July 19 to July 23, 2004
American Institute of Mathematics,
Palo Alto, California
Gene H. Golub,
Tamara G. Kolda,
James G. Nagy,
and Charles F. Van Loan
This workshop will consider mathematical problems of tensor decomposition.
Though higher-order tensor (also known as multidimensional, multi-way,
or n-way array) decompositions have been around for more than three
decades, the door is now opening on greater mathematical understanding
and new applications. Previously, this topic has been the domain of
researchers in psychometrics and chemometrics. Now, however,
computationally oriented mathematicians have begun to take an interest
and envision many more potential applications ranging from image and
signal processing to data mining and more. The challenge is to find
ways to extend these methods to larger data sets, i.e., data sets with
thousands to millions of entries. This will require advances in the
theory and computation of higher-order tensor decompositions.
The workshop will bring together researchers on
this topic with specialists in scientific computing, linear algebra, and
applications. The goal of the workshop is to develop the theoretical
and computational tools necessary to tackle larger problems and new
applications. Some of the specific issues to be addressed are
- Mathematical Properties of Tensor Decompositions.
- Computing with Tensor Decompositions.
- Applications of Tensor Decompositions.
Material from the workshop
A list of participants.
Reading List and Bibliography:
A reading list and bibliography, compiled by
the organizers from participant contributions:
Carla Martin - Basics of Tensors
(PDF, 258 kb)
Pieter Kroonenberg - Applications of
three-mode techniques: Overview, problems, and prospects (PDF, 427 kb)
Tammy Kolda and Brett Bader
- Tensor Notation and a MATLAB Tensor Class for Fast Algorithm Prototyping
(PDF, 1.9 MB)
Piere Comon (impromptu) (PDF, 62 kb)
Rasmus Bro - Practical Problems in Multiway Analysis (PDF, 31 MB)
Vince Fernando - 3D SVD (PDF, 117 kb)
Lieven De Lathauwer - Independent Component Analysis
(PDF, 111 kb)
Alex Vasilescu - TensorFaces
Eugene Tyrtyshnikov - Tensor Approximation and
Its Use in the Computation of the Matrix Inverse (PDF, 799 kb)
Cleve Moler (impromptu) - PCA and Human Gait
Michael Mahoney (impromptu) - Extracting Structure from Matrices and Tensors
by Random Sampling (PDF, 1.3 MB; PPT, 1.2MB)
Ed D'Azevdeo (impromptu) - Questions on
Fast Solvers for Kronecker Decompositions (PDF, 82 kb)
Marko Huhtanen (impromptu) - Real Linear SVD Framework [external
link to related paper]
Lek-Heng Lim (impromptu) - What's Possible
and What's Impossible in Tensor Decompositions/Approximation (PDF, 103
Berkant Savas - Handwritten Digit Recognition by HOSVD
Orly Alter - Genomic Signal Processing (PDF, 29MB)
Richard Harshman - Nature of Degenerate Solutions
(PDF, 654 kb) plus
annotated bibliography (PDF, 113kb)
Links to Related Papers
B. W. Bader and T. G. Kolda,
Classes for Fast Algorithm Prototyping, Technical Report SAND2004-5187,
Sandia National Laboratories, Livermore, California, October 2004. [pdf,
Canonical Tensor Decompositions,'
I3S Report, RR-2004-17, June 17, 2004.
- P. Comon and B. Mourrain,
Decomposition of quantics in sums
of powers of linear forms,
Signal Processing, Elsevier, 53(2):93--107, September 1996
- P. Comon,
in J. G. McWhirter and I. K. Proudler, editors, Mathematics in Signal
Processing V, pp. 1--24. Clarendon Press, Oxford, UK, 2002.
- L. De Lathauwer, B. De Moor, J. Vandewalle, A multilinear singular value decomposition, SIAM J. Matrix Anal. Appl., vol. 21, no. 4, Apr. 2000, pp. 1253-1278.
- L. De Lathauwer, B. De Moor, J. Vandewalle, Independent component analysis and (simultaneous) third-order tensor diagonalization, IEEE Transactions on Signal Processing, vol. 49, no. 10, Oct. 2001, pp. 2262-2271.
- L. De Lathauwer, B. De Moor, J. Vandewalle, Computation of the Canonical Decomposition by Means of a Simultaneous Generalized Schur Decomposition, SIAM Journal on Matrix Analysis and Applications, vol. 26, no. 2, 2004, pp. 295-327.
- L. De Lathauwer, J. Vandewalle, Dimensionality reduction in higher-order signal processing and rank-(R_1, R_2,...,R_N) reduction in multilinear algebra, Linear Algebra and its Applications, Special Issue on Linear Algebra in Signal and Image Processing, vol. 391, Nov. 2004, pp. 31-55.
W. S. Hodge and C.-F. Westin,
Identification of translational displacements between N-dimensional data
sets using the high order SVD and phase correlation, to appear in IEEE
Trans. on Image Processing.
Misha Elena Kilmer and
Carla D. Moravitz Martin,
Tensor, SIAM News, 37(9): November 2004.
V. Pereyra and G. Scherer,
Efficient Computer Manipulation of Tensor Products with Applications to
Multidimensional Approximation, Mathematics of Computation,
27(123):595-605, July 1973 (PDF, 735 kb)
N.D. Sidiropoulos, Low-Rank Decomposition of Multi-Way
Arrays: A Signal Processing Perspective, companion paper for
plenary lecture in Proceedings of
2004 IEEE Workshop on Sensor Array and Multichannel processing (SAM2004),
July 18-21, Sitges, Barcelona, Spain. ("Copyright (c) 2004 Institute of
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Hongcheng Wang, Narendra Ahuja,
Expression Decomposition, International Conference on Computer Vision (ICCV),
Hongcheng Wang, Narendra Ahuja ,
Compact Representation of Multidimensional Data Using Tensor Rank-One
Decomposition, International Conference on Pattern Recognition (ICPR),
To add a paper here, please send email to
The workshop schedule.
A report on the workshop activities.
A list of open problems.