Introduction Mappings Generalized Sufficient Condition Maximum Persistency Experiments References Maximum Persistency in Energy Minimization Alexander Shekhovtsov, TU Graz June 25, 2014 1/26 Alexander Shekhovtsov, TU Graz Maximum Persistency in Energy Minimization
Introduction Mappings Generalized Sufficient Condition Maximum Persistency Experiments References Discrete Energy Minimization f st (0 , 1) f st (1 , 1) f t (1) x f s (0) f st (0 , 0) f (0 , 0) s t ′ t f st (1 , 0) Minimize partially separable function � � E f ( x ) = f 0 + f s ( x s ) + f st ( x s , x t ) , s ∈V st ∈E over assignments (labelings): x = ( x s ∈ L s | s ∈ V ) studied as MAP MRF/CRF, WCSP NP-hard to approximate ( e.g . Orponen 1990 for TSP) This work: reduce domains (sets of labels L s ) while retaining some/all optimal solutions, in polynomial time 2/26 Alexander Shekhovtsov, TU Graz Maximum Persistency in Energy Minimization
Introduction Mappings Generalized Sufficient Condition Maximum Persistency Experiments References Partial Optimality Example to illustrate what is the hope here: Stereo Reconstruction Partial Optimality (Model of Alahari et al . 2010) (Method of Kovtun, 2010) Can find a partial assignment that holds for any global optimum, which is unknown 3/26 Alexander Shekhovtsov, TU Graz Maximum Persistency in Energy Minimization
Introduction Mappings Generalized Sufficient Condition Maximum Persistency Experiments References Several Different Methods There were proposed several substantially different methods: Dead End Elimination (DEE) Persistency in Quadratic Pseudo-Boolean Optimization (QPBO) MQPBO Methods of Kovtun 04, 10 Methods of Swoboda et al . 13 (14) What do they have in common? 4/26 Alexander Shekhovtsov, TU Graz Maximum Persistency in Energy Minimization
Introduction Mappings Generalized Sufficient Condition Maximum Persistency Experiments References Improving Mappings 5/26 Alexander Shekhovtsov, TU Graz Maximum Persistency in Energy Minimization
Introduction Mappings Generalized Sufficient Condition Maximum Persistency Experiments References Improving Mapping t ′ s t Definition Mapping p : L → L is improving if ( ∀ x ∈ L ) E f ( p ( x )) ≤ E f ( x ) strictly improving if x � = p ( x ) ⇒ E f ( p ( x )) < E f ( x ) If x is optimal then p ( x ) is optimal For strictly improving all optimal solutions are in p ( L ) Composition: if p , q are improving ⇒ p ◦ q is improving: E f ( p ( q ( x ))) ≤ E f ( q ( x )) ≤ f ( x ) 6/26 Alexander Shekhovtsov, TU Graz Maximum Persistency in Energy Minimization
Introduction Mappings Generalized Sufficient Condition Maximum Persistency Experiments References Dead End Elimination (DEE) Family of methods by Desmet et al . 1992, Goldstein 1994, etc . y α t ′′ β t ′ t ′ s s t Apply mapping in a single pixel s Improving iff f s ( α ) − f s ( β ) + � x t ∈L t [ f st ( α, x t ) − f st ( β, x t )] ≥ 0 min t ∈N ( s ) (worst case energy change over neighbours assignment) Compose many such mappings 7/26 Alexander Shekhovtsov, TU Graz Maximum Persistency in Energy Minimization
Introduction Mappings Generalized Sufficient Condition Maximum Persistency Experiments References Quadratic Pseudo-Boolean Optimization (QPBO) Nemhauser and Trotter 75, Hammer et al . 84, Boros et al . 02, Rother et al . 07 1 1 1 1 2 2 2 1 1 1 0 2 1 2 2 Integral part of the LP relaxation is globally optimal A ⊂ V , y = ( y s | s ∈ A ) ”Autarky”: replace x with y on A ( x [ A ← y ]) is guaranteed not to increase the energy mapping x �→ x [ A ← y ] is improving 8/26 Alexander Shekhovtsov, TU Graz Maximum Persistency in Energy Minimization
Introduction Mappings Generalized Sufficient Condition Maximum Persistency Experiments References Multilabel QPBO (MQPBO) Kohli et al . 08, Windheuser et al . 12 max x min x Fixed linear ordering Reduction to pseudo-Boolean + QPBO guarantees ”Autarky”: mapping x �→ ( x ∨ x min ) ∧ x max is improving 9/26 Alexander Shekhovtsov, TU Graz Maximum Persistency in Energy Minimization
Introduction Mappings Generalized Sufficient Condition Maximum Persistency Experiments References Kovtun one vs. all Method Kovtun 2004, 2010 test labeling y Builds auxiliary submodular 2-label energy for given y ”Autarky”: mapping x �→ x [ A ← y ] is improving 10/26 Alexander Shekhovtsov, TU Graz Maximum Persistency in Energy Minimization
