A Predual Proximal Point Algorithm solving a Non Negative Basis Pursuit Denoising model

(1)

HAL Id: hal-00133050

https://hal.archives-ouvertes.fr/hal-00133050

Submitted on 23 Feb 2007

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A Predual Proximal Point Algorithm solving a Non Negative Basis Pursuit Denoising model

François Malgouyres, Tieyong Zeng

To cite this version:

François Malgouyres, Tieyong Zeng. A Predual Proximal Point Algorithm solving a Non Negative

Basis Pursuit Denoising model. International Journal of Computer Vision, Springer Verlag, 2009, 83

(3), pp.294-311. �hal-00133050�

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²¡¥²µX;ÞéXµ¼äzÞBg»¥²!Xº}Xµr»¥¼è Ý »¥²$º}µ}¶e ²¡FMêXg²¶Æz ºXXµg»¥¼ämg -¡¥²²x¶¥gê;²!m½;²r²¶Tè

¬{z ²!»x¨Tê²xm¶e²g»¥²¶e²X¼½;}gr}¡®X¡¶2µ²X¼½Fmg X¡¨g»¥²X¡³2¶eò1¹¥s³2¡g»¥²6 ¼6½²¼6²¡F}¡

zÞ»¥²$zê;æX²®Xº}Xµr»¥¼ g»²X¼½¥¹ej}¡LXÞ-»¥²®º}µ}¶e ²¡F

∇fu^m

¨e¡Æs²½£¦X¨ez¡¶1g»¥²)µ²gX ¹eg X¡1XÞg»²mg²½

ôeè

ñ®¹¥µ ¡Fg²µ²mÞé}µ{»¥²Xº}Xµr»¥¼zê;æX²X¼²x3ÞéµX¼ g»¥²°ÞåXg»¨?X®´3 Bê²6m»´3¡¡g»¥²¡¥²óc)g²g X¡¨

g»¥Fm²Ms´MX¼½¥¹ej}¡*z¡6ê;²3½²µmÞé}µg¼²¶²ó¥}.g ³Xègg²¡Fgz ³X¨g»²{}m*zÞ;g»¥²²æX¹zg X¡6zÞ

∇fu^m

*}¡¥²3¶e²

X¼½;}gr}¡ìz¡¶X¡¥²®µg²x}¼6½;}gg X¡1 ¡

(ψi)i∈I

z¡¶1g»¥²)}mLÞé}µMX¼½¥¹e¡º

arg minw∈RNL⁰(w,(λ^m_i )i∈I, u^m)

3}¡¥²µg²x}¼6½;}gg X¡ì ¡

(ψi)i∈I

è

2Xµ²æ}²µx¨´M²3´3 g»¥´çg»z

∇fu^m

½j»¥gù{z¡¶´M²3´3 ½¥µgæc¶e²rj -½j»¥gù{X¡mX¡}{îå´3»¥ j»1z¡ê;²

X¼½¥¹¥g²¶¡F¹¼6²µgX ³¥ï.è

Ý

»¥{´3 *ºX¹zµjz¡Fg²²g»¥²6X¡cæ}²µºX²¡²zÞ»¥²º}µ}¶e ²¡F{êXg²¶z ºXXµg»¥¼{X¡g ¶¥²µ²¶

¡1mg²½Æ¦}èX·M³$g»²{´M³}¨´²3´3 }¡m¶e²µ*s´æzµ X¡F¢})½¥µs²x.²¶6ºXµjX¶e ²¡F8X²¡F*XÞ?X¡mX¡Fmg²½1m ù²$îúFg²½

¦ì!»¥²¡ç {ùx´M2Xº}Xµr»¥¼+gX æc¡º

(Pu^m)ïz¡¶¬{²mg²µgæo{ ºX}µgg»¼ îå´3»¥ j»ç 6zµºX¹¥²x¶g£ê²[X½e¼z

z¼X¡¥ºº}µ}¶e²¡FMê}m²x¶[Xº}Xµr»¥¼1ï)îåg²²ß¦§xá¨c²g X¡2ôeètôXïè

·²ÞéXµ²X¨ºXX ¡¥º[¡Fgìg»¥Fm²6¶e²jz ¨;²)¹®µjm)m²!»¥²°Þé} ´3 ¡¥ºì½¥µX½;}gg X¡2´3»¥j»º}¹zµjz¡Fg²²»X¹µ

