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A Tightrope Walk Between Convexity and Non-convexity in Computer Vision Thomas Pock Institute for Computer Graphics and Vision, Graz University of Technology, 8010 Graz, Austria Qualcomm Augmented Reality Lecture, 29.11.2013 Joint work with

  1. Introduction Non-convex Optimization Convex Optimization Local Convexification Convex Envelopes Conclusion Convex versus non-convex ◮ From optimization, it is well known that the great watershed in optimization is between convexity and non-convexity [Rockafellar] ◮ In recent years, machine learning and computer vision has suffered from a convexivitis epidemic [LeCun] ◮ Nowadays, convexity is considered to be a virtue for new models in machine learning and vision ◮ Sure, convexity is a very useful property, but it can also be a limitation ◮ Most interesting problems are non-convex (optical flow, stereo, image restoration, segmentation, classification, ...) ◮ To solve complicated tasks, we will not be able to avoid non-convexity

  2. Introduction Non-convex Optimization Convex Optimization Local Convexification Convex Envelopes Conclusion Convex versus non-convex ◮ From optimization, it is well known that the great watershed in optimization is between convexity and non-convexity [Rockafellar] ◮ In recent years, machine learning and computer vision has suffered from a convexivitis epidemic [LeCun] ◮ Nowadays, convexity is considered to be a virtue for new models in machine learning and vision ◮ Sure, convexity is a very useful property, but it can also be a limitation ◮ Most interesting problems are non-convex (optical flow, stereo, image restoration, segmentation, classification, ...) ◮ To solve complicated tasks, we will not be able to avoid non-convexity Strategies for bridging the gap between convex and non-convex approaches

  3. Introduction Non-convex Optimization Convex Optimization Local Convexification Convex Envelopes Conclusion Strategies to solve non-convex problems 1 Work directly with the non-convex problem ◮ Sometimes works well ◮ Sometimes, does not work at all ◮ Consider functions with a small degree of non-convexity

  4. Introduction Non-convex Optimization Convex Optimization Local Convexification Convex Envelopes Conclusion Strategies to solve non-convex problems 1 Work directly with the non-convex problem ◮ Sometimes works well ◮ Sometimes, does not work at all ◮ Consider functions with a small degree of non-convexity 2 Local convexification of the problem ◮ Majorization- and minimization of the problem ◮ Linearization of the source of non-convexity ◮ We can solve a sequence of convex problems ◮ Can work very well, but often no guarantees

  5. Introduction Non-convex Optimization Convex Optimization Local Convexification Convex Envelopes Conclusion Strategies to solve non-convex problems 1 Work directly with the non-convex problem ◮ Sometimes works well ◮ Sometimes, does not work at all ◮ Consider functions with a small degree of non-convexity 2 Local convexification of the problem ◮ Majorization- and minimization of the problem ◮ Linearization of the source of non-convexity ◮ We can solve a sequence of convex problems ◮ Can work very well, but often no guarantees 3 Minimize the (approximated) convex envelope ◮ Compute the convex envelope of the problem ◮ We can solve a single convex optimization problem ◮ Often allows to give a-priori approximation guarantees ◮ Restricted to relatively simple models

  6. Introduction Non-convex Optimization Convex Optimization Local Convexification Convex Envelopes Conclusion Overview 1 Introduction 2 Non-convex Optimization 3 Convex Optimization 4 Local Convexification 5 Convex Envelopes 6 Conclusion

  7. Introduction Non-convex Optimization Convex Optimization Local Convexification Convex Envelopes Conclusion Non-convex optimization problems ◮ Efficiently finding solutions to the whole class of Lipschitz continuous problems is a hopeless case [Nesterov ’04] ◮ Can take several million years for small problems with only 10 unknowns

  8. Introduction Non-convex Optimization Convex Optimization Local Convexification Convex Envelopes Conclusion Non-convex optimization problems ◮ Efficiently finding solutions to the whole class of Lipschitz continuous problems is a hopeless case [Nesterov ’04] ◮ Can take several million years for small problems with only 10 unknowns ◮ Smooth non-convex problems can be solved via generic nonlinear numerical optimization algorithms (SD, CG, BFGS, ...) ◮ Often hard to generalize to constraints, or non-differentiable functions ◮ Line-search procedure can be time intensive

  9. Introduction Non-convex Optimization Convex Optimization Local Convexification Convex Envelopes Conclusion Non-convex optimization problems ◮ Efficiently finding solutions to the whole class of Lipschitz continuous problems is a hopeless case [Nesterov ’04] ◮ Can take several million years for small problems with only 10 unknowns ◮ Smooth non-convex problems can be solved via generic nonlinear numerical optimization algorithms (SD, CG, BFGS, ...) ◮ Often hard to generalize to constraints, or non-differentiable functions ◮ Line-search procedure can be time intensive ◮ A reasonable idea is to develop algorithms for special classes of structured non-convex problems ◮ A promising class of problems that has a moderate degree of non-convexity is given by the sum of a smooth non-convex function and a non-smooth convex function [Sra ’12], [Chouzenoux, Pesquet, Repetti ’13]

  10. Introduction Non-convex Optimization Convex Optimization Local Convexification Convex Envelopes Conclusion Smooth plus convex problems ◮ We consider the problem of minimizing a function h : X → R ∪ { + ∞} min x ∈ X h ( x ) = f ( x ) + g ( x ) , where X is a finite dimensional real vector space. ◮ We assume that h is coercive, i.e. � x � 2 → + ∞ ⇒ h ( x ) → + ∞ and bounded from below by some value h > −∞

