## GPU accelerated greedy algorithms for compressed sensing.(English)Zbl 1300.65038

Summary: For appropriate matrix ensembles, greedy algorithms have proven to be an efficient means of solving the combinatorial optimization problem associated with compressed sensing. This paper describes an implementation for graphics processing units (GPU) of hard thresholding, iterative hard thresholding, normalized iterative hard thresholding, hard thresholding pursuit, and a two-stage thresholding algorithm based on compressive sampling matching pursuit and subspace pursuit. The GPU acceleration of the former bottleneck, namely the matrix-vector multiplications, transfers a significant portion of the computational burden to the identification of the support set. The software solves high-dimensional problems in fractions of a second which permits large-scale testing at dimensions currently unavailable in the literature. The GPU implementations exhibit up to $$70\times$$ acceleration over standard Matlab central processing unit implementations using automatic multi-threading.

### MSC:

 65K05 Numerical mathematical programming methods 90C27 Combinatorial optimization 65Y05 Parallel numerical computation

### Software:

CoSaMP; GAGA; Matlab; Thrust; CUDA; SPGL1; ggks
Full Text:

### References:

 [1] Alabi, T., Blanchard, J., Gordon, B., Steinbach, R.: Fast k-selection algorithms for graphics processing units. ACM J. Exp. Algorithmics 17(2), 4.2:1–4.2:29 (2012) (Article 4.2) · Zbl 1284.68637 [2] Allison, D., Noga, M.: Selection by distributive partitioning. Inf. Process. Lett. 11(1), 7–8 (1980) · Zbl 0444.68051 [3] Beliakov, G.: Parallel calculation of the median and order statistics on GPUs with application to robust regression. Computing Research Repository abs/1104.2 (2011) [4] Berg, E.v.d., Friedlander, M.P.: Probing the Pareto frontier for basis pursuit solutions. SIAM J. Sci. Comput. 31(2), 890–912 (2008) · Zbl 1193.49033 [5] Blanchard, J.D., Tanner, J.: GAGA: GPU Accelerated Greedy Algorithms (2013). www.gaga4cs.org . Version 1.0.0 [6] Blanchard, J.D., Tanner, J.: Performance comparisons of greedy algorithms in compressed sensing. www.math.grinnell.edu/$$\sim$$blanchaj/PCGACS_preprint.pdf (2013, submitted) · Zbl 1300.65038 [7] Blumensath, T., Davies, M.E.: Iterative hard thresholding for compressed sensing. Appl. Comput. Harmon. Anal. 27(3), 265–274 (2009) · Zbl 1174.94008 [8] Blumensath, T., Davies, M.E.: Normalised iterative hard thresholding; guaranteed stability and performance. IEEE Select. Topics Signal Process. 4(2), 298–309 (2010) [9] Candès, E.J.: Compressive sampling. In: International Congress of Mathematicians, Vol. III, pp. 1433–1452. Eur. Math. Soc., Zürich (2006) · Zbl 1130.94013 [10] Candès, E.J., Romberg, J.K., Tao, T.: Stable signal recovery from incomplete and inaccurate measurements. Commun. Pure Appl. Math. 59(8), 1207–1223 (2006) · Zbl 1098.94009 [11] Candès, E.J., Tao, T.: Decoding by linear programming. IEEE Trans. Inform. Theory 51(12), 4203–4215 (2005) · Zbl 1264.94121 [12] Cevher, V.: An ALPS view of sparse recovery. In: 2011 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 5808–5811. IEEE (2011) [13] Dai, W., Milenkovic, O.: Subspace pursuit for compressive sensing signal reconstruction. IEEE Trans. Inform. Theory 55(5), 2230–2249 (2009) · Zbl 1367.94082 [14] Donoho, D.L.: Neighborly polytopes and sparse solution of underdetermined linear equations: Technical Report. Stanford University, Department of Statistics (2004) [15] Donoho, D.L.: Compressed sensing. IEEE Trans. Inform. Theory 52(4), 1289–1306 (2006) · Zbl 