Analysis of Block OMP using Block RIP

Transcription

1 Anayss of ock OMP usng ock RIP Jun Wang, Gang L, Hao Zhang, Xqn Wang Department of Eectronc Engneerng, snghua Unversty, eng 00084, Chna Emas: {gang, haozhang, Abstract Orthogona matchng pursut (OMP) s a canonca greey agorthm for sparse sgna reconstructon. When the sgna of nterest s bock sparse,.e., t has nonzero coeffcents occurrng n custers, the bock verson of OMP agorthm (.e., ock OMP) outperforms the conventona OMP. In ths paper, we emonstrate that a new noton of bock restrcte sometry property (ock RIP), whch s ess strngent than stanar restrcte sometry property (RIP), can be use for a very straghtforwar anayss of ock OMP. It s emonstrate that ock OMP can eacty recover any bock K-sparse sgna n no more than K steps f the ock RIP of orer K+ wth a suffcenty sma sometry constant s satsfe. Usng ths resut t can be prove that ock OMP can ye better reconstructon propertes than the conventona OMP when the sgna s bock sparse. Ine erms ock sparsty, ock OMP, ock RIP I. Introucton Recenty, the compresse sensng (CS) theory [, ] has been use n many areas because of ts eceent performance on reconstructon of sparse sgnas. Conser an equaton y D,where s a K-sparse sgna of ength to be etermne, D s a L measurement matr (where L typcay), y s the measurement vector of ength L. Many agorthms have been propose to etermne n a stabe an effcent manner. Orthogona matchng pursut (OMP) s one of conventona greey agorthms, whch s commony use n CS area for ts smpcty. wo theoretca toos have been use to anayze OMP agorthm [3, 4]. One s note as coherence

2 [3]: ma, matr satsfes that, where enotes the th coumn of the matr D. When the coherence of measurement u K, t has been shown that OMP can recover any K-sparse sgna from the measurements y [3]. he other one s restrcte sometry property (RIP), t has been shown that OMP can eacty recover any K-sparse sgna n no more than K steps f the RIP of orer K+ wth sometry constant s 3 K satsfe [4]. In ths paper we conser bock sparse sgnas that have nonzero coeffcents occurrng n custers. he probem of nterest s that whether epcty takng the bock sparsty nto account can ye better reconstructon propertes than treatng the sgna as a conventona sparse sgna. hs probem s treate n [5], where t s shown that, a me / -norm agorthm for recoverng bock-sparse sgnas can ye eact sparse recovery f the bock restrcte sometry property (ock RIP) [5] s satsfe. In [6], a noton of the bock coherence s propose, an the eact sparse recovery can be acheve by usng the ock OMP agorthm when the bock coherence s suffcenty sma. For the more genera settng of moe-base compresse sensng where bock-sparsty s ncue as a speca case, t s shown n [7] that a mofe vson of CoSaMP agorthm [8] can ye eceent recovery reconstructon propertes for bock sparse sgnas. In ths paper we anayze the recovery performance of ock OMP usng ock RIP, whch has not been stue n estng terature. Our approach s party base on [4] an [9] (an the mathematca technques use theren). he man contrbuton n ths paper s provng that, f the ctonary matr D satsfes the ock-rip of orer K+ wth a suffcenty sma sometry constant, the ock OMP can eacty recover any bock K-sparse sgna n no more than K steps, whch s ooser than the eact recovery conton usng the conventona OMP. Moreover, t can aso be ncate that the teraton steps usng ock OMP are ess than usng OMP. hs paper s organze as foows. Founatona concepts of bock sparse sgna reconstructon,.e., bock sparsty, ock RIP an ock OMP are revewe n Secton II. he anayss of ock OMP usng ock RIP s performe n Secton III. Fnay, a bref concuson s gven n Secton IV. otaton: Vectors an matrces are enote by boface etters. A vectors are coumn vectors. () enotes the

3 transpose operaton; an enotes an norms, respectvey. II. Revew of bock sparse sgna reconstructon In ths secton, we revew some founatona concepts of bock sparse sgna reconstructon, ncung bock sparsty, ock RIP an ock OMP. A. ock Sparsty Assume that conssts of bocks wth gven bock ength, an M wth an nteger M, then can be epresse as: [,,,,,,,,, ] () [] [] M [ ] where [] enotes a vector whch starts from the ( ) th eement an ens to the th eement of, s an nteger, [] s cae the th bock of. he sgna s cae a -bock K-sparse, f the number of [] whch has nonzero norm s no more than K. Usng the efnton of me / 0 -norm [6], bock sparsty can aso be epresse as: M I( [ ] 0), (),0 where I( [ ] 0) s a ncator functon, whch s f [ ] 0 an 0 otherwse.. ock RIP Frst we revew the conventona RIP [, ]. A matr L D satsfes the RIP of orer K f there s a constant (0,) such that: ( ) D ( ), (3) for a K-sparse. Ref. [5] etens ths property to bock-sparse vectors an ea to the foowng efnton. he matr D has the ock RIP of orer K wth sometry constant (0,), f for a -bock K-sparse, we have ( ) D ( ). (4) ote that a -bock K-sparse vector s K-sparse n the conventona sense. If D satsfes the RIP of orer K

