General Tensor Decomposition, Moment Matrices and Applications

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

1 GALAAD - Geometry, algebra, algorithms
2 GIPSA-CICS - GIPSA - Communication Information and Complex Systems

The tensor decomposition addressed in this paper may be seen as a generalisation of Singular Value Decomposition of matrices. We consider general multilinear and multihomogeneous tensors. We show how to reduce the problem to a truncated moment matrix problem and give a new criterion for flat extension of Quasi-Hankel matrices. We connect this criterion to the commutation characterisation of border bases. A new algorithm is described. It applies for general multihomogeneous tensors, extending the approach of J.J. Sylvester to binary forms. An example illustrates the algebraic operations involved in this approach and how the decomposition can be recovered from eigenvector computation.

Domaines: Géométrie algébrique [math.AG]

Alessandra Bernardi, Jerome Brachat, Pierre Comon, Bernard Mourrain. General Tensor Decomposition, Moment Matrices and Applications. Journal of Symbolic Computation, 2013, 52 (May), pp.51-71. ⟨10.1016/j.jsc.2012.05.012⟩. ⟨inria-00590965v3⟩