A Fully Probabilistic Tensor Network for Regularized Volterra System Identification

Modeling nonlinear systems with Volterra series is challenging because the number of kernel coefficients grows exponentially with the model order. This work introduces Bayesian Tensor Network Volterra kernel machines (BTN-V), extending the Bayesian Tensor Network framework to Volterra system identification. BTN-V represents Volterra kernels using canonical polyadic decomposition, reducing model complexity from O(I^D) to O(DIR). By treating all tensor components and hyperparameters as random variables, BTN-V provides predictive uncertainty estimation at no additional computational cost. Sparsity-inducing hierarchical priors enable automatic rank determination and the learning of fading-memory behavior directly from data, improving interpretability and preventing overfitting. Empirical results demonstrate competitive accuracy, enhanced uncertainty quantification, and reduced computational cost.