Sparse Representations in Unions of Bases

Transcription

1 330 IEEE TRANSACTIONS ON INFORMATION THEORY, VO 49, NO, DECEMBER 003 rates taen so high that further increasing them roduced no visibe changes in the figure As can be seen, the a;b obtained in that way turns from a we-behaved function for the vaues b =:5, :5 into a quite irreguary behaved one when b aroaches or 3 REFERENCES [] I Daubechies, Ten ectures on Waveets Phiadehia, PA: SIAM, 99 [] K Gröchenig, Foundations of Time-Frequency Anaysis Boston, MA: Birhäuser, 00 [3] A Ron and Z Shen, Wey-Heisenberg frames and Riesz bases in (R ), Due Math J, vo 89, no, 37 8, 997 [4] A J E M Janssen, Za transforms with few zeros and the tie, in Advances in Gabor Anaysis, H G Feichtinger and T Strohmer, Eds Boston, MA: Birhäuser, 003, 3 70 [5] Y I yubarsiĭ, Frames in the Bargmann sace of entire functions, in Entire and Subharmonic Functions Providence, RI: Amer Math Soc, 99, [6] K Sei, K Sei, and R Wastén, Density theorems for saming and interoation in the Bargmann-Foc sace I; II, J Reine Angew Math, vo 49, 9 06, 99 [7] A J E M Janssen and T Strohmer, Hyerboic secants yied Gabor frames, A Comut Harmon Ana, vo, 59 67, 00 [8] A J E M Janssen, On generating tight Gabor frames at critica density, J Fourier Ana A, vo 9, no, 75 4, 003 [9] H G Feichtinger and N Kaibinger, Varying the time-frequency attice of Gabor frames, Trans Amer Math Soc, to be ubished [0] A Cavaretta, W Dahmen, and C A Micchei, Stationary Subdivision, Mem Amer Math Soc, vo 53, no 453, 86, 99 [] A J E M Janssen, The duaity condition for Wey-Heisenberg frames, in Gabor Anaysis and Agorithms, H G Feichtinger and T Strohmer, Eds Boston, MA: Birhäuser, 998, [], From continuous to discrete Wey-Heisenberg frames through saming, J Fourier Ana A, vo 3, , 997 [3] N Kaibinger, Aroximation of the Fourier transform and the Gabor dua function from sames, rerint, 003 Sarse Reresentations in Unions of Bases Rémi Gribonva, Member, IEEE, and Morten Niesen Abstract The urose of this corresondence is to generaize a resut by Donoho and Huo and Ead and Brucstein on sarse reresentations of signas in a union of two orthonorma bases for We consider genera (redundant) dictionaries for, and derive sufficient conditions for having uniquesarsereresentations ofsignasinsuchdictionaries Theseciacase where the dictionary is given by the union of orthonorma bases for isstudiedinmoredetaiinarticuar,itisrovedthattheresutofdonoho and Huo, concerning the reacement of the otimization robem with a inear rogramming robem when searching for sarse reresentations, has an anaog for dictionaries that may be highy redundant Index Terms Dictionaries, Grassmannian frames, inear rogramming, mutuay incoherent bases, noninear aroximation, sarse reresentations I INTRODUCTION We consider vectors (aso referred to as signas) in H = N (res, H = N ) The goa is to find an efficient reresentation of a signa s Manuscrit received November 3, 00; revised August 4, 003 R Gribonva is with IRISA-INRIA, Camus de Beauieu, F-3504 Rennes Cedex, France (remigribonva@inriafr) M Niesen is with the Deartment of Mathematica Sciences, Aaborg University, DK-90 Aaborg East, Denmar (mniesen@mathaucd) Communicated by G Battai, Associate Editor At arge Digita Object Identifier 009/TIT H One we-nown way to do this is to tae an orthonorma basis 8= f; ; N g for H and use the Fourier coefficients fhs; ig N = to reresent s This aroach is sime and wors reasonaby we in many cases However, one can aso consider a more genera tye of exansion where the orthonorma basis is reaced by a so-caed dictionary for H Definition : A dictionary in H = N (res, H = N ) is a famiy of K N unit (coumn) vectors fg g that sans H We wi use the matrix