Continuous-time DC kernel --- a stable generalized first-order spline kernel

TL;DR

This paper presents a new interpretation of the continuous-time DC kernel as a stable generalized first-order spline kernel, offering insights into its properties and deriving new mathematical expressions and properties.

Contribution

It introduces a novel interpretation of the DC kernel, derives a new orthonormal basis expansion, and explores its maximum entropy property and matrix inverse structure.

Findings

Derived a new orthonormal basis expansion for the DC kernel.

Established the maximum entropy property of the non-uniformly sampled DC kernel.

Proved the kernel matrix has a tridiagonal inverse.

Abstract

The stable spline (SS) kernel and the diagonal correlated (DC) kernel are two kernels that have been applied and studied extensively for kernel-based regularized LTI system identification. In this note, we show that similar to the derivation of the SS kernel, the continuous-time DC kernel can be derived by applying the same "stable" coordinate change to a "generalized" first-order spline kernel, and thus can be interpreted as a stable generalized first-order spline kernel. This interpretation provides new facets to understand the properties of the DC kernel. In particular, we derive a new orthonormal basis expansion of the DC kernel, and the explicit expression of the norm of the RKHS associated with the DC kernel. Moreover, for the non-uniformly sampled DC kernel, we derive its maximum entropy property and show that its kernel matrix has tridiagonal inverse.