Related papers: On fast multiplication of a matrix by its transpos…
In this work an efficient algorithm to perform a block decomposition (and so to compute the rank) of large dense rectangular matrices with entries in $\mathbb{F}_2$ is presented. Depending on the way the matrix is stored, the operations…
One of the most famous conjectures in computer algebra is that matrix multiplication might be feasible in not much more than quadratic time. The best known exponent is 2.376, due to Coppersmith and Winograd. Many attempts to solve this…
We improve the current best running time value to invert sparse matrices over finite fields, lowering it to an expected $O\big(n^{2.2131}\big)$ time for the current values of fast rectangular matrix multiplication. We achieve the same…
This work formalizes efficient Fast Fourier-based multiplication algorithms for polynomials in quotient rings such as $\mathbb{Z}_{m}[x]/\left<x^{n}-a\right>$, with $n$ a power of 2 and $m$ a non necessarily prime integer. We also present a…
We describe an efficient algorithm for computing the matrix vector products that appear in the numerical resolution of boundary integral equations in 2 space dimension. This work is an extension of the so-called Sparse Cardinal Sine…
Large language models have been widely adopted but require significant GPU memory for inference. We develop a procedure for Int8 matrix multiplication for feed-forward and attention projection layers in transformers, which cut the memory…
Advanced optimization algorithms such as Newton method and AdaGrad benefit from second order derivative or second order statistics to achieve better descent directions and faster convergence rates. At their heart, such algorithms need to…
Consider the special linear group of degree $2$ over an arbitrary finite field, acting on the full space of $2 \times 2$-matrices by transpose. We explicitly construct a generating set for the corresponding modular matrix invariant ring,…
In this paper, we consider the matrices approximated in H2 format. The direct solution, as well as the preconditioning, of systems with such matrices is a challenging problem. We propose a non-extensive sparse factorization of the H2 matrix…
A Fast algorithm for the Discrete Hartley Transform (DHT) is presented, which resembles radix-2 fast Fourier Transform (FFT). Although fast DHTs are already known, this new approach bring some light about the deep relationship between fast…
We show that Boolean matrix multiplication, computed as a sum of products of column vectors with row vectors, is essentially the same as Warshall's algorithm for computing the transitive closure matrix of a graph from its adjacency matrix.…
We consider a multi-block separable convex optimization problem with the linear constraints, where the objective function is the sum of m individual convex functions without overlapping variables. The linearized version of the generalized…
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…
Interested in formalizing the generation of fast running code for linear algebra applications, the authors show how an index-free, calculational approach to matrix algebra can be developed by regarding matrices as morphisms of a category…
We present the asymptotically fastest known algorithms for some basic problems on univariate polynomial matrices: rank, nullspace, determinant, generic inverse, reduced form. We show that they essentially can be reduced to two computer…
Matrix multiplication is a fundamental task in almost all computational fields, including machine learning and optimization, computer graphics, signal processing, and graph algorithms (static and dynamic). Twin-width is a natural complexity…
Fast approximations to matrix multiplication have the potential to dramatically reduce the cost of neural network inference. Recent work on approximate matrix multiplication proposed to replace costly multiplications with table-lookups by…
The matrix LU factorization algorithm is a fundamental algorithm in linear algebra. We propose a generalization of the LU and LEU algorithms to accommodate the case of a commutative domain and its field of quotients. This algorithm…
We use the well-known observation that the solutions of Jacobi's differential equation can be represented via non-oscillatory phase and amplitude functions to develop a fast algorithm for computing multi-dimensional Jacobi polynomial…
We propose a more accurate variant of an algorithm for multiplying 4x4 matrices using 48 multiplications over any ring containing an inverse of 2. This algorithm has an error bound exponent of only log 4 $\gamma$$\infty$,2 $\approx$ 2.386.…