(i) {Î± - Convex Optimization

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Class of problems in CS Recall y = Hx + ε, ε iid and Var (ε) =σε 2 < +∞, x = Φα, α (nearly-)sparse. Typical (equivalent) minimization problems: (P eq ) : min Ψ (α) s.t. y = HΦα (P σ ) : min Ψ (α) s.t. ‖y − HΦα‖ l2 ≤ σ (P τ ) : min 1 2 ‖y − HΦα‖2 l 2 s.t. Ψ (α) ≤ τ (P λ ) : min 1 2 ‖y − HΦα‖2 l 2 +λΨ(α) (P eq ) is (P σ ) when no noise. Ψ (α) = ∑ γ∈Γ ψ (α γ). ψ a sparsity-promoting penalty: non-negative, continuous, even-symmetric, and non-decreasing on R + , not necessarily smooth at point zero to produce sparse solutions. e.g. Ψ(α) =‖α‖ l1 . Stanford seminar 08-7
Class of problems in CS (cont’d) (P σ ), (P τ ) and (P λ ) can all be cast as: (P1) : min α f 1(α) + f 2 (α) f 1 and f 2 are proper lsc convex functions. Stanford seminar 08-8
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: Convex analysis and operator splitt
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 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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