Abstract

The canonical polyadic decomposition (CPD) allows one to extract compact and interpretable representations of tensors. Several optimization-based methods exist to fit the CPD of a tensor for the standard least-squares (LS) cost function. Extensions have been proposed for more general cost functions such as beta-divergences as well. For these non-LS cost functions, a generalized Gauss–Newton (GGN) method has been developed. This is a second-order method that uses an approximation of the Hessian of the cost function to determine the next iterate and with this algorithm, fast convergence can be achieved close to the solution. Unfortunately, for large tensors, the exact GGN approach becomes too expensive. In this paper, we therefore propose an inexact GGN method. The approximation of the Hessian is only used implicitly, which greatly improves the scalability of the method. Next, we show that by using a compressed instance of the GGN Hessian approximation, the computation time of the inexact GGN method can be lowered even more, with only limited influence on the convergence speed. Further, the maximum likelihood estimator for Rician distributed data is examined in detail as an example of an alternative cost function. This cost function is useful for the analysis of the moduli of complex data, as in functional magnetic resonance imaging, for instance. Finally, we compare the proposed method to the existing CPD methods and demonstrate the method’s speed and effectiveness on synthetic and simulated real-life data.

Code description

This package provides an implementation of the canonical polyadic decomposition methods for general cost functions discussed in the paper, as well as a tutorial on how to use these methods and files to generate the experiments from the papers.

Reference

M. Vandecappelle, N. Vervliet, and L. De Lathauwer, "Inexact generalized Gauss-Newton for scaling the canonical polyadic decomposition with non-least-squares cost function," IEEE Journal of Selected Topics on Signal Processing, vol. 15, nr. 3, pp. 491–505, Apr. 2021.

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