Partial Optimality by Pruning for MAP-inference with General - PowerPoint PPT Presentation

Partial Optimality by Pruning for MAP-inference with General Graphical Models Paul Swoboda, Bogdan Savchynskyy, J¨ org Kappes, Christoph Schn¨ orr Heidelberg University, Germany 1/1

Segment the image... 2/1

Optimal labeling 3/1

Optimal labeling NP hard 3/1

Solve convex relaxation (LP) Round relaxed solution NO optimality guarantees Optimal labeling NP hard 3/1

Partial labeling Optimal labeling Polynomially solvable NP hard Optimality guaranteed 3/1

Energy Minimization - MAP-Inference with Graphical Models ÿ arg min x P X J p x q : “ θ f p x ne p f q q f P F x i P t 1 , . . . , N u - variable Factor graph f P F Ă V n - factor G “ p V , F , E q - potential of x ne p f q P t 1 , . . . , N u | ne p f q| θ f p x ne p f q q 4/1

Easy Examples Figure : Color segmentation [Lellman 2010] 5/1

Difficult Examples (a) color segmentation (b) stereo (c) panorama stitching Figure : [Lellman 2010],[Szeliski et al. 2008],[Agarwala et al. 2004] 6/1

Related Work Aux. problem higher order non-binary non-Potts Work Boros & Hammer 2002 ´ ´ ` QPBO Kovtun 2003 ` ´ ´ submodular Rother et al. 2007 ´ ´ ` QPBO Kohli et al. 2008 ` ´ ` QPBO Kovtun 2005 ` ´ ` submodular Fix et al. 2011 ´ ` ` QPBO Kahl & Strandmark 2012 ´ ` ` bi-submodular Windheuser et al. 2012 ` ` ` bi-submodular ` ´ ´ Swoboda et al. 2013 LP Shekhovtsov 2014 ` ´ ` LP Ours ` ` ` any relaxation 7/1

Algorithm Outline Initialize: Generate labeling proposal repeat Verify the proposal on a current graph Shrink the graph until verification succeeds 8/1

Algorithm Proposed Partial Labeling ÿ ÿ J p x q “ θ v p x v q ` θ uv p x v , x u q v P V uv P E Labeling: x P X Partial optimality: χ P t 0 , 1 u | V | 9/1

Algorithm Proposed Partial Labeling Perturbed Problem ÿ ÿ ÿ ¯ ÿ ÿ J χ p x q “ θ v p x v q` θ uv p x v , x u q` θ v p x v q J p x q “ θ v p x v q ` θ uv p x v , x u q v P V Y V uv P E v P V v P V uv P E Labeling: x P X Partial optimality: χ P t 0 , 1 u | V | 9/1

Algorithm Proposed Partial Labeling Perturbed Problem ÿ ÿ ÿ ¯ ÿ ÿ J χ p x q “ θ v p x v q` θ uv p x v , x u q` θ v p x v q J p x q “ θ v p x v q ` θ uv p x v , x u q v P V Y V uv P E v P V v P V uv P E x “ arg min x P X J χ p x q Labeling: x P X ˆ NP-hard Ñ Relaxation Partial optimality: χ P t 0 , 1 u | V | 9/1

Algorithm Proposed Partial Labeling Perturbed Problem ÿ ÿ ÿ ¯ ÿ ÿ J χ p x q “ θ v p x v q` θ uv p x v , x u q` θ v p x v q J p x q “ θ v p x v q ` θ uv p x v , x u q v P V Y V uv P E v P V v P V uv P E x “ arg min x P X J χ p x q Labeling: x P X ˆ NP-hard Ñ Relaxation x i ‰ x i Ñ χ i “ 0 Partial optimality: χ P t 0 , 1 u | V | ˆ Shrinking Rule 9/1

Algorithm Proposed Partial Labeling Perturbed Problem ÿ ÿ ÿ ¯ ÿ ÿ J χ p x q “ θ v p x v q` θ uv p x v , x u q` θ v p x v q J p x q “ θ v p x v q ` θ uv p x v , x u q v P V Y V uv P E v P V v P V uv P E Labeling: x P X x “ arg min x P X J χ p x q ˆ NP-hard Ñ Relaxation x i ‰ x i Ñ χ i “ 0 ˆ Shrinking Rule Partial optimality: χ P t 0 , 1 u | V | 9/1

Use of Approximate Solvers Do we need to solve the relaxed problem x “ arg min x P X J χ p x q ˆ exactly? NO! Approximate solvers with optimality certificate (like TRW-S [Kolmogorov 2005]) are allowed here 10/1

Results Experiment (N) MQPBO Kovtun GRD Fix Ours 0 : : : 0.4423 teddy 0 : : : 0.0009 venus 0.0432 : : : 0.0611 family 0.1247 : : : 0.5680 pano : : Potts (12) 0.1839 0.7475 0.9231 0.0247 : : : 0.6513 side-chain (21) protein (8) : : 0.2603 0.2545 0.7799 : : : 0.1771 0.9992 cell-tracking : : : : 0.8407 geo-surf (50) Table : Percentage of persistent variables; : - method inapplicable. We used local polytope relaxation and TRW-S and CPLEX as solvers. Benchmarks: [Szeliski et al. 2008],[Kappes et al. 2013], [PIC 2011] 11/1

Potts models: Results Ours Kovtun’s method 12/1

Take Home Message and Outlook We presented: A generic method for partial optimality from MAP-Inference, which can employ any relaxation can use certain approximate solvers in the loop (e.g. TRW-S) scales as well as the used MAP-inference solver 17/1

Take Home Message and Outlook We presented: A generic method for partial optimality from MAP-Inference, which can employ any relaxation can use certain approximate solvers in the loop (e.g. TRW-S) scales as well as the used MAP-inference solver Code preliminary research code at http://paulswoboda.net revised code will be included to OpenGM library soon. 17/1

Partial Optimality by Pruning for MAP-inference with General - PowerPoint PPT Presentation

Partial Optimality by Pruning for MAP-inference with General Graphical Models Paul Swoboda, Bogdan Savchynskyy, J org Kappes, Christoph Schn orr Heidelberg University, Germany 1/1 Segment the image... 2/1 Optimal labeling 3/1

Partial Optimality via Iterative Pruning for the Potts Model Paul Swoboda, Bogdan Savchynskyy,

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