On the Role of the Properties of the Nonzero Entries on Sparse Signal Recovery

Transcription

1 On the Role of the Properties of the Nonzero Entries on Sparse Signal Recovery Yuzhe Jin and Bhaskar D. Rao Department of Electrical and Computer Engineering, University of California at San Diego, La Jolla, CA , USA {yujin, Abstract We study the role of the nonzero entries on the performance limits in support recovery of sparse signals. The key to our results is the recently studied connection between sparse signal recovery and multiple user communication. By leveraging the concept of outage capacity in information theory, we explicitly characterize the impact of the probability distribution imposed on the nonzero entries of the sparse signal on support recovery. When Multiple Measurement Vectors MMV) are available, we show that the identification of the nonzero rows of the signal is closely connected to decoding the messages from multiple users over a Single-Input Multiple-Output channel. Necessary and sufficient conditions for support indices of nonzero rows) recovery are provided, and the results allow us to understand the role of correlation of the nonzero entries as well as the role of the rank of the matrix formed from the non-zero entries. I. INTRODUCTION Suppose the signal of interest is X R m l, and X is said to be sparse when only a few of its rows contain nonzero elements whereas the rest consist of zero elements. One wishes to estimate X via the noisy measurements Y = AX + Z 1) where A R n m is the measurement matrix and Z R n l is the measurement noise. Specifically, when l = 1, this problem is usually termed as sparse signal recovery with single measurement vector SMV); when l > 1, it is commonly referred to as sparse signal recovery with multiple measurement vectors MMV) [1]. This problem has received much attention from many applications such as compressed sensing [], [3], biomagnetic inverse problems [4], [5], image processing [6], [7], bandlimited extrapolation and spectral estimation [8], robust regression and outlier detection [9], speech processing [10], echo cancellation [11], and wireless communication [1]. A. Brief Background on the SMV Problem For the problem of sparse signal recovery with SMV, computationally efficient algorithms have been proposed to find or approximate the sparse solution X R m in various settings. A partial list includes matching pursuit [13], orthogonal matching pursuit OMP) [14], lasso [15], basis pursuit [16], FOCUSS [4], iteratively reweighted l 1 minimization [17], iteratively reweighted l minimization [18], and sparse Bayesian learning SBL) [19], [0]. Analysis has been developed to shed light on the performances of these practical algorithms. For example, Donoho [], Donoho, Elad, and Temlyakov [1], Candès and Tao [], and Candès, Romberg, and Tao [3] presented sufficient conditions for l 1 -norm minimization algorithms, including basis pursuit and its variant in the noisy setting, to successfully recover the sparse signals with respect to different performance metrics. Tropp [4], Tropp and Gilbert [5], and Donoho, Tsaig, Drori, and Starck [6] studied the performances of greedy sequential selection methods such as matching pursuit and its variants. Wainwright [7] and Zhao and Yu [8] provided sufficient and necessary conditions for lasso to recover the support of the sparse signal, i.e., the set of indices of the nonzero entries. On the other hand, from an information theoretic perspective, a series of papers, for instance, Wainwright [9], Fletcher, Rangan, and Goyal [30], Wang, Wainwright, and Ramchandran [31], Akçakaya and Tarokh [3], Jin, Kim, and Rao [33], provided sufficient and necessary conditions to indicate the performance limits of optimal algorithms for support recovery, regardless of computational complexity. B. Brief Background on the MMV Problem A fast emerging trend is the capability of collecting multiple measurements in an increasing number of applications, such as magnetoencephalography MEG) and electroencephalography EEG) [34], [35], multivariate regression [36], and direction of arrival estimation [37]. This gives rise to the problem of sparse signal recovery with multiple measurement vectors. Practical algorithms have been developed to address the new challenges in this scenario. One class of algorithms for solving the MMV problem can be viewed as straightforward extensions based on their counterparts for the SMV problem. To sample a few, M-OMP [1], [38], M-FOCUSS [1], l 1 /l minimization method 1 [39], multivariate group lasso [36], and M-SBL [40] can be all viewed as examples of this kind. Another class of algorithms additionally make explicit effort to exploit the structure underlying the sparse signal X, such as the temporal correlation of the non-zero entries of a row which would be otherwise unavailable when l = 1, to aim for better performance of sparse signal recovery. For instance, the improved M-FOCUSS algorithms [35] and the auto-regressive sparse Bayesian learning AR-SBL) [41] both have the capability of explicitly taking advantage of the structural properties of X to 1 This method is sometimes referred to as l /l 1 minimization, due to the naming convention in a specific paper. In this paper, we use l 1 /l p to indicate the cost of a matrix B which is define as i j b i,j q ) 1/q.

