Low Complexity Damped Gauss--Newton Algorithms for CANDECOMP/PARAFAC

The damped Gauss--Newton (dGN) algorithm for CANDECOMP/PARAFAC (CP) decomposition can handle the challenges of collinearity of factors and different magnitudes of factors; nevertheless, for factorization of an order-$N$ tensor of size $I_1\times\cdots\times I_N$ with rank $R$, the algorithm is compu...

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Bibliographic Details
Published in:SIAM journal on matrix analysis and applications Vol. 34; no. 1; pp. 126 - 147
Main Authors: Phan, Anh-Huy, Tichavský, Petr, Cichocki, Andrzej
Format: Journal Article
Language:English
Published: Philadelphia Society for Industrial and Applied Mathematics 01.01.2013
Subjects:
Algorithms
Approximation
Complexity
Computation
Decomposition
Inverse
Inversions
Mathematical analysis
Tensors
ISSN:0895-4798, 1095-7162
Online Access:Get full text
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Summary:The damped Gauss--Newton (dGN) algorithm for CANDECOMP/PARAFAC (CP) decomposition can handle the challenges of collinearity of factors and different magnitudes of factors; nevertheless, for factorization of an order-$N$ tensor of size $I_1\times\cdots\times I_N$ with rank $R$, the algorithm is computationally demanding due to construction of a large approximate Hessian of size $(RT \times RT)$ and its inversion where $T = \sum_n I_n$. In this paper, we propose a fast implementation of the dGN algorithm which is based on novel expressions of the inverse approximate Hessian in block form. The new implementation has lower computational complexity, besides computation of the gradient (this part is common to both methods), requiring the inversion of a matrix of size $NR arrow up \times NR arrow up $, which is much smaller than the whole approximate Hessian, if $T \gg NR$. In addition, the implementation has lower memory requirements, because neither the Hessian nor its inverse never needs to be stored in its entirety. A variant of the algorithm working with complex-valued data is proposed as well. Complexity and performance of the proposed algorithm is compared with those of dGN and ALS with line search on examples of difficult benchmark tensors.
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ISSN:0895-4798
1095-7162
DOI:10.1137/100808034