相关论文: Givens Rotations for QR Decomposition, SVD and PCA…

We address the task of higher-order derivative evaluation of computer programs that contain QR decompositions and real symmetric eigenvalue decompositions. The approach is a combination of univariate Taylor polynomial arithmetic and matrix…

A new algorithm to compute the restricted singular value decomposition of dense matrices is presented. Like Zha's method \cite{Zha92}, the new algorithm uses an implicit Kogbetliantz iteration, but with four major innovations. The first…

While the proper orthogonal decomposition (POD) is optimal under certain norms it's also expensive to compute. For large matrix sizes, it is well known that the QR decomposition provides a tractable alternative. Under the assumption that it…

A standard approach to compute the roots of a univariate polynomial is to compute the eigenvalues of an associated \emph{confederate} matrix instead, such as, for instance the companion or comrade matrix. The eigenvalues of the confederate…

We derive algorithms for higher order derivative computation of the rectangular $QR$ and eigenvalue decomposition of symmetric matrices with distinct eigenvalues in the forward and reverse mode of algorithmic differentiation (AD) using…

Matrix decompositions in dual number representations have played an important role in fields such as kinematics and computer graphics in recent years. In this paper, we present a QR decomposition algorithm for dual number matrices,…

This paper presents a portable, GPU-accelerated implementation of a QR-based singular value computation algorithm in Julia. The singular value ecomposition (SVD) is a fundamental numerical tool in scientific computing and machine learning,…

Principal Component Analysis (PCA) is a foundational technique in machine learning for dimensionality reduction of high-dimensional datasets. However, PCA could lead to biased outcomes that disadvantage certain subgroups of the underlying…

The QR-algorithm is one of the most important algorithms in linear algebra. Its several variants make feasible the computation of the eigenvalues and eigenvectors of a numerical real or complex matrix, even when the dimensions of the matrix…

Many statistical problems involve the estimation of a $\left(d\times d\right)$ orthogonal matrix $\textbf{Q}$. Such an estimation is often challenging due to the orthonormality constraints on $\textbf{Q}$. To cope with this problem, we…

Second-order optimization approaches like the generalized Gauss-Newton method are considered more powerful as they utilize the curvature information of the objective function with preconditioning matrices. Albeit offering tempting…

The graphical lasso \citep{FHT2007a} is an algorithm for learning the structure in an undirected Gaussian graphical model, using $\ell_1$ regularization to control the number of zeros in the precision matrix ${\B\Theta}={\B\Sigma}^{-1}$…

Mining groups of genes that consistently co-express is an important problem in biomedical research, where it is critical for applications such as drug-repositioning and designing new disease treatments. Recently, Cooley et al. modeled this…

We consider the problem of computing a QR (or QZ) decomposition of a real, dense, tall and very skinny matrix. That is, the number of columns is tiny compared to the number of rows, rendering most computations completely or partially…

The need to compute the intersections between a line and a high-order curve or surface arises in a large number of finite element applications. Such intersection problems are easy to formulate but hard to solve robustly. We introduce a…

The Graphical Lasso (GLasso) algorithm is fast and widely used for estimating sparse precision matrices (Friedman et al., 2008). Its central role in the literature of high-dimensional covariance estimation rivals that of Lasso regression…

Product quantization (PQ) coupled with a space rotation, is widely used in modern approximate nearest neighbor (ANN) search systems to significantly compress the disk storage for embeddings and speed up the inner product computation.…

EigenDecomposition (ED) is at the heart of many computer vision algorithms and applications. One crucial bottleneck limiting its usage is the expensive computation cost, particularly for a mini-batch of matrices in the deep neural networks.…

We present efficient realization of Generalized Givens Rotation (GGR) based QR factorization that achieves 3-100x better performance in terms of Gflops/watt over state-of-the-art realizations on multicore, and General Purpose Graphics…

Reconstructing DNA sequences without a reference, known as de novo assembly, is a complex computational task involving the alignment of overlapping fragments. To address this problem, a usual strategy is to map the assembly to a Quadratic…