4 wth sometry constant, (3) must ho for a -bock K-sparse. On the contrary, f D satsfes the ock RIP of orer K wth sometry constant, (4) may not ho for a K-sparse. As a resut, the RIP of orer K s the suffcent conton of ock RIP of orer K for the same constant. In the other wor, the ock RIP constant s typcay smaer than the conventona RIP constant for the same matr. C. ock OMP agorthm ock OMP s the bock vson of OMP, an accompshe n reconstruct bock sparse sgnas [6]. he entre agorthm s specfe n Agorthm. Agorthm ock Orthogona Matchng Pursut nput: measurement matr D, measurements y, stoppng teraton ne L ntaze: resua error r 0 y, sgna 0 0, support set 0, teraton ne 0 whe L. h D r. h {arg ma [ ] } 3. arg mn y - Dz z:supp ( z) r y D en output: ˆ L where supp ( z) { {,,, M} z [ ] 0}, an h r enotes, h, r, at the th teraton, respectvey.,,, Compare wth the conventona OMP agorthm, the man fference of ock OMP s that ock OMP chooses the bock ne accorng to arg ma h [ ], whe the conventona OMP chooses the ne accorng to arg ma h, where h [ ] s a vector an h s a scaar. III. Anayss of ock OMP usng ock RIP

5 In ths secton, we begn wth some observatons regarng ock OMP whch set the stage for our man resuts. hese resuts ncue four emmas, two coroares an a theorem. Lemma an Lemma s the bock vson of Lemma 3. an Lemma 3. n [4], hen we prove Lemma 3, Lemma 4, Coroary an Coroary, an at ast prove heorem, whch nuces the man concuson of ths paper. Smary to anayss of OMP n [4], our goa s to fn how to guarantee that the bock ne chosen at each teraton step s correct. Let ( D ) enote the spannng space of coumns of D, where D s a L matr an s efne as D [ D[ ()], D[ ()],, D [ ( )]] (here s the ength of ). efore the teraton stops, t s accepte that be enote by D s a matr of fu coumn rank ( L), then the Moore-Penrose pseuonverse of D can D ( D D ) D. It s cear to see that the orthogona proecton operator onto ( D ) s P D D, whe the orthogona proecton operator onto the orthogona compement of ( D ) s P ( I P ). hen we efne A as A P D, whch s the resut of orthogonazng the coumns of D aganst ( D ). It s easy to prove that bocks of ow we conser how A nee by are equa to zeros. s obtane at the th teraton n Agorthm. s sove as a east squares probem, so t s gven by: D y, an 0, (5) c ( ) where s efne as [ [ ()], [ ()],, [ ( )]] c, an {,,, M}/. Snce P s orthogona proecton operator, P ( P ) ( P ). hen we have r y D y D D y ( I P ) y P y P D A, (6) an r P y P P y ( P ) P y, (7) so h D r D ( P ) P y A r A A. (8) We epect that h can ho most of nformaton of, whch means that h s epecte not too arge. Lemma 3 beow w prove the correctness of ths hypothess.

6 We begn wth two emmas whch are ust the bock verson of Lemma 3. an Lemma 3. n [4]. Lemma [4]: Suppose u, v, f D has the ock RIP of orer K ma( u v, u v ) (see ()) wth,0,0 sometry constant (0,), then: Du, Dv u, v u v. (9) hs resut emonstrates that bock sparse vectors that are orthogona reman neary orthogona after the appcaton of D. Lemma [4]: Suppose that D has the ock RIP of orer K wth sometry constant (0,), {,,..., M}. If K, then u wth u an supp ( u ), we have that: K,0 ( ) u A ( ) u u, (0) whch means that, f D satsfes the ock RIP of orer K wth sometry constant, then A satsfes a restrcte ock RIP of orer K wth sometry constant restrcton s supp ( u ). (t s cear to see that for (0,) ). Here the From Lemma an Lemma, we prove Coroary whch s crtca to our anayss beow. Coroary : Let be gven. If L D satsfes the ock RIP of orer supp ( ) wth sometry constant (0,), where M, then for a (, M), we have D D[ ] [ ]. () Proof: Let supp ( ) { }, so supp ( )+. Let I enotes the entty operator on enotes that the space consste of the coumns whose bock ne beongs to, the menson of, where s. From the proof of Proposton 3. n [0], we know that y, y D D I sup D y y, where D D I s an operator an efne as D D I ( y) D D y y, y. It s cear to see that, Dy Dy Dy Dy, y y y, y, where y s the contnuaton of y on,.e., y[ ( )] y [ ],,,, y[] 0 f.

