Maximum Persistency in Energy Minimization

We consider discrete pairwise energy minimization problem (weighted constraint satisfaction, max-sum labeling) and methods that identify a globally optimal partial assignment of variables. When finding a complete optimal assignment is intractable, determining optimal values for a part of variables is an interesting possibility. Existing methods are based on different sufficient conditions. We propose a new sufficient condition for partial optimality which is: (1) verifiable in polynomial time (2) invariant to reparametrization of the problem and permutation of labels and (3) includes many existing sufficient conditions as special cases. We pose the problem of finding the maximum optimal partial assignment identifiable by the new sufficient condition. A polynomial method is proposed which is guaranteed to assign same or larger part of variables than several existing approaches. The core of the method is a specially constructed linear program that identifies persistent assignments in an arbitrary multi-label setting.Comment: Extended technical report for the CVPR 2014 paper. Update: correction to the proof of characterization theore

Maximum Persistency via Iterative Relaxed Inference with Graphical Models

We consider the NP-hard problem of MAP-inference for undirected discrete graphical models. We propose a polynomial time and practically efficient algorithm for finding a part of its optimal solution. Specifically, our algorithm marks some labels of the considered graphical model either as (i) optimal, meaning that they belong to all optimal solutions of the inference problem; (ii) non-optimal if they provably do not belong to any solution. With access to an exact solver of a linear programming relaxation to the MAP-inference problem, our algorithm marks the maximal possible (in a specified sense) number of labels. We also present a version of the algorithm, which has access to a suboptimal dual solver only and still can ensure the (non-)optimality for the marked labels, although the overall number of the marked labels may decrease. We propose an efficient implementation, which runs in time comparable to a single run of a suboptimal dual solver. Our method is well-scalable and shows state-of-the-art results on computational benchmarks from machine learning and computer vision.Comment: Reworked version, submitted to PAM

A discriminative view of MRF pre-processing algorithms

While Markov Random Fields (MRFs) are widely used in computer vision, they present a quite challenging inference problem. MRF inference can be accelerated by pre-processing techniques like Dead End Elimination (DEE) or QPBO-based approaches which compute the optimal labeling of a subset of variables. These techniques are guaranteed to never wrongly label a variable but they often leave a large number of variables unlabeled. We address this shortcoming by interpreting pre-processing as a classification problem, which allows us to trade off false positives (i.e., giving a variable an incorrect label) versus false negatives (i.e., failing to label a variable). We describe an efficient discriminative rule that finds optimal solutions for a subset of variables. Our technique provides both per-instance and worst-case guarantees concerning the quality of the solution. Empirical studies were conducted over several benchmark datasets. We obtain a speedup factor of 2 to 12 over expansion moves without preprocessing, and on difficult non-submodular energy functions produce slightly lower energy.Comment: ICCV 201

Combinatorial persistency criteria for multicut and max-cut

In combinatorial optimization, partial variable assignments are called persistent if they agree with some optimal solution. We propose persistency criteria for the multicut and max-cut problem as well as fast combinatorial routines to verify them. The criteria that we derive are based on mappings that improve feasible multicuts, respectively cuts. Our elementary criteria can be checked enumeratively. The more advanced ones rely on fast algorithms for upper and lower bounds for the respective cut problems and max-flow techniques for auxiliary min-cut problems. Our methods can be used as a preprocessing technique for reducing problem sizes or for computing partial optimality guarantees for solutions output by heuristic solvers. We show the efficacy of our methods on instances of both problems from computer vision, biomedical image analysis and statistical physics

Potts model, parametric maxflow and k-submodular functions

The problem of minimizing the Potts energy function frequently occurs in computer vision applications. One way to tackle this NP-hard problem was proposed by Kovtun [19,20]. It identifies a part of an optimal solution by running $k$ maxflow computations, where $k$ is the number of labels. The number of "labeled" pixels can be significant in some applications, e.g. 50-93% in our tests for stereo. We show how to reduce the runtime to $O(\log k)$ maxflow computations (or one {\em parametric maxflow} computation). Furthermore, the output of our algorithm allows to speed-up the subsequent alpha expansion for the unlabeled part, or can be used as it is for time-critical applications. To derive our technique, we generalize the algorithm of Felzenszwalb et al. [7] for {\em Tree Metrics}. We also show a connection to {\em $k$-submodular functions} from combinatorial optimization, and discuss {\em $k$-submodular relaxations} for general energy functions.Comment: Accepted to ICCV 201

Foam: A General-Purpose Cellular Monte Carlo Event Generator

A general purpose, self-adapting, Monte Carlo (MC) event generator (simulator) is described. The high efficiency of the MC, that is small maximum weight or variance of the MC weight is achieved by means of dividing the integration domain into small cells. The cells can be $n$-dimensional simplices, hyperrectangles or Cartesian product of them. The grid of cells, called ``foam'', is produced in the process of the binary split of the cells. The choice of the next cell to be divided and the position/direction of the division hyper-plane is driven by the algorithm which optimizes the ratio of the maximum weight to the average weight or (optionally) the total variance. The algorithm is able to deal, in principle, with an arbitrary pattern of the singularities in the distribution. As any MC generator, it can also be used for the MC integration. With the typical personal computer CPU, the program is able to perform adaptive integration/simulation at relatively small number of dimensions ($\leq 16$). With the continuing progress in the CPU power, this limit will get inevitably shifted to ever higher dimensions. {\tt Foam} is aimed (and already tested) as a component in the MC event generators for the high energy physics experiments. A few simple examples of the related applications are presented. {\tt Foam} is written in fully object-oriented style, in the C++ language. Two other versions with a slightly limited functionality, are available in the Fortran77 language. The source codes are available from http://jadach.home.cern.ch/jadach

Efficient Semidefinite Branch-and-Cut for MAP-MRF Inference

We propose a Branch-and-Cut (B&C) method for solving general MAP-MRF inference problems. The core of our method is a very efficient bounding procedure, which combines scalable semidefinite programming (SDP) and a cutting-plane method for seeking violated constraints. In order to further speed up the computation, several strategies have been exploited, including model reduction, warm start and removal of inactive constraints. We analyze the performance of the proposed method under different settings, and demonstrate that our method either outperforms or performs on par with state-of-the-art approaches. Especially when the connectivities are dense or when the relative magnitudes of the unary costs are low, we achieve the best reported results. Experiments show that the proposed algorithm achieves better approximation than the state-of-the-art methods within a variety of time budgets on challenging non-submodular MAP-MRF inference problems.Comment: 21 page