Iteratively reweighted least squares minimization for sparse recovery
Under certain conditions (known as the restricted isometry property, or RIP) on the m × N matrix Φ (where m < N), vectors x ∈ ℝN that are sparse (i.e., have most of their entries equal to 0) can be recovered exactly from y := Φx even though Φ−1(y) is typically an (N − m)—dimensional hyperplane; i...
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| Published in | Communications on pure and applied mathematics Vol. 63; no. 1; pp. 1 - 38 |
|---|---|
| Main Authors | , , , |
| Format | Journal Article |
| Language | English |
| Published |
Hoboken
Wiley Subscription Services, Inc., A Wiley Company
01.01.2010
Wiley John Wiley and Sons, Limited |
| Subjects | |
| Online Access | Get full text |
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| Abstract | Under certain conditions (known as the restricted isometry property, or RIP) on the m × N matrix Φ (where m < N), vectors x ∈ ℝN that are sparse (i.e., have most of their entries equal to 0) can be recovered exactly from y := Φx even though Φ−1(y) is typically an (N − m)—dimensional hyperplane; in addition, x is then equal to the element in Φ−1(y) of minimal 𝓁1‐norm. This minimal element can be identified via linear programming algorithms. We study an alternative method of determining x, as the limit of an iteratively reweighted least squares (IRLS) algorithm. The main step of this IRLS finds, for a given weight vector w, the element in Φ−1(y) with smallest 𝓁2(w)‐norm. If x(n) is the solution at iteration step n, then the new weight w(n) is defined by w i(n) := [|x i(n)|2 + ε n2]−1/2, i = 1, …, N, for a decreasing sequence of adaptively defined εn; this updated weight is then used to obtain x(n + 1) and the process is repeated. We prove that when Φ satisfies the RIP conditions, the sequence x(n) converges for all y, regardless of whether Φ−1(y) contains a sparse vector. If there is a sparse vector in Φ−1(y), then the limit is this sparse vector, and when x(n) is sufficiently close to the limit, the remaining steps of the algorithm converge exponentially fast (linear convergence in the terminology of numerical optimization). The same algorithm with the “heavier” weight w i(n) = [|x i(n)|2 + ε n2]−1+τ/2, i = 1, …, N, where 0 < τ < 1, can recover sparse solutions as well; more importantly, we show its local convergence is superlinear and approaches a quadratic rate for τ approaching 0. © 2009 Wiley Periodicals, Inc. |
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| AbstractList | Under certain conditions (known as the restricted isometry property, or RIP) on the m x N matrix ... (where m < N), vectors x ... that are sparse (i.e., have most of their entries equal to 0) can be recovered exactly from y := ...x even though ...(y) is typically an (N - m) - dimensional hyperplane; in addition, x is then equal to the element in ...(y) of minimal ...-norm. This minimal element can be identified via linear programming algorithms. We study an alternative method of determining x, as the limit of an iteratively reweighted least squares (IRLS) algorithm. The main step of this IRLS finds, for a given weight vector w, the element in ...(y) with smallest ...-norm. If x... is the solution at iteration step n, then the new weight w... is defined by w...: ..., i = 1, ..., N, for a decreasing sequence of adaptively defined ...; this updated weight is then used to obtain x... and the process is repeated. We prove that when satisfies the RIP conditions, the sequence x... converges for all y, regardless of whether ...(y) contains a sparse vector. If there is a sparse vector in ...(y), then the limit is this sparse vector, and when x... is sufficiently close to the limit, the remaining steps of the algorithm converge exponentially fast (linear convergence in the terminology of numerical optimization). The same algorithm with the heavier weight w... = ..., i = 1, ..., N, where 0 < < 1, can recover sparse solutions as well; more importantly, we show its local convergence is superlinear and approaches a quadratic rate for ... approaching 0. (ProQuest: ... denotes formulae/symbols omitted.) Under certain conditions (known as the restricted isometry property , or RIP) on the m × N matrix Φ (where m < N ), vectors x ∈ ℝ N