  Algorithms based on the hard thresholding principle have been well studied
with sounding theoretical guarantees in the compressed sensing and more general
sparsity-constrained optimization. It is widely observed in existing empirical
studies that when a restricted Newton step was used (as the debiasing step),
the hard-thresholding algorithms tend to meet halting conditions in a
significantly low number of iterations and are very efficient. Hence, the thus
obtained Newton hard-thresholding algorithms call for stronger theoretical
guarantees than for their simple hard-thresholding counterparts. This paper
provides a theoretical justification for the use of the restricted Newton step.
We build our theory and algorithm, Newton Hard-Thresholding Pursuit (NHTP), for
the sparsity-constrained optimization. Our main result shows that NHTP is
quadratically convergent under the standard assumption of restricted strong
convexity and smoothness. We also establish its global convergence to a
stationary point under a weaker assumption. In the special case of the
compressive sensing, NHTP effectively reduces to some of the existing
hard-thresholding algorithms with a Newton step. Consequently, our fast
convergence result justifies why those algorithms perform better than without
the Newton step. The efficiency of NHTP was demonstrated on both synthetic and
real data in compressed sensing and sparse logistic regression.
