Tensor subspace Tracking via Kronecker structured projections (TeTraKron) for time-varying multidimensional harmonic retrieval

Abstract

We present a framework for Tensor-based subspace Tracking via Kronecker-structured projections (TeTraKron). TeTraKron allows to extend arbitrary matrix-based subspace tracking schemes to track the tensor-based subspace estimate. The latter can be computed via a structured projection applied to the matrix-based subspace estimate which enforces the multi-dimensional structure in a computationally efficient fashion. This projection is tracked by considering all matrix rearrangements of the signal tensor jointly, which can be efficiently realized via parallel processing. In addition, we incorporate forward-backward-averaging and find a similar link between the real-valued matrix-based and tensor-based subspace estimation. This enables the tracking of the real-valued tensor-based subspace estimate via a similar Kronecker-structured projection applied to the real-valued matrix-based subspace estimate. In time-varying multidimensional harmonic retrieval problems, the TeTraKron-based subspace tracking schemes outperform the original matrix-based subspace tracking algorithms as well as the batch solutions provided by the SVD and the HOSVD. Moreover, incorporating forward-backward-averaging leads to an improved accuracy of the subspace tracking, and only real-valued processing is involved. Furthermore, we evaluate the performances of ESPRIT-type parameter estimation schemes where the subspace estimates obtained by the proposed TeTraKron-based subspace tracking algorithms are used for the tracking of spatial frequencies in time-varying scenarios.