Research Article Support Recovery of Greedy Block Coordinate Descent Using the Near Orthogonality Property

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1 Hindawi Mathematical Problems in Engineering Volume 17, Article ID , 7 pages Research Article Support Recovery of Greedy Block Coordinate Descent Using the Near Orthogonality Property Haifeng i College of Mathematics and Information Science, Henan Normal University, Xinxiang 4537, China Correspondence should be addressed to Haifeng i; lihaifengxx@16com Received 3 November 16; Accepted 15 March 17; Published 7 April 17 Academic Editor: Bogdan Dumitrescu Copyright 17 Haifeng i This is an open access article distributed under the Creative Commons Attribution icense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited In this paper, using the near orthogonal property, we analyze the performance of greedy block coordinate descent GBCD) algorithm when both the measurements and the measurement matrix are perturbed by some errors An improved sufficient condition is presented to guarantee that the support of the sparse matrix is recovered exactly A counterexample is provided to show that GBCD fails It improves the existing result By experiments, we also point out that GBCD is robust under these perturbations 1 Introduction Greedy block coordinate descent GBCD) algorithm was presented by [1] for direction of arrival DOA) estimation In the work of [1], the DOA estimation is treated as the multiple measurement vectors MMV) model that recovers a common support shared by multiple unknown vectors from multiple measurements The authors provided a sufficient condition, based on mutual coherence, to guarantee that GBCD exactly recover the nonzero supports with noiseless measurements Recently, the work of [] discussed the following method: min X X,1 st Ŷ = AX + N,  = A + E, with inputs Ŷ R m and  R m N N denotes the measurement noise and E denotes the system perturbation The perturbations E and N are quantified with the following relative bounds: N F ε, Y F E K) ε A K) A, 1) ) where A K) and Y F are nonzero Here, A K) denotes the largest spectral norm taken over all K-column submatrices of A Throughout the paper, we are only interested in the case where ε and ε A are far less than 1 In 1), X is a K-group sparse matrix; that is, it has no more than K nonzero rows, and X,1 = N xi, x i is the ith row of X It is assumed that all columns of  arenormalizedtobeofunit-norm [3] Both Y = AX and A are totally perturbed in 1) This case can be found in source separation [4], radar [5], remote sensing [6], and countless other problems In addition, the total perturbations have also been discussed in [7 9] One of the most commonly known conditions is the restricted isometry property RIP) A matrix A satisfies RIP of the order K if there exists a constant δ, 1) such that 1 δ) h Ah 1+δ) h 3) for all K-sparse vector h In particular, the minimum of all constants δ satisfying 3) is called the restricted isometry constant RIC) δ K There are many papers [8, 1 14] discussing the sufficient condition for orthogonal matching pursuit OMP) that is one of the widely greedy algorithms for sparse recovery In [3], using the near orthogonality property, the authors improved the sufficient condition of OMP As cited in [3], the near orthogonality property can further develop the orthogonality characterization of columns in A; it will play a fundamental role in the study of the signal reconstruction performance in

