Related papers: Faster Algorithms for Structured Matrix Multiplica…
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.…
An open-source C++ framework for discovering fast matrix multiplication schemes using the flip graph approach is presented. The framework supports multiple coefficient rings -- binary ($\mathbb{Z}_2$), modular ternary ($\mathbb{Z}_3$) and…
The flip graph algorithm is a method for discovering new matrix multiplication schemes by following random walks on a graph. We introduce a version of the flip graph algorithm for matrix multiplication schemes that admit certain symmetries.…
This study proposes the "adaptive flip graph algorithm", which combines adaptive searches with the flip graph algorithm for finding fast and efficient methods for matrix multiplication. The adaptive flip graph algorithm addresses the…
We introduce a new method for discovering matrix multiplication schemes based on random walks in a certain graph, which we call the flip graph. Using this method, we were able to reduce the number of multiplications for the matrix formats…
Matrix multiplication optimization remains a fundamental challenge in computational mathematics. This work introduces a novel approach that discovers matrix multiplication schemes whose coefficients are restricted to the set $\{-1, 0, 1\}$…
Continuing recent investigations of bounding the tensor rank of matrix multiplication using flip graphs, we present here improved rank bounds for about thirty matrix formats.
It is known that the multiplication of an $N \times M$ matrix with an $M \times P$ matrix can be performed using fewer multiplications than what the naive $NMP$ approach suggests. The most famous instance of this is Strassen's algorithm for…
Fast matrix multiplication can be described as searching for low-rank decompositions of the matrix--multiplication tensor. We design a neural architecture, \textsc{StrassenNet}, which reproduces the Strassen algorithm for $2\times 2$…
This paper presents a new state-of-the-art algorithm for exact $3\times3$ matrix multiplication over general non-commutative rings, achieving a rank-23 scheme with only 58 scalar additions. This improves the previous best additive…
We propose a strategy for the generation of fast and accurate versions of non-commutative recursive matrix multiplication algorithms. To generate these algorithms, we consider matrix and tensor norm bounds governing the stability and…
The quest for non-commutative matrix multiplication algorithms in small dimensions has seen a lot of recent improvements recently. In particular, the number of scalar multiplications required to multiply two $4\times4$ matrices was first…
In this paper, we present an improved algorithm for the All Pairs Non-decreasing Paths (APNP) problem on weighted simple digraphs, which has running time $\tilde{O}(n^{\frac{3 + \omega}{2}}) = \tilde{O}(n^{2.686})$. Here $n$ is the number…
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…
Algebraic matrix multiplication algorithms are designed by bounding the rank of matrix multiplication tensors, and then using a recursive method. However, designing algorithms in this way quickly leads to large constant factors: if one…
We show how to construct highly symmetric algorithms for matrix multiplication. In particular, we consider algorithms which decompose the matrix multiplication tensor into a sum of rank-1 tensors, where the decomposition itself consists of…
We discuss a generalization of the Cohn-Umans method, a potent technique developed for studying the bilinear complexity of matrix multiplication by embedding matrices into an appropriate group algebra. We investigate how the Cohn-Umans…
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…
We multiply two $n \times n$ matrices $S,T$ over semirings in the Congested Clique model, where $n$ fully connected nodes communicate synchronously using $O(\log n)$-bit messages, within $O(nz(S)^{1/3} nz(T)^{1/3}/n + 1)$ rounds of…
We present a non-commutative algorithm for the multiplication of a 2 x 2 block-matrix by its adjoint, defined by a matrix ring anti-homomorphism. This algorithm uses 5 block products (3 recursive calls and 2 general products)over C or in…