Real root finding for low rank linear matrices

Abstract : The problem of finding m × s matrices (with m ≥ s) of rank r in a real affine subspace of dimension n has many applications in information and systems theory, where low rank is synonymous of structure and parsimony. We design computer algebra algorithms 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 essentially polynomial in n+m(s−r) ; it improves on the state-of-the-art in the field. Moreover, computer experiments show the practical efficiency of our approach.
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Submitted on : Wednesday, October 25, 2017 - 10:54:59 AM
Last modification on : Saturday, July 27, 2019 - 1:18:12 AM

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Didier Henrion, Simone Naldi, Mohab Safey El Din. Real root finding for low rank linear matrices. Applicable Algebra in Engineering, Communication and Computing, Springer Verlag, 2019, ⟨10.1007/s00200-019-00396-w⟩. ⟨hal-01159210v2⟩

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