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Noise-resilient penalty operators based on statistical differentiation schemes

Marc Vidal, Yves Rosseel

TL;DR

The paper develops a grid-based penalized smoothing framework for discretized trajectories by penalizing local roughness with statistically calibrated discrete difference operators. It introduces decorrelated difference operators under a white-noise model, constructs convex and multi-order smoothing schemes, and analyzes asymptotic properties using Hellinger differentiability rather than function-space differentiability. Convergence rates are established under a source-condition and eigenvalue decay, and model-based GCV is proposed for curve-wise tuning of smoothing parameters. Through simulations on locally irregular and sinusoidal curves, the proposed discrete smoothers demonstrate robust, high-quality performance across diverse noise structures, highlighting practical advantages for discrete functional data with limited or irregular smoothness. The approach provides a principled, grid-focused alternative to basis- or kernel-based smoothing with strong theoretical guarantees and competitive empirical performance.

Abstract

Penalized smoothing is a standard tool in regression analysis. Classical approaches often rely on basis or kernel expansions, which constrain the estimator to a fixed span and impose smoothness assumptions that may be restrictive for discretely observed data. We introduce a class of penalized estimators that operate directly on the data grid, denoising sampled trajectories under minimal smoothness assumptions by penalizing local roughness through statistically calibrated difference operators. Some distributional and asymptotic properties of sample-based contrast statistics associated with the resulting linear smoothers are established under Hellinger differentiability of the model, without requiring Fréchet differentiability in function space. Simulation results confirm that the proposed estimators perform competitively across both smooth and locally irregular settings.

Noise-resilient penalty operators based on statistical differentiation schemes

TL;DR

The paper develops a grid-based penalized smoothing framework for discretized trajectories by penalizing local roughness with statistically calibrated discrete difference operators. It introduces decorrelated difference operators under a white-noise model, constructs convex and multi-order smoothing schemes, and analyzes asymptotic properties using Hellinger differentiability rather than function-space differentiability. Convergence rates are established under a source-condition and eigenvalue decay, and model-based GCV is proposed for curve-wise tuning of smoothing parameters. Through simulations on locally irregular and sinusoidal curves, the proposed discrete smoothers demonstrate robust, high-quality performance across diverse noise structures, highlighting practical advantages for discrete functional data with limited or irregular smoothness. The approach provides a principled, grid-focused alternative to basis- or kernel-based smoothing with strong theoretical guarantees and competitive empirical performance.

Abstract

Penalized smoothing is a standard tool in regression analysis. Classical approaches often rely on basis or kernel expansions, which constrain the estimator to a fixed span and impose smoothness assumptions that may be restrictive for discretely observed data. We introduce a class of penalized estimators that operate directly on the data grid, denoising sampled trajectories under minimal smoothness assumptions by penalizing local roughness through statistically calibrated difference operators. Some distributional and asymptotic properties of sample-based contrast statistics associated with the resulting linear smoothers are established under Hellinger differentiability of the model, without requiring Fréchet differentiability in function space. Simulation results confirm that the proposed estimators perform competitively across both smooth and locally irregular settings.
Paper Structure (16 sections, 4 theorems, 43 equations, 1 figure, 2 tables)

This paper contains 16 sections, 4 theorems, 43 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Difference operators with decorrelation properties
  3. Discrete smoothing operators
  4. Discrete penalization with finite differences
  5. Smoothing with convex roughness penalties (fixed order)
  6. Sequential and simultaneous multi-resolution smoothing
  7. Sequential (adaptive) smoothing.
  8. Simultaneous (multi-order) smoothing.
  9. Local stability and asymptotic linearity under Hellinger differentiability
  10. Convergence rates
  11. Curve-wise selection of $\alpha$ by GCV
  12. Numerical experiments
  13. Locally irregular curve.
  14. Sinusoidal curve.
  15. Proofs of formal statements
  16. ...and 1 more sections

Key Result

Proposition 2.1

Let $L \in \mathbb{N}$, and let $\bar{D}^{(0)}, \ldots, \bar{D}^{(r-1)} \in \mathbb{R}^{2L + 1}$ be previously constructed, normalized stencil vectors following Mizuta's scheme. Each vector is indexed symmetrically by offsets $\ell = -L, \ldots, L$, and centred at $\ell = 0$. Then the $r$th order un Let $\mathcal{V}_r \subset \mathbb{R}^{2L+1}$ denote the subspace defined by conditions (1)–(3). Th

Figures (1)

  • Figure 1: Empirical convergence of discrete penalized smoothing. Left:Log--log plots of the mean squared error (MSE), squared bias, and variance of the sample-averaged smoothed estimator ($r=2, \eta = 0.5$) as functions of the sample size $n$, with unsmoothed (raw) counterparts shown in grey for comparison.Right:Log--log plot of $\mathrm{MSE}\times\mathrm{Bias}^2$, illustrating the bias--variance trade-off and the associated decay behaviour.

Theorems & Definitions (9)

  • Proposition 2.1
  • Corollary 2.2
  • Theorem 5.4
  • proof : Proof of Proposition \ref{['prop:1']}
  • proof : Proof of Corollary \ref{['col:1']}
  • Corollary A.1
  • proof
  • Remark A.2
  • proof : Proof of Theorem \ref{['thm:rate']}