Compressed Sensing of Block-Sparse Signals: Uncertainty Relations and Efficient Recovery

Transcription

1 Compessed Sensng of Bock-Spase Sgnas: Uncetanty Reatons and Effcent Recovey Yonna C. Eda, Seno Membe, IEEE, Patck Kuppnge, Student Membe, IEEE, and Hemut Böcske, Feow, IEEE axv: v2 [cs.it] 8 Jun 2009 Abstact We consde compessed sensng of bock-spase sgnas,.e., spase sgnas that have nonzeo coeffcents occung n custes. An uncetanty eaton fo bock-spase sgnas s deved, based on a bock-coheence measue, whch we ntoduce. We then show that a bock-veson of the othogona matchng pusut agothm ecoves bock k-spase sgnas n no moe than k steps f the bock-coheence s suffcenty sma. The same condton on bock-coheence s shown to guaantee successfu ecovey though a mxed 2/ -optmzaton appoach. Ths compements pevous ecovey esuts fo the bock-spase case whch eed on sma bock-estcted somety constants. The sgnfcance of the esuts pesented n ths pape es n the fact that makng expct use of bock-spasty can povaby yed bette econstucton popetes than teatng the sgna as beng spase n the conventona sense, theeby gnong the addtona stuctue n the pobem. I. INTRODUCTION The famewok of compessed sensng s concened wth the ecovey of an unknown vecto fom an undedetemned system of nea equatons [], [2]. The key popety expoted fo ecovey of the unknown data s the assumpton of spasty. Moe concetey, denotng by x an unknown vecto that s obseved though a measuement matx D accodng to y = Dx, t s assumed that x has ony a few nonzeo entes. A fundamenta obsevaton s that f D s chosen popey and x s suffcenty spase, then x can be ecoveed fom y = Dx, espectvey of the ocatons of the nonzeo entes of x, even f D has fa fewe ows than coumns. Ths esut has gven se to a muttude of dffeent ecovey agothms whch can be poven to ecove a spase vecto x unde a vaety of dffeent condtons on D [3], [4], [5], [], [6]. Two wdey studed ecovey agothms ae the bass pusut (BP), o -mnmzaton appoach [7], [], and the othogona matchng pusut (OMP) agothm [8]. One of the man toos fo the chaactezaton of the ecovey abtes of BP s the estcted somety popety (RIP) [], [9]. Specfcay, f the measuement matx D satsfes the RIP wth appopate estcted somety constants, then x can be ecoveed by BP. Unfotunatey, detemnng the RIP constants of a gven Y. Eda s wth the Depatment of Eectca Engneeng, Technon, Hafa, Isae, Ema: yonna@ee.technon.ac. H. Böcske and P. Kuppnge ae wth the Communcaton Technoogy Laboatoy, ETH Zuch, Zuch, Swtzeand, Ema: {boecske,patcku}@na.ee.ethz.ch Ths wok was suppoted n pat by the Euopean Commsson FP7 Netwok of Exceence n Weess Communcatons NEWCOM++ and by the Isae Scence Foundaton. Ths pape was pesented n pat at IEEE ICASSP 2009, Tape, Tawan, Ap matx s n genea an NP-had pobem. A moe smpe and convenent way to chaacteze ecovey popetes of a dctonay s va the coheence measue [0], [], [5]. It was shown n [5], [2] that appopate condtons on the coheence guaantee that both BP and OMP ecove the spase vecto x. The coheence aso pays an mpotant oe n uncetanty eatons fo spase sgnas [0], [], [3]. In ths pape, we consde compessed sensng of spase sgnas that exhbt addtona stuctue n the fom of the nonzeo coeffcents occung n custes. Such sgnas ae efeed to as bock-spase [4], [5]. Ou goa s to expcty take ths bock stuctue nto account, both n tems of the ecovey agothms and n tems of the measues that ae used to chaacteze the pefomance. The sgnfcance of the esuts we obtan es n the fact that makng expct use of bock-spasty can povaby yed bette econstucton popetes than teatng the sgna as beng spase n the conventona sense, theeby gnong the addtona stuctue n the pobem. Bock-spasty ases natuay, e.g., when deang wth mut-band sgnas [6], [7], [8] o n measuements of gene expesson eves [9]. Anothe nteestng speca case of the bock-spase mode appeas n the mutpe measuement vecto (MMV) pobem, whch deas wth the measuement of a set of vectos that shae a ont spasty patten [20], [2], [22], [4], [23]. Futhemoe, t was shown n [4], [5] that the bock-spasty mode can be used to teat the pobem of sampng sgnas that e n a unon of subspaces [24], [25], [4], [26], [3], [6], [7]. One appoach to expotng bock-spasty s by sutaby extendng the BP method, esutng n a mxed 2 / -nom ecovey agothm [4], [27]. It was shown n [4] that f D has sma bock-estcted somety constants, whch geneazes the conventona RIP noton, then the mxed nom method s guaanteed to ecove any bock-spase sgna, espectvey of the ocatons of the nonzeo bocks. Futhemoe, ecovey w be obust n the pesence of nose and modeng eos (.e., when the vecto s not exacty bock-spase). It was aso estabshed n [4] that cetan andom matces satsfy the bock RIP wth ovewhemng pobabty, and that ths pobabty s substantay age than that of satsfyng the standad RIP. In [28] extensons of the CoSaMP agothm [29] and of teatve had theshodng [30] to the mode-based settng, whch ncudes bock-spasty as a speca case, ae poposed and shown to exhbt povabe ecovey guaantees and obustness popetes. The focus of the pesent pape s on deveopng a paae

