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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 show that assuming the availability of the processor with variable precision arithmetic, we can compute matrix-by-matrix multiplications in $O(N^2log_2N)$ computational complexity. We replace the standard matrix-by-matrix multiplications…
Matrix multiplication (hereafter we use the acronym MM) is among the most fundamental operations of modern computations. The efficiency of its performance depends on various factors, in particular vectorization, data movement and arithmetic…
In response to a recent Nature article which announced an algorithm for multiplying $5\times5$-matrices over $\mathbb{Z}_2$ with only 96 multiplications, two fewer than the previous record, we present an algorithm that does the job with…
Multiplying matrices is among the most fundamental and compute-intensive operations in machine learning. Consequently, there has been significant work on efficiently approximating matrix multiplies. We introduce a learning-based algorithm…
To multiply astronomic matrices using parallel workers subject to straggling, we recommend interleaving checksums with some fast matrix multiplication algorithms. Nesting the parity-checked algorithms, we weave a product code flavor…
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 consider the techniques behind the current best algorithms for matrix multiplication. Our results are threefold. (1) We provide a unifying framework, showing that all known matrix multiplication running times since 1986 can be achieved…
We present a non-commutative algorithm for the multiplication of a 2x2-block-matrix by its transpose using 5 block products (3 recursive calls and 2 general products) over C or any finite field.We use geometric considerations on the space…
Fast algorithms for matrix multiplication, namely those that perform asymptotically fewer scalar operations than the classical algorithm, have been considered primarily of theoretical interest. Apart from Strassen's original algorithm, few…
This paper presents a new fast, highly scalable distributed matrix multiplication algorithm on Apache Spark, called Stark, based on Strassen's matrix multiplication algorithm. Stark preserves Strassen's 7 multiplications scheme in a…
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…
For every $p\leq n$ positive integer we obtain the lower bound $(3-\frac{1}{p+1})n^2-\big(2\binom{2p}{p+1}-\binom{2p-2}{p-1}+2\big)n$ for the rank of the $n\times n$ matrix multiplication. This bound improves the previous one…
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…
This article presents new properties of the mesh array for matrix multiplication. In contrast to the standard array that requires 3n-2 steps to complete its computation, the mesh array requires only 2n-1 steps. Symmetries of the mesh array…
Hierarchical matrices approximate a given matrix by a decomposition into low-rank submatrices that can be handled efficiently in factorized form. $\mathcal{H}^2$-matrices refine this representation following the ideas of fast multipole…
In this overview article we present several methods for multiplying matrices and the implementation of these methods in C. Also a little test program is given to compare their running time and the numerical stability. The methods are: naive…
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…
Matrix multiplication is a fundamental computation in many scientific disciplines. In this paper, we show that novel fast matrix multiplication algorithms can significantly outperform vendor implementations of the classical algorithm and…
Matrix multiplication consumes a large fraction of the time taken in many machine-learning algorithms. Thus, accelerator chips that perform matrix multiplication faster than conventional processors or even GPU's are of increasing interest.…