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Exact Support Recovery for Sparse Spikes Deconvolution

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Talk at SAMPTA 2013 conference, 2013/07/02.

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Exact Support Recovery for Sparse Spikes Deconvolution

  1. 1. Gabriel Peyré www.numerical-tours.com Exact Support Recovery for Sparse Spikes Deconvolution Vincent Duval VISI N
  2. 2. Overview • Sparse Spikes Deconvolution • Robust Support Recovery • Vanishing Derivative Certificate • Ideal Low Pass Filter
  3. 3. Radon measure m on T = R/Z. Discrete measure: ma,x = PN i=1 ai xi , a 2 RN , x 2 TN Deconvolution of Measures ma,x
  4. 4. y = (m) Convolution measurements: ' 2 C2 (T) Radon measure m on T = R/Z. Discrete measure: ma,x = PN i=1 ai xi , a 2 RN , x 2 TN y(t) = R T '(x t)dm(x) Deconvolution of Measures ma,x '
  5. 5. y = (m) Convolution measurements: ' 2 C2 (T) Radon measure m on T = R/Z. Discrete measure: ma,x = PN i=1 ai xi , a 2 RN , x 2 TN Example: ideal low-pass ˆ' = 1[ fc,fc] '(t) = sin((2fc+1)⇡t) sin(⇡t) y(t) = R T '(x t)dm(x) Deconvolution of Measures ma,x ' = 0.5/fc y
  6. 6. y = (m) Convolution measurements: ' 2 C2 (T) Radon measure m on T = R/Z. Discrete measure: ma,x = PN i=1 ai xi , a 2 RN , x 2 TN Example: ideal low-pass ˆ' = 1[ fc,fc] '(t) = sin((2fc+1)⇡t) sin(⇡t) y(t) = R T '(x t)dm(x) Minimum separation: = mini6=j |xi xj| ! Signal-dependent recovery criteria. Deconvolution of Measures ma,x ' = 0.5/fc y y = 2/fc y = 1/fc
  7. 7. Not to be confounded with total variation of functions. Sparse Deconvolution of Measures Total variation of measures: ||m||TV = sup R dm 2 C(T), || ||1 6 1
  8. 8. Not to be confounded with total variation of functions. Discrete measures: Sparse Deconvolution of Measures Total variation of measures: ||m||TV = sup R dm 2 C(T), || ||1 6 1 ma,x = PN i=1 ai xi ||ma,x||TV = ||a||1 = PN i=1 |ai|
  9. 9. Not to be confounded with total variation of functions. Sparse recovery: (P0(y)) (P (y)) Discrete measures: Sparse Deconvolution of Measures min (m)=y ||m||TV min m 1 2 || (m) y||2 + ||m||TV Total variation of measures: ||m||TV = sup R dm 2 C(T), || ||1 6 1 ma,x = PN i=1 ai xi ||ma,x||TV = ||a||1 = PN i=1 |ai|
  10. 10. Not to be confounded with total variation of functions. Sparse recovery: (P0(y)) (P (y)) ! Algorithms: [Bredies, Pikkarainen, 2010] (proximal-based) [Cand`es, Fernandez-G. 2012] (root finding) Discrete measures: Sparse Deconvolution of Measures min (m)=y ||m||TV min m 1 2 || (m) y||2 + ||m||TV Total variation of measures: ||m||TV = sup R dm 2 C(T), || ||1 6 1 ma,x = PN i=1 ai xi ||ma,x||TV = ||a||1 = PN i=1 |ai|
  11. 11. = 0.45/fc = 0.55/fc= 0.3/fc = 0.1/fc Yes if > 1.85/fc. Robustness and Support-stability Is m0 solution of (P0(y)) for y = m0 ? ! [Cand`es, Fernandez-G. 2012] ˆ' = 1[ fc,fc]
  12. 12. = 0.45/fc = 0.55/fc= 0.3/fc = 0.1/fc How close is the solution m of (P (y + w)) to m0 ? Yes if > 1.85/fc. ! [Cand`es, Fernandez-G. 2012] Robustness and Support-stability Is m0 solution of (P0(y)) for y = m0 ? ! [Cand`es, Fernandez-G. 2012] ˆ' = 1[ fc,fc]
  13. 13. = 0.45/fc = 0.55/fc= 0.3/fc = 0.1/fc Support localization. How close is the solution m of (P (y + w)) to m0 ? Yes if > 1.85/fc. ! [Cand`es, Fernandez-G. 2012] ! [Fernandez-G. 2012][de Castro 2012] Robustness and Support-stability Is m0 solution of (P0(y)) for y = m0 ? ! [Cand`es, Fernandez-G. 2012] ˆ' = 1[ fc,fc]
