Variational Splines and Generalized Gaussian Processes
Abstract
Tikhonov regularization, kriging, and Gaussian-process regression share the same algorithm because, for a linear inverse problem with a Gaussian prior and Gaussian noise, the minimum-mean-square-error (MMSE) estimator is the solution of a regularized least-squares problem. We show that this equivalence survives the passage from vectors to generalized functions: the regularized least-squares estimators become generalized splines, and the Gaussian random vectors become generalized Gaussian processes (GGPs) on the dual of a nuclear space. A whitening operator L: ๐ฎ โ L2 specifies both the native Hilbert space โL โ ๐ฎ' of the splines and a family of GGPs, which solve a stochastic differential equation subject to boundary conditions. We prove that the Gaussian spaces of these processes are congruent to the core Lโ(L2) of the native space, and that their MMSE estimators from finitely many linear measurements are generalized splines. Our framework extends the classical correspondences of Kimeldorf-Wahba, Parzen and Kailath in two respects: our estimators form a broader class that includes generalized functions, and we give a proper Bayesian treatment of the case where the null space of L is non-trivial. The framework gives a precise meaning to statements such as ``fractional splines are the optimal estimators of fractional Brownian motion,'' and it provides a rigorous Bayesian formulation of infinite-dimensional linear inverse problems.
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BibTeX
@misc{unser2026variational,
title = {{Variational Splines and Generalized Gaussian Processes}},
author = {Michael Unser},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.06233},
eprint = {2611.06233},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.06233},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Variational Splines and Generalized Gaussian Processes
AU - Michael Unser
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.06233
UR - https://arxiv.org/abs/2611.06233
AB - Tikhonov regularization, kriging, and Gaussian-process regression share the same algorithm because, for a linear inverse problem with a Gaussian prior and Gaussian noise, the minimum-mean-square-error (MMSE) estimator is the solution of a regularized least-squares problem. We show that this equivalence survives the passage from vectors to generalized functions: the regularized least-squares estimators become generalized splines, and the Gaussian random vectors become generalized Gaussian processes (GGPs) on the dual of a nuclear space. A whitening operator L: ๐ฎ โ L_2 specifies both the native Hilbert space โ_L โ ๐ฎ' of the splines and a family of GGPs, which solve a stochastic differential equation subject to boundary conditions. We prove that the Gaussian spaces of these processes are congruent to the core L^โ(L_2) of the native space, and that their MMSE estimators from finitely many linear measurements are generalized splines. Our framework extends the classical correspondences of Kimeldorf-Wahba, Parzen and Kailath in two respects: our estimators form a broader class that includes generalized functions, and we give a proper Bayesian treatment of the case where the null space of L is non-trivial. The framework gives a precise meaning to statements such as ``fractional splines are the optimal estimators of fractional Brownian motion,'' and it provides a rigorous Bayesian formulation of infinite-dimensional linear inverse problems.
ER -