2 2 ne of esuts by geneazng the noton of coheence to the bock settng. Ths can be seen as extendng the pogam ad out n [5], [2] to the bock-spase case. Specfcay, we defne two sepaate notons of coheence: coheence wthn a bock, efeed to as sub-coheence and captung oca popetes of the dctonay, and bock-coheence, descbng goba dctonay popetes. We w show that both coheence notons ae necessay to chaacteze the essence of bockspasty. We pesent extensons of the BP, the matchng pusut (MP), and the OMP agothms to the bock-spase case and pove coespondng pefomance guaantees. We pont out that the tem bock-coheence was used pevousy n [3] n the context of quantfyng the ecovey pefomance of the MP agothm n bock-ncoheent dctonaes. Ou defnton petans to bock-vesons of the MP and the OMP agothm and s dffeent fom that used n [3]. We begn, n Secton II, by ntoducng ou defntons of bock-coheence and sub-coheence. In Secton III, we estabsh an uncetanty eaton fo bock-spase sgnas, and show how the bock-coheence measue defned pevousy occus natuay n ths uncetanty eaton. In Secton IV, we ntoduce a bock veson of the OMP agothm, temed BOMP, and of the MP agothm [8], temed BMP, and fnd a suffcent condton on bock-coheence that guaantees ecovey of bock k-spase sgnas though BOMP n no moe than k steps as we as exponenta convegence of BMP. The same condton on bock-coheence s shown to guaantee successfu ecovey though the mxed 2 / optmzaton appoach. The BOMP agothm can be vewed as an extenson of the subspace OMP method fo MMV systems [23]. The poofs of ou man esuts ae contaned n Secton V. A dscusson on the pefomance mpovements that can be obtaned though expotng bock-spasty s povded n Secton VI. Coespondng numeca esuts ae epoted n Secton VII. We concude n Secton VIII. Thoughout the pape, we denote vectos by bodface owecase ettes, e.g., x, and matces by bodface uppecase ettes, e.g., A. The dentty matx s wtten as I o I d when the dmenson s not cea fom the context. Fo a gven matx A, A T, A H, and T(A) denote ts tanspose, conugate tanspose, and tace, espectvey, A s the pseudo nvese, R(A) denotes the ange space of A, A, s the eement n the th ow and th coumn of A, and a stands fo the th coumn of A. The th eement of a vecto x s denoted by x. The Eucdean nom of the vecto x s x 2 = x H x, x = x s the -nom, x = max x s the -nom, and x 0 desgnates the numbe of nonzeo entes n x. The Konecke poduct of the matces A and B s wtten as A B. The specta nom of A s denoted by ρ(a) = λ /2 max(a H A), whee λ max (B) s the agest egenvaue of the postve-semdefnte matx B. II. BLOCK-SPARSITY AND BLOCK-COHERENCE A. Bock-spasty We consde the pobem of epesentng a vecto y C L n a gven dctonay D of sze L N wth L < N, so that y = Dx () fo a coeffcent vecto x C N. Snce the system of equatons () s undedetemned, thee ae, n genea, many possbe choces of x that satsfy () fo a gven y. Theefoe, futhe assumptons on x ae needed to guaantee unqueness of the epesentaton. Hee, we consde the case of spase vectos x,.e., x has ony a few nonzeo entes eatve to ts dmenson. The standad spasty mode consdeed n compessed sensng [], [2] assumes that x has at most k nonzeo eements, whch can appea anywhee n the vecto. As dscussed n [28], [4], [5] thee ae pactca scenaos that nvove vectos x wth nonzeo entes appeang n bocks (o custes) athe than beng abtay spead thoughout the vecto. Specfc exampes ncude sgnas that e n unons of subspaces [25], [24], [4], [26], and mut-band sgnas [6], [7], [8]. The ecovey of bock-spase vectos x fom measuements y = Dx s the focus of ths pape. To defne bock-spasty, we vew x as a concatenaton of bocks assumed thoughout the pape to be of ength d wth x[] denotng the th bock,.e., x = [x... x }{{ d } x T [] x d+... x 2d... x N d+... x N ] T (2) }{{}}{{} x T [2] x T [M] whee N = Md. We futhemoe assume that L = Rd wth R ntege. Smay to (2), we can epesent D as a concatenaton of coumn-bocks D[] of sze L d: D = [d... d }{{ d } D[] d d+... d 2d... d N d+... d N ]. (3) }{{}}{{} D[2] D[M] A vecto x C N s caed bock k-spase f x[] has nonzeo Eucdean nom fo at most k ndces. When d =, bock-spasty educes to conventona spasty as defned n [], [2]. Denotng x 2,0 = M I( x[] 2 > 0) (4) = wth the ndcato functon I( ), a bock k-spase vecto x s defned as a vecto that satsfes x 2,0 k. In the emande of the pape conventona spasty w be efeed to smpy as spasty, n contast to bock-spasty. We ae nteested n povdng condtons on the dctonay D ensung that the bock-spase vecto x can be ecoveed fom measuements y of the fom () though computatonay effcent agothms. Ou appoach s paty based on [5], [], [2] (and the mathematca technques used theen) whee equvaent esuts ae povded fo the spase case. The two agothms nvestgated ae BOMP and a mxed 2 / - optmzaton pogam (efeed to as L-OPT [4]). It was shown n [4] that L-OPT yeds pefect ecovey f the dctonay D satsfes appopate estcted somety popetes. The pupose of ths pape s to povde ecovey condtons fo BOMP and L-OPT based on a sutaby defned measue of bock-coheence. We w see that bock-coheence pays a oe sma to coheence n the case of conventona spasty. Befoe defnng bock-coheence, we note that n ode to have a unque bock k-spase x satsfyng () t s cea that we need R > k and the coumns wthn each bock D[], =

3 3, 2,..., M, need to be neay ndependent. Moe geneay, we have the foowng poposton taken fom [4]. Poposton. The epesentaton () s unque f and ony f Dg 0 fo evey g 0 that s bock 2k-spase. Fom Poposton the coumns of D[] ae neay ndependent fo a. Thoughout the pape, we assume that the dctonaes we consde satsfy the condton of Poposton, and, futhemoe, d 2 =, =, 2,..., N. B. Bock-coheence The coheence of a dctonay D measues the smaty between bass eements, and s defned by [0], [], [5] µ = max, dh d. (5) Ths defnton was ntoduced n [8] to heustcay chaacteze the pefomance of the MP agothm, and was ate shown to pay a fundamenta oe n quantfyng ecovey theshods fo the OMP agothm and fo BP [5]. The coheence µ futhemoe occus n -uncetanty eatons eevant n the context of decomposng a vecto nto two othonoma bases [0], []. A defnton of coheence fo anaog sgnas, aong wth a coespondng uncetanty eaton, s povded n [3]. It s natua to seek a geneazaton of coheence to the bock-spase settng wth the esutng bock-coheence measue havng the same opeatona sgnfcance as the coheence µ n the spase case. Beow, we popose such a geneazaton, whch s shown n Sectons III and