Equivalent polyadic decompositions of matrix multiplication tensors

Decompositions of moment tensors

Asymptotic tensor rank of graph tensors: beyond matrix multiplication

We present an upper bound on the exponent of the asymptotic behaviour of the tensor rank of a family of tensors defined by the complete graph on k vertices. For k ≥ 4, we show that the exponent per edge is at most 0.77, outperforming the best known upper bound on the exponent per edge for matrix multiplication (k = 3), which is approximately 0.79. We raise the question whether for some k the ex...

Matrix Multiplication, Trilinear Decompositions, APA Algorithms, and Summation

Matrix multiplication (hereafter we use the acronym MM) is among the most fundamental operations of modern computations. The efficiency of its performance depends on various factors, in particular vectorization, data movement and arithmetic complexity of the computations, but here we focus just on the study of the arithmetic cost and the impact of this study on other areas of modern computing. ...

Orthogonal Rank Decompositions for Tensors

The theory of orthogonal rank decompositions for matrices is well understood, but the same is not true for tensors. For tensors, even the notions of orthogonality and rank can be interpreted several diierent ways. Tensor decompositions are useful in applications such as principal component analysis for multiway data. We present two types of orthogonal rank decompositions and describe methods to...

Computing the polyadic decomposition of nonnegative third order tensors

Computing the minimal polyadic decomposition (also often referred to as canonical decomposition, or sometimes Parafac) amounts to finding the global minimum of a coercive polynomial in many variables. In the case of arrays with nonnegative entries, the low-rank approximation problem is well posed. In addition, due to the large dimension of the problem, the decomposition can be rather efficientl...