(i) {Î± - Convex Optimization

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Characterizing Problem (Peq) min Ψ(α) s.t. y = HΦα. Proposition 1 (i) Existence: (P eq ) has at least one solution. (ii) Uniqueness: (P eq ) has a unique solution if ψ is strictly convex. (iii) If α solves (P eq ), then it solves (P σ=0 ). Stanford seminar 08-20
DR to solve Problem (Peq) Theorem 1 Let A = HΦ. Let µ ∈ (0, +∞), let {β t } be a sequence in (0, 2), and let {a t } and {b t } be sequences in H such that ∑ t β t(2 − β t ) = +∞ and ∑ t β t (‖a t ‖ + ‖b t ‖) < +∞. Fix α 0 and define the sequence of iterates, α t+1/2 = α t + A ∗ (AA ∗ ) −1 ( y − Aα t) + b t ( α t+1 = α t + β t (prox µΨ ◦ 2α t+1/2 − α t) + a t − α t+1/2) . Then {α t , t ≥ 0} converges weakly to some point α and α+A ∗ (AA ∗ ) −1 (y − Aα) is a solution to (P eq ). As before Stanford seminar 08-21
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 and 22: Operator splitting schemes Idea: re
Page 23 and 24: Proximity operators Some properties
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: DR to solve Problem (P σ ) Theorem
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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