Xi'an
January 13, 2014
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slides of Anouar Seghir for the Reading Classics Seminar

presentation of the paper Adapting to unknown smoothness via wavelet shrinkage by Donoho and Johnstone, JASA 1995

January 13, 2014

Transcript

1. Adapting  to  unknown   smoothness  via  wavelet   shrinkage

David  L.  Donoho  and  Iain  M.  Johnstone     J.  American  Statistical  Assoc  Vol.  90,  No.432    (Dec  1995).       Presented  by  :  Anouar  Seghir   Master  TSI   January  13,2014
2. Outline Introduction I.  Discret  Wavelet  Transform II.  Threshold III. Adaptive  Estimation

IV. Examples Conclusion 2

5. Introduction Ø Problem  formulation:            Given  N

noisy  samples  of  a  function  f: Ø Goal:              To  estimate  the  vector         5 y i = f (t i )+ z i !!!!!i =1,..., N t i = (i !1) N !!!!!!!z i !iid!N(0,! 2 ) f = ( f (t i ) i=1 N )
6. Introduction Ø Estimation:   Ø Usual  approach:

We  specify  a  class  of  function            :  Minimax  Risk   We  seek  an  estimator                aJaining    the  minimax  risk. 6 f ^ = arg!min f ^ !R( f ^ , f ) R( f ^ , f ) = N !1.!E || f ^ ! f || 2 2 R(N,!) = inf f ^ !sup f !R( f ^ , f ) f ^ !
7. Introduction Ø Many  theoritical  developpements: •  Stone(1982) •  Nemirovsky •  Polyak

and  Tsybakov(1985) Ø In  Practice: •  No  knowledge  of  an  a  priori  class   •  Unknown  smoothness 7 !
8. Introduction Ø Donoho  and  Johnstone    Sureshrink  approach: 1.  Apply  DWT(discret

wavelet  transform)  to  noisy  data 2.  Thresholding  of  Noisy  Wavelet  Coeﬃcients 3.  Stein’s  Unbiased  Estimate  of  Risk  for  Threshold  Choice Ø  Sureshrink  is  asymptotically  near-­‐‑minimax  over  large   intervals  of  the  besov,  Sobolev  and  Triebel  Scales. 8
9. I.  Discret  Wavelet  Transform 1.  Wavelet  recall 2.  Discret  wavelet

transform  (DWT) 9

11. I.  Discret  Wavelet  transform Ø  Continious  wavelet  transform: •  Ψ

is  the  mother  wavelet •  a  is  the  scale  parameter •  b  is  the  translation  parameter •   Љhas  mean  equal  to  zero 11 1  Wavelet  recall !! !"# !! ! ! ! ! !! ! ! ! ! !! !! !! ! !"!!!!!!!! ! !!!!!! ! ! ! !"# !! ! ! ! !!!! ! ! !!!! ! ! ! ! ! ! ! ! ! ! ! ! !
12. I.  Discret  Wavelet  transform Ø  Discret  wavelet  transform: •  One

restricts  the  parameters  a,b  to  only  discrete  values:   •  Dyadic  transform:   12 2  Wavelet  recall ! ! ! ! !!!!!!! ! !! ! !!!!!!!! ! !! ! ! !"# !!! ! ! !!!! ! ! ! ! ! ! !!! ! !!! ! !! !! !!! !! ! !"!! ! ! ! ! !!!! ! ! !!!! !!!! ! ! ! ! ! !!!! ! ! ! ! !!!!! !! !!! ! !!! ! ! ! ! !! ! !!!!!!!! ! !!! ! !!!! ! ! !!!!!! !!!! ! ! ! ! ! !
13. I.  Discret  Wavelet  transform •  Localized  in  time  an  frequency

•  Many  choice  of  mother  wavelet: §  Continuous  case:  Meyer,  Mexican  hat §  Discrete  case:  Haar  ,  daubechies 13 3  Wavelet  recall
14. I.  Discret  Wavelet  Transform 1.  Wavelet  recall 2.  Discret  wavelet

transform  (DWT) 14
15. I.  Discret  Wavelet  transform Ø  Orthonormal  Family  of  Wavlet: 15

