Recovery of Sparse Signals Using Multiple Orthogonal Least Squares

Abstract

Sparse recovery aims to reconstruct sparse signals from compressed linear measurements. In this paper, we propose a sparse recovery algorithm called multiple orthogonal least squares (MOLS), which extends the well-known orthogonal least squares (OLS) algorithm by allowing multiple <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$L$</tex-math> </inline-formula> indices to be selected per iteration. Owing to its ability to catch multiple support indices in each selection, MOLS often converges in fewer iterations and hence improves the computational efficiency over the conventional OLS algorithm. Theoretical analysis shows that MOLS ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$L &gt; 1$</tex-math></inline-formula> ) performs exact recovery of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$K$</tex-math> </inline-formula> -sparse signals ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$K &gt; 1$</tex-math></inline-formula> ) in at most <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$K$</tex-math></inline-formula> iterations, provided that the sensing matrix obeys the restricted isometry property with isometry constant <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\delta _{LK} &lt; {\sqrt{L}}/({\sqrt{K} + 2 \sqrt{L}}).$</tex-math></inline-formula> When <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$L = 1,$</tex-math></inline-formula> MOLS reduces to the conventional OLS algorithm and our analysis shows that exact recovery is guaranteed under <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX"> $\delta_{K +1} &lt; 1 / (\sqrt{K} + 2)$</tex-math></inline-formula> . This condition is nearly optimal with respect to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\delta _{K+1}$</tex-math></inline-formula> in the sense that, even with a small relaxation (e.g., <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\delta_{K + 1} = 1 / \sqrt{K}$</tex-math> </inline-formula> ), exact recovery with OLS may not be guaranteed. The recovery performance of MOLS in the noisy scenario is also studied. It is shown that stable recovery of sparse signals can be achieved with the MOLS algorithm when the signal-to-noise ratio scales linearly with the sparsity level of input signals.