  14. 14. Overview • Sparse Spikes Deconvolution • Robust Support Recovery • Vanishing Derivative Certificate • Ideal Low Pass Filter
  15. 15. Dual Certificates S0(y) = ⌘ 2 L2 (T) ⌘ 2 Im( ⇤ ) @||m? ||TV Dual certificates of (P0(y)): for any solution m? , @||ma,x||TV = {⌘ 2 C(T) ||⌘||1 6 1, 8 i, ⌘(xi) = sign(ai)}
  16. 16. ⌘0 ⌘= 0.6/fc ⌘0 2 S0 ⌘ 2 S0 ⌘0 ⌘= 0.7/fc ⌘ 2 Im( ⇤ ) ⌘ /2 S0 Minimal-norm certificate: p0 = argmin ||p|| s.t. ⇤ p 2 S0(y) ⌘0 = ⇤ p0 where Dual Certificates S0(y) = ⌘ 2 L2 (T) ⌘ 2 Im( ⇤ ) @||m? ||TV Dual certificates of (P0(y)): for any solution m? , @||ma,x||TV = {⌘ 2 C(T) ||⌘||1 6 1, 8 i, ⌘(xi) = sign(ai)}
  17. 17. ⌘0 ⌘= 0.6/fc ⌘0 2 S0 ⌘ 2 S0 ⌘0 ⌘= 0.7/fc ⌘ 2 Im( ⇤ ) ⌘ /2 S0 Minimal-norm certificate: p0 = argmin ||p|| s.t. ⇤ p 2 S0(y) ⌘0 = ⇤ p0 where Dual Certificates Proposition: For any solution m of (P (y)), p = y m !0+ ! p0 in L2 (T) S0(y) = ⌘ 2 L2 (T) ⌘ 2 Im( ⇤ ) @||m? ||TV Dual certificates of (P0(y)): for any solution m? , @||ma,x||TV = {⌘ 2 C(T) ||⌘||1 6 1, 8 i, ⌘(xi) = sign(ai)}
  18. 18. Non-degenerate Certificates 9p, ⇤ p 2 @||m0||TV In finite dimension: SC(m0) () m0 solution of P0( m0). Source condition SC(m0):
  19. 19. Non-degenerate Certificates ⌘0= 1/fc ⌘0 = 0.6/fc 9p, ⇤ p 2 @||m0||TV In finite dimension: SC(m0) () m0 solution of P0( m0). 8 s 2 T {x0,1, . . . x0,N }, |⌘0(s)| < 1 8 i 2 {1, . . . N}, ⌘00 0 (x0,i) 6= 0 Non Degenerate Source Condition NDSC(m0): Source condition SC(m0): ('(· x0,i), '0 (· x0,i))N i=1 has full rank.
  20. 20. ↵, 0 = {( , w) 0 6 6 0, ||w||2 6 ↵ } Noise Robustness PN i=1 ˜a ,i ˜x ,i ( , w) 2 7! (˜a , ˜x ) is C1where Let m0 = ma0,x0 satisfying NDSC(m0). is unique and reads Theorem: For ( , w) 2 ↵, 0 , the solution of (P (y + w)) 0 ||w||
  21. 21. ↵, 0 = {( , w) 0 6 6 0, ||w||2 6 ↵ } In particular: Noise Robustness |˜x ,i x0,i| = O(||w||2) |˜a ,i a0,i| = O(||w||2) PN i=1 ˜a ,i ˜x ,i ( , w) 2 7! (˜a , ˜x ) is C1where Let m0 = ma0,x0 satisfying NDSC(m0). is unique and reads Theorem: For ( , w) 2 ↵, 0 , the solution of (P (y + w)) 0 ||w||
  22. 22. ↵, 0 = {( , w) 0 6 6 0, ||w||2 6 ↵ } In particular: Noise Robustness |˜x ,i x0,i| = O(||w||2) |˜a ,i a0,i| = O(||w||2) PN i=1 ˜a ,i ˜x ,i ( , w) 2 7! (˜a , ˜x ) is C1where Let m0 = ma0,x0 satisfying NDSC(m0). is unique and reads Theorem: For ( , w) 2 ↵, 0 , the solution of (P (y + w)) ˜xi, Noiseless w = 0. ||w||/↵ 0 0 ˜xi, 0 ||w||
  23. 23. Overview • Sparse Spikes Deconvolution • Robust Support Recovery • Vanishing Derivative Certificate • Ideal Low Pass Filter
  24. 24. Vanishing Derivative Pre-Certificate ¯⌘0 = ⇤ ¯p0 where ¯p0 = argmin ||p|| s.t. 8 i ⇢ ( ⇤ p)(x0,i) = sign(a0,i), ( ⇤ p)0 (x0,i) = 0. Vanishing Derivative (Pre-)Certificate ⌘0= 1/fc ⌘0 = 0.6/fc Intuition: 8 i, ⌘0 0(x0,i) = 0, where ma0,x0 solves (P0(y)).
  25. 25. Vanishing Derivative Pre-Certificate ¯⌘0 = ⇤ ¯p0 where ¯p0 = argmin ||p|| s.t. 8 i ⇢ ( ⇤ p)(x0,i) = sign(a0,i), ( ⇤ p)0 (x0,i) = 0. Vanishing Derivative (Pre-)Certificate ⌘0= 1/fc ⌘0 = 0.6/fc ¯⌘0 = ⇤ +,⇤ x0 (sign(a0), 0)⇤ Proposition: if x0 has full rank, x0 = ('(· x0,i), '0 (· x0,i))N i=1 Intuition: 8 i, ⌘0 0(x0,i) = 0, where ma0,x0 solves (P0(y)).
