Lieven De Lathauwer
KU Leuven, Belgium
Canonical Polyadic Decomposition and Sets of Polynomial Equations
12 December 2023, 5pm
KU Leuven, Department of Electrical Engineering (ESAT)
Aula C (ELEC B91.300)
Streaming link: https://eu.bbcollab.com/guest/e72c11871a8b40f9a2db610f16c9393f
Abstract
Solving systems of polynomial equations is important in engineering practice. In this talk, we present a novel strategy to find all (approximate) common roots of an overdetermined polynomial system corrupted by noise (e.g., caused by measurement error). In comparison to earlier frameworks that reduce the problem to a generalized eigenvalue problem, we introduce a multi-pencil approach that reduces the problem to the computation of a canonical polyadic decomposition (CPD) of a tensor (or a block term decomposition (BTD) in the case of coinciding roots) [2, 3]. The tensor is obtained from the null space of a Macaulay matrix. For high polynomial degrees, the Macaulay matrix suffers from the curse of dimensionality. However, its algebraic structure can be exploited for efficiency [1]. The benefits of this approach, both from a conceptual and numerical standpoint, are analyzed. The technique is illustrated with an application involving the localization of two transmitters from the power received in arbitrary antenna configurations [4].
[1] Govindarajan N., Widdershoven R., Chandrasekaran S., De Lathauwer L., “A fast algorithm for computing Macaulay nullspaces of bivariate polynomial systems“, Tech. Report 23-16, ESAT-STADIUS, KU Leuven (Leuven, Belgium), 2023. SIAM Journal on Matrix Analysis and Applications, to appear.
[2] Vanderstukken J., Kürschner P., Domanov I., De Lathauwer L., “Systems of polynomial equations, higher-order tensor decompositions and multidimensional harmonic retrieval: A unifying framework. Part II: The block-term decomposition“, SIAM Journal on Matrix Analysis and Applications, vol. 42, no. 2, Jun. 2021, pp. 913-953.
[3] Vanderstukken J., De Lathauwer L., “Systems of polynomial equations, higher-order tensor decompositions and multidimensional harmonic retrieval: A unifying framework. Part I: The canonical polyadic decomposition“, SIAM Journal on Matrix Analysis and Applications, vol. 42, no. 2, Jun. 2021, pp. 883-912.
[4] Widdershoven R., Govindarajan N., De Lathauwer L., “Overdetermined systems of polynomial equations: tensor-based solution and application“, in Proc. of the 31st European Signal Processing Conference (EUSIPCO 2023), Helsinki, Finland, Sep. 2023, pp. 650-654.
