Multihomogeneous Polynomial Decomposition using Moment Matrices

Alessandra Bernardi (1) , Jérôme Brachat (1) , Pierre Comon (2) , Bernard Mourrain (1)

1 GALAAD - Geometry, algebra, algorithms
2 Laboratoire d'Informatique, Signaux, et Systèmes de Sophia-Antipolis (I3S) / Equipe SIGNAL

In the paper, we address the important problem of tensor decomposition which can be seen as a generalisation of Sin- gular Value Decomposition for matrices. We consider gen- eral multilinear and multihomogeneous tensors. We show how to reduce the problem to a truncated moment matrix problem and we give a new criterion for flat extension of Quasi-Hankel matrices. We connect this criterion to the commutation characterisation of border bases. A new algo- rithm is described: it applies for general multihomogeneous tensors, extending the approach of J.J. Sylvester on binary forms. An example illustrates the algebraic operations in- volved in this approach and how the decomposition can be recovered from eigenvector computation.

Domaines: Géométrie algébrique [math.AG]

Alessandra Bernardi, Jérôme Brachat, Pierre Comon, Bernard Mourrain. Multihomogeneous Polynomial Decomposition using Moment Matrices. 36th international symposium on Symbolic and algebraic computation (ISSAC 11), Jun 2011, San Jose, United States. pp.35-42, ⟨10.1145/1993886.1993898⟩. ⟨inria-00638837⟩