  26. 26. Vanishing Derivative Pre-Certificate ¯⌘0 = ⇤ ¯p0 where ¯p0 = argmin ||p|| s.t. 8 i ⇢ ( ⇤ p)(x0,i) = sign(a0,i), ( ⇤ p)0 (x0,i) = 0. Vanishing Derivative (Pre-)Certificate ⌘0= 1/fc ⌘0 = 0.6/fc ¯⌘0 = ⇤ +,⇤ x0 (sign(a0), 0)⇤ Proposition: if x0 has full rank, x0 = ('(· x0,i), '0 (· x0,i))N i=1 Theorem: if m0 satisfies NDSC(m0), ⌘0 = ¯⌘0. Intuition: 8 i, ⌘0 0(x0,i) = 0, where ma0,x0 solves (P0(y)).
  27. 27. Overview • Sparse Spikes Deconvolution • Robust Support Recovery • Vanishing Derivative Certificate • Ideal Low Pass Filter
  28. 28. [Cand`es, Fernandez-G. 2012] K(t) = ✓ sin((fc 2 +1)⇡t) (fc 2 +1)sin(⇡t) ◆4 Fejer Kernel Pre-certificate: ˆ⌘0(t) = PN i=1 (↵iK(t x0,i) + iK0 (t x0,i)) ⇢ ˆ⌘0(x0,i) = sign(a0,i), ˆ⌘0 0(x0,i) = 0. 8 i, For m0 = ma0,x0 . = 0.6/fc = 0.7/fc(ˆ⌘0, ¯⌘0) Fejer Kernel Pre-certificates ˆ' = 1[ fc,fc] '(t) = sin((2fc+1)⇡t) sin(⇡t) (ˆ⌘0, ¯⌘0)
  29. 29. [Cand`es, Fernandez-G. 2012] K(t) = ✓ sin((fc 2 +1)⇡t) (fc 2 +1)sin(⇡t) ◆4 Fejer Kernel Pre-certificate: ˆ⌘0(t) = PN i=1 (↵iK(t x0,i) + iK0 (t x0,i)) ⇢ ˆ⌘0(x0,i) = sign(a0,i), ˆ⌘0 0(x0,i) = 0. 8 i, For m0 = ma0,x0 . = 0.6/fc = 0.7/fc(ˆ⌘0, ¯⌘0) Fejer Kernel Pre-certificates ˆ' = 1[ fc,fc] '(t) = sin((2fc+1)⇡t) sin(⇡t) Theorem: [C. F-G. 2012] If > 1.85/fc, ˆ⌘0 2 @||m0||TV. (ˆ⌘0, ¯⌘0)
  30. 30. [Cand`es, Fernandez-G. 2012] K(t) = ✓ sin((fc 2 +1)⇡t) (fc 2 +1)sin(⇡t) ◆4 Fejer Kernel Pre-certificate: ˆ⌘0(t) = PN i=1 (↵iK(t x0,i) + iK0 (t x0,i)) ⇢ ˆ⌘0(x0,i) = sign(a0,i), ˆ⌘0 0(x0,i) = 0. 8 i, For m0 = ma0,x0 . = 0.6/fc = 0.7/fc(ˆ⌘0, ¯⌘0) Fejer Kernel Pre-certificates ˆ' = 1[ fc,fc] '(t) = sin((2fc+1)⇡t) sin(⇡t) Theorem: [C. F-G. 2012] If > 1.85/fc, ˆ⌘0 2 @||m0||TV. (ˆ⌘0, ¯⌘0) Conjecture: ˆ⌘0 2 @||m0||TV ) ¯⌘0 2 @||m0||TV
  31. 31. Deconvolution of measures: ! L2 errors are not well-suited. Weak-* convergence. Optimal transport distance. Exact support estimation. ... Conclusion
  32. 32. Low-noise behavior: ! dictated by ⌘0. Deconvolution of measures: ! L2 errors are not well-suited. Weak-* convergence. Optimal transport distance. Exact support estimation. ... Conclusion = 1/fc ⌘0 ⌘0= 0.6/fc
  33. 33. Extends to: Arbitrary dimension. Non-stationary “convolutions”. Low-noise behavior: ! dictated by ⌘0. Deconvolution of measures: ! L2 errors are not well-suited. Weak-* convergence. Optimal transport distance. Exact support estimation. ... Conclusion = 1/fc ⌘0 ⌘0= 0.6/fc
  34. 34. Extends to: Arbitrary dimension. Non-stationary “convolutions”. Lasso on discrete grids: Low-noise behavior: ! dictated by ⌘0. Deconvolution of measures: ! L2 errors are not well-suited. Weak-* convergence. Optimal transport distance. Exact support estimation. ... similar ⌘0-analysis applies. ! Relate discrete and continuous recoveries. Conclusion = 1/fc ⌘0 ⌘0= 0.6/fc

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