相关论文: Randomly pivoted Cholesky: Practical approximation…

We study the problem of computing a rank-$k$ approximation of a matrix using randomized block Krylov iteration. Prior work has shown that, for block size $b = 1$ or $b = k$, a $(1 + \varepsilon)$-factor approximation to the best rank-$k$…

We present a sublinear query algorithm for outputting a near-optimal low-rank approximation to any positive semidefinite Toeplitz matrix $T \in \mathbb{R}^{d \times d}$. In particular, for any integer rank $k \leq d$ and $\epsilon,\delta >…

Binary quadratic programming problems have attracted much attention in the last few decades due to their potential applications. This type of problems are NP-hard in general, and still considered a challenge in the design of efficient…

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…

Geostatistics represents one of the most challenging classes of scientific applications due to the desire to incorporate an ever increasing number of geospatial locations to accurately model and predict environmental phenomena. For example,…

We derive a new adaptive leverage score sampling strategy for solving the Column Subset Selection Problem (CSSP). The resulting algorithm, called Adaptive Randomized Pivoting, can be viewed as a randomization of Osinsky's recently proposed…

Low-rank approximation of a matrix by means of random sampling has been consistently efficient in its empirical studies by many scientists who applied it with various sparse and structured multipliers, but adequate formal support for this…

Randomized SVD has become an extremely successful approach for efficiently computing a low-rank approximation of matrices. In particular the paper by Halko, Martinsson, and Tropp (SIREV 2011) contains extensive analysis, and has made it a…

We investigate the application of randomized quasi-Monte Carlo (RQMC) methods in random feature approximations for kernel-based learning. Compared to the classical Monte Carlo (MC) approach \citep{rahimi2007random}, RQMC improves the…

In this paper, we present a general, multistage framework for graphical model approximation using a cascade of models such as trees. In particular, we look at the problem of covariance matrix approximation for Gaussian distributions as…

Most kernel-based methods, such as kernel or Gaussian process regression, kernel PCA, ICA, or $k$-means clustering, do not scale to large datasets, because constructing and storing the kernel matrix $\mathbf{K}_n$ requires at least…

The classical Krasnoselskii-Mann iteration is broadly used for approximating fixed points of nonexpansive operators. To accelerate the convergence of the Krasnoselskii-Mann iteration, the inertial methods were received much attention in…

We introduce the $k$-banded Cholesky prior for estimating a high-dimensional bandable precision matrix via the modified Cholesky decomposition. The bandable assumption is imposed on the Cholesky factor of the decomposition. We obtained the…

This paper studies the estimation of large precision matrices and Cholesky factors obtained by observing a Gaussian process at many locations. Under general assumptions on the precision and the observations, we show that the sample…

It is known that standard stochastic Galerkin methods face challenges when solving partial differential equations (PDEs) with random inputs. These challenges are typically attributed to the large number of required physical basis functions…

Low rank approximation is an important tool used in many applications of signal processing and machine learning. Recently, randomized sketching algorithms were proposed to effectively construct low rank approximations and obtain approximate…

Randomized orthogonal projection methods (ROPMs) can be used to speed up the computation of Krylov subspace methods in various contexts. Through a theoretical and numerical investigation, we establish that these methods produce…

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…

The Matrix-based Renyi's entropy enables us to directly measure information quantities from given data without the costly probability density estimation of underlying distributions, thus has been widely adopted in numerous statistical…

Matrices arising in scientific applications frequently admit linear low-rank approximations due to smoothness in the physical and/or temporal domain of the problem. In large-scale problems, computing an optimal low-rank approximation can be…