Related papers: A more accurate rational non-commutative algorithm…
Recently, reinforcement algorithms discovered new algorithms that really jump-started a wave of excitements and a flourishing of publications. However, there is little on implementations, applications, and, especially, no absolute…
We present a non-commutative algorithm for the product of 3x5 by 5x5 matrices using 58 multiplications. This algorithm allows to construct a non-commutative algorithm for multiplying 5x5 (resp. 10x10, 15x15) matrices using 98 (resp. 686,…
Numerical approximate computation can solve large and complex problems fast. It has the advantage of high efficiency. However it only gives approximate results, whereas we need exact results in many fields. There is a gap between…
Recently, Pagh presented a randomized approximation algorithm for the multiplication of real-valued matrices building upon work for detecting the most frequent items in data streams. We continue this line of research and present new {\em…
Renewed interest in mixed-precision algorithms has emerged due to growing data capacity and bandwidth concerns, as well as the advancement of GPUs, which enable significant speedup for low precision arithmetic. In light of this, we propose…
We explore new approaches for finding matrix multiplication algorithms in the commutative setting by adapting the flip graph technique: a method previously shown to be effective for discovering fast algorithms in the non-commutative case.…
The main purpose of this paper is to develop new algorithms for computing invariant rings in a general setting. This includes invariants of nonreductive groups but also of groups acting on algebras over certain rings. In particular, we…
We give two algorithms for output-sparse matrix multiplication (OSMM), the problem of multiplying two $n \times n$ matrices $A, B$ when their product $AB$ is promised to have at most $O(n^{\delta})$ many non-zero entries for a given value…
In this paper, we describe new methods to compute the radical (resp. real radical) of an ideal, assuming it complex (resp. real) variety is finite. The aim is to combine approaches for solving a system of polynomial equations with dual…
We present a non-commutative algorithm for multiplying 5x5 matrices using 99 multiplications. This algorithm is a minor modification of Makarov's algorithm which exhibit the previous best known bound with 100 multiplications.
An algorithm for solving smooth nonconvex optimization problems is proposed that, in the worst-case, takes $\mathcal{O}(\epsilon^{-3/2})$ iterations to drive the norm of the gradient of the objective function below a prescribed positive…
We present an optimized single-precision implementation of the Sparse Approximate Matrix Multiply (\SpAMM{}) [M. Challacombe and N. Bock, arXiv {\bf 1011.3534} (2010)], a fast algorithm for matrix-matrix multiplication for matrices with…
Classic cache-oblivious parallel matrix multiplication algorithms achieve optimality either in time or space, but not both, which promotes lots of research on the best possible balance or tradeoff of such algorithms. We study modern…
We demonstrate that, even when there are moderate overlaps in the inputs of sloppy or accurate double-word addition algorithms in the QD library, these algorithms still guarantee error bounds of $O(u^2(|a|+|b|))$ in faithful rounding.…
A robust algorithm for non-negative matrix factorization (NMF) is presented in this paper with the purpose of dealing with large-scale data, where the separability assumption is satisfied. In particular, we modify the Linear Programming…
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…
In this paper we analyze several inexact fast augmented Lagrangian methods for solving linearly constrained convex optimization problems. Mainly, our methods rely on the combination of excessive-gap-like smoothing technique developed in…
Matrix Completion is the problem of recovering an unknown real-valued low-rank matrix from a subsample of its entries. Important recent results show that the problem can be solved efficiently under the assumption that the unknown matrix is…
In 1969, Strassen shocked the world by showing that two n x n matrices could be multiplied in time asymptotically less than $O(n^3)$. While the recursive construction in his algorithm is very clear, the key gain was made by showing that 2 x…
In prior work, Gupta et al. (SPAA 2022) presented a distributed algorithm for multiplying sparse $n \times n$ matrices, using $n$ computers. They assumed that the input matrices are uniformly sparse--there are at most $d$ non-zeros in each…