Jouhayna Harmouch - Decomposition of Hankel matrices and tensors
19 May 2017 –
Decomposition of Hankel matrices and tensors
We decompose the multivariate Hankel operator of low rank into sum of indecomposable Hankel operators of rank one which is equivalent to the decomposition of its symbol sigma into sum of polynomial exponential series. This problem has been studied before in the univariate case using the solution of a vandermonde system of a polynomial in the kernel of the Hankel matrix $H_\sigma$ and after in the multivariate case by projecting it into univariate case, we compute the frequencies using the minimization of least square problem. Now we use the algebraic properties of the quotient algebra of the set of multivariate polynomials into the kernel of this operator $H_\sigma$ to compute the multiplication operators by a variable and deduce the weights and the frequencies from the eigenvectors of them. We use the singular value decomposition of $H_\sigma$ to compute bases of $\cA_\sigma$ and we deduce then the multiplication operator in theses bases. We propose a rescaling technique which improves the reconstruction where the frequencies are of high amplitude and a Newton method which converges locally to the solution. The problem is studied with noisy input.
This problem is related to the symmetric tensor decomposition problem when the coefficients of the tensor are deduced from the moments of the symbol of the Hankel operator. We adapt our algorithm to this case with the constraint of $r<dim(col(H_\sigma))$ and we give some examples with the ODF tensor which shows the reconstruction of the angle between the frequencies(directions) up to a small error. Once the number of moments are not enough to satisfy the constraints of the symmetric tensor decomposition problem, the problem of completion of multivariate Hankel matrices arises, we propose SDP and SVT to solve this problem. We also show the theoretical relationship between the low rank multivariate Hankel decomposition problem and the partial symmetric tensor decomposition problem.
Acces: visio.inria.fr (IP: 188.8.131.52), Num: 310, PIN: 9136