60 research outputs found

    Maximum Persistency via Iterative Relaxed Inference with Graphical Models

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    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

    Accelerated Consensus via Min-Sum Splitting

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    We apply the Min-Sum message-passing protocol to solve the consensus problem in distributed optimization. We show that while the ordinary Min-Sum algorithm does not converge, a modified version of it known as Splitting yields convergence to the problem solution. We prove that a proper choice of the tuning parameters allows Min-Sum Splitting to yield subdiffusive accelerated convergence rates, matching the rates obtained by shift-register methods. The acceleration scheme embodied by Min-Sum Splitting for the consensus problem bears similarities with lifted Markov chains techniques and with multi-step first order methods in convex optimization

    A Message Passing Algorithm for the Minimum Cost Multicut Problem

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    We propose a dual decomposition and linear program relaxation of the NP -hard minimum cost multicut problem. Unlike other polyhedral relaxations of the multicut polytope, it is amenable to efficient optimization by message passing. Like other polyhedral elaxations, it can be tightened efficiently by cutting planes. We define an algorithm that alternates between message passing and efficient separation of cycle- and odd-wheel inequalities. This algorithm is more efficient than state-of-the-art algorithms based on linear programming, including algorithms written in the framework of leading commercial software, as we show in experiments with large instances of the problem from applications in computer vision, biomedical image analysis and data mining.Comment: Added acknowledgment

    A dual ascent framework for Lagrangean decomposition of combinatorial problems

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    We propose a general dual ascent framework for Lagrangean decomposition of combinatorial problems. Although methods of this type have shown their efficiency for a number of problems, so far there was no general algorithm applicable to multiple problem types. In this work, we propose such a general algorithm. It depends on several parameters, which can be used to optimize its performance in each particular setting. We demonstrate efficacy of our method on graph matching and multicut problems, where it outperforms state-of-the-art solvers including those based on subgradient optimization and off-the-shelf linear programming solvers

    A dual ascent framework for Lagrangean decomposition of combinatorial problems

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    We propose a general dual ascent framework for Lagrangean decomposition of combinatorial problems. Although methods of this type have shown their efficiency for a number of problems, so far there was no general algorithm applicable to multiple problem types. In this work, we propose such a general algorithm. It depends on several parameters, which can be used to optimize its performance in each particular setting. We demonstrate efficacy of our method on graph matching and multicut problems, where it outperforms state-of-the-art solvers including those based on subgradient optimization and off-the-shelf linear programming solvers

