Novel Algorithm for Sparse Solutions to Linear Inverse. Problems with Multiple Measurements

Transcription

1 Novel Algorithm for Sarse Solutions to Linear Inverse Problems with Multile Measurements Lianlin Li, Fang Li Institute of Electronics, Chinese Acaemy of Sciences, Beijing, China ABSRAC: In this reort, a novel efficient algorithm for recovery of jointly sarse signals (sarse matrix) from multile incomlete measurements has been resente, in articular, the NESA-base MMV otimization metho. In a nutshell, the jointly sarse recovery is obviously suerior to alying stanar sarse reconstruction methos to each channel iniviually. Moreover several efforts have been mae to imrove the NESA-base MMV algorithm, in articular, (1) the NESA-base MMV algorithm for artially nown suort to greatly imrove the convergence rate, () the etection of artial (or all) locations of unnown jointly sarse signals by using so-calle MUSIC algorithm; (3) the iterative NESA-base algorithm by combing har thresholing technique to ecrease the numbers of measurements. It has been shown that by using roose aroach one can recover the unnown sarse matrix X with Sar ( A) -sarsity from ( ) Sar A measurements, reicte in Ref. [1], where the measurement matrix enote by A satisfies the so-calle restricte isometry roerty (RIP). Uner a very mil conition on the sarsity of X an characteristics of the A, the iterative har threshol (IH)-base MMV metho has been shown to be also a very goo caniate. INDEX ERMS: comressive sensing, SMV (single measurement vector), MMV (multile measurement vector), Nesterov s metho, iterative har threshol algorithm, MUSIC, restricte isometry roerty (RIP) I. INRODUCION Recovery of sarse signals from a small number of measurements is a

2 funamental roblem in many ractical alications such as meical imaging, seismic exloration, communication, image enoising, analog-to-igital conversion, an so on. he well-nown comresse sensing, eveloe by Canes, ao an Donoho et al, stuies information acquisition methos as well as efficient comutational algorithms. By exloiting colorful results eveloe within the framewor of comressive sensing, we can reconstruct a sarse vector x by solving the highly uneretermine linear equations y = Ax uner minimal 1 l -norm constraint only if the measurement matrix A satisfies some roerties such as restricte isometry roerty (RIP), null-sace roerty (NSP), an so on. hough etermining the sarest vector x consistent with the ata y = Ax is generally an NP-har roblem, many subotimal algorithms have been formulate to attac this roblem, for examle, greey algorithm, basis ursuit (BP), Bayesian algorithm, an so on. he single measurement sarse solution roblem has been extensively stuie in the ast. In many ractical alications such as ynamic meical imaging, neromagnetic inverse roblem, beam forming, electromagnetic inverse source, communication, an so on, the recovery of jointly sarse signal or MMV roblem, the variation of the comressive sensing or sarse linear inverse roblem, is an imortant toic; in articular, to eal with the comutation of sarse solution when there are multile measurement vectors (MMV) an the solutions are assume to have a common sarsity rofile or jointly sarse. he most wiely stuie aroaches to the MMV roblem are base on solving the convex otimization roblem min X X, subject to Bn L= An NXN L (1) q, where the mixe l q, norm of X is efine as X = X q, = q q q with X X ( 1,: ), X (,: ),, X ( N,: ), (,:) X j is the jth row of X. U to now, many efforts have mae to attac this roblem. Cotter et al. consiere the minimization roblem of []

3 min X B = A X. X, subject to n L n N N L Chen an Huo consiere the uniqueness uner = 0 via the sar of measurement matrix A an equivalence between the minimization roblem with = 1an = 0 [1]. Further, the orthogonal matching ursuit (OMP) algorithm for MMV has also been eveloe [1]. ro ealt with (1) for = an q =. Mishali an Elar roose the ReMBo algorithm which reuces MMV to a series of SMV roblems. Elar an Rauhut roose the OMP algorithm with har threshol technique an analyze the average case for jointly sarse signal recovery [4]. Berg an Frielaner stuie erformance of 1,1 l an 1, l for ifferent structure of sarse X [5]. In this resentation, we consier in eth the extension of a class of algorithm NESA algorithm to the multile measurement vectors available, an solutions with a common sarsity structure must be comute, esecially, NESA-base MMV algorithm. Insire by recent breathroughs in the eveloment of novel first-orer methos in convex otimization, the cost functions aroriate to NESA-base MMV are eveloe, an algorithms are erive base on their minimization. Further several aroaches to imrove the NESA-base MMV algorithm to ecrease the number of measurements an increase the convergence rate have been roose. his reort emonstrates that this aroach is ieally suite for solving large-scale MMV reconstruction roblems. II. ALGORIHMS In this section, the basic iea of NESA-base MMV algorithm has been rovie; moreover, several aroaches to imrove it have been iscusse. We will refer the reaer to [7] for etaile iscussions about roose aroaches. II.1 NESA-base MMV Algorithm Similar one by Nesterov s metho for single-measurement roblem [9] [10], the NESA-base MMV algorithm minimize the smooth convex function f on the convex