¶e¹z8X½¥½¥µ}}j»£Xg¹X ³2½¥µæc ¶e²xz¡oz½¥½¥µóe¼zg X¡XÞLg»²1X.¹z8gX ¹e}¡´M²Xµg²6 F}÷c¡¥º[ÞéXµxè õ !½¥µgcXÞL

ºX æX²¡Æ ¡½¥½;²¡¶eóTè

! #"%$'&)(*&,+.-/&*0%1,&2(

(u^m)m∈N

3

0%4 ((λ^m_i )i∈I)_m_∈N

45&60&,48790:"

35;*<

&>=8(

3?

7@(BA,C

=5D (u^m)m∈N

1,E0F&.GHI&2(

? E ?

$'&)(*E

<- ? 7E0

w^∗ ^EJA (P)^D

K D

limm→+∞inf_(λ^∗

i)i∈I∈Sk(λ^m_i −λ^∗_i)i∈Ik= 0^LNM $/&*GO&

LNM

&PGO&,1

3<9<Q?

$

3?

L S ^7@( ^?$'&RETS

?

79U

3<

(.&

?

EJA (D)^D

(6)

µjsx¨?}®¹m¹z*´3r»»¥²ºXµjX¶e ²¡FXÞ8Þé¹¥¡g X¡¶e²¡¥²¶£X)ì¼ ¡¥ ¼°¹¥¼¨?¼z¡c³g²µg¼)z¡²®X¹e!X¡¶

´²)¡X ³ì»æX²

∇fu^m((λi)i∈I) = (hw^∗, ψii −1)_i_∈_I,

´3»¥²µ²

w^∗= arg min

w∈RNαkw−u^mk²+kwk − hw,1 τv−X

i∈I

λiψii. ^îúëXï

3}¡m²xíF¹¥²¡²X¨c¼e¶e¹¥ 6¶e²xX¼½;}gr}¡[ ¡

(ψi)i∈I

¨c»¥²$X¼½¥¹¥}¡[XÞ

∇fu^m

X¡¶ìg»¥²µ²gX ¹eg X¡[zÞ

s²½ôê}{¶e´3¡[g6g»²!gX¼6²½¥µXê²¼ ¢ Ý »¥²µ²gX ¹eg X¡ÆzÞ!îúëXï.è

²3¹3µm{m ¼½¥ r²xg»²$¡¥zjg X¡3z¡¶Æ}¡m¶e²µ»¥²$½¥µXê¥ ²¼

w^∗= arg min

w∈RNαkw−uk²+kwk+hw, ri,

´3»¥²µ²

u^z¡¶ r^Xµg² ¡ R^N

è

²¹8ê;²º}¡´3g»»¥²gr¹}¡´3»¥²µ²

kw^∗k= 0è}ª®;²µg²¡} zg ¡¥º¨}´²÷F¡´ç»

2α(w^∗−u) +r∈∂F(0)^¨

´3»¥²µ²

F(w) =kwk^è g¡¥ºîú§Xï¨¥´²»æ}²

w^∗ = 0⇒ kr−2αuk ≤1.

ñ®¡ì»¥²$z»¥²µ3»X¡¶T¨¥Þ*´²$Xg¹¥¼²®»

kw^∗k 6= 0¨e´M²÷F¡´Jg»

2α(w^∗−u) + w^∗

kw^∗k+r= 0.

Ý »¥ {ºX æX²

kw^∗k(2αu−r) = (2αkw^∗k+ 1)w^∗.

îåðFï

Ý

X÷F ¡¥º6g»²$¡¥Xµ¼äzÞg»¥²!Xêæ}²®²xíF¹z rs³}¨e´²$}êeX¡

k2αu−rk= 2αkw^∗k+ 1, ^îå«}ï

´3»¥ j»º}¹zµjz¡Fg ²Lg»

k2αu−rk>1^è

3}¡ ¹g}¡¨

w^∗ = 0⇔ k2αu−rk ≤1,

z¡¶Æ´3»¥²¡

w^∗6= 0^¨ w^∗ z¡Æê²!X¼½¥¹e²¶T¨¹m ¡¥ºî=ð}ïz¡¶£î=«}ï.¨z¡¶[

w^∗=k2αu−rk −1

2αk2αu−rk (2αu−r)

