Related papers: Adaptive multiplication of rank-structured matrice…
Kernel matrices are ubiquitous in computational mathematics, often arising from applications in machine learning and scientific computing. In two or three spatial or feature dimensions, such problems can be approximated efficiently by a…
Sparse matrix multiplication is an important component of linear algebra computations. Implementing sparse matrix multiplication on an associative processor (AP) enables high level of parallelism, where a row of one matrix is multiplied in…
Coded matrix multiplication is a technique to enable straggler-resistant multiplication of large matrices in distributed computing systems. In this paper, we first present a conceptual framework to represent the division of work amongst…
Square matrices appear in many machine learning problems and models. Optimization over a large square matrix is expensive in memory and in time. Therefore an economic approximation is needed. Conventional approximation approaches factorize…
Several researchers have developed a rich toolbox of matrix compression techniques that exploit structure and redundancy in large matrices. Classical methods such as the block low-rank format and the Fast Multipole Method make it possible…
We present new algorithms for the randomized construction of hierarchically semi-separable matrices, addressing several practical issues. The HSS construction algorithms use a partially matrix-free, adaptive randomized projection scheme to…
Matrix factorization exploits the idea that, in complex high-dimensional data, the actual signal typically lies in lower-dimensional structures. These lower dimensional objects provide useful insight, with interpretability favored by sparse…
The parallel algorithm for loading large sparse matrices from files into distributed memories of high performance computing (HPC) systems is presented. This algorithm was designed specially for matrices stored in files in the space-effcient…
Algebraic matrix multiplication algorithms are designed by bounding the rank of matrix multiplication tensors, and then using a recursive method. However, designing algorithms in this way quickly leads to large constant factors: if one…
We show how to construct highly symmetric algorithms for matrix multiplication. In particular, we consider algorithms which decompose the matrix multiplication tensor into a sum of rank-1 tensors, where the decomposition itself consists of…
We propose HAMSI (Hessian Approximated Multiple Subsets Iteration), which is a provably convergent, second order incremental algorithm for solving large-scale partially separable optimization problems. The algorithm is based on a local…
This text investigates relations between two well-known family of algorithms, matrix factorisations and recursive linear filters, by describing a probabilistic model in which approximate inference corresponds to a matrix factorisation…
We comment on two randomized algorithms for constructing low-rank matrix decompositions. Both algorithms employ the Subsampled Randomized Hadamard Transform [14]. The first algorithm appeared recently in [9]; here, we provide a novel…
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…
Factorization of large dense matrices are ubiquitous in engineering and data science applications, e.g. preconditioners for iterative boundary integral solvers, frontal matrices in sparse multifrontal solvers, and computing the determinant…
It is widely known that the lower bound for the algorithmic complexity of square matrix multiplication resorts to at least $n^2$ arithmetic operations. The justification builds upon the following reasoning: given that there are $2 n^2$…
A method for the kernel-independent construction of $\mathcal{H}^2$-matrix approximations to non-local operators is proposed. Special attention is paid to the adaptive construction of nested bases. As a side result, new error estimates for…
In this letter, we propose an algorithm for recovery of sparse and low rank components of matrices using an iterative method with adaptive thresholding. In each iteration, the low rank and sparse components are obtained using a thresholding…
This paper introduces a dynamic, error-bounded hierarchical matrix (H-matrix) compression method tailored for Physics-Informed Neural Networks (PINNs). The proposed approach reduces the computational complexity and memory demands of…
A matrix algorithm runs superfast (aka at sublinear cost) if it involves much fewer flops and memory cells than an input matrix has entries. Big Data are frequently represented by matrices of immense sizes that cannot be handled directly…