New Bounds for Restricted Isometry Constants
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sparse matrices
compressed sensing
k-sparse signal
minimization
positive integer
real matrix
restricted isometry constant
computer aided instruction
linear matrix inequalities
mathematics
measurement errors
minimization methods
noise
noise measurement
signal processing
statistics
upper bound
vectors
restricted isometry property
sparse signal recovery
Applied Mathematics
Computer Sciences
Statistics and Probability
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Abstract
This paper discusses new bounds for restricted isometry constants in compressed sensing. Let Φ be an n × p real matrix and A; be a positive integer with k ≤ n. One of the main results of this paper shows that if the restricted isometry constant δk of Φ satisfies δk <; 0.307 then k-sparse signals are guaranteed to be recovered exactly via ℓ1 minimization when no noise is present and k-sparse signals can be estimated stably in the noisy case. It is also shown that the bound cannot be substantially improved. An explicit example is constructed in which δk = k-1/2k-1 <; 0.5, but it is impossible to recover certain k-sparse signals.