  11. Introduction Non-convex Optimization Convex Optimization Local Convexification Convex Envelopes Conclusion Smooth plus convex problems ◮ We consider the problem of minimizing a function h : X → R ∪ { + ∞} min x ∈ X h ( x ) = f ( x ) + g ( x ) , where X is a finite dimensional real vector space. ◮ We assume that h is coercive, i.e. � x � 2 → + ∞ ⇒ h ( x ) → + ∞ and bounded from below by some value h > −∞ ◮ The function f ∈ C 1 , 1 is possibly non-convex but has a Lipschitz continuous L gradient, i.e. �∇ f ( x ) − ∇ f ( y ) � 2 ≤ L � x − y � 2 , ∀ x, y ∈ dom f .

  12. Introduction Non-convex Optimization Convex Optimization Local Convexification Convex Envelopes Conclusion Smooth plus convex problems ◮ We consider the problem of minimizing a function h : X → R ∪ { + ∞} x ∈ X h ( x ) = f ( x ) + g ( x ) , min where X is a finite dimensional real vector space. ◮ We assume that h is coercive, i.e. � x � 2 → + ∞ ⇒ h ( x ) → + ∞ and bounded from below by some value h > −∞ ◮ The function f ∈ C 1 , 1 is possibly non-convex but has a Lipschitz continuous L gradient, i.e. �∇ f ( x ) − ∇ f ( y ) � 2 ≤ L � x − y � 2 , ∀ x, y ∈ dom f . ◮ The function g is a proper lower semi-continuous convex function with an efficient to compute proximal map x � 2 � x − ˆ ( I + α∂g ) − 1 (ˆ 2 x ) := arg min + αg ( x ) , 2 x ∈ X where α > 0 .

  13. Introduction Non-convex Optimization Convex Optimization Local Convexification Convex Envelopes Conclusion Forward-backward splitting ◮ We aim at seeking a critical point x ∗ , i.e. a point satisfying 0 ∈ ∂h ( x ∗ ) which in our case becomes −∇ f ( x ∗ ) ∈ ∂g ( x ∗ ) . ◮ A critical point can also be characterized via the proximal residual r ( x ) := x − ( I + ∂g ) − 1 ( x − ∇ f ( x )) , where I is the identity map. ◮ Clearly r ( x ∗ ) = 0 implies that x ∗ is a critical point. ◮ The norm of the proximal residual can be used as a (bad) measure of optimality

  14. Introduction Non-convex Optimization Convex Optimization Local Convexification Convex Envelopes Conclusion Forward-backward splitting ◮ We aim at seeking a critical point x ∗ , i.e. a point satisfying 0 ∈ ∂h ( x ∗ ) which in our case becomes −∇ f ( x ∗ ) ∈ ∂g ( x ∗ ) . ◮ A critical point can also be characterized via the proximal residual r ( x ) := x − ( I + ∂g ) − 1 ( x − ∇ f ( x )) , where I is the identity map. ◮ Clearly r ( x ∗ ) = 0 implies that x ∗ is a critical point. ◮ The norm of the proximal residual can be used as a (bad) measure of optimality ◮ The proximal residual already suggests an iterative method of the form x n +1 = ( I + ∂g ) − 1 ( x n − ∇ f ( x n )) ◮ For f convex, this algorithm is well studied [Lions, Mercier ’79], [Tseng ’91], [Daubechie et al. ’04], [Combettes, Wajs ’05], [Raguet, Fadili, Peyr´ e ’13]

  15. Introduction Non-convex Optimization Convex Optimization Local Convexification Convex Envelopes Conclusion Inertial methods ◮ Introduced by Polyak in [Polyak ’64] as a special case of multi-step algorithms for minimizing a function f ∈ S 1 , 1 µ,L x n +1 = x n − α ∇ f ( x n ) + β ( x n − x n − 1 ) ◮ Optimal convergence rate on strongly convex problems ◮ Close relations to the conjugate gradient method ◮ Can be seen as a discrete variant of the heavy-ball with friction dynamic system ◮ Hence, the inertial term acts as an acceleration term ◮ Can help to avoid suprious critical points ◮ We propose a generalization to minimize the sum of a smooth and a convex function

  16. Introduction Non-convex Optimization Convex Optimization Local Convexification Convex Envelopes Conclusion iPiano (inertial Proximal algoriothm for non-convex optimization) For minimizing the sum of a smooth and a convex function, we propose the following algorithm: ◮ Initialization: Choose c 1 , c 2 > 0 , x 0 ∈ dom h and set x − 1 = x 0 . ◮ Iterations ( n ≥ 0) : Update x n +1 = ( I + α n ∂g ) − 1 ( x n − α n ∇ f ( x n ) + β n ( x n − x n − 1 )) , where L n > 0 is the local Lipschitz constant satisfying + L n ∇ f ( x n ) , x n +1 − x n � 2 � x n +1 − x n � 2 f ( x n +1 ) ≤ f ( x n ) + � 2 , and α n ≥ c 1 , β n ≥ 0 are chosen such that δ n ≥ γ n ≥ c 2 defined by δ n := 1 − L n 2 − β n γ n := 1 − L n 2 − β n and . α n 2 α n α n α n and ( δ n ) ∞ n =0 is monotonically decreasing.

