On Recovery of Sparse Signals With Prior Support Information via Weighted ℓₚ-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.