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Beyond Trees: MRF Inference via OuterPlanar Decomposition
, 2010
"... Maximum a posteriori (MAP) inference in Markov Random Fields (MRFs) is an NPhard problem, and thus research has focussed on either finding efficiently solvable subclasses (e.g. trees), or approximate algorithms (e.g. Loopy Belief Propagation (BP) and Treereweighted (TRW) methods). This paper prese ..."
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Cited by 8 (1 self)
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Maximum a posteriori (MAP) inference in Markov Random Fields (MRFs) is an NPhard problem, and thus research has focussed on either finding efficiently solvable subclasses (e.g. trees), or approximate algorithms (e.g. Loopy Belief Propagation (BP) and Treereweighted (TRW) methods). This paper presents a unifying perspective of these approximate techniques called “Decomposition Methods”. These are methods that decompose the given problem over a graph into tractable subproblems over subgraphs and then employ message passing over these subgraphs to merge the solutions of the subproblems into a global solution. This provides a new way of thinking about BP and TRW as successive steps in a hierarchy of decomposition methods. Using this framework, we take a principled first step towards extending this hierarchy beyond trees. We leverage a new class of graphs amenable to exact inference, called outerplanar graphs, and propose an approximate inference algorithm called OuterPlanar Decomposition (OPD). OPD is a strict generalization of BP and TRW, and contains both of them as special cases. Our experiments show that this extension beyond trees is indeed very powerful – OPD outperforms current stateofart inference methods on hard nonsubmodular synthetic problems and is competitive on real computer vision applications.
Planar Cycle Covering Graphs
"... We describe a new variational lowerbound on the minimum energy configuration of a planar binary Markov Random Field (MRF). Our method is based on adding auxiliary nodes to every face of a planar embedding of the graph in order to capture the effect of unary potentials. A ground state of the resulti ..."
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We describe a new variational lowerbound on the minimum energy configuration of a planar binary Markov Random Field (MRF). Our method is based on adding auxiliary nodes to every face of a planar embedding of the graph in order to capture the effect of unary potentials. A ground state of the resulting approximation can be computed efficiently by reduction to minimumweight perfect matching. We show that optimization of variational parameters achieves the same lowerbound as dualdecomposition into the set of all cycles of the original graph. We demonstrate that our variational optimization converges quickly and provides highquality solutions to hard combinatorial problems 10100x faster than competing algorithms that optimize the same bound. 1
Fast Planar Correlation Clustering for Image Segmentation
"... Abstract. We describe a new optimization scheme for finding highquality clusterings in planar graphs that uses weighted perfect matching as a subroutine. Our method provides lowerbounds on the energy of the optimal correlation clustering that are typically fast to compute and tight in practice. We ..."
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Abstract. We describe a new optimization scheme for finding highquality clusterings in planar graphs that uses weighted perfect matching as a subroutine. Our method provides lowerbounds on the energy of the optimal correlation clustering that are typically fast to compute and tight in practice. We demonstrate our algorithm on the problem of image segmentation where this approach outperforms existing global optimization techniques in minimizing the objective and is competitive with the state of the art in producing highquality segmentations. 1 1
Tightening MRF Relaxations with Planar Subproblems
"... We describe a new technique for computing lowerbounds on the minimum energy configuration of a planar Markov Random Field (MRF). Our method successively adds large numbers of constraints and enforces consistency over binary projections of the original problem state space. These constraints are repr ..."
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Cited by 1 (0 self)
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We describe a new technique for computing lowerbounds on the minimum energy configuration of a planar Markov Random Field (MRF). Our method successively adds large numbers of constraints and enforces consistency over binary projections of the original problem state space. These constraints are represented in terms of subproblems in a dualdecomposition framework that is optimized using subgradient techniques. The complete set of constraints we consider enforces cycle consistency over the original graph. In practice we find that the method converges quickly on most problems with the addition of a few subproblems and outperforms existing methods for some interesting classes of hard potentials. 1
www.ece.cmu.edu/˜dbatra
"... This paper deals with Dynamic MAP Inference, where the goal is to solve an instance of the MAP problem given that we have already solved a related instance of the problem. We propose an algorithm for Dynamic MAP Inference in planar Ising models, called Dynamic PlanarCuts. As an application of our p ..."
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This paper deals with Dynamic MAP Inference, where the goal is to solve an instance of the MAP problem given that we have already solved a related instance of the problem. We propose an algorithm for Dynamic MAP Inference in planar Ising models, called Dynamic PlanarCuts. As an application of our proposed approach, we show that we can extend the MAP inference algorithm of Schraudolph and Kamenetsky [14] to efficiently compute minmarginals for all variables in the same time complexity as the MAP inference algorithm, which is an O(n) speedup over a naïve approach. 1
The idiots guide to Quantum Error Correction.
, 2009
"... Contents Quantum Error Correction and faulttolerant quantum computation represent arguably the most vital theoretical aspect of quantum information processing. It was well known from the early developments of this exciting field that the fragility of coherent quantum systems would be a catastrophic ..."
