Structured low rank decomposition of multivariate Hankel matrices
This work addresses the problem of multivariate exponential analysis from moments, offering a numerically robust algorithm for signal processing and system identification.
The paper presents a new algorithm for decomposing multivariate Hankel matrices into low-rank components, enabling the recovery of polynomial-exponential series from noisy moment data. The method generalizes the Pencil method to multivariate settings and includes a Newton iteration for error correction.
We study the decomposition of a multivariate Hankel matrix H\_$σ$ as a sum of Hankel matrices of small rank in correlation with the decomposition of its symbol $σ$ as a sum of polynomial-exponential series. We present a new algorithm to compute the low rank decomposition of the Hankel operator and the decomposition of its symbol exploiting the properties of the associated Artinian Gorenstein quotient algebra A\_$σ$. A basis of A\_$σ$ is computed from the Singular Value Decomposition of a sub-matrix of the Hankel matrix H\_$σ$. The frequencies and the weights are deduced from the generalized eigenvectors of pencils of shifted sub-matrices of H $σ$. Explicit formula for the weights in terms of the eigenvectors avoid us to solve a Vandermonde system. This new method is a multivariate generalization of the so-called Pencil method for solving Prony-type decomposition problems. We analyse its numerical behaviour in the presence of noisy input moments, and describe a rescaling technique which improves the numerical quality of the reconstruction for frequencies of high amplitudes. We also present a new Newton iteration, which converges locally to the closest multivariate Hankel matrix of low rank and show its impact for correcting errors on input moments.