IV to occu natuay n uncetanty eatons and n ecovey theshods fo the bock-spase case. We defne the bock-coheence of D as µ B = max ρ(m[, ]) (6), d wth M[, ] = D H []D[]. (7) Note that M[, ] s the (, )th d d bock of the N N matx M = D H D. When d =, as expected, µ B = µ. Whe µ B quantfes goba popetes of the dctonay D, oca popetes ae descbed by the sub-coheence of D, defned as ν = max max, dh d, d, d D[]. (8) We defne ν = 0 fo d =. In addton, f the coumns of D[] ae othonoma fo each, then ν = 0. Snce the coumns of D have unt nom, the coheence µ n (5) satsfes µ [0, ] and theefoe, as a consequence of ν [0, µ], we have ν [0, ]. The foowng poposton estabshes the same mts fo the bock-coheence µ B, whch expans the choce of nomazaton by /d n the defnton (6). In the emande of the pape conventona coheence w be efeed to smpy as coheence, n contast to bock-coheence and sub-coheence. Poposton 2. The bock-coheence µ B satsfes 0 µ B µ. Poof: Snce the specta nom s non-negatve, ceay µ B 0. To pove that µ B µ, note that the entes of M[, ] fo have absoute vaue smae than o equa to µ. It then foows that µ B = max, = max, max, max, ρ(m[, ]) d λ max (M d H [, ]M[, ]) d max (M d H [, ]M[, ]), (9) = d max dµ d 2 = = µ (0) whee (9) s a consequence of Gešgon s dsc theoem ([32, Cooay 6..5]). Fom µ, wth Poposton 2, t now foows tvay that µ B. When the coumns of D[] ae othonoma fo each, we can futhe bound µ B. Poposton 3. If D conssts of othonoma bocks,.e., D H []D[] = I d fo a, then µ B /d. Poof: Usng the submutpcatvty of the specta nom, we have µ B = max, = max, max, = d ρ(m[, ]) d d ρ(dh []D[]) d ρ(dh [])ρ(d[]) () whee () foows fom D H []D[] = I d, fo a, λ max (D H []D[]) = λ max (D[]D H []), and λ max (I d ) = combned wth the defnton of the specta nom. III. UNCERTAINTY RELATION FOR BLOCK-SPARSE SIGNALS We next show how the bock-coheence µ B defned above natuay appeas n an uncetanty eaton fo bock-spase sgnas. Ths uncetanty eaton geneazes the coespondng esut fo the spase case deved n [0], []. Uncetanty eatons fo spase sgnas ae concened wth epesentatons of a vecto x C L n two dffeent othonoma bases fo C L : {φ, L} and {ψ, L} [0], []. Any vecto x C L can be expanded unquey n tems of each one of these bases accodng to: x = L a φ = = L b ψ. (2) =

4 4 The uncetanty eaton sets mts on the spasty of the decompostons (2) fo any x C L. Specfcay, denotng A = a 0 and B = b 0, t s shown n [] that 2 (A + B) AB (3) µ(φ, Ψ) whee µ(φ, Ψ) s the coheence between Φ and Ψ, defned as µ(φ, Ψ) = max, φh ψ. (4) It s easy seen that fo D consstng of the othonoma bases Φ and Ψ,.e., D = [Φ Ψ], we have µ(φ, Ψ) = µ, whee µ s as defned n (5) and assocated wth D = [Φ Ψ]. In [0] t s shown that / L µ(φ, Ψ). The uppe bound foows fom the Cauchy-Schwaz nequaty and the fact that the bass eements have nom. The owe bound s obtaned as foows: The matx M = Φ H Ψ s untay so that L L = = φh ψ 2 = T(M H M) = T(I L ) = L. Consequenty, we have L 2 max, φ H ψ 2 L whch mpes µ(φ, Ψ) / L. Ths owe bound can be acheved, fo exampe, by choosng the two othonoma bases Φ and Ψ as the spke (dentty) and Foue bases [0]. Wth ths choce, the uncetanty eaton (3) becomes A + B 2 AB 2 L. (5) When L s an ntege, the eatons n (5) can a be satsfed wth equaty by choosng x as a Dac comb δ L wth spacng L, esutng n L nonzeo eements. Ths foows fom the fact that the Foue tansfom of δ L s aso δ L. We now deveop an uncetanty eaton fo bock-spase decompostons. Specfcay, we deve a esut that s equvaent to (3) wth A and B epaced by bock-spasty eves as defned n (4), and µ(φ, Ψ) epaced by the bock-coheence between the othonoma bases consdeed, and defned beow n (8). Theoem. Let Φ, Ψ be two untay L L matces wth L d bocks {Φ[], Ψ[], R} and et x C L satsfy R R x = Φ[]a[] = Ψ[]b[]. (6) = = Let A = a 2,0 and B = b 2,0. Then, whee 2 (A + B) AB dµ B (Φ, Ψ) µ B (Φ, Ψ) = max, (7) d ρ(φh []Ψ[]). (8) Note that fo D consstng of the othonoma bases Φ and Ψ,.e., D = [Φ Ψ], we have µ B (Φ, Ψ) = µ B, whee µ B s as defned n (6) and assocated wth D = [Φ Ψ]. Poof: Wthout oss of geneaty, we assume that x 2 2 =. Then, R = x 2 2 = a H []A[, ]b[] (9),= R a H []A[, ]b[] (20),= whee we set A[, ] = Φ H []Ψ[]. Now, fom the Cauchy- Schwaz nequaty, fo any a, b, a H A[, ]b b 2 A H [, ]a 2 λ /2 max(a[, ]A H [, ]) b 2 a 2 dµ B b 2 a 2 (2) whee, fo bevty, we wote µ B = µ B (Φ, Ψ). Substtutng nto (9), we get R dµ B b[] 2 = R = Appyng the Cauchy-Schwaz nequaty yeds a[] 2. (22) R b[] 2 ( R /2 B b[] 2) 2 = B (23) = = whee we used the fact that R = b[] 2 2 = b 2 2 = snce x 2 2 = and Ψ s untay. Smay, we have that R = a[] 2 A. Substtutng nto (22) and usng the nequaty of athmetc and geometc means competes the poof. The bound povded by Theoem can be tghte than that obtaned by appyng the conventona uncetanty eaton (3) to the bock-spase case. Ths can be seen by usng a 0 d a 2,0 and b 0 d b 2,0 n (3) to obtan a 2,0 b 2,0 dµ. (24) Snce µ B µ, ths bound may be oose than (7). A. Bock-ncoheent dctonaes As aeady noted, n the spase case (.e., d = ) fo any two othonoma bases Φ and Ψ, we have µ / L. We next show that the bock-coheence satsfes a sma nequaty, namey µ B / dl. Poposton 4. The bock-coheence (8) satsfes µ B / dl. Poof: Let Φ and Ψ be two othonoma bases fo C L and et A = Φ H Ψ wth A[, ] denotng the (, )th d d bock of A. Wth R = L/d, we have R 2 µ 2 B Now, t hods that R = = R R d 2 λ max(a H [, ]A[, ]) ( R ) d 2 λ R max A H [, ]A[, ]. (25) = = R A H [, ]A[, ] = = = ( R R ) Ψ H [] Φ[]Φ H [] Ψ[]. = = (26) Snce Φ s a squae matx consstng of othonoma coumns, we have R = Φ[]ΦH [] = ΦΦ H = I L. Futhemoe, snce

