An Algebraic Solution for the Candecomp/PARAFAC Decomposition with Circulant Factors

• Published in: SIAM journal on matrix analysis and applications Vol. 35; no. 4; pp. 1543 - 1562
• Main Authors: Goulart, J. H. de M., Favier, G.
• Format: Journal Article
• Language: English
• Published: Society for Industrial and Applied Mathematics 01.01.2014
• Subjects: Algebra; Compounding; Compounds; Computer Science; Decomposition; Engineering Sciences; Iterative methods; Mathematical analysis; Mathematical models; Signal and Image Processing; Tensors; homogeneous monomial equations; circulant matrices; canonical polyadic decomposition; Candecomp/PARAFAC decomposition; tensor decomposition
• ISSN: 0895-4798; 1095-7162
• DOI: 10.1137/140955963

The Candecomp/PARAFAC decomposition (CPD) is an important mathematical tool used in several fields of application. Yet, its computation is usually performed with iterative methods which are subject to reaching local minima and to exhibiting slow convergence. In some practical contexts, the data tensors of interest admit decompositions constituted by matrix factors with particular structure. Often, such structure can be exploited for devising specialized algorithms with superior properties in comparison with general iterative methods. In this paper, we propose a novel approach for computing a circulant-constrained CPD, i.e., a CPD of a hypercubic tensor whose factors are all circulant (and possibly tall). To this end, we exploit the algebraic structure of such tensor, showing that the elements of its frequency-domain counterpart satisfy homogeneous monomial equations in the eigenvalues of square circulant matrices associated with its factors, which we can therefore estimate by solving these equations. Then, we characterize the sets of solutions admitted by such equations under Kruskal's uniqueness condition. Simulation results are presented, validating our approach and showing that it can help avoiding typical disadvantages of iterative methods.