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A random matrix is likely to be well conditioned, and motivated by this well known property we employ random matrix multipliers to advance some fundamental matrix computations. This includes numerical stabilization of Gaussian elimination…
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…
This paper extends the framework of randomised matrix multiplication to a coarser partition and proposes an algorithm as a complement to the classical algorithm, especially when the optimal probability distribution of the latter one is…
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 present a new improvement on the laser method for designing fast matrix multiplication algorithms. The new method further develops the recent advances by [Duan, Wu, Zhou FOCS 2023] and [Vassilevska Williams, Xu, Xu, Zhou SODA 2024].…
We introduce a new approach to evaluate the largest Lyapunov exponent of a family of nonnegative matrices. The method is based on using special positive homogeneous functionals on $R^{d}_+,$ which gives iterative lower and upper bounds for…
In recent years, several algorithms, which approximate matrix decomposition, have been developed. These algorithms are based on metric conservation features for linear spaces of random projection types. We show that an i.i.d sub-Gaussian…
The multiplication of matrices is an important arithmetic operation in computational mathematics. In the context of hierarchical matrices, this operation can be realized by the multiplication of structured block-wise low-rank matrices,…
In this paper, we study quantum algorithms of matrix multiplication from the viewpoint of inputting quantum/classical data to outputting quantum/classical data. The main target is trying to overcome the input and output problem, which are…
This paper introduces a new Monte Carlo algorithm to invert large matrices. It is based on simultaneous coupled draws from two random vectors whose covariance is the required inverse. It can be considered a generalization of a previously…
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…
Numerous algorithms call for computation over the integers modulo a randomly-chosen large prime. In some cases, the quasi-cubic complexity of selecting a random prime can dominate the total running time. We propose a new variant of the…
Distributed-memory matrix multiplication (MM) is a key element of algorithms in many domains (machine learning, quantum physics). Conventional algorithms for dense MM rely on regular/uniform data decomposition to ensure load balance. These…
We make an in-depth study of the known border rank (i.e. approximate) algorithms for the matrix multiplication tensor encoding the multiplication of an n x 2 matrix by a 2 x 2 matrix.
Matrix multiplication is a fundamental kernel in high performance computing. Many algorithms for fast matrix multiplication can only be applied to enormous matrices ($n>10^{100}$) and thus cannot be used in practice. Of all algorithms…
We propose a hyperpower iteration for numerical computation of the outer generalized inverse of a matrix which achieves the 18th order of convergence by using only seven matrix multiplication per iteration loop. This is the record high…
We propose the arbitrary precision approximate (APA) bilinear algorithm of length 46 for multiplication of 4 x 4 and 4 x 4 matrices. The algorithm has polynomial order 3 and 352 nonzero coefficients from total 2208.
We give an algorithm that generates a uniformly random contingency table with specified marginals, i.e. a matrix with non-negative integer values and specified row and column sums. Such algorithms are useful in statistics and combinatorics.…
As the ratio between the rate of computation and rate with which data can be retrieved from various layers of memory continues to deteriorate, a question arises: Will the current best algorithms for computing matrix-matrix multiplication on…
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 $\{-1, 0, 1\}$…