Paper:
Constraints Based Convex Belief Propagation
Inference in Markov random fields subject to consistency structure is a fundamental problem that arises in many real-life applications. In order to enforce consistency, classical approaches utilize consistency potentials or encode constraints over feasible instances. Unfortunately this comes at the price of a tremendous computational burden. In this paper we suggest to tackle consistency by incorporating constraints on beliefs. This permits derivation of a closed-form message-passing algorithm which we refer to as the Constraints Based Convex Belief Propagation (CBCBP). Experiments show that CBCBP outperforms the conventional consistency potential based approach, while being at least an order of magnitude faster.

Review:
The authors propose belief propagation for discrete graphical models with certain consistency constraints, related to PN-potentials. The contribution lies in deriving closed-form solutions to the belief-propagation operations in this more general model and showing empirically that it is advantageous over naively doing belief-propagation on general factors explicitly modeling the consistency constraints. 
General comments:  (i) The authors only solve a new special kind of higher order consistency constraints, generalizing soft PN-potentials, but not a truly general class of constraints, as indicated in the title or in the abstract.  (ii) I do not agree with what the authors say in lines 72 - 78. In case of MAP-inference, which is normally desired, the goal is to obtain a single assignment which satisfies all given linear constraints. The proposed model (i.e. computing marginals) is then less desirable. The relaxed model the authors optimize is simply a byproduct of looking for marginals instead of MAP-assignments (the added entropy is responsible for this). In case of vanishing entropy one gets the same model. Hence there certainly remains the disadvantage of a parameter in the PN-potential, but now hidden in the entropy. Additionally, when a MAP-solution is wanted, the proposed algorithm is in fact disadvantageous, as inference is not done w.r.t. the energy of a single MAP-solution and rounding results in some arbitrariness of the obtained solution.  (iii) Experimental comparison: - No comparison against solving MAP-inference with PN-potentials and existing inference algorithms is given. - Experiments are very small scale, e.g. image segmentation is only done on superpixels. I do not consider such microbenchmarks very informative. One can usually just plug everything into an off-the-shelf LP-solver in such cases. - To be able to really judge the algorithm, convergence plots would be helpful, but none are given. In general, I deem experimental comparison unconvincing: while many experiments have been performed, no real comparison against any algorithm other than CBP is performed.  (iv) Related work: Many references to work on inference with higher order potentials are lacking. To name a few: - Komodakis: Beyond pairwise energies: Efficient optimization for higher-order MRFs  - Tarlow et al.: HOP-MAP: Efficient Message Passing with High Order Potentials. - Kappes: Higher-order Segmentation via Multicuts E.g., the work of Kappes shows how to very efficiently include PN-potentials in an LP-relaxation. Generally, efficient ways for PN-potentials have been explored before, which is not acknowledged in the author's text.  (v) The work is rather incremental: The main contribution is the derivation of a new updating formula for convex belief propagation and consistency constraints.  Detailed comments: - The entropy H is never defined. - Notation is scattered around: Some is defined in section 2 and some is defined in section 3, but only informally in the text, making the article harder to read. - Line 8: What is the standard approach? - Line 71: which is expect -> which is expected - Line 96: One can also derive duals when the primal is not strictly convex (duals exist even for non-convex programs).