  17. Introduction Non-convex Optimization Convex Optimization Local Convexification Convex Envelopes Conclusion Convergence Analysis We can give the following convergence result: Theorem (a) The sequence ( h ( x n )) ∞ n =0 converges. (b) There exists a converging subsequence ( x n k ) ∞ k =0 . (c) Any limit point x ∗ := lim k →∞ x n k is a critical point of h . ◮ Convergence of the whole sequence can be obtained by assuming that the so-called Kurdyka-� Lojasiewicz property holds, wich is true for most reasonable functions

  18. Introduction Non-convex Optimization Convex Optimization Local Convexification Convex Envelopes Conclusion Convergence rate in the non-convex case ◮ Absence of convexity makes live hard

  19. Introduction Non-convex Optimization Convex Optimization Local Convexification Convex Envelopes Conclusion Convergence rate in the non-convex case ◮ Absence of convexity makes live hard ◮ We can merely establish the following very weak convergence rate

  20. Introduction Non-convex Optimization Convex Optimization Local Convexification Convex Envelopes Conclusion Convergence rate in the non-convex case ◮ Absence of convexity makes live hard ◮ We can merely establish the following very weak convergence rate Theorem The iPiano algorithm guarantees that for all N ≥ 0 � h ( x 0 ) − h 2 0 ≤ n ≤ N � r ( x n ) � 2 ≤ min c 1 c 2 N + 1 √ i.e. the smallest proximal residual converges with rate O (1 / N ) . ◮ Similar bound for β = 0 is shown in [Nesterov ’12]

  21. Introduction Non-convex Optimization Convex Optimization Local Convexification Convex Envelopes Conclusion Application to image compression based on linear diffusion ◮ A new image compression methodology introduced in [Galic, Weickert, Welk, Bruhn, Belyaev, Seidel ’08] ◮ The idea is to select a subset of image pixels such that the reconstruction of the whole image via linear diffusion yields the best reconstruction [Hoeltgen, Setzer, Weickert ’13] ◮ Is written as the following bilevel optimization problem 1 2 � u − u 0 � 2 min 2 + λ � c � 1 u,c s.t. C ( u − u 0 ) − ( I − C ) Lu = 0 , where C = diag( c ) ∈ R N × N and L is the Laplace operator ◮ We can transform the problem into an non-convex single-level problem of the form 1 2 � A − 1 Cu 0 − u 0 � 2 min 2 + λ � c � 1 , A = C + ( C − I ) L c

  22. Introduction Non-convex Optimization Convex Optimization Local Convexification Convex Envelopes Conclusion ◮ Perfectly fits to the framework of iPiano 2 � A − 1 Cu 0 − u 0 � 2 ◮ We choose f = 1 2 and g = λ � c � 1 ◮ The gradient of f is given by ∇ f ( c ) = diag( − ( I + L ) u + u 0 )( A ⊤ ) − 1 ( u − u 0 ) , u = A − 1 Cu 0 ◮ Lipschitz, if at least one entry of c is non-zero ◮ One evaluation of the gradient requires to solve two linear systems ◮ Proximal map with respect to g is standard

  23. Introduction Non-convex Optimization Convex Optimization Local Convexification Convex Envelopes Conclusion Results Comparison with the successive primal-dual (SPD) algorithm proposed in [Hoeltgen, Setzer, Weickert ’13] Test Algorithm Iterations Energy Density MSE image iPiano 1000 21.574011 4.98% 17.31 Trui SPD 200/4000 21.630280 5.08% 17.06 iPiano 1000 20.631985 4.84% 19.50 Peppers SPD 200/4000 20.758777 4.93% 19.48 iPiano 1000 10.246041 4.82% 8.29 Walter SPD 200/4000 10.278874 4.93% 8.01

  24. Introduction Non-convex Optimization Convex Optimization Local Convexification Convex Envelopes Conclusion Results for Trui

  25. Introduction Non-convex Optimization Convex Optimization Local Convexification Convex Envelopes Conclusion Results for Trui

  26. Introduction Non-convex Optimization Convex Optimization Local Convexification Convex Envelopes Conclusion Results for Trui

  27. Introduction Non-convex Optimization Convex Optimization Local Convexification Convex Envelopes Conclusion Results for Walter

  28. Introduction Non-convex Optimization Convex Optimization Local Convexification Convex Envelopes Conclusion Results for Walter

  29. Introduction Non-convex Optimization Convex Optimization Local Convexification Convex Envelopes Conclusion Results for Walter

  30. Introduction Non-convex Optimization Convex Optimization Local Convexification Convex Envelopes Conclusion Overview 1 Introduction 2 Non-convex Optimization 3 Convex Optimization 4 Local Convexification 5 Convex Envelopes 6 Conclusion

  31. Introduction Non-convex Optimization Convex Optimization Local Convexification Convex Envelopes Conclusion A class of problems Let us consider the following class of structured convex optimization problems min x ∈ X F ( Kx ) + G ( x ) , ◮ K : X → Y is a linear and continuous operator from a Hilbert space X to a Hilbert space Y and F , G are convex, (non-smooth) proper, l.s.c. functions.

