Real root finding for low rank linear matrices

We consider m × s matrices (with m ≥ 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, ba...

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Bibliographic Details
Published in:Applicable algebra in engineering, communication and computing Vol. 31; no. 2; pp. 101 - 133
Main Authors: Henrion, Didier, Naldi, Simone, Din, Mohab Safey El
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.03.2020
Springer Nature B.V
Springer Verlag
Subjects:
Algebraic Geometry
Algorithms
Artificial Intelligence
Computer algebra
Computer Hardware
Computer Science
Mathematical analysis
Mathematics
Matrix methods
Original Paper
Parameterization
Polynomials
Symbolic and Algebraic Manipulation
Symbolic Computation
System theory
Systems theory
Theory of Computation
13-XX
68W30
Polynomial system solving
14Q20
12Y05
Low rank matrices
Real algebraic geometry
Symbolic computation
ISSN:0938-1279, 1432-0622
Online Access:Get full text
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Summary:We consider m × s matrices (with m ≥ 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 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.
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ISSN:0938-1279
1432-0622
DOI:10.1007/s00200-019-00396-w