Generalized Additive Decompositions of Symmetric Tensors

TL;DR

This paper introduces a geometric and algebraic framework for the generalized additive decomposition of symmetric tensors, providing methods to measure, compute, and ensure minimality and uniqueness of such decompositions.

Contribution

It offers a new linear algebra approach to determine GAD-rank, characterizes the apolar scheme, and develops a stable numerical algorithm for symmetric tensor decomposition.

Findings

GAD-rank coincides with Catalecticant matrix rank under regularity.

The apolar scheme is explicitly described as a polynomial-exponential annihilator.

The numerical algorithm effectively computes Waring and tangential decompositions.

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.