5 5 Ψ[] conssts of othonoma coumns, fo each, we have Ψ H []Ψ[] = I d. Theefoe, (25) becomes µ 2 B d 2 R = (27) dl whch concudes the poof. We now constuct a pa of bases that acheves the owe bound n (27) and theefoe has the smaest possbe bockcoheence. Let F be the DFT matx of sze R = L/d wth F, = (/ R) exp(2π/r). Defne Φ = I L and Ψ = F U d (28) whee U d s an abtay d d untay matx. Fo ths choce, Φ H []Ψ[] = F, U d. Snce ρ(u d ) = and F, = / R, we get µ B = d R =. (29) dl When d =, ths bass pa educes to the spke-foue pa whch s we known to be maxmay ncoheent [0]. When µ B satsfes (29) the uncetanty eaton becomes A + B 2 AB 2 R. (30) If R s ntege, the nequates n (30) ae met wth equaty fo the sgna x = δ R c whee c s an abtay nonzeo ength-d vecto. Indeed, n ths case, the epesentaton of x n the spke bass eques R bocks (of sze d), so that a 2,0 = R. The epesentaton of x n the bass Ψ n (28) s obtaned as b = (F H U H d )(δ R c) = δ R UH d c (3) whee we used the fact that the Foue tansfom of δ R s aso δ R. Theefoe, b has R nonzeo bocks so that b 2,0 = R and hence A = B = R, whch mpes that a nequates n (30) ae met wth equaty. IV. EFFICIENT RECOVERY ALGORITHMS We now gve opeatona meanng to bock-coheence by showng that f t s sma enough, then a bock-spase sgna x can be ecoveed fom the measuements y = Dx usng computatonay effcent agothms. We consde two dffeent ecovey methods, namey the mxed 2 / -optmzaton pogam (L-OPT) poposed n [4]: mn x M x[] 2 s. t. y = Dx (32) = and an extenson of the OMP agothm [8] to the bockspase case descbed beow and temed bock-omp (BOMP). We then deve theshods on the bock-spasty eve as a functon of µ B and ν fo both methods to ecove the coect bock-spase x. Fo L-OPT ths compements the esuts n [4] that estabsh the ecovey capabtes of L-OPT unde the condton that D satsfes a bock-rip wth a sma enough estcted somety constant. Fo the speca case of the coumns of D[] beng othonoma fo each, we suggest a bock-veson of the MP agothm [8], temed bock-mp (BMP). A. Bock OMP and bock MP The BOMP agothm begns by ntazng the esdua as 0 = y. At the th stage ( ) we choose the bock that s best matched to accodng to: = ag max D H [] 2. (33) Once the ndex s chosen, we fnd x [] as the souton to mn y D[]x [] (34) I whee I s the set of chosen ndces,. The esdua s then updated as 2 = y I D[]x []. (35) In the speca case of the coumns of D[] beng othonoma fo each (the eements acoss dffeent bocks do not have to be othonoma), we consde an extenson of the MP agothm to the bock-case. The esutng agothm, temed BMP, stats by ntazng the esdua as 0 = y and at the th stage ( ) chooses the bock that s best matched to accodng to (33). Then, howeve, the agothm does not pefom a east-squaes mnmzaton ove the bocks that have aeady been seected, but decty updates the esdua accodng to B. Recovey condtons = D[ ]D H [ ]. (36) Ou man esut, summazed n Theoems 2 and 3 beow, s that any bock k-spase vecto x can be ecoveed fom measuements y = Dx usng ethe the BOMP agothm o L-OPT f the bock-coheence satsfes kd < (µ B + d (d )νµ B )/2. In the speca case of the coumns of D[] beng othonoma fo each, we have ν = 0 and theefoe the ecovey condton becomes kd < (µ B + d)/2. In ths case BMP exhbts exponenta convegence ate (see Theoem 4). If the bock-spase vecto x was teated as a (conventona) kd-spase vecto wthout expotng knowedge of the bockspasty stuctue, a suffcent condton fo pefect ecovey usng OMP [5] o (32) fo d = (known as BP) s kd < (µ + )/2. Compang wth kd < (µ B + d)/2, we can see that, thanks to µ B µ, makng expct use of bock-spasty eads to guaanteed ecovey fo a potentay hghe spasty eve. Late, we w estabsh condtons fo such a esut to hod even when ν 0. To fomay state ou man esuts, suppose that x 0 s a ength-n bock k-spase vecto, and et y = Dx 0. Let D 0 denote the L (kd) matx whose bocks coespond to the nonzeo bocks of x 0, and et D 0 be the matx of sze L (N kd) whch contans the L d bocks of D that ae not n D 0. We then have the foowng theoem poved n Secton V. Theoem 2. Let x 0 C N be a bock k-spase vecto wth bocks of ength d, and et y = Dx 0 fo a gven L N

6 6 matx D. A suffcent condton fo the BOMP and the L-OPT agothm to ecove x 0 s that whee ρ c (D 0 D 0) < (37) ρ c (A) = max ρ(a[, ]) (38) and A[, ] s the (, )th d d bock of A. In ths case, BOMP pcks up a coect new bock n each step, and consequenty conveges n at most k steps. Note that ρ c (D 0 D 0) = max ρ c (D 0 D 0[]). (39) Theefoe, (37) mpes that fo a, ρ c (D 0 D 0[]) <. (40) The suffcent condton (37) depends on D 0 and theefoe on the ocaton of the nonzeo bocks n x 0, whch, of couse, s not known n advance. Nonetheess, as the foowng theoem, poved n Secton V, shows, (37) hods unvesay unde cetan condtons on µ B and ν assocated wth the dctonay D. Theoem 3. Let µ B be the bock-coheence and ν the subcoheence of the dctonay D. Then (37) s satsfed f kd < ( ) µ ν B + d (d ). (4) 2 µ B Fo d =, and theefoe ν = 0, we ecove the coespondng condton k < (µ +)/2 epoted n [5], [2]. In the speca case whee the coumns of D[] ae othonoma fo each, we have ν = 0 and (4) becomes kd < 2 (µ B + d). (42) The next theoem shows that unde condton (42), BMP exhbts exponenta convegence ate n the case whee each bock D[] conssts of othonoma coumns. Theoem 4. If D H []D[] = I d, fo a, and kd < (µ B + d)/2, then we have: ) BMP pcks up a coect bock n each step. 2) The enegy of the esdua decays exponentay,.e., 2 2 β wth β = (k )dµ B. (43) k V. PROOFS OF THEOREMS 2, 3, AND 4 Befoe poceedng wth the actua poofs, we stat wth some defntons and basc esuts that w be used thoughout ths secton. Fo x C N, we defne the genea mxed 2 / p -nom (p =, 2, hee and n the foowng): x 2,p = v p, whee v = x[] 2 (44) and the x[] ae consecutve ength-d bocks. Fo an L N matx A wth L = Rd and N = Md, whee R and M ae nteges, we defne the mxed matx nom (wth bock sze d) as Ax 2,p A 