Real root finding for low rank linear matrices

TL;DR

This paper introduces an efficient, exact computer algebra algorithm for finding low-rank matrices within real affine subspaces, with applications in systems theory, improving on existing methods in algebraic geometry.

Contribution

The paper develops a novel polynomial-time algorithm for low-rank matrix detection in affine spaces, combining advanced polynomial system solving with practical efficiency.

Findings

Algorithm is polynomial in key parameters.

Improves on state-of-the-art in algebraic geometry.

Practical experiments confirm efficiency.

Abstract

We consider $m \times s$ matrices (with $m\geq s$) in a real affine subspace of dimension $n$. The problem of finding elements of low rank in such spaces finds many applications in information and systems theory, where low rank is synonymous of structure and parsimony. We design computer algebra algorithms, based on advanced methods for polynomial system solving, to solve this problem efficiently and exactly: the input are the rational coefficients of the matrices spanning the affine subspace as well as the expected maximum rank, and the output is a rational parametrization encoding a finite set of points that intersects each connected component of the low rank real algebraic set. The complexity of our algorithm is studied thoroughly. It is polynomial in $\binom{n+m(s-r)}{n}$. It improves on the state-of-the-art in computer algebra and effective real algebraic geometry. Moreover, computer experiments show the practical efficiency of our approach.