Marginal MAP inference involves making MAP predictions in systems defined with latent variables or missing information. It is significantly more difficult than pure marginalization and MAP tasks, for which a large class of efficient and convergent variational algorithms, such as dual decomposition, exist. In this work, we generalize dual decomposition to a generic power sum inference task, which includes marginal MAP, along with pure marginalization and MAP, as special cases. Our method is based on a block coordinate descent algorithm on a new convex decomposition bound, that is guaranteed to converge monotonically, and can be parallelized efficiently. We demonstrate our approach on marginal MAP queries defined on real-world problems from the UAI approximate inference challenge, showing that our framework is faster and more reliable than previous methods.
Thus, most algorithms do not satisfy all the desirable properties listed in the introduction. For example, many works have developed convergent message-passing algorithms for convex free energies [e.g.,][] hazan08, hazan10 . However, by optimizing the dual they do not provide a bound until convergence, and the representation and constraints on the counting numbers do not facilitate optimizing the bound over these parameters. To optimize counting numbers, @cite_14 adopt a more restrictive free energy form requiring positive counting numbers on the entropies; but this cannot represent marginal MAP, whose free energy involves conditional entropies (equivalent to the difference between two entropy terms).
@cite_14
2103498186
We introduce a new technique that can reduce any higher-order Markov random field with binary labels into a first-order one that has the same minima as the original. Moreover, we combine the reduction with the fusion-move and QPBO algorithms to optimize higher-order multi-label problems. While many vision problems today are formulated as energy minimization problems, they have mostly been limited to using first-order energies, which consist of unary and pairwise clique potentials, with a few exceptions that consider triples. This is because of the lack of efficient algorithms to optimize energies with higher-order interactions. Our algorithm challenges this restriction that limits the representational power of the models, so that higher-order energies can be used to capture the rich statistics of natural scenes. To demonstrate the algorithm, we minimize a third-order energy, which allows clique potentials with up to four pixels, in an image restoration problem. The problem uses the fields of experts model, a learned spatial prior of natural images that has been used to test two belief propagation algorithms capable of optimizing higher-order energies. The results show that the algorithm exceeds the BP algorithms in both optimization performance and speed.