Title: Low rank solutions to differentiable systems over matrices and applications,
Abstract: Differentiable systems in this paper means systems of equations that are
described by differentiable real functions in real matrix variables. This paper
proposes algorithms for finding minimal rank solutions to such systems over
(arbitrary and/or several structured) matrices by using the Levenberg-Marquardt
method (LM-method) for solving least squares problems. We then apply these
algorithms to solve several engineering problems such as the low-rank matrix
completion problem and the low-dimensional Euclidean embedding one. Some
numerical experiments illustrate the validity of the approach.
On the other hand, we provide some further properties of low rank solutions
to systems linear matrix equations. This is useful when the differentiable
function is linear or quadratic.