The importance of aggregated count data, which is calculated from the data of multiple individuals, continues to increase. Collective Graphical Model (CGM) is a probabilistic approach to the analysis of aggregated data. One of the most important operations in CGM is maximum a posteriori (MAP) inference of unobserved variables under given observations. Because the MAP inference problem for general CGMs has been shown to be NP-hard, an approach that solves an approximate problem has been proposed. However, this approach has two major drawbacks. First, the quality of the solution deteriorates when the values in the count tables are small, because the approximation becomes inaccurate. Second, since continuous relaxation is applied, the integrality constraints of the output are violated. To resolve these problems, this paper proposes a new method for MAP inference for CGMs on path graphs. First we show that the MAP inference problem can be formulated as a (non-linear) minimum cost flow problem. Then, we apply Difference of Convex Algorithm (DCA), which is a general methodology to minimize a function represented as the sum of a convex function and a concave function. In our algorithm, important subroutines in DCA can be efficiently calculated by minimum convex cost flow algorithms. Experiments show that the proposed method outputs higher quality solutions than the conventional approach.

Solving constrained combinatorial optimisation problems via MAP inference is often achieved by introducing extra potential functions for each constraint. This can result in very high order potentials, e.g. a 2nd-order objective with pairwise potentials and a quadratic constraint over all N variables would correspond to an unconstrained objective with an order-N potential. This limits the practicality of such an approach, since inference with high order potentials is tractable only for a few special classes of functions. We propose an approach which is able to solve constrained combinatorial problems using belief propagation without increasing the order. For example, in our scheme the 2nd-order problem above remains order 2 instead of order N. Experiments on applications ranging from foreground detection, image reconstruction, quadratic knapsack, and the M-best solutions problem demonstrate the effectiveness and efficiency of our method. Moreover, we show several situations in which our approach outperforms commercial solvers like CPLEX and others designed for specific constrained MAP inference problems.

Maximum a posteriori (MAP) inference over discrete Markov random fields is a fundamental task spanning a wide spectrum of real-world applications, which is known to be NP-hard for general graphs. In this paper, we propose a novel semidefinite relaxation formulation (referred to as SDR) to estimate the MAP assignment. Algorithmically, we develop an accelerated variant of the alternating direction method of multipliers (referred to as SDPAD-LR) that can effectively exploit the special structure of the new relaxation. Encouragingly, the proposed procedure allows solving SDR for large-scale problems, e.g., problems on a grid graph comprising hundreds of thousands of variables with multiple states per node. Compared with prior SDP solvers, SDPAD-LR is capable of attaining comparable accuracy while exhibiting remarkably improved scalability, in contrast to the commonly held belief that semidefinite relaxation can only been applied on small-scale MRF problems. We have evaluated the performance of SDR on various benchmark datasets including OPENGM2 and PIC in terms of both the quality of the solutions and computation time. Experimental results demonstrate that for a broad class of problems, SDPAD-LR outperforms state-of-the-art algorithms in producing better MAP assignment in an efficient manner.

Dual decomposition provides a tractable framework for designing algorithms for finding the most probable (MAP) configuration in graphical models. However, for many real-world inference problems, the typical decomposition has a large integrality gap, due to frustrated cycles. One way to tighten the relaxation is to introduce additional constraints that explicitly enforce cycle consistency. Earlier work showed that cluster-pursuit algorithms, which iteratively introduce cycle and other higherorder consistency constraints, allows one to exactly solve many hard inference problems. However, these algorithms explicitly enumerate a candidate set of clusters, limiting them to triplets or other short cycles. We solve the search problem for cycle constraints, giving a nearly linear time algorithm for finding the most frustrated cycle of arbitrary length. We show how to use this search algorithm together with the dual decomposition framework and clusterpursuit. The new algorithm exactly solves MAP inference problems arising from relational classification and stereo vision.