146 research outputs found
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
Submodular relaxation for inference in Markov random fields
In this paper we address the problem of finding the most probable state of a
discrete Markov random field (MRF), also known as the MRF energy minimization
problem. The task is known to be NP-hard in general and its practical
importance motivates numerous approximate algorithms. We propose a submodular
relaxation approach (SMR) based on a Lagrangian relaxation of the initial
problem. Unlike the dual decomposition approach of Komodakis et al., 2011 SMR
does not decompose the graph structure of the initial problem but constructs a
submodular energy that is minimized within the Lagrangian relaxation. Our
approach is applicable to both pairwise and high-order MRFs and allows to take
into account global potentials of certain types. We study theoretical
properties of the proposed approach and evaluate it experimentally.Comment: This paper is accepted for publication in IEEE Transactions on
Pattern Analysis and Machine Intelligenc
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
Block Stability for MAP Inference
To understand the empirical success of approximate MAP inference, recent work
(Lang et al., 2018) has shown that some popular approximation algorithms
perform very well when the input instance is stable. The simplest stability
condition assumes that the MAP solution does not change at all when some of the
pairwise potentials are (adversarially) perturbed. Unfortunately, this strong
condition does not seem to be satisfied in practice. In this paper, we
introduce a significantly more relaxed condition that only requires blocks
(portions) of an input instance to be stable. Under this block stability
condition, we prove that the pairwise LP relaxation is persistent on the stable
blocks. We complement our theoretical results with an empirical evaluation of
real-world MAP inference instances from computer vision. We design an algorithm
to find stable blocks, and find that these real instances have large stable
regions. Our work gives a theoretical explanation for the widespread empirical
phenomenon of persistency for this LP relaxation
Complexity of Discrete Energy Minimization Problems
Discrete energy minimization is widely-used in computer vision and machine
learning for problems such as MAP inference in graphical models. The problem,
in general, is notoriously intractable, and finding the global optimal solution
is known to be NP-hard. However, is it possible to approximate this problem
with a reasonable ratio bound on the solution quality in polynomial time? We
show in this paper that the answer is no. Specifically, we show that general
energy minimization, even in the 2-label pairwise case, and planar energy
minimization with three or more labels are exp-APX-complete. This finding rules
out the existence of any approximation algorithm with a sub-exponential
approximation ratio in the input size for these two problems, including
constant factor approximations. Moreover, we collect and review the
computational complexity of several subclass problems and arrange them on a
complexity scale consisting of three major complexity classes -- PO, APX, and
exp-APX, corresponding to problems that are solvable, approximable, and
inapproximable in polynomial time. Problems in the first two complexity classes
can serve as alternative tractable formulations to the inapproximable ones.
This paper can help vision researchers to select an appropriate model for an
application or guide them in designing new algorithms.Comment: ECCV'16 accepte
Structured Prediction Problem Archive
Structured prediction problems are one of the fundamental tools in machinelearning. In order to facilitate algorithm development for their numericalsolution, we collect in one place a large number of datasets in easy to readformats for a diverse set of problem classes. We provide archival links todatasets, description of the considered problems and problem formats, and ashort summary of problem characteristics including size, number of instancesetc. For reference we also give a non-exhaustive selection of algorithmsproposed in the literature for their solution. We hope that this centralrepository will make benchmarking and comparison to established works easier.We welcome submission of interesting new datasets and algorithms for inclusionin our archive.<br
Getting Feasible Variable Estimates From Infeasible Ones: MRF Local Polytope Study
This paper proposes a method for construction of approximate feasible primal
solutions from dual ones for large-scale optimization problems possessing
certain separability properties. Whereas infeasible primal estimates can
typically be produced from (sub-)gradients of the dual function, it is often
not easy to project them to the primal feasible set, since the projection
itself has a complexity comparable to the complexity of the initial problem. We
propose an alternative efficient method to obtain feasibility and show that its
properties influencing the convergence to the optimum are similar to the
properties of the Euclidean projection. We apply our method to the local
polytope relaxation of inference problems for Markov Random Fields and
demonstrate its superiority over existing methods.Comment: 20 page, 4 figure
A Comparative Study of Modern Inference Techniques for Structured Discrete Energy Minimization Problems
International audienceSzeliski et al. published an influential study in 2006 on energy minimization methods for Markov Random Fields (MRF). This study provided valuable insights in choosing the best optimization technique for certain classes of problems. While these insights remain generally useful today, the phenomenal success of random field models means that the kinds of inference problems that have to be solved changed significantly. Specifically , the models today often include higher order interactions, flexible connectivity structures, large label-spaces of different car-dinalities, or learned energy tables. To reflect these changes, we provide a modernized and enlarged study. We present an empirical comparison of more than 27 state-of-the-art optimization techniques on a corpus of 2,453 energy minimization instances from diverse applications in computer vision. To ensure reproducibility, we evaluate all methods in the OpenGM 2 framework and report extensive results regarding runtime and solution quality. Key insights from our study agree with the results of Szeliski et al. for the types of models they studied. However, on new and challenging types of models our findings disagree and suggest that polyhedral methods and integer programming solvers are competitive in terms of runtime and solution quality over a large range of model types
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