A filtered bucket-clustering method for projection onto the simplex and the $$\ell _1$$ ℓ 1 ball

Guillaume Perez (1) , Michel Barlaud (2) , Lionel Fillatre (3) , Jean-Charles Régin (1)

1 Laboratoire d'Informatique, Signaux, et Systèmes de Sophia-Antipolis (I3S) / Equipe CEP
2 Laboratoire I3S - SIS - Signal, Images et Systèmes
3 I3S - Laboratoire d'Informatique, Signaux, et Systèmes de Sophia Antipolis

We propose in this paper a new method processing the projection of an arbitrary size vector onto the probabilistic simplex or the 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. Mathematics Subject Classification 49M30 · 65C60 · 65K05 · 90C25

Domaines: Traitement des images [eess.IV]; Machine Learning [stat.ML]

Guillaume Perez, Michel Barlaud, Lionel Fillatre, Jean-Charles Régin. A filtered bucket-clustering method for projection onto the simplex and the $$\ell _1$$ ℓ 1 ball. Mathematical Programming, 2019, ⟨10.1007/s10107-019-01401-3⟩. ⟨hal-02131738⟩