MSE Analysis of the Projected Gradient Algorithm on Unions of Subspaces

We present a mean-squared-error (MSE) analysis of the projected gradient algorithm when applied to linear inverse problems with constraint sets that are unions of subspaces. This algorithm has been recently presented as a generalization of the iterative hard thresholding algorithm. The main difficulty with the MSE analysis is to calculate the expected value of the supremum of a continuous random process. We present a simple approach to calculate the expected supremum, which builds upon the concept of Lipschitz random processes.