Solving relaxations of MAP-MRF problems: Combinatorial in-face Frank-Wolfe directions

We consider the problem of solving LP relaxations of MAP-MRF inference problems, and in particular the method proposed recently in [16, 35]. As a key computational subroutine, it uses a variant of the Frank-Wolfe (FW) method to minimize a smooth convex function over a combinatorial polytope. We propose an efficient implementation of this subroutine based on in-face Frank-Wolfe directions, introduced in [4] in a different context. More generally, we define an abstract data structure for a combinatorial subproblem that enables in-face FW directions, and describe its specialization for tree-structured MAP-MRF inference subproblems. Experimental results indicate that the resulting method is the current state-of-art LP solver for some classes of problems. Our code is available at pub.ist. ac.at/similar to vnk/papers/IN-FACE- FW.html.