notation D =[g; ;g K ] for a dictionary By a reresentation of s in D we mean a (coumn) vector = ( ) K (res, in K ) such that s = D We notice that when K>N, the vectors of D are no onger ineary indeendent and the reresentation of s is not unique The hoe is that among a ossibe reresentations of s there is a very sarse reresentation, ie, a reresentation with few nonzero coefficients The tradeoff is that we have to search a ossibe reresentations of s to find the sarse reresentations, and then determine whether there is a unique sarsest reresentation Foowing [] and [], we wi measure the sarsity of a reresentation s = D by two quantities: the `0 and the ` norm of, resectivey (the `0-norm simy counts the number of nonzero entries of a vector) This eads to the foowing two minimization robems to determine the sarsest reresentation of s: and minimize 0 subject to s = D () minimize subject to s = D: () It turns out that the otimization robem () is much easier to hande than () through the use of inear rogramming (P), so it is imortant to now the reationshi between the soution(s) of () and (), and to determine sufficient conditions for the two robems to have the same unique soution This robem has been studied in detai in [] and ater has been refined in [] in the secia case where the dictionary D is the union of two orthonorma bases In what foows, we generaize the resuts of [] and [] to arbitrary dictionaries The case where D is the union of orthonorma bases for H is studied in detai This eads to a natura generaization of the recent resuts from [] vaid for = In Section II, we rovide conditions for a soution of the robem minimize subject to s = D (3) to be indeed the unique soution, with 0 and an arbitrary dictionary D We ut a secia emhasis on sufficient conditions of the tye 0 <f(d) and rove a sufficient condition for f0; g with f (D) =( + =)= where :=max jhg ;g ij (4) 6= is the coherence of the dictionary The secia case where D is the union of bases is studied in Section III, eading to exicit sufficient conditions for =0with and for f0; g with f (D) = =+ f (D) = ( 0 ) 0 + ( 0 ) = =: Parae wor done indeendenty by Donoho and Ead [3] aso addresses the question of generaizing revious resuts to genera dictionaries Though there are some simiarities between this wor to the wor in [3], a somewhat different ersective on the robem is adoted and the roofs use different techniques /03$ IEEE

2 IEEE TRANSACTIONS ON INFORMATION THEORY, VO 49, NO, DECEMBER In the case =we simy recover the main resut from [], and for 6 we obtain a condition that is ess restrictive than the condition for arbitrary dictionaries In Section IV, we construct highy redundant dictionaries where the resuts of the resent corresondence give fairy reaxed conditions for () and () to have a unique soution II SPARSE ` REPRESENTATIONS, 0 Any vector s Hhas (ossiby many) reresentations s = D with coefficient vector K (res, K ) Definition : The suort of a coefficient vector =( ) (res, K )is S() :=f; 6=0g: (5) The erne of the dictionary wi ay a secia roe Ker (D) :=fx; Dx =0g (6) as we as the integer quantity (caed sar of the dictionary in [3]) Z(D) := min x 0 : (7) xker (D);x6=0 By refining ideas from [] we have the foowing emma emma : et D be a (ossiby redundant) dictionary and S f; ;Kg a set of indexes For 0 define P (S; D) := jx j S max xker (D);x6=0 jx j (8) where we use the convention 0 0 =0and x 0 =;x 6= 0 ) If P (S; D) < = then, for a such that S() S, is the unique soution to the robem (3) with s := D ) If P (S; D) == then, for a such that S() S, is a soution to the robem (3) with s := D 3) If P (S; D) > = there exists such that S() S and such that < and D = D Proof: The emma was used without being stated exicity