    Lifted edges as connectivity priors for multicut and disjoint paths

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    This work studies graph decompositions and their representation by 0/1 labeling of edges. We study two problems. The first is multicut (MC) which represents decompositions of undirected graphs (clustering of nodes into connected components). The second is disjoint paths (DP) in directed acyclic graphs where the clusters correspond to node- disjoint paths. Unlike an alternative representation by node labeling, the number of clusters is not part of the input but is fully determined by the costs of edges. Our main interest is to study connectivity priors represented by so-called lifted edges in the two problems. The cost of a lifted edge expresses whether its endpoints should belong to the same cluster (path) in the optimal decomposition. We call the resulting problems lifted multicut (LMC) and lifted disjoint paths (LDP). The extension of MC to LMC was originally motivated by image segmentation where the information about the connectivity between non-neighboring pixels or superpixels led to a significant quality improvement. After that, LMC was successfully applied to other problems like multiple object tracking (MOT) which is also the main application of our proposed LDP model. Our study of lifted multicut concentrates on partial LMC represented by labeling of a subset of (lifted) edges. Given partial labeling, we conclude that deciding whether a complete LMC consistent with the partial labels exists is NP-complete. Similarly, we conclude that deciding whether an unlabeled edge exists such that its label is determined by the labels of other edges is NP-hard. After that, we present metrics for comparing (partial) graph decompositions. Finally, we study the properties of the LMC polytope. The largest part of this work is dedicated to the proposed LDP problem. We prove that this problem is NP-hard and propose an optimal integer linear programming (ILP) solver. In order to enable its global optimization, we formulate several classes of linear inequalities that produce a high-quality LP relaxation. Additionally, we propose efficient cutting plane algorithms for separating the proposed linear inequalities. Despite the advanced constraints and efficient separation routines, the general time complexity of our optimal ILP solver remains exponential. In order to solve even larger instances, we introduce an approximate LDP solver based on Lagrange decomposition. LDP is a convenient model for MOT because the underlying disjoint paths model naturally leads to trajectories of objects. Moreover, lifted edges encode long-range temporal interactions and thus help to prevent id switches and re-identify persons. Our tracker using the optimal LDP solver achieves nearly optimal assignments w.r.t. input detections. Consequently, it was a leading tracker on three benchmarks of the MOT challenge MOT15/16/17, improving significantly over state-of-the-art at the time of its publication. Our approximate LDP solver enables us to process the MOT15/16/17 benchmarks without sacrificing solution quality and allows for solving large and dense instances of a challenging dataset MOT20. On all these four standard MOT benchmarks we achieved performance comparable or better than state-of-the-art methods (at the time of publication) including our tracker based on the optimal LDP solver.Diese Arbeit studiert Graphenzerlegungen und ihre ReprĂ€sentation durch 0/1-wertige Kantenbelegungen. Das erste Problem ist das Mehrfachschnittproblem. Es reprĂ€sentiert Zerlegungen von ungerichteten Graphen (Cluster von Knoten sodass jeder Cluster eine Zusammenhangskomponente reprĂ€sentiert). Das zweite Problem ist die Suche von disjunkten Pfaden in einem gerichteten azyklischen Graph in dem die Cluster knotendisjunkten Pfaden entsprechen. Im Unterschied zu der alternativen ReprĂ€sentation durch Knotenbelegungen ist die Zahl von Clustern nicht im Voraus gegeben, sondern sie ist abhĂ€ngig von den Kosten der Kanten. Der Fokus dieser Arbeit ist die Erforschung von hochgezogenen Kannten, die eine apriori Information ĂŒber Verbundenheit von Knoten in Clustern respektive durch Pfade in den zwei Problemen darstellen. Die Kosten einer hochgezogenen Kante drĂŒcken aus, ob ihre Knoten zu dem gleichen Cluster (Pfad) in der optimalen Zerlegung gehören sollten. Wir bezeichnen diese neuen Probleme als das hochgezogene Mehrfachschnittproblem und das Problem der hochgezogenen disjunkten Pfade. Die Erweiterung des Mehrfachschnittproblems zu dem hochgezogenen Mehrfachschnittproblem wurde ursprĂŒnglich durch die Bildsegmentierung motiviert, fĂŒr die die Information ĂŒber Verbundenheit von nicht benachbarten Pixeln oder Superpixeln zu einer bedeutenden Verbesserung der QualitĂ€t fĂŒhrte. Danach wurde das hochgezogene Mehrfachschnittproblem zu der Lösung von anderen Problemen wie zum Beispiel der Verfolgung von mehreren Objekten in einem Video angewendet. Diese Aufgabe ist auch die Hauptanwendung des vorgeschlagenen Problems der hochgezogenen disjunkte Pfade. In unserer Untersuchung des hochgezogenen Mehrfachschnittproblems konzentrieren wir uns auf das teilweise hochgezogene Mehrfachschnittproblem. Das Problem wird durch eine Belegung einer Teilmenge der (hochgezogenen) Kanten reprĂ€sentiert. Wir beweisen, dass es NP-vollstĂ€ndig ist zu entscheiden, ob ein kompletter hochgezogener Mehrfachschnitt existiert, der einer gegebenen teilweisen Kantenbezeichnung entspricht. In analogerWeise beweisen wir, dass es NP-schwer ist zu entscheiden, ob eine nicht belegte Kante existiert, deren Belegung durch die Belegungen anderer Kanten entschieden ist. Danach prĂ€sentieren wir Metriken zum Vergleich von (teilweisen) Graphenzerlegungen. Schließlich untersuchen wir Eigenschaften des hochgezogenen Mehrfachschnitt-Polytops. Der grĂ¶ĂŸte Teil dieser Arbeit widmet sich dem von uns vorgeschlagenen Problem der hochgezogenen disjunkten Pfade. Wir beweisen, dass es NP-schwer ist. Wir formulieren es als ein ganzzahliges lineares Optimierungsproblem und implementieren ein Programm fĂŒr dessen optimale Lösung. Um die globale Optimierung zu ermöglichen, formulieren wir mehrere Klassen von linearen Ungleichungen, die zu einer linearen Relaxierung mit einer hohen QualitĂ€t fĂŒhren. ZusĂ€tzlich prĂ€sentieren wir ein effektives Schnittebenenverfahren fĂŒr die Separierung der vorgeschlagenen Ungleichungen. Trotz der fortgeschrittenen Ungleichungen und der Effizienz der Schnittebenenseparierung in unserem optimalen Löser bleibt die allgemeine KomplexitĂ€t des Algorithmus exponentiell. Um noch kompliziertere Instanzen zu lösen, prĂ€sentieren wir einen approximativen Löser, der auf Lagrange-DualitĂ€t aufbaut. Hochgezogene disjunkte Pfade sind ein praktisches Modell fĂŒr die Verfolgung von mehreren Objekten, weil die disjunkten Pfade eine natĂŒrliche ReprĂ€sentation von Trajektorien der Objekten darstellen. Außerdem reprĂ€sentieren die hochgezogenen Kanten Interaktionen einer langen zeitlichen Reichweite. Deswegen helfen sie dieselbe Person in zeitlich weiter auseinander liegenden Zeitpunkten wieder zu identifizieren und Verwechselungen ihrer IdentitĂ€t zu verhindern. Aus diesem Grund war unsere Methode zur Zeit ihrer Publikation die beste fĂŒr drei VergleichsdatensĂ€tzen MOT Challenge MOT15/16/17 fĂŒr die Verfolgung von mehreren Objekten. Im Vergleich zu den bisherigen besten Methoden war ihre Leistung sogar bedeutend höher. Unsere approximative Methode fĂŒr hochgezogene disjunkte Pfade ermöglicht uns die VergleichsdatensĂ€tzen MOT15/16/17 zu verarbeiten ohne die QualitĂ€t der Lösungen zu vermindern und erlaubt uns, die großen Instanzen mit hoher Personendichte des anspruchsvolleren Datensatzes MOT20 zu lösen. Zur Zeit ihrer Publikation erreichte die Methode vergleichbare oder bessere Ergebnisse als die bisherigen besten Methoden einschließlich unseres optimalen Löser fĂŒr hochgezogene disjunkte Pfade

    New Convex Relaxations and Global Optimality in Variational Imaging

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    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
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