An iterative algorithm for singular value decomposition on noisy incomplete matrices

In this paper, we propose a simple iterative algorithm, called iSVD, for estimating the singular value decomposition (SVD) of a noisy incomplete given matrix. The iSVD relies on first order optimization over orthogonal manifolds and automatically estimates the rank of the SVD. The main goal here is to estimate the singular vectors through optimization in the right space, which is the space of the orthogonal matrix manifolds. The rank estimation is based on the ratio between estimated large singular values and the sum of all singular values. We empirically evaluate the iSVD on synthetic matrices and image reconstruction tasks. The evaluation shows that the iSVD is comparable to the recently introduced methods for matrix completion such as singular value thresholding (SVT) and fixed-point iteration with approximate SVD (FPCA).