in [] and [], in the secia case = and with D a union of two orthonorma bases The roof foows the same stes as in [] and [] Under the assumtion P (S; D) < = and S() S, what we need to rove is that for a x Ker(D), This is equivaent to showing =S jx j + j + x j > S j j : j + x j 0j j > 0: For 0, we have the quasi-triange inequaity ja+bj jaj + jbj, from which we can derive the inequaity j +yj 0jj 0jyj It is thus sufficient to rove that for a x Ker (D) or equivaenty =S S jx j 0 S jx j > 0 jx j < jx j : But this is exacty the assumtion P (S; D) < = To rove the resut for P (S; D) = = we coy the above ine of arguments and K simy reace strict inequaities with arge ones To rove the resut for P (S; D) > =, it is sufficient to tae some x Ker (D) so that S jx j > jx j = and to consider for S, := 0x, := 0 and for = S, := 0, := x Because x = 0 Ker (D) one easiy checs that D = D Obviousy and S() S j j = =S jx j < S jx j = j j emma wi be most usefu to oo for sufficient conditions on S that ensure uniqueness of the sarsest ` exansion, ie, conditions such that P (S; D) < = Of articuar interest are sufficient conditions that tae the form if card (S) <f(d); then P (S; D) < = (9) which corresond to uniqueness resuts of the form if 0 <f(d); then is the unique soution to (3): The foowing emma shows that such conditions are intimatey reated to the sar Z(D) We denote by dxe the smaest integer not smaer than x, ie, dxe 0 <xdxe emma : ) If 0 <Z(D)=, then is the unique soution to () In other words, (9) hods for =0with f (D) =Z(D)= ) If (9) hods true for some 0 with some f (D), then it hods true with the same constant f (D) for =0 3) If (9) hods true for = 0 with some f (D), then f (D) dz(d)=e Proof: For any S we observe that P 0 (S; D) card(s) max xker(d);x6=0 x 0 card(s) : Z The first statement immediatey foows The second statement is amost trivia Assuming that condition (9) hods true with and f, we now that when 0 <f, for a 6= such that D = D we have > Assume satisfies D = D and 0 0: then in articuar 0 <fso is aso the unique minimizer of the same ` robem, hence, = It foows that is indeed the unique minimizer of the `0 robem To concude, et us rove the third statement By definition, there exists x Ker (D) such that x 0 = Z We can sit its suort S(x) into two disjoint sets S and S of same cardinaity Z= =dz=e (if Z is even) or with card (S )=(Z 0 )= and card (S )=(Z +)= =dz=e (if Z is odd) Obviousy, P 0 (S ;D) =, hence (9) cannot hod true for =0with f>card (S )=dz=e There are two consequences of this emma The first one is that we need to estimate Z(D) The second is that if we are abe to rove that (9) hods for some 0 with some constant f (D), then it wi aso hod for =0with the same constant This fact wi be extensivey used to find sufficient conditions so that a soution to the ` robem aso soves uniquey the `0 robem In [] and [], the case of D = [B ;B ] was considered where B and B are two orthonorma matrices corresonding to orthonorma bases Donoho and Huo roved an uncertainty rincie

3 33 IEEE TRANSACTIONS ON INFORMATION THEORY, VO 49, NO, DECEMBER 003 Z([B ;B ]) + =M [, Theorem VII3] and obtained the sufficient condition (9) for f0; g with f ([B ;B ]) = ( + =): (0) Ead and Brucstein imroved the uncertainty rincie by getting [, Theorem ] Z([B ;B ]) =M Thus, they obtained the sufficient condition (9) for =0with the ess restrictive constant f ([B ;B ]) = =: () Eventuay, Ead and Brucstein used another technique to obtain condition (9) for =(and, thus, for =0) with f ([B ;B ]) = 0 = = 0:94=: () Feuer and Nemirovsy [4] recenty roved that the above constant is essentiay the best one to get condition (9) for =, in articuar it cannot be reaced with the ess restrictive constant () Next we show