Section: New Results

Structured low rank decomposition of multivariate Hankel matrices

Participants : Jouhayna Harmouch, Bernard Mourrain.

In [11], we study the decomposition of a multivariate Hankel matrix $H_{\sigma }$ as a sum of Hankel matrices of small rank in correlation with the decomposition of its symbol $\sigma$ 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 $A_{\sigma }$. A basis of $A_{\sigma }$ is computed from the Singular Value Decomposition of a sub-matrix of the Hankel matrix $H_{\sigma }$. The frequencies and the weights are deduced from the generalized eigenvectors of pencils of shifted sub-matrices of $H_{\sigma }$. 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 analyze its numerical behavior 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.

This is a joint work with Houssam Khalil.