A Sharp Condition for Exact Support Recovery With Orthogonal Matching Pursuit

Support recovery of sparse signals from noisy measurements with orthogonal matching pursuit (OMP) has been extensively studied. In this paper, we show that for any <inline-formula><tex-math notation="LaTeX">$K$</tex-math> </inline-formula>-sparse signal <inline-formula><tex-math notation="LaTeX">${\boldsymbol{x}}$</tex-math> </inline-formula>, if a sensing matrix <inline-formula><tex-math notation="LaTeX">$\boldsymbol{A}$</tex-math> </inline-formula> satisfies the restricted isometry property (RIP) with restricted isometry constant <inline-formula> <tex-math notation="LaTeX">$\delta _{K+1} < 1/\sqrt{K+1}$</tex-math></inline-formula>, then under some constraints on the minimum magnitude of nonzero elements of <inline-formula><tex-math notation="LaTeX">${\boldsymbol{x}}$ </tex-math></inline-formula>, OMP exactly recovers the support of <inline-formula><tex-math notation="LaTeX"> ${\boldsymbol{x}}$</tex-math></inline-formula> from its measurements <inline-formula><tex-math notation="LaTeX"> ${\boldsymbol{y}}=\boldsymbol{A}{\boldsymbol{x}}+\boldsymbol{v}$</tex-math></inline-formula> in <inline-formula> <tex-math notation="LaTeX">$K$</tex-math></inline-formula> iterations, where <inline-formula><tex-math notation="LaTeX"> $\boldsymbol{v}$</tex-math></inline-formula> is a noise vector that is <inline-formula><tex-math notation="LaTeX">$\ell _2$</tex-math></inline-formula> or <inline-formula><tex-math notation="LaTeX">$\ell _{\infty }$</tex-math> </inline-formula> bounded. This sufficient condition is sharp in terms of <inline-formula><tex-math notation="LaTeX"> $\delta _{K+1}$</tex-math></inline-formula> since for any given positive integer <inline-formula> <tex-math notation="LaTeX">$K$</tex-math></inline-formula> and any <inline-formula><tex-math notation="LaTeX"> $1/\sqrt{K+1}\leq \delta <1$</tex-math></inline-formula>, there always exists a matrix <inline-formula> <tex-math notation="LaTeX">$\boldsymbol{A}$</tex-math></inline-formula> satisfying the RIP with <inline-formula> <tex-math notation="LaTeX">$\delta _{K+1}=\delta$</tex-math></inline-formula> for which OMP fails to recover a <inline-formula><tex-math notation="LaTeX">$K$</tex-math></inline-formula>-sparse signal <inline-formula> <tex-math notation="LaTeX">${\boldsymbol{x}}$</tex-math></inline-formula> in <inline-formula><tex-math notation="LaTeX"> $K$</tex-math></inline-formula> iterations. Also, our constraints on the minimum magnitude of nonzero elements of <inline-formula><tex-math notation="LaTeX">${\boldsymbol{x}}$</tex-math></inline-formula> are weaker than existing ones. Moreover, we propose worst case necessary conditions for the exact support recovery of <inline-formula> <tex-math notation="LaTeX">${\boldsymbol{x}}$</tex-math></inline-formula>, characterized by the minimum magnitude of the nonzero elements of <inline-formula><tex-math notation="LaTeX">${\boldsymbol{x}}$</tex-math></inline-formula>.

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