On Recovery of Sparse Signals with Prior Support Information via Weighted ℓ -Minimization

A complete characterization for the restricted isometry constant (RIC) bounds on delta {{{ tk}}} for all {t}>0 is an important problem on recovery of sparse signals with prior support information via weighted ell {{p}} -minimization ( 0 < {p} leqslant 1 ). In this paper, new bounds on the restricted isometry constants delta {{{ tk}}} ( 0 < {t} < frac {4}{3}{d} ), where d is a key constant determined by prior support information, are established to guarantee the sparse signal recovery via the weighted ell {{p}} minimization in both noiseless and noisy settings. This result fills a vacancy on delta {{{ tk}}} with 0 < {t} < frac {4}{3}{d} , compared with previous works on delta {{{ tk}}} ( {t} geqslant frac {4}3{d} ). We show that, when the accuracy of prior support estimate is at least 50%, the new recovery condition in terms of delta {{{ tk}}} ( 0 < {t} < frac {4}{3}{d} ) via weighted ell {1} minimization is weaker than the condition required by classical ell {1} minimization without weighting. Our weighted ell {1} minimization gives better recovery error bounds in noisy setting. Similarly, the new recovery condition in terms of delta {{{ tk}}} ( 0 < {t} < frac {4}{3}{d} ) is extended to weighted ell {{p}} ( 0 < {p} < 1 ) minimization, and it is also weaker than the condition obtained by standard non-convex ell {{p}} ( 0 < {p} < 1 ) minimization without weighting. Numerical illustrations are provided to demonstrate our new theoretical results.