Related papers: Improving the numerical stability of fast matrix m…
We implement an Augmented Lagrangian method to minimize a constrained least-squares cost function designed to find polyadic decompositions of the matrix multiplication tensor. We use this method to obtain new discrete decompositions and…
We deliver a call to arms for probabilistic numerical methods: algorithms for numerical tasks, including linear algebra, integration, optimization and solving differential equations, that return uncertainties in their calculations. Such…
We present a novel class of methods to compute functions of matrices or their action on vectors that are suitable for parallel programming. Solving appropriate simple linear systems of equations in parallel (or computing the inverse of…
We survey and unify recent results on the existence of accurate algorithms for evaluating multivariate polynomials, and more generally for accurate numerical linear algebra with structured matrices. By "accurate" we mean that the computed…
Matrix multiplication is a foundational operation in scientific computing and machine learning, yet its computational complexity makes it a significant bottleneck for large-scale applications. The shift to parallel architectures, primarily…
A key trait of stochastic optimizers is that multiple runs of the same optimizer in attempting to solve the same problem can produce different results. As a result, their performance is evaluated over several repeats, or runs, on the…
If several independent algorithms for a computer-calculated quantity exist, then one can expect their results (which differ because of numerical errors) to follow approximately Gaussian distribution. The mean of this distribution,…
Computation of the large sparse matrix exponential has been an important topic in many fields, such as network and finite-element analysis. The existing scaling and squaring algorithm (SSA) is not suitable for the computation of the large…
This paper extends the framework of randomised matrix multiplication to a coarser partition and proposes an algorithm as a complement to the classical algorithm, especially when the optimal probability distribution of the latter one is…
It is widely known that the lower bound for the algorithmic complexity of square matrix multiplication resorts to at least $n^2$ arithmetic operations. The justification builds upon the following reasoning: given that there are $2 n^2$…
It is well known that Strassen and Winograd algorithms can reduce the computational costs associated with dense matrix multiplication. We have already shown that they are also very effective for software-based multiple precision…
We present an algorithm to reduce the computational effort for the multiplication of a given matrix with an unknown column vector. The algorithm decomposes the given matrix into a product of matrices whose entries are either zero or integer…
A tight $\Omega((n/\sqrt{M})^{\log_2 7}M)$ lower bound is derived on the \io complexity of Strassen's algorithm to multiply two $n \times n$ matrices, in a two-level storage hierarchy with $M$ words of fast memory. A proof technique is…
Matrix powering is a fundamental computational primitive in linear algebra. It has widespread applications in scientific computing and engineering, and underlies the solution of time-homogeneous linear ordinary differential equations,…
We survey classical and recent developments in numerical linear algebra, focusing on two issues: computational complexity, or arithmetic costs, and numerical stability, or performance under roundoff error. We present a brief account of the…
Communication lower bounds have long been established for matrix multiplication algorithms. However, most methods of asymptotic analysis have either ignored the constant factors or not obtained the tightest possible values. Recent work has…
We survey the numerical stability of some fast algorithms for solving systems of linear equations and linear least squares problems with a low displacement-rank structure. For example, the matrices involved may be Toeplitz or Hankel. We…
Currently, many machine learning algorithms contain lots of iterations. When it comes to existing large-scale distributed systems, some slave nodes may break down or have lower efficiency. Therefore traditional machine learning algorithm…
A fast algorithm for the approximate multiplication of matrices with decay is introduced; the Sparse Approximate Matrix Multiply (SpAMM) reduces complexity in the product space, a different approach from current methods that economize…
The multiplication of matrices is an important arithmetic operation in computational mathematics. In the context of hierarchical matrices, this operation can be realized by the multiplication of structured block-wise low-rank matrices,…