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

TL;DR

This paper introduces a Kronecker-based approach for identifying large-scale vector autoregressive models, significantly reducing computational complexity and enabling efficient modeling of spatial-temporal sensor array data.

Contribution

The paper proposes a novel Kronecker network framework for VAR models with an efficient ALS algorithm, improving scalability and allowing structured matrix integration.

Findings

Achieves high data compression for large sensor arrays.

Reduces computational complexity from O(N^6) to O(N^3 N_t).

Performs comparably to unstructured methods with fewer parameters.

Abstract

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 $N \times N$, 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 $\mathcal{O}(N^3N_t)$, where $N_t$ is the number of temporal samples, instead of $\mathcal{O}(N^6)$ 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.