40 research outputs found
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
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
Scalable Semidefinite Relaxation for Maximum A Posterior Estimation
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.Comment: accepted to International Conference on Machine Learning (ICML 2014
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Tightness of LP relaxations for almost balanced models
This is the author accepted manuscript. The final version is available from MIcrotome Publishing via http://www.jmlr.org/proceedings/papers/v51/weller16b.html.Linear programming (LP) relaxations are widely used to attempt to identify a most likely configuration of a discrete graphical model. In some cases, the LP relaxation attains an optimum vertex at an integral location and thus guarantees an exact solution to the original optimization problem. When this occurs, we say that the LP relaxation is tight. Here we consider binary pairwise models and derive sufficient conditions for guaranteed tightness of (i) the standard LP relaxation on the local polytope LP+LOC, and (ii) the LP relaxation on the triplet-consistent polytope LP+TRI (the next level in the Sherali-Adams hierarchy). We provide simple new proofs of earlier results and derive significant novel results including that LP+TRI is tight for any model where each block is balanced or almost balanced, and a decomposition theorem that may be used to break apart complex models into smaller pieces. An almost balanced (sub-)model is one that contains no frustrated cycles except through one privileged variable.MR acknowledges support by the UK Engineering and Physical Sciences Research Council (EPSRC) grant EP/L016516/1 for the University of Cambridge Centre for Doctoral Training, the Cambridge Centre for Analysis. DS was supported by NSF CAREER award #1350965
New Convex Relaxations and Global Optimality in Variational Imaging
Variational methods constitute the basic building blocks for solving many image analysis tasks, be it segmentation, depth estimation, optical flow, object detection etc. Many of these problems can be expressed in the framework of Markov Random Fields (MRF) or as continuous labelling problems. Finding the Maximum A-Posteriori (MAP) solutions of suitably constructed MRFs or the optimizers of the labelling problems give solutions to the aforementioned tasks. In either case, the associated optimization problem amounts to solving structured energy minimization problems.
In this thesis we study novel extensions applicable to Markov Random Fields and continuous labelling problems through which we are able to incorporate statistical global constraints. To this end, we devise tractable relaxations of the resulting energy minimization problem and efficient algorithms to tackle them. Second, we propose a general mechanism to find partial optimal solutions to the problem of finding a MAP-solution of an MRF, utilizing only standard relxations
Cutting plane algorithms for variational inference in graphical models
Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2007.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Includes bibliographical references (leaves 65-66).In this thesis, we give a new class of outer bounds on the marginal polytope, and propose a cutting-plane algorithm for efficiently optimizing over these constraints. When combined with a concave upper bound on the entropy, this gives a new variational inference algorithm for probabilistic inference in discrete Markov Random Fields (MRFs). Valid constraints are derived for the marginal polytope through a series of projections onto the cut polytope. Projecting onto a larger model gives an efficient separation algorithm for a large class of valid inequalities arising from each of the original projections. As a result, we obtain tighter upper bounds on the logpartition function than possible with previous variational inference algorithms. We also show empirically that our approximations of the marginals are significantly more accurate. This algorithm can also be applied to the problem of finding the Maximum a Posteriori assignment in a MRF, which corresponds to a linear program over the marginal polytope. One of the main contributions of the thesis is to bring together two seemingly different fields, polyhedral combinatorics and probabilistic inference, showing how certain results in either field can carry over to the other.by David Alexander Sontag.S.M
Large-scale Binary Quadratic Optimization Using Semidefinite Relaxation and Applications
In computer vision, many problems such as image segmentation, pixel
labelling, and scene parsing can be formulated as binary quadratic programs
(BQPs). For submodular problems, cuts based methods can be employed to
efficiently solve large-scale problems. However, general nonsubmodular problems
are significantly more challenging to solve. Finding a solution when the
problem is of large size to be of practical interest, however, typically
requires relaxation. Two standard relaxation methods are widely used for
solving general BQPs--spectral methods and semidefinite programming (SDP), each
with their own advantages and disadvantages. Spectral relaxation is simple and
easy to implement, but its bound is loose. Semidefinite relaxation has a
tighter bound, but its computational complexity is high, especially for large
scale problems. In this work, we present a new SDP formulation for BQPs, with
two desirable properties. First, it has a similar relaxation bound to
conventional SDP formulations. Second, compared with conventional SDP methods,
the new SDP formulation leads to a significantly more efficient and scalable
dual optimization approach, which has the same degree of complexity as spectral
methods. We then propose two solvers, namely, quasi-Newton and smoothing Newton
methods, for the dual problem. Both of them are significantly more efficiently
than standard interior-point methods. In practice, the smoothing Newton solver
is faster than the quasi-Newton solver for dense or medium-sized problems,
while the quasi-Newton solver is preferable for large sparse/structured
problems. Our experiments on a few computer vision applications including
clustering, image segmentation, co-segmentation and registration show the
potential of our SDP formulation for solving large-scale BQPs.Comment: Fixed some typos. 18 pages. Accepted to IEEE Transactions on Pattern
Analysis and Machine Intelligenc