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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 and is used to prove many theorems in the field of compressed sensing. There are no known large matrices with bounded restricted isometry constants (and computing these constants is NP-hard), 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. The current smallest upper bounds for any large rectangular matrices are for those of Gaussian matrices. Web forms to evaluate bounds for the Gaussian ensemble are available at the Edinburgh Compressed Sensing RIC page.

Definition

Let A be an m × p matrix and let 1 ≤ s ≤ p be an integer. Suppose that there exists a constant ${\delta }_s$ such that, for every m × s submatrix A_s of A and for every vector y,

$(1-{\delta }_s)\Vert y{\Vert }_{{\ell }_2}^2\le \Vert A_{s}y{\Vert }_{{\ell }_2}^2\le (1+{\delta }_s)\Vert y{\Vert }_{{\ell }_2}^2.$

Then, the matrix A is said to satisfy the s-restricted isometry property with restricted isometry constant ${\delta }_s$.

See also

• Compressed sensing
• Mutual coherence (linear algebra)
• Terence Tao's 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

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