(i) {α - Convex Optimization
(i) {α - Convex Optimization (i) {α - Convex Optimization
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. (P λ ) 1 2 ‖y − HΦα‖2 l 2 + λΨ (α) f 1 (α) = 1 2 ‖y − HΦα‖2 l 2 ,f 2 (α) =λΨ (α) Stanford seminar 08-8
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. (P λ ) 1 2 ‖y − HΦα‖2 l 2 + λΨ (α) f 1 (α) = 1 2 ‖y − HΦα‖2 l 2 ,f 2 (α) =λΨ (α) (P σ ) Ψ (α) s.t. ‖y − HΦα‖ l2 ≤ σ f 1 (α) =Ψ (α) ,f 2 (α) =ı Bl2 ,σ (α) 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 and 14: Convex analysis and operator splitt
- Page 15: 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
Class of problems in CS (cont’d)<br />
(P σ ), (P τ ) and (P λ ) can all be cast as:<br />
(P1) : min<br />
α<br />
f 1(α) + f 2 (α)<br />
f 1 and f 2 are proper lsc convex functions.<br />
(P λ )<br />
1<br />
2 ‖y − HΦα‖2 l 2<br />
+ λΨ (α)<br />
f 1 (α) = 1 2 ‖y − HΦα‖2 l 2<br />
,f 2 (α) =λΨ (α)<br />
(P σ )<br />
Ψ (α) s.t. ‖y − HΦα‖ l2<br />
≤ σ<br />
f 1 (α) =Ψ (α) ,f 2 (α) =ı Bl2 ,σ (α)<br />
Stanford seminar 08-8
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