On the Maximum Entropy Property of the First-Order Stable Spline Kernel and its Implications

TL;DR

This paper provides a new proof and explicit formulas for the inverse and factorization of the first-order stable spline kernel, enhancing understanding of its maximum entropy properties and improving computational efficiency in system identification.

Contribution

It offers an independent proof of the maximum entropy property, derives a closed-form inverse, and introduces a computationally efficient factorization for the first-order stable spline kernel.

Findings

Derived a closed-form inverse of the kernel

Established a common factorization structure for all first-order kernels

Highlighted maximum likelihood properties of the kernel

Abstract

A new nonparametric approach for system identification has been recently proposed where the impulse response is seen as the realization of a zero--mean Gaussian process whose covariance, the so--called stable spline kernel, guarantees that the impulse response is almost surely stable. Maximum entropy properties of the stable spline kernel have been pointed out in the literature. In this paper we provide an independent proof that relies on the theory of matrix extension problems in the graphical model literature and leads to a closed form expression for the inverse of the first order stable spline kernel as well as to a new factorization in the form $UWU^\top$ with $U$ upper triangular and $W$ diagonal. Interestingly, all first--order stable spline kernels share the same factor $U$ and $W$ admits a closed form representation in terms of the kernel hyperparameter, making the factorization computationally inexpensive. Maximum likelihood properties of the stable spline kernel are also highlighted. These results can be applied both to improve the stability and to reduce the computational complexity associated with the computation of stable spline estimators.