Investigation into Matrix Factorization when Elements are Unknown Technical Report

The problem of low-rank matrix factorization has seen significant attention in recent computer vision research. Problems that use factorization to find solutions include structure from motion, non-rigid object tracking and illumination based reconstructions. Matrix decomposition algorithms, such as singular value decomposition, can be used to obtain the factorizations when all the input data are known, reliably finding the global minimum of a certain cost function. However, in practice, missing data leads to incomplete matrices that prevent the application of standard factorization algorithms. To date, many algorithms have been proposed to deal with the missing data problem. This report presents the results of an investigation into these algorithms and discusses their effectiveness. It is seen that they rarely find the global minimum. Newton based methods, which have not been previously applied to this problem, are investigated and shown to find the global minimum more reliably, but do not fulfil the expectations one may have of optimization routines. However, it is argued that they are more easily extended to overcome the shortfalls of the basic approach than the other algorithms reviewed. Furthermore, the suitability of the global minimum as a solution is covered, creating more avenues of investigation for the improvement of factorization schemes. Future research looking into such extensions is described with the aim of engineering a successful algorithm.