The LASSO Risk for Gaussian Matrices
We consider the problem of learning a coefficient vector x ο ∈ R N from noisy linear observation y = Ax o + ∈ R n . In many contexts (ranging from model selection to image processing), it is desirable to construct a sparse estimator x̂. In this case, a popular approach consists in solving an ℓ 1 -pe...
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| Vydáno v: | IEEE transactions on information theory Ročník 58; číslo 4; s. 1997 - 2017 |
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| Hlavní autoři: | , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
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New York, NY
IEEE
01.04.2012
Institute of Electrical and Electronics Engineers The Institute of Electrical and Electronics Engineers, Inc. (IEEE) |
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| ISSN: | 0018-9448, 1557-9654 |
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| Abstract | We consider the problem of learning a coefficient vector x ο ∈ R N from noisy linear observation y = Ax o + ∈ R n . In many contexts (ranging from model selection to image processing), it is desirable to construct a sparse estimator x̂. In this case, a popular approach consists in solving an ℓ 1 -penalized least-squares problem known as the LASSO or basis pursuit denoising. For sequences of matrices A of increasing dimensions, with independent Gaussian entries, we prove that the normalized risk of the LASSO converges to a limit, and we obtain an explicit expression for this limit. Our result is the first rigorous derivation of an explicit formula for the asymptotic mean square error of the LASSO for random instances. The proof technique is based on the analysis of AMP, a recently developed efficient algorithm, that is inspired from graphical model ideas. Simulations on real data matrices suggest that our results can be relevant in a broad array of practical applications. |
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| AbstractList | We consider the problem of learning a coefficient vector $x_{0}in{BBR}^{N}$ from noisy linear observation $y=Ax_{0}+win{BBR}^{n}$ . In many contexts (ranging from model selection to image processing), it is desirable to construct a sparse estimator ${widehat x}$. In this case, a popular approach consists in solving an $ell_{1}$ -penalized least-squares problem known as the LASSO or basis pursuit denoising. [PUBLICATION ABSTRACT] We consider the problem of learning a coefficient vector x ο ∈ R N from noisy linear observation y = Ax o + ∈ R n . In many contexts (ranging from model selection to image processing), it is desirable to construct a sparse estimator x̂. In this case, a popular approach consists in solving an ℓ 1 -penalized least-squares problem known as the LASSO or basis pursuit denoising. For sequences of matrices A of increasing dimensions, with independent Gaussian entries, we prove that the normalized risk of the LASSO converges to a limit, and we obtain an explicit expression for this limit. Our result is the first rigorous derivation of an explicit formula for the asymptotic mean square error of the LASSO for random instances. The proof technique is based on the analysis of AMP, a recently developed efficient algorithm, that is inspired from graphical model ideas. Simulations on real data matrices suggest that our results can be relevant in a broad array of practical applications. |
| Author | Montanari, A. Bayati, M. |
| Author_xml | – sequence: 1 givenname: M. surname: Bayati fullname: Bayati, M. email: bayati@stanford.edu organization: Grad. Sch. of Bus., Stanford Univ., Stanford, CA, USA – sequence: 2 givenname: A. surname: Montanari fullname: Montanari, A. email: montanari@stanford.edu organization: Dept. of Electr. Eng., Stanford Univ., Stanford, CA, USA |
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| Keywords | state evolution Range finding statistical learning Random matrix message passing algorithms Image processing Noise reduction Model selection Least squares problem Algorithm Mean square error Learning Message passing Algorithm performance Simulation graphical models Signal processing random matrix theory Compressed sensing |
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| References_xml | – year: 2005 ident: ref4 publication-title: Spectral Analysis of Large Dimensional Random Matrices – volume: 2569 start-page: 564 year: 1995 ident: ref7 article-title: Examples of basis pursuit publication-title: Proc SPIE doi: 10.1117/12.217610 – ident: ref11 doi: 10.1214/aos/1024691081 – ident: ref25 doi: 10.1111/j.1467-9868.2011.00771.x – ident: ref5 doi: 10.1137/080716542 – start-page: 33 year: 2007 ident: ref1 article-title: Scalable training of <formula formulatype="inline"> <tex Notation="TeX">$l^{1}$</tex></formula>-regularized log-linear models publication-title: Proc 24th Int Conf Mach Learn – start-page: 6920 year: 2011 ident: ref13 publication-title: The Noise Sensitivity Phase Transition in Compressed Sensing – ident: ref6 doi: 10.1214/aop/1176989118 – ident: ref22 doi: 10.1093/acprof:oso/9780198570837.001.0001 – ident: ref3 doi: 10.1109/TIT.2010.2094817 – volume: 234 start-page: 383 year: 2003 ident: ref27 article-title: On the rigorous solution of Gardners problem publication-title: Commun Math Phys doi: 10.1007/s00220-002-0783-3 – ident: ref23 doi: 10.1109/TIT.2011.2177575 – year: 2010 ident: ref14 publication-title: CVX Matlab Software for Disciplined Convex Programming Version 1 21 – ident: ref10 doi: 10.1007/BF01199026 – ident: ref19 doi: 10.1088/1742-5468/2009/09/L09003 – year: 2001 ident: ref20 publication-title: The Concentration of Measure Phenomenon – ident: ref24 doi: 10.1002/ett.4460100604 – start-page: 145 year: 2010 ident: ref2 article-title: The LASSO risk: Asymptotic results and real world examples publication-title: Advances in neural information processing systems – ident: ref17 doi: 10.1070/IM1977v011n02ABEH001719 – ident: ref8 doi: 10.1002/cpa.20124 – volume: 7 start-page: 2541 year: 2006 ident: ref26 article-title: On model selection consistency of Lasso publication-title: J Mach Learn Res – year: 2010 ident: ref28 publication-title: Mean Field Models for Spin Glasses Volume I Basic Examples – ident: ref21 doi: 10.1016/j.aim.2004.08.004 – year: 2009 ident: ref15 article-title: A single-letter characterization of optimal noisy compressed sensing publication-title: 47th Annu Allerton Conf Communication Control and Computing – ident: ref9 doi: 10.1214/009053606000001523 – ident: ref16 doi: 10.4171/022-1/13 – ident: ref18 doi: 10.1109/TIT.2011.2112231 – ident: ref12 doi: 10.1073/pnas.0909892106 |
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| SubjectTerms | Algorithm design and analysis Applied sciences Calibration Compressed sensing Detection, estimation, filtering, equalization, prediction Equations Exact sciences and technology graphical models Image processing Image processing systems Information theory Information, signal and communications theory Mean square error methods message passing algorithms Noise Noise measurement Normal distribution random matrix theory Services and terminals of telecommunications Signal and communications theory Signal processing Signal, noise Sparsity state evolution statistical learning Systems, networks and services of telecommunications Telecommunications Telecommunications and information theory Telemetry. Remote supervision. Telewarning. Remote control Vectors |
| Title | The LASSO Risk for Gaussian Matrices |
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