相关论文: Low-Rank Kernel Matrix Approximation Using Skeleto…

This survey describes probabilistic algorithms for linear algebra computations, such as factorizing matrices and solving linear systems. It focuses on techniques that have a proven track record for real-world problem instances. The paper…

Efficient exploitation of exascale architectures requires rethinking of the numerical algorithms used in many large-scale applications. These architectures favor algorithms that expose ultra fine-grain parallelism and maximize the ratio of…

Stochastic First-Order (SFO) methods have been a cornerstone in addressing a broad spectrum of modern machine learning (ML) challenges. However, their efficacy is increasingly questioned, especially in large-scale applications where…

We introduce a novel kernel learning framework toward efficiently solving nonlinear partial differential equations (PDEs). In contrast to the state-of-the-art kernel solver that embeds differential operators within kernels, posing…

The least-squares support vector machine is a frequently used kernel method for non-linear regression and classification tasks. Here we discuss several approximation algorithms for the least-squares support vector machine classifier. The…

Computing low-rank approximations of kernel matrices is an important problem with many applications in scientific computing and data science. We propose methods to efficiently approximate and store low-rank approximations to kernel matrices…

First-order methods in convex optimization offer low per-iteration cost but often suffer from slow convergence, while second-order methods achieve fast local convergence at the expense of costly Hessian inversions. In this paper, we…

Matrix decompositions are fundamental tools in the area of applied mathematics, statistical computing, and machine learning. In particular, low-rank matrix decompositions are vital, and widely used for data analysis, dimensionality…

For a given matrix subspace, how can we find a basis that consists of low-rank matrices? This is a generalization of the sparse vector problem. It turns out that when the subspace is spanned by rank-1 matrices, the matrices can be obtained…

A classical problem in matrix computations is the efficient and reliable approximation of a given matrix by a matrix of lower rank. The truncated singular value decomposition (SVD) is known to provide the best such approximation for any…

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…

This paper describes practical randomized algorithms for low-rank matrix approximation that accommodate any budget for the number of views of the matrix. The presented algorithms, which are aimed at being as pass efficient as needed, expand…

With the dramatic growth in the number of application domains that generate probabilistic, noisy and uncertain data, there has been an increasing interest in designing algorithms for geometric or combinatorial optimization problems over…

This article gives a new insight of kernel-based (approximation) methods to solve the high-dimensional stochastic partial differential equations. We will combine the techniques of meshfree approximation and kriging interpolation to extend…

The Nystr\"om method is a convenient heuristic method to obtain low-rank approximations to kernel matrices in nearly linear complexity. Existing studies typically use the method to approximate positive semidefinite matrices with low or…

We introduce a new numerical method based on machine learning to approximate the solution of elliptic partial differential equations with collocation using a set of sigmoidal functions. We show that a feedforward neural network with a…

In this study, we consider the numerical solution of large systems of linear equations obtained from the stochastic Galerkin formulation of stochastic partial differential equations. We propose an iterative algorithm that exploits the…

Recent hardware-aware matrix-free algorithms for higher-order finite-element (FE) discretized matrix-vector multiplications reduce floating point operations and data access costs compared to traditional sparse matrix approaches. This work…

We study the problem of optimally projecting the transition matrix of a finite ergodic multivariate Markov chain onto a lower-dimensional state space, as well as the problem of finding an optimal partition of coordinates such that the…

The accuracy and complexity of machine learning algorithms based on kernel optimization are determined by the set of kernels over which they are able to optimize. An ideal set of kernels should: admit a linear parameterization (for…