Multidimensional harmonic retrieval based on Vandermonde tensor train

Abstract

Multidimensional Harmonic Retrieval (MHR) is at the heart of important signal-based applications. The exploitation of the large number of available measurement diversities for data fusion increases inexorably the tensor order/dimensionality. The need to mitigate the “curse of dimensionality” in this case is crucial. To efficiently cope with this massive data processing problem, a new scheme called JIRAFE (Joint dImensionality Reduction And Factors rEtrieval) is proposed to estimate the parameters of interest in the MHR problem, namely, the MP angular-frequencies, of the associated P-order rank-M Canonical Polyadic Decomposition (CPD). Our methodology consists of two main steps. The first one breaks the high-order measurement tensor into a collection of graph-connected 3-order tensors, each following a 3-order CPD of rank-M, also called Tensor Train (TT)-cores. This result is based on a model property equivalence between the CPD and the Tensor Train decomposition (TTD) with coupled TT-cores. The second step makes use of a Vandermonde based rectified Alternating Least Squares (RecALS) algorithm to estimate the factors of interest, by enforcing the desired matrix structure. We show that our methodology has several advantages in terms of flexibility, robustness to noise, computational cost and automatic pairing of the parameters of interest with respect to the tensor order.