4 setq, ( ) min f X X Q () where the rimal feasible setq is efine by X Q : = x: AX B ε. o exloit fully the structure of unnown matrix X, we introuce X =Ψα with sarse transformation matrix Ψ. he smoothe version of convex function f ( X ) in () is fμ ( α) = max U, α μ U Q ( U ) (3) where Q : { u : U 1} = is the ual feasible set. o control flexibly the inherent structure of X, the smoothe convex function (3) is roose to be rewritten as where ( ) = max u Q, ( ) fμ α u α μ u (4) ( ( )) ( ) ( ) ( ( )) α = m α 1,:, m α,:,, m α N,:, ( α ( )) is a homogeneous function of jth row of ( j,: ) m j,: ( ) u is a rox-function for ual feasible set α, Q = u u. Q enote by : { : 1} As one by stanar NESA s metho for recovery of single-measurement sarse signal, one has the roceure of NESA-base MMV algorithm shown in able 1 From able 1, it is note that (1) f μ ( α ) can be easily comute in the close form, () the roose algorithm belongs to the first-orer metho for constraint otimization roblem,(3) if the row of A is orthogonal, which is often the case in comresse sensing alications [9], the comutational cost is very low, in articular, each iteration is extremely fast. o ecrease the number of measurements an increase the convergence rate, the following aroaches are carrie: (1) As one in [9], the homotoy technique can be exloite to accelerate it.

5 () It has been emirically shown if artial suort of unnown sarse matrixα, the convergent see will be imrove; moreover, the number of measurements can be greatly ecrease. If the artial common locations ofα enote by are nown, the function (3) is moifie as ( ) max, c U Q ( ) f U U μ α = α μ (3m) Of course, (4) can be moifie as { } ( ) = max u Q, c ( ) fμ α u α μ u (4m) Of course, the size of u in (4m) will be smaller than one in (4). (3) o estimate the artial suort of jointly sarse matrixα, the so-calle MUSIC algorithm can be exloite. As we nown, if the more column-ran ofα is, the more suort of α can be obtaine. (4)o ecrease the number of measurements, the iterative NESA-base MMV algorithm by combing har threshol technique is carrie out, see able, ABLE1. he roceure of NESA-base MMV algorithm Initialize α. For 0 Ste1. Comute Ste. Comute y : 0 f ( α ) L z = arg min ( ) ( ), α α Q i f αi x α α + i σ i= 0 i + 1 αi = Ste4. Uate α : ( 1 ) α = τ z + τ τ = + 3 Sto when a given criterion is vali L y = arg min α ( ), Q α α + f α α α Ste3. Comute z : y

6 able. Iterative NESA-base MMV algorithm Initialize α. Ste1. Carrying NESA base MMV algorithm with artially nown suor rovie in able1 Ste Ιf a given criterion is vali; sto esle ( ) ( ) Choose suort shown in Eq.3 m or Eq.4 m via har threshol, Goo SEP1 en II. IH-base Algorithm As a matter of fact, the iterative har thresholing algorithm roose by Blumensath an Davies can be easily generalize to eal with MMV roblem (see able 3). Further, the theoretical analysis about erformance guarantee can be carrie out along the same line as one in [8] able 3. Iterative har threshol base MMV algorithm Initialize α = Φ y, where, Φ = AΨ DO R n n = α α K n ( a ) ( ) a = α + μφ y Φx α n+ 1 n n = H n K Sto a given criterion is vali III. CONCLUSIONS In this reort, we focus on the NESA-base algorithm to eal with the recovery of jointly sarse signal. Numerical exeriences tell us that the NESA-base MMV algorithm can be use to eal with the large-scale MMV roblem. Moreover, the roose aroach outerforms the stat-of-art algorithms for MMV roblem. Detaile iscussion about our algorithms, with necessary theoretical analysis will aear in [7].

7 REFERENCES [1]J. Chen an X. Huo, heoretical results on sarse reresentations of multi-measurements vectors, IEEE rans. Signal Processing, 54(1), , 006 []S. Cotter, B. D. Rao, K. Engan an K. Kreutz-Delgao, Sarse solutions to linear inverse roblems with multile measurement vectors, IEEE rans. Signal Processing, 53(7), , 005 [3]Y. C. Elar, Comresse sensing of analog signals, submitte to IEEE rans. Signal Processing. [4]Y. C. Elar, an H. Rauhut, Average case analysis for multichannel sarse recovery using convex relaxation, rerint, 009 [5]E. van en Berg an M. P. Freielaner, Joint-sarse recovery from multile measurements, echnical reort from Deartment of Comuter Science, University of British Columbia, 009 [6]D. M. Malioutov, M. Cetin an A. S. Willsy, Source localization by enforcing sarsity through a Lalacian rior: an SVD-base aroach, [7]Lianlin Li an Fang Li, Novel Algorithm for Sarse Solutions to Linear Inverse Problems with Multile Measurements, submitte to IEEE rans. Signal Processing for ossible ublication, 009 [8]. Blumensath an M. E. Davies, Iterative thresholing for sarse aroximations, Journal of Fourier Analysis an Alications, 14(5), , 008 [9]S. Becer, J. Bobin an E. Canes, NESA: A fast an accurate first-orer metho for sarse recovery, Prerint, 009 [10]Y. Nesterov, Smooth minimization of non-smooth functions, Math. Program., Serie A., 103, 17-15, 007