Related papers: Fast Matrix Multiplication Without Tears: A Constr…
Although reliable long precision floating-point arithmetic libraries such as QD and MPFR/GMP are necessary to solve ill-conditioned problems in numerical simulation, long precision BLAS-level computation such as matrix multiplication has…
There have been several algorithms designed to optimise matrix multiplication. From schoolbook method with complexity $O(n^3)$ to advanced tensor-based tools with time complexity $O(n^{2.3728639})$ (lowest possible bound achieved), a lot of…
Motivated by the philosophy and phenomenal success of compressed sensing, the problem of reconstructing a matrix from a sampling of its entries has attracted much attention recently. Such a problem can be viewed as an information-theoretic…
We analyze rank decompositions of the $3\times 3$ matrix multiplication tensor over $\mathbb{Z}/2\mathbb{Z}$. We restrict our attention to decompositions of rank $\le 21$, as only those decompositions will yield an asymptotically faster…
We study the capability of the Fast Fourier Transform (FFT) to accelerate exact and approximate matrix multiplication without using Strassen-like divide-and-conquer. We present a simple exact algorithm running in $O(n^{2.89})$ time, which…
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…
The well known algorithm of Volker Strassen for matrix multiplication can only be used for $(m2^k \times m2^k)$ matrices. For arbitrary $(n \times n)$ matrices one has to add zero rows and columns to the given matrices to use Strassen's…
This paper addresses the problem of finding the closest generalized essential matrix from a given $6\times 6$ matrix, with respect to the Frobenius norm. To the best of our knowledge, this nonlinear constrained optimization problem has not…
We present the submatrix method, a highly parallelizable method for the approximate calculation of inverse p-th roots of large sparse symmetric matrices which are required in different scientific applications. We follow the idea of…
Optimized multiple precision basic linear computation, especially matrix multiplication, is crucial for solving ill-conditioned problems. The recently proposed Ozaki scheme, which implements accurate matrix multiplication using existing…
We give a short proof for Strassen's result that the rank of the 2 by 2 matrix multiplication tensor is at most 7. The proof requires no calculations and also no pattern matching or other type of nontrivial verification, and is based solely…
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 non-commutative algorithm for multiplying (7x7) matrices using 250 multiplications and a non-commutative algorithm for multiplying (9x9) matrices using 520 multiplications. These algorithms are obtained using the same…
Cryptographic primitives have been used for various non-cryptographic objectives, such as eliminating or reducing randomness and interaction. We show how to use cryptography to improve the time complexity of solving computational problems.…
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…
Nonnegative Matrix Factorization consists in (approximately) factorizing a nonnegative data matrix by the product of two low-rank nonnegative matrices. It has been successfully applied as a data analysis technique in numerous domains, e.g.,…
We develop an automated framework for proving lower bounds on the bilinear complexity of matrix multiplication over finite fields. Our approach systematically combines orbit classification of the restricted first matrix and dynamic…
We study the problem of estimating precision matrices in Gaussian distributions that are multivariate totally positive of order two ($\mathrm{MTP}_2$). The precision matrix in such a distribution is an M-matrix. This problem can be…
Matrix factorization (MF) is a widely used collaborative filtering (CF) algorithm for recommendation systems (RSs), due to its high prediction accuracy, great flexibility and high efficiency in big data processing. However, with the…
Nonnegative matrix factorization (NMF) is a powerful technique for dimension reduction, extracting latent factors and learning part-based representation. For large datasets, NMF performance depends on some major issues: fast algorithms,…