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Contents Quantum Error Correction and faulttolerant quantum computation represent arguably the most vital theoretical aspect of quantum information processing. It was well known from the early developments of this exciting field that the fragility of coherent quantum systems would be a catastrophic obstacle to the development of large scale quantum computers. The introduction of
UC Santa Barbara,
"... Abstract. We address the problem of segmenting an image into a previously unknown number of segments from the perspective of graph partitioning. Specifically, we consider minimum multicuts of superpixel affinity graphs in which all affinities between nonadjacent superpixels are negative. We propose ..."
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Abstract. We address the problem of segmenting an image into a previously unknown number of segments from the perspective of graph partitioning. Specifically, we consider minimum multicuts of superpixel affinity graphs in which all affinities between nonadjacent superpixels are negative. We propose a relaxation by Lagrangian decomposition and a constrained set of reparameterizations for which we can optimize exactly and efficiently. Our contribution is to show how the planarity of the adjacency graph can be exploited if the affinity graph is nonplanar. We demonstrate the effectiveness of this approach in userassisted image segmentation and show that the solution of the relaxed problem is fast and the relaxation is tight in practice. 1
Approximation Algorithms and Heuristics for a Heterogeneous Traveling Salesman
, 2011
"... Unmanned Vehicles (UVs) are developed for several civil and military applications. For these applications, there is a need for multiple vehicles with different capabilities to visit and monitor a set of given targets. In such scenarios, routing problems arise naturally where there is a need to plan ..."
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Unmanned Vehicles (UVs) are developed for several civil and military applications. For these applications, there is a need for multiple vehicles with different capabilities to visit and monitor a set of given targets. In such scenarios, routing problems arise naturally where there is a need to plan paths in order to optimally use resources and time. The focus of this thesis is to address a basic optimization problem that arises in this setting. We consider a routing problem where some targets have to be visited by specific vehicles. We approach this problem by dividing the routing into two sub problems: partitioning the targets while satisfying vehicle target constraints and sequencing. We solve the partitioning problem with the help of a minimum spanning tree algorithm. We use 3 different approaches to solve the sequencing problem; namely, the 2 approximation algorithm, Christofide’s algorithm and the Lin Kernighan Heuristic (LKH). The approximation algorithms were implemented in MATLAB R ○. We also developed an integer programming (IP) model and a relaxed linear programming (LP) model in C++ with the help of Concert Technology for CPLEX, to obtain lower bounds.
This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. IEEE TRANSACTIONS ON VERY LARGE SCALE INTEGRATION (VLSI) SYSTEMS 1 The Optimal FanOut of Clock Network for Po
"... Abstract—Gating of the clock signal in VLSI chips is nowadays a mainstream design methodology for reducing switching power consumption. In this paper we develop a probabilistic model of the clock gating network that allows us to quantify the expected power savings and the implied overhead. Expressio ..."
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Abstract—Gating of the clock signal in VLSI chips is nowadays a mainstream design methodology for reducing switching power consumption. In this paper we develop a probabilistic model of the clock gating network that allows us to quantify the expected power savings and the implied overhead. Expressions for the power savings in a gated clock tree are presented and the optimal gater fanout is derived, based on flipflops toggling probabilities and process technology parameters. The resulting clock gating methodology achieves 10 % savings of the total clock tree switching power. The timing implications of the proposed gating scheme are discussed. The grouping of FFs for a joint clocked gating is also discussed. The analysis and the results match the experimental data obtained for a 3D graphics processor and a 16bit microcontroller, both designed at 65nanometer technology. Index Terms—Clock gating, clock networks, clock tree, dynamic power minimization, optimal fanout.
Learning with DegreeBased Subgraph Estimation
"... Networks and their topologies are critical to nearly every aspect of modern life, with social networks governing human interactions and computer networks governing global informationflow. Network behavior is inherently structural, and thus modeling data from networks benefits from explicitly modeli ..."
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Networks and their topologies are critical to nearly every aspect of modern life, with social networks governing human interactions and computer networks governing global informationflow. Network behavior is inherently structural, and thus modeling data from networks benefits from explicitly modeling structure. This thesis covers methods for and analysis of machine learning from network data while explicitly modeling one important measure of structure: degree. Central to this work is a procedure for exact maximum likelihood estimation of a distribution over graph structure, where the distribution factorizes into edgelikelihoods for each pair of nodes and degreelikelihoods for each node. This thesis provides a novel method for exact estimation of the maximum likelihood edge structure under the distribution. The algorithm solves the optimization by constructing an augmented graph containing, in addition to the original nodes, auxiliary nodes whose edges encode the degree potentials. The exact solution is then recoverable by finding the maximum weight bmatching on the augmented graph, a wellstudied combinatorial optimization. To solve the combinatorial optimization, this thesis focuses in particular on a belief propagationbased approach to finding the optimal bmatching and provides a novel proof of convergence for belief propagation on the loopy graphical model representing the bmatching objective. Additionally,