Tucker1 model algorithms for fast solutions to large PARAFAC problems

We describe a method of performing trilinear analysis on large data sets using a modification of the PARAFAC‐ALS algorithm. Our method iteratively decomposes the data matrix into a core matrix and three loading matrices based on the Tucker1 model. The algorithm is particularly useful for data sets that are too large to upload into a computer's main memory. While the performance advantage in utilizing our algorithm is dependent on the number of data elements and dimensions of the data array, we have seen a significant performance improvement over operating PARAFAC‐ALS on the full data set. In one case of data comprising hyperspectral images from a confocal microscope, our method of analysis was approximately 60 times faster than operating on the full data set, while obtaining essentially equivalent results. Copyright © 2008 by John Wiley & Sons, Ltd.

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

[2]  C. Loan.  The ubiquitous Kronecker product , 2000 .

[3]  Paul Geladi,et al.  Techniques and applications of hyperspectral image analysis , 2007 .

[4]  Howland D. T. Jones,et al.  Hyperspectral confocal microscope. , 2006, Applied optics.

[5]  Rasmus Bro,et al.  Multi-way Analysis with Applications in the Chemical Sciences , 2004 .

[6]  R. Tauler,et al.  Multivariate curve resolution applied to liquid chromatography—diode array detection , 1993 .

[7]  Ben C. Mitchell,et al.  Slowly converging parafac sequences: Swamps and two‐factor degeneracies , 1994 .

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

[9]  Joos Vandewalle,et al.  A Multilinear Singular Value Decomposition , 2000, SIAM J. Matrix Anal. Appl..

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

[11]  R. Tauler.  Multivariate curve resolution applied to second order data , 1995 .

[12]  Rasmus Bro,et al.  Improving the speed of multi-way algorithms:: Part I. Tucker3 , 1998 .

[13]  M. V. Van Benthem,et al.  Application of equality constraints on variables during alternating least squares procedures , 2002 .

[14]  Michael R. Keenan,et al.  Accounting for Poisson noise in the multivariate analysis of ToF‐SIMS spectrum images , 2004 .

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

[16]  H. Kiers.  Towards a standardized notation and terminology in multiway analysis , 2000 .

[17]  Anders Berglund,et al.  New and old trends in chemometrics. How to deal with the increasing data volumes in R&D&P (research, development and production)—with examples from pharmaceutical research and process modeling , 2002 .

[18]  M. V. Van Benthem,et al.  Fast algorithm for the solution of large‐scale non‐negativity‐constrained least squares problems , 2004 .

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

[20]  Bruce R. Kowalski,et al.  Recent developments in multivariate calibration , 1991 .