Related papers: A recursive skeletonization factorization based on…
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…
We present a fast direct solver for structured linear systems based on multilevel matrix compression. Using the recently developed interpolative decomposition of a low-rank matrix in a recursive manner, we embed an approximation of the…
This paper presents a general method for applying hierarchical matrix skeletonization factorizations to the numerical solution of boundary integral equations with possibly rank-deficient integral operators. Rank-deficient operators arise in…
Integral equations are commonly encountered when solving complex physical problems. Their discretization leads to a dense kernel matrix that is block or hierarchically low-rank. This paper proposes a new way to build a low-rank…
This paper introduces the hierarchical interpolative factorization for elliptic partial differential equations (HIF-DE) in two (2D) and three dimensions (3D). This factorization takes the form of an approximate generalized LU/LDL…
The factorization of skew-symmetric matrices is a critically understudied area of dense linear algebra, particularly in comparison to that of general and symmetric matrices. While some algorithms can be adapted from the symmetric case, the…
We propose a localized divide and conquer algorithm for inverse factorization $S^{-1} = ZZ^*$ of Hermitian positive definite matrices $S$ with localized structure, e.g. exponential decay with respect to some given distance function on the…
This paper introduces the hierarchical interpolative factorization for integral equations (HIF-IE) associated with elliptic problems in two and three dimensions. This factorization takes the form of an approximate generalized LU…
Matrix factorization is a well-studied task in machine learning for compactly representing large, noisy data. In our approach, instead of using the traditional concept of matrix rank, we define a new notion of link-rank based on a…
We present a new approach to RSA factorization inspired by geometric interpretations and square differences. This method reformulates the problem in terms of the distance between perfect squares and provides a recurrence relation that…
When solving partial differential equations (PDEs) using finite difference or finite element methods, efficient solvers are required for handling large sparse linear systems. In this paper, a recursive sparse LU decomposition for matrices…
We present a fast algorithm for linear least squares problems governed by hierarchically block separable (HBS) matrices. Such matrices are generally dense but data-sparse and can describe many important operators including those derived…
Boundary value problems involving elliptic PDEs such as the Laplace and the Helmholtz equations are ubiquitous in mathematical physics and engineering. Many such problems can be alternatively formulated as integral equations that are…
Robust principal component analysis (RPCA) has drawn significant attentions due to its powerful capability in recovering low-rank matrices as well as successful appplications in various real world problems. The current state-of-the-art…
Many applications in scientific computing and data science require the computation of a rank-revealing factorization of a large matrix. In many of these instances the classical algorithms for computing the singular value decomposition are…
We address the problem of efficient sparse fixed-rank (S-FR) matrix decomposition, i.e., splitting a corrupted matrix $M$ into an uncorrupted matrix $L$ of rank $r$ and a sparse matrix of outliers $S$. Fixed-rank constraints are usually…
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.…
Recovering low-rank and sparse matrices from incomplete or corrupted observations is an important problem in machine learning, statistics, bioinformatics, computer vision, as well as signal and image processing. In theory, this problem can…
What learning algorithms can be run directly on compressively-sensed data? In this work, we consider the question of accurately and efficiently computing low-rank matrix or tensor factorizations given data compressed via random projections.…
We present a fast direct algorithm for computing symmetric factorizations, i.e. $A = WW^T$, of symmetric positive-definite hierarchical matrices with weak-admissibility conditions. The computational cost for the symmetric factorization…