2 improve the recovery performance. Along side the algorithmic advancement, a series of work have been focusing on the theoretic analysis to support the effectiveness of existing algorithms for the MMV problem. We briefly divide these results into two categories. The first category of theoretic analysis aims at specific practical algorithms for sparse signal recovery with MMV. For example, Chen and Huo [4] discovered the sufficient conditions for l 1 /l p norm minimization method and orthogonal matching pursuit to exactly recover every sparse signal within certain sparsity level in the noiseless setting. Eldar and Rauhut [43] also analyzed the performance of sparse recovery using the l 1 /l norm minimization method in the noiseless setting, but the sparse signal was assumed to be randomly distributed according to certain probability distribution and the performance was averaged over all possible realizations of the sparse signal. Obozinski, Wainwright, and Jordan [36] provided sufficient and necessary conditions for multivariate group lasso to successfully recover the support of the sparse signal in the presence of measurement noise. The second category of performance analysis bears an information theoretic nature, and it explores the performance limits that any algorithm, regardless of computational complexity, could possibly achieve. In this regard, Tang and Nehorai [37] employed a hypothesis testing framework with the likelihood ratio test as the optimal decision rule to study how fast the error probability decays. Sufficient and necessary conditions are further identified for successful support recovery in the asymptotic sense. C. Contributions of this Paper In this paper, we develop sharp performance tradeoffs involving the signal dimension m, the number of nonzero rows k, the number of measurements per measurement vector n, the number of measurement vectors l, and especially the nonzero entries for exact support recovery in the noisy setting. Specifically, we consider two cases. The first case is the support recovery problem with SMV, and the nonzero entries are modeled as random quantities. In this case, due to the randomness of the nonzero entries, we provide an probability lower bound for successful support recovery in an asymptotic sense. The second case is the support recovery problem with MMV, where the nonzero entries are assumed to be fixed. In this case, we show that n = )/cx) is sufficient and necessary. We give a complete characterization of cx) that explicitly depends on the elements of the nonzero rows of X. Together with the interpretations we provide, we demonstrate the potential performance improvement enabled by having MMV, and hence bolster its usage in practical applications. Our main results are inspired by the analogy to wireless communication over the additive white Gaussian noise AWGN) single-input multiple-output SIMO) multiple access channel MAC). According to this connection, the columns of the measurement matrix form a common codebook for all senders. Codewords from the senders are individually multiplied by unknown channel gains, which correspond to nonzero entries of X. Then, the noise corrupted linear combinations of these codewords are observed by multiple receivers, which correspond to the multiple measurement vectors. Thus, the problem of support recovery can be interpreted as multiple receivers joint decoding messages sent by multiple senders. With appropriate modifications, the techniques for deriving the capacity of a SIMO MAC channel and outage capacity for slow fading channel can be leveraged to provide performance tradeoffs for support recovery. D. Notation R m denotes the m-dimensional real Euclidean space. [k] denotes the set {1,,..., k}. The notation S denotes the cardinality of set S, x denotes the l -norm of a vector x. For a matrix A, A T denotes the submatrix formed by the rows of A indexed by the set T. Let 1 denote a column vector whose elements are all 1 s, and its length can be determined from the context. II. PROBLEM FORMULATION Let W R k l, where w i,j 0 for all i, j. Note that W can be either fixed or random, where in the latter case every element of W has bounded support. Let S = [S 1,..., S k ] [m] k be such that S 1,..., S k are chosen uniformly at random from [m] without replacement. In particular, {S 1,..., S k } is uniformly distributed over all size-k subsets of [m]. Then, the signal of interest X = XW, S) is generated as { wj,i if s = S X s,i = j, ) 0 if s / {S 1,..., S k }. The support of X, denoted by suppx), is defined as the set of indices corresponding to the nonzero rows of X, i.e., suppx) = {S 1,..., S k }. According to the signal model ), suppx) = k. Throughout this paper, we assume k is known. We measure X through the linear operation 1). We assume that the elements of A are independently and identically distributed i.i.d.) according to the Gaussian distribution N 0, σ a), and the noise Z i,j are i.i.d. according to N 0, σ z). Upon observing the noisy measurement Y, the goal is to recover the support of X. A support recovery map is defined as d : R n l [m]. 3) Given the signal model ), the measurement model 1), and the support recovery map 3), we define the average probability of error by P{dY ) suppxw, S))} Note that the probability is averaged over the locations of the nonzero rows S, the measurement matrix A, the measurement noise Z, and the possible randomness of the nonzero signal matrix W. III. CONNECTION TO MULTIUSER COMMUNICATION We introduce an important interpretation of the sparse signal recovery problem using a communication problem over the Gaussian SIMO MAC, which extends our earlier work [33]. This analogy motivates the intuition behind our main results and facilities the development of the proof techniques.