2,p = max. (45) x 0 x 2,p The foowng emma povdes bounds on A 2,p, whch w be used n the seque. Lemma. Let A be an L N matx wth L = Rd and N = Md. Denote by A[, ] the (, )th d d bock of A. Then, A 2, max A 2, max In patcua, ρ (A) = ρ c (A H ). Poof: See Appendx A. ρ(a[, ]) = ρ (A) (46) ρ(a[, ]) = ρ c (A). (47) Lemma 2. ρ c (A) as defned n (38) s a matx nom and as such satsfes the foowng popetes: Nonnegatve: ρ c (A) 0 Postve: ρ c (A) = 0 f and ony f A = 0 Homogeneous: ρ c (αa) = α ρ c (A) fo a α C Tange nequaty: ρ c (A + B) ρ c (A) + ρ c (B) Submutpcatve: ρ c (AB) ρ c (A)ρ c (B). Poof: See Appendx B. A. Poof of Theoem 2 fo BOMP We begn by povng that (37) s suffcent to ensue ecovey usng the BOMP agothm. We fst show that f s n R(D 0 ), then the next chosen ndex w coespond to a bock n D 0. Assumng that ths s tue, t foows mmedatey that s coect snce ceay 0 = y es n R(D 0 ). Notng that es n the space spanned by y and D[], I, whee I denotes the ndces chosen up to stage, t foows that f I coesponds to coect ndces,.e., D[] s a bock of D 0 fo a I, then aso es n R(D 0 ) and the next ndex w be coect as we. Thus, at evey step a coect L d bock of D s seected. As we w show beow no ndex w be chosen twce snce the new esdua s othogona to a the pevousy chosen subspaces; consequenty the coect x 0 w be ecoveed n k steps. We fst show that f R(D 0 ), then unde (37) the next chosen ndex coesponds to a bock n D 0. Ths s equvaent to equng that z( ) = DH 0 2, D H 0 2, <. (48) Fom the popetes of the pseudo-nvese, t foows that D 0 D 0 s the othogona poecto onto R(D 0). Hence, t hods that D 0 D 0 =. Snce D 0 D 0 s Hemtan, we have (D 0 )H D H 0 =. (49)

7 7 Substtutng (49) nto (48) yeds z( ) = DH 0 (D 0 )H D H 0 2, D H 0 2, ρ (D H 0 (D 0 )H ) = ρ c (D 0 D 0) (50) whee we used Lemma. It emans to show that BOMP n each step chooses a new bock patcpatng n the (unque) epesentaton y = Dx. We stat by defnng D = [D[ ] D[ ]] whee I,. It foows that the souton of the mnmzaton pobem n (34) s gven by ˆx = (D H D ) D H y whch upon nsetng nto (35) yeds = (I D (D H D ) D H )y. Now, we note that D (D H D ) D H s the othogona poecto onto the ange space of D. Theefoe D H [] 2 = 0 fo a bocks D[] that e n the span of the matx D. By the assumpton n Poposton () we ae guaanteed that as ong as < k thee exsts at east one bock (n D 0 ) whch does not e n the span of D. Snce ths bock (o these bocks) w ead to stcty postve D H [] 2 the esut s estabshed. Ths concudes the poof. B. Poof of Theoem 2 fo L-OPT We next show that (37) s aso suffcent to ensue ecovey usng L-OPT. To ths end we ey on the foowng emma: Lemma 3. Suppose that v C kd wth v[] 2 > 0, fo a, and that A s a matx of sze L (kd), wth L = Rd and the d d bocks A[, ]. Then, Av 2, ρ c (A) v 2,. If n addton the vaues of ρ c (AJ ) ae not a equa, then the nequaty s stct. Hee, J s a (kd) d matx that s a zeo except fo the th d d bock whch equas I d. Poof: See Appendx C. To pove that L-OPT ecoves the coect vecto x 0, et x x 0 be anothe ength-n bock k-spase vecto fo whch y = Dx. Denote by c 0 and c the ength-kd vectos consstng of the nonzeo eements of x 0 and x, espectvey. Let D 0 and D denote the coespondng coumns of D so that y = D 0 c 0 = D c. Fom the assumpton n Poposton, t foows that thee cannot be two dffeent epesentatons usng the same bocks D 0. Theefoe, D must contan at east one bock, Z, that s not ncuded n D 0. Fom (40), we get ρ c (D 0Z) <. Fo any othe bock U n D, we must have that ρ c (D 0U). (5) Indeed, f U D 0, then U = D 0 [] = D 0 J whee J was defned n Lemma 3. In ths case, D 0 D 0[] = J and theefoe ρ c (D 0 U) = ρ c(j ) =. If, on the othe hand, U = D[] fo some, then t foows fom (40) that ρ c (D 0U) <. Now, suppose fst that the (kd) d bocks n D 0 D do not a have the same ρ c. Then, c 0 2, = D 0 D 0c 0 2, (52) = D 0 D c 2, < ρ c (D 0 D ) c 2, (53) c 2, (54) whee the fst equaty s a consequence of the coumns of D 0 beng neay ndependent (a consequence of the assumpton n Poposton ), the fst nequaty foows fom Lemma 3 snce c [] 2 > 0, fo a, and the ast nequaty foows fom (5). If a the (kd) d bocks n D 0 D have dentca ρ c, then the nequaty (53) s no onge stct, but the second nequaty (54) becomes stct nstead as a consequence of ρ c (D 0 Z) < ; theefoe c 0 2, < c 2, st hods. Snce x 0 2, = c 0 2, and x 2, = c 2,, we concude that unde (40), any set of coeffcents used to epesent the ogna sgna that s not equa to x 0 w esut n a age 2 / -nom. C. Poof of Theoem 3 We stat by devng an uppe bound on ρ c (D 0D) n tems of µ B and ν. Wtng D 0 out, we have that ρ c (D 0 D) = ρ c((d H 0 D 0 ) D H 0 D). (55) Submutpcatvty of ρ c (A) (Lemma 2) mpes that ρ c (D 0 D) ρ c((d H 0 D 0 ) )ρ c (D H 0 D) = ρ c ((D H 0 D 0 ) ) max ρ(d H []D[]) (56) / Λ 0 Λ 0 whee Λ 0 s the set of ndces fo whch D[] s n D 0. Snce Λ 0 contans k ndces, the ast tem n (56) s bounded above by kdµ B, whch aows us to concude that ρ c (D 0 D) ρ c((d H 0 D 0 ) )kdµ B. (57) It emans to deveop a bound on ρ c ((D H 0 D 0 ) ). To ths end, we expess D H 0 D 0 as D H 0 D 0 = I + A, whee A s a (kd) (kd) matx wth bocks A[, ] of sze d d such that A, = 0, fo a. Ths foows fom the fact that the coumns of A ae nomazed. Snce A[, ] = D H 0 []D 0 [], fo a, and A[, ] = D H 0 []D 0 [] I d, we have ρ c (A) = max max ρ(a[, ]) ρ(a[, ]) + max ρ(a[, ]) (58) (d )ν + (k )dµ B (59) whee the fst tem n (59) s obtaned by appyng Gešgon s dsc theoem ([32, Cooay 6..5]) togethe wth the defnton of ν; the second tem n (59) foows fom the fact that the summaton n the second tem of (58) s ove k eements and ρ(a[, ]), fo a, can be uppe-bounded by dµ B. Note that fo an (sd) d matx A, ρ c(a) = P ρ(a[]), whee A[], =, 2,..., s, denotes the d d bock of A made up of the ows {( )d +,..., d}.

