On degeneration of tensors and algebras

Abstract

An important building block in all current asymptotically fast algorithms for matrix multiplication are tensors with low border rank, that is, tensors whose border rank is equal or very close to their size. To find new asymptotically fast algorithms for matrix multiplication, it seems to be important to understand those tensors whose border rank is as small as possible, so called tensors of minimal border rank. We investigate the connection between degenerations of associative algebras and degenerations of their structure tensors in the sense of Strassen. It allows us to describe an open subset of $n\times n\times n$ tensors of minimal border rank in terms of smoothability of commutative algebras. We describe the smoothable algebra associated to the Coppersmith-Winograd tensor and prove a lower bound for the border rank of the tensor used in the "easy construction" of Coppersmith and Winograd.

11 pages; accepted at MFCS 2016

Cited by in corpus (26)

• Hyperspectral Super-Resolution with Coupled Tucker Approximation: Recoverability and SVD-based algorithms
• Determinantal equations for secant varieties and the Eisenbud-Koh-Stillman conjecture
• Ideals of varieties parameterized by certain symmetric tensors
• A New Sampling Technique for Tensors
• An extension of the Moore-Penrose Inverse of a Tensor via the Einstein product
• Report on "Geometry and representation theory of tensors for computer science, statistics and other areas."
• Grassmann secants, identifiability, and linear systems of tensors
• On the typical rank of real binary forms
• Tensor decompositions and tensor equations over quaternion algebra
• Waring-like decompositions of polynomials - 1
• On the geometry of border rank algorithms for matrix multiplication and other tensors with symmetry
• Tensor Transpose and Its Properties
• Solving Tensor Structured Problems with Computational Tensor Algebra
• Prospects of tensor-based numerical modeling of the collective electrostatic potential in many-particle systems
• Tensor and Matrix Inversions with Applications
• The landscape of software for tensor computations
• Randomized algorithms for the low multilinear rank approximations of tensors
• Equations for the third secant variety of the Segre product of n projective spaces
• A Super-Resolution Framework for Tensor Decomposition
• On the geometry of geometric rank
• Complex tensor factorisation with PARAFAC2 for the estimation of brain connectivity from the EEG
• Tensor Completion through Total Variationwith Initialization from Weighted HOSVD
• Collineation varieties of tensors
• On the typical rank of real polynomials (or symmetric tensors) with a fixed border rank
• Canonical forms of Order-$k$ ($k=2,3,4$) Symmetric Tensors of Format $3\times \cdots \times 3$ Over Prime Fields
• Riemannian optimization on tensor products of Grassmann manifolds: Applications to generalized Rayleigh-quotients