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

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 discretetime 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.

Matrix output extension of the tensor network Kalman filter with an application in MIMO Volterra system identification

“…For example, A(:, i) denotes the ith column of the matrix A, while A(:, :, i) denotes the ith matrix slice of a third-order tensor A. A more detailed description of the tensor concepts and operations used in this article can be found in [1,2].…”

“…Fortunately, it is possible to derive an efficient algorithm that exploits the repeated Kronecker product structure to construct an exact tensor network representation of C(t). Before providing the constructive derivation of the main algorithm, we first revisit the result for the row vector c(t) as described in [2] and explain why it fails for the matrix output case.…”

“…Numerical experiments in Section 6 demonstrate the efficacy of our proposed conversion algorithm and compare the performance of the tensor network Kalman filter described in [2] with the newly proposed matrix output tensor network Kalman filter. It will be shown that using a matrix output can double the convergence speed of the estimated Volterra kernel coefficients at practically no additional cost.…”

Tensor network subspace identification of polynomial state space models

“…Most of the notation on subspace methods is adopted from [5] and the notation on tensors from [1,2] is also used. Tensors are multi-dimensional arrays that generalize the notions of vectors and matrices to higher orders.…”

“…to denote scalars. The elements of a set of d tensors, in particular in the context of tensor networks, are denoted A (1) , A (2) , . .…”