8 8 Assumpton (4) now mpes that (d )ν + (k )dµ B < and theefoe, fom (59), we have ρ c (A) <. We next use the foowng esut. Lemma 4. Suppose that ρ c (A) <. Then (I + A) = k=0 ( A)k. Poof: Foows mmedatey by usng the fact that ρ c (A) s a matx nom (cf. Lemma 2) and appyng [32, Cooay 5.6.6]. Thanks to Lemma 4, we have that ( ) ρ c ((D H 0 D 0 ) ) = ρ c ( A) k k=0 (ρ c (A)) k (60) k=0 = ρ c (A). (6) (d )ν (k )dµ B Hee, (60) s a consequence of ρ c (A) satsfyng the tange nequaty and beng submutpcatve and (6) foows by usng (59). Combnng (6) wth (57), we get ρ c (D 0 D) kdµ B (d )ν (k )dµ B < (62) whee the ast nequaty s a consequence of (4). D. Poof of Theoem 4 The poof of the fst pat of Theoem 4 foows fom the aguments n the poofs of Theoems 2 and 3 fo ν = 0. As a consequence of the fst statement of Theoem 4, we get that the esdua n each step of the agothm w be n R(D 0 ). Fo the poof of the second statement n Theoem 4, we mmc the coespondng poof n [33]. We fst need the foowng emma, whch s an extenson of [34, Lemma 3.5] to the bock-spase case. Ths emma w povde us wth a owe bound on the amount of enegy that can be emoved fom the esdua n one step of the BMP agothm. Lemma 5. Let D 0 denote the L (kd) matx whose bocks coespond to the nonzeo bocks of x 0. Then, we have max D H 0 [] (63) c 2, whee c s the coeffcent vecto coespondng to 0,.e., = D 0 c. Poof: We stat by notng that = D 0 c = k = D 0[]c [], whee c [] 0 fo at east one ndex {, 2,..., k}. It foows that 2 2 = k c H []D H 0 [] = k c H []D H 0 [] = k c [] 2 D H 0 [] 2 = ( ) k max D H 0 [] 2 c [] 2. (64) The esut then foows by notng that k = c [] 2 = c 2,. Next, we compute an uppe bound on c 2,. Usng M[, ] = D H 0 []D 0 [], whee, {,..., k}, we get 2 2 = c H D H 0 D 0 c k k = c H []M[, ]c [] = = = k c H []M[, ]c [] + = k c [] 2 2 = k = k = = k c H []M[, ]c [] = k c H []M[, ]c [] (65) = whee we used the fact that M[, ] = I d, fo a, as a consequence of each of the bocks of D 0 consstng of othonoma vectos. Appyng the Cauchy-Schwaz nequaty to the second tem n (65), we get 2 2 k c [] 2 2 = c 2 2,2 k = k = k c [] 2 M[, ]c [] 2 (66) = k c [] 2 c [] 2 dµ B (67) = k = c 2 2,2 dµ B s= = k c [] 2 c [( + s) k ] 2 (68) whee ( + s) k stands fo ( + s) moduo k, (67) foows fom M[, ]c [] 2 dµ B c [] 2, and (68) s obtaned by meey eaangng tems n the summaton n (67). Appyng the Cauchy-Schwaz nequaty to the nne poduct k = c [] 2 c [( + s) k ] 2, we obtan k 2 2 c 2 2,2 dµ B c 2 2,2 (69) s= = ( (k )dµ B ) c 2 2,2 ( (k )dµ B) c 2 2, (70) k

9 9 whee (70) foows by the same agument as used n (23). Thus, combnng (64) wth (70), we get ( (k max D H )dµb ) 0 [] 2 2. (7) k Snce, by the fst statement n Theoem 4, BMP pcks a bock n D 0 n each step, we can bound the enegy of the esdua n the ( + )st step as = 2 2 D H [ + ] 2 2 (72) = 2 2 max D H 0 [] 2 2 ( ( (k )dµ ) B) 2 2 (73) k whee n (72) we used the fact that + s othogona to D[ + ]D H [ + ]. Ths concudes the poof. ecovey theshod (µ B + d)/2 (µ + )/ bock sze d Fg.. Recovey theshods fo both bock-spasty and conventona spasty fo R = 0 as a functon of d. VI. DISCUSSION Theoem 3 ndcates unde whch condtons expotng bock-spasty eads to hghe ecovey theshods than teatng the bock-spase sgna as a (conventonay) spase sgna. Fo dctonaes D whee the ndvdua bocks D[] consst of othonoma coumns, fo each, we have ν = 0 and hence, thanks to µ B µ, ecovey though expotng bock-spasty s guaanteed fo a potentay hghe spasty eve. If the ndvdua bocks D[] ae, howeve, not othonoma, we have ν > 0, and (4) shows that ν has to be sma fo bockspase ecovey to esut n hghe ecovey theshods than spase ecovey. It s now natua to consde the case whee one stats wth a genea dctonay D and othogonazes the ndvdua bocks D[] so that ν = 0. The compason that s meanngfu hee s between the ecovey theshod of the ogna dctonay D wthout expotng bock-spasty and the ecovey theshod of the othogonazed dctonay takng bock-spasty nto account. To ths end, we stat by notng that the assumpton n Poposton mpes that the coumns of D[] ae neay ndependent, fo each. We can theefoe wte D[] = A[]W whee A[] conssts of othonoma coumns that span R(D[]) and W s nvetbe. The othogonazed dctonay s gven by the L N matx A wth bocks A[]. Snce D = AW wth the N N bock-dagona matx W wth bocks W, we concude that c = Wx s bock-spase and thanks to the nvetbty of the W of the same bock-spasty eve as x,.e., othogonazaton peseves the bock-spasty eve. It s easy to see that the defnton of bock-coheence n (6) s nvaant to the choce of othonoma bass A[] fo R(A[]). Ths s because any othe bass has the fom A[]U fo some untay matx U, and fom the popetes of the specta nom ρ(m[, ]) = ρ(u H M[, ]U ) (74) fo any untay matces U, U. Unfotunatey, t seems dffcut to deve genea esuts on the eaton between µ befoe and µ B afte othogonazaton. Nevetheess, we can estabsh a mnmum bock sze d above whch othogonazaton foowed by bock-spase ecovey eads to a guaanteed mpovement n the ecovey theshods. We fst note that the coheence µ of a dctonay consstng of N = Md eements n a vecto space of dmenson L = Rd can be owe-bounded as [35] M R Md M R µ R(Md ) RMd. (75) Usng ths owe bound togethe wth Poposton 3 and the fact that afte othogonazaton we