Coarse-to-Fine Minimization of Some Common Nonconvexities

The continuation method is a popular heuristic in computer vision for nonconvex optimization. The idea is to start from a simplified problem and gradually deform it to the actual problem while tracking the solution. There are many choices for how to map the nonconvex objective to some convex task. One popular principle for such construction is Gaussian smoothing of the objective function. This involves an integration which may be expensive to compute numerically. We argue that often simple tricks at the problem formulation plus some mild approximations can make the resulted task amenable to closed form integral.