LOW COMPLEXITY DAMPED GAUSS-NEWTON ALGORITHMS FOR CANDECOMP/PARAFAC

作者：Anh Huy Phan^*; Tichavsky Petr; Cichocki Andrzej

来源：SIAM Journal on Matrix Analysis and Applications, 2013, 34(1): 126-147.

DOI：10.1137/100808034

• CANDECOMP/PARAFAC
• tensor factorization
• canonical decomposition
• complex-valued tensor factorization
• low-rank approximation
• alternating least squares
• line search
• Gauss-Newton
• Levenberg-Marquardt
• inverse problems

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 x ... x I-N with rank R, the algorithm is computationally demanding due to construction of a large approximate Hessian of size (RT x RT) and its inversion where T = Sigma(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 N R-2 x N R-2, which is much smaller than the whole approximate Hessian, if T %26gt;%26gt; N R. 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.

• 出版日期2013