Estimation of rank deficient matrices from partial observations: Two-step iterative algorithms

Abstract

Several computer vision applications require estimating a rank deficient matrix from noisy observations of its entries. When the observation matrix has no missing data, the LS solution of such problem is known to be given by the SVD. However, in practice, when several entries of the matrix are not observed, the problem has no closed form solution; In this paper, we study two iterative algorithms for minimizing the non-linear LS cost function obtained when estimating rank deficient matrices from partial observations. In the first algorithm, the iterations are the well known Expectation and Maximization (EM) steps that have succeeded in several estimation problems with missing data. The second algorithm, which we call Row-Column (RC), estimates, in alternate steps, the row and column spaces of the solution matrix. Our conclusions are that RC performs better than EM in what respects to the robustness to the initialization and to the convergence speed. We also demonstrate the algorithms when inferring 3D structure from video sequences. © Springer-Verlag Berlin Heidelberg 2003.