Related papers: HODLR3D: Hierarchical matrices for $N$-body proble…
This article introduces HODLR2D, a new hierarchical low-rank representation for a class of dense matrices arising out of $N$ body problems in two dimensions. Using this new hierarchical framework, we propose a new fast matrix-vector product…
In this article, we prove new theorems bounding the rank of different sub-matrices arising from these kernel functions. Bounds like these are often useful for analyzing the complexity of various hierarchical matrix algorithms. We also plot…
Hierarchical matrix computations have attracted significant attention in the science and engineering community as exploiting data-sparse structures can significantly reduce the computational complexity of many important kernels. One…
We are interested in solving linear systems arising from three applications: (1) kernel methods in machine learning, (2) discretization of boundary integral equations from mathematical physics, and (3) Schur complements formed in the…
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…
Randomized sampling has recently been proven a highly efficient technique for computing approximate factorizations of matrices that have low numerical rank. This paper describes an extension of such techniques to a wider class of matrices…
This article presents a fast solver for the dense "frontal" matrices that arise from the multifrontal sparse elimination process of 3D elliptic PDEs. The solver relies on the fact that these matrices can be efficiently represented as a…
We describe a randomized algorithm for producing a near-optimal hierarchical off-diagonal low-rank (HODLR) approximation to an $n\times n$ matrix $\mathbf{A}$, accessible only though matrix-vector products with $\mathbf{A}$ and…
We present two new algebraic multilevel hierarchical matrix algorithms to perform fast matrix-vector product (MVP) for $N$-body problems in $d$ dimensions, namely efficient $\mathcal{H}^2_{*}$ (fully nested algorithm, i.e., $\mathcal{H}^2$…
Obtaining lightweight and accurate approximations of Hessian applies in inverse problems governed by partial differential equations (PDEs) is an essential task to make both deterministic and Bayesian statistical large-scale inverse problems…
Matrices with hierarchical low-rank structure, including HODLR and HSS matrices, constitute a versatile tool to develop fast algorithms for addressing large-scale problems. While existing software packages for such matrices often focus on…
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…
We study the two-dimensional hierarchical rectangle packing problem, motivated by applications in analog integrated circuit layout, facility layout, and logistics. Unlike classical strip or bin packing, the dimensions of the container are…
In this paper, we apply the hierarchical modeling technique and study some numerical linear algebra problems arising from the Brownian dynamics simulations of biomolecular systems where molecules are modeled as ensembles of rigid bodies.…
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…
We consider the discretization of time-space diffusion equations with fractional derivatives in space and either 1D or 2D spatial domains. The use of implicit Euler scheme in time and finite differences or finite elements in space, leads to…
Three-dimensional magnetic recording (3DMR) is a highly promising approach to achieving ultra-large data storage capacity in hard disk drives. One of the greatest challenges for 3DMR lies in performing sequential and correct writing of bits…
In this paper we establish links between, and new results for, three problems that are not usually considered together. The first is a matrix decomposition problem that arises in areas such as statistical modeling and signal processing:…
Disentangling attributes of various sensory signals is central to human-like perception and reasoning and a critical task for higher-order cognitive and neuro-symbolic AI systems. An elegant approach to represent this intricate…
With the daily influx of 3D data on the internet, text-3D retrieval has gained increasing attention. However, current methods face two major challenges: Hierarchy Representation Collapse (HRC) and Redundancy-Induced Saliency Dilution…