Solving Ill-Posed Linear Systems With Constraints on Statistical Moments

Abstract-The problem of finding a solution to an ill-posed linear problem Ax = b, with specific statistical properties is addressed, constraining the statistical moments of the N elements in x up to a given order d. It is reformulated as a higher dimension minimization problem with Nd variables, whose objective function is the composition of a convex function and a projection. Although convergence to a local minima is possible in rare cases, simulations show that in the vast majority of the cases, global convergence is attained with a standard descent algorithm. By extension, this opens the possibility to efficiently solve a larger class of problems that are linear in the powers of x. The tomographic reconstruction from a limited number of projection angles, constrained by centered moments, is considered as an example.