On Minimal Subspaces in Tensor Representations

• Published in: Foundations of computational mathematics Vol. 12; no. 6; pp. 765 - 803
• Main Authors: Falcó, Antonio, Hackbusch, Wolfgang
• Format: Journal Article
• Language: English
• Published: New York Springer-Verlag 01.12.2012; Springer; Springer Nature B.V
• Subjects: Algebra; Applications of Mathematics; Approximation; Calculus; Computational mathematics; Computer Science; Economics; Linear and Multilinear Algebras; Math Applications in Computer Science; Mathematical analysis; Mathematical models; Mathematics; Mathematics and Statistics; Matrix Theory; Norms; Numerical Analysis; Representations; Subspaces; Tensors; Topological manifolds; Topology; 46B28; Minimal subspace; Tensor product; Tensor Banach space; 46A32; Weak closedness; 15A69; Numerical tensor calculus
• ISSN: 1615-3375; 1615-3383; 1615-3383
• DOI: 10.1007/s10208-012-9136-6

In this paper we introduce and develop the notion of minimal subspaces in the framework of algebraic and topological tensor product spaces. This mathematical structure arises in a natural way in the study of tensor representations. We use minimal subspaces to prove the existence of a best approximation, for any element in a Banach tensor space, by means of a tensor given in a typical representation format (Tucker, hierarchical, or tensor train). We show that this result holds in a tensor Banach space with a norm stronger than the injective norm and in an intersection of finitely many Banach tensor spaces satisfying some additional conditions. Examples using topological tensor products of standard Sobolev spaces are given.