  32. Introduction Non-convex Optimization Convex Optimization Local Convexification Convex Envelopes Conclusion A class of problems Let us consider the following class of structured convex optimization problems min x ∈ X F ( Kx ) + G ( x ) , ◮ K : X → Y is a linear and continuous operator from a Hilbert space X to a Hilbert space Y and F , G are convex, (non-smooth) proper, l.s.c. functions. ◮ Main assumption: F , G are “simple” in the sense that they have easy to compute resolvent operators: p � 2 � p − ˆ ( I + ∂F ) − 1 (ˆ p ) = arg min + F ( p ) 2 λ p x � 2 � x − ˆ ( I + ∂G ) − 1 (ˆ x ) = arg min + G ( x ) 2 λ x

  33. Introduction Non-convex Optimization Convex Optimization Local Convexification Convex Envelopes Conclusion A class of problems Let us consider the following class of structured convex optimization problems min x ∈ X F ( Kx ) + G ( x ) , ◮ K : X → Y is a linear and continuous operator from a Hilbert space X to a Hilbert space Y and F , G are convex, (non-smooth) proper, l.s.c. functions. ◮ Main assumption: F , G are “simple” in the sense that they have easy to compute resolvent operators: p � 2 � p − ˆ ( I + ∂F ) − 1 (ˆ p ) = arg min + F ( p ) 2 λ p x � 2 � x − ˆ ( I + ∂G ) − 1 (ˆ x ) = arg min + G ( x ) 2 λ x ◮ It turns out that many standard problems can be cast in this framework.

  34. Introduction Non-convex Optimization Convex Optimization Local Convexification Convex Envelopes Conclusion Some examples ◮ The ROF model u �∇ u � 2 , 1 + λ 2 � u − f � 2 min 2 ,

  35. Introduction Non-convex Optimization Convex Optimization Local Convexification Convex Envelopes Conclusion Some examples ◮ The ROF model u �∇ u � 2 , 1 + λ 2 � u − f � 2 min 2 , ◮ Basis pursuit problem (LASSO) x � x � 1 + λ 2 � Ax − b � 2 min 2

  36. Introduction Non-convex Optimization Convex Optimization Local Convexification Convex Envelopes Conclusion Some examples ◮ The ROF model u �∇ u � 2 , 1 + λ 2 � u − f � 2 min 2 , ◮ Basis pursuit problem (LASSO) x � x � 1 + λ 2 � Ax − b � 2 min 2 ◮ Linear support vector machine n λ 2 � w � 2 � min 2 + max (0 , 1 − y i ( � w, x i � + b )) w,b i =1

  37. Introduction Non-convex Optimization Convex Optimization Local Convexification Convex Envelopes Conclusion Some examples ◮ The ROF model u �∇ u � 2 , 1 + λ 2 � u − f � 2 min 2 , ◮ Basis pursuit problem (LASSO) x � x � 1 + λ 2 � Ax − b � 2 min 2 ◮ Linear support vector machine n λ 2 � w � 2 � min 2 + max (0 , 1 − y i ( � w, x i � + b )) w,b i =1 ◮ General linear programming problems � Ax = b min x � c, x � , s.t. ≥ 0 x

  38. Introduction Non-convex Optimization Convex Optimization Local Convexification Convex Envelopes Conclusion Primal, dual, primal-dual The real power of convex optimization comes through duality Recall the convex conjugate: F ∗ ( y ) = max x ∈ X � x, y � − F ( x ) , we can transform our initial problem min x ∈ X F ( Kx ) + G ( x ) (Primal)

  39. Introduction Non-convex Optimization Convex Optimization Local Convexification Convex Envelopes Conclusion Primal, dual, primal-dual The real power of convex optimization comes through duality Recall the convex conjugate: F ∗ ( y ) = max x ∈ X � x, y � − F ( x ) , we can transform our initial problem min x ∈ X F ( Kx ) + G ( x ) (Primal) y ∈ Y � Kx, y � + G ( x ) − F ∗ ( y ) min x ∈ X max (Primal-Dual)

  40. Introduction Non-convex Optimization Convex Optimization Local Convexification Convex Envelopes Conclusion Primal, dual, primal-dual The real power of convex optimization comes through duality Recall the convex conjugate: F ∗ ( y ) = max x ∈ X � x, y � − F ( x ) , we can transform our initial problem min x ∈ X F ( Kx ) + G ( x ) (Primal) y ∈ Y � Kx, y � + G ( x ) − F ∗ ( y ) min x ∈ X max (Primal-Dual) y ∈ Y − ( F ∗ ( y ) + G ∗ ( − K ∗ y )) max (Dual)

  41. Introduction Non-convex Optimization Convex Optimization Local Convexification Convex Envelopes Conclusion Primal, dual, primal-dual The real power of convex optimization comes through duality Recall the convex conjugate: F ∗ ( y ) = max x ∈ X � x, y � − F ( x ) , we can transform our initial problem min x ∈ X F ( Kx ) + G ( x ) (Primal) y ∈ Y � Kx, y � + G ( x ) − F ∗ ( y ) min x ∈ X max (Primal-Dual) y ∈ Y − ( F ∗ ( y ) + G ∗ ( − K ∗ y )) max (Dual) There is a primal-dual gap: G ( x, y ) = F ( Kx ) + G ( x ) + ( F ∗ ( y ) + G ∗ ( − K ∗ y )) that vanishes if and only if ( x, y ) is optimal

  42. Introduction Non-convex Optimization Convex Optimization Local Convexification Convex Envelopes Conclusion Optimality conditions We focus on the primal-dual saddle-point formulation: y ∈ Y � Kx, y � + G ( x ) − F ∗ ( y ) min x ∈ X max The optimal solution is a saddle-point (ˆ x, ˆ y ) ∈ X × Y which satisfies the Euler-Lagrange equations � x ) + K ∗ ˆ � ∂G (ˆ y 0 ∈ ∂F ∗ (ˆ y ) − K ˆ x 3 2.5 2 1.5 1 0.5 0 1 −0.5 0.5 0 −1 −0.5 | x | + | x − f | 2 / 2 −1.5 −1 −0.5 0 0.5 −1 1 1.5 How can we find a saddle-point (ˆ x, ˆ y ) ?