²!z¡Æµg²½¥»¥µjXg²»¥ X

w^∗=

( 0 ^¨eÞ k2αu−rk ≤1

k2αu−rk−1

2αk2αu−rk(2αu−r) ¨ezg»²µ´3 g²Xè

!2X¡¹g X¡¨T ¡g»¥²[cg²½¦XÞ»¥²1z ºX}µgg»¥¼ ¶e²µê;²¶ ¡ Ý Xê¥²2¦}¨Tg»²ºXµjX¶e ²¡F!X¡£ê;²ì}¼½¥¹eg²x¶

´3r»¢

∇fu^m((λi)i∈I) = (hw^∗, ψii −1)_i_∈_I,

îm¦¯Fï

´3»¥²µ²

w^∗=

( 0 ^¨rÞ ktk ≤1

ktk−1

2αktk t ¨zg»²µ´3 g²X¨

îm¦X¦ï

´3r»

t= 2αu^m+v τ −X

i∈I

λiψi.

2Xµ²æ}²µx¨X»¥²!s²½2ôzÞBg»¥²!Xº}Xµr»¥¼ÔzÞ

Ý

Xê¥ ²6¦{gX æX²¶ìêc³ìz½¥½³c ¡¥ºîs¦}¦xï

(λ^m_i )^è

9 ?B* B *?! )? BI*,D!

L⁰^B? ^E ^.=, w ^E ^?! (λi)i∈I^{C ?)5} ^?!././?!

B "B

max min ^E ^?,./ ^B ^. ^∂L_∂w⁰ ^0"I, ^H? ^, ^B ^./ _∂λ^∂w

i

BB

EE

B,",*?! ?F

∇fum((λi)i∈I)^C6I?! ^. Ê ^,?F"B ^J,",^*?! ÎB Ê ^?0?F ^?F8 ^B ^{C C}^C^"!

(7)

Ý »¥²$z¹¹XÞ»¥²$½¥µ²²¶e ¡¥ºm²x.}¡[½;²µ¼rjMg½¥µæX²®g»¥²Þé} ´3 ¡¥ºµ²g¹¥è

QEG

3

0/C

(λi)i∈I ³

0%4

(λ⁰i)i∈I

790

R^IL

k∇fu^m((λi)i∈I)− ∇fu^m((λ⁰_i)i∈I)k ≤Ck(λi−λ⁰_i)i∈Ik,

M 7?$

C=³^√^M_2α¹^M²^LM ⁷^?^$

M1=X

i∈I

kψik²,

3

04

M2

7@(

?

$'&

<3 GH5&*(

?

&.7H5&.0/F

3<

-/&REJA

?

$/& G 3 U#U

3?

G27 (hψi, ψji)i,j∈I

D

GOE E A è-²

(λi)i∈I

z¡¶

(λ⁰_i)i∈I

ê;²$¡

R^I^è ²$»æX²}¨}ÞéµX¼+îs¦x¯}ïj¨

∇fu^m((λi)i∈I)− ∇fu^m((λ⁰i)i∈I) = (hw^∗−w^0∗, ψii)_i_∈_I,

´3r»

w^∗=

( 0 ^¨¥Þ ktk ≤1

ktk−1

2αktk t ¨¥z»¥²µ´3m²}¨

X¡¶

w^0∗=

( 0 ^¨¥Þ kt⁰k ≤1

kt⁰k−1 2αkt⁰k t⁰

¨¥z»¥²µ´3m²}¨

´3»¥²µ²

t= 2αu^m+^v_τ −P

i∈Iλiψi

X¡¶

t⁰= 2αu^m+^v_τ −P

i∈Iλ⁰_iψi.

