We provide a tutorial introduction to certain numerical computations both in linear algebra and linear systems in the context of bounded arithmetic. The essential characteristics of bounded arithmetic are discussed in an introductory section followed by a review of the fundamental concepts of numerical stability and conditioning. The singular value decomposition (SVD) is then presented along with some related comments on the numerical determination of rank. A variety of applications of the SVD in linear algebra and linear systems is then outlined. A final section discusses some details of the implementation of the SVD on a digital computer. An Appendix is provided which contains a number of useful illustrative example.

The singular value decomposition: Its computation and some applications

Unlabeled document collections are becoming increasingly common and availables mining such data sets represents a major contemporary challenge. Using words as features, text documents are often represented as high-dimensional and sparse vectors–a few thousand dimensions and a sparsity of 95 to 99% is typical. In this paper, we study a certain spherical k-means algorithm for clustering such document vectors. The algorithm outputs k disjoint clusters each with a concept vector that is the centroid of the cluster normalized to have unit Euclidean norm. As our first contribution, we empirically demonstrate that, owing to the high-dimensionality and sparsity of the text data, the clusters produced by the algorithm have a certain “fractal-like” and “self-similar” behavior. As our second contribution, we introduce concept decompositions to approximate the matrix of document vectorss these decompositions are obtained by taking the least-squares approximation onto the linear subspace spanned by all the concept vectors. We empirically establish that the approximation errors of the concept decompositions are close to the best possible, namely, to truncated singular value decompositions. As our third contribution, we show that the concept vectors are localized in the word space, are sparse, and tend towards orthonormality. In contrast, the singular vectors are global in the word space and are dense. Nonetheless, we observe the surprising fact that the linear subspaces spanned by the concept vectors and the leading singular vectors are quite close in the sense of small principal angles between them. In conclusion, the concept vectors produced by the spherical k-means algorithm constitute a powerful sparse and localized “basis” for text data sets.

/pdf/concept-decompositions-for-large-sparse-text-data-using-mtm6tp1gzn.pdf

Concept Decompositions for Large Sparse Text Data Using Clustering

We addressthe question: How should N n-dimensional subspaces of m-dimensional Euclidean space be arranged so that they are as far apart as possible? The resuIts of extensive computations for modest values of N, n, m are described, as well as a reformulation of the problem that was suggested by these computations The reformulation gives a way to describe n-dimensional subspaces of m-space as points on a sphere in dimension ½(m–l)(m+2), which provides a (usually) lowerdimensional representation than the Plucker embedding, and leads to a proof that many of the new packings are optimal The results have applications to the graphical display of multidimensional data via Asimov's grand tour method

/pdf/packing-lines-planes-etc-packings-in-grassmannian-spaces-59ara3fv0c.pdf

Packing lines, planes, etc.: packings in Grassmannian spaces

This paper addresses the question: how should N n-dimensional subspaces of m-dimensional Euclidean space be arranged so that they are as far apart as possible? The results of extensive computations for modest values of N, n, m are described, as well as a reformulation of the problem that was suggested by these computations. The reformulation gives a way to describe n-dimensional subspaces of m-space as points on a sphere in dimension (m-1)(m+2)/2, which provides a (usually) lower-dimensional representation than the Pluecker embedding, and leads to a proof that many of the new packings are optimal. The results have applications to the graphical display of multi-dimensional data via Asimov's "Grand Tour" method.

Packing Lines, Planes, etc.: Packings in Grassmannian Space

We consider the numerical calculation of several matrix eigenvalue problems which require some manipulation before the standard algorithms may be used. This includes finding the stationary values of a quadratic form subject to linear constraints and determining the eigenvalues of a matrix which is modified by a matrix of rank one. We also consider several inverse eigenvalue problems. This includes the problem of determining the coefficients for the Gauss–Radau and Gauss–Lobatto quadrature rules. In addition, we study several eigenvalue problems which arise in least squares.

/pdf/some-modified-matrix-eigenvalue-problems-13itvdbat3.pdf

Some modified matrix eigenvalue problems

Assume that two subspaces F and G of unitary space are defined as the ranges (or nullspaces) of given rectangular matrices A and B. Accurate numerical methods are developed for computing the principal angles $\theta_k (F,G)$ and orthogonal sets of principal vectors $u_k\ \epsilon\ F$ and $v_k\ \epsilon\ G$, k = 1,2,..., q = dim(G) $\leq$ dim(F). An important application in statistics is computing the canonical correlations $\sigma_k\ = cos \theta_k$ between two sets of variates. A perturbation analysis shows that the condition number for $\theta_k$ essentially is max($\kappa (A),\kappa (B)$), where $\kappa$ denotes the condition number of a matrix. The algorithms are based on a preliminary QR-factorization of A and B (or $A^H$ and $B^H$), for which either the method of Householder transformations (HT) or the modified Gram-Schmidt method (MGS) is used. Then cos $\theta_k$ and sin $\theta_k$ are computed as the singular values of certain related matrices. Experimental results are given, which indicates that MGS gives $\theta_k$ with equal precision and fewer arithmetic operations than HT. However, HT gives principal vectors, which are orthogonal to working accuracy, which is not in general true for MGS. Finally the case when A and/or B are rank deficient is discussed.

/pdf/numerical-methods-for-computing-angles-between-linear-4nm02rs89y.pdf

Numerical methods for computing angles between linear subspaces

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Numerical methods for computing angles between linear subspaces