1288.94016 [16] Donoho, D.L.: For most large underdetermined systems of equations, the minimal $$l_1$$ l 1 -norm near-solution approximates the sparsest near-solution. Commun. Pure Appl. Math. 59(7), 907–934 (2006) · Zbl 1105.90068 [17] Donoho, D.L.: High-dimensional centrally symmetric polytopes with neighborliness proportional to dimension. Discrete Comput. Geom. 35(4), 617–652 (2006) · Zbl 1095.52500 [18] Donoho, D.L., Tanner, J.: Sparse nonnegative solution of underdetermined linear equations by linear programming. Proc. Natl. Acad. Sci. USA 102(27), 9446–9451 (2005) (electronic) · Zbl 1135.90368 [19] Donoho, D.L., Tsaig, Y.: Fast solution of l1 minimization problems when the solution may be sparse. IEEE Trans. Inform. Theory 54(11), 4789–4812 (2008) · Zbl 1247.94009 [20] Donoho, D.L., Tsaig, Y., Drori, I., Stark, J.L.: Sparse solution of underdetermined linear equations by stagewise orthogonal matching pursuit. IEEE Trans. Inform. Theory 58(2), 1094–1121 (2012) · Zbl 1365.94069 [21] Figueiredo, M.A.T., Nowak, R.D., Wright, S.J.: Gradient projection for sparse reconstruction: application to compressed sensing and other inverse problems. IEEE Select. Topics Signal Process. 1(4), 586–597 (2007) [22] Foucart, S.: Hard thresholding pursuit: an algorithm for compressive sensing. SIAM J. Numer. Anal. 49(6), 2543–2563 (2011) · Zbl 1242.65060 [23] Hoberock, J., Bell, N.: Thrust: A parallel template library (2010). http://www.meganewtons.com/ . Version 1.3.0, http://www.meganewtons.com/ [24] Kyrillidis, A., Cevher, V.: Recipes on hard thresholding methods. In: 2011 4th IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP), pp. 353–356. IEEE (2011) · Zbl 1311.90141 [25] Lee, S., Wright, S.J.: Implementing algorithms for signal and image reconstruction on graphical processing units (2008). http://pages.cs.wisc.edu/swright/GPUreconstruction/gpu_image.pdf [26] Mallat, S., Zhang, Z.: Matching pursuits with time-frequency dictionaries. IEEE Trans. Signal Process. 41(12), 3397–3415 (1993) · Zbl 0842.94004 [27] Merrill, D., Grimshaw, A.: High performance and scalable radix sorting: a case study of implementing dynamic parallelism for GPU computing. Parallel Process. Lett. 21(02), 245–272 (2011) · Zbl 06115252 [28] Monroe, L., Wendelberger, J., Michalak, S.: Randomized selection on the GPU. In: Proceedings of the ACM SIGGRAPH Symposium on High Performance Graphics, HPG ’11, pp. 89–98. ACM, New York (2011) [29] Natarajan, B.K.: Sparse approximate solutions to linear systems. SIAM J. Comput. 24(2), 227–234 (1995) · Zbl 0827.68054 [30] Needell, D., Tropp, J.: CoSaMP: iterative signal recovery from incomplete and inaccurate samples. Appl. Comput. Harmon. Anal. 26(3), 301–321 (2009) · Zbl 1163.94003 [31] NVIDIA: Cuda toolkit 4.0 (2011). http://developer.nvidia.com/cuda-toolkit-40 [32] Tropp, J.A.: Greed is good: algorithmic results for sparse approximation. IEEE Trans. Inform. Theory 50(10), 2231–2242 (2004) · Zbl 1288.94019 [33] Tropp, J.A., Gilbert, A.C.: Signal recovery from random measurements via orthogonal matching pursuit. IEEE Trans. Inform. Theory 53(12), 4655–4666 (2007) · Zbl 1288.94022 [34] Wright, S.J., Nowak, R.D., Figueiredo, M.A.T.: Sparse reconstruction by separable approximation. In: Proceedings of the International Conference on Acoustics, Speech, and, Signal Processing (2008) · Zbl 1391.94442 [35] Yin, W., Osher, S., Goldfarb, D., Darbon, J.: Bregman iterative algorithms for $$$$\backslash$$ell \^1$$ 1 -minimization with applications to compressed sensing. SIAM J. Imaging Sci. 1(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.