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In [[linear algebra]], the '''restricted isometry property''' characterizes [[matrix (mathematics)|matrices]] which are nearly orthonormal, at least when operating on [[sparse]] vectors. The concept was introduced by [[Emmanuel Candès]] and [[Terence Tao]]<ref>E. J. Candes and T. Tao, "Decoding by Linear Programming," IEEE Trans. Inf. Th., 51(12): 4203&ndash;4215 (2005).</ref> and is used to prove many theorems in the field of [[compressed sensing]].<ref>E. J. Candes, J. K. Romberg, and T. Tao, "Stable Signal Recovery from Incomplete and Inaccurate Measurements," Communications on Pure and Applied Mathematics, Vol. LIX, 1207–1223 (2006).</ref> There are no known large matrices with bounded restricted isometry constants (and computing these constants is [[NP-hard]]<ref>A. S. Bandeira, E. Dobriban, D. G. Mixon, W. F. Sawin, "Certifying the Restricted Isometry Property is Hard," IEEE Trans. Inf. Th., 59(6): 3448-3450 (2013)</ref>), but many random matrices have been shown to remain bounded. In particular, it has been shown that with exponentially high probability, random Gaussian, Bernoulli, and partial Fourier matrices satisfy the RIP with number of measurements nearly linear in the sparsity level.<ref>F. Yang, S. Wang, and C. Deng, "''Compressive sensing of image reconstruction using multi-wavelet transform''", IEEE 2010</ref> The current smallest upper bounds for any large rectangular matrices are for those of Gaussian matrices.<ref>B. Bah and J. Tanner "Improved Bounds on Restricted Isometry Constants for Gaussian Matrices"</ref> Web forms to evaluate bounds for the Gaussian ensemble are available at the Edinburgh Compressed Sensing RIC page.<ref>http://ecos.maths.ed.ac.uk/ric_bounds.shtml</ref>
 
== Definition ==
Let ''A'' be an ''m''&nbsp;&times;&nbsp;''p'' matrix and let ''1''&nbsp;&le;&nbsp;''s''&nbsp;&le;&nbsp;''p'' be an integer. Suppose that there exists a constant <math>\delta_s</math> such that, for every ''m''&nbsp;&times;&nbsp;''s'' submatrix ''A''<sub>''s''</sub> of ''A'' and for every vector&nbsp;''y'',
 
:<math>(1-\delta_s)\|y\|_{\ell_2}^2 \le \|A_s y\|_{\ell_2}^2 \le (1+\delta_s)\|y\|_{\ell_2}^2. \, </math>
 
Then, the matrix ''A'' is said to satisfy the ''s''-restricted isometry property with restricted isometry constant <math>\delta_s</math>.
 
== See also ==
*[[Compressed sensing]]
*[[Mutual coherence (linear algebra)]]
*Terence Tao's [http://www.math.ucla.edu/~tao/preprints/sparse.html website on compressed sensing] lists several related conditions, such as the 'Exact reconstruction principle' (ERP) and 'Uniform uncertainty principle' (UUP)
*[[Nullspace property]], another sufficient condition for sparse recovery
 
== References ==
{{reflist}}
 
[[Category:Signal processing]]
[[Category:Linear algebra]]
 
 
{{signal-processing-stub}}
{{Linear-algebra-stub}}

Revision as of 15:05, 25 January 2014

In linear algebra, the restricted isometry property characterizes matrices which are nearly orthonormal, at least when operating on sparse vectors. The concept was introduced by Emmanuel Candès and Terence Tao[1] and is used to prove many theorems in the field of compressed sensing.[2] There are no known large matrices with bounded restricted isometry constants (and computing these constants is NP-hard[3]), but many random matrices have been shown to remain bounded. In particular, it has been shown that with exponentially high probability, random Gaussian, Bernoulli, and partial Fourier matrices satisfy the RIP with number of measurements nearly linear in the sparsity level.[4] The current smallest upper bounds for any large rectangular matrices are for those of Gaussian matrices.[5] Web forms to evaluate bounds for the Gaussian ensemble are available at the Edinburgh Compressed Sensing RIC page.[6]

Definition

Let A be an m × p matrix and let 1 ≤ s ≤ p be an integer. Suppose that there exists a constant δs such that, for every m × s submatrix As of A and for every vector y,

(1−δs)‖y‖ℓ22≤‖Asy‖ℓ22≤(1+δs)‖y‖ℓ22.

Then, the matrix A is said to satisfy the s-restricted isometry property with restricted isometry constant δs.

See also

References

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Template:Signal-processing-stub Template:Linear-algebra-stub

  1. ↑ E. J. Candes and T. Tao, "Decoding by Linear Programming," IEEE Trans. Inf. Th., 51(12): 4203–4215 (2005).
  2. ↑ E. J. Candes, J. K. Romberg, and T. Tao, "Stable Signal Recovery from Incomplete and Inaccurate Measurements," Communications on Pure and Applied Mathematics, Vol. LIX, 1207–1223 (2006).
  3. ↑ A. S. Bandeira, E. Dobriban, D. G. Mixon, W. F. Sawin, "Certifying the Restricted Isometry Property is Hard," IEEE Trans. Inf. Th., 59(6): 3448-3450 (2013)
  4. ↑ F. Yang, S. Wang, and C. Deng, "Compressive sensing of image reconstruction using multi-wavelet transform", IEEE 2010
  5. ↑ B. Bah and J. Tanner "Improved Bounds on Restricted Isometry Constants for Gaussian Matrices"
  6. ↑ http://ecos.maths.ed.ac.uk/ric_bounds.shtml
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