Least square projection: A fast high-precision multidimensional projection technique and its application to document mapping

The problem of projecting multidimensional data into lower dimensions has been pursued by many researchers due to its potential application to data analyses of various kinds. This paper presents a novel multidimensional projection technique based on least square approximations. The approximations compute the coordinates of a set of projected points based on the coordinates of a reduced number of control points with defined geometry. We name the technique Least Square Projections ( LSP). From an initial projection of the control points, LSP defines the positioning of their neighboring points through a numerical solution that aims at preserving a similarity relationship between the points given by a metric in mD. In order to perform the projection, a small number of distance calculations are necessary, and no repositioning of the points is required to obtain a final solution with satisfactory precision. The results show the capability of the technique to form groups of points by degree of similarity in 2D. We illustrate that capability through its application to mapping collections of textual documents from varied sources, a strategic yet difficult application. LSP is faster and more accurate than other existing high-quality methods, particularly where it was mostly tested, that is, for mapping text sets.

Near threshold vibrational excitation of molecules by positron impact: A projection operator approach

We report vibrational excitation (v(i) = 0 -> v(f) = 1) cross-sections for positron scattering by H(2) and model calculations for the (v(i) = 0 -> v(f) = 1) excitation of the C-C symmetric stretch mode of C(2)H(2). The Feshbach projection operator formalism was employed to vibrationally resolve the fixed-nuclei phase shifts obtained with the Schwinger multichannel method. The near threshold behavior of H(2) and C(2)H(2) significantly differ in the sense that no low lying singularity (either virtual or bound state) was found for the former, while a e(+)-acetylene virtual state was found at the equilibrium geometry (this virtual state becomes a bound state upon stretching the molecule). For C(2)H(2), we also performed model calculations comparing excitation cross-sections arising from virtual (-i kappa(0)) and bound (+i kappa(0)) states symmetrically located around the origin of the complex momentum plane (i.e. having the same kappa(0)). The virtual state is seen to significantly couple to vibrations, and similar cross-sections were obtained for shallow bound and virtual states. (c) 2007 Elsevier B.V. All rights reserved.