Finding a sparse signal solution of an underdetermined linear system of measurements is commonly solved in compressed sensing by convexly relaxing the sparsity requirement with the help of the l1 norm. Here, we tackle instead the original nonsmooth nonconvex l0-problem formulation using projected gradient methods. Our interest is motivated by a recent surprising numerical find that despite the perceived global optimization challenge of the l0-formulation, these simple local methods when applied to it can be as effective as first-order methods for the convex l1-problem in terms of the degree of sparsity they can recover from similar levels of undersampled measurements. We attempt here to give an analytical justification in the language of asymptotic phase transitions for this observed behaviour when Gaussian measurement matrices are employed. Our approach involves novel convergence techniques that analyse the fixed points of the algorithm and an asymptotic probabilistic analysis of the convergence conditions that derives asymptotic bounds on the extreme singular values of combinatorially many submatrices of the Gaussian measurement matrix under matrix-signal independence assumptions.
This work is joint with Andrew Thompson (Duke University, USA).