2 Mathematical Problems in Engineering compressed sensing In the noiseless case, the work of [15] analyzed the performance of GBCD using near orthogonality property and improved the results in [] In this paper, under the total perturbations, we use near orthogonality property to improve the theoretical guarantee for the GBCD algorithm In [], the authors stated that δ K+1 < 1/ K +1)is a sufficient condition for GBCD We improve this condition to δ K+1 < 4K + 1 1)/KWealsopresent a counterexample to show that GBCD fails The example is superior to that in [] Under the total perturbations, the robustness of GBCD is shown by experiments Nowwegivesomenotationsthatwillbeusedinthis paper a i denotes the ith column of a matrix A A denotes the transpose of A I M denotes an M Midentity matrix The symbol vec denotes the vectorization operator by stacking the columns of a matrix one underneath the other The cardinality of a finite set Γ is denoted by Γ etω fl {1,,,N} Γ c = Ω \ Γ = {i i Ω, and i Γ}The support of X is denoted by suppx) suppx) ={i x i = }) A K) denotes the largest spectral norm taken over all Kcolumn submatrices of Aet A, denote the imum l norm of the rows of AWewriteA Γ for the column submatrix of A whoseindicesarelistedinsetofγand X Γ for the row submatrix of X whoseindicesarelistedinthesetγ e i R N denotes the ith unit standardvector Problem Formulation Analogous to [1], 1) can be rewritten as 1 min X Ŷ ÂX +λ X F,1 4) Assume that Γ fl suppx) Obviously, Γ = K The objective function in 4) can be written as F X) =GX) +HX), 5) where GX) = 1/) Ŷ ÂX F = 1/) vecŷ) I  vecx) F with denoting the Kronecker product and HX) = λ X,1 = λ N xi Combiningthequadratic approximation for GX) and standard BCD algorithm, the solution to the ith subproblem can be given by a softthresholding operator The authors in [1] only update the block that yields the greatest descent distance Now, we list GBCD algorithm Algorithm 1) Suppose that A satisfies the Kth order RIC δ K, 1) Recall that X has no more than K nonzero rows According to the fact X F = x i,wecanobtain 1 δ K ) X F AX F 1+δ K) X F 6) from 3) Combining emma 4 in [3] and 6), we have A Γ AX F = A Γ Ax i 7) 1 δ K ) Ax i ) 8) =1 δ K ) Ax i 9) =1 δ K ) AX F 1) emma 1 near orthogonality property, see [3]) et u and v be two orthogonal sparse vectors with supports T u and T V fulfilling T u T V KSupposethatA satisfies RIP of order K with RIC δ K Thenwehave cos Au, Ak) K, 11) where Au, Ak) denotes the angle between Au and Ak emma see [3]) Under the same assumptions as in emma 1, we have Au, Ak δ Au Ak 1) emma 3 For finite sets Γ and Γ, letsuppx) = Γ and supp X) = Γ Here, Γ Γ =,and Γ Γ K IfA satisfies the RIP condition 3) with δ K,1),thenwehave AX, A X F δ K AX F A X 13) F Proof Note that the Frobenius norm of A is derived from the Frobenius inner product AX, A X F = Ax i, A x i δ K =δ K δ K 14) Ax i A x i ) 15) Ax i A x i ) Ax i A x i 16) 17) =δ K AX F A X, 18) F where15)and17)followfromemmaandcauchy- Schwarz inequality, respectively 3 RIP Based Recovery Condition In this section, we firstly present the upper bound of the noise matrix EX + N and provide the recovery condition for GBCD emma 4 see []) Suppose that  satisfies the Kth order RIC δ K, 5)Thenwehave ε A EX + N F < +ε) Ŷ F 1/ 3 1+1/ 3) ε A 1 ε 19)