Introduction Mappings Generalized Sufficient Condition Maximum Persistency Experiments References Kovtun general Method Kovtun 2004, 2010 found y Builds auxiliary submodular multilabel energy and y Mapping x �→ ( x ∨ y ) is improving 11/26 Alexander Shekhovtsov, TU Graz Maximum Persistency in Energy Minimization
Introduction Mappings Generalized Sufficient Condition Maximum Persistency Experiments References Iterative Pruning Swoboda et al . 2013, 2014 found y Iteratively builds auxiliary energy and solves its LP relaxation ”Autarky”: mapping x �→ x [ A ← y ] is improving 12/26 Alexander Shekhovtsov, TU Graz Maximum Persistency in Energy Minimization
Introduction Mappings Generalized Sufficient Condition Maximum Persistency Experiments References Verification Problem Verifying whether p : L → L is improving is NP-hard e.g ., Boros et al . 2006 Determining whether a partial assignment is an autarky is NP-hard How do these methods find one? – Finer sufficient conditions. 13/26 Alexander Shekhovtsov, TU Graz Maximum Persistency in Energy Minimization
Introduction Mappings Generalized Sufficient Condition Maximum Persistency Experiments References Generalized Sufficient Condition 14/26 Alexander Shekhovtsov, TU Graz Maximum Persistency in Energy Minimization
Introduction Mappings Generalized Sufficient Condition Maximum Persistency Experiments References LP Relaxation Schlesinger 76, Koster et al . 98, 99, Chekuri et al . 01, Wainwright et al . 02, Werner 08. (1,1) ¹ 12 (1 ; 1) ¹ 1 (1) ¹ 12 (1 ; 1) M ¹ 2 (1) ¹ 2 (1) mapping ± (0,1) ¹ 1 (1) (0,0) (1,0) Embedding: δ ( x ) ∈ R I E f ( x ) = � f , δ ( x ) � Relaxation: min x ∈L � f , δ ( x ) � ≥ min µ ∈ Λ � f , µ � Λ ⊃ conv( δ ( L )) 15/26 Alexander Shekhovtsov, TU Graz Maximum Persistency in Energy Minimization
Introduction Mappings Generalized Sufficient Condition Maximum Persistency Experiments References Relaxed Improving Mapping (1,1) ¹ 1 2 (1 ; 1) M mapping ± P ( M ) ¹ 2 (1) (0,1) p s p t ¹ 1 (1) (0,0) (1,0) Linear Extension ( ∀ x ∈ L ) δ ( p ( x )) = P δ ( x ) 16/26 Alexander Shekhovtsov, TU Graz Maximum Persistency in Energy Minimization
Introduction Mappings Generalized Sufficient Condition Maximum Persistency Experiments References Relaxed Improving Mapping (1,1) ¹ 1 2 (1 ; 1) M mapping ± P ( M ) ¹ 2 (1) (0,1) p s p t ¹ 1 (1) (0,0) (1,0) Linear Extension ( ∀ x ∈ L ) δ ( p ( x )) = P δ ( x ) Definition Mapping p : L → L is Relaxed-improving if ( ∀ µ ∈ Λ) � f , P µ � ≤ � f , µ � 16/26 Alexander Shekhovtsov, TU Graz Maximum Persistency in Energy Minimization
Introduction Mappings Generalized Sufficient Condition Maximum Persistency Experiments References Relaxed Improving Mapping Improving Relaxed-Improving ( ∀ x ) E f ( p ( x )) ≤ E f ( x ) ( ∀ µ ∈ Λ) � f , P µ � ≤ � f , µ � Λ ⊃ conv( δ ( L )) Sufficient condition 17/26 Alexander Shekhovtsov, TU Graz Maximum Persistency in Energy Minimization
Introduction Mappings Generalized Sufficient Condition Maximum Persistency Experiments References Relaxed Improving Mapping Improving Relaxed-Improving ( ∀ x ) E f ( p ( x )) ≤ E f ( x ) ( ∀ µ ∈ Λ) � f , P µ � ≤ � f , µ � Λ ⊃ conv( δ ( L )) Sufficient condition Can be verified via LP: min µ ∈ Λ � f , ( I − P ) µ � ≤ 0 17/26 Alexander Shekhovtsov, TU Graz Maximum Persistency in Energy Minimization
Introduction Mappings Generalized Sufficient Condition Maximum Persistency Experiments References Relaxed Improving Mapping Improving Relaxed-Improving ( ∀ x ) E f ( p ( x )) ≤ E f ( x ) ( ∀ µ ∈ Λ) � f , P µ � ≤ � f , µ � Λ ⊃ conv( δ ( L )) Sufficient condition Can be verified via LP: min µ ∈ Λ � f , ( I − P ) µ � ≤ 0 Theorem Relaxed-improving condition is satisfied for all methods: Goldstein’s General DEE QPBO MQPBO (prev. work, Shekhovtsov et al . 07) Methods of Kovtun (prev. work, Shekhovtsov et al . 12) Methods of Swoboda et al . 13 (14*) 17/26 Alexander Shekhovtsov, TU Graz Maximum Persistency in Energy Minimization
Introduction Mappings Generalized Sufficient Condition Maximum Persistency Experiments References Maximum Persistency 18/26 Alexander Shekhovtsov, TU Graz Maximum Persistency in Energy Minimization
Introduction Mappings Generalized Sufficient Condition Maximum Persistency Experiments References Maximum Persistency Given that verification problem is polynomially solvable, Which method is better? Proposition Pose ”the best partial optimality” as optimization problem 19/26 Alexander Shekhovtsov, TU Graz Maximum Persistency in Energy Minimization
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