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Multicuts and Perturb & MAP for Probabilistic Graph Clustering

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Abstract

We present a probabilistic graphical model formulation for the graph clustering problem. This enables us to locally represent uncertainty of image partitions by approximate marginal distributions in a mathematically substantiated way, and to rectify local data term cues so as to close contours and to obtain valid partitions. We exploit recent progress on globally optimal MAP inference by integer programming and on perturbation-based approximations of the log-partition function, in order to sample clusterings and to estimate marginal distributions of node-pairs both more accurately and more efficiently than state-of-the-art methods. Our approach works for any graphically represented problem instance. This is demonstrated for image segmentation and social network cluster analysis. Our mathematical ansatz should be relevant also for other combinatorial problems.

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Notes

  1. For \(\theta \in \mathbb {R}^N\) we have \(\mu \in {{\mathrm{{\mathcal {MC}}}}}^\circ (G)\). Boundary points of \({{\mathrm{{\mathcal {MC}}}}}(G)\) are only reached when at least one entry of \(\theta \) is diverging to infinity.

  2. To overcome this problem we will restrict the number of possible labels and exclude thereby some partitions. If the number of used labels is greater or equal than the chromatic number of the graph all partitions are representable.

  3. A graph is called chordal, if each cycle of length strictly larger than 3 have a chord, which is an edge that is not part of the cycle but connects two vertices of the cycle.

  4. http://dblp.uni-trier.de/db/.

  5. For reasons of anonymity, however, we show anonymized results.

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Acknowledgments

This work has been supported by the German Research Foundation (DFG) within the programme “Spatio-/Temporal Graphical Models and Applications in Image Analysis”, Grant GRK 1653.

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Authors and Affiliations

  1. HCI, Heidelberg University, Berliner Str. 43, 69120, Heidelberg, Germany

    Jörg Hendrik Kappes

  2. Image & Pattern Analysis Group, Heidelberg University, Im Neuenheimer Feld 205, 69120, Heidelberg, Germany

    Paul Swoboda & Christoph Schnörr

  3. Technische Universität Dresden, 01062, Dresden, Germany

    Bogdan Savchynskyy

  4. Technion - Israel Institute of Technology, Bloomfield Building, Room 503, Technion City, Haifa, 32000, Israel

    Tamir Hazan

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  1. Jörg Hendrik Kappes
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  2. Paul Swoboda
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Correspondence to Jörg Hendrik Kappes.

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Kappes, J.H., Swoboda, P., Savchynskyy, B. et al. Multicuts and Perturb & MAP for Probabilistic Graph Clustering. J Math Imaging Vis 56, 221–237 (2016). https://doi.org/10.1007/s10851-016-0659-3

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