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21
On brambles, gridlike minors, and parameterized intractability of monadic secondorder logic
"... Brambles were introduced as the dual notion to treewidth, one of the most central concepts of the graph minor theory of Robertson and Seymour. Recently, Grohe and Marx showed that there are graphs G, in which every bramble of order larger than the square root of the treewidth is of exponential size ..."
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Cited by 12 (4 self)
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Brambles were introduced as the dual notion to treewidth, one of the most central concepts of the graph minor theory of Robertson and Seymour. Recently, Grohe and Marx showed that there are graphs G, in which every bramble of order larger than the square root of the treewidth is of exponential size in G. On the positive side, they show the existence of polynomialsized brambles of the order of the square root of the treewidth, up to log factors. We provide the first polynomial time algorithm to construct a bramble in general graphs and achieve this bound, up to logfactors. We use this algorithm to construct gridlike minors, a replacement structure for gridminors recently introduced by Reed and Wood, in polynomial time. Using the gridlike
Learning Robot Grasping from 3D Images with Markov Random Fields
"... Abstract — Learning to grasp novel objects is an essential skill for robots operating in unstructured environments. We therefore propose a probabilistic approach for learning to grasp. In particular, we learn a function that predicts the success probability of grasps performed on surface points of a ..."
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Abstract — Learning to grasp novel objects is an essential skill for robots operating in unstructured environments. We therefore propose a probabilistic approach for learning to grasp. In particular, we learn a function that predicts the success probability of grasps performed on surface points of a given object. Our approach is based on Markov Random Fields (MRF), and motivated by the fact that points that are geometrically close to each other tend to have similar grasp success probabilities. The MRF approach is successfully tested in simulation, and on a real robot using 3D scans of various types of objects. The empirical results show a significant improvement over methods that do not utilize the smoothness assumption and classify each point separately from the others. Fig. 1. Barrett hand equipped with a SwissRanger timeofflight camera I.
Learning Bounded Treewidth Bayesian Networks using Integer Linear Programming
, 2014
"... In many applications one wants to compute conditional probabilities given a Bayesian network. This inference problem is NPhard in general but becomes tractable when the network has low treewidth. Since the inference problem is common in many application areas, we provide a practical algorithm for ..."
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Cited by 8 (0 self)
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In many applications one wants to compute conditional probabilities given a Bayesian network. This inference problem is NPhard in general but becomes tractable when the network has low treewidth. Since the inference problem is common in many application areas, we provide a practical algorithm for learning bounded treewidth Bayesian networks. We cast this problem as an integer linear program (ILP). The program can be solved by an anytime algorithm which provides upper bounds to assess the quality of the found solutions. A key component of our program is a novel integer linear formulation for bounding treewidth of a graph. Our tests clearly indicate that our approach works in practice, as our implementation was able to find an optimal or nearly optimal network for most of the data sets.
Convergent Decomposition Solvers for Treereweighted Free Energies
"... We investigate minimization of treereweighted free energies for the purpose of obtaining approximate marginal probabilities and upper bounds on the partition function of cyclic graphical models. The solvers we present for this problem work by directly tightening treereweighted upper bounds. As a re ..."
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We investigate minimization of treereweighted free energies for the purpose of obtaining approximate marginal probabilities and upper bounds on the partition function of cyclic graphical models. The solvers we present for this problem work by directly tightening treereweighted upper bounds. As a result, they are particularly efficient for treereweighted energies arising from a small number of spanning trees. While this assumption may seem restrictive at first, we show how small sets of trees can be constructed in a principled manner. An appealing property of our algorithms, which results from the problem decomposition, is that they are embarrassingly parallel. In contrast to the original message passing algorithm introduced for this problem, we obtain global convergence guarantees. 1
Approximate Counting via Correlation Decay on Planar Graphs
"... We show for a broad class of counting problems, correlation decay (strong spatial mixing) implies FPTAS on planar graphs. The framework for the counting problems considered by us is the Holant problems with arbitrary constantsize domain and symmetric constraint functions. We define a notion of regu ..."
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Cited by 5 (1 self)
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We show for a broad class of counting problems, correlation decay (strong spatial mixing) implies FPTAS on planar graphs. The framework for the counting problems considered by us is the Holant problems with arbitrary constantsize domain and symmetric constraint functions. We define a notion of regularity on the constraint functions, which covers a wide range of natural and important counting problems, including all multistate spin systems, counting graph homomorphisms, counting weighted matchings or perfect matchings, and all counting CSPs and Holant problems with symmetric constraint functions of constant arity. The core of our algorithm is a fixedparameter tractable algorithm which computes the exact values of the Holant problems with regular constraint functions on graphs of bounded treewidth. By utilizing the locally treelike property of apexminorfree families of graphs, the parameterized exact algorithm implies an FPTAS for the Holant problem on these graph families whenever the Gibbs measure defined by the problem exhibits strong spatial mixing. We further extend the recursive coupling technique to establish the strong spatial mixing on Holant problems. As consequences, we have new deterministic approximation algorithms on planar graphs for several counting problems. 1
Approximating the Bethe partition function
"... When belief propagation (BP) converges, it does so to a stationary point of the Bethe free energy F, and is often strikingly accurate. However, it may converge only to a local optimum or may not converge at all. An algorithm was recently introduced by Weller and Jebara for attractive binary pairwise ..."
