Fast Decomposition of Large Nonnegative Tensors

In signal processing, tensor decompositions have gained in popularity this last decade. In the meantime, the volume of data to be processed has drastically increased. This calls for novel methods to handle Big Data tensors. Since most of these huge data are issued from physical measurements, which are intrinsically real nonnegative, being able to compress nonnegative tensors has become mandatory. Following recent works on HOSVD compression for Big Data, we detail solutions to decompose a nonnegative tensor into decomposable terms in a compressed domain.

[1]  H. Sebastian Seung,et al.  Algorithms for Non-negative Matrix Factorization , 2000, NIPS.

[2]  J. Cullum,et al.  Lanczos algorithms for large symmetric eigenvalue computations , 1985 .

[3]  Chih-Jen Lin,et al.  Projected Gradient Methods for Nonnegative Matrix Factorization , 2007, Neural Computation.

[4]  Pierre Comon,et al.  Tensors : A brief introduction , 2014, IEEE Signal Processing Magazine.

[5]  P. Paatero The Multilinear Engine—A Table-Driven, Least Squares Program for Solving Multilinear Problems, Including the n-Way Parallel Factor Analysis Model , 1999 .

[6]  Andrzej Cichocki,et al.  Nonnegative Matrix and Tensor Factorization T , 2007 .

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

[8]  Andrzej Cichocki,et al.  Fast Nonnegative Matrix/Tensor Factorization Based on Low-Rank Approximation , 2012, IEEE Transactions on Signal Processing.

[9]  Rasmus Bro,et al.  MULTI-WAY ANALYSIS IN THE FOOD INDUSTRY Models, Algorithms & Applications , 1998 .

[10]  Nikos D. Sidiropoulos,et al.  Parallel Randomly Compressed Cubes : A scalable distributed architecture for big tensor decomposition , 2014, IEEE Signal Processing Magazine.

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

[12]  P. Comon,et al.  Tensor decompositions, alternating least squares and other tales , 2009 .

[13]  Nikos D. Sidiropoulos,et al.  Parallel factor analysis in sensor array processing , 2000, IEEE Trans. Signal Process..

[14]  Pierre Comon,et al.  Computing the polyadic decomposition of nonnegative third order tensors , 2011, Signal Process..

[15]  Carl Tim Kelley,et al.  Iterative methods for optimization , 1999, Frontiers in applied mathematics.

[16]  Pierre Comon,et al.  Nonnegative approximations of nonnegative tensors , 2009, ArXiv.

[17]  Stephen P. Boyd,et al.  Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers , 2011, Found. Trends Mach. Learn..

[18]  Pierre Comon,et al.  Enhanced Line Search: A Novel Method to Accelerate PARAFAC , 2008, SIAM J. Matrix Anal. Appl..

[19]  Lu Wang,et al.  Nonnegative compression for Semi-Nonnegative Independent Component Analysis , 2014, 2014 IEEE 8th Sensor Array and Multichannel Signal Processing Workshop (SAM).

[20]  Andrzej Cichocki,et al.  Nonnegative Matrix and Tensor Factorizations : An algorithmic perspective , 2014, IEEE Signal Processing Magazine.

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

[22]  Nathan Halko,et al.  Finding Structure with Randomness: Probabilistic Algorithms for Constructing Approximate Matrix Decompositions , 2009, SIAM Rev..