  43. Introduction Non-convex Optimization Convex Optimization Local Convexification Convex Envelopes Conclusion A first-order primal-dual algorithm Proposed in a series of papers: [P., Cremers, Bischof, Chambolle, ’09], [Chambolle, P., ’10], [P., Chambolle, ’11] ◮ Initialization: Choose T , Σ ∈ S ++ , θ ∈ [0 , 1] , ( x 0 , y 0 ) ∈ X × Y .

  44. Introduction Non-convex Optimization Convex Optimization Local Convexification Convex Envelopes Conclusion A first-order primal-dual algorithm Proposed in a series of papers: [P., Cremers, Bischof, Chambolle, ’09], [Chambolle, P., ’10], [P., Chambolle, ’11] ◮ Initialization: Choose T , Σ ∈ S ++ , θ ∈ [0 , 1] , ( x 0 , y 0 ) ∈ X × Y . ◮ Iterations ( n ≥ 0) : Update x n , y n as follows: x n +1 = ( I + T ∂G ) − 1 ( x n − T K ∗ y n ) � y n +1 = ( I + Σ ∂F ∗ ) − 1 ( y n + Σ K ( x n +1 + θ ( x n +1 − x n )))

  45. Introduction Non-convex Optimization Convex Optimization Local Convexification Convex Envelopes Conclusion A first-order primal-dual algorithm Proposed in a series of papers: [P., Cremers, Bischof, Chambolle, ’09], [Chambolle, P., ’10], [P., Chambolle, ’11] ◮ Initialization: Choose T , Σ ∈ S ++ , θ ∈ [0 , 1] , ( x 0 , y 0 ) ∈ X × Y . ◮ Iterations ( n ≥ 0) : Update x n , y n as follows: x n +1 = ( I + T ∂G ) − 1 ( x n − T K ∗ y n ) � y n +1 = ( I + Σ ∂F ∗ ) − 1 ( y n + Σ K ( x n +1 + θ ( x n +1 − x n ))) ◮ T , Σ are preconditioning matrices

  46. Introduction Non-convex Optimization Convex Optimization Local Convexification Convex Envelopes Conclusion A first-order primal-dual algorithm Proposed in a series of papers: [P., Cremers, Bischof, Chambolle, ’09], [Chambolle, P., ’10], [P., Chambolle, ’11] ◮ Initialization: Choose T , Σ ∈ S ++ , θ ∈ [0 , 1] , ( x 0 , y 0 ) ∈ X × Y . ◮ Iterations ( n ≥ 0) : Update x n , y n as follows: x n +1 = ( I + T ∂G ) − 1 ( x n − T K ∗ y n ) � y n +1 = ( I + Σ ∂F ∗ ) − 1 ( y n + Σ K ( x n +1 + θ ( x n +1 − x n ))) ◮ T , Σ are preconditioning matrices ◮ Alternates gradient descend in x and gradient ascend in y

  47. Introduction Non-convex Optimization Convex Optimization Local Convexification Convex Envelopes Conclusion A first-order primal-dual algorithm Proposed in a series of papers: [P., Cremers, Bischof, Chambolle, ’09], [Chambolle, P., ’10], [P., Chambolle, ’11] ◮ Initialization: Choose T , Σ ∈ S ++ , θ ∈ [0 , 1] , ( x 0 , y 0 ) ∈ X × Y . ◮ Iterations ( n ≥ 0) : Update x n , y n as follows: x n +1 = ( I + T ∂G ) − 1 ( x n − T K ∗ y n ) � y n +1 = ( I + Σ ∂F ∗ ) − 1 ( y n + Σ K ( x n +1 + θ ( x n +1 − x n ))) ◮ T , Σ are preconditioning matrices ◮ Alternates gradient descend in x and gradient ascend in y ◮ Linear extrapolation of iterates of x in the y step

  48. Introduction Non-convex Optimization Convex Optimization Local Convexification Convex Envelopes Conclusion Convergence Theorem Let θ = 1 , T and Σ symmetric positive definite maps satisfying 2 � 2 < 1 , 1 1 2 K T � Σ then the primal-dual algorithm converges to a saddle-point.

  49. Introduction Non-convex Optimization Convex Optimization Local Convexification Convex Envelopes Conclusion Convergence Theorem Let θ = 1 , T and Σ symmetric positive definite maps satisfying 2 � 2 < 1 , 1 1 2 K T � Σ then the primal-dual algorithm converges to a saddle-point. The algorithm gives different convergence rates on different problem classes [Chambolle, P., ’10] ◮ F ∗ and G non-smooth: O (1 /n )

  50. Introduction Non-convex Optimization Convex Optimization Local Convexification Convex Envelopes Conclusion Convergence Theorem Let θ = 1 , T and Σ symmetric positive definite maps satisfying 2 � 2 < 1 , 1 1 2 K T � Σ then the primal-dual algorithm converges to a saddle-point. The algorithm gives different convergence rates on different problem classes [Chambolle, P., ’10] ◮ F ∗ and G non-smooth: O (1 /n ) ◮ F ∗ or G uniformly convex: O (1 /n 2 )