3 Mathematical Problems in Engineering 3 Input: Â, Ŷ, X) =, n = 1, λ >, β > 1) Repeat until stopping criterion is met ) for :N 3) 4) p i n 1) = x i n 1) βâ i ÂXn 1) Ŷ) x i n) = p i n 1)/ p i n 1) ), p i n 1) λβ) 5) compi) = x i n) x i n 1) 6) end for 7) Choose the index i such that compi )=comp) 8) Xn) [x 1 n 1); ; x i 1 n 1); 9) x i n); x i+1 n 1); ; x N n 1)] 1) n n+1 11) End Repeat Algorithm 1: GBCD: greedy block coordinate descent algorithm [1] According to steps 7) and8) of Algorithm 1, at the nth iteration, GBCD can obtain a correct index if i Γ Xi n) X i n 1) > j Γ c Xj n) X j n 1) ) Theorem 5 Consider model 4) et t = min i Γ x i Ifthe matrix  satisfies RIP of order K+1with δ K+1 < 4K + 1 1, 1) K t > 1+ δ K+1 + K) ε 1 δ K+1 1 δ K+1 K δ K+1, ) where ε =ε A /1/ 3 1+1/ 3)ε A )+ε) Ŷ F /1 ε)),then GBCD can exactly recover the support set Γ Proof Consider n = 1The initial value is X) = In order to guarantee that GBCD selects a correct index i Γ, combining step 4) of Algorithm 1 and ), we should verify the following inequality: = i Γ = j Γ c i Γ xi 1) x i ) 3) p i ), pi ) pi ) λβ) 4) > j Γ c xj 1) x j ) 5) p j ), pj ) pj ) λβ) 6) If p j ) λβ j Γ c ), the right-hand-side is Then inequality 6) holds Thus, we only consider p j ) λβ> UsingRemark1 in [], inequality 6) is true when i Γ pi ) > ) j Γ c pj 7) Now, it is sufficient to verify 7) et us construct an upper bound for j Γ c p j ) Bystep3) of Algorithm 1, we have j Γ c pj ) 8) = j Γ c xj ) βâ j ÂX ) Ŷ) 9) = j Γ c βâ jŷ 3) = βâ Γ c ÂX EX + N), 31) j Γ c βâ jâx +β Â Γ c, EX + N F 3) =β j Γ c Âe j, ÂX F +β Â Γ c, EX + N F 33) β δ K+1 ÂX F +βε, 34) where 3) is from the property of norm and 34) follows from each column of  which is of unit-norm, emmas 3 and 4 To prove 7), we only need to prove i Γ pi ) >β δ K+1 ÂX +βε F 35) We then go on to show by contradiction that 35) is true For alli Γ, assume that Thenwehave pi ) β δ K+1 ÂX F +βε 36) PΓ ) = F pi ) i T β K δ K+1 ÂX F +ε ) 37)

4 4 Mathematical Problems in Engineering Using the triangle inequality, we can get PΓ ) F = XΓ ) βâ Γ ÂX ) Ŷ) F = βâ ΓŶ F β  ΓÂX F β Â Γ EX + N) F 38) 39) β 1 δ K+1 ÂX F 1+ δ K+1 ε ), 4) where4)isfrom1)andthepropertyofnorm After straightforward manipulations, we have PΓ ) F β K δ K+1 ÂX F +ε ) +β 1 δ K+1 K δ K+1 ) ÂX F 41) It is sufficient to prove that 48) holds Note that suppxn 1)) Γ;wehave Now, we only need to prove j Γ c â j ÂX n 1) Ŷ) 49) δ K+1  X n 1) X) F +ε 5) i Γ â i ÂX n 1) Ŷ) 51) > δ K+1  X n 1) X) F +ε 5) We then show that 5) is true by contradiction For all i Γ, assume that â i ÂX n 1) Ŷ) δ K+1  X n 1) X) F +ε Using the definition of Frobenius norm, we have 53) β 1+ δ K+1 + K) ε β K δ K+1 ÂX F +ε ) Â Γ ÂX n 1) Ŷ) F = i Γ â i ÂX n 1) Ŷ) 54) +β 1 δ K+1 K δ K+1 ) 1 δ K+1 X F β 1+ δ K+1 + K) ε 4) >β K δ K+1 ÂX F +ε )+K 1) 1+ δ K+1 ε +K K) ε 43) β K δ K+1 ÂX F +ε ), 44) where 41) follows from 1) and 43) follows from X F Kt and ) Obviously, 44) contradicts 37), so this fact guarantees 7) Assume that GBCD always picks up indices from the support Γ for n kk 1is an integer) Consider n=k+1in order to prove that GBCD can choose a correct index i Γ, analogous to [], inequality 46) should be verified i Γ pi n 1) x i n 1) 45) > j Γ c pj n 1) x j n 1) 46) Combining step 3) of Algorithm 1 with 46) yields i Γ â i ÂX n 1) Ŷ) 47) > j Γ c â j ÂX n 1) Ŷ) 48) K δ K+1  X n 1) X) F +ε ) 55) Combining Xn 1) X F t,1),and),wehave Â Γ ÂX n 1) Ŷ) F 56) 1 δ K+1  X n 1) X) F 1+ δ K+1 ε 57) = K δ K+1  X n 1) X) F +ε ) + 1 δ K+1 K δ K+1 )  X n 1) X) F 1+ δ K+1 + K) ε 58) > K δ K+1  X n 1) X) F +ε ), 59) where 59) follows from 1 δ K+1 K δ K+1 )  X n 1) X) F 6) > 1+ δ K+1 + K) ε 61) This contradicts 53) Thus, 48) is true Remark 6 The weaker the RIC bound is, the less required number of measurements we need, and the improved RIC results can be used in many CS-based applications [16] In theworkof[],theauthorsprovidedthattheconditionfor GBCD is δ K+1 <1/ K+1) Obviously, it is smaller than the bound 4K+1 1)/Kin 1)

