A Fast-Converging Algorithm for Nonlinear Mapping of High-Dimensional Data to a Plane

IEEE Transactions on ComputersPublished 1 February 1979

Niemann, Weiss

DOI10.1109/tc.1979.1675303

Source URL

Citations28

SJR quartileQ1

SJR score1.16

SNIP1.61

TL;DR

An iterative algorithm for nonlinear mapping of high-dimensional data is developed and the step size of the descent algorithm is chosen to assure convergence.

Abstract

An iterative algorithm for nonlinear mapping of high-dimensional data is developed. The step size of the descent algorithm is chosen to assure convergence. Steepest descent and Coordinate descent are treated. The algorithm is applied to artificial and real data to demonstrate its excellent convergence properties.

Keywords

Computer Science

Students Quarterly JournalOptimization by Vector Space Methods

6,128 Citations1970

This book shows engineers how to use optimization theory to solve complex problems with a minimum of mathematics and unifies the large field of optimization with a few geometric principles.

PsychometrikaNonmetric Multidimensional Scaling: A Numerical Method

4,882 Citations1964Joseph B. Kruskal

The numerical methods required in the approach to multi-dimensional scaling are described and the rationale of this approach has appeared previously.

IEEE Transactions on ComputersA Nonlinear Mapping for Data Structure Analysis

3,381 Citations1969John W. Sammon

An algorithm for the analysis of multivariate data is presented along with some experimental results that is based upon a point mapping of N L-dimensional vectors from the L-space to a lower-dimensional space such that the inherent data "structure" is approximately preserved.

Journal of the Royal Statistical Society Series A (General)Pattern Classification and Scene Analysis.

3,240 Citations1974Mairi Clarke, Richard O. Duda +1 more

PsychometrikaThe Analysis of Proximities: Multidimensional Scaling with an Unknown Distance Function. I.

2,553 Citations1962Roger N. Shepard

The program is proposed as a tool for reductively analyzing several types of psychological data, particularly measures of interstimulus similarity or confusability, by making explicit the multidimensional structure underlying such data.

Pattern RecognitionDistance functions on digital pictures

835 Citations1968Amir Rosenfeld, John L. Pfaltz

Algorithms for computing various functions on a digital picture which depend on the distance to a given subset of the picture, which involve local operations which are performed repeatedly, "in parallel”, on every picture element and its immediate neighbors are described.

IEEE Transactions on Systems Man and CyberneticsA Heuristic Relaxation Method for Nonlinear Mapping in Cluster Analysis

73 Citations1973C. L. Chang, R. C. T. Lee

A relaxation method mapping high-dimensional sample points to low- dimensional sample points is discussed, which tries to preserve the local interdistance of sample points.

IEEE Transactions on ComputersA Nonlinear Feature Extraction Algorithm Using Distance Transformation

41 Citations1972Warren Koontz, Keinosuke Fukunaga

Feature extraction has been recognized as a useful technique for pattern recognition by constructing a mapping from the measurement space to a feature space by optimizing the parameters with respect to a separability criterion.

IEEE Transactions on ComputersA Two-Dimensional Display for the Classification of Multivariate Data

20 Citations1971Keinosuke Fukunaga, D.R. Olsen

The properties of a two-dimensional display whose coordinates are the Euclidean distances from two points in a multivariate space are presented and some geometric structure is preserved by this mapping.

IEEE Transactions on ComputersRepresentation of Nonlinear Data Surfaces

11 Citations1973D.R. Olsen, Keinosuke Fukunaga

This paper is concerned with the use of "intrinsic dimensionality" in the representation of multivariate data sets that lie on nonlinear surfaces by transforming the data to a linear space of mI dimensions using a new noniterative mapping procedure.