  Iterative hard thresholding (IHT) has gained in popularity over the past
decades in large-scale optimization. However, convergence properties of this
method have only been explored recently in non-convex settings. In matrix
completion, existing works often focus on the guarantee of global convergence
of IHT via standard assumptions such as incoherence property and uniform
sampling. While such analysis provides a global upper bound on the linear
convergence rate, it does not describe the actual performance of IHT in
practice. In this paper, we provide a novel insight into the local convergence
of a specific variant of IHT for matrix completion. We uncover the exact linear
rate of IHT in a closed-form expression and identify the region of convergence
in which the algorithm is guaranteed to converge. Furthermore, we utilize
random matrix theory to study the linear rate of convergence of IHTSVD for
large-scale matrix completion. We find that asymptotically, the rate can be
expressed in closed form in terms of the relative rank and the sampling rate.
Finally, we present various numerical results to verify the aforementioned
theoretical analysis.