that the resut with the most restrictive of the constants, that is, (0), extends to the case of arbitrary dictionaries In the next section, we wi consider resuts for dictionaries buit by taing the union of orthonorma bases Theorem : For any dictionary, if 0 < ( + =) (3) then is the (unique) soution to both the `0 and the ` minimization robems Proof: As aready noticed, we wi just need to show that (9) hods for =with f := ( + =M )= Consider x Ker (D) For every we have III SPARSE REPRESENTATIONS IN UNIONS OF BASES We now switch to the secia case of D a union of orthonorma bases, ie, D =[B ; ;B ] where B is an orthonorma matrix, First, we concentrate on getting a resut of the tye (9) for = 0 This wi corresond to getting a sharer generaized uncertainty rincie by getting a ower bound on Z(D) emma 3: et D be a union of orthonorma bases et with x Consequenty N (res, = x = x x Ker (D) N ) and assume x 6= 0 Then +x 0 0 : (7) Z(D) + 0 : (8) Proof: Because x Ker (D), for every we have B x = 0 6= B x, hence, x = 0 6= B T B x Denoting X N the vector having as entries the absoute vaues of those of the origina vector x, we have for a x Ker (D) and X 6= X N (9) where N N is a coumn vector with a entries equa to one For each, summing over the nonzero coordinates of X we obtain X M X 0 6= X It foows that x g = 0 6= x g hence, taing the inner roduct of both hand sides with g, jx j 6= jx j : It foows that ( + M ) jx jm x : (4) Summing over S we get P (S; D) x card(s)m +M x,so hence ( + M X 0 ) X M X 0 Summing over we obtain x we get x M x 0 +M x 0 x : Mx +Mx X x from which P (S; D) as soon as card (S) < ( + =M )= card (S) +=M < = Note that the above ine of arguments can be modified sighty to rove that for arbitrary dictionaries we have the generaized uncertainty rincie Z(D) +=: (5) Notice that, as soon as the dictionary contains an orthonorma basis and an additiona unit vector, the vaue of M is at east = N To see that, et us assume, without oss of generaity, that the orthonorma basis corresonds to the first N vectors of D By N = jhg N+;g ij = g N+ = we see that max N = jhg N+;g ij =N, hence the inequaity = N: (6) = M x 0 +M x 0 : This is easiy rewritten g(m = x 0 ) 0 with g(y) := =( + y) and gives (7) By the convexity of g and the fact that M x 0 = M = x 0,wehaveg(Mx 0=) ( 0 )= hence M x 0 and (8) foows 0 Notice that for = the condition (7) can be rewritten x 0 x 0 =M as in [, Theorem ] There are exames of airs of bases with Z ==M, so the generaized uncertainty rincie (8) is shar for =For 3 it is an oen robem whether there exist exames of orthonorma bases for which MZ is arbitrariy cose to +=( 0 ) Using (8) together with emma we have the foowing coroary Coroary : et D be a union of orthonorma bases If 0 < + ( 0 ) then the unique soution to the `0 robem is (0)

4 IEEE TRANSACTIONS ON INFORMATION THEORY, VO 49, NO, DECEMBER For =, we find again the east restrictive condition () of Ead and Brucstein As we increase the number of bases whie eeing M constant (we wi see in Section IV that it is indeed ossibe to have u to = N +orthonorma bases with erfect searation M = = N, for N a ower of two), Condition (0) gets more and more restrictive It is ony natura that we have to ay a rice for increasing the redundancy of the dictionary For sma enough vaues of, (0) is ess restrictive than (3) For +=M, however, the bound in Coroary becomes more restrictive than the genera resut from Theorem, so the atter shoud be used in this case et us now consider the ` minimization robem with unions of orthonorma bases For airs of bases, the genera resut of Theorem was imroved in [] to get the ess restrictive sufficient condition 0 < ( 0 0:5)= The authors in [] indeed roved a stronger resut which can