QUARKS: Identification of large-scale Kronecker Vector-AutoRegressive models

In this paper we propose a Kronecker-based modeling for identifying the spatial-temporal dynamics of large sensor arrays. 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. For a two-dimensional array of size N × N, and when the number of terms in the sum is small compared to N, exploiting the Kronecker structure leads to high data compression. We propose an Alternating Least Squares algorithm to identify the coefficient matrices with O(N3Nt), where Nt is the number of temporal samples, instead of O(N6) in the unstructured case. This framework moreover allows for a convenient integration of more structure (e.g sparse, banded, Toeplitz) on the factor 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.