A Pseudo-Inverse-Based Hard Thresholding Algorithm for Sparse Signal Recovery
Jinming Wen, Hongyu Hè, Zihao He, Fumin Zhu
Abstract
Acquiring a sparse signal from an underdetermined linear system arises from numerous applications. Several hard thresholding algorithms have been developed for the sparse reconstruction. In particular, the so- called Newton-Step-based Iterative Hard Thresholding (NSIHT) and Newton-Step-based Hard Thresholding Pursuit (NSHTP) algorithms were developed by Meng <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">et al.</i> recently. Although they are efficient, a faster sparse recovery algorithm with a better reconstruction performance is still needed. In this paper, we first propose a Pseudo-inverse-based Hard Thresholding sparse signal recovery algorithm called PHT for short. Unlike the Iterative Hard Thresholding (IHT) algorithm which uses the gradient of the objective function to iteratively update the solution, our proposed algorithm utilizes the pseudo-inverse of the sensing matrix to iteratively update the solution. We then analyze the computational complexity of PHT and show that it is <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$O(n^{2}/m^{2})$ </tex-math></inline-formula> times faster than both NSIHT and NSHTP if they perform the same number of iterations, where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$m$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> are the number of rows and columns of the sensing matrix <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\boldsymbol {A}$ </tex-math></inline-formula> . Furthermore, we establish a sufficient condition of stable recovery of the sparse signal with PHT by using the restricted isometry property (RIP) of the sensing matrix. Finally, extensive experiments are conducted which indicate that our proposed algorithm PHT is much faster than both NSIHT and NSHTP with overall better recovery performance.