Interest zone matrix approximation

An algorithm for matrix approximation, when only some of its entries are taken into consideration, is described. The approximation constraint can be any whose approximated solution is known for the full matrix. For low rank approximations, this type of algorithms appears recently in the literature under different names, where it usually uses the Expectation-Maximization algorithm that maximizes the likelihood for the missing entries. In this paper, the algorithm is extended to different cases other than low rank approximations under Frobenius norm, such as minimizing the Frobenius norm under nuclear norm constraint, spectral norm constraint, orthogonality constraint and more. The geometric interpretation of the proposed approximation process along with its optimality for convex constraints is also discussed. In addition, it is shown how the approximation algorithm can be used for matrix completion as well, under a variety of spectral regularizations. Its applications to physics, electrical engineering and data interpolation problems are also described.