The Field Equations of Penalized non-Parametric Regression

Sven Pappert

Home/Archive/arXiv

arXiv · 2503.14763

The Field Equations of Penalized non-Parametric Regression

Sven Pappert

View original source ↗Read source document ↗Export metadata ↓

Abstract

We view penalized risks through the lens of the calculus of variations. We consider risks comprised of a fitness-term (e.g. MSE) and a gradient-based penalty. After establishing the Euler-Lagrange field equations as a systematic approach to finding minimizers of risks involving only first derivatives, we proceed to exemplify this approach to the MSE penalized by the integral over the squared l2-norm of the gradient of the regression function. The minimizer of this risk is given as the solution to a second order inhomogeneous PDE, where the inhomogeneity is given as the conditional expectation of the target variable conditioned on the features. We discuss properties of the field equations and practical implications thereof, which also apply to the classical Ridge penalty for linear models, and embed our findings into the existing literature. In particular, we find that we can recover the Rudin-Osher-Fatemi model for image-denoising, if we consider the features as deterministic and evenly distributed. Last, we outline several directions for future research.

Explore related subjects

math.ST math.AP math.FA stat.TH

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Sven Pappert. 2026-09-17. The Field Equations of Penalized non-Parametric Regression. https://arxiv.org/abs/2503.14763

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Variance Reduction for Independent Metropolis

Assume that we would like to estimate the expected value of a function $F$ with respect to an intractable density $π$, which is specified up to some unknown normalising constant. We prove that if $π$ is close enough under KL divergence to another density $q$, an independent Metropolis sampler estimator that obtains samples from $π$ with proposal density $q$, enriched with a variance reduction computational strategy based on control variates, achieves smaller asymptotic variance than i.i.d. sampling from $π$. The control variates construction requires no extra computational effort but assumes that the expected value of $F$ under $q$ is analytically available. We illustrate this result by calculating the marginal likelihood in a linear regression model with prior-likelihood conflict and a non-conjugate prior. Furthermore, we propose an adaptive independent Metropolis algorithm that adapts the proposal density such that its KL divergence with the target is being reduced. We demonstrate its applicability in a Bayesian logistic and Gaussian process regression problems and we rigorously justify our asymptotic arguments under easily verifiable and essentially minimal conditions.

math.ST↗

arXiv2026

Edgeworth corrections for the spiked eigenvalues of non-Gaussian sample covariance matrices with applications

Yang and Johnstone (2018) established an Edgeworth correction for the largest sample eigenvalue in a spiked covariance model under the assumption of Gaussian observations, leaving the extension to non-Gaussian settings as an open problem. In this paper, we address this issue by establishing first-order Edgeworth expansions for spiked eigenvalues in both single-spike and multi-spike scenarios with non-Gaussian data. Leveraging these expansions, we construct more accurate confidence intervals for the population spiked eigenvalues and propose a novel estimator for the number of spikes. Simulation studies demonstrate that our proposed methodology outperforms existing approaches in both robustness and accuracy across a wide range of settings, particularly in low-dimensional cases.

math.ST↗

arXiv2026

Measures of Dependence based on Wasserstein distances

Measuring dependence between random variables is a fundamental problem in Statistics, with applications across diverse fields. While classical measures such as Pearson's correlation have been widely used for over a century, they have notable limitations, particularly in capturing nonlinear relationships and extending to general metric spaces. In recent years, the theory of Optimal Transport and Wasserstein distances has provided new tools to define measures of dependence that generalize beyond Euclidean settings. This survey explores recent proposals, outlining two main approaches: one based on the distance between the joint distribution and the product of marginals, and another leveraging conditional distributions. We discuss key properties, including characterization of independence, normalization, invariances, robustness, sample, and computational complexity. Additionally, we propose an alternative perspective that measures deviation from maximal dependence rather than independence, leading to new insights and potential extensions. Our work highlights recent advances in the field and suggests directions for further research in the measurement of dependence using Optimal Transport.

math.ST↗