Related papers: On fast multiplication of a matrix by its transpos…
This paper presents an efficient technique for matrix-vector and vector-transpose-matrix multiplication in distributed-memory parallel computing environments, where the matrices are unstructured, sparse, and have a substantially larger…
Randomized sampling has recently been demonstrated to be an efficient technique for computing approximate low-rank factorizations of matrices for which fast methods for computing matrix vector products are available. This paper describes an…
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.…
Frugal computing is becoming an important topic for environmental reasons. In this context, several techniques have been proposed to reduce the storage of scientific data by dedicated compression methods specially tailored for arrays of…
The Strassen $2\times2$ matrix multiplication algorithm arises from the volume form on the 3-dimensional quotient space of the $2\times 2$ matrices by the multiples of identity.
We present new algorithms to detect and correct errors in the lower-upper factorization of a matrix, or the triangular linear system solution, over an arbitrary field. Our main algorithms do not require any additional information or…
While existing algorithms may be used to solve a linear system over a general field in matrix-multiplication time, the complexity of constructing a symmetric triangular factorization (LDL) has received relatively little formal study. The…
This paper introduces a new fast algorithm for the 8-point discrete cosine transform (DCT) based on the summation-by-parts formula. The proposed method converts the DCT matrix into an alternative transformation matrix that can be decomposed…
The polynomial multiplication problem has attracted considerable attention since the early days of computer algebra, and several algorithms have been designed to achieve the best possible time complexity. More recently, efforts have been…
We show a new algorithm and its implementation for multiplying bit-polynomials of large degrees. The algorithm is based on evaluating polynomials at a specific set comprising a natural set for evaluation with additive FFT and a high order…
In 1969 Strassen showed surprisingly that it is possible to multiply two 2 x 2 matrices using seven multiplications and 18 additions, instead of the naive eight multiplications and four additions. The number of additions was later reduced…
Fast linear transforms are ubiquitous in machine learning, including the discrete Fourier transform, discrete cosine transform, and other structured transformations such as convolutions. All of these transforms can be represented by dense…
We study the complexity of polynomial multiplication over arbitrary fields. We present a unified approach that generalizes all known asymptotically fastest algorithms for this problem. In particular, the well-known algorithm for…
Additive Fourier Transform is sdudied. A fast multiplication algorithm for polynomials over the binary field is given. The bit complexity of the algorithm is $O(n(log n)(\log\log n)^2)$.
We present a new algorithm for fast matrix multiplication using tensor decompositions which have special features. Thanks to these features we obtain exponents lower than what the rank of the tensor decomposition suggests. In particular for…
We present an algorithm to perform a simultaneous modular reduction of several residues. This algorithm is applied fast modular polynomial multiplication. The idea is to convert the $X$-adic representation of modular polynomials, with $X$…
Hierarchical matrices provide a highly memory-efficient way of storing dense linear operators arising, for example, from boundary element methods, particularly when stored in the H^2 format. In such data-sparse representations, iterative…
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…
The hierarchical matrix framework partitions matrices into subblocks that are either small or of low numerical rank, enabling linear storage complexity and efficient matrix-vector multiplication. This work focuses on the $H^2$-matrix format…
We present two simple ways of reducing the number of parameters and accelerating the training of large Long Short-Term Memory (LSTM) networks: the first one is "matrix factorization by design" of LSTM matrix into the product of two smaller…