7 From Lemma, we have Snce supp ( ), we have D D I sup Dy, Dy y, y y. y, y ( D D) D D. Snce, we have D D[ ] [ ] ( D D). In what foows, we w prove Lemma 3 usng Coroary, an from Lemma 3 the reatonshp between h an can be etermne. Lemma 3: Suppose that {,,..., M}, R an supp ( ). Defne h A A. () If D satsfes the bock RIP of orer wth sometry constant,0, then we have: h[ ] [ ]. (3) Proof: From Lemma, we have that for a, A satsfes the restrcte bock RIP of orer wth sometry constant,0,0. hen from Coroary an (), we have (3) at once. It s cear to see that h efne n () s amost the same as h n the th teraton of ock OMP (see Agorthm an (8)) Snce the reatonshp between h an s etermne, we may erve a suffcent conton uner whch the entfcaton step of ock OMP w succee. Coroary : Suppose that, D, meet the assumptons specfe n Lemma 3, an et h be as efne n (3). If, (4), where ma [ ], then we have, arg ma h[ ] supp ( ). (5)

8 Proof: If supp ( ), [ ] 0.If supp ( ) an, then from (3), we have that h[ ]. If supp ( ) but, then from () an the efnton of A, we have that h [ ] 0. So supp ( ), h[ ]. From (4), we have that 0supp ( ), s. t. [ 0].From (3) an the trange nequaty, we have that h[ 0 ], so arg ma h[ ] supp ( ). y choosng hep us to fn an approprate. sma enough, t s possbe to guarantee that the conton (5) s satsfe. Lemma 4 beow may Lemma 4: R,,,0, where [ ], ma. M Proof:,0 ma [ ] [ ],0 M ma [ ], M,0,0,0, so,,0. Fnay, usng above resuts, we have the foowng man concuson. heorem : Suppose that D satsfes the ock RIP of orer K+ wth sometry constant K, then R an K, ock OMP can eacty recover from y,0 D n K steps. Proof: he theorem s prove by nucton. At the frst teraton,.e., 0, 0 0 h D r D y D D. From the efnton of A we know that D A. Snce K, from Lemma 4, we have that,0, K. Snce K, we have K, an hence. From Coroary, we have that, 0 arg ma [ ] supp ( ) h. Suppose that at the th step,.e.,, the concuson n heorem s correct,.e., supp ( ) an,. hen when h D r, from the efnton of A an (8), we have that h A A A A,where s efne as: 0, c c. Snce that supp ( ) ( ) ( ), supp ( ), supp ( ) supp ( ), K an,0 the ock-rip of orer K+ wth sometry constant an K, we have that,0 K.Snce D satsfes K ( K ), from,0

9 Lemma 4, we have K K,,0. From Coroary, we have 0 arg ma h [ ] supp ( ) supp ( ), an snce 0 supp ( ) an supp ( ), we have 0. { 0}, so. When, we have that an r A. From the efnton of,0,0, we have 0. So r 0, y D. Suppose that,, supp ( ), supp ( ) s.t. y = D = D, an we know that supp ( ), so K. From the ock RIP of D, we know that,0 D( ) 0, so, whch means the souton of ock OMP s unque. If s treate as a conventona K sparse vector wthout epotng knowege of the bock sparse structure, a suffcent conton for eact recovery usng OMP s that D satsfes the RIP of orer K+ wth sometry constant 3 K. It s cear to see that the conton for ock OMP s ooser compare wth that for OMP. Moreover, ock OMP nee K steps to recover whe OMP nee K steps, so ock OMP s faster thanks to the pror nformaton of bock sparsty. When = (.e., the bock sze s equa to ), the bock sparse sgna recovery probem becomes genera sparse sgna recovery probem. Accorng to heorem, we have that D s neee to satsfy the RIP of orer K+ wth sometry constant. hs s fferent from the conton n [4] where D s neee to satsfy the RIP of K orer K+ wth sometry constant. he reason s that n the proof of heorem 3. n [4], s 3 K 3 K requre to guarantee, whereas s enough actuay as a guarantee. K K IV. Concuson A greey agorthm for bock sparse sgna, ock OMP agorthm s scusse n ths paper. We anayze the recovery performance of ock OMP usng ock RIP, an we prove Lemma 3, Lemma 4, Coroary an Coroary n Secton III an at ast prove heorem whch states that f the ctonary matr D satsfes the ock RIP of orer

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