Efficient Projection onto the ${\mathrm{\ell }}_{\mathrm{\infty },1}$ Mixed-Norm Ball using a Newton root search method

Mixed norms that promote structured sparsity have numerous applications in signal processing and machine learning problems. In this work, we present a new algorithm, based on a Newton root search technique, for computing the projection onto the ${\mathrm{\ell }}_{\mathrm{\infty },1}$ ball, which has found application in cognitive neuroscience and classification tasks. Numerical simulations show that our proposed method is between 8 and 10 times faster on average, and up to 20 times faster for very sparse solutions, than the previous state of the art. Tests on real functional magnetic resonance image data show that, for some data distributions, our algorithm can obtain speed improvements by a factor of between 10 and 100, depending on the implementation.