# zbMATH — the first resource for mathematics

Revisiting compressed sensing: exploiting the efficiency of simplex and sparsification methods. (English) Zbl 1353.65057
Summary: We propose two approaches to solve large-scale compressed sensing problems. The first approach uses the parametric simplex method to recover very sparse signals by taking a small number of simplex pivots, while the second approach reformulates the problem using Kronecker products to achieve faster computation via a sparser problem formulation. In particular, we focus on the computational aspects of these methods in compressed sensing. For the first approach, if the true signal is very sparse and we initialize our solution to be the zero vector, then a customized parametric simplex method usually takes a small number of iterations to converge. Our numerical studies show that this approach is 10 times faster than state-of-the-art methods for recovering very sparse signals. The second approach can be used when the sensing matrix is the Kronecker product of two smaller matrices. We show that the best-known sufficient condition for the Kronecker compressed sensing (KCS) strategy to obtain a perfect recovery is more restrictive than the corresponding condition if using the first approach. However, KCS can be formulated as a linear program with a very sparse constraint matrix, whereas the first approach involves a completely dense constraint matrix. Hence, algorithms that benefit from sparse problem representation, such as interior point methods (IPMs), are expected to have computational advantages for the KCS problem. We numerically demonstrate that KCS combined with IPMs is up to 10 times faster than vanilla IPMs and state-of-the-art methods such as $$\ell_1\_\ell_s$$ and Mirror Prox regardless of the sparsity level or problem size.

##### MSC:
 65K05 Numerical mathematical programming methods 90C05 Linear programming 90C51 Interior-point methods 90C06 Large-scale problems in mathematical programming 94A12 Signal theory (characterization, reconstruction, filtering, etc.)
##### Software:
ALPO; CoSaMP; l1_ls; LOQO; LPbook; Matlab; PDCO
Full Text:
##### References:
  Adler, I; Karp, RM; Shamir, R, A simplex variant solving an $$m× d$$ linear program in $${O}(\min (m_2, d_2)$$ expected number of pivot steps, J. Complex., 3, 372-387, (1987) · Zbl 0641.65054  Belloni, A; Chernozhukov, V, $$ℓ _1$$-penalized quantile regression in high-dimensional sparse models, Ann. Stat., 39, 82-130, (2011) · Zbl 1209.62064  Cai, T.T., Zhang, A.: Sharp RIP bound for sparse signal and low-rank matrix recovery. Appl. Comput. Harmonic Anal. 35, 74-93 (2012) · Zbl 1310.94021  Candès, E; Romberg, J; Tao, T, Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information, IEEE Trans. Inf. Theory, 52, 489-509, (2006) · Zbl 1231.94017  Candès, EJ, The restricted isometry property and its implications for compressed sensing, C. R. Math., 346, 589-592, (2008) · Zbl 1153.94002  Chen, SS; Donoho, DL; Saunders, MA, Atomic decomposition by basis pursuit, SIAM J. Sci. Comput., 20, 33-61, (1998) · Zbl 0919.94002  Cohen, A; Dahmen, W; Devore, R, Compressed sensing and best $$k$$-term approximation, J. Am. Math. Soc., 22, 211-231, (2009) · Zbl 1206.94008  Dantzig, G.B.: Linear Programming and Extensions. Princeton University Press, Princeton (1998) · Zbl 0997.90504  Donoho, DL, Compressed sensing, IEEE Trans. Inf. Theory, 52, 1289-1306, (2006) · Zbl 1288.94016  Donoho, DL; Elad, M, Optimally sparse representation in general (nonorthogonal) dictionaries via $$ℓ _1$$-minimization, Proc. Natl. Acad. Sci USA, 100, 2197-2202, (2003) · Zbl 1064.94011  Donoho, DL; Elad, M; Temlyakov, VN, Stable recovery of sparse overcomplete representations in the presence of noise, IEEE Trans. Inf. Theory, 52, 6-18, (2006) · Zbl 1288.94017  Donoho, DL; Huo, X, Uncertainty principles and ideal atomic decomposition, IEEE Trans. Inf. Theory, 47, 2845-2862, (2001) · Zbl 1019.94503  Donoho, DL; Maleki, A; Montanari, A, Message passing algorithms for compressed sensing, Proc. Natl. Acad. Sci. USA, 106, 18914-18919, (2009)  Donoho, DL; Stark, PB, Uncertainty principles and signal recovery, SIAM J. Appl. Math., 49, 906-931, (1989) · Zbl 0689.42001  Donoho, D.L., Tanner, J.: Neighborliness of randomly projected simplices in high dimensions. Proc. Natl. Acad. Sci. 102, 9452-9457 (2005) · Zbl 1135.60300  Donoho, D. L., Tanner, J.: Sparse nonnegative solutions of underdetermined linear equations by linear programming. Proc. Natl. Acad. Sci. 