By Enrica Barrilli (Inria)
Classical tensor decomposition—Waring’s model—seeks to express a symmetric tensor as a minimal sum of pure powers of linear forms. While elegant and foundational, this model is also inherently limited. In both geometry and applications, many tensors exhibit a far richer internal structure: multiplicities, tangent directions, and other subtle data that simple points cannot capture.
In our upcoming work, Generalized Additive Decomposition of Symmetric Tensors (with B. Mourrain, Inria d’Université Cote d’Azur, and D. Taufer, KU Leuven), we propose a new algorithm that computes Generalized Additive Decompositions (GADs)—a robust extension of Waring decompositions that embraces this higher-order behavior.
What sets GADs apart is their deep geometric and algebraic foundation. At the heart of our method is the Artinian Gorenstein algebra associated to the tensor via apolarity. We show that its decomposition into local algebras mirrors the decomposition of the tensor itself.
A key technical insight is the use of Schur decompositions of multiplication operators. When applied to the algebra, they reveal a block upper triangular structure, where each block isolates a local component of the tensor. This gives a precise, computable description of how the tensor is supported—and how it behaves—around each point.
The algorithm we introduce combines duality, truncated Hankel operators, and spectral techniques to compute GADs efficiently and stably. It generalizes classical methods while capturing a richer picture of the tensor’s geometry.
We believe this approach offers not just a new algorithm, but a new language for under-standing symmetric tensors.
The full paper is coming soon—stay tuned!