  Hard thresholding pursuit (HTP) is a recently proposed iterative sparse
recovery algorithm which is a result of combination of a support selection step
from iterated hard thresholding (IHT) and an estimation step from the
orthogonal matching pursuit (OMP). HTP has been seen to enjoy improved recovery
guarantee along with enhanced speed of convergence. Much of the success of HTP
can be attributed to its improved support selection capability due to the
support selection step from IHT. In this paper, we propose a generalized HTP
algorithm, called regularized HTP (RHTP), where the support selection step of
HTP is replaced by a IHT-type support selection where the cost function is
replaced by a regularized cost function, while the estimation step continues to
use the least squares function. With decomposable regularizer, satisfying
certain regularity conditions, the RHTP algorithm is shown to produce a
sequence dynamically equivalent to a sequence evolving according to a HTP-like
evolution, where the identification stage has a gradient premultiplied with a
time-varying diagonal matrix. RHTP is also proven, both theoretically, and
numerically, to enjoy faster convergence vis-a-vis HTP with both noiseless and
noisy measurement vectors.
