A filtered bucket-clustering method for projection onto the simplex and the ℓ _1 ball

We propose in this paper a new method processing the projection of an arbitrary size vector onto the probabilistic simplex or the ℓ _1 ball. Our method merges two principles. The first one is an original search of the projection using a bucket algorithm. The second one is a filtering, on the fly, of the values that cannot be part of the projection. The combination of these two principles offers a simple and efficient algorithm whose worst-case complexity is linear with respect to the vector size. Furthermore, the proposed algorithm exploits the representation of numeric values in digital computers to define the number of buckets and to accelerate the filtering.