(i) {Î± - Convex Optimization

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Operator splitting schemes Idea: replace explicit evaluation of the resolvent of ∂ (f 1 + f 2 ) ( i.e. J ∂(f1 +f 2 )) , by a sequence of calculations involving only ∂f 1 and ∂f 2 at a time. An extensive literature essentially divided into three classes: Splitting method Assumptions Iteration Forward-Backward f 2 has a Lipschitzcontinuous gradient Backward-Backward f 1 ,f 2 proper lsc convex but ... Douglas/Peaceman-Rachford All f 1 ,f 2 proper lsc convex α t+1 = J µ∂f1 (I − µ∇f 2 )(α t ) α t+1 = J ∂f1 J ∂f2 (α t ) α t+1 =(J ∂f1 (2J ∂f2 − I)+I − J ∂f2 )(α t ) J ∂fi = (I + ∂f i ) −1 Moreau(-Yosida) proximity operators Stanford seminar 08-10
Proximity operators Some properties The prox is uniquely valued. The Moreau envelope is Fréchet-differentiable with Lipschitz continuous gradient, hence the name Moreau-Yosida regularization. Many useful rules in proximal calculus (scaling, translation, quadratic perturbation, etc). Stanford seminar 08-11
Page 1 and 2: Optimization problems in compressed
Page 3 and 4: Compressed/ive Sensing Stanford sem
Page 5 and 6: Compressed/ive Sensing Common wisdo
Page 7 and 8: Compressed/ive Sensing Common wisdo
Page 9 and 10: Compressed/ive Sensing (cont’d) C
Page 11 and 12: Compressed/ive Sensing (cont’d) C
Page 13 and 14: Convex analysis and operator splitt
Page 15 and 16: Class of problems in CS (cont’d)
Page 17 and 18: Class of problems in CS (cont’d)
Page 19 and 20: Characterization Theorem 1 (i) Exis
Page 21: Operator splitting schemes Idea: re
Page 25 and 26: Example of proximity operator Stanf
Page 27 and 28: Compressed sensing optimization pro
Page 29 and 30: Characterizing Problem (P τ ) Stan
Page 31 and 32: Proximity operators of Ψ Conclusio
Page 33 and 34: DR to solve Problem (P σ ) Theorem
Page 35 and 36: DR to solve Problem (Peq) Theorem 1
Page 37 and 38: Pros and cons (P σ ) and (P eq ) h
Page 39 and 40: CS reconstruction (1) H = Fourier,
Page 41 and 42: Inpainting and CS H = Dirac, Φ = C
Page 43 and 44: Inpainting and CS H = Dirac, Φ = C
Page 45 and 46: Computation time CS H = Fourier, Φ
Page 47 and 48: Ongoing and future work Beyond the

(i) {Î± - Convex Optimization

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