  51. Introduction Non-convex Optimization Convex Optimization Local Convexification Convex Envelopes Conclusion Convergence Theorem Let θ = 1 , T and Σ symmetric positive definite maps satisfying 2 � 2 < 1 , 1 1 2 K T � Σ then the primal-dual algorithm converges to a saddle-point. The algorithm gives different convergence rates on different problem classes [Chambolle, P., ’10] ◮ F ∗ and G non-smooth: O (1 /n ) ◮ F ∗ or G uniformly convex: O (1 /n 2 ) ◮ F ∗ and G uniformly convex: O ( ω n ) , ω < 1

  52. Introduction Non-convex Optimization Convex Optimization Local Convexification Convex Envelopes Conclusion Convergence Theorem Let θ = 1 , T and Σ symmetric positive definite maps satisfying 2 � 2 < 1 , 1 1 2 K T � Σ then the primal-dual algorithm converges to a saddle-point. The algorithm gives different convergence rates on different problem classes [Chambolle, P., ’10] ◮ F ∗ and G non-smooth: O (1 /n ) ◮ F ∗ or G uniformly convex: O (1 /n 2 ) ◮ F ∗ and G uniformly convex: O ( ω n ) , ω < 1 ◮ Coincide with lower complexity bounds for first-order methods [Nesterov, ’04]

  53. Introduction Non-convex Optimization Convex Optimization Local Convexification Convex Envelopes Conclusion α -preconditioning ◮ It is important to choose the preconditioner such that the prox-operators are still easy to compute ◮ Restrict the preconditioning matrices to diagonal matrices

  54. Introduction Non-convex Optimization Convex Optimization Local Convexification Convex Envelopes Conclusion α -preconditioning ◮ It is important to choose the preconditioner such that the prox-operators are still easy to compute ◮ Restrict the preconditioning matrices to diagonal matrices Lemma Let T = diag ( τ 1 , ...τ n ) and Σ = diag ( σ 1 , ..., σ m ) . 1 1 τ j = i =1 | K i,j | 2 − α , σ i = � m � n j =1 | K i,j | α then for any α ∈ [0 , 2] 1 1 2 K T 2 x � 2 � Σ 2 � 2 = 1 1 2 K T � Σ sup ≤ 1 . � x � 2 x ∈ X, x � =0 [P., Chambolle, ’11]

  55. Introduction Non-convex Optimization Convex Optimization Local Convexification Convex Envelopes Conclusion α -preconditioning ◮ It is important to choose the preconditioner such that the prox-operators are still easy to compute ◮ Restrict the preconditioning matrices to diagonal matrices Lemma Let T = diag ( τ 1 , ...τ n ) and Σ = diag ( σ 1 , ..., σ m ) . 1 1 τ j = i =1 | K i,j | 2 − α , σ i = � m � n j =1 | K i,j | α then for any α ∈ [0 , 2] 1 1 2 K T 2 x � 2 � Σ 2 � 2 = 1 1 2 K T � Σ sup ≤ 1 . � x � 2 x ∈ X, x � =0 [P., Chambolle, ’11] ◮ The parameter α can be used to vary between pure primal ( α = 0 ) and pure dual ( α = 2 ) preconditioning

  56. Introduction Non-convex Optimization Convex Optimization Local Convexification Convex Envelopes Conclusion Parallel computing? ◮ The algorithm basically computes matrix-vector products

  57. Introduction Non-convex Optimization Convex Optimization Local Convexification Convex Envelopes Conclusion Parallel computing? ◮ The algorithm basically computes matrix-vector products ◮ The matrices are usually very sparse

  58. Introduction Non-convex Optimization Convex Optimization Local Convexification Convex Envelopes Conclusion Parallel computing? ◮ The algorithm basically computes matrix-vector products ◮ The matrices are usually very sparse ◮ Well suited for highly parallel architectures

  59. Introduction Non-convex Optimization Convex Optimization Local Convexification Convex Envelopes Conclusion Parallel computing? ◮ The algorithm basically computes matrix-vector products ◮ The matrices are usually very sparse ◮ Well suited for highly parallel architectures ◮ Gives high speedup factors ( ∼ 30-50)

  60. Introduction Non-convex Optimization Convex Optimization Local Convexification Convex Envelopes Conclusion Overview 1 Introduction 2 Non-convex Optimization 3 Convex Optimization 4 Local Convexification 5 Convex Envelopes 6 Conclusion

  61. Introduction Non-convex Optimization Convex Optimization Local Convexification Convex Envelopes Conclusion Local Convexification ◮ The local convexification uses the structure of the problem ◮ Identify the source of non-convexity ◮ Locally approximate the non-convex function by a convex one ◮ Solve the resulting non-convex problem and repeat the convexification

  62. Introduction Non-convex Optimization Convex Optimization Local Convexification Convex Envelopes Conclusion Non-convex potential functions ◮ The choice of the potential function in image restoration is motivated by the statistics of natural images ◮ Let us record a histogram of the filter-response of a DTC5 filter on natural images [Huang and Mumford ’99] 3.5 -log PDF 3 2.5 2 1.5 1 0.5 0 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 ◮ A good fit is obtanied for the family of non-convex functions log(1 + x 2 )