that are sparse (i.e., have most of their entries equal to 0) can be recovered exactly from y := Φ x even though Φ −1 ( y ) is typically an ( N − m )—dimensional hyperplane; in addition, x is then equal to the element in Φ −1 ( y ) of minimal 1 ‐norm. This minimal element can be identified via linear programming algorithms. We study an alternative method of determining x , as the limit of an iteratively reweighted least squares (IRLS) algorithm. The main step of this IRLS finds, for a given weight vector w , the element in Φ −1 ( y ) with smallest 2 ( w )‐norm. If x ( n ) is the solution at iteration step n , then the new weight w ( n ) is defined by w := [| x | 2 + ε ] −1/2 , i = 1, …, N , for a decreasing sequence of adaptively defined ε n ; this updated weight is then used to obtain x ( n + 1) and the process is repeated. We prove that when Φ satisfies the RIP conditions, the sequence x ( n ) converges for all y , regardless of whether Φ −1 ( y ) contains a sparse vector. If there is a sparse vector in Φ −1 ( y ), then the limit is this sparse vector, and when x ( n ) is sufficiently close to the limit, the remaining steps of the algorithm converge exponentially fast ( linear convergence in the terminology of numerical optimization). The same algorithm with the “heavier” weight w = [| x | 2 + ε ] −1+τ/2 , i = 1, …, N , where 0 < τ < 1, can recover sparse solutions as well; more importantly, we show its local convergence is superlinear and approaches a quadratic rate for τ approaching 0. © 2009 Wiley Periodicals, Inc. Under certain conditions (known as the restricted isometry property, or RIP) on the m × N matrix Φ (where m < N), vectors x ∈ ℝN that are sparse (i.e., have most of their entries equal to 0) can be recovered exactly from y := Φx even though Φ−1(y) is typically an (N − m)—dimensional hyperplane; in addition, x is then equal to the element in Φ−1(y) of minimal 𝓁1‐norm. This minimal element can be identified via linear programming algorithms. We study an alternative method of determining x, as the limit of an iteratively reweighted least squares (IRLS) algorithm. The main step of this IRLS finds, for a given weight vector w, the element in Φ−1(y) with smallest 𝓁2(w)‐norm. If x(n) is the solution at iteration step n, then the new weight w(n) is defined by w i(n) := [|x i(n)|2 + ε n2]−1/2, i = 1, …, N, for a decreasing sequence of adaptively defined εn; this updated weight is then used to obtain x(n + 1) and the process is repeated. We prove that when Φ satisfies the RIP conditions, the sequence x(n) converges for all y, regardless of whether Φ−1(y) contains a sparse vector. If there is a sparse vector in Φ−1(y), then the limit is this sparse vector, and when x(n) is sufficiently close to the limit, the remaining steps of the algorithm converge exponentially fast (linear convergence in the terminology of numerical optimization). The same algorithm with the “heavier” weight w i(n) = [|x i(n)|2 + ε n2]−1+τ/2, i = 1, …, N, where 0 < τ < 1, can recover sparse solutions as well; more importantly, we show its local convergence is superlinear and approaches a quadratic rate for τ approaching 0. © 2009 Wiley Periodicals, Inc. |
| Author | Güntürk, C. Si̇nan Daubechies, Ingrid Fornasier, Massimo DeVore, Ronald |
| Author_xml | – sequence: 1 givenname: Ingrid surname: Daubechies fullname: Daubechies, Ingrid email: ingrid@math.princeton.edu organization: Princeton University, Department of Mathematics and Program in Applied and Computational Mathematics, Fine Hall, Washington Road, Princeton, NJ 08544 – sequence: 2 givenname: Ronald surname: DeVore fullname: DeVore, Ronald email: devore@math.sc.edu organization: University of South Carolina, Department of Mathematics, Columbia, SC 29208 – sequence: 3 givenname: Massimo surname: Fornasier fullname: Fornasier, Massimo email: massimo.fornasier@oeaw.ac.at organization: Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Altenbergerstrasse 69, A-4040 Linz, Austria – sequence: 4 givenname: C. Si̇nan surname: Güntürk fullname: Güntürk, C. Si̇nan email: gunturk@courant.nyu.edu organization: Courant Institute, 251 Mercer Street, New York, NY 10012 |
| BackLink | http://pascal-francis.inist.fr/vibad/index.php?action=getRecordDetail&idt=22200554$$DView record in Pascal Francis |
