Improving the speed of multi-way algorithms:: Part I. Tucker3

Abstract In an attempt to improve the speed of multi-way algorithms, this paper investigates several different implementations of the Tucker3 algorithm. The interest is specifically aimed at developing a fast algorithm in the MATLAB™ environment that is suitable for large data arrays. Nine different implementations are developed and tested on real and simulated data. In a subsequent paper, it will be demonstrated that a fast algorithm for the Tucker3 model provides a perfect basis for improving the speed of other multi-way algorithms. From the Internet address http:\\newton.mli.kvl.dk\foodtech.html, the developed algorithms can be downloaded.

[1]  Rasmus Bro,et al.  Improving the speed of multiway algorithms: Part II: Compression , 1998 .

[2]  J. Leeuw,et al.  Some additional results on principal components analysis of three-mode data by means of alternating least squares algorithms , 1987 .

[3]  H. Kiers.  An alternating least squares algorithm for fitting the two- and three-way dedicom model and the idioscal model , 1989 .

[4]  H. Neudecker,et al.  An approach ton-mode components analysis , 1986 .

[5]  Pieter M. Kroonenberg,et al.  Gram-Schmidt versus Bauer-Rutishauser in alternating least-squares algorithms for three-mode principal component analysis , 1989 .

[6]  Vladimir Simeon,et al.  Assessment of the effective rank of a (co)variance matrix: A non‐parametric goodness‐of‐fit test , 1993 .

[7]  E. Davidson,et al.  Strategies for analyzing data from video fluorometric monitoring of liquid chromatographic effluents , 1981 .

[8]  Pieter M. Kroonenberg,et al.  An efficient algorithm for TUCKALS3 on data with large numbers of observation units , 1992 .

[9]  James W. Longley.  Least squares computations using orthogonalization methods , 1984 .

[10]  J. Leeuw,et al.  Principal component analysis of three-mode data by means of alternating least squares algorithms , 1980 .

[11]  F. L. Bauer.  Das Verfahren der Treppeniteration und verwandte Verfahren zur Lösung algebraischer Eigenwertprobleme , 1957 .

[12]  Edmund R. Malinowski,et al.  Factor Analysis in Chemistry , 1980 .

[13]  J. Hamilton,et al.  Principal component analysis, trace elements, and blue crab shell disease. , 1992, Analytical chemistry.

[14]  Pieter M. Kroonenberg,et al.  External analysis with three-mode principal component models , 1985 .

[15]  H. Rutishauser.  Computational aspects of F. L. Bauer's simultaneous iteration method , 1969 .

[16]  R. Harshman,et al.  Relating two proposed methods for speedup of algorithms for fitting two- and three-way principal component and related multilinear models , 1997 .

[17]  S. Wold.  Cross-Validatory Estimation of the Number of Components in Factor and Principal Components Models , 1978 .

[18]  Jiri Janata,et al.  Sensitive layer for electrochemical detection of hydrogen cyanide , 1992 .

[19]  J. C. van Houwelingen,et al.  Three-mode factor analysis of data on retention in normal-phase high-performance liquid chromatography , 1984 .

[20]  Olav M. Kvalheim,et al.  Speed improvement of multivariate algorithms by the method of postponed basis matrix multiplication: Part II. Three-mode principal component analysis , 1994 .

[21]  H. Kiers.  Weighted least squares fitting using ordinary least squares algorithms , 1997 .

[22]  Henk A. L. Kiers,et al.  An efficient algorithm for PARAFAC of three-way data with large numbers of observation units , 1991 .

[23]  L. Tucker,et al.  Some mathematical notes on three-mode factor analysis , 1966, Psychometrika.

[24]  S. Leurgans,et al.  A Decomposition for Three-Way Arrays , 1993, SIAM J. Matrix Anal. Appl..

[25]  Lars Nørgaard,et al.  Classification and prediction of quality and process parameters of thick juice and beet sugar by fluorescence spectroscopy and chemometrics , 1995 .

[26]  J. Kruskal,et al.  Candelinc: A general approach to multidimensional analysis of many-way arrays with linear constraints on parameters , 1980 .

[27]  R. Harshman,et al.  Uniqueness proof for a family of models sharing features of Tucker's three-mode factor analysis and PARAFAC/candecomp , 1996 .

[28]  B. Everitt,et al.  Three-Mode Principal Component Analysis. , 1986 .