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
as a sum of Hankel matrices of small rank in correlation
with the decomposition of its symbol 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 basis
of is computed from the Singular Value Decomposition of a
sub-matrix of the Hankel matrix . The frequencies and the
weights are deduced from the generalized eigenvectors of pencils of
shifted sub-matrices of . 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.