the proposedalgorithm is shown to converge uniquely under mild conditions and has aworst-case $O(1/\sqrt{k})$ convergence rate. Numerical experiments demonstratethe efficiency of our proposed algorithm.

Towards An Efficient Approach for the Nonconvex $\ell_p$ Ball Projection: Algorithm and Analysis