õ¡ìXµj¶e²µ½µgæ}²{g»²)½µg}½Fmg X¡¨F´M²)Xµg²)ºX}¡¥º°g¶es¡¥º}¹¥m»[g»¥²$¶e?²µ²¡FM½;}gê¥ ²)ezg X¡[zÞ

ktk^z¡¶

kt⁰k´3g»µ²g½²x.3

1^è

µjsx¨erÞ*êXg»

ktk ≤1^X¡¶ kt⁰k ≤1 ^¢ ²$Xêcæc}¹m ³»æ}²

k∇fu^m((λi)i∈I)− ∇fu^m((λ⁰i)i∈I)k= 0,

´3»¥ j»Æµg æc X ³1gzgs²Mg»²!sjg²¼²¡F{zÞg»¥²$½¥µX½;}gg X¡è

Ý »¥²$m²xX¡¶ì}m²)´3»¥²¡

ktk>1^z¡¶ kt⁰k ≤1îég»¥²$Xg²

ktk ≤1^X¡¶ kt⁰k>13m ¼zµ3z¡¶ì´3 T¡z3ê²

gµ²²¶ï)¢ ²g»²¡»æX²

k∇fu^m((λi)i∈I)− ∇fu^m((λ⁰_i)i∈I)k² = X

i∈I

hw^∗−w^0∗, ψii²

≤ kw^∗−w^0∗k²X

i∈I

kψik²

≤ M1

ktk −1 2αktk

2

ktk²

≤ M1

4α²(ktk −1)²,

´3»¥²µ²

M1

º}æ}²¡Æ¡Æ»¥²$½¥µX½;}gr}¡è ²!X g»æX²

ktk = k2αu^m+v τ −X

i∈I

λ⁰_iψi−X

i∈I

(λi−λ⁰_i)ψik

≤ kt⁰k+kX

i∈I

(λi−λ⁰_i)ψik

≤ 1 +kX

i∈I

(λi−λ⁰_i)ψik.

c¨¥´M²»æX²

(ktk −1)²≤M2k(λi−λ⁰_i)i∈Ik²,

´3»¥²µ²

M2

º}æ}²¡Æ¡Æ»¥²$½¥µX½;}gr}¡è

(8)

k∇fu^m((λi)i∈I)− ∇fu^m((λ⁰i)i∈I)k²≤ M1M2

4α² k(λi−λ⁰_i)i∈Ik²,

´3»¥ j»¼²X¡

k∇fu^m((λi)i∈I)− ∇fu^m((λ⁰i)i∈I)k ≤

√M1M2

2α k(λi−λ⁰_i)i∈Ik.

Ý »¥²µ²Þé}µg²}¨cg»¥²!mzg²¼²¡}3XÞ»¥²½¥µg}½Fmg X¡Æ»¥} ¶¥ ¡Æg»z}m²}è

²¡z ³1¡²²¶[mg¹¶e³1»¥²!}m²)´3»¥²µ²êXg»

ktk>1^X¡¶ kt⁰k>1 ^¢ ²g»¥²¡»æX²

k∇fu^m((λi)i∈I)− ∇fu^m((λ⁰i)i∈I)k² = X

i∈I

hw^∗−w^0∗, ψii²

≤ M1kw^∗−w^0∗k²

≤ M1

ktk −1

2αktk t−kt⁰k −1 2αkt⁰k t⁰

2

≤ M1

(2αktkkt⁰k)²

(ktk −1)kt⁰kt−(kt⁰k −1)ktkt⁰

2

≤ M1

(2αktkkt⁰k)²

ktkkt⁰k(t−t⁰)− kt⁰kt+ktkt⁰

2

.

c¨¥´M²)¡X ³1Xê¥z ¡

k∇fu^m((λi)i∈I)− ∇fu^m((λ⁰_i)i∈I)k ≤

√M1

2α kt−t⁰k+

√M1

2αktkkt⁰k

ktkt⁰− kt⁰kt .

·¹¥¨¥´M²$zm»æ}²X¨

ktkt⁰− kt⁰kt =

ktk(t⁰−t) + (ktk − kt⁰k)t

≤ ktkkt⁰−tk+

ktk − kt⁰k ktk

≤ 2ktkkt⁰−tk

c¡²X¨

kt⁰−tk ≤p

M2k(λi−λ⁰_i)i∈Ik,

´²)¡X ³ì»æX²

k∇fu^m((λi)i∈I)− ∇fu^m((λ⁰_i)i∈I)k ≤ √

M1M2

2α +

√M1M2

αkt⁰k

k(λi−λ⁰_i)i∈Ik

≤ 3√ M1M2

2α k(λi−λ⁰_i)i∈Ik.

