Continuous-time DC kernel --- a stable generalized first-order spline kernel
This work offers theoretical unification and new properties for the DC kernel, which is used in kernel-based system identification, but the results are primarily analytical and incremental.
The authors show that the DC kernel can be derived as a stable generalized first-order spline kernel, providing new insights and derivations including an orthonormal basis expansion, explicit RKHS norm, maximum entropy property, and tridiagonal inverse for non-uniform sampling.
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.