Introduction Proximal IP method Proximity operator of the barrier Proposed architecture Network stability Numerical experiments Conclusion Deep Unfolding of a Proximal Interior Point Method for Image Restoration M.-C. Corbineau 1 in collaboration with C. Bertocchi 2 , E. Chouzenoux 1 , J.C. Pesquet 1 , M. Prato 2 1Université Paris-Saclay, CentraleSupélec, Inria, Centre de Vision Numérique, Gif-sur-Yvette, France 2Università di Modena e Reggio Emilia, Modena, Italy 19 November 2019 Workshop on Regularisation for Inverse Problems and Machine Learning Jussieu, Paris Corbineau et al. Deep Unfolding of a Proximal Interior Point Method Workshop Jussieu, 2019 1 / 35
Introduction Proximal IP method Proximity operator of the barrier Proposed architecture Network stability Numerical experiments Conclusion Motivation Inverse problem in imaging y = D ( Hx ) where y ∈ R m observed image, D degradation model, H ∈ R m × n linear observation model, x ∈ R n original image Corbineau et al. Deep Unfolding of a Proximal Interior Point Method Workshop Jussieu, 2019 2 / 35
Introduction Proximal IP method Proximity operator of the barrier Proposed architecture Network stability Numerical experiments Conclusion Motivation Inverse problem in imaging y = D ( Hx ) where y ∈ R m observed image, D degradation model, H ∈ R m × n linear observation model, x ∈ R n original image Variational methods minimize f ( Hx , y ) + λ R ( x ) x ∈C where f : R m × R m → R data-fitting term, R : R n → R regularization function, λ > 0 regularization weight ✓ Incorporate prior knowledge about solution and enforce desirable constraints ✗ No closed-form solution → advanced algorithms ✗ Estimation of λ and tuning of algorithm parameters → time-consuming Corbineau et al. Deep Unfolding of a Proximal Interior Point Method Workshop Jussieu, 2019 2 / 35
Introduction Proximal IP method Proximity operator of the barrier Proposed architecture Network stability Numerical experiments Conclusion Motivation Inverse problem in imaging y = D ( Hx ) where y ∈ R m observed image, D degradation model, H ∈ R m × n linear observation model, x ∈ R n original image Variational methods minimize f ( Hx , y ) + λ R ( x ) x ∈C where f : R m × R m → R data-fitting term, R : R n → R regularization function, λ > 0 regularization weight ✓ Incorporate prior knowledge about solution and enforce desirable constraints ✗ No closed-form solution → advanced algorithms ✗ Estimation of λ and tuning of algorithm parameters → time-consuming Deep-learning methods ✓ Generic and very efficient architectures ✗ Pre-processing step : solve optimization problem → estimate regularization parameter ✗ Black-box, no theoretical guarantees Corbineau et al. Deep Unfolding of a Proximal Interior Point Method Workshop Jussieu, 2019 2 / 35
Introduction Proximal IP method Proximity operator of the barrier Proposed architecture Network stability Numerical experiments Conclusion Motivation Inverse problem in imaging y = D ( Hx ) where y ∈ R m observed image, D degradation model, H ∈ R m × n linear observation model, x ∈ R n original image Variational methods minimize f ( Hx , y ) + λ R ( x ) x ∈C where f : R m × R m → R data-fitting term, R : R n → R regularization function, λ > 0 regularization weight ✓ Incorporate prior knowledge about solution and enforce desirable constraints ✗ No closed-form solution → advanced algorithms ✗ Estimation of λ and tuning of algorithm parameters → time-consuming Deep-learning methods ✓ Generic and very efficient architectures ✗ Pre-processing step : solve optimization problem → estimate regularization parameter ✗ Black-box, no theoretical guarantees → Combine benefits of both approaches : unfold proximal interior point algorithm Corbineau et al. Deep Unfolding of a Proximal Interior Point Method Workshop Jussieu, 2019 2 / 35
Introduction Proximal IP method Proximity operator of the barrier Proposed architecture Network stability Numerical experiments Conclusion Deep Unfolding • Examples • Sparse coding : FISTA [Gregor and LeCun, 2010] , ISTA [Kamilov and Mansour, 2016] • Compressive sensing : ISTA [Zhang and Ghanem, 2018] , ADMM [Sun et al., 2016] • Principle Iterative solver Unfolded algorithm for k = 0 , 1 , . . . for k = 0 , 1 , . . . , K − 1 � � x k , L ( θ ) x k +1 = A ( x k , θ k ) = ⇒ x k +1 = A k ( x k ) ↓ ↓ hyperparameters layer estimating hyperparameters Estimate : x ∗ = lim Estimate : x ∗ = x K k →∞ x k • Operators and functions included in A can be learned ✓ Gradient backpropagation and training are simpler ✗ Link to the original algorithm is weakened Corbineau et al. Deep Unfolding of a Proximal Interior Point Method Workshop Jussieu, 2019 3 / 35
