Restricted isometry property
In linear algebra, the restricted isometry property (RIP) 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 (computing these constants is strongly NP-hard, and is hard to approximate as well), 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 formula_1 such that, for every "m" × "s" submatrix "A" of "A" and for every "s"-dimensional vector "y", Then, the matrix "A" is said to satisfy the "s"-restricted  isometry property with restricted isometry constant formula_3. This condition is equivalent to the statement that for every "m" × "s" submatrix "A" of "A" we have where formula_5 is the formula_6 identity matrix and formula_7 is the operator norm. See for example for a proof. Finally this is equivalent to stating that all eigenvalues of formula_8 are in the interval formula_9. Restricted Isometric Constant (RIC). The RIC Constant is defined as the infimum of all possible formula_10 for a given formula_11. It is denoted as formula_13. Eigenvalues. For any matrix that satisfies the RIP property with a RIC of formula_13 , the following condition holds: The tightest upper bound on the RIC can be computed for Gaussian matrices. This can be achieved by computing the exact probability that all the eigenvalues of Wishart matrices lie within an interval.