The Multilinear Engine—A Table-Driven, Least Squares Program for Solving Multilinear Problems, Including the n-Way Parallel Factor Analysis Model

Abstract A technique for fitting multilinear and quasi-multilinear mathematical expressions or models to two-, three-, and many-dimensional data arrays is described. Principal component analysis and three-way PARAFAC factor analysis are examples of bilinear and trilinear least squares fit. This work presents a technique for specifying the problem in a structured way so that one program (the Multilinear Engine) may be used for solving widely different multilinear problems. The multilinear equations to be solved are specified as a large table of integer code values. The end user creates this table by using a small preprocessing program. For each different case, an individual structure table is needed. The solution is computed by using the conjugate gradient algorithm. Non-negativity constraints are implemented by using the well-known technique of preconditioning in opposite way for slowing down changes of variables that are about to become negative. The iteration converges to a minimum that may be local or ...

[1]  M. Hestenes,et al.  Methods of conjugate gradients for solving linear systems , 1952 .

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

[3]  J. Chang,et al.  Analysis of individual differences in multidimensional scaling via an n-way generalization of “Eckart-Young” decomposition , 1970 .

[4]  Richard A. Harshman,et al.  Foundations of the PARAFAC procedure: Models and conditions for an "explanatory" multi-model factor analysis , 1970 .

[5]  Pieter M. Kroonenberg,et al.  Three-mode principal component analysis : theory and applications , 1983 .

[6]  P. Kroonenberg.  Annotated bibliography of three-mode factor analysis. , 1983 .

[7]  Gene H. Golub,et al.  Matrix computations , 1983 .

[8]  R. A. Harshman,et al.  Data preprocessing and the extended PARAFAC model , 1984 .

[9]  J. Carroll,et al.  Fitting of the Latent Class model via iteratively reweighted least squares CANDECOMP with nonnegativity constraints , 1989 .

[10]  S. Leurgans,et al.  Multilinear Models: Applications in Spectroscopy , 1992 .

[11]  H. Kiers.  An alternating least squares algorithms for PARAFAC2 and three-way DEDICOM , 1993 .

[12]  R. Harshman,et al.  PARAFAC: parallel factor analysis , 1994 .

[13]  Sue Leurgans,et al.  [27] Component resolution using multilinear models , 1995 .

[14]  Charles L. Lawson,et al.  Solving least squares problems , 1976, Classics in applied mathematics.

[15]  P. Paatero.  Least squares formulation of robust non-negative factor analysis , 1997 .

[16]  R. Bro,et al.  A fast non‐negativity‐constrained least squares algorithm , 1997 .

[17]  P. Paatero.  A weighted non-negative least squares algorithm for three-way ‘PARAFAC’ factor analysis , 1997 .

[18]  P. Paatero,et al.  THREE-WAY (PARAFAC) FACTOR ANALYSIS : EXAMINATION AND COMPARISON OF ALTERNATIVE COMPUTATIONAL METHODS AS APPLIED TO ILL-CONDITIONED DATA , 1998 .

[19]  P. Paatero,et al.  Identification of source nature and seasonal variations of Arctic aerosol by the multilinear engine , 1999 .