be stated as foows: if we denote = with N (res, N ) and K := 0, =;, then a sufficient condition to ensure P (S();D) < = is that M (D)K K + max(k ;K ) 0 < 0: () Next we generaize this resut to a union of bases Theorem : et D be a union of orthonorma bases Denote = with N (res, N ) Without oss of generaity, we can assume that the bases B have been numbered so that 0 0 If M 0 +M 0 < ( + M 0 ) () then is the (unique) soution to the ` minimization robem Proof: We foow [, roof of Theorem 3] and start simiary to the roof of emma 3 Consider with x hence, N (res, x = x x Ker (D) N ) For every we have B x = 0 x = 0 6= 6= B x B T B x : Denoting X N the vector having as entries the absoute vaues of those of the origina vector x, we have for a x Ker (D) and X 6= NN X (3) where NN is an N -by-n matrix with a entries equa to one By definition, X aso satisfies X 0 In addition, for a x Ker (D) with x = = x =,wehave = T NX = (4) and = T S( ) X = S() jx j ; (5) where N N is a vector with a entries equa to one and S N is a vector with ones on the index set S and zeroes esewhere Thus, it is sufficient to show that under the condition () and the constraints (3) (4) and X 0 we have max X ;;X = T S( ) X < : et us roceed as in []: by reacing the equaity constraints (4) with two inequaities, we now have a cassica inear rogramming robem, which can be ut into canonica form with min rima := min CT Z subject to AZ B; Z 0 Z = C T =[0 S( ) ; ; 0 S( ) ] and A = X X 0I N M NN M NN M NN 0I N M NN 0I N M NN M NN M NN 0I N T N T N 0 T N 0 T N B T =[0: T N; ; 0: T N; ; 0]: What we need to rove is min rima > 0 The dua inear rogramming robem is max := max dua BT U subject to A T U C; U 0 and we now [5] that max dua = min rima, so the desired resut wi be obtained if we can rove that there exists some U 0 that satisfies A T U C and B T U> 0= We wi oo for such a U in a arametric form U T := [a T S( ); ;a T S( ) ;b;c] with a ;b;c 0 Noticing that B T U = b 0 c, the goa wi be to choose a ;b;cso that b 0 c>0= and A T U C Straightforward comutations show that the condition A T U C is equivaent to the inequaities for (b 0 c) N + M NN 6= a T S( ) +(0 a ) T S( ) 0: By the equaity NN T S = card (S) T N this becomes b0c+m a 0 T N 0Ma 0 T N +(0a ) T S( ) 0 where we used the fact that 0 := card (S( )) Denoting y + = max(y; 0) and y 0 = min(y; 0), the ositive and negative arts of

5 334 IEEE TRANSACTIONS ON INFORMATION THEORY, VO 49, NO, DECEMBER 003 y, the constraint is eventuay exressed, for such that 0 6=0,as b 0 c + M a 0 Ma 0 +(a 0 ) 0 : (6) The constraint for a such that 0 =0(if there is any) is b 0 c + M a 0 0: Now that the constraints have been estabished, et us buid a, b, and c We define a = when 0 = 0 and use a threshod arameter to define a () := = 0 when 0 < and a () := ( +M)=( + M 0 ) when 0 et us aso define 6() := a () 0 One can chec case-by-case that for a, when 0 6=0, the constraint (6) becomes b 0 c + M 6() M: (7) If there is some vaue of for which 0 =0, the associated constraint is stronger and becomes b 0 c + M 6() 0: (8) Obviousy, (7) (res, (8)) can aways be satisfied by taing b = g() + and c = 0g() 0 with g() :=M ( 0 6()) (res, g() := 0M 6()) For U = U () buit in this arametric form, U () satisfies the constraints U 0 and AU C, and we have B T U () =g() Thus, the robem is now whether max g() > 0=: (9) et us dea first with the case where 0 6= 0for a It is easy to chec that g( 0)=0 M 0 +M 0 ( + M 0) so the theorem is roved Sime (but tedious) comutations woud show that indeed max g() = max(g(0);g( 0)), and that when the maximum is g(0) it does not satisfy the constraint (9) So, in this case, the sufficient condition g( 0) > 0= is, in a sense, otima for the tye of argument we have resented