Abstract This paper is about the efficient solution of large-scale compressed sensing problems.

Transcription

1 Noname manuscript No. (will be inserted by the editor) Optimization for Compressed Sensing: New Insights and Alternatives Robert Vanderbei and Han Liu and Lie Wang Received: date / Accepted: date Abstract This paper is about the efficient solution of large-scale compressed sensing problems. We have two main results. First, we propose a reformulation of the vector sensing problem in which the vector signal is stacked into a matrix. We call this the Kronecker reformulation. We show that the Kronecker reformulation requires stronger assumptions for perfect recovery compared to the original vector problem. However, the Kronecker reformulation, modeled correctly, is a much sparser linear optimization problem. Hence, an interior-point method applied to solve the linear programming problem solves the Kronecker formulation much faster than it solves the equivalent vector problem. In our numerical studies, we see a ten-fold improvement in the computation time. Secondly, we note that the vector (or matrix) of zeros is a basic (infeasible) solution for the linear programming problem and therefore, if the sensing signal is very sparse, some variants Department of Operations Research and Financial Engineering, Princeton University, Princeton, NJ 08544, USA; rvdb@princeton.edu Department of Operations Research and Financial Engineering, Princeton University, Princeton, NJ 08544, USA; hanliu@princeton.edu Research supported by NSF Grant III Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA; liewang@math.mit.edu Research supported by NSF Grant DMS Robert J. Vanderbei Department of Ops. Res. and Fin. Eng., Princeton University, Princeton, NJ Tel.: rvdb@princeton.edu

2 2 Robert Vanderbei and Han Liu and Lie Wang of the simplex method can be expected to take only a small number of pivots to arrive at a solution. We implemented one such variant and demonstrate a further ten-fold improvement in computation time. Keywords Linear Programming Compressive sensing parametric simplex method sparse signals interior-point methods Mathematics Subject Classification (2010) MSC 65K05 62P99 Compressed sensing aims to correctly recover a sparse signal from very few measurements. The theoretical foundation of compressed sensing was first laid out by Donoho (2006) and Candès et al. (2006) and can be traced further back to the sparse recovery work of Donoho and Stark (1989); Donoho and Huo (2001); Donoho and Elad (2003). More recent progress in the area of compressed sensing is summarized in Kutyniok (2012) and Elad (2010). In a typical compressed sensing problem, we let x 0 := (x 0 1,..., x 0 n) T R n be a signal where n is assumed to be large. We either assume that x 0 itself is sparse, i.e., it has very few non-zero components, which we can express mathematically by saying that x 0 0 := #{i : x i 0} is small, or that there exists a basis or a frame (a frame is a set of vectors that spans R n but does not have to be linearly independent) Φ such that x 0 = Φc with c being sparse (Kutyniok and Lim, 2011; Aharon et al., 2006; Rauhut et al., 2008; Candès et al., 2010). Let A be an m n sensing matrix with m < n. The compressed sensing problem is to recover x 0 from the knowledge of y = Ax 0, or recover c from the knowledge of y = AΦc. In both cases, we face an underdetermined linear system of equations with sparsity as prior information about the target signal. In the current paper, we mainly consider the case where x 0 is itself sparse. Since x 0 is a sparse vector, one can hope that it is the sparsest solution to the underdetermined linear system and therefore can be recovered from y by solving (P 0 ) min x x 0 subject to Ax = y. This problem is NP-hard due to the nonconvexity of the 0-pseudo-norm. To avoid the NPhardness, Chen et al. (1998) propose the basis pursuit approach in which x 1 replaces x 0 : (P 1 ) min x x 1 subject to Ax = y. (1)

