Bottleneck Potentials in {Markov Random Fields}

We consider general discrete Markov Random Fields(MRFs) with additional bottleneck potentials which penalize the maximum (instead of the sum) over local potential value taken by the MRF-assignment. Bottleneck potentials or analogous constructions have been considered in (i) combinatorial optimization (e.g. bottleneck shortest path problem, the minimum bottleneck spanning tree problem, bottleneck function minimization in greedoids), (ii) inverse problems with $L_{\infty}$-norm regularization, and (iii) valued constraint satisfaction on the $(\min,\max)$-pre-semirings. Bottleneck potentials for general discrete MRFs are a natural generalization of the above direction of modeling work to Maximum-A-Posteriori (MAP) inference in MRFs. To this end, we propose MRFs whose objective consists of two parts: terms that factorize according to (i) $(\min,+)$, i.e. potentials as in plain MRFs, and (ii) $(\min,\max)$, i.e. bottleneck potentials. To solve the ensuing inference problem, we propose high-quality relaxations and efficient algorithms for solving them. We empirically show efficacy of our approach on large scale seismic horizon tracking problems

Empirical evaluation of Soft Arc Consistency algorithms for solving Constraint Optimization Problems

A large number of problems in Artificial Intelligence and other areas of science can be viewed as special cases of constraint satisfaction or optimization problems. Various approaches have been widely studied, including search, propagation, and heuristics. There are still challenging real-world COPs that cannot be solved using current methods. We implemented and compared several consistency propagation algorithms, which include W-AC*2001, EDAC, VAC, and xAC. Consistency propagation is a classical method to reduce the search space in CSPs, and has been adapted to COPs. We compared several consistency propagation algorithms, based on the resemblance between the optimal value ordering and the approximate value ordering generated by them. The results showed that xAC generated value orderings of higher quality than W-AC*2001 and EDAC. We evaluated some novel hybrid methods for solving COPs. Hybrid methods combine consistency propagation and search in order to reach a good solution as soon as possible and prune the search space as much as possible. We showed that the hybrid method which combines the variant TP+OnOff and branch-and-bound search performed fewer constraint checks and searched fewer nodes than others in solving random and real-world COPs

Computing a partition function of a generalized pattern-based energy over a semiring

Valued constraint satisfaction problems with ordered variables (VCSPO) are a special case of Valued CSPs in which variables are totally ordered and soft constraints are imposed on tuples of variables that do not violate the order. We study a restriction of VCSPO, in which soft constraints are imposed on a segment of adjacent variables and a constraint language $\Gamma$ consists of $\{0,1\}$-valued characteristic functions of predicates. This kind of potentials generalizes the so-called pattern-based potentials, which were applied in many tasks of structured prediction. For a constraint language $\Gamma$ we introduce a closure operator, $\overline{\Gamma^{\cap}}\supseteq \Gamma$, and give examples of constraint languages for which $|\overline{\Gamma^{\cap}}|$ is small. If all predicates in $\Gamma$ are cartesian products, we show that the minimization of a generalized pattern-based potential (or, the computation of its partition function) can be made in ${\mathcal O}(|V|\cdot |D|^2 \cdot |\overline{\Gamma^{\cap}}|^2 )$ time, where $V$ is a set of variables, $D$ is a domain set. If, additionally, only non-positive weights of constraints are allowed, the complexity of the minimization task drops to ${\mathcal O}(|V|\cdot |\overline{\Gamma^{\cap}}| \cdot |D| \cdot \max_{\rho\in \Gamma}\|\rho\|^2 )$ where $\|\rho\|$ is the arity of $\rho\in \Gamma$. For a general language $\Gamma$ and non-positive weights, the minimization task can be carried out in ${\mathcal O}(|V|\cdot |\overline{\Gamma^{\cap}}|^2)$ time. We argue that in many natural cases $\overline{\Gamma^{\cap}}$ is of moderate size, though in the worst case $|\overline{\Gamma^{\cap}}|$ can blow up and depend exponentially on $\max_{\rho\in \Gamma}\|\rho\|$