V-Splines and Bayes Estimate

TL;DR

This paper explores the Bayesian interpretation of V-splines, deriving their form via reproducing kernel Hilbert spaces and extending cross-validation methods to optimize parameters.

Contribution

It introduces a Bayesian formulation of V-splines using RKHS and develops an extended GCV method for parameter selection.

Findings

Bayesian V-splines derived using RKHS

Extended GCV formula for parameter optimization

Connections between smoothing splines and Gaussian process regression

Abstract

Smoothing splines can be thought of as the posterior mean of a Gaussian process regression in a certain limit. By constructing a reproducing kernel Hilbert space with an appropriate inner product, the Bayesian form of the V-spline is derived when the penalty term is a fixed constant instead of a function. An extension to the usual generalized cross-validation formula is utilized to find the optimal V-spline parameters.