2  Discret  Wavelet  transform • !"#\$%&'()\$"#%)*+,*-\$.(%(+)/+(0-1'++%(+ !! !!! ("12+3+ o !!! ! !!!! ! !!!! + o !"#!!!! !! ! ! + o !"#!!!! !! ! !+ o !!!+("12+#2-#!! !!! ! !! !!!! +%(+-*+)#2)*)&4-\$+5-(%(+)/++!! ++ o !!! ! !! ! ! !! ! !!!! ! ! !!!! + o !! !! ! ! !!! ! ! !! ! !!! !!!!! ! !! + ! ! ! ! ! ! !!!! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! !!!! ! !! ! ! ! ! ! !!!!!! !!! ! !!!! !
16. I.  Discret  Wavelet  transform Ø  Mallat’s  Algorithm:

•  H  is  a  high  pass  ﬁlter •  G  is  a  low  pass  ﬁlter 16 2  Discret  Wavelet  transform
17. I.  Discret  Wavelet  transform Ø  DWT  in  the  problem:

17 2  Discret  Wavelet  transform • !"#"\$\$! ! !!! ! !!! !!!!!! ! !!\$ • !!\$%&'()('\$*+,\$ o !!!"#\$%&!!"!!"!#\$%&\$ o -\$\$./0012#\$3(&4#5\$ • 612\$7\$"&'\$-\$8%)('\$9(\$4(#\$"\$#2"&.812:"#%1&\$:"#2%)\$;,\$ o ! ! !"\$ o ! ! !!!\$\$\$ \$ !! ! !!!! !!!! ! !!! \$\$\$\$ \$ \$\$\$\$ !! ! !!!!! ! !!!!!! ! !!!!!!!! ! !!!!!!!"!#!"##\$\$ \$

Sure 18
19. II.  Thresholding Ø  W  transforms  whithe  noise  into  white  noise:

Ø  W  transforms  an  estimator  in  one  domain  into  an  estimator  in  the   other  domain:   19 1  Noisy  Wavelet  Coeﬃcients !!!! ! !!!! ! !!!! !!!!!!!!!!!!!!! !!!"!!!!!!!!! ! ! !!!!
20. II.  Thresholding Ø   Most  of  the  coeﬃcients  in  a  noiseless

wavelet  transform  are   eﬀectively  zero. Ø  Recovering  f  is  equivalent  to  recover  the  signiﬁcantly  nonzero   coeﬃcients  of  f. Ø  Thresholding  scheme: 20 1  Noisy  Wavelet  Coeﬃcients • !"#\$\$%&'%()\$\$'!!!! ' • !*++,%&'\$)-.+'!!!! '
21. II.  Thresholding Ø  Hard  Thresholding  :   Ø  Plot  du

Hard  Treshold  pr  un  certain  t Ø  Soft  Thresholding:     Ø  Plot  du  soft  threshold  pr  le  meme  t  que  above 21 1  Noisy  Wavelet  Coeﬃcients !! !!!! ! !!!!!"! !!!! ! ! !!!! !!!!"! !!!! ! ! ! !! !!!! ! !"#!!!!! !! !!!! ! !!!!

Sure 23
24. II.  Thresholding Ø Recall  on  SURE  Estimator: •  SURE:  Stein’s  Unbiased

risk  estimator •  Unbiased  estimator  of  the  MSE  of  an  arbitrary  estimator 24 2  Threshold  Selection  by  Sure • ! ! !!!!"!!"#!!"#"\$%"!!"#"\$%&%#! • ! ! !!!!!"#\$!!"#!!! ! !!"!!!!!!!!! • ! ! !!"#!\$%&'("&)*!)+!!!!%,-.!&."&!!! ! ! ! ! !!!!!! • !!!"#\$%&!!"##\$%\$&'"()*\$!! !"#\$ ! ! !!! ! ! ! ! ! !!! ! !!! !! !!! ! !!! ! !
25. II.  Thresholding Ø  We  consider  the  soft  threshold  estimator: •