have ν = 0, t can be shown that f d > RM/(M R), then the ecovey theshod obtaned fom takng bock-spasty nto account n the othogonazed dctonay s hghe than the ecovey theshod coespondng to conventona spasty n the ogna dctonay. Ths s tue espectvey of the dctonay we stat fom as ong as the dctonay satsfes the condtons of Poposton. Fnay, we note that fndng dctonaes that ead to sgnfcant mpovements n the ecovey theshods when expotng bock-spasty seems to be a dffcut desgn pobem. Fo exampe, pattonng the eazatons of..d. Gaussan matces nto bocks w, n genea, not ead to satsfactoy esuts. Nevetheess, thee do exst dctonaes whee sgnfcant mpovements ae possbe. Consde, fo exampe, the pa of bases Φ = I L and Ψ = F U d shown n Secton III-A to acheve the owe bound n (27). Fo the coespondng dctonay D = [Φ Ψ], we have M = 2R, µ B = /(d R), wth the ecovey theshod, assumng that bock-spasty s expoted, gven by kd < d( R + )/2. The coheence of the dctonay s µ = vec(u d ) / R. Fg., obtaned by aveagng ove andomy chosen untay matces U d, shows that the ecovey theshods obtaned by takng bockspasty nto account can be sgnfcanty hghe than those fo conventona spasty. In patcua, fo U d = I d, we obtan the conventona ecovey theshod as k = kd < ( R+)/2, whch aows us to concude that expotng bock-spasty can esut n guaanteed ecovey fo a spasty eve that s d tmes hghe than what woud be obtaned n the (conventona) spase case. VII. NUMERICAL RESULTS The am of ths secton s to quantfy the mpovement n the ecovey popetes of OMP and BP obtaned by takng bock-

10 BOMP-O BOMP-O success ate BOMP success ate OMP BOMP OMP bock-spasty eve bock-spasty eve Fg. 2. Pefomance of OMP, BOMP, and BOMP-O fo a dctonay wth L = 40, N = 400, and d = 4. Fg. 3. Pefomance of OMP, BOMP, and BOMP-O fo a dctonay wth L = 80, N = 60, and d = 8. spasty expcty nto account and pefomng ecovey usng BOMP and L-OPT, espectvey. In a smuaton exampes beow, we andomy geneate dctonaes by dawng fom..d. Gaussan matces and nomazng the esutng coumns to. The dctonay s dvded nto consecutve bocks of ength d. The spase vecto to be ecoveed has..d. Gaussan entes on the andomy chosen suppot set (accodng to a unfom po). In Fgs. 2 and 3, we pot the ecovey success ate 2 as a functon of the bock-spasty eve of the sgna to be ecoveed. Fo each bock-spasty eve we aveage ove 000 pas of eazatons of the dctonay and the bock-spase sgna. We can see that BOMP outpefoms OMP sgnfcanty and BOMP wth othogonazed bocks, denoted as BOMP- O, yeds sghty bette pefomance than BOMP. We aso evauate the pefomance of L-OPT compaed to BP, as we as L-OPT un on othogonazed bocks, temed L-OPT-O. Fo each bock-spasty eve we aveage ove 200 pas of eazatons of the dctonay and the bock-spase sgna. The coespondng esuts, depcted n Fgs. 4 and 5, show that L-OPT outpefoms BP, and L-OPT-O sghty outpefoms L-OPT. Futhemoe, we can see that BOMP-O sgnfcanty outpefoms L-OPT-O. VIII. CONCLUSION Ths pape extends the concepts of uncetanty eatons, coheence, and ecovey theshods fo matchng pusut and bass pusut to the case of spase sgnas that have addtona stuctue, namey bock-spasty. The extenson s made possbe by an appopate defnton of bock-coheence. The motvaton fo consdeng bock-spase sgnas s twofod. Fst, n many appcatons the nonzeo eements of spase vectos tend to custe n bocks; sevea exampes ae gven n [4]. Second, t s shown n [4] that sampng pobems ove unons of subspaces can be conveted nto bock-spase ecovey pobems. Specfcay, ths s tue when the unon has a dect-sum decomposton, whch s the case 2 Success s decaed f the ecoveed vecto s wthn a cetan sma Eucdean dstance of the ogna vecto. success ate BP L-OPT L-OPT-O bock-spasty eve BOMP-O Fg. 4. Pefomance of BP, L-OPT, L-OPT-O, and BOMP-O fo a dctonay wth L = 40, N = 400, and d = 4. n many appcatons ncudng mutband sgnas [26], [6], [7], [8]. Reducng unon of subspaces pobems to bockspase ecovey pobems aows fo the fst genea cass of concete ecovey methods fo unon of subspace pobems. Ths was the man contbuton of [4] togethe wth equvaence and obustness poofs fo L-OPT based on a sutaby modfed defnton of the estcted somety popety. Hee, we compement ths contbuton by deveopng sma esuts usng the concept of bock-coheence. We fst pove (46): Ax 2, = max APPENDIX A PROOF OF LEMMA max max A[, ]x[] 2 A[, ]x[] 2 x[] 2 ρ(a[, ]) x 2, max ρ(a[, ]). (76)

11 success ate BP L-OPT bock-spasty eve L-OPT-O BOMP-O Fg. 5. Pefomance of BP, L-OPT, L-OPT-O, and BOMP-O fo a dctonay wth L = 80, N = 60, and d = 8. Theefoe, fo any x C N wth x 0, we have Ax 2, x 2, ρ (A) (77) whch estabshes (46). The poof of (47) s sma: Ax 2, = A[, ]x[] 2 A[, ]x[] 2 x[] 2 ρ(a[, ]) ρ c (A) x 2, (78) fom whch the esut foows. Fnay, we have ρ c (A H ) = max ρ(ah [, ]) = max ρ(a[, ]) = ρ (A). APPENDIX B PROOF OF LEMMA 2 Nonnegatvty and postvty foow mmedatey fom the fact that the specta nom s a matx nom [32, p. 295]. Homogenety foows by notng that ρ c (αa) = max = max ρ(αa[, ]) α ρ(a[, ]) = α ρ c (A). (79) The tange nequaty s obtaned as foows: ρ c (A + B) = max ρ(a[, ] + B[, ]) ( max ρ(a[, ]) + ) ρ(b[, ]) max ρ(a[, ]) + max ρ(b[, ]) = ρ c (A) + ρ c (B) whee the fst nequaty s a consequence of the specta nom satsfyng the tange nequaty. Fnay, to