Identification of Large-Scale Vector-AutoRegressive models with Kronecker modeling

In this work we address the identification of (2D) spatial-temporal dynamical systems described by the Vector-AutoRegressive (VAR) form. Modeling large-scale networks has been studied so far assuming different structures to alleviate the computational requirements, for example using sparsity in the interconnection pattern, [1]. A data-sparse structure has been investigated in [2] to model 1D strings of interconnected subsystems. The so-called Sequentially Semi-Separable (SSS) structure represents each of the subsystem in a 1D array with a mixed causal anti-causal linear time varying model which shares unknown interconnections with the closest neighbors. A data-sparse representation may not be achieved as the dimension grow as modeling multi-dimensional systems within the SSS framework requires an exponential number of parameters.
To overcome this curse of dimensionality, we present in this work a new class of
structured matrices to model 2D spatial-temporal systems. The class of Kronecker networks is first introduced and the identification of Vector Auto-Regressive models is subsequently addressed within this framework. Such a Kronecker structure leads to high data compression and benefits from the efficient linear algebra operations as described in [3]. The challenge lies in deriving algorithms that are, on the one hand, scalable in terms of data storage as well as in terms of computational complexity in identifying and using these models, e.g in subsequent control design, and on the other hand, that still ensures similar prediction performances compared to the unstructured least-squares estimates.