5 Mathematical Problems in Engineering 5 4 The Counterexample Consider the measurements Ŷ = E) X + N = ÂX EX + N 6) In this section, giving a matrix Â, whosericisaslight relaxation of 1/ K+1, we will verify that GBCD can fail to recoverthesupportofsparsematrixfrom6) et t t t t X = ) t t ) K+1) 5 E = ) N = 4 t 4 t ε ε ) ) K+1) K+1) ) K+1),,, 63) where suppx) = {1,,,K} = Γ, Γ c ={K+1}and t /ε > 1+1/Kthe value of ε is far less than 1; this is reasonable) The matrix  is constructed as where a s a s a s  = ) a s 1 s= δ K, a= 1 δ ) K+1) K+1), 64) 65) Set The eigenvalues {λ i } K+1 δ= 1 ε/t K+1 66) λ i =1 δ, λ K =1 δ, λ K+1 =1+δ of   are 1 i K 1, 67) Thus, the RIC of  is δ K+1 Â) =δ Recall that condition 7) is the criterion of recovery for GBCD Note that p i ) = βâ iŷ Onecanobtain i Γ â iŷ = t a, t a =a ) t 68) On the other hand, we have j Γ c â jŷ = t Kas + ε, t Kas + ε) =Kast +ε Itcanbederivedthat 69) Kast +ε t a =t aks a )+ε 7) =t 1 δ Kδ 1+δ )+ε 71) >, 7) where 71) and 7) follow from 65) and 66) It is obviously in contradiction to 7) Thus, GBCD fails to recover support Γ Remark 7 Intheworkof[],theauthorspresentedamatrix A whose RIC is δ K+1 A) = 1/ K ε / Kt K+ε ) They showed that the GBCD algorithm fails when using A as measurement matrix After a simple calculation, we can get 1 ε/t K+1 < 1 K ε Kt K+ε ) 73) Thus, our result improves this existing result 5 Experimental Results In this section, under the total perturbations, we test the performance of the GBCD algorithm for solving the DOA estimation problem Consider K narrowband far-field point source signals impinging on an m-element uniform linear array The steering vector of the matrix A is a i =[1 e jπ cos θ i 1 e jm 1)π cos θ i 1 ], 74) where 1 i N is the number of snapshots

6 6 Mathematical Problems in Engineering RMSE SNR1 Figure 1: For SNR = 1 The RMSE of GBCD versus input SNR1 RMSE SNR Figure : For SNR1 = RMSE of GBCD versus input SNR Using the sparse optimization approach in [1], the DOA estimation problem can be rewritten as model 1) Then the aim is hence to find out which row of the matrix X is nonzero, thatis,thesupportofthematrixx Analogous to the simulation of [1], we have the following assumptions: i) The number of the array elements is m=11 ii) The number of snapshots is = iii) The grid spacing is 1 from to 18 ThenN = 181 iv) Five K =5) uncorrelated signals impinge from θ l1 = 3, θ l =8, θ l3 =1, θ l4 =1,andθ l5 =145 v) Both the signals and the noise are white and follow Gaussian distributions The power of nonzero entries of X is σ, and the power of each entry of N is σ N vi) Use the following SNR1 and SNR to measure noises E and N,respectively: SNR1 =1log 1 σ σn ) 75) SNR = A F E F 76) Define the root mean square error RMSE) of 5 Monte Carlo trials as the performance index: K θ RMSE = 5 lk i) θ lk ), 77) 5K k=1 where θ lk i) istheestimateofθ lk at the ith trial Figure 1, fixing matrix E, describes the performance of GBCD The results show that RMSE decreases as SNR1 