Related papers: Matrix multiplication algorithms from group orbits

The (asymptotic) complexity of matrix multiplication (over the complex field) is measured by a real parameter w > 0, called the exponent of matrix multiplication (over the complex field), which is defined to be the smallest real number w >…

We study the symmetric outer product decomposition which decomposes a fully (partially) symmetric tensor into a sum of rank-one fully (partially) symmetric tensors. We present iterative algorithms for the third-order partially symmetric…

We solve tensor balancing, rescaling an Nth order nonnegative tensor by multiplying N tensors of order N - 1 so that every fiber sums to one. This generalizes a fundamental process of matrix balancing used to compare matrices in a wide…

We propose a new algorithm for tensor decomposition, based on Jennrich's algorithm, and apply our new algorithmic ideas to blind deconvolution and Gaussian mixture models. Our first contribution is a simple and efficient algorithm to…

We consider the problem of estimating a rank-one nonsymmetric matrix under additive white Gaussian noise. The matrix to estimate can be written as the outer product of two vectors and we look at the special case in which both vectors are…

We present explicit algorithms for computing structured matrix-vector products that are optimal in the sense of Strassen, i.e., using a provably minimum number of multiplications. These structures include Toeplitz/Hankel/circulant,…

In 1969, V. Strassen improves the classical~2x2 matrix multiplication algorithm. The current upper bound for 3x3 matrix multiplication was reached by J.B. Laderman in 1976. This note presents a geometric relationship between Strassen and…

For a given matrix subspace, how can we find a basis that consists of low-rank matrices? This is a generalization of the sparse vector problem. It turns out that when the subspace is spanned by rank-1 matrices, the matrices can be obtained…

We perform forward error analysis for a large class of recursive matrix multiplication algorithms in the spirit of [D. Bini and G. Lotti, Stability of fast algorithms for matrix multiplication, Numer. Math. 36 (1980), 63--72]. As a…

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…

Fast matrix multiplication algorithms may be useful, provided that their running time is good in practice. Particularly, the leading coefficient of their arithmetic complexity needs to be small. Many sub-cubic algorithms have large leading…

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 present an approximate algorithm for matrix multiplication based on matrix sketching techniques. First one of the matrix is chosen and sparsified using the online matrix sketching algorithm, and then the matrix product is calculated…

An equivalence relation in the symmetric group, where is a positive integer has been considered. An algorithm for calculation of the number of the equivalence classes by this relation for arbitrary integer has been described.

We give explicit low-rank bilinear non-commutative schemes for multiplying structured $n \times n$ matrices with $2 \leq n \leq 5$, which serve as building blocks for recursive algorithms with improved multiplicative factors in asymptotic…

We further develop the group-theoretic approach to fast matrix multiplication introduced by Cohn and Umans, and for the first time use it to derive algorithms asymptotically faster than the standard algorithm. We describe several families…

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…

We show that the product of an nx3 matrix and a 3x3 matrix over a commutative ring can be computed using 6n+3 multiplications. For two 3x3 matrices this gives us an algorithm using 21 multiplications. This is an improvement with respect to…

We investigate the problem of factorizing a matrix into several sparse matrices and propose an algorithm for this under randomness and sparsity assumptions. This problem can be viewed as a simplification of the deep learning problem where…

We propose a new numerical algorithm for computing the tensor rank decomposition or canonical polyadic decomposition of higher-order tensors subject to a rank and genericity constraint. Reformulating this computational problem as a system…