Introduction Proximal IP method Proximity operator of the barrier Proposed architecture Network stability Numerical experiments Conclusion Notation and Assumptions Proximity operator Let Γ 0 ( R n ) be the set of proper lsc convex functions from R n to R ∪ { + ∞} . The proximal operator [ http://proximity-operator.net/ ] of g ∈ Γ 0 ( R n ) at x ∈ R n is uniquely defined as � 2 � z − x � 2 � g ( z ) + 1 prox g ( x ) = argmin . z ∈ R n Assumptions P 0 : minimize f ( Hx , y ) + λ R ( x ) x ∈C We assume that f ( · , y ) and R are twice-differentiable, f ( H · , y ) + λ R ∈ Γ 0 ( R n ) is either coercive or C is bounded. The feasible set is defined as C = { x ∈ R n | ( ∀ i ∈ { 1 , . . . , p } ) c i ( x ) ≥ 0 } where ( ∀ i ∈ { 1 , . . . , p } ), − c i ∈ Γ 0 ( R n ). The strict interior of the feasible set is nonempty. Existence of a solution to P 0 Twice-differentiability : training using gradient descent B : logarithmic barrier − � p � i =1 ln( c i ( x )) if x ∈ int C ( ∀ x ∈ R n ) B ( x ) = + ∞ otherwise . Corbineau et al. Deep Unfolding of a Proximal Interior Point Method Workshop Jussieu, 2019 4 / 35
Introduction Proximal IP method Proximity operator of the barrier Proposed architecture Network stability Numerical experiments Conclusion Logarithmic barrier method Constrained Problem P 0 : minimize f ( Hx , y ) + λ R ( x ) x ∈C Corbineau et al. Deep Unfolding of a Proximal Interior Point Method Workshop Jussieu, 2019 5 / 35
Introduction Proximal IP method Proximity operator of the barrier Proposed architecture Network stability Numerical experiments Conclusion Logarithmic barrier method Constrained Problem P 0 : minimize f ( Hx , y ) + λ R ( x ) x ∈C ⇓ Unconstrained Subproblem P µ : minimize f ( Hx , y ) + λ R ( x ) + µ B ( x ) x ∈ R n where µ > 0 is the barrier parameter. P 0 is replaced by a sequence of subproblems ( P µ j ) j ∈ N . Subproblems solved approximately for a sequence µ j → 0 Main advantages : feasible iterates, superlinear convergence for NLP ✗ Inversion of an n × n matrix at each step Corbineau et al. Deep Unfolding of a Proximal Interior Point Method Workshop Jussieu, 2019 5 / 35
Introduction Proximal IP method Proximity operator of the barrier Proposed architecture Network stability Numerical experiments Conclusion Proximal interior point strategy → Combine interior point method with proximity operator Exact version of the proximal IPM in [Kaplan and Tichatschke, 1998] . Let x 0 ∈ int C , γ > 0, ( ∀ k ∈ N ) γ ≤ γ k and µ k → 0 ; for k = 0 , 1 , . . . do x k +1 = prox γ k ( f ( H · , y )+ λ R + µ k B ) ( x k ) end for ✗ No closed-form solution for prox γ k ( f ( H · , y )+ λ R + µ k B ) Corbineau et al. Deep Unfolding of a Proximal Interior Point Method Workshop Jussieu, 2019 6 / 35
Introduction Proximal IP method Proximity operator of the barrier Proposed architecture Network stability Numerical experiments Conclusion Proximal interior point strategy → Combine interior point method with proximity operator Exact version of the proximal IPM in [Kaplan and Tichatschke, 1998] . Let x 0 ∈ int C , γ > 0, ( ∀ k ∈ N ) γ ≤ γ k and µ k → 0 ; for k = 0 , 1 , . . . do x k +1 = prox γ k ( f ( H · , y )+ λ R + µ k B ) ( x k ) end for ✗ No closed-form solution for prox γ k ( f ( H · , y )+ λ R + µ k B ) Proposed forward–backward proximal IPM. Let x 0 ∈ int C , γ > 0, ( ∀ k ∈ N ) γ ≤ γ k and µ k → 0 ; for k = 0 , 1 , . . . do � � H ⊤ ∇ 1 f ( Hx k , y ) + λ ∇R ( x k ) �� x k +1 = prox γ k µ k B x k − γ k end for ✓ Only requires prox γ k µ k B Corbineau et al. Deep Unfolding of a Proximal Interior Point Method Workshop Jussieu, 2019 6 / 35
Introduction Proximal IP method Proximity operator of the barrier Proposed architecture Network stability Numerical experiments Conclusion Proximity operator of the barrier Let ϕ : ( x , α ) �→ prox α B ( x ). A neural network obtained by unfolding an iterative solver A • requires to compute A ( x , θ ). → expression for the proximity operator ϕ ( x , α ) ? Corbineau et al. Deep Unfolding of a Proximal Interior Point Method Workshop Jussieu, 2019 7 / 35
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