  63. Introduction Non-convex Optimization Convex Optimization Local Convexification Convex Envelopes Conclusion Application to non-convex image denoising ◮ Approximately minimize a non-convex energy based on Student-t potential functions log(1 + | ( K i x ) p | 2 ) + 1 � � 2 � x − f � 2 min α i 2 , x p i ◮ The application of the linear operators K i are realized via convolution with filters k i K i x ⇔ k i ∗ x ◮ Parameters α i and filters k i are learned using bilevel optimization [Chen et al. ’13]

  64. Introduction Non-convex Optimization Convex Optimization Local Convexification Convex Envelopes Conclusion The filters (5.21,0.33) (5.03,0.22) (4.96,0.29) (4.88,0.13) (4.87,0.22) (4.84,0.01) (4.83,0.13) (4.83,0.02) (4.83,0.03) (4.82,0.27) (4.81,0.25) (4.81,0.07) (4.81,0.08) (4.81,0.02) (4.80,0.05) (4.78,0.06) (4.77,0.02) (4.77,0.05) (4.75,0.02) (4.75,0.13) (4.75,0.25) (4.74,0.02) (4.74,0.18) (4.73,0.02) (4.73,0.01) (4.73,0.02) (4.71,0.01) (4.71,0.03) (4.70,0.13) (4.68,0.23) (4.68,0.20) (4.68,0.01) (4.65,0.02) (4.65,0.23) (4.63,0.02) (4.61,0.01) (4.60,0.10) (4.56,0.02) (4.53,0.01) (4.51,0.19) (4.50,0.42) (4.48,0.10) (4.46,0.10) (4.42,0.01) (4.39,0.03) (4.34,0.01) (4.32,0.34) (4.32,0.23) (4.29,0.01) (4.17,0.34) (4.09,0.14) (4.03,0.29) (4.02,0.25) (4.00,0.41) (3.99,0.27) (3.97,0.13) (3.96,0.24) (3.94,0.50) (3.89,0.44) (3.72,0.60) (3.64,0.32) (3.58,0.27) (3.53,0.23) (3.41,0.29) (3.40,0.23) (3.24,0.70) (3.22,0.59) (3.15,0.43) (3.09,0.45) (2.90,0.59) (2.88,0.24) (2.74,0.58) (2.71,0.69) (2.59,0.44) (2.59,0.39) (2.37,0.63) (2.15,1.17) (2.14,0.78) (1.90,0.79) (1.51,0.56)

  65. Introduction Non-convex Optimization Convex Optimization Local Convexification Convex Envelopes Conclusion Iterated Huber for Student-t ◮ Majorize-Minimize strategy: ◮ Minimize a sequence a of convex weighted Huber- ℓ 1 problems w i ( x n ) p | ( K i x ) p | ε + 1 x n +1 = arg min � � 2 � x − f � 2 α i 2 x p i where w i ( x n ) = 2 max { ε, | K i x n |} and | · | ε denotes the Huber function 1+ | K i x n | 2 50 50 log(1 + t 2 ) log(1 + t 2 ) w | t | ε + c w | t | ε + c 40 40 30 30 20 20 10 10 0 0 −5 0 5 −5 0 5

  66. Introduction Non-convex Optimization Convex Optimization Local Convexification Convex Envelopes Conclusion Iterated Huber for Student-t ◮ Majorize-Minimize strategy: ◮ Minimize a sequence a of convex weighted Huber- ℓ 1 problems w i ( x n ) p | ( K i x ) p | ε + 1 x n +1 = arg min � � 2 � x − f � 2 α i 2 x p i where w i ( x n ) = 2 max { ε, | K i x n |} and | · | ε denotes the Huber function 1+ | K i x n | 2 50 50 log(1 + t 2 ) log(1 + t 2 ) w | t | ε + c w | t | ε + c 40 40 30 30 20 20 10 10 0 0 −5 0 5 −5 0 5 ◮ Best fit for ε = 1

  67. Introduction Non-convex Optimization Convex Optimization Local Convexification Convex Envelopes Conclusion Iterated Huber for Student-t ◮ Majorize-Minimize strategy: ◮ Minimize a sequence a of convex weighted Huber- ℓ 1 problems w i ( x n ) p | ( K i x ) p | ε + 1 x n +1 = arg min � � 2 � x − f � 2 α i 2 x p i where w i ( x n ) = 2 max { ε, | K i x n |} and | · | ε denotes the Huber function 1+ | K i x n | 2 50 50 log(1 + t 2 ) log(1 + t 2 ) w | t | ε + c w | t | ε + c 40 40 30 30 20 20 10 10 0 0 −5 0 5 −5 0 5 ◮ Best fit for ε = 1 ◮ The primal-dual algorithm has a linear convergence rates on the convex sub-problems

  68. Introduction Non-convex Optimization Convex Optimization Local Convexification Convex Envelopes Conclusion Iterated Huber for Student-t ◮ Majorize-Minimize strategy: ◮ Minimize a sequence a of convex weighted Huber- ℓ 1 problems w i ( x n ) p | ( K i x ) p | ε + 1 x n +1 = arg min � � 2 � x − f � 2 α i 2 x p i where w i ( x n ) = 2 max { ε, | K i x n |} and | · | ε denotes the Huber function 1+ | K i x n | 2 50 50 log(1 + t 2 ) log(1 + t 2 ) w | t | ε + c w | t | ε + c 40 40 30 30 20 20 10 10 0 0 −5 0 5 −5 0 5 ◮ Best fit for ε = 1 ◮ The primal-dual algorithm has a linear convergence rates on the convex sub-problems

