Planar Cycle Covering Graphs for inference in MRFS The Typhon Algorithm A New Variational Approach to Ground State Computation in Binary Planar Markov Random Fields by Julian Yarkony, Charless Fowlkes Alexander Ihler 1

Foreground/Background Segmentation 22 Use appearance, edges, and prior information to segment image into foreground and background regions. Edge Information can be very useful when good models for foreground and background are unavailable. Foreground Background

Binary MRFs and segmentation 3 Cost to take on foreground Cost to disagree with neighbors Use a real delta or a 1 Min_X E(,\theta)

When can we find the exact minimum? Sub-modular Problems ( > 0) – Solve by reduction to graph cut [Boykov 2002] Planar Problems without unary potentials ( =0) – Solve using a reduction to minimum cost perfect matching. [Kastyln 1969, Fourtin 1969, Schauldolph 2007] 4

Trick for eliminating unary potentials 5

6 Problem: transformed graph may no longer be planar

Planarity lost Recall: perfect matching solution requires – No unary potentials – Planar 7

Idea: duplicate field node to maintain planarity 8 relaxation min_Xi E(Xi,theta) >= min_Xi,Xf Elb(Xi,Xf,theta,theta_if)

TYPHON: Optimizing the Lower Bound Solve using projected sub-gradient To solve alternate between gradient step in and optimizing X This optimization is CONVEX so this procedure is guaranteed to find global optima 9

Sub-Gradient Update 10 Use (1/N) Add parentheses Call this X_f3 etc. Old value New value Step size Disagreement Mean disagreement Each x i neighbors several copies of the field node Optimization drives the x f towards agreement Preserve  µ if = µ i

Sub-Gradient Update Each x i neighbors several copies of the field node Optimization drives the x f towards agreement Preserve  µ if = µ i 11 Use (1/N) Add parentheses Call this X_f3 etc. 0 1 Old value New value Step size Disagreement Mean disagreement 0 01

Sub-Gradient Update Each duplicated edge is modified to encourage that all copies agree with x, or all copies disagree Nodes that disagree have their cost increase Nodes that agree have their cost decrease 12 Use (1/N) Add parentheses Call this X_f3 etc. 0 1 Old value New value Step size Disagreement Mean disagreement 0 01

Convergence of Upper and Lower Bounds during sub-gradient optimization 13 Energy MAP Time Lower Bound Upper Bound

Computing Upper Bound at Each Step 14 Ground State, Lower Bound Upper Bound II Upper Bound I Upper bounds are obtained by using the configuration produced at any given time for all non-field nodes.

Dual Decomposition TRW decomposes MRF into a sum of trees [Wainwright 2005] – How many trees are needed? – Sufficient to choose a set of trees which cover each edge in the original graph at least once. 15

Cycle Decomposition 16 - Cycles give a tighter bound than trees - Collection of Cycles provides a tighter bound than trees. -How many cycles? - Lots!! - e.g. one way to ensure all cycles are covered is to include all triplets - [Sontag 2008] uses cutting plane techniques to iteratively add cycles

Lemma: Relaxation is tight for a single cycle 17 Reverse equality =

TYPHON relaxation covers all cycles Every cycle of G is present somewhere in the new graph, with copies of the field node That cycle and its field node copies are tight TYPHON is at least as tight as the set of all cycle subproblems 18

Experimental Results Synthetic problem test set – “Easy”, “Medium”, and “Hard” parameters – Pairwise potentials drawn from uniform, U[-R,R] – Unary drawn from Easy: 3.2*[-R,R] – strong local information Medium: 0.8*[-R,R] Hard: 0.2*[-R,R] – very weak local information Compare to state of the art algorithms: – MPLP, [Sontag 2008] – RPM, [Schraudolph 2010] 19 (R = 500)

Duality Gap as a function of time Size: 36x36 grids Easy Medium Hard 20

Time Until Convergence Easy Medium Hard 21 Runs which did not converge to the required tolerance are left off

Conclusions New variational bound for binary planar MRF’s Equal to cycle decomposition. Currently Applying to segmentation and extending to non-planar MRF’s and non-binary MRF’s 22

Thank You 23