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When belief propagation (BP) converges, it does so to a stationary point of the Bethe free energy F, and is often strikingly accurate. However, it may converge only to a local optimum or may not converge at all. An algorithm was recently introduced by Weller and Jebara for attractive binary pairwise MRFs which is guaranteed to return an ɛapproximation to the global minimum of F in polynomial time provided the maximum degree ∆ = O(log n), where n is the number of variables. Here we extend their approach and derive a new method based on analyzing first derivatives of F, which leads to much better performance and, for attractive models, yields a fully polynomialtime approximation scheme (FPTAS) without any degree restriction. Further, our methods apply to general (nonattractive) models, though with no polynomial time guarantee in this case, demonstrating that approximating log of the Bethe partition function, log ZB = − min F, for a general model to additive ɛaccuracy may be reduced to a discrete MAP inference problem. This allows the merits of the global Bethe optimum to be tested.
Lower Bounds on the Complexity of MSO1 ModelChecking
"... One of the most important algorithmic metatheorems is a famous result by Courcelle which states that any graph problem definable in monadic secondorder logic with edgeset quantifications (MSO2) is decidable in linear time on any class of graphs of bounded treewidth. In the parlance of parameteri ..."
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Cited by 3 (0 self)
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One of the most important algorithmic metatheorems is a famous result by Courcelle which states that any graph problem definable in monadic secondorder logic with edgeset quantifications (MSO2) is decidable in linear time on any class of graphs of bounded treewidth. In the parlance of parameterized complexity, this means that MSO2 is FPTtractable wrt. the treewidth as parameter. Recently, Kreutzer and Tazari [13] proved a corresponding complexity lowerbound— that MSO2 modelchecking is not even in XP wrt. the formula size as parameter for graph classes that are subgraphclosed and whose treewidth is polylogarithmically unbounded. Of course, this is not an unconditional result but holds modulo a certain complexitytheoretic assumption, namely, the Exponential Time Hypothesis (ETH). In this paper we present a closely related result. We show that even MSO1 modelchecking with a fixed set of vertex labels, but without edgeset quantifications, is not in XP wrt. the formula size as parameter for graph classes which are subgraphclosed and whose treewidth is polylogarithmically unbounded unless the nonuniform ETH fails. In comparison to Kreutzer and Tazari, (1) we use a stronger prerequisite, namely nonuniform instead of uniform ETH, to avoid
Sensor Selection in HighDimensional Gaussian Trees with Nuisances
"... We consider the sensor selection problem on multivariate Gaussian distributions where only a subset of latent variables is of inferential interest. For pairs of vertices connected by a unique path in the graph, we show that there exist decompositions of nonlocal mutual information into local informa ..."
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We consider the sensor selection problem on multivariate Gaussian distributions where only a subset of latent variables is of inferential interest. For pairs of vertices connected by a unique path in the graph, we show that there exist decompositions of nonlocal mutual information into local information measures that can be computed efficiently from the output of message passing algorithms. We integrate these decompositions into a computationally efficient greedy selector where the computational expense of quantification can be distributed across nodes in the network. Experimental results demonstrate the comparative efficiency of our algorithms for sensor selection in highdimensional distributions. We additionally derive an onlinecomputable performance bound based on augmentations of the relevant latent variable set that, when such a valid augmentation exists, is applicable for any distribution with nuisances. 1
Nonlinearly Constrained MRFs: Exploring the Intrinsic Dimensions of HigherOrder Cliques
"... This paper introduces an efficient approach to integrating nonlocal statistics into the higherorder Markov Random Fields (MRFs) framework. Motivated by the observation that many nonlocal statistics (e.g., shape priors, color distributions) can usually be represented by a small number of parameter ..."
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This paper introduces an efficient approach to integrating nonlocal statistics into the higherorder Markov Random Fields (MRFs) framework. Motivated by the observation that many nonlocal statistics (e.g., shape priors, color distributions) can usually be represented by a small number of parameters, we reformulate the higherorder MRF model by introducing additional latent variables to represent the intrinsic dimensions of the higherorder cliques. The resulting new model, called NCMRF, not only provides the flexibility in representing the configurations of higherorder cliques, but also automatically decomposes the energy function into less coupled terms, allowing us to design an efficient algorithmic framework for maximum a posteriori (MAP) inference. Based on this novel modeling/inference framework, we achieve stateoftheart solutions to the challenging problems of classspecific image segmentation and templatebased 3D facial expression tracking, which demonstrate the potential of our approach. 1.
1Hierarchical Graphical Models for Simultaneous Tracking and Recognition in WideArea Scenes
"... Abstract—We present a unified framework to track multiple people, as well localize and label their activities, in complex longduration video sequences. To do this, we focus on two aspects the influence of tracks on the activities performed by the corresponding actors and the structural relationshi ..."
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Abstract—We present a unified framework to track multiple people, as well localize and label their activities, in complex longduration video sequences. To do this, we focus on two aspects the influence of tracks on the activities performed by the corresponding actors and the structural relationships across activities. We propose a twolevel hierarchical graphical model which learns the relationship between tracks, relationship between tracks and their corresponding activity segments, as well as the spatiotemporal relationships across activity segments. Such contextual relationships between tracks and activity segments are exploited at both the levels in the hierarchy for increased robustness. An L1regularized structure learning approach is proposed for this purpose. While it is well known that availability of the labels and locations of activities can help in determining tracks more accurately and viceversa, most current approaches have dealt with these problems separately. Inspired by research in the area of biological vision, we propose a bidirectional approach that integrates both bottomup and topdown processing, i.e., bottomup recognition of activities using computed tracks and topdown computation of tracks using the obtained recognition. We demonstrate our results on the recent and publicly available UCLA and VIRAT datasets consisting of realistic indoor and outdoor surveillance sequences. I.