3 A. A Brief Review on SIMO MAC Consider the following wireless communication scenario. Suppose k users wish to transmit information to a set of l common receivers. Each sender i has access to a codebook C i) = {c i) 1, ci),..., ci) }, where c i) m i) j R n is a codeword and m i) is the number of codewords in the codebook. The rate for the ith sender is R i) = i) )/n. To transmit information, each sender chooses a codeword from its codebook, and all senders transmit their codewords simultaneously over a SIMO MAC: Y j,i = h j,1 X 1,i + h j, X,i + + h j,k X k,i + Z j,i 4) i = 1,,..., n, and j = 1,,..., l where X q,i denotes the input symbol from the qth sender to the channel at the ith use of the channel, h j,q denotes the channel gain between with the qth sender and the jth receiver, Z j,i is the additive Gaussian noise i.i.d. according to N 0, ), and Y j,i is the channel output at the jth receiver at the ith use of the channel. After receiving Y j,1,,..., Y j,n at each receiver j [l], the receivers work jointly to determine the codewords transmitted by each sender. Since the senders interfere with each other, there is an inherent tradeoff among their operating rates. The notion of capacity region is introduced to capture this tradeoff by characterizing all possible rate tuples R 1), R ),..., R k) ) at which reliable communication can be achieved with diminishing error probability of decoding. We discuss two different cases based on the assumption on the channel gains. In the first case, we assume the channel gains are fixed and known at the receivers. By assuming each sender obeys the power constraint c i) j /n σc for all j [m i) ] and all i [k], the capacity region of a SIMO MAC [44], [45] is { R 1),..., R k) ) : ) } R i) 1 log I + σ c h i h i, T [k] i T where h i [h 1,i,..., h l,i ] for i [k]. In the second case, we assume the channel gains are random according to certain distribution. Further, the channel gains are realized once and keep fixed during the entire channel use. As a result, there could be a nontrivial possibility that the channel gains are realized too poor to support the target rate. In this case, the channel model 4) is recognized as a slow fading channel, and outage capacity is employed to characterize the performance of this channel [45]. i T B. Similarities and Differences to Sparse Signal Recovery Based on the measurement model 1), we can remove the columns in A which correspond to the zero rows of X, and obtain the following effective form of the measurement procedure 5) Y j = X S1,jA S1 + + X Sk,jA Sk + Z j 6) for j [l]. By contrasting 6) to a SIMO MAC 4), we can draw the following key connections that relate the two problems. i) A nonzero entry as a sender: We can view the existence of a nonzero row index S i as sender i that accesses the MAC. ii) A measurement vector as a receiver: We can view the existence of a measurement vector Y j as receiver j. iii) X Si,j as the channel gain: The nonzero entry X Si,j, i.e., w i,j, plays the role of the channel gain h j,i from the ith user to the jth receiver. When X Si,j is assumed to be fixed, it corresponds to the channel with fixed gain. When X Si,j is assume to be random, it corresponds to a random channel gain which is realized once and fixed during the entire channel use. In the latter case, it is conceivable that the outage capacity is useful in analyzing the performance limit of sparse signal recovery. iv) A i as the codeword: We treat the measurement matrix A as a codebook with each column A i, i [m], as a codeword. Each element of A Si is fed one by one through the channel as input symbols for the ith sender to the l receivers, resulting in n uses of the channel. v) Similarity of objectives: In the problem of sparse signal recovery, we focus on finding the support {S 1,..., S k } of the signal. In the problem of MAC communication, the receiver needs to determine the indices of codewords, i.e., S 1,..., S k, that are transmitted by senders. Based on the abovementioned aspects, the two problems share significant similarities which enable leveraging the information theoretic methods for performance analysis of support recovery of sparse signals. However, there are domain specific differences, namely the problems of common codebook and unknown channel gains [33], between the support recovery problem and the channel coding problem that should be addressed accordingly to rigorously apply the information theoretic approaches. Based on techniques that are rooted in channel capacity results, but suitably modified to deal with the differences, performance tradeoffs for support recovery of sparse signals can be obtained. IV. PERFORMANCE TRADEOFFS FOR SUPPORT RECOVERY We state the main results of the paper. For a fixed W, define the auxiliary constant [ )] 1 cw ) min log det I + σ a T [k] T W T W T. 7) A. SMV with Random Nonzero Entries We consider the SMV case, i.e., l = 1, and the nonzero entries are randomly drawn according to certain probability distribution. The following theorem states the performance of support recovery of sparse signals with random signal activities. The proof is presented in [33]. Theorem 1: Suppose W R k has bounded support, and lim sup n m = r for some constant r > 0. Then, there

