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This paper presents new methodology for computationally efficient kernel density estimation. It is shown that a large class of kernels allows for exact evaluation of the density estimates using simple recursions. The same methodology can be…
The matrix element method utilizes ab initio calculations of probability densities as powerful discriminants for processes of interest in experimental particle physics. The method has already been used successfully at previous and current…
We develop a sparse hierarchical $hp$-finite element method ($hp$-FEM) for the Helmholtz equation with variable coefficients posed on a two-dimensional disk or annulus. The mesh is an inner disk cell (omitted if on an annulus domain) and…
Standard numerical algorithms like the fast multipole method or $\mathcal{H}$-matrix schemes rely on low-rank approximations of the underlying kernel function. For high-frequency problems, the ranks grow rapidly as the mesh is refined, and…
We propose a new technique for constructing low-rank approximations of matrices that arise in kernel methods for machine learning. Our approach pairs a novel automatically constructed analytic expansion of the underlying kernel function…
Matrix-vector multiplication forms the basis of many iterative solution algorithms and as such is an important algorithm also for hierarchical matrices which are used to represent dense data in an optimized form by applying low-rank…
For some kernel matrices, low-rank approximations can be quickly obtained via analytic techniques. One important class of analytic methods that has received attention in recent years is based on the use of proxy points. Accuracy analysis…
The kernel-based multi-scale method has been proven to be a powerful approximation method for scattered data approximation problems which is computationally superior to conventional kernel-based interpolation techniques. The multi-scale…
This paper introduces a new kernel-based classifier by viewing kernel matrices as generalized graphs and leveraging recent progress in graph embedding techniques. The proposed method facilitates fast and scalable kernel matrix embedding,…
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…
A combination of hierarchical tree-like data structures and data access patterns from fast multipole methods and hierarchical low-rank approximation of linear operators from H-matrix methods appears to form an algorithmic path forward for…
Matrices resulting from the discretization of a kernel function, e.g., in the context of integral equations or sampling probability distributions, can frequently be approximated by interpolation. In order to improve the efficiency, a…
Matrix data sets are common nowadays like in biomedical imaging where the Diffusion Tensor Magnetic Resonance Imaging (DT-MRI) modality produces data sets of 3D symmetric positive definite matrices anchored at voxel positions capturing the…
We present two fast algorithms for matrix-vector multiplication $y=Ax$, where $A$ is a Hankel matrix. The current asymptotically fastest method is based on the Fast Fourier Transform (FFT), however in multiprecision arithmetics with very…
We consider fast kernel summations in high dimensions: given a large set of points in $d$ dimensions (with $d \gg 3$) and a pair-potential function (the {\em kernel} function), we compute a weighted sum of all pairwise kernel interactions…
In order to fully utilize "big data", it is often required to use "big models". Such models tend to grow with the complexity and size of the training data, and do not make strong parametric assumptions upfront on the nature of the…
We consider the acoustic source imaging problems using multiple frequency data. Using the data of one observation direction/point, we prove that some information (size and location) of the source support can be recovered. A non-iterative…
We present an efficient numerical method for computing Hamiltonian matrix elements between non-orthogonal Slater determinants, focusing on the most time-consuming component of the calculation that involves a sparse array. In the usual case…
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 presents an efficient parallel direct algorithm with near-optimal complexity for the compact fourth and sixth-order approximation of the three-dimensional Helmholtz equations [1] with the problem coefficient depending on only one…