  We describe several algorithms for matrix completion and matrix approximation
when only some of its entries are known. The approximation constraint can be
any whose approximated solution is known for the full matrix. For low rank
approximations, similar algorithms appears recently in the literature under
different names. In this work, we introduce new theorems for matrix
approximation and show that these algorithms can be extended to handle
different constraints such as nuclear norm, spectral norm, orthogonality
constraints and more that are different than low rank approximations. As the
algorithms can be viewed from an optimization point of view, we discuss their
convergence to global solution for the convex case. We also discuss the optimal
step size and show that it is fixed in each iteration. In addition, the derived
matrix completion flow is robust and does not require any parameters. This
matrix completion flow is applicable to different spectral minimizations and
can be applied to physics, mathematics and electrical engineering problems such
as data reconstruction of images and data coming from PDEs such as Helmholtz
equation used for electromagnetic waves.