vefy submutpcatvty, note that, Theefoe, f we pove that ρ c (AB) = max ρ c (AB[]). (80) ρ c (AB[]) ρ c (A)ρ c (B[]) (8) the esut foows fom (80) and the fact that max ρ c (B[]) = ρ c (B). To pove (8), note that ρ c (AB[]) = ρ A[, ]B[, ] ρ (A[, ]B[, ]) ρ(a[, ])ρ(b[, ]) (82) whee we used the tange nequaty fo, and the submutpcatvty of, the specta nom. Now, we have ρ(a[, ]) max ρ(a[, ]) = ρ c (A). (83) Substtutng nto (82) yeds ρ c (AB[]) ρ c (A) whch competes the poof. ρ(b[, ]) = ρ c (A)ρ c (B[]) (84) APPENDIX C PROOF OF LEMMA 3 The poof of the statement Av 2, ρ c (A) v 2, foows decty fom (78) by epacng A by an L (kd) matx and x C N by v C kd wth v[] 2 > 0, fo a. If the a = ρ(a[, ]) ae not a equa, then the ast nequaty n (78) s stct. Snce a = ρ c (AJ ) the esut foows. REFERENCES [] E. J. Candès, J. K. Rombeg, and T. Tao, Robust uncetanty pncpes: Exact sgna econstucton fom hghy ncompete fequency nfomaton, IEEE Tans. Inf. Theoy, vo. 52, no. 2, pp , Feb [2] D. L. Donoho, Compessed sensng, IEEE Tans. Inf. Theoy, vo. 52, no. 4, pp , Ap [3] G. Davs, S. Maat, and M. Aveaneda, Adaptve geedy appoxmatons, Const. Appox., vo. 3, no., pp , 997. [4] B. Efon, T. Haste, I. Johnstone, and R. Tbshan, Least ange egesson, Ann. Statst., vo. 32, no. 2, pp , [5] J. Topp, Geed s good: Agothmc esuts fo spase appoxmaton, IEEE Tans. Inf. Theoy, vo. 50, no. 0, pp , Oct [6] E. J. Candès and T. Tao, Decodng by nea pogammng, IEEE Tans. Inf. Theoy, vo. 5, no. 2, pp , Dec [7] S. S. Chen, D. L. Donoho, and M. A. Saundes, Atomc decomposton by bass pusut, SIAM J. Sc. Comput., vo. 20, no., pp. 33 6, 999. [8] S. G. Maat and Z. Zhang, Matchng pusuts and tme-fequency dctonaes, IEEE Tans. Sg. Poc., vo. 4, no. 2, pp , Dec [9] E. J. Candès, The estcted somety popety and ts mpcatons fo compessed sensng, C. R. Acad. Sc. Pas, Se. I, vo. 346, pp , 2008.

12 [0] D. L. Donoho and X. Huo, Uncetanty pncpes and dea atomc decompostons, IEEE Tans. Inf. Theoy, vo. 47, no. 7, pp , Nov [] M. Ead and A. M. Bucksten, A geneazed uncetanty pncpe and spase epesentaton n pas of bases, IEEE Tans. Inf. Theoy, vo. 48, no. 9, pp , Sept [2] D. L. Donoho and M. Ead, Optmay spase epesentaton n genea (nonothogona) dctonaes va mnmzaton, Poc. Nat. Acad. Sc., vo. 00, no. 5, pp , Ma [3] Y. C. Eda, Uncetanty eatons fo anaog sgnas, IEEE Tans. Inf. Theoy, Sept. 2008, submtted. [4] Y. C. Eda and M. Msha, Robust ecovey of sgnas fom a stuctued unon of subspaces, IEEE Tans. Inf. Theoy, 2008, submtted. [5], Bock-spasty and sampng ove a unon of subspaces, to appea n DSP2009. [6] M. Msha and Y. C. Eda, Bnd mut-band sgna econstucton: Compessed sensng fo anaog sgnas, IEEE Tans. Sg. Poc., vo. 57, no. 3, pp , Ma [7], Fom theoy to pactce: Sub-Nyqust sampng of spase wdeband anaog sgnas, axv ; submtted to IEEE Jouna of Seected Topcs n Sgna Pocessng, [8] H. J. Landau, Necessay densty condtons fo sampng and ntepoaton of cetan ente functons, Acta Math., vo. 7, no., pp , 967. [9] F. Pavaesh, H. Vkao, S. Msa, and B. Hassb, Recoveng spase sgnas usng spase measuement matces n compessed DNA mcoaays, IEEE Jouna of Seected Topcs n Sgna Pocessng, vo. 2, no. 3, pp , Jun [20] S. F. Cotte, B. D. Rao, K. Engan, and K. Keutz-Degado, Spase soutons to nea nvese pobems wth mutpe measuement vectos, IEEE Tans. Sg. Poc., vo. 53, no. 7, pp , Ju [2] J. Chen and X. Huo, Theoetca esuts on spase epesentatons of mutpe-measuement vectos, IEEE Tans. Sg. Poc., vo. 54, no. 2, pp , Dec [22] M. Msha and Y. C. Eda, Reduce and boost: Recoveng abtay sets of onty spase vectos, IEEE Tans. Sg. Poc., vo. 56, no. 0, pp , Oct [23] Y. C. Eda and H. Rauhut, Aveage case anayss of mutchanne spase ecovey usng convex eaxaton, submtted to IEEE Tans. Inf. Theoy. [24] Y. M. Lu and M. N. Do, Sampng sgnas fom a unon of subspaces, IEEE Sg. Poc. Mag., vo. 25, no. 2, pp. 4 47, Ma [25] T. Bumensath and M. E. Daves, Sampng theoems fo sgnas fom the unon of fnte-dmensona nea subspaces, IEEE Tans. Inf. Theoy, vo. 55, no. 4, pp , Ap [26] Y. C. Eda, Compessed sensng of anaog sgnas n shft-nvaant spaces, to appea n IEEE Tans. Sg. Poc. [27] M. Stonc, F. Pavaesh, and B. Hassb, On the econstucton of bock-spase sgnas wth an optma numbe of measuements, IEEE Tans. Sg. Poc., 2009, to appea. [28] R. G. Baanuk, V. Cevhe, M. F. Duate, and C. Hegde, Mode-based compessve sensng, IEEE Tans. Inf. Theoy, 2008, submtted. [29] D. Neede and J. A. Topp, CoSaMP: Iteatve sgna ecovey fom ncompete and naccuate sampes, Apped and Computatona Hamonc Anayss, vo. 26, pp , [30] T. Bumensath and M. E. Daves, Iteatve had theshodng fo compessed sensng, Jan. 2009, submtted to Eseve. [3] L. Peotta and P. Vandegheynst, Matchng pusut wth bock ncoheent dctonaes, IEEE Tans. Sg. Poc., vo. 55, no. 9, pp , [32] R. A. Hon and C. R. Johnson, Matx Anayss. New Yok, NY: Cambdge Pess, 985. [33] R. Gbonva and P. Vandegheynst, On the exponenta convegence of matchng pusuts n quas-ncoheent dctonaes, IEEE Tans. Infom. Theoy, vo. 52, no., pp , Jan [34] R. A. DeVoe and V. N. Temyakov, Some emaks on geedy agothms, Adv. n Comp. Math., vo. 5, pp , 996. [35] T. Stohme and R. W. Heath J., Gassmannan fames wth appcatons to codng and communcaton, Apped and Computatona Hamonc Anayss, vo. 4, pp ,