  The sparse spike estimation problem consists in estimating a number of
off-the-grid impulsive sources from under-determined linear measurements.
Information theoretic results ensure that the minimization of a non-convex
functional is able to recover the spikes for adequately chosen measurements
(deterministic or random). To solve this problem, methods inspired from the
case of finite dimensional sparse estimation where a convex program is used
have been proposed. Also greedy heuristics have shown nice practical results.
However, little is known on the ideal non-convex minimization method. In this
article, we study the shape of the global minimum of this non-convex
functional: we give an explicit basin of attraction of the global minimum that
shows that the non-convex problem becomes easier as the number of measurements
grows. This has important consequences for methods involving descent algorithms
(such as the greedy heuristic) and it gives insights for potential improvements
of such descent methods.
