Related papers: Partial and weighted matrix multiplication
Whereas matrix rank is additive under direct sum, in 1981 Sch\"onhage showed that one of its generalizations to the tensor setting, tensor border rank, can be strictly subadditive for tensors of order three. Whether border rank is additive…
This paper presents a method to analyze the powers of a given trilinear form (a special kind of algebraic constructions also called a tensor) and obtain upper bounds on the asymptotic complexity of matrix multiplication. Compared with…
In the past few years, successive improvements of the asymptotic complexity of square matrix multiplication have been obtained by developing novel methods to analyze the powers of the Coppersmith-Winograd tensor, a basic construction…
We present a polynomial time algorithm to approximately scale tensors of any format to arbitrary prescribed marginals (whenever possible). This unifies and generalizes a sequence of past works on matrix, operator and tensor scaling. Our…
We consider the techniques behind the current best algorithms for matrix multiplication. Our results are threefold. (1) We provide a unifying framework, showing that all known matrix multiplication running times since 1986 can be achieved…
Since the seminal works of Strassen and Valiant it has been a central theme in algebraic complexity theory to understand the relative complexity of algebraic problems, that is, to understand which algebraic problems (be it bilinear maps…
Determining the asymptotic algebraic complexity of matrix multiplication, succinctly represented by the matrix multiplication exponent $\omega$, is a central problem in algebraic complexity theory. The best upper bounds on $\omega$, leading…
We show that the border support rank of the tensor corresponding to two-by-two matrix multiplication is seven over the complex numbers. We do this by constructing two polynomials that vanish on all complex tensors with format…
We establish basic information about border rank algorithms for the matrix multiplication tensor and other tensors with symmetry. We prove that border rank algorithms for tensors with symmetry (such as matrix multiplication and the…
We study the algorithmic problem of multiplying large matrices that are rectangular. We prove that the method that has been used to construct the fastest algorithms for rectangular matrix multiplication cannot give algorithms with…
In this paper, the canonical polyadic (CP) decomposition of tensors that corresponds to matrix multiplications is studied. Finding the rank of these tensors and computing the decompositions is a fundamental problem of algebraic complexity…
We introduce Bayesian multi-tensor factorization, a model that is the first Bayesian formulation for joint factorization of multiple matrices and tensors. The research problem generalizes the joint matrix-tensor factorization problem to…
We define tensors, corresponding to cubic polynomials, which have the same exponent $\omega$ as the matrix multiplication tensor. In particular, we study the symmetrized matrix multiplication tensor $sM_n$ defined on an $n\times n$ matrix…
We develop a notion of {\em inner rank} as a tool for obtaining lower bounds on the rank of matrix multiplication tensors. We use it to give a short proof that the border rank (and therefore rank) of the tensor associated with $n\times n$…
Moment polytopes of tensors, the study of which is deeply rooted in invariant theory, representation theory and symplectic geometry, have found relevance in numerous places, from quantum information (entanglement polytopes) and algebraic…
Multilinear and tensor decompositions are a popular tool in linear and multilinear algebra and have a wide range of important applications to modern computing. Our paper of 1972 presented the first nontrivial application of such…
The commutation matrix was first introduced in statistics as a transposition matrix by Murnaghan in 1938. In this paper, we first investigate the commutation matrix which is employed to transform a matrix into its transpose. We then extend…
A generalization of recent group-theoretic matrix multiplication algorithms to an analogue of the theory of partial matrix multiplication is presented. We demonstrate that the added flexibility of this approach can in some cases improve…
The tensor decomposition addressed in this paper may be seen as a generalisation of Singular Value Decomposition of matrices. We consider general multilinear and multihomogeneous tensors. We show how to reduce the problem to a truncated…
A well studied problem in algebraic complexity theory is the determination of the complexity of problems relying on evaluations of bilinear maps. One measure of the complexity of a bilinear map (or 3-tensor) is the optimal number of…