Shortest paths in spaces of certain IIR-filters

There are several notions of distance between (stable) IIR-filters. The mathematically most promising definition is based on Riemannian geometry. The underlying metric is typically highly warped which makes navigating in such spaces a complicated task. In particular, shortest paths between two given filters can be regarded as geodesics. We present a novel approach that is a combination of three ideas. (1) Riemannian geometry is used to describe the underlying metric. (2) A finite set of well-distributed IIR-filters (in the sense of the metric) of a given order is chosen that represents the geometry of all filters well. (3) Graph-theoretical shortest path algorithms are used to determine approximations of the real geodesic.