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L^1 data fitting for Inverse Problems yields optimal rates of convergence in case of discretized white Gaussian noise

Kristina Bätz, Frank Werner

TL;DR

This work analyzes generalized Tikhonov regularization with $\mathbf{L}^1$ data fidelity for inverse problems $F(u)=g$ in a setting that includes discretization and forward-operator errors. By developing a general error-bounds framework based on the effective noise level, a variational source condition, and smoothing/admissibility concepts, the authors derive convergence rates that remain valid under discretized noises and impulsive perturbations, and specialize these results to the Gaussian-discrete data case where rates become order-optimal. In particular, for $a$-times smoothing operators and Gaussian white noise, the achievable rates match the classical $\mathcal{O}(\sigma^{\frac{2s}{2a+2s+d}})$, showing that $\mathbf{L}^1$ data fitting does not sacrifice asymptotic efficiency while offering enhanced robustness. The paper also provides practical ADMM-based algorithms and numerical experiments confirming the theoretical rates and robustness advantages in discretized settings.

Abstract

It is well-known in practice, that L^1 data fitting leads to improved robustness compared to standard L^2 data fitting. However, it is unclear whether resulting algorithms will perform as well in case of regular data without outliers. In this paper, we therefore analyze generalized Tikhonov regularization with L^1 data fidelity for Inverse Problems F(u) = g in a general setting, including general measurement errors and errors in the forward operator. The derived results are then applied to the situation of discretized Gaussian white noise, and we show that the resulting error bounds allow for order-optimal rates of convergence. These findings are also investigated in numerical simulations.

L^1 data fitting for Inverse Problems yields optimal rates of convergence in case of discretized white Gaussian noise

TL;DR

This work analyzes generalized Tikhonov regularization with data fidelity for inverse problems in a setting that includes discretization and forward-operator errors. By developing a general error-bounds framework based on the effective noise level, a variational source condition, and smoothing/admissibility concepts, the authors derive convergence rates that remain valid under discretized noises and impulsive perturbations, and specialize these results to the Gaussian-discrete data case where rates become order-optimal. In particular, for -times smoothing operators and Gaussian white noise, the achievable rates match the classical , showing that data fitting does not sacrifice asymptotic efficiency while offering enhanced robustness. The paper also provides practical ADMM-based algorithms and numerical experiments confirming the theoretical rates and robustness advantages in discretized settings.

Abstract

It is well-known in practice, that L^1 data fitting leads to improved robustness compared to standard L^2 data fitting. However, it is unclear whether resulting algorithms will perform as well in case of regular data without outliers. In this paper, we therefore analyze generalized Tikhonov regularization with L^1 data fidelity for Inverse Problems F(u) = g in a general setting, including general measurement errors and errors in the forward operator. The derived results are then applied to the situation of discretized Gaussian white noise, and we show that the resulting error bounds allow for order-optimal rates of convergence. These findings are also investigated in numerical simulations.

Paper Structure

This paper contains 18 sections, 11 theorems, 77 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Error bounds for generalized Tikhonov regularization
  3. Effective noise level
  4. Variational source condition
  5. Bounding the effective noise level
  6. Error bounds
  7. Impulsive noise and resulting error bounds
  8. Impulsive noise
  9. Resulting error bounds
  10. Results for finitely smoothing operators under classical smoothness conditions
  11. Application to discretized noises and optimality in the Gaussian case
  12. General setup
  13. Gaussian case and optimal rates of convergence
  14. Numerical simulations
  15. Numerical algorithm
  16. ...and 3 more sections

Key Result

Lemma 2.2

Suppose $\mathbb M \subset \mathbb R^d$ is a bounded Lipschitz domain and $F : D(F) \subset X \to \mathbf W^{k,p} \left(\mathbb M\right)$ for $k > \frac{d}{p}$ with the standard $\mathbf L^p$-based Sobolev space $\mathbf W^{k,p} \left(\mathbb M\right)$ is Lipschitz continuous in the sense that Then $F$ is $\left(\mathbf L^\infty\left(\mathbb M\right), \gamma,\ell\right)$-smoothing with $\gamma\le

Figures (1)

  • Figure 1: Simulations results for \ref{['eq:tikl2']} and \ref{['eq:primal']}, the latter solved either by ADMM or AD-LPMM. The different lines in color depict the error $\left(\mathbb E \left[\left\Vert \hat{u}_\alpha -u^\dagger \right\Vert_{\mathbf L^2}^2\right]\right)^{\frac{1}{2}}$ approximated by $M = 100$ Monte Carlo runs, where the optimal $\alpha_{\mathrm{opt}}$ is chosen out of a set of $40$ candidates $\alpha \in \left[10^{-8}, 10^{-2}\right]$. The black line illustrates the optimal rate of convergence $\mathcal{O} \left(\sigma^{\frac{3}{8}}\right)$.

Theorems & Definitions (25)

  • Example 2.1
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Theorem 2.4
  • Definition 3.1: Impulsive noise
  • Theorem 3.2
  • proof
  • Corollary 3.3
  • Remark 3.4
  • ...and 15 more