  69. Introduction Non-convex Optimization Convex Optimization Local Convexification Convex Envelopes Conclusion Example

  70. Introduction Non-convex Optimization Convex Optimization Local Convexification Convex Envelopes Conclusion Example

  71. Introduction Non-convex Optimization Convex Optimization Local Convexification Convex Envelopes Conclusion Example

  72. Introduction Non-convex Optimization Convex Optimization Local Convexification Convex Envelopes Conclusion Evaluation ◮ Comparison with five state-of-the-art approaches: K-SVD [Elad and Aharon ’06], FoE [Q. Gao and Roth ’12], BM3D [Dabov et al. ’07], GMM [D. Zoran et al. ’12], LSSC [Mairal et al. ’09] ◮ We report the average PSNR on 68 images of the Berkeley image data base σ KSVD FoE BM3D GMM LSSC ours 15 30.87 30.99 31.08 31.19 31.27 31.22 25 28.28 28.40 28.56 28.68 28.70 28.70 50 25.17 25.35 25.62 25.67 25.72 25.76 ◮ Performs as well as state-of-the-art ◮ A GPU implementation is significantly faster ◮ Can be used as a prior for general inverse problems

  73. Introduction Non-convex Optimization Convex Optimization Local Convexification Convex Envelopes Conclusion Optical flow ◮ Optical Flow is a central topic in computer vision [Horn, Schunck, 1981], [Shulman, Herv´ e ’89], [Bruhn, Weickert, Schn¨ orr ’02], [Brox, Bruhn, Papenberg, Weickert ’04], [Zach, P., Bischof, DAGM’07] ... ◮ Computes a vector field, describing the aparent motion of pixel intensities ◮ Numerous applications ◮ TV- L 1 optical flow min u �∇ u � 2 , 1 + λ � I 2 ( x + u ) − I 1 ( x ) � 1 ◮ The source of non-convexity lies in the expression I 2 ( x + u )

  74. Introduction Non-convex Optimization Convex Optimization Local Convexification Convex Envelopes Conclusion Optical flow ◮ Optical Flow is a central topic in computer vision [Horn, Schunck, 1981], [Shulman, Herv´ e ’89], [Bruhn, Weickert, Schn¨ orr ’02], [Brox, Bruhn, Papenberg, Weickert ’04], [Zach, P., Bischof, DAGM’07] ... ◮ Computes a vector field, describing the aparent motion of pixel intensities ◮ Numerous applications ◮ TV- L 1 optical flow min u �∇ u � 2 , 1 + λ � I 2 ( x + u ) − I 1 ( x ) � 1 ◮ The source of non-convexity lies in the expression I 2 ( x + u )

  75. Introduction Non-convex Optimization Convex Optimization Local Convexification Convex Envelopes Conclusion Optical flow ◮ Optical Flow is a central topic in computer vision [Horn, Schunck, 1981], [Shulman, Herv´ e ’89], [Bruhn, Weickert, Schn¨ orr ’02], [Brox, Bruhn, Papenberg, Weickert ’04], [Zach, P., Bischof, DAGM’07] ... ◮ Computes a vector field, describing the aparent motion of pixel intensities ◮ Numerous applications ◮ TV- L 1 optical flow min u �∇ u � 2 , 1 + λ � I 2 ( x + u ) − I 1 ( x ) � 1 ◮ The source of non-convexity lies in the expression I 2 ( x + u )

  76. Introduction Non-convex Optimization Convex Optimization Local Convexification Convex Envelopes Conclusion Optical flow ◮ Optical Flow is a central topic in computer vision [Horn, Schunck, 1981], [Shulman, Herv´ e ’89], [Bruhn, Weickert, Schn¨ orr ’02], [Brox, Bruhn, Papenberg, Weickert ’04], [Zach, P., Bischof, DAGM’07] ... ◮ Computes a vector field, describing the aparent motion of pixel intensities ◮ Numerous applications ◮ TV- L 1 optical flow min u �∇ u � 2 , 1 + λ � I 2 ( x + u ) − I 1 ( x ) � 1 ◮ The source of non-convexity lies in the expression I 2 ( x + u )

  77. Introduction Non-convex Optimization Convex Optimization Local Convexification Convex Envelopes Conclusion Optical Flow ◮ Convexification via linearization: � I 2 ( x + u ) − I 1 ( x ) � 1 ≈ � I t + ∇ I 2 ( u − u 0 ) � 1 ◮ Only valid in a small neighborhood around u 0 ◮ Minimized via the primal-dual algorithm

  78. Introduction Non-convex Optimization Convex Optimization Local Convexification Convex Envelopes Conclusion Real-time implementation ◮ Due to the strong non-convexity, the algorithm has to be integrated into a coarse-to-fine / warping framework ◮ Works well in case of small displacements but can fail in case of large displacements [Brox, Bregler, Malik ’09] ◮ GPU-implementation yields real-time performance ( > 20 fps) for 854 × 480 images using a recent Nvidia graphics card [Zach, P., Bischof, ’07] [Werlberger, P., Bischof, ’10] ◮ GLSL shader implementation on a mobile GPU (Adreno 330 in Nexus 5) implementation currently yields 10 fps on 320 × 240 images (implemented by Christoph Bauernhofer). ◮ The performance is expected to increase in near future.

  79. Introduction Non-convex Optimization Convex Optimization Local Convexification Convex Envelopes Conclusion Resolution: 854 × 480

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