By  Stein’s  result: •  Threshold  selection:     25 2  Threshold  Selection  by  Sure !! ! !! ! !! !!! !! !"#\$ !!! ! ! ! !!! !! !! ! ! ! ! !! ! !!! ! !!! ! !! ! !"#!"# !!!! !!"#!!!! !"#\$ !!! !
26. II.  Thresholding Ø Donoho  &  Johnoston’s  VisuShrink: •  Use  a  global

threshold:     •  No  computational  complexity  but  less  adaptive  to  unknow   smoothness.       26 2  Threshold  Selection  by  Sure !! ! !"#\$!!!!
27. II.  Thresholding Ø Extreme  Sparsity  Situation: •  Hybrid  Scheme: o  Measure

of  sparsity: o  Threshold  choice:     27 2  Threshold  Selection  by  Sure !! ! ! !! ! ! !! ! ! ! !! ! !!! !"# ! ! !!!! ! !!"#\$!%#&'( ! ! !!!!!!!!!!"!!! ! ! ! !!!!!!!!!!!!!!!"#\$%&'(\$ !
28. II.  Thresholding Ø Threshold  estimator:     28 2  Threshold  Selection

by  Sure !!!"#\$%!!"#\$ !! ! ! !! !!! !!!!!!!!"!!! ! ! ! !!!!! !! !!! !!!!!!!!!"#\$%&'(\$ ! !"#\$%&#'()!!*+(\$,+")-('%)!!),\$&)\$.,+("+)*\$"/(+#+)0&,+1+#)#(&."-'(%)
29. II.  Thresholding Ø Computational  eﬀort: •  Threshold  selection     •

To  apply  Threshold     29 2  Threshold  Selection  by  Sure !!!"#\$ ! !! !!!!!

over  Besov  Bodies 30
31. III.  Adaptive  Estimation Ø   Besov  space: •  De  Voor  and

Popov  (1988)     31 1  The  Besov  Scale • ! ! ! !!"!#\$%&!'())*#*+,*! ! ! ! ! ! ! ! !! !!!! ! !"! ! !!! ! ! • !!!! !!! !!!#\$%&!-.'/0/1!.)!1-..%&+*11!.)!!! ! !!! !! ! !! ! ! ! ! ! !!!! !! ! ! ! ! ! ! !!! !!!!!" ! ! ! • 2&*!3*1.4!5*-(+.#-!.)!(+'*6!!!! !! !!!).#!! ! !"! ! ! !!!! ! ! !!!! !! ! !! ! !" ! ! ! !!!!!!!"!! ! ! !"#!!!!! !!!! !!! !! !! !!!!!!!!!!!"!! ! ! ! ! !
32. III.  Adaptive  Estimation Ø   Besov  space: Ø Besov  Ball:

32 1  The  Besov  Scale !!!! ! ! !! !!! ! !!! ! !!! !!! !! ! !!!! ! ! ! ! !!!! ! !!! ! !! !!! ! !!! ! !!! !!! !! ! !!!! ! ! ! !
33. III.  Adaptive  Estimation Ø   Besov  space  and  others  spaces:

33 1  The  Besov  Scale ! • !!!! ! !"#\$%"\$&'!(\$)*!)*'!+,!!-#.#/'0!123"'!!!! !! ! • 4#5'!6'%'53//7!3//!)*'!!+8!-#.#/'0!123"'1!!! !!35'!"#%)'\$%'&!\$%!! !!!! ! !!! ! • +')91!! ! !!!:#/&'5!;/311!)*'%!!!!! ! !! !!! ! !!!! !!! ! !!!! ! • <*'!1')!#=!.#>%&'&!035\$3)\$#%!\$1!!3!1>.1')!#=!!!!! ! !!! ! !" ! !"# ! !! ! ! !!!! ! ! ! !! ! !! ! ! ! !!!! ! !! !! ! ! !

