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 & Goals Experimental Results

Quadratic Assignment

{jyarkony, fowlkes, ihler}@ics.uci.eduJulian Yarkony, Charless Fowlkes, Alexander IhlerUniversity 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, 2006.L. Torresani, V. Kolmogorov, and C. Rother. Feature correspondence via graph matching: Models and global optimization. In ECCV, pages 596–609, 2008.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, 2005.

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• Xi 2 1…L specifies the location of part i• Minimize assignment cost

: cost of assigning part i to location Xi

: pairwise assignment cost i→Xi, j→Xj

: 1-to-1 assignment constraint

• Select a set of spanning trees• Decompose problem µ into {µT} s.t. µT = 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 µCT 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