3 Optimization for Compressed Sensing 3 Donoho and Elad (2003) and Cohen et al. (2009) characterize conditions under which the solutions to P 0 and P 1 are unique. The key question in compressed sensing research is to provide conditions under which they are the same. Various sufficient conditions have been discovered: see, e.g., Donoho and Huo (2001); Elad and Bruckstein (2002); Tropp (2004); Candès et al. (2006); Borup et al. (2008); Cohen et al. (2009); Donoho and Kutyniok (2010); Foucart (2010). More specifically, letting A S denote the submatrix of a matrix A with columns indexed by a subset S {1,..., n}, we say that A has the k-restricted isometry property (RIP) with RIP constant δ k if for any S with cardinality k, (1 δ k ) v 2 2 A S v 2 2 (1 + δ k ) v 2 2 for any v R k. (2) We denote by δ k (A) the smallest value of δ k for which the matrix A has the RIP property. Under the assumption that k := x 0 0 n and that the matrix A satisfies the RIP condition, Cai and Zhang (2012) prove that whenever δ k (A) < 1/3, the solutions to P 0 and P 1 are the same. In addition to these sufficient conditions, Donoho and Tanner (2005a,b, 2009) use the theory of convex polytopes to study necessary conditions for P 0 and P 1 to produce the same answer. Existing algorithms for solving the convex program P 1 include interior-point methods (Candès et al., 2006), projected gradient methods (Figueiredo et al., 2008), and Bregman iterations (Yin et al., 2008). Besides solving the convex program P 1, several greedy algorithms have been proposed, including matching pursuit (Mallat and Zhang, 1993) and its many variants (Tropp, 2004; Donoho et al., 2006; Needell and Vershynin, 2009; Needell and Tropp, 2010; Donoho et al., 2009). To achieve more scalability, combinatorial algorithms such as HHS pursuit (Gilbert et al., 2007) and a sub-linear Fourier transform (Iwen, 2010) have also been developed. In this paper, we propose to solve a linear programming formulation of P 1 using a variant of the simplex method. In particular, we use the parametric simplex method (see, e.g., Vanderbei (2007)) applied to a parametrically relaxed problem. Our approach is motivated by the fact that the desired solution is sparse and therefore should require only a relatively small number of simplex pivots from an appropriately chosen starting point the vector of all zeros. If the equality constraints in the model are relaxed and a parametrically-weighted penalty term is added to the objective function to gauge the deviation from equality, then the parametric simplex method can be employed to solve the problem in a small fraction of the number of pivots one would expect a naive implementation to require.

4 4 Robert Vanderbei and Han Liu and Lie Wang We also consider two other methods. These methods involve stacking the signal vector x into a matrix X and then multiplying the matrix signal on both the left and the right to get a compressed matrix signal. Using ideas analogous to one step of a fast Fourier transform, we essentially make a dramatic sparsification of the constraint matrix in the linear programming problem which allows us to solve the problem much faster than we could with the equivalent dense formulation. For our third method, we break up the left- and right-hand multiplications into separate smaller linear programming problems. This last method is very fast but requires much stricter assumptions of sparsity for the method to correctly recover the signal. We refer to these two new methods as Kronecker compressed sensing (KCS). Theoretically, KCS involves a tradeoff between computational complexity and informational complexity: it gains computational advantages at the price of requiring more measurements. More specifically, in later sections, we show that, using sub-gaussian random sensing matrices, whenever m 225k 2 (log(n/k 2 )) 2, (3) we recover the true signal with probability at least 1 4 exp( 0.1 m). It is easy to see that this scaling of (m, n, k) is tight by considering the special case when all the nonzero entries of x form a continuous block. The rest of this paper is organized as follows. In the next section, we describe the main idea the Kronecker compressed sensing (KCS). In particular, we introduce two variants: coupled KCS and decoupled KCS. In section 3, we describe the parametric simplex method and explain how to apply it to solve the original compressed sensing problem and the Kronecker compressed sensing problems. Numerical comparisons and discussion are provided in Section 4. 1 Kronecker Compressed Sensing In this section, we introduce the Kronecker compressed sensing (Duarte and Baraniuk, 2012). Unlike the classical compressed sensing which mainly focus on one-dimensional vector signals, the Kronecker compressed sensing can be used for sensing multidimensional signals (e.g., matrix or tensor). For example, given a sparse matrix signal X 0 R n1 n2, we can use two sensing matrices A R m1 n2 and B R m2 n2 and try to recover X 0 from the knowledge of Y = AX 0 B T. It is clear that when the signal is multidimensional, Kronecker compressed sensing

