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Let $A$ be a $d\times d$ complex self-adjoint matrix, let $\mathcal{X},\mathcal{Y}\subset \mathbb{C}^d$ be $k$-dimensional subspaces, and let $X$ be a $d\times k$ complex matrix whose columns form an orthonormal basis of $\mathcal{X}$; that is, $\mathcal{...

A priori, a posteriori, and mixed type upper bounds for the absolute change in Ritz values of self-adjoint matrices in terms of submajorization relations are obtained. Some of our results prove recent conjectures by Knyazev, Argentati, and Zhu, which extend ...

Low-rank approximations to a real matrix $\mathbf{A}$ can be computed from $\mathbf{Z}\mathbf{Z}^T\mathbf{A}$, where $\mathbf{Z}$ is a matrix with orthonormal columns, and the accuracy of the approximation can be estimated from some norm of $\mathbf{A}-\...

Given a real symmetric positive semidefinite matrix $E$, and an approximation $S$ that is a sum of $n$ independent matrix-valued random variables, we present bounds on the relative error in $S$ due to randomization. The bounds do not depend on the matrix ...

This paper is concerned with approximating the dominant left singular vector space of a real matrix $A$ of arbitrary dimension, from block Krylov spaces generated by the matrix ${A}{A}^T$ and the block vector $A{X}$. Two classes of results are presented. ...

One of the most efficient interior-point methods for some classes of block-angular structured problems solves the normal equations by a combination of Cholesky factorizations and preconditioned conjugate gradient for, respectively, the block and linking ...

The leverage scores of a full-column rank matrix $A$ are the squared row norms of any orthonormal basis for $\mathrm{range}\,(A)$. We show that corresponding leverage scores of two matrices $A$ and $A+\Delta A$ are close in the relative sense if they have ...

We study the eigenvalues and eigenspaces of the quadratic matrix polynomial $M\lambda^2+sD\lambda+K$ as $s\rightarrow\infty$, where $M$ and $K$ are symmetric positive definite and $D$ is symmetric positive semidefinite. This work is motivated by its ...

This article introduces the canonical decomposition of the vector space of multivariate polynomials for a given monomial ordering. Its importance lies in solving multivariate polynomial systems, computing Gröbner bases, and solving the ideal membership ...

We present a fast algorithm for approximate canonical correlation analysis (CCA). Given a pair of tall-and-thin matrices, the proposed algorithm first employs a randomized dimensionality reduction transform to reduce the size of the input matrices, and then ...

Many real-world problems deal with collections of high-dimensional data, such as images, videos, text, and web documents, DNA microarray data, and more. Often, such high-dimensional data lie close to low-dimensional structures corresponding to several ...

Multivariate polynomials are usually discussed in the framework of algebraic geometry. Solving problems in algebraic geometry usually involves the use of a Gröbner basis. This article shows that linear algebra without any Gröbner basis computation suffices ...

This paper investigates the general problem of pattern change discovery between high-dimensional data sets. Current methods either mainly focus on magnitude change detection of low-dimensional data sets or are under supervised frameworks. In this paper, ...

Cameras are ubiquitous everywhere and hold the promise of significantly changing the way we live and interact with our environment. Human activity recognition is central to understanding dynamic scenes for applications ranging from security surveillance,...

Several important design problems in signal processing for communications can be cast as optimization problems in which the objective is a function of the subspaces spanned by tall complex matrix variables with orthonormal columns. Such problems can be ...

This paper considers the accuracy of sensor node location estimates from self-calibration in sensor networks. The total parameter space is shown to have a natural decomposition into relative and centroid transformation components. A linear ...

We address the problem of comparing sets of images for object recognition, where the sets may represent variations in an object's appearance due to changing camera pose and lighting conditions. Canonical Correlations (also known as principal or ...

Many inequality relations between real vector quantities can be succinctly expressed as “weak (sub)majorization” relations using the symbol ${\prec}_{w}$. We explain these ideas and apply them in several areas, angles between subspaces, Ritz values, and ...

This paper considers a natural error correcting problem with real valued input/output. We wish to recover an input vector f∈Rⁿ from corrupted measurements y=Af+e. Here, A is an m by n (coding) matrix and e is an arbitrary and unknown vector of ...

Computation of principal angles between subspaces is important in many applications, e.g., in statistics and information retrieval. In statistics, the angles are closely related to measures of dependency and covariance of random variables. When applied to ...