A novel Multiplicative Polynomial Kernel for Volterra series identification

Abstract

Volterra series are especially useful for nonlinear system identification, also thanks to their capability to approximate a broad range of input-output maps. However, their identification from a finite set of data is hard, due to the curse of dimensionality. Recent approaches have shown how regularization strategies can be useful for this task. In this paper, we propose a new regularization network for Volterra models identification. It relies on a new kernel given by the product of basic building blocks. Each block contains some unknown parameters that can be estimated from data using marginal likelihood optimization or cross-validation. In comparison with other algorithms proposed in the literature, numerical experiments show that our approach allows to better select the monomials that really influence the system output, much increasing the prediction capability of the model.

A data-efficient geometrically inspired polynomial kernel for robot inverse dynamics

“…A valid alternative to ( 9) is represented by the MPK, recently introduced in [15]. When considering the space of inhomogeneous polynomial with maximum degree p, the MPK is defined as the product of p linear kernels,…”

“…Observe that the RKHSs identified by (10) and ( 9) contains the same basis functions. However, as discussed in [15], (10) is equipped with a richer set of hyperparameters, that can be tuned by ML maximization, and allows a better selection of the monomials that highly influence the system output.…”

“…The main idea supporting our approach is related to the existence of a suitable transformation of the standard inputs, that are, positions, velocities, and accelerations, of the generalized coordinates, into an augmented space, where the inverse dynamics map derived with the Lagrangian equations is a polynomial function. Inspired by this property, we propose a model based on the Multiplicative Polynomial Kernel (MPK), recently introduced in [15], which is a reparameterization of the standard polynomial kernel. As shown in [15], compared to the standard polynomial kernel, the MPK parametrization allows for greater flexibility in neglecting eventual unnecessary basis functions of the corresponding Reproducing Kernel Hilbert Space (RKHS), leading to higher generalization performance.…”

“…Inspired by this property, we propose a model based on the Multiplicative Polynomial Kernel (MPK), recently introduced in [15], which is a reparameterization of the standard polynomial kernel. As shown in [15], compared to the standard polynomial kernel, the MPK parametrization allows for greater flexibility in neglecting eventual unnecessary basis functions of the corresponding Reproducing Kernel Hilbert Space (RKHS), leading to higher generalization performance.…”