# Low rank solutions to differentiable systems over matrices and applications

## TL;DR

This paper introduces algorithms based on the Levenberg-Marquardt method for finding minimal rank solutions to differentiable matrix systems, with applications in matrix completion and Euclidean embedding, supported by numerical experiments.

## Contribution

It develops novel algorithms for low-rank solutions to differentiable matrix systems and explores their applications and properties, especially for linear and quadratic functions.

## Key findings

- Algorithms effectively find low-rank solutions in practical problems.
- Numerical experiments validate the approach.
- Properties of low-rank solutions are characterized for specific cases.

## 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.