Related papers: Transformations of Matrix Structures Work Again II
Hierarchical matrices (usually abbreviated ${\mathcal H}$-matrices) are frequently used to construct preconditioners for systems of linear equations. Since it is possible to compute approximate inverses or $LU$ factorizations in ${\mathcal…
Over the past two decades, topological data analysis has emerged as a field of applied mathematics with new applications and algorithmic developments appearing rapidly. Two fundamental computations in this field are persistent homology and…
Many quantum algorithms for numerical linear algebra assume black-box access to a block-encoding of the matrix of interest, which is a strong assumption when the matrix is not sparse. Kernel matrices, which arise from discretizing a kernel…
This paper introduces and analyses the new grid-based tensor approach to approximate solution of the elliptic eigenvalue problem for the 3D lattice-structured systems. We consider the linearized Hartree-Fock equation over a spatial…
This paper studies the problem of multichannel spectral super-resolution with either constant amplitude (CA) or not. We propose two optimization problems based on low-rank Hankel-Toeplitz matrix factorization. The two problems effectively…
A new algorithm to compute the restricted singular value decomposition of dense matrices is presented. Like Zha's method \cite{Zha92}, the new algorithm uses an implicit Kogbetliantz iteration, but with four major innovations. The first…
Recently, a two-matrix-model with a new type of interaction [1] has been introduced and analyzed using bi-orthogonal polynomial techniques. Here we present the complete 1/N^2 expansion for the formal version of this model, following the…
For matrices with displacement structure, basic operations like multiplication, inversion, and linear system solving can all be expressed in terms of the following task: evaluate the product $\mathsf{A}\mathsf{B}$, where $\mathsf{A}$ is a…
In this paper, a hierarchical Tucker low-rank (HTLR) matrix is proposed to approximate non-oscillatory kernel functions in linear complexity. The HTLR matrix is based on the hierarchical matrix, with the low-rank blocks replaced by Tucker…
We use lookup tables to design faster algorithms for important algebraic problems over finite fields. These faster algorithms, which only use arithmetic operations and lookup table operations, may help to explain the difficulty of…
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,…
Matrix-valued optimization tasks, including those involving symmetric positive definite (SPD) matrices, arise in a wide range of applications in machine learning, data science and statistics. Classically, such problems are solved via…
Since its introduction by Gauss, Matrix Algebra has facilitated understanding of scientific problems, hiding distracting details and finding more elegant and efficient ways of computational solving. Today's largest problems, which often…
Efficient estimation of high-dimensional matrices-including covariance and precision matrices-is a cornerstone of modern multivariate statistics. Most existing studies have focused primarily on the theoretical properties of the estimators…
In this work, two fast multipole boundary element formulations for the linear time-harmonic acoustic analysis of finite periodic structures are presented. Finite periodic structures consist of a bounded number of unit cell replications in…
We consider the simplest gauge theories given by one- and two- matrix integrals and concentrate on their stringy and geometric properties. We remind general integrable structure behind the matrix integrals and turn to the geometric…
The class of quasiseparable matrices is defined by the property that any submatrix entirely below or above the main diagonal has small rank, namely below a bound called the order of quasiseparability. These matrices arise naturally in…
Matrix multiplication (MatMul) is the computational backbone of modern machine learning, yet its classical complexity remains a bottleneck for large-scale data processing. We propose a hybrid quantum-classical algorithm for matrix…
The efficient and accurate QR decomposition for matrices with hierarchical low-rank structures, such as HODLR and hierarchical matrices, has been challenging. Existing structure-exploiting algorithms are prone to numerical instability as…
Fast matrix multiplication can be described as searching for low-rank decompositions of the matrix--multiplication tensor. We design a neural architecture, \textsc{StrassenNet}, which reproduces the Strassen algorithm for $2\times 2$…