Ý »¥²zê;æX²½¥µX½;}gr}¡,2 ¼6½;XµgX¡Fm ¡²ÚºX¹zµjz¡Fg²²[g»zm}¼²ÚºXµjX¶e ²¡FêXg²¶ z ºX}µgg»¼ ´3r»

X¡sjz¡Fmg²½âgù²X¨*¹g²¶â¡çg»¥²2Xº}Xµr»¥¼°XÞ Ý zê¥ ²£¦}¨X¡cæ}²µºX²$ÞéXµm}¼²Æmg²½Ígù²î=m²²[¡²ócg²g X¡jï.è

Ý }ºX²»¥²µ3´3g»¸8µX½;}gg X¡¦}¨eg» 3²¡g¹¥µg²xg»z3g»¥²$´3»¥}²$Xº}Xµr»¥¼ }¡cæX²µºX²xg»¥²!¶e²xm µ²¶Æm}¹e}¡è

´M²æX²µ¨¡£Xµj¶e²µ®gj»¥Fm²g»¥²mg²½m ù²6 ¡»¥²g²6Xº}Xµr»¥¼´M²6¡¥²²x¶2gÆ»æ}²z¡£²mg ¼g²6zÞLg»¥²ê;²m

½Fggê²)X¡sjz¡F ½j»¥gù®X¡mX¡Fè

Ý

» MX¡¨}XÞ-}¹¥µg²{ê;²®¶e}¡¥²²óe½²µg ¼²¡FX ³ê;²µ¹¥¡¥¡¡¥º°g»¥²)z ºX}µgg»¥¼

ÞéXµ{g²æX²µXTs²½wm ù²}¨e´3»¥²¡2X?»¥²$z»¥²µ3½XµX¼²g²µXµg²®¥óe²¶Tè

/¼6}µg²²óeê²2´³Úgj»}g²[»¥²mg²½g ù²2gÚ¹g²g»²2ÞéXµ¼°¹¥²óe½¥µ²m ¡¥ºg»²2ê}¹¥¡¶

C ºX æX²¡ ¡

¸8µg}½Fmg X¡2ôeè

g»»¥3µg²º}Xµ¶¥¨}ÞéXµ3¼}m{¶e g X¡Xµg ²¨¥z T3²²¼6²¡Fê¥¹e

M2

zµ²®²xXg³g1z¹¥²Xè

õ

¡Æ}µ¶¥²µg²xs¼zg²

M2

¨¥X¡¥²!z¡¹g²!sjz¡¶¥Xµ¶Æz ºX}µgg»¼ÔÞéXµX¼½¥¹e¡ºg»¥²$zµºX²xs3² ºX²¡cæX¹²)XÞ*X¡

µX¼ ¼zgµró

(hψi, ψji)i,j∈I

g16æX²x.g}µ

(λi)i∈I

µg²xí}¹µ²MX¡²)µ²X¼½;}gr}¡[z¡¶[}¡¥²$¶e²X¼½;}gr}¡[ ¡Æg»¥²!¶e.}¡zµ³

(ψi)i∈I

è

(9)

²µjm»¥º}Xµ¡2µj ² Ý »¥²}µg²¼1ï

M2≤

maxi∈I kψik

X

i∈I

kψik

!

. ^îm¦xô}ï

¡X ³X¨;X{z¡²}m ³[ê²m²²¡ÆÞéµg}¼+îm¦¯Fïz¡¶Úîm¦X¦xï.¨

fu^m

¶ec²{¡zgmÞé³ÆX¡F³Æm}µm3XÞ*² ½egs³[½µg}½²µms³}è

õ

¡Æ½zµgg¹¥zµx¨cr{3¡¥X² ½egzè

Ê T×o×LÌÜØFÒmÏLÓ Ï1ÑÌ,×Ð LÏ8ÜÒwÑ &

õ

¡g»¥Mm²x.g X¡-¨}´M²{½¥µ²g²¡FL»¥²®Xº}Xµr»¥¼ Xêejz ¡¥²¶´3»²¡»¥²º}µ}¶e²¡FLêXg²¶Xº}Xµr»¥¼ ¹m²x¶g6m}æ}²3g»²

s²½¦®zÞ»¥²z ºXXµg»¥¼'¶e²µ ê²x¶1 ¡ Ý zê¥ ²°¦® °g ¼6½²)½¥µs²g²x¶ºXµjX¶e ²¡F}g²¡F´3g»Æ}¡sjz¡FL¼²s²½è

