Tensor decomposition and homotopy continuation

Alessandra Bernardi (1) , Noah S. Daleo (2) , Jonathan D. Hauenstein (3) , Bernard Mourrain (4)

1 Dipartimento di Matematica [Bologna]
2 Department of Mathematics [Worcester]
3 Department of Applied and Computational Mathematics and Statistics [Notre Dame]
4 AROMATH - AlgebRe, geOmetrie, Modelisation et AlgoriTHmes

A computationally challenging classical elimination theory problem is to compute polyno-mials 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 X1,. .. , Xk ⊂ P N defined over C. After computing ranks over C, we also explore computing real ranks. Various examples are included to demonstrate this numerical algebraic geometric approach.

Mots clés: tensor rank; homotopy continuation; witness sets; numerical algebraic geometry; joins; secant varieties; numerical elimination theory

Domaines: Géométrie algébrique [math.AG]

Alessandra Bernardi, Noah S. Daleo, Jonathan D. Hauenstein, Bernard Mourrain. Tensor decomposition and homotopy continuation. Differential Geometry and its Applications, 2017, 55, pp.78-105. ⟨10.1016/j.difgeo.2017.07.009⟩. ⟨hal-01250398⟩