In the case where 0 =0for some (ie, 0 =0), we notice that 6() is a iecewise-inear increasing function, so max g() = g(0) Because 0 =0we concude by estimating g(0) as 0 = M 0 +M 0 = 0 M 0 +M 0 ( + M 0 ): In the case of =bases, () is exacty the condition () roved in [] where it is roved that a simer sufficient condition is 0 ( 0 =)=M The genera condition () is sime to chec for any given However, in order to benefit from emma and get a sufficient condition for to simutaneousy minimize the `0 and the ` robems, et us oo for a sufficient condition 0 <f(d) Coroary : For a dictionary that is the union of orthonorma bases, if 0 < 0 + ( 0 ) (30) then is the (unique) soution to both the `0 and the ` minimization robems With the notations of Theorem, the same concusion is reached if the above inequaity is arge but there exists an index such that ( + M 0 )=( + M 0 ) 6= Proof: Denoting y := M 0 and y =(y ) =, condition () can be rewritten y g(y) := 0 +y = 0: +y For any c>0consider the set H c := fy j y 0; et us comute G(c) := su yh g(y): = y = cg and Using agrange mutiiers, we now that any y? that corresonds to an extremum of g under the constraint y = c wi satisfy the equaities =, For =, this becomes ( + y )? whie for, this corresonds to ( + y? ) 0 = ooing at the second artia derivatives of g we easiy chec that a extrema are indeed maxima The ony maximum that satisfies the additiona constraint y? 0 is given by y? = 0= = 0 y? = 0= 0 ; ; and we can chec that y? y? = = y? et us exress as a function of c By using the constraint we get c = y? = 0= = +( 0 ) 0 and it foows that ( + y? ) = ( + y? )= 0= =( + c)(= +( 0 )): Then we get by direct comutations g(y? )= ( 0 )( c + ( + c) 0 ) 0 = so the condition G(c) =g(y? ) 0 is equivaent to ( 0 )( c + 0 ) =, that is, c 0 + ( 0 ) : To concude, et us consider y := (M 0 ) = and assume the strict inequaity (30) is satisfied Then by the above comutations g(y) G(M 0) < 0 hence the strict inequaity () is satisfied If (30) is satisfied as a arge inequaity and there exists some index such that ( + M 0 )=( + M 0 ) 6= (this is generay the case!), then y 6= y? so we have g(y) < G(M 0) 0 and we get the same resut In both cases, we reach the concusion using Theorem The sufficient conditions in Coroary and are very simiar, but the atter is a bit more restrictive, with a ga = 0 ( 0 ) 0:086 in the constant in front of =M Tabe I ists the vaues of the constant in front of = in Coroary For =, we recover the constant 0 = from [, Theorem 3] For arger vaues of, we get more restrictive constraints, ie, with smaer constants Indeed, for 7, one can chec that for any vaue of M, 0 + =M < ( + =M )= ( 0 ) so the genera sufficient condition in Theorem is ess restrictive than the seciaized one in Coroary For 6 and sma vaues of M (ie, because of (6), in arge dimension N ), the condition in Coroary is ess restrictive than that of Theorem, and we get an imroved resut For arge vaues of M and 6, one has to chec on a case-by-case basis which resut is stronger

6 IEEE TRANSACTIONS ON INFORMATION THEORY, VO 49, NO, DECEMBER TABE I NUMERICA VAUES OF THE CONSTANT + IN COROARY FOR SMA VAUES OF IV HIGHY REDUNDANT DICTIONARIES et us show how to ay the extended resut (Theorem ) to highy redundant dictionaries It is erhas not obvious that one can have a arge number of orthonorma bases in N with a sma coherence factor, but this is ossibe (for certain vaues of N ), and we wi use the foowing theorem to buid exames of such dictionaries We refer to [6] and [7] for a roof of Theorem 3 Theorem 3: et N = j+, j 0, and consider H = N There exists a dictionary D in H consisting of the union of = j = N= orthonorma bases for H, such that for any air u; v of distinct vectors beonging to D: