Analysis of Singular Value Thresholding Algorithm for Matrix Completion

This paper provides analysis for convergence of the singular value thresholding algorithm for solving matrix completion and affine rank minimization problems arising from compressive sensing, signal processing, machine learning, and related topics. A necessary and sufficient condition for the convergence of the algorithm with respect to the Bregman distance is given in terms of the step size sequence {δ_k}_k∈N as ∑^∞_k=1 δ_k = ∞.  Concrete convergence rates in terms of Bregman distances and Frobenius norms of matrices are presented. Our novel analysis is carried out by giving an identity for the Bregman distance as the excess gradient descent objective function values and an error decomposition after viewing the algorithm as a mirror descent algorithm with a non-differentiable mirror map.