3,326 research outputs found
Tensor decomposition and homotopy continuation
A computationally challenging classical elimination theory problem is to
compute polynomials which vanish on the set of tensors of a given rank. By
moving away from computing polynomials via elimination theory to computing
pseudowitness sets via numerical elimination theory, we develop computational
methods for computing ranks and border ranks of tensors along with
decompositions. More generally, we present our approach using joins of any
collection of irreducible and nondegenerate projective varieties
defined over . After computing
ranks over , we also explore computing real ranks. Various examples
are included to demonstrate this numerical algebraic geometric approach.Comment: We have added two examples: A Coppersmith-Winograd tensor, Matrix
multiplication with zeros. (26 pages, 1 figure
On deflation and multiplicity structure
This paper presents two new constructions related to singular solutions of
polynomial systems. The first is a new deflation method for an isolated
singular root. This construction uses a single linear differential form defined
from the Jacobian matrix of the input, and defines the deflated system by
applying this differential form to the original system. The advantages of this
new deflation is that it does not introduce new variables and the increase in
the number of equations is linear in each iteration instead of the quadratic
increase of previous methods. The second construction gives the coefficients of
the so-called inverse system or dual basis, which defines the multiplicity
structure at the singular root. We present a system of equations in the
original variables plus a relatively small number of new variables that
completely deflates the root in one step. We show that the isolated simple
solutions of this new system correspond to roots of the original system with
given multiplicity structure up to a given order. Both constructions are
"exact" in that they permit one to treat all conjugate roots simultaneously and
can be used in certification procedures for singular roots and their
multiplicity structure with respect to an exact rational polynomial system.Comment: arXiv admin note: substantial text overlap with arXiv:1501.0508
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