jhu; vijf0;n 0= g For N = j, j 0, and H = N, one can find a dictionary D in H consisting of the union of = N +orthonorma bases for H, again with the erfect searation roerty that for any air u; v of distinct vectors beonging to D: jhu; vijf0;n 0= g The dictionaries from Theorem 3 are caed Grassmannian dictionaries due to the fact that their construction is cosey reated to the Grassmannian acing robem, see [6] and [7] for detais For N = j+, Theorem 3 tes us that we can tae a dictionary D consisting of the union of N +orthonorma bases in N, that is, D contains the arge number N (N +)=of eements, but we sti have coherence =N 0= We can extract from such a dictionary many exames of unions D of bases ( N +)with the same coherence For each exame, we can ay Theorem or Coroary to concude that is the unique sarsest `0 and ` reresentation of s := D as soon as 0 < max N=+=; N V CONCUSION 0 + ( 0 ) We have studied sarse reresentations of signas using an arbitrary dictionary D in H = N (res, H = N ) For any dictionary D, f0; g, and a given signa s we rove that, with s := D, is the unique soution to the otimization robem : We aso roved an uncertainty rincie for unions of orthonorma bases for H and derived a sighty ess restrictive sufficient condition 0 < + ( 0 ) to ensure that the (most difficut) `0 minimization robem admits as a unique soution The roofs of the above resuts are based on the techniques introduced in [] and [] so the main contribution of the resent corresondence is to oint out that we are not restricted to dictionaries that are the union of two orthonorma bases We can consider more genera dictionaries and sti enjoy a the ractica benefits from restating the robem as a inear rogramming minimization robem and get the `0 minimizer for free in cases where the outut from the P agorithm has few nonzero entries Finay, we shoud note that many natura and usefu redundant dictionaries such as the discrete Gabor dictionary, unions of bi-orthogona discrete waveet dictionaries, etc, cannot be written as a union of two orthonorma bases and thus were not covered by the resuts in [] and [] REFERENCES [] D Donoho and X Huo, Uncertainty rincies and idea atomic decomositions, IEEE Trans Inform Theory, vo 47, , Nov 00 [] M Ead and A Brucstein, A generaized uncertainty rincie and sarse reresentations in airs of bases, IEEE Trans Inform Theory, vo 48, , Set 00 [3] D Donoho and M Ead, Otimay sarse reresentation in genera (nonorthogona) dictionaries via minimization, Proc Nat Acad Sci, vo 00, no 5, 97 0, Mar 003 [4] A Feuer and A Nemirovsy, On sarse reresentations in airs of bases, IEEE Trans Inform Theory, vo 49, , June 003 [5] D Bertseas, Non-inear Programming, nd ed Bemont, MA: Athena Scientific, 995 [6] A R Caderban, P J Cameron, W M Kantor, and J J Seide, -Kerdoc codes, orthogona sreads, and extrema eucidean ine-sets, Proc ondon Math Soc (3), vo 75, no, , 997 [7] T Strohmer and R Heath, Grassmannian frames with aications to coding and communications, A Com Harm Ana, vo 4, no 3, 57 75, 003 minimize subject to D = s (3) rovided that 0 < ( + =) So this condition on 0 ensures that the more difficut `0 minimization robem has exacty the same unique soution as the ` robem This is of ractica imortance since (3) can be restated and soved as a inear rogramming minimization robem, thus giving us a feasibe way to actuay comute the minimizer When D is a union of orthonorma bases for H, wehave derived the sufficient condition 0 < 0 + ( 0 ) for with s := D, to be the simutaneous unique minimizer in (3) for f0; g When 6, this condition is generay ess restrictive (and the resut thus covers more cases) than the estimate for arbitrary dictionaries For =, we simy recover the main resut from []