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

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