over  Besov  Bodies 34
35. III.  Adaptive  Estimation Ø   Problem  formulation  to  theory:

35 2  Adaptive  Estimation  over  Besov  Bodies !!!! ! !!!! ! !!!! !!!!!!!!! !!!"!!!!! !!!!!!!!! ! ! ! ! ! • "#\$%&!!! ! !!!! ! • '#&()!'(*+#!! !!!! ! ! !!! ! !!!! !! ! !!!!! ! ! ! ! o !!,-+&#!.-(/!!! ! !!!! ! !! o ! ! ! ! ! ! ! ! ! ! o ! !!!!! ! ! !!! !!!! ! !!!!!! !!! ! !!! !!!!! • 0+1+/,2!-+&3!+&4! ! ! ! !! !!!! ! ! !"# ! !!"!!!! ! ! ! ! ! ! ! ! o 5/(16!\$7#!\$7-#&7(8*4! ! ! ! !! !!!! ! ! !"# ! !!"!!!! ! ! ! !!!"#\$!!!"#\$! ! ! ! ! ! o ! !!!"#\$!!!"#\$! ! ! !!"#\$!!!"#\$ !!!!! !!!! !
36. III.  Adaptive  Estimation Ø   Fundamental  theorem: Ø  Using  this  result

we  the  have  the  following  Theorem:     36 2  Adaptive  Estimation  over  Besov  Bodies !"#\$!! ! ! ! ! ! ! \$\$#%"&'\$ !"#! !!!!!! ! !! ! !!!"#\$!!!"#\$! ! ! ! ! ! !! ! !! !!!! ! ! ! ! ! ! !!!!!! ! !\$ • !"#\$#%"\$&'()*"#"\$+,-"."#\$,/,.0('(\$)1**"(21/&\$#1\$,\$+,-"."#\$!\$%,-'/3\$*\$/4..\$ 515"/#(\$,/&\$*\$)1/#'/4"(\$&"*'-,#'-"(6\$! ! !"# !! ! !\$ • !"#\$#%"\$5'/'5,7\$*'(8\$9"\$&"/1#"&\$90\$\$ ! !! !!!! ! !!! ! !"# ! !!" !!!!!!! ! ! ! ! ! ! ! \$ • :%"/\$;4*";%*'/8\$'(\$(45.#,/"14(\$/",*.0\$5'/'5,7<\$ \$ !!" ! !!! ! ! ! !!"#\$!!!"#\$! ! ! ! !! !!!! ! ! !!!!!!!! ! !\$ \$ =1*\$,..\$!! ! ! !!! !!6\$=1*\$,..\$! ! !!! !!6\$,/&\$=1*\$,..\$!! ! ! ! !\$
37. IV.  Examples 1.  VisuShrink  versus  Sureshrink 37 • !"#\$%&'''''''''''''''! !

! !! ! ! ! !! !'''''''! ! ! !!!!"# ! ! ! ' • !()*&'''''''''! ! ! !! ! !!!!! ! !! !''''''''''! ! ! !! ! ! !!' • +#**",-'''! ! ! !!! ! !! !"# !! !!! !!! !!!!!! ! !!!"' • .,/0&12,'''! ! ! ! !"# !!! ! !"# ! ! !!! ! !"#!!!!" ! !!' ' TBlock=[0.1,0.13,0.15,0.23,0.25,0.40,0.44,0.65,0.76, 0.78,0.81]; HBlock=[4,-5,3,-4,5,-4.2,2.1,4.3,-3.1,2.1,-4.2]; HBump=[4,5,3,4,5,4.2,2.1,4.3,3.1,5.1,4.2]; WBump=[0.005,0.005,0.006,0.01,0.01,0.03,0.01,0.01,0. 005,0.008,0.005]; • ! ! !"#!!"#! ! ! !!"!#!!"#\$% '