Critical point theory for sparse recovery

We study the problem of sparse recovery in the context of compressed sensing. This is to minimize the sensing error of linear measurements by sparse vectors with at most $s$ non-zero entries. We develop the so-called critical point theory for sparse recovery. This is done by introducing nondegenerate M-stationary points which adequately describe the global structure of this nonconvex optimization problem. We show that all M-stationary points are generically nondegenerate. In particular, the sparsity constraint is active at all local minimizers of a generic sparse recovery problem. Additionally, the equivalence of strong stability and nondegeneracy for M-stationary points is shown. We claim that the appearance of saddle points - these are M-stationary points with exactly $s-1$ non-zero entries - cannot be neglected. For this purpose we derive a so-called Morse relation, which gives a lower bound on the number of saddle points in terms of the number of local minimizers. The relatively involved structure of saddle points can be seen as a source of well-known difficulty by solving the problem of sparse recovery to global optimality.