102, 9446-9451 (2005) · Zbl 1135.90368  Donoho, DL; Tanner, J, Observed universality of phase transitions in high-dimensional geometry, with implications for modern data analysis and signal processing, Philos. Trans. Roy. Soc. S. A, 367, 4273-4273, (2009) · Zbl 1185.94029  Duarte, MF; Baraniuk, RG, Kronecker compressive sensing, IEEE Trans. Image Process., 21, 494-504, (2012) · Zbl 1372.94379  Elad, M.: Sparse and Redundant Representations—From Theory to Applications in Signal and Image Processing. Springer, New York (2010) · Zbl 1211.94001  Figueiredo, M; Nowak, R; Wright, S, Gradient projection for sparse reconstruction: application to compressed sensing and other inverse problems, IEEE J. Sel. Top. Signal Process., 1, 586-597, (2008)  Forrest, JJ; Goldfarb, D, Steepest-edge simplex algorithms for linear programming, Math. Program., 57, 341-374, (1992) · Zbl 0787.90047  Foucart, S, Hard thresholding pursuit: an algorithm for compressive sensing, SIAM J. Numer. Anal., 49, 2543-2563, (2011) · Zbl 1242.65060  Foucart, S., Rauhut, H.: A Mathematical Introduction to Compressive Sensing. Springer, New York (2013) · Zbl 1315.94002  Gilbert, A.C., Strauss, M.J., Tropp, J.A., Vershynin, R.: One sketch for all: fast algorithms for compressed sensing. In: Proceedings of the thirty-ninth annual ACM symposium on Theory of computing, pp. 237-246. ACM, New York (2007) · Zbl 1232.94008  Gill, P.E., Murray, W., Ponceleon, D.B., Saunders, M.A.: Solving reduced KKT systems in barrier methods for linear and quadratic programming. Tech. rep, DTIC Document (1991) · Zbl 0815.65080  Iwen, MA, Combinatorial sublinear-time Fourier algorithms, Found. Comut. Math., 10, 303-338, (2010) · Zbl 1230.65145  Juditsky, A., Karzan, F.K., Nemirovski, A.: $$ℓ _1$$ minimization via randomized first order algorithms. Université Joseph Fourier, Tech. rep. (2014) · Zbl 1282.90128  Kim, S; Koh, K; Lustig, M; Boyd, S; Gorinevsky, D, An interior-point method for large-scale $$l_1$$-regularized least squares, IEEE Trans. Sel. Top. Signal Process., 1, 606-617, (2007)  Klee, V., Minty, G. J.: How good is the simplex method? Inequalities-III, pp. 159-175 (1972) · Zbl 0297.90047  Kutyniok, G.: Compressed sensing: theory and applications. CoRR . arXiv:1203.3815 (2012) · Zbl 1064.94011  Lustig, IJ; Mulvey, JM; Carpenter, TJ, Formulating two-stage stochastic programs for interior point methods, Oper. Res., 39, 757-770, (1991) · Zbl 0739.90048  Mallat, S; Zhang, Z, Matching pursuits with time-frequency dictionaries, Signal Process. IEEE Trans., 41, 3397-3415, (1993) · Zbl 0842.94004  Needell, D; Tropp, JA, Cosamp: iterative signal recovery from incomplete and inaccurate samples, Commun. ACM, 53, 93-100, (2010) · Zbl 1163.94003  Needell, D; Vershynin, R, Uniform uncertainty principle and signal recovery via regularized orthogonal matching pursuit, Found. Comut. Math., 9, 317-334, (2009) · Zbl 1183.68739  Pan, P-Q, A largest-distance pivot rule for the simplex algorithm, Eur. J. Oper. Res., 187, 393-402, (2008) · Zbl 1149.90101  Post, I; Ye, Y, The simplex method is strongly polynomial for deterministic Markov decision processes, Math. Oper. Res., 40, 859-868, (2015) · Zbl 1329.90084  Spielman, DA; Teng, S-H, Smoothed analysis of algorithms: why the simplex algorithm usually takes polynomial time, J. ACM (JACM), 51, 385-463, (2004) · Zbl 1192.90120  Tropp, JA, Greed is good: algorithmic results for sparse approximation, IEEE Trans. Inf. Theory, 50, 2231-2242, (2004) · Zbl 1288.94019  Vanderbei, R, Splitting dense columns in sparse linear systems, Linear Algebra Appl., 152, 107-117, (1991) · Zbl 0727.65034  Vanderbei, R, LOQO: an interior point code for quadratic programming, Optim. Methods Softw., 12, 451-484, (1999) · Zbl 0973.90518  Vanderbei, R.: Linear Programming: Foundations and Extensions, 3rd edn. Springer, New York (2007) · Zbl 0874.90133  Vanderbei, RJ, Alpo: another linear program optimizer, ORSA J. Comput., 5, 134-146, (1993) · Zbl 0777.90031  Vanderbei, R. J.: Linear programming. Foundations and extensions, International Series in Operations Research & Management Science, vol. 37 (2001) · Zbl 1043.90002  Vanderbei, RJ, Fast Fourier optimization, Math. Prog. Comp., 4, 1-17, (2012) · Zbl 1257.90049  Yin, W; Osher, S; Goldfarb, D; Darbon, J, Bregman iterative algorithms for $$ℓ _1$$-minimization with applications to compressed sensing, SIAM J. Imaging Sci., 1, 143-168, (2008) · Zbl 1203.90153
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.