Fast projection onto the ℓ_∞,1-Mixed norm ball using steffensen root search

Mixed norms that promote structured sparsity have broad application in signal processing and machine learning problems. In this work we present a new algorithm for computing the projection onto the ℓ_∞,1 ball, which has found application in cognitive neuroscience and classification tasks. This algorithm is based on a Steffensen type root search technique, with a number of improvements over prior root search methods for the same problem. First, we theoretically derive an initial guess for the root search algorithm that helps to reduce the number of iterations to be performed. Second, we change the root search method, and through an analysis of the root search function, we construct a pruning strategy that significantly reduces the number of operations. Numerical simulations show that, compared to the state-of-the-art, our algorithm is between 4 and 5 times faster on average, and of up to 14 times faster for very sparse solutions.