Convergence of projected subgradient method with sparse or low-rank constraints

Many problems in data science can be treated as recovering structural signals from a set of linear measurements, sometimes perturbed by dense noise or sparse corruptions. In this paper, we develop a unified framework of considering a nonsmooth formulation with sparse or low-rank constraint for meeting the challenges of mixed noises—bounded noise and sparse noise. We show that the nonsmooth formulations of the problems can be well solved by the projected subgradient methods at a rapid rate when initialized at any points. Consequently, nonsmooth loss functions (\(\ell _1\)-minimization programs) are naturally robust against sparse noise. Our framework simplifies and generalizes the existing analyses including compressed sensing, matrix sensing, quadratic sensing, and bilinear sensing. Motivated by recent work on the stochastic gradient method, we also give some experimentally and theoretically preliminary results about the projected stochastic subgradient method.