Related papers: A non-commutative algorithm for multiplying (7 $\t…
We present a matrix-factorization algorithm that scales to input matrices with both huge number of rows and columns. Learned factors may be sparse or dense and/or non-negative, which makes our algorithm suitable for dictionary learning,…
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…
Fast matrix multiplication algorithms are asymptotically faster than the classical cubic-time algorithm, but they are often slower in practice. One important obstacle is the use of complex coefficients, which increases arithmetic overhead…
Optimized multiple precision basic linear computation, especially matrix multiplication, is crucial for solving ill-conditioned problems. The recently proposed Ozaki scheme, which implements accurate matrix multiplication using existing…
Many important applications across science, data analytics, and AI workloads depend on distributed matrix multiplication. Prior work has developed a large array of algorithms suitable for different problem sizes and partitionings including…
The multiplicative update (MU) algorithm has been extensively used to estimate the basis and coefficient matrices in nonnegative matrix factorization (NMF) problems under a wide range of divergences and regularizers. However, theoretical…
In this study, we explore the arithmetic properties of $b_{7^k}(n)$ for any $k\geq1$, which enumerates the partitions of $n$ where no part is divisible by $7^k$. By constructing generating functions for $b_{7^k}(n)$ over specific arithmetic…
Modern Neural Network (NN) architectures heavily rely on vast numbers of multiply-accumulate arithmetic operations, constituting the predominant computational cost. Therefore, this paper proposes a high-throughput, scalable and energy…
The distributed matrix multiplication problem with an unknown number of stragglers is considered, where the goal is to efficiently and flexibly obtain the product of two massive matrices by distributing the computation across N servers.…
The division operation is important for many areas of data processing. Especially considering today's demand for hardware accelerators for machine learning algorithms, there is a high demand for an efficient calculation of the division…
This paper describes a new accumulate-and-add multiplication algorithm. The method partitions one of the operands and re-combines the results of computations done with each of the partitions. The resulting design turns-out to be both…
This paper considers the problem of calculating the matrix multiplication of two massive matrices $\mathbf{A}$ and $\mathbf{B}$ distributedly. We provide a modulo technique that can be applied to coded distributed matrix multiplication…
We describe a quantum algorithm for preparing states that encode solutions of non-homogeneous linear partial differential equations. The algorithm is a continuous-variable version of matrix inversion: it efficiently inverts differential…
Ootomo, Ozaki, and Yokota [Int. J. High Perform. Comput. Appl., 38 (2024), p. 297-313] have proposed a strategy to recast a floating-point matrix multiplication in terms of integer matrix products. The factors A and B are split into integer…
In this paper, we propose a general framework to accelerate significantly the algorithms for nonnegative matrix factorization (NMF). This framework is inspired from the extrapolation scheme used to accelerate gradient methods in convex…
Quantum computing and modern tensor-based computing have a strong connection, which is especially demonstrated by simulating quantum computations with tensor networks. The other direction is less studied: quantum computing is not often…
One of prospective ways to find new fast algorithms of matrix multiplication is to study algorithms admitting nontrivial symmetries. In the work possible algorithms for multiplication of $3\times3$ matrices, admitting a certain group $G$…
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 $\{-1, 0, 1\}$…
Efficient multiple precision linear numerical computation libraries such as MPLAPACK are critical in dealing with ill-conditioned problems. Specifically, there are optimization methods for matrix multiplication, such as the Strassen…
A parallel algorithm has perfect strong scaling if its running time on P processors is linear in 1/P, including all communication costs. Distributed-memory parallel algorithms for matrix multiplication with perfect strong scaling have only…