increases Figure, fixing matrix N, describes the performance of GBCD The results show that RMSE decreases as SNR increases Thus, the performance of GBCD still is robust under the total perturbations 6 Conclusion In this paper, using the near orthogonality property, we provide a recovery condition for GBCD under the total perturbations A counterexample is presented to show that GBCD fails By experiments, we point out that GBCD is robust under the total perturbations Conflicts of Interest The author declares that there are no conflicts of interest regarding the publication of this paper Acknowledgments This work was supported by National Natural Science Foundation of China nos , , , and U14463), the Scientific Research Foundation for PhD of Henan Normal University no qd1414), and the Key Scientific Research Project of Colleges and Universities in Henan Province no 17A118) References [1] X Wei, Y Yuan, and Q ing, DOA estimation using a greedy block coordinate descent algorithm, IEEE Transactions on Signal Processing,vol6,no1,pp ,1 []Hi,YFu,RHu,andRRong, Perturbationanalysisof greedy block coordinate descent under RIP, IEEE Signal Processing etters,vol1,no5,pp518 5,14 [3] -H Chang and J-Y Wu, An improved RIP-based performance guarantee for sparse signal recovery via orthogonal matching pursuit, IEEE Transactions on Information Theory, vol 6, no 9, pp , 14 [4] T Blumensath and M Davies, Compressed sensing and source separation, in Proceedings of the International Conference on Independent Component Analysis and Signal Separation ICA 7), pp , 7 [5] M A Herman and T Strohmer, High-resolution radar via compressed sensing, IEEE Transactions on Signal Processing, vol 57, no 6, pp 75 84, 9

7 Mathematical Problems in Engineering 7 [6] ACFannjiang,TStrohmer,andPYan, Compressedremote sensing of sparse objects, SIAM Journal on Imaging Sciences, vol 3, no 3, pp , 1 [7] MAHermanandTStrohmer, Generaldeviants:ananalysis of perturbations in compressed sensing, IEEE Journal on Selected Topics in Signal Processing, vol4,no,pp34 349, 1 [8] JDing,Chen,andYGu, Perturbationanalysisoforthogonal matching pursuit, IEEE Transactions on Signal Processing, vol 61, no, pp , 13 [9] B i, Y Shen, Z Wu, and J i, Sufficient conditions for generalized Orthogonal Matching Pursuit in noisy case, Signal Processing,vol18,pp111 13,15 [1] T Zhang, Sparse recovery with orthogonal matching pursuit under RIP, IEEE Transactions on Information Theory, vol 57, no 9, pp , 11 [11] J Wen, X Zhu, and D i, Improved bounds on restricted isometry constant for orthogonal matching pursuit, Electronics etters,vol49,no3,pp ,13 [1] RWu,WHuang,andD-RChen, Theexactsupportrecovery of sparse signals with noise via orthogonal matching pursuit, IEEE Signal Processing etters,vol,no4,pp43 46,13 [13] Q Mo and Y Shen, A remark on the restricted isometry property in orthogonal matching pursuit, IEEE Transactions on Information Theory,vol58,no6,pp ,1 [14] J Wang and B Shim, On the recovery limit of sparse signals using orthogonal matching pursuit, IEEE Transactions on Signal Processing, vol 6, no 9, pp , 1 [15] Hi,YMa,Wiu,andYFu, Improvedanalysisofgreedy block coordinate descent under RIP, Electronics etters, vol 51, no 6, pp , 15 [16] CBSong,STXia,andXJiu, Improvedanalysisforsubspace pursuit algorithm in terms of restricted isometry constant, IEEE Signal Processing etters, vol 1, no 11, pp , 14

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