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In this paper, we present a fast implementation of the Singular ValueThresholding (SVT) algorithm for matrix completion. A rank-revealing randomizedsingular value decomposition (R3SVD) algorithm is used to adaptively carry outpartial singular value decomposition (SVD) to fast approximate the SVT operatorgiven a desired, fixed precision. We extend the R3SVD algorithm to a recyclingrank revealing randomized singular value decomposition (R4SVD) algorithm byreusing the left singular vectors obtained from the previous iteration as theapproximate basis in the current iteration, where the computational cost forpartial SVD at each SVT iteration is significantly reduced. A simulatedannealing style cooling mechanism is employed to adaptively adjust the low-rankapproximation precision threshold as SVT progresses. Our fast SVTimplementation is effective in both large and small matrices, which isdemonstrated in matrix completion applications including image recovery andmovie recommendation system.

A Fast Implementation of Singular Value Thresholding Algorithm using Recycling Rank Revealing Randomized Singular Value Decomposition