Generalized Additive Decompositions of Symmetric Tensors
Enrica Barrilli, Bernard Mourrain, Daniele Taufer
TL;DR
This work advances the Generalized Additive Decomposition ($GAD$) framework for symmetric tensors by tying $GAD$-rank to Catalecticant-rank under regularity and by providing a novel apolar-scheme description via polynomial-exponential duals. It introduces a geometric interpretation of $GAD$s as secant representations to osculating varieties of the Veronese, alongside an explicit algebraic structure using Artinian algebras, Hankel operators, and Schur factorizations. A numerically robust algorithm, gad_decompose, is developed to compute unique minimal $GAD$s (when regularity is favorable) and is demonstrated on Waring and tangential decompositions with strong stability, extending eigen-based methods to multiple points. The results yield a tight link between $GAD$-rank, cactus rank, and apolar schemes, enabling reliable minimal decompositions under regularity conditions and suggesting paths for handling obstructions via moment-extension techniques in future work.
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.