5 Optimization for Compressed Sensing 5 is more natural than the classical vector compressed sensing. Here, we would like to point out that, sometimes even when facing vector signals, it is still beneficial to use Kronecker compressed sensing due to its added computational efficiency. More specifically, even though the target signal is a vector x 0 R n, we may first stack it into a matrix X 0 R n1 n2 by putting each length n 1 sub-vector of x 0 into a column of X 0. Here we assume n = n 1 n 2. We then multiply the matrix signal X 0 on both the left and the right by sensing matrices A and B to get a compressed matrix signal Y 0. In the next section, we will show that using ideas analogous to one step of a fast Fourier transform, we are able to solve this Kronecker compressed sensing problem much more efficiently than the classical vector compressed sensing problem. In this section, we introduce two variants of the Kronecker compressed sensing: a coupled version versus a decoupled version. The decoupled KCS is computationally easier than the coupled KCS, but require more stringent conditions for successful recovery. In the sequel, we adopt the following notation. Let X j denote the j th column of the matrix X and X i be the i th row of the matrix X. Also, we define X 0 = j,k I(X jk 0) and X 1 := j,k X jk. 1.1 Coupled KCS Given a matrix Y R m1 m2 and the sensing matrices A and B, our goal is to recover the original sparse signal X 0, we first consider a coupled Kronecker sensing method, which solves the following optimization problem: (P 2 ) X = argmin X 1 subject to AXB T = Y. (4) Here, A and B are sensing matrices of size m 1 n 1 and m 2 n 2, respectively. Let x = vec(x) and y = vec(y), where the vec() operator takes a matrix and concatenates its elements columnby-column to build one large column-vector containing all the elements of the matrix. In terms of x and y, problem P 2 can be rewritten as vec( X) = argmin x 1 subject to Ux = y, (5)

6 6 Robert Vanderbei and Han Liu and Lie Wang where U is given by the (m 1 m 2 ) (n 1 n 2 ) Kronecker product of A and B: AB 11 AB 1n2 U := A B = AB m21 AB m2n 2 In this way, (4) becomes an ordinary compressed sensing problem. To analyze the properties of this coupled Kronecker sensing approach, we recall the definition of the restricted isometry constant for a matrix. For any m n matrix U, the k-restricted isometry constant δ k (U) is defined as the smallest nonnegative number such that for any k- sparse vector h R n, (1 δ k (U)) h 2 2 Uh 2 2 (1 + δ k (U)) h 2 2. (6) Based on the results in Cai and Zhang (2012), we have Lemma 1 (Cai and Zhang (2012)) Suppose k = X 0 0 is the sparsity of matrix X 0. Then if δ k (U) < 1/3, we have vec( X) = x 0 or equivalently X = X 0. For the value of δ k (U), by lemma 2 of Duarte and Baraniuk (2012), we know that 1 + δ k (U) (1 + δ k (A))(1 + δ k (B)). (7) In addition, we define strictly sub-gasusian distribution as follows: Definition 1 (Strictly Sub-Gaussian Distribution) We say a mean-zero random variable X follows a strictly sub-gaussian distribution with variance 1/m if it satisfies EX 2 = 1 m, ( ) t 2 E exp (tx) exp for all t R. 2m It is obvious that the Gaussian distribution with mean 0 and variance 1/m 2 satisfies the above definition. The next theorem provides sufficient conditions that guarantees perfect recovery of the coupled KCS with a desired probability. Theorem 1 Suppose matrices A and B are both generated by independent strictly sub-gaussian entries with variance 1/m. Let C > 28.1 be a constant. Whenever m 1 C k log (n 1 /k) and m 2 C k log (n 2 /k), (8)

