Stable signal recovery from incomplete and inaccurate measurements. Candès, E. J., Romberg, J. K., & Tao, T. Communications on Pure and Applied Mathematics, 59(8):1207–1223, August, 2006.
Stable signal recovery from incomplete and inaccurate measurements [link]Paper  doi  abstract   bibtex   
Abstract Suppose we wish to recover a vector x 0 ∈ ℝ 𝓂 (e.g., a digital signal or image) from incomplete and contaminated observations y = A x 0 + e ; A is an 𝓃 × 𝓂 matrix with far fewer rows than columns (𝓃 ≪ 𝓂) and e is an error term. Is it possible to recover x 0 accurately based on the data y ? To recover x 0 , we consider the solution x # to the 𝓁 1 ‐regularization problem where ϵ is the size of the error term e . We show that if A obeys a uniform uncertainty principle (with unit‐normed columns) and if the vector x 0 is sufficiently sparse, then the solution is within the noise level As a first example, suppose that A is a Gaussian random matrix; then stable recovery occurs for almost all such A 's provided that the number of nonzeros of x 0 is of about the same order as the number of observations. As a second instance, suppose one observes few Fourier samples of x 0 ; then stable recovery occurs for almost any set of 𝓃 coefficients provided that the number of nonzeros is of the order of 𝓃/(log 𝓂) 6 . In the case where the error term vanishes, the recovery is of course exact, and this work actually provides novel insights into the exact recovery phenomenon discussed in earlier papers. The methodology also explains why one can also very nearly recover approximately sparse signals. © 2006 Wiley Periodicals, Inc.
@article{candes_stable_2006,
	title = {Stable signal recovery from incomplete and inaccurate measurements},
	volume = {59},
	issn = {0010-3640, 1097-0312},
	url = {https://onlinelibrary.wiley.com/doi/10.1002/cpa.20124},
	doi = {10.1002/cpa.20124},
	abstract = {Abstract
            
              Suppose we wish to recover a vector
              x
              0
              ∈ ℝ
              𝓂
              (e.g., a digital signal or image) from incomplete and contaminated observations
              y
              =
              A x
              0
              +
              e
              ;
              A
              is an 𝓃 × 𝓂 matrix with far fewer rows than columns (𝓃 ≪ 𝓂) and
              e
              is an error term. Is it possible to recover
              x
              0
              accurately based on the data
              y
              ?
            
            
              To recover
              x
              0
              , we consider the solution
              x
              \#
              to the 𝓁
              1
              ‐regularization problem
              
              where ϵ is the size of the error term
              e
              . We show that if
              A
              obeys a uniform uncertainty principle (with unit‐normed columns) and if the vector
              x
              0
              is sufficiently sparse, then the solution is within the noise level
              
            
            
              As a first example, suppose that
              A
              is a Gaussian random matrix; then stable recovery occurs for almost all such
              A
              's provided that the number of nonzeros of
              x
              0
              is of about the same order as the number of observations. As a second instance, suppose one observes few Fourier samples of
              x
              0
              ; then stable recovery occurs for almost any set of 𝓃 coefficients provided that the number of nonzeros is of the order of 𝓃/(log 𝓂)
              6
              .
            
            In the case where the error term vanishes, the recovery is of course exact, and this work actually provides novel insights into the exact recovery phenomenon discussed in earlier papers. The methodology also explains why one can also very nearly recover approximately sparse signals. © 2006 Wiley Periodicals, Inc.},
	language = {en},
	number = {8},
	urldate = {2024-03-19},
	journal = {Communications on Pure and Applied Mathematics},
	author = {Candès, Emmanuel J. and Romberg, Justin K. and Tao, Terence},
	month = aug,
	year = {2006},
	keywords = {/unread},
	pages = {1207--1223},
}
Downloads: 0
About
Documentation
Keywords
Search
Users
Terms and Conditions of Use
Privacy Policy
Sitemap
© BibBase.org 2021