  This paper concerns the problem of matrix completion, which is to estimate a
matrix from observations in a small subset of indices. We propose a calibrated
spectrum elastic net method with a sum of the nuclear and Frobenius penalties
and develop an iterative algorithm to solve the convex minimization problem.
The iterative algorithm alternates between imputing the missing entries in the
incomplete matrix by the current guess and estimating the matrix by a scaled
soft-thresholding singular value decomposition of the imputed matrix until the
resulting matrix converges. A calibration step follows to correct the bias
caused by the Frobenius penalty. Under proper coherence conditions and for
suitable penalties levels, we prove that the proposed estimator achieves an
error bound of nearly optimal order and in proportion to the noise level. This
provides a unified analysis of the noisy and noiseless matrix completion
problems. Simulation results are presented to compare our proposal with
previous ones.
