# A gradient system approach for Hankel structured low-rank approximation

A Hankel matrix H is a structured matrix whose entry on the i-th row and the j-th column depends only on the sum i + j. Hankel matrices can be associated in a natural way to vectors or time series.

Linear dependence among the rows of Hankel matrices plays an important role in several applications in system theory and identification. However, in practice, the coefficients of these matrices are noisy due to e.g. measure- ments errors and computational errors, so generically the involved matrices are full rank. This motivates the problem of Hankel structured low-rank approximation. Structured low-rank approximation problems, in general, do not have a global and efficient solution technique. In [1] we propose a local optimization approach based on a two-levels iteration which describes a descent direction for the smallest singular value via the stationary points of ordinary differential equations. Experimental results show that the proposed algorithm usually achieves good accuracy and shows a higher robustness with respect to the initial approximation, compared to alternative approaches.

### Reference

- A. Fazzi, N. Guglielmi, I. Markovsky. A gradient system approach for Hankel structured low-rank approximation. Linear Algebra Its Appl. 2021. In press