  We propose a new iterative greedy algorithm for reconstructions of sparse
signals with or without noisy perturbations in compressed sensing. The proposed
algorithm, called \emph{subspace thresholding pursuit} (STP) in this paper, is
a simple combination of subspace pursuit and iterative hard thresholding.
Firstly, STP has the theoretical guarantee comparable to that of $\ell_1$
minimization in terms of restricted isometry property. Secondly, with a tuned
parameter, on the one hand, when reconstructing Gaussian signals, it can
outperform other state-of-the-art reconstruction algorithms greatly; on the
other hand, when reconstructing constant amplitude signals with random signs,
it can outperform other state-of-the-art iterative greedy algorithms and even
outperform $\ell_1$ minimization if the undersampling ratio is not very large.
In addition, we propose a simple but effective method to improve the empirical
performance further if the undersampling ratio is large. Finally, it is showed
that other iterative greedy algorithms can improve their empirical performance
by borrowing the idea of STP.
