Update any set S of nodes simultaneously with step-size We show fixed point update is monotone for · 1/|S| Covering Trees and Lower-bounds on Quadratic Assignment Overview & GoalsExperimental Results Quadratic Assignment {jyarkony, fowlkes, Julian Yarkony, Charless Fowlkes, Alexander Ihler University of California, Irvine TRW and Dual Decomposition: Data Set V. Kolmogorov. Convergent tree-reweighted message passing for energy minimization. IEEE Trans. Pattern Anal. Machine Intell., 28(10):1568– 1583, L. Torresani, V. Kolmogorov, and C. Rother. Feature correspondence via graph matching: Models and global optimization. In ECCV, pages 596– 609, M. J. Wainwright, T. Jaakkola, and A. S. Willsky. MAP estimation via agreement on (hyper)trees: message-passing and linear programming approaches. IEEE Trans. Inform. Theory, 51(11):3697–3717, References Optimizing the Covering Tree Bound Convergence of various optimization algorithms on 10 by 10 Ising grids with repulsive potentials Develop new algorithms for determining corresponding points between images Covering Trees A new variational optimization of the tree- reweighted (TRW) bound BAR (Bottleneck Assignment Rounding) use the min-marginals from covering tree to construct an assignment Problem Formulation P parts, L locations X i 2 1…L specifies the location of part i Minimize assignment cost : cost of assigning part i to location X i : pairwise assignment cost i→X i, j→X j : 1-to-1 assignment constraint Select a set of spanning trees Decompose problem µ into { µ T } s.t.   µ T = µ Gives a lower bound on the optimum TRW algorithm optimizes the lower bound over µ Original Graph Collection of trees Constructing a Covering Tree Create a single tree which covers all edges of graph Duplicate nodes as required Achieves the same lower- bound as TRW Minimal representation; no updates of edge parameters Parameters of the covering tree µ CT distribute unary potentials from original problem across copies of the node Bottleneck Assignment Rounding “House” data set 110 views, 30 marked points for correspondence Unary costs: shape context features Pairwise costs: deviation in distance and angle Only consider pairwise potentials between neighboring parts (2,3,4 neighbors per part) We compare our algorithm to "Graph Matching" proposed by Toressani et al. which also uses a dual- decomposition approach. Two parameter optimization updates: Subgradient update Fixed point update : set of copies of node i : indicator of copy t taking on state k : min-marginal of copy t Optimizing state X in the covering tree May not be consistent across all copies May not be a 1-1 assignment “Assignment Rounding” find a valid assignment using CT estimates Construct bound lower bounds cost when part i in location k Find the best assignment consistent with bounds Solving Bottleneck assignment problem Search over all possible bottleneck values Restrict to states less than bottleneck value Check feasibility of value with bipartite matching Correspondence accuracy vs. view separation Focus on large baselines (grey region) Compare CT+BAR to CT alone BAR improves estimate quality Comparing accuracy & timing Fix the number of neighbors in graph Similar accuracy, CT+BAR faster 2 nbr Graph Matching vs. 4 nbr CT+BAR Similar timing, CT+BAR more accurate