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

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
IV. Examples
Conclusion
2

3. Introduction
Ø Noise:

3

4. Introduction
Ø Denoising  Signal:

4

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

10. I.  Discret  Wavelet  transform

10
1  Wavelet  recall

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:

12
2  Wavelet  recall
! ! !
!
!!!!!!! ! !!
!
!!!!!!!! ! !!
!
!
!"# !!! ! ! !!!!
! !
!
!
!
! !!!
!
!!! !
!!
!!
!!!
!! ! !"!!
! ! ! ! !!!!
!
!
!!!!
!!!!
!
!
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!!!!
! ! !
!
!!!!! !!
!!! ! !!!
!
!
!
!
!!
! !!!!!!!!
! !!!
!
!!!!
! ! !!!!!! !!!! ! ! !
!
!
!

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 ! ! !!!\$\$\$
\$
!!
! !!!!
!!!!
!
!!!
\$\$\$\$ \$ \$\$\$\$
!! ! !!!!! ! !!!!!! ! !!!!!!!! ! !!!!!!!"!#!"##\$\$
\$

18. II.  Thresholding
1.  Noisy  Wavelet  Coeﬃcients

2.  Threshold  Selection  by  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
!!
!!!!
!
!!!!!"! !!!!
! !
!!!!
!!!!"! !!!!
! !
!
!!
!!!!
! !"#!!!!!
!! !!!!
! !!!!

22. II.  Thresholding

22
1  Noisy  Wavelet  Coeﬃcients

23. II.  Thresholding
1.  Noisy  Wavelet  Coeﬃcients

2.  Threshold  Selection  by  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
!!!"#\$ ! !!
!!!!!

1.  The  Besov  Scale

2.  Adaptive  Estimation  over  Besov  Bodies

30

Ø   Besov  space:
•  De  Voor  and  Popov  (1988)

31
1  The  Besov  Scale
• !
!
! !!"!#\$%&!'())*#*+,*!
!
!
! ! !
!
!
!! !!!! ! !"!
!
!!!
!
!
• !!!!
!!! !!!#\$%&!-.'/0/1!.)!1-..%&+*11!.)!!! ! !!! !! ! !!
! ! ! ! !
!!!!
!! ! ! !
!
! !
!!! !!!!!" !
!
!
• 2&*!3*1.4!5*-(+.#-!.)!(+'*6!!!! !! !!!).#!! ! !"!
!
! !!!!
!
!
!!!!
!! !
!!
!
!"
!
!
!
!!!!!!!"!! ! !
!"#!!!!!
!!!!
!!! !!
!!
!!!!!!!!!!!"!! ! !
!
!
!

Ø   Besov  space:
Ø Besov  Ball:

32
1  The  Besov  Scale
!!!!
! ! !! !!! ! !!! ! !!! !!! !! ! !!!!
! ! ! !
!!!!
! !!! ! !! !!! ! !!! ! !!! !!! !! ! !!!!
! ! ! !

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')!#=!!!!!
! !!!
!
!" ! !"# ! !!
! ! !!!!
! ! ! !!
! !!
! ! ! !!!!
! !!
!! ! ! !

1.  The  Besov  Scale

2.  Adaptive  Estimation  over  Besov  Bodies

34

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 !
!!!"#\$!!!"#\$!
! !
!!"#\$!!!"#\$
!!!!!
!!!!
!

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];
• ! ! !"#!!"#! ! ! !!"!#!!"#\$%
'

38
2  Adaptive  Estimation  over  Besov  Bodies

39. Conclusion
1.  SureShrink  is  nearly  Minimax  over  a  large  range  of
Besov  Spaces.

2.     Low  computational  complexity

3.  More  important  is  the  choose  of  the  wavelet  family  and
the  number  of  resolution.

39

40. References
1.  Ten  Lectures  on  Wavelet  -­‐‑Daubechies

2.     Ideal  Spatial  adaptation  by  wavelet  shrinkage-­‐‑
Donoho&Johnstone

3.  Multiresolution  Analysis  and  fast  algorithms  on  an  interval  –
Daubechies-­‐‑Cohen-­‐‑Vial-­‐‑Jawerth

4.  OndeleJes  sur  l’intervalle-­‐‑Meyer

40