Basis (linear algebra)

About: Basis (linear algebra) is a research topic. Over the lifetime, 14069 publications have been published within this topic receiving 278522 citations. The topic is also known as: Hamel basis & algebraic basis.

Producing high-dimensional semantic spaces from lexical co-occurrence

Abstract: A procedure that processes a corpus of text and produces numeric vectors containing information about its meanings for each word is presented. This procedure is applied to a large corpus of natural language text taken from Usenet, and the resulting vectors are examined to determine what information is contained within them. These vectors provide the coordinates in a high-dimensional space in which word relationships can be analyzed. Analyses of both vector similarity and multidimensional scaling demonstrate that there is significant semantic information carried in the vectors. A comparison of vector similarity with human reaction times in a single-word priming experiment is presented. These vectors provide the basis for a representational model of semantic memory, hyperspace analogue to language (HAL).

A novel discrete variable representation for quantum mechanical reactive scattering via the S-matrix Kohn method

Abstract: A novel discrete variable representation (DVR) is introduced for use as the L2 basis of the S‐matrix version of the Kohn variational method [Zhang, Chu, and Miller, J. Chem. Phys. 88, 6233 (1988)] for quantum reactive scattering. (It can also be readily used for quantum eigenvalue problems.) The primary novel feature is that this DVR gives an extremely simple kinetic energy matrix (the potential energy matrix is diagonal, as in all DVRs) which is in a sense ‘‘universal,’’ i.e., independent of any explicit reference to an underlying set of basis functions; it can, in fact, be derived as an infinite limit using different basis functions. An energy truncation procedure allows the DVR grid points to be adapted naturally to the shape of any given potential energy surface. Application to the benchmark collinear H+H2→H2+H reaction shows that convergence in the reaction probabilities is achieved with only about 15% more DVR grid points than the number of conventional basis functions used in previous S‐matrix Kohn...

An introduction to nonharmonic Fourier series

Abstract: Bases in Banach Spaces - Schauder Bases Schauder's Basis for C[a,b] Orthonormal Bases in Hilbert Space The Reproducing Kernel Complete Sequences The Coefficient Functionals Duality Riesz Bases The Stability of Bases in Banach Spaces The Stability of Orthonormal Bases in Hilbert Space Entire Functions of Exponential Type The Classical Factorization Theorems - Weierstrass's Factorization Theorem Jensen's Formula Functions of Finite Order Estimates for Canonical Products Hadamard's Factorization Theorem Restrictions Along a Line - The "Phragmen-Lindelof" Method Carleman's Formula Integrability on a line The Paley-Wiener Theorem The Paley-Wiener Space The Completeness of Sets of Complex Exponentials - The Trigonometric System Exponentials Close to the Trigonometric System A Counterexample Some Intrinsic Properties of Sets of Complex Exponentials Stability Density and the Completeness Radius Interpolation and Bases in Hilbert Space - Moment Sequences in Hilbert Space Bessel Sequences and Riesz-Fischer Sequences Applications to Systems of Complex Exponentials The Moment Space and Its Relation to Equivalent Sequences Interpolation in the Paley-Wiener Space: Functions of Sine Type Interpolation in the Paley-Wiener Space: Stability The Theory of Frames The Stability of Nonharmonic Fourier Series Pointwise Convergence Notes and Comments References List of Special Symbols Index

Concept Decompositions for Large Sparse Text Data Using Clustering

Abstract: 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.