Generalized Additive Decompositions of Symmetric Tensors

Abstract: This article addresses the Generalized Additive Decomposition (GAD) of symmetric tensors, that is, degree-$d$ forms $f \in \mathcal{S}_d$. From a geometric perspective, a GAD corresponds to representing a point on a secant of osculating varieties to the Veronese variety, providing a compact and structured description of a tensor that captures its intrinsic algebraic properties. We provide a linear algebra method for measuring the GAD size and prove that the minimal achievable size, which we call the GAD-rank of the considered tensor, coincides with the rank of suitable Catalecticant matrices, under certain regularity assumptions. We provide a new explicit description of the apolar scheme associated with a GAD as the annihilator of a polynomial-exponential series. We show that if the Castelnuovo-Mumford regularity of this scheme is sufficiently small, then both the GAD and the associated apolar scheme are minimal and unique. Leveraging these results, we develop a numerical GAD algorithm for symmetric tensors that effectively exploits the underlying algebraic structure, extending existing algebraic approaches based on eigen computation to the treatment of multiple points. We illustrate the effectiveness and numerical stability of such an algorithm through several examples, including Waring and tangential decompositions.