Efficient Approximation of Volterra Series for High-Dimensional Systems
This work addresses scalability issues for researchers and practitioners in system identification, offering a theoretically grounded method for high-dimensional systems, though it is incremental as it builds on existing Tensor Network techniques.
The paper tackles the computational and memory bottlenecks in identifying high-dimensional nonlinear dynamical systems using Volterra series with Tensor Network methods by introducing the Tensor Head Averaging (THA) algorithm, which reduces complexity through an ensemble of localized models and provides theoretical error bounds and performance analysis.
The identification of high-dimensional nonlinear dynamical systems via the Volterra series has significant potential, but has been severely hindered by the curse of dimensionality. Tensor Network (TN) methods such as the Modified Alternating Linear Scheme (MVMALS) have been a breakthrough for the field, offering a tractable approach by exploiting the low-rank structure in Volterra kernels. However, these techniques still encounter prohibitive computational and memory bottlenecks due to high-order polynomial scaling with respect to input dimension. To overcome this barrier, we introduce the Tensor Head Averaging (THA) algorithm, which significantly reduces complexity by constructing an ensemble of localized MVMALS models trained on small subsets of the input space. In this paper, we present a theoretical foundation for the THA algorithm. We establish observable, finite-sample bounds on the error between the THA ensemble and a full MVMALS model, and we derive an exact decomposition of the squared error. This decomposition is used to analyze the manner in which subset models implicitly compensate for omitted dynamics. We quantify this effect, and prove that correlation between the included and omitted dynamics creates an optimization incentive which drives THA's performance toward accuracy superior to a simple truncation of a full MVMALS model. THA thus offers a scalable and theoretically grounded approach for identifying previously intractable high-dimensional systems.