Abstract
We consider the problem of finding a linear combination of at most t out of K column vectors in a matrix, such that a target vector is approximated as closely as possible. The motivation of the model is to find a lower-dimensional representation of a given signal vector (target) while minimizing loss of accuracy. We point out the computational intractability of this problem, and suggest some local search heuristics for the unit norm case. The heuristics, all of which are based on pivoting schemes in a related linear program, are compared experimentally with respect to speed and accuracy.
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References
Barrodale I, Roberts FDK (1973) An improved algorithm for discrete ℓ1 linear approximation. SIAM Journal on Numerical Analysis 10(5):839–848
Bartels RH, Conn AR, Sinclair JW (1978) Minimization techniques for piecewise differentiable functions: the ℓ1 solution to an overdetermined linear system. SIAM Journal on Numerical Analysis 15(2):224–241
Chen SS, Donoho DL, Saunders MA (1998) Atomic decomposition by Basis Pursuit. SIAM Journal on Scientific Computing 20(1):33–61
Coifman RR, Wickerhauser MV (1992) Entropy-based algorithms for best-basis selection. IEEE Transactions on Information Theory 38:713–718
Couvreur C, Bresler Y (2000) On the optimality of the backward greedy algorithm for the subset selection problem. SIAM Journal on Matrix Analysis and Applications 21(3):797–808
Daubechies I (1988) Time-frequency localization operators: a geometric phase space approach. IEEE Transactions on Information Theory 34:605–612
Engan K (2000) Frame based signal representation and compression. Dr.Ing. Thesis, Stavanger University College
Gharavi-Alkhansari M, Huang TS (May 1998) A fast orthogonal matching pursuit algorithm. In Proc. ICASSP '98, Seattle, Washington, USA, pp 1389–1392
Golub GH, van Loan CF (1989) Matrix computations, 2nd. ed., The Johns Hopkins University Press, Baltimore, MD
Haugland D (2001) An edge-traversal algorithm for the subspace selection problem. Paper presented at The 7th Meeting of the Nordic MPS Section, Copenhagen, Denmark
Mallat S, Zhang Z (1993) Matching pursuit in a time-frequency dictionary. IEEE Transactions on Signal Processing 41:3397–3415
Moody G (1992) MIT-BIH arrhythmia database cd-rom (second edition) overview. Massachusetts Institute of Technology
Natarajan BK (1995) Sparse approximate solutions to linear systems. SIAM Journal on Computing 24(2):227–234
Reeves SJ (1999) An efficient implementation of the backward greedy algorithm for sparse signal reconstruction. IEEE Signal Processing Letters 6:266–268
Ryen T (June 19–21, 2001) Simultaneous selection and quantization in frame based compression. Presented at 2nd IEEE R8-EURASIP Symposium on Image and Signal Processing and Analysis (ISPA'01), Pula, Croatia
Watson GA (1980) Approximation theory and numerical methods. Wiley, Binghamton, New York
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Haugland, D., Storøy, S. Local search methods for ℓ1-minimization in frame based signal compression. Optim Eng 7, 81–96 (2006). https://doi.org/10.1007/s11081-006-6592-3
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DOI: https://doi.org/10.1007/s11081-006-6592-3