Compressed Sensing and Discrete Optimization
The goal of this talk is to give an overview on compressed sensing from an discrete optimization/geometry point of view. The main problem in compressed sensing – the sparse representation problem – is to find the sparsest solutions of underdetermined linear equation systems. This problem is computationally hard, but can be efficiently solved by a linear program if certain conditions are satisfied. The talk will review some well known conditions and show that they are computationally hard to check. Moreover, the solution of this problem to global optimality will be discussed. The next step consists of presenting results on the unique recovery of integer solutions. Of course, even obtaining some feasible solution is hard in this case. Nevertheless, the potential of investing computational resources to obtain optimal solutions will be discussed.