7 Optimization for Compressed Sensing 7 the convex program P 2 attains perfect recovery with probability ( ) P X = X 0 ( 1 2 exp ( ) ( )m 1 2 exp C }{{} ρ(m 1,m 2) Proof From Equation (7) and Lemma 1, it suffices to show that ( ) )m 2. (9) C ( P δ k (A) < 2 1 and δ k (B) < 2 ) 1 1 ρ(m 1, m 2 ). (10) 3 3 This result directly follows from Theorem 3.6 of Baraniuk et al. (2010) with a careful calculation of constants. From the above theorem, we see that for m 1 = m 2 = m and n 1 = n 2 = n, whenever the number of measurements satisfies m 225k 2 (log(n/k 2 )) 2, (11) we have X = X 0 with probability at least 1 4 exp( 0.1 m). Here we compare the above result to that of vector compressed sensing, i.e., instead of stacking the original signal x 0 R n into a matrix, we directly use a strictly sub-gaussian sensing matrix A R m n to multiply on x 0 to get y = Ax 0. We then plug y and A into the convex program P 1 in Equation (1) to recover x 0. Following the same argument as in Theorem 1, whenever m 30k log (n/k), (12) we have x = x 0 with probability at least 1 2 exp( 0.1m). Compare (12) with (11), we see that coupled KCS requires more stringent conditions for perfect recovery. 1.2 Decoupled KCS In this section, we introduce a decoupled KCS method. More specifically, we consider the following two-stage procedure (P 3, Stage 1) Ŵ := argmin W 1 subject to AW = Y, (13) W R n 1 m 2 (P 3, Stage 2) X := argmin X 1 subject to XB T = Ŵ. (14) X R n 1 n 2

8 8 Robert Vanderbei and Han Liu and Lie Wang It is easy to see that the Stage 1 problem in (13) decomposes into m 2 subproblems, each of which is a linear program: Ŵ j = argmin w 1 subject to Aw = Y j for j = 1,..., m 2. (15) w R n 1 Similarly, the Stage 2 problem in (14) decomposes into n 2 subproblems, each of which is also a linear program: X i = argmin x 1 subject to Bx = ŴT i for i = 1,..., n 1. (16) x R n 2 To solve each individual linear programming problem, we again use the parametric simplex method (which will be introduced in the next section). Since the above problem can be decoupled into many smaller linear program subproblems, it is named decoupled KCS. The next theorem provides theoretical analysis of the decoupled KCS algorithm. Theorem 2 Suppose matrices A and B are both generated by independent strictly sub-gaussian entries with variance 1/m. Let C > 22.5 be a constant. Whenever m 1 C k log (n 1 /k) and m 2 C k log (n 2 /k), (17) The convex program P 3 attains perfect recovery with probability ( ) P X = X 0 ( 1 2 exp ( ) ( )m 1 2 exp C }{{} ξ(m 1,m 2) ( ) )m 2. (18) C Proof Let W 0 = X 0 B T. It is straightforward to show that W 0 j 0 k. We define the event E 1 := {Ŵ = W 0 }. (19) From Lemma 1 and Theorem 3.6 of Baraniuk et al. (2010), we have, whenever n 1 C k log (m 1 /k), ( P(E 1 ) P δ k (A) < 1 ) ( ( 1 2 exp C )m 1 ). (20) Conditioning on E 1 and following the same reasoning, we have, whenever n 2 C k log (m 2 /k): We then get the desired result. ( ) ( ( P X = X 0 E exp C )m 1 ). (21)

9 Optimization for Compressed Sensing 9 From Theorem 2, if we set m 1 = m 2 = m and n 1 = n 2 = n, whenever the number of measurements satisfies m 126k 2 (log(n/k 2 )) 2, (22) we have X = X 0 with probability at least 1 4 exp( 0.1 m). This scaling in (22) is the same as that for coupled KCS in (11). At the first sight, it seems that the decoupled KCS is a better approach than the coupled KCS. This is not necessarily true! The main caveat lies in the fact that the proof of Theorem 2 crucially hinges on the fact that we can perfectly solve the optimization problem (13) in the first stage of the decoupled KCS method. This is practically impossible since all the numerical solvers only deliver solutions that are close to the theoretical minimizer up to a controlled precision. In the numerical study section, we will show that such a small numerical inaccuracy may severely degrade the recovery performance in the second stage. To the best of our knowledge, this is the first time such a phenomenon has been pointed out. 2 Computational Algorithm In this section, we show how to formulate the convex programs P 1, P 2, and P 3 as linear programming problems and we show how to ensure that the LP formulations of the two Kronecker problems involve sparse constraint matrices. 2.1 Vector-Sensor Method The classical vector-sensor method P 1 can be formulated as a linear programming problem: (P 1:LP ) min x +,x 1 T (x + + x ) (23) subject to A(x + x ) = y x +, x 0. (24) The components of the vectors x + and x represent the positive and negative parts of x. Of course, to be the positive and negative parts it is required that one or the other of each component vanish. It is easy to see that this condition is satisfied automatically at optimality.

