相关论文: Simple non-extensive sparsification of the hierarc…

Inversion of sparse matrices with standard direct solve schemes is robust, but computationally expensive. Iterative solvers, on the other hand, demonstrate better scalability; but, need to be used with an appropriate preconditioner (e.g.,…

Solving sparse linear systems from discretized PDEs is challenging. Direct solvers have in many cases quadratic complexity (depending on geometry), while iterative solvers require problem dependent preconditioners to be robust and…

In this paper we consider linear systems with dense-matrices which arise from numerical solution of boundary integral equations. Such matrices can be well-approximated with $\mathcal{H}^2$-matrices. We propose several new preconditioners…

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…

We propose a new approximate factorization for solving linear systems with symmetric positive definite sparse matrices. In a nutshell the algorithm is to apply hierarchically block Gaussian elimination and additionally compress the fill-in.…

Sparse matrix factorization is the problem of approximating a matrix $\mathbf{Z}$ by a product of $J$ sparse factors $\mathbf{X}^{(J)} \mathbf{X}^{(J-1)} \ldots \mathbf{X}^{(1)}$. This paper focuses on identifiability issues that appear in…

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…

We describe a second-order accurate approach to sparsifying the off-diagonal blocks in the hierarchical approximate factorizations of sparse symmetric positive definite matrices. The norm of the error made by the new approach depends…

Nonnegative Matrix Factorization consists in (approximately) factorizing a nonnegative data matrix by the product of two low-rank nonnegative matrices. It has been successfully applied as a data analysis technique in numerous domains, e.g.,…

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…

While quantum algorithms for solving large scale systems of linear equations offer potentially exponential speedups, their application has largely been confined to sparse matrices. This work extends the scope of these algorithms to a broad…

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…

In solving a linear system with iterative methods, one is usually confronted with the dilemma of having to choose between cheap, inefficient iterates over sparse search directions (e.g., coordinate descent), or expensive iterates in…

We derive analytical expression of matrix factorization/completion solution by variational Bayes method, under the assumption that observed matrix is originally the product of low-rank dense and sparse matrices with additive noise. We…

Hierarchical low-rank approximation of dense matrices can reduce the complexity of their factorization from O(N^3) to O(N). However, the complex structure of such hierarchical matrices makes them difficult to parallelize. The block size and…

This is the second of two papers to describe a matrix sparsification algorithm that takes a general real or complex matrix as input and produces a sparse output matrix of the same size. The first paper presented the original algorithm, its…

This is the first of two papers to describe a matrix sparsification algorithm that takes a general real or complex matrix as input and produces a sparse output matrix of the same size. The non-zero entries in the output are chosen to…

A new fast algebraic method for obtaining an $\mathcal{H}^2$-approximation of a matrix from its entries is presented. The main idea behind the method is based on the nested representation and the maximum-volume principle to select…

Many matrices appearing in numerical methods for partial differential equations and integral equations are rank-structured, i.e., they contain submatrices that can be approximated by matrices of low rank. A relatively general class of…

The computational cost of many signal processing and machine learning techniques is often dominated by the cost of applying certain linear operators to high-dimensional vectors. This paper introduces an algorithm aimed at reducing the…