4 exists a sequence of support recovery maps {d m) } m=k, dm) : R n m [m], such that lim sup P{d m) A m) XW, S) + Z) suppx)} P{cW) r} 8) Theorem 1 implies that, in general, rather than having a diminishing error probability, we have to tolerate certain error probability which is upper-bounded by PcW) r). From a channel coding viewpoint as discussed in Section III, we can view r as the rate which all senders are operating at, and view cw ) as the channel capacity for a specific realization W. Hence, PcW) r) represents the probability that the channel realization is too poor to support the target rate, which corresponds to the event of channel outage. B. MMV with Fixed Nonzero Entries We consider the support recovery of a sequence of sparse signals generated with the same fixed W. The following theorems state the performance tradeoff for this case. The proof can be found in [46]. Theorem : If lim sup n m < cw ) 9) then there exists a sequence of support recovery maps {d m) } m=k, dm) : R n m l [m], such that Theorem 3: If lim P{dY ) suppxw, S))} = 0. 10) lim sup n m > cw ) 11) then for any sequence of support recovery maps {d m) } m=k, dm) : R nm l [m], lim inf P{dY ) suppxw, S))} > 0. 1) Theorems and 3 together indicate that n = 1 cw )±ϵ is the sufficient and necessary number of measurements per measurement vector to ensure asymptotically successful support recovery. The constant cw ) explicitly captures the role of the nonzero entries in the performance tradeoff. C. Performance Improvement via MMV It have been observed in existing literature that having MMV can improve the performance of sparse signal recovery [1], [35], [39]. Our analysis provides theoretical support to the improvement enabled by MMV. At this point it is useful to note that the SIMO MAC is related to point to point MIMO with full spatial multiplexing, i.e independent data streams from each transmit antenna, and the sum rate is equal to the capacity of the point to point MIMO problem, a well studied problem [44], [45]. In particular, the capacity depends on the rank of the MIMO channel and scales as minimum of the number of transmit and receive antennas in the full rank case. Relating to the sparse recovery problem, cw ) depends on the rank of W and scales as mink, l), the minimum of the number of nonzero entries and the number of measurement vectors. For the purpose of illustration, let us consider three special cases. Let k be an even number. The first case deals with an SMV problem, where w = 1 R k. The second case is concerned about an MMV problem, where W = [1, 1] R k. The third case considers an MMV [ 1 1 ] problem, where W = R 1 1 k, and the second column of W consists of equal number of 1 and 1. According to Theorem, we calculate the upper bound on m such that successful support recovery is attained asymptotically as long as m is within that upper bound. The following table summarizes the results. Structure of nonzero matrix Upper bound on m 1) SMV: W = 1 R k m < 1 + kσ a 1 + kσ a ) MMV: W = [1, 1] R k m < [ ] 1 1 3) MMV: W = 1 1 R k m < 1 + kσ a k k k From this table, we have the following observations. First, the cases with MMV enjoy a larger upper bound on m than that of the SMV case. This means that more positions can be monitored in the sparse signal, which improves the performance of sparse signal recovery. Second, the upper bound in case 3) is larger than that of case ). By inspecting the nonzero entries, we can see that case ) has two identical nonzero signal source vectors. It leads to a gain equivalent to doubling the signal to noise ratio. On the contrary, the two nonzero source vectors in case 3) are orthogonal. Therefore, the structure of the nonzero signal matrix W, especially the rank, plays an important role in the performance of support recovery. Furthermore, since the performance limit of support recovery is closely related to the nonzero signal matrix W, a practical algorithm should explore the structure of W in order to achieve better performance. The smooth overlapped blocks M- FOCUSS [35] and the autoregressive sparse Bayesian learning AR-SBL) [41] can be viewed as this type of algorithms. For example, the AR-SBL models the nonzero entries as a first order autoregressive process and it attempts to learn the correlation coefficient along with the other parameters. Experimental studies have shown that these algorithms achieve better performance than algorithms that do not explicitly take into account the structure of W. V. SUMMARY This paper discussed the performance tradeoffs for support recovery of sparse signals. The key to our results is the connection between sparse signal recovery and SIMO multiuser communication systems. The SIMO MAC capacity and outage capacity proved useful in unveiling the performance limits of

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