A Tensor Network Kalman filter with an application in recursive MIMO Volterra system identification

TL;DR

This paper presents a Tensor Network Kalman filter that efficiently estimates large state vectors in recursive MIMO Volterra system identification, transforming exponential complexity into linear, enabling fast computations on standard hardware.

Contribution

It introduces a novel Tensor Network Kalman filter that handles exponentially large state vectors without explicit construction, applicable to high-order nonlinear system identification.

Findings

Efficiently estimates a state vector of length 10^9 in 0.007s.

Transforms nonlinear system identification into a linear state estimation problem.

Demonstrates robustness and accuracy through numerical experiments.

Abstract

This article introduces a Tensor Network Kalman filter, which can estimate state vectors that are exponentially large without ever having to explicitly construct them. The Tensor Network Kalman filter also easily accommodates the case where several different state vectors need to be estimated simultaneously. The key lies in rewriting the standard Kalman equations as tensor equations and then implementing them using Tensor Networks, which effectively transforms the exponential storage cost and computational complexity into a linear one. We showcase the power of the proposed framework through an application in recursive nonlinear system identification of high-order discrete-time multiple-input multiple-output (MIMO) Volterra systems. The identification problem is transformed into a linear state estimation problem wherein the state vector contains all Volterra kernel coefficients and is estimated using the Tensor Network Kalman filter. The accuracy and robustness of the scheme are demonstrated via numerical experiments, which show that updating the Kalman filter estimate of a state vector of length $10^9$ and its covariance matrix takes about 0.007s on a standard desktop computer in Matlab.