Ý »¥²6mg²½â¦ ®»¥²¡oz¡ {ù´ìXº}Xµr»¥¼Öm}æc ¡¥º1»¥²¶e¹z*XÞ

(Pu^m) îég»c¹®g»²6¡z¼²XÞ8g»²æ}²µjm X¡ïè

æX²¡

¸8µg}½Fmg X¡Úôe¨?´M²÷F¡´ îåg²²ß ¦x§xá¨Xµxèôeè ¦Xètôe¨?½¥½è*ë¯¥¨TX¡¶ Ý »-è-ôeètôeèð¥¨?½¥½è-ð}ð}ï{g»z!r!X¡cæX²µgº}²)Xmc}¡}

g»¥²$¼²6s²½ ¡»¥²!µjz¡¥º}²

(0,_C²)^{¨´3»¥²µg²} C ºX æX²¡Æ ¡2¸8µX½;}gr}¡ôeè2}µg²æX²µ¨Fg»².ê;²m{g ¼²°mg²½ )

ρ= _C¹^è

²!X g÷F¡´ îåg²²ß ¦x§xáéïM»x¨eÞéXµ

ρ= _C¹ ^z¡¶ u^m∈R^N¨¥g»¥²µg²$²ócm3}¡sjz¡F

C1>0 îå´3»¥ j»2¶e²½²¡¶¥

X¡Æg»¥² ¡¥gz ùxg X¡2íF¹Xs³¥ïg¹j»[g»z

fu^m (λ^ki)i∈I

−f_u^∗m ≤C1

2C k+ 4,

´3»¥²µ²

(λ^k_i)i∈I

g»²)µ²g¹¥g»²

k

g»

r²µj}¡[XÞ»¥²!z ºX}µgg»¥¼äX¡¶

f_u^∗m= min

(λi)i∈I∈R+Ifu^m (λ^k_i)i∈I ^îm¦Fï

Ý »¥²)¡zz ºXXµg»¥¼Ô{¶e²xgµg ê²x¶ì ¡ Ý Xê¥ ²$ô¥è

õ¡¥gz ù²

(λ⁰_i)i∈I

¨

u⁰∈R^N ^z¡¶ ρ= _C¹^è

²½²x{¹¥¡Fg -X¡cæX²µgº}²¡²îé cX½Æ¡

m^ï

²½;²z{¹¥¡Fg -}¡Fæ}²µºX²¡²6îé F}½[ ¡

k^ï

¦}è®X¼½¥¹e²

w^k = 2αu^m−P

i∈Iλ^k_iψi+¹_τv

ô¥è3rÞ

(kw^kk ≤1)^¨g² w^k = 0

z»¥²µ´3 g²X¨eg²

w^k ←a w^k^¨e´3r» a= ^k_2α^w_k^k_w^k−kk¹

è {½?¶¥²

λ^k+1

¨

∀i∈I, λ^k+1_i = max 0, λ^ki +ρ(hw^k, ψii −1)

¹¥½;¶g²

u^m+1=w^k z¡¶T¨cÞé}µ{z

i∈I^¨ λ⁰_i =λ^k+1_i ^è

Q ô

{ùx´M°æX²µg}¡ìXÞg»¥²$Xº}Xµr»¥¼Ö¢

Ý

»¥²s²½¦)XÞg»²Xº}Xµr»¥¼ä¶e²xgµg ê²x¶1 ¡

Ý

zê²¦) gX æX²¶ìêc³1

½¥µg s²g²¶[ºXµjX¶¥²¡}3¶e²xg²¡F{´3r»2X¡sjz¡Fmg²½2m ù²}è

õ¡Ú½¥µ}. ²îåz¡¶£ ¡og»²1²óe½;²µ¼²¡Fj$½¥µ²g²¡Fg²x¶£ ¡çc²x.g X¡Fï.¨´² ¡¥gz ù²¶

(λ⁰_i)i∈I

z¡¶

u⁰ ^z 0^è ^Ý ^»²

X¡sjz¡F

C ´XM²mg ¼²¶Æ´3r»¸8µg}½Fmg X¡2ô6z¡¶oîm¦xôXï.è Ý »¥ºX æX²¢

C=3q P

i∈Ikψik² P

i∈Ikψik

(maxi∈Ikψik)

2α . ^îm¦¤cï

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