38 research outputs found
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
The Power of Linear Programming for Valued CSPs
A class of valued constraint satisfaction problems (VCSPs) is characterised
by a valued constraint language, a fixed set of cost functions on a finite
domain. An instance of the problem is specified by a sum of cost functions from
the language with the goal to minimise the sum. This framework includes and
generalises well-studied constraint satisfaction problems (CSPs) and maximum
constraint satisfaction problems (Max-CSPs).
Our main result is a precise algebraic characterisation of valued constraint
languages whose instances can be solved exactly by the basic linear programming
relaxation. Using this result, we obtain tractability of several novel and
previously widely-open classes of VCSPs, including problems over valued
constraint languages that are: (1) submodular on arbitrary lattices; (2)
bisubmodular (also known as k-submodular) on arbitrary finite domains; (3)
weakly (and hence strongly) tree-submodular on arbitrary trees.Comment: Corrected a few typo
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
Speeding up weighted constraint satisfaction using redundant modeling.
Woo Hiu Chun.Thesis (M.Phil.)--Chinese University of Hong Kong, 2006.Includes bibliographical references (leaves 91-99).Abstracts in English and Chinese.Chapter 1 --- Introduction --- p.1Chapter 1.1 --- Constraint Satisfaction Problems --- p.1Chapter 1.2 --- Weighted Constraint Satisfaction Problems --- p.3Chapter 1.3 --- Redundant Modeling --- p.4Chapter 1.4 --- Motivations and Goals --- p.5Chapter 1.5 --- Outline of the Thesis --- p.6Chapter 2 --- Background --- p.8Chapter 2.1 --- Constraint Satisfaction Problems --- p.8Chapter 2.1.1 --- Backtracking Tree Search --- p.9Chapter 2.1.2 --- Local Consistencies --- p.12Chapter 2.1.3 --- Local Consistencies in Backtracking Search --- p.17Chapter 2.1.4 --- Permutation CSPs --- p.19Chapter 2.2 --- Weighted Constraint Satisfaction Problems --- p.20Chapter 2.2.1 --- Branch and Bound Search --- p.23Chapter 2.2.2 --- Local Consistencies --- p.26Chapter 2.2.3 --- Local Consistencies in Branch and Bound Search --- p.32Chapter 2.3 --- Redundant Modeling --- p.34Chapter 3 --- Generating Redundant WCSP Models --- p.37Chapter 3.1 --- Model Induction for CSPs --- p.38Chapter 3.1.1 --- Stated Constraints --- p.39Chapter 3.1.2 --- No-Double-Assignment Constraints --- p.39Chapter 3.1.3 --- At-Least-One-Assignment Constraints --- p.40Chapter 3.2 --- Generalized Model Induction for WCSPs --- p.43Chapter 4 --- Combining Mutually Redundant WCSPs --- p.47Chapter 4.1 --- Naive Approach --- p.47Chapter 4.2 --- Node Consistency Revisited --- p.51Chapter 4.2.1 --- Refining Node Consistency Definition --- p.52Chapter 4.2.2 --- Enforcing m-NC* c Algorithm --- p.55Chapter 4.3 --- Arc Consistency Revisited --- p.58Chapter 4.3.1 --- Refining Arc Consistency Definition --- p.60Chapter 4.3.2 --- Enforcing m-AC*c Algorithm --- p.62Chapter 5 --- Experiments --- p.67Chapter 5.1 --- Langford's Problem --- p.68Chapter 5.2 --- Latin Square Problem --- p.72Chapter 5.3 --- Discussion --- p.75Chapter 6 --- Related Work --- p.77Chapter 6.1 --- Soft Constraint Satisfaction Problems --- p.77Chapter 6.2 --- Other Local Consistencies in WCSPs --- p.79Chapter 6.2.1 --- Full Arc Consistency --- p.79Chapter 6.2.2 --- Pull Directional Arc Consistency --- p.81Chapter 6.2.3 --- Existential Directional Arc Consistency --- p.82Chapter 6.3 --- Redundant Modeling and Channeling Constraints --- p.83Chapter 7 --- Concluding Remarks --- p.85Chapter 7.1 --- Contributions --- p.85Chapter 7.2 --- Future Work --- p.87List of Symbols --- p.88Bibliograph
The power of linear programming for general-valued CSPs
Let , called the domain, be a fixed finite set and let , called
the valued constraint language, be a fixed set of functions of the form
, where different functions might have
different arity . We study the valued constraint satisfaction problem
parametrised by , denoted by VCSP. These are minimisation
problems given by variables and the objective function given by a sum of
functions from , each depending on a subset of the variables.
Finite-valued constraint languages contain functions that take on only rational
values and not infinite values.
Our main result is a precise algebraic characterisation of valued constraint
languages whose instances can be solved exactly by the basic linear programming
relaxation (BLP). For a valued constraint language , BLP is a decision
procedure for if and only if admits a symmetric fractional
polymorphism of every arity. For a finite-valued constraint language ,
BLP is a decision procedure if and only if admits a symmetric
fractional polymorphism of some arity, or equivalently, if admits a
symmetric fractional polymorphism of arity 2.
Using these results, we obtain tractability of several novel classes of
problems, including problems over valued constraint languages that are: (1)
submodular on arbitrary lattices; (2) -submodular on arbitrary finite
domains; (3) weakly (and hence strongly) tree-submodular on arbitrary trees.Comment: A full version of a FOCS'12 paper by the last two authors
(arXiv:1204.1079) and an ICALP'13 paper by the first author (arXiv:1207.7213)
to appear in SIAM Journal on Computing (SICOMP
The complexity of finite-valued CSPs
We study the computational complexity of exact minimisation of
rational-valued discrete functions. Let be a set of rational-valued
functions on a fixed finite domain; such a set is called a finite-valued
constraint language. The valued constraint satisfaction problem,
, is the problem of minimising a function given as
a sum of functions from . We establish a dichotomy theorem with respect
to exact solvability for all finite-valued constraint languages defined on
domains of arbitrary finite size.
We show that every constraint language either admits a binary
symmetric fractional polymorphism in which case the basic linear programming
relaxation solves any instance of exactly, or
satisfies a simple hardness condition that allows for a
polynomial-time reduction from Max-Cut to