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| References_xml | – reference: Chen, S. S.; Donoho, D. L.; Saunders, M. A. Atomic decomposition by basis pursuit. SIAM J Sci Comput 20 (1998), no. 1, 33-61 (electronic). – reference: Fornasier, M.; March, R. Restoration of color images by vector valued BV functions and variational calculus. SIAM J Appl Math 68 (2007), no. 2, 437-460. – reference: Tibshirani, R. Regression shrinkage and selection via the lasso. J Roy Statist Soc Ser B 58 (1996), no. 1, 267-288. – reference: Chartrand, R. Exact reconstructions of sparse signals via nonconvex minimization. IEEE Signal Process Lett 14 (2007), no. 10, 707-701. – reference: Taylor, H. L.; Banks, S. C.; McCoy, J. F. Deconvolution with the 1 norm. Geophysics 44 (1979), no. 1, 39-52. – reference: Candès, E. J.; Tao, T. Decoding by linear programming. IEEE Trans Inform Theory 51 (2005), no. 12, 4203-4215. – reference: Li, Y. A globally convergent method for lp problems. SIAM J Optim 3 (1993), no. 3, 609-629. – reference: Candès, E. J.; Romberg, J. K.; Tao, T. Stable signal recovery from incomplete and inaccurate measurements. Comm Pure Appl Math 59 (2006), no. 8, 1207-1223. – reference: Rudelson, M.; Vershynin, R. On sparse reconstruction from Fourier and Gaussian measurements. Comm Pure Appl Math 61 (2008), no. 8, 1025-1045. – reference: Claerbout, J. F.; Muir, F. Robust modeling with erratic data. Geophysics 38 (1973), no. 5, 826-844. – reference: Pinkus, A. M. On L1-approximation. Cambridge Tracts in Mathematics, 93. Cambridge University Press, Cambridge, 1989. – reference: Candès, E. J.; Wakin, M. B.; Boyd, S. P. Enhancing sparsity by reweighted l1 minimization. J Fourier Anal Appl 14 (2008), no. 5-6, 877-905. – reference: Baraniuk, R. G. Compressive sensing [Lecture notes]. IEEE Signal Processing Magazine 24 (2007), no. 4, 118-121. – reference: O'Brien, M. S.; Sinclair, A. N.; Kramer, S. M. Recovery of a sparse spike time series by 1 norm deconvolution. IEEE Trans Signal Process 42 (1994), no. 12, 3353-3365. – reference: Donoho, D. L. Compressed sensing. IEEE Trans Inform Theory 52 (2006), no. 4, 1289-1306. – reference: Donoho, D. L.; Tsaig, Y. Fast solution of 1-norm minimization problems when the solution may be sparse. IEEE Trans Inform Theory 54 (2008), no. 11, 4789-4812. – reference: Gorodnitsky, I. F.; Rao, B. D. Sparse signal reconstruction from limited data using FOCUSS: a re-weighted minimum norm algorithm. IEEE Trans Signal Process 45 (1997), no. 3, 600-616. – reference: Gribonval, R.; Nielsen, M. Highly sparse representations from dictionaries are unique and independent of the sparseness measure. Appl Comput Harmon Anal 22 (2007), no. 3, 335-355. – reference: DeVore, R. A. Nonlinear approximation. Acta Numer 7 (1998), 51-150. – reference: Donoho, D. L.; Tanner, J. Sparse nonnegative solution of underdetermined linear equations by linear programming. Proc Natl Acad Sci USA 102 (2005), no. 27, 9446-9451 (electronic). – reference: Figueiredo, M. A. T.; Bioucas-Dias, J. M.; Nowak, R. D. Majorization-minimization algorithms for wavelet-based image restoration. IEEE Trans Image Process 16 (2007), no. 12, 2980-2991. – reference: Cline, A. K. Rate of convergence of Lawson's algorithm. Math Comp 26 (1972), 167-176. – reference: Donoho, D. L. High-dimensional centrally symmetric polytopes with neighborliness proportional to dimension. Discrete Comput Geom 35 (2006), no. 4, 617-652. – reference: Donoho, D. L.; Logan, B. F. Signal recovery and the large sieve. SIAM J Appl Math 52 (1992), no. 2, 577-591. – reference: Baraniuk, R.; Davenport, M.; DeVore, R.; Wakin, M. B. A simple proof of the restricted isometry property for random matrices (aka "The Johnson-Lindenstrauss lemma meets compressed sensing"). Constr Approx 28 (2008), no. 3, 253-263. – reference: Candès, E. J.; Romberg, J.; Tao, T. Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information. IEEE Trans Inform Theory 52 (2006), no. 2, 489-509. – reference: Candès, E. J.; Tao, T. Near-optimal signal recovery from random projections: universal encoding strategies? IEEE Trans Inform Theory 52 (2006), no. 12, 5406-5425. – reference: Donoho, D. L.; Tanner, J. Counting faces of randomly projected polytopes when the projection radically lowers dimension. J Amer Math Soc 22 (2009), no. 1, 1-53. – reference: Donoho, D. L.; Stark, P. B. Uncertainty principles and signal recovery. SIAM J Appl Math 49 (1989), no. 3, 906-931. – reference: Chartrand, R.; Staneva, V. Restricted isometry properties and nonconvex compressive sensing. Inverse Problems 24 (2008), no. 035020, 1-14. – reference: Cohen, A.; Dahmen, W.; DeVore, R. Compressed sensing and best k-term approximation. J Amer Math Soc 22 (2009), no. 1, 211-231. – reference: Santosa, F.; Symes, W. W. Linear inversion of band-limited reflection seismograms. SIAM J Sci Statist Comput 7 (1986), no. 4, 1307-1330. – reference: Chambolle, A.; Lions, P.-L. Image recovery via total variation minimization and related problems. Numer Math 76 (1997), no. 2, 167-188. – year: 1985 – volume: 52 start-page: 577 issue: 2 year: 1992 end-page: 591 article-title: Signal recovery and the large sieve publication-title: SIAM J Appl Math – volume: 52 start-page: 489 issue: 2 year: 2006 end-page: 509 article-title: Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information publication-title: IEEE Trans Inform Theory – volume: 93 year: 1989 – start-page: 1433 year: 2006 end-page: 1452 – volume: 26 start-page: 167 year: 1972 end-page: 176 article-title: Rate of convergence of Lawson's algorithm publication-title: Math Comp – volume: 16 start-page: 2980 issue: 12 year: 2007 end-page: 2991 article-title: Majorization‐minimization algorithms for wavelet‐based image restoration publication-title: IEEE Trans Image Process – year: 2007 – volume: 7 start-page: 1307 issue: 4 year: 1986 end-page: 1330 article-title: Linear inversion of band‐limited reflection seismograms publication-title: SIAM J Sci Statist Comput – volume: 24 start-page: 1 issue: 035020 year: 2008 end-page: 14 article-title: Restricted isometry properties and nonconvex compressive sensing publication-title: Inverse Problems – start-page: 3885 year: 2008 end-page: 3888 – year: 1979 – volume: 59 start-page: 1207 issue: 8 year: 2006 end-page: 1223 article-title: Stable signal recovery from incomplete and inaccurate measurements publication-title: Comm Pure Appl Math – volume: 14 start-page: 707 issue: 10 year: 2007 end-page: 701 article-title: Exact reconstructions of sparse signals via nonconvex minimization publication-title: IEEE Signal Process Lett – volume: 20 start-page: 33 issue: 1 year: 1998 end-page: 61 article-title: Atomic decomposition by basis pursuit publication-title: SIAM J Sci Comput – volume: 51 start-page: 4203 issue: 12 year: 2005 end-page: 4215 article-title: Decoding by linear programming publication-title: IEEE Trans Inform Theory – volume: 52 start-page: 5406 issue: 12 year: 2006 end-page: 5425 article-title: Near‐optimal signal recovery from random projections: universal encoding strategies publication-title: IEEE Trans Inform Theory – volume: 52 start-page: 1289 issue: 4 year: 2006 end-page: 1306 article-title: Compressed sensing publication-title: IEEE Trans Inform Theory – year: 1961 – volume: 102 start-page: 9446 issue: 27 year: 2005 end-page: 9451 article-title: Sparse nonnegative solution of underdetermined linear equations by linear programming publication-title: Proc Natl Acad Sci USA – volume: 22 start-page: 211 issue: 1 year: 2009 end-page: 231 article-title: Compressed sensing and best ‐term approximation publication-title: J Amer Math Soc – volume: 22 start-page: 335 issue: 3 year: 2007 end-page: 355 article-title: Highly sparse representations from dictionaries are unique and independent of the sparseness measure publication-title: Appl Comput Harmon Anal – volume: 38 start-page: 826 issue: 5 year: 1973 end-page: 844 article-title: Robust modeling 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| SubjectTerms | Calculus of variations and optimal control Convergence Exact sciences and technology Linear programming Mathematical analysis Mathematics Numerical analysis Numerical analysis. Scientific computation Numerical methods in mathematical programming, optimization and calculus of variations Numerical methods in optimization and calculus of variations Optimization algorithms Sciences and techniques of general use Vector space |
| Title | Iteratively reweighted least squares minimization for sparse recovery |
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