Related papers: A note on computing range space bases of rational …
This paper is concerned with the analysis of the randomized subspace iteration for the computation of low-rank approximations. We present three different kinds of bounds. First, we derive both bounds for the canonical angles between the…
For scientific computations on a digital computer the set of real number is usually approximated by a finite set F of "floating-point" numbers. We compare the numerical accuracy possible with difference choices of F having approximately the…
This paper develops new methods to recover the missing entries of a high-rank or even full-rank matrix when the intrinsic dimension of the data is low compared to the ambient dimension. Specifically, we assume that the columns of a matrix…
In exploratory factor analysis, rotation techniques are employed to derive interpretable factor loading matrices. Factor rotations deal with equality-constrained optimization problems aimed at determining a loading matrix based on measure…
Coherent, continuous spatial representations are critical for synthesizing physical and perceptual phenomena into a single representational space. Radial basis kernels provide a path forward for this type of distributed representation. In…
By a tensor we mean an element of a tensor product of vector spaces over a field. Up to a choice of bases in factors of tensor products, every tensor may be coordinatized, that is, represented as an array consisting of numbers. This note is…
We consider the problem of computing the rank of an m x n matrix A over a field. We present a randomized algorithm to find a set of r = rank(A) linearly independent columns in \~O(|A| + r^\omega) field operations, where |A| denotes the…
The kernel matrix used in kernel methods encodes all the information required for solving complex nonlinear problems defined on data representations in the input space using simple, but implicitly defined, solutions. Spectral analysis on…
The identity testing of rational formulas (RIT) in the free skew field efficiently reduces to computing the rank of a matrix whose entries are linear polynomials in noncommuting variables\cite{HW15}. This rank computation problem has…
Gr\"obner bases can be used for computing the Hilbert basis of a numerical submonoid. By using these techniques, we provide an algorithm that calculates a basis of a subspace of a finite-dimensional vector space over a finite prime field…
Motivated by the philosophy and phenomenal success of compressed sensing, the problem of reconstructing a matrix from a sampling of its entries has attracted much attention recently. Such a problem can be viewed as an information-theoretic…
We propose a very simple preconditioning method for integer programming feasibility problems: replacing the problem b' <= Ax <= b, x \in Z^n with b' <= AUy <= b, y \in Z^n, where U is a unimodular matrix computed via basis reduction, to…
Pseudospectra and structured pseudospectra are important tools for the analysis of matrices. Their computation, however, can be very demanding for all but small matrices. A new approach to compute approximations of pseudospectra and…
The recently developed theory of higher--rank numerical ranges originated in problems of error correction in quantum information theory but its mathematical implications now include a quite satisfactory understanding of \emph{scalar}…
An effective exact method is proposed for computing generalized eigenspaces of a matrix of integers or rational numbers. Keys of our approach are the use of minimal annihilating polynomials and the concept of the Jourdan-Krylov basis. A new…
We contribute a new algebraic method for computing the orthogonal projections of a point onto a rational algebraic surface embedded in the three dimensional projective space. This problem is first turned into the computation of the finite…
We present a class of fast subspace tracking algorithms based on orthogonal iterations for structured matrices/pencils that can be represented as small rank perturbations of unitary matrices. The algorithms rely upon an updated data sparse…
We study categories of matrix factorizations. These categories are defined for any regular function on a suitable regular scheme. Our paper has two parts. In the first part we develop the foundations; for example we discuss derived direct…
As a typical application, the Lenstra-Lenstra-Lovasz lattice basis reduction algorithm (LLL) is used to compute a reduced basis of the orthogonal lattice for a given integer matrix, via reducing a special kind of lattice bases. With such…
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…