10 10 Robert Vanderbei and Han Liu and Lie Wang 2.2 Kronecker Sensor Method The coupled Kronecker sensing problem P 2 can also be converted to a linear programming problem: (P 2:LP ) min X +,X 1 T (X + + X )1 subject to A(X + X )B T = Y X +, X 0 where, as with vectors, the matrix inequality denotes component-wise inequality. As before, the components of the matrices X + and X become the component-wise positive and negative parts of X at optimality. As formulated, this linear programming problem is no easier to solve than the vector-sensor formulation P 1. However, it is easy to give an equivalent formulation that is computationally more tractable. The trick is to break the constraint A(X + X )B T = Y into a pair of constraints: (P 2:LPsparse ) min X +,X 1T (X + + X )1 subject to A(X + X ) Z = 0 ZB T = Y X +, X 0 Despite the introduction of new variables, Z, this formulation can be solved more efficiently because the constraint relations have been made sparse. Indeed, the matrix Y has m = m 1 m 2 variables so there are m constraints in the equation system A(X + X )B T = Y. And, each of these constraints involves every one of the n = n 1 n 2 variables in X + and in X. On the other hand, there are m 1 n 2 constraints in the equation A(X + X ) Z = 0 but each column in this matrix equation relates just that column of X + (or X ) to that column of Z. Hence, these equations are highly sparse. The same is true for ZB T = Y. It turns out that this sparsification trick is closely related to the fundamental idea underlying the fast Fourier optimization; see Vanderbei (2012) for more detailed discussions. Sparsity is also well-known to be a key factor in efficiently solving linear programming problems: see Vanderbei (1991). In a similar manner, the decoupled Kronecker sensing method can also be expressed as a linear programming problem.

11 Optimization for Compressed Sensing Vector-Sensor Method Solved by Parametric Simplex Method Finally, we consider a completely different way to improve the computational efficiency for solving the linear programming problems presented above. Consider the following parametric modification to P 1 : x := argmin x µ x T ɛ 1 (25) subject to Ax + ɛ = y. We have relaxed the fundamental equality and we have added a penalty term to the objective function that penalizes in proportion to the degree to which the original equations are not met. For large values of µ, the optimal solution has x = 0 and ɛ = y. For values of µ close to zero, the situation reverses: ɛ = 0. Our aim is to reformulate this problem as a parametric linear programming problem and solve it using the parametric simplex method (see, e.g., Vanderbei (2007)). In particular, we set parameter µ to start at µ = and successively reducing the value of µ for which the current basic solution is optimal until arriving at a value of µ for which the optimal solution has ɛ = 0 at which point we will have solved the original problem. If the number of pivots are few, then the final vector x will be mostly zero. It turns out that the best way to reformulate (25) as a linear programming problem is, as before, to split each variable into a difference between two nonnegative variables, x = x + x and ɛ = ɛ + ɛ. The next step is to replace x 1 with 1 T (x + +x ) and to make a similar substitution for ɛ 1. In general, the sum does not represent the absolute value but it is easy to see that it does at optimality. Here is the linear programming problem: min µ1t (x + + x ) + 1 T (ɛ + + ɛ ) x +,x,ɛ +,ɛ subject to A(x + x ) + (ɛ + ɛ ) = y x +, x, ɛ +, ɛ 0. For µ large, the optimal solution has x + = x = 0, and ɛ + ɛ = y. And, given that these latter variables are required to be nonnegative, it follows that y i > 0 = ɛ + i > 0 and ɛ i = 0

12 12 Robert Vanderbei and Han Liu and Lie Wang whereas y i < 0 = ɛ i > 0 and ɛ + i = 0 (the equality case can be decided either way). With these choices for variable values, the solution is feasible for all µ and is optimal for large µ. Furthermore, declaring the nonzero variables to be basic variables and the zero variables to be nonbasic, we see that this optimal solution is also a basic solution and can therefore serve as a starting point for the parametric simplex method. The parametric simplex algorithm described in Vanderbei (2007) can be downloaded from It is easy to modify this code to generate and solve problems discussed here. In fact, we have done this. The solution time for problems have the size characteristics described above solve in 5 seconds. 3 Numerical Results For the vector sensor, we generated random problems using m = 1,000 and n = 20,000. We varied the number of nonzeros k in signal x 0 from zero to 100. We solved the straightforward linear programming formulations of these instances using an interior-point solver called loqo (Vanderbei (1999)). We also solved a large number of instances of the parametrically formulated problem using the parametric simplex method as outlined above. We followed a similar plan for the Kronecker (Matrix) sensor. For these problems, we used m 1 = 33, m 2 = 34, n 1 = 141, n 2 = 142, and, again, various values of k. Again, the straightforward linear programming problems were solved by loqo and the parametrically formulated versions were solved by a custom developed parametric simplex method. We also made an implementation of the decoupled matrix sensor and implemented a parametric simplex method for that too. The results are shown in Figure 1. In terms of solving the linear programming problems P 1:LP and P 2:LP directly using an interior-point solver, the latter solves more than an order of magnitude faster. Also, the solution time is virtually independent of the signal sparsity parameter k. When the signal is very sparse, say k 70, both of the parametric simplex codes beat the interior-point solver. And, the improvement increases with sparsity. With k = 10, the improvement is yet another order of magnitude. So, for k = 10, the improvement over the

13 Optimization for Compressed Sensing Solution Time vs. Sparsity 10 3 Solution Time (seconds) LOQO Vector LOQO Matrix LOQO Decoupled Param Simplex Vector Param Simplex Matrix Param Simplex Decoupled Number of nonzeros in signal Fig. 1 Solution times for a large number of problem instances having m 1,000, n 20,000, and various degrees of sparsity in the underlying signal. The horizontal axis shows the number of nonzeros in the signal. The vertical axis gives a semi-log scale of solution times. original LP formulation of the vector sensor is almost a factor of 200. This improvement is dramatic and warrants further investigation. Finally, the decoupled problems solve extremely fast in both the regular LP method and in the parametric method. However, beyond k = 8, they don t correctly recover the original signal. The main reason is that in the first stage of P 3 we need to solve m 2 linear program subproblems to recover the columns of W. Though each subproblem admits a perfect recovery with high probability, among all the m 2 subproblems, several of them can not be recovered well. These errors in the first stage are significantly manifested in the second stage. This error cascading phenomenon leads to the inferior performance of the decoupled KCS compared to the coupled one.

14 14 Robert Vanderbei and Han Liu and Lie Wang 4 Conclusions We revisit compressed sensing from an optimization perspective. We advocate the usage of parametric simplex algorithm for solving large-scale compressed sensing problem. The parametric simplex is a homotopy algorithm and enjoys many good computational properties. We also propose two alternative ways for compressed sensing which illustrate a tradeoff between computing and statistics. In future work, we plan to extend the proposed method to the setting of 1-bit compressed sensing. References Aharon, M., Elad, M. and Bruckstein, A. (2006). K-SVD: An Algorithm for Designing Overcomplete Dictionaries for Sparse Representation. IEEE Transactions on Signal Processing, Baraniuk, R., Davenport, M. A., Duarte, M. F. and Hegde, C. (2010). An Introduction to Compressive Sensing. CONNEXIONS, Rice University, Houston, Texas. Borup, L., Gribonval, R. and Nielsen, M. (2008). Beyond coherence: Recovering structured time-frequency representations. Applied and Computational Harmonic Analysis, Cai, T. T. and Zhang, A. (2012). Sharp rip bound for sparse signal and low-rank matrix recovery. Applied and Computational Harmonic Analysis. URL sciencedirect.com/science/article/pii/s Candès, E., Romberg, J. and Tao, T. (2006). Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information. IEEE Transactions on Information Theory, Candès, E. J., Eldar, Y. C., Needell, D. and Randall, P. (2010). Compressed sensing with coherent and redundant dictionaries. Applied and Computational Harmonic Analysis. Chen, S. S., Donoho, D. L. and Saunders, M. A. (1998). Atomic Decomposition by Basis Pursuit. SIAM Journal on Scientific Computing, Cohen, A., Dahmen, W. and Devore, R. (2009). Compressed sensing and best k-term approximation. J. Amer. Math. Soc

15 Optimization for Compressed Sensing 15 Donoho, D. L. (2006). Compressed sensing. IEEE Transactions on Information Theory, Donoho, D. L. and Elad, M. (2003). Optimally sparse representation in general (nonorthogonal) dictionaries via l-minimization. Proc. Natl. Acad. Sci. USA, 100. Donoho, D. L., Elad, M. and Temlyakov, V. N. (2006). Stable recovery of sparse overcomplete representations in the presence of noise. IEEE Transactions on Information Theory, Donoho, D. L. and Huo, X. (2001). Uncertainty principles and ideal atomic decomposition. IEEE Transactions on Information Theory, Donoho, D. L. and Kutyniok, G. (2010). Microlocal analysis of the geometric separation problem. CoRR, abs/ Donoho, D. L., Maleki, A. and Montanari, A. (2009). Message passing algorithms for compressed sensing. Proc. Natl. Acad. Sci. USA, Donoho, D. L. and Stark, P. B. (1989). Uncertainty principles and signal recovery. SIAM J. Appl. Math., Donoho, D. L. and Tanner, J. (2005a). Neighborliness of randomly projected simplices in high dimensions. Proceedings of the National Academy of Sciences of the United States of America, Donoho, D. L. and Tanner, J. (2005b). Sparse nonnegative solutions of underdetermined linear equations by linear programming. In Proceedings of the National Academy of Sciences Donoho, D. L. and Tanner, J. (2009). Observed universality of phase transitions in highdimensional geometry, with implications for modern data analysis and signal processing. Philos. Trans. Roy. Soc. S.-A, Duarte, M. F. and Baraniuk, R. G. (2012). Kronecker compressive sensing. IEEE Transactions on Image Processing, Elad, M. (2010). Sparse and Redundant Representations - From Theory to Applications in Signal and Image Processing. Springer. Elad, M. and Bruckstein, A. M. (2002). A generalized uncertainty principle and sparse representation in pairs of bases. IEEE Transactions on Information Theory, Figueiredo, M., Nowak, R. and Wright, S. (2008). Gradient projection for sparse reconstruction: Application to compressed sensing and other inverse problems. IEEE Journal of

16 16 Robert Vanderbei and Han Liu and Lie Wang Selected Topics in Signal Processing, Foucart, S. (2010). A note on guaranteed sparse recovery via l 1 -minimization. Appl. Comput. Harmon. Anal., Gilbert, A. C., Strauss, M. J., Tropp, J. A. and Vershynin, R. (2007). One sketch for all: fast algorithms for compressed sensing. In STOC (D. S. Johnson and U. Feige, eds.). ACM, Iwen, M. A. (2010). Combinatorial sublinear-time fourier algorithms. Foundations of Computational Mathematics, Kutyniok, G. (2012). Compressed sensing: Theory and applications. CoRR, abs/ Kutyniok, G. and Lim, W.-Q. (2011). Compactly supported shearlets are optimally sparse. Journal of Approximation Theory, Mallat, S. and Zhang, Z. (1993). Matching pursuits with time-frequency dictionaries. Signal Processing, IEEE Transactions on, Needell, D. and Tropp, J. A. (2010). Cosamp: iterative signal recovery from incomplete and inaccurate samples. Commun. ACM, Needell, D. and Vershynin, R. (2009). Uniform uncertainty principle and signal recovery via regularized orthogonal matching pursuit. Foundations of Computational Mathematics, Rauhut, H., Schnass, K. and Vandergheynst, P. (2008). Compressed sensing and redundant dictionaries. IEEE Transactions on Information Theory, Tropp, J. A. (2004). Greed is good: algorithmic results for sparse approximation. IEEE Transactions on Information Theory, Vanderbei, R. (1991). Splitting dense columns in sparse linear systems. Lin. Alg. and Appl., Vanderbei, R. (1999). LOQO: An interior point code for quadratic programming. Optimization Methods and Software, Vanderbei, R. (2007). Linear Programming: Foundations and Extensions. 3rd ed. Springer. Vanderbei, R. J. (2012). Fast fourier optimization. Math. Prog. Comp., Yin, W., Osher, S., Goldfarb, D. and Darbon, J. (2008). Bregman iterative algorithms for l1-minimization with applications to compressed sensing. SIAM J. Img. Sci.,