Kronecker-based modeling of networks with unknown communication links

In this paper we propose a Kronecker-based modeling of large networks with unknown interconnection links. The class of Kronecker networks is defined for which we formulate a Vector Autoregressive model. Its coefficient-matrices are decomposed into a sum of Kronecker products. When the network is labeled such that the number of terms in the sum is small compared to the size of the matrix, exploiting this Kronecker structure leads to high data compression. Two algorithms were designed for an efficient estimation of the coefficientmatrices, namely a non-iterative and overparametrized algorithm as well as an Alternating Least Squares minimization. We prove that the latter always converges to the true parameters for non-zero initial conditions. This framework moreover allows for a convenient integration of more structure (e.g sparse, banded, Toeplitz) on smaller-size matrices. Numerical examples on atmospheric turbulence data has shown comparable performances with the unstructured least-squares estimation while the number of parameters is growing only linearly w.r.t. the number of nodes instead of quadratically in the full unstructured matrix case.