Structured low rank decomposition of multivariate Hankel matrices

Jouhayna Harmouch (1, 2) , Houssam Khalil (2) , Bernard Mourrain (1)

1 AROMATH - AlgebRe, geOmetrie, Modelisation et AlgoriTHmes
2 LARIFA - Faculty of Sciences Libanese University Lebanon

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.

Mots clés: Singular Value Decomposition; eigenvector; low rank decomposition; Hankel; polynomial; exponential series

Domaines: Géométrie algébrique [math.AG]

Jouhayna Harmouch, Houssam Khalil, Bernard Mourrain. Structured low rank decomposition of multivariate Hankel matrices. Linear Algebra and its Applications, 2017, ⟨10.1016/j.laa.2017.04.015⟩. ⟨hal-01440063⟩