The Hyperbolic Region for Restricted Isometry Constants in Compressed Sensing

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1 INTERNATIONAL JOURNAL OF CIRCUITS SYSTEMS AND SIGNAL PROCESSING Volume 8 Te Hyeroli Region for Restrited Isometry Constants in Comressed Sensing Siqing Wang Yan Si and Limin Su Astrat Te restrited isometry onstants (RIC lay an imortant role in omressed sensing sine if RIC satisfy some ounds ten sarse signals an e reovered exatly in te noiseless ase and estimated staly in te noisy ase During te last few years some ounds of RIC ave otained Te ounds of RIC among tem were introdued y Candes (8 Fouart and Lai (9 Fouart ( Cai et al ( Mo and Li ( In te aer we otain a yeroli region on and It omletely inludes te regions of te ounds on otained y te autors aove and if and elong to te yeroli region ten sarse signals an e reovered exatly in te noiseless ase Keywords Comressed sensing L minimization restrited isometry roerty sarse signal reovery I INTRODUCTION Comressed sensing aims to reover ig-dimensional sarse signals ased on onsideraly fewer linear measurements We onsider y unnown signal Φ β were te matrix n Φ wit n te β Let β e te numer of nonzero elements of β and β β i i Te signal β is alled sarse if β Our goal is to reonstrut β ased on y and Φ A naive aroa for solving tis rolem is to onsider L minimization were te goal is to find te sarsest solution in te feasile set of ossile solutions However tis is NP ard and tus is omutationally infeasile It is ten natural to onsider te metod of L minimization wi an e viewed as Siqing Wang is wit te College of Matematis and Information Sienes Nort Cina University of Water Resoures and Eletri Power 55 Zengzou Cina ( wangsiqing@nwuedun Yan Si is wit Institute of Environmental and Muniial Engineering Nort Cina University of Water Resoures and Eletri Power 55 Zengzou Cina ( siyan@nwuedun Limin Su is wit College of Matematis and Information Sienes Nort Cina University of Water Resoures and Eletri Power 55 Zengzou Cina ( suliminlove@63om a onvex relaxation of L minimization Te L minimization metod in tis ontext is { y } ˆ β arg min γ sujet to Φγ ( γ Tis metod as een suessfully used as an effetive way for reonstruting a sarse signal in many settings See e g Donoo and Huo [] Donoo [3] Candes et al [8-] and Cai et al [ 3] Reovery of ig dimensional sarse signals is losely onneted wit Lasso and Dantzig seletors e g see Candes et al [] Biel et al [] Wang and Su [9-] One of te most ommonly used framewors for sarse reovery via L minimization is te restrited isometry roerty wit a RIC introdued y Candes and Tao [9] For an n matrix Φ and an integer te restrited isometry onstant ( Φ is te smallest onstant su tat ( Φ u Φu + ( Φ u for every sarse vetor u If + ' te ' restrited ortogonality onstant θ ' ( Φ is te smallest numer tat satisfies Φu Φu' θ ( Φ u u' ' for all u and u ' su tat u and u ' are sarse and ' sarse resetively and ave disjoint suorts For notational simliity we sall write for ( Φ and θ ' for θ ' Φ ereafter It as een sown tat L minimization an reover a sarse signal wit a small or zero error under various onditions on and θ ' So a great deal of attention as een foused ere during te last few year for examle te onditions involving a and θ were a 355 and 55 see Candes et al [8-] and Cai et al [ 5]; te onditions involving only see Candes [7] Fouart and Lai [6] Fouart [5] Cai et al [] Mo and Li [8] and only see Cai et al [] Ji and Peng [7] Cai and zang [6] ISSN:

2 INTERNATIONAL JOURNAL OF CIRCUITS SYSTEMS AND SIGNAL PROCESSING Volume 8 It is ovious tat and are two of te most imortant and asi arameters In tis aer we otain te suffiient onditions involving only and It is a yeroli region Tis yeroli region omletely inludes te regions of te ounds on in te literature [ ] and if and elong to te yeroli region ten sarse signals an e reovered exatly in te noiseless ase Te rest of te aer is organized as follows In Setion some asi notations are introdued and te funtions on and are given Our yeroli region on and is resented in Setion 3 In Setion we disuss te rolem tat te yeroli region omletely inludes te regions of te ounds on in te literature [ ] Oter meaningful rolems are also disussed II THE FUNCTIONS OF THE RESTRICTED ISOMETRY CONSTANTS We onsider te simle setting were no noise is resent In tis ase te goal is to reover te signal β exatly wen it is sarse Tis ase is of signifiant interest in its own rigt as it is also losely onneted to te rolem of deoding of linear odes See for examle Candes and Tao [9] Te ideas used in treating tis seial ase an e easily extended to treat te general ase were noise is resent Let ˆβ e te minimizer to te rolem ( Let ˆ β β For any suset Q { } we define Q IQ were I Q denotes te indiator funtion of te set Q ie IQ ( j if j Q and if j Q Let S e te index set of te largest elements (in asolute value Rearrange te indies of S if neessary aording to te desending order of i i S Partition S in order into S S i i were Si l te last Si satisfies Si For simliity wen tere is no amiguity we write i S i i l Let t ( i i Tus ( t ( i i i i It is ovious from ( and ( tat Furter i i i i i ( t i i i (5 3 t / i i (6 i i In fat from Cai et al [] ( and ( i i + i i 3 t/ i + i i i Let Φ ten Φ ( + Φ i i Φ ( + Φ i i Te following need to use some asi fats: < < < θ < see Candes et al [7 9] It is easy to see y Candes and Tao [9] tat ten tere must e t [/ l] In fat y te definition of S i we ave l tl (3 i i i i If i ten t / l if i ten te elements of i S are all zero Te following we suose tat l sine t wen l From ( we ave i ( Φ ( + + ( ( ( + + / By te definition of and (5 and (6 we ave Φ Φ Φ i i j i i j (7 ISSN:

3 INTERNATIONAL JOURNAL OF CIRCUITS SYSTEMS AND SIGNAL PROCESSING Volume 8 From (7 i i j i j> i Φ + Φ Φ ( + + i i j i j> i ( + t ( t + ( 5 t i i ( ( + ( + t( t + ( 5t i i Deriving f( t wit reset to t ten ave ( ( f '( t t Te stationary oint is t If 5 < ten t > Te funtion f( t is monotone inreasing wen t t and monotone dereasing wen t t So f( t reaes te maximum wen t t and te maximum is From ( t( t 5t t 6 t t i i t 6( ( i i Teorem If 5 < ten ξ i i Hene Tat is f( t f( t i 6( i ξ i i Remar It is ovious tat < t < sine were ( ξ Proof Consider te funtion ( ( f( t t t were t ( + We firstly sow tat > Davies and Grionval [] onstruted examles wi sowed tat if exat reovery of ertain -sarse signal an fail in te noiseless ase Tis imlies tat tere must e < in order to guarantee stale reovery of -sarse signals So < < 3 < < Te inequality in Teorem is equality wen t t if / l t < ut te strit inequality in Teorem olds if < t < / l Similar to te roof of Teorem it follows tat Teorem 3 y Mo and Li [8] is a seial ase of Teorem Corollary If < 3 ten were η η i i ( + 5 ( ( 3 5 III THE HYPERBOLIC REGION OF THE RESTRICTED ISOMETRY CONSTANTS Lemma If < / and < / ten ISSN:

4 INTERNATIONAL JOURNAL OF CIRCUITS SYSTEMS AND SIGNAL PROCESSING Volume 8 < i i Proof From < / and (8 ave t( t + 5t + t 6 + tt i i 6 + t( 36 i 6( i Let gt 6 + t 36 t were t ( + Similar to te roof of Teorem te stationary oint is t 36 were is a yeroli region < i i < 8 of te origin at ( u v ( v v ( u+ u > u v see Aendix Proof Rewrite x and y for te sae of onveniene From Teorem we only need to rove ξ < It is ovious tat ξ < if and only if x 57 y xy 8x 85y 8 Tat is + + < (9 If < / ten t > Te funtion gt ( is monotone inreasing wen t t and monotone dereasing wen t t So gt ( reaes te maximum wen t t and te maximum is Write 6 x x < y y ( xy gt ( 36 6 A a a ot θ 6 57 a Hene were < θ < π / Ten gt i 6( i osθ sinθ a + a a + Tat is were ζ i i were a tanθ By trigonometri identity we ave 6a 6a 6 + ( ζ ( ( a ( It is ovious tat ζ < if and only if < / Lemma If 5 < and ten < 8 Let x osθ sinθ u ( y sin θ os θ v Ten ISSN:

5 INTERNATIONAL JOURNAL OF CIRCUITS SYSTEMS AND SIGNAL PROCESSING Volume 8 if and only if x 57 y + xy 8x + 85y < 8 ( v v ( u+ u > It is a yeroli region of te origin at ( u v seifi alulations see Aendix Write sets Te U ( : < < < U : < g( < 8 were g U ( {( : 5 < g( < 8 } 3 U ( : < < 5 < Teorem If ( U+ U ten < i i Proof From Lemma and Lemma if ( ten < i Note tat sets i U + U 3 { 5 < } { g ( < 8} { < / } wen < < / and wen / < / So Tis imlies tat { 5 < } { g ( < 8} U3 U + U U+ U U+ U3 U+ U + U U+ U Remar Teorem is intuitive U+ U is te oen region enlosed y straigt lines x x y y / and yerola x 57 y + xy 8x + 85y 8 IV DISCUSSION AND CONCLUSION Candes [7] Fouart and Lai [6] Fouart [5] Cai et al [] Mo and Li [8] gave te onditions involving only Mo and Li [8] sowed tat if < 93 ten < i Tis is te est result on so far We illustrate tat Teorem omletely imroves te result y Mo and Li [8] elow < 93 in fat orresonds to set U : < 93 < We only need to {( } 5 sow U5 U+ U In order to reision we tae te exat < instead of aroximate value value < 93 We will rove te intersetion etween straigt line y ( and yerola is If use x 57 y + xy 8x + 85y 8 ( x y i y 3y y x 8 8 3y ( 5y536 8 an't get exat value We use anoter metod Note tat y 77y+ 8 sine y ( if and only if y < So if ( y ten Tat is x y x 57 y + xy 8x + 85y 8 ( x y( x y sine x / Note tat U 5 is roer suset of U+ U In fat U+ U U5 is te region enlosed y straigt lines x y ( y / and yerola x 57 y xy 8x 85y Davies and Grionval [] onstruted examles wi sowed tat if exat reovery of ertain -sarse ISSN:

6 INTERNATIONAL JOURNAL OF CIRCUITS SYSTEMS AND SIGNAL PROCESSING Volume 8 signal an fail in te noiseless ase Cai et al [] onstruted an examle wi sowed tat if / it is imossile to reover ertain -sarse signals Tese imly tat tere must e < and < / in order to guarantee stale reovery of -sarse signals In addition Cai and Zang [6] sowed tat if < /3 ten -sarse signals an e reovered exatly in te noiseless ase Terefore te remaining wor are to resear reovery of -sarse signals in region U {( : 3 < < g( 8} 6 By te way te result if and < 6569 ten < (see Corollary 3 in Mo and Li [8] is i i inorret In fat te rigt side of te equation (3 in Mo and Li [8] is equal to zero wen l sine t [/ l] In te roof of Teorem 33 in Mo and Li [8] t( + t ( + / 8 if and only if ( + / / l sine t [/ l] Tis imlies tat wen l and /3 wen l 3 We disuss seond rolem Wy te demaration oint is < / in Lemma? As we will see elow tis disussion is meaningful If < d from (8 similar to te roofs of Teorem and Lemma ten ave 6 + 8( + d 5 t ( 3 + 6d i 6( i d d d i + i ( ( 3 6d l( d i i Our goal is to aieve maximum wen ( seifi alulating ( l d < By l d < if and only if < / sine < and < / wen < / So te demaration oint is < / in Lemma Tis imly tat using te metod of Teorem and Lemma annot ave / Terefore to resear reovery of -sarse signals in region U 6 must use new metods V APPENDIX: COMPLETION OF THE PROOF OF LEMMA We give seifi alulation in order to get te reise yeroli equation as follows By ( and ( we ave x y xy x y Tus if and only if ( a a + + a57a a a 85 8 a + u ( a a a a + + v + a + ( a a57 ( 85a 8 ( 8a a57a a a57 ( a a + ( a 83a u + 6 a 83 6 ( a a ( a 58a v 6 a ( 85a 8 3( 8a a 6 58a A D a + u B v E a + + C A C B D 3E A + B x 57 y + xy 8x + 85y < 8 3E a + D a + v u+ B A > 3E D C 3E D C 8 8 B A B B A 3A By ( and ( ten ave D were 3865 Tus E 3A 53 C B + 53 C + 6 C ( ( ISSN:

7 INTERNATIONAL JOURNAL OF CIRCUITS SYSTEMS AND SIGNAL PROCESSING Volume E C E C B B CB ( 53 ( + 33 ( D C D A B C ( ( C 53 B 3 Based on te aove results y diret alulation we ave Similarly ave 3 E D 8 C B A B E D 8 C B B A 3A 3A v u 3E a + 3E C B CB D a 3D + C A 3A C Te aroximate values are v 73 and u 35 resetively REFERENCES [] P J Biel Y Ritov and A B Tsyaov Simultaneous analysis of Lasso and Dantzig seletor Ann Stat vol [] T Cai L Wang and G Xu Sifting inequality and reovery of sarse signals IEEE Trans Signal Proess vol [3] T Cai L Wang and G Xu Stale reovery of sarse signals and an orale inequality IEEE Trans Inf Teory vol [] T Cai L Wang and G Xu New ounds for restrited isometry onstants IEEE Trans Inf Teory vol [5] T Cai G Xu and J Zang On reovery of sarse signals via L minimization IEEE Trans Inf Teory vol [6] T Cai A Zang Sar RIP ound for sarse signal and low-ran matrix reovery Al Comut Harmon Anal vol [7] E J Candes Te restrited isometry roerty and its imliations for omressed sensing Comtes Rendus Matematique vol [8] E J Candes J Romerg and T Tao Stale signal reovery from inomlete and inaurate measurements Comm Pure Al Mat vol [9] E J Candes T Tao Deoding y linear rogramming IEEE Trans Inf Teory vol [] E J Candes T Tao Near-otimal signal reovery from random rojetions: Universal enoding strategies IEEE Trans Inf Teory vol [] E J Candes T Tao Te Dantzig seletor: Statistial estimation wen is mu larger tan n (wit disussion Ann Stat vol [] M E Davies R Grionval Restrited isometry onstants were L sarse reovery an fail for < IEEE Trans Inf Teory vol [3] D L Donoo Comressed sensing IEEE Trans Inf Teory vol [] D L Donoo X Huo Unertainty riniles and ideal atomi deomosition IEEE Trans Inf Teory vol [5] S Fouart A note on guaranteed sarse reovery via L minimization Al Comut Harmon Anal vol [6] S Fouart M Lai Sarsest solutions of underdetermined linear systems via L q minimization for < q Al Comut Harmon Anal vol [7] J Ji J Peng Imroved ounds for restrited isometry onstants Disrete Dyn Nat So vol Availale: tt://wwwindawiom/journals/ddns//86/as/ [8] Q Mo S Li New ounds on te restrited isometry onstant Al Comut Harmon Anal vol [9] S Q Wang L M Su Te orale inequalities on simultaneous Lasso and Dantzig seletor in ig dimensional nonarametri regression Mat Prol Eng vol 3 Availale: tt://wwwindawiom/journals/me/3/5736/as/ [] S Q Wang L M Su Reovery of ig-dimensional sarse signals via L -minimization J Al Mat vol 3 Availale: tt://wwwindawiom/journals/jam/3/6369/ [] S Q Wang L M Su Simultaneous Lasso and Dantzig seletor in ig dimensional nonarametri regression Int J Al Mat Stat vol 3-8 [] S Q Wang L M Su New ounds of mutual inoerene roerty on sarse signals reovery Int J Al Mat Stat vol Siqing Wang is a Professor of Matematis at Nort Cina University of Water Resoures and Eletri Power Zengzou Cina His field of interest is Matematis Information Sienes and Statistis He as more tan 8 resear artiles in journal of Matematis Information Sienes and Statistis Yan Si is a Dotor at Institute of Environmental and Muniial Engineering Nort Cina University of Water Resoures and Eletri Power Limin Su is a Master' of Alied Matematis at College of Matematis and Information Sienes Nort Cina University of Water Resoures and Eletri Power ISSN:

Shifting Inequality and Recovery of Sparse Signals

Shifting Inequality and Recovery of Sparse Signals T. Tony Cai Lie Wang